ABSTRACT AND CONCRETE CATEGORIESTHE JOY OF CATS
Dedicated to Bernhard Banaschewski
JIŘÍ ADÁMEK, HORST HERRLICH, GEORGE E. STRECKER

Originally published as: Abstract and Concrete Categories, John Wiley and Sons, 1990
The newest edition of the file of the present book can be downloaded from http://katmat.math.
uni-bremen.de/acc. The authors are grateful for any improvements, corrections, and remarks.
Received by the editors 2006-05-06.
Transmitted by R. Rosebrugh, J. Rosicky and W. Tholen. Reprint published on 2006-07-14 as TAC
Reprints, No. 17..

1

9th July 2006

PREFACE to the ONLINE EDITION
Abstract and Concrete Categories was published by John Wiley and Sons, Inc, in 1990,
and after several reprints, the book has been sold out and unavailable for several years.
We now present an improved and corrected version as an open access file. This was madepossible due to the return of copyright to the authors, and due to many hours of hard
work and the exceptional skill of Christoph Schubert, to whom we wish to express our
profound gratitude. The illustrations of Edward Gorey are unfortunately missing in thecurrent version (for copyright reasons), but fortunately additional original illustrations
by Marcel Erné, to whom additional special thanks of the authors belong, counterbalance
the loss.
Open access includes the right of any reader to copy, store or distribute the book orparts of it freely. (See the GNU Free Documentation License at the end of the text.)
Besides the acknowledgements appearing at the end of the original preface (below),
we wish to thank all those who have helped to eliminate mistakes that survived thefirst printing of the text, particularly H. Bargenda, J. Jürjens W. Meyer, L. Schröder,A. M. Torkabud, and O. Wyler.
January 12, 2004
J. A., H. H., and G. E. S.

3

PREFACE
Sciences have a natural tendency toward diversification and specialization. In particular,
contemporary mathematics consists of many different branches and is intimately related
to various other fields. Each of these branches and fields is growing rapidly and is itself
diversifying. Fortunately, however, there is a considerable amount of common ground— similar ideas, concepts, and constructions. These provide a basis for a general theoryof structures.
The purpose of this book is to present the fundamental concepts and results of such atheory, expressed in the language of category theory — hence, as a particular branch ofmathematics itself. It is designed to be used both as a textbook for beginners and as areference source. Furthermore, it is aimed toward those interested in a general theory of
structures, whether they be students or researchers, and also toward those interested inusing such a general theory to help with organization and clarification within a special
field. The only formal prerequisite for the reader is an elementary knowledge of set
theory. However, an additional acquaintance with some algebra, topology, or computerscience will be helpful, since concepts and results in the text are illustrated by various
examples from these fields.
One of the primary distinguishing features of the book is its emphasis on concrete categories. Recent developments in category theory have shown this approach to be particularly useful. Whereas most terminology relating to abstract categories has beenstandardized for some time, a large number of concepts concerning concrete categories
has been developed more recently. One of the purposes of the book is to provide a reference that may help to achieve standardized terminology in this realm. Another featurethat distinguishes the text is the systematic treatment of factorization structures, which
gives a new unifying perspective to many earlier concepts and results and summarizes
recent developments not yet published in other books.
The text is organized and written in a “pedagogical style”, rather than in a highly
economical one. Thus, in order to make the flow of topics self-motivating, new concepts
are introduced gradually, by moving from special cases to the more general ones, rather
than in the opposite direction. For example,• equalizers (§7) and products (§10) precede limits (§11),• factorizations are introduced first for single morphisms (§14), then for sources(§15), and finally for functor-structured sources (§17),• the important concept of adjoints (§18) comes as a common culmination of three
separate paths: 1. via the notions of reflections (§4 and §16) and of free objects(§8), 2. via limits (§11), and 3. via factorization structures for functors (§17).
Each categorical notion is accompanied by many examples — for motivation as well as
clarification. Detailed verifications for examples are usually left to the reader as implied
exercises. It is not expected that every example will be familiar to or have relevance4Sec. 0]for each reader. Thus, it is recommended that examples that are unfamiliar should
be skipped, especially on the first reading. Furthermore, we encourage those who are
working through the text to carry along their favorite category and to keep in mind
a “global exercise” of determining how each new concept specializes in that particularsetting. The exercises that appear at the end of each section have been designed both asan aid in understanding the material, e.g., by demonstrating that certain hypotheses areneeded in various results, and as a vehicle to extend the theory in different directions.
They vary widely in their difficulty. Those of greater difficulty are typically embellishedwith an asterisk (∗).
The book is organized into seven chapters that represent natural “clusters” of topics,
and it is intended that these be covered sequentially. The first five chapters containthe basic theory, and the last two contain more recent research results in the realm ofconcrete categories, cartesian closed categories, and quasitopoi. To facilitate references,each chapter is divided into sections that are numbered sequentially throughout thebook, and all items within a given section are numbered sequentially throughout it. We
use the symbol
to indicate either the end of a proof or that there is a proof that is
sufficiently straightforward that it is left as an exercise for the reader. The symbol Dmeans that a proof of the dual result has already been given. Symbols such as A 4.19
are used to indicate that no proof is given, since a proof can be obtained by analogy
to the one referenced (i.e., to item 19 in Section 4). Two tables of symbols appear
at the end of the text. One contains a list (in alphabetical order) of the abbreviatednames for special categories that are dealt with in the text. The other contains a list(in order of appearance in the text) of special mathematical symbols that are used. The
bibliography contains only books and monographs. However, each section of the textends with a (chronologically ordered) list of suggestions for further reading. These lists
are designed to aid those readers with a particular interest in a given section to “strikeout on their own” and they often contain material that can be used to solve the moredifficult exercises. They are intended as merely a sampling, and (in view of the vast
literature) there has been no attempt to make them complete1 or to provide detailed
historical notes.

Acknowledgements
We are grateful for financial support from each of our “home universities” and from the
Natural Sciences and Engineering Research Council of Canada, the National Academiesof Sciences of Czechoslovakia and the United States, the U.S. National Science Foundation, and the U.S. Office of Naval Research. We particularly appreciate that suchsupport made it possible for us to meet on several occasions to work together on the
manuscript.
Our special thanks go to Marcel Erné for several original illustrations that have been
incorporated in the text and to Volker Kühn for his efforts on a frontispiece that the
1

Indeed, although some could serve as a suggested reading for more than one section, none appears inmore than one.

9th July 2006

Publisher decided not to use. We also express our special thanks to Reta McDermott
for her expert typesetting, to Jürgen Koslowski for his valuable TEXnical assistance, andto Y. Liu for assistance in typesetting diagrams. We were also assisted by D. Bressler
and Y. Liu in compiling the index and by G. Feldmann in transferring electronic files
between Manhattan and Bremen. We are especially grateful to J. Koslowski for carefully
analyzing the entire manuscript, to P. Vopěnka for fruitful discussions concerning themathematical foundations, and to M. Erné, H.L. Bentley, D. Bressler, H. Andréka,
I. Nemeti, I. Sain, J. Kincaid, and B. Schröder, each of whom has read parts of earlier
versions of the manuscript, has made suggestions for improvements, and has helped toeliminate mistakes. Naturally, none of the remaining mistakes can be attributed to anyof those mentioned above, nor can such be blamed on any single author — it is always
the fault of the other two.
srohtua eht

9th July 2006

Contents

Preface to the Online Edition

3

Preface

40 Introduction
91
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
I Categories, Functors, and Natural Transformations
3
Categories and functors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5Concrete categories and concrete functors . . . . . . . . . . . . . . . . . .
6
Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17
18
45
58
80

II Objects and Morphisms
95
7
Objects and morphisms in abstract categories . . . . . . . . . . . . . . . . 96
8
Objects and morphisms in concrete categories . . . . . . . . . . . . . . . . 127
9
Injective objects and essential embeddings . . . . . . . . . . . . . . . . . . 147
III Sources and Sinks
163
10 Sources and sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
11 Limits and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
12 Completeness and cocompleteness . . . . . . . . . . . . . . . . . . . . . . 206
13 Functors and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
IV Factorization Structures
231
14 Factorization structures for morphisms . . . . . . . . . . . . . . . . . . . . 232
15 Factorization structures for sources . . . . . . . . . . . . . . . . . . . . . . 250
16 E-reflective subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
17 Factorization structures for functors . . . . . . . . . . . . . . . . . . . . . 283
V Adjoints and Monads
295
18 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
19 Adjoint situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
20 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

7

VI Topological and Algebraic Categories
349
21 Topological categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
22 Topological structure theorems . . . . . . . . . . . . . . . . . . . . . . . . 373
23 Algebraic categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
24 Algebraic structure theorems . . . . . . . . . . . . . . . . . . . . . . . . . 399
25 Topologically algebraic categories . . . . . . . . . . . . . . . . . . . . . . . 408
26 Topologically algebraic structure theorems . . . . . . . . . . . . . . . . . . 420
VII Cartesian Closedness and Partial Morphisms427
27 Cartesian closed categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
28 Partial morphisms, quasitopoi, and topological universes . . . . . . . . . . 442
Bibliography

459

Tables463
Functors and morphisms: Preservation properties . . . . . . . . . . . . . . . . . 463
Functors and morphisms: Reflection properties . . . . . . . . . . . . . . . . . . 463
Functors and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
Functors and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
Stability properties of special epimorphisms . . . . . . . . . . . . . . . . . . . . 464
Table of Categories

465

Table of Symbols

470

Index

474

9th July 2006

9

Chapter 0

INTRODUCTION
There’s a tiresome young man in Bay Shore.
When his fiancée cried, ‘I adore
The beautiful sea’,He replied, ‘I agree,
It’s pretty, but what is it for?’Morris Bishop

9th July 2006

1 Motivation
Why study categories? Some reasons are these:
1.1 ABUNDANCE
Categories abound in mathematics and in related fields such as computer science. Such
entities as sets, vector spaces, groups, topological spaces, Banach spaces, manifolds,
ordered sets, automata, languages, etc., all naturally give rise to categories.
1.2 INSIGHT INTO SIMILAR CONSTRUCTIONS
Constructions with similar properties occur in completely different mathematical fields.
For example,(1) “products” for vector spaces, groups, topological spaces, Banach spaces, automata,etc.,(2) “free objects” for vector spaces, groups, rings, topological spaces, Banach spaces,etc.,(3) “reflective improvements” of certain objects, e.g., completions of partially orderedsets and of metric spaces, Čech-Stone compactifications of topological spaces, symmetrizations of relations, abelianizations of groups, Bohr compactifications of topological groups, minimalizations of reachable acceptors, etc.
Category theory provides the means to investigate such constructions simultaneously.
1.3 USE AS A LANGUAGE
Category theory provides a language to describe precisely many similar phenomena that
occur in different mathematical fields. For example,(1) Each finite dimensional vector space is isomorphic to its dual and hence also to its
second dual. The second correspondence is considered “natural”, but the first isnot. Category theory allows one to precisely make the distinction via the notionof natural isomorphism.
(2) Topological spaces can be defined in many different ways, e.g., via open sets, via
closed sets, via neighborhoods, via convergent filters, and via closure operations.
Why do these definitions describe “essentially the same” objects? Category theory
provides an answer via the notion of concrete isomorphism.
(3) Initial structures, final structures, and factorization structures occur in many different situations. Category theory allows one to formulate and investigate suchconcepts with an appropriate degree of generality.

10Sec. 1]

Motivation

11

1.4 CONVENIENT SYMBOLISM
Categorists have developed a symbolism that allows one quickly to visualize quite complicated facts by means of diagrams.

1.5 TRANSPORTATION OF PROBLEMS
Category theory provides a vehicle that allows one to transport problems from one areaof mathematics (via suitable functors) to another area, where solutions are sometimes
easier. For example, algebraic topology can be described as an investigation of topological problems (via suitable functors) by algebraic methods.
1.6 DUALITY
The concept of category is well balanced, which allows an economical and useful duality.
Thus in category theory the “two for the price of one” principle holds: every concept is
two concepts, and every result is two results.
The reasons given above show that familiarity with category theory will help those whoare confronted with a new field to detect analogies and connections to familiar fields, toorganize the new field appropriately, and to separate the general concepts, problems andresults from the special ones, which deserve special investigations. Categorical knowledge
thus helps to direct and to organize one’s thoughts.

9th July 2006

2 Foundations
Before delving into categories per se, we need to briefly discuss some foundational aspects. In §1 we have seen that in category theory we are confronted with extremely
large collections such as “all sets”, “all vector spaces”, “all topological spaces”, or “allautomata”. The reader with some set-theoretical background knows that these entities
cannot be regarded as sets. For instance, if U were the set of all sets, then the subset
A = {x | x ∈ U and x ∈/ x} of U would have the property that A ∈ A if and only if A ∈/A(Russell’s paradox). Someone working, for example, in algebra, topology, or computerscience usually isn’t (and needn’t be) bothered with such set-theoretical difficulties. But
it is essential that those who work in category theory be able to deal with “collections”
like those mentioned above. To do so requires some foundational restrictions. Nevertheless, certain naturally arising categorical constructions should not be outlawed simply
because of the foundational safeguards. Hence, what is needed is a foundation that, on
the one hand, is sufficiently flexible so as not to unduly inhibit categorical inquiry and, on
the other hand, is sufficiently rigid to give reasonable assurance that the resulting theory
is consistent, i.e., does not lead to contradictions. We also require that the foundation be
sufficiently close to those foundational systems that are used by most mathematicians.
Below we provide a brief outline of the features such a foundation should have.
The basic concepts that we need are those of “sets” and “classes”. On a few occasions
we will need to go beyond these and also use “conglomerates”.
2.1 SETS
Sets can be thought of as the usual sets of intuitive set theory (or of some axiomatic
set theory). In particular, we require that the following constructions can be performed
with sets.
(1) For each set X and each “property” P , we can form the set {x ∈ X | P (x)} of all
members of X that have the property P .
(2) For each set X, we can form the set P(X) of all subsets of X (called the power
set of X).
(3) For any sets X and Y , we can form the following sets:(a) the set {X, Y } whose members are exactly X and Y ,(b) the (ordered) pair (X, Y ) with first coordinate X and second coordinate Y ,[likewise for n-tuples of sets, for any natural number n > 2],(c) the union X ∪ Y = {x | x ∈ X or x ∈ Y },(d) the intersection X ∩ Y = {x | x ∈ X and x ∈ Y },(e) the cartesian product X × Y = {(x, y) | x ∈ X and y ∈ Y },(f) the relative complement X \ Y = {x | x ∈ X and x 6∈ Y },12

Sec. 2]

Foundations

13(g) the set Y X of all functions2 f : X → Y from X to Y .
(4) For any set I and any family3 (Xi )i∈I of sets, we can form the following sets:(a) the image {Xi | i ∈ I} of the indexing function,
S(b) the union i∈I Xi = {x | x ∈ Xi for some i ∈ I},T(c) the intersection i∈I Xi = {x | x ∈ Xi for all i ∈ I}, provided that I 6= ∅,QS(d) the cartesian product i∈I Xi = {f : I → i∈I Xi | f (i) ∈ Xi for eachi ∈ I},US(e) the disjoint union i∈I Xi = i∈I (Xi × {i}).
(5) We can form the following sets:

N of all natural numbers,
Z of all integers,
Q of all rational numbers,
R of all real numbers, andC of all complex numbers.

The above requirements imply that each topological space is a set. [It is a pair (X, τ ),where X is its (underlying) set and τ is a topology (that is the set of all open subsetsof X); i.e., τ ∈ P(P(X)).] Analogously, each vector space and each automaton is a set.
However, by means of the above constructions, we cannot form “the set of all sets”, or“the set of all vector spaces”, etc.
2.2 CLASSES
The concept of “class” has been created to deal with “large collections of sets”. In
particular, we require that:(1) the members of each class are sets,(2) for every “property” P one can form the class of all sets with property P .
Hence there is the largest class: the class of all sets, called the universe and denotedby U. Classes are precisely the subcollections of U. Thus, given classes A and B, one
may form such classes as A ∪ B, A ∩ B, and A × B. Because of this, there is no problem
in defining functions between classes, equivalence relations on classes, etc. A family4(Ai )i∈I of sets is a function A : I → U (sending i ∈ I to A(i) = Ai ). In particular, if I
is a set, then (Ai )i∈I is said to be set-indexed [cf. 2.1(4)].
For convenience we require further
2

A function with domain X and codomain Y is a triple (X, f, Y ), where f ⊆ X × Y is a relation suchthat for each x ∈ X there exists a unique y ∈ Y with (x, y) ∈ f [notation: y = f (x) or x 7→ f (x)].
ffgFunctions are denoted by f : X → Y or X −−→ Y . Given functions X −−→ Y and Y −→ Z, theg◦fcomposite function X −−−→ Z is defined by x 7→ g(f (x)).
3
For a formal definition of families of sets see 2.2(2).
4
One should be aware that a family and its image are different entities and that, moreover, a family
is not determined by its image for essentially the same reason that a sequence (i.e., an -indexedfamily) is not determined by its set of values. A family (Ai )i∈I is sometimes denoted by (Ai )I .

N

9th July 2006

14

Introduction[Chap. 0(3) if X1 , X2 , . . . , Xn are classes, then so is the n-tuple (X1 , X2 , . . . , Xn ), and(4) every set is a class (equivalently: every member of a set is a set).
Hence sets are special classes. Classes that are not sets are called proper classes. They
cannot be members of any class. Because of this, Russell’s paradox now translates into
the harmless statement that the class of all sets that are not members of themselves
is a proper class. Also the universe U, the class of all vector spaces, the class of all
topological spaces, and the class of all automata are proper classes.
Notice that in this setting condition 2.1(4)(a) above gives us the Axiom of Replacement :(5) there is no surjection from a set to a proper class.
This means that each set must have “fewer” elements than any proper class.
Therefore sets are also called small classes, and proper classes are called large classes.
This distinction between “large” and “small” turns out to be crucial for many categoricalconsiderations.5
The framework of sets and classes described so far suffices for defining and investigatingsuch entities as the category of sets, the category of vector spaces, the category of
topological spaces, the category of automata, functors between these categories, andnatural transformations between such functors. Thus for most of this book we need not
go beyond this stage. Therefore we advise the beginner to skip from here, go directly to§3, and return to this section only when the need arises.
The limitations of the framework described above become apparent when we try to perform certain constructions with categories; e.g., when forming “extensions” of categories
or when forming categories that have categories or functors as objects. Since membersof classes must be sets and U is not a set, we can’t even form a class {U} whose only
member is U, much less a class whose members are all the subclasses of U or all functions
from U to U. In order to deal effectively with such “collections” we need a further levelof generality:
2.3 CONGLOMERATESThe concept of “conglomerate” has been created to deal with “collections of classes”. In
particular, we require that:(1) every class is a conglomerate,(2) for every “property” P , one can form the conglomerate of all classes with propertyP,(3) conglomerates are closed under analogues of the usual set-theoretic constructions
outlined above (2.1); i.e., they are closed under the formation of pairs, unions,products (of conglomerate-indexed families), etc.
5See, for example, Remark 10.33.

9th July 2006

Sec. 2]

Foundations

15

Thus we can form the conglomerate of all classes as well as such entities as functions
between conglomerates and families of conglomerates.
Furthermore, we require(4) the Axiom of Choice for Conglomerates ; namely for each surjection between conglomerates f : X → Y , there is an injection g : Y → X with f ◦ g = idY .
In other words, every equivalence relation on a conglomerate has a system of representatives. Notice that this Axiom of Choice implies an Axiom of Choice for Classes andalso the familiar Axiom of Choice for Sets.
Conglomerates

Classes =subcollections of
the universe U

Sets =
small classes =elements of U

The hierarchy of “collections”

A conglomerate X is said to be codable by a conglomerate Y provided that there exists
a surjection Y → X (equivalently: provided that there exists an injection X → Y ).
Conglomerates that are codable by a class (resp. by a set) are called legitimate (resp.
small) and will sometimes be treated like classes (resp. sets). For example, {U} is a small
conglomerate, and U ∪ {U } is a legitimate one. Conglomerates that are not legitimate
are called illegitimate. For example, P(U) is an illegitimate conglomerate.
Since our main interest lies with such categories as the category of all sets, the categoryof all vector spaces, the category of all topological spaces, the category of all automata,and possible “extensions” of these, no need arises to consider any “collections” beyond
the level of conglomerates, such as the entity of “all conglomerates”.
For a set-theoretic model of the above foundation, see e.g., the Appendix of the monograph of Herrlich and Strecker (see Bibliography), where, in view of the requirement
2.2(3), the familiar Kuratowski definition of an ordered pair (A, B) = {{A}, {A, B}}
needs to be replaced by a more suitable one; e.g., by (A, B) = {{{a}, {a, 0}} | a ∈A} ∪ {{{b}, {b, 1}} | b ∈ B}.

9th July 2006

16

Introduction[Chap. 0

Suggestions for Further Reading
Lawvere, F. W. The category of categories as a foundation for mathematics. Proceedings of the Conference on Categorical Algebra (La Jolla, 1965), Springer, Berlin–
Heidelberg–New York, 1966, 1–20.
Mac Lane, S. One universe as a foundation for category theory. Springer Lect. Notes
Math. 106 (1969): 192–200.
Feferman, S. Set-theoretical foundations of category theory. Springer Lect. Notes Math.
106 (1969): 201–247.
Bénabou, J. Fibred categories and the foundations of naive category theory. J. Symbolic
Logic 50 (1985): 10–37.

9th July 2006

17

Chapter I

CATEGORIES,FUNCTORS,AND
NATURAL TRANSFORMATIONSIn this chapter we introduce the most fundamental concepts of category theory, as well
as some examples that we will find to be useful in the remainder of the text.

9th July 2006

3 Categories and functors
CATEGORIES
Before stating the formal definition of category, we recall some of the motivating examples from §1. The notion of category should be sufficiently broad that it encompasses(1) the class of all sets and functions between them,(2) the class of all vector spaces and linear transformations between them,(3) the class of all groups and homomorphisms between them,(4) the class of all topological spaces and continuous functions between them, and(5) the class of all automata and simulations between them.
3.1 DEFINITION
A category is a quadruple A = (O, hom, id, ◦) consisting of(1) a class O, whose members are called A-objects,(2) for each pair (A, B) of A-objects, a set hom(A, B), whose members are called
A-morphisms from A to B — [the statement “f ∈ hom(A, B)” is expressedmore graphically6 by using arrows; e.g., by statements such as “f : A → B is afmorphism” or “A −→ B is a morphism”],
idA(3) for each A-object A, a morphism A −−−→ A, called the A-identity on A,f(4) a composition law associating with each A-morphism A −→ B and each A-morgg◦fphism B −→ C an A-morphism A −−−→ C, called the composite of f and g,
subject to the following conditions:fgh(a) composition is associative; i.e., for morphisms A −→ B, B −→ C, and C −→ D, theequation h ◦ (g ◦ f ) = (h ◦ g) ◦ f holds,(b) A-identities act as identities with respect to composition; i.e., for A-morphismsfA −→ B, we have idB ◦ f = f and f ◦ idA = f ,(c) the sets hom(A, B) are pairwise disjoint.
3.2 REMARKS
If A = (O, hom, id, ◦) is a category, then
(1) The class O of A-objects is usually denoted by Ob(A).
6

Notice that although we use the same notation f : A → B for a function from A to B (2.1) and for amorphism from A to B, morphisms are not necessarily functions (see Examples 3.3(4) below).

18

Sec. 3]Categories and functors

19(2) The class of all A-morphisms (denoted by M or(A)) is defined to be the union of all
the sets hom(A, B) in A.
f(3) If A −→ B is an A-morphism, we call A the domain of f [and denote it by dom(f )]and call B the codomain of f [and denote it by cod(f )]. Observe that condition(c) guarantees that each A-morphism has a unique domain and a unique codomain.
However, this condition is given for technical convenience only, because whenever allother conditions are satisfied, it is easy to “force” condition (c) by simply replacing
each morphism f in hom(A, B) by a triple (A, f, B) (as we did when defining functions in 2.1). For this reason, when verifying that an entity is a category, we will
disregard condition (c).
(4) The composition, ◦, is a partial binary operation on the class M or(A). For a pair(f, g) of morphisms, f ◦ g is defined if and only if the domain of f and the codomainof g coincide.
(5) If more than one category is involved, subscripts may be used (as in homA (A, B))
for clarification.
3.3 EXAMPLES(1) The category Set whose object class is the class of all sets; hom(A, B) is the setof all functions from A to B, idA is the identity function on A, and ◦ is the usualcomposition of functions.
(2) The following constructs; i.e., categories of structured sets and structure-preserving
functions between them (◦ will always be the composition of functions and idA will
always be the identity function on A):(a) Vec with objects all real vector spaces and morphisms all linear transformations
between them.
(b) Grp with objects all groups and morphisms all homomorphisms between them.
(c) Top with objects all topological spaces and morphisms all continuous functions
between them.
(d) Rel with objects all pairs (X, ρ), where X is a set and ρ is a (binary) relation
on X. Morphisms f : (X, ρ) → (Y, σ) are relation-preserving maps; i.e., maps
f : X → Y such that xρ x0 implies f (x) σf (x0 ).
(e) Alg(Ω) with objects all Ω-algebras and morphisms all Ω-homomorphisms
between them. Here Ω = (ni )i∈I is a family of natural numbers ni , indexed bya set I. An Ω-algebra is a pair (X, (ωi )i∈I ) consisting of a set X and a family offunctions ωi : X ni → X, called ni -ary operations on X. An Ω-homomorphismf : (X, (ωi )i∈I ) → (X̂, (ω̂i )i∈I ) is a function f : X → X̂ for which the diagram
X ni

f ni

/

X̂ ni

ωi



X

9th July 2006

f

/



ω̂i

X̂

20

Categories, Functors, and Natural Transformations[Chap. I

commutes (i.e., f ◦ ωi = ω̂i ◦ f ni ) for each i ∈ I. In case ni = 1 for each i ∈ I,
the symbol Σ = (ni )i∈I is usually used instead of Ω.
(f) Σ-Seq with objects all (deterministic, sequential) Σ-acceptors, where Σ is a
finite set of input symbols. Objects are quadruples (Q, δ, q0 , F ), where Q is a
finite set of states, δ : Σ × Q → Q is a transition map, q0 ∈ Q is the initial state,and F ⊆ Q is the set of final states.
A morphism f : (Q, δ, q0 , F ) → (Q0 , δ 0 , q00 , F 0 ) (called a simulation) is a function
f : Q → Q0 that preserves(i) transitions, i.e., δ 0 (σ, f (q)) = f (δ(σ, q)),(ii) the initial state, i.e., f (q0 ) = q00 , and(iii) the final states, i.e., f [F ] ⊆ F 0 .
(3) For constructs, it is often clear what the morphisms should be once the objects aredefined. However, this is not always the case. For instance:(a) there are at least three natural constructs each having as objects all metricspaces; namely,
Met with morphisms all non-expansive maps (= contractions),7
Metu with morphisms all uniformly continuous maps,
Metc with morphisms all continuous maps.
(b) there are at least two natural constructs each having as objects all Banach
spaces; namely,
Ban with morphisms all linear contractions,8Banb with morphisms all bounded linear maps (= continuous linear maps =uniformly continuous linear maps).
(4) The following categories where the objects and morphisms are not structured sets
and structure-preserving functions:(a) Mat with objects all natural numbers, and for which hom(m, n) is the set of
all real m × n matrices, idn : n → n is the unit diagonal n × n matrix, andcomposition of matrices is defined by A ◦ B = BA, where BA denotes the usual
multiplication of matrices.
(b) Aut with objects all (deterministic, sequential, Moore) automata. Objects are
sextuples (Q, Σ, Y, δ, q0 , y), where Q is the set of states, Σ and Y are the sets of
input symbols and output symbols, respectively, δ : Σ × Q → Q is the transition
map, q0 ∈ Q is the initial state, and y : Q → Y is the output map. Morphisms
from an automaton (Q, Σ, Y, δ, q0 , y) to an automaton (Q0 , Σ0 , Y 0 , δ 0 , q00 , y 0 ) are
triples (fQ , fΣ , fY ) of functions fQ : Q → Q0 , fΣ : Σ → Σ0 , and fY : Y → Y 0
satisfying the following conditions:
ˆ is called non-expansive (or a contraction) provided that
A function f : (X, d) → (Y, d)ˆ (a), f (b)) ≤ d(a, b), for all a, b ∈ X.
d(f8
For Banach spaces the distance between a and b is given by k a − b k.

7

9th July 2006

Sec. 3]Categories and functors

21(i) preservation of transition: δ 0 (fΣ (σ), fQ (q)) = fQ (δ(σ, q)),(ii) preservation of outputs: fY (y(q)) = y 0 (fQ (q)),(iii) preservation of initial state: fQ (q0 ) = q00 .
(c) Classes as categories:
Every class X gives rise to a category C(X) = (O, hom, id, ◦) — the objects ofwhich are the members of X, and whose only morphisms are identities — asfollows:(∅
if x 6= y,
O = X, hom(x, y) =idx = x, and x ◦ x = x.
{x} if x = y,C(∅) is called the empty category. C({0}) is called the terminal category
and is denoted by 1.
(d) Preordered classes as categories:
Every preordered class, i.e., every pair (X, ≤) with X a class and ≤ a reflexive
and transitive relation on X, gives rise to a category C(X, ≤) = (O, hom, id, ◦)— the objects of which are the members of X — as follows:({(x, y)} if x ≤ y,
O = X, hom(x, y) =idx = (x, x),∅otherwise,and (y, z) ◦ (x, y) = (x, z).
(e) Monoids as categories:
Every monoid (M, •, e), i.e., every semigroup (M, •) with unit, e, gives rise to acategory C(M, •, e) = (O, hom, id, o) — with only one object — as follows:
O = {M },hom(M, M ) = M,

idM = e,and y ◦ x = y • x.

(f) Set×Set is the category that has as objects all pairs of sets (A, B), as morphisms
fgfrom (A, B) to (A0 , B 0 ) all pairs of functions (f, g) with A −→ A0 and B −→ B 0 ,identities given by id(A,B) = (idA , idB ), and composition defined by(f2 , g2 ) ◦ (f1 , g1 ) = (f2 ◦ f1 , g2 ◦ g1 ).
Similarly, for any categories A and B one can form A×B, or, more generally, forfinitely many categories C1 , C2 , . . . , Cn , one can form the product category
C1 × C2 × · · · × Cn .
3.4 REMARKS(1) In the cases of classes, preordered classes, and monoids, for notational convenience
we will sometimes not distinguish between them and the categories they determine
in the sense of Examples 3.3(4)(c), (d), and (e) above. Thus, we might speak of apreordered class (X, ≤) or of a monoid (M, •, e) as a category.

9th July 2006

22

Categories, Functors, and Natural Transformations[Chap. I(2) Morphisms in a category will usually be denoted by lowercase letters, with uppercase
letters reserved for objects. The morphism h = g ◦ f will sometimes be denoted byfg

A −→ B −→ C or by saying that the triangle
f

/B
@@
@@
g
h @@ A@

C

commutes. Similarly, the statement that the square
A
h

f

/B
g



C



k

/D

commutes means that g ◦ f = k ◦ h.
(3) The order of writing the compositions may seem backwards. However, it comesfrom the fact that in many of the familiar examples (e.g., in all constructs) thecomposition law is the composition of functions.
(4) Notice that because of the associativity of composition, the notationfghA −→ B −→ C −→ D is unambiguous.

THE DUALITY PRINCIPLE
3.5 DEFINITION
For any category A = (O, homA , id, ◦) the dual (or opposite) category of A is thecategory Aop = (O, homAop , id, ◦op ), where homAop (A, B) = homA (B, A) and f ◦op g =g ◦ f . (Thus A and Aop have the same objects and, except for their direction, the samemorphisms.)
3.6 EXAMPLES(1) If A = (X, ≤) is a preordered class, considered as a category [3.3(4)(d)], thenAop = (X, ≥).
(2) If A = (M, •, e) is a monoid, considered as a category [3.3(4)(e)], thenAop = (M, ˆ•, e), where a ˆ• b = b • a.
3.7 REMARK
Because of the way dual categories are defined, every statement SAop (X) concerning an
object X in the category Aop can be translated into a logically equivalent statement
op(X) concerning the object X in the category A. This observation allows one toSA

9th July 2006

Sec. 3]Categories and functors

23

associate (in two steps) with every property P concerning objects in categories, a dual
property concerning objects in categories, as demonstrated by the following example:
Consider the property of objects X in A:
PA (X) ≡ For any A-object A there exists exactly one A-morphism f : A → X.
Step 1: In PA (X) replace all occurrences of A by Aop , thus obtaining the property
PAop (X) ≡ For any Aop -object A there exists exactly one Aop -morphism f : A → X.
Step 2: Translate PAop (X) into the logically equivalent statementopPA(X) ≡ For any A-object A there exists exactly one A-morphism f : X → A.
op
Observe that, roughly speaking, PA(X) is obtained from PA (X) by reversing the direction of each arrow and the order in which morphisms are composed. Naturally, inopgeneral, PA(X) is not equivalent to PA (X). For example, the above property PSet (X)opholds if and only if X is a singleton set, whereas the dual property PSet
(X) holds if andonly if X is the empty set.

In a similar manner any property about morphisms9 in categories gives rise to a dual
property concerning morphisms in categories, as demonstrated by the following example:fConsider the property of morphisms A −→ B in A:
g

f

g

id

A
QA (f ) ≡ There exists an A-morphism B −→ A with A −→ B −→ A = A −−−→ A (i.e.,g ◦ f = idA ) in A.

Step 1: Replace in QA (f ) all occurrences of A by Aop , thus obtaining the property
g

f

g

id

A
QAop (f ) ≡There exists an Aop -morphism B −→ A with A −→ B −→ A = A −−−→ A(i.e., g ◦ f = idA ) in Aop .

Step 2: Translate QAop (f ) into the logically equivalent statement
g

g

f

id

A
Qop
−−
→ A (i.e.,
A (f ) ≡ There exists an A-morphism A −→ B with A −→ B −→ A = A −f ◦ g = idA ) in A.

For example, the above property QSet (f ) holds if and only if f is an injective function
with nonempty domain or is the identity on the empty set, whereas the dual property
Qop
Set (f ) holds if and only if f is a surjective function.
More complex properties PA (A, B, . . . , f, g, . . .) that involve objects A, B, . . . and morphisms f, g, . . . in a category A can be dualized in a similar way.
If P = PA (A, B, . . . , f, g, . . .) holds for all A-objects A, B, . . . and all A-morphisms
f, g, . . ., then we say that A has the property P or that P(A) holds.
9ff

Observe that if a property concerns morphisms A −−→ B, then its dual concerns morphisms B −−→ A.
In particular if a property concerns dom(f ), then its dual concerns cod(f ).

9th July 2006

24

Categories, Functors, and Natural Transformations[Chap. I

The Duality Principle for Categories states
Whenever a property P holds for all categories,
then the property P op holds for all categories.
The proof of this (extremely useful) principle follows immediately from the facts thatfor all categories A and properties P(1) (Aop )op = A, and(2) P op (A) holds if and only if P(Aop ) holds.
For example, consider the property R = RA (f ) ≡ if PA (dom(f )), then QA (f ), where
P and Q are the properties defined above. One can easily show that R(A) holds for
all categories A, so that by the Duality Principle Rop (A) holds for all categories A,
op
op
where10 Rop
A (f ) ≡ if PA (cod(f )) then QA (f ).

The duality principle

Because of this principle, each result in category theory has two equivalent formulations(which at first glance might seem to be quite different). However, only one of them needs
to be proved, since the other one follows by virtue of the Duality Principle.
Often the dual concept P op of a concept P is denoted by “co-P” (cf. equalizers andcoequalizers (7.51 and 7.68), wellpowered and co-wellpowered (7.82 and 7.87), products
and coproducts (10.19 and 10.63), etc.). A concept P is called self-dual if P = P op .
An example11 of a self-dual concept is that of “identity morphism”.
10
11

See footnote 4.
“Live dual, laud evil” — E.H.

9th July 2006

Sec. 3]Categories and functors

25

Formulation of the duals of definitions and results will be an implied exercise throughout
the remainder of the book. However, we find that it is sometimes instructive to provide
such formulations. When we do so for results, we usually will conclude with the symbol
D to indicate that the dual result has been stated and proved at an earlier point, sothat (by the Duality Principle) no proof is needed.

ISOMORPHISMS
3.8 DEFINITION
A morphism f : A → B in a category12 is called an isomorphism provided that there
exists a morphism g : B → A with g ◦ f = idA and f ◦ g = idB . Such a morphism g iscalled an inverse of f .
3.9 REMARK
It is clear from the above definition that the statement “f is an isomorphism” is self-dual;i.e., f is an isomorphism in A if and only if f is an isomorphism in Aop .
3.10 PROPOSITION
If f : A → B, g : B → A and h : B → A are morphisms such that g ◦ f = idA andf ◦ h = idB , then g = h.
Proof: h = idA ◦ h = (g ◦ f ) ◦ h = g ◦ (f ◦ h) = g ◦ idB = g.
3.11 COROLLARY
If g1 and g2 are inverses of a morphism f , then g1 = g2 .



3.12 REMARK
Due to the above corollary we may speak of the inverse of an isomorphism f . It will bedenoted by f −1 .
3.13 EXAMPLES(1) Every identity idA is an isomorphism and id−1
A = idA .
(2) In Set the isomorphisms are precisely the bijective maps, in Vec they are precisely
the linear isomorphisms, in Grp they are precisely the group-theoretic isomorphisms,
in Top they are precisely the homeomorphisms, and in Rel they are precisely the
relational isomorphisms. Observe that in all of these cases every isomorphism is abijective morphism, but that the converse to this statement, namely, “every bijective
morphism is an isomorphism”, is true for Set, Vec, and Grp, but not for Rel orTop.
12From now on, when making a definition or stating a result that is valid for any category, we will
not name the category. Also whenever we speak about morphisms or objects without specifying acategory, we usually mean that they belong to the same category. When more than one category
is involved and confusion may occur, we will use a hyphenated notation, such as A-identity or Aisomorphism.

9th July 2006

26

Categories, Functors, and Natural Transformations[Chap. I(3) In Banb [3.3(3)] the isomorphisms are precisely the linear homeomorphisms, whereas
in Ban isomorphisms are precisely the norm-preserving linear bijections.
(4) In Mat the isomorphisms are precisely the regular matrices; i.e., the square matrices
with nonzero determinant.
(5) A morphism (fQ , fΣ , fY ) in Aut is an isomorphism if and only if each of the maps
fQ , fΣ , and fY is bijective.
(6) In a monoid, considered as a category, every morphism is an isomorphism if andonly if the monoid is a group.
3.14 PROPOSITIONff −1(1) If A −→ B is an isomorphism, then so is B −−−→ A and (f −1 )−1 = f .
f

g

g◦f

(2) If A −→ B and B −→ C are isomorphisms, then so is A −−−→ C, and(g ◦ f )−1 = f −1 ◦ g −1 .
Proof:(1). Immediate from the definitions of inverse and isomorphism (3.8).
(2). By associativity and the definition of inverse, we have: (g ◦ f ) ◦ (f −1 ◦ g −1 ) =g◦(f ◦f −1 )◦g −1 = g◦idB ◦g −1 = g◦g −1 = idC . Similarly, (f −1 ◦g −1 )◦(g◦f ) = idA .
3.15 DEFINITION
Objects A and B in a category are said to be isomorphic provided that there is an
isomorphism f : A → B.
3.16 REMARKFor any category, A, “is isomorphic to” clearly yields an equivalence relation on Ob(A).
[Reflexivity follows from the fact that identities are isomorphisms, and symmetry andtransitivity are immediate from the proposition above.] Isomorphic objects are frequently regarded as being “essentially” the same.

FUNCTORS
In category theory it is the morphisms, rather than the objects, that have the primaryrole. Indeed, we will see that it is even possible to define “category” without using thenotion of objects at all (3.53). Now, we take a more global viewpoint and consider categories themselves as structured objects. The “morphisms” between them that preserve
their structure are called functors.
3.17 DEFINITION
If A and B are categories, then a functor F from A to B is a function that assignsfto each A-object A a B-object F (A), and to each A-morphism A −→ A0 a B-morphism
F (f )

F (A) −−−−→ F (A0 ), in such a way that

9th July 2006

Sec. 3]Categories and functors

27(1) F preserves composition; i.e., F (f ◦ g) = F (f ) ◦ F (g) whenever f ◦ g is defined, and(2) F preserves identity morphisms; i.e., F (idA ) = idF (A) for each A-object A.
3.18 NOTATIONF
Functors F from A to B will be denoted by F : A → B or A −−→ B. We frequently use
the simplified notations F A and F f rather than F (A) and F (f ). Indeed, we sometimes
denote the action on both objects and morphisms byf

Ff

F (A −→ B) = F A −−→ F B.

3.19 REMARK
Notice that a functor F : A → B is technically a family of functions; one from Ob(A) to
Ob(B), and for each pair (A, A0 ) of A-objects, one from hom(A, A0 ) to hom(F A, F A0 ).
Since functors preserve identity morphisms and since there is a bijective correspondence
between the class of objects and the class of identity morphisms in any category, theobject-part of a functor actually is determined by the morphism-parts. Indeed, we will
see later that if we choose the “object-free” definition of category (3.53), then a functor
between categories can be defined simply as a function between their morphism classes
that preserves identities and composition (3.55).
3.20 EXAMPLES(1) For any category A, there is the identity functor idA : A → A defined byffidA (A −→ B) = A −→ B.
(2) For any categories A and B and any B-object B, there is the constant functor
CB : A → B with value B, defined byfid

B
CB (A −→ A0 ) = B −−−→ B.

(3) For any of the constructs A mentioned above [3.3(2)(3)] there is the forgetful
functor (or underlying functor) U : A → Set, where in each case U (A) is theunderlying set of A, and U (f ) = f is the underlying function of the morphism f .
(4) For any category A and any A-object A, there is the covariant hom-functor
hom(A, −) : A → Set, defined byfhom(A,f )hom(A, −)(B −→ C) = hom(A, B) −−−−−−−→ hom(A, C)
where hom(A, f )(g) = f ◦ g.

9th July 2006

28

Categories, Functors, and Natural Transformations[Chap. I(5) For any category A and any A-object A, there is the contravariant13 hom-functorfhom(−, A) : Aop → Set defined on any Aop -morphism14 B −→ C byhom(f,A)fhom(−, A)(B −→ C) = homA (B, A) −−−−−−→ homA (C, A)
with hom(f, A)(g) = g ◦ f , where the composition is the one in A.

A forgetful functor(6) If A and B are monoids considered as categories [3.3(4)(e)], then functors from A
to B are essentially just monoid homomorphisms from A to B.
(7) If A and B are preordered sets considered as categories [3.3(4)(d)], then functors
from A and B are essentially just order-preserving maps from A to B.
(8) The covariant power-set functor P : Set → Set is defined byf

Pf

P(A −→ B) = PA −−→ PB
where PA is the power-set of A; i.e., the set of all subsets of A; and for each X ⊆ A,
Pf (X) is the image f [X] of X under f .
13Functors are sometimes called covariant functors. A contravariant functor from A to B means afunctor from Aop to B.

14

Recall that Aop -objects are precisely the same as A-objects, and B −−→ C is an Aop -morphism meansfthat C −−→ B is an A-morphism.

f

9th July 2006

Sec. 3]Categories and functors

29(9) The contravariant8 power-set functor Q : Setop → Set is defined by
Qff

Q(A −→ B) = QA −−−→ QB
where QA is the power-set of A, and for each X ⊆ A, Qf (X) is the preimage f −1 [X]of X under the function f : B → A.
(10) For any positive integer n the nth power functor S n : Set → Set is given byfnf

S n (X −→ Y ) = X n −−→ Y n ,where f n (x1 , . . . , xn ) = (f (x1 ), . . . , f (xn )).
(11) The Stone-functor S : Topop → Boo (where Boo is the construct of boolean
algebras and boolean homomorphisms) assigns to each topological space the booleanfalgebra of its clopen subsets, and for any continuous map X −→ Y ; i.e., for anyfmorphism Y −→ X in Topop , Sf : S(Y ) → S(X) is given by Sf (Z) = f −1 [Z].
(12) The duality functor for vector spaces (ˆ) : Vecop → Vec associates with any
vector space V its dual V̂ (i.e., the vector space hom(V, ) with operations defined
op

f

R

f

pointwise) and with any Vec -morphism V −→ W , i.e., any linear map W −→ V ,the morphism fˆ : V̂ → Ŵ , defined by fˆ(g) = g ◦ f .
(13) If M = (M, •, e) is a monoid, then functors from M (regarded as a one-object
category) into Set are essentially just M -actions; i.e., pairs (X, ∗), where X is a set
and ∗ is a map from M × X to X such that e ∗ x = x and (m • m̂) ∗ x = m ∗ (m̂ ∗ x).
[Associate with any such M -action (X, ∗) the functor F : M → Set, defined by
m

F (m)

F (M −−→ M ) = X −−−−→ X, where F (m)(x) = m ∗ x.]

PROPERTIES OF FUNCTORS
3.21 PROPOSITIONk
All functors F : A → B preserve isomorphisms; i.e., whenever A −→ A0 is an Aisomorphism, then F (k) is a B-isomorphism.
Proof: F (k)◦F (k −1 ) = F (k◦k −1 ) = F (idA0 ) = idF A0 . Similarly, F (k −1 )◦F (k) = idF A .
3.22 REMARKS(1) Although the above proposition has a trivial proof, it has interesting consequences.
In particular, it can be used to show that certain objects in a category are notisomorphic. For example, the fundamental group functor can be used to prove thatcertain topological spaces are not homeomorphic by showing that their fundamentalgroups are not isomorphic.

9th July 2006

30

Categories, Functors, and Natural Transformations[Chap. I(2) Even though all functors preserve isomorphisms, they need not reflect isomorphisms (in the sense that if F (k) is an isomorphism, then k must be an isomorphism). For example, consider the forgetful functor U : Top → Set. The identity
function from the set of real numbers, with the discrete topology, to , with its
usual topology, is not a homeomorphism (i.e., isomorphism in Top), although its
underlying function is an identity, and thus is an isomorphism in Set.

R

3.23 PROPOSITION
If F : A → B and G : B → C are functors, then the composite G ◦ F : A → C
defined byf

G(F f )(G ◦ F )(A −→ A0 ) = G(F A) −−−−−→ G(F A0 )is a functor.15



3.24 DEFINITION(1) A functor F : A → B is called an isomorphism provided that there is a functor
G : B → A such that G ◦ F = idA and F ◦ G = idB .
(2) The categories A and B are said to be isomorphic provided that there is an isomorphism F : A → B.
3.25 REMARKS(1) Obviously the functor G in the above definition is uniquely determined by F . It will
be denoted by F −1 .
(2) Clearly, “is isomorphic to” is an equivalence relation on the conglomerate of all
categories. Isomorphic categories are considered to be essentially the same.
3.26 EXAMPLES(1) For any pair of classes (X, Y ), the categories C(X) and C(Y ) [3.3(4)(c)] are isomorphic if and only if there exists a bijection from X to Y . A category is isomorphic
to a category of the form C(X) if and only if each of its morphisms is an identity.
Such categories are called discrete.
(2) For any pair ((X, ≤), (Y, ≤)) of preordered classes, the categories C(X, ≤) andC(Y, ≤) [3.3(4)(d)] are isomorphic if and only if (X, ≤) and (Y, ≤) are order-isomorphic. A category is isomorphic to a category of the form C(X, ≤) if and only if foreach pair of objects (A, B), hom(A, B) has at most one member. Such categories
are called thin.
(3) For any pair (M, N ) of monoids, the categories C(M ) and C(N ) [3.3(4)(e)] areisomorphic if and only if M and N are isomorphic monoids. A category is isomorphic
to a category of the form C(M ) if and only if it has precisely one object.
15

Occasionally, for typographical efficiency, we will use juxtaposition to denote composition of functors,i.e., GF rather than G ◦ F . When F is an endofunctor, i.e., when its domain and codomain are thesame, we may even use F 2 or F 3 to denote F ◦ F or F ◦ F ◦ F , respectively.

9th July 2006

Sec. 3]Categories and functors

31(4) The construct Boo of boolean algebras [3.20(11)] is isomorphic to the construct
BooRng of boolean rings16 and ring homomorphisms.
(5) For any commutative ring R let R-Mod (resp. Mod-R) denote the construct of left
(resp. right) R-modules, and module homomorphisms.17 Then:(a) R-Mod is isomorphic to Mod-R, for any ring R.

ZZ(b) Ifdenotes the ring of integers, then -Mod is isomorphic to the construct
Ab of abelian groups and group homomorphisms.
(6) For any monoid, M , let M -Act be the category of all M -actions [3.20(13)] andaction homomorphisms [f (m ∗ x) = m ∗ f (x)]. If Σ∗ is the free monoid of all words
over Σ, then Σ∗ -Act is isomorphic to Alg(Σ) [3.3(2)(e)].
3.27 DEFINITION
Let F : A → B be a functor.
(1) F is called an embedding provided that F is injective on morphisms.
(2) F is called faithful provided that all the hom-set restrictions
F : homA (A, A0 ) → homB (F A, F A0 )are injective.
(3) F is called full provided that all hom-set restrictions are surjective.
(4) F is called amnestic provided that an A-isomorphism f is an identity wheneverF f is an identity.
3.28 REMARK
Notice that a functor is:(1) an embedding if and only if it is faithful and injective on objects, and(2) an isomorphism if and only if it is full, faithful, and bijective on objects.
3.29 EXAMPLES(1) The forgetful functor U : Vec → Set is faithful and amnestic, but is neither full nor
an embedding. This is the case for all of the constructs mentioned above (except, ofcourse, for Set itself).
(2) The covariant power-set functor P : Set → Set and the contravariant power-set
functor Q : Setop → Set [3.20(8)(9)] are both embeddings that are not full.
16

A boolean ring is a ring, with unit, in which each element is idempotent with respect to multiplication.
17
Notice that for the ringof real numbers, -Mod = Vec.

R

9th July 2006

R

32

Categories, Functors, and Natural Transformations[Chap. I(3) The functor U : Metc → Top defined byff

U ((X, d) −→ (X 0 , d0 )) = (X, τd ) −→ (X 0 , τd0 )(where τd denotes the topology induced on X by the metric d) is full and faithful,
but not an embedding.
(4) For any category A, the unique functor from A to 1[3.3(4)(c)] is faithful if and only
if A is thin.
(5) The discrete space functor D : Set → Top defined byff

D(X −→ Y ) = (X, δX ) −→ (Y, δY )(where δZ denotes the discrete topology on the set Z) is a full embedding.
(6) The indiscrete space functor N : Set → Top defined byff

N (X −→ Y ) = (X, ιX ) −→ (Y, ιY )(where ιZ denotes the indiscrete topology on the set Z) is a full embedding.
3.30 PROPOSITION
Let F : A → B and G : B → C be functors.
(1) If F and G are both isomorphisms (resp. embeddings, faithful, or full), then so is
G ◦ F.
(2) If G ◦ F is an embedding (resp. faithful), then so is F .
(3) If F is surjective on objects and G ◦ F is full, then G is full.
3.31 PROPOSITION
If F : A → B is a full, faithful functor, then for every B-morphism f : F A → F A0 ,there exists a unique A-morphism g : A → A0 with F g = f .
Furthermore, g is an A-isomorphism if and only if f is a B-isomorphism.
Proof: The morphism exists by fullness, and it is unique by faithfulness. Since by
Proposition 3.21 functors preserve isomorphisms, f is an isomorphism if g is. If
f : F A → F A0 is a B-isomorphism, let g 0 : A0 → A be the unique A-morphism withF (g 0 ) = f −1 . Then F (g 0 ◦ g) = F g 0 ◦ F g = f −1 ◦ f = idF A = F (idA ), so that byfaithfulness g 0 ◦ g = idA . Likewise g ◦ g 0 = idA0 . Hence g is an isomorphism.

3.32 COROLLARY
Functors F : A → B that are full and faithful reflect isomorphisms; i.e., whenever g isan A-morphism such that F (g) is a B-isomorphism, then g is an A-isomorphism. 

9th July 2006

Sec. 3]Categories and functors

33

Recall that isomorphic categories are considered as being essentially the same. This
concept of “sameness” is very restrictive. The following slightly weaker and more flexible
notion of “essential sameness” called equivalence of categories is much more frequently
satisfied. It will turn out that equivalent categories have the same behavior with respect
to all interesting categorical properties.
3.33 DEFINITION(1) A functor F : A → B is called an equivalence provided that it is full, faithful,and isomorphism-dense in the sense that for any B-object B there exists someA-object A such that F (A) is isomorphic to B.
(2) Categories A and B are called equivalent provided that there is an equivalencefrom A to B.
3.34 REMARK
It is shown in Proposition 3.36 that “is equivalent to” is an equivalence relation on theconglomerate of all categories.
3.35 EXAMPLES(1) Each isomorphism between categories is an equivalence. Hence isomorphic categories
are equivalent.
(2) The category Mat [3.3(4)(a)] is equivalent to the construct of finite-dimensional
vector spaces (and linear transformations), but is not isomorphic to it. The factthat there is no isomorphism can be deduced from the observation that in Mat
different objects cannot be isomorphic. An equivalence is given by the functor thatassigns to each natural number n ∈ Ob(Mat) the vector space n and to eachn × m matrix A ∈ Mor(Mat) the linear map from n to m that assigns to each(x1 , x2 , . . . , xn ) ∈ n the 1×m matrix [x1 x2 . . . xn ]A (given by matrix multiplication)
considered as an m-tuple in m .

R

R

R

R

R

(3) The constructs Metc of metric spaces and continuous maps and Topm of metrizable
topological spaces and continuous maps are equivalent. The functor that associateswith each metric space its induced topological space is an equivalence that is not an
isomorphism.
(4) Posets,18 considered as categories, are equivalent if and only if they are isomorphic.
However, preordered sets, considered as categories, can be equivalent without being
isomorphic (cf. Exercise 3H).
(5) The category of all minimal acceptors (i.e., those with a minimum number of states
for accepting the given language), as a full subcategory of Σ-Seq, is equivalent to
the poset of all recognizable languages (ordered by inclusion and considered as athin category). In fact, for two minimal acceptors A and A0 , there exists at mostone simulation A → A0 , and such a simulation exists if and only if A0 accepts eachword accepted by A.
18

A poset (or partially ordered set) is a pair (X, ≤) that consists of a set X and a transitive, reflexive,
and antisymmetric relation ≤ on X.

9th July 2006

34

Categories, Functors, and Natural Transformations[Chap. I

3.36 PROPOSITIONFG(1) If A −−→ B is an equivalence, then there exists an equivalence B −−→ A.
F

H

H◦F

(2) If A −−→ B and B −−→ C are equivalences, then so is A −−−−→ C.
Proof:(1) For each object B of B, choose an object G(B) of A and a B-isomorphism
εB : F (G(B)) → B. Since F is full and faithful, for each B-morphism g : B → B 0
there is a unique A-morphism G(g) : G(B) → G(B 0 ) with
0
F (G(g)) = ε−1
B 0 ◦ g ◦ εB : F (G(B)) → F (G(B )).

Hence G(g) is the unique A-morphism for which the diagram
F (G(B))F (G(g))/ F (G(B 0 ))

εB





B

g

(∗)

εB 0

/ B0

commutes. That G preserves identities follows immediately from the uniquenessrequirement in the above diagram. That G preserves composition follows from theuniqueness, the commutativity of the diagram
F (G(B))F (G(g))

εB





B

/ F (G(B 0 ))g

F (G(h))

εB 0

/ B0

/ F (G(B 00 ))


h

εB 00

/ B 00and the fact that F preserves composition. Thus G is a functor. G is full because0
for each A-morphism f : G(B) → G(B 0 ), the morphism εB 0 ◦ F (f ) ◦ ε−1
B : B →B
(which we denote by g) has the property that g ◦ εB = εB 0 ◦ F (f ), and this implies[by uniqueness for (∗)] that f = G(g). G is faithful since given B

g1
g2

//

B 0 withG(g1 ) = G(g2 ) = f , an application of (∗) yields−1−1
g1 = εB 0 ◦ F (G(g1 )) ◦ ε−1
B = εB 0 ◦ F (f ) ◦ εB = εB 0 ◦ F (G(g2 )) ◦ εB = g2 .

Finally, G is isomorphism-dense because in view of Proposition 3.31 for each Aobject A, the B-isomorphism εF A : F (G(F A)) → F A is the image of some A-isomorphism GF A → A.
(2) By Proposition 3.30 it suffices to show that H ◦ F is isomorphism-dense. Given a
C-object C, the fact that both F and H are isomorphism-dense gives a B-object
B, an isomorphism h : H(B) → C, and an A-object A, with an isomorphism
k : F (A) → B. Thus h ◦ H(k) : (H ◦ F )(A) → C is an isomorphism.


9th July 2006

Sec. 3]Categories and functors

35

3.37 REMARKThe concept of equivalence is especially useful when duality is involved. There are
numerous examples of pairs of familiar categories where each category is equivalent to
the dual of the other.
3.38 DEFINITION
Categories A and B are called dually equivalent provided that Aop and B are equivalent.
3.39 EXAMPLES(1) The construct Boo of boolean algebras is dually equivalent to the construct BooSpa
of boolean spaces (i.e., to the construct of zero-dimensional compact Hausdorff spaces
and continuous maps). An equivalence can be obtained by associating with eachboolean space its boolean algebra of clopen subsets (Stone Duality).
(2) The category of finite-dimensional real vector spaces is dually equivalent to itself.
An equivalence can be obtained by associating with each finite-dimensional vector
space its dual space [cf. 3.20(12)].
(3) Set is dually equivalent to the category of complete atomic boolean algebras andcomplete boolean homomorphisms. An equivalence can be obtained by associatingwith each set its power-set, considered as a complete atomic boolean algebra.
(4) The category of compact Hausdorff abelian groups is dually equivalent to Ab. An
equivalence can be obtained by associating with each compact Hausdorff abeliangroup G its group of characters hom(G, / ) (Pontrjagin Duality).

RZ(5) The category of locally compact abelian groups is dually equivalent to itself. An
equivalence can be obtained as in (4) above.
(6) The category HComp of compact Hausdorff spaces (and continuous functions) isdually equivalent to the category of C ∗ -algebras and algebra homomorphisms. An
equivalence can be obtained by associating with each compact Hausdorff space X
the C ∗ -algebra C(X, ) of complex-valued continuous functions (Gelfand-Naimark
Duality).

C

3.40 REMARK
Recall that we have formulated a duality principle related to objects, morphisms, andcategories (3.7). We now extend this to functors; i.e., we introduce for any functor
F : A → B, the concept of its dual functor that can be used to formulate the duals of
categorical statements involving functors. The reader may guess as to what the dualof a categorical statement involving a functor might be. Two common mistakes are toeither do too little (by dualizing just one of the categories A or B) or to do too much(by even reversing the arrow representing the functor F ). In either of these cases, one
generally does not obtain a new functor. The proper dual concept is the following:

9th July 2006

36

Categories, Functors, and Natural Transformations[Chap. I

3.41 DEFINITION
Given a functor F : A → B, the dual (or opposite) functor F op : Aop → Bop is thefunctor defined byf

Ff

F op (A −→ A0 ) = F A −−→ F A0 .
3.42 REMARK
Obviously (F op )op = F . To form the dual of a categorical statement that involves
functors, make the same statement, but with each category and each functor replacedby its dual. Then translate this back into a statement about the original categories andfunctors.
3.43 PROPOSITION
Each of the following properties of functors is self-dual: “isomorphism”, “embedding”,“faithful”, “full”, “isomorphism-dense”, and “equivalence”.
CATEGORIES OF CATEGORIES
We have seen above that functors act as morphisms between categories; they are closed
under composition, which is associative (since it is just the composition of functions between classes) and the identity functors act as identities with respect to the composition.
Because of this, one is tempted to consider the “category of all categories”. However,there are two difficulties that arise when we try to form this entity. First, the “categoryof all categories” would have objects such as Vec and Top, which are proper classes,
so that since proper classes cannot be elements of classes, the conglomerate of all objects would not be a class (thus violating condition 3.1(1) in the definition of category).
Second, given any categories A and B, it is not generally true that the conglomerateof all functors from A to B forms a set. This violates condition 3.1(2) in the definitionof category. However, if we restrict our attention to categories that are sets, then bothproblems are eliminated.
3.44 DEFINITION
A category A is said to be small provided that its class of objects, Ob(A), is a set.
Otherwise it is called large.
3.45 REMARK
Notice that when Ob(A) is a set, then M or(A) must be a set, so that the category
A = (Ob(A), hom, id, ◦) must also be a set (cf. Exercise 3M).
3.46 EXAMPLES
Mat is small; so are all preordered sets considered as categories, and all monoids considered as categories. However, Mon, the category of all monoids and monoid homomorphisms between them, is not small.

9th July 2006

Sec. 3]Categories and functors

37

3.47 DEFINITION
The category Cat of small categories has as objects all small categories, as morphisms
from A to B all functors from A to B, as identities the identity functors, and as composition the usual composition of functors.
3.48 REMARKS(1) That Cat is indeed a category follows immediately from the facts that(a) since each small category is a set, the conglomerate of all small categories is aclass, and(b) for each pair (A, B) of small categories, the conglomerate of all functors from
A to B is a set.
(2) Cat itself is not small. In fact, there are full embeddings from each of the constructs
Set and Mon into Cat (3.3(4)(c) and 3I).
Since (because of size restrictions) we can’t form the category of all categories, andother naturally occurring entities that we will want to investigate later, we introduce
the concept of “quasicategories”. This is done by freeing the concept of category fromits set-theoretical restrictions:3.49 DEFINITION
A quasicategory is a quadruple A = (O, hom, id, ◦) defined in the same way as acategory except that the restrictions that O be a class and that each conglomeratehom(A, B) be a set are removed. Namely,(1) O is a conglomerate, the members of which are called objects,(2) for each pair (A, B) of objects, hom(A, B) is a conglomerate called the conglomerateof all morphisms from A to B (with f ∈ hom(A, B) denoted by f : A → B),(3) for each object A, idA : A → A is called the identity morphism on A,(4) for each pair of morphisms (f : A → B , g : B → C) there is a composite morphismg ◦ f : A → C,
subject to the following conditions:(a) composition is associative,(b) identity morphisms act as identities with respect to composition,(c) the conglomerates hom(A, B) are pairwise disjoint.
3.50 DEFINITION
The quasicategory19 CAT of all categories has as objects all categories, as morphisms
from A to B all functors from A to B, as identities the identity functors, and as composition the usual composition of functors.
19

Frequently proper quasicategories (see 3.51(2)) will be denoted by all capital letters to distinguishthem from categories.

9th July 2006

38

Categories, Functors, and Natural Transformations[Chap. I

3.51 REMARK(1) Clearly each category is a quasicategory.
(2) CAT is a proper quasicategory in the sense that it is not a category. [Notice thathom(Set, Set) is not a set.](3) Virtually every categorical concept has a natural analogue or interpretation for quasicategories. The names for such quasicategorical concepts will be the same as thoseof their categorical analogues. Thus we have, for example, the notions of functor
between quasicategories, equivalence of quasicategories, discrete and thin quasicategories, etc. Because the main object of our study is categories, most notions will
only be specifically formulated for categories. However, we will freely make use of
implied quasicategorical analogues, especially when it allows clearer or more convenient expression. For example, at this point it is clear that an isomorphism between
categories (3.24) is precisely the same as an isomorphism in CAT (3.8). Not every
categorical concept has a reasonable quasicategorical interpretation. An outstanding example of this is the fact that quasicategories in general lack hom-functors intoSet.
(4) Dealing with quasicategories and forming CAT gives us the possibility of applying
category theory to itself. There are advantages to doing this (some of which areindicated above) as well as certain dangers. One danger is the tendency to want
to form something like the “quasicategory of all quasicategories”. However, to doso causes a Russell-like paradox that cannot be salvaged within our foundational
system, as outlined in §2. Because our main interest is in categories, as opposed toquasicategories, we will never need to consider such an entity as the “quasicategoryof all quasicategories”.

OBJECT-FREE DEFINITION OF CATEGORIES
Because of the bijection between the class of objects and the class of identity morphisms
in any category (given by A 7→ idA ) and the fact that identities in a category can becharacterized by their behavior with respect to composition, it is possible to obtain an“object-free” definition of category. This definition, given below, is formally simpler
than the original one and is “essentially” equivalent to it (3.55). The reason for choosing
the definition given in 3.1 is that it is more closely associated with standard examples
of categories.
3.52 DEFINITION(1) A partial binary algebra is a pair (X, ∗) consisting of a class X and a partial
binary operation ∗ on X; i.e., a binary operation defined on a subclass of X × X.
[The value of ∗(x, y) is denoted by x ∗ y.](2) If (X, ∗) is a partial binary algebra, then an element u of X is called a unit of (X, ∗)provided that x ∗ u = x whenever x ∗ u is defined, and u ∗ y = y whenever u ∗ y isdefined.

9th July 2006

Sec. 3]Categories and functors

39

3.53 DEFINITION
An object-free category is a partial binary algebra C = (M, ◦), where the membersof M are called morphisms, that satisfies the following conditions:(1) Matching Condition: For morphisms f , g, and h, the following conditions are equivalent:(a) g ◦ f and h ◦ g are defined,(b) h ◦ (g ◦ f ) is defined, and(c) (h ◦ g) ◦ f is defined.
(2) Associativity Condition: If morphisms f , g, and h satisfy the matching conditions,then h ◦ (g ◦ f ) = (h ◦ g) ◦ f .
(3) Unit Existence Condition: For every morphism f there exist units uC and uD of(M, ◦) such that uC ◦ f and f ◦ uD are defined.
(4) Smallness Condition: For any pair of units (u1 , u2 ) of (M, ◦) the class hom(u1 , u2 ) ={f ∈ M | f ◦ u1 and u2 ◦ f are defined} is a set.
3.54 PROPOSITION
If A is a category, then(1) (M or(A), ◦) is an object-free category, and(2) an A-morphism is an A-identity if and only if it is a unit of (M or(A), ◦).
Proof: (M or(A), ◦) is clearly a partial binary algebra, where f ◦ g is defined if and onlyu
if the domain of f is the codomain of g. Thus each A-identity is a unit. If A −→ B is aunit in (M or(A), ◦), then u = u ◦ idA = idA , where the first equality holds since idA is
a A-identity and the second one holds since u is a unit. Thus (2) is established. From
this, (1) is immediate.

3.55 REMARK
We now have two versions of the concept of category, the “standard” one (3.1), which
is more intuitive and is more easily associated with familiar examples, and the “objectfree” one (3.53), which is more succinctly stated and so, in many cases, more convenient
to use. Next we will see that these two concepts are equivalent. Proposition 3.54 showsthat with every category we can associate an object-free category. Even though thiscorrespondence is neither injective nor surjective, it provides an equivalence between
the “standard” and the “object-free” definitions of category. This claim can be madeprecise as follows:(1) One can define functors between object-free categories to be functions between theirclasses of morphisms that preserve both units (= identities) and composition.
(2) Parallel to the definition of the quasicategory CAT of all categories one can define
the quasicategory CATof of all object-free categories and functors between them.

9th July 2006

40

Categories, Functors, and Natural Transformations[Chap. I(3) The correspondence from Proposition 3.54 is the object part of a functor between
the quasicategories CAT and CATof that can be shown to be an equivalence in thesense of Definition 3.33.
In this sense the two concepts of category are essentially the same; i.e., essentially thesame “category theory” will result if one proceeds from either of the two formulations.

EXERCISES
3A. Graphs of Categories
A graph is a quadruple (V, E, d, c) consisting of a set V (of vertices), a set E (of(directed) edges), and functions d, c : E → V (giving the domain and codomain of anedge). A large graph is the same concept except that V and E are allowed to beclasses. The graph G(A) of a category A is the obvious large graph with V = Ob(A)and E = M or(A).
(a) Verify that a thin category is determined up to isomorphism by its graph.
(b) Find two non-isomorphic categories with the same graph.
(c) Determine which of the following graphs are of the form G(A) for some category A(where vertices and identity edges are indicated by dots, and non-identity edges areindicated by arrows).
(1) • -•(3) • 

-

•

@
R
@

- •

(2) • 



•



- •

(4) • - ••(d) Show that for each of the following graphs G there exists up to isomorphism precisely
one category A with G(A) = G.
(1) • 

-

• 



(2) • - •6(e) The free category generated by a graph (V, E, d, c) is the category A with Ob(A) =V , M or(A) = all paths (= all finite sequences in E in which the domain of eachedge is the codomain of the preceding one), composition is the obvious compositionof paths, and identity morphisms are the empty paths. Verify that A is indeed acategory.

9th July 2006

Sec. 3]Categories and functors

41

3B. Pointed Categories(a) Show that there is a category whose objects are all pairs of the form (A, a), where
A is a set and a ∈ A andhom((A, a), (B, b)) = { f | f : A → B and f (a) = b }.
This is called the category of pointed sets (and base-point-preserving functions).
It is denoted by pSet.
(b) Show that there is a faithful functor U : pSet → Set with the property that U (f ) =f for each base-point-preserving function, f . Does U reflect isomorphisms?
(c) Prove that Set and pSet are not equivalent.
(d) Mimic the above construction of the category of pointed sets to obtain the categories
pTop of pointed topological spaces and pGrp of pointed groups. Determine
whether or not Top and pTop (resp. Grp and pGrp) are equivalent or isomorphic.
3C. Alternative Definition of Category
Define a category of type 2 to be a quintuple A = (O, M, dom, cod, ◦) consisting of(1) a class O, of A-objects,(2) a class M, of A-morphisms,(3) functions dom : M → O and cod : M → O, assigning to each morphism its domain
and codomain, and(4) a function ◦ from D = {(f, g) | f, g ∈ M and dom(f ) = cod(g)} to M [with ◦(f, g)
written f ◦ g],subject to the following conditions:(a) If (f, g) ∈ D, then dom(f ◦ g) = dom(g) and cod(f ◦ g) = cod(f ).
(b) If (f, g) and (h, f ) belong to D, then h ◦ (f ◦ g) = (h ◦ f ) ◦ g.
(c) For each A ∈ O there exists a morphism e such that dom(e) = A = cod(e) and(1) f ◦ e = f whenever (f, e) ∈ D, and(2) e ◦ g = g whenever (e, g) ∈ D.
(d) For any (A, B) ∈ O × O, the class{f | f ∈ M, dom(f ) = A, and cod(f ) = B}
is a set.
Compare the definition of category of type 2 with that of category (3.1) and objectfree category (3.53) and determine in which sense these definitions can be considered“equivalent”.
If functors between categories of type 2 are defined to be functions between their morphism classes that preserve identities and composition, show that the quasicategory
CAT2 of categories of type 2 and functors between them is equivalent to CAT and to
CATof .

9th July 2006

42

Categories, Functors, and Natural Transformations[Chap. I

3D. Identities(a) Show that whenever A = (O, hom, id, ◦) and A∗ = (O, hom, id∗ , ◦) are categories,
then A = A∗ .
(b) A functor F is said to reflect identities provided that if F (k) is an identity thenk must be an identity. Show that every isomorphism between categories reflects
identities, but that not every equivalence does.
3E. Duality(a) Show that none of the following categories is dually equivalent to itself: Set, Vec(but compare 3.39(2)), Grp, Top, Rel, Pos.
(b) Determine whether or not the category of finite sets is dually equivalent to itself.
(c) Establish the following consequence of the duality principle:
If S is a categorical statement, then S holds for all categories satisfying
property P if and only if S op holds for all categories satisfying P op .
3F. Isomorphisms(a) Describe the isomorphisms in each of the categories Met, Metu , and Metc .
(b) Show that a monoid is a group if and only if, considered as a category, each of its
morphisms is an isomorphism.
3G. Functors(a) Consider the category A with exactly one object A, with hom(A, A) = {a, b}, andwith composition defined by a ◦ a = a and a ◦ b = b ◦ a = b ◦ b = b. Consider further
the category B with exactly one object B, with hom(B, B) = {b}, and with b ◦ b = b.
Let F be defined by F (B) = A, F (b) = b. Is F : B → A a functor?
(b) Show that a category A is thin (resp. empty) if and only if every functor with domain
A is faithful (resp. full).
3H. Equivalences(a) Show that none of the following categories is equivalent to any of the others: Set,
Vec, Grp, Top, Rel, Pos.
(b) Show that two posets (resp. monoids), considered as categories, are equivalent if andonly if they are isomorphic.
(c) Show that, considered as categories, every preordered set is equivalent to a poset,
and that a preordered set is isomorphic to a poset if and only if it is a poset itself.
(d) Show that an equivalence is an embedding if and only if it reflects identities.
(e) Prove that categories A and B are isomorphic if and only if there exists an equivalence E : A → B such that for each A-object A the number of isomorphic copies of
A in A coincides with the number of isomorphic copies of E(A) in B.

9th July 2006

Sec. 3]Categories and functors

43

3I.
Show that there is a full embedding from the category Mon of monoids into Cat.
3J.
Show that there is a covariant power-set functor R : Set → Set that is different fromthe one given in Example 3.20(8): such that on objects A, RA is the power set of A,fand on functions A −→ B, (Rf )(X) = { b ∈ B | f −1 (b) ⊆ X }.
3K. Comma CategoriesFG

If A −−→ C and B −−→ C are functors, then the comma category (F ↓ G) is thecategory whose objects are triples (A, f, B) with A ∈ Ob(A), B ∈ Ob(B), andf

F A −→ GB ∈ Mor(C); whose morphisms from (A, f, B) to (A0 , f 0 , B 0 ) are pairs (a, b)abwith A −→ A0 ∈ Mor(A), and B −→ B 0 ∈ Mor(B) such that the square
FA
f

Fa





GB

/ F A0Gb

f0/ GB 0commutes; whose identities are id(A,f,B) = (idA , idB ); and whose composition is definedcomponentwise, i.e., by (a, b) ◦ (a, b) = (a ◦ a, b ◦ b).
(a) Verify that (F ↓ G) is indeed a category.
(b) For any category A and any A-object K, denote by CK : 1 → A the constantfunctor with value K. Give explicit descriptions of• the comma category (idA ↓ CK ), also denoted by (A ↓ K) and called thecategory of objects over K,• the comma category (CK ↓ idA ), also denoted by (K ↓ A) and called thecategory of objects under K,• the comma category (idA ↓ idA ), also denoted by A2 and called the arrow
category of A.
(c) Show that (A ↓ K)op = (K ↓ Aop ).
(d) Show that if P is a singleton set, then (P ↓ Set) is isomorphic with pSet (cf.
Exercise 3B).
3L. Quasicategories as Objects
Show that one may not form the “quasicategory of all quasicategories”. [Hint: Russell’sparadox].
3M. Small Categories
Let A be a category whose object-class is a set. Show that (a) – (d) below are true.
[Hint: see 2.1.]

9th July 2006

44

Categories, Functors, and Natural Transformations(a) M or(A) =S[Chap. I{hom(A, B) | (A, B) ∈ Ob(A) × Ob(A)} is a set.

(b) hom : Ob(A) × Ob(A) → P(M or(A)) is a set.
(c) ◦ ⊆ M or(A) × M or(A) × M or(A) is a set.
(d) A is a set.
3N. Decompositions of Functors
F

Let A −−→ B be a functor. Show thatGH(a) There exist functors A −−→ C and C −−→ B with the following properties:(1) F = H ◦ G,(2) G is full and bijective on objects,(3) H is faithful,FGH(4) whenever A −−→ B = A −−→ C −−→ B and H is faithful, then there exists aK

unique functor C −−→ C, such that the diagram
/C
~
K ~~
H
G
~~
 ~~

/B
C

A

G

(∗)

H

commutes, i.e., G = K ◦ G and H = H ◦ K.
GH(b) There exist functors A −−→ C and C −−→ B with the following properties:(1) F = H ◦ G,(2) G is bijective on objects,(3) H is full and faithful,FGH(4) whenever A −−→ B = A −−→ C −−→ B and H is full and faithful, then thereK

exists a unique functor C −−→ C such that the diagram (∗) above commutes.

9th July 2006

4 Subcategories
In §3 we have seen several instances where one category is included in another (Abin Grp, HComp in Top, etc.). In this section we investigate this phenomenon morethoroughly.
4.1 DEFINITION(1) A category A is said to be a subcategory of a category B provided that thefollowing conditions are satisfied:(a) Ob(A) ⊆ Ob(B),(b) for each A, A0 ∈ Ob(A), homA (A, A0 ) ⊆ homB (A, A0 ),(c) for each A-object A, the B-identity on A is the A-identity on A,(d) the composition law in A is the restriction of the composition law in B to themorphisms of A.
(2) A is called a full subcategory of B if, in addition to the above, for eachA, A0 ∈ Ob(A), homA (A, A0 ) = homB (A, A0 ).
4.2 REMARKS(1) Because of the nature of full subcategories, a full subcategory of a category B canbe specified by merely specifying its object class within B.
(2) Notice that conditions (a), (b), and (d) of part (1) of the definition do not imply(c). (See Exercise 4A.)
(3) If F : A → B is a full functor or is injective on objects, then the image of A under
F is a subcategory of B. However, for arbitrary functors F : A → B, the image of• /•A under F need not be a subcategory of B. Consider, e.g., a functor from• /•• @@ / •@  .
to•
4.3 EXAMPLES(1) For any category A, the empty category and A itself are full subcategories of A.
(2) The class of all Hausdorff spaces specifies the full subcategory Haus of Top; likewise,
all Tychonoff spaces (i.e., completely regular T1 spaces) yields a full subcategory
Tych of Haus; and HComp is a full subcategory of Tych.
(3) The class of all preordered sets (i.e., all sets supplied with a reflexive and transitive
relation) determines a full subcategory Prost of Rel. The class of all partiallyordered sets (i.e., all sets supplied with a reflexive, transitive and antisymmetric
relation) determines a full subcategory Pos of Prost. The category Lat that consistsof all lattices (i.e., all partially ordered sets for which each pair of elements has a

45

46

Categories, Functors, and Natural Transformations[Chap. I

meet and a join) together with all lattice homomorphisms (i.e., all maps preserving
meets and joins of pairs) is a nonfull subcategory of Pos. The category JCPos ofcomplete lattices and join-preserving maps is a nonfull subcategory of Pos. The
category CLat of complete lattices and meet- and join-preserving maps is a nonfull
subcategory of JCPos that has the same objects as JCPos.
(4) The category Grp of groups is a full subcategory of the construct Mon that consists of all monoids (i.e., semigroups with unit) and monoid homomorphisms (i.e.,
unit-preserving semigroup homomorphisms). Mon is a nonfull subcategory of theconstruct Sgr of all semigroups and semigroup homomorphisms.
(5) Ban is a nonfull subcategory of Banb that has the same class of objects.
(6) The subcategories of a monoid M , considered as a category, are precisely the empty
category and the submonoids of M .
4.4 REMARKFor every subcategory A of a category B there is a naturally associated inclusion
functor E : A ,→ B. Moreover, each such inclusion is(1) an embedding;(2) a full functor if and only if A is a full subcategory of B.
As the next proposition shows, inclusions of subcategories are (up to isomorphism)
precisely the embedding functors and (up to equivalence) precisely the faithful functors.

A full embedding
4.5 PROPOSITION(1) A functor F : A → B is a (full) embedding if and only if there exists a (full)subcategory C of B with inclusion functor E : C → B and an isomorphism
G : A → C with F = E ◦ G.
(2) A functor F : A → B is faithful if and only if there exist embeddings E1 : D → B
and E2 : A → C and an equivalence G : C → D such that the diagram
A
E2

F

E1



C

/B
O

G

/D
9th July 2006

Sec. 4]

Subcategories

47

commutes.
Proof:(1). One direction is immediate and the other is a consequence of the fact that (full)embeddings are closed under composition.
(2). One direction follows from the compositive nature of faithful functors. To show,
conversely, that every faithful functor can be decomposed as stated, let E1 : D → B
be the inclusion of the full subcategory D of B that has as objects all images (underF ) of A-objects. Let C be the category with Ob(C) = Ob(A), withhomC (A, A0 ) = homB (F A, F A0 ),and with identities and composition defined as in B. C is easily seen to be acategory. Now define functors E2 : A → C and G : C → D byf

Ff

E2 (A −→ A0 ) = A −−→ A0gg

and G(C −→ C 0 ) = F C −→ F C 0 .

Then E2 is an embedding, G is an equivalence, and F = E1 ◦ G ◦ E2 .



4.6 DEFINITION
A category A is said to be fully embeddable into B provided that there exists a fullembedding A → B, or, equivalently, provided that A is isomorphic to a full subcategoryof B.
4.7 EXAMPLES
Although it is far from easy to prove (see the monograph Pultr-Trnková in the References), each category of the form Alg(Ω) is fully embeddable into each of the following
constructs:(a) Sgr,(b) Rel,(c) Alg(1, 1), i.e., the construct of unary algebras on two operations.
Under an additional set-theoretical hypothesis (the non-existence of measurable cardinals), every construct is fully embeddable into Sgr (or Rel or Alg(1, 1)).
4.8 REMARK
Because full subcategories are determined by their object classes, they are often regarded
as “properties of objects”. Since most of the interesting properties P satisfy the condition
that whenever an object A has property P then every object isomorphic to A also has
P , we often require that full subcategories have the property (defined below) of being
isomorphism-closed.
4.9 DEFINITION
A full subcategory A of a category B is called

9th July 2006

48

Categories, Functors, and Natural Transformations[Chap. I(1) isomorphism-closed provided that every B-object that is isomorphic to someA-object is itself an A-object,(2) isomorphism-dense provided that every B-object is isomorphic to some A-object.
4.10 REMARK
If A is a full subcategory of B, then the following conditions are equivalent:(1) A is an isomorphism-dense subcategory of B,(2) the inclusion functor A ,→ B is isomorphism-dense,(3) the inclusion functor A ,→ B is an equivalence.
4.11 EXAMPLE
The full subcategory of Set with the single object(of natural numbers) is neither
isomorphism-closed nor isomorphism-dense in Set. It is equivalent to the isomorphismclosed full subcategory of Set consisting of all countable infinite sets.

N

There are instances when one wishes to consider full subcategories in which different
objects cannot be isomorphic:
4.12 DEFINITION
A skeleton of a category is a full, isomorphism-dense subcategory in which no two
distinct objects are isomorphic.
4.13 EXAMPLES(1) The full subcategory of all cardinal numbers is a skeleton for Set.
(2) The full subcategory determined by the powers
cardinal numbers, is a skeleton for Vec.

Rm, where m runs through all

4.14 PROPOSITION(1) Every category has a skeleton.
(2) Any two skeletons of a category are isomorphic.
(3) Any skeleton of a category C is equivalent to C.
Proof:(1). This follows from the Axiom of Choice [2.3(4)] applied to the equivalence relation“is isomorphic to” on the class of objects of the category.
(2). Let A and B be skeletons of C. Then each A-object A is isomorphic in C to aunique B-object. Denote the latter by F (A) and choose for each A-object A a
C-isomorphism fA : A → F (A). Then the functor F : A → B defined by:
h

f −1

h

f 0

F (A −→ A0 ) = F A −−A−→ A −→ A0 −−A−→ F A0
is an isomorphism.

9th July 2006

Sec. 4]

Subcategories

49(3). The inclusion of a skeleton of C into C is an equivalence. (See 4.10(2).)
4.15 COROLLARY
Two categories are equivalent if and only if they have isomorphic skeletons.



REFLECTIVE AND COREFLECTIVE SUBCATEGORIES
4.16 DEFINITION
Let A be a subcategory of B, and let B be a B-object.
r(1) An A-reflection (or A-reflection arrow) for B is a B-morphism B −→ A from B
to an A-object A with the following universal property:
ffor any B-morphism B −→ A0 from B into some A-object A0 , there exists a uniqueA-morphism f 0 : A → A0 such that the triangle
/A
AA
AA
f0
f AA BAr

A0

commutes.
By an “abuse of language” an object A is called an A-reflection for B provided thatr
there exists an A-reflection B −→ A for B with codomain A.
(2) A is called a reflective subcategory of B provided that each B-object has an
A-reflection.
4.17 EXAMPLES
Several familiar constructions in mathematics such as certain “completions”, certain
formations of “quotients”, and certain “modifications” of structures can be regarded ina natural way as reflections. Here we list a few such examples of reflective subcategories
A of categories B. In every example except the last two, A is a full subcategory of B.
A. Modifications of the Structure(1) Making a relation symmetric: B = Rel, A = Sym, the full subcategory of symidX
metric relations. (X, ρ) −−−→ (X, ρ ∪ ρ−1 ) is an A-reflection20 for (X, ρ).

(2) Making a topological space completely regular: B = Top, A = the full subcategoryof completely regular (not necessarily T1 ) spaces. If (X, τ ) is a topological space,then the collection of all cozero sets21 in (X, τ ) is a base for a topology τc on X.
idX(X, τ ) −−−→ (X, τc ) is an A-reflection for (X, τ ).
20 −1
21

ρ = {(y, x) | (x, y) ∈ ρ}.
A is called a cozero set in (X, τ ) provided that there is a continuous map f : (X, τ ) →A = X − f −1 [{0}].

9th July 2006

R with50

Categories, Functors, and Natural Transformations[Chap. I

B. Improving Objects by Forming Quotients(3) Making a preordered set partially ordered: B = Prost (preordered sets and orderpreserving maps), A = Pos. If (X, ≤) is a preordered set, define an equivalence
relation ≈ on X by: x ≈ y ⇔ (x ≤ y and y ≤ x). Let p : X → X/ ≈ be thepcanonical map. Then (X, ≤) −→ (X/ ≈ , (p × p)[≤]) is an A-reflection for (X, ≤).
(4) Making a group abelian: B = Grp, A = Ab. Let G be a group and let G0 be thecommutator subgroup of G. Then the canonical map G → G/G0 is an A-reflection
for G.
(5) Making a topological space T0 : B = Top, A = Top0 , the full subcategory consisting of all T0 spaces. Let X be a topological space, and let ≈ be the equivalence
relation on X, given by: x ≈ y ⇔ the closure of {x} = the closure of {y}. Thenthe canonical map X → X/ ≈ is an A-reflection for X.
(6) Making an abelian group torsion-free: B = Ab, A = TfAb, the full subcategoryof torsion-free abelian groups. Let G be an abelian group and let T G be the torsion
subgroup of G. Then the canonical map G → G/T G is an A-reflection for G.
(7) Making a reachable acceptor minimal: B = the full subcategory of Σ-Seq consisting of all reachable acceptors (i.e., those for which each state can be reachedfrom the initial one by an input word), A = the full subcategory of B consisting of
all minimal acceptors (i.e., those reachable acceptors with the property that notwo different states are observably equivalent. The observability equivalence ≈on a reachable acceptor B is given by: q ≈ q 0 provided that whenever the initialstate of B is changed to q, the resulting acceptor recognizes the same language as it
does when the initial state is changed to q 0 ). Then the canonical map B → B/ ≈,where B/ ≈ is the induced acceptor on the set of all ≈-equivalence classes of statesof B, is an A-reflection for B.
C. Completions(8) B = Metu (metric spaces and uniformly continuous maps) or Met(metric spaces
and non-expansive maps), A = the full subcategory of complete metric spaces. In
either case the metric completion (X, d) ,→ (X ∗ , d∗ ) is an A-reflection for (X, d).
(9) B = Tych, A = HComp. If X is a Tychonoff space, then the Čech-Stone compactification X ,→ βX is an A-reflection for X.
(10) B = JPos (posets and join-preserving maps), A = JCPos. Let (B, ≤) be a poset,
and let B ∗ be the collection of all subsets S of B that satisfy(a) S is an lower-set, i.e., b ∈ S and b0 ≤ b ⇒ b0 ∈ S,(b) S is closed under the formation of all existing joins, i.e., if A ⊆ S and b is thejoin of A in (B, ≤), then b ∈ S.
Then (B ∗ , ⊆) is a complete lattice, and the function(B, ≤) → (B ∗ , ⊆),defined by

b 7→ {x ∈ B | x ≤ b}

9th July 2006

Sec. 4]

Subcategories

51

is an A-reflection for (B, ≤). Note that these A-reflections differ, in general, from
the Mac Neille completions. For example, an A-reflection for a 3-element discrete
poset is an 8-element complete lattice, whereas the Mac Neille completion is a5-element one.
(11) B = Pos, A = the nonfull (!) subcategory JCPos. Let (B, ≤) be a poset and let
B̃ be the collection of all lower-sets S of B [cf. (10)(a)]. Then (B̃, ⊆) is a complete
lattice and the function (B, ≤) → (B̃, ⊆) defined as in (10) is an A-reflection.
(12) B = Sgr, A = the nonfull (!) subcategory Mon. If (X, •) is a semigroup, thenthe extension (X, •) ,→ (X ∪ {e}, ˆ•, e), obtained by adding a unit element e 6∈ Xof the operation ˆ•, is an A-reflection for (X, •).
4.18 REMARK
Observe that in the last two examples given above, the reflection arrows are neversurjective. This fact may seem surprising since in each of the other examples each
A-reflection arrow for an A-object is an isomorphism. It is not surprising, however, inview of Proposition 4.20.
4.19 PROPOSITION
Reflections are essentially unique, i.e.,rr̂(1) if B −→ A and B −→ Â are A-reflections for B, then there exists an A-isomorphismk : A → Â such that the triangle
r /
A
??
??
? k
r̂ ?? B?

Âcommutes,rk(2) if B −→ A is an A-reflection for B and A −→ Â is an A-isomorphism, thenk◦r

B −−−→ Â is an A-reflection for B.
Proof:(1). The existence of a morphism k with r̂ = k ◦r follows from the definition of reflection
and the fact that Â is an A-object. Similarly there is a morphism k̂ with r = k̂ ◦ r̂.
Now (k̂ ◦ k) ◦ r = r = idA ◦ r, so that by the uniqueness requirement in the definitionof reflection, k̂ ◦ k = idA . Analogously, one can see that k ◦ k̂ = idÂ , so that k is an
isomorphism.
(2). Obvious.
4.20 PROPOSITION
If A is a reflective subcategory of B, then the following conditions are equivalent:(1) A is a full subcategory of B,

9th July 2006



52

Categories, Functors, and Natural Transformations[Chap. I

idA(2) for each A-object A, A −−−→ A is an A-reflection,r(3) for each A-object A, A-reflection arrows A −−A→ A∗ are A-isomorphisms,r(4) for each A-object A, A-reflection arrows A −−A→ A∗ are A-morphisms.
f

Proof: That (1) ⇒ (2) ⇒ (3) ⇒ (4) is clear. To see that (4) ⇒ (1), let A −→ A0 be a
B-morphism between A-objects. By the definition of reflection there is an A-morphismfA∗ −→ A0 with f = f ◦ rA . Thus f is the composite of A-morphisms, and so must be
an A-morphism.

4.21 REMARKThere exist nonfull reflective subcategories A of B such that every A-reflection arrow isa B-isomorphism; e.g., let A = • → • and B = •  •.
4.22 PROPOSITION
Let A be a reflective subcategory of B, and for each B-object B let rB : B → AB bean A-reflection arrow. Then there exists a unique functor R : B → A such that thefollowing conditions are satisfied:(1) R(B) = AB for each B-object B,(2) for each B-morphism f : B → B 0 , the diagramrB

B/ R(B)

f



B0



R(f )

0
/
rB 0 R(B )commutes.
Proof: There exists a unique function R on objects satisfying (1) and (by the definitionof A-reflection arrows) a unique function R on morphisms satisfying (2). It remains tobe shown that R is a functor, i.e., that R preserves identities and composition. The firstfact follows from the commutativity of the diagram
B
idB

rB



B

/ RB


rB

idRB

/ RB

and the second one from the commutativity of the diagram
B
g ◦f

rB



B 00

/ RB


rB 00

Rg ◦Rf/ RB 00fg

obtained by pasting together the corresponding diagrams for B −→ B 0 and B 0 −→ B 00 .

9th July 2006

Sec. 4]

Subcategories

53

4.23 DEFINITION
A functor R : B → A constructed according to the above proposition is called a reflector for A.
4.24 REMARK
If A is a reflective subcategory of B, then a reflector for A depends upon the choiceof the reflection arrows. Hence, there are usually many different reflectors for A. In
Proposition 6.7 we will see that any two such reflectors are essentially the same; i.e.,they will turn out to be “naturally isomorphic”.
The dual of the concept reflective subcategory is coreflective subcategory. That is, A
is a coreflective subcategory of B if and only if Aop is a reflective subcategory of Bop .
Although each of the above statements is an adequate definition, to aid the reader
we provide a detailed dual formulation that doesn’t involve the dual categories. We
occasionally provide such explicit dual formulations although, strictly speaking, to do so
is redundant.
4.25 DEFINITION
Let A be a subcategory of B and let B be a B-object.
c(1) An A-coreflection (or A-coreflection arrow) for B is a B-morphism A −→ Bfrom an A-object A to B with the following universal property:
ffor any B-morphism A0 −→ B from some A-object A0 to B there exists a uniqueA-morphism f 0 : A0 → A such that the triangle
A0 A

AA f
AA
AA

A c /Bf0

commutes.
By an “abuse of language” an object A is called an A-coreflection for B providedcthat there exists an A-coreflection A −→ B for B with domain A.
(2) A is called a coreflective subcategory of B provided that each B-object has an
A-coreflection.
4.26 EXAMPLES
Several mathematical constructions such as certain “modifications” of structures andcertain “selections” of convenient subobjects can be regarded in a natural way as coreflections. Here we list a few examples of full coreflective subcategories A of categories
B.
A. Modifications of the Structure

9th July 2006

54

Categories, Functors, and Natural Transformations[Chap. I
idX(1) Making a relation symmetric: B = Rel, A = Sym. (X, ρ ∩ ρ−1 ) −−−→ (X, ρ) is
an A-coreflection for (X, ρ).
This example shows one of the rare instances where a subcategory is simultaneously
reflective and coreflective [cf. 4.17(1)].

(2) Making a topological space sequential: B=Top, A= the full subcategory of sequential spaces (i.e., spaces in which every sequentially closed set is closed). If (X, τ )is a topological space, then τ 0 = {A ⊆ X | X \ A is sequentially closed in (X, τ )}id

X
is a topology on X and (X, τ 0 ) −−−→ (X, τ ) is an A-coreflection for (X, τ ).

B. Sorting Out Convenient Subobjects(3) Making an abelian group a torsion group: B = Ab, A = the full subcategory ofabelian torsion groups. For any abelian group G the canonical embedding T G ,→ Gof the torsion-subgroup T G of G into G is an A-coreflection for G.
(4) Making a sequential acceptor reachable: B = Σ-Seq, A = the full subcategory ofreachable acceptors. For any acceptor A the canonical embedding RA ,→ A of theacceptor RA, formed by removing from the state set of A all states that cannot bereached from the initial state, is an A-coreflection for A. Thus a minimalizationof sequential acceptors is obtained in two steps: first the coreflection, which yields
a reachable acceptor, and then the reflection, which gives the minimal quotient of
the reachable part [cf. 4.17(7)].
4.27 PROPOSITIONc
If A is a coreflective subcategory of B and for each B-object B, AB −−B→ B is an Acoreflection arrow, then there exists a unique functor C : B → A (called a coreflector
for A) such that the following conditions are satisfied:(1) C(B) = AB for each B-object B,(2) for each B-morphism f : B → B 0 the diagram
C(B)
C(f )



cB

/B
f



C(B 0 ) c 0 / B 0
Bcommutes.

D 22

Suggestions for Further Reading
Herrlich, H. Topologische Reflexionen und Coreflexionen. Lecture Notes in Mathematics, Vol. 78, Springer, Berlin–Heidelberg–New York, 1968.
22

See Remark 3.7.

9th July 2006

Sec. 4]

Subcategories

Banaschewski, B., and E. Nelson. Completions of partially ordered sets.
Comp. 11 (1982): 521–528.

55
SIAM J.

Kelly, G. M. On the ordered set of reflective subcategories. Bull. Austral. Math. Soc.
36 (1987): 137–152.
Tholen, W. Reflective subcategories. Topol. Appl. 27 (1987): 201–212.

EXERCISES
4A. Subcategories and Identities
Consider the categories A and B described in Exercise 3G(a). If A = B, then is B a
subcategory of A?
4B. Isomorphism-Closed Subcategories
A (not necessarily full) subcategory A of a category B is called isomorphism-closed
provided that every B-isomorphism with domain in A belongs23 to A. Show that every
subcategory A of B can be embedded into a smallest isomorphism-closed subcategory
A0 of B that contains A. The inclusion functor A ,→ A0 is an equivalence iff all Bisomorphisms between A-objects belong to A.
4C. Full Subcategories(a) Show that a category is discrete if and only if each of its subcategories is full.
(b) Show that in a poset, considered as a category,• every subcategory is isomorphism-closed,• every (co)reflective subcategory is full.
4D. Reflective Subcategories of Special Categories* (a) Show that Set has• precisely three full, isomorphism-closed, reflective subcategories,• precisely two full, isomorphism-closed, coreflective subcategories,• infinitely many reflective subcategories,• infinitely many coreflective subcategories.
* (b) Show that HComp has precisely two full, isomorphism-closed, coreflective subcategories.
(c) Show that a full subcategory A of a poset B, considered as category, is reflectivein B if and only if for each element b of B the set {a ∈ A | b ≤ a} has a smallest
element.
23

Observe that if a morphism belongs to A, so does its domain and its codomain.

9th July 2006

56

Categories, Functors, and Natural Transformations[Chap. I(d) Consider the poset of natural numbers as a category, B. Verify that a subset A ofB, considered as full subcategory of B, is• reflective in B if and only if A is infinite,• coreflective in B if and only if 0 ∈ A.
(e) Show that no finite monoid, considered as a category, has a proper reflective subcategory. However, if A is the monoid of all maps from
into , considered asa category, then the subcategory of A, consisting of all maps f :→withf (0) = 0, is reflective in A.

N

N

N

N

(f) Show that the (nonfull) subcategory A of Pos, consisting of those posets for which
every nonempty subset has a meet, and those morphisms that preserve meets ofidnonempty subsets, is reflective in Pos. For a poset A, A −−→ A is an A-reflection if
and only if A is well-ordered.
(g) Verify that the category •// •has no proper reflective subcategory.

4E. Subcategories That Are Simultaneously Reflective and Coreflective(a) Show that in Rel the full subcategory of symmetric relations is both reflective andcoreflective. What about the full subcategory of reflexive relations, . . ., or of transitive relations?
(b) Verify that in Alg(1), the category of unary algebras on one operation, all idempotent algebras form a full subcategory that is both reflective and coreflective.
(c) Show that the poset of natural numbers, considered as a category, has infinitely many
isomorphism-closed subcategories that are simultaneously reflective and coreflective.
[Cf. 4D(c).]*(d) Show that neither Set nor Top has a proper isomorphism-closed full subcategorythat is both reflective and coreflective. What about nonfull isomorphism-closed
subcategories of Set?
4F. Intersections of Reflective Subcategories(a) Show that in the poset of natural numbers, considered as a category B, the following hold:• the intersection of any nonempty family of coreflective subcategories of B iscoreflective in B,• every full subcategory of B is an intersection of two reflective full subcategories of B.
* (b) Let BiTop be the construct of bitopological spaces (i.e., triples (X, τ1 , τ2 ), where τ1and τ2 are topologies on X) and bicontinuous maps (i.e., maps that are separately
continuous with respect to the first topologies and with respect to the second
topologies). Verify that the full subcategory B1 that consists of all bitopologicalspaces with τ1 compact Hausdorff is reflective in BiTop. By symmetry the full

9th July 2006

Sec. 4]

Subcategories

57

subcategory B2 that consists of all bitopological spaces with τ2 compact Hausdorffis also reflective in BiTop. Show, however, that BiComp = B1 ∩ B2 is notreflective in BiTop. [Hint: A space (X, τ1 , τ2 ), where X is infinite and τ1 and τ2
are discrete, has no reflection arrow.]* (c) Let PsTop be the construct of pseudotopological spaces (i.e., pairs (X, α), where
X is a set and α is a function that assigns to each ultrafilter on X a subset of··
X, in such a way that for each point x ∈ X, x ∈ α(x) [where x is the ultrafilterfof all supersets of {x}]) and continuous maps (i.e., functions (X, α) −→ (Y, β)having the property that for each x ∈ α(F) it follows that f (x) ∈ β(f (F)), wheref (F) = { M ⊆ Y | f −1 [M ] ∈ F}). Verify that the full subcategory of PsTop
consisting of all compact Hausdorff spaces, i.e., all spaces (X, α) with the property
that each α(F) is a singleton set, is not reflective even though it is an intersectionof a class of full reflective subcategories. [Hint: Let Bk be the subcategory thatconsists of all (X, α) such that α(F) is a singleton whenever some member of F isa set of cardinality less than k.]
4G. Subcategories of Subcategories
Let A be a subcategory of B and B be a subcategory of C. Prove that(a) A is a subcategory of C,(b) if A is reflective in B and B is reflective in C, then A is reflective in C,(c) if A is reflective in C and B is a full subcategory of C, then A is reflective in B,(d) if A is reflective in C, then A need not be reflective in B. [Construct an example.]
4H. Reflectors(a) Let A be a reflective subcategory of B. Show that if A is full in B, then there exists
a reflector R : B → A with R ◦ E ◦ R = R. Does the converse hold?
(b) Let A be the full subcategory of Vec consisting of all one-element vector spaces.
Verify that A is simultaneously reflective and coreflective in Vec, and every functorfrom Vec to A is simultaneously a reflector and a coreflector.
4I. Skeletons
Given two skeletons of a category A, show that there is an isomorphism A → A that
restricts to an isomorphism between the skeletons.
* 4J. Universal Categories
Does there exist a category A such that every category can be fully embedded into A?
4K. Full Embeddability
Show that(a) Prost is fully embeddable into Top as a coreflective subcategory,(b) Alg(1) is fully embeddable into Rel as a reflective subcategory,(c) Σ-Seq is fully embeddable into Aut.

9th July 2006

5 Concrete categories and concrete functors
As we have seen, many familiar categories such as Vec and Top are constructs (i.e.,
categories of structured sets and structure-preserving functions between them). If weregard such constructs as purely abstract categories (as we have done so far), somevaluable information (concerning underlying sets of objects and underlying functions ofmorphisms) is lost. Fortunately, category theory enables us to retain this information by
providing a means for a formal definition of construct — a construct being a pair (A, U )consisting of a category A and a faithful functor U : A → Set. A careful analysis reveals
that, for instance, many of the interesting properties of the constructs of vector spaces
and topological spaces are not properties of the corresponding abstract categories Vec
and Top but rather of the corresponding constructs (Vec, U ) and (Top, V ), where U
and V denote the obvious underlying functors. In fact, they are often properties of justthe underlying functors U and V . For example, the facts that the construct of vector
spaces is “algebraic” and the construct of topological spaces is “topological” are very
conveniently expressed by specific properties of the underlying functors, rather than byproperties of the abstract categories Vec and Top (see Chapter VI).
If, instead of vector spaces or topological spaces, we want to investigate objects with more
complex structures, e.g., topological vector spaces, we may supply the corresponding
abstract category TopVec with forgetful functors not only into Set (forgetting boththe topology and the linear structure), but also into Top (just forgetting the linear
structure) or into Vec (just forgetting the topology). Hence we can consider TopVec
as a concrete category over Set or over Top or over Vec. This leads to the concept ofconcrete categories over a category X as pairs (A, U ) consisting of a category A and afaithful functor U : A → X. In this way we may decompose complex structures into
simpler ones, or, conversely, construct more complex structures out of simpler ones bycomposing forgetful functors. The concept of concrete categories over arbitrary basecategories provides a suitable language to carry out such investigations.
5.1 DEFINITION(1) Let X be a category. A concrete category over X is a pair (A, U ), where A
is a category and U : A → X is a faithful functor. Sometimes U is called theforgetful (or underlying) functor of the concrete category and X is called thebase category for (A, U ).
(2) A concrete category over Set is called a construct.
5.2 EXAMPLES(1) Every category A can be regarded via the identity functor as a concrete category(A, idA ) over itself.

58

Sec. 5]

Concrete categories and concrete functors

59(2) If A is a construct in the sense of Example 3.3(2) and U is its naturally associatedunderlying functor [3.20(3)], then the pair (A, U ) is a construct in the sense of
Definition 5.1. Frequently we will denote the construct (A, U ) (by an “abuse ofnotation”) by A alone. For example, the abstract category of vector spaces and theconstruct of vector spaces will both be denoted by Vec. It will always be clear fromthe context which of these entities is being considered.
(3) Ban [3.3(3)(b)] can be considered as a construct in two natural ways:(a) via the obvious forgetful functor U : Ban → Set,(b) via the less obvious but useful “unit ball” functor O : Ban → Set, where
O(X) = { x ∈ X | k x k ≤ 1 } and O(f ) is the corresponding restriction of
U (f ).
[It will turn out that, as constructs, (Ban, O) has more pleasant properties than(Ban, U ).](4) Whenever (A, U ) is a construct, then (Aop , Q ◦ U op ) is also a construct (where
Q : Setop → Set is the contravariant power-set functor [cf. 3.20(9)]). In particular,(Setop , Q) is a construct.
(5) The category TopVec of topological vector spaces and continuous linear transformations can be regarded naturally via the obvious forgetful functors as(a) an abstract category,(b) a construct,(c) a concrete category over Top,(d) a concrete category over Vec.
A similar situation occurs for TopGrp, the category of topological groups (andcontinuous homomorphisms).
(6) The category Aut of deterministic sequential Moore automata [3.3(4)(b)] can naturally be considered as a concrete category over Set × Set × Set [cf. 3.3(4)(f)]. Thisis the case since for any object (Q, Σ, Y, δ, q0 , y) of Aut, each of Q, Σ, and Y is a set
and Aut-morphisms (i.e., simulations) are triples of functions that carry Q, Σ, andY into the corresponding sets of the codomain.
(7) Since for any category A the unique functor A → 1 is faithful if and only if A isthin, the concrete categories over 1 are essentially just the thin categories; i.e., thepreordered classes [3.26(2)].
5.3 REMARK
We adopt the following conventions:(1) Since faithful functors are injective on hom-sets, we usually assume [in view of 3.2(3)]that homA (A, B) is a subset of homX (U A, U B) for each pair (A, B) of A-objects.
This familiar convention allows one to express the property that “for A-objects A and9th July 2006

60

Categories, Functors, and Natural Transformations[Chap. If

B and an X-morphism U A −→ U B there exists a (necessarily unique) A-morphismf

A → B with U (A → B) = U A −→ U B” much more succinctly, by statingf“U A −→ U B

is an A-morphism (from A to B)”.24

Observe, however, that since U doesn’t need to be injective on objects, the expression
id

X“U A −−−→ UB

is an A-morphism (from A to B)”

does not imply that A = B or that idX = idA , although it does imply that U A =U B = X. (This is the case, for example, for the morphism in Pos from ({0, 1}, =)to ({0, 1}, ≤) that is the identity function on {0, 1}.) To avoid possible confusion infsuch a circumstance, we call an A-morphism A −→ B identity-carried if U f = idX .
(2) Sometimes we will write simply A for the concrete category (A, U ) over X, when U
is clear from the context; e.g., in the example above. In these cases the underlying
object of an A-object A will sometimes be denoted by |A|; i.e., “| |” will serve as astandard notation for underlying functors.
(3) If P is a property of categories (or of functors), then we will say that a concrete
category (A, U ) has property P provided that A (or U ) has property P .

FIBRES IN CONCRETE CATEGORIES
5.4 DEFINITION
Let (A, U ) be a concrete category over X.
(1) The fibre of an X-object X is the preordered class consisting of all A-objects Awith U (A) = X ordered by:(2) A ≤ B if and only if idX : U A → U B is an A-morphism.
(3) A-objects A and B are said to be equivalent provided that A ≤ B and B ≤ A.
(4) (A, U ) is said to be amnestic provided that its fibres are partially ordered classes;
i.e., no two different A-objects are equivalent.
(5) (A, U ) is said to be fibre-small provided that each of its fibres is small, i.e., apreordered set.
5.5 EXAMPLES
If A is a construct in the sense of 3.3(2)(3) and X is a set, then the A-fibre of X is (upto order-isomorphism) the class of all A-structures on X, ordered appropriately, e.g.,
24

Observe that analogues of this expression are frequently used in “concrete situations”, e.g., in sayingthat a certain function between vector spaces “is linear” or that a certain function between topologicalspaces “is continuous”, etc.

9th July 2006

Sec. 5]

Concrete categories and concrete functors

61(1) if A = Rel, then for relations ρ1 and ρ2 on X
ρ1 ≤ ρ2 ⇔ ρ 1 ⊆ ρ2 ,(2) if A = Met, then for metrics d1 and d2 on X
d1 ≤ d2 ⇔ ∀(x, y) ∈ X × X, d2 (x, y) ≤ d1 (x, y),(3) if A = Σ-Seq, then for acceptors (Q, δ, q0 , F ) and (Q, δ 0 , q00 , F 0 ) on a set Q(δ, q0 , F ) ≤ (δ 0 , q00 , F 0 ) ⇔ δ = δ 0 , q0 = q00 , and F ⊆ F 0 ,(4) if A = Top, then for topologies τ1 and τ2 on X
τ1 ≤ τ 2 ⇔ τ 2 ⊆ τ 1 ,(5) if A = Vec, then for vector space structures ν1 and ν2 on X
ν1 ≤ ν2 ⇔ ν1 = ν2 .
Observe that for Top the fibres are complete lattices (with smallest element the discrete
topology and largest element the indiscrete topology), whereas for Vec the fibres areordered by the equality relation, i.e., no two different vector space structures on thesame set are related to each other. These properties of the fibres of Top (resp. Vec) are
typical for “topological” (resp. “algebraic”) constructs. See Chapter VI.
5.6 REMARKS
A concrete category (A, U ) is anmestic if and only if the functor U is amnestic(cf. 3.27(4)). Most of the familiar concrete categories are both amnestic and fibre-small.
However:(1) No proper class, considered as a concrete category over 1, is fibre-small. Fibresof the quasiconstruct25 of quasitopological spaces26 are not even preordered classes,
but rather are proper conglomerates. The latter statement follows from the fact thatfor each set X with at least two points, the fibre of X cannot be put in bijective
correspondence with any subclass of the universe U.
(2) The constructs Banb and Metc are not amnestic. In Proposition 5.33 we will seethat each concrete category (A, U ) over X has an associated “amnestic modification”
that in most respects behaves like (A, U ).
25

A quasiconstruct is a pair (A, U ), where A is a quasicategory and U is a faithful functor from U to
Set.
26
See, e.g., E. Spanier, Quasi-topologies, Duke Math. J. 30 (1963): 1–14.

9th July 2006

62

Categories, Functors, and Natural Transformations[Chap. I

5.7 DEFINITION
A concrete category is called(1) fibre-complete provided that its fibres are (possibly large) complete lattices,27(2) fibre-discrete provided that its fibres are ordered by equality.
5.8 PROPOSITION
A concrete category (A, U ) over X is fibre-discrete if and only if U reflects identities(i.e., if U (k) is an X-identity, then k must be an A-identity).


CONCRETE FUNCTORS
Just as functors can be considered as the natural “morphisms” between abstract categories, and are a useful tool for investigating them, concrete functors are the natural“morphisms” between concrete categories.
5.9 DEFINITION
If (A, U ) and (B, V ) are concrete categories over X , then a concrete functor from
(A, U ) to (B, V ) is a functor F : A → B with U = V ◦ F . We denote such a functor
by F : (A, U ) → (B, V ).
5.10 PROPOSITION(1) Every concrete functor is faithful.
(2) Every concrete functor is completely determined by its values on objects.
(3) Objects that are identified by a full concrete functor are equivalent.
(4) Every full concrete functor with amnestic domain is an embedding.
Proof: Let F : (A, U ) → (B, V ) be a concrete functor.
(1). This follows from Proposition 3.30(2).
(2). Suppose that G : (A, U ) → (B, V ) is a concrete functor with G(A) = F (A) forfeach A-object A. Then for any A-morphism A −→ A0 we have the B-morphisms
Ff

GA = F A

Gf//

F A0 = GA0

with V (F f ) = U (f ) = V (Gf ). Since V is faithful, F f = Gf . Hence F = G.
(3). Let A and A0 be A-objects with F A = F A0 (= B). Then, by using Proposition
3.31, idB : F A → F A0 can be lifted to an A-isomorphism g : A → A0 . Hence A
and A0 are equivalent.
27

A partially ordered class (X, ≤) is called a large complete lattice provided that every subclass of
X has a join and a meet.

9th July 2006

Sec. 5]

Concrete categories and concrete functors(4). F is full, faithful, and injective on objects; hence it is an embedding.

63

5.11 EXAMPLES
Because of part (2) of the above proposition, when specifying concrete functors, we need
only describe how they map objects.
(1) The forgetful functor Rng → Ab that “forgets” multiplication is a concrete functorfrom the category of rings (and ring homomorphisms) to the category of abeliangroups, where both categories are considered as constructs. In a similar way, theforgetful functors Met → Top (that assigns to each metric space (X, d) the topological space determined by the distance function d ) and TopVec → Vec (that assigns
to each topological vector space its underlying vector space) are concrete functors.
(2) There are many functors from Set to itself, but there is only one concrete functor
between the construct (Set, idSet ) and itself — namely, the identity functor.
(3) The “discrete-space functor” and the “indiscrete-space functor” [3.29(5),(6)] are examples of concrete functors from Set to Top.
(4) Any functor between concrete categories over 1 is already concrete.
5.12 REMARKA concrete isomorphism F : (A, U ) → (B, V ) between concrete categories over X isa concrete functor that is an isomorphism of categories. All of the isomorphisms given
in Examples 3.26(2), (4), (5), and (6) are concrete. That such a concrete isomorphism
exists means, informally, that each structure in A, i.e., each object A of A, can becompletely substituted by a structure in B, namely F (A) (keeping, of course, the samemorphisms). For example, the standard descriptions of topological spaces by means of• neighborhoods,• open sets,• closure operators, or• convergent filters,
give technically different constructs, all of which are concretely isomorphic. This is why
the differences between the various descriptions are regarded as inessential and we canin good conscience call each of them “Top”. The concept of concretely isomorphic
concrete categories gives rise to an equivalence relation that is stronger than the relationof isomorphism of categories. For example, assuming that no measurable cardinals
exist, Top (and, indeed, any construct) can be thought of as being isomorphic to afull subcategory of Rel (cf. 4.7). However, Top is not concretely isomorphic to suchℵa subcategory, because there are more topologies on(namely, 22 0 ) than there arebinary relations on(namely, 2ℵ0 ).

N

9th July 2006

N

64

Categories, Functors, and Natural Transformations[Chap. I

5.13 REMARK
If F : (A, U ) → (B, V ) is a concrete isomorphism, then its inverse F −1 : B → A
is concrete from (B, V ) to (A, U ). Unfortunately, the corresponding result does nothold for concrete equivalences (i.e., concrete functors that are equivalences) since if
F : (A, U ) → (B, V ) is a concrete equivalence, then it may happen that there is noconcrete equivalence from (B, V ) to (A, U ) even though there are equivalences from
B to A; cf. Proposition 3.36. For example, the embedding of the skeleton of cardinal
numbers into Set is such a concrete equivalence of constructs that is not invertible. Thus,
even though it makes sense to say that two concrete categories over X are concretely
isomorphic, it makes little sense to say that they are concretely equivalent since therelation between concrete categories of being concretely equivalent is not symmetric.
The following proposition provides the basis for forming the quasicategory of all concrete
categories over a given base category.
5.14 PROPOSITION(1) The identity functor on a concrete category is a concrete isomorphism.
(2) Any composite of concrete functors over X is a concrete functor over X.



5.15 DEFINITION
The quasicategory that has as objects all concrete categories over X and as morphisms all concrete functors between them is denoted by CAT(X). In particular,
CONST = CAT(Set) is the quasicategory of all constructs.
5.16 REMARK
Notice that even when X is small, CAT(X) is never an actual category (unless X is theempty category). CAT(1) is essentially the quasicategory of all preordered classes andorder-preserving functions.
5.17 EXAMPLE
There is a natural embedding F : Monop → CONST that assigns to any monoid M
the category M -Act. If f : M → N is a monoid homomorphism, thenF (f ) : N -Act → M -Act is the concrete functor that assigns to any N -action (X, ∗N )the M -action (X, ∗M ), where m ∗M x is defined to be f (m) ∗N x.
5.18 DEFINITION
If F and G are both concrete functors from (A, U ) to (B, V ), then F is finer than G(or G is coarser than F ), denoted by: F ≤ G, provided that F (A) ≤ G(A) for eachA-object A [5.4(1)].
5.19 EXAMPLES(1) For order-preserving functions considered as concrete functors over 1, f ≤ g if andonly if this relation holds pointwise.

9th July 2006

Sec. 5]

Concrete categories and concrete functors

65(2) Among the concrete functors from Set to Top there is a finest one (the discrete-spacefunctor), and a coarsest one (the indiscrete-space functor).

DUALITY FOR CONCRETE CATEGORIES
5.20 REMARKFor every concrete category (A, U ) over X, its dual (Aop , U op ) is a concrete category
over Xop . Moreover, for every concrete functor F : (A, U ) → (B, V ) over X its dual
functor F op : (Aop , U op ) → (Bop , V op ) is a concrete functor over Xop . Thus, there is aduality principle for concrete categories and concrete functors. Observe, however, thatunless X = Xop there is no duality for concrete categories over a fixed base category X.
In particular, we don’t have a duality principle for constructs. However, since 1 = 1op ,there is a duality principle for concrete categories over 1 (i.e., for preordered classes).
This is indeed a familiar duality.

CONCRETE SUBCATEGORIES
5.21 CONVENTION
If (B, U ) is a concrete category over X and A is a subcategory of B with inclusion
E : A ,→ B, then A will often be regarded (via the functor U ◦E) as a concrete category(A, U ◦E) over X. In such cases we will call (A, U ◦E) a concrete subcategory of(B, U ). In the case that the base category is Set, we will call (A, U◦E) a subconstructof (B, U ).
5.22 DEFINITION(1) A concrete subcategory (A, U ) of (B, V ) is called concretely reflective in (B, V )(or a reflective modification of (B, V )) provided that for each B-object there
exists an identity-carried A-reflection arrow.
(2) Reflectors induced by identity-carried reflection arrows are called concrete reflectors.28
Dual Notions: Concretely coreflective subcategory (or coreflective modification); concrete coreflectors.
5.23 EXAMPLES(1) The construct of symmetric relations is simultaneously a concretely reflective anda concretely coreflective subconstruct of the construct Rel of (binary) relations[cf. 4.17(1) and 4.26(1)].
(2) The construct of completely regular spaces is a concretely reflective subconstruct ofTop [cf. 4.17(2)].
28

Caution: Concrete functors that are reflectors need not be concrete reflectors. See 5.23(6).

9th July 2006

66

Categories, Functors, and Natural Transformations[Chap. I(3) The construct of sequential topological spaces is a concretely coreflective subconstruct of Top [cf. 4.26(2)].
(4) None of the reflective subconstructs described in Examples 4.17(3)–(12) is concretelyreflective.
(5) None of the coreflective subconstructs described in Examples 4.26(3) and (4) is
concretely coreflective.
(6) Let X be a category consisting of a single object X and two morphisms idX and swith s ◦ s = idX . Let A be the concrete category over X, consisting of two objects
A0 and A1 and the morphism sets(
{idX }
if i = jhomA (Ai , Aj ) ={s}
if i 6= j.
Consider A as a concretely reflective subcategory of itself. Then idA : A → A is aconcrete reflector, and the concrete functor R : A → A, defined by R(Ai ) = A1−i , isa reflector that is not a concrete reflector.
5.24 PROPOSITION
Every concretely reflective subcategory of an amnestic concrete category is a full subcategory.
Proof: Let (A, U ) be a concretely reflective subcategory of an amnestic (B, V ), let A
be an A-object, and let r : A → A∗ be an identity-carried A-reflection arrow for A. We
wish to show that r = idA so that Proposition 4.20 can be applied. By reflectivity there
exists a unique A-morphism s : A∗ → A such that the diagram
r / ∗
A
AA
AA
s
A
idA AA AA

A

commutes.
Since r is identity-carried, V (r) = idVA . Since also V (idA ) = idVA , we conclude that
V (s) = idVA as well. Faithfulness of V gives us r ◦ s = idA∗ . Hence r is a B-isomorphismwith V (r) = idVA . Amnesticity of (B, V ) yields r = idA . Therefore, by Proposition 4.20,
A is full in B.

5.25 REMARKA concretely reflective subcategory of a nonamnestic concrete category need not be full,as Example 4.21 shows. (The categories given there should be considered as concrete
categories over 1).
5.26 PROPOSITION
For a concrete full subcategory (A, U ) of a concrete category (B, V ) over X, with inclusion functor E : (A, U ) ,→ (B, V ), the following are equivalent:

9th July 2006

Sec. 5]

Concrete categories and concrete functors

67(1) (A, U ) is concretely reflective in (B, V ),(2) there exists a concrete functor R : (B, V ) → (A, U ) that is a reflector withR ◦ E = idA and idB ≤ E ◦ R,29
(3) there exists a concrete functor R : (B, V ) → (A, U ) with R ◦ E ≤ idA andidB ≤ E ◦ R.
r

Proof: (1) ⇒ (2). For each B-object B choose a reflection arrow B −−B→ AB such thatrid

B
rB is identity-carried, and in case that B is an A-object B −−B→ AB = B −−−→ B (4.20).
The associated reflector (4.23) has the properties required in (2).

(2) ⇒ (3). Immediate.
(3) ⇒ (1). If R is a concrete functor that satisfies the requirements of (3), then for eachr
B-object B, we have B ≤ RB; i.e., there is an identity-carried B-morphism B −−B→ RB.
f

To show that rB is an A-reflection arrow for B, let A be an A-object and B −→ A befa B-morphism. Since R is a concrete functor, RB −→ RA is an A-morphism. In viewfof the fact that RA ≤ A, this implies that RB −→ A is an A-morphism. Hence it is theunique A-morphism that makes the triangle
rB

/ RB
CC
CC
f
C
f CC 
!

BC

A

commute.



5.27 REMARK
By the above proposition every full concretely reflective subcategory has a reflector thatis a concrete functor. The converse holds (Proposition 5.31) for isomorphism-closed fullconcrete subcategories of transportable concrete categories, defined below. However, fornonfull subcategories [cf. Exercise 5E(d)] or non-transportable categories [cf. Exercise
5E(c)] the converse need not hold.

TRANSPORTABILITY
We have seen that many of the familiar constructs have the extra property of being
amnestic. Another frequently encountered convenient property is that of transportability.
5.28 DEFINITION
A concrete category (A, U ) over X is said to be (uniquely) transportable providedkthat for every A-object A and every X-isomorphism U A −→ X there exists a (unique)kA-object B with U B = X such that A −→ B is an A-isomorphism.
29

Observe that R ◦ E = idA just means that RA = A for each A-object A and that idB ≤ E ◦ R justmeans that B ≤ RB for each B-object B [5.4(1)].

9th July 2006

68

Categories, Functors, and Natural Transformations[Chap. I

Transportability

5.29 PROPOSITION
A concrete category is uniquely transportable if and only if it is transportable and amnestic.
Proof: Let (A, U ) be uniquely transportable and let f : A → A0 be an A-isomorphism
such that U f = idX . Since idA is also an A-isomorphism with domain A and U (idA ) =idX , by uniqueness it follows that f = idA .
k

1
Conversely, if (A, U ) is transportable and amnestic, A is an A-object, and A −−→
A1k2 ◦k−1k2
and A −−→
A2 are A-isomorphisms with U k1 = U k2 , then A1 −−−−1−→ A2 is an identity
carried A-isomorphism. Hence, by amnesticity k2 ◦ k1−1 = idA1 , so that k1 = k2 .

5.30 EXAMPLES(1) For a preordered class (A, ≤) considered as a concrete category over 1 [cf. 5.2(7)],the following conditions are equivalent:(a) (A, ≤) is uniquely transportable,(b) (A, ≤) is amnestic,(c) ≤ is antisymmetric, i.e., ≤ is a partial-order on A.
(2) Familiar constructs such as Vec, Rel, Grp, and Top are uniquely transportable.
(3) The skeleton of Vec formed by all spaces
transportable.

Rm [see 4.13(2)] is amnestic, but not

5.31 PROPOSITION
If (A, U ) is an isomorphism-closed full concrete subcategory of a transportable concrete
category (B, V ) over X , then the following are equivalent:
1. (A, U ) is concretely reflective in (B, V ),9th July 2006

Sec. 5]

Concrete categories and concrete functors

69

2. there exists a reflector R : B → A that is concrete from (B, V ) to (A, U ).
Proof: (1) ⇒ (2). Immediate (cf. 5.26).
r(2) ⇒ (1). Let R be as in (2) and for each B-object B, let B −−B→ RB be an A-reflection
arrow. Then the diagram
B
rB



RB

rB

/

/ RB


RrB

rRB R(RB)commutes (cf. 4.22). This implies (by uniqueness in the definition of reflection arrows)that RrB = rRB . Since RB is an A-object and A is full in B, rRB is an A-isor =RrB
morphism (4.20). Hence U (RB −−RB−−−−−→ R(RB)) is an X-isomorphism. By the
r

Rr

r −1

concreteness of R, V (B −−B→ RB) = U (RB −−−B→ R(RB)). Hence V RB −−B−→ V B
is an X-isomorphism. By the transportability of (B, V ) there exists an isomorphismr−1

RB −−B−→ B̂ in B, so that since A is isomorphism-closed in B, B̂ is an A-object. Conrr−1

sequently, the composite B −−B→ RB −−B−→ B̂ is an identity-carried A-reflection arrow
for B.

5.32 REMARKFor concretely reflective subcategories there usually exist (even in the uniquely transportable cases) several reflectors that are concrete functors [see Example 5.23(6)]. However, in the amnestic case there exists only one concrete reflector.
Although transportability and amnesticity are quite useful and convenient properties forconcrete categories, they are not especially strong ones. The next propositions show that
if a concrete category fails to have one of them, it can be replaced via slight modifications
by concrete categories that do have them.
5.33 PROPOSITION
For every concrete category (A, U ) over X , there exists an amnestic concrete category(B, V ) over X that has the following properties:(1) there exists an (injective) concrete equivalence E : (B, V ) → (A, U ),(2) there exists a surjective concrete equivalence P : (A, U ) → (B, V ).
Moreover, if (A, U ) is transportable, then so is (B, V ).
Proof: The relation for A-objects, given by “A is equivalent to Â if and only if
A ≤ Â and Â ≤ A”, is an equivalence relation on the class of A-objects. If B is a fullsubcategory of A that contains, as objects, precisely one member from each equivalenceclass, E : B ,→ A is the inclusion, and V = U ◦ E, then (B, V ) is an amnestic concrete
category and E : (B, V ) → (A, U ) satisfies condition (1).

9th July 2006

70

Categories, Functors, and Natural Transformations[Chap. IP

If (B, V ) is constructed as above, then the unique concrete functor (A, U ) −−→ (B, V )that sends each A-object A to its unique equivalent B-object satisfies condition (2).
That transportability is not destroyed is obvious from the construction.



5.34 REMARKPThe construction of (A, U ) −−→ (B, V ), given above, is unique in the sense that for everyP̂

surjective concrete equivalence (A, U ) −−→ (B̂, V̂ ) with (B̂, V̂ ) amnestic, there exists a
concrete isomorphism K : (B, V ) → (B̂, V̂ ) such that P̂ = K ◦ P . In view of this theabove (B, V ) can be called “the” amnestic modification of (A, U ). With respect toalmost every interesting categorical property30 a concrete category is indistinguishable
from its amnestic modification. If one takes the view that in non-amnestic categories
there are “too many objects floating around in the fibres”, then the amnestic modifications can be considered to be more natural than the concrete categories that give riseto them. For an example of this phenomenon, notice that the amnestic modification of
the construct Metc of metric spaces and continuous maps is31 the construct Topm of
metrizable topological spaces and continuous maps.
5.35 LEMMA
For every concrete category (A, U ) over X, there exists a transportable concrete category(B, V ) over X and a concrete equivalence E : (A, U ) → (B, V ).
Proof: Define B as follows: Each B-object is a triple (A, a, X), with A ∈ Ob(A),X ∈ Ob(X), and a : U A → X an X-isomorphism.
homB ((A, a, X), (Â, â, X̂)) = homA (A, Â).
Identities and composition are as in A. Define V : B → X byâ◦ Uf ◦a−1f

V ((A, a, X) −→ (Â, â, X̂)) = X −−−−−−−→ X̂;i.e., such that the square
UA
Uf

a



U Â

â

/X
/



Vf

X̂

commutes.
f

Clearly, V is a faithful functor, and if V (A, a, X) −→ Y is an X-isomorphism, thenidA : (A, a, X) → (A, f ◦ a, Y ) is a B-isomorphism with V (idA ) = f . Hence (B, V ) istransportable. E : (A, U ) → (B, V ), defined by: E(A) = (A, idU A , U A), is a concrete
equivalence.

30
31

Amnesticity obviously is not one of them.
with respect to the canonical forgetful functor Metc → Topm

9th July 2006

Sec. 5]

Concrete categories and concrete functors

71

5.36 PROPOSITION
For every concrete category (A, U ) over X there exists a uniquely transportable concrete
category (B, V ) over X and a concrete equivalence E : (A, U ) → (B, V ) that is uniquelydetermined32 up to concrete isomorphism.
Proof: According to Lemma 5.35 there is a transportable concrete category (C, W )over X and a concrete equivalence F : (A, U ) → (C, W ). By Proposition 5.33 there
is an amnestic concrete category (B, V ) over X and a surjective concrete equivalenceP : (C, W ) → (B, V ). Consequently, E = P ◦ F : (A, U ) → (B, V ) is a concrete
equivalence, and by Proposition 5.33 (B, V ) is uniquely transportable.
E

To show the essential uniqueness, let (A, U ) −−→ (B, V ) be a concrete equivalence,with (B, V ) uniquely transportable. For each B-object B choose an A-object ABhand a B-isomorphism EAB −−B→ B such that whenever B ∈ E[Ob(A)] then hB is a
h

id

h

h

B
B-identity, EAB −−B→ B = B −−−→ B. Then V (EAB −−B→ B) = U AB −−B→ V B =h

V (EAB ) −−B→ V B is an X-isomorphism. Since (B, V ) is uniquely transportable, therehis a unique B-isomorphism EAB → CB with V (EAB → CB ) = V (EAB ) −−B→ V B.
A simple computation establishes that there exists a unique full concrete functor
H : (B, V ) → (B, V ) with HB = CB for each B-object B. By Proposition 5.10, H is aconcrete embedding. To show that E = H ◦E, let A be an A-object and let B = EA. Byhour construction EAB −−B→ B is a B-identity, which implies that EAB = EA. Hence,by Proposition 5.10, AB and A are equivalent A-objects. Consequently, EAB and EA
are equivalent B-objects. Since (B, V ) is amnestic, this implies that EAB = EA. Henceid

EAB −−→ EA is a B-isomorphism withhid

V (EAB −−→ EA) = V (EAB ) −−B→ V B,
so that by the definition of CB we have CB = EA; i.e., (H ◦ E)A = HB = CB = EA.
It remains to be shown that H is surjective. Let C be a B-object. Since E ishisomorphism-dense, there exists some A-object A and some B-isomorphism EA −→ C.
hhhThen V (EA −→ C) = U A −→ V C = V (EA) −→ V C is an X-isomorphism, sothat by the unique transportability of (B, V ), there exists a unique B-isomorphismh

EA → B, with V (EA → B) = V (EA) −→ V C. We claim that HB = C. To see this,hhconsider the chosen B-isomorphism EAB −−B→ B. Since EA −→ B is a B-isomorphism,
h−1 ◦hB
EAB −−−−−−→ EA is a B-isomorphism. By the fullness and faithfulness of E, thish−1 ◦hh−1 ◦hB
B
implies that AB −−−−−−→ A is an A-isomorphism (3.31), and hence EAB −−−−−−→ EA
is a B-isomorphism. Consequently,hh−1 ◦hhB
EAB −−B→ C = EAB −−−−−−→ EA −→ C
32

E

That is, whenever (A, U ) −−→ (B, V ) is a concrete equivalence to a uniquely transportable concrete
category over X, then there exists a concrete isomorphism H : (B, V ) → (B, V ) such that E = H ◦E.

9th July 2006

72

Categories, Functors, and Natural Transformations[Chap. I

is a B-isomorphism withhh

V (EAB −−B→ C) = V (EAB ) −−B→ V C = V B,
which, by the definition of HB, implies that HB = C. Thus H : (B, V ) → (B, V ) is aconcrete isomorphism with E = H ◦ E.


FUNCTORS “INDUCING” CONCRETE CATEGORIES
Many familiar constructs of an “algebraic” or “topological” nature have natural descriptions that can be accomplished in two steps. The first step, which will be formalized
below (separately for algebraic and topological cases), consists of defining algebraic (resp.
topological) categories by means of certain functors. The second step consists of singlingout full, concrete subcategories by imposing certain axioms on the objects (cf. 16.11 and22.1).
5.37 DEFINITION
Let T : X → X be a functor. Alg(T ) is the concrete category over X, the objects ofwhich (called T -algebras) are pairs (X, h) with X an X-object and h : T (X) → X
an X-morphism. Morphisms f : (X, h) → (X 0 , h0 ) (called T -homomorphisms) are
X-morphisms f : X → X 0 such that the diagram
T (X)
T (f )hf

T (X 0 )

/X


h0

/ X0
f

f

commutes. The underlying functor to X is given by: (X, h) −→ (X 0 , h0 ) = X −→ X 0 .
5.38 EXAMPLES(1) Consider the squaring functor [3.20(10)] S 2 : Set → Set. The construct Alg(S 2 )is obviously the construct of binary algebras Alg(Ω0 ), where Ω0 consists of thesingleton natural number 2 [3.3(2)(e)]. Observe that such familiar constructs as Sgr,Mon, Grp, and Ab can be considered as subcategories of Alg(S 2 ). [Later we shallsee that each Alg(Ω) is isomorphic to Alg(T ) for some T (10U).]h(2) A unary Σ-algebra [3.3(2)(e)] (X, σ(−))σ∈Σ gives rise to a function X × Σ −→ X
that maps (x, σ) to σ(x). Define a functor (− × Σ) : Set → Set as follows. On
objects: (− × Σ)(X) = X × Σ, and on morphisms: (− × Σ)(f ) = f × idΣ . Then thepassage from (X, σ(−))σ∈Σ to (X, h) defines a concrete isomorphism
F : Alg(Σ) → Alg(− × Σ)that is the identity function on morphisms.

9th July 2006

Sec. 5]

Concrete categories and concrete functors

73(3) Let Ŝ 2 : Pos → Pos be the squaring functor for Pos; i.e., Ŝ 2 (X, ≤) = (X × X, ≤),where the order relation on X × X is defined coordinatewise, and Ŝ 2 (f ) = f × f ,for any order-preserving map f . Then the concrete category Alg(Ŝ 2 ) over Pos is
concretely isomorphic to the category of ordered binary algebras, considered as aconcrete category over Pos.
(4) For any category X and X-object, X, the constant functor CX : X → X [3.20(2)]yields the category Alg(CX ) which is the comma category (X ↓ X) of objects underX (see Exercise 3K).
5.39 PROPOSITION
Each concrete category of the form Alg(T ) is fibre-discrete.



5.40 DEFINITION
Let T : X → Set be a functor. Spa(T ) is the concrete category over X, the objects offwhich (called T -spaces) are pairs (X, α) with α ⊆ T (X). Morphisms (X, α) −→ (Y, β)(called T -maps) are X-morphisms f : X → Y such that T (f )[α] ⊆ β. The underlyingfffunctor to X is given by: (X, α) −→ (Y, β) = X −→ Y. Concrete categories of the form
Spa(T ) are called functor-structured categories.
5.41 EXAMPLES(1) Consider the squaring functor S 2 : Set → Set [3.20(10)]. Then the construct
Spa(S 2 ) is equal to the construct Rel. Observe that such familiar constructs as
Prost and Pos can be considered as full subcategories of Spa(S 2 ).
(2) Any functor T : 1 → Set consists of choosing a distinguished set A, and so, for thiscase, Spa(T ) is just the power-set of A regarded as a poset or as a concrete category
over 1.
5.42 PROPOSITION
Each concrete category of the form Spa(T ) is fibre-complete.



5.43 REMARK
We will see later (22.3) that every small-fibred “topological” category (A, U ) over X
is isomorphic to a full concrete subcategory of Spa(T̂ ) for a suitable T̂ : X → Set.
A more natural representation of (A, U ) is often obtained in terms of a full concrete
subcategory of (Spa(T ))op considered as a concrete category over X = (Xop )op forsome functor T : Xop → Set. Concrete categories of the form (Spa(T ))op are called
functor-costructured categories.
5.44 EXAMPLE
Consider the contravariant power-set functor Q : Setop → Set. Then the construct(Spa(Q))op has as objects all pairs (X, τ ) consisting of a set X and a set τ of subsets of
X, where a morphism f : (X, τ ) → (Y, σ) is a function f : X → Y such that Qf [σ] ⊆ τ .
In particular, Top is a full subconstruct of (Spa(Q))op .

9th July 2006

74

Categories, Functors, and Natural Transformations[Chap. I

5.45 REMARKConcrete categories of the form Spa(T ) can be very large. In fact, every construct A
can be fully embedded in(a) Spa(P), where P is the covariant power-set functor;(b) (Spa(Q))op , where Q is the contravariant power-set functor;(c) the concrete category of topological spaces and open, continuous functions.
These and related results can be found in the monograph Pultr-Trnková (see References).
However, the corresponding embeddings usually will not be concrete.

Suggestions for Further Reading
Katětov, M. Allgemeine Stetigkeitsstrukturen. Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963,
473–479.
Isbell, J. R. Two set-theoretical theorems in categories. Fund. Math. 53 (1963): 43–49.
Barr, M. Coequalizers and free triples. Math. Z. 116 (1970): 307–322.
Ehresmann, A., and C. Ehresmann. Categories of sketched structures. Cahiers Topol.
Geom. Diff. 13 (1972): 105–214.
Kučera, L., and A. Pultr. On a mechanism of defining morphisms in concrete categories. Cahiers Topol. Geom. Diff. 13 (1972): 397–410.
Freyd, P. J. Concreteness. J. Pure Appl. Algebra 3 (1973): 171–191.
Rosický, J., and M. Sekanina. Realizations of topologies by set systems. Colloquium
on Topology (Keszthely, 1972), Colloq. Math. Soc. János Bolyai 8, North Holland,
Amsterdam, 1974, 535–555.
Koubek, V. Each concrete category has a representation by T2 paracompact topologicalspaces. Comment. Math. Univ. Carolinae 15 (1976): 655–664.
Porst, H.-E., and W. Tholen Concrete Dualities. Category Theory at Work (eds. H.
Herrlich and H.-E. Porst). Heldermann Verlag 1991, 111–136.

EXERCISES
5A. Fibres(a) Show that the category TopGrp, as a concrete category over Top, is fibre-discrete
and, as a concrete category over Grp, is fibre-complete. What about the construct
TopGrp? What about the constructs HComp and Comp (of compact spaces)?
(b) Which concrete categories have the property that each fibre is a singleton?

9th July 2006

Sec. 5]

Concrete categories and concrete functors

75

5B. Non-concrete Isomorphisms
Show that there is an illegitimate conglomerate of isomorphisms of Set onto itself, although the only concrete isomorphism of the construct Set onto itself is id.
5C. Categories of Concrete Categories
Verify that for each small category X there is a category whose objects are all fibre-small
concrete categories over X and whose morphisms are all concrete functors. Nevertheless,show that the conglomerate of all fibre-small constructs is illegitimate. [In fact, theconglomerate of all full subcategories of Set is illegitimate.]
5D. Concrete Functors Between Constructs(a) Show that there is precisely one concrete functor from Rel to Set (the forgetful
functor) and that there are precisely three concrete functors from Set to Rel, thediscrete functor, the diagonal functor (F X = (X, ∆), where ∆ = { (x, x) | x ∈ X }),and the indiscrete functor.
(b) Show that there is precisely one concrete functor from Set to Pos.
(c) Show that there is no concrete functor from Set to Vec. Generalize this to otheralgebraic constructs, and to Σ-Seq.
* (d) Show that there are precisely two concrete functors from Set to Top, but a properclass of concrete functors from Top into itself. [Hint: For each cardinal number α
let Fα : Top → Top assign to each topological space X the space Fα X generated
by intersections of α-indexed families of X-open sets.]
5E. Concrete Reflections(a) Show that no proper subconstruct of Grp is concretely reflective (or coreflective).
Generalize this to all fibre-discrete concrete categories.
(b) Show that the full subconstruct of Rel formed by all objects (X, ρ) without proper
cycles (i.e., if x1 ρx2 , x2 ρx3 , · · · , xn−1 ρxn , and x1 = xn , then x1 = x2 = · · · = xn ) isreflective, but not concretely reflective.
(c) Show that a reflective, isomorphism-closed, full concrete subcategory (A, U ) ofa (non-transportable) concrete category (B, V ) over X with a reflector functor
R(B, V ) −−→ (A, U ) that is concrete need not be concretely reflective. [Hint: Consider the additive monoid of integers( as base category X; let) B be the category with∅,
if m = 0 < nOb(B) = and homB (n, m) =; let the functor{(n, m)}, otherwiseNV

B −−→ X be defined by V (n, m) = n − m; and let (A, U ) be the full concrete
subcategory of (B, V ) obtained by deleting the object 0.] Cf. Proposition 5.31.
* (d) Show that a reflective (nonfull) concrete subcategory (A, U ) of a transportable conR

crete category (B, V ) over X with a reflector (B, V ) −−→ (A, U ) that is a concrete
functor need not be concretely reflective. [Hint: Consider the compositive monoidof all order-preserving endomorphisms of the set , with its natural order, as baseN

9th July 2006

76

Categories, Functors, and Natural Transformations[Chap. I

category X, and let (A, U ) (resp. (B, V )) be the concrete category over X withN

N, and whose morphisms (N, n) −f→ (N, m)f
are all X-morphisms N −→ N that satisfy f (n + p) = m + p for all p ∈ N
objects all pairs ( , n), with n ∈(resp. f (n + p) = m + p for all p ≥ 1 and f (n) ≤ m).] Cf. Proposition 5.31.

5F. Amnestic Modification(a) Describe the amnestic modification of the construct Metu of metric spaces anduniformly continuous maps.
(b) What are the amnestic modifications for concrete categories over 1?
5G. Categories of T -Algebras(a) Describe Alg(T ) for the constant functor T : Set → Set(1) with value ∅,(2) with value 1 = {0}.
(b) Find a functor T : Set → Set such that Alg(T ) is concretely isomorphic to thecategory of commutative binary algebras (i.e., algebras with a binary operation ·satisfying x · y = y · x).
(c) Describe Alg(id).
(d) Let T : Pos → Pos be the functor that assigns to each poset (X, ≤) the discretelyordered set, and is defined on morphisms by: T f = f . Describe Alg(T ).
5H. Categories of T -Spaces(a) Describe Spa(T ) for the constant functor T : Set → Set(1) with value ∅,(2) with value 1 = {0}.
(b) Find a functor T : Set → Set such that Spa(T ) is concretely isomorphic to theconstruct of symmetric binary relations (i.e., if x is related to y, then y must be
related to x).
(c) Show that there is no functor T : Set → Set such that Spa(T ) is concretely isomorphic to the construct of reflexive binary relations.
5I. Cat as a Concrete Category
Cat can be viewed in various ways as a concrete category. Depending on the chosen
definition of category (cf. 3.1, 3.53 and 3C) and on the chosen forgetful functor, various
situations arise:(a) Show that by assuming any of the two definitions of categories 3.1 or 3C, Cat canbe considered as a concrete category over Set × Set via the functor
U : Cat → Set × Set, defined by
F

F

F

M
U (A −−→ B) = (Ob(A) −−O→ Ob(B) , M or(A) −−−→ M or(B)),

9th July 2006

Sec. 5]

Concrete categories and concrete functors

77(where FO is the restriction of F to objects and FM is its restriction to morphisms),
and that in both cases U is uniquely transportable.
(b) Show that by assuming any of the three definitions of categories, Cat can be conF

sidered as construct via the functor U : Cat → Set, defined by U (A −−→ B) =FM
M or(A) −−−→ M or(B). In the cases of Definition 3.1 and of Exercise 3C, U istransportable, but not amnestic; and in the case of Definition 3.53, U is amnestic
and transportable. [Thus, when considered as a construct, Catof is usually preferred
to Cat.]5J. Concretizable Categories
A category A is called concretizable over X provided that there exists a faithful
functor from A to X. Verify that(a) A is concretizable over 1 if and only if A is thin.
(b) Setop is concretizable over Set.
(c) If A is concretizable over Set, then so is Aop .
* (d) There exist categories that are not concretizable over Set, e.g., the category hTop,
whose objects are topological spaces and whose morphisms are homotopy equivalenceclasses of continuous maps.
(e) A is concretizable over Set if and only if A is embeddable into Set.
(f) There exist categories A that are not concretizable over Aop .
* 5K. Subconstructs That Are Simultaneously Reflective andCoreflective ModificationsShow that the construct Rel has precisely six subconstructs that are simultaneously
reflective and coreflective modifications and that the five proper ones can be describedby the following implications:(1) xρy ⇒ yρx (symmetry),(2) xρy ⇒ xρx,(3) xρy ⇒ yρy,(4) xρy ⇒ (xρx and yρy),(5) xρy ⇒ (xρx and yρx).
5L. The Constructs Topop
0 and Fram(a) If Top0 denotes the construct of T0 topological spaces, show that Topop
0 can beopconsidered as a construct via the functor L : Top0 → Set that forgets not the“structure” but rather the underlying set; i.e., for a continuous function
f

f

Lf

(Y, σ) −→ (X, τ ), L((X, τ ) −→ (Y, σ)) = τ −−→ σ, where Lf (A) = f −1 [A].

9th July 2006

78

Categories, Functors, and Natural Transformations[Chap. I(b) Let Fram be the construct
whoseW
W objects are frames, i.e., complete lattices thatsatisfy the law a ∧ i∈I bi = i∈I (a ∧ bi ), and whose morphisms are framehomomorphisms, i.e., functions that preserve finite meets and arbitrary joins.
TShow that there is a unique concrete functor Topop−→ Fram over Set that sends
0 −each topological space to its set of open subsets, ordered by inclusion.
[Loc=Framop is called the category of locales. The functor T op : Top0 → Loc is a
nonfull coreflective embedding, and its restriction to sober spaces is a full coreflective
embedding].
5M. HCompop as a Construct
Show that HCompop can be considered as a construct via the contravariant hom-functor
hom(−, [0, 1]) : HCompop → Set.
5N. Concrete Isomorphisms(a) Show that the following constructs are concretely isomorphic:(1) MCPos (complete lattices and meet-preserving maps),(2) JCPos,(3) the full subconstruct of Alg(P) consisting of those P-algebras (X, h) that satisfy
the following two conditions:(i) h({x}) = x for each x ∈ X,(ii) h(∪A) = h({h(A) A ∈ A}) for each A ⊆ PX.
(b) Show that the three constructs defined below are pairwise concretely isomorphic andtherefore provide three different approaches to topology.
(1) Approach via closure; the construct Clos:
Objects are closure spaces; i.e., pairs (X, cl) with cl : P(X) → P(X) a function that satisfies:(i) cl(∅) = ∅,(ii) A ⊆ cl(A) for each A ∈ P(X),(iii) cl(A ∪ B) = cl(A) ∪ cl(B) for each A, B ⊆ X.
fMorphisms are closure-preserving maps; i.e., functions (X, cl) −→ (X 0 , cl0 )such that f [cl(A)] ⊆ cl0 (f [A]) for each A ⊆ X.
(2) Approach via convergence; the construct PrTop:
Objects are pretopological spaces; i.e., pairs (X, conv), where conv is therelation that shows which filters converge to which points of X, subject to thefollowing conditions:(i) for every x ∈ X the fixed ultrafilter at x (ẋ) converges to x,(ii) if a filter converges to x, then so does every finer filter,

9th July 2006

Sec. 5]

Concrete categories and concrete functors

79(iii) if each member of a family of filters converges to x, then so does theintersection of the family.
Morphisms are convergence-preserving maps; i.e., functions
f(X, conv) −→ (X 0 , conv 0 ) such that whenever F converges to x, then the filter
generated by f [F] converges to f (x).
(3) Approach via neighborhoods; the construct Neigh:Objects are neighborhood spaces; i.e., pairs (X, N ), where N associates withany x ∈ X a filter N (x), the neighborhood filter of x, subject to the condition
that if U is a neighborhood of x then x is a member of U .
fMorphisms are all functions (X, N ) −→ (X 0 , N 0 ) such that whenever U is aneighborhood of f (x), then f −1 [U ] is a neighborhood of x.
5O. Realizations
Full concrete embeddings are called realizations. Show that(a) There are precisely two realizations from Pos to Rel.
* (b) There are precisely two realizations from Top to Spa(Q)op .
(c) There is a realization from Alg(1) to Rel.
(d) There is a realization from Sgr to Spa(S 3 ).
(e) For each Ω, there is a realization from Alg(Ω) to some functor-structured category.
(f) There is a realization from Prost to Top.
(g) There is no realization from Rel to Sgr.
(h) There is a proper class of realizations from Tych to Spa(Q)op .
(i) There are at least two realizations from Haus to Spa(Q)op . [It is unknown whether
there are more than two.](j) There is a no fibre-small construct (A, U ) such that every fibre-small construct hasa realization to (A, U ).
* (k) There is a construct (A, U ) such that every construct has a realization to (A, U ).
5P. Amnesticity
Show that a concrete category (A, U ) over X is amnestic if and only if each A-isomorphism f is an A-identity whenever U f is an X-identity.

9th July 2006

6 Natural transformations
Let V be a finite-dimensional real vector space and let V̂ be its dual (i.e., the set of
all linear functionals V →with vector-space operations defined pointwise). V andˆ
V̂ are known to be isomorphic. Hence V and its second dual V̂ are isomorphic as
well. However, there is a fundamental difference between these two situations. Thereˆ
is a “natural” isomorphism τ : V → V̂ which to every vector x assigns the “evaluate
at x” functional τ (x) : V̂ → . But there is no “natural” isomorphism between V
and V̂ . This section provides a formal definition for the intuitive notion of “natural
isomorphism” and (more generally) of “natural transformation”.

RR

6.1 DEFINITION
Let F, G : A → B be functors. A natural transformation τ from F to G (denotedτ
by τ : F → G or F −→ G) is a function that assigns to each A-object A a B-morphism
τA : F A → GA in such a way that the following naturality condition holds: for eachfA-morphism A −→ A0 , the square
τA

FA
Ff





F A0

/ GA

τA0

Gf

/ GA0

commutes.
6.2 EXAMPLESr(1) Let A be a reflective subcategory of B with inclusion functor E, let B −−B→ RB bean A-reflection arrow for each B-object B, and let R : B → A be the associatedrreflector (4.23). Then r = (rB )B∈Ob(B) is a natural transformation: idB −→ E ◦ R.
(2) Let U : Grp → Set be the forgetful functor, and let S : Grp → Set be thef2f“squaring-functor”, defined by S(G −→ H) = G2 −−→ H 2 . For each group G, its
multiplication is a function τG : G2 → G. The family τ = (τG ) is a natural transformation from S to U . The naturality condition simply means that f (x·y) = f (x)·f (y)ffor any group homomorphism G −→ H and any x, y ∈ G. Thus “multiplication” ingroups can be regarded as a natural transformation. Likewise, for any type of algebras, each of the defining operations can be considered as a natural transformation
between suitable functors.
(3) Let ( ˆˆ) : Vec → Vec be the second-dual functor for vector spaces defined by(ˆˆ)( ˆ )op(ˆ)

Vec −−−→ Vec = (Vecop )op −−−−→ Vecop −−−→ Vec,

80

Sec. 6]Natural transformations

81

where (ˆ)op is the dual of the duality functor for vector spaces (cf. 3.20(12)and 3.41), and let idVec be the identity functor on Vec. Then the linear transforˆ
mations τV : V → V̂ , defined by (τV (x))(f ) = f (x), yield a natural transformation
τ
idVec −→ ( ˆˆ).
(4) The assignment of the Hurewicz homomorphism πn (X) → Hn (X) to each topological space X is a natural transformation from the nth homotopy functor
πn : Top → Grp to the nth homology functor Hn : Top → Grp.
f(5) If B −→ C is an A-morphism, thenτfhomA (C, −) −−→ homA (B, −),defined by τf (g) = g ◦ f , andσfhomA (−, B) −−→ homA (−, C),
defined by σf (g) = f ◦ g, are natural transformations.
(6) Let U : Σ-Seq → Set be the forgetful functor. For each σ ∈ Σ, and each acceptor
A = (Q, δ, q0 , F ), let σ̂A be the function δ(−, σ) : Q → Q. Then σ̂ = (σ̂A ) : U → U
is a natural transformation.
6.3 DEFINITIONτ
If G, G0 : A → B are functors and G −→ G0 is a natural transformation, then(1) for each functor F : C → A, the natural transformation τ F : G ◦ F → G0 ◦ F isdefined by(τ F )C = τF C ,(2) for each functor H : B → D, the natural transformation Hτ : H ◦ G → H ◦ G0 is
defined by(Hτ )A = H(τA ).
τ op

Likewise the natural transformation G0op −−−→ Gop is defined byτAop = τA ,6.4 EXAMPLE
If S 2 : Set → Set is the squaring functor [3.20(10)] and ∆ : id → S 2 is the naturaltransformation that associates with every set X the diagonal map ∆X : X → X 2 given
by x 7→ (x, x), then(1) S 2 ∆ : S 2 → S 2 ◦ S 2 is given by (x, y) 7→ ((x, x), (y, y)),(2) ∆S 2 : S 2 → S 2 ◦ S 2 is given by (x, y) 7→ ((x, y), (x, y)).

9th July 2006

82

Categories, Functors, and Natural Transformations[Chap. I

NATURAL ISOMORPHISMS
6.5 DEFINITION
Let F, G : A → B be functors.
τ(1) A natural transformation F −→ G whose components τA are isomorphisms is called
a natural isomorphism from F to G.
More generally, a natural transformation from F to G whose components belong tosome specified class M of B-morphisms is called an M -transformation.
(2) F and G are said to be naturally isomorphic (denoted by F ∼= G) provided that33
there exists a natural isomorphism from F to G.
6.6 EXAMPLES(1) For each functor F : A → B we have the identity natural transformation on F ,idF : F → F given by (idF )A = idF A , which is clearly a natural isomorphism.
(2) Let F : Set → Vec be a functor that assigns to each set X a vector space F X withfFfbasis X, and to each function X −→ Y the unique linear extension F X −−→ F Y
of f . This actually is not a correct definition of a functor, since there are manydifferent vector spaces with the same basis. However, the definition is “correct up to
natural isomorphism”. Whenever we choose, for each set X, a specific vector space
F X with basis X, we do obtain a functor F : Set → Vec (since the above condition
determines the action of F on functions uniquely). Furthermore, any two functorsthat are obtained in this way are naturally isomorphic.
(3) The natural transformation τ : idVec → ( ˆˆ ) from the identity functor on Vec tothe second-dual functor for vector spaces ( ˆˆ ) given in Example 6.2(3) becomes anatural isomorphism when the above functors are restricted to the full subcategoryof finite-dimensional vector spaces.
(4) For any 2-element set A, hom(A, −) is naturally isomorphic to the squaring functor
S 2 [3.20(10)] and hom(−, A) is naturally isomorphic to the contravariant power-set
functor Q [3.20(9)].
(5) If f is a morphism and τf and σf are the associated natural transformations for thehom-functors [6.2(5)], then the following are equivalent:(a) f is an isomorphism,(b) σf is a natural isomorphism,(c) τf is a natural isomorphism.
Thus, if A and B are isomorphic objects, then hom(A, −) and hom(B, −) are naturally isomorphic functors, and so are hom(−, A) and hom(−, B). The converse holds
as well (cf. Exercise 6M).
33

Observe that the relation ∼= is an equivalence relation on the conglomerate of all functors from A toB.

9th July 2006

Sec. 6]Natural transformations

83

6.7 PROPOSITION
If A is a reflective subcategory of B, then any two reflectors for A are naturally isomorphic.
r

Proof: Let R and S be reflectors for A with associated reflection arrows B −−B→ RBfgs
and B −−B→ SB. Then there exist A-morphisms RB −−B→ SB and SB −−B→ RB suchthat the diagrams
rB

BC

/ RB

CC
CC
fB
sB CCC
! 

sB

/ SB
CC
CC
gB
rB CCC
! 

BC
and

SB

RB

commute. Uniqueness in the definition of reflection arrows implies that gB ◦ fB = idRB
and fB ◦ gB = idSB . Hence the fB ’s are A-isomorphisms. That (fB )B∈Ob(A) is anatural transformation (hence a natural isomorphism) follows from the fact that for anyfB-morphism B −→ B 0 the following diagrams 1 and 2 — hence, by the uniquenessr
property for B −−B→ RB, also 3 — commute:
B
f



B0

sB

1
sB 0

/ SB


Sf

/ SB 0

B
=

f



B0

rB

/ RB

fB

Rf

3

2
rB 0



/ RB 0

fB 0

/ SB


Sf

/ SB 0FIn Proposition 3.36 we have seen that if A −−→ B is an equivalence, then there is anG
equivalence B −−→ A. Now that we have the notion of natural isomorphism, we are able
to sharpen this result:
6.8 PROPOSITIONFG
A functor A −−→ B is an equivalence if and only if there exists a functor B −−→ A suchthat idA ∼= G ◦ F and F ◦ G ∼= idB .
Proof: Let F be an equivalence. By following the proof of Proposition 3.36 we seethat there is a functor G and a natural isomorphism ε : F ◦ G → idB . Now for eachA-object A, ε−1
F A : F A → (F ◦ G ◦ F )(A) is an isomorphism. Since F is full and faithful,
Proposition 3.31 implies that there is a unique isomorphism ηA : A → (G ◦ F )(A) such−1 (cf. Exercise 6L) andthat F (ηA ) = ε−1
F A . The naturality of η follows from that of εthe faithfulness of F .
ηεConversely, let G : B → A be a functor and let idA −→ G ◦ F and F ◦ G −→ idB beεnatural isomorphisms. Since F (GB) −−B→ B is an isomorphism for any B-object B,f

it follows that F is isomorphism-dense. F is faithful, since for any pair Ag// A0 ofB-morphisms the equality F f = F g implies ηA0 ◦ f = GF f ◦ ηA = GF g ◦ ηA = ηA0 ◦ g;−1
hence f = g. F is full, since for any B-morphism f : F A → F A0 , ηA
0 ◦ Gf ◦ ηA is an0
A-morphism g : A → A with F g = f .


9th July 2006

84

Categories, Functors, and Natural Transformations[Chap. I

6.9 DEFINITION
A functor F : A → Set is called representable (by an A-object A) provided that F
is naturally isomorphic to the hom-functor hom(A, −) : A → Set.
6.10 REMARKObjects that represent the same functor (or two naturally isomorphic functors) are isomorphic. (Cf. Exercise 6M.)

6.11 EXAMPLES(1) Forgetful functors are often representable. For example,(a) Vec → Set is represented by the vector spaceR,(b) Grp → Set is represented by the group of integers

Z,(c) Top → Set is represented by any one-point topological space.
(2) The underlying functor U for the construct Ban [5.2(3)] is not representable (seeExercise 10J). However, the faithful unit ball functor O : Ban → Set is represented
in the complex case by the Banach space of complex numbers.

C(3) The forgetful functor for Σ-Seq is not representable (since for each (finite) acceptor
A there exists a nonempty acceptor B with hom(A, B) = ∅).
6.12 REMARKFor constructs (A, U ) the forgetful functor is represented by an object A if and only
if A is a free object over a singleton set [see Definition 8.22(2)]. This provides many
additional examples of representations.

FUNCTOR CATEGORIES
6.13 DEFINITIONστ
If F, G, H : A → B are functors and F −→ G and G −→ H are natural transformations,then the composition of natural transformations τ ◦ σ : F → H is the naturaltransformation that assigns to each A-object A the morphism τA ◦ σA : F (A) → H(A).
6.14 REMARK
It is obvious that the composition of natural transformations is a natural transformation,that this composition is associative, and that the identity natural transformations act
as units.
6.15 DEFINITION
For categories A and B the functor quasicategory [A, B] has as objects all functors
from A to B, as morphisms from F to G all natural transformations from F to G, as
identities the identity natural transformations, and as composition the composition ofnatural transformations given above.

9th July 2006

Sec. 6]Natural transformations85

6.16 REMARKS(1) If A and B are small categories, then [A, B] is a category. If A is small and
B is large, then [A, B], though being a proper quasicategory, is isomorphic to acategory. Quasicategories that are isomorphic to categories are called legitimate
quasicategories and are treated as categories. If A and B are both large, then[A, B] will generally fail to be isomorphic to a category. Such quasicategories arecalled illegitimate.
(2) A natural transformation between functors from A to B is a natural isomorphism
if and only if it is an isomorphism in [A, B].
6.17 EXAMPLES(1) If A is a discrete category with one object, then [A, B] is isomorphic to B.
(2) If A is a discrete category with two objects, then [A, B] is isomorphic to B × B.
(3) If 2 is a category of the form • → • with two objects and one non-identity morphism,
then [2, B] is isomorphic to the arrow category of B (cf. Exercise 3K).
(4) If M is a monoid, considered as a category, then [M, Set] is isomorphic to thecategory M -Act of M -actions.
6.18 PROPOSITION
For any functor F : A → Set, any A-object A and any element a ∈ F (A), there exists
a unique natural transformation τ : hom(A, −) → F with τA (idA ) = a.
Proof: Let τB (f ) = (F (f ))(a). Pointwise evaluations establish that τ is a naturaltransformation. If δ : hom(A, −) → F is such that δA (idA ) = a, then by the naturalityof δ, δB (f ) = δB (f ◦idA ) = (δB ◦hom(A, f ))(idA ) = (F (f )◦δA )(idA ) = F (f )(a) = τB (f ).
6.19 COROLLARY (YONEDA LEMMA)
If F : A → Set is a functor and A is an A-object, then the following function
Y : [hom(A, −), F ] → F (A)defined by

Y (σ) = σA (idA ),is a bijection (where [hom(A, −), F ] is the set of all natural transformations fromhom(A, −) to F ).

6.20 THEOREM
For any category A, the functor E : A → [Aop , Set], defined byfσfE(A −→ B) = hom(−, A) −−→ hom(−, B),
where σf (g) = f ◦ g, is a full embedding.
Proof: The described assignment clearly preserves identities and composition. Thus it
is a functor. If f and f 0 are distinct members of hom(A, B), then σf and σf 0 clearly differ
on idA . Hence E is faithful. Fullness follows from Corollary 6.19 with F = hom(B, −).

9th July 2006

86

Categories, Functors, and Natural Transformations[Chap. I

6.21 EXAMPLES(1) If M is a monoid considered as a category, then the unique object assigned to it by
the above embedding is the M op -action on the set M defined by right translationsin M .
(2) If A is a poset considered as a thin category, then the above embedding is therepresentation of A by all the principal ideals in A.

CONCRETE NATURAL TRANSFORMATIONSAND GALOIS CORRESPONDENCES
6.22 REMARK
From now on, when investigating concrete categories, we typically will use the notational
conventions described in Remark 5.3. In particular,(1) we will denote a concrete category (A, U ) over X by A alone and denote the underlying functor U by | |,f(2) the expression “|A| −→ |B| is an A-morphism” means that for the X-morphismf|A| −→ |B| there exists a (necessarily unique) A-morphism A → B, which will alsof

be denoted by f , with |A → B| = |A| −→ |B|.
6.23 DEFINITION
If A and B are concrete categories over X and F, G : A → B are concrete functors (5.9),
then a natural transformation τ : F → G is called concrete (or identity-carried)
provided that |τA | = id|A| for each A-object A.
6.24 PROPOSITION
If F, G : A → B are concrete functors, then the following are equivalent:(1) F ≤ G (5.18),(2) there exists a (necessarily unique) concrete natural transformation τ : F → G.



6.25 DEFINITION
Let A and B be concrete categories over X. If G : A → B and F : B → A are concrete
functors over X, then the pair (F, G) is called a Galois correspondence (between A
and B over X) provided that F ◦ G ≤ idA and idB ≤ G ◦ F .
6.26 EXAMPLES(1) Galois isomorphisms: If K : A → B is a concrete isomorphism, then (K −1 , K)
is a Galois correspondence, called a Galois isomorphism.

9th July 2006

Sec. 6]Natural transformations

87(2) Galois reflections and coreflections:(a) If E : A → B is a concrete embedding and R : B → A is a concrete reflector,
then (R, E) is a Galois correspondence, called a Galois reflection.
(b) If E : A → B is a concrete embedding and C : B → A is a concrete coreflector,
then (E, C) is a Galois correspondence, called a Galois coreflection.
(3) Galois correspondences for constructs:(a) Let U : Top → Set be the forgetful functor, let D : Set → Top be the discrete
functor, and let N : Set → Top be the indiscrete functor [3.29(5) and (6)].
Then (D, U ) and (U, N ) are both Galois correspondences.
(b) Let G : Top → Rel be the concrete functor (over Set) defined on objects by:(X, τ ) 7→ (X, ρτ ), where xρτ y if and only if x is in the τ -closure of {y}. Let
F : Rel → Top be the concrete functor defined on objects by (X, ρ) 7→ (X, τρ ),where A is τρ -closed if and only if A is a lower-set for ρ [i.e., a ∈ A anda0 ρa ⇒ a0 ∈ A]. Then (F, G) is a Galois correspondence.
(c) If U : Unif → Top is the forgetful functor that assigns to any uniform space
the completely regular space underlying it, and F : Top → Unif is the finefunctor that assigns to any topological space the fine uniform space determinedby it, then (F, U ) is a Galois correspondence.
(4) Galois connections:(a) Recall that when X = 1, then concrete categories over X are essentially preordered classes and concrete functors between them are essentially order-preserving functions. Historically, a Galois connection between preordered classes A andB has been defined as a pair (f, g) of order-preserving34 functions g : A → B
and f : B → A with the property that for all a ∈ A and b ∈ B, f (b) ≤ a if andonly if b ≤ g(a). Notice that the latter condition is equivalent to: (f ◦ g)(a) ≤ a
for all a ∈ A and b ≤ (g ◦ f )(b) for all b ∈ B. Thus Galois connections are just
Galois correspondences with the base category X = 1. (See also Proposition
6.28 below.)(b) Important special cases of Galois connections arise from (binary) relations:Let ρ be a relation from the set X to the set Y , i.e., ρ ⊆ X × Y . Denote by A
the poset of all subsets of X ordered by inverse inclusion and by B the poset of
all subsets of Y ordered by inclusion. Then the following maps yield a Galois
connection:

34

g : A → B,

defined byg(S) = {y ∈ Y | sρy for all s ∈ S}

f : B → A,

defined byf (T ) = {x ∈ X | xρt for all t ∈ T }.

Many of the earlier definitions of Galois connections deal only with the case of partially ordered sets and they usually adopt a “contravariant” formulation as follows: If
g : A → B and f : B → A are order-reversing functions between posets, then (f, g) is a (contravariant) Galois connection provided that a ≤ f (b) if and only if b ≤ g(a). Notice that this is thesame as the formulation above if the order on A is reversed. The order-preserving version corresponds
to what are frequently called residuated-residual pairs or (sometimes) Galois connections of mixed
type.

9th July 2006

88

Categories, Functors, and Natural Transformations[Chap. I

6.27 PROPOSITION(1) If (F, G) is a Galois correspondence between A and B and (F̂ , Ĝ) is a Galois correspondence between B and C, then (F ◦ F̂ , Ĝ ◦ G) is a Galois correspondence between
A and C [sometimes denoted by (F̂ , Ĝ) ◦ (F, G)].
(2) If (F, G) is a Galois correspondence between A and B over X, then (Gop , F op ) is a
Galois correspondence between Bop and Aop over Xop .
Proof:(1). For any A-object A we obtain F̂ Ĝ(GA) ≤ GA. Application of the functor F yields
F F̂ ĜG(A) ≤ F GA. Since F GA ≤ A, transitivity of ≤ yields (F F̂ )(ĜG)(A) ≤ A.
Likewise B ≤ (ĜG)(F F̂ )B for each B-object B.
(2). Obvious.



6.28 PROPOSITION
Let G : A → B and F : B → A be concrete functors over X. Then the following are
equivalent:(1) (F, G) is a Galois correspondence,ff(2) an X-morphism |F (B)| −→ |A| is an A-morphism if and only if |B| −→ |G(A)| is a
B-morphism.
Proof:(1) ⇒ (2).

f

If |F B| −→ |A| is an A-morphism, then by applying G and using
id|B|

f

idB ≤ G ◦ F , one has that |B| −−−→ |(G ◦ F )(B)| −→ |GA| is a B-morphism.
fConversely, the facts that |B| −→ |GA| is a B-morphism and that F ◦ G ≤ idA imply
f

id|A|

that |F B| −→ |(F ◦ G)(A)| −−−→ |A| is an A-morphism.
id|F B|(2) ⇒ (1). Since each |F B| −−−−→ |F B| is an A-morphism, from (2) we see thatB ≤ (G ◦ F )(B) for each B-object B. Similarly, (F ◦ G)(A) ≤ A for each A-object A.
Thus (F, G) is a Galois correspondence.

6.29 PROPOSITION
The functors in a Galois correspondence between amnestic concrete categories determine
each other uniquely; in particular, if (F, G) and (F 0 , G) are such Galois correspondences,then F = F 0 .
Proof: Let B be a B-object. If (F, G) is a Galois correspondence, B ≤ (G ◦ F )(B),
so that by the above proposition, if (F 0 , G) is a Galois correspondence, F 0 B ≤ F B.
Similarly, F B ≤ F 0 B, so that by amnesticity, F = F 0 . [Dually, it can be shown that if
each of (F, G) and (F, G0 ) are Galois correspondences, then G = G0 .]

9th July 2006

Sec. 6]Natural transformations89

6.30 PROPOSITION
If (F, G) is a Galois correspondence between amnestic concrete categories, then G ◦ F ◦G = G and F ◦ G ◦ F = F .
Proof: Clearly, idB ≤ G ◦ F implies F = F ◦ idB ≤ F ◦ G ◦ F . Similarly, F ◦ G ≤ idA
implies F ◦ G ◦ F ≤ idA ◦ F = F , so that by amnesticity F = F ◦ G ◦ F . The otherequation holds by duality.

6.31 COROLLARY
If (F, G) is a Galois correspondence between amnestic concrete categories, then (G ◦ F ) ◦(G ◦ F ) = G ◦ F and (F ◦ G) ◦ (F ◦ G) = F ◦ G.

6.32 COROLLARY
Let G : A → B and F : B → A be concrete functors between amnestic concrete categories such that (F, G) is a Galois correspondence, and let A∗ be the full subcategoryof A with objects: {F (B) | B ∈ Ob(B)} and B∗ the full subcategory of B with objects:{G(A) | A ∈ Ob(A)}. Then
(1) A∗ is coreflective in A, and A ∈ Ob(A∗ ) if and only if A = (F ◦ G)(A).
(2) B∗ is reflective in B, and B ∈ Ob(B∗ ) if and only if B = (G ◦ F )(B).
(3) The restrictions of G and F to A∗ and B∗ are concrete isomorphisms, G∗ : A∗ → B∗
and F ∗ : B∗ → A∗ , that are inverse to each other.

6.33 EXAMPLES(1) If (R, E) is a Galois reflection between amnestic concrete categories with E : A → B,then A∗ = B∗ = A. Similarly for Galois coreflections.
(2) For the Galois correspondence (D, U ) of Example 6.26(3)(a) Set∗ = Set and Top∗is the full subcategory of discrete spaces.
(3) For the Galois correspondence (U, N ) of Example 6.26(3)(a) Set∗ = Set and Top∗is the full subcategory of indiscrete spaces.
(4) For the Galois correspondence (F, G) of Example 6.26(3)(b) Rel∗ = Prost and Top∗consists of those topological spaces for which arbitrary intersections of open sets are
open.
(5) For the Galois correspondence (F, U ) of Example 6.26(3)(c) Top∗ is the full subcategory of completely regular (= uniformizable) spaces, and Unif ∗ is the full subcategory of fine uniform spaces.
6.34 PROPOSITION
Let G : A → B and F : B → A be concrete functors between amnestic concrete categories such that (F, G) is a Galois correspondence. Then the following are equivalent:(1) G is a full embedding,

9th July 2006

90

Categories, Functors, and Natural Transformations[Chap. I(2) G is full,(3) G is injective on objects,(4) F is surjective on objects,(5) F ◦ G = idA ,(6) up to Galois isomorphism, (F, G) is a Galois reflection; i.e., there exists a Galois reflection (R, E) and a Galois isomorphism (K −1 , K) with (F, G) = (R, E)◦(K −1 , K).
Proof:(1) ⇒ (2). Trivial.
(2) ⇒ (3). If GA = GA0 , then by fullness, A ≤ A0 and A0 ≤ A. Hence, by amnesticity,
A = A0 .
(3) ⇒ (4). By Proposition 6.30 for any A-object A, we have (G ◦ F ◦ G)(A) = G(A), sothat by (3), F (GA) = A.
(4) ⇒ (5). For any A-object A there exists, by (4), a B-object B with F (B) = A.
Hence, by Proposition 6.30, (F ◦ G)(A) = (F ◦ G ◦ F )(B) = F (B) = A.
(5) ⇒ (6). Let B∗ be the full subcategory of B with objects {G(A) | A ∈ Ob(A)};let K be the codomain restriction of G, i.e., K : A → B∗ ; let E : B∗ ,→ B be theinclusion; and let R = K ◦ F . Then by Corollary 6.32 K −1 = F ◦ E and (K −1 , K) is
a Galois isomorphism. Clearly, G = E ◦K and F = K −1 ◦R. It remains to be shown that
(R, E) is a Galois reflection; i.e., that R is a concrete reflector. For each B-object B,
id|B|

B ≤ (G ◦ F )(B). But (G ◦ F )(B) = R(B). Thus |B| −−−→ |R(B)| is a B-morphism sothat the reflection arrows are identity-carried.
(6) ⇒ (1). By amnesticity the embedding E must be full (5.24). Thus G = E ◦ K is thecomposition of full embeddings and so must be one too.

6.35 DECOMPOSITION THEOREM FOR GALOISCORRESPONDENCES
Every Galois correspondence (F, G) between amnestic concrete categories is a composite(F, G) = (R, EB ) ◦ (K −1 , K) ◦ (EA , C) of(1) a Galois coreflection, (EA , C),(2) a Galois isomorphism, (K −1 , K) and(3) a Galois reflection, (R, EB ).
A
C

G

EB



A∗

/B
O

K

/ B∗

AO o

F

EA

A∗ o

B


K −1

R

B∗

9th July 2006

Sec. 6]Natural transformations

91

Proof: If G : A → B and F : B → A are concrete functors such that (F, G) is a Galois
correspondence, then let A∗ and B∗ be the full subcategories of A and B determined by
the images of F and G, respectively (Corollary 6.32). Let C : A → A∗ and R : B → B∗
be the codomain restrictions of F ◦ G and G ◦ F , respectively, let K : A∗ → B∗ andK −1 : B∗ → A∗ be the corresponding restrictions of G and F , and let EA : A∗ → A
and EB : B∗ → B be the full embedding of these subcategories. The result then follows
immediately from Proposition 6.34 and its dual.

6.36 REMARK
Notice that the above theorem actually gives a characterization of Galois correspondences
between amnestic concrete categories since the composition of Galois correspondences is
a Galois correspondence.

Suggestions for Further Reading
Eilenberg, S., and S. Mac Lane. General theory of natural equivalences. Trans. Amer.
Math. Soc. 58 (1945): 231–294.
Yoneda, N. On the homology theory of modules. J. Fac. Sci. Tokyo 7 (1954): 193–227.
Herrlich, H., and M. Hušek. Galois connections categorically. J. Pure Appl. Algebra
68 (1990): 165–180.

EXERCISES
6A. Composition of Natural Transformations
τσ

Let F, F 0 : A → B and G, G0 : B → C be functors and let F −→ F 0 and G −→ G0 be
natural transformations. Show that(a) σF 0 ◦ Gτ = G0 τ ◦ σF . [This natural transformation is called the star product of τσ∗τand σ and is denoted by G ◦ F −−−→ G0 ◦ F 0 .](b) σF = σ ∗ idF and Gτ = idG ∗ τ .
(c) idG ∗ idF = idG◦F .
δ(d) If H, H 0 : C → D are functors and H −→ H 0 is a natural transformation, thenδ ∗ (σ ∗ τ ) = (δ ∗ σ) ∗ τ.
F 00

G00

τ0

σ0

(e) If A −−→ B and B −−→ C are functors and F 0 −−→ F 00 and G0 −−→ G00 are naturaltransformations, then(σ 0 ◦ σ) ∗ (τ 0 ◦ τ ) = (σ 0 ∗ τ 0 ) ◦ (σ ∗ τ ).

9th July 2006

92

Categories, Functors, and Natural Transformations[Chap. I

H(f) If C −−→ D is a functor, then (Hσ)F = H(σF ).
H(g) If C −−→ D is a functor, then (H ◦ G)τ = H(Gτ ).
K(h) If D −−→ A is a functor, then σ(F ◦ K) = (σF )K.
G00

σ0

H

(i) If B −−→ C and C −−→ D are functors and G0 −−→ G00 is a natural transformation,then H(σ 0 ◦ σ)F = (Hσ 0 F ) ◦ (HσF ).
6B. Counting Natural Transformations(a) Show that there is precisely one natural transformation idSet → idSet .
(b) Let M be a monoid considered as a category. Show that an element x of M yields
a natural transformation idM → idM if and only if x ◦ y = y ◦ x for each y ∈ M .
(c) How many natural transformations are there from S 2 to P? Cf. 3.20(8) and 3.20(10).
[Hint: Observe that S 2 is naturally isomorphic to hom(2, −) and use Corollary 6.19.]
6C. Functor-Structured Categories
Let S, T : X → Set be functors. Show that Spa(S) and Spa(T ) are concretely isomorphic if and only if S and T are naturally isomorphic.
6D. Functors Naturally Isomorphic to idAShow that
F(a) If a functor A −−→ A is naturally isomorphic to idA , then F is an equivalence.
F(b) A functor Set −−→ Set is naturally isomorphic to idSet if and only if F is an equivalence.
F(c) If G is a group considered as a category, then a functor G −−→ G (i.e., a group
endomorphism F ) is naturally isomorphic to idG if and only if F is an inner automorphism of G.
(d) If A is a discrete category, then idA is the only functor from A to A that is naturally
isomorphic to idA .
O

U

6E. Ban −−→ Set and Ban −−→ SetShow that O is representable, but U is not. Cf. 5.2(3).
6F. Representability of Power-Set Functors
Q

Show that the contravariant power-set functor Setop −−→ Set is representable (by anyPtwo-element set), but that the covariant power-set functor Set −−→ Set is not representable.
6G. Maps Induce Galois Connections
f

Pf

Qf

Let A −→ B be a map. Consider the functions PA −−→ PB and PB −−−→ PA[cf. 3.20(8) and (9)] as concrete functors between the power-sets PA and PB, orderedby inclusion, and show that (Pf, Qf ) is a Galois connection.

9th July 2006

Sec. 6]Natural transformations

93

6H. Legitimate Functor Quasicategories.
(a) Prove that for each small category A all quasicategories [A, B] are legitimate.
(b) Prove that 1 and ∅ are the only categories B with the property that each quasicategory [A, B] is legitimate.
(c) Prove that [Set, Set] is illegitimate.
6I. Total Categories
A category A is called total provided that the natural embedding E : A → [Aop , Set]of Theorem 6.20 maps it onto a reflective subcategory E[A] of [Aop , Set].
(a) Prove that Set is total. [Hint: A reflection of F : Setop → Set is given byτ : F → hom(−, F (1)), where 1 = {0} and τA maps a ∈ F (A) to the function
τA (a) : A → F (1) given by t 7→ F f (a) for f : 1 → A, with f (0) = t.](b) Prove that Spa(T ) is total for each T : Set → Set. [Hint: Analogous to (a) with(1, ∅) substituted for 1.](c) Prove that if A is a total category then every full reflective subcategory of A is total.
(d) Prove that a category isomorphic to a total category is total. Conclude that Pos,
Vec, Sgr, and Alg(Ω) are total categories. [Hint: Combine 6I(c) and 4K.]
6J. Yoneda Embedding
Show that the bijective function Y of Corollary 6.19 is “natural in the variables A andF ”, i.e., define functors H, G : [A, Set] × A → Set on objects by H(F, A) = F (A) andG(F, A) = [hom(A, −), F ] such that Y becomes a natural isomorphism from G to H.
6K. Representable Functors
If F : A → Set is a functor, then a universal point of F is a pair (A, a) consistingof an A-object A and a point a ∈ F A with the following (universal) property: for eachA-object B and each point b ∈ F B there exists a unique A-morphism f : A → B withF f (a) = b.
(a) Prove that a functor is representable if and only if it has a universal point.
(b) Find a universal point of each of the forgetful functors of Vec, Top, and Pos. Show
that the forgetful functor of Σ-Seq has no universal point.
(c) Find a universal point of Q : Setop → Set.
(d) For which sets M does the functor − × M : Set → Set have a universal point?
(e) Does P : Set → Set have a universal point?
6L. “Naturally Isomorphic” is an Equivalence Relationστ

Let F , G, and H be functors from A to B and let F −→ G and G −→ H be natural
isomorphisms. Show that:−1(a) σ −1 = (σA) : G → F is a natural isomorphism.

9th July 2006

94

Categories, Functors, and Natural Transformations[Chap. I(b) τ ◦ σ : F → H is a natural isomorphism.
6M. Naturally Isomorphic Hom-Functors
Let τ : hom(A, −) → hom(B, −) be a natural isomorphism. Show that τA (idA ) : B → A
is an isomorphism.

9th July 2006

95

Chapter II

OBJECTS AND MORPHISMS

9th July 2006

7 Objects and morphisms in abstract categories
INITIAL AND TERMINAL OBJECTS
7.1 DEFINITION
An object A is said to be an initial object provided that for each object B there isexactly one morphism from A to B.
7.2 EXAMPLES(1) The empty set ∅ is the unique initial object for Set. Likewise, the empty partiallyordered set (resp. the empty topological space) is the unique initial object for Pos(resp. Top).
(2) Every one-element group is an initial object for Grp; likewise for Vec.
(3) The empty category (i.e., the category with no objects and no morphisms) is the
only initial object for Cat. It is also the only initial object for the quasicategory
CAT, and, considered as a concrete category over X (via the inclusion), is the only
initial object in CAT(X).
(4) For any category of the form Spa(T ) (Definition 5.40) an object (X, α) is initial if
and only if X is an initial object in X and α = ∅.
(5) For any category of the form Alg(Ω) [Example 3.3(2)] let Ωn = {i ∈ I | ni = n}. IfΩ0 = ∅, then the unique initial object is the empty algebra. If Ω0 6= ∅, then an initial
object in Alg(Ω) is the term algebra. Its members are terms, defined inductively
as follows:(a) each element of Ω0 is a term;(b) if i ∈ Ωn and if ω1 , ω2 , . . . , ωn are terms, then iω1 ω2 . . . ωn is a term; and(c) all terms are obtained by iterative application of (a) and (b) above.
Each term algebra operation is concatenation via rule (b) resp. (a).
(6) The ring of integers is an initial object in the construct Rng of rings with unit andunitary ring homomorphisms. For a ring R with unit e, the unique homomorphism
f : → R is defined by f (n) = e + e + · · · + e [n summands], and f (−n) = −f (n).

Z(7) The two-element boolean algebras are initial objects in Boo.
(8) Σ-Seq has no initial object.
(9) In a poset considered as a category, an object is an initial object if and only if it isa smallest element.

96Sec. 7]

Objects and morphisms in abstract categories

97

7.3 PROPOSITION
Initial objects are essentially unique, i.e.,(1) if A and B are initial objects, then A and B are isomorphic,(2) if A is an initial object, then so is every object that is isomorphic to A.
Proof:kh(1). By definition, there are morphisms A −→ B and B −→ A. Furthermore, h ◦ k = idA
since idA is the unique morphism from A to A. Analogously, k ◦ h = idB . Thus k is
an isomorphism.
(2). Let k : A0 → A be an isomorphism. For each object B there is a unique morphism
f : A → B. Then f ◦ k : A0 → B is a morphism from A0 to B. It is unique since if
g : A0 → B, then g ◦ k −1 : A → B. So g ◦ k −1 must be f ; i.e., g must be f ◦ k. 
Next we define terminal objects. They are dual to initial objects; i.e., A is terminal in
A if and only if A is initial in Aop .
7.4 DEFINITION
An object A is called a terminal object provided that for each object B there is exactly
one morphism from B to A.

A terminal object

7.5 EXAMPLES(1) Every singleton set is a terminal object for Set.
(2) Frequently for constructs, there is only one structure on the singleton set {0}, andin these cases the corresponding object is a terminal object. This is the case, forexample, in Vec, Pos, Grp, Top, and Catof .

9th July 2006

98

Objects and Morphisms[Chap. II(3) In Rel there are two structures on the set {0}. Of these, the pair ({0}, {(0, 0)}) isa terminal object.
(4) For any category of the form Spa(T) (Definition 5.40) an object (X, α) is terminal
if and only if X is a terminal object object in X and α = T (X).
(5) In Σ-Seq the acceptor with exactly one state that is both an initial and a final stateis a terminal object.
(6) In a poset considered as a category an object is a terminal object if and only if it isa largest element.
(7) In CAT(X) an object (A, U ) is terminal if and only if U : A → X is an isomorphism.
In particular, (X, idX ) is a terminal object.
7.6 PROPOSITION
Terminal objects are essentially unique.

D

ZERO OBJECTS
7.7 DEFINITION
An object A is called a zero object provided that it is both an initial object and aterminal object.
7.8 REMARK
Notice that since “terminal object” is dual to “initial object”, the notion of zero object
is self-dual; i.e., A is a zero object in A if and only if it is a zero object in Aop .
7.9 EXAMPLES(1) Set and Top don’t have zero objects, but pSet and pTop (cf. Exercise 3B) do havezero objects — the “singletons”.
(2) Vec, Ban, Banb , TopVec, and Mon have zero objects, but Sgr doesn’t.
(3) Ab and Grp have zero objects, but Rng doesn’t.
(4) Pos and Cat don’t have zero objects.

SEPARATORS AND COSEPARATORS
7.10 DEFINITION
An object S is called a separator provided that whenever A

f
g

//

B are distinct mor-hphisms, there exists a morphism S −→ A such that
h

f

h

g

S −→ A −→ B 6= S −→ A −→ B.

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

99

7.11 EXAMPLES(1) In Set the separators are precisely the nonempty sets.
(2) In Top (resp. Pos) the separators are precisely the nonempty spaces (resp. nonempty
posets).
(3) In Vec the separators are precisely the nonzero vector spaces.

Z(4) The group of integers
under addition is a separator for Grp and for Ab. The
monoid of natural numbers
under addition is a separator for Mon.

N

(5) (X, ρ) is a separator in Rel if and only if X 6= ∅ = ρ.
7.12 PROPOSITION
An object S of a category A is a separator if and only if hom(S, −) : A → Set is afaithful functor.
Proof: The faithfulness of hom(S, −) means, by definition, that given distinct A-morf

phisms Ag//

B , their hom(S, −) images are distinct; i.e., they differ in at least one

element h ∈ hom(S, A). In other words, f ◦ h 6= g ◦ h, and this is precisely the definitionof separator.

7.13 REMARKAs we will see later, the existence of a separator in A often serves as a useful “smallness”
condition for A that guarantees that there are not “too many” A-objects. A slightly
weaker condition that serves the same purpose is the existence of a separating set.
7.14 DEFINITIONA set T of objects is called a separating set provided that for any pair A

f
g

//

B of

h

distinct morphisms, there exists a morphism S −→ A with domain S a member of Tsuch that f ◦ h 6= g ◦ h.
7.15 EXAMPLES(1) The empty set is a separating set for A if and only if A is thin.
(2) A one-element set {S} is a separating set if and only if S is a separator.
(3) Set × Set has no separators, but the set consisting of the two objects (∅, {0}) and({0}, ∅) is a separating set.
(4) Aut has no separator, but {A1 , A2 } is a separating set, where A1 has states {q0 , q1 },with q0 initial, no input, and output set {0, 1, 2}, with y(q0 ) = 0 and y(q1 ) = 1;
and A2 has states {qi | i ∈ }, with q0 initial, one input σ, with δ(σ, qi ) = qi+1 , andy(qi ) = i.

N

9th July 2006

100

Objects and Morphisms[Chap. II

Next we introduce coseparators — the dual concept to separators. From the list of
examples one can discern that in familiar categories coseparators are more rare thanseparators.
7.16 DEFINITION
An object C is a coseparator provided that whenever B

f
g

//

A are distinct morphisms,hthere exists a morphism A −→ C such that
f

g

h

h

B −→ A −→ C 6= B −→ A −→ C.
7.17 PROPOSITION
C is a coseparator for A if and only if hom(−, C) : Aop → Set is faithful.

D7.18 EXAMPLES(1) In Set the coseparators are precisely those sets that have at least two elements. [IffC has at least two elements and if Bg//

A differ on b ∈ B, let h : A → C be any

function with h(f (b)) 6= h(g(b)).](2) In Vec the coseparators are precisely the nonzero vector spaces.
(3) The coseparators in Pos are precisely the nondiscrete posets. To see this, supposefthat C has elements x < y and Bg// A differ on b ∈ B, say, f (b) 6≤ g(b). Defineh : A → C by(xh(a) =
yif a ≤ g(b)if a ≤
6 g(b).

(4) The coseparators in Top are precisely the non-T0 -spaces. [If C is a non-T0 -space,then C has an indiscrete subspace with at least two points. Using this fact, proceed
as in (1) above.](5) The coseparators in Top0 are precisely the non-T1 -spaces. To see this, suppose that
C is a T0 -space that is not T1 . Then there exist elements x and y with y ∈ c`{x}
and x 6∈ c`{y}. If B

fg// A differ on b ∈ B, then there exists an open set W in A

that contains precisely one of f (b) and g(b). Define h : A → C by(x
if a ∈ W
h(a) =
yif a 6∈ W.
(6) In a preordered class considered as a category, every element (= object) is simultaneously a separator and a coseparator.

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

101(7) Any two-element boolean algebra is a coseparator for Boo. The closed unit interval
is a coseparator for Tych. The Banach space of complex numbers is a coseparator for both Banb and Ban (Hahn-Banach Theorem). The circle group / is a
coseparator for Ab and, considered as a compact group, it is a coseparator for the
category of locally compact abelian groups. Set is a coseparator for the quasicategory CAT. [If F, G : B → A, with F 6= G, then hom(A, −) ◦ F 6= hom(A, −) ◦ G
for some A-object A.]

RZ(8) None of the categories Rng, Grp, Sgr, or Haus has a coseparator. (This follows forrings from the existence of arbitrarily large fields, for groups and semigroups fromthe existence of arbitrarily large simple groups, and for Hausdorff spaces from thefact that for each Hausdorff space X there exists a Hausdorff space Y with morethan one point such that every continuous map from Y to X is constant.) Indeed,none of these categories even has a coseparating set.

SECTIONS AND RETRACTIONS
7.19 DEFINITIONf
A morphism A −→ B is called a section provided that there exists some morphismgB −→ A such that g ◦ f = idA (i.e., provided that f has a “left-inverse”).

Sections

7.20 EXAMPLES(1) A morphism in Set is a section if and only if it is an injective function and is not
the empty function from the empty set to a nonempty set.
(2) In Vec the sections are exactly the injective linear transformations.
(3) In Ab a homomorphism f : A → B is a section if and only if it is injective and f [A]is a direct summand of B.

9th July 2006

102

Objects and Morphisms[Chap. II(4) If X and Y are sets (resp. topological spaces) and if a ∈ Y , then the function
f : X → X × Y defined by f (x) = (x, a) is a section in Set (resp. Top). [Note thatthe image of f is a “cross-section” of the product, which is in one-to-one correspondence (resp. homeomorphic) to X. This motivates our use of the word “section” inDefinition 7.19.](5) The sections described in (4) are just special cases of the following situation: Letf : X → Y be a morphism in Set (resp. Top, Grp, R−Mod). Consider the graphof f as a subset (resp. subspace, subgroup, submodule) of the product X × Y . Thenthe embedding of X into X × Y defined by x 7→ (x, f (x)) is a section in the category
in question.
(6) If T is a terminal object, then every morphism with domain T is a section.
(7) In a thin category, sections are precisely the isomorphisms.
7.21 PROPOSITIONfgfg(1) If A −→ B and B −→ C are sections, then A −→ B −→ C is a section.
fg(2) If A −→ B −→ C is a section, then f is a section.
Proof:(1). Given h with h ◦ f = idA and k with k ◦ g = idB , then (h ◦ k) ◦ (g ◦ f ) = idA .
(2). Given h with h ◦ (g ◦ f ) = idA , we have (h ◦ g) ◦ f = idA .



7.22 PROPOSITION
Every functor preserves sections (i.e., if F : A → B is a functor and f is an Asection, then F (f ) is a B-section).
Proof: If h ◦ f = idA , then F h ◦ F f = F (h ◦ f ) = F (idA ) = idF A .



7.23 PROPOSITION
Every full, faithful functor reflects sections (i.e., if F : A → B is full and faithful andF (f ) is a B-section, then f is an A-section).
Proof: Given h : F B → F A with h ◦ F (f ) = idF A , by fullness there is k : B → A with
h = F (k). Thus F (k ◦ f ) = idF A = F (idA ), so that by faithfulness k ◦ f = idA .
The dual concept for “section” is “retraction”. The name comes from topology, where asubspace Y of a space X is called a retract if there is a continuous function f : X → Y
with f (y) = y for each y ∈ Y ; so that if e : Y → X is the inclusion, f ◦ e = idY .
7.24 DEFINITIONf
A morphism A −→ B is called a retraction provided that there exists some morphismgB −→ A such that f ◦ g = idB (i.e., provided that f has a “right-inverse”). If there exists
such a retraction, then B will be called a retract of A.

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

103

7.25 EXAMPLES(1) The retractions in Set are precisely the surjective functions. [Notice that this statement is equivalent to the Axiom of Choice for sets.](2) In Vec the retractions are exactly the surjective linear transformations.
(3) A morphism f in Top is a retraction if and only if there is a topological retractionr and a homeomorphism h such that f = h ◦ r. In other words, the retractions in
Top are (up to homeomorphism) exactly the topological retractions.
(4) In any thin category the concepts “retraction”, “section”, and “isomorphism” agree.
(5) The usual underlying functor U : Top → Set is a retraction in the quasicategory
CAT. It has precisely two right inverses, the discrete functor and the indiscrete
functor.
(6) In Ab retractions are (up to isomorphism) the projection homomorphisms, i.e.,f : A → B is a retraction if and only if there is an abelian group C such thatfhp

A −→ B = A −→ B × C −→ B, where h is an isomorphism and p is a projection.
7.26 PROPOSITION
For a morphism f the following are equivalent:(1) f is an isomorphism,(2) f is a section and a retraction.
Proof: This follows immediately from Proposition 3.10.



7.27 PROPOSITIONfgg◦f(1) If A −→ B and B −→ C are retractions, then A −−−→ C is a retraction.
g◦f(2) If A −−−→ C is a retraction, then g is a retraction.

D

7.28 PROPOSITION
Every functor preserves retractions.

D

7.29 PROPOSITION
Every full, faithful functor reflects retractions.

D

7.30 REMARKS(1) Notice that by combining Proposition 7.23 and its dual (7.29) we get the result(already proved [3.32]) that every full, faithful functor reflects isomorphisms.
(2) Notice also that we did not need to define “sect” as the dual of retract, since “retract”
is self-dual.

9th July 2006

104

Objects and Morphisms[Chap. II

7.31 PROPOSITION
An isomorphism-closed full reflective subcategory A of a category B contains with any
object A each retract of A in B.
r

Proof: Let A −→ B be a B-retraction of an A-object A. Then there exist a sectionus : B → A such that r ◦ s = idB and an A-reflection B −→ AB . By the definition oftreflection there is an A-morphism AB −→ A such that s = t ◦ u. Thus (r ◦ t) ◦ u = r ◦ s =idB . Hence u ◦ (r ◦ t) ◦ u = u = idAB ◦ u, so that by uniqueness u ◦ (r ◦ t) = idAB . Thusu is an isomorphism, and since A is isomorphism-closed in B, B must belong to A. 

MONOMORPHISMS AND EPIMORPHISMS
7.32 DEFINITIONf
A morphism A −→ B is said to be a monomorphism provided that for all pairsChk// A of morphisms such that f ◦ h = f ◦ k, it follows that h = k (i.e., f is “left-

cancellable” with respect to composition).
7.33 EXAMPLES(1) A function is a monomorphism in Set if and only if it is injective. [To show thata monomorphism f : A → B must be injective, take a, b ∈ A and consider theconstant functions â, b̂ : {p} → A.](2) For any morphism f in Vec, the following are equivalent:(a) f is a monomorphism,(b) f is a section,(c) f is injective.
(3) In many constructs, monomorphisms are precisely those morphisms that have injective underlying functions; e.g., this is the case for Pos, Top, Grp, Ab, Sgr,Rng, Rel, and Alg(Ω). In fact this is true for any construct with a representable
underlying functor (7.37 and 7.38).
(4) In the construct {σ}-Seq (whose forgetful functor is not representable), monomorphisms are precisely the injective simulations. [If f : A → B is not injective, thenthere exist distinct states x1 and x2 with f (x1 ) = f (x2 ), and there exist n, m, and ksuch that σ n+k x1 = σ k x1 and σ m+k x2 = σ k x2 . Let C be the acceptor obtained fromA by adding k +nm nonfinal states q1 , . . . , qk+nm such that σqi = qi+1 for i < k +nm
and σqk+nm = qk+1 . For i = 1, 2, there is a unique simulation gi : C → A that is
the inclusion on A-states and such that gi (q1 ) = xi . Then f ◦ g1 = f ◦ g2 .](5) In the category DivAb of divisible abelian groups and group homomorphisms there
are monomorphisms that have non-injective underlying functions. Consider thenatural quotient → / , whereis the additive group of rational numbers andis the additive group of integers.

Z

Q QZ

Q

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories
f

105
f˜

(6) There is a monomorphism X −→ Y in Top such that the homotopy class X −→ Yof f is not a monomorphism in the category hTop, whose objects are topologicalspaces and whose morphisms are homotopy equivalence classes of continuous maps.
[Consider the usual embedding of the bounding circle of a disc into the disc.](7) The monomorphisms in CAT are precisely the embedding functors.
(8) In the category of all fields35 and homomorphisms between them, every morphism
is a monomorphism.
(9) In any thin category every morphism is a monomorphism.
7.34 PROPOSITIONfgfg(1) If A −→ B and B −→ C are monomorphisms, then A −→ B −→ C is a monomorphism.
fg(2) If A −→ B −→ C is a monomorphism, then f is a monomorphism.
Proof: Let h, k : D → A.
(1). (g ◦ f ) ◦ h = (g ◦ f ) ◦ k ⇒ g ◦ (f ◦ h) = g ◦ (f ◦ k) ⇒ f ◦ h = f ◦ k ⇒ h = k.
(2). f ◦ h = f ◦ k ⇒ (g ◦ f ) ◦ h = (g ◦ f ) ◦ k ⇒ h = k.

7.35 PROPOSITION
Every section is a monomorphism.
Proof: Suppose that g ◦ f = id and f ◦ h = f ◦ k. Then h = g ◦ f ◦ h = g ◦ f ◦ k = k.
7.36 PROPOSITION
For any morphism f the following are equivalent:(1) f is an isomorphism,(2) f is a retraction and a monomorphism.
Proof: (1) ⇒ (2) is clear from Propositions 7.26 and 7.35. To show that (2) ⇒ (1), let
f be a monomorphism with f ◦ g = id. Then f ◦ (g ◦ f ) = (f ◦ g) ◦ f = id ◦ f = f ◦ id, sothat by left-cancellation, g ◦ f = id. Hence f is an isomorphism.

7.37 PROPOSITION(1) Every representable functor preserves monomorphisms, i.e., if F : A → Set isrepresentable and if f is a monomorphism in A, then F (f ) is a monomorphism in
Set (i.e., an injective function).
(2) Every faithful functor reflects monomorphisms, i.e., if F : A → B is faithfuland F (f ) is a B-monomorphism, then f is an A-monomorphism.
35

Recall that in each field 0 6= 1.

9th July 2006

106

Objects and Morphisms[Chap. II

Proof:(1). Simple computations show that(a) hom-functors hom(A, −) : A → Set preserve monomorphisms,(b) whenever functors F and G are naturally isomorphic and F preserves monomorphisms, then so does G.
(2). Suppose that f ◦ h = f ◦ k. Then F f ◦ F h = F f ◦ F k implies that F h = F k, sothat by faithfulness h = k.
7.38 COROLLARY
In any construct all morphisms with injective underlying functions are monomorphisms.
When the underlying functor is representable, the monomorphisms are precisely the morphisms with injective underlying functions.

The categorical dual of “monomorphism” is “epimorphism”.
7.39 DEFINITIONf
A morphism A −→ B is said to be an epimorphism provided that for all pairs B

h
k

//

C

of morphisms such that h ◦ f = k ◦ f , it follows that h = k (i.e., f is “right-cancellable”
with respect to composition).
7.40 EXAMPLES(1) In both Set and Vec the following are equivalent for any morphism f :(a) f is an epimorphism,(b) f is a retraction,(c) f is surjective.
f[To show that an epimorphism A −→ B in Set is surjective, consider two functions
from B to {0, 1}, one of them mapping every point of B to 0 and the other mappingf

precisely the points of f [A] to 0. To show that A −→ B in Vec is surjective, use theabove idea with the quotient B/f [A] replacing {0, 1}.]
(2) In a number of constructs the epimorphisms are precisely the morphisms with surjective underlying functions. This is the case, for instance, in Top, Rel, each Alg(Ω),
Lat, Σ-Seq, Pos, Ab, Grp, and HComp. [For Top, argue as in Set, where {0, 1}
is given the indiscrete topology; similarly for Rel. For Ab and HComp, argue asin Vec.] However, this situation occurs less frequently than that of monomorphisms
being precisely those morphisms with injective underlying functions. In quite a few
familiar constructs, epimorphisms fail to be surjective (see below), and even when
they are surjective, the proof may be far from obvious; e.g., for Grp see Exercise
7H.

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

107(3) In Haus the epimorphisms are precisely the continuous functions with dense images.
Also in Banb , and in Ban (with either of the two natural forgetful functors) [5.2(3)],the epimorphisms are precisely the morphisms with dense images. In the categoryof Hausdorff topological groups there exist epimorphisms with non-dense images.
f(4) In the category of torsion-free abelian groups, a morphism A −→ B is an epimorphism if and only if the factor group B/f [A] is a torsion group. Thus, in thiscategory, the inclusion 2 ,→ is a non-surjective epimorphism.

Z

Z(5) In Rng and Sgr there are epimorphisms that are not surjective; e.g., the usual

Z f Q

embedding −→ of the integers into the rationals is an epimorphism in Rng andin Sgr. [If h and k are homomorphisms such that h ◦ f = k ◦ f and if n/m ∈ ,thenQh(n/m) = h(n) · h(1/m) · h(1) = k(n) · h(1/m) · k(1)= k(n) · h(1/m) · k(m) · k(1/m) = k(n) · h(1/m) · h(m) · k(1/m)= k(n) · h(1) · k(1/m) = k(n) · k(1) · k(1/m) = k(n/m).](6) In Cat there are epimorphisms that are not surjective. Consider the epimorphismg
F : A → B, where A = • −→ •, B is the additive monoid of natural numbers, andF is the unique functor from A and B with F (g) = 1.
f

f˜(7) There is an epimorphism X −→ Y in Top such that the homotopy class X −→ Yof f is not an epimorphism in the category hTop. Consider the covering projectionof the real line onto the circle, defined by: x 7→ eix .
(8) In a thin category, each morphism is an epimorphism.
7.41 PROPOSITIONfgfg(1) If A −→ B and B −→ C are epimorphisms, then A −→ B −→ C is an epimorphism.
fg(2) If A −→ B −→ C is an epimorphism, then g is an epimorphism.

D

7.42 PROPOSITION
Every retraction is an epimorphism.

D

7.43 PROPOSITION
For a morphism, f , the following are equivalent:(1) f is an isomorphism.
(2) f is a section and an epimorphism.

D

7.44 PROPOSITION
Every faithful functor reflects epimorphisms.

D

9th July 2006

108

Objects and Morphisms[Chap. II

7.45 COROLLARY
In any construct all morphisms with surjective underlying functions are epimorphisms. D
7.46 REMARK
Although faithful functors reflect epimorphisms and monomorphisms, they need notpreserve them (as the above examples show). In fact, even full embeddings may failto do so.36 For example, the full embedding E : Haus ,→ Top doesn’t preserve epimorphisms [7.40(2) and (3)] and so the full embedding E op : Hausop ,→ Topop doesn’t
preserve monomorphisms. However, if such functors are also isomorphism-dense, thenthey preserve monomorphisms and epimorphisms, as the following shows:
7.47 PROPOSITION
Every equivalence functor preserves and reflects each of the following: monomorphisms,
epimorphisms, sections, retractions, and isomorphisms.
Proof: By duality and Propositions 7.22, 7.23, and 7.37, we need only show preserf

vation of monomorphisms. Let F : A → B be an equivalence, let A0 −→ A be an
A-monomorphism, and let B

r /s/ F A0 be morphisms with F f ◦ r = F f ◦ s. Since F iskisomorphism-dense, there exists an A-object A00 and a B-isomorphism F A00 −→ B. By
fullness there are A-morphisms A00

r0
s0

//

A0 with F r0 = r ◦ k and F s0 = s ◦ k. Thus

F (f ◦ r0 ) = F f ◦ F r0 = F f ◦ r ◦ k = F f ◦ s ◦ k = F f ◦ F s0 = F (f ◦ s0 ).
Faithfulness and the fact that f is a monomorphism imply that r0 = s0, from which it
follows readily that r = s.

7.48 REMARKThe above is typical of equivalences. They preserve and reflect virtually all properties
that are considered to be categorical. In fact, a reasonable way to define a “categoricalproperty” would be as “a property of categories that is preserved and reflected by all
equivalences”. This is why equivalent categories are considered to be almost as muchalike as isomorphic ones.
7.49 DEFINITION(1) A morphism is called a bimorphism provided that it is simultaneously a monomorphism and an epimorphism.
(2) A category is called balanced provided that each of its bimorphisms is an isomorphism.
36

However, we will see later [Proposition 18.6 and Example 19.12(1)] that embeddings of reflective
subcategories must preserve monomorphisms.

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

109

7.50 EXAMPLES(1) Set, Vec, Grp, Ab, and HComp are balanced categories.
(2) Rel, Pos, Top, Mon, Sgr, Rng, Cat, Ban, and Banb are not balanced categories.
The inclusion ,→ is a non-isomorphic bimorphism in Mon, Sgr, Rng, and Cat.

Z

Qf
The function `∞ −→ c0 ,37 defined by f (xn ) = ( xnn ), is a non-isomorphic bimorphism

in Ban and Banb .

REGULAR AND EXTREMAL MONOMORPHISMS
Originally it was believed that monomorphisms would constitute the correct categoricalabstraction of the notion “embeddings of substructures” that exists in various constructs.
However, in many instances the concept of monomorphism is too weak; e.g., in Top
monomorphisms are just injective continuous maps and thus need not be embeddings.
Below we introduce two stronger notions (and there are several others in current use— see Exercises 7D and 14C) that more frequently correspond with embeddings in
categories. However, a satisfactory concept of “embeddings” seems to be possible onlyin the setting of constructs (see 8.6).
7.51 DEFINITION
f
e
//
Let A
B be a pair of morphisms. A morphism E −→ A is called an equalizer ofgf and g provided that the following conditions hold:(1) f ◦ e = g ◦ e,(2) for any morphism e0 : E 0 → A with f ◦ e0 = g ◦ e0, there exists a unique morphisme : E 0 → E such that e0 = e ◦ e, i.e., such that the triangle
E0 A

AA 0
AAe
AA

E e /A

e

f
g

// B

commutes.
7.52 EXAMPLES(1) Let A be one of the categories Set, Vec, Pos, Top, or Grp. If A

f
g

//

B are A-morphisms, and if E denotes the set {a ∈ A | f (a) = g(a)} considered as a subset(resp. linear subspace, subposet, subspace, subgroup) of A, then the inclusion frome0

E to A is an equalizer of f and g. [If E 0 −−→ A is such that f ◦ e0 = g ◦ e0 , then e isthe codomain restriction of e0 .]37 ∞`(resp. c0 ) is the classical Banach space of all bounded sequences in the field K (resp. all sequencesin K that converge to 0) with the sup-norm.

9th July 2006

110

Objects and Morphisms[Chap. II(2) If A is the full subcategory of Top that consists of sequential spaces (resp. compactly
generated spaces) and if E (as in (1)) is supplied with the coarsest sequential (resp.
ecompactly generated) topology for which the inclusion E −→ A is continuous, thene is an equalizer of f and g in A.
7.53 PROPOSITION
Equalizers are essentially unique; i.e., given A

f
g

//

B in a category, then the following

hold:
e0

e

(1) if each of E −→ A and E 0 −−→ A is an equalizer of f and g, then there is an
isomorphism k : E 0 → E with e0 = e ◦ k,ek(2) if E −→ A is an equalizer of f and g, and if E 0 −→ E is an isomorphism, thene◦k

E 0 −−−→ A is also an equalizer of f and g.
Proof:(1). Since f ◦ e0 = g ◦ e0, there is a k with e0 = e ◦ k. Analogously, there is an h withe = e0 ◦ h. Thus e ◦ id = e = e0 ◦ h = e ◦ (k ◦ h), so that by the uniquenessrequirement in the definition of equalizer, id = k ◦ h. Similarly, h ◦ k = id, so thatk is an isomorphism.
(2). This is clear, since whenever e is a unique morphism with e ◦ e = e0 , then k −1 ◦ ewill be a unique morphism with (e ◦ k) ◦ (k −1 ◦ e) = e0 .
7.54 PROPOSITIONe
If E −→ A is an equalizer of A

f
g

//

B , then the following are equivalent:(1) f = g,(2) e is an epimorphism,(3) e is an isomorphism,(4) idA is an equalizer of f and g.



7.55 PROPOSITIONgf
idA
If A −→ B −→ A = A −−−→ A, then g is an equalizer of g ◦ f and idB .
Proof: Clearly (g ◦ f ) ◦ g = g ◦ (f ◦ g) = g = idB ◦ g. Given h : C → B with (g ◦ f ) ◦ h =idB ◦ h, then f ◦ h : C → A is the unique morphism k with h = g ◦ k.
7.56 DEFINITIONe
A morphism E −→ A is called a regular monomorphism provided that it is anequalizer of some pair of morphisms.

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

111

7.57 REMARKS(1) It is clear from the uniqueness requirement in the definition of equalizer that regularmonomorphisms must be monomorphisms.
(2) One should be aware of the difference between the concepts of equalizer and regularmonomorphism. “Equalizer” is defined relative to pairs of morphisms, whereas “regular monomorphism” is absolute. The difference is more than just a technical one.
For example, later we will see that it is possible for a functor to preserve regularmonomorphisms without preserving equalizers (13.6).
7.58 EXAMPLES(1) In Set the regular monomorphisms are the injective functions, i.e., up to isomore
phism, precisely the inclusions of subsets. [If E −→ A is an inclusion of a subset,
consider functions f and g from A to {0, 1}, where f maps every point of A to 1
and g maps precisely the points of E to 1. Then e is an equalizer of f and g.](2) In Top the regular monomorphisms are, up to isomorphism, precisely the embede
dings of subspaces. [If E −→ A is an inclusion of a subspace, proceed as in (1), andsupply the set {0, 1} with the indiscrete topology.](3) In Haus the regular monomorphisms are, up to isomorphism, precisely the inclusionsof closed subspaces. [If e : E → A is an embedding of a closed subspace in Haus,
let A1 = A2 = A, and let Ã be the topological sum of A1 and A2 with embeddingsµ1 and µ2 . Let q : Ã → B be the quotient map that for each x ∈ E identifies µ1 (x)and µ2 (x), but identifies nothing else. Then B is Hausdorff and e is an equalizer ofq ◦ µ1 and q ◦ µ2 .] Similarly, in Metc (resp. Banb ) the regular monomorphisms are
the topological embeddings of closed (linear) subspaces. In Met (resp. Ban) the
regular monomorphisms are the isometric embeddings of closed (linear) subspaces.
(4) In many “algebraic” categories, e.g., in Vec, Grp, and Alg(Ω), and also in HComp,all monomorphisms are regular. [If e : E → A is a monomorphism in Vec, consider
the quotient space B = A/e[E], the zero map f : A → B, and the natural quotient
map g : A → B. Then e is an equalizer of f and g.] However, in Sgr and Rng
monomorphisms need not be regular, e.g., the inclusion,→
is a non-regularmonomorphism. [If it were regular, then since it is an epimorphism [7.40(5)] by
Corollary 7.63 and Proposition 7.66 it would necessarily be an isomorphism.]ZQ(5) In {σ}-Seq the regular monomorphisms are precisely the injective simulations that
map each nonfinal state to a nonfinal state.
7.59 PROPOSITION(1) Every section is a regular monomorphism.
(2) Every regular monomorphism is a monomorphism.
Proof: (1) is immediate from Proposition 7.55 and (2) follows from the uniquenessrequirement in the definition of equalizer.


9th July 2006

112

Objects and Morphisms[Chap. II

7.60 REMARKNeither implication of the previous proposition can be reversed. For example, ,→is a monomorphism in Sgr that is not regular, and every embedding of a non-connected
topological space into a connected one is a regular monomorphism in Top that is not asection. Also the embedding of the unit circle into the unit disc is not a section in Top.
(This is the crucial lemma for Brouwer’s fixed-point theorem.)Z

Q

Next, we introduce another type of monomorphism that is particularly useful because(as is the case with regular monomorphisms) it often coincides with embeddings inconstructs, and, since it is defined by intrinsic properties, it is sometimes easier tohandle than regular monomorphisms.
7.61 DEFINITION
A monomorphism m is called extremal provided that it satisfies the following extremal
condition: If m = f ◦ e, where e is an epimorphism, then e must be an isomorphism.
7.62 PROPOSITIONfg
Let A −→ B and B −→ C be morphisms.
(1) If f is an extremal monomorphism and g is a regular monomorphism, then g ◦ f isan extremal monomorphism.
(2) If g ◦ f is an extremal monomorphism, then f is an extremal monomorphism.
(3) If g ◦ f is a regular monomorphism and g is a monomorphism, then f is a regularmonomorphism.
Proof:(1). Let g be an equalizer of r and s. Let g ◦ f = h ◦ e, where e is an epimorphism. Thenr ◦ h ◦ e = r ◦ g ◦ f = s ◦ g ◦ f = s ◦ h ◦ e implies that r ◦ h = s ◦ h. Thus there exists
a unique morphism k with h = g ◦ k. Hence g ◦ k ◦ e = h ◦ e = g ◦ f . Consequently,k ◦ e = f , which implies that e is an isomorphism.
(2). Immediate.
(3). If g ◦ f is an equalizer of r and s, then f is an equalizer of r ◦ g and s ◦ g.
7.63 COROLLARY
Every regular monomorphism is extremal.



7.64 REMARK
A composite of extremal monomorphisms may fail to be extremal. However, if all extremal monomorphisms in a category are regular, then, by Proposition 7.62, in thatcategory the class of extremal (= regular) monomorphisms is closed under composition
and is cancellable from the left; i.e., its compositive behavior is similar to that of theclass of all monomorphisms (cf. Proposition 7.34). This is the case for most of the familiar categories, e.g., for Set, Vec, Rel, Pos, Top, Haus, Grp, and Alg(Ω). (See also
the dual of Proposition 14.14.) However, this is not always the case, as the following
example and Exercises 7J and 14I show.

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

113

7.65 EXAMPLELet FHaus denote the full subcategory of Top consisting of the functionally Hausdorfffspaces.38 Then a morphism A −→ B in FHaus is(a) a monomorphism if and only if f is injective,(b) an extremal monomorphism if and only if f is an embedding such that for everygsubspace C of B with f [A] ( C, there exists a non-constant morphism C −→with the restriction g|f [A] constant,R(c) a regular monomorphism if and only if f is an embedding of a subspace of B thatis an intersection of zerosets39 in B (cf. 7J).
7.66 PROPOSITION
For any morphism f the following are equivalent:(1) f is an isomorphism,(2) f is an extremal monomorphism and an epimorphism.
Proof: (1) ⇒ (2) is immediate from Proposition 7.59, Corollary 7.63, and duality. If fsatisfies (2), the trivial factorization f = id ◦ f shows that f is an isomorphism.

7.67 PROPOSITION
For any category A, the following are equivalent:(1) A is balanced,(2) in A each monomorphism is extremal.
Proof: (1) ⇒ (2) is immediate from Proposition 7.34(2), and (2) ⇒ (1) follows fromthe proposition above (7.66).


REGULAR AND EXTREMAL EPIMORPHISMS
One of the nice insights that can be gleaned from category theory is that the formationof quotient structures (such as groups of cosets and identification topologies) can beviewed as the dual of the formation of substructures. Analogous to the situation wheremonomorphisms frequently are too weak to represent substructures, epimorphisms are
frequently too weak to represent quotient structures. The notions of regular epimorphismand extremal epimorphism, which are the duals of the notions of regular monomorphismand extremal monomorphism, frequently coincide with natural quotients in abstractcategories. However, as is the case with subobjects, a truly satisfactory concept of“quotient” seems to be possible only in the setting of constructs (see 8.10).
38

A topological space A is called a functionally Hausdorff space provided that for any two distinct
points a and b of A there exists a continuous map from A to the real numbers that has different
values at a and b.
39
Z is called a zeroset in B provided that there exists a continuous map from B to the real numbers
such that Z is the inverse image of 0.

9th July 2006

114

Objects and Morphisms[Chap. II

7.68 DEFINITION
f
c
//
Let A
B be a pair of morphisms. A morphism B −→ C is called a coequalizer ofgf and g provided that the following conditions hold:(1) c ◦ f = c ◦ g,(2) for any morphism c0 : B → C 0 with c0 ◦ f = c0 ◦ g, there exists a unique morphism
c : C → C 0 such that c0 = c ◦ c; i.e., such that the triangle
f

A

g

//

c /
C
AA
AA
A c
c0 A BA

C0

commutes.
7.69 EXAMPLES
1. Given two functions A

fg// B in Set, let ∼ be the smallest equivalence relation on

B such that f (a) ∼ g(a) for all a ∈ A. Then the natural map q : B → B/∼, whichassigns to each b ∈ B the equivalence class to which b belongs, is a coequalizer off and g.
f

2. If Ag// B are continuous functions in Top, then the procedure given above yields

a coequalizer of f and g, provided that B/ ∼ is assigned the finest topology thatmakes q continuous (i.e., the final (or identification) topology).
3. Coequalizers in Rel and Prost are formed analogously to the way that they areformed in Top. In Pos they are formed by first forming the coequalizers in Prost
and then taking the Pos-reflection of the resulting preordered set.
f

4. If Ag// B are Ab-morphisms, let B0 be the subgroup {f (a) − g(a) | a ∈ A} of

B. Then the natural map q : B → B/B0 to the group of cosets is a coequalizer off and g.
f

5. If Ag//

B are homomorphisms in Alg(Ω), then a coequalizer for them is ob-

tained analogously as in Set by using, instead of equivalence relations, the smallest
congruence relation ∼ such that f (a) ∼ g(a) for all a ∈ A.
f

6. If Ag// B are simulations in Σ-Seq, then a coequalizer for them is obtained

analogously as in Alg(Ω) with final states precisely the congruence classes that
contain some final state of B.

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

115

7.70 REMARK
Since the concept of coequalizer is dual to that of equalizer, the general results about
equalizers can be translated (via the Duality Principle) into results about coequalizers.
For example,(1) Coequalizers are essentially unique.
g

f

id

A
(2) If A −→ B −→ A = A −−−→ A, then f is the coequalizer of g ◦ f and idB .

c(3) If B −→ C is a coequalizer of A

f
g

//

B , then the following are equivalent:(a) f = g,(b) c is a monomorphism,(c) c is an isomorphism,(d) idB is a coequalizer of f and g.
7.71 DEFINITIONc
A morphism B −→ C is called a regular epimorphism provided that it is a coequalizer
of some pair of morphisms.
7.72 EXAMPLES(1) In Set the regular epimorphisms are the surjective functions. [Given e : A → B
surjective, consider the two projections from D = {(a, a0 ) ∈ A × A | e(a) = e(a0 )} to
A.] In fact, in Set,
Retr = RegEpi = Surj = Epi.
(2) In Top the regular epimorphisms are precisely the topological quotient maps; i.e.,the surjective continuous maps onto spaces with the final topology. [For such a map
e proceed as in Set with D considered as a subspace of A × A.] In Σ-Seq regularepimorphisms are precisely the surjective simulations that have the property thateach final state in the codomain has a final preimage in the domain. Thus, in bothof these categories,
RegEpi ( Surj = Epi.
(3) In each Alg(Ω), Vec, Grp(cf. 7H), Ab, Lat, and HComp,RegEpi = Surj = Epi.
(4) In Sgr, Mon, and Rng,
RegEpi = Surj ( Epi.
(5) In the constructs Cat and (Ban, O) there exist regular epimorphisms that areF

not surjective. In Cat, consider the regular epimorphism A −−→ B of Example
7.40(6). In (Ban, O), consider the regular epimorphism e : c0 → K, where K is
the field of real (or complex) numbers considered as a Banach space, c0 is the classical Banach
P space nof all sequences in K converging to 0 with the sup-norm, ande(xn ) = ∞n=1 xn /2 .

9th July 2006

116

Objects and Morphisms[Chap. II

7.73 REMARKEach of the forgetful functors from Vec, Grp, and Top1 (T1 -topological spaces) into
Set preserves regular epimorphisms. However, none of them preserves coequalizers.
(Cf. Remark 13.6.)
7.74 DEFINITION
An epimorphism e is called extremal provided that it satisfies the following extremal
condition: If e = m◦f, where m is a monomorphism, then m must be an isomorphism.
7.75 PROPOSITION(1) Every retraction is a regular epimorphism.
(2) Every regular epimorphism is an extremal epimorphism.

D7.76 REMARK(1) In most of the familiar categories the regular epimorphisms and the extremal epimorphisms coincide. See, e.g., Proposition 14.14. However, in Cat there exist extremal
epimorphisms that are not regular epimorphisms. An example is the composite G◦Fof the regular epimorphism (functor) F : A → B of Example 7.40(6) with the regular epimorphism (functor) G : B → C, where C is the multiplicative submonoid{0, 1} of (considered as a category), G(n) = 0 for n > 0, and G(0) = 1.

Z(2) In general, “between” regular epimorphisms and extremal epimorphisms there areseveral other commonly used types of epimorphisms, the “strong epimorphisms” (thedual of which is introduced in Exercise 14C), the “swell epimorphisms” (introducedin 15A), and the “strict epimorphisms” (introduced in Exercise 7D). Thus we havethe following diagram that summarizes the relative strengths of the various notions
introduced. Notice that the notions in the boxes frequently coincide (see (1) above),but none of the implications can be reversed in general.
isomorphism






Q
Q
QQ
Q
Q

section⇓retraction⇓

regular mono⇓

regular epi⇓

strict mono⇓swell mono⇓strong mono⇓
extremal monostrict epi⇓
swell epi⇓
strong epi⇓
extremal epi∨
bimorphism






∨
monomorphism

Q
Q
QQ
Q
Q

∨
epimorphism
9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

117

SUBOBJECTSFor most of the familiar constructs, a canonical notion of “subobject” is generally understood. Category theory has made it apparent that “A being a subobject of B” isnot only a property of the object A, but also of the naturally associated “inclusion morphism” from A into B. Hence we will define subobjects of B to be pairs (A, m), where
A is an object and m is an “inclusion morphism”.
Since, as we have seen, for different categories one may need different concepts of morphism40 to characterize (up to isomorphism) the “inclusion morphisms of subobjects”,
the concept of subobject will be made dependent upon a chosen class M of morphismsthat represents (in each case) such inclusions.
7.77 DEFINITION
Let M be a class of monomorphisms. An M -subobject of an object B is a pair (A, m),mwhere A −−→ B belongs to M . In case M consists of all (regular, extremal) monomorphisms, M -subobjects are called (regular, extremal) subobjects.
7.78 REMARK
By the above definition, a subobject of a set B in Set is a pair (A, m), where m : A → B
is an injective function. In order for the notion of subobject to correspond more closely
to the notion of subset, two subobjects (A, m) and (A0 , m0 ) should be considered to be
essentially the same if m[A] = m0 [A0 ]. Furthermore a subobject (A, m) of B in Set
should be considered to be “smaller than” the subobject (A0 , m0 ) of B provided thatm[A] ⊆ m0 [A0 ]. The following definitions capture these ideas.
7.79 DEFINITION
Let (A, m) and (A0 , m0 ) be subobjects of B.
(1) (A, m) and (A0 , m0 ) are called isomorphic provided that there exists an isomorphism
h : A → A0 with m = m0 ◦ h.
(2) (A, m) is said to be smaller than (A0 , m0 ) — denoted by (A, m) ≤ (A0 , m0 ) —provided that there exists some (necessarily unique) morphism41 h : A → A0 withm = m0 ◦ h
h / 0
A@
A
@@
@@
@@ m0
@ 

m

B

7.80 REMARK
Observe that for the class of all subobjects of a given object
40

For example, in Rng the class of monomorphisms is appropriate, whereas the class of regular (=extremal) monomorphisms is not. However, for Top the class of regular (= extremal) monomorphisms
is appropriate, whereas the class of monomorphisms is not.
41
Observe that by Proposition 7.34, h must be a monomorphism.

9th July 2006

118

Objects and Morphisms[Chap. II(1) the relation of being isomorphic is an equivalence relation, and(2) the relation ≤ is a preorder; i.e., it is reflexive and transitive. In general, it fails tobe antisymmetric, but (A, m) ≤ (A0 , m0 ) and (A0 , m0 ) ≤ (A, m) imply that these two
subobjects are isomorphic.
7.81 EXAMPLES(1) Let B be a set, let PB be the set of all subsets of B, let SB be the class ofall subobjects of B in Set, and let H : SB → PB be the function defined by:H(A, m) = m[A]. Then(a) (A, m) and (A0 , m0 ) are isomorphic if and only if H(A, m) = H(A0 , m0 ), and(b) (A, m) ≤ (A0 , m0 ) if and only if H(A, m) ⊆ H(A0 , m0 ).
Hence, up to isomorphism, subsets of B correspond bijectively to subobjects of B,
where the correspondence is order-preserving.
(2) Similarly, in the categories Vec, Sgr, Mon, Grp, Ab, Rng, Alg(Ω), andHComp, subobjects correspond to linear subspaces, subsemigroups, submonoids,
subgroups, subrings, subalgebras, and compact subspaces, respectively.
(3) Analogously, in Top, Rel, and Pos, regular (= extremal) subobjects correspond to
subspaces, subrelations, and subposets, respectively.
7.82 DEFINITION
Let M be a class of monomorphisms of a category A.
(1) A is called M -wellpowered provided that no A-object has a proper class of pairwise42 non-isomorphic M -subobjects.
(2) In case M is the class of all (regular, extremal) monomorphisms, then M -wellpowered is called (regular, extremally) wellpowered.
7.83 REMARK
Every wellpowered category must be regular wellpowered and extremally wellpowered,but not conversely. [The thin category Ord of all ordinal numbers (with its usual
order) is wellpowered. Ordop is extremally wellpowered, but is not wellpowered. In
fact, each of its objects has a proper class of pairwise non-isomorphic subobjects, but
only one extremal subobject.] Thus one should be aware that even though the phrase“extremally wellpowered” sounds stronger than “wellpowered”, it is actually weaker.

QUOTIENT OBJECTS
Often in mathematics one has a “quotient” or “identification” procedure that maps
points to equivalence classes or “collapses” parts of a structure. Frequently, both of
42“Pairwise” in this context means that any pair consisting of distinct members are assumed to be
non-isomorphic.

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

119

these processes are considered to be essentially the same, and are used interchangeably.
For example, collapsing a subgroup of an abelian group to the identity element is understood either as the homomorphism onto the group of cosets or as the homomorphism
onto a particular set of representatives of the cosets. The reason that mathematicians
don’t distinguish these procedures is because they both give rise to essentially the same
quotient object — where “quotient object” is the dual concept to that of subobject.
7.84 DEFINITION
Let E be a class of epimorphisms. An E-quotient object of an object A is a pair (e, B),ewhere A −→ B belongs to E. In case E consists of all (regular, extremal) epimorphisms,
E-quotient objects are called (regular, extremal) quotient objects.
7.85 DEFINITION
Let (e, B) and (e0 , B 0 ) be quotient objects of A.
(1) (e, B) and (e0 , B 0 ) are called isomorphic provided that there exists an isomorphismh : B → B 0 with e0 = h ◦ e.
(2) (e, B) is said to be larger than (e0 , B 0 ) — denoted by (e, B) ≥ (e0 , B 0 ) — provided
that there exists some (necessarily unique) morphism h : B → B 0 with e0 = h ◦ e.
e /
B
AA
AA
A h
e0 A 

AA

B0

7.86 EXAMPLES(1) Let A be a set, let EA be the set of all equivalence relations on A, let KA be the setof all quotient objects of A in Set, and let H : KA → EA be the function definedby: H(e, B) = {(x, y) ∈ A × A | e(x) = e(y)}. Then(a) (e, B) and (e0 , B 0 ) are isomorphic if and only if H(e, B) = H(e0 , B 0 ), and(b) (e, B) ≥ (e0 , B 0 ) if and only if H(e, B) ⊆ H(e0 , B 0 ).
Hence, up to isomorphism, quotient objects of A correspond bijectively to equivalence relations on A, where the correspondence is order-reversing.
(2) Similarly, in the constructs Top, Rel, and Pos the regular (= extremal) quotient
objects of an object correspond (up to isomorphism) bijectively to the equivalence
relations on its underlying set.
(3) Analogously, in HComp the quotient objects (= regular quotient objects) of an
object A correspond (up to isomorphism) bijectively to those equivalence relations
ρ on the underlying set of A that, considered as subsets of A × A, are closed.
(4) In algebraic constructs such as Vec, Sgr, Mon, Grp, Ab, Rng, and Alg(Ω), theregular (= extremal) quotient objects of an object A correspond (up to isomorphism)
bijectively to the congruence relations on A; i.e., to the equivalence relations on theunderlying set of A that respect the operations.

9th July 2006

120

Objects and Morphisms[Chap. II

7.87 DEFINITION
Let E be a class of epimorphisms of a category A.
(1) A is called E-co-wellpowered provided that no A-object has a proper class ofpairwise non-isomorphic E-quotient objects.
(2) In case E is the class of all (regular, extremal) epimorphisms, then E-co-wellpoweredis called (regular, extremally) co-wellpowered.
7.88 THEOREM
Every construct is regular wellpowered and regular co-wellpowered.
Proof: Let (A, U ) be a construct.
(1). To show that A is a regular co-wellpowered category, let A be an A-object andlet {(ei , Bi ) | i ∈ I} be a class of pairwise non-isomorphic regular quotient objectsof A. Let EA be the set of all equivalence relations on U A and let the function
H : I → EA be defined by: H(i) = {(a, b) ∈ U A × U A | ei (a) = ei (b)}. To seethat H is injective, let i1 , i2 ∈ I with H(i1 ) = H(i2 ). For k = 1, 2, let (eik , Bik ) bea coequalizer of Ak

rk
sk

//

A . The equality ei1 ◦ r1 = ei1 ◦ s1 implies that for eacha ∈ U A1 (r1 (a), s1 (a)) ∈ H(i1 ) = H(i2 ). Consequently, ei2 ◦ r1 = ei2 ◦ s1 , so that by
the definition of coequalizer (ei1 , Bi1 ) ≥ (ei2 , Bi2 ). By symmetry, and by using thedual of Remark 7.80(2), we have that (ei1 , Bi1 ) and (ei2 , Bi2 ) are isomorphic andhence, by assumption, i1 = i2 . Thus H : I → EA is an injective function, so thatsince EA is a set, I must be a set.
(2). That A is regular wellpowered follows immediately from (1) and the fact that
(Aop , Q ◦ U op ) is a construct [5.2(4)].

7.89 COROLLARY
Every category with a separator or a coseparator is regular wellpowered and regular cowellpowered.
Proof: If S is a separator in A, then (A, hom(S, −)) is a construct. If C is a coseparator in A, then (A, Q ◦ hom(−, C)op ) is a construct, where Q : Setop → Set is thecontravariant power-set functor.

7.90 EXAMPLES(1) Fibre-small transportable constructs for which the epimorphisms are precisely themorphisms with surjective underlying functions must be co-wellpowered.
(2) Constructs are frequently co-wellpowered, even when they have non-surjective epimorphisms; e.g., Sgr, Rng, and Haus. However, in these cases establishing cowellpoweredness is more involved.

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

121(3) The construct of Urysohn spaces43 is not co-wellpowered. The proof is nontrivial.
Another non-co-wellpowered subcategory of Top is described in Exercise 7L. Bothof these constructs are regular co-wellpowered (since they are constructs). Ord is
obviously not co-wellpowered.
(4) Constructs need not be extremally co-wellpowered: consider the construct whose
objects are all sets and whose morphisms are all identities and all constant functions
from sets A to sets B, where the cardinality of B is greater than that of A.

Suggestions for Further Reading
Grothendieck, A. Sur quelque points d’algébre homologique. Tohoku Math. J. 9 (1957):
119–221.
Isbell, J. R. Some remarks concerning categories and subspaces. Canad. J. Math. 9(1957): 563–577.
Burgess, W. The meaning of mono and epi in some familiar categories. Canad. Math.
Bull. 8 (1965): 759–769.
Baron, S. Note on epi in T0 . Canad. Math. Bull. 11 (1968): 503–504.
Isbell, J. R. Epimorphisms and dominions IV. J. London Math. Soc. 1 (1969): 265–273.
Poguntke, D. Epimorphisms of compact groups are onto. Proc. Amer. Math. Soc., 26(1970): 503–504.
Pareigis, B., and M. E. Sweedler On generators and cogenerators. Manuscr. Math. 2(1970): 49–66.
Schröder, J. The category of Urysohn spaces is not cowellpowered. Topol. Appl. 16(1983): 237–241.

EXERCISES
7A. Zero Morphismsf

A morphism A −→ B is called a constant morphism provided that for any pair/A0 s / A of morphisms we have f ◦ r = f ◦ s. Morphisms that are simultaneously
constant and coconstant are called zero morphisms. Show thatr(a) If f is a constant morphism then so is any composite h ◦ f ◦ g.
(b) If f can be factored through a terminal object, then f is constant.
(c) If A has a zero object, 0, then an A-morphism is a zero-morphism if and only if it
can be factored through 0.
43

A topological space is called a Urysohn space provided that any two distinct points have disjoint
closed neighborhoods.

9th July 2006

122

Objects and Morphisms[Chap. II

7B. Pointed Categories
A category is called pointed provided that each of its morphism sets hom(A, B) contains
a zero morphism. Show that a category is pointed if and only if it is a full subcategoryof a category with a zero object.
7C. Kernels and Normal Monomorphismsf

In a pointed category an equalizer of A −→ B and a zero morphism A → B is called akernel of f . A morphism is called a normal monomorphism provided that it is thekernel of some morphism. Show that(a) Every normal monomorphism is regular.
(b) In Grp the normal monomorphisms are the embeddings of normal subgroups,
whereas the regular monomorphisms are the embeddings of subgroups. [Use 7H(a)].
fg(c) If A −→ B −→ C are morphisms in a pointed category and g is a monomorphism,
then f is a kernel of g if and only if A is a zero object.
(d) Each nonempty pointed category has a zero object if each of its morphisms has akernel.
(e) If A is a pointed category such that each of its morphisms has a kernel and a cokernel,then the following hold:(1) A morphism f is a normal monomorphism if and only if it is a kernel of acokernel of f .
(2) For each object A the preordered class of all non-isomorphic normal subobjectsof A is anti-isomorphic with the preordered class of all non-isomorphic normal
quotient objects of A.
7D. Strict Monomorphismsf

A morphism A −→ B is called a strict monomorphism provided that wheneverf0

A0 −−→ B is a morphism with the property that for all morphisms B

rs// C , r ◦ f = s ◦ ffimplies that r ◦ f 0 = s ◦ f 0 , then there exists a unique morphism A0 −→ A with f 0 = f ◦ f .
Show that:(a) Every regular monomorphism is a strict monomorphism, but not vice versa.
(b) Every strict monomorphism is an extremal monomorphism, but not vice versa(cf. 14I).
fg(c) If A −→ B is an extremal monomorphism and B −→ C is a strict monomorphism,fgthen A −→ B −→ C is an extremal monomorphism.
f

g

f

g

(d) If A −→ B is a strict monomorphism and B −→ C is a section, then A −→ B −→ C
is a strict monomorphism.

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories

123(e) The composite of two strict monomorphisms is an extremal but not necessarily strict
monomorphism.
fgg(f) If A −→ B −→ C is a strict monomorphism and B −→ C is a monomorphism, thenfA −→ B is a strict monomorphism.
7E. SectionsShow that(a) Every morphism with a terminal domain is a section.
f

g

f

g

(b) If A −→ B −→ C is a section and A −→ B is an epimorphism, then B −→ C is asection.
f(c) If A −→ B is a function and A has at least two elements, then in Set the following
are equivalent:(1) f is an isomorphism,g(2) there exists precisely one function B −→ A with g ◦ f = idA .
(d) If A is a full subcategory of B, then an A-reflection arrow is a section if and only if
it is an isomorphism.
(e) In a construct each section (but not necessarily each regular monomorphism) is
injective.
(f) If a natural transformation τ = (τA ) regarded as a morphism in the functor quasicategory [A, B] is a section, then each τA is a B-section (but the converse need not
be true).
7F. Full Embeddings of Reflective Subcategories
Let A be a full reflective subcategory of B. Show that the inclusion functor A ,→ B(a) preserves and reflects monomorphisms, sections, retractions, and isomorphisms,(b) preserves but need not reflect regular monomorphisms, extremal monomorphisms,
and strict monomorphisms,(c) reflects but need not preserve regular epimorphisms, extremal epimorphisms, strict
epimorphisms, and epimorphisms.
7G. Faithful Functors(a) Show that a faithful functor that reflects extremal epimorphisms must reflect isomorphisms.
e(b) Show that a functor F : A → B that reflects equalizers (i.e., whenever E −→ A
f

and A

g

//

B are A-morphisms such that F e is an equalizer of F f and F g, then eis an equalizer of f and g) is faithful.

9th July 2006

124

Objects and Morphisms[Chap. II

7H. Epimorphisms for Groups* (a) Show that if K is a subgroup of the (finite) group H, then there exists a (finite)group G and group homomorphisms f1 , f2 : H → G such thatK = { h ∈ H f1 (h) = f2 (h) }.
[Hint: Consider the set X obtained from the set { hK h ∈ H } of all left K-cosetsof H by adjoining a single new element K̂. Let G be the permutation group of X,
and let ρ : X → X be the permutation that interchanges the elements eK (= K)
and K̂, and leaves all other elements of X fixed.
Define f1 , f2 : H → G by(
hh0 K
f1 (h)(S) =K̂

if S = h0 K
if S = K̂

f2 (h) = ρ ◦ f1 (h) ◦ ρ−1 .](b) Show that the epimorphisms in Grp are precisely the surjective homomorphisms.
(c) Show that the epimorphisms in the category of finite groups are precisely thesurjective homomorphisms.
f(d) Show that in the category of commutative cancellative semigroups, A −→ B is an
epimorphism if and only if for each b ∈ B there exist a1 , a2 ∈ A with f (a1 ) + b =f (a2 ). [Hint: Embed each commutative cancellative semigroup in an abelian group.
There the epimorphisms are surjective.]f(e) Show that in the category of torsion-free abelian groups, a morphism A −→ B isan epimorphism if and only if the factor group B/f [A] is a torsion group.
7I. Epimorphisms in FHaus
f(a) Show that in FHaus a morphism A −→ B is an epimorphism if and only if eachgmorphism B −→ is constant whenever g ◦ f is constant.

R

R

R Q

(b) Let B be the space with underlying set that has τ ∪ { \ } as a subbase for its
topology (where τ is the usual topology on ). Let A be the subspace of B with
underlying set . Show that the (closed) embedding A ,→ B is an epimorphism in
FHaus. Notice that this implies that A ,→ B is not an extremal monomorphism in
FHaus.

Q

R

7J. Regular Monomorphisms in FHaus
In the category of functionally Hausdorff spaces (cf. 7.65) consider the discrete spaces
A and B with underlying sets { n1 | n ∈ + } and {0} ∪ { n1 | n ∈ + }, the space C with
underlying set , having τ ∪ { \ A} as a subbase for its topology (where τ is the usual
topology on ), and the space D obtained as a quotient of C by identifying the points
0 and 1.

R

R

R

N

N

9th July 2006

Sec. 7]

Objects and morphisms in abstract categories
f

125

g

(a) Show that the natural inclusions A −→ B and B −→ C are regular monomorphisms,
but that g ◦ f is not regular.
h(b) Show that if C −→ D is the natural quotient map, then h ◦ (g ◦ f ) is a regularmonomorphism, but that g ◦ f is not regular.
7K. Subobjects
Show that it is possible for (A, m) and (A0 , m0 ) to be non-isomorphic subobjects of an
object B even though A and A0 are isomorphic.
* 7L. A Non-co-wellpowered Category of Topological Spaces
Let A be the full subcategory of Top that consists of those topological spaces in which
every compact subspace is Hausdorff. Show that(a) A is a reflective subcategory of Top, and(b) A is not co-wellpowered.
7M. Coequalizerse

Let A −→ B be an epimorphism and let B

f
g

//

C be a pair of morphisms. Show that c

is a coequalizer of f and g if and only if c is a coequalizer of f ◦ e and g ◦ e.
7N. Extremal Monomorphisms
Show that the composite of two extremal monomorphisms need not be extremal.
7O. Epi = RegEpi and Mono = RegMono
Show that the above equations hold in any of the following categories: Set, Vec, Ab,
Grp, Alg(Ω), and HComp.
* 7P. A Characterization of Monomorphisms by Sections
Let f be a morphism in a category A. Show that f is a monomorphism in A if and only
if A can be embedded into a category B such that f is a section in B.
7Q. Separating Sets and Concretizable Categories(a) Show that if A has a separating set { Ai | i ∈ I } and I is the discrete category
associated with I, then A is concretizable over [I, Set]. (Cf. Proposition 7.12.)(b) Show that if A has a separating set, then A is concretizable over Set.
(c) Exhibit a construct that has no separating set.
7R. Epi-Transformations and Mono-Transformations
Let S, T : X → Set be functors. Show that(a) If there exists an Epi-transformation from S to T , then Spa(T ) is concretely isomorphic to a reflective modification of Spa(S).

9th July 2006

126

Objects and Morphisms[Chap. II(b) If there exists a Mono-transformation from S to T , then Spa(S) is concretely isomorphic to a coreflective modification of Spa(T ).
7S. Regular Epimorphisms in CatFG(a) Let A −−→ B be the functor described in 7.40(6), let B −−→ C be the functor
described in 7.76(1), let D be the category •

f
g

// •

H

, let A −−→ D be the inclusionKfunctor, and let D −−→ B be the functor defined by K(g) = 1 and K(f ) = 2. Show
that
(1) F and G are regular epimorphisms in Cat, but G ◦ F is not a regular epimorphism,(2) F = K ◦H is a regular epimorphism in Cat, but K is not a regular epimorphism,(b) Consider the functors:
A1

f1

/ B1

A2

f2

/ B2

A1

f1

F

−−→
A3

f3

/ B3

/ B1 = A2 f2 /5 B2

f3

f1

/•

f2

/ B2

G

−−→

f

A3

A2

/ B3

A3

f =f3

/ B3

Show that in Cat:(1) F is a regular epimorphism,(2) G is a retraction,(3) G ◦ F is not a regular epimorphism.
[Contrast this with Exercise 10M.]

9th July 2006

8 Objects and morphisms in concrete
categories
In this section we study special objects and morphisms in concrete categories. It will
become apparent that these concepts depend heavily upon the underlying functors, and,hence, can be defined in the context of arbitrary (not necessarily faithful) functors.
The notational conventions of Remark 6.22 will be used throughout this section. In
particular, forgetful functors will be denoted by | |.

DISCRETE AND INDISCRETE OBJECTS
8.1 DEFINITION
An object A in a concrete category A over X is called discrete whenever, for eachobject B, every X-morphism |A| → |B| is an A-morphism.
8.2 EXAMPLES(1) In the construct Pos the discrete objects are precisely the sets ordered by equality.
(2) In Cat the discrete objects are the discrete small categories.
(3) In Top the discrete objects are the discrete topological spaces.
(4) Let PMet denote the construct of pseudometric spaces44 and contractions. For each
set X, the object (X, d), where d is defined by(
0, if x = yd(x, y) =∞, if x 6= y,is a discrete object in PMet. In contrast, in the construct Met no space with atleast two points is discrete.
(5) The constructs Vec, Grp, and Σ-Seq have no discrete objects. For Ω 6= ∅, Alg(Ω)
has no discrete object, with the possible exception of the empty Ω-algebra.
(6) In TopGrp, considered as a concrete category over Grp, the discrete objects are
the topological groups with discrete topology. In TopGrp, considered as a concrete
category over Top, no object is discrete.
44

A pseudometric on a set X is a function d : X × X → [0, ∞], such that:(1) d(x, x) = 0 for all x ∈ X,(2) d(x, y) = d(y, x) for all x, y ∈ X,(3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.

127

128

Objects and Morphisms[Chap. II(7) In an abstract category considered as concrete over itself (w.r.t. the identity functor)
every object is discrete. In a poset A, considered as a concrete category over 1, an
object is discrete if and only if it is the smallest element of A.
Next we consider the concrete dual (5.20) of the concept of discrete object.
8.3 DEFINITION
An object A in a concrete category A over X is called indiscrete whenever, for eachobject B, every X-morphism |B| → |A| is an A-morphism.
8.4 REMARK
Notice that if A is a discrete (resp. indiscrete) object in the concrete category A, thenA is the smallest (resp. largest) element in the fibre of |A| (cf. 5.4). The converse neednot hold, as seen below [8.5(2)].
8.5 EXAMPLES(1) In the construct Pos the only indiscrete objects are the empty poset and the singleton posets; similarly in Cat.
(2) In Top the indiscrete objects are precisely the indiscrete topological spaces. In thefull subcategory Top1 of Top, consisting of all T1 -spaces, only the empty space andthe singleton spaces are indiscrete. However, each fibre has a largest element — thecofinite topology.
(3) In PMet (8.2) the indiscrete objects are precisely those for which every pair ofpoints has distance 0. The only indiscrete objects in Met are the metric spaces
with at most one point.
(4) In Vec, Grp, and Mon only the trivial objects with precisely one element are
indiscrete.
(5) In Σ-Seq an acceptor is indiscrete if and only if it has only one state and that stateis final.
(6) If TopGrp is considered as a concrete category over Grp, then the indiscrete objects
are the topological groups with indiscrete topology. If TopGrp is considered as aconcrete category over Top or Set, then the indiscrete objects are trivial.

EMBEDDINGSIn §7 several classes of monomorphisms have been introduced in order to formalize theintuitive concept of “embeddings of subobjects”. However, none of these concepts works
in all situations. (Recall that monomorphisms fail in Top [7.33(3)] and extremal andregular monomorphisms fail in Sgr [7.58(4), 7I].) Below, we introduce the concept ofembedding, which agrees with the intuitive notion of “embeddings of subobjects” in every
familiar construct.

9th July 2006

Sec. 8]

Objects and morphisms in concrete categories

129

8.6 DEFINITION
Let A be a concrete category over X.
f(1) An A-morphism A −→ B is called initial45 provided that for any A-object C an Xf ◦ggmorphism |C| −→ |A| is an A-morphism whenever |C| −−−→ |B| is an A-morphism.
f(2) An initial morphism A −→ B that has a monomorphic underlying X-morphismf|A| −→ |B| is called an embedding.
f(3) If A −→ B is an embedding, then (f, B) is called an extension of A and (A, f ) is
called an initial subobject of B.
8.7 PROPOSITION
For any concrete category the following hold:(1) Each embedding is a monomorphism.
(2) Each section (and in particular each isomorphism) is an embedding.
(3) If the forgetful functor preserves regular monomorphisms, then each regular monomorphism is an embedding.
Proof:(1). Immediate from Proposition 7.37(2).
sr(2). Suppose that A −→ B and B −→ A are A-morphisms with r ◦ s = idA . Letgs◦g|C| −→ |A| be an X-morphism for which |C| −−−→ |B| is an A-morphism. Thensg = r ◦ (s ◦ g) is an A-morphism. Thus A −→ B is initial, hence an embedding.
m(3). Suppose that A −−→ B is an equalizer of a pair of morphisms (u, v), and let
g : |C| → |A| be an X-morphism such that m ◦ g is an A-morphism. Then sinceu ◦ (m ◦ g) = v ◦ (m ◦ g), there exists an A-morphism k : C → A with m ◦ g = m ◦ k.
Since, by assumption, m is a X-monomorphism, this implies that k = g. Thusg = k : C → A is an A-morphism. Hence m is initial, and so is an embedding. 
8.8 EXAMPLES46(1) If an abstract category A is considered to be concrete over itself via the identity
functor, then every morphism is initial. In particular,
Emb(A) = Mono(A).
(2) In the construct A = Top a continuous map f : (X, τ ) → (Y, σ) is initial if andonly if τ is the “initial topology” with respect to f and σ, i.e., τ = { f −1 [S] | S ∈ σ }.
Thus embeddings are precisely the “topological embeddings”, i.e., homeomorphisms
onto subspaces. In particular,
RegMono(A) = ExtrMono(A) = Emb(A) ( Mono(A).
45
46

Notwithstanding their names, the concepts of initial object (7.1) and initial morphism are unrelated.
Emb(A) is the class of all A-embeddings. Similarly, Mono(A) is the class of all A-monomorphisms,
etc. See the Table of Symbols.

9th July 2006

130

Objects and Morphisms[Chap. II(3) In the construct A = Haus the initial morphisms are precisely the topologicalembeddings. Hence
ExtrMono(A) ( Init(A) = Emb(A) ( Mono(A).
(4) In the construct A = FHaus of functionally Hausdorff spaces the initial morphisms
are precisely the topological embeddings. Hence (cf. 7.65, 7I and 7J)
RegMono(A) ( ExtrMono(A) ( Init(A) = Emb(A) ( Mono(A).
(5) In some “algebraic” constructs A such as Vec and Grp the initial morphisms coincide with each of the other “reasonable” families of monomorphisms; i.e.,
RegMono(A) = Init(A) = Emb(A) = Mono(A).
However, if A is any of the constructs Rng, Banb , or (Ban, O), then
ExtrMono(A) ( Init(A) = Emb(A) = Mono(A).
(6) In the construct A = (Ban, U ) injective morphisms usually fail to be initial; instead the initial morphisms are precisely the isometric embeddings of (automaticallyclosed) subspaces. Thus
ExtrMono(A) = Init(A) = Emb(A) ( Mono(A).
(7) In the construct A = Met the initial morphisms are precisely the isometric embeddings of (not necessarily closed) subspaces. Thus
ExtrMono(A) ( Init(A) = Emb(A) ( Mono(A).
(8) In the constructs Rel and Pos a morphism f : (X, ρ) → (Y, σ) is initial if and only
if the equivalence: xρy ⇔ f (x)σf (y) holds. Thus in Pos initial morphisms are
precisely the embeddings, and in both cases
ExtrMono(A) = Emb(A) ( Mono(A).
(9) Extremal monomorphisms need not be embeddings. In the full subconstruct A =DRail of Top consisting of all discrete spaces and all infinite indiscrete spaces,
every injective map with finite discrete domain and indiscrete range is an extremal
monomorphism, but (for non-trivial domain) not an embedding. Hence
ExtrMono(A) 6⊆ Emb(A) = RegMono(A) ( ExtrMono(A).
(10) In the construct Cat the embeddings are precisely the embedding functors.
f(11) In the construct Σ-Seq embeddings are those injective simulations A −→ B for
which any state a ∈ A, with f (a) final in B, must itself be final in A.

9th July 2006

Sec. 8]

Objects and morphisms in concrete categories

131(12) In a preordered class A considered as a concrete category over 1, initial morphisms
and isomorphisms coincide. Thus, if A is a nondiscrete poset, thenIso(A) = Init(A) = Emb(A) = ExtrMono(A) ( Mono(A).
(13) If TopGrp is considered as a concrete category over Grp (resp. over Top), then afmorphism G −→ Ĝ is initial if and only if G has the initial topology with respect tof (resp. if and only if it is injective).
8.9 PROPOSITIONfgg◦f(1) If A −→ B and B −→ C are initial morphisms (resp. embeddings), then A −−−→ C
is an initial morphism (resp. an embedding).
g◦f(2) If A −−−→ C is an initial morphism (resp. an embedding), then f is initial (resp. anembedding).


QUOTIENT MORPHISMS
The concepts of final morphism and quotient morphism are dual to the concepts of initialmorphism and embedding, respectively.
8.10 DEFINITION
Let A be a concrete category over X.
f(1) An A-morphism A −→ B is called final provided that for any A-object C, an Xgg◦fmorphism |B| −→ |C| is an A-morphism whenever |A| −−−→ |C| is an A-morphism.
ff(2) A final morphism A −→ B with epimorphic underlying X-morphism |A| −→ |B| iscalled a quotient morphism.
f(3) If A −→ B is a quotient morphism, then (f, B) is called a final quotient object
of A.
8.11 EXAMPLES(1) In the construct Top a continuous function f : (X, τ ) → (Y, σ) is final if and only
if σ = {A ⊆ Y | f −1 [A] ∈ τ }; i.e., σ is the “final topology” on Y with respect to f
and τ . Thus in Top the quotient morphisms are the topological quotient maps.
(2) In the construct Rel a morphism f : (X, ρ) → (Y, σ) is final if and only if σ is the
final relation on Y with respect to f and ρ; i.e., σ = {(f (x), f (y)) | (x, y) ∈ ρ}.
(3) For any of the constructs Grp, Ab, Vec, Boo, Lat, HComp, BooSp, Mon, Sgr,and Σ-Seq, we have47
Final(A) = Quot(A) = RegEpi(A).
47

Final(A) is the class of all final morphisms in A. Similarly, Quot(A) is the class of all A-quotientmorphisms, etc. See the Table of Symbols.

9th July 2006

132

Objects and Morphisms[Chap. II(4) For the constructs Cat and (Ban, O) there exist regular epimorphisms that are not
final. Some examples of such morphisms are the non-surjective regular epimorphisms
described in Example 7.72(5).
(5) In the construct Cat there exist surjective extremal epimorphisms that are not final.
An example of this is the functor G ◦ F : A → C described in Remark 7.76(1).
fˆ is final in PMet if and only if(6) A morphism (X, d) −→ (Y, d)ˆ y 0 ) = inf
d(y,

 n−1
X

distd (f −1 (yi ),f −1 (yi+1 )) | (y1 , . . . , yn ) is a finite sequence in Y

i=1

with y1 = y and yn = y0.

8.12 PROPOSITION
For any concrete category the following hold:(1) Each quotient morphism is an epimorphism.
(2) Each retraction (and in particular each isomorphism) is a quotient morphism.
(3) If the forgetful functor preserves regular epimorphisms, then each regular epimorphism is a quotient morphism.
D8.13 PROPOSITIONfg(1) If A −→ B and B −→ C are final morphisms (resp. quotient morphisms), theng◦f

A −−−→ C is final (resp. a quotient morphism).
g◦f(2) If A −−−→ C is a final morphism (resp. a quotient morphism), then g is final (resp. aDquotient morphism).
8.14 PROPOSITION
In a concrete category A over X, the following conditions are equivalent for each Amorphism f :(1) f is an A-isomorphism.
(2) f is an initial morphism and an X-isomorphism.
(3) f is a final morphism and an X-isomorphism.
Proof: (1) ⇒ (2) follows from Proposition 3.21 and Proposition 8.7(2).
f −1

f

f

(2) ⇒ (1). If A −→ B is an initial X-isomorphism, then |B| −−−→ |A| −→ |B| =
id

f −1

f

B
|B| −−−→ |B| implies, by initiality, that |B| −−−→ |A| is an A-morphism. Thus A −→ B
is an A-isomorphism.

(1) ⇔ (3) follows by duality from (1) ⇔ (2).



9th July 2006

Sec. 8]

Objects and morphisms in concrete categories

133

STRUCTURED ARROWS
For concrete categories A over X, we have introduced properties of A-objects and of
A-morphisms. Next we will investigate properties of pairs (f, A) consisting of an Xmorphism f and a suitably related A-object A. First we define three concepts of “generation”. Whereas for “topological” constructs the concept of generation works better
than that of extremal generation, and for “algebraic” constructs the concept of extremal
generation works better than that of generation, the concept of concrete generation
works well in either setting.
8.15 DEFINITION
Let A be a concrete category over X.
(1) A structured arrow with domain X is a pair (f, A) consisting of an A-object A
fand an X-morphism X −→ |A|. Such a structured arrow will often be denoted byf

X −→ |A|.
(2) A structured arrow (f, A) is said to be generating provided that for any pair of
A-morphisms r, s : A → B the equality r ◦ f = s ◦ f implies that r = s.
(3) A generating arrow (f, A) is called extremally generating (resp. concretely generating) provided that each A-monomorphism (resp. A-embedding) m : A0 → A,
through which f factors (i.e., f = m ◦ g for some X-morphism g), is an A-isomorphism.
(4) In a construct, an object A is (extremally resp. concretely) generated by asubset X of |A| provided that the inclusion map X ,→ |A| is (extremally resp.
concretely) generating.
8.16 PROPOSITION
In a concrete category A over X the following hold for each structured arrowf : X → |A|:
(1) If (f, A) is extremally generating, then (f, A) is concretely generating.
(2) If (f, A) is concretely generating, then (f, A) is generating.
f(3) If X −→ |A| is an X-epimorphism, then (f, A) is generating.
f(4) If X −→ |A| is an extremal epimorphism in X, and if | | preserves monomorphisms,
then (f, A) is extremally generating.

8.17 EXAMPLES48(1) If an abstract category A is considered to be concrete over itself via the identityffunctor, then an A-morphism A −→ B, considered as a structured arrow (f, B), is
48

Gen(A) is the class of all generating structured arrows in A, ExtrGen(A) is the class of all extremallygenerating structured arrows in A, ConcGen(A) is the class of all concretely generating structured
arrows in A, etc. See the Table of Symbols.

9th July 2006

134

Objects and Morphisms[Chap. II

generating (resp. extremally or concretely generating) if and only if f is an epimorphism (resp. an extremal epimorphism). That is,Gen(A) = Epi(A)andExtrGen(A) = ConcGen(A) = ExtrEpi(A).

(2) In Vec, Grp, Sgr, Rng, and other algebraic constructs, the concepts of concrete
generation and of extremal generation coincide with the familiar (non-categorical)
concept of generation.
In the constructs Sgr and Rng the inclusion map ,→is generating, but is notconcretely generating [cf. 7.40(5)].

Z

Q(3) In the construct A = Top we have
ConcGen(A) = Gen(A) = Surjective maps, and
ExtrGen(A) = Surjective maps with discrete codomain.
(4) In the construct A = Haus we have
Gen(A) = Dense maps,
ConcGen(A) = Surjective maps, and
ExtrGen(A) = Surjective maps with discrete codomain.
(5) In a partially ordered set A, considered as a concrete category over 1, every structured arrow is concretely generating. A structured arrow (f, a) is extremally generating if and only if a is a minimal element of A.
(6) In Σ-Seq, given an acceptor A, the inclusion map of {q0 }, where q0 is the initialstate, concretely generates A if and only if A is reachable.
8.18 REMARKf
In a concrete category A, if an A-morphism A −→ B is regarded as a structured arrow(f, B) with domain |A|, then the following hold:f(1) A −→ B is an epimorphism if and only if (f, B) is generating.
(2) If (f, B) is extremally generating and the forgetful functor preserves monomorf

phisms, then A −→ B is an extremal epimorphism.
f(3) If A −→ B is an extremal epimorphism, then (f, B) is concretely generating.
8.19 DEFINITION
Let A be a concrete category over X.
(1) Structured arrows (f, A) and (g, B) in A with the same domain are said to be isomorphic provided that there exists an A-isomorphism k : A → B with k ◦ f = g.

9th July 2006

Sec. 8]

Objects and morphisms in concrete categories

135(2) A is said to be concretely co-wellpowered provided that for each X-object X
any class of pairwise non-isomorphic concretely generating arrows with domain X
is a set.
8.20 EXAMPLES
Most of the familiar constructs such as Vec, Grp, Top, HComp, Pos, Alg(Ω) for eachΩ, and Σ-Seq for each Σ are concretely co-wellpowered. The construct with objects
all sets and morphisms all identity mappings is fibre-small and co-wellpowered, but
not concretely co-wellpowered. The constructs CLat and CBoo are not concretely cowellpowered (cf. Exercise 8E). A proper class, ordered by equality and considered as aconcrete category over 1, is co-wellpowered, but is not concretely co-wellpowered.
8.21 PROPOSITION
Each concretely co-wellpowered concrete category is extremally co-wellpowered.
Proof: This follows immediately from the following two facts:(1) Each extremal quotient (f, B) of A, considered as a structured arrow with domain|A|, is concretely generating. Cf. Remark 8.18(3).
(2) Extremal quotients (f1 , B1 ) and (f2 , B2 ) are isomorphic if and only if the structured
arrows (f1 , B1 ) and (f2 , B2 ) are isomorphic.


UNIVERSAL ARROWS AND FREE OBJECTS
8.22 DEFINITION
In a concrete category A over Xu(1) a universal arrow over an X-object X is a structured arrow X −→ |A| with domainfX that has the following universal property: for each structured arrow X −→ |B|with domain X there exists a unique A-morphism fˆ : A → B such that the triangle
/ |A|
@@
@@
@
fˆ
f @@ 

X@u|B|

commutes,(2) a free object over an X-object X is an A-object A such that there exists a universal
arrow (u, A) over X.
8.23 EXAMPLES(1) In a construct, an object A is a free object(a) over the empty set if and only if A is an initial object.

9th July 2006

136

Objects and Morphisms[Chap. II(b) over a singleton set if and only if A represents the forgetful functor (6.9).
(2) In the construct Vec each object is a free object over any basis for it.
(3) In the constructs Top and Pos the free objects are precisely the discrete ones.
(4) In the construct Ab free objects over X are the free abelian groups generated by X.
They can be constructed as the group of all functions p : X → with finite carrier,where addition is carried out componentwise, and, if A is such a group of functions,a universal arrow u : X → |A| is given by(
1, if z = x(u(x))(z) =0, if z 6= x.

Z

Similarly, the familiar free group generated by a set X is a free object over X in theconstruct Grp.

A non-free group and a free object(5) In the construct Mon a free monoid X ∗ over a set X is a free object over X;
elements are words (= finite sequences, including the empty one) formed from members of X, with operation • of concatenation: (x1 , x2 , . . . , xn ) • (y1 , y2 , . . . , ym ) =(x1 , x2 , . . . , xn , y1 , y2 , . . . , ym ), and u : X → |A| is given by u(x) = (x). Analogously,in the construct Sgr a free semigroup generated by X consists of all nonempty words
formed from members of X.
(6) In each construct Alg(Ω) the free objects over a set X can be described in a way
similar to the description of initial objects [cf. 7.2(5)] except that in step (a) eachelement of Ω0 ] X should be required to be a “term”. Notice that in any Alg(Σ) a

9th July 2006

Sec. 8]

Objects and morphisms in concrete categories

137

free unary algebra over a singleton set is, by coincidence, the algebra Σ∗ of all words
with unary operations given by concatenations. A free algebra over a set X is the
disjoint union of card(X) copies of Σ∗ (cf. Exercise 8H).
(7) In the construct CLat an initial object is the two-point chain, a free object over a
one-element set is the three-point chain, and a free object over a two-element set isthe six-element lattice formed by replacing the middle member of a three-element
chain by a four-element boolean algebra. There are no free objects over sets thathave more than two elements (cf. Exercise 8G). In the construct CBoo an initial
object is the two-point chain, the four-element boolean algebra is a free object over anone-element set, and a free object over n (< ℵ0 ) generators has 22 elements. Thereis no free object over any infinite set. (See Exercise 8F.)
(8) In JCPos the free objects are, up to isomorphism, precisely the power sets (regardeduas JCPos objects). For each set X, the structured arrow X −→ |P(X)|, defined byu(x) = {x}, is a universal arrow.
However, in the construct Λ-JCPos (with objects triples (X, ≤, λ) consisting of acomplete lattice (X, ≤) and a unary operation λ on X, and morphisms the joinpreserving {λ}-homomorphisms) there do not exist free objects over any set. In
particular, Λ-JCPos has no initial object [even though Λ-JCPos is extremally cowellpowered, the empty set extremally generates arbitrarily large objects!], and theforgetful functor is not representable.
g(9) In the construct Cat a free object over the set {0} is a category of the form • −→ •(where the universal arrow u sends 0 to g).
(10) In the construct Rng the polynomial ringobject over M .

Z[M ] over a set M of variables is a free(11) Let Y be a nonempty set and consider Set as a construct via the forgetful functoru
U = hom(Y, −). Then the structured arrow X −→ U (X × Y ) = hom(Y, X × Y ),defined by (u(x))(y) = (x, y), is universal.
(12) To construct a universal arrow in (Ban, O) over a set X, let `1 (X) be the subspaceof the vector space K X consisting of all r = (rx )x∈X in K X whose norm krk =Pux∈X |rx | is finite. Then `1 (X) is a Banach space. Define X −→ O(`1 (X)) at yby the Dirac function u(y) = (δyx )x∈X . Then (u, `1 (X)) is a universal arrow over
X. Observe, for comparison, that for the construct (Ban, U ) the only set having
a universal arrow is the empty set, and that for the construct Banb the only setshaving universal arrows are the finite ones.
(13) In an abstract category, considered as a concrete category over itself via the identity
functor, every object is free over itself.
(14) In an preordered set A, considered as a concrete category over 1, an object a is freeover the single object of 1 if and only if a is the smallest element of A.
(15) In the concrete category of Rng over Mon in which the forgetful functor “forgets
addition” a universal arrow over a monoid M is given by the monoid ring [M ] of M

Z

9th July 2006

138

Objects and Morphisms

Z[Chap. IIZover the additive group ( , +) and the arrow u : M → | [M ]| defined by u(x) = x.
Likewise in the concrete category of Rng over Ab, where multiplication is forgotten,
a universal arrow over an abelian group G is given by the tensor ring over G.
(16) No acceptor is free in the construct Σ-Seq. (In fact, if A is an acceptor on n statesand if B is the (n + 1)-state acceptor in which the next-state maps δ(−, σ) are(n + 1)-cycles, then there is no simulation from A to B.)
8.24 PROPOSITION
Every universal arrow is extremally generating.
uProof: Let X −→ |A| be a universal arrow. By the uniqueness requirement, (u, A) isgmgenerating. Let A0 −−→ A be an A-monomorphism, and let X −→ |A0 | be an X-morphismĝwith u = m ◦ g. Since (u, A) is universal, there exists an A-morphism A −→ A0 with
id

m◦ĝ

A
g = ĝ ◦ u. Hence A −−−→ A and A −−−→ A are A-morphisms with idA ◦ u = (m ◦ ĝ) ◦ u.
By the uniqueness requirement in the definition of universal arrow, this implies that
idA = m ◦ ĝ. Hence m is simultaneously an A-retraction and an A-monomorphism, andso is an A-isomorphism (7.36).


8.25 PROPOSITION
For any X-object X, universal arrows over X are essentially unique; i.e., any two uniu
versal arrows with domain X are isomorphic, and conversely, if X −→ |A| is a universalkk◦uarrow and A −→ A0 is an A-isomorphism, then X −−−→ |A0 | is also universal. A 4.19 49
8.26 DEFINITION
A concrete category over X is said to have free objects provided that for each X-object
X there exists a universal arrow over X.
8.27 EXAMPLES
By Examples 8.23 the constructs Vec, Grp, Ab, Mon, Sgr, Alg(Ω), Top, Pos, and(Ban, O) have free objects; but the constructs CLat, CBoo, and (Ban, U ) don’t. Neither do the constructs Banb and Met; however, PMet does. A partially ordered set,considered as a concrete category over 1, has free objects if and only if it has a smallest
element.
8.28 PROPOSITION
If a concrete category A over X has free objects, then an A-morphism is an A-monomorphism if and only if it is an X-monomorphism.
49

The symbol A 4.19 indicates that a proof of the preceding result can be obtained as a straightforward
analogue of the proof of Proposition 4.19.

9th July 2006

Sec. 8]

Objects and morphisms in concrete categories

139

Proof: The sufficiency holds for all concrete categories [cf. Proposition 7.37(2)]. To seefthe necessity, let A −→ B be an A-monomorphism, and let X

r /s/ |A| be a pair ofuX-morphisms with f ◦ r = f ◦ s. If X −→ |C| is a universal arrow over X, then there
exist A-morphisms Cr̂ŝ// A with r̂ ◦ u = r and ŝ ◦ u = s. Thus (f ◦ r̂) ◦ u = (f ◦ ŝ) ◦ u.

By the uniqueness requirement in the definition of universal arrow, this implies thatf ◦ r̂ = f ◦ ŝ. Since f is an A-monomorphism, it follows that r̂ = ŝ, so that r = s. Hencef is an X-monomorphism.

8.29 PROPOSITION
If a construct A has a free object over a singleton set, then the monomorphisms in A
are precisely those morphisms that are injective functions.
Proof: By Example 8.23(1) the forgetful functor for A is representable. Hence theresult follows from Corollary 7.38.


OBJECTS AND MORPHISMS
WITH RESPECT TO A FUNCTOR
8.30 DEFINITION
Let G : A → B be a functor, and let B be a B-object.
(1) A G-structured arrow with domain B is a pair (f, A) consisting of an
A-object A and a B-morphism f : B → GA.
(2) A G-structured arrow (f, A) with domain B is called(a) generating provided that for any pair of A-morphisms A

r
s

//

Â , the equality

Gr ◦ f = Gs ◦ f implies that r = s,m(b) extremally generating provided that it is generating and whenever A0 −−→ A
is an A-monomorphism and (g, A0 ) is a G-structured arrow with f = G(m) ◦ g,then m is an A-isomorphism,(c) G-universal for B provided that for each G-structured arrow (f 0 , A0 ) withfˆ
domain B there exists a unique A-morphism A −→ A0 with f 0 = G(fˆ) ◦ f , i.e.,such that the triangle
f
/ GA
DD
DD
D
Gfˆ
f 0 DD! BD

GA0commutes.

9th July 2006

140

Objects and Morphisms[Chap. II

8.31 EXAMPLES(1) For concrete categories (A, U ) over X, we have the following:
U -structured arrow = structured arrow,U -generating arrow = generating arrow,extremally U -generating arrow = extremally generating arrow,
U -universal arrow = universal arrow.
(2) If A is a subcategory of B and E : A → B is the associated inclusion functor, thenuan E-structured arrow (u, A) with domain B is E-universal if and only if B −→ A
is an A-reflection arrow for B.
(3) Let A be a category and let G be the unique functor from A to 1. Then a Gstructured arrow (u, A) is G-universal if and only if A is an initial object in A.
u(4) If G = hom(A, −) : A → Set, then a G-structured arrow X −→ G(B) = hom(A, B)
fx

is G-universal if and only if for every A-object C and every family (A −−→ C)x∈X off

A-morphisms there exists a unique A-morphism B −→ C with fx = f ◦u(x) for eachx ∈ X. [By the definition of coproducts in §10, this is equivalent to ((u(x))x∈X , B)
being a coproduct of the family consisting of X copies of A; i.e., to B being anu(x)X-copower of A with injections A −−−→ B.](5) Minimal realization is universal. This means that for the category Beh of behavb

iors (i.e., triples (Σ, Y, b), where Σ∗ −→ Y is a function) and behavior morphismsfg[i.e., (f, g) : (Σ, Y, b) → (Σ0 , Y 0 , b0 ), where Σ −→ Σ0 and Y −→ Y 0 are functions
with (g ◦ b)(σ1 σ2 · · · σn ) = b0 (f (σ1 )f (σ2 ) · · · f (σn ))], the minimal realization functor
M : Beh → Autr has universal arrows. Here Autr denotes the full subcategoryof Aut [3.3(4)(b)] formed by those automata for which each state can be reachedfrom the initial one, and M assigns to each behavior its minimal realization. AnηM -universal arrow A −→ M (Σ, Y, bA ) (where bA : Σ∗ → Y is the external behaviorof A) is the unique simulation onto the minimal realization of bA .
8.32 PROPOSITION
If G : A → B is a functor, then the following are equivalent:(1) G is faithful,(2) each A-epimorphism, considered as a G-structured arrow, is generating,(3) each A-identity, considered as a G-structured arrow, is generating.
8.33 PROPOSITION
Every G-universal arrow is extremally generating.


A 8.24

9th July 2006

Sec. 8]

Objects and morphisms in concrete categories

141

8.34 DEFINITION
G-structured arrows (f, A) and (f 0 , A0 ) with the same domain are said to be isomorphic
provided that there exists an A-isomorphism k : A → A0 with G(k) ◦ f = f 0 .
8.35 PROPOSITION
For any functor G : A → B and any B-object B, G-universal arrows for B are essentially unique; i.e., any two G-universal arrows with domain B are isomorphic, and,ukconversely, if B −→ GA is a G-universal arrow and A −→ A0 is an isomorphism, thenGk◦u
A 4.19B −−−−→ GA0 is also G-universal.
8.36 PROPOSITION
Let G : A → B be a functor. If the triangle
/ GA
DD
DD
D
Gfˆ
f DD! 

XD

u

GB

fˆ

commutes, where (u, A) is a G-universal arrow and A −→ B is an A-morphism, thenthe following hold:(1) (f, B) is generating if and only if fˆ is an epimorphism.
(2) (f, B) is extremally generating if and only if fˆ is an extremal epimorphism.
Proof: It is clearly sufficient to show that whenever fˆ is an extremal epimorphism, thengm(f, B) is extremally generating. Let X −→ GC be a G-structured arrow and let C −−→ Bĝbe an A-monomorphism with f = Gm ◦ g. Then there exists an A-morphism A −→ Cwith g = Gĝ ◦ u. Thus the equality G(m ◦ ĝ) ◦ u = f = Gfˆ ◦ u implies that m ◦ ĝ = fˆ.
Hence m is an isomorphism.

8.37 DEFINITION(1) A functor G : A → B is called (extremally) co-wellpowered provided that for
any B-object B, any class of pairwise non-isomorphic (extremally) generating Gstructured arrows with domain B is a set.
(2) A faithful functor G : A → B is called concretely co-wellpowered provided thatthe concrete category (A, G) is concretely co-wellpowered.
8.38 PROPOSITION
If a faithful functor G : A → B is co-wellpowered, then so is A.

9th July 2006

A 8.21

142

Objects and Morphisms[Chap. II

8.39 REMARK
If a faithful functor G : A → B is extremally co-wellpowered, then A need not be
extremally co-wellpowered. Conversely, if A is co-wellpowered, then G : A → B need
not even be extremally co-wellpowered. Consider, e.g., the unique functor from a large
discrete category to 1.
More can be said if, e.g., for every B-object B there exists a universal G-structured
arrow for B. See 18.11 and 18B.

COSTRUCTURED ARROWS
All of the concepts relating to G-structured arrows have duals. In particular:
8.40 DEFINITION
Let G : A → B be a functor and let B be a B-object.
(1) A G-costructured arrow with codomain B is a pair (A, f ) consisting of anfA-object A and a B-morphism GA −→ B.
(2) A G-costructured arrow (A, f ) with codomain B is called G-co-universal for B
provided that for each G-costructured arrow (A0 , f 0 ) with codomain B there existsfˆ
a unique A-morphism A0 −→ A with f 0 = f ◦ G(fˆ).
8.41 EXAMPLES(1) If A is a subcategory of B and E : A → B is the associated inclusion functor, thenan E-costructured arrow (A, u) with codomain B is E-co-universal if and only ifu
A −→ B is an A-coreflection arrow for B.
(2) For forgetful functors U of familiar concrete categories, U-co-universal arrows are
relatively rare. For example, Grp has U-co-universal arrows only for one-point sets,
and Pos has U-co-universal arrows only for sets with at most one point. However,(a) in the construct Top for every set X there exists a co-universal arrowid(X, τ ) −−→ X, where τ is the indiscrete topology,(b) if T : X → Set is a functor and Spa(T ) is the associated concrete category overX, then for every X-object X there exists a co-universal arrow
idX(X, T (X)) −−−→ X,(c) in the constructs Alg(Σ) of unary algebras there exist co-universal arrows for∗each set X: consider the Σ-algebra X Σ of all functions from Σ∗ (the set of allwords over Σ) into X, with the operation σ ∈ Σ sending a function g : Σ∗ → X
to the function g(σ− ), defined by: g(σ− )((σ1 · · · σn )) = g(σσ1 · · · σn ). Then∗ε : X Σ → X given by ε(g) = g(∅) is co-universal. [In fact, given a Σ-algebra∗
A and a function f : A → X, the unique homomorphism fˆ : A → X Σ withf = ε ◦ fˆ is given by fˆ(a)(σ1 · · · σn ) = f (σ1 · · · σn a).]9th July 2006

Sec. 8]

Objects and morphisms in concrete categories

143

Suggestions for Further Reading
Samuel, P. On universal mappings and free topological groups. Bull. Amer. Math. Soc.
54 (1948): 591–598.
Sonner, J. Universal and special problems. Math. Z. 82 (1963): 200–211.
Hales, A. W. On the non-existence of free complete Boolean algebras. Fund. Math. 54(1964): 45–66.
Gaifman, H. Infinite Boolean polynomials. Fund. Math. 54 (1964): 230–250.
Solovay, R. New proof of a theorem of Gaifman and Hales. Bull. Amer. Math. Soc. 72(1966): 282–284.
Ehresmann, C. Construction de structures libres.
(1969): 74–104.

Springer Lect. Notes Math.

92

Pumplün, D. Universelle und spezielle Probleme. Math. Ann. 198 (1972): 131–146.

EXERCISES
8A. Regular Monomorphisms vs. Embeddings(a) Show that in a construct a regular monomorphism need not be an embedding. [Cf.
Proposition 8.7.](b) Prove that each regular monomorphism in a construct that has a free object over anonempty set must be an embedding.
(c) Prove that each embedding in a construct that has a two-element indiscrete object
must be a regular monomorphism.
8B. Initial Monomorphisms vs. Embeddings
Show that in constructs the embeddings are precisely the initial monomorphisms, butthat in concrete categories initial monomorphismss may fail to be embeddings.
8C. An Initial, Non-injective Morphism in Sgr
Show that if A resp. B are the semigroups with underlying sets {0, 1, 2} resp. {0, 1} andfmultiplication defined by x · y = 0 for all x and y, then the non-injective map A −→ B,
defined by f (x) = Min{x, 1}, is an initial morphism in Sgr.
8D. A Characterization of Concretely Co-wellpowered Constructs
Show that a uniquely transportable construct is concretely co-wellpowered if and only if
it is fibre-small and for every cardinal number k there exists a cardinal number k suchthat every object that is concretely generated by a set of cardinality not exceeding k has
an underlying set with cardinality not exceeding k.

9th July 2006

144

Objects and Morphisms[Chap. II* 8E. CBoo is Extremally Co-wellpowered,
but Not Concretely Co-wellpowered
Let CBoo be the construct of complete boolean algebras and boolean homomorphismsthat preserve arbitrary meets and joins. Show that:(a) CBoo is wellpowered and extremally co-wellpowered,(b) CBoo is not concretely co-wellpowered,(c) the forgetful functor CBoo → Set is not extremally co-wellpowered.
[Hint for (b) and (c): Let X be a topological space. A subset A of X is called regular
open provided that int(clA) = A (where “int” designates “interior” and “cl” designates“closure”). The set R(X) of all regular open subsets of X is a complete boolean algebrawith respect to the following operations:_[
M = int(cl( M )),for M ⊆ R(X)^\M = int( M ),for ∅ =
6 M ⊆ R(X)A0 = int(X − A),

for A ∈ R(X).

Let K be an infinite cardinal number. Let X be the set of all ordinal numbers withcardinality less than K, considered as a discrete topological space. Let P = X N be thetopological product of countably many copies of X, with projections πn : X N → X. The
complete boolean algebra R(X N ) is extremally generated by the family{πn−1 (ξ) n ∈ , ξ ∈ X}, hence by the countable set {Am,n m, n ∈ }, where
Am,n = {x ∈ X N πm (x) ≤ πn (x)}.]N

N

8F. Free Objects in CBooShow that in the construct CBoo(a) there exists a free object over each finite set.
(b) there does not exist a free object over any infinite set. [Hint: 8E.]
8G. Free Objects in JPos, JCPos, CLat, and FramShow that* (a) the constructs JCPos and Fram have free objects.
(b) in the construct JPos there exists a free object over X if and only if card X 6= 1.
* (c) in the construct CLat there exists a free object over X if and only if card X ≤ 2.
8H. Free Objects in Alg(Ω)
As outlined in Example 8.23(6) free Ω-algebras can be constructed as algebras of terms.
Another, more graphic, description can be achieved via labeled trees: Consider theinfinite regular tree ω ∗ of all words in ω (with root ∅, the empty word; the first levelconsists of all natural numbers i ∈ ω; in the second level, the successors of i are all words
ij with j ∈ ω, etc.). An Ω-labeled tree in a set X is defined to be a “labeling” partial
function t : ω ∗ → Ω ] X such that

9th July 2006

Sec. 8]

Objects and morphisms in concrete categories

145(a) the domain of definition Dt of t is finite, and it has ∅ as a member;(b) if t(i1 i2 · · · in ) is an operation symbol of arity k, then i1 i2 · · · in in+1 belongs to Dt if
and only if in+1 ∈ {0, 1, . . . , k − 1};(c) if t(i1 i2 · · · in ) is an element of X, then i1 i2 · · · in in+1 does not belong to Dt for any
in+1 .
Show that the set of all Ω-labeled trees is an Ω-algebra that is isomorphic to the term
algebra, and hence is free.
[Given σ ∈ Ω of arity k, and given Ω-labeled trees t0 , t1 , . . . , tk−1 , then the treeσ(t0 , t1 , . . . , tk−1 ) = t is defined on i0 i1 · · · in byt(i0 i1 · · · in ) = ti0 (i1 · · · in )fori0 < k,i1 · · · in ∈ Dti0 ;otherwise it is undefined.]
op
8I. Free Objects in Setop , Topop0 , and HComp
op
Show that the constructs Setop (see 5.2(4)), Topop(see 5M)
0 (see 5L), and HComp
have free objects and, in each case, describe them explicitly.

8J. Free Objects in BooSp and HComp
Show that the constructs BooSp and HComp have free objects and, in each case,describe them explicitly.
8K. Free Objects in (Ban, O) and (Ban, U )Show that (Ban, O) has free objects (cf. 8.23(12)), but that (Ban, U ) has free objects
only over the empty set.
* 8L. Isomorphic Free Objects
Show that it can happen that a construct A has free objects in such a way that any
two free objects over finite, nonempty sets are isomorphic (as objects). [Hint: Consider
the concrete full subcategory of Alg(Ω), where Ω = (1, 1, 2), consisting of those Ωalgebras (X, (ω1 , ω2 , ω3 )) that satisfy the equations ω1 (ω3 (x, y)) = x, ω2 (ω3 (x, y)) =y and ω3 (ω1 (x), ω2 (x)) = x. Show that whenever an object (X, (ω1 , ω2 , ω3 )) of theconstruct A is free over a set Y ] {x, y}, then it is free over Y ] {ω3 (x, y)}.]8M. Discrete Objects
Show that for concrete categories the following hold:(a) Retracts of discrete objects are discrete.
id(b) An object A is discrete if and only if the structured arrow |A| −−→ |A| is universal.
8N. A Characterization of Faithfulnessf
G
Show that a functor A −−→ B is faithful if and only if every A-epimorphism A −→ Â,
considered as a G-structured arrow (Gf, Â), is generating.

9th July 2006

146

Objects and Morphisms[Chap. II

8O. Regular Epimorphisms and Finality
Let A be a concrete category over X that has free objects, and let f be an A-morphism
such that U f is a regular epimorphism in X. Show that f is a regular epimorphism in
A if and only if f is final.
8P. Free Automata(a) Show that the construct Σ-Seq has free objects if and only if Σ = ∅.
(b) Show that Aut considered as a concrete category over Set × Set × Set has only the(trivial) free objects over (Σ, Q, Y ) with Q = Y = ∅.

9th July 2006

9 Injective objects and essential embeddings
Earlier it has been shown that many familiar constructions, particularly “completions”
such as the completion of a metric space or the Čech-Stone compactification of a Tychonoff space, can be naturally regarded as reflections. However, there also exist familiar
completions that cannot be (or only artificially can be) regarded as such. Examples are
the Mac Neille completion of a poset and the algebraic closure of a field.50 In both cases,
and in several others, the construction in question can be regarded rather naturally as an
injective hull, a concept that will be studied in this section. Roughly speaking, an object
C is called injective provided that for any object A, any morphism from a subobjectof A into C can be extended to a morphism from A into C. Since a satisfactory concept of subobjects is available for concrete categories, but not for arbitrary categories,
we will first define injective objects for concrete categories only. Later, for arbitrary
categories A and arbitrary classes M of A-morphisms, M -injective objects will be introduced in such a way that for concrete categories the injective objects are precisely the
Emb(A)-injective objects.

INJECTIVITY IN CONCRETE CATEGORIESFor concrete categories we will use the notational conventions described in Remark 6.22.
9.1 DEFINITION
In a concrete category an object C is called injective provided that for any embeddingmfg

A −−→ B and any morphism A −→ C there exists a morphism B −→ C extending f , i.e.,such that the triangle
m /
A@
B
@@
@@
g
f @@ C

commutes.
9.2 REMARK
The morphism g in the above definition is not required to be uniquely determined by m
and f . This contrasts sharply with many categorical definitions in which existence anduniqueness requirements are coupled (see 4.16 for a typical example).
9.3 EXAMPLESIn Examples (1)–(5) below, we consider injective objects in various constructs:
50

Observe, e.g., that there exist two different automorphisms of the field of complex numbers (thealgebraic closure of the field of real numbers) that keep the reals pointwise fixed.

147

148

Objects and Morphisms[Chap. II(1) In Set the injective objects are precisely the nonempty sets.
(2) In Pos the injective objects are precisely the complete lattices. [Injectivity of acomplete lattice C follows from the fact that, for given m and f as in Definition
9.1, the map g : B → C defined by g(b) = sup{f (a) | a ∈ A and m(a) ≤ b} is orderpreserving. Completeness of an injective object C follows from 9.5 and the fact that
C is a retract of its Mac Neille completion.] Similarly the injective objects(a) in Boo (and in the construct DLat of distributive lattices) are precisely thecomplete Boolean algebras,(b) in the construct SLat (posets with finite meets and maps that preserve finite
meets) are precisely the frames [see Exercise 5L(b)],(c) in JCPos are precisely the completely distributive complete lattices.
(3) (Algebra)(a) In Vec every object is injective.
(b) In Ab the injective objects are precisely the divisible abelian groups.
(c) In Alg(1) the injective objects are precisely those unary algebras whose single
unary operation is surjective and has a fixed point. Analogously, a {σ}-acceptor
is injective in {σ}-Seq if and only if every state is final and its next state function
is a permutation with a fixed point.
(d) In Grp only the terminal objects are injective [since each group can be properly
embedded into a simple group]. Analogously, terminal objects are the only
injective objects in Lat, Mon, Sgr, and Rng.
(4) (Topology)(a) In HComp the injective objects are precisely the retracts of powers [0, 1]I of theunit interval [0, 1]. In particular, [0, 1] is injective (Tietze-Urysohn Theorem).
(b) In BooSp the injective objects are precisely the retracts of Cantor spaces, i.e.,of powers of the two-element Boolean space B = ({0, 1}, P{0, 1}).
(c) In Top0 the injective objects are precisely the retracts of powers of the Sierpinski
space S = ({0, 1}, {∅, {0}, {0, 1}}).
(d) In Top the injective objects are precisely the retracts of powers C I of the space
C = ({0, 1, 2}, {∅, {0, 1}, {0, 1, 2}}).
(e) In Met an object (X, d) is injective if and only if it is hyperconvex, i.e., if forany superadditive map51 f : X → + there exists z ∈ X with d(z, x) ≤ f (x)for all x ∈ X.

R(f) In the construct Unif of uniform spaces and uniformly continuous functions,the unit interval [0, 1] is injective, but the real line
is not.

R

51

f: X→

R is superadditive provided that d(x, y) ≤ f (x) + f (y) for all x, y ∈ X.
+

9th July 2006

Sec. 9]Injective objects and essential embeddings

149(g) In some constructs (e.g., for Top1 , Haus, and Tych) only the terminal objects
are injective [since a map f from \ {0} into a T1 -space X, sending all r < 0
to x and all r > 0 to y, can be extended to a continuous map f :→ X only
if x = y].

RR(5) (Analysis)
In Ban(K) (Banach spaces over K) the injective objects are (up to isomorphism)
precisely the function spaces C(X, K) for extremally disconnected compact Hausdorff spaces X. [In case K is the fieldof real numbers, these are precisely thehyperconvex spaces (see (4)(e) above). In case K is the field of complex numbers,
the zero-space is the only hyperconvex injective object.] In particular, K itself is
injective (Hahn-Banach Theorem).

RC(6) In a partially ordered set, considered as a concrete category over 1, every object isinjective.
9.4 PROPOSITION
Every terminal object is injective.



9.5 PROPOSITION
Every retract of an injective object is injective.
r

Proof: Let C −→ D be a retraction with C an injective object. Then there existssfm

D −→ C with r ◦ s = idD . Let A −−→ B be an embedding and A −→ D be a morphism.
gs◦fSince C is an injective object, there exists an extension B −→ C of A −−−→ C; i.e., amorphism g such that
m /
A
B
f

g





D

s

commutes. Thus r ◦ g is an extension of f .

/C

9.6 DEFINITION
In a concrete category an object C is called an absolute retract provided that any
embedding with domain C is a section.
9.7 PROPOSITION
Every injective object is an absolute retract.



9.8 REMARKOne can easily provide constructs in which absolute retracts fail to be injective. However,as we shall see below (9.10), under reasonable assumptions, injective objects are precisely
the absolute retracts. From Examples 9.17 we see that this is the case in the constructs
Set, Vec, Pos, Ab, Met, (Ban, O), and Field (= fields and algebraic field extensions).

9th July 2006

150

Objects and Morphisms[Chap. II

9.9 DEFINITION
A concrete category has enough injectives provided that each of its objects is an
initial subobject of an injective object.
9.10 PROPOSITION
If a concrete category A has enough injectives, then in A injective objects are precisely
the absolute retracts.
m

Proof: Let C be an absolute retract in A, and let C −−→ D be an initial monomorphismwith D an injective object. Since C is an absolute retract, m is a section, i.e., thererexists a retraction D −→ C with r ◦ m = idC . By Proposition 9.5, C is injective.

9.11 REMARK
If a concrete category has enough injectives, one may ask whether for every object Amthere exists a distinguished injective extension (i.e., an embedding A −−→ B with B
an injective object), e.g., one which is in some sense minimal or smallest. The crucial
concept needed in order to describe and analyze such “injective hulls” is that of essential extensions, i.e., extensions that are in a certain sense “dense” (but not necessarily
epimorphic).
9.12 DEFINITIONm
In a concrete category an embedding A −−→ B is called essential provided that af ◦mfmorphism B −→ C is an embedding, whenever A −−−→ C is an embedding.
9.13 EXAMPLESIn Examples (1)–(8) below, we consider essential embeddings for various constructs:(1) In Vec the essential embeddings are the isomorphisms.
(2) In Set the only essential embeddings are the bijective functions and the maps
∅ → {a} with empty domain and one-element codomain.
(3) In Pos the essential embeddings are the embeddings that are meet-dense and joindense (e.g., the embedding of
into ).

Q

R(4) In Boo the essential embeddings are the join-dense embeddings (which are automatically meet-dense).
m(5) In Ab an embedding A −−→ B is essential if and only if every nontrivial subgroupof B meets m[A] nontrivially (e.g., the embedding of ( , +) into the rationals).

Z(6) In Tych the essential embeddings are precisely the homeomorphisms, the one-pointcompactifications of locally compact, noncompact spaces, and the embeddings of theempty space into one-point spaces. In particular this example shows that epimorphic
embeddings need not be essential.
m(7) In Met an embedding A −−→ B is essential if and only if it is tight, i.e., if and only
if it satisfies the following two conditions:

9th July 2006

Sec. 9]Injective objects and essential embeddings(a) dB (b1 , b2 ) = sup{ dB (m(a), b1 ) − dB (m(a), b2 )151

a ∈ A } for all b1 , b2 ∈ B.

R(b) For each b ∈ B, the map f : A → + , defined by f (a) = dB (m(a), b), is aminimal superadditive map [cf. 9.3(4)(e)].
VW
In particular, if A is a subspace of with A = a and A = b, then the embedding
A → [a, b] is essential.

R(8) In Lat every object has arbitrarily large essential extensions (see Exercise 9H).
9.14 PROPOSITION(1) Every isomorphism is essential.
(2) The composite of essential embeddings is essential.
(3) If f and g are embeddings with g ◦ f essential, then g is essential.
(4) If f and g ◦ f are essential embeddings, then g is an essential embedding.

9.15 PROPOSITION
Injective objects have no proper essential extensions.
m

Proof: If C −−→ D is an extension of an injective object C, then there is a retraction gwith idC = g ◦ m. If m is essential, then g is an embedding, and hence an isomorphism.
Thus m is an isomorphism as well.

9.16 DEFINITIONm
An injective hull of A is an extension A −−→ B of A such that B is injective and m isessential.
9.17 EXAMPLESIn Examples (1)–(6) below, we describe injective hulls for various constructs:
idA(1) In Vec every object A has an injective hull, namely, A −−−→ A.

(2) In Set every object A has an injective hull, namely,
( id
A
A −−−
→ A, in case A 6= ∅;A −−→ {a}, in case A = ∅.
(3) In Pos every object has an injective hull, namely, its Mac Neille completion (=completion by cuts). Likewise in Boo every object has an injective hull, its Mac
Neille completion.
(4) In Ab every object has an injective hull. The embedding

Z ,→ Q is an example.

(5) In the construct Field every object has an injective hull, namely, its algebraic closure. The embedding ,→ is an example.

R

9th July 2006

C

152

Objects and Morphisms[Chap. II(6) In Met every object hasVan injective W
hull, namely, its hyperconvex envelope. If A
is a subspace ofwith A = a and A = b, then the embedding A ,→ [a, b] is aninjective hull. Likewise in (Ban, O) every object has an injective hull.

R

9.18 REMARK
It is sometimes difficult to decide whether or not every object in a given concrete category
has an injective hull. The existence of enough injectives is a necessary but not sufficientcondition. For example, the construct BooSp has enough injectives, since every Boolean
space is a subspace of some Cantor space [cf. 9.3(4)(b)], but no nonempty Boolean space
has a proper essential extension. A similar situation occurs for HComp (cf. Exercise
9D). In Exercise 9C we will formulate conditions that guarantee the existence of injective
hulls.
9.19 PROPOSITION
Injective hulls are essentially unique, i.e.,(1) if (m, B) and (m0 , B 0 ) are injective hulls of A, then there exists an isomorphismkB −→ B 0 with m0 = k ◦ m,k(2) if (m, B) is an injective hull of A, and if B −→ B 0 is an isomorphism, then (k◦m, B 0 )is an injective hull of A.
Proof:k(1). Since m is an embedding and B 0 is injective, there exists a morphism B −→ B 0 with
m0 = k ◦ m. By Proposition 9.14(4), k is an essential embedding. Since B has noproper essential extension, k is an isomorphism.
(2). Obvious.



9.20 PROPOSITION
If an object A has an injective hull, then for any extension (m, B) of A the following
conditions are equivalent:(1) (m, B) is an injective hull of A,(2) (m, B) is a maximal essential extension of A (i.e., (m, B) is an essential extension of A, and B has no proper essential extension),(3) (m, B) is a largest essential extension of A (i.e., (m, B) is an essential extension of A, and for every essential extension (m0 , B 0 ) of A there exists an essentialmembedding B 0 −−→ B with m = m ◦ m0 ),(4) (m, B) is a smallest injective extension of A (i.e., (m, B) is an injective extension of A, and for every injective extension (m0 , B 0 ) of A there exists an embeddingmB −−→ B 0 with m0 = m ◦ m),(5) (m, B) is a minimal injective extension of A (i.e., (m, B) is an injective extenm

m0msion of A and whenever A −−→ B = A −−→ B 0 −−→ B with m0 and m embeddings,
and B 0 an injective object, then m is an isomorphism).

9th July 2006

Sec. 9]Injective objects and essential embeddings

153

Proof: Let (m0 , B0 ) be an injective hull of A.
(1) ⇒ (2). Immediate from Proposition 9.15.
(2) ⇒ (1). Let (m, B) be a maximal essential extension of A. Since B0 is injective,
there exists a morphism f with m0 = f ◦ m. By Proposition 9.14(4), f is an essential
embedding. Since (m, B) is a maximal essential extension, f must be an isomorphism.
Hence (m, B) is an injective hull of A.
(1) ⇒ (3). Let (m, B) be an essential extension of A. By the injectivity of B0 , there exists
a morphism f with m0 = f ◦ m. By Proposition 9.14(4), f is an essential embedding.
(3) ⇒ (1). Let (m, B) be a largest essential extension of A. Then there exists a morphism
g with m = g ◦ m0 . By Proposition 9.14(4) g is an essential embedding, and so by
Proposition 9.15 is an isomorphism. Hence (m, B) is an injective hull of A.
(1) ⇒ (4). Immediate.
(4) ⇒ (1). If (m, B) is a smallest injective extension of A, then there exists an embedding
m with m0 = m ◦ m. By Proposition 9.14(3), m is essential and hence, since B has noproper essential extension, m is an isomorphism. Thus (m, B) is an injective hull of A.
(1) ⇒ (5). Let (m, B) be an injective extension of A and let (m, B0 ) be an extension of
B with m0 = m ◦ m. Then by Proposition 9.14(3), m is an essential embedding; hence,since B has no proper essential extensions, m is an isomorphism.
(5) ⇒ (1). Let (m, B) be a minimal injective extension of A. Since B is injective, there
exists a morphism f with m = f ◦ m0 . Since m0 is essential, f is an embedding; hence,by (5) an isomorphism. Thus (m, B) is an injective hull of A.

9.21 REMARK
If A has no injective hull, then the concepts mentioned in the above proposition may
fall apart. For example, in Tych every object has a simultaneously largest and maximal essential extension, but no space with more than one point has an injective hull[cf. 9.13(6) and 9.3(4)]. For further “negative” examples see Exercises 9F and 9G.

M-INJECTIVES IN ABSTRACT CATEGORIES
9.22 DEFINITION
Let M be a class of morphisms in a category A.
m(1) An object C is called M -injective provided that for every morphism A −−→ B infg

M and every morphism A −→ C there exists a morphism B −→ C with f = g ◦ m.
m(2) A morphism A −−→ B in M is called M -essential provided that a morphismfB −→ C belongs to M whenever f ◦ m does.
(3) An M -injective hull of an object A is a pair (m, B) consisting of an M -injectivemobject B and an M -essential morphism A −−→ B.

9th July 2006

154

Objects and Morphisms[Chap. II(4) A has enough M -injectives provided that for each object A there exists an M m
injective object C and a morphism A −−→ C in M .
9.23 REMARK
If M is the class of all embeddings in a concrete category A, then the concepts defined
above specialize to those of injective objects, essential embeddings, and injective hulls.
Also the previous results of this section carry over to the more general context of M injectivity, provided that suitable assumptions are imposed on M , e.g.,(a) Iso(A) ⊆ M ⊆ M ono(A); i.e., M is a class of A-monomorphisms that contains all
A-isomorphisms,(b) M ◦ M ⊆ M ; i.e., M is closed under composition.
We refrain from explicitly formulating the corresponding results.
9.24 EXAMPLES(1) In any category, if M consists of all sections, then every object is M -injective.
(2) In Met, if M consists of all dense embeddings, then the M -injective objects are thecomplete metric spaces, and the M -injective hulls are the metric completions.
(3) In Top0 , if M consists of all front-dense embeddings, then the M -injective objects
are the sober spaces, and the M -injective hulls are the sober reflections.
(4) In Top, if M consists of all embeddings X → X ∪ {p} of infinite discrete spaces into
ultrafilter spaces (i.e., p is a point of the Čech-Stone compactification of X), thenthe M -injective objects are the compact spaces.
(5) In Top, if M consists of the single embedding {0, 1} → [0, 1], then the M -injective
objects are the pathwise connected spaces. Similarly, if M consists of the single
embedding of the unit circle into the unit disc, then a complex domain is an M injective object if and only if it is simply connected.
(6) In Top a metrizable space X satisfies dim X ≤ n if and only if the n-sphere S n is
an M -injective object, where M consists of all embeddings of closed subspaces of X
into X.
(7) In Top, if M consists of all embeddings of closed subspaces of normal spaces, then[0, 1] andare M -injective objects [Tietze-Urysohn Theorem].

R(8) In Ab, if M consists of all pure embeddings, then the M -injective objects are thealgebraically compact abelian groups, or, equivalently, are all the direct summandsof direct products of cocyclic abelian groups.
(9) In CAT(X), if M consists of all concrete full embeddings, then the M -injective
objects are the “topological” concrete categories over X. Such categories will bedefined and studied in detail in §21. In particular, the constructs Top, Unif , Rel,
Prost, and each Spa(T ) are topological.

9th July 2006

Sec. 9]Injective objects and essential embeddings

155(10) In a (meet) semilattice C, considered as a category, an object is Mono-injective if
and only if it is the largest element of C.
9.25 PROPOSITION
If B is a reflective, isomorphism-closed, full subcategory of A and M is the class of all
B-reflection arrows, then
(1) the M -injective objects of A are precisely the B-objects, and(2) the M -injective hulls are precisely the B-reflections.
Proof:(1). Clearly each B-object is M -injective. Let C be an M -injective object, and letfr

C −→ B be an B-reflection arrow for C. Then there exists some B −→ C for which
idC = f ◦ r. The fact that idB ◦ r = (r ◦ f ) ◦ r and the uniqueness in the definitionof reflection arrow show that idB = r ◦ f . Thus r is an isomorphism, so that since
B is isomorphism-closed, C is a B-object.
r(2). By (1) we need only show that each B-reflection arrow A −→ B is M -essential.
rfSuppose that A −→ B −→ B 0 belongs to M , i.e., is also a B-reflection arrow.
Since B-reflection arrows for A are essentially unique (4.19), we can conclude thatf

B −→ B 0 is an isomorphism, and hence is a member of M .



9.26 REMARK
We may call a class B of A-objects (resp. the associated full subcategory B of A) an
injectivity class in A, whenever there exists a class M of A-morphisms such that
B is precisely the class of M -injective objects. The above proposition shows that every
reflective, isomorphism-closed, full subcategory of A is an injectivity class. The converse
is not true. By Example 9.24(4) compact spaces form an injectivity class in Top, but
the associated subcategory is not reflective in Top. A characterization of injectivity
classes by suitable stability properties is unknown even for “nice” categories (e.g., forTop). Connected spaces, which have stability properties similar to those of compact
spaces (e.g., they are closed under the formation of products and of continuous images),
do not form an injectivity class in Top. In fact the only injectivity class in Top that
contains all connected spaces is Top itself.

9th July 2006

156

Objects and Morphisms[Chap. II

PROJECTIVITY
9.27 TABLE OF DUAL CONCEPTS
The following table provides the names for the concepts dual to those investigated inthis section:
Concept

Dual Concept

embedding
injective object
essential embedding
injective hull
M -injective object
M -essential morphism
M -injective hull

quotient morphism
projective object
coessential quotient morphism
projective cover
M -projective object
M -coessential morphism
M -projective cover

9.28 EXAMPLESIn Examples (1)–(4) below, we consider projective objects in various constructs:(1) In Set and in Vec every quotient morphism is a retraction. Hence in either construct every object is projective, and the coessential quotient morphisms are theisomorphisms. Thus, in these constructs, each object has a projective cover.
(2) In any of the constructs Top, Pos, Ab, and Grp the projective objects are precisely
the free objects. Projective covers generally fail to exist.
(3) In BooSp and in HComp the projective objects are the extremally disconnectedcompact Hausdorff spaces (= the retracts of Čech-Stone compactifications of discrete spaces), and the coessential quotient morphisms are the irreducible quotientmorphisms (= those continuous surjections f : X → Y that map no proper closed
subset of X onto Y ). Projective covers exist for each object (and are called projective
resolutions).
(4) In Alg(Ω) the projective objects are precisely the retracts of the free objects. This
follows from the next proposition:
9.29 PROPOSITION
If (A, U ) is a concrete category over X that has free objects, and E is the class of all
A-morphisms f for which U f is a retraction, then the following are equivalent:(1) A is an E-projective object,(2) A is a retract of a free object.

9th July 2006

Sec. 9]Injective objects and essential embeddings

157

Proof: (1) ⇒ (2). Let (u, B) be a universal arrow over U A. Then there exists a uniqueA-morphism e : B → A with idU A = U e ◦ u. So e belongs to E. Hence, by (1), there
exists an A-morphism f : A → B with idA = e ◦ f . Consequently A is a retract of thefree object B.
(2) ⇒ (1). Let A be a free object (with universal arrow (u, A)), let e : B → C bea morphism in E, and let f : A → C be a A-morphism. Since in X every object is
U E-projective there exists an X-morphism g such that the diagram
X

u

/ UA

g



UB



Ue

Uf

/ UC

commutes. Since (u, A) is universal, there exists a unique A-morphism k : A → B withg = U k ◦ u. Hence U f ◦ u = U e ◦ g = U e ◦ U k ◦ u = U (e ◦ k) ◦ u, which implies thatf = e ◦ k. Hence A is E-projective. By the dual of Proposition 9.5 every retract of A is
also E-projective.
Projectivity and retracts of a free object

9.30 COROLLARY
If in a construct with free objects every surjective morphism is a quotient morphism,
then the projective objects are precisely the retracts of the free objects.


9th July 2006

158

Objects and Morphisms[Chap. II

9.31 EXAMPLES
The above corollary applies to such constructs as Grp, Lat, HComp (and many others).

Suggestions for Further Reading
Baer, R. Abelian groups that are direct summands of every containing abelian group.
Bull. Amer. Math. Soc. 46 (1940): 800–806.
Eckmann, B., and A. Schopf. Über injektive Moduln. Archiv Math. 9 (1953): 75–78.
Gleason, A. M. Projective topological spaces. Illinois J. Math. 2 (1958): 482–489.
Isbell, J. R. Six theorems about injective metric spaces. Comment. Math. Helv. 39(1964): 65–76.
Cohen, H. B. Injective envelopes of Banach spaces. Bull. Amer. Math. Soc. 70 (1964):
723–726.
Maranda, J. M. Injective structures. Trans. Amer. Math. Soc. 110 (1964): 98–135.
Banaschewski, B., and G. Bruns. Categorical characterization of the Mac Neille completion. Archiv Math. 18 (1967): 369–377.
Banaschewski, B. Projective covers in categories of topological spaces and topological algebras. Proceedings of the Kanpur Topological Conference 1968, Academia,
Prague, 1971, 63–91.
Banaschewski, B. Injectivity and essential extensions in equational classes of algebras.
Queen’s Papers Pure Appl. Math. 25 (1970): 131–147.
Scott, D. Continuous lattices. Springer Lect. Notes Math. 274 (1972): 97–136.
Banaschewski, B. Essential extensions of T0 -spaces. Gen. Topol. Appl. 7 (1977): 233–
246.
Blass, A. Injectivity and projectivity of abelian groups and the axiom of choice. Trans.
Amer. Math. Soc. 255 (1979): 31–59.
Tholen, W. Injective objects and cogenerating sets. J. Algebra 73 (1981): 139–155.
Porst, H.-E. Characterization of injective envelopes. Cahiers Topol. Geom. Diff. 22(1981): 399–406.

9th July 2006

Sec. 9]Injective objects and essential embeddings

159

EXERCISES* 9A. The Axiom of Choice
Many results of this book can be expressed (in localized form) in the realm of ZermeloFraenkel set theory (ZF). But if the axiom of choice for sets (AC) is not assumed, several
results fail to be true. Consider the following:(ET) In Set every epimorphism is a retraction.
(PT) In Set every product of injective objects is injective.
(BT) The injective objects in Ab are precisely the divisible abelian groups.
(ST) The injective objects in Boo are precisely the complete Boolean algebras.
(UT) In Boo the two-element Boolean algebra is injective.
(GT) The projective objects in HComp are the extremally disconnected compact Hausdorff spaces.
(a) that in ZF the following implications hold:(AC) ⇐⇒ (ET) ⇐⇒ (PT) ⇐⇒ (BT) =⇒ (ST) =⇒ (UT).
(b) Show that in ZF the following holds: (ST) ⇐⇒ [(GT) and (UT)].
(c) Does (ST) imply (AC)? [Unsolved.]
9B. A Characterization of Injective Objects
Let A be a construct satisfying(1) A has enough injectives, andfm(2) for every embedding A −−→ B there exists a morphism B −→ C such thatf ◦m

A −−−→ C is an essential embedding.
Show that in A injective objects are precisely those objects that have no proper essential
extension.
9C. Existence of M -Injective Hulls
Let M be a class of morphisms in a category A and let M ∗ be the class of M -essential
morphisms. Assume that the following conditions are satisfied:(1) Iso(A) ⊆ M ⊆ M ono(A),(2) M ◦ M ⊆ M ,(3) for any m ∈ M , there exists an A-morphism f such that f ◦ m ∈ M ∗ ,mf(4) for every 2-source B ←−− A −→ C with m ∈ M , there exists a 2-sinkfm

B −→ D ←−− C with m ∈ M and f ◦ m = m ◦ f ,(5) every well-ordered system in M has an upper bound in M ,9th July 2006

160

Objects and Morphisms[Chap. II(6) A is M ∗ -co-wellpowered (in the obvious sense; M ∗ need not consist of epimorphisms
only).
Show that(a) For A-objects A the following conditions are equivalent:(a1) A is M -injective,m(a2) every A −−→ B in M is a section,m(a3) every A −−→ B in M ∗ is an isomorphism.
(b) Every A-object has an M -injective hull.
* 9D. Enough Injectives and Injective Hulls in HComp
Show that HComp has enough injectives, but that the only compact Hausdorff spacethat has a proper essential extension is the empty space.
9E. Regular Projective Objects
Show that the RegEpi-projective objects* (a) in Top are the discrete topological spaces,* (b) in Haus are the discrete topological spaces,(c) in Set2 are the injective functions,(d) in (Set2 )op are the surjective functions with nonempty domain.
9F. Injective Objects and Maximal Essential Extensions in Top1
Show that(a) In Top1 the following conditions are equivalent:(1) X is injective,(2) X is an absolute retract,(3) X is a terminal object.
(b) If a T1 -space X has a point such that each of its neighborhoods is cofinite in X,
then X has no proper essential extension. If a T1 -space X has no such point, then it
has an essentially unique proper essential extension. [Add a point p to X with eachneighborhood of p cofinite in X ∪ {p}.](c) In Top1 every object has an essentially unique maximal essential extension.
(d) In Top1 the only objects that have injective hulls are the initial object and theterminal objects.
(e) If M is the class of embeddings in Top1 , then all the conditions of Exercise 9C aresatisfied except condition (3).

9th July 2006

Sec. 9]Injective objects and essential embeddings

161* 9G. Minimal and Smallest Injective Extensions
Show that(a) Minimal injective extensions need not be smallest injective extensions. [In the construct that consists of all sets X with card(X) 6= 1 together with all identities
and constant maps as morphisms, every ∅ → B with B 6= ∅ is a minimal injective
extension. However, ∅ has no smallest injective extension.](b) Smallest injective extensions need not be minimal. [In the full subconstruct of Set
that contains ∅ and all infinite sets, ∅ → N is a smallest injective extension. However,∅ has no minimal injective extension.](c) Injective extensions that are simultaneously smallest and minimal need not be injective hulls. [In the full subconstruct of Set that consists of all sets X withcard(X) 6= 1, the inclusion ∅ ,→ {1, 2} is simultaneously a smallest and a minimal injective extension. However, ∅ has no M -injective hull.]
9H. Essential Extensions in Lat
Show that every lattice that has at least two elements has arbitrarily large essential
extensions. [Hint: First assume that L is a lattice with smallest element 0 and largest
element 1. Show that for any set A that has at least three elements and is disjoint fromL, if A is ordered by equality, then the embedding
v1
vv
v
vv
vv
vv

L \ {0, 1}

HH
HH
HH
HH
HH

0


/

v 1 ==
vv
==
v
==
vv
vv
==
v
v

L \ {0, 1}

HH
HH
HH
HH
HH

0







A

is an essential extension of L.]
9I. Injective Automata(a) Prove that the following are equivalent for each object A of {σ}-Seq:(1) A is injective,(2) A is an absolute retract,(3) all states in A are final, and δ(−, σ) is a permutation with a fixed point.
Whenever Σ has more than one element, show that there exists an object of Σ-Seq
that is not an absolute retract, although every state is final and each δ(−, σ) is apermutation with a fixed point.
(b) Characterize those objects of Σ-Seq that have an injective hull.

9th July 2006

163

Chapter III

SOURCES AND SINKS

A source

A sink

9th July 2006

10 Sources and sinks
SOURCESA basic categorical concept that simultaneously generalizes the concepts of objects andof morphisms is that of sources.
10.1 DEFINITION
A source is a pair (A, (fi )i∈I ) consisting of an object A and a family of morphisms
fi : A → Ai with domain A, indexed by some class I. A is called the domain of thesource and the family (Ai )i∈I is called the codomain of the source.
10.2 REMARK(1) Whenever convenient we use more concise notations, such as (A, fi )I , (A, fi ) orfi

(A −−→ Ai )I .
(2) The indexing class I of a source (A, fi )I may be a proper class, a nonempty set, orthe empty set. In case I = ∅, the source is determined by A. In case I 6= ∅, thesource is determined by the family (fi )I .
(3) Sources indexed by the empty set are called empty sources and are denoted by(A, ∅). Whenever convenient, objects may be regarded as empty sources.
(4) Sources that are indexed by a set are called set-indexed or small.
(5) Sources that are indexed by the set {1, . . . , n} are called n-sources and are denotedby (A, (f1 , . . . , fn )). Whenever convenient, morphisms f : A → B may be regarded
as 1-sources (A, f ).
(6) There are properties of sources that depend heavily on the fact that (fi )I is a family,i.e., an indexed collection (e.g., the property of being a product). There are otherproperties of sources (A, fi ), depending on the domain A and the associated class{ fi | i ∈ I } only (e.g., the property of being a mono-source). In order to avoid
a clumsy distinction between indexed and non-indexed sources, we will sometimes
regard classes as families (indexed by themselves via the corresponding identity
function). Hence for any object A and any class S of morphisms with domain A,
the pair (A, S) will be considered as a source. A particularly useful example is thetotal source (A, SA ), where SA is the class of all morphisms with domain A.
10.3 DEFINITION
gij
fi
If S = (A −−→ Ai )I is a source and, for each i ∈ I, Si = (Ai −−→ Aij )Ji is a source, thenthe source
gij ◦fi(Si ) ◦ S = (A −−−−→ Aij )i∈I, j∈Ji
is called the composite of S and the family (Si )I .

164

Sec. 10]Sources and sinks

165

10.4 REMARKS
fi(1) For a source S = (A −−→ Ai )I and a morphism f : B → A we use the notationfi ◦f

S ◦ f = (B −−−→ Ai )I .
(2) The composition of morphisms can be regarded as a special case of the compositionof sources.

MONO-SOURCES
10.5 DEFINITION
A source S = (A, fi )I is called a mono-source provided that it can be cancelled fromthe left, i.e., provided that for any pair B

r
s

//

A of morphisms the equation S ◦ r = S ◦ s(i.e., fi ◦ r = fi ◦ s for each i ∈ I) implies r = s.
10.6 EXAMPLES(1) An empty source (A, ∅) is a mono-source if and only if for each object B there is at
most one morphism from B to A.
(2) A 1-source (A, f ) is a mono-source if and only if f is a monomorphism.
(3) In Set mono-sources are precisely the point-separating sources (A, fi )I , i.e.,sources (A, fi )I such that for any two different elements a and b of A there exists
some i ∈ I with fi (a) 6= fi (b).
(4) In many familiar constructs, e.g., in Vec, Grp, Top, and Pos, a source is a monosource if and only if it is point-separating. [See Corollary 10.8.](5) In every preordered class, considered as a category, every source is a mono-source.
This property characterizes thin categories. [Consider the empty sources.]
10.7 PROPOSITION(1) Representable functors preserve mono-sources (i.e., if G : A → Set is a representable functor and S is a mono-source in A, then GS is a mono-source in Set).
(2) Faithful functors reflect mono-sources (i.e., if G : A → B is a faithful functor,
S = (A, fi ) is a source in A, and GS = (GA, Gfi ) is a mono-source in B, then S
is a mono-source in A).
Proof: 52(1). If a functor preserves mono-sources, then, clearly, so does every functor that is natfi

urally isomorphic to it. Thus it suffices to show that each mono-source (B −−→ Bi )I
is sent by each hom-functor hom(A, −) : A → Set into a point-separating source:hom(A,fi )(hom(A, B) −−−−−−−→ hom(A, Bi ))I .
52

Even though the proof is immediate by arguments analogous to those used in the proof of Proposition
7.37, we nevertheless sketch a proof so that the reader may gain some familiarity with notation
concerning sources.

9th July 2006

166

Sources and Sinks[Chap. III

But this is immediate from the definition of mono-source.
(2). Let G and S be as described. If B

r
s

//

A is a pair of A-morphisms with S ◦ r =S ◦ s, then GS ◦ Gr = G(S ◦ r) = G(S ◦ s) = GS ◦ Gs. Since GS is a mono-source,
this implies Gr = Gs. Since G is faithful, this gives r = s.
10.8 COROLLARY
In a construct (A, U ) every point-separating source is a mono-source. The converse
holds whenever U is representable.

10.9 PROPOSITION
Let T = (Si ) ◦ S be a composite of sources.
(1) If S and all Si are mono-sources, then so is T .
(2) If T is a mono-source, then so is S.

10.10 PROPOSITION
Let (A, fi )I be a source.
(1) If (A, fj )J is a mono-source for some J ⊆ I, then so is (A, fi )I .
(2) If fj is a monomorphism for some j ∈ I, then (A, fi )I is a mono-source.



10.11 DEFINITION
A mono-source S is called extremal provided that whenever S = S ◦ e for some epimorphism e, then e must be an isomorphism.
10.12 EXAMPLES(1) A 1-source (A, f ) is an extremal mono-source if and only if f is an extremal monomorphism.
(2) In balanced categories (e.g. in Set, Vec, and Grp) every mono-source is extremal.
[This follows immediately from Proposition 10.9(2).] Conversely, if every monosource in C is extremal, then C is balanced (cf. Proposition 7.67).
fi(3) A source (A −−→ Ai )I in Pos is an extremal mono-source provided that the following
equivalence holds:a ≤ b ⇔ ∀ i ∈ I fi (a) ≤ fi (b).
fi(4) A mono-source (A −−→ Ai )I in Top is extremal if and only if A carries the initial(= weak) topology with respect to (fi ).
fi(5) A mono-source (A −−→ Ai )I in Σ-Seq is extremal if and only if a state q of A is final
whenever each state fi (q) is final in Ai .
(6) In a poset, considered as a category, a source (A → Ai )I is an extremal mono-source
if and only if A is a maximal lower bound of { Ai | i ∈ I }.

9th July 2006

Sec. 10]Sources and sinks

167

10.13 PROPOSITION(1) If a composite source (Si ) ◦ S is an extremal mono-source, then so is S.
(2) If S ◦ f is an extremal mono-source, then f is an extremal monomorphism.



10.14 REMARKIf S and each Si are extremal mono-sources, then (Si ) ◦ S need not be extremal. See
Exercise 7N.
10.15 PROPOSITION
Let (A, fi )I be a source.
(1) If (A, fj )J is an extremal mono-source for some J ⊆ I, then so is (A, fi )I .
(2) If fj is an extremal monomorphism for some j ∈ I, then (A, fi )I is an extremal
mono-source.

10.16 REMARKThe concept of source allows a simple description of coseparators: namely, A is a coseparator if and only if, for any object B, the source (B, hom(B, A)) is a mono-source. Thissuggests the following definition:
10.17 DEFINITION
An object A is called an extremal coseparator provided that for any object B thesource (B, hom(B, A)) is an extremal mono-source.
10.18 EXAMPLES(1) In a balanced category every coseparator is extremal.
(2) In the nonbalanced category Pos every coseparator is extremal [cf. 7.18(3)].
(3) A topological space is an extremal coseparator in Top if and only if it contains an
indiscrete subspace with two elements and a Sierpinski subspace (i.e., a nondiscrete
T0 -space with two elements) [cf. 7.18(4)].
(4) The unit interval [0, 1] is an extremal coseparator in HComp.
(5) The category Tych of Tychonoff spaces and continuous maps has no extremal coseparator [cf. 7.18(7)].
(6) Let Σ-Seq0 denote the category of sequential Σ-acceptors that have no initial state
and that are observable (i.e., the observability equivalence of Example 4.17(7) isequality). Then Σ-Seq0 has an extremal coseparator: the acceptor (Rat, δ, F ) of
all rational languages in Σ, where δ(L, σ) = { σσ1 · · · σn | σ1 · · · σn ∈ L } and F ={ L ∈ Rat | ∅ ∈ L }. [For each acceptor A consider the simulation f : A → (Rat, δ, F )that assigns to each state q the language f (q) ∈ Rat accepted by A in the initialstate q. Then f is an extremal monomorphism.]

9th July 2006

168

Sources and Sinks[Chap. III

PRODUCTS
Cartesian products of pairs of sets or, more generally, of families of sets (resp. direct
products of families of vector spaces, resp. topological products of families of topologicalspaces) can be regarded as objects together with families of (projection) morphisms
emanating from them, i.e., as sources. As such — but not as objects alone — they canbe characterized, up to isomorphism, by the following categorical property:
10.19 DEFINITIONpi
A source P = (P −−→ Ai )I is called a product provided that for every source S =fif(A −−→ Ai )I with the same codomain as P there exists a unique morphism A −→ P with
S = P ◦ f . A product with codomain (Ai )I is called a product of the family (Ai )I .
10.20 EXAMPLES(1) In the category Set, given sets A1 and A2 , the projections from the cartesian productπ1 : A1 × A2 → A1 and π2 : A1 × A2 → A2 (given by: πi (x1 , x2 ) = xi ) form afi

πi
product source (A1 × A2 −−→
Ai )i=1,2 . Indeed, given a source (A −−→ Ai )i=1,2 ,fthere is a unique A −→ A1 × A2 with fi = πi ◦ f, namely, f (a) = (f1Q(a), f2 (a)).
More generally, let (Ai )I be a family of sets indexed by a set
I,
and
let
i∈I Ai beS
its cartesian product, i.e., the set of all functions g : I → I A
Qi with the property
that g(i) ∈ Ai . Then the family of projection functions πj :i∈I Ai → Aj , given
by: g 7→ g(j), is a product in Set.
(2) Likewise in the categories Vec, Ab, and Grp the “direct products”, in Pos the“ordinal products”, and in Top the “topological products”, considered as sources
via the projections, are products.
(3) If (Ai )i∈I is a set-indexed familyof objects in the category AbTor of abelian torsion
Q
groups, its direct product i∈I Ai need not be a torsion group. However,Q a productof the family does exist in AbTor. Let P be the torsion-subgroup of i∈I Ai (i.e.,the subgroup consisting of all torsion-elements), and for each j ∈ I let pj : P → Aj
Qpi−→ Ai )i∈I is
be the restriction of the jth projection πj :i∈I Ai → Aj . Then (P −fia product in AbTor. [Indeed, given a source (A −−→ Ai )I in AbTor,
Q each a ∈ A
is a torsion-element of A, so that (fi (a))I is a torsion-element of
Ai . Thus theffunction A −→ P , given by f (a) = (fi (a))I , satisfies the above definition.](4) Similarly, in Ban products can be obtained as the subspaces of the direct productsof the corresponding vector spaces, consisting of those elements a = (ai )i∈I withkak = supi∈I kai k < ∞, supplied with the restrictions of the projection-maps.
(5) In Σ-Seq the product of two acceptors A1 × A2 is their parallel connection (thestate of which is determined by knowing the state of both A1 and A2 ). Thus finite
products have a clear interpretation in Σ-Seq, although infinite products usually
don’t exist, since acceptors are by definition finite.

9th July 2006

Sec. 10]Sources and sinks

169pi(6) In a partially ordered class, considered
as a category [3.3(4)(d)], a source (P −−→ Ai )IV
is a product if and only if P = i∈I Ai .
(7) An empty source (P, ∅) is a product if and only if P is a terminal object. Thus,
regarding empty sources as objects, one might say: terminal objects are empty
products.
(8) A 1-source (P, p) is a product if and only if p is an isomorphism. Thus, regarding
1-sources as morphisms, one might say: isomorphisms are 1-products.
10.21 PROPOSITION
Every product is an extremal mono-source.
Proof: If P = (P, pi ) is a product and Ars// P is a pair of morphisms with P ◦ r =P ◦ s, then S = P ◦ r = P ◦ s is a source with the same codomain as P. The uniquenessrequirement in the definition of product implies that r = s. Hence P is a mono-source.
To show that P is extremal, let P = Q ◦ e for some epimorphism e. Since P and Q havethe same codomain, there exists a unique morphism f with Q = P ◦ f . Since P is amono-source, the equation P ◦ idp = P = Q ◦ e = P ◦ (f ◦ e) implies that idp = f ◦ e.
Consequently, e is a section and an epimorphism, hence an isomorphism.

10.22 PROPOSITION
For any family (Ai )I of objects, products of (Ai )I are essentially unique; i.e., if P =pi(P −−→ Ai )I is a product of (Ai )I , then the following hold:qih(1) for each product Q = (Q −−→ Ai )I there exists an isomorphism Q −→ P with
Q = P ◦ h,h(2) for each isomorphism A −→ P the source P ◦ h is a product of (Ai )I .
Proof:(1). Since P and Q are products with the same codomain, there exist unique morphisms
h and k with Q = P ◦ h and P = Q ◦ k. Therefore Q ◦ idQ = Q ◦ (k ◦ h) andP ◦ idP = P ◦ (h ◦ k). Since P and Q are mono-sources, these equations imply that
idQ = k ◦ h and idP = h ◦ k. Hence h is an isomorphism.
(2). Obvious.



10.23 REMARKThe above uniqueness result allows us to introduce special notations for products (provided that they exist):Qπj(1) Products of (Ai )I will be denoted by ( i∈I Ai −−→ Aj )j∈I , or, more simply, byQ
πj( Ai −−→ Aj )I , and the morphisms πj will be called projections.

9th July 2006

170

Sources and Sinks[Chap. IIIQπjfj(2) If ( Ai −−→ Aj )I is a product and (A Q−−→ Aj )I is a source with the same codomain,then the unique morphism f : A → Ai with fj = πj ◦ f for each j ∈ I will bedenoted by hfi i:
hfi i Q
/
(∗)
Ai
AC
CC
CC
πj
C
fj CC! Aj(3) In case I = {1, 2, . . . , n} the following notation will often be used instead of thatabove:
hf1 ,f2 ,...,fn i/ A1 × A2 × · · · × An
TTTT
TTTT
πj
TTTT
fj
TTTT T*

A TTTTT(∗∗)

Aj

10.24 REMARKThe above diagram (∗) makes visible why products are useful. The correspondence(A, fi ) 7→ hfi i provides a bijection from the collection of all sources with domain
AQ
and codomain (Ai )I to the set of all morphisms with domain A and codomain i∈I Ai .
Hence products allow one to treat sources as if they were morphisms. Propositions 10.26
and 10.38 below show how this correspondence works.
10.25 PROPOSITION
Let Q = (Pi ) ◦ P be a composite of sources.
(1) If P and all Pi are products, then so is Q.
(2) If Q is a product and all Pi are mono-sources, then P is a product.
Proof: This follows immediately from the definition of products and the fact that products are mono-sources (10.21).

10.26 PROPOSITION
Considerhfi i Q

/
Ai
CC
CC
πj
C
fj CC! 

AC

Aj

Then
(1) (A, fi )I is a mono-source if and only if hfi i is a monomorphism.
(2) (A, fi )I is an extremal mono-source if and only if hfi i is an extremal monomorphism.
(3) (A, fi )I is a product if and only if hfi i is a product; i.e., an isomorphism.

9th July 2006

Sec. 10]Sources and sinks

171

Proof: (1) follows from Proposition 10.9, (3) follows from Propositions 10.21 and 10.25,
and the “only if ” part of (2) follows from Proposition 10.13. To show the “if ” partof (2), let hfi i be an extremal monomorphism. By (1), (A, fi ) is a mono-source. LeteA −→ B be an epimorphism and (B, gi )I be a source with fi = gi ◦ e for each i ∈ I.
Then, since the product is a mono-source,
πi ◦ hfi i = gi ◦ e = πi ◦ hgi i ◦ e implies that hfi i = hgi i ◦ e.
Hence e is an isomorphism.



10.27 REMARK
Whereas each product is an extremal mono-source, the single projections are usually
retractions, as the following result shows. That this need not always be the case isdemonstrated in Set by the projection: ∅ × → .

N N

10.28 PROPOSITIONpi
If (P −−→ Ai )I is a product and if i0 ∈ I is such that hom(Ai0 , Ai ) 6= ∅ for each i ∈ I,
then pi0 is a retraction.
Proof: For each i ∈ I choose fi ∈ hom(Ai0 , Ai ) with fi0 = idAi0 . Then hfi i : Ai0 → P
is a morphism with pi0 ◦ hfi i = fi0 = idAi0.

10.29 DEFINITION(1) A category has products provided that for every set-indexed family (Ai )I of objectsQ
πjthere exists a product ( Ai −−→ Aj )I .
(2) A category has finite products provided that for every finite family (Ai )I of objectsQ
πjthere exists a product ( Ai −−→ Aj )I .
10.30 PROPOSITION
A category has finite products if and only if it has terminal objects and products of pairsof objects.
Proof: The result follows from the observations that(1) empty products are terminal objects [10.20(7)],(2) products of singleton families always exist [10.20(8)],(3) products of n-indexed families for n ≥ 3 can be constructed via induction by composing products of pairs [cf. 10.25(1)]:A1 × A2 × · · · × An = (· · · ((A1 × A2 ) × A3 ) × · · · × An ).



10.31 EXAMPLES(1) The categories Set, Vec, Grp, Top, Rel, Pos, Alg(Ω), and Aut have products.

9th July 2006

172

Sources and Sinks[Chap. III(2) Met and Σ-Seq have finite products, Metu and Metc have countable products,
and PMet has products. [In fact, a set-indexed family (Xi )i∈I of nonempty metricspaces each having finite diameter diam(Xi ) has a product in Met if and only if theset { diam(Xi ) | i ∈ I } is bounded.](3) Banb has finite products, and Ban has products.
(4) A poset, considered as a category, has products if and only if it is a complete lattice.
10.32 THEOREM(1) A category that has products for all class-indexed families must be thin.
(2) A small category has products if and only if it is equivalent to a complete lattice.
Proof:(1). Assume that A has all class-indexed products, but that the set hom(A, B) has atleast two elements. Consider the family (Bi )i∈I with I = M or(A) and Bi = B
for each i ∈ I. Since hom(A, B) has at least two members, there are at least as
many distinct sourcesQwith domain A and codomain (Bi )I as there are subclassesof I. Hence hom(A, Bi ) containsQat least as many members as this (cf. 10.24),
contradicting the fact that hom(A, Bi ) is contained in I.
(2). Since A is small, M or(A) is a set, so that if A has products, the above proof showsthat it is thin. Thus it is a preordered set with meets of all subsets, and, hence, isequivalent to a complete lattice [cf. 10.31(4)]. The converse is clear.

10.33 REMARKThe above theorem shows why in Definition 10.29(1) we didn’t require the existence of
products for families of objects indexed by arbitrary (hence also proper) classes. Sucha requirement would be far too strong. None of our familiar constructs [e.g., from
Examples 10.31 (1)–(3)] satisfies this condition, yet many of them do satisfy the weakercondition of having products of all set -indexed families. This observation demonstrates
strikingly that when working with categories one needs to distinguish carefully between
sets and proper classes (resp. between “small” and “large” collections).
10.34 DEFINITION
Q
Q
πj
pj
fi
If (Ai −−→ Bi )I is a family of morphisms
and
if
(
A
−
−
→
A
)
and
(
B
−
−
→ Bj )I are
i
j
i
I
Q
Q
products, then the unique morphismAi →Bi that makes the following diagram
commute for each j ∈ I
Q
Πfi Q
/
Ai
Bi
πj





Aj

fj

pj

/ Bj

is denoted by Πfi and is called the product of the family (fi )I . If I = {1, . . . , n} thenΠfi is usually written as f1 × f2 × · · · × fn .

9th July 2006

Sec. 10]Sources and sinks

173

10.35 PROPOSITION
Let (fi )I be a set-indexed family of morphisms with product Πfi . If each fi has any of
the following properties, then so does Πfi :(1) isomorphism,(2) section,(3) retraction,(4) monomorphism,(5) regular monomorphism (provided that the category in question has products).
Proof: (1), (2), and (3) follow immediately from the observation that Πgi ◦ Πfi =Π(gi ◦ fi ). (4) follows from Propositions 10.9 and 10.21.
(5) follows from the next proposition. (Observe that products of regular monomorphisms
are always extremal monomorphisms. Cf. also 10D)

10.36 PROPOSITIONei
In a category with products, if I is a set and if Ei −−→
Ai is an equalizer of Ai
for each i ∈ I, then

Q

Πe

i
Ei −−−→Q

Q

Ai is an equalizer of

Ai

Πfi Q
Πgi

//

fi
gi

//

Bi

Bi .

Proof:
Q
πj

Ei

Πei



Ej

/

Q


Ai

ej

Πfi
Πgi

pj

/ Aj

fj
gj

//

Q


Bi
qj

// B

j

Q
qj
That Πfi ◦ Πei = Πgi ◦ Πei follows from the fact that the product ( Bi −−→ Bj )h Q
is a mono-source. If C −→Ai is a morphism such that Πfi ◦ h = Πgi ◦ h, thenfj ◦ pj ◦ h = gj ◦ pj ◦ h for each j ∈ I. Hence for each j ∈ I Q
there exists a morphism
hj : C → Ej with pj ◦ h = ej ◦ hj . Consequently, hhi i : C → Ei is a morphism withpj ◦ Πei ◦ hhi i = ej ◦ πj ◦ hhi i = ej ◦ hj = pj ◦ h; hence with Πei ◦ hhi i = h. Therefore hfactors through Πei . Since, by Proposition 10.35(4), Πei is a monomorphism, h factors
uniquely.

10.37 DEFINITIONQ
Q
πj
If I is a set and ( Ai −−→ Aj )I is a product with Ai = A for each i ∈ I, then Ai is
denoted by AI and called an Ith power of A.
10.38 PROPOSITION
In a category that has products, an object A is an (extremal) coseparator if and only ifevery object is an (extremal) subobject of some power AI of A.

9th July 2006

174

Sources and Sinks[Chap. III

Proof: Let A and B be objects. Consider the source S = (B, hom(B, A)) and themmorphism B −−→ Ahom(B,A) defined by f = πf ◦ m for each f ∈ hom(B, A). By
Proposition 10.26, S is an (extremal) mono-source if and only if m is an (extremal)mmonomorphism. Hence A is an (extremal) coseparator if and only if B −−→ Ahom(B,A)
gis an (extremal) monomorphism for each object B. Finally, B −→ AI is an (extremal)monomorphism for some I [i.e., the source (B, πi ◦ g)i∈I is an (extremal) mono-source] if
and only if S = (B, hom(B, A)) is an (extremal) mono-source. Cf. Propositions 10.10(1)and 10.15(1).

10.39 REMARKS(1) In a category A that has products and an (extremal) coseparator the above result
provides a useful description of the A-objects. For example, since the two-element
chain 2 is a coseparator for Pos, posets are precisely the subobjects of powers of
2 in Pos, and since the unit interval [0, 1] is an extremal coseparator for HComp,
compact Hausdorff spaces are precisely the extremal subobjects (= closed subspaces)of powers of [0, 1] in HComp.
(2) In a category that has products, M -coseparators may be defined (for any class M
of monomorphisms) as objects A such that each object is an M -subobject of somepower of A.
10.40 PROPOSITION
For any class M of morphisms, every product of M -injective objects is M -injective.53 

SOURCES IN CONCRETE CATEGORIESNext we turn our attention to sources in concrete categories. As before, in the context
of concrete categories, we use the notational conventions of Remark 6.22.

INITIAL SOURCES
10.41 DEFINITIONfi
Let A be a concrete category over X. A source (A −−→ Ai ) in A is called initial
provided that an X-morphism f : |B| → |A| is an A-morphism whenever each composite
fi ◦ f : |B| → |Ai | is an A-morphism.
10.42 EXAMPLES(1) An empty source (A, ∅) is initial if and only if A is indiscrete.
(2) A 1-source (A, f ) is initial if and only if f is an initial morphism (cf. 8.6).
53

The special case for products that are empty sources yields Proposition 9.4.

9th July 2006

Sec. 10]Sources and sinks

175(3) A source (A, fi )I in Top is initial if and only if A carries the initial (= weak) topology
with respect to the family (fi )I . In particular, a topological space X is completely
regular if and only if the source S(X, ), consisting of all continuous maps from X
to the real line, is initial (in the construct Top); and X is a Tychonoff space if andonly if S(X, ) is an initial mono-source.
Tfi(4) In Spa(T ) a source ((X, α) −−→ (Xi , αi ))I is initial if and only if α = i∈I (T f )−1 [αi ].

R

R

fi

In particular, a source ((X, ρ) −−→ (Xi , ρi ))I in Rel is initial if and only if theequivalence: xρy ⇔ ∀i ∈ I fi (x)ρi fi (y) holds.
fi(5) In PMet a source ((X, d) −−→ (Xi , di ))I is initial if and only ifd(x, y) = sup di (fi (x), fi (y)).
i∈I(6) In Pos and in Σ-Seq the initial sources are precisely the extremal mono-sources(cf. 10.12).
(7) In any of the constructs Vec, Ab, Grp, Mon, Rng, Boo, and Alg(Ω) the initialsources are precisely the mono-sources.
(8) In a preordered class, considered as a concrete category over 1, the initial sources are
precisely the products [cf. 10.20(6)]. This fact is not as accidental as it may seem.
As we will see later in this section (cf. 10.58), a source in an arbitrary category A isa product if and only if it is T -initial, where T : A → 1 is the unique functor from
A to 1.
10.43 PROPOSITIONfi
If (A −−→ Ai )I is an initial source in A, thenA = max{ B ∈ Ob(A)fi|B| = |A| and all |B| −−→ |Ai | are A-morphisms }.54

10.44 REMARKThe above property often characterizes initial sources, e.g., in such constructs as Top or
Spa(T ). However, in the construct Top1 , there are non-initial sources with the above
property. In fact, as shown in Example 8.5(2), for each set X the fibre of X has a largest
element AX , hence the empty source (AX , ∅) satisfies the above property. But (AX , ∅)is initial (i.e., AX is indiscrete) only for card(X) ≤ 1.
10.45 PROPOSITION
Let T = (Si ) ◦ S be a composite of sources in a concrete category.
(1) If S and all Si are initial, then so is T .
(2) If T is initial, then so is S.
54

Recall the order on the fibre of |A| [5.4(1)].

9th July 2006



176

Sources and Sinks[Chap. III

10.46 PROPOSITION
Let (A, fi )I be a source in a concrete category. If (A, fi )J is initial for some J ⊆ I, thenso is (A, fi )I .

10.47 DEFINITION
A concrete functor F : A → B over X is said to preserve initial sources provided
that for every initial source S in A, the source F S is initial in B.
10.48 EXAMPLES(1) The forgetful concrete functor Rng → Ab over Set, which forgets multiplication,
preserves initial sources [cf. 10.42(7)].
(2) The concrete functor Top → Rel over Set, which assigns to each topological space(X, τ ) the object (X, ρ) with xρy if and only if x ∈ cl{y}, preserves initial sources.
(3) An order-preserving map between preordered classes, considered as a concrete functor over 1, preserves initial sources if and only if it preserves all existing meets.
10.49 PROPOSITION
If (F, G) is a Galois correspondence, then G preserves initial sources.
Proof: Let G : A → B and F : B → A be concrete functors over X such that (F, G)
fi

is a Galois correspondence over X. Let (A −−→ Ai )I be an initial source in A and let
h

h

fi

B −→ GA be an X-morphism such that all B −→ GA −−→ GAi are B-morphisms. Then
h

fi

h

by Proposition 6.28, all F B −→ A −−→ Ai are A-morphisms. Hence F B −→ A is anhA-morphism. Again by Proposition 6.28, B −→ GA is a B-morphism.

10.50 COROLLARY
Embeddings of concretely reflective subcategories preserve initial sources.



10.51 REMARKLet G : A → B be a monotone map between posets, considered as a concrete functor
over 1. If A is a complete lattice, then G preserves initial sources (= meets) if andonly if there exists a (necessarily unique) monotone
map F : B → A such that (F, G)
V
is a Galois connection. [Namely, F (b) = { a ∈ A | b ≤ G(a) }]. Theorem 21.24 is acorresponding result for concrete functors over arbitrary base categories X.

CONCRETE PRODUCTS
10.52 DEFINITION
Let A be a concrete category over X. A source S in A is called a concrete productin A if and only if S is a product in A and |S| is a product in X.

9th July 2006

Sec. 10]Sources and sinks

177

10.53 PROPOSITION
A source S in a concrete category A over X is a concrete product if and only if it isinitial in A and |S| is a product in X.

10.54 DEFINITION
A concrete category A has concrete products if and only if for every set-indexedpifamily (Ai )I of A-objects there exists a concrete product (P −−→ Ai )I in A, i.e., if andonly if A has products and the forgetful functor preserves them.
10.55 EXAMPLES(1) Many familiar constructs, e.g., Vec, Grp, Ab, Rng, Top, Rel, Pos, and Alg(Ω),
have concrete products. According to Proposition 10.53 they can be constructed in
two steps: Given a family of objects (Ai )I in A, first form the cartesian productQ
Q
πj( |Ai | −−→ |Aj |)I of the underlying sets, and then supply the set |Ai | with theinitial structure with respect to (πj )I .
(2) The construct (Ban, O) has concrete products, but the products in (Ban, U ) generally fail to be concrete [cf. 10.20(4)].
(3) Products in the construct AbTor generally fail to be concrete [cf. 10.20(3)].
(4) In a concrete category over 1 every product is concrete.
10.56 PROPOSITION
Let Q = (Pi ) ◦ P be a composite of sources in a concrete category A.
(1) If P and all Pi are concrete products, then so is Q.
(2) If Q is a concrete product and each |Pi | is a mono-source, then P is a concrete
product.


G-INITIAL SOURCES
Now we investigate sources not only in relation to forgetful functors of concrete categories, but also in relation to arbitrary functors. This will throw additional light on
products.
10.57 DEFINITION
fi
Let G : A → B be a functor. A source S = (A −−→ Ai )I in A is called G-initial
gi
provided that for each source T = (B −−→ Ai )I in A with the same codomain as S andheach B-morphism GB −→ GA with GT = GS ◦ h there exists a unique A-morphismhB −→ A with T = S ◦ h and h = Gh.
BA

AA g
AA i
h
AA

/ Ai
A
fi

9th July 2006

GB F

FF Gg
FF i
FF
F"

/ GAi
GA

Gh=h

Gfi

178

Sources and Sinks[Chap. III

10.58 EXAMPLES(1) If (A, U ) is a concrete category, then U -initial sources are precisely the initial sourcesin (A, U ).
(2) If A is a category and G : A → 1 is the unique functor from A to 1, then G-initialsources are precisely the products in A.
10.59 PROPOSITION
For a functor G : A → B the following conditions are equivalent:(1) G is faithful,(2) for each A-object A the 2-source (A, (idA , idA )) is G-initial,(3) whenever (A, fi )I is a source in A and (A, fj )J is G-initial for some J ⊆ I, then(A, fi )I is G-initial.
Proof: Obviously (1) ⇒ (3) ⇒ (2). To show that (2) implies (1) let A

r
s

//

B be apair of A-morphisms with Gr = Gs. Consider the 2-sources S = (B, (idB , idB )) andT = (A, (r, s)) in A and the B-morphism h = Gr : GA → GB. Then GT = GS ◦ h.
Hence, by G-initiality of S, there exists an A-morphism h : A → B with T = S ◦ h, i.e.,with r = idB ◦ h = s.
10.60 PROPOSITION
If G : A → B is a functor such that each mono-source in A is G-initial, then thefollowing hold:(1) G is faithful,(2) G reflects products,(3) G reflects isomorphisms.
Proof:(1). follows from Proposition 10.59, since in A each 2-source (A, (idA , idA )) is a monosource.
(2). If P is a source in A such that GP is a product in B, then GP is a mono-sourcein B. Hence, by (1), P is a mono-source in A and, consequently, is G-initial. This,
together with faithfulness of G, immediately implies that P is a product in A.
(3). follows from (2), since isomorphisms are 1-products.



10.61 REMARKThe property that all mono-sources be initial, is not unfamiliar. As we will see in §23,
it is typical for “algebraic” categories.

9th July 2006

Sec. 10]Sources and sinks

179

SINKS
The concept dual to that of source is called sink. Whereas the concepts of sources
and sinks are dual to each other, frequently sources occur more naturally than sinks(cf. §15 and §17). However, there are cases where the opposite is true. Cf. in particularDefinition 10.69 and Proposition 10.71 below.
10.62 DEFINITIONfi
A sink is a pair ((fi )i∈I , A) [sometimes denoted by (fi , A)I or (Ai −−→ A)I ] consistingof an object A (the codomain of the sink) and a family of morphisms fi : Ai → A indexed by some class I. The family (Ai )i∈I is called the domain of the sink. Compositionof sinks is defined in the (obvious) way dual to that of composition of sources.
10.63 TABLE OF DUAL CONCEPTS
The following table provides the names for the concepts dual to those investigated inthis section:
Concept

Dual Concept

sourcemono-source
extremal mono-source
extremal coseparator
initial source
G-initial sourcesink
epi-sink
extremal epi-sinkextremal separator
final sinkG-final sinkQ
product ( Ai , πj )j∈I
projection πjAChfi i Q

/

Ai

CC
CC
πj
C
fj CC! `Ai )j∈I

injection µj
A oaC[fi ] `

A

O i
CC
CC
µj
C
fj CC

Aj

Aj

power AI

copower I A

hf,giC −−−→ A × B
Q

9th July 2006

coproduct (µj ,fi ; f × g[f,g]

C ←−−− A + B`

fi ; f + g

180

Sources and Sinks[Chap. III

10.64 EXAMPLES OF EPI-SINKS
fi(1) In Set a sink (AS−→ A)I is an epi-sink if and only if it is jointly surjective, i.e., ifi −and only if A = i∈I fi [Ai ].
In every construct, all jointly surjective sinks are epi-sinks. The converse implication
holds, e.g., in Vec, Pos, Top, and Σ-Seq.
(2) In Sgr we have seen that there are epimorphisms that are not surjective [7.40(5)].
Thus, there are epi-sinks that are not jointly surjective.
(3) A category A is thin if and only if every sink in A is an epi-sink.
10.65 EXAMPLES OF EXTREMAL EPI-SINKS(1) Every epi-sink (= jointly surjective sink) is an extremal epi-sink in Set, Vec, andAb.
fi(2) In Top an epi-sink (Ai −−→ A) is extremal if and only if A carries the final topology
with respect to (fi )i∈I .
fi(3) In Pos an epi-sink (Ai −−→ A)I is extremal if and only if the ordering of A is thetransitive closure of the relation consisting of all pairs (fi (x), fi (y)) with i ∈ I andx ≤ y in Ai .
fi(4) In Σ-Seq an epi-sink (Ai −−→ A)I is extremal if and only if each final state of A has
the form fi (q) for some i ∈ I and some final state q of Ai .
10.66 EXAMPLES OF EXTREMAL SEPARATORS(1) Every separator is extremal in Set, Vec, and Ab.
(2) In Pos the separators are precisely the nonempty posets, whereas the extremal
separators are precisely the nondiscrete posets.
(3) Top has no extremal separator.
10.67 EXAMPLES OF COPRODUCTS
Many familiar categories have coproducts. However, as opposed to the situation for
products, coproducts in familiar constructs often fail to be concrete.
(1) If (Ai )I is a pairwise-disjoint family of sets, indexed by a set I, then the sink ofSµj
inclusion maps (Aj −−→ i∈I Ai )j∈I is a coproduct in Set. If (Ai )I is an arbitrary set-indexed family of sets, then it can be “made disjoint” by pairing eachmember of S
Ai with the index i, i.e., by working with Ai × {i}, rather than Ai .
The union i∈I (Ai × {i}) is called the disjoint union of the family (Ai )I and isUUµjdenoted by i∈I Ai . The sink of natural injections (Aj −−→ i∈I Ai )j∈I [whereµj (a) = (a, j)] is a coproduct55 in Set of the the family (Ai )I . [Indeed, given a sinkUfi
(Aj −−→ B)j∈I in Set, the unique function f :i∈I Ai → B satisfying fi = f ◦ µifor all i is defined by: f (a, i) = fi (a).]55

For a disjoint family of sets, the union and the disjoint union (together with associated injections)each form a coproduct. This is not a contradiction since coproducts are determined only up toisomorphism. Compare this with the corresponding result for products: Proposition 10.22.

9th July 2006

Sec. 10]Sources and sinks

181(2) In the construct Top coproducts are called “topological sums” and can be constructed as for sets by supplying the disjoint union with the final topology. Thus,
Top has concrete coproducts.
(3) In the construct Pos coproducts are called “cardinal sums” and can be constructed
as for sets by supplying the disjoint union of the underlying sets with the orderthat agrees on each Ai × {i} with the order on Ai , and where members of distinct
summands are incomparable. Thus, Pos has concrete coproducts.
L(4) Vec has nonconcrete coproducts called direct sums. The direct Q
sum i∈I Ai of afamily (Ai )I of vector spaces is the subspace of the direct product i∈I Ai consistingof all elements (ai )i∈I with finite
L carrier (i.e., { i ∈ I | ai 6= 0 } is finite), together
with the injections µj : Aj →Ai given by:µj (a) = (ai )i∈I(a, if i = j
with ai =0, if i 6= j.

The description of coproducts in Ab is analogous.
(5) HComp has nonconcrete coproducts, namely the Čech-Stone compactifications of
the topological sums.
(6) Grp has nonconcrete coproducts, called “free products”.
(7) The constructs Alg(Ω) all have coproducts. They are concrete if and only if all theoperations are unary.
(8) In a poset A, considered as a category, coproducts are joins. Thus A has coproducts
if and only if it is a complete lattice.
id(9) In every category every 1-indexed family (A) has a coproduct A −−→ A; an empty
sink with codomain A is a coproduct if and only if A is an initial object.
10.68 EXAMPLES OF FINAL SINKS(1) An empty sink (∅, A) in a concrete category is final if and only if A is discrete.
f(2) A singleton sink A −→ B in a concrete category is final if and only if f is a final
morphism.
fi(3) A sink ((Xi , τi ) −−→ (X, τ ))I in Top is final if and only if τ is the final topology withrespect to the maps (fi )I , i.e., τ = { U ⊆ X | for all i ∈ I, fi−1 [U ] ∈ τi }.
fi(4) A sink ((Xi , ≤i ) −−→ (X, ≤))I in Pos
S is final if and only if ρ is the transitive closureof the relation { (x, x) | x ∈ X } ∪ I { (fi (x), fi (y)) | x ≤i y }.
fi(5) A sink ((Xi , αi ) −−→ (X, α))I in Spa(T ) is final if and only if α =9th July 2006

S

I T fi [αi ].

182

Sources and Sinks[Chap. III

10.69 DEFINITION
A full concrete subcategory A of a concrete category B is said to be finally dense infi

B provided that for every B-object B there is a final sink (Ai −−→ B)I in B with Ai in
A for all i ∈ I.
Dual Notion: initially dense.
10.70 EXAMPLES
A. Final density in constructs(1) In Met the full subcategory consisting of all two-element metric spaces is finallydense.
(2) In Pos the full subcategory whose only object is a 2-chain is finally dense.

R(3) In Vec the full subcategory whose only object is 2 is finally dense. In general, ina construct of algebras with operations of arity at most n, any free algebra on n
generators, considered as a full subcategory with one object, is finally dense.
(4) Top does not have any small, finally dense subcategory.
(5) Σ-Seq does not have any finite, finally dense subcategory. [Consider an automaton
whose transitions form a cycle larger than the number of objects of the given finite
subcategory.]
B. Final density in concrete categories(6) For a poset B, considered as a concrete category over 1, a subset A is finally dense
if and only if it is join-dense (i.e., each b ∈ B is a join of a subset of A).
(7) For a category A considered as concrete over itself, the empty subcategory is finallydense.
10.71 PROPOSITION
If A is a finally dense full concrete subcategory of a concrete category B, then every
initial source in A is initial in B.
fi

Proof: Let A = (A −−→ Ai )I be an initial source in A, let B be a B-object, and
fi ◦f

f

|B| −→ |A| be an X-morphism, such that each |B| −−−→ |Ai | is a B-morphism. To show
gjfthat |B| −→ |A| is a B-morphism, let B = (Cj −−→ B)J be a final sink in B with each
Cj belonging to A. Then each
gj

f

fi

|Cj | −−→ |B| −→ |A| −−→ |Ai |is an A-morphism. Since A is initial in A and each Cj belongs to A, each
gj

f

f

|Cj | −−→ |B| −→ |A| is an A-morphism. Since B is final, |B| −→ |A| is a B-morphism.

9th July 2006

Sec. 10]Sources and sinks

183

10.72 REMARK
A full concrete embedding E : A → B is called finally dense if its image is a finallydense subcategory of B. The above proposition states, more succinctly, that finallydense embeddings preserve initiality.

Suggestions for Further Reading
Mac Lane, S. Duality for groups. Bull. Amer. Math. Soc. 56 (1950): 485–516.
Taylor, J. C. Weak families of maps. Canad. Math. Bull. 8 (1965): 771–781.
Pumplün, D. Initial morphisms and monomorphisms. Manuscr. Math. 32 (1980): 309–
333.

EXERCISES
10A. A Characterization of (Extremal) Mono-Sourcesf

Let A be a category such that for any pair Ag//

B of morphisms there exists a co-equalizer in A. Show that(a) A source S in A is a mono-source provided that whenever S = S ◦ e for some regularepimorphism e, it follows that e is an isomorphism.
(b) A source S in A is an extremal mono-source provided that whenever S = S ◦ e forsome epimorphism e, it follows that e is an isomorphism.
10B. A Characterization of Extremal (Co)Separators
Show that an object S of a category A ishom(S,−)(a) an extremal separator if and only if the functor A −−−−−−−→ Set is faithful andreflects isomorphisms, andhom(−,S)(b) an extremal coseparator if and only if the functor Aop −−−−−−−→ Set is faithful andreflects isomorphisms.
10C. Extremal (Co)Separators in HComp, BooSpa, and Tych
Show that(a) In HComp every nonempty space is an extremal separator, and [0, 1] is an extremal
coseparator.
(b) In BooSpa every nonempty space is an extremal separator, and every space withat least two points is an extremal coseparator.
* (c) Tych has neither an extremal separator nor an extremal coseparator.

9th July 2006

184

Sources and Sinks[Chap. III

10D. Products of Special Morphisms
Show that(a) Products of regular monomorphisms are extremal monomorphisms.
(b) In a category with products, products of regular monomorphisms are regular monomorphisms.
(c) Products of regular monomorphisms may fail to be regular.
(d) Products of extremal monomorphisms may fail to be extremal.
(e) Products of epimorphisms may fail to be epimorphisms.
10E. Hom-Functors Preserve and “Collectively Detect” Products
Show that a source S is a product in A if and only if for each A-object A the sourcehom(A, −)(S) is a product in Set.
10F. Concrete Products and Hom-Functors
Let A be a free object over a nonempty set in a construct A. Prove that a source S in
A is a concrete product if and only if S is initial and hom(A, −)(S) is a product in Set.
10G. Products in Ab
Let (G2 , (π1 , π2 )) be the cartesian product of an abelian group G with itself (where the+group operation is given by G2 −−→ G). Show that each of (G2 , (π1 , π2 )), (G2 , (π1 , +))and (G2 , (π2 , +)) is a product of the pair (G, G) in Ab.
10H. Products with Terminal Objectst

In A let T be a terminal object, A be an arbitrary object, and A −−A→ T be the uniquemorphism from A to T . Show that (A, (idA , tA )) is a product of the pair (A, T ) in A.
10I. No (Co)Products in Fields or in a GroupShow that neither Field nor a nontrivial group (nor a nontrivial finite monoid), considered as a category, has finite products or finite coproducts.
10J. Products for Banach Spaces
Show that(a) Banb has finite products, but Banb does not have products.
(b) Ban has products, but (Ban, U ) does not have concrete products.
(c) (Ban, O) has concrete products.
10K. Comparison of Powers
Let I be a subset of K and let AI and AK be powers of A in A. Show that:(a) If I 6= ∅, then AI is a retract of AK.
(b) If I = ∅, then AI need not even be a subobject of AK. [Hint: Consider A = ∅ inSet.]

9th July 2006

Sec. 10]Sources and sinks

185* 10L. A Characterization of Concretizable Categories Over Set
Show that a category that has equalizers and finite products is concretizable over Set if
and only if it is regular wellpowered.
10M. Composites of Regular Monomorphisms With SectionsShow that in a category that has finite products the composite s ◦ m of a regular monomorphism m and a section s is a regular monomorphism. [Hint: If r ◦ s = id and m isan equalizer of f and g, then s ◦ m is an equalizer of hf ◦ r, idi and hg ◦ r, s ◦ ri.] Contrast
this with Exercise 7S.
10N. Dualities and Representability
Let (A, U ) and (B, V ) be concrete categories that are dually equivalent; i.e., there arecontravariant functors G : A → B and F : B → A such that F ◦G ∼= 1A and G◦F ∼= 1B .
Suppose that the A-object A represents U and the B-object B represents V , and letÃ = F (B) and B̃ = G(A). Prove that(a) U (Ã) ∼= V (B̃).
(b) V ◦ G ∼= homA (−, Ã) and U ◦ F ∼= homB (−, B̃).
If, moreover, B has products, then show that(c) for each A-object X there exists a monomorphism mX : G(X) → B̃ U (X) such that
V (mX ) is the embedding of V (G(X)) ∼= homA (X, Ã) into∼
homSet (U (X), U (Ã)) = homSet (U (X), V (B̃)) = (V (B̃))U (X) ∼= V (B̃ U (X) ).
10O. Composites of F -Initial Sources
Let F : A → B be a functor and let T = (Si ) ◦ S be a composite of sources in A. Show
that(a) If S and Si are F -initial, then so is T .
(b) If T is F -initial and all Si are mono-sources, then S is F -initial.
(c) If T is F -initial, then S need not be F -initial. [Hint: Consider the unique functor
F from
1O >NNN
1O >NNN0>> NNN>> NNN>> NNN
NN'

/
@ @2
pp7 3
p
p
pp
ppp
p
p
ppp

to

0

>> NNN>> NNN>> NNN
NN'

/
2
@
pp7 3
p
p
pp
ppp
p
p
ppp

which leaves the four objects fixed. Then 1 → 2 is not F -initial, but 1 → 2 → 3 is.]10P. Universally Initial Morphisms
Let f be a morphism in a category A. Show that
* (a) f is an isomorphism if and only if f is F -initial with respect to every functor Fthat has domain A.

9th July 2006

186

Sources and Sinks[Chap. IIIF* (b) If f is a section, then f is F -initial with respect to every faithful functor A −−→ B.
Does the converse hold?
(c) f is initial with respect to every full and faithful functor with domain A.
10Q. A Characterization of Products
Show that a source in A is a product if and only if it is initial with respect to everyFfunctor A −−→ B that preserves mono-sources.
10R. Copowers Versus Free Objects in Constructs
Let (A, U ) be a construct such that U is representable by an object A. Show that forany set I and any A-object B the following conditions are equivalent:(1) B is a free object over I,(2) B is an Ith copower IA of A.
10S. Copowers in CBoo and CLatShow that neither CBoo nor CLat has small copowers. [Hint: Use 10R, 8.23(1)(b),8.23(7), and 8F resp. 8G.]
10T. Mono-Sources in Σ-Seq
Prove that mono-sources in {σ}-Seq are precisely the point-separating sources — although the forgetful functor is not representable (cf. Corollary 10.8).
10U. Coproducts of Functors
Given functors Fi : A → B, where B is a category with coproducts, then F =denotes the functor of “pointwise” coproducts, i.e.,f

F (A −→ A0 ) =

a

qFi f

Fi A −−−−→

a

`

Fi

F i A0 .

(a) Verify that F is a coproduct of the family (Fi ) in the functor quasicategory [A, B](6.15).
(b) Verify that for each type Ω there exists a coproduct T of functors S n [3.20(10)] suchthat Alg(Ω) is concretely isomorphic to Alg(T ).
10V. The Diagonal-Morphism ∆A∆A = hidA , idA i : A → A × A is called the diagonal morphism of A. Consider theππ2
product A ←−1− A × A −−→
A and show that (A, ∆A ) is an equalizer of (π1 , π2 ).

9th July 2006

Sec. 10]Sources and sinks

187

10W. Products and Constant Morphisms
Consider the diagram
AO

f

/B
O
p1

π1

A×A


f ×f

/B×B

π2

A



f

p2

/B

Show that the following are equivalent:(1) f is constant,(2) f ◦ π1 = f ◦ π2 ,(3) p1 ◦ (f × f ) = p2 ◦ (f × f ),(4) f × f factors through ∆B , i.e., for some g:
f ×f

g

∆

B
A × A −−−→ B × B = A × A −→ B −−−→ B × B.

9th July 2006

11 Limits and colimits
Many basic constructions in mathematics can be described as limits (or dually as colimits). Such constructions associate with a given diagram a distinguished object together with morphisms connecting this object with the objects of the diagram; i.e., acertain source called the limit of the diagram. This limit can be characterized, up toisomorphism, by a purely categorical property. We have encountered two special casesof limits already: equalizers and products. The main new type of limit introduced below
is that of pullback. Unlike products, the usefulness of pullbacks was generally recognized
only after the emergence of category theory.

LIMITS11.1 DEFINITION(1) A diagram in a category A is a functor D : I → A with codomain A. The domain,I, is called the scheme of the diagram.56(2) A diagram with a small (or finite) scheme is said to be small (or finite).
11.2 EXAMPLES(1) A diagram in A with discrete scheme is essentially just a family of A-objects.
(2) A diagram in A with scheme • ⇒ • is essentially just a pair of A-morphisms with
common domain and common codomain.
11.3 DEFINITION
Let D : I → A be a diagram.
fi(1) An A-source (A −−→ Di )i∈Ob(I) is said to be natural for D provided that for eachdI-morphism i −→ j, the triangle
AB

BB f
BB j
BB
B

/ Dj
Di

fi

Dd

commutes.
56

Although there is technically no difference between a diagram and a functor, or between a scheme anda category, we use the alternate terminology when treating limits and colimits, for reasons of both its
historical development and to indicate a slight change of perspective. For example, we often denotethe image of an object i under a diagram D by Di rather than D(i). This produces notation that ismore consistent with that introduced earlier for sources and sinks.

188

Sec. 11]

Limits and colimits

189

`

i
(2) A limit of D is a natural source (L −−→
Di )i∈Ob(I) for D with the (universal )fiproperty that each natural source (A −−→ Di )i∈Ob(I) for D uniquely factors throughit; i.e., for every such source there exists a unique morphism f : A → L with fi =`i ◦ f for each i ∈ Ob(I).
11.4 EXAMPLES(1) For a diagram D : I → A with a discrete scheme, every source with codomain(Di )i∈Ob(I) is natural. A source is a limit of D if and only if it is a product of thefamily (Di )i∈Ob(I) . Expressed briefly: products are limits of diagrams with discrete
schemes. In particular, an object, considered as an empty source, is a limit of theempty diagram (i.e., the one with empty scheme) if and only if it is a terminal object.
f(2) For a pair of A-morphisms A
e

//

g

B , considered as a diagram D with scheme • ⇒ •,ha source (A ←− C −→ B) is natural provided that g ◦ e = h = f ◦ e. Observe thatein this case h is determined by e. Hence, C −→ A is an equalizer of Aefg// B if andf ◦eonly if the source (A ←− C −−−→ B) is a limit of D. Thus we may say (imprecisely)that equalizers are limits of diagrams with scheme • ⇒ •. If, in the above scheme,
the two arrows are replaced by an arbitrary set of arrows, then limits of diagramswith such schemes are called multiple equalizers.
(3) If I is a down-directed57 poset (considered as a category), then limits of diagramswith scheme I are called projective (or inverse) limits. If, e.g., I = op is theposet of all non-negative integers with the opposite of the usual ordering, a diagram
D : I → A with this scheme is essentially a sequence

N

d

d

d

2
1
0
· · · −−→
D2 −−→
D1 −−→D0dof A-morphisms (where D(n + 1 → n) = Dn+1 −−n→ Dn , D(n + 2 → n) = dn ◦ dn+1 ,fnetc.). A natural source for D is a source (A −−→ Dn )n∈N with fn = dn ◦ fn+1
for each n. In Set a projective limit of a diagram D with scheme op is a source`(L −−n→ Dn )n∈N , where L is the set of all sequences (xn )n∈N with xn ∈ Dn anddn (xn+1 ) = xn for
Q each n ∈ ; and where each `m is a restriction of the mth
projection πm :n∈N Dn → Dm.

N

N

`

(4) If D : A → A is the identity functor, then a source L = (L −−A→ A)A∈ObA is alimit of D if and only if L is an initial object of A. The sufficiency is obvious.
f

For the necessity let L be a limit of D and let L −→ A be a morphism. By thenaturality of L for D we obtain f ◦ `L = `A . Application of this to f = `A yields`A ◦ `L = `A = `A ◦ idL for each object A. Hence, by the uniqueness requirement
57

Down-directed means that every pair of elements has a lower bound. The dual notion is updirected.

9th July 2006

190

Sources and Sinks[Chap. III

in the definition of limit, `L = idL . Consequently, f = f ◦ idL = f ◦ `L = `A . Thus
hom(L, A) = {`A } for each object A.
(5) If the category I has an initial object i0 with hom(i0 , i) = {mi } for each i ∈ Ob(I),Dmithen every diagram D : I → A with scheme I has a limit: (Di0 −−−−→
Di )i∈Ob(I) .

11.5 REMARKIf D : I → A is a diagram, then natural sources for D can be regarded as naturaltransformations from constant functors C : I → A to the functor D.
11.6 PROPOSITION
Every limit is an extremal mono-source.

A 10.2111.7 PROPOSITION`i
Limits are essentially unique; i.e., if L = (L −−→
Di )i∈Ob(I) is a limit of D : I → A,then the following hold:khi(1) for each limit K = (K −−→
Di )i∈Ob(I) of D, there exist an isomorphism K −→ Lwith K = L ◦ h,h(2) for each isomorphism A −→ L, the source L ◦ h is a limit of D.

A 10.22

PULLBACKS
11.8 DEFINITION(1) A square
P

f

/B

(∗)

g

g



A



/C

f

is called a pullback square provided that it commutes and that for any commuting
square of the form
fˆ

P̂
ĝ

/B
g



A



/C

f

k

there exists a unique morphism P̂ −→ P for which the following diagram commutes
P̂ ??

ˆ

f
??
??
??k


ĝ

P

 

f

g

g

A

"
/B


f

/C

9th July 2006

Sec. 11]

Limits and colimits
g

191
f

(2) If (∗) is a pullback square, then the 2-source A ←− P −→ B is called a pullback offg

the 2-sink A −→ C ←− B and f is called a pullback of f along g.
11.9 REMARKS(1) A square (∗) is a pullback square if and only if the 3-source (P, (g, f ◦ g, f )) is afglimit of the 2-sink A −→ C ←− B, considered as a diagram in A with scheme• −→ • ←− •. Shortly (and imprecisely) pullbacks are limits of diagrams with
scheme • −→ • ←− •.
(2) If the square (∗) is a pullback, then (P, (f , g)) is a extremal mono-source. This
follows from (1) and Proposition 11.6.
11.10 PROPOSITION
Let
•

/•

/•




/•


/•

•
commute in A. Then

(1) if the squares are pullback squares, then so is the outer rectangle; i.e., pullbacks canbe composed by “pasting” them together,(2) if the outer rectangle and right-hand square are pullback squares, then so is theleft-hand square.


RELATIONSHIP OF PULLBACKS TO OTHER LIMITS
11.11 PROPOSITION (Canonical Construction of Pullbacks)fg

Let A −→ C ←− B be a pair of morphisms with common codomain.
ππe

If A ←−A− A × B −−B→ B is a product of A and B, and E −→ A × B is an equalizer of
A×B

f ◦πA
g◦πB

//

C , then
E
πB ◦e



B
is a pullback square.

9th July 2006

πA ◦e

/A


g

f

/C

192

Sources and Sinks

Proof: Let
r

/A

s

P




B

g

[Chap. III

f

/C

hs,ri

be a commuting square. Then P −−−→ A × B is a morphism for which f ◦ πA ◦ hs, ri =kg ◦ πB ◦ hs, ri. Since e is an equalizer of (f ◦ πA , g ◦ πB ), there is a morphism P −→ E forwhich hs, ri = e ◦ k. Thus πA ◦ e ◦ k = s and πB ◦ e ◦ k = r. That k is the only morphism
π ◦e
π ◦e
with this property follows from the fact that A ←−A−−− E −−B−−→ B is, as a compositionof mono-sources, a mono-source.

11.12 EXAMPLES
Application of the above construction provides concrete descriptions of pullbacks in manycases:fg(1) Let A −→ C ←− B be a pair of maps in Set, let P = {(a, b) ∈ A × B | f (a) = g(b)},gfand let P −→ A and P −→ B be the domain restrictions of the projections fromA × B. Then
g

P
f



/A


B

g

f

/C

is a pullback square.
(2) In any of the constructs Pos, Top, or Vec pullbacks can be constructed as for sets
by supplying P in each case with the initial structure with respect to its inclusion
into the product A × B; i.e.,(a) for Pos, by the pointwise order,(b) for Top, by the subspace topology,(c) for Vec, by defining operations coordinatewise.
11.13 PROPOSITION
If T is a terminal object, then the following are equivalent:
(1)
P
pB



B

pA

/A

/T

is a pullback square,(2) (P, (pA , pB )) is a product of A and B.



9th July 2006

Sec. 11]

Limits and colimits

193

11.14 PROPOSITION (Construction of Equalizers via Products and Pullbacks)
f

If A

g

//

B are morphisms, (A × B, πA , πB ) is a product of (A, B), and

P
p2

p1

/A




hidA ,f i

A hid ,gi/ A × B
A

is a pullback square, then p1 = p2 is an equalizer of f and g.
Proof: Since hidA , f i ◦ p1 = hidA , gi ◦ p2 , we have
p1 = πA ◦ hidA , f i ◦ p1 = πA ◦ hidA , gi ◦ p2 = p2 .
Similarly, f ◦ p1 = πB ◦ hidA , f i ◦ p1 = πB ◦ hidA , gi ◦ p2 = g ◦ p2 = g ◦ p1 .
kSuppose that K −→ A is a morphism such that f ◦ k = g ◦ k. ThenπB ◦ hidA , f i ◦ k = πB ◦ hidA , gi ◦ k,andπA ◦ hidA , f i ◦ k = πA ◦ hidA , gi ◦ k.
Since the product is a mono-source, we obtain hidA , f i ◦ k = hidA , gi ◦ k, so that since
the square is a pullback square, there is a unique h : K → P such that k = p1 ◦ h. 

PULLBACKS RELATED TO SPECIAL MORPHISMS
11.15 LEMMA
Suppose that the diagram
/•
~
~
~
1
~~
~
~
? • @@
~
@@h
~~
@@
~~
@ 
~
 ~
/•
•

•

commutes.
(1) If the outer square is a pullback square, then so is 1 .
(2) If 1 is a pullback square and h is a monomorphism, then the outer square is apullback square.


9th July 2006

194

Sources and Sinks[Chap. III

11.16 PROPOSITIONf
A −→ B is a monomorphism if and only if
A
idA

idA





A

/A

f

f

/B

is a pullback square.



11.17 DEFINITION
A class M of morphisms in a category is called pullback stable (or closed under theformation of pullbacks) provided that for each pullback square
P

f

/B

(∗)

g

g





A

f

/C

with f ∈ M , it follows that f ∈ M .
11.18 PROPOSITION
Monomorphisms, regular monomorphisms, and retractions are pullback stable.
Proof: Let f be a monomorphism in the above pullback square (∗).
(1). If h, k : Q → P are morphisms such that f ◦ h = f ◦ k, thenf ◦ (g ◦ h) = g ◦ (f ◦ h) = g ◦ (f ◦ k) = f ◦ (g ◦ k),so that g ◦ h = g ◦ k. Since pullbacks are mono-sources, this implies that h = k.
(2). Suppose that f is an equalizer of p and q. Then (p ◦ g) ◦ f = (q ◦ g) ◦ f . To showthat f is an equalizer of p ◦ g and q ◦ g, let t : Q → B be a morphism such that(p ◦ g) ◦ t = (q ◦ g) ◦ t. Then, by the definition of equalizer, there is some u : Q → A
with f ◦ u = g ◦ t. Thus, by the definition of pullback, there is some h : Q → P suchthat t = f ◦ h. Uniqueness of h follows from the fact that f is a monomorphism.
s(3). If f is a retraction, then there is some C −→ A with f ◦ s = idC . Hence f ◦ (s ◦g) = g ◦ idB , so that by the definition of pullback there is some h : B → P with
idB = f ◦ h.
11.19 REMARKg
If (A, m) is a subobject of B, C −→ B is an arbitrary morphism and
P
g

m

/C
g



A



m

/B

9th July 2006

Sec. 11]

Limits and colimits

195

is a pullback square, then by the above proposition (P, m) is a subobject of C. It is called
an inverse image of (A, m) under g, since it corresponds to the concept of “inverseg
image” in familiar constructs. In particular, in Set, if C −→ B is a function and A isa subset of B, considered as a subobject of B via its inclusion function, then g −1 [A],considered as a subobject of C via its inclusion function, is an inverse image of A underg.

CONGRUENCES
If f : A → B is a group homomorphism, then the group-theoretic congruence relation
determined by f is the subset C of A × A consisting of all pairs (a, b) with f (a) = f (b).
Obviously, C can be regarded as a subgroup of A × A, and if m : C ,→ A × A is theinclusion and A × A

π1
π2

//

A are the projections, then (according to Proposition 11.11)

the square
C
π2 ◦m

π1 ◦m



A

/A


f

f

/B

is a pullback square.
This motivates our next definition.
11.20 DEFINITION(1) If
•

p

/•

q





•

f

f

/•

is a pullback square, then the pair (p, q) is called a congruence relation of f .
(2) A pair (p, q) of morphisms is called a congruence relation provided that there
exists some morphism f such that (p, q) is a congruence relation of f .
11.21 LEMMAf
Let (p, q) be a congruence relation of A −→ B. Thenm◦fm(1) (p, q) is a congruence relation of A −−−→ C, for each monomorphism B −−→ C,(2) if f = g ◦ h and h ◦ p = h ◦ q, then (p, q) is a congruence relation of h.
Proof: Apply Lemma 11.15.

9th July 2006



196

Sources and Sinks[Chap. III

11.22 PROPOSITION(1) If (p, q) is a congruence relation and c is a coequalizer of p and q, then (p, q) is acongruence relation of c.
(2) If c is a regular epimorphism and (p, q) is a congruence relation of c, then c is acoequalizer of p and q.
Proof:(1). Immediate from Lemma 11.21(2).
(2). Let c be a coequalizer of r and s. By the pullback property there exists a morphism
k with p ◦ k = r and q ◦ k = s. To show that c is a coequalizer of p and q, let f bea morphism with f ◦ p = f ◦ q. Then f ◦ r = f ◦ s. Hence, since c is a coequalizerof r and s, there exists a unique morphism f with f = f ◦ c.


INTERSECTIONS
11.23 DEFINITION
Let A be a family of subobjects (Ai , mi ) of an object B, indexed by a class I. A subobject(A, m) of B is called an intersection of A provided that the following two conditions
are satisfied:(1) m factors through each mi ; i.e., for each i there exists an fi with m = mi ◦ fi ,f(2) if a morphism C −→ B factors through each mi , then it factors through m.58
11.24 EXAMPLES(1) Let B be a set and let (Ai )I be a family
T of subsets of B, considered (via inclusions)as subobjects of B in Set. Then i∈I Ai , considered as a subobject of B, is anintersection in the sense of Definition 11.23. Similarly for Vec, Top, Σ-Seq, etc.
(2) In a poset, considered as a category, intersections are meets.
(3) For two subobjects (A1 , m1 ) and (A2 , m2 ) of B, an intersection is the diagonal morm1
mphism of a pullback of A1 −−→
B ←−2− A2 .
11.25 REMARKS(1) Intersections can be regarded as limits (cf. Exercises 11F and 11L).
(2) Any two intersections of a family of subobjects of B are isomorphic subobjects ofB.
h(3) Let A −→ B be a morphism. Then (A, h) is an intersection of the empty family of
subobjects of B if and only if h is an isomorphism.
58

Observe that any intersection (A, m) of a family of subobjects (Ai , mi ) of an object B, indexed bya class I, depends only on the class {(Ai , mi ) | i ∈ I} of subobjects of B and not on the indexing
function. Hence we sometimes also speak of an intersection of a class of subobjects of B.

9th July 2006

Sec. 11]

Limits and colimits

197

11.26 DEFINITION
A class M of monomorphisms is said to be closed under the formation of intersections provided that whenever (A, m) is an intersection of a family of subobjects (Ai , mi )of B and each mi belongs to M , then m belongs to M .

COLIMITSThe notion of colimit is dual to that of limit; namely,
11.27 DEFINITION
Let D : I → A be a diagram.
fi(1) An A-sink (Di −−→ A)i∈Ob(I) is said to be natural for D provided that for eachdI-morphism i −→ j, the triangle
Dd /
Dj
BB
BB
fj
B
fi BB


Di B

A

commutes.
ci(2) A colimit of D is a natural sink (Di −−→K)i∈Ob(I) for D with the (universal )property that each natural sink for D uniquely factors through it.

11.28 EXAMPLES(1) Coproducts are precisely the colimits of diagrams with discrete schemes. In particular, initial objects are the colimits of empty diagrams.
f(2) For a pair of morphisms Acg// B , considered as a diagram D with scheme • ⇒ •,cc◦f

B −→ C is a coequalizer of f and g if and only if (B −→ C ←−−− A) is a colimit ofD.
(3) The dual of the concept of an intersection of a family of subobjects of B is theconcept of a cointersection of a family of quotient objects of B. In particular,(a) If B is a set and if (ρi )I is a family of equivalence relations on B considered (as in7.86) as quotientSobjects of B in Set, then the smallest equivalence relation ρ on
B generated by i∈I ρi , considered as a quotient object of B, is a cointersectionof the family in Set.
(b) In a poset, considered as a category, cointersections are joins.

9th July 2006

198

Sources and Sinks[Chap. III(4) The dual of the concept of projective limit is that of directed colimit (also called
inductive limit). That is, directed colimits are colimits of diagrams whose schemes
are up-directed posets.
Every diagram D : I → Set, whose scheme is an up-directed poset, has a canonicalS
dijcolimit. Denote D(i → j) by Di −−→ Dj , let C = i∈I (Di × {i}) be the disjointµjunion of the family (Di )i∈I , and let (Dj −−→ C)j∈Ob(I) be the canonical coproductof the family (Di )i∈Ob(I) in Set, as constructed in 10.67(1). Define an equivalence
relation ∼ on C by: (x, i) ∼ (y, j) if and only if there exists k ≥ i, j with dik (x) =q
djk (y). If C −→ Q is the natural map from C onto the set Q = C/∼ of equivalenceq◦µiclasses under ∼, then (Di −−−→ Q)i∈Ob(I) is a colimit of D.
Observe that directed colimits in any construct of the form Alg(Ω) can be constructed as in Set: Let Ω = (nk )k∈K and let D : I → Alg(Ω) be a diagram fordij

an up-directed poset I. Denote D(i → j) by (Di , (ωki )k∈K ) −−→ (Dj , (ωkj )k∈K ) and
q◦µi

U ◦D

form (as above) the colimit (Di −−−→ Q)i∈Ob(I) of the diagram I −−−−→ Set, whereU

Alg(Ω) −−→ Set is the forgetful functor. Since I is up-directed, for each k ∈ Kand each (x1 , . . . , xnk ) ∈ Qnk there exists some i ∈ Ob(I) and (y1 , . . . , ynk ) ∈ Dink
with (q ◦ µi )nk (y1 , . . . , ynk ) = (x1 , . . . , xnk ). Define ωk (x1 , . . . , xnk ) =q ◦ µi ◦ ωki (y1 , . . . , ynk ). Then up-directedness of I implies that the so-defined mapω
Qnk −−k→ Q is the unique function that makes the following diagram commute foreach i ∈ Ob(I):(q◦µi )nk

Dink
ωki

/ Qk




Di

ωk

/Q

q◦µi
q◦µi

It follows that the sink ((Di , (ωki )k∈K ) −−−→ (Q, (ωk )k∈K ))i∈Ob(I) is a colimit of D.
(5) The dual of the concept of pullback is that of pushout, explicitly described below.

11.29 PROPOSITION
Colimits are essentially unique and each colimit is an extremal epi-sink.

D

11.30 DEFINITION(1) The square
C

f

/A

g



B



f

(∗)

g

/P

9th July 2006

Sec. 11]

Limits and colimits

199

is called a pushout square provided that it commutes and that for any commuting
square of the form
C

f

/A

g



B

fˆ

/



ĝ

P̂

k

there exists a unique morphism P −→ P̂ for which the diagram
C

f

g



B

/A
g

f


/P

fˆ

ĝ

?? k
??
??
??
 
+

P̂

commutes.
(2) If (∗) is a pushout square, the 2-sink ((g, f ), P ) is called a pushout of the 2-source(C, (f, g)), and f is called a pushout of f along g.

Pullback and Pushout

9th July 2006

200

Sources and Sinks[Chap. III

11.31 REMARK
Since pushouts are dual to pullbacks, there is a canonical construction of pushouts via
coequalizers and coproducts (cf. 11.11).
11.32 DEFINITION
A square that is simultaneously a pullback square and pushout square is called a pulation square.
11.33 PROPOSITION
Consider a commuting square
p

•

/•

q





•

c

(∗)
c

/•

(1) If (∗) is a pushout square, then c is a coequalizer of p and q.
(2) If (p, q) is a congruence relation of c, and c is a regular epimorphism, then (∗) is apulation square.
Proof:(1). Immediate.
(2). Let
•

p

/•

q





•

p̂

q̂

/•

be a commutative square. Since (∗) is a pullback square and c ◦ id = c ◦ id, there
exists a morphism k such that id = p ◦ k = q ◦ k. Hence p̂ ◦ q = q̂ ◦ p implies thatp̂ = p̂ ◦ q ◦ k = q̂ ◦ p ◦ k = q̂.
Hence p̂ ◦ p = p̂ ◦ q. Since by Proposition 11.22, c is a coequalizer of p and q, there
exists a unique morphism h with p̂ = h ◦ c; i.e., with p̂ = h ◦ c and q̂ = h ◦ c. Hence(∗) is a pushout square.


Suggestions for Further Reading
Maranda, J. M. Some remarks on limits in categories. Canad. Math. Bull. 5 (1962):
133–146.
Eckmann, B., and P. J. Hilton. Group-like structures in general categories II: Equalizers, limits, length. Math. Ann. 151 (1962): 150–186.

9th July 2006

Sec. 11]

Limits and colimits

201

Kelly, G. M. Monomorphisms, epimorphisms, and pullbacks. J. Austral. Math. Soc. 9(1969): 124–142.
Tholen, W. Amalgamations in categories. Algebra Univ. 14 (1982): 391–397.

EXERCISES
11A. Schemes With Initial Objects
Show that for categories A the following conditions are equivalent:(1) A has an initial object,idA(2) the diagram A −−−→ A has a limit,(3) each diagram with scheme A has a limit.
11B. Products as Projective Limits of Finite Products
Let (Ai )i∈J be a family of objects in A, indexed by an infinite set J. Let I be theset of all finite subsets of J, ordered by: M ≤ N ⇔ N ⊆ M . For each M ∈ I let
{i}
Q
Q
pi
id
A
−
−
−→ Ai = Ai −−→ Ai for))beaproduct
of
(A
)
in
A
(where( M Ai , (pM
i i∈M
i
{i} i
each i ∈ J). Show that(a) There exists a diagram D : I → A that associates with M ≤ N in I the unique
Q
f N Q
M
−−→ N Ai that satisfies pNmorphism M Ai −−M
i ◦ fM N = pi for each i ∈ N .
(b) If (P, (pN )N ∈I ) is a (projective) limit of D, then (P, (p{i} )i∈J ) is a product of (Ai )i∈I .
11C. Pullbacks as Products in Comma Categories
Show that a commuting square
P
pB

pA



B

/A


g

f

with f ◦ pA = g ◦ pB = p/C

is a pullback square in A if and only if ((P, p), (pA , pB )) is a product of f and g in thecomma category (A ↓ C).
11D. Products of Morphisms as Pullbacks
Show that each diagram of the form
A×B
idA ×f



A×C
is a pullback square.

9th July 2006

πB

/B


πC

f

/C

202

Sources and Sinks[Chap. III

11E. Kernels as Pullbacks
Show that in a category A with a zero-object 0, k is a kernel of f if and only if
•

/0




/•

k

•

f

is a pullback square.
11F. Intersections as Pullbacks
Let m1 and m2 be monomorphisms and let
A
m1

m2



A2

/ A1
m1



m2

/B

be a commuting square with m = m1 ◦m2 = m2 ◦m1 . Show that the square is a pullbacksquare if and only if (A, m) is an intersection of the family ((Ai , mi ))i∈{1,2} of subobjects
of B.
11G. Pushouts as Pullbacks
Show that in a group, considered as a category, the pushout squares are precisely the
pullback squares.
11H. Pullback Stability of Special Monomorphisms
Show that(a) in any category the class of strict monomorphisms is pullback stable,(b) in Set the class of sections is not pullback stable,(c) in DRail the class of extremal monomorphisms is not pullback stable.
11I. Pullback Stability of Epi-Sinks in Set
For each i ∈ I let
Bi
gi

ki



B

/ Ai


f

fi

/A
fi

be a pullback square in Set. Show that whenever (Ai −−→ A)I is an epi-sink, then so isgi(Bi −−→ B)I .
11J. Stable Epimorphisms
A morphism f is called a stable epimorphism provided every pullback of f is an
epimorphism. Show that

9th July 2006

Sec. 11]

Limits and colimits

203(a) Every retraction is a stable epimorphism.
(b) Every stable epimorphism is an epimorphism.
* (c) In Cat not all regular epimorphisms are stable.
* (d) In Haus and in Met: ExtrEpi ( StableEpi = SurjMorph ( Epi.
(e) Stable epimorphisms are pullback stable.
(f) If A has pullbacks of all 2-sinks, then in A stable epimorphisms are closed undercomposition.
11K. IntersectionsShow that(a) in Set an intersection of two sections need not be a section,(b) if A has products, then in A an intersection of a set-indexed family of regularsubobjects is a regular subobject.
[Hint: If (A, m) is an intersection of the family (Ai , mi )I , where each mi is an
hri i Qri////equalizer of B
Ci .]Ci , then m is an equalizer of B
si

hsi i

11L. Multiple Pullbacksffi

A pair (f, S), consisting of a morphism A −→ B and a source S = (A −−→ Ai )I , is calledgi
a multiple pullback of a sink (Ai −−→ B)I provided that(1) f = gi ◦ fi for each i ∈ I, and
f0

f0

i
Ai )I a source(2) for each pair (f 0 , S 0 ), with A0 −−→ B a morphism and S 0 = (A0 −−→g
00
for which f = gi ◦ fi for each i ∈ I, there exists a unique morphism A0 −→ A with
f 0 = f ◦ g and fi0 = fi ◦ g for each i ∈ I.

(a) Interpret multiple pullbacks as limits.
(b) Interpret pullbacks as multiple pullbacks of 2-sinks.
(c) Interpret intersections as multiple pullbacks. In particular, show that wheneverf

R = ((Ai , mi ))I is a family of subobjects of B and A −→ B is a morphism, thenthe following are equivalent:(1) (A, f ) is an intersection of R,(2) there exists a (unique) source S such that (f, S) is a multiple pullback ofmithe sink (Ai −−→B)i∈I .
(d) Show that each sink that consists of isomorphisms alone has a multiple pullback.

9th July 2006

204

Sources and Sinks[Chap. III

11M. Products as Multiple Pullbacks
Let A have a terminal object T , and let (Ai )I be a family of A-objects. Show that apisource P = (P −−→ Ai )I is a product of (Ai )I if and only if (P → T, P) is a multiple
pullback of the sink (Ai → T )I .
11N. A Characterization of Monomorphisms by Multiple PullbacksmShow that a morphism A −−→ B is a monomorphism if and only if, for each class I, themisink (Ai −−→B)I , defined by mi = m for each i ∈ I, has a multiple pullback.
11O. Directed Colimits in Ab
Show that an abelian group is torsion free if and only if it is a directed colimit in Ab of
free abelian groups.
11P. Pushout Stability of Monomorphisms and Mono-Sources in Set(a) Show that in Set the class of monomorphisms is pushout stable.
(b) For each i ∈ I let
A
mi

f



Ai

/B


gi

ni

/ Bi
m

i
Ai )I is a mono-source
be a pushout square in Set. Show that the fact that (A −−→nidoes not imply that the source (B −−→
Bi )I is a mono-source. [Contrast this with
11I.]

11Q. Epimorphisms and Pulation Squares(a) Let (p, q) be a congruence relation of e. Show that e is a regular epimorphism if andonly if the square
•

p

/•

q



•



e

e

/•

is a pulation square (cf. 11.32).
(b) Dualize part (a).
11R. A Characterization of Monomorphisms
Show that a morphism is a monomorphism if and only if it has a congruence relation of
the form (p, p).

9th July 2006

Sec. 11]

Limits and colimits

11S. A Construction of Equalizers
Show that if
•
p1



p2

/B


hid,idiA hf,gi / B × B
is a pullback square, then p1 is an equalizer of f and g.

9th July 2006

205

12 Completeness and cocompletenessIn this section we consider the existence of limits in a given category A. Only rarely
does every diagram in A have a limit. [Recall that any such A must be thin (cf. 10.32
and 10.33).] However, in many familiar categories A, every small diagram has a limit,
and for each object A in A every (possibly large) family of subobjects of A has anintersection. This leads to the following definitions:12.1 DEFINITION
A category is said to(1) have (finite) products provided that for each (finite) set-indexed family of objects
there exists a product (cf. Definition 10.29),(2) have equalizers provided that for each parallel pair of morphisms there exists anequalizer,(3) have pullbacks provided that for each 2-sink there exists an pullback,(4) have (finite) intersections provided that for each object A, and every (finite)family of subobjects of A, there exists an intersection.
Dual Notions: have (finite) coproducts, have coequalizers, and have (finite)
cointersections.
12.2 DEFINITION
A category A is said to be(1) finitely complete if for each finite diagram in A there exists a limit,(2) complete if for each small diagram in A there exists a limit,(3) strongly complete if it is complete and has intersections.
Dual Notions: finitely cocomplete, cocomplete, and strongly cocomplete categories.
12.3 THEOREM
For each category A the following conditions are equivalent:(1) A is complete,(2) A has products and equalizers,(3) A has products and finite intersections.

206

Sec. 12]Completeness and cocompleteness

207

Proof:(1) ⇒ (3) is immediate.
(3) ⇒ (2) follows from Proposition 11.14 and Example 11.24(3).
(2) ⇒ (1). Suppose that A has products and equalizers, and let D : I → A be asmall diagram. For each morphism t : i → j in I let d(t) = i and c(t) = j. Form the
Q
Q
πj
π̂s
products ( i∈Ob(I) Di −−→ Dj )j∈Ob(I) and ( t∈Mor(I) Dc(t)Q−−→ Dc(s) )s∈Mor(I) . For eacht ∈ Mor(I) there is a pair of morphisms πc(t) , Dt ◦ πd(t) :i∈Ob(I) Di → Dc(t) , and thesedefine a pair of morphisms
YY
Dc(t) .
Di −→hπc(t) i, hDt ◦ πd(t) i :i∈Ob(I)t∈Mor(I)Q
π ◦e
Let e : E → Di be an equalizer of this pair. We claim that (E −−i−→ Di )i∈Ob(I) is alimit of D in A. Indeed,s(a) It is a natural source for D because for each morphism i −→ j in I, we haveπj ◦ e = πc(s) ◦ e = π̂s ◦ hπc(t) i ◦ e = π̂s ◦ hDt ◦ πd(t) i ◦ e = Ds ◦ πd(s) ◦ e = Ds ◦ πi ◦ e.
Qfi(b) Suppose that (A −−→ Di ) is a natural source for D and let f = hfi i : A → Di .
For each morphism s of I, naturality of the source implies that fc(s) = Ds ◦ fd(s) .
Consequently,π̂s ◦ hπc(t) i ◦ f = πc(s) ◦ f = fc(s) = Ds ◦ fd(s) = Ds ◦ πd(s) ◦ f = π̂s ◦ hDt ◦ πd(t) i ◦ f.
Since the product is a mono-source, hπc(t) i ◦ f = hDt ◦ πd(t) i ◦ f . Thus there is af0

unique A −−→ E such that f = e ◦ f 0 . Thus fi = πi ◦ e ◦ f 0 for each i, and f 0 is clearlyseen to be unique with respect to the latter property.

12.4 THEOREM
For each category A the following conditions are equivalent:(1) A is finitely complete,(2) A has finite products and equalizers,(3) A has finite products and finite intersections,(4) A has pullbacks and a terminal object.
Proof: (1) ⇔ (2) ⇔ (3) can be proved as in Theorem 12.3.
(1) ⇒ (4) is immediate.
(4) ⇒ (3). Recall that products of pairs of objects can be formed via pullbacks ofmorphisms to a terminal object (11.13) and that intersections of pairs of subobjects areformed via pullbacks [11.24(3)].


9th July 2006

208

Sources and Sinks[Chap. III

12.5 PROPOSITION
Each complete and wellpowered category is strongly complete.
mi
Proof: Let A be complete and wellpowered. For each family (Ai −−→A)I of subobjects
miof some object A, there exists a subset J ⊆ I such that each subobject Ai −−→
A, i ∈ I,
mj
is isomorphic to some Aj −−→ A, j ∈ J. Obviously, any intersection (B, m) of the small
mjmisink (Aj −−→ A)J is an intersection of the original sink (Ai −−→A)I .


12.6 EXAMPLES(1) Each of the categories Set, Vec, Top, HComp, Pos, Grp, Cat, and Σ-Seq isstrongly complete and strongly cocomplete.
Observe however that the quasicategory CAT fails to have coequalizers.
(2) Met has equalizers, but not products. JPos has products, but not equalizers.
(3) Every non-trivial group, considered as a category, has pullbacks and pushouts, but
not equalizers or coequalizers, nor products or coproducts of pairs, nor terminal or
initial objects.
(4) A poset, considered as a category, is (co)complete if and only if it is a complete
lattice. Thus, for posets, completeness and cocompleteness coincide.
However, the partially ordered class Ord of all ordinal numbers is not complete (ithas no terminal object) even though it is cocomplete (but not strongly cocomplete).
(5) The categories CLat and CBoo (= complete boolean algebras and boolean homomorphisms) are strongly complete, but not cocomplete (cf. Exercise 12I).
(6) The category of finite sets is finitely complete and finitely cocomplete, but is neither
complete nor cocomplete.
(7) The category Field is neither finitely complete nor finitely cocomplete.

COCOMPLETENESS ALMOST IMPLIES COMPLETENESS
Although completeness and cocompleteness are not equivalent, the constructions of limits
and colimits are intimately related. In fact, as is shown below, under suitable “smallness
conditions”, completeness and cocompleteness are equivalent.
12.7 THEOREM
A small category is complete if and only if it is cocomplete.
Proof: Immediate from Theorem 10.32(2).



12.8 PROPOSITION (Canonical Construction of Limits via Large Colimits)
For a small diagram D : I → A, let SD be the category whose objects are all naturalgsources (A, fi ) for D, whose morphisms (A, fi ) −→ (A0 , fi0 ) are all those A-morphisms

9th July 2006

Sec. 12]Completeness and cocompleteness209gA −→ A0 with (A, fi ) = (A0 , fi0 ) ◦ g, and whose identity morphisms and composition law
are as in A. If D∗ : SD → A is the diagram given by:ggD∗ ((A, fi ) −→ (A0 , fi0 )) = A −→ A0 ,then for each A-object L, the following conditions are equivalent:`i(1) D has a limit L = (L −−→
Di )i∈Ob(I) ,k(2) D∗ has a colimit K = (D∗ (S) −−S→ L)S∈Ob(SD ) ,`i(3) SD has a terminal object L = (L −−→
Di )i∈Ob(I) .

Proof: (1) ⇔ (3) is immediate.
(3) ⇒ (2) follows from the dual of Example 11.4(5).
f(2) ⇒ (3). Given an object j of I and an object S = (A, fi ) in SD let D∗ (S) −−S→ Dj be
fj

the morphism A −−→ Dj in the source S that is indexed by j. Then for each j ∈ Ob(I)f(D∗ (S) −−S→ Dj )S∈Ob(SD )is a natural sink for D∗ , so that by the definition of colimit there is a unique morphism`j

L −−→ Dj with the property that fj = fS = `j ◦ kS for each S in SD .
`di
To see that the source L = (L −−→
Di )i∈Ob(I) is natural for D, let i −→ j be any morphism
in I. If S is any natural source for D, then fj = D(d) ◦ fi , so that `j ◦ kS = D(d) ◦ `i ◦ kS .
Since K, being a colimit, is an epi-sink, it follows that `j = D(d) ◦ `i . Consequently, L
is an object of SD .

fi

To see that L is a terminal object, first notice that if S = (A −−→ Di )i∈Ob(I) is a naturalkksource for D, then the A-morphism A −−S→ L is also a morphism S −−S→ L in SD .
g
Moreover if S −→ L is any SD -morphism, then since K is a natural sink for D∗ , this
implies that kL ◦ g = kS . In particular, if for each object S in SD one lets g take on therole of kS , this yields:kL ◦ kS = kS = idL ◦ kS

for each S in SD .

Since K is an epi-sink, this gives kL = idL . Consequently, g = idL ◦ g = kL ◦ g = kS , sothat kS is the unique morphism from S to L.

12.9 PROPOSITION
A cocomplete category A has a terminal object if and only if it has a weak terminal
object K; i.e., for each A-object A, there exists at least one morphism from A to K.

9th July 2006

210

Sources and Sinks[Chap. III

Proof: Let K be a weak terminal object in a cocomplete category A. Then let K be
the full subcategory of A that consists of the single object K, let D : K → A be thekinclusion functor, and let (K −→ T ) be a colimit of D. We shall prove that T is aterminal object of A.
For each A-object A there is some morphism f : A → K, and hence a morphism g =k ◦ f : A → T . Suppose that g 0 : A → T . Then to show that g = g 0 , form a coequalizercT −→ C of A

g
g0

//

T . One need only show that c is an monomorphism. Indeed, there

exists a morphism f 0 : C → K, and since f 0 ◦ c ◦ k is an element of hom(K, K), thenaturality of the colimit of D implies that k ◦ f 0 ◦ c ◦ k = k. Furthermore, k is an
epimorphism (since it is a singleton colimit sink) and, hence, k ◦ f 0 ◦ c = idT ; so c is asection, and thus a monomorphism.
12.10 DEFINITION
A full subcategory B of a category A with embedding E : B → A is called colimitdense in A provided that for every A-object A there exists a diagram D : I → B suchthat the diagram E ◦ D : I → A has a colimit with codomain A.
12.11 EXAMPLES(1) A full subcategory of Set is colimit-dense in Set if and only if it contains at least
one nonempty set as an object. [Observe that every set X is a coproduct of card(X)copies of {0}.](2) A full subcategory of Pos is colimit-dense in Pos if and only if it contains at least
one nondiscrete poset as an object.
(3) A full subcategory of Vec is colimit-dense in Vec if and only if it contains at least
one nonzero vector space as an object.
(4) Any full subcategory of Ab that contains

Z × Z × Z is colimit-dense in Ab.

(5) The full subcategory of Top that consists of all zero-dimensional Hausdorff spaces iscolimit-dense in Top. [Observe that every topological space is a (regular) quotientof a zero-dimensional Hausdorff space.]
12.12 THEOREM
Every cocomplete category with a small colimit-dense subcategory is complete.
Proof: Let B be a small, colimit-dense subcategory of a cocomplete category A. For
each small diagram D : I → A, consider the category SD of all natural sources for D(cf. 12.8).
First we establish that SD is cocomplete. Let D0 : J → SD be a small diagram (suchthat for each object j of J, D0 (j) = (Aj , fji )I ) and let D∗ : SD → A be the diagramcjof Proposition 12.8. Form the colimit (Aj −−→ K)j∈Ob(J) of the composite diagram
D

D∗

fji

0
J −−→
SD −−→ A. Since for each object i of I (Aj −−→ Di )Ob(J) is a natural sink

9th July 2006

Sec. 12]Completeness and cocompleteness211

for D∗ ◦ D0 , we have the existence of a unique morphism gi : K → Di such that gi ◦cj = fji . Since (cj , K)Ob(J) is an epi-sink, (K, gi )Ob(I) is a natural source for D andcj((Aj , fji ) −−→ (K, gi ))Ob(J) is a colimit of D0 . Thus SD is cocomplete.
Now to show that the small diagram D : I → A has a limit, by Propositions 12.8 and12.9 it is sufficient to prove that the category SD has a weak terminal object. Let I∗
be the full subcategory of SD given by all those natural sources S = (A, fi )Ob(I) whose
domain A is a member of B. Since B is small, so is I∗ , and since SD is cocomplete, thecinclusion I∗ ,→ SD has a colimit (S −−S→ K)S∈Ob(I∗ ) in SD . We claim that K = (K, pi )is a weak terminal object in SD .
To see this, let Ŝ = (A, fi )Ob(I) be any object of SD . By the colimit density, there existsaa small diagram D̂ : K → B with a colimit (D̂k −−k→ A)Ob(K) in A. Now for each object
k of K, Sk = (D̂k , fi ◦ak )Ob(I) is an object of I∗ . Thus cSk : Sk → K is an SD -morphism,gand since (cS , K) is a colimit, for each K-morphism k −→ k 0 we have cSk = cSk0 ◦ D̂(g).
Thus (cSk , K)k∈Ob(K) is natural for D̂, so that since (ak , A)Ob(K) is a colimit of D̂, theretis a unique A −−A→ K such that tA ◦ ak = cSk . Since (ak , A) is an epi-sink, it follows thattA is a morphism in SD . Thus K is a weak terminal object.

12.13 THEOREM
Every co-wellpowered cocomplete category with a separator is wellpowered and complete.
Proof: Let A be a co-wellpowered cocomplete category with a separator S. First we
prove the following two facts about A:(1) For every source S there exist an epimorphism e and a mono-source M with S =M ◦ e.
mi(2) If M = (A −−→
Ai )I is a small mono-source, then A is a quotient object ofΠI hom(S,Ai ) S or a quotient object of ∅ S.

(1). Let (ej )J be the collection of those epimorphisms ej , for which there exists a source
Sj with S = Sj ◦ ej . If e is a cointersection of (ej )J , then there exists a source M
with S = M ◦ e. To show that M is a mono-source, let r and s be morphisms withM ◦ r = M ◦ s. If c is a coequalizer of r and s, then there exists a source T with
M = T ◦ c. This implies that S = T ◦ (c ◦ e). Thus there exists some ̂ ∈ J withc ◦ e = e̂ . By the definition of e, there exists a morphism f̂ with e = f̂ ◦ e̂ . Since e isan epimorphism, the equations (f̂ ◦ c) ◦ e = f̂ ◦ e̂ = e = id ◦ e imply that f̂ ◦ c = id.
Thus c is a section and, consequently, an isomorphism. Hence r = s.
Q(2). Since M is a small mono-source, the map ϕ : hom(S, A) → i∈I hom(S, Ai ),given by (ϕ(f ))(i) = mi ◦ f , is injective. Thus either hom(S, A) = ∅ or there exists
an epimorphism ΠI hom(S,Ai ) S → hom(S,A)S. (Cf. the dual of Exercise 10K.) Since S isa separator, there exists an epimorphism hom(S,A) S → A (cf. the proof of Proposition
10.38). Thus A is a quotient object of ∅ S or a quotient object of ΠI hom(S,Ai ) S.

9th July 2006

212

Sources and Sinks[Chap. III

Since A is co-wellpowered, (2) immediately implies that A is wellpowered. It remains tobe shown that every small diagram D : I → A has a limit. Let D be such a diagram andlet Z be a set that represents all quotient objects of ∅ S and of Πi∈Ob(I) hom(S,D(i)) S. Thenfji

the collection G = { Sj | j ∈ J } of all natural sources` Sj = (Qj −−→ D(i))i∈Ob(I) for D,
whose domain Qj belongs to Z, is a set. Let (µj , Qj )j∈J be a coproduct of the small`fifamily (Qj )j∈J . Then there exists a unique source T = ( Qj −−→ D(i))i∈Ob(I) suchthat Sj = T ◦ µj for each j ∈ J. By (1) there exist an epimorphism e and a mono-source
L with T = L ◦ e. Since each Sj is natural for D, so are T and L. To verify that L isa limit of D it remains to be shown that every natural source S for D factors uniquely
through L. By (1) there exist an epimorphism e and a mono-source M with S = M ◦ e.
Since S is natural for D, so is M. By (2) there exist ̂ ∈ J and an isomorphism h with
M = S̂ ◦ h. Thus f = e ◦ µ̂ ◦ h ◦ e is a morphism with S = L ◦ f . That f is uniquelydetermined by S = L ◦ f follows from the fact that L is a mono-source.

12.14 REMARKFor slight modifications of the last two theorems see Exercise 12N.

Suggestions for Further Reading
Isbell, J. R. Structure of categories. Bull. Amer. Math. Soc. 72 (1966): 619–655.
Trnková, V. Limits in categories and limit preserving functors. Comment. Math. Univ.
Carolinae 7 (1966): 1–73.
Trnková, V. Completions of small categories.
(1967): 581–633.

Comment. Math. Univ. Carolinae 8Isbell, J. R. Small subcategories and completeness. Math. Systems Theory 2 (1968):
27–50.
Kennison, J. Normal completions of small categories. Canad. J. Math.
196–201.

21 (1969):

Pumplün, D., and W. Tholen. Covollständigkeit vollständiger Kategorien. Manuscr.
Math. 11 (1974): 127–140.
Börger, R., W. Tholen, M. B. Wischnewsky, and H. Wolff. Compact and hypercomplete categories. J. Pure Appl. Algebra 21 (1981): 129–144.
Adámek, J., and V. Koubek. Completion of concrete categories. Cahiers Topol. Geom.
Diff. 22 (1981): 209–228.
Herrlich, H. Universal completions of concrete categories. Springer Lect. Notes Math.
915 (1982): 127–135.
Kelly, G. M. A survey of totality for enriched and ordinary categories. Cahiers Topol.
Geom. Diff. Cat. 27 (1986): 109–132.

9th July 2006

Sec. 12]Completeness and cocompleteness213

EXERCISES
12A. Regular Monomorphisms vs. Strict Monomorphisms
Show that(a) The class of strict monomorphisms is closed under the formation of intersections.
Neither the class of regular monomorphisms nor the class of extremal monomorphisms need be closed under the formation of intersections.
(b) Intersections of regular monomorphisms are strict monomorphisms.
(c) If A has equalizers, then in A strict subobjects are precisely the intersections offamilies of regular subobjects.
(d) If A is strongly complete, then in A StrictMono = RegMono.
(e) If A has pushouts, then in A StrictMono = RegMono.
12B. Regular Monomorphisms via Extremal Monomorphisms
Show that(a) If A is strongly complete and in A regular monomorphisms are closed under composition, then in A RegMono = ExtrMono [cf. Corollary 14.20].
(b) If A has equalizers and pushouts and in A regular monomorphisms are closed undercomposition, then in A RegMono = ExtrMono [cf. Proposition 14.22].
12C. Multiple Pullbacks and CompletenessShow that:(a) A category is complete if and only if it has a terminal object and multiple pullbacksof small sinks.
(b) A group, considered as a category, has multiple pullbacks for all (even large) sinks,
but — if nontrivial — it fails to have equalizers or finite products.
12D. Dense Subcategories
A full subcategory B of a category A is called dense provided that each A-object A isa colimit of its “canonical diagram” with scheme B ↓ A (see 3K) given on objects by(X → A) 7→ X and on morphisms by h 7→ h (with the canonical colimit sink).
(a) Verify that the colimit-dense subcategories of Example 12.11(1), (2), (4), and (5)are in fact dense.
(b) Verify that

R is colimit-dense in Vec, but is not dense in Vec.

(c) Let B be a small subcategory of A. Define E : A → [Bop , Set] by assigning to eachA-object A the restriction of hom(−, A) : Aop → Set to Bop . Prove that E is fulland faithful if and only if B is dense.
12E. Limit-Dense and Codense Subcategories(a) Formulate the concepts of limit-density (dual to 12.10) and codensity (dual to 12D).

9th July 2006

214

Sources and Sinks[Chap. III(b) Let A be the full subcategory of Set that consists of all finite sets, and let B be thefull subcategory of A that consists of ∅ and of all finite sets that have cardinality apower of 2. Show that:(1) A has no proper limit-closed subcategory (cf. 13.26) that contains {0, 1}. However, {0, 1} is not limit-dense in A.
(2) {0, 1} is limit-dense in B, but is not codense in B.
* (c) Prove the equivalence of the following statements:(1) the monoid of all functions fromN to N is a codense subcategory of Set,

(2) no cardinal is measurable; i.e., every ultrafilter that is closed under the formationof countable meets is closed under the formation of all meets.
[Hints: For (1) ⇒ (2), let F be an ultrafilter on X that is closed under the formationof countable meets. For each function f : X → define fˆ : {0} → by: fˆ(0) = n,where n is the unique member ofwith f −1 (n) ∈ F. For each natural sourcefi(X −−→ )I the source (fˆi )I is also
T natural. Thus, it factors through (fi )I by aunique g : {0} → X. Then g(0) ∈ F.
For (2) ⇒ (1), notice that given a set X, a natural source of the canonical diagramX ↓ { } → Set is a set A and a function ϕ : hom(X, ) → hom(A, ) withϕ(p ◦ f ) = p ◦ ϕ(f ) for all p :→ . Given a ∈ A, prove that the collection of all
subsets Y ⊆ X with ϕ(fY )(a) = 1 (where fY (x) = 1 if x ∈ Y , elseT 0) is an ultrafilter
closed under countable meets. Define h : A → X by {h(a)} = { Y | ϕ(fY )(t) = 1 },then ϕ(f ) = f ◦ h for each f .]

N

N

N

N

N N

N

N

N

* (d) Prove that the following statements are equivalent:(1) Set has a small codense subcategory,(2) there do not exist arbitrarily large measurable cardinals; i.e., for some n, every
ultrafilter closed under n-meets is closed under all meets.
12F. Weakly Terminal Sets of Objects
Define the concept of weakly terminal set of objects. Prove that a cocomplete
category has a terminal object if and only if it has a weakly terminal set of objects.
12G. Existence of Coproducts
Let A be complete and wellpowered. Show that:(a) If an A-object A has an Ith copower I A, then it has a Kth copower K A for eachnonempty subset K of I.
``(b) If A is connected and I Xi exists for some family (Xi )I , then K Xi exists foreach subset K of I (including the case K = ∅).
* 12H. Copowers and Cocompleteness
Let A be complete, wellpowered, co-wellpowered, and have a separator S. Prove thatthe following are equivalent:

9th July 2006

Sec. 12]Completeness and cocompleteness215(1) A is cocomplete,(2) for each set I there exists an Ith copower I S of S in A.
* 12I. Strongly Complete but not Cocomplete(a) Show that the categories CBoo and CLat are each complete, wellpowered, andextremally co-wellpowered, but not cocomplete. (Cf. 8E and 10S.)
λ(b) Show that the construct A, whose objects are triples (X, λ, µ), where X −→ X andµf

PX \ {∅} −→ X are functions, and whose morphisms (X, λ, µ) −→ (X 0 , λ0 , µ0 ) aref

those functions X −→ X 0 that make the diagram
PX \ {∅}
f[ ]

µ

/X

λ

/X

f



PX 0 \ {∅}

f



/ X0

µ0



λ0

/ X0

commute, is complete, wellpowered, and co-wellpowered, but not cocomplete.
12J. Completeness and Coequalizers
Show that a category A has coequalizers whenever A is complete, wellpowered, andextremally co-wellpowered. (Cf. 15.25 and 15.7.)* 12K. Cocomplete Subcategories of HComp
Show that a full subcategory of HComp is reflective in HComp if and only if it iscocomplete.
* 12L. Completions of Abstract Categories(a) Show that the following category A cannot be fully embedded into a category B
such that the pair (A, B) has a product in B and the embedding preserves productsof pairs:
GF

@A

A

GF

/ Aα
O

AO
Cβ


@A

ED

rβ

B

fαβ

/ Aα × Bα

gα


/ Bα
sβ

(
rβ , if α = βwhere α and β are ordinals and gα ◦ fαβ =sβ , if α 6= β.

9th July 2006


/D
O

BC

216

Sources and Sinks[Chap. III[Hint: If (P, (πA , πB )) were a product of (A, B) in B, then homB (P, D) would be aproper class.](b) Show that the following category A cannot be fully embedded into a category B suchthat the pair (r, s) has an equalizer in B and the embedding preserves equalizers:
o

ED
//
?B
AA
s
@@ qβ
~
~~~?
AA
@@
~
AA
rα ~~~~@@ f~AAfα ~~~gα @@ αβ~~~ sαA@@~AA ~~~~~
/

D `@o
A

pβ

GF

Cβ

/A

Eα

Aα

r

where α and β are ordinals, r = rα ◦ fα , s = sα ◦ fα , and(pβ , if α = βgα ◦ fαβ =qβ , if α 6= β.
e[Hint: If E −→ A were an equalizer of r and s in B, then homB (E, D) would be aproper class.](c) Show that for every category A that has a separating set there exists a full limitE

dense embedding, A −−→ B, where B is complete and E preserves small limits.
(d) Show that every category can be fully embedded into a complete quasicategory.
* 12M. Universal Completions of Concrete Categories
E

A full concrete embedding (A, U ) −−→ (B, V ) is called a universal completion of(A, U ) provided that the following conditions are satisfied:(1) (B, V ) is concretely complete [i.e., every small diagram has a concrete limit in(B, V )] and is uniquely transportable,(2) E preserves concrete limits of small diagrams,(3) E is concrete limit-dense, i.e., for every B-object B there exists a small diagram
D : I → A, such that B is (the object part of) a concrete limit of E ◦ D,(4) for every concretely complete and uniquely transportable category (C, W ) and everyG

concrete functor (A, U ) −−→ (C, W ) that preserves concrete limits of small diagrams,G

there exists a unique concrete functor (B, V ) −−→ (C, W ) that preserves concrete
limits of small diagrams, with G = G ◦ E.
Show that(a) Every amnestic concrete category over a complete category has a universal completion that is uniquely determined up to concrete isomorphism.
(b) If A is the construct of finite Boolean spaces (equivalently: the construct of finitesets), then the inclusion A ,→ BooSp is a universal completion.

9th July 2006

Sec. 12]Completeness and cocompleteness217* 12N. Re: Theorems 12.12 and 12.13(a) Show that every cocomplete category with a small colimit-dense subcategory isstrongly complete.
(b) Show that a strongly cocomplete category with a separating set is strongly complete.
(c) Construct a strongly cocomplete category with a separator that is neither wellpowered nor co-wellpowered.
12O. Completeness of Functor-Categories(a) Show that limits and colimits in the category [A, B] are formed componentwise.
(b) Show that for each small category A and each strongly (co)complete category B,[A, B] is strongly (co)complete.

9th July 2006

13 Functors and limitsF

In this section the behavior of limits with respect to functors A −−→ B is investigated.
The following problems receive special attention:(A) Does F “preserve” limits?
F

The information that a functor A −−→ B preserves limits is particularly useful if the domain category A is known to have enough limits. See, e.g., the adjoint functor theoremsin §18.
In many cases, particularly for embedding functors of full subcategories and for forgetful
functors of concrete categories, it is not immediately clear whether or not the domainF
category A of the functor A −−→ B in question has enough limits, but it is known thatthe codomain category B does. In these cases the following questions arise naturally:(B) Can limits in B be lifted along F to limits of A?
If so, to which extent is the lifting unique?
If not, are there suitable properties of F that at least guarantee the existence oflimits in A provided that limits exist in B?

PRESERVATION OF LIMITS
13.1 DEFINITION`i
Di ) of a diagram(1) A functor F : A → B is said to preserve a limit L = (L −−→F`i
D : I → A provided that F L = (F L −−−→F Di ) is a limit of the diagram
F ◦ D : I → B.

(2) F is said to preserve limits over a scheme I provided that F preserves all limitsof diagrams D : I → A with scheme I.
(3) F preserves equalizers if and only if F preserves all limits over the scheme • ⇒ •;F preserves products if and only if F preserves all limits over small discrete
schemes; F preserves small limits if and only if F preserves limits over all small
schemes; F preserves strong limits if and only if F preserves small limits andarbitrary intersections; etc.
Dual Notions: F preserves colimits (over a scheme I), coequalizers, coproducts, small colimits, strong colimits, etc.
13.2 EXAMPLES(1) For the constructs Top, Rel, and Alg(Σ), the forgetful functors preserve limits andcolimits.

218

Sec. 13]

Functors and limits

219(2) For constructs of the form Alg(Ω) the forgetful functors preserve limits and directed
colimits [see Example 11.28(4)], but generally fail to preserve coproducts or coequalizers. Also for the following constructs, the forgetful functors preserve limits anddirected colimits, but neither coproducts nor coequalizers: Vec, Grp, Sgr, Mon,
Rng, and Σ-Seq.
(3) For the constructs AbTor and (Ban,U ) the forgetful functors preserve equalizers,
but not products (cf. 10.55).
(4) The full embeddings Haus → Top and Pos → Rel preserve limits and coproducts,
but not coequalizers.
(5) The covariant power-set functor P : Set → Set preserves neither products (of pairs),
nor coproducts (of pairs), nor equalizers, nor coequalizers.
13.3 PROPOSITION
If F : A → B is a functor and A is finitely complete, then the following conditions areequivalent:(1) F preserves finite limits,(2) F preserves finite products and equalizers,(3) F preserves pullbacks and terminal objects.
Proof: Immediate from the characterizations of finite limits (12.4).



13.4 PROPOSITION
For a complete category A, a functor F : A → B preserves small limits if and only if it
preserves products and equalizers.
Proof: This follows immediately from the construction of small limits via products andequalizers, presented in the proof of Theorem 12.3.

13.5 PROPOSITION(1) If a functor preserves finite limits, then it preserves monomorphisms and regularmonomorphisms.
(2) If a functor preserves (small) limits, then it preserves (small) mono-sources.
Proof: (1) follows from Proposition 11.16.
fi(2) follows from the observation that a source (A −−→ Ai )I is a mono-source if and only
if the sourceA

9th July 2006

idA

/A
@@ f
@ i
idA @@@A@

Ai(i ∈ I)

220

Sources and Sinks[Chap. III

is a limit of the diagram
A


A

fi

fi

(i ∈ I)/ Ai



13.6 REMARK
If a functor F preserves equalizers, then F (obviously) preserves regular monomorphisms.
The converse is not true: the forgetful functors from the categories Vecop , Grpop , Posop ,and Hausop to Setop preserve regular monomorphisms, but don’t preserve equalizers.
13.7 PROPOSITION
Hom-functors preserve limits.
Proof: Let F = hom(A, −) : A → Set be a hom-functor, let D : I → A be a diagram,`iDi ) be a limit of D. Then F L is obviously a natural source forand let L = (L −−→fiF ◦ D. Let S = (X −−→ hom(A, Di )) be an arbitrary natural source for F ◦ D. Then,fi (x)

for each element x of X, (A −−−−→ Di ) is a natural source for D. Hence for each x ∈ X
there exists a unique morphism f (x) : A → L with fi (x) = `i ◦ f (x) for each i ∈ Ob(I).
This defines a function f : X → hom(A, L) which is the unique function f : X → F L
that satisfies S = F L ◦ f .

13.8 PROPOSITION
If F and G are naturally isomorphic functors, then F preserves limits over a scheme I
if and only if G does.
Proof: Straightforward computation.
13.9 COROLLARY
Representable functors preserve limits.



13.10 REMARK
Since the forgetful functors for constructs are quite often representable, the above corollary explains why limits in constructs are much more often concrete (Definition 13.12)
than are colimits. In §18 we will show that adjoint functors (in particular embeddingsof reflective subcategories) preserve limits. Next we will show that embeddings of “sufficiently big” subcategories preserve limits as well.
13.11 PROPOSITION
Embeddings of colimit-dense subcategories preserve limits.

9th July 2006

Sec. 13]

Functors and limits

221

Proof: Let A be a colimit-dense subcategory of B with embedding E : A → B, and let`i
D : I → A be a diagram with a limit L = (L −−→
Di ). Then L is a natural source forfi

E ◦ D. Let S = (B −−→ Di ) be an arbitrary natural source for E ◦ D. By colimit-density
cj
there exists a diagram G : J → A and a colimit (Gj −−→ B) of E ◦ G. For each object
fi ◦cjj of J, (Gj −−−−→ Di )i∈ObI is a natural source for D. Hence for each j ∈ Ob(J) there
exists a unique morphism gj : Gj → L with fi ◦ cj = `i ◦ gj for each i ∈ I.
Gj

cj

gj





L

/B

`i

fi

/ Di
gj

Since L is a mono-source in A, it follows that (Gj −−→ L) is a natural sink for G andhence for E◦G. Consequently, there exists a unique morphism f : B → L with gj = f ◦cj
cj
for each j. Since (Gj −−→ B) is an epi-sink, this implies that fi = `i ◦ f for each i. To
show that f is the unique morphism with this property, let f 0 : B → L be a morphism`iwith fi = `i ◦ f 0 for each i. Since (L −−→
Di ) is a mono-source in A, this implies thatgj = f 0 ◦ cj for each j, and hence that f 0 = f .
CONCRETE LIMITS
13.12 DEFINITION(1) Let (A, U ) be a concrete category. A limit L of a diagram D : I → A is called aconcrete limit of D in (A, U ) provided that it is preserved by U.
(2) A concrete category (A, U ) has (small) concrete limits, resp. concrete products, etc., if and only if A has (small) limits, resp. products, etc., and U preserves
them.
Dual Notions: (has) (small) concrete colimits, concrete coproducts, etc.
13.13 EXAMPLES(1) The constructs Top, Rel, Prost, and Alg(Σ) have small concrete limits and small
concrete colimits. So have all constructs of the form Spa(T).
(2) The constructs Vec, Grp, Pos, and Haus have small concrete limits and nonconcrete small colimits. Alg(Ω) has small concrete limits; it has small concrete
colimits only if the defining operations are all unary.
(3) The construct Σ-Seq has finite concrete limits and finite colimits that are not concrete.
(4) The concrete category Aut over Set × Set × Set has small concrete limits and small
colimits that are not concrete.

9th July 2006

222

Sources and Sinks[Chap. III

13.14 PROPOSITION
A concrete category has small concrete limits if and only if it has concrete products andA 12.3
concrete equalizers.
13.15 PROPOSITION`i
If (A, U ) is a concrete category and D : I → A is a diagram, then L = (L −−→
Di )i∈Ob(I)is a concrete limit in (A, U ) if and only if U (L) is a limit of U ◦ D and L is an initial
A 10.53
source in (A, U ).
13.16 REMARK
By the above, a concrete limit can be constructed in two steps: first form the limitof the underlying diagram in the base category, and then provide an initial lift of this(underlying) limit.

LIFTING OF LIMITS
13.17 DEFINITION
A functor F : A → B is said to(1) lift limits (uniquely) provided that for every diagram D : I → A and every limit
L of F ◦ D there exists a (unique) limit L0 of D with F (L0 ) = L,
(2) create limits provided that for every diagram D : I → A and every limit L offi

F ◦ D there exists a unique source S = (L −−→ Di ) in A with F (S) = L, and that,
moreover, S is a limit of D.
Similarly, one has lifts small limits, lifts products, creates equalizers, creates
finite limits, etc.
Dual Notions: lift colimits (uniquely), create colimits, etc.
13.18 EXAMPLES
The forgetful functors for the constructs(1) of the form Alg(Σ) create limits and colimits,(2) of the form Alg(Ω) create limits and directed colimits [see Example 11.28(4)],(3) Vec, Grp, Sgr, Mon, Lat, CATof , and HComp create limits,(4) Top, Rel, and Prost lift limits and colimits uniquely, but create neither,(5) Metc and Banb lift finite limits, but do not lift them uniquely,(6) Σ-Seq lifts finite limits uniquely.

9th July 2006

Sec. 13]

Functors and limits

223

13.19 THEOREMF
If a functor A −−→ B lifts limits and B is (strongly) complete, then A is (strongly)
complete and F preserves small limits (and arbitrary intersections).
Proof: If F lifts limits and B is complete, then obviously A is complete and F preserves
small limits. Hence, by Proposition 13.5, F preserves monomorphisms. Thus, if B has
arbitrary intersections, so does A, and F preserves them.
Lifting of limits

13.20 REMARKS(1) The obvious implications
F creates limits ⇒ F lifts limits uniquely ⇒ F lifts limits
cannot be reversed.
(2) If B has certain limits and F : A → B lifts them, then A has these limits andF preserves them. Hence the concepts of lifting and creating limits are useful forFfunctors A −−→ B with range categories B that have enough limits. In general,
functors that create limits need not even preserve them. An example of this is theembedding of the terminal category 1 = {•} = A into B = • ⇒ ◦.
13.21 PROPOSITION
For functors F : A → B the following conditions are equivalent:

9th July 2006

224

Sources and Sinks[Chap. III(1) F lifts limits uniquely,(2) F lifts limits and is amnestic.
f

Proof: (1) ⇒ (2). To show amnesticity, let A0 −→ A be an isomorphism in A withf

idAF f = idB . The 1-indexed sources (A0 −→ A) and (A −−−→ A) are products of theidB
1-indexed family (A). Since they are mapped via F to the product (B −−−→ B) of the
1-indexed family (B) = (F A), uniqueness implies that f = idA .

`0

`

i
(2) ⇒ (1). If L = (L −−→
Di ) and L0 = (L0 −−i→ Di ) are limits of a diagram D : I → A,then there exists an isomorphism h : L → L0 with L = L0 ◦ h. If F (L) = F (L0 ) is a limitof F ◦ D, then F h = idF L . Hence, by amnesticity, h = idL . Consequently, L = L0 .


13.22 DEFINITION
A functor F : A → B is said to(1) reflect limits provided that for each diagram D : I → A an A-source S =fi(A −−→ Di )i∈Ob(I) is a limit of D whenever F (S) is a limit of F ◦ D,(2) detect limits, provided that a diagram D : I → A has a limit whenever F ◦ D has
one.
Similarly, one has reflect equalizers, detect products, etc.
Dual Notions: reflect colimits, detect colimits, etc.
13.23 EXAMPLES(1) Every functor that lifts limits, detects them.
(2) The forgetful functor U : Top → Set lifts limits uniquely (and detects and preserves
them), but does not reflect them.
(3) Full embeddings reflect limits, but they need not lift, preserve, nor detect them.
13.24 PROPOSITION
A functor that reflects equalizers is faithful.
Proof: Let F : A → B be a functor and let A

f
g

//

B be a pair of A-morphisms withidA
F f = F g. Then S = (A −−−→ A) is a 1-source and F S is an equalizer of F f and F g. If
F reflects equalizers, then S is an equalizer of f and g. Hence f = idA ◦ f = idA ◦ g = g.

13.25 PROPOSITION
For any functor F : A → B the following conditions are equivalent:(1) F creates limits,(2) F lifts limits uniquely and reflects limits,

9th July 2006

Sec. 13]

Functors and limits

225(3) F lifts limits, is faithful and amnestic, and reflects isomorphisms in the sense thatwhenever F f is a B-isomorphism, then f is an A-isomorphism.
Proof: (1) ⇔ (2) is obvious.
(2) ⇒ (3) follows from Propositions 13.21, 13.24, and Example 10.20(8).
(3) ⇒ (2). To show that F reflects limits, let D : I → A be a diagram and let S =fi(A −−→ Di )i∈Ob(I) be a source in A such that F (S) is a limit of F ◦ D. Since F lifts`

i
limits, there exists a limit L = (L −−→
Di ) of D with F (L) = F (S). Since F isffaithful, S is a natural source for D. Hence there exists a morphism A −→ L with
S = L ◦ f . Consequently, F (L) = F (S) = F (L) ◦ F f , which implies that F f = idF L .
Since F reflects isomorphisms, f is an isomorphism. Hence, by amnesticity, f = idL .
Consequently, S = L. Thus S is a limit of D.

13.26 REMARK
Embeddings E : A → B of full subcategories obviously reflect limits. Hence they lift
limits if and only if they create them. A more suggestive term for such full subcategories
is that they are closed under the formation of limits (or just limit-closed) in B.
13.27 PROPOSITION
A full reflective subcategory A of B is limit-closed in B if and only if A is isomorphismclosed in B.
Proof: If A is limit-closed in B, then obviously A is isomorphism-closed in B. For
the converse, consider a diagram D : I → A such that E ◦ D : I → B has a limit`irDi ), where E : A → B denotes the embedding. Let L −→ A be an
L = (L −−→
A-reflection arrow for L. Then for each i there exists a morphism fi : A → Di with`i = fi ◦r. By the uniqueness property of reflection arrows and the fact that all Di belongfi

to A, S = (A −−→ Di ) must be a natural source for D resp. for E ◦ D. Consequently,
there exists a morphism f : A → L with S = L ◦ f . Hence L = S ◦ r = L ◦ f ◦ r, which
implies that f ◦ r = idL . Therefore (r ◦ f ) ◦ r = r ◦ idL = idA ◦ r, which implies thatr ◦ f = idA . Hence r is an isomorphism, and, consequently, L belongs to A.

13.28 COROLLARY
If a category has certain limits, then so does each of its isomorphism-closed full reflective
subcategories.

13.29 REMARK
Isomorphism-closed full reflective subcategories A of B usually fail to be colimit-closed.
However, the following proposition shows that the associated inclusion functors detect
colimits.

9th July 2006

226

Sources and Sinks[Chap. III

13.30 PROPOSITION
Let A be a full subcategory of B with embedding E : A → B, and let D : I → A be acir
diagram. If C = (Di −−→C) is a colimit of E ◦ D, and if C −→ A is an A-reflection
arrow for C, then C 0 = r ◦ C is a colimit of D.
fi

Proof: Obviously, C 0 is a natural sink for D. Let S = (Di −−→ A0 ) be an arbitrary
natural sink for D. Then S is a natural sink for E ◦ D. Hence there exists a uniquefmorphism C −→ A0 with S = f ◦ C. Since r is an A-reflection arrow, there exists agunique morphism A −→ A0 with f = g ◦ r. Hence S = f ◦ C = g ◦ r ◦ C = g ◦ C 0 .
Uniqueness of g is obtained by retracing the steps of the above construction.

13.31 EXAMPLES(1) HComp is a full reflective subcategory of Top. The construction of coproducts in
HComp given in Example 10.67(5) is a special case of the above result.
(2) AbTor is a full coreflective subcategory of Ab. The construction of products in
AbTor given in Example 10.20(3) is a special case of the dual of the above result.
13.32 COROLLARY
Embeddings of full reflective subcategories detect colimits.

13.33 REMARKS(1) Embeddings of nonfull reflective subcategories may be quite awkward. See Exercise
13K.
(2) As will be seen in Chapter VI, all “reasonable” forgetful functors of concrete categories lift limits uniquely and detect colimits.
13.34 PROPOSITIONF
If a functor A −−→ B preserves limits, then the following conditions are equivalent:(1) F lifts limits (uniquely),(2) F detects limits and is (uniquely) transportable.
Proof: (1) ⇒ (2) follows immediately from the fact that isomorphisms are products of
1-indexed families.
`i(2) ⇒ (1). Let D : I → A be a diagram and let S = (L −−→
F Di )I be a limit offi

F ◦ D. Since F detects limits, D has a limit L = (A −−→ Di )I . Since F preserves
F fi

limits, F L = (F A −−−→ F Di )I is a limit of F ◦ D. Hence there exists a B-isomorphismh

F A −→ L with S = h ◦ F L. By transportability, h can be lifted to an A-isomorphismkfi ◦k−1

A −→ B. Hence (B −−−−−→ Di )I is a limit of D that lifts S. The uniqueness part follows
from Proposition 13.21.


9th July 2006

Sec. 13]

Functors and limits

227

CREATION AND REFLECTION OF ISOMORPHISMS
Since isomorphisms can be regarded as the limits of diagrams with scheme 1, the conceptsof creation and reflection of limits specialize to the following concepts:
13.35 DEFINITIONG
A functor A −−→ B is said to(1) create isomorphisms provided that whenever h : X → GA is a G-structured Bisomorphism, there exists precisely one A-morphism ĥ : B → A with G(ĥ) = h,and, moreover, ĥ is an isomorphism,(2) reflect isomorphisms provided that an A-morphism f is an A-isomorphism whenever Gf is a B-isomorphism.
13.36 PROPOSITION(1) If G creates (resp. reflects) limits, then G creates (resp. reflects) isomorphisms.
(2) G creates isomorphisms if and only if G reflects isomorphisms and is uniquely transportable.
(3) If G creates isomorphisms, then G reflects identities.



13.37 EXAMPLES(1) Let E : A → B be the embedding of a full subcategory. Then E reflects isomorphisms. E creates isomorphisms if and only if A is isomorphism-closed in B.
G(2) Let A be a monoid, considered as a category. Then the unique functor A −−→ 1reflects isomorphisms if and only if A is a group. G creates isomorphisms if andonly if A is a trivial (one-element) group.
(3) Each of the forgetful functors from Vec, Grp, and Mon to Set creates isomorphisms. None of the forgetful functors from Top, Rel, and Pos to Set reflects
isomorphisms.
13.38 REMARKThe following diagram summarizes some of the relationships between functors and limits.
creates limits





reflects limits↓

HH
jreflects identities
amnestic

lifts limits uniquely

H
HH
j

↓
↓

9th July 2006

HH

creates isomorphisms

reflects isos

↓





↓

H
HH
j
H

uniquely transportable↓transportable






lifts limits↓
detects limits

228

Sources and Sinks[Chap. III

Suggestions for Further Reading
Trnková, V. Some properties of set functors. Comment. Math. Univ. Carolinae 10(1969): 323–352.
Trnková, V. When the product-preserving functors preserve limits. Comment. Math.
Univ. Carolinae 11 (1970): 365–378.
Adámek, J., J. Rosický, and V. Trnková. Are all limit-closed subcategories of locallypresentable categories reflective? Springer Lect. Notes Math. 1348 (1988): 1–18.

EXERCISES
13A. Preservation of Multiple Pullbacks
F

Let A be a complete category. Show that a functor A −−→ B preserves small limits if
and only if F preserves small multiple pullbacks and terminal objects.
13B. Preservation of Limits by Set-Valued Functors
Show that
G(a) If a functor A −−→ Set has a G-universal arrow with a nonempty domain, then G
preserves limits. Deduce Corollary 13.9.
(b) If A is a full subcategory of Set that contains at least one nonempty set, then theembedding A → Set preserves limits.
* (c) Every functor Set → Set preserves nonempty finite intersections.
(d) A functor Set → Set preserves products if and only if it is representable.
(e) A functor Set → Set preserves coproducts if and only if it is naturally equivalent,for some set A, to the functor A × − : Set → Set, defined by
f

id ×f

(A × −)(X −→ Y ) = A × X −−A−−→ A × Y.
13C. Reflection of Concrete Limits
Let (A, U ) have small concrete limits. Show that U reflects small limits if and only if U
reflects isomorphisms.
13D. Lifting of Limits and Faithfulness
Show that(a) A functor that lifts equalizers is faithful if and only if it reflects epimorphisms.
(b) A functor that lifts limits need not be faithful.
13E. Lifting of Limits and Transportability
Show that a functor that lifts limits (uniquely) is (uniquely) transportable.

9th July 2006

Sec. 13]

Functors and limits

229

13F. Reflection of Isomorphisms
Show that a functor that reflects equalizers (or finite products) reflects isomorphisms.
13G. Initial Mono-Sources and Reflection of Limits
Show that if mono-sources are G-initial, then G reflects limits. [Cf. Proposition 10.60.]
13H. Hom-Functors and Limits
Show that`i(a) If D : I → A is a diagram, then a source S = (L −−→
Di )i∈Ob(I) is a limit of D if
and only if, for each A-object A, hom(A, −)(S) is a limit of hom(A, −) ◦ D.

(b) If A is a nonempty set, then hom(A, −) : Set → Set reflects limits.
13I. Limit- and Colimit-Closed Full Subcategories(a) Show that
(1) Set has no proper full subcategory that is both limit-closed and colimit-closed.
* (2) Top has no proper full subcategory that is both limit-closed and colimit-closed.
(3) Vec has precisely two full subcategories that are both limit-closed and colimitclosed.
(b) Determine all full subcategories of Rel that are both limit-closed and colimit-closed.
13J. Limit-Closed vs. Reflective Full Subcategories
Show thatZ(a) Every full subcategory of the partially ordered set , considered as a category, islimit- and colimit-closed in , but generally fails to be either reflective or coreflective
in .

Z

Z

(b) A full subcategory of a complete lattice A, considered as a category, is reflective in
A if and only if it is limit-closed in A.
* (c) There exist limit-closed full subcategories of Top that are not reflective in Top.
[Cf. 16D.]
13K. A Misbehaved Nonfull Reflective EmbeddingE

Consider the following nonfull embedding A −−→ B of preordered sets, considered as
categories:• _@• _@o @@?•~/ ? •
@@
@@
@@

•

~~
~~
~
~~

Show that(a) A is reflective in B,(b) B is complete and cocomplete,

9th July 2006



E

/

@@@@
~~
@@@@
~~~~
@@@ ~~~~~~
~

•

230

Sources and Sinks[Chap. III(c) A is neither complete nor cocomplete,(d) E detects neither limits nor colimits.
13L. Colimit-Dense Full Embeddings
Show that colimit-dense full embeddings preserve limits.
13M. Creation of Isomorphisms
Show that(a) G creates limits if and only if G is faithful, lifts limits, and creates isomorphisms.
(b) G creates isomorphisms if and only if G creates 1-indexed products.
(c) If G creates isomorphisms, then G reflects identities.
(d) If G reflects identities, then G is amnestic.
13N. Creation of Limits of T -Algebras
Prove that for each functor T : X → X the forgetful functor of Alg(T ) creates limits.

9th July 2006

231

Chapter IV

FACTORIZATION STRUCTURES
f

Every function A −→ B can be factored through its image, i.e., written as a composite
f

e

m

e

A −→ B = A −→ f [A] −−→ B, where A −→ f [A] is the codomain-restriction of f andm
f [A] −−→ B is the inclusion. This fact, though simple, is often useful. Similarly, inconstructs such as Vec, Grp, and Top every morphism can be factored through its“image”. Since(a) for categories in general, no satisfactory concept of “embedding of subobjects” andhence of “image of a morphism” is available,
and(b) for certain constructs, factorizations of morphisms different from the one throughthe image are of interest (e.g., in Top the one through the closure of the image:f

A −→ B = A → clB f [A] ,→ B),
categorists have created an axiomatic theory of factorization structures (E, M ) for morphisms of a category A. Here E and M are classes of A-morphisms59 such that
f

e∈E

m∈M

each A-morphism has an (E, M )-factorization A −→ B = A −−−→ C −−−−→ B.
Naturally, without further assumptions on E and M such factorizations might be quite
useless. A careful analysis has revealed that the crucial requirement that causes (E, M )factorizations to have appropriate characteristics is the so-called “unique (E, M )-diagonalization” condition, described in Definition 14.1. Such factorization structures formorphisms have turned out to be useful, especially for “well-behaved” categories (e.g.,
those having products and satisfying suitable smallness conditions). They have beentransformed into powerful categorical tools by two successive generalizations(a) factorization structures for sources in a category, and(b) factorization structures for G-structured sources with respect to a functor G.
Instead of describing the most general concept first and then specializing to the others,
our presentation will follow the historical development described above.

59

The requirements E ⊆ Epi(A) and M ⊆ Mono(A) were originally included, but later dropped.

9th July 2006

14 Factorization
phismsstructures

for

mor-

In this section factorization structures for morphisms are defined and investigated. In
particular, it is shown that every strongly complete category is simultaneously (ExtrEpi,
Mono)-structured and (Epi, ExtrMono)-structured (see 14.21).
14.1 DEFINITION
Let E and M be classes of morphisms in a category A.
(E, M ) is called a factorization structure for morphisms in A and A is called(E, M )-structured provided that(1) each of E and M is closed under composition with isomorphisms,60(2) A has (E, M )-factorizations (of morphisms); i.e., each morphism f in A has afactorization f = m ◦ e, with e ∈ E and m ∈ M, and(3) A has the unique (E, M )-diagonalization property; i.e., for each commutativesquare
e /
(∗)
A
B
f

g





C

/D

m

with e ∈ E and m ∈ M there exists a unique diagonal, i.e., a morphism d such thatthe diagram
e /
B
A
f

~~
 ~~ ~

C

~

d ~~
m

g



/D

commutes (i.e., such that d ◦ e = f and m ◦ d = g).
14.2 EXAMPLES(1) For any category, (Iso, Mor) and (Mor, Iso)61 are (trivial) factorization structures
for morphisms.
60

Condition (1) can be replaced by the following conditions (1a) and (1b):(1a) if e ∈ E and h ∈ Iso(A), and h ◦ e exists, then h ◦ e ∈ E,(1b) if m ∈ M and h ∈ Iso(A), and m ◦ h exists, then m ◦ h ∈ M.
This follows from Proposition 14.6 below since for its proof not the full strength of (1) but only thatof (1a) and (1b) is used. When generalizing factorization structures to sources or to functors, theformulation in terms of (1a) and (1b) turns out to be more appropriate.
61

Recall that Iso resp. Iso(A) is the class of all isomorphisms (in the category A); similarly for Mor,Mono, etc. See the Table of Symbols.

232Sec. 14]

Factorization structures for morphisms

233(2) For Set, Vec, Grp, Mon, Σ-Seq, each Alg(Ω) and many other categories of algebras, (RegEpi, Mono) is a factorization structure for morphisms.
(3) For Cat, (RegEpi, Mono) is not a factorization structure for morphisms. [Regular epimorphisms are not closed under composition (cf. 14.6 and 7.76).] However,(ExtrEpi, Mono) is a factorization structure for morphisms in Cat.
(4) Set has precisely four factorization structures for morphisms: besides the three
mentioned above, namely, (Epi, Mono) = (RegEpi, RegMono) and the two trivial
ones, the following (pathological) (E, M ), wheree

E = {X −→ Y | X = ∅ ⇒ Y = ∅};m

M = {X −−→ Y | mandis a bijection or X = ∅}.

(5) Top has a proper class (even an illegitimate conglomerate) of factorization structures
for morphisms. Each of(Epi, RegMono) = (surjection, embedding),(RegEpi, Mono) = (quotient, injection),(dense, closed embedding), and(front-dense, front-closed embedding),
is a factorization structure for morphisms in Top, but (Epi, Mono) is not.
(6) (dense C ∗ -embedding, perfect map) is a factorization structure for morphisms in
Tych.
14.3 PROPOSITION
A is (E, M )-structured if and only if Aop is (M, E)-structured.



14.4 PROPOSITION
If A is (E, M )-structured, then (E, M )-factorizations are essentially unique, i.e.,
e

f

m

i
i
(1) if A −−→
Ci −−→
B are (E, M )-factorizations of A −→ B for i = 1, 2, then there
exists a (unique) isomorphism h, such that the diagram

e1

/ C1
|
h |||
m1
e2
||
|
 ~|

/B
C2

A

m2

commutes,
f

e

m

h

(2) if A −→ B = A −→ C −−→ B is an (E, M )-factorization and C −→ D is an
f

h◦e

m◦h−1

isomorphism, then A −→ B = A −−−→ D −−−−−→ B is also an (E, M )-factorization.

9th July 2006

234

Factorization Structures[Chap. IV

Proof:(1). By the diagonalization-property there exist morphisms h and k such that the diagrams
e1

/ C1
|
h |||
m1
e2
||
|
 ~|

/B
C2

A

and

m2

e2

/ C2
|
k |||
m2
e1
||
|
 ~|

/B
C1

A

m1

commute. Hence, also the diagrams
e1

/ C1
|
k◦h |||
m1
e1
||
 ~||

/B
C1

A

and

m1

e2

/ C2
|
h◦k |||
m2
e2
||
 ~||

/B
C2

A

m2

commute. By uniqueness we conclude k ◦ h = idC1 and h ◦ k = idC2 . Therefore h is
an isomorphism.
(2). This follows directly from the assumption that each of E and M is closed undercomposition with isomorphisms.

14.5 LEMMA
Let A be (E, M )-structured and let e ∈ E and m ∈ M. If the diagram
/•
~
~
d ~
m
id
~~
 ~~

/•
•

•

e

f

commutes, then e is an isomorphism and f ∈ M.
Proof: The diagram
/•
~
~
x ~
e
m
~~
 ~~

/•
•

•

e

m

commutes for x = id and for x = e ◦ d. Hence, by uniqueness, e ◦ d = id. Consequently,
e is an isomorphism. Thus f = m ◦ e belongs to M by our assumptions on M.

14.6 PROPOSITION
If A is (E, M )-structured, then the following hold:(1) E ∩ M = Iso(A),(2) each of E and M is closed under composition,

9th July 2006

Sec. 14]

Factorization structures for morphisms

235(3) E and M determine each other via the diagonalization-property;62 in particular, amorphism m belongs to M if and only if for each commutative square of the form(∗) (see Definition 14.1) with e ∈ E there exists a diagonal.
Proof:(1). For any f ∈ E ∩ M there exists a diagonal d that makes the diagram
f

/•
~
d ~~
id
id
~~
~
 ~

/•
•

•

f

commute. Hence f ∈ Iso(A). Conversely, if f ∈ Iso(A) and f = m ◦ e is an(E, M )-factorization of f , then for d = f −1 ◦ m the diagram
/•
~
~
d ~
m
id
~~
 ~~

/•
•
e

•

f

commutes. Hence, by Lemma 14.5, f ∈ M. By duality f ∈ E.
m

m

1
2
(2). Let A −−→
B and B −−→
C be morphisms in M. If m2 ◦ m1 = m ◦ e is an (E, M )factorization, then diagonals d1 and d2 can be constructed successively such thatthe diagrams

/•
~
~
d1 ~~
m1
m
~~

 ~~~
/C
B
e

A

m2

/•
~
~
d2 ~~
d1
id
~~
 ~~

/B
A

A

and

e

m1

/•
~
~
d2 ~~
m
id
~~
 ~~

/C
A

A

and henceem2 ◦m1

commute. By Lemma 14.5 this implies m2 ◦ m1 ∈ M. By duality, E is also closed
under composition.
(3). This follows for M immediately from Lemma 14.5, and for E via duality.



14.7 PROPOSITION
If E and M are classes of morphisms in A, then A is (E, M )-structured if and only if
the following conditions are satisfied(1) Iso(A) ⊆ E ∩ M,(2) each of E and M is closed under composition,
62

Here the diagonal needn’t be required to be unique.

9th July 2006

236

Factorization Structures[Chap. IV(3) A has the (E, M )-factorization property, unique in the sense that for any pair of(E, M )-factorizations m1 ◦ e1 = f = m2 ◦ e2 of a morphism f there exists a unique
isomorphism h, such that the diagram
•
e2

e1

/•
~

h ~~

~~
 ~~

•



m1

/•

m2

commutes.
Proof: By Propositions 14.4 and 14.6, conditions (1)–(3) are necessary. To show thatthey suffice, the unique (E, M )-diagonalization property must be established. Let
•

e

/•

(∗)
g

f



•



m

/•

be a commutative square with e ∈ E and m ∈ M. Let f = m0 ◦ e0 and g = m00 ◦ e00
be (E, M )-factorizations. Then there exists a unique isomorphism h that makes thediagram• e /•
(∗∗)
e0



•o
m0

h



•

e00



•


m

m00

/•

commute. Hence d = m0 ◦ h ◦ e00 is a diagonal for (∗). To show uniqueness, let d˜ be a
diagonal for (∗). If d˜ = m̃ ◦ ẽ is an (E, M )-factorization, then there exist isomorphisms
h0 and h00 such that the diagrams
/•
~
0
~
h ~
m̃
e0
~~
 ~~

/•
•

•

ẽ◦e

m0

•
and

ẽ

e00

~~
 ~~

•

/•
~

h00 ~~
m◦m̃



m00

/•

commute. Hence in (∗∗) h can be replaced by h0 ◦ h00 , which implies that h = h0 ◦ h00 .
Consequently, d˜ = m̃ ◦ ẽ = m0 ◦ h0 ◦ h00 ◦ e00 = m0 ◦ h ◦ e00 = d.
14.8 REMARK
If condition (3) in the above proposition would be weakened by just requiring theexistence but not the uniqueness of h, then (1)–(3) would not imply that (E, M )factorizations are unique. To see this, let A be a category with three objects A, B,

9th July 2006

Sec. 14]

Factorization structures for morphisms

237

and C, and four non-identity morphisms, three of which are depicted in the commutative diagram
e /
B
@@
@@
e
@ m
 f @ 
/C
B

A@

m

h

and for which the fourth is B −→ B with h ◦ h = h. If E = Iso(A) ∪ {e} and M =Iso(A) ∪ {m}, then both h and idB serve as diagonals of the above square. See also
Exercise 14B.
14.9 PROPOSITION
Let A be (E, M )-structured and let f ◦ g ∈ M.
(1) If f ∈ M, then g ∈ M.
(2) If f is a monomorphism, then g ∈ M.
(3) If g is a retraction, then f ∈ M.
Proof:(1). Let g = m ◦ e be an (E, M )-factorization. Then (f ◦ m) ◦ e and (f ◦ g) ◦ id are(E, M )-factorizations of f ◦ g. By Proposition 14.4 there is an isomorphism h withh ◦ e = id. Consequently, e is an isomorphism, and thus g = m ◦ e belongs to M.
(2). Let g = m◦e be an (E, M )-factorization. Then there exists a diagonal d that makes
the diagram• e /•
id

~

d ~~

~~
 ~~

•

f ◦g



f ◦m

/•

commute. Since f is a monomorphism, the diagram
/•
~
~
d ~
m
id
~~
~

 ~
/•
•

•

e

g

commutes as well. Hence Lemma 14.5 implies that g ∈ M.
(3). Let f = m ◦ e be an (E, M )-factorization, and let s be a morphism with id = g ◦ s.
Then there exists a diagonal d that makes the diagram
/•
~
d ~~
s
m
~~
~
 ~

/•
•

•

e

f ◦g

9th July 2006

238

Factorization Structurescommute. Hence[Chap. IV

/•
~
~
g◦d ~
m
id
~~
 ~~

/•
•

•

e

f

commutes, so that, by Lemma 14.5, f ∈ M.

RELATIONSHIP TO SPECIAL MORPHISM CLASSES
14.10 PROPOSITION
If A is (E, M )-structured, then the following hold:(1) E ⊆ Epi(A) implies that ExtrMono(A) ⊆ M.
If, moreover, A has (Epi,Mono)-factorizations, then(2) Epi(A) ⊆ E implies that M ⊆ ExtrMono(A).
(3) Epi(A) = E implies that M = ExtrMono(A).
Proof:(1). If f = m ◦ e is an (E, M )-factorization of an extremal monomorphism, thene ∈ Epi(A) implies that e ∈ Iso(A). Hence f ∈ M.
(2). We will show that whenever m ∈ M is factored as m = f ◦ e, where e is an epimorphism, then e is an isomorphism. This will imply that: (a) m is a monomorphism(consider an (Epi, Mono)-factorization of m), and (b) m is extremal. It is clearlysufficient to show that e is a section, and this follows from the existence of a diagonal
d for the diagram• e /•
id

~~
 ~~

•
(3) follows from (1) and (2).

~

d ~~
m



f

/•



14.11 PROPOSITION
If A is (E, M )-structured and has products of pairs, then the following conditions areequivalent:(1) E ⊆ Epi(A),(2) ExtrMono(A) ⊆ M,
(3) Sect(A) ⊆ M,(4) for each object A, the diagonal morphism ∆A = hidA , idA i : A → A × A belongs toM,

9th July 2006

Sec. 14]

Factorization structures for morphisms

239(5) f ◦ g ∈ M implies that g ∈ M,(6) f ◦ e ∈ M and e ∈ E imply that e ∈ Iso(A),(7) M = {f ∈ Mor(A) | f = g ◦ e and e ∈ E imply that e ∈ Iso(A)}.
Proof: The implication (1) ⇒ (2) follows from Proposition 14.10(1). The implications(2) ⇒ (3) ⇒ (4), are obvious.
e(4) ⇒ (1). Consider A −→ B in E and B// C with r ◦ e = s ◦ e (= h). Then therersexists a diagonal morphism d that makes the diagram
/
wB
w
d ww
hr,si
h
ww
w
w

 {w
/C ×C
C
e

A

∆C

commute. Hence r = d = s, so that e is an epimorphism.
Therefore conditions (1)–(4) are equivalent.
(1) ⇒ (6). If e ∈ E and f ◦ e ∈ M, then there exists a diagonal morphism d that makes
the diagram• e /•
id

~

d ~~

~~
 ~~

•

f ◦e



f

/•

commute. Hence e is a section and an epimorphism, i.e., an isomorphism.
(6) ⇒ (7) is obvious.
(7) ⇒ (5). If f ◦g ∈ M and g = m◦e is an (E, M )-factorization, then e is an isomorphism.
Hence g ∈ M.
(5) ⇒ (3) follows from Iso(A) ⊆ M.

14.12 COROLLARY
If A is (E, Mono)-structured and has products of pairs, then E = ExtrEpi(A).
Proof: Immediate from the Proposition 14.11 and the dual of Proposition 14.10(3). 
14.13 REMARKS(1) If A does not have products of pairs, the conditions of the above proposition needno longer be equivalent. Consider, e.g., a category A with Ob(A) = , and
 



 0

 m<n
2
elements if
homA (n, m) having exactly
m=n+1


 


1
otherwiseZ

9th July 2006

240

Factorization Structures[Chap. IV

If E = Mor(A) and M = Iso(A) = Mono(A) = Epi(A), then A is (E, M )-structured
and conditions (2), (3), (5), (6), and (7) of Proposition 14.11 are satisfied, but (1)is not.
The fact that A is (M, E)-structured also shows that the results (2) and (3) of
Proposition 14.10 need not hold for factorization structures for morphisms in arbitrary categories.
(2) As will be seen in Theorem 15.4, factorization structures (E, M) for sources in a
category A always satisfy E ⊆ Epi(A).
14.14 PROPOSITION
If A has (RegEpi, Mono)-factorizations, then the following hold:(1) A is (RegEpi, Mono)-structured,(2) RegEpi(A) = ExtrEpi(A),(3) the class of regular epimorphisms in A is closed under composition,(4) if f ◦ g is a regular epimorphism in A, then so is f .
Proof:(1). Obviously, every category has the unique (RegEpi, Mono)-diagonalization property.
(2). This follows from the dual of Proposition 14.10(3).
(3). This follows from (1) and Proposition 14.6(2).
(4). This follows from (2).

RELATIONSHIP TO LIMITS
14.15 PROPOSITION
If A is (E, M )-structured, then M is closed under the formation of products and pullbacks, and M ∩ Mono(A) is closed under the formation of intersections.63
Proof: Q
Πmi Q
Bi be a product of morphisms mi in M, and let(1). Let Ai −−−→
Q
Q
Πmi Q
e
m Q
Ai −−−→Bi =Ai −→ C −−→
Bi be an (E, M )-factorization. Then foreach index j there exists a diagonal morphism dj that makes the following diagram
commute (where πj and ρj are the projection morphisms):
Q
πj

Ai

/C
z

dj zzz

z
zz
 }zz

Aj
63

e

mj



ρj ◦m

/ Bj

In fact, M is closed under the formation of multiple pullbacks (cf. Exercise 11L).

9th July 2006

Sec. 14]

Factorization structures for morphisms

241

This implies that hdi i ◦ e = idΠAi and Πmi ◦ hdi i = m. Therefore Lemma 14.5
implies that Πmi ∈ M.
(2). Let the diagram

f

/B

m

A





D

f

/C

m
m

e

m̃

be a pullback with m ∈ M, and let A −−→ B = A −→ A0 −−→ B be an (E, M )factorization. Then there exists a diagonal morphism d that makes the diagram
/ A0
}
}
d }
f ◦m̃
f
}}
}
}
 ~

/C
D

A

e

m

g

commute. By the pullback-property there exists a morphism A0 −→ A with m̃ =m ◦ g and d = f ◦ g. This implies that m ◦ (g ◦ e) = m and f ◦ (g ◦ e) = f and hence
that g ◦ e = idA . By Lemma 14.5 this implies that m ∈ M.
f

fi

m

i
(3). Let A −→ B = A −−→ Ai −−→
B be an intersection with each mi being a monomor-

f

e

m

phism in M, and let A −→ B = A −→ C −−→ B be an (E, M )-factorization. Thenfor each index i there exists a diagonal morphism di that makes the diagram
/C
}
di }}
m
fi
}}
 ~}}

/B
Ai

A

e

mi

d

commute. Thus m factors through each mi . Hence there exists a morphism C −→ A
with m = f ◦ d. Consequently, f ◦ d ◦ e = f , which implies that d ◦ e = idA . So, byLemma 14.5, f ∈ M.

We conclude this section with several results showing that the existence of suitable
limits or colimits in a category guarantees the existence of distinguished factorization
structures for morphisms.
14.16 FACTORIZATION LEMMAf
Let A have intersections and equalizers, let C −→ D be an A-morphism, and let M ⊆Mono(A) satisfy the following conditions:(a) intersections of families of M -subobjects of D belong to M,(b) if f = m̂ ◦ g ◦ h with m̂ ∈ M and g ∈ RegMono(A), then m̂ ◦ g ∈ M.
Then there exist m ∈ M and e ∈ Epi(A), such that

9th July 2006

242

Factorization Structures[Chap. IV(1) f = m ◦ e,(2) if f = m ◦ g with m ∈ M, then there exists a diagonal d that makes the diagram
/•
~
~
d ~
g
m
~~
 ~~

/•
•
e

•

m

commute,(3) if e = m ◦ g, where m ◦ m ∈ M, then m ∈ Iso(A).
miProof: Let S be the sink consisting of all M -subobjects Di −−→
D of D through which
f factors (i.e., f = mi ◦ gi for some gi ), and let (B, m) be the intersection of S. Thenm ∈ M and there exists a morphism e with f = m◦e. By construction, (2) is immediate.
To establish (3) consider a factorization e = m ◦ g of e with m ◦ m ∈ M. Then, by (2),there exists a diagonal d that makes the diagram

•
g

e

~~
 ~~

•

/•
~

d ~~

m◦m

m



/•

commute. Since m is a monomorphism, this implies that m ◦ d = id. Hence m is aretraction and, being the first factor of the monomorphism m ◦ m, a monomorphism;
thus an isomorphism. Hence (3) holds. It remains to be shown that e is an epimorphism.
Consider a pair (r, s) of morphisms with r ◦ e = s ◦ e, and let g be an equalizer of (r, s).
Then there exists a morphism h with e = g ◦ h. Hence f = m ◦ g ◦ h. By (b) m ◦ g ∈ M,
so that by (3) g is an isomorphism. Thus r = s.
14.17 THEOREM
If A has finite limits and intersections, then A is (ExtrEpi, Mono)-structured.
Proof: Application of Lemma 14.16 to M = Mono(A) yields that A is (ExtrEpi,
Mono)-factorizable. To show that A has the (automatically unique) (ExtrEpi, Mono)diagonalization property, consider a commutative diagram
•

e

/•
g

f



•



m

/•

with e an extremal epimorphism and m a monomorphism. Let
•
ĝ

m̂

/•
g





•

m

/•

9th July 2006

Sec. 14]

Factorization structures for morphisms

243

be a pullback. Then there exists a unique morphism h such that the diagram• @ e / •O
@
f



•o

@@
@@
@

m̂

h

ĝ

•

commutes. Since e is an extremal epimorphism and m̂ (being a pullback of a monomorphism) is a monomorphism, m̂ must be an isomorphism. Hence d = ĝ ◦ m̂−1 is thedesired diagonal.

14.18 PROPOSITION
If A has the (Epi, ExtrMono)-diagonalization property, then the class of extremal monomorphisms in A is closed under composition and intersections.
Proof: Since monomorphisms are closed under composition and intersections, only thestability of the extremal-property needs to be verified. If m and m are extremal monomorphisms and m ◦ m = f ◦ e for some epimorphism e, then by assumption there exists
a diagonal d that makes the diagram
•
m

e

~~
 ~~

•

/•
~

d ~~
m


/•

f

commute. Hence m = d ◦ e implies that e is an isomorphism.
mdi
Now let A = (Ai −−→A)I be a family of extremal subobjects of A. If D −→ A =

d

m

i
i
Ai −−→A is an intersection of A, and if d = f ◦ e for some epimorphism e, thenD −−→
for each i ∈ I there exists a diagonal ki that makes the diagram

/•
~
~
ki ~
f
di
~~
 ~~

/•
•

•

e

mi

commute. Since the intersection (being a limit) is an extremal mono-source, the epimorphism e must be an isomorphism.

14.19 THEOREM
If A has equalizers and intersections, then A is (Epi, ExtrMono)-structured.
Proof: First we show that A has the (Epi, ExtrMono)-diagonalization property. Let
A

c

/B

g



C

9th July 2006



f

h

/D

244

Factorization Structures[Chap. IV

be a commutative diagram, where c is an epimorphism and f is an extremal monomorphism. Let M be the class of all subobjects (Dm , m) of D with the property that there
exist (necessarily unique) morphisms fm and hm that make the diagram
c

A

g

C

D
= mC

fm |||

|
||
 ||

/B
{
hm {{{
{{
}{{

f

h

CC
CC
m CCC
! 
/D

commute. Then M satisfies the conditions (a) and (b) of Lemma 14.16, so that there
exists an (Epi, M )-factorization f = m ◦ e. Since f is an extremal monomorphism, this
implies that e (= fm ) is an isomorphism. Thus d = e−1 ◦ hm is the desired diagonal.
Consequently, A is (Epi, ExtrMono)-diagonalizable. This fact, together with Proposition
14.18, implies that the class M = ExtrMono(A) satisfies conditions (a) and (b) of Lemma
14.16 for every A-morphism f . Hence A is (Epi, M )-factorizable.

14.20 COROLLARY
In a category with equalizers and intersections the class of extremal monomorphisms isthe smallest class of morphisms that contains all regular monomorphisms and is closed
under composition and intersections.
Proof: By the above results (14.18 and 14.19) and Corollary 7.63, ExtrMono(A) contains all regular monomorphisms and is closed under composition and intersections. If
M̂ is a class of morphisms with these properties, then M = M̂ ∩ M ono(A) satisfies
conditions (a) and (b) of Lemma 14.16 for every A-morphism f , which implies that
A is (Epi, M )-factorizable. If f = m ◦ e is an (Epi, M )-factorization of an extremal
monomorphism f , then e is an isomorphism, and so belongs to M. Thus f = m◦e ∈ M.
14.21 COROLLARY
Every strongly complete category is (ExtrEpi, Mono)-structured and (Epi, ExtrMono)structured.

14.22 PROPOSITION
A category with pullbacks and coequalizers is (RegEpi, Mono)-structured if and only if
regular epimorphisms are closed under composition.
Proof: The composition closure is a necessary condition, by Proposition 14.6(2). To
show that it is sufficient, observe that by Proposition 14.14 it need only be shown thatevery morphism f has a (RegEpi, Mono)-factorization. Let (p, q) be a congruencerelation of f , and let c be a coequalizer of (p, q). Then there exists a unique morphism
m with f = m ◦ c. To show that m is a monomorphism, let (r, s) be a pair of morphisms
with m ◦ r = m ◦ s and let c be a coequalizer of (r, s). Then there exists a unique

9th July 2006

Sec. 14]

Factorization structures for morphisms

245

morphism g with m = g ◦ c. Since (p, q) is a congruence relation of f = g ◦ c ◦ c and sincec ◦ p = c ◦ q, (p, q) is a congruence relation of c and of c ◦ c. Since c and c ◦ c are both
regular epimorphisms, each (by Proposition 11.22) is a coequalizer of (p, q). Hence, by
essential uniqueness of coequalizers, c must be an isomorphism. So r = s. Thus m is amonomorphism.

14.23 EXAMPLES(1) Many familiar categories are (RegEpi, Mono)- and (Epi, RegMono)-structured, e.g.,
Set, Vec, Grp, Pos, Top, Aut, Σ-Seq, and HComp. So are any of the constructs
Alg(Ω) and Spa (T).
(2) Cat is (ExtrEpi, Mono)-structured, but it has extremal epimorphisms that are notregular (cf. 7.76).
(3) Sgr is (Epi, ExtrMono)-structured, but it has extremal monomorphisms that are
not regular (cf. 14I).

Suggestions for Further Reading
Kennison, J. F. Full reflective subcategories and generalized covering spaces. Illinois J.
Math. 12 (1968): 353–365.
Ringel, C. M. Diagonalisierungspaare I, II. Math. Z. 117 (1970): 249-266, and 122(1971): 10–32.
Freyd, P. J., and G. M. Kelly. Categories of continuous functors I. J. Pure Appl. Algebra 2 (1972): 169–191.
Bousfield, A. K. Constructions of factorization systems in categories. J. Pure Appl.
Algebra 9 (1977): 207–220.
Manes, E. G. Compact Hausdorff objects. Topol. Appl. 4 (1979): 341–360.
Cassidy, C., M. Hébert, and G. M. Kelly. Reflective subcategories, localizations andfactorization systems. J. Austral. Math. Soc. 38 (1985): 287–329, Corrigenda
41 (1986): 286.

EXERCISES
14A. Factorization Structures for Morphisms in Special Categoriesf

Let A −→ B be a morphism in one of the categories Set, Vec, Top, Pos, Sgr, or Rng.
femShow that the familiar “image”-factorization A −→ B = A −→ f [A] −−→ B is(a) an (Epi, ExtrMono)-factorization in Set, Vec, Top, and Pos,(b) an (ExtrEpi, Mono)-factorization in Set, Vec, Sgr, and Rng.

9th July 2006

246

Factorization Structures[Chap. IV

14B. Diagonals
1. Show that Set has(1) (Mono, Epi)-factorizations and -diagonalizations, but not unique ones,(2) (Section, Projection)-factorizations and -diagonalizations, but not unique ones.
2. Show that every category that has pullbacks has the unique (ExtrEpi, Mono)diagonalization property.
14C. Strong Monomorphisms
A monomorphism m in A is called strong provided A has the unique (Epi, {m})diagonalization property. Show that(a) Every strict monomorphism is strong, but not vice versa.
(b) Every strong monomorphism is extremal, but not vice versa.
(c) The class of strong monomorphisms is closed under composition, intersections, pullbacks, products, and left-cancellation.
(d) If A has pushouts, then in A StrongMono = ExtrMono.
(e) If A has equalizers and intersections, then in A StrongMono = ExtrMono [cf. Theorem 14.19].
(f) If A is (Epi, M )-structured for some class M of monomorphisms, thenM = StrongMono = ExtrMono.
(g) If A is (Epi, M )-structured, then M need not consist of monomorphisms alone [cf.
Remark 14.13(1)].
14D. (RegEpi, Mono)-Structured Categories
Let A have pullbacks and coequalizers. Show that(a) A is (RegEpi, Mono)-structured if and only if RegEpi(A) = ExtrEpi(A). [Cf. Propositions 14.22 and 7.62(1).](b) If regular epimorphisms are stable in A(cf. Exercise 11J), then A is (RegEpi, Mono)structured.
(c) If there exists a faithful functor from A into a (RegEpi, Mono)-structured category that preserves and reflects regular epimorphisms, then A is (RegEpi, Mono)structured.
14E. Regular Categories
A category A is called regular provided that it satisfies the following conditions:(1) A has finite limits,(2) A is (RegEpi, Mono)-structured,(3) In A regular epimorphisms are stable (cf. 11J).

9th July 2006

Sec. 14]

Factorization structures for morphisms

247

Show that(a) If A has coequalizers, then A is regular provided that it satisfies (1) and (3) above.
(b) If A is regular and X is small, then [X, A] is regular.
14F. Exact Categories
Pointed categories that are (NormalEpi, NormalMono)-structured are called exact.
Show that(a) A pointed category is exact if and only if it has (NormalEpi, NormalMono)-factorizations.
(b) If A is exact, then so is Aop .
(c) In an exact category every monomorphism is normal.
(d) An exact category has kernels and finite intersections.
(e) A nonempty exact category has a zero-object.
(f) An exact category is wellpowered if and only if it is co-wellpowered.
(g) Exact constructs are wellpowered and co-wellpowered.
(h) Vec and Ab are exact, but none of Grp, Mon, or pSet is exact.
14G. Closure Properties
Let A be (E, M )-structured. Show that(a) M is closed under the formation of multiple pullbacks (cf. Proposition 14.15).
(b) M is closed under formation of retracts in the arrow category A2 (cf. Exercise
3K(b)).
* 14H. (E, −)-structured Categories
A is called (E, −)-structured provided that there exists some M such that A is
(E, M )-structured. Show that a cocomplete category A is (E, −)-structured if and only
if E ⊆ M or(A) satisfies the following conditions:(1) Iso(A) ⊆ E,
(2) E is closed under composition,(3) E is closed under the formation of pushouts,(4) E is closed under the formation of colimits,(5) if e = f ◦ e with e and e in E, then f ∈ E,

9th July 2006

248

Factorization Structures[Chap. IV(6) (solution set condition) for each A-morphism f there exists a set of factorizations
f = gi ◦ ei , i ∈ I, with ei ∈ E such that for each factorization f = g ◦ e with e ∈ E
there is some i ∈ I and some morphism h such that the following diagram commutes:
/•
~
~
h ~
g
ei
~~
 ~~

/•
•
e

•

gi

14I. Regular Monomorphisms in the Category of Semigroups
Let C = {0, a, b, c, d, e} and consider the following multiplication table:
·
0
a
b
c
d
e

0
0
0
0
0
0
0

a
0
0
0
0
b
c

b
0
0
0
0
c
0

c
0
0
0
0
0
0

d
0
b
c
0
e
0

e
0
c
0
0
0
0

(a) Prove that C with the above multiplication · is a semigroup.
(b) A = {0, a, b} and B = {0, a, b, c} are subsemigroups of C. Let f : A ,→ B andg : B ,→ C be the inclusion homomorphisms. Prove that f and g are regularmonomorphisms and that g ◦ f is an extremal monomorphism in Sgr.
(c) Prove that if Ch /k

/ D are morphisms in Sgr that coincide on A, then h and k

coincide on B. [Hint: Use the equalities b = da and c = bd.] Conclude that g ◦ f is
not a regular monomorphism in Sgr.
(d) Let B̂ be the free semigroup on three generators {â, b̂, ĉ}; let Â be the subsemigroupof B̂ generated by {â, b̂â, âĉ}; let Ĉ be the quotient semigroup obtained by identifying
the words â and b̂âĉ; and let i : Â ,→ B̂ and p : B̂ → Ĉ be the inclusion map andnatural map, respectively.
(1) Construct a semigroup D̂ and homomorphisms Ĉ

r
s

//

D̂ such that p ◦ i is anequalizer of r and s.
(2) Show that i is not a regular monomorphism in Sgr.
(e) Conclude that for the complete, wellpowered category Sgr, the following hold:(1) The class of regular monomorphisms coincides with the class of strict monomorphisms.
(2) There exist extremal monomorphisms that are not regular.
(3) The class of regular monomorphisms is not closed under composition.
(4) Sgr is not (Epi, RegMono)-factorizable.

9th July 2006

Sec. 14]

Factorization structures for morphisms

249(5) The first factor of a regular monomorphism is not necessarily regular.
14J. Dominionsf
Let A be a wellpowered complete category, and let X −→ Y be an A-morphism. Prove
that
f

g

d

(a) X −→ Y has a factorization X −→ D −→ Y , where d is a regular monomorphismthat is characterized uniquely by any of the following equivalent conditions:(1) for all morphisms r and s, r ◦ f = s ◦ f implies that r ◦ d = s ◦ d,(2) (D, d) is the smallest regular subobject of Y through which f can be factored,(3) d is the intersection of all regular subobjects of Y through which f can befactored.
(D, d) is called a dominion of f .
(b) Any two dominions of f are isomorphic subobjects of Y .
(c) In A the extremal monomorphisms are precisely the regular monomorphisms if andonly if the (Epi, ExtrMono)-factorization of any morphism is the same as its dominion factorization.
(d) f is an epimorphism if and only if (Y, idY ) is a dominion of f .
(e) f is a regular monomorphism if and only if (X, f ) is a dominion of f .
(f) If f = r ◦ s, where s is an epimorphism, then the dominions of f and r coincide.
(g) Consider the pushout square
•
f

f

/•
p



•



q

/•

Then (D, d) is a dominion of f if and only if d is an equalizer of p and q.

9th July 2006

15 Factorization structures for sources
Because of their generality, factorization structures for sources are frequently a more
powerful categorical tool than factorizations for morphisms. However, in two respects
they are more restrictive. Namely, if (E, M) is a factorization structure for sources inA, then(1) E must be contained in the class of epimorphisms of A (15.4), and(2) certain colimits of diagrams in A involving morphisms in E must exist (15.14).
Such source factorization structures occur quite frequently (15.10 and 15.25), and forco-wellpowered categories with products there is no essential difference between these
two approaches to factorization since for such categories A every factorization structure
for morphisms, with E ⊆ Epi(A), has a unique extension to a factorization structure
for sources (15.21). In particular, every strongly complete, co-wellpowered category has
both (Epi, ExtrMono-Source) and (ExtrEpi, Mono-Source) as factorization structures
for sources (15.25).
15.1 DEFINITION
Let E be a class of morphisms and let M be a conglomerate of sources in a category A.
(E, M) is called a factorization structure on A, and A is called an (E, M)-category
provided that(1) each of E and M is closed under compositions with isomorphisms in the following
sense:(1a) if e ∈ E and h ∈ Iso(A) and h ◦ e exists, then h ◦ e ∈ E,(1b) if S ∈ M and h ∈ Iso(A) and S ◦ h exists, then S ◦ h ∈ M,
(2) A has (E, M)-factorizations (of sources); i.e., each source S in A has a factorization S = M ◦ e with e ∈ E and M ∈ M, ande(3) A has the unique (E, M)-diagonalization property; i.e., whenever A −→ B and
f

gi

m

i
A −→ C are A-morphisms with e ∈ E, and S = (B −−→ Di )I and M = (C −−→Di )I
are sources in A with M ∈ M, such that M ◦ f = S ◦ e, then there exists a uniqueddiagonal, i.e., a morphism B −→ C such that for each i ∈ I the diagram

/B
}
}
d }}
gi
f
}}
}

 ~}
/ Di
C

A

e

mi

commutes.

250

Sec. 15]

Factorization structures for sources

251

Factorization of sources

15.2 REMARKS(1) As opposed to the concept of factorization structures for morphisms the conceptof factorization structures (for sources) is not self-dual. The dual concept, that offactorization structures for sinks, will not be explicitly formulated here.
(2) Another distinction from factorization structures for morphisms is the fact that theuniqueness requirement for diagonals in the definition of factorization structures (forsources) is redundant. Cf. Exercise 14B on the one hand, and the proof of Theorem
15.4 on the other.
(3) If (E, M) is a factorization structure on A and M is the class of those A-morphismsthat (considered as 1-sources) belong to M, then (E, M ) is a factorization structure
for morphisms on A. (Cf. footnote to Definition 14.1) Hence all the results of §14
apply.
(4) Factorization of sources applies to empty sources too. If (E, M) is a factorization
structure on A, and AM is the class of all A-objects A such that the empty source(A, ∅) with domain A belongs to M, then conditions (2) and (3) in Definition 15.1
translate into:e(2∅ ) for every object A there exists a morphism A −→ B in E with B ∈ AM ,fe(3∅ ) for every morphism A −→ B in E and every morphism A −→ C with C ∈ AMdthere exists a unique morphism B −→ C with f = d ◦ e.

9th July 2006

252

Factorization Structures[Chap. IV

15.3 EXAMPLES(1) Every category is an (Iso, Source)-category.64 This factorization structure is called
trivial. Also, every category has the (unique) (RegEpi, Mono-Source)-diagonalization property, although not all categories are (RegEpi, Mono-Source)-categories(cf. Proposition 15.13).
(2) Set is an (Epi, Mono-Source)-category. For Set this is the only nontrivial factorfi

ization structure. (Epi, Mono-Source)-factorizations of a source (A −−→ Ai )I in Set
can be obtained via either of the following two constructions:(a) Define an equivalence relation on A by: “a ∼ b if and only if fi (a) = fi (b) foreeach i ∈ I”, and let A −→ A/∼ be the naturally associated surjection. Then formieach i ∈ I there exists a unique map A/∼ −−→
Ai with fi = mi ◦ e. Then
fi

m

e

i
A −−→ Ai = A −→ A/∼ −−→
Ai

is an (Epi, Mono-Source)-factorization of (fi ) in Set.
Q(b) Consider the cartesian product ( Ai , πi )I of the codomain (Ai )I — that is, theconglomerate65 of all functions x with domain I such that x(i) ∈ Ai for eachf Q
i ∈ I. The function A −→ Ai defined by a 7→ f (a), where (f (a))(i) = fi (a)
f Q
e
m Q
for each i ∈ I, has a factorization A −→Ai = A −→ B −−→
Ai , with eea surjective function and m an injective one. Since A is a set and A −→ B
is surjective, B is codable by a set; i.e., there exists a set B 0 and a bijectionmib
Ai defined by mi (b) = (m(b))(i), it follows thatB 0 −→ B. Then for B −−→fib−1 ◦ e

m ◦b

i
−→ Ai
A −−→ Ai = A −−−−→ B 0 −−−

is an (Epi, Mono-Source)-factorization of (fi ) in Set.
For various constructs, one of the above constructions can be used to obtain (Epi,Mono-Source)-factorizations.
(3) Many algebraic constructs such as Vec, Grp, Mon, Rng, and Alg(Ω) are(RegEpi, Mono-Source)-categories.
(4) Cat is an (ExtrEpi, Mono-Source)-category.
64

Source (resp. Source(A)) is the conglomerate of all sources (in the category A). Similarly, Mono-Sourceis the conglomerate of all mono-sources in the given category, etc. See the Table of Symbols.
65
One might be tempted to describe this construction more categorically by forming the quasicategory
Class of all classes and all functions, and by letting ( Ai , πi )I be the product of (Ai )I in thatquasicategory. This, however, is not possible, since the conglomerate ( Ai )I frequently fails to be aclass and, hence, to be an object of the quasicategory Class. In fact, the quasicategory Class behaves
badly with respect to many familiar constructions (recall, e.g., that there is a largest class, U). This
problem could be resolved, e.g., by introducing in addition to “sets”, “classes”, and “conglomerates”,
one higher level of entities, say, “collections”. Then the collection of all conglomerates and allfunctions between them would form a rather well-behaved “quasi-quasicategory”. Since there are
only a few occasions in this text where the use of something like this would be advantageous, we haverefrained from complicating the foundations by introducing it.

Q

Q

9th July 2006

Sec. 15]

Factorization structures for sources

253(5) Aut is an (RegEpi, Mono-Source)-category.
(6) Top has a proper class (even an illegitimate conglomerate) of factorization structures(cf. Exercise 15L). In particular, Top is an (Epi, ExtrMono-Source)-category, an(ExtrEpi, Mono-Source)-category, and a (Bimorphism, Initial-Source)-category.
15.4 THEOREM
If A is an (E, M)-category, then E ⊆ Epi(A).
Proof: Consider e ∈ E and a pair (r, s) of morphisms with r ◦ e = s ◦ e. The source(hf )f ∈Mor(A) , defined by hf = r ◦ e, has an (E, M)-factorization (hf ) = (mf ) ◦ e. The
source (gf )f ∈Mor(A) , defined by(r, if mf ◦ f = s
gf =s, otherwise,
satisfies the equation (mf ) ◦ e = (gf ) ◦ e. Hence there exists a diagonal d that makes thediagram• e /•
e

~

d ~~

~~
 ~~

•

mf



gf

/•

commute for each f ∈ Mor(A). In particular,(r, if md ◦ d = s
md ◦ d = gd =s, otherwise.
This is possible only for r = s. Thus e is a epimorphism.
15.5 PROPOSITION
If A is an (E, M)-category, then the following hold:(1) (E, M)-factorizations are essentially unique,(2) E ⊆ Epi(A) and ExtrMono-Source(A) ⊆ M,
(3) E ∩ M = Iso(A),(4) each of E and M is closed under composition,(5) if f ◦ g ∈ E and g ∈ Epi(A), then f ∈ E,(6) if f ◦ g ∈ E and f ∈ Sect(A), then g ∈ E,(7) if (Si ) ◦ S ∈ M, then S ∈ M,(8) if a subsource of S belongs to M, then S belongs to M,(9) E and M determine each other via the diagonalization-property; moreover,

9th July 2006



254

Factorization Structures[Chap. IV(a) a source belongs to M if and only if every E-morphism through which it factors
is an isomorphism,(b) if M consists of mono-sources only, then a morphism f belongs to E if and only
if f = m ◦ g with m ∈ M implies that m ∈ Iso(A).
Proof: (8) follows from (9). All other conditions are proved as in §14 by use of Theorem
15.4.


RELATIONSHIP TO SPECIAL MORPHISMS
AND SPECIAL SOURCES
15.6 PROPOSITION
If A is a (RegEpi, M)-category, then M contains all mono-sources of A.
Proof: Let S be a mono-source and S = M ◦ e be a (RegEpi, M)-factorization of S.
Let e be a coequalizer of r and s. Then S ◦ r = M ◦ e ◦ r = M ◦ e ◦ s = S ◦ s. Since S
is a mono-source, this implies that r = s. Hence e is an isomorphism, so that S ∈ M.
15.7 PROPOSITION
For (E, M)-categories A, the following are equivalent:(a) M ⊆ Mono-Source(A),(b) A has coequalizers and RegEpi(A) ⊆ E.
Proof: (a) ⇒ (b). Let Ars// B be a pair of morphisms. Consider the source S =fi(B −−→ Bi )I that consists of all morphisms fi with fi ◦ r = fi ◦ s. If S = M ◦ e isan (E, M)-factorization, then, since M is a mono-source, e is a coequalizer of (r, s).
Hence A has coequalizers. If c is a coequalizer of some pair (r, s), and c = m ◦ e is an(E, M)-factorization, then e is a coequalizer of (r, s). Hence m ∈ Iso(A), so that c ∈ E.
(b) ⇒ (a). Let M belong to M and let (r, s) be a pair of morphisms with M ◦ r = M ◦ s.
Then M factors through the coequalizer c of (r, s). Hence c ∈ E implies, by Proposition
15.5(9)(a), that c is an isomorphism. So r = s.
15.8 PROPOSITION
For (E, M)-categories A the following hold:(1) if M = Mono-Source(A), then E = ExtrEpi(A),(2) if M = ExtrMono-Source(A), then E = Epi(A),(3) if E = Epi(A), then the following conditions are equivalent:(a) M = ExtrMono-Source(A),9th July 2006

Sec. 15]

Factorization structures for sources

255(b) A has coequalizers,(4) if E = ExtrEpi(A) or E = RegEpi(A), then the following conditions are equivalent:(a) M = Mono-Source(A),(b) A has coequalizers.
Proof:(1). Apply Theorem 15.4 and Proposition 15.5(9)(b).
(2). E ⊆ Epi(A) by Theorem 15.4. If f is an epimorphism and f = m ◦ e is an (E, M )factorization, then m is an extremal monomorphism and (as the second factor ofan epimorphism) an epimorphism; hence an isomorphism. Thus f ∈ E.
(3a) ⇒ (3b) and (4a) ⇒ (4b) follow from Proposition 15.7.
(3b) ⇒ (3a) follows from Propositions 15.5(9)(a) and 15.7.
(4b) ⇒ (4a). By Proposition 15.7, M ⊆ Mono-Source(A). If S is a mono-source andS = M ◦ e is an (E, M)-factorization, then e is an extremal epimorphism and a monomorphism; hence an isomorphism. Thus S ∈ M.
15.9 EXAMPLE
The category A described in Example 14.13(1) is simultaneously an (Epi, Source)category, an (ExtrEpi, Source)-category and a (RegEpi, Source)-category. But in A
not every source is a mono-source.
Also, every group, considered as a category, is an (Epi, Source)-category; but for anonzero group the empty source is not a mono-source.

EXISTENCE OF FACTORIZATION STRUCTURES
15.10 THEOREM
Every category that has (Epi, Mono-Source)-factorizations is an (ExtrEpi, MonoSource)-category.
fi

Proof: Let A have (Epi, Mono-Source)-factorizations and let S = (A −−→ Ai )I be aejsource in A. Consider the source T = (A −−→ Bj )J that consists of all epimorphisms
mij
ej for which there exists a (necessarily unique) mono-source Mj = (Bj −−−→ Ai )I withnje
S = Mj ◦ ej . Let T = N ◦ e = (A −→ B −−→ Bj )J be an (Epi, Mono-Source)factorization. Then the sources (Mj ) ◦ nj do not depend upon j and thus each canmibe denoted by M = (B −−→
Ai )I . Hence S = M ◦ e is an (Epi, Mono-Source)factorization. To show that e is an extremal epimorphism, consider a factorizatione = m ◦ g with monomorphic m. Let g = n ◦ e be an (Epi, Mono)-factorization ofg. Then S = (M ◦ m ◦ n) ◦ e is an (Epi, Mono-Source)-factorization of S. Hence
there exists some j in J with M ◦ m ◦ n = Mj and e = ej . For this j, the equations

9th July 2006

256

Factorization Structures[Chap. IVm◦n◦nj ◦e = m◦n◦ej = m◦n◦e = m◦g = e hold. Since e is an epimorphism, we conclude
that m ◦ n ◦ nj = id. Hence m is a monomorphic retraction, i.e., an isomorphism. Thus
A is (ExtrEpi, Mono-Source)-factorizable. To show that A has the (ExtrEpi, MonoSource)-diagonalization property, consider morphisms ej and sources Sj , for j = 1, 2,
such that e1 is an extremal epimorphism, S2 is a mono-source, and S1 ◦ e1 = S2 ◦ e2 . Ifejmje
(A −−→ Bj )j∈{1,2} = (A −→ B −−→ Bj )j∈{1,2} is an (Epi, Mono-Source)-factorization,then S1 ◦ m1 ◦ e = S1 ◦ e1 = S2 ◦ e2 = S2 ◦ m2 ◦ e. Hence S1 ◦ m1 = S2 ◦ m2 .
e1
/•
@@
~?
~
m
1 ~
@@e
@@ ~~~
 ~
e2
S1
•
~
m2 ~~
~~
 ~~

/•
•

•@

S2

Let C

r
s

//

B be a pair of morphisms with m1 ◦ r = m1 ◦ s. Then
S2 ◦ m2 ◦ r = S1 ◦ m1 ◦ r = S1 ◦ m1 ◦ s = S2 ◦ m2 ◦ s.
mj

Since S2 is a mono-source, this implies that m2 ◦ r = m2 ◦ s. Since (B −−→ Bj )j∈{1,2} isa mono-source, this implies that r = s. Thus m1 is a monomorphism. Since e1 = m1 ◦ eis an extremal epimorphism, this implies that m1 is an isomorphism. Consequently,
d = m2 ◦ m−1
1 is the desired diagonal.
15.11 REMARKAs the above result shows, every (Epi, Mono-Source)-factorizable category is an(E, Mono-Source)-category for a suitable class E. However, it need not be an (Epi,M)-category for any conglomerate M:
Every poset, considered as a category, is an (ExtrEpi, Mono-Source)-category. But a
poset with a smallest element is an (Epi, M)-category for some M if and only if it is acomplete lattice.
15.12 DEFINITION
A category is said to have regular factorizations provided that it is (RegEpi, MonoSource)-factorizable.
15.13 PROPOSITION
If a category has regular factorizations, then it is a (RegEpi, Mono-Source)-category.
Proof: Every category has the (RegEpi, Mono-Source)-diagonalization property.



15.14 THEOREM
If E is a class of morphisms in A, then A is an (E, M)-category for some M if andonly if the following conditions are satisfied:

9th July 2006

Sec. 15]

Factorization structures for sources

257(1) Iso(A) ⊆ E ⊆ Epi(A),(2) E is closed under composition,fe(3) for every A −→ B in E and every morphism A −→ C there exists a pushout square

f

/B

e

A




C

f

/D

e

for which e ∈ E,ei(4) for every source (A −−→
Ai )I that consists of E-morphisms, there exists a cointersectionpieieA −→ B = A −−→
Ai −−→ B

for which e ∈ E.
Proof: Necessity: (1) and (2) follow from Theorem 15.4 and Proposition 15.5. Tofesee (3), let A −→ B and A −→ C be morphisms with e ∈ E. Consider the sourcefi

S = (C −−→ Ci )I consisting of those morphisms fi for which there exists a (necessarily
unique) morphism gi with fi ◦ f = gi ◦ e. Let S = M ◦ e be an (E, M)-factorization.
Then there exists a diagonal that makes the diagram/B~d ~~~ g
e◦f
i
~
~

 ~~~
/ Ci
•

A

e

M

commute. As can be seen easily, the diagram
A
f

e



C

/B


e

d

/•

is a pushout.
eiTo establish (4), let (A −−→
Ai )I be a source consisting of E-morphisms. Consider thefjsource S = (A −−→ Bj )J that consists of those morphisms fj that have the property
that for each i ∈ I there exists a (necessarily unique) morphism fij with fj = fij ◦ ei ,and proceed as in the proof of part (3).
Sufficiency: Define M to be the conglomerate of all sources that do not factor througha non-isomorphic morphism in E. In order to show that A has (E, M)-factorizations,fiej

let (A −−→ Ai )I be a source in A. Consider the source S = (A −−→ Bj )J that consists

9th July 2006

258

Factorization Structures[Chap. IVof those morphisms ej ∈ E that have the property that for every i ∈ I there exists a(necessarily unique) morphism fij with fi = fij ◦ ej . Let
ej

e

pj

A −→ B = A −−→ Bj −−→ B
be a cointersection of S such that e ∈ E. Then for each i ∈ I there exists a morphism
mi that makes the diagram
fi

/ Ai
>F
}
//??? ejfij }} 
// ??

}} 
// ??}} /mie // Bj
//pj

// 
/  

A/ ?

B

m

i
commute for each j ∈ J. To show that the source M = (B −−→
Ai )I belongs to M, let
M = T ◦ ê be a factorization with ê ∈ E. Then there exists some j ∈ J with ê ◦ e = ej .
Hence, for this j the equality e = pj ◦ ej = pj ◦ ê ◦ e implies that pj ◦ ê = id. Thus ê isan epimorphic section; hence an isomorphism.

To show that A has the (E, M)-diagonalization property, let e and f be morphisms withe ∈ E, and let M = (mi )I and S = (fi )I be sources with M ∈ M and M ◦ f = S ◦ e.
Let
• e /•
f





•

ê

fˆ

/•

be a pushout square for which ê ∈ E. Then for each i ∈ I there exists a morphism githat makes the diagrame/••fˆ ~~~

~~
~~
f
fi
•
~? @@@
ê ~~
@@
~~
gi @@
 ~~
 
/•
•
mi

commute. Hence M factors through ê ∈ E, which implies that ê is an isomorphism.
Thus ê−1 ◦ fˆ is the desired diagonal.

15.15 COROLLARY
Let A be a category with pushouts and cointersections. Then a class E of A-morphisms
is part of a factorization structure on A (i.e., A is an (E, M)-category for some M)
if and only if Iso(A) ⊆ E ⊆ Epi(A) and E is closed under composition, pushouts, andcointersections.


9th July 2006

Sec. 15]

Factorization structures for sources259

15.16 COROLLARY(1) A is an (Epi, M)-category for some M if and only if A has cointersections and has
fea pushout for every 2-source of the form • ←− • −→ • with epimorphic e.
(2) A is an (Epi, ExtrMono-Source)-category if and only if A has cointersections,fepushouts for 2-sources of the form • ←− • −→ • with epimorphic e, and has coequalizers.

15.17 COROLLARY
Every strongly cocomplete category is an (Epi, ExtrMono-Source)-category.



15.18 EXAMPLE
The partially ordered class of all ordinals, considered as a category, is a cocomplete(ExtrEpi, Mono-Source)-category, but is not (Epi, ExtrMono-Source)-factorizable. [Consider empty sources.] See also Exercise 15D.
Further results concerning the existence of factorization structures will follow after somenecessary preparations, cf., e.g., Theorem 15.25.

EXTENSIONS OF FACTORIZATION STRUCTURES
Factorization structures may be considered for certain specified sources only; e.g., for(1) empty sources (= objects), see Exercise 15G,
(2) 1-sources (= morphisms), see §14,(3) 2-sources, see Exercise 15I,(4) set-indexed sources (= small sources), see Exercise 15J.
Here we are concerned with the question: Under which conditions is it possible toextend factorization structures (E, M ) for morphisms to factorization structures (E, M)
for sources, or at least for small sources?
15.19 PROPOSITION(1) If A has products, then every factorization structure (E, M ) for morphisms on A
can be uniquely extended to a factorization structure (E, M) for small sources.
(2) Conversely, if A has an initial object and each factorization structure (E, M ) for
morphisms on A can be extended to a factorization structure (E, M) for small
sources, then A has products.
Proof:(1). Let (E, M ) be a factorization structure for morphisms. If M is the class of all
small sources of the form P ◦ m, where P is a product and m ∈ M , then (E, M)

9th July 2006

260

Factorization Structures[Chap. IVfi

is a factorization structure for small sources.
Indeed, if S = (A −−→ Ai )I is a setem Q
indexed source, let hfi i = A −→ B −−→
Ai be an (E, M ) factorization. Thenπ ◦m
S = (B −−i−−→ Ai )I ◦ e is an (E, M)-factorization.
Uniqueness of the extension follows from the observation that M is determined by
E as in Proposition 15.5(9).
(2). Let (Mor(A), M) be an extension of the factorization structure (Mor(A), Iso(A))
for morphisms to a factorization structure for small sources. Then M consists
precisely of the set-indexed products in A. In fact, if (Ai )I is a set-indexed familyof objects, Q is an initial object, S = (Q → Ai )I is the associated source, andS = M ◦ e is a (Mor(A), M)-factorization, then M is a product of (Ai )I .

15.20 PROPOSITION
If (E, M) is a factorization structure for small sources on A and A is co-wellpowered,then the following conditions are equivalent:(1) (E, M) can be uniquely extended to a factorization structure (E, N) on A,(2) E ⊆ Epi(A),(3) Sect(A) ⊆ M,(4) for each object A the 2-source (idA , idA ) belongs to M,(5) whenever a subsource of a small source S belongs to M, then so does S.
Proof: (1) ⇔ (2). By Theorem 15.4, (1) implies (2). For the converse it suffices toverify that E satisfies the conditions (1)–(4) of Theorem 15.14. That 15.14(1) and15.14(2) hold follows from Proposition 14.6. The validity of 15.14(3) can be established
as in the corresponding part of the proof of Theorem 15.14 by replacing the sourcefi

S = (C −−→ Ci )I by a small subsource that contains a representative for each fi ∈ E.
Likewise the validity of 15.14(4) can be established as in the corresponding part of the
fj

proof of Theorem 15.14 by replacing the source S = (A −−→ Bj )J by a small subsourcethat contains a representative for each fj ∈ E.
The equivalence of the conditions (2)–(5) can be established as in Proposition 14.11 byusing the fact that the assumption made there that products of pairs exist is not needed
here, since we have (E, M)-factorizations for 2-sources. Details are left to the reader. 
15.21 COROLLARY
In a co-wellpowered category A with products, every factorization structure (E, M ) for
morphisms with E ⊆ Epi(A) can be uniquely extended to a factorization structure (E, M)
for sources.

15.22 EXAMPLE
The factorization structure (dense maps, closed embeddings) for morphisms• cannot be extended to a factorization structure (dense maps, M) on Top, but

9th July 2006

Sec. 15]

Factorization structures for sources261• can be uniquely extended to a factorization structure (dense maps, M) on Haus.
The latter is one of the most important factorization structures on Haus.
15.23 REMARK
Proposition 15.19 shows that products are a suitable tool for extending factorization
structures for morphisms to those for small sources. Proposition 15.20 shows that cowellpoweredness is a suitable tool for extending factorization structures for small sources
to those for sources. The next proposition shows that factorization structures themselves
can be used to extend factorization structures of morphisms to those for sources.
15.24 PROPOSITION
In an (E, M)-category A every factorization structure (C, N ) for morphisms withC ⊆ E can be uniquely extended to a factorization structure (C, N) for sources.
Proof: Let N be the conglomerate of all sources of the form M ◦ n with M ∈ M andn ∈ N . If S is a source with (E, M)-factorization S = M ◦ e and e = n ◦ c is a (C, N )factorization, then S = (M ◦ n) ◦ c is a (C, N)-factorization. If f and c are morphisms
with c ∈ C, and N = M ◦ n and S are sources with n ∈ N and M ∈ M (i.e., N ∈ N)
and if N ◦ f = S ◦ c, then diagonals d1 and d2 can be successively constructed such thatthe diagrams
/•
~
d1 ~~
n◦f
S
~~
 ~~

/•
•

•

c

M

/•
~
d2 ~~
f
d1
~~
 ~~

/•
•

•

and

c

•
and hence

n

f

c

~~
 ~~

•

/•
~

d2 ~~
N



S

/•

commute. Uniqueness follows from c ∈ C ⊆ E ⊆ Epi(A).



FACTORIZATION STRUCTURES AND LIMITSThe fact that in an (E, M)-category every extremal mono-source belongs to M
[cf. 15.5(2)] implies that all limit sources belong to M (cf. Proposition 11.6). Below, we
further explore the relationship between factorization structures and limits.
15.25 THEOREM
Let A be a strongly complete, extremally co-wellpowered category. Then the following
hold:(1) A is an (ExtrEpi, Mono-Source)-category.
(2) If A is co-wellpowered, then A is an (Epi, ExtrMono-Source)-category.
(3) If in A regular epimorphisms are closed under composition, then A is a
(RegEpi, Mono-Source)-category.

9th July 2006

262

Factorization Structures[Chap. IV

Proof:
fi(1). Let S = (A −−→ Ai )I be a source. By Theorem 14.17, for each i ∈ I there is
fi

e

m

i
i
an (ExtrEpi, Mono)-factorization A −−→ Ai = A −−→
Bi −−→
Ai . Since A is
extremally co-wellpowered, there exists a subset J of I, and for each i ∈ I there isa j(i) ∈ J and an isomorphism hi with ei = hi ◦ ej(i) (where for i ∈ J we can choosepjj(i) = i and hi = id). The family (Bj )j∈J has a product (P −−→ Bj )J . Let
hej i

e

m

A −−−→ P = A −→ B −−→ P
be an (Epi, Mono)-factorization. Thenfiemi ◦hi ◦pj(i) ◦m

A −−→ Ai = A −→ B −−−−−−−−−−→ Ai
is an (Epi, Mono-Source)-factorization of S. Hence (1) follows from Theorem 15.10.
(2). Since A has equalizers and intersections, it is (Epi, ExtrMono)-structured (14.19).
By Corollary 15.21, A is an (Epi, M)-category for a suitable conglomerate M ofsources. Moreover, by (1), A is an (ExtrEpi, Mono-Source)-category. Hence by
Proposition 15.8(4), A has coequalizers. Thus by Proposition 15.8(3), M is theconglomerate of all extremal mono-sources.
(3). By (1) and Proposition 15.7, A has coequalizers. Thus by Proposition 14.22, A is
(RegEpi, Mono)-structured. By Proposition 15.24 this implies that A is a (RegEpi,M)-category for some conglomerate M. By 15.8(4) it follows that M is the conglomerate of all mono-sources in A.

15.26 REMARK
If A is a strongly complete, extremally co-wellpowered category, there need not be any
M for which A is an (Epi, M)-category. See Exercise 15D.

Suggestions for Further Reading
Strecker, G. E. Perfect sources. Springer Lect. Notes Math. 540 (1976): 605–624.
Hoffmann, R.-E. Factorization of cones. Math. Nachr. 87 (1979): 221–238.
Herrlich, H., G. Salicrup, and R. Vazquez. Dispersed factorization structures. Canad.
J. Math. 31 (1979): 1059–1071.
Tholen, W. Factorizations, localizations, and the orthogonal subcategory problem.
Math. Nachr. 114 (1983): 63–85.

9th July 2006

Sec. 15]

Factorization structures for sources

263

EXERCISES
15A. Swell Epimorphisms
A morphism f in A is called a swell epimorphism provided that A has the unique({f }, Mono-Source)-diagonalization property. Show that(a) Every strict epimorphism is swell, but not vice versa.
(b) Every swell epimorphism is an epimorphism.
(c) Every swell epimorphism is a strong epimorphism (cf. 14C), but not vice versa.
(d) The class of swell epimorphisms is closed under right-cancellation, composition, andthe formation of cointersections, pushouts, and coproducts.
(e) Under each of the following assumptions the equality SwellEpi = ExtrEpi holds inA:(e1) In A 2-sources are (Epi, Mono-Source)-factorizable [but cf. Exercise 15B(b)].
(e2) A is strongly complete and extremally co-wellpowered.
(e3) A is complete and in A every mono-source contains a small mono-source.
(e4) A is strongly cocomplete. [Cf. (d), Corollary 15.15, Proposition 15.7, and (f).](f) If A is an (E, Mono-Source)-category, then E = SwellEpi = ExtrEpi.
15B. (E, −)-Categories
A is called an (E, −)-category provided it is an (E, M)-category for some M. Showthat(a) Every (Epi,−)-category is a (SwellEpi,−)-category [cf. Theorem 15.14].
(b) The category/•@@@@ O
@@@@
@@@


o

• @@@
•

•

is a (SwellEpi, −)-category, but neither an (Epi, −)-category nor an (ExtrEpi, −)category. In particular, SwellEpi 6= ExtrEpi.
(c) Every partially ordered set, considered as a category, is an (ExtrEpi, Mono-Source)category [and SwellEpi = ExtrEpi]. But a partially ordered set (X, ≤) is an(Epi, −)-category if and only if for each x ∈ X the set {y ∈ X | x ≤ y} is a complete
lattice. [E.g.,
• 99


9

• LL s •
•

Ls
sssLL

•

is an (Epi, −)-category, but the addition of a smallest element destroys this property.]

9th July 2006

264

Factorization Structures[Chap. IV(d) Let A be an (E, −)-category. Show that for any D ⊆ E, A is a (D, −)-category if andonly if Iso(A)⊆ D and D is closed under composition, pushouts, and cointersections.
15C. (ExtrEpi, Mono-Source)- vs. (Epi, ExtrMono-Source)-Categories
Show that(a) Every (Epi, ExtrMono-Source)-category is an (ExtrEpi, Mono-Source)-category [cf.
Ex. 15B(a) and 15A(e)].
(b) Not every (ExtrEpi, Mono-Source)-category is an (Epi, ExtrMono-Source)-category[cf. Ex. 15B(c), 15D, or 16A(b)].
* 15D. A Strongly Complete, Extremally Co-wellpowered,Non-(Epi, –)-Category
Let Λ-CCPos be the construct that has as objects all triples (X, ≤, λ) with (X, ≤) apartially ordered set in which each nonempty chain has a join and λ : X → X a unary(not necessarily order-preserving) operation, and as morphisms all λ-homomorphismsthat preserve joins of nonempty chains. Show that(a) Λ-CCPos is complete, cocomplete, and wellpowered, hence strongly complete.
(b) Λ-CCPos is extremally co-wellpowered but not concretely co-wellpowered, hence
not co-wellpowered. [Hint: Consider for each limit ordinal α the natural injectione
B0 −−α→ Aα , where B0 = ( , =, λ) with λ(n) = n + 1 and where Aα = (Xα , ≤, λα )with (Xα , ≤) the ordered set of all ordinals less than or equal to α and λα : Xα → Xα
defined by λα (β) = M in{β + 1, α}.]N(c) Λ-CCPos is an (ExtrEpi, Mono-Source)-category [cf. Theorem 15.25].
(d) Λ-CCPos does not have cointersections. [Hint: Consider for each ordinal α theenatural injection B0 −−α→ Bα , where Bα = (Yα , ≤, λα ) with (Yα , ≤) the set of all
ordinals less than α + ω, ordered by β < γ ⇔ (β < γ ≤ α w.r.t. the natural order ofordinals), and λα (β) = β + 1.](e) Λ-CCPos is not an (Epi, −)-category [cf. 15.16(2)], but (Epi, Small ExtrMonoSource) is a factorization-structure for small sources on Λ-CCPos [cf. Proposition
15.19 and Corollary 14.21].
15E. Re: Theorem 15.14
Consider the three-element chain {0, 1, 2} with its natural order as a category A and let
E consist of all A-morphisms except 0 → 2. Show that A and E satisfy conditions (1),(3), and (4) of Theorem 15.14, but not condition (2).
15F. Existence of Coequalizers(a) Show that each category A that satisfies one of the following conditions, has coequalizers:(1) A has (Epi, Mono-Source)-factorizations,(2) A is strongly complete and extremally co-wellpowered [cf. Theorem 15.25].

9th July 2006

Sec. 15]

Factorization structures for sources

265(b) Construct a strongly complete category that does not have coequalizers.
15G. Factorization Structures for Empty Sources(a) Let (E, M) be a factorization structure for empty sources on A. Show that
(1) An empty source (A, ∅) belongs to M if and only if A is E-injective.
(2) The full subcategory of A that consists of all E-injective objects is reflective inA.
(b) Let E be a class of epimorphisms in A that is closed under composition with isomorphisms. Show that the following are equivalent:(1) There exists a class M of empty sources such that (E, M) is a factorization
structure for empty sources on A.
(2) A has enough E-injectives, i.e., for each A-object A there exists an Ee
injective object B and a morphism A −→ B in E.
(c) Let (E, M ) be a factorization structure for morphisms on a category A with aterminal object T . Let N be the class of all empty sources (A, ∅) with A → T ∈ M .
Show that (E, N ) is a factorization structure for empty sources on A.
15H. E-Injectives
Show that every (E, M)-category has enough E-injectives.
15I. Factorization Structures for 2-Sources and 1-Sources
Let (E, M) be a factorization structure for 2-sources and 1-sources on A. Show that(a) Conditions (1), (2), (3), (5), and (6) of Proposition 14.11 are equivalent to eachother and to each of the following conditions:(7a) a morphism f belongs to M iff f = g ◦ e and e ∈ E imply e ∈ Iso(A).
(7b) a 1-source or 2-source F belongs to M iff F = G◦e and e ∈ E imply e ∈ Iso(A).
idid(8) for each object A the 2-source A ←−− A −−→ A belongs to M,mmm(9) for each 1-source A −−→ B in M the 2-source B ←−− A −−→ B belongs to M,mf(10) for each 1-source A −−→ B in M and each morphism A −→ C the 2-sourcefm

B ←−− A −→ C belongs to M.
(b) The full subcategory B of A that consists of those objects A for which the 2-sourceididA ←−− A −−→ A belongs to M is closed under the formation of products and Msubobjects.
15J. Factorization Structures for Small Sources
Show that(a) (Mor, Small Product) is a factorization structure for small sources on Set, but Set
is not a (Mor, −)-category.

9th July 2006

266

Factorization Structures[Chap. IV(b) (Epi, Small ExtrMono-Source) is a factorization structure for small sources on
Λ-CCPos, but Λ-CCPos is not an (Epi, −)-category.
15K. Source-Sink-Diagonals
mj

fi

e

i
Show that if (Xi −−→X)I is an epi-sink, (Xi −−→ Y )I is an arbitrary sink, (Y −−→ Yj )J

gj

is a mono-source, and (X −−→ Yj )J is an arbitrary source in Set, then there exists adunique function X −→ Y such that the following diagram commutes:
ei

/X
~
~
d ~~
gj
fi
~~
~

 ~~
/ Yj
Y

Xi

mj

* 15L. Dispersed Factorization Structures
Let (E, M) be a fixed factorization structure on A. For a full subcategory B of A callan A-morphism f B-concentrated provided that f ∈ E and each B-object is {f }injective, and call an A-source S with domain A B-dispersed provided that the source
obtained from S by adding all morphisms with domain A and codomain in B belongs
to M. Show that(a) A is a (B-Concentrated, B-Dispersed)-category.
(b) If B is E-reflective, then the following are equivalent for each A-object A:(1) A belongs to B,(2) the empty source with domain A is B-dispersed,(3) A is B-Concentrated-injective.
(c) There are at least as many factorization structures on A as there are E-reflective
subcategories of A.
(d) Each of the categories Top and Rere has a proper class of factorization structures.
(e) If (C, D) is a factorization structure on A with C ⊆ E, then the following are
equivalent:(1) there exists a full subcategory B of A with C = B-Concentrated morphisms
and D = B-Dispersed sources,(2) if g ◦ f ∈ C and f ∈ E, then f ∈ C.
15M. Factorization Structures on Cat
Show that(a) Cat is an (Epi, ExtrMono-Source)-category.
(b) Cat is an (ExtrEpi, Mono-Source)-category.
(c) Cat is not a (RegEpi, −)-category.

9th July 2006

Sec. 15]

Factorization structures for sources267

15N. Factorization Structures on Set2 and (Set2 )op
Show that Set2 and (Set2 )op have regular factorizations.
15O. Re: Proposition 15.19
Show that one cannot omit the hypothesis in Proposition 15.19(2) that an initial object
exists. [Consider discrete categories.]

9th July 2006

16 E-reflective subcategories
In this section we will demonstrate that factorization structures provide a convenient
tool to investigate reflective subcategories. The typical situation is this: B is an (E, M)category and A is an isomorphism-closed full E-reflective subcategory of B.
16.1 DEFINITION
Let B be a category and let E be a class of B-morphisms. An isomorphism-closed, fullsubcategory A of B is called E-reflective in B provided that each B-object has an
A-reflection arrow in E. In particular, we use the terms epireflective (resp. monoreflective, bireflective) in case E is the class of epimorphisms (resp. monomorphisms, bimorphisms) in B. Likewise, we use the terms regular epireflective (resp. extremallyepireflective) in case E is the class of regular (resp. extremal) epimorphisms in B.
Dual Notion: E-coreflective subcategory.
16.2 EXAMPLES(1) In Met the full subcategory of complete metric spaces is bireflective [cf. 4.17(8)].
(2) HComp is reflective but not epireflective in Top, even though HComp is epireflective in Haus, and Haus is epireflective in Top.
(3) Ab is regular epireflective in Grp [cf. 4.17(4)] and Pos is regular epireflective in
Prost [cf. 4.17(3)].
(4) An isomorphism-closed full concrete subcategory of a concrete category B is (identity
carried)-reflective in B if and only if it is concretely reflective in B (cf. 5.22).
(5) The construct of minimal acceptors is regular epireflective in the construct of reachable acceptors [cf. 4.17(7)].

Haus is epireflective in Top
268

Sec. 16]

E-reflective subcategories

269

16.3 PROPOSITION
Every monoreflective subcategory of B is bireflective in B.
Proof: Let A be a monoreflective (hence by Definition 16.1 a full) subcategory of B.
rConsider an A-reflection-arrow B −→ A for some B-object B and a pair A

p
q

//

B 0 ofr0

B-morphisms with p ◦ r = q ◦ r. Let B 0 −−→ A0 be an A-reflection arrow for B 0 . Then(r0 ◦ p) ◦ r = (r0 ◦ q) ◦ r and A0 ∈ Ob(A) imply that r0 ◦ p = r0 ◦ q. Since r0 is assumed to
be a monomorphism, this implies that p = q. Hence r is an epimorphism.

16.4 PROPOSITION
Every coreflective isomorphism-closed full subcategory of B that contains a B-separator
is bicoreflective in B.
Proof: Let S be a separator in B, let A be a coreflective full subcategory of B thatccontains S, and let A −→ B be an A-coreflection arrow. By the dual of Proposition 16.3
p

it suffices to show that c is an epimorphism. Let Bq//

B 0 be distinct morphisms. Thenhthere exists a morphism S −→ B with p ◦ h 6= q ◦ h. Since S belongs to A, there exists ah0unique morphism S −−→ A with h = c◦h0 . Consequently, p◦c◦h0 = p◦h 6= q◦h = q◦c◦h0 ,which implies that p ◦ c 6= q ◦ c.
16.5 EXAMPLES
The above proposition immediately implies that
(1) Every coreflective isomorphism-closed full subcategory of Top (resp. Pos) that contains a nonempty space (resp. a nonempty poset) is bicoreflective in Top (resp. Pos);
hence it is a coreflective modification.
(2) Every reflective isomorphism-closed full subcategory of Top0 (resp. Pos) that contains the Sierpinski space (resp. a two-element chain) is bireflective in Top0 (resp.
Pos); hence it is a reflective modification.
(3) Set (resp. Vec) contains precisely one proper coreflective, isomorphism-closed, fullsubcategory — namely, the one consisting of the empty set (resp. the zero vector
spaces).
(4) Set contains precisely two proper reflective, isomorphism-closed, full subcategories— namely, the one consisting of all one-element sets and the one consisting of all setswith at most one element. Vec contains precisely one proper reflective, isomorphismclosed, full subcategory — namely, the one consisting of all zero vector spaces.
16.6 REMARK
It is not always easy to determine whether or not a given subcategory is reflective. (Forexample, which of the full subcategories consisting of regular spaces, or of connected
spaces, or of compact spaces is reflective in Top?) Fortunately, for E-reflectivity wehave powerful and easily applied criteria, as will be seen in the next results.

9th July 2006

270

Factorization Structures[Chap. IV

16.7 DEFINITION
Let M be a conglomerate of sources in a category B. A subcategory A of B is said tofi

be closed under the formation of M-sources provided that whenever (B −−→ Ai )I
is a source in M such that all Ai belong to A, then B belongs to A.
16.8 THEOREM
If A is a full subcategory of an (E, M)-category B, then the following conditions areequivalent:(1) A is E-reflective in B.
(2) A is closed under the formation of M-sources in B.
In the case that B has products and is E-co-wellpowered, the above conditions are equivalent to:(3) A is closed under the formation of products and M-subobjects66 in B.
mi
Proof: (1) ⇒ (2). Let S = (B −−→
Ai )I be a source in M such that all Ai belong toe
A. If B −→ A is an A-reflection arrow for B in E, then S factors through e. Hence, by
Proposition 15.5(9)(a), e is an isomorphism. Thus B belongs to A.

(2) ⇒ (1). For any B-object B, consider the source S with domain B, consisting of allmorphisms with domain B and codomain in A. If S = M ◦ e is an (E, M)-factorization,then e is an A-reflection arrow for B.
(2) ⇒ (3). Obvious.
mi(3) ⇒ (2). Consider a source M = (B −−→
Ai )I in M such that all Ai belong to A.
einimi
For each i ∈ I let B −−→ Ai = B −−→ Bi −−→ Ai be an (E, M)-factorization. Then each
Bi , being an M-subobject of Ai , belongs to A. Since B is E-co-wellpowered, we canselect a subset J of I and for each i ∈ I a j(i) ∈ J and an isomorphism hi : Bj(i) → Biwith ei = hi ◦ ej(i) . If (P, πj )J is a product of the family (Bj )j∈J , then P belongs to A.
If
hej i
e
m
B −−−→ P = B −→ C −−→ P

is an (E, M )-factorization, then C belongs to A, and we have commutativity of the
diagram
B 5FF
zz 55 FF
zz
55 FF
z
z
55 FFF mzze zej(i) 55ei FFFi
z
z
hej i
FF
55
zz
FF
55
zz
FF
z
z
5
F"


|zz
/
/
/ Ai
/
B
Bi
P
C
j(i)
m

πj(i)

hi

ni

Since M factors through e, and e ∈ E, by Proposition 15.5(9)(a), e is an isomorphism.
Thus B belongs to A.

66

An M-subobject is simply a singleton M-source. It need not be a monomorphism.

9th July 2006

Sec. 16]

E-reflective subcategories

271

16.9 COROLLARY
A full subcategory of a co-wellpowered, strongly complete category B is epireflective in B
if and only if it is closed under the formation of products and extremal subobjects in B.
Proof: Immediate from Theorems 16.8 and 15.25(2).



16.10 EXAMPLE
The full subcategory B of BiTop that consists of all bitopological spaces with both
topologies Hausdorff is strongly complete (but not co-wellpowered). The full subcategory BiComp of B [cf. 4F(b)] is closed under the formation of products and extremal
subobjects in B, but is not reflective in B.

SUBCATEGORIES DEFINED BYEQUATIONS AND IMPLICATIONS
16.11 MOTIVATING REMARK
Many familiar mathematical objects (e.g., semigroups, monoids, groups, abelian groups,
rings, lattices, boolean algebras, vector spaces, etc.) can be defined by means of operations and equations. Moreover, the corresponding categories (Sgr, Mon, Grp, etc.) canbe obtained as full subcategories of categories of the form Alg(Ω), consisting of thoseobjects that satisfy suitable equations. This singling out of subcategories by equations(or, more generally, by implications) will be described here in some detail in order tomotivate the much simpler and far more elegant categorical concepts introduced below.
As an example, consider Sgr, which is the full subcategory of the category Alg(2) ofalgebras with one binary operation (usually written as multiplication) consisting of those
algebras that satisfy the equation:∀x ∀y ∀zx · (y · z) = (x · y) · z.

(e)

In order to find a categorical description of the above equation (e) and of those algebras that satisfy (e), observe that the expressions x · (y · z) and (x · y) · z can beinterpreted as elements t and s of the free algebra F on the set {x, y, z} of generators. If
η : {x, y, z} → F is the corresponding universal arrow, then, for any algebra A in Alg(2),the following conditions are equivalent:(1) A is a semigroup,(2) A satisfies equation (e),(3) for every map f : {x, y, z} → |A| (i.e., for every interpretation of the variables x,y,and z), the unique homomorphism f : F → A, determined by f = f ◦ η, satisfies
f (t) = f (s).
There is a further characterization: if ρ is the congruence on F generated by (t, s), (i.e.,ethe smallest congruence relation on the algebra F such that tρ s) and F −→ F/ρ is thenatural map onto the quotient algebra F/ρ, then (3) is equivalent to

9th July 2006

272

Factorization Structures[Chap. IV(4) A is {e}-injective (i.e., every homomorphism from F to A factors through e).
Hence an algebra satisfies equation (e) if and only if it is injective with respect to asuitable morphism e. In general, an equation in Alg(Ω) is a pair (t, s) of elements insome free Ω-algebra F . An Ω-algebra A satisfies the equation (t, s) if and only if A ise{e}-injective, where F −→ F/ρ is the quotient map corresponding to the congruence
relation ρ on F generated by (t, s).
Slightly more complicated — but in its categorical formulation even simpler (!) — is theconcept of implications. An implication P ⇒ K consist of a set P of equations, called
premises, and a set K of equations, called conclusions. For example, the implication∀x ∀y ∀zx·y =x·z

⇒

y=z

(i)

describes, together with above equation (e), the left-cancellative semigroups. It is clear
what it means to satisfy implication (i): whenever an interpretation of variables satisfies
the premise, then it also satisfies the conclusion. (In this example, both the set ofXpremises and the set of conclusions are singletons.) Formally, an implication P ⇒ K in
Alg(Ω) consists of a set X (of “variables”), a subset P of |F | × |F | (called the set of“premises”), and a subset K of |F |×|F | (called the set of “conclusions”), where F denotes
the canonical free Ω-algebra on X. (Cf. Example 8.23(6).) An Ω-algebra A satisfies theXimplication P ⇒ K provided that under each interpretation of variables (i.e., for each
map f : X → |A|) such that each premise (p, q) ∈ P becomes true (i.e., such that thehomomorphic extension f : F → A of f satisfies f (p) = f (q) for each (p, q) ∈ P ), eachconclusion (t, s) ∈ K must be true (i.e., f (t) = f (s) for each (t, s) ∈ K). If π resp.
eγ are the congruence relations on F generated by P resp. by P ∪ K, if F −−P→ F/πeKeand F −−→ F/γ are the associated quotient maps, and if F/π −→ F/γ is the unique
homomorphism satisfying eK = e ◦ eP , then the following conditions are easily seen tobe equivalent for any Ω-algebra A:
X(1) A satisfies the implication P ⇒ K,
(2) A is {e}-injective in Alg(Ω).
eSince in Alg(Ω) morphisms of the form F/π −→ F/γ are, up to isomorphism, precisely
the surjective homomorphisms (= epimorphisms = regular epimorphisms), we are led to
the following definition:
16.12 DEFINITION(1) (Regular) epimorphisms are called (regular) implications.
e(2) An object Q satisfies the implication A −→ B provided that Q is {e}-injectiveff(i.e., provided that for each morphism A −→ Q there exists a morphism B −→ Q
with f = f ◦ e).
(3) A full subcategory A of B is called implicational provided that there exists a class
C of implications in B such that A consists precisely of those B-objects that satisfy

9th July 2006

Sec. 16]

E-reflective subcategories

273

each implication in C. Constructs that are concretely isomorphic to implicational
subconstructs of Alg(Ω) for some Ω are called finitary quasivarieties.67
In case C can be chosen to be a subclass of some class E of implications in B, A is called
E-implicational.
16.13 EXAMPLES(1) Pos is an implicational subcategory of Rel. Consider the following implications:
id

REFLEXIVITY: ({x}, ∅) −−→ ({x}, {(x, x)}),id

TRANSITIVITY: ({x, y, z}, {(x, y), (y, z)}) −−→ ({x, y, z}, {(x, y), (y, z), (x, z)}),e

ANTISYMMETRY: ({x, y}, {(x, y), (y, x)}) −→ ({x}, {(x, x)}),where e(x) = e(y) = x.
A binary relation satisfies the above implications if and only if it is a poset.
(2) T1 -spaces form an implicational subcategory of Top. If P is the Sierpinski spaceand P 0 is a singleton space, then a topological space is T1 if and only if it satisfies
the implication P → P 0 .
(3) In Σ-Seq, Σ = {σ}, consider all the automata with permutation transition (i.e.,δ(σ, x) = δ(σ, y) ⇒ x = y). These form an implicational class given by Pn → Pn0(n = 1, 2, . . .), where Pn has states x, y, 1, . . . , n such that δ(σ, x) = δ(σ, y) = 1 and
1, . . . , n form a cycle, and Pn0 is the quotient of Pn obtained by merging x with y.
(4) The constructs Vec, R-Mod, Ab, Grp, Mon, Sgr, Rng, Lat, DLat, SLat, Boo,pSet, and AbTop (= abelian topological groups) are finitary quasivarieties. However, Field is not a finitary quasivariety.
16.14 THEOREM
A full subcategory of an (E, M)-category B is E-implicational if and only if it is Ereflective in B.
Proof: Let C be a subclass of E and let A be the full subcategory of B determined
by C; i.e., given by those objects that satisfy each implication in C. Consider a sourcemi(B −−→Ai )I in M such that all Ai belong to A. By Theorem 16.8, it is sufficient tofeshow that B belongs to A. Let Q −→ R be an element of C and let Q −→ B be agimorphism. Then for each i ∈ I there exists a morphism R −−→ Ai with gi ◦ e = mi ◦ f .
Hence there exists a diagonal d that makes the diagram
/R
~
~
d ~~
gi
f
~~
~

 ~
/ Ai
B

Q

e

mi

commute. In particular f = d ◦ e. Hence B belongs to A.
67

Recall that in Alg(Ω), Epi = RegEpi = surjective homomorphisms.

9th July 2006

274

Factorization Structures[Chap. IV

Conversely, let A be a E-reflective full subcategory of B. Consider the class C of all
A-reflection arrows for B-objects, and let C be the implicational subcategory of B
determined by C. Obviously, A ⊆ C. For the converse, let D be an object of C andclet D −→ A be an A-reflection arrow for D. Since D satisfies the implication c, there
exists a morphism f with idD = f ◦ c. Thus c is an epimorphic section (cf. 15.4); hence
an isomorphism. Since A is isomorphism-closed, this implies that D belongs to A.
Therefore C = A.

16.15 REMARKAs indicated in Remark 16.11, equations in Alg(Ω) are those implications that havee
the empty set P of premises; equivalently, they are those implications A −→ B whose
domain A is a free Ω-algebra. The concept of free Ω-algebras, unfortunately, refers to theconstruct Alg(Ω) and not to the abstract category Alg(Ω). Fortunately, however, as we
will see below, a full subcategory A of Alg(Ω) is E-implicational, where E is the class ofall (regular) implications in Alg(Ω) with free domain, if and only if A is E-implicational,
where E is the class of all (regular) implications in Alg(Ω) with (regular-) projective
domain. This motivates the following definition:
16.16 DEFINITION(1) Let E be a class of epimorphisms in a category B. An implication in E with Eprojective domain is called an E-equation. Regular epimorphic equations are called
regular equations. A full subcategory A of B is called E-equational provided
that there exists a class C of E-equations in B such that A consists precisely of
those B-objects that satisfy each E-equation in C.
(2) Let B be a construct. Regular implications with free domain are called equations.
A full subcategory A of B is called equational provided that it can be defined asabove by a class C of equations in B.
(3) Constructs that are concretely isomorphic to equational subconstructs of Alg(Ω)
for some Ω are called finitary varieties.
16.17 THEOREM
Let B be an (E, M)-category with enough E-projectives (9.22 dual). Then the following
conditions are equivalent for any full subcategory A of B:(1) A is E-equational in B.
(2) A is closed under the formation of M-sources and E-quotients in B.
In the case that B has products and is E-co-wellpowered, the above conditions are equivalent to:(3) A is closed under the formation of products, M-subobjects, and E-quotients in B.

9th July 2006

Sec. 16]

E-reflective subcategories

275

Proof: (1) ⇒ (2). By Theorems 16.8 and 16.14, A is closed under the formation ofe
M-sources in B. Let A be in A and be A −→ B be in E. To show that B is in A we willcverify that B satisfies any E-equation C −→ D that is satisfied by all A-objects. Sincef

C is E-projective, for each morphism C −→ B there exists a morphism f with f = e ◦ f .
Since A is in A, there exists a morphism f˜ with f = f˜ ◦ c. Hence f = (e ◦ f˜) ◦ c. Thus
B belongs to A.
c /
C@
D
f



Bo

@@
@@
@@
f

e



f˜

A

(2) ⇒ (1). By Theorem 16.8, A is E-reflective in B. Let C be the class of all A-reflection
arrows for E-projective objects P , and let C be the subcategory of B determined by C.
Obviously, A ⊆ C. For the converse, let D be an object of C. Then there exists anecE-projective object P and a morphism P −→ D in E. Let P −→ A be an A-reflectionfarrow for P . Since D satisfies the E-equation c, there exists a morphism A −→ D withf

e = f ◦ c. Since c is an epimorphism, this implies, by Proposition 15.5(5), that A −→ D
belongs to E. Thus D is an E-quotient of an A-object, and so belongs to A.
(2) ⇔ (3). Immediate from Theorem 16.8.



16.18 THEOREM
Let B be a fibre-small, transportable, complete construct that has free objects and forwhich the surjective morphisms, extremal epimorphisms and regular epimorphisms coincide. Then for full subconstructs A of B the following conditions are equivalent:(1) A is equational in B,(2) A is regular-equational in B,(3) A is regular epireflective and closed under the formation of regular quotients (=homomorphic images) in B,(4) A is closed under the formation of products, subobjects, and homomorphic images
in B.
Proof: In B the extremal epimorphisms are precisely the surjective morphisms, and, by
Proposition 8.28 the monomorphisms are precisely the injective morphisms. Thus fibresmallness and transportability imply wellpoweredness and extremal co-wellpoweredness.
Hence completeness implies strong completeness. Since surjective morphisms are closed
under composition, by Theorem 15.25(3), B is a (RegEpi, Mono-Source)-category. Consequently, Theorem 16.8 and Theorem 16.17 imply the equivalence of the conditions (2),(3), and (4). Since by Proposition 9.29 every free object is regular projective, (1) impliesc(2). To show that (3) implies (1), let C be the class of all A-reflection arrows F −→ A
for free objects F in B, and let C be the subcategory of B determined by C. Since in
B every object is a homomorphic image of a free object, the same argument as in the
proof of Theorem 16.17 shows that C = A.


9th July 2006

276

Factorization Structures[Chap. IV

16.19 COROLLARY
For full subcategories A of Alg(Ω), the following hold:(1) A is implicational in Alg(Ω) if and only if A is closed under the formation of
products and subalgebras.
(2) A is equational in Alg(Ω) if and only if A is closed under the formation of products,
subalgebras and homomorphic images.


E-REFLECTIVE HULLS
16.20 PROPOSITION
For (E, M)-categories B, the following hold:(1) The intersection of any conglomerate of E-reflective subcategories of B is E-reflective in B.
(2) For every full subcategory A of B there exists a smallest E-reflective subcategory ofB that contains A.
Proof: (1) follows from Theorem 16.8. (2) follows from (1).



16.21 DEFINITION
If A is a full subcategory of an (E, M)-category B, then the smallest E-reflective subcategory of B that contains A is called the E-reflective hull of A in B.
16.22 PROPOSITION
If A is a full subcategory of an (E, M)-category B, then a B-object B belongs to thefi

E-reflective hull of A in B if and only if there exists a source (B −−→ Ai )I in M with
all Ai in A.

16.23 Examples
B

A

Epireflective hullof A in B

Top
Top
Haus
Rel
Set
Vec

S (= Sierpinski-space)[0, 1][0, 1]({0, 1}, ≤){0, 1}

Top0
Tych
HComp
Pos
Set
Vec

R

9th July 2006

Sec. 16]

E-reflective subcategories277

16.24 PROPOSITION (Reflectors as Composites of Epireflectors)
If A is a full reflective subcategory of an (Epi, Mono-Source)-factorizable category B,
and if C is the extremally epireflective hull of A in B, then A is epireflective in C and
C is epireflective in B.
Proof: By Theorem 15.10, B is an (ExtrEpi, Mono-Source)-category. Hence A has an
ExtrEpi-reflective hull C. Obviously, C is epireflective in B, and A is reflective in C. For
miany C in C there exists, by Proposition 16.22, a mono-source M = (C −−→
Ai )I with allr
Ai in A. Let C −→ A be an A-reflection arrow for C. Then M factors through r, which
implies that r is a monomorphism in B, hence also in C. Thus A is monoreflective inC, so that by Proposition 16.3 it is epireflective in C.
16.25 REMARK
If we generalize the situation that we are working in just slightly, our results may break
down completely. Here are two examples, the first very simple, the second very deep:(1) Consider the ordered set of all natural numbers as a category A. Then A almost
satisfies the assumptions of Theorem 15.25: A is well-powered, co-wellpowered, and“almost” strongly complete (every nonempty diagram has a limit). But A is not an(Epi, M)-category for any M. If B (resp. C) is the isomorphism-closed, epireflective
subcategory consisting of all even (resp. all odd) numbers, then B ∩ C is empty —hence not even reflective.
(2) The category Top is strongly complete, wellpowered, and co-wellpowered. However,there exist two isomorphism-closed full reflective subcategories of Top whose intersection is not reflective in Top. [For a corresponding example in the category ofbitopological spaces see Exercise 4F(b).]
If in a category B the intersection of two (epi)reflective isomorphism-closed full subcategories is not (epi)reflective, then this intersection obviously has no (epi)reflective hullin B.
If there is an isomorphism-closed full subcategory of B that does not have an (epi)reflective hull in B, then (epi)reflective subcategories of B cannot be characterized among theisomorphism-closed full subcategories of B by suitable closure properties.

Suggestions for Further Reading
Kennison, J. F. Reflective functors in general topology and elsewhere. Trans. Amer.
Math. Soc. 118 (1965): 303–315.
Baron, S. Reflectors as compositions of epi-reflectors. Trans. Amer. Math. Soc. 136(1969): 499–508.
Hatcher, W. S. Quasiprimitive subcategories. Math. Ann. 190 (1970): 93–96.
Ringel, C. M. Monofunctors as reflectors. Trans. Amer. Math. Soc. 161 (1971): 293–306.

9th July 2006

278

Factorization Structures[Chap. IV

Banaschewski, B., and H. Herrlich. Subcategories defined by implications. Houston J.
Math. 2 (1976): 149–171.
Hušek, M. Lattices of reflections and coreflections in continuous structures. Springer
Lect. Notes Math. 540 (1976): 404–424.
Nyikos, P. Epireflective categories of Hausdorff spaces. Springer Lect. Notes Math. 540(1976): 452–481.
Marny, T. On epireflective subcategories of topological categories. Gen. Topol. Appl.
10 (1979): 175–181.
Hoffmann, R.-E. Reflective hulls of finite topological spaces. Archiv Math. 33 (1979):
258–262.
Németi, I., and I. Sain. Cone-implicational subcategories and some Birkhoff-type theorems. Universal Algebra (Esztergom, 1977), Colloq. Math. Soc. János Bolyai 29,
North Holland, Amsterdam, 1982, 535–578.
Hoffmann, R.-E. Cowell-powered reflective subcategories. Proc. Amer. Math. Soc. 90(1984): 45–46.
Giuli, E., and M. Hušek. A diagonal theorem for epireflective subcategories of Top andcowell-poweredness. Anal. Math. Pure Appl. 145 (1986): 337–346.

EXERCISES
16A. Epireflective Subcategories with Bad Behavior(a) Let I be a proper class, and let the category B have as objects A00 , A11 , Ai2 for i ∈ I,
A33 , and A44 ; and morphism-sets such that
no element, if m < n or (m = n and i 6= j)ijhomB (An , Am ) has
precisely two elements, if (n, m) = (0, 1) or (n, m) = (2, 3)
precisely one element, otherwise.
Let A be the full subcategory obtained from B by removing object A33 .
Show that(1) A is regular epireflective in B.
(2) B is co-wellpowered, but A is not even extremally co-wellpowered.
(3) The inclusion functor A ,→ B sends certain extremal epimorphisms to nonepimorphisms.

NN(b) Let the category B have as objects A02n for n ∈ , and A1n , for n ∈ ; and morphismsets such that
precisely two elements, if n = m and i < jijhomB (An , Am ) has
no elements, if m < n or j < i
precisely one element, otherwise.

9th July 2006

Sec. 16]

E-reflective subcategories

279

Let A be the full subcategory of B whose objects are all A02n and all A12n .
Show that(1) A is epireflective in B.
(2) B has regular factorizations, but A is not even (Epi, Mono-Source)-factorizable.
[Consider empty sources.](3) The inclusion functor A ,→ B sends certain regular epimorphisms to nonextremal epimorphisms.
16B. Inheritance of Factorization Structures
Let E : A ,→ B be the embedding of a subcategory. Show that(a) If B has regular factorizations and A is regular epireflective in B, then A has regularfactorizations and E preserves and reflects regular epimorphisms.
(b) If B is an (ExtrEpi, Mono-Source)-category and A is extremally epireflective in B,
then A is an (ExtrEpi, Mono-Source)-category and E preserves and reflects extremal
epimorphisms.
(c) If B is an (Epi, ExtrMono-Source)-category and A is epireflective in B, then E need
not preserve epimorphisms [cf. Haus and Top].
(d) If B has regular factorizations and A is epireflective in B, then A need not be an(ExtrEpi, Mono-Source)-category and E need neither preserve regular nor preserve
extremal epimorphisms [cf. Exercise 16A(b)].
16C. Co-wellpoweredness and (Epi)Reflective Hulls
Let A be an isomorphism-closed full subcategory of a complete, well-powered, and cowellpowered category C, and let B be the epireflective hull of A in C. Show that(a) A is epireflective in C if and only if A is closed under the formation of products andregular subobjects in C.
(b) The embedding A ,→ B preserves epimorphisms.
(c) If D is an isomorphism-closed full subcategory of B that contains A, then thefollowing are equivalent:(1) D is reflective in C,(2) D is epireflective in B.
(d) B is strongly complete.
(e) If B is co-wellpowered, then the epireflective hull of A in B is a reflective hull of
A in C, i.e., the smallest isomorphism-closed full reflective subcategory of C that
contains A.
(f) The following are equivalent:(1) A is reflective in C and co-wellpowered,

9th July 2006

280

Factorization Structures[Chap. IV(2) A is limit-closed in C and B is co-wellpowered.
(g) The following are equivalent:(1) A has a co-wellpowered reflective hull in C,(2) A has a co-wellpowered epireflective hull in C.
* 16D. Subcategories of Top(a) Show that a full reflective subcategory of Top is co-wellpowered if and only if its
epireflective hull in Top is co-wellpowered.
(b) Construct an epireflective, non-co-wellpowered subcategory of Top. [Cf. Example
7.90(3).](c) Construct a full, limit-closed, non-reflective subcategory of Top.
16E. Smallest E-Reflective Subcategories
Let (E, M ) be a factorization structure for empty sources (resp. for morphisms) on acategory A (with a terminal object T ). Show that(a) A is E-injective if and only if (A, ∅) ∈ M (resp. A → T ∈ M ).
(b) The full subcategory B of A that consists of all E-injective objects is the smallestr
E-reflective subcategory of A. Moreover, a morphism A −→ B is a B-reflectionrarrow if and only if (A, ∅) = A −→ (B, ∅) is an (E, M )-factorization [resp. if andr
only if A → T = A −→ B −→ T is an (E, M )-factorization].
16F. The Čech-Stone Compactification
Show that if E is the class of dense C ∗ -embeddings and M is the class of perfect maps,
then (E, M ) is a factorization structure for morphisms on Tych. Apply the results of
Exercise 16E.
16G. A Generalized Birkhoff Theorem
Let A be a full subcategory of an (E, M)-category B and let Q be a class of epimorphisms
in B such that(α) B has enough Q-projectives,(β) if g ◦ e ∈ Q and e ∈ E, then g ∈ Q.
(a) Show that the following conditions are equivalent:(1) A is closed under the formation of M-sources and Q-quotients.
e(2) A is definable by implications B −→ C in E with Q-projective domain B.
(b) Show that in the case that A is co-wellpowered and has products, the above conditions are equivalent to(3) A is closed under the formation of products, M-subobjects, and Q-quotients.

9th July 2006

Sec. 16]

E-reflective subcategories

281

16H. Epireflective Subcategories Are Closed Under Extremal Mono-Sourcesmi
Let A be an epireflective subcategory of B and let (B −−→
Bi )I be an extremal monosource in B. Show that whenever all Bi belong to A, then so does B.

16I. The Lattice of Regular Epireflective Subcategories
Show that the lattice of regular epireflective subcategories of(a) Set has precisely two elements,(b) Set2 has the form
3

1

0

5
xxx
4
xxx
2
xxx

(c) (Set2 )op has the form4::  :

3


1/

//
//
/

0




2

* 16J. Injective Hulls for Finitary Varieties
Show that in a finitary variety (A, U ) every object has an injective hull if and only if
the following conditions (cf. 9C) are satisfied:(1) Monomorphisms are pushout stable,(2) A is M -co-wellpowered, where M is the class of essential monomorphisms.
* 16K. Finitary Quasivarieties Isomorphic to Finitary Varieties
Show that a finitary quasivariety (A, U ) is a finitary variety if and only if there exists aHfinitary variety (B, V ) and an isomorphism A −−→ B.
* 16L. Strongly Limit-Closed vs. Epireflective Subcategories
A full subcategory A of B is called strongly limit-closed in B provided that wheneverfi(L −−→ Di )Ob(I) is a limit of a diagram D : I → B and for each i ∈ Ob(I) there exists aj ∈ Ob(I) with Dj in A and homI (j, i) 6= ∅, then L belongs to A. Show that:(a) If A is epireflective in B, then A is strongly limit-closed in B.

9th July 2006

282

Factorization Structures[Chap. IV(b) If B is complete, wellpowered, and co-wellpowered, then a full subcategory A of B
is epireflective in B if and only if A is strongly limit-closed in B.
16M. A Characterization of Extremally Epireflective Subcategories
Show that a full subcategory of an extremally co-wellpowered, strongly complete category
B is extremally epireflective in B if and only if it is closed under the formation of products
and subobjects in B.
16N. Finitary Varieties and Finitary Quasivarieties
Show that(a) All the finitary quasivarieties exhibited in Example 16.13(4) are finitary varieties.
(b) The full subconstruct TfAb of Ab is a finitary quasivariety, but is not a finitaryvariety. [Cf. 20.41(2) and 24.7.](c) Every regular epireflective subconstruct of a finitary quasivariety is a finitary quasivariety.
(d) The finitary variety Grp is a full, isomorphism-closed, reflective subconstruct of thefinitary variety Sgr. However, Grp is not even implicational in Sgr.

9th July 2006

17 Factorization structures for functors
In §8 we have generalized the concept of morphisms to that of G-structured arrows,
where G is a functor. Here we will generalize, in an analogous way, the concept ofsources to that of G-structured sources. Many (but by no means all) of the results on
factorization structures of sources can be carried over to the context of G-structured
sources. Sometimes even proofs of the more general results can be readily obtained byadapting the corresponding proofs from §15.
This section contains the basic results on factorization structures for functors. Specialclasses of functors, defined by means of particular factorization properties, will be studiedin greater detail in the remaining chapters; especially: adjoint functors (Chapter V); andtopological, algebraic, and topologically algebraic functors (Chapter VI).

Domain and codomain of a source

17.1 DEFINITION
Let G : A → X be a functor. A G-structured source S is a pair (X, (fi , Ai )i∈I ) thatfi

consists of an X-object X and a family of G-structured arrows X −−→ GAi with domain
X, indexed by some class I.
X is called the domain of S and the family (Ai )i∈I is called the codomain of S.
17.2 REMARKS(1) Whenever it is convenient, we use notations for G-structured sources such asfifi(X −−→ GAi )i∈I or (X −−→ GAi )I (which is reminiscent of notations used for sourcesin §15).

283

284

Factorization Structures[Chap. IV(2) As with sources, we use, whenever convenient, such expressions as(a) G-structured empty sources (i.e., X-objects),(b) G-structured 1-sources (i.e., G-structured arrows),(c) G-structured 2-sources (when the cardinality of the index class is 2), and(d) G-structured small sources (when the index class is a set).
(3) As with sources, every pair (X, S) that consists of an X-object X and a class S of
G-structured arrows with domain X can be considered to be a G-structured source
via indexing S by itself.
17.3 DEFINITION
Let G : A → X be a functor, let E be a class of G-structured arrows, and let M be aconglomerate of A-sources. (E, M) is called a factorization structure for G, and G
is called an (E, M)-functor provided that
(1) E and M are closed under composition with isomorphisms in the following sense:
e

h

e

Gh

(1a) if X −→ GA ∈ E and A −→ B ∈ Iso(A), then X −→ GA −−→ GB ∈ E,
m

h

h

m

i
i
Ai )I ∈ M.
Ai )I ∈ M and B −→ A ∈ Iso(A), then (B −→ A −−→(1b) if (A −−→fi(2) G has (E, M)-factorizations, i.e., for each G-structured source (X −−→ GAi )Imie
there exists X −→ GA ∈ E and M = (A −−→
Ai )I ∈ M such that
fi

Gm

e

i
GAi
X −−→ GAi = X −→ GA −−−→

for each i ∈ I.
f(3) G has the unique (E, M)-diagonalization property, i.e., whenever X −→ GA
mi
e
Ai )I
and X −→ GB are G-structured arrows with (e, B) ∈ E, and M = (A −−→fi

and S = (B −−→ Ai )I are A-sources with M ∈ M, such that (Gmi ) ◦ f = (Gfi ) ◦ e
dfor each i ∈ I, then there exists a unique diagonal, i.e., an A-morphism B −→ A
with f = Gd ◦ e and S = M ◦ d, which will be expressed (imprecisely) by sayingthat the following diagram commutes:
/ GB
x
x
Gd xx
f
Gfi
xx

 |xx
/ GAi
GA

X

e

Gmi

17.4 REMARKS(1) A category A is an (E, M)-category if and only if the identity functor on A is ane(E, M)-functor (provided that we identify each A-morphism A −→ B with the idA structured arrow (e, B)). Hence (E, M)-categories can be considered as special casesof (E, M)-functors.

9th July 2006

Sec. 17]

Factorization structures for functors

285(2) The dual of a factorization structure on G is a factorization structure on Gop , hence
it concerns factorizations of G-costructured sinks.
(3) A functor G is called an (E, −)-functor provided that there exists some M for which
G is an (E, M)-functor. Analogously for (−, M)-functors.
(4) Whereas every category is an (E, M)-category for suitably chosen E and M, afunctor is an (E, M)-functor for suitably chosen E and M if and only if it is anadjoint functor (cf. 18.3).
17.5 EXAMPLES(1) The forgetful functor U : Top → Set is a (Generating, Initial Mono-Source)fifunctor. The desired factorizations of U -structured sources of the form (X −−→(Xi , τi ))I
fi

m

e

i
can be obtained in two steps: First, let X −−→ Xi = X −→ Y −−→
Xi be an(Epi, Mono-Source)-factorization in Set. Second, let τ be the initial topology on Y
with respect to the mi and τi . Then

fi

m

e

i
(Xi , τi )X −−→ (Xi , τi ) = X −→ (Y, τ ) −−→

provides a factorization with the desired properties. By using similar arguments, U
can be seen to be an (Extremally Generating, Mono-Source)-functor and a (Bijection,
Initial Source)-functor.
(2) The forgetful functor U : Grp → Set is an (Extremally Generating, Mono-Source)efunctor. (Recall that X −→ U G is extremally generating if and only if the set e[X]generates the group G in the familiar algebraic sense.) The desired factorizations offi

U -structured sources (X −−→ U Gi )I can be obtained by either of the following two
constructions, each being of independent interest:u(a) Let X −→ U F be a universal arrow in Grp. Then each fi can be uniquelyfiextended to a homomorphism F −−→
Gi . Let

f

e

m

i
i
F −−→
Gi = F −→ G −−→
Gi ,i ∈ I,

be an (Epi, Mono-Source)-factorization of the source (f i )I in the category Grp.
Then the factorization
fi

U e◦u

Um

i
X −−→ U Gi = X −−−−→ U G −−−→U Gi ,i ∈ I,

has the desired properties.
Q(b) Form the cartesian product conglomerate I Gi and consider this as a (possibly
illegitimate)coordinate wise. [That is, the memQ group by defining the operationSbers of I Gi are functions x : I → Gi with x(i) ∈Q
Gi , and (x·y)(i) = x(i)·y(i)−1−1and x (i) = (x(i)) .] Define
Q a function f : X → I Gi by (f (x))(i) = fi (x).
Then f [X] is a subset of I Gi , and it clearly generates a (small) subgroup

9th July 2006

286

Factorization Structures[Chap. IV

Q
H of the (possibly illegitimate) group I Gi . Thus the codomain restriction
e : X → U H of f together with the functions mi : H → Gi , defined bymi (x) = x(i), form the desired factorization.68
Observe that the second construction
Qis more elementary than the first one, eventhough it uses the generalized group I Gi (which often fails to be a group because
it often fails to be a set).
(3) The forgetful functors U of many familiar constructs are (Extremally Generating,Mono-Source)-functors, e.g., those of Top, Haus, Pos, Rel, Vec, Grp, Rng, Sgr,Mon, Lat, Boo, HComp, Cat, and Alg(Ω). In each case a construction analogous to one of those given in (1) and (2) yields the desired factorizations. The
diagonalization property holds automatically: cf. Theorem 17.10.
17.6 THEOREM
If G is an (E, M)-functor, then each member of E is generating.

A 15.417.7 PROPOSITION
If G : A → B is an (E, M)-functor, then the following hold:(1) (E, M)-factorizations are essentially unique,(2) M determines E via the unique diagonalization property,ẽ(3) if A is an (Ẽ, M)-category, (e, A) ∈ E and A −→ B ∈ Ẽ, then ((Gẽ) ◦ e, B) ∈ E.
A 15.517.8 REMARKEven though (E, M)-functors are analogous to (E, M)-categories, not all of the properties of (E, M)-categories carry over to (E, M)-functors. In particular, for (E, M)functors(1) M need not be determined by E [cf. 15.5(9)]. [The canonical forgetful functor
U : Top → Set is simultaneously an (E, Mi )-functor for i = 1, 2, where E consistsof all U -structured arrows (e, A) with e bijective and A discrete, M1 consists of all
sources in Top, and M2 consists of all sources in Top with discrete domain.](2) Extremal mono-sources need not all belong to M[cf. 15.5(2)]. [In (1) above, there arelimit sources — even isomorphisms (considered as 1-sources) — that do not belong
to M2 .]
(3) M need not be closed under composition [cf. 15.5(4)]. [If U and E are as in (1)above, and M3 = M2 ∪ { ,→ , ,→ }, then U is an (E, M3 )-functor, but,→ does not belong to M3 .]

Q

68

C

Q

RR

C

If I is a set, then the entire construction above is just the image factorization of f = hfi i (as f = m ◦ e)composed with the product source (πi ), i.e., mi = πi ◦ m. Even if I is a proper class, the construction
could be performed in the quasicategory of “large groups”, but in order to introduce this entity wewould need to enrich our foundations.

9th July 2006

Sec. 17]

Factorization structures for functors

287

17.9 PROPOSITION
If G is an (E, Mono-Source)-functor, then E consists precisely of those structured arrows
A 15.8(1)that are extremally generating.
17.10 THEOREM
If a functor G has (Generating, Mono-Source)-factorizations, then G is an(Extremally Generating, Mono-Source)-functor.

A 15.10

FACTORIZATION STRUCTURES AND LIMITS
17.11 THEOREM
GLet A be a strongly complete category and let A −−→ X be a functor that preserves strong
limits.
(1) If G is extremally co-wellpowered or if A has a coseparator, then G is an (ExtrGen,
Mono-Source)-functor.
(2) If G is faithful and concretely co-wellpowered, then G is a (ConGen, Initial MonoSource)-functor.
Proof:(1). In case G is extremally co-wellpowered, the result is a straightforward generalization of Theorem 15.25(1). In case A has a coseparator C, we conclude as abovethat G has (ExtrGen, Mono-Source)-factorizations for set-indexed G-structured
sources. As in Theorem 15.10 this implies that G has the (ExtrGen, Mono-Source)diagonalization property (for arbitrary sources). Let X be an arbitrary X-object.
h

For each G-structured morphism X −→ G(C hom(X,GC) ) select an (ExtrGen, Mono)factorization:
g

h

Gm

X −→ G(C hom(X,GC) ) = X −−h→ GAh −−−−h→ G(C hom(X,GC) ).
k

Likewise for each G-structured morphism X −→ G(C ∅ ) select an (ExtrGen, Mono)factorization:
k

g

Gm

X −→ G(C ∅ ) = X −−k→ GAk −−−−k→ G(C ∅ ).
Then M = {(gh , Ah ) h ∈ hom(X, G(C hom(X,C) ))} ∪ {(gk , Ak ) k ∈ hom(X, G(C ∅ ))}f

is a set. Next we show that every G-structured morphism X −→ GA has an(M , Mono)-factorization. Analogous to Theorem 15.25(1) this implies that G is
f

e

Gm

(ExtrGen, Mono Source)-factorizable. Let X −→ GA = X −→ GB −−−→ GA be a(Gen, Mono)-factorization. Since C is a coseparator, there exists a monomorphismn
B −→ C homA (B,C) . In case homA (B, C) = ∅, choose k = Gn ◦ e. Then there exists

9th July 2006

288

Factorization Structures[Chap. IVda diagonal Ak −→ B such that the diagram
gk

/ GAk
w
Gd wwwew
Gmkww w{ w/ G(C ∅ )
GB

X

Gn

commutes. Since mk is a monomorphism, so is d. Thusf

G(m◦d)g

X −→ GA = X −−k→ GAk −−−−−→ GA
is an (M , Mono)-factorization. In case homA (B, C) 6= ∅, there exists a monomorñphism C homA (B,C) −→ C homX (X,GC) , since the map
ϕ : homA (B, C) → homX (X, GC),
defined by ϕ(`) = G` ◦ e, is injective (cf. Exercise 10K). By choosing h = G(ñ ◦ n) ◦ ef

and proceeding as before, one obtains an (M , Mono)-factorization of X −→ GA.
(2). This follows as a ready adaptation of the proof of Theorem 15.25 since regularmonomorphisms are G-initial [cf. Proposition 8.7(3)].


FACTORIZATIONS OF STRUCTURED 2-SOURCES
17.12 PROPOSITION
If G-structured 2-sources have (Generating, −)-factorizations, then G preservesmono-sources.
mGiAi )I be a mono-source in A, andProof: Let A −−→ X be a functor. Let M = (A −−→

let X

r1
r2

// GA be a pair of X-morphisms with GM ◦ r1 = GM ◦ r2 . If
r

e

Gs

i
i
GA,
GA = X −→ GB −−−→X −−→

i ∈ {1, 2}
ri
is an (Generating, −)-factorization of the structured 2-source (X −−→GA)i∈{1,2} , thenG(mi ◦ s1 ) ◦ e = Gmi ◦ Gs1 ◦ e = Gmi ◦ r1 = Gmi ◦ r2 = Gmi ◦ Gs2 ◦ e = G(mi ◦ s2 ) ◦ e
for each i ∈ I. Since (e, B) is generating, this implies that mi ◦ s1 = mi ◦ s2 for eachmii ∈ I. Since (A −−→
Ai )I is a mono-source, this implies that s1 = s2 . Thus r1 = r2 . 

17.13 PROPOSITION
If G : A → B is a functor such that G-structured 2-sources have (Generating, MonoSource)-factorizations, then the following conditions are equivalent:(1) G reflects isomorphisms,

9th July 2006

Sec. 17]

Factorization structures for functors

289(2) each mono-source is G-initial,(3) G is faithful and reflects extremal epimorphisms,(4) G reflects limits,(5) G reflects equalizers.
fi

mi
Proof: (1) ⇒ (2). Let M = (A −−→
Ai )I be a mono-source in A, let S = (B −−→ Ai )If

be a source in A, and let GB −→ GA be an B-morphism with GS = GM ◦ f . Considera (Generating, Mono-Source)-factorization:
GBx< O

idGB xx

x
Gm
xx
xx g
/ GC
GB F
FF
FF
Gn
F
f FF" GAfidof the G-structured 2-source GB ←−− GB −→ GA. Then G(M ◦ n) ◦ g = G(S ◦ m) ◦ g
implies that M ◦ n = S ◦ m. Thus, since (n, m) and M are mono-sources, this implies
that m is a monomorphism. Hence, by Proposition 17.12, Gm is a monomorphism and aretraction, so an isomorphism. By (1), m is an isomorphism. Thus f = n◦m−1 : B → A
is an A-morphism with Gf = f and S = M ◦ f . It is the unique morphism with theseproperties, since M is a mono-source.
(2) ⇒ (3). By Proposition 10.60 G is faithful and reflects isomorphisms. Let e bean A-morphism such that Ge is an extremal epimorphism. By faithfulness, e is an
epimorphism. If e = m ◦ g is a (−, Mono)-factorization of e, then, by Proposition 17.12,
Ge = Gm ◦ Gg is a (−, Mono)-factorization of Ge. Hence Gm is an isomorphism in B.
Thus m is an isomorphism in A.
(3) ⇒ (1). G reflects extremal epimorphisms and, by faithfulness, monomorphisms.
Hence, by the dual of Proposition 7.66, G reflects isomorphisms.
(2) ⇒ (4). By Proposition 10.60 G is faithful. Let D : I → A be a diagram and let`i
L = (L −−→
Di )I be a source in A such that GL is a limit of G ◦ D. By faithfulness L

fi

is a natural source for D. Let S = (A −−→ Di )I be an arbitrary natural source for D.
g
Then GS is a natural source for G ◦ D. Hence there exists a B-morphism GA −→ GLwith GS = GL ◦ g. Since GL, being a limit, is a mono-source, the faithfulness of G
implies that L is a mono-source; hence it is G-initial. Thus there exists an A-morphismfA −→ L with Gf = g and S = L ◦ f . Since L is a mono-source, f is uniquely determinedby the latter property. Hence L is a limit of D.
(4) ⇒ (5). Obvious.
f(5) ⇒ (1). Immediate, since a morphism A −→ B is an equalizer of (idB , idB ) if andonly if f is an isomorphism.


9th July 2006

290

Factorization Structures[Chap. IV

17.14 REMARK
Observe that in the diagram below, all arrows indicate implications that hold without any
assumptions, whereas those labeled 1, 2, 3, and 4 are equivalences under the hypothesisof the preceding Proposition (cf. 17G).
monosources
are initial1?

reflects limits
2?

reflects equalizers- faithful
63

faithful andreflects extremal
epimorphisms?

reflects
isomorphisms4

17.15 PROPOSITIONG
A functor A −−→ X is faithful if and only if for each A-object A the G-structured source
id

id(GA ←−− GA −−→ GA) is (Generating, Initial Source)-factorizable.
Proof: If G is faithful, thenGAy< O

idGA yyyGidA
yyyyidGA
/ GA
GAE
EE
EE
GidA
E
idGA EE" 

GA

is, by Proposition 8.16(3) and Proposition 10.59, a (Generating, Initial Source)-factorization. Conversely, if A

r
s

//

B is a pair of A-morphisms with Gr = Gs andGAy< O

idGA yy

y
Gm
yy
yy g
/ GC
GAF
FF
FF
Gn
F
idGA FF" GA

is a (Generating, Initial Source)-factorization, then there exists an A-morphismgA −→ C with Gg = g and m ◦ g = idA = n ◦ g. Since (g, C) is generating, the equalities

9th July 2006

Sec. 17]

Factorization structures for functors

291

G(r ◦ m) ◦ g = Gr ◦ Gm ◦ g = Gr = Gs = Gs ◦ Gn ◦ g = G(s ◦ n) ◦ g imply thatr ◦ m = s ◦ n. Thus r = r ◦ idA = r ◦ m ◦ g = s ◦ n ◦ g = s ◦ idA = s.
17.16 COROLLARY
If G-structured 2-sources are (Generating, Initial Source)-factorizable, then G is faithful.
Suggestions for Further Reading
Tholen, W. Factorizations of cones along a functor. Quaest. Math. 2 (1977): 335–353.
Herrlich, H., and G. E. Strecker. Semi-universal maps and universal initial completions. Pacific J. Math. 82 (1979): 407–428.
Nakagawa, R. A note on (E, M )-functors. Springer Lect. Notes Math. 719 (1979):
250–258.

EXERCISES
17A. A Characterization of Generating Arrows
Let (A, U ) be a concrete category that has concrete equalizers. Show that a structured
arrow is(a) generating if and only if it does not factor through a non-isomorphic regular monomorphism,(b) concretely generating if and only if it does not factor through a non-isomorphic
embedding,(c) extremally generating if and only if it does not factor through a non-isomorphic
monomorphism.
17B. Composites of (Generating,−)-Factorizable Functors
F

GShow that whenever A −−→ B is (Generating, Mono-Source)-factorizable and B −−→ CG◦Fis (Generating, −)-factorizable, then A −−−→ C is (Generating, Mono-Source)-factorizable.
Conclude that a functor with an (Epi, Mono-Source)-factorizable domain is (Generating,Mono-Source)-factorizable if and only if it is (Generating,−)-factorizable.
17C. Existence of Factorization Structures
Let (A, U ) be a concrete category over X. Show that(a) If (A, U ) has concrete equalizers and concrete intersections, then structured arrows
have (Extremally Generating, Mono)-factorizations.

9th July 2006

292

Factorization Structures[Chap. IV(b) If X is strongly complete and if U lifts limits and is concretely co-wellpowered, then
U is a (Concretely Generating, Initial Mono-Source)-functor.
17D. Uniqueness of Diagonals
Show that a functor is an (E, M)-functor if it has (E, M)-factorizations and the (notnecessary unique) (E, M )-diagonalization property.
17E. Properties of (E, M)-Functors
G

Consider the following properties of an (E, M)-functor A −−→ B:e(1) If GA −→ GA0 is a B-morphism and (e, A0 ) belongs to E, then there exists aneA-morphism A −→ A0 with Ge = e.
(2) E is closed under composition.
(3) (idGA , A) ∈ E for each A-object A.
(4) Any source in M is G-initial.
mi(5) A source (A −−→
Ai )I belongs to M if and only if for every commutative diagram

B
f

/ GA

e





Gfi

GA Gm / GAiiwith (e, A) ∈ E there exists a unique diagonal.
Show that(a) (1) implies (2), but not conversely.
(b) (2) implies (3) if each A-object is the codomain of a member of E.
(c) (3) and (4) are equivalent.
(d) (4) implies (5), but not conversely.
(e) (5) implies that every A-isomorphism, considered as source, belongs to M.
(f) (3) implies that G is faithful and that all extremal mono-sources belong to M.
17F. Factorizations of Structured 2-Sources
Show that(a) If G has (Generating, Mono-Source)-factorizations for structured 2-sources, then G
has the unique (Extremally Generating, Mono-Source)-diagonalization property (forall structured sources).
(b) If G has (Concretely Generating, Initial Mono-Source)-factorizations for structured
2-sources, then G has the unique (Concretely Generating, Initial Mono-Source)diagonalization property.

9th July 2006

Sec. 17]

Factorization structures for functors

293(c) If G has the (E, Extremal Mono-Source)-diagonalization property for structured
2-sources, then each (e, A) in E is generating.
(d) If G has the (E, Mono-Source)-diagonalization property for structured 2-sources,
then each (e, A) in E is extremally generating.
(e) If G is faithful and G has the (E, Initial Mono-Source)-diagonalization property for
2-sources, then each (e, A) in E is concretely generating.
17G. Re: Remark 17.14
Show that(a) A functor need not be faithful even though it reflects every type of monomorphismand every type of epimorphism introduced above (extremal, regular, strict, strong,
section, retraction, isomorphism, etc.). [Hint: Consider the functor G from 0 ⇒ 1
to 0 → 1 with G(0) = 0 and G(1) = 1.](b) A faithful functor that reflects isomorphisms and extremal epimorphisms need notreflect equalizers. Hint: Consider the inclusion functor from
0 >NNN

0 >NNN

1>> NNN>> NNN>> NNNNN'/@ 2 ppp/7 3
pp
ppp
p
p
p
ppp

>> NNN>> NNN>> NNNNN'/@ 2 ppp/7 3
pp
ppp
p
p
 ppppinto

1(c) A functor that reflects equalizers need reflect neither limits nor regular monomorphisms. Hint: For limits consider the inclusion functor from the discrete category
with objects 0 and 1 into 0 → 1; for regular monomorphisms consider the inclusion
functor from
0
0>
into


1

>>
>>
>>

//

1

2

G

(d) If a functor A −−→ B reflects limits, then mono-sources in A need not be G-initial.
Hint: Consider the inclusion functor from
/1
>>
>>
>>
 

1

0>

>>
>>
>>
 

0>
into

2

2

(e) If mono-sources are G-initial, then G need not reflect extremal epimorphisms. Hint:
Consider the inclusion functor from0
1

9th July 2006

//
0
>>
>>
>>
 

2>
into

1

294

Factorization Structures[Chap. IV* 17H. A Generalization of Theorem 17.11
Let A be a strongly complete category with a coseparating set. Show that every functorG

A −−→ B that preserves strong limits is an (ExtrGen, Mono-Source)-functor.
17I.
Prove that if A has equalizers and intersections and G : A → B preserves these limits, then G-structured arrows have (Extremal Generating, Mono)-factorizations. Cf.
Theorem 14.19.
17J.
Let (A, U ) be a concrete category over a strongly complete category X. Show that if
U is concretely co-wellpowered and preserves and detects limits, then U is (Concretely
Generating, Initial Mono-Source)-factorizable, thus a (−, −)-functor.
17K.
Let (A, U ) be a fibre-small, transportable concrete category over a wellpowered category
X. Prove that if (A, U ) has concrete limits and bounded generation, then U is an(Extremal Generating, Mono-Source)-functor.
17L.
Show that the nonfull embedding


3

3 ::

:

1 ::

: 

0

2


/

2
1
0

satisfies all of the conditions of Proposition 17.13 except condition (2). Cf. also Proposition 10.60.

9th July 2006

295

Chapter V

ADJOINTS AND MONADS

9th July 2006

18 Adjoint functors
Perhaps the most successful concept of category theory is that of adjoint functor. Adjoint
functors occur frequently in many branches of mathematics and the “adjoint functor
theorems” have a surprising range of applications.
18.1 DEFINITION
A functor G : A → B is said to be adjoint provided that for every B-object B there
exists a G-universal arrow with domain B (cf. 8.30).
Dually, a functor G : A → B is said to be co-adjoint provided that for every B-object
B there exists a G-co-universal arrow with codomain B (cf. 8.40).
18.2 EXAMPLES(1) A subcategory A of a category B is (co)reflective in B if and only if the associatedinclusion A ,→ B is a (co-)adjoint functor [cf. 8.31(2) and 8.41(1)].
(2) The forgetful functor U of a concrete category (A, U ) over X is adjoint if and only
if for each X-object X there exists a free object over X. In particular, the forgetful
functors:(a) of the constructs Rel, Top, and Alg(Σ) are both adjoint and co-adjoint (cf. 8.23
and 8.41),(b) of the constructs Vec, Grp, Pos, and Cat are adjoint but not co-adjoint,(c) of the constructs CLat and CBoo are neither adjoint nor co-adjoint,(d) of the concrete categories of the form Spa(T ) are both adjoint and co-adjoint,(e) TopGrp → Grp, TopGrp → Top, and TopGrp → Set are adjoint.
(3) O : Ban → Set is adjoint, but U : Ban → Set is not [cf. 8.23(12)].
(4) Each equivalence is adjoint and co-adjoint.
(5) If A is a category, then the unique functor A → 1 is adjoint if and only if A has an
initial object [cf. 8.31(3)].
(6) A hom-functor hom(A, −) : A → Set is adjoint if and only if there exist arbitrary
copowers of A in A [cf. 8.31(4)].
f(7) For each set M , the endofunctor (M × −) : Set → Set that sends A −→ B to
id

×f

M
M × A −−−−−→ M × B is co-adjoint. “Evaluation” ev : M × AM → A defined by(m, f ) 7→ f (m) is an (M × −)-co-universal arrow for A (cf. §27).

(8) The minimal realization functor M : Beh → Autr is adjoint [cf. 8.31(5)].

296Sec. 18]

Adjoint functors

297

18.3 PROPOSITION
For a functor G : A → B the following conditions are equivalent:(1) G is adjoint,(2) G has (Generating69,−)-factorizations,(3) G is an (E, M)-functor for some E and M,(4) G is a (Universal69, Source)-functor.
Proof: (1) ⇒ (4) ⇒ (3). Obvious.
(3) ⇒ (2). Theorem 17.6.
fi

e

Gm

i
(2) ⇒ (1). For any B-object B let B −−→ GAi = B −→ GA −−−→
GAi be a(Generating,−) factorization of the G-structured source that consists of all G-structured
arrows (fi , Ai ) with domain B. Then the G-structured arrow (e, A) is universal for B.

18.4 PROPOSITION
If A is an (Epi, M)-category, then for any functor G : A → B the following are equivalent:(1) G is adjoint,(2) G is a (Generating,M)-functor.
fiuProof: (1) ⇒ (2). Let (B −−→ GAi )I be a G-structured source. If B −→ GA isfia universal arrow, then for each i ∈ I there exists an A-morphism A −−→
Ai with

f

m

e

i
i
fi = Gf i ◦ u. If A −−→
Ai = A −→ A −−→
Ai is an (Epi,M)-factorization, thenfi(Ge)◦uGmi
GAi is a (Generating,M)-factorization. To show
B −−→ GAi = B −−−−−→ GA −−−→
the (Generating,M)-diagonalization property, consider a commutative diagram

B
f

g

/ GA




(∗)

Gfi

GA0 Gm / GAi
i

m

u

i
Ai )I in M. If B −→ GA is a universal arrow, thenwith (g, A) generating and (A0 −−→fg

there exist A-morphisms A −→ A0 and A −→ A with f = Gf ◦ u and g = Gg ◦ u. Thisimplies that mi ◦ f = fi ◦ g for each i ∈ I. Since (g, A) is generating, g must be andepimorphism. Hence there exists an A-morphism A −→ A0 that makes
g

/A
~
~
d ~~
fi
f
~~
~

 ~~
/ Ai
A0

A

mi

69

“Generating” in this context denotes the class of all generating G-structured arrows; likewise, “Universal” denotes the class of all universal G-structured arrows.

9th July 2006

298

Adjoints and Monads[Chap. V

commute. Obviously, d is a diagonal for (∗). Uniqueness follows from the fact that (g, A)
is generating.
(2) ⇒ (1). Proposition 18.3.



PROPERTIES OF ADJOINT FUNCTORS
18.5 PROPOSITION
G1G2
G2 ◦G1
If A −−→
B and B −−→
C are adjoint, then so is A −−−−−→ C.
uv

Proof: If C −→ G2 B is universal for C, and B −→ G1 A is universal for B, then(G2 v)◦uC −−−−−−→ G2 G1 A is universal for C.



18.6 PROPOSITION
Adjoint functors preserve mono-sources.
miAi )I be a mono-source in
Proof: Let G : A → B be an adjoint functor, let S = (A −−→

A, and let B

r /
s

u

/ GA be a pair of B-morphisms with GS ◦ r = GS ◦ s. If B −→ GA is auniversal arrow, then there exist A-morphisms A

r
s

//

A with r = Gr ◦ u and s = Gs ◦ u.

This implies that G(S ◦ r) ◦ u = G(S ◦ s) ◦ u; hence S ◦ r = S ◦ s, so that r = s, and thusr = s.
18.7 COROLLARY
Embeddings of reflective subcategories preserve and reflect mono-sources.



18.8 REMARK(1) As shown in Example 7.33(5) embeddings of full subcategories need not preservemonomorphisms, hence they need not preserve mono-sources.
(2) Full, reflectiveembeddings preserve extremal monomorphisms. (Cf. 7F.)
18.9 PROPOSITION
Adjoint functors preserve limits.
Proof: Let G : A → B be an adjoint functor, let D : I → A be a diagram, and let`i
L = (L −−→
Di )I be a limit of D. Then GL is a natural source for G ◦ D. Let S =fiu(B −−→ GDi )I be a natural source for G ◦ D. If B −→ GA is a universal arrow for B,
then for each i ∈ I there exists an A-morphism f i : A → Di with fi = Gf i ◦ u. Sincefi(u, A) is generating, the source S = (A −−→
Di ) is natural for D. Hence there exists
an A-morphism f : A → L with S = L ◦ f . Consequently, f = Gf ◦ u : B → GL is a
B-morphism with S = GL ◦ f . Uniqueness follows from the fact that, by Proposition
18.6, GL is a mono-source.


9th July 2006

Sec. 18]

Adjoint functors

299

18.10 COROLLARY
If (A, U ) is a concrete category over X that has free objects, then the following hold:(1) all limits in (A, U ) are concrete,(2) U preserves and reflects mono-sources,(3) if (A, U ) is fibre-small and transportable, then wellpoweredness of X implies wellpoweredness of A.

18.11 PROPOSITION
If G : A → B is an adjoint functor and A is co-wellpowered, or extremally co-wellpowered, then so is G.
Proof: Immediate from Proposition 8.36.



ADJOINT FUNCTOR THEOREMS
We have seen above that adjoint functors preserve limits. Next we will see that thisproperty in conjunction with certain completeness and smallness conditions actuallycharacterizes such functors. We have seen before that both general constructions andsmallness conditions are essential ingredients of category theory. The adjoint functor
theorems of this section are prime examples of this fact. Below we will see that each ofthem contains a completeness condition and a smallness condition, neither of which canbe eliminated.
18.12 ADJOINT FUNCTOR THEOREM
A functor G : A → B, whose domain A is complete, is adjoint if and only if G satisfies
the following conditions:(1) G preserves small limits,(2) for each B-object B there exists a G-solution set, i.e., a set-indexed G-structuredfisource (B −−→ GAi )I through which each G-structured arrow factors (in the sensefgthat given any B −→ GA, there exists a j ∈ I and a Aj −→ A in A such that
fj

/ GAj
CC
CC
Gg
C
f CC
! 

BC

GA

commutes).
Proof: If G is an adjoint functor, then, by Proposition 18.9, G preserves limits. IfuB −→ GA is a G-universal arrow for B, then the G-structured source SB consistingof (u, A) alone is a G-solution set for B. Hence conditions (1) and (2) are necessary.

9th July 2006

300

Adjoints and Monads[Chap. V

To show the converse, suppose that (1) and (2) are satisfied. Let B be a B-object
and let SB = ((ui , Ai ))I be a G-solution set for B. If (P, πi )I is a product of thefamily (Ai )I in A, then by (1) (GP, Gπi )I is a product of (GAi )I in B. Thus u =hui i : B → GP is a G-structured arrow such that the G-structured source, consistingof (u, P ) alone, is a G-solution set for B. Consider the family (gj )J consisting of all
gje
A-morphisms P −−→ P with (Ggj ) ◦ u = u. If E −→ P is a multiple equalizer ofGe(gj )J , then by (1) GE −−→ GP is a multiple equalizer of (Ggj )J . Thus there is av
B-morphism B −→ GE with u = Ge ◦ v. The G-structured source, consisting of the
G-structured arrow (v, E) alone, is a G-solution set for B. Thus to prove that (v, E) isa G-universal arrow, it need only be shown that (v, E) is G-generating. Let E

r
s

// A

e

be a pair of A-morphisms with Gr ◦ v = Gs ◦ v. If E −→ E is an equalizer of r and s,Gethen by (1) GE −−→ GE is an equalizer of Gr and Gs. Thus there exists a B-morphismfB −→ GE with v = Ge ◦ f . Since ((u, P )) is a G-solution set for B, there exists an
e◦e◦g

g

A-morphism P −→ E with f = Gg ◦ u. Hence P −−−−→ P is an A-morphism withG(e ◦ e ◦ g) ◦ u = G(e ◦ e) ◦ f = Ge ◦ v = u. Therefore there exists some j ∈ J with
gj = e ◦ e ◦ g. There also exists some k ∈ J with gk = idP . Thus gj ◦ e = gk ◦ e impliese ◦ e ◦ g ◦ e = e. Since e is a monomorphism, this implies that e ◦ g ◦ e = idE . Thus e
is a retraction, which by Proposition 7.54 implies that r = s. Consequently, (v, E) is a
G-universal arrow for B. Thus G is an adjoint functor.

18.13 EXAMPLES
The following functors have complete domains and preserve limits, but fail to be adjoint:(1) the unique functor Ordop → 1, where Ord is the partially ordered class of all ordinal
numbers, considered as a category,(2) the forgetful functors CLat → Set and CBoo → Set [cf. 8.23(7)].
Whereas the “limit-preservation” condition in the above theorem is usually easy to check,the “solution set” condition is often rather cumbersome, particularly since the theorem
gives no idea of how to find such a set. In the next results, attention is focused on“canonical” candidates for solution sets.
18.14 THEOREM
If A is strongly complete and (extremally) co-wellpowered, then the following conditions
G

are equivalent for any functor A −−→ B:(1) G is adjoint,(2) G preserves small limits and is (extremally) co-wellpowered.
Moreover the implication (2) ⇒ (1) holds without the assumption that A be (extremally)
co-wellpowered.
Proof: (1) ⇒ (2). Immediate from Propositions 18.9 and 18.11.
(2) ⇒ (1). Immediate from Proposition 18.3 and Theorem 17.11(1).



9th July 2006

Sec. 18]

Adjoint functors

301

18.15 COROLLARY
Fibre-small, concretely co-wellpowered constructs that are concretely complete have freeobjects.
An adjoint functor is co-wellpowered and preserves small limits

18.16 EXAMPLES
The conditions in the above theorem are carefully balanced as the following examples
show:G

1. There exist full embeddings A −−→ B such that A is complete and co-wellpowered
and G preserves limits and is extremally co-wellpowered, but G is neither adjoint
nor co-wellpowered. [Let C be the partially ordered class of all ordinals, considered
as a category, let A = Cop, let B be obtained from A by adding an initial object,Gand let A −−→ B be the inclusion.]
2. There exist strongly complete categories A that are not co-wellpowered, evenGthough the unique functor A −−→ 1 is adjoint and co-wellpowered. [Consider
Λ-CCPos.]
For suitable categories A the somewhat cumbersome solution set condition completely
vanishes, as the following result shows:
18.17 SPECIAL ADJOINT FUNCTOR THEOREM
If A is a strongly complete category with a coseparator, then for any functor G : A → B,
the following conditions are equivalent:(1) G is adjoint,

9th July 2006

302

Adjoints and Monads[Chap. V(2) G preserves strong limits.
Proof: Immediate from Theorem 17.11, Proposition 18.3, and Proposition 18.9.



18.18 REMARKThe categories Set, Vec, Pos, Top, and HComp are complete, wellpowered, and havecoseparators (cf. 7.18), so that the above theorem applies to them. Since many familiar
categories have separators but fail to have coseparators, the dual of the Special Adjoint
Functor Theorem is applicable even more often than the theorem itself.
18.19 CONCRETE ADJOINT FUNCTOR THEOREM
Let G : (A, U ) → (B, V ) be a concrete functor. If (A, U ) is complete, wellpowered,co-wellpowered, and has free objects, then G is adjoint if and only if G preserves small
limits.
Proof: As before, it suffices to show that the above assumptions imply that G is cou
wellpowered. Let B be a B-object and let V B −→ U AB be a (fixed) universal U structured arrow with domain V B. It suffices to show that for every generating Gg
structured arrow B −→ GA, A can be considered (via a suitable epimorphism) as aquotient-object of AB . By universality there exists a unique A-morphism g : AB → A
such that the triangleu/ U AB
VB M
MMM
MMM
M
V g MMM&Ug

V GA = U A

commutes. Since (g, A) is a generating G-structured arrow, g is an epimorphism; hence
A is a quotient-object of AB .

18.20 REMARKS(1) Faithfulness of U is not needed for the above result, but faithfulness of V is essential.
(2) If (A, U ) and (B, V ) satisfy the assumptions of the above result and if V reflects
limits, then every concrete functor (A, U ) → (B, V ) preserves limits, and hencemust be adjoint. Similar situations will be investigated in §23. (Observe that thereflection of limits is typical for forgetful functors of “algebraic” concrete categories.)

Suggestions for Further Reading
Kan, D. N. Adjoint functors. Trans. Amer. Math. Soc. 87 (1958): 294–329.
Freyd, P. Functor theory. Ph. D. dissertation, Princeton University, 1960.
Felscher, W. Adjungierte Funktoren und primitive Klassen. Sitzungsberichte Heidelberger Akad. Wiss. 4 (1965): 447–509.

9th July 2006

Sec. 18]

Adjoint functors

303

Ehresmann, C. Sur l’existence de structures libres et de foncteurs adjoints. Cahiers
Topol. Geom. Diff. 9 (1967): 33–180.

EXERCISES
18A. Functors that are Simultaneously Adjoint and Co-adjointG

Let A −−→ B be a functor. Show that(a) If G is an equivalence then G is adjoint and co-adjoint.
(b) If G is adjoint and co-adjoint and A = B = Set, then G is an equivalence.
(c) If A and B are monoids, considered as categories, then G is adjoint iff G is co-adjoint.
(d) If A has an initial and a terminal object and B = 1, then G is adjoint and co-adjoint.
(e) If A = 1 and G maps the single object of A to a zero object in B, then G is adjoint
and co-adjoint.
(f) The forgetful functor Top → Set is adjoint and co-adjoint.
(g) The forgetful functor Alg(1) → Set is adjoint and co-adjoint, where the η : X →(X × , λ), defined by η(x) = (x, 0) and λ(x, n) = (x, n + 1), are universal arrows,
and the ε : (X N , µ) → X, defined by ε(xn ) = x0 and µ(xn ) = (xn+1 ) are couniversal
arrows.

N(h) If A is small and C is a category that is complete and cocomplete, then the functor[G,id][B, C] −−−−→ [A, C] is adjoint and co-adjoint.
18B. Smallness Conditions For Adjoints
G(a) Let A −−→ B be a faithful adjoint functor. Show that A is co-wellpowered if andonly if G is co-wellpowered.
G(b) Let A −−→ B be an adjoint functor such that mono-sources are G-initial. Show that
A is extremally co-wellpowered if and only if G is extremally co-wellpowered.
(c) Show that the unique functor Λ-CCPos → 1 is adjoint and co-wellpowered, eventhough Λ-CCPos is not co-wellpowered. Cf. Exercise 15D.
18C. Adjoints via Representable Functors
Show that
G(a) A functor A −−→ Set is adjoint if and only if G is representable by an object for
which arbitrary copowers exist (cf. Exercise 10R).
G(b) A functor A −−→ B is adjoint if and only if for each B-object B the functorGhom(B,−)

A −−→ B −−−−−−−→ Set is representable.

9th July 2006

304

Adjoints and Monads[Chap. V

18D. (Co-)Adjoints and Colimits(a) Show that a functor Set → A is co-adjoint if and only if it preserves coproducts.
G(b) Let A −−→ Set be a faithful and representable functor whose domain is a complete,
wellpowered, and co-wellpowered category A. Show that G is adjoint if and only if
A is cocomplete.
18E. Power-Set Functors
Show that the covariant power-set functor P : Set → Set is neither adjoint nor coadjoint, but that the contravariant power-set functor Q : Setop → Set is adjoint.
18F. Upper Semicontinuity as Adjointness
Let A be a complete totally ordered set and let R be the ordered set of real numbers.
Let G : A → R be an order-preserving map, considered as a functor. Show that G isadjoint if and only if G is upper semicontinuous.
18G. Complete Boolean Algebras
Show that (cf. Exercises 8E, 8F, and 10R):(a) CBoo is complete, well-powered, and extremally co-wellpowered.
(b) CBoo is not cocomplete.
(c) The forgetful functor CBoo −→ Set is not adjoint.
18H. Adjoint Functors between PosetsfConsider order-preserving functions A −→ B between posets as functors and show that(a) f is adjoint if and only if for each b ∈ B, the set { a ∈ Aelement.
(b) If A = B =b ≤ f (a) } has a smallest

N (with the usual order), then(1) f is adjoint if and only if it is unbounded.
(2) f is co-adjoint if and only if it is unbounded and f (0) = 0.
(c) If A = B =Z (with the usual order), then f is adjoint if and only if it is co-adjoint.

18I. Stable Epimorphisms and Adjoints
Show that faithful adjoints reflect stable epimorphisms.
* 18J. Extremal Monomorphisms and Adjoints
Show that adjoint functors need not preserve extremal monomorphisms.
18K. Compact Categories
A category A is called compact provided that each functor with domain A that preserves colimits is co-adjoint. Prove that(a) Each cocomplete co-wellpowered category that has a separator is compact.

9th July 2006

Sec. 18]

Adjoint functors

305(b) Each compact category is complete. [Hint: For each small diagram D in A thefunctor F : A → Setop that assigns to each A-object A the set of all natural sources
for D with domain A preserves colimits. Thus F op is representable.]
18L. Coseparating Sets(a) Prove that a set S of objects of a category A that has products is an (extremally)
coseparating set for A if and only if every A-object is an (extremal) subobject ofsome product of objects in S (cf. Proposition 10.38).
(b) Using (a) prove that the Special Adjoint Functor Theorem (18.17) holds also for all
strongly complete categories A that have a coseparating set.

9th July 2006

19 Adjoint situations
G

With every adjoint functor A −−→ B there can be naturally associated (in an essentiallyηFunique way) a functor B −−→ A and two natural transformations idB −→ G ◦ F andε
F ◦ G −→ idA . There is an intricate web of relationships between the functors G and F
and the natural transformations η and ε, including an inherent duality, that is largely
responsible for the importance of adjoint functors.
19.1 THEOREM
Let G : A → B be an adjoint functor, and for each B-object B let ηB : B → G(AB ) be a
G-universal arrow. Then there exists a unique functor F : B → A such that F (B) = AB
η=(ηB )

for each B-object B, and idB −−−−−→ G ◦ F is a natural transformation.
εMoreover, there exists a unique natural transformation F ◦ G −→ idA that satisfies thefollowing conditions:
ηG

Gε

id

Fη

εF

id

G
(1) G −−→ GF G −−→ G = G −−−→ G,F(2) F −−→ F GF −−→ F = F −−−→ F.

Proof:(a). The existence of a unique F with the required properties follows analogously to theproof of Proposition 4.22.
ε(b). For each A-object A, there exists a unique A-morphism F GA −−A→ A such thatηGA

/ GF GA
II
II
II
GεA
idGA II$ 

GA I

GA

commutes. Thus if ε = (εA ) is a natural transformation from F G to idA , it is thefunique one that satisfies (1). To show naturality, consider an A-morphism A −→ A0 .
Then it follows that G(f ◦ εA ) ◦ ηGA = Gf ◦ GεA ◦ ηGA = Gf = GεA0 ◦ ηGA0 ◦ Gf =GεA0 ◦ GF Gf ◦ ηGA = G(εA0 ◦ F Gf ) ◦ ηGA by the naturality of η. Hence f ◦ εA =
εA0 ◦ F Gf .
(c). Since η : idB → GF is natural, we get ηGF ◦ η = GF η ◦ η. Hence (1) implies that
G(idF ) ◦ η = idG F ◦ η = GεF ◦ ηGF ◦ η = GεF ◦ GF η ◦ η = G(εF ◦ F η) ◦ η. Thus
idF = εF ◦ F η, i.e., (2) holds.

19.2 REMARK
Below (cf. 19.7) it is shown that conditions (1) and (2) of Theorem 19.1 play a crucialGF

role. In particular, given functors A −−→ B and B −−→ A and natural transformationsηε
idB −→ GF and F G −→ idA that satisfy the above conditions, the following hold:306

Sec. 19]

Adjoint situations

307• G is adjoint.
• F is co-adjoint.
η• Each B −−B→ G(F B) is a G-universal arrow.
ε• Each F (GA) −−A→ A is an F -co-universal arrow.
19.3 DEFINITION
An adjoint situation (η, ε) : F
G

G : A → B consists of functors A −−→ B andηFε

B −−→ A and natural transformations idB −→ GF (called the unit) and F G −→ idA(called the co-unit) that satisfy the following conditions:
ηG

Gε

id

Fη

εF

id

G
(1) G −−→ GF G −−→ G = G −−−→ G,F(2) F −−→ F GF −−→ F = F −−−→ F.

19.4 EXAMPLES
By Theorem 19.1 every adjoint functor gives rise to an adjoint situation. (Cf. Examples
18.2 of adjoint functors.) In particular:E(1) every embedding A −−→ B of a full reflective subcategory gives rise to an adjointsituation (η, ε) : R
E : A → B, where R is a reflector, the ηB ’s are A-reflection
arrows, and the εA ’s are isomorphisms,(2) every concrete category (A, U ) over X with free objects gives rise to an adjointsituation (η, ε) : F
U : A → X, where F is called a free functor, the ηX ’s areuniversal arrows (also called insertions of generators), and each εA expresses A
as a retract of the free object generated by U A,(3) for each set M , (η, ε) : (M ×−)hom(M, −) : Set → Set is an adjoint situation,
ηwhere (M × −) is the endofunctor of Example 18.2(7), each B −−B→ hom(M, M × B)ε
is defined by (ηB (b))(m) = (m, b), and each M × hom(M, A) −−A→ A is defined byεA (m, f ) = f (m).

19.5 REMARK
We have seen that every adjoint functor gives rise to an adjoint situation. Below it will
be shown that(1) every adjoint situation arises in this way (cf. 19.7),
(2) every adjoint situation (η, ε) : F
G : A → B is, up to a natural isomorphism,
uniquely determined by G (cf. 19.9).

First, however, we exhibit a duality that is inherent in the concept of adjoint situations.
19.6 DUALITY THEOREM FOR ADJOINT SITUATIONS
If (η, ε) : F
G : A → B is an adjoint situation, then op
(εop , η op ) : Gop
F : Bop → Aop is an adjoint situation.

9th July 2006

308

Adjoints and Monads
G[Chap. V
Gop

F

Proof: If A −−→ B and B −−→ A are functors, then so are Aop −−−→ Bop andηF opεBop −−−→ Aop . If idB −→ GF and F G −→ idA are natural transformations, thenη op
εop
so are Gop F op −−−→ idBop and idAop −−−→ F op Gop . If Gε ◦ ηG = idG , it follows thatη op Gop ◦ Gop εop = idGop , and if εF ◦ F η = idF , it follows that F op η op ◦ εop F op = idF op .
19.7 PROPOSITION
If (η, ε) : F
G : A → B is an adjoint situation, then the following hold:(1) G is an adjoint functor,
η(2) for each B-object B, B −−B→ GF B is a G-universal arrow,(3) F is a co-adjoint functor,ε(4) for each A-object A, F GA −−A→ A is a F -co-universal arrow.
Proof: It suffices to prove (2), since (2) implies (1), and (3) and (4) follow by thefDuality Theorem 19.6 from (1) and (2). Let B −→ GA be a G-structured arrow. Thenthe commutative diagram
f

B
ηB



/ GA

GF B GF f

II
IIidGA
II
ηGA
II
I$

/ GF GA/ GA
GεAε ◦F fshows that F B −−A−−−→ A is an A-morphism f with f = Gf ◦ ηB . Conversely, iff

F B −→ A is an A-morphism with f = Gf ◦ ηB , then the commutative diagram
FBJ

Ff
JJ
JJ
JJF ηBJJ$

idB

F GF B! 

FBshows that f = εA ◦ F f .

F Gf

εF B

'
/ F GA


f

εA

/A



19.8 REMARKS(1) Observe that Theorem 19.1 and Proposition 19.7 imply that G : A → B is an adjointfunctor if and only if there exists an adjoint situation (η, ε) : F
G : A → B.
(2) A functor G : A → B is an equivalence if and only if there exists an adjoint situation(η, ε) : F
G : A → B with natural isomorphisms η and ε. Cf. Proposition 6.8.

9th July 2006

Sec. 19]

Adjoint situations

309(3) If G : A → B and F : B → A are concrete functors over X, then (F, G) is a Galois
correspondence if and only if there exist concrete natural transformations η and εfor which (η, ε) : F
G : A → B is an adjoint situation. (Cf. 6.24 and 6.25.)
19.9 PROPOSITION
Adjoint situations associated with a given adjoint functor G : A → B are essentially
unique, i.e., for each adjoint situation (η, ε) : F
G : A → B, the following hold:(1) if (η, ε) : F
G : A → B is an adjoint situation, then there exists a naturalτisomorphism F −→ F for which η = Gτ ◦ η and ε = ε ◦ τ −1 G,τ(2) if F : B → A is a functor and F −→ F is a natural isomorphism, then(Gτ ◦ η , ε ◦ τ −1 G) : F
G : A → B is an adjoint situation.
Proof:
ηη
(1). By Proposition 19.7, for each B-object B, B −−B→ GF B and B −−B→ GF B are
G-universal arrows for B. Hence by Proposition 8.25 there exists an isomorphismττ =(τB )F B −−B→ F B with GτB ◦ ηB = η B . It follows that F −−−−−→ F is a natural
isomorphism with η = Gτ ◦ η. For each A-object A the equalities GεA ◦ ηGA =idGA = GεA ◦η GA = GεA ◦GτGA ◦ηGA = G(εA ◦τGA )◦ηGA imply that εA = εA ◦τGA .
Hence ε = ε ◦ τ G.
(2). Straightforward computations show that G(ε ◦ τ −1 G) ◦ (Gτ ◦ η)G = idG and that(ε ◦ τ −1 G)F ◦ F (Gτ ◦ η) = idF .
19.10 DEFINITIONGF
Let A −−→ B and B −−→ A be functors. Then F is called a co-adjoint for G and
G is called an adjoint for F (in symbols: FG) provided that there exist naturaltransformations η and ε such that (η, ε) : F
G : A → B is an adjoint situation.
19.11 REMARKS(1) Every adjoint functor G has, by Theorem 19.1, a co-adjoint F ; and any two coadjoints F and F of G are, by Proposition 19.9, naturally equivalent. Thus every
adjoint functor has, up to natural equivalence, a unique co-adjoint. Conversely, if afunctor G has a co-adjoint, then, by Proposition 19.7, G is an adjoint functor. Thusa functor is adjoint if and only if it has a co-adjoint. Moreover, by Proposition 19.7,
the co-adjoint of a functor is a co-adjoint functor.
(2) By duality, a functor is co-adjoint if and only if it has an (essentially unique) adjoint.
A functor F is a co-adjoint for G if and only if G is an adjoint for F . Thus adjoint
functors and co-adjoint functors naturally come in pairs, and these pairs are partsof adjoint situations.
(3) The reader should be aware that the following alternative terminology is also used:
G is right adjoint = G has a left adjoint = G is adjoint.
F is left adjoint = F has a right adjoint = F is co-adjoint.

9th July 2006

310

Adjoints and Monads[Chap. V

19.12 EXAMPLES
R(1) Let A be a reflective subcategory of B. Then a functor B −−→ A is a co-adjoint for
the inclusion functor A ,→ B if and only if R is a reflector for A.
C(2) Let A be a coreflective subcategory of B. Then a functor B −−→ A is an adjoint for
the inclusion functor A ,→ B if and only if C is a coreflector for A.
(3) If a construct (A, U ) has free objects, i.e., if U is an adjoint functor, then co-adjoints
for U are the free functors for (A, U ).
(4) The discrete functor D : Set → Top is a co-adjoint for the forgetful functor
U : Top → Set, and the indiscrete functor I : Set → Top is an adjoint for U .
(5) If a functor G is an isomorphism, then G−1 is simultaneously an adjoint and a coG

adjoint for G. More generally, for every equivalence A −−→ B there exists a functorF
B −−→ A that is simultaneously an adjoint and a co-adjoint for G. Cf. Proposition
6.8.
(6) Let A be a category with a zero-object. Then the zero-functor 1 → A is simultaneously an adjoint and a co-adjoint for the functor A → 1. Cf. Exercise 4H(b).
(7) A co-adjoint for the minimal realization functor [cf. 18.2(8)] is the behavior functor
B : Autr → Beh that assigns to each reachable automaton its behavior function[cf. 8.31(5)].
19.13 PROPOSITION
Adjoint situations can be composed; specifically, if (η, ε) : F(η, ε) : F
G : B → C are adjoint situations, then so is(GηF ◦ η , ε ◦ F εG) : F ◦ FG : A → B andG ◦ G : A → C.

19.14 THEOREM
If (η, ε) : F
G : A → B is an adjoint situation, then the following hold:(1) The following are equivalent:(a) G is faithful,(b) G reflects epimorphisms,(c) ε is an Epi-transformation.
(2) The following are equivalent:(a) G is faithful and reflects isomorphisms,(b) G reflects extremal epimorphisms,(c) ε is an (Extremal Epi)-transformation.
(3) G is full if and only if ε is a Section-transformation.

9th July 2006

Sec. 19]

Adjoint situations

311(4) G is full and faithful if and only if ε is a natural isomorphism.
(5) If G reflects regular epimorphisms, then each mono-source is G-initial.70
Proof: (1a) ⇒ (1b) ⇒ (1c) is immediate, since Gε is a Retraction-transformation.
(1c) ⇒ (1a). If A

f
g

//

Â is a pair of A-morphisms with Gf = Gg, then the equationsf ◦ εA = εÂ ◦ F Gf = εÂ ◦ F Gg = g ◦ εA imply that f = g.
(2a) ⇒ (2b) ⇒ (2c) is immediate, since G, being adjoint, preserves monomorphisms.
f(2c) ⇒ (2a). By (1), G is faithful. Let A −→ Â be an A-morphism such that Gf
is an isomorphism. Since f ◦ εA = εÂ ◦ F Gf is a composite of an isomorphism withan extremal epimorphism, and so is an extremal epimorphism, the dual of Proposition
7.62(2) implies that f is an extremal epimorphism. By the faithfulness of G, f is amonomorphism as well; hence it is an isomorphism.
s(3). Let G be full. Then for each A-object A there exists an A-morphism A −−A→ F GA
with GsA = ηGA . Hence the equations G(sA ◦ εA ) ◦ ηGA = GsA ◦ GεA ◦ ηGA = GsA =G(idF GA ) ◦ ηGA imply that sA ◦ εA = idF GA . Thus εA is a section. Conversely, let εfbe a Section-transformation, let A and Â be A-objects, and let GA −→ GÂ be a Br
morphism. Then there exists an A-morphism A −→ F GA with r ◦ εA = idF GA . Hence
fˆ

r

ε

Ff

A −→ Â = A −→ F GA −−→ F GÂ −−Â→ Â is an A-morphism. Since εÂ is an F -couniversal arrow, the equalities εÂ ◦ F Gfˆ = fˆ ◦ εA = εÂ ◦ F f ◦ r ◦ εA = εÂ ◦ F f imply
that Gfˆ = f . Thus G is full.
(4). This follows from (1) and (3).
(5). If G reflects regular epimorphisms, then ε is a (Regular Epi)-transformation. Let
f

fi

m

i
Ai )I be a mono-source, (Â −−→ Ai )I be a source in A, and GÂ −→ GA be a
(A −−→
B-morphism such that Gfi = Gmi ◦ f for each i ∈ I. Then the equalities fi ◦ εÂ =εAi ◦ F Gfi = εAi ◦ F Gmi ◦ F f = mi ◦ εA ◦ F f imply that for each i ∈ I the diagramF GÂ
εA ◦F f

εÂ

Â




A

/

mi

fi

/ Ai

commutes. Since every category has the (Regular Epi, Mono-Source)-diagonalizationdproperty, there exists a diagonal Â −→ A. In particular fi = mi ◦ d. Since, by (1), G ismifaithful, this implies that the mono-source (A −−→
Ai )I is G-initial.
70

See Exercise 19B(c) for a characterization of adjoint functors G with the property that each monosource is G-initial.

9th July 2006

312

Adjoints and Monads[Chap. V

Suggestions for Further Reading
Booth, P. I. Sequences of adjoint functors. Archiv Math. 23 (1972): 489–493.
Hoffmann, R.-E. Sequences of adjoints for Ens-valued functors. Manuscr. Math. 32(1980): 191–210.

EXERCISES
19A. Alternative Description of Adjoint Situations
G

F

Show that functors A −−→ B and B −−→ A yield an adjoint situation if and only if there
exists an isomorphism hom(F A, B) ∼= hom(A, GB) that is natural in the variables A
and B; more precisely,GF

Let A −−→ B and B −−→ A be functors. Define functors hom(F− , − ) : Bop × A → Set
and hom(− , G− ) : Bop × A → Set byhom(F− , − )(B, A) = homA (F B, A)
hom(F− , − )(f, g) = hom(F f, g), where hom(F f, g)(k) = g ◦ k ◦ F fhom(− , G− )(B, A) = homB (B, GA)hom(− , G− )(f, g) = hom(f, Gg), where hom(f, Gg)(k) = Gg ◦ k ◦ f.
(a) Let (η, ε) : F
G be an adjoint situation.
Define τ(B,A) : homA (F B, A) → homB (B, GA) by τ(B,A) (f ) = Gf ◦ ηB and showthat τ = (τB,A ) : hom(F− , − ) → hom(− , G− ) is a natural isomorphism.

(b) Let τ : hom(F− , − ) → hom(− , G− ) be a natural isomorphism. Define η = (ηB ) by−1
ηB = τ(B,F B) (idF B ) and ε = (εA ) by εA = τ(GA,A)(idGA ) and show that
G is an adjoint situation.
(η, ε) : F(c) The constructions described in (a) and (b) are essentially inverse to each other.
19B. Adjoints Reflecting Special Epimorphisms
Let (η, ε) : F



G : A → B be an adjoint situation. Show that(a) If f is G-final and Gf is an extremal (resp. a regular) epimorphism, then f is anextremal (resp. a regular) epimorphism.
(b) G reflects regular epimorphisms if and only if every A-morphism f , for which Gf isa regular epimorphism, is G-final.
(c) The following are equivalent:(1) mono-sources are G-initial,(2) G reflects swell epimorphisms,(3) ε is a (Swell Epi)-transformation.

9th July 2006

Sec. 19]

Adjoint situations

313(d) Each of the following conditions implies but is not equivalent to its immediate successor:(1) G reflects regular epimorphisms,(2) G reflects swell epimorphisms,(3) G reflects extremal epimorphisms,(4) G reflects epimorphisms.
(e) If G reflects swell epimorphisms, then G reflects limits.
(f) If G reflects isomorphisms and A has equalizers, then G is faithful.
(g) If G reflects isomorphisms, then G need not be faithful.
19C. Hom-functors Reflecting Epimorphisms
Let hom(A, −) : A → Set be a covariant hom-functor that reflects epimorphisms. Show
that(a) If in A arbitrary copowers IA of A exist, then hom(A, −) is faithful.
(b) In general hom(A, −) need not be faithful.
19D. Galois Adjunctions
An adjoint situation (η, ε) : F
G is called a Galois adjunction provided that ηG
is a natural isomorphism. Show that(a) For adjoint situations the following conditions are equivalent:(1) (η, ε) : F

G is a Galois adjunction,(2) F η is a natural isomorphism,(3) ηG is an Epi-transformation,(4) F η is an Epi-transformation,(5) GF η = ηGF .
(b) If (η, ε) : F

G is a Galois-adjunction, then so is (εop , η op ) : Gop



F op .

(c) If (η, ε) : F
G is an adjoint situation and if η is an Epi-transformation or if Gis full, then (η, ε) : F
G is a Galois-adjunction.
(d) If (η, ε) : F

G is a Galois-adjunction, then G and GF G are naturally isomorphic.

19E. Galois Correspondences
GF(a) Let (A, U ) −−→ (B, V ) and (B, V ) −−→ (A, U ) be concrete functors over X. Show
that (F, G) is a Galois correspondence if and only if there exists an adjoint situation(η, ε) : F
G with concrete natural transformations η and ε.
* (b) Construct a concrete functor G that has a concrete co-adjoint, but is not a part ofa Galois correspondence (F, G). Cf. Exercise 21E(b).

9th July 2006

314

Adjoints and Monads[Chap. V* 19F. Adjoint Sequences

N as a category and show that(a) There is a sequence of functors Gn : N → N, no two of which are naturally isomorphic, such that G0 (x) = x + 1 and Gn+1
Gn for each n ∈ N.
(b) If Gn : N → N is a sequence of functors such that Gn
Gn+1 for each n ∈ N,
then Gn = idN for each n ∈ N.
Consider the poset

19G. Self-Adjoints
G(a) Let A −−→ A be an endofunctor of A with G

G. Show that

(1) If A is a poset, considered as a category, then G is an isomorphism.
(2) If A = Set, then G is an equivalence.
(b) Let A be a category with a zero object A and let G : A → A be the constant functorwith value A. Show that G
G.
19H. Adjoint Situations and Equivalences
G(a) Show that a functor A −−→ B is an equivalence if and only if there exists an adjointsituation (η, ε) : F
G : A → B with natural isomorphisms η and ε.
(b) Let (A, U ) be a construct. Show that F
lence.
* 19I.
Show that if F
faithful.


G





U



F implies that F is an equiva-

H, then F is full and faithful if and only if H is full and

19J. Units and Co-unitsLet (η, ε) : F
G : A → B be an adjoint situation. Show that(a) GF η ◦ η = ηGF ◦ η.
(b) ε ◦ εF G = ε ◦ F Gε.
f(c) If B −→ GA is a G-structured arrow, then f = εA ◦ F f is the unique A-morphismf

F B −→ A with f = Gf ◦ ηB .
19K. Swell Separators
In A an object S is called a swell separator provided that mono-sources in A arehom(S, −)-initial. Show that(a) Every swell separator is an extremal separator.
(b) If S is a swell separator, then hom(S, −) reflects limits.
(c) If in A arbitrary copowers I S of S exist, then the following hold:(1) S is a swell separator if and only if hom(S, −) reflects swell epimorphisms.

9th July 2006

Sec. 19]

Adjoint situations

315(2) S is an extremal separator if and only if hom(S, −) reflects extremal epimorphisms.
(3) S is a separator if and only if hom(S, −) reflects epimorphisms. [But cf. 19C(b).](4) If hom(S, −) reflects regular epimorphisms, then S is a swell separator.
(d) In Set every separator is swell.

9th July 2006

20 Monads
Algebraic constructs (A, U ), such as Vec, Grp, Mon, and Lat, can be fully describedby the following data, called the monad associated with (A, U ):1. the functor T : Set → Set, where T = U ◦ F and F : Set → A is the associatedfree functor (19.4(2)),
2. the natural transformation η : idSet → T formed by universal arrows, and3. the natural transformation µ : T ◦ T → T given by the unique homomorphismµX : T (T X) → T X that extends idT X.
In fact, in the above cases, there is a canonical concrete isomorphism K between (A, U )xand the full concrete subcategory of Alg(T ) consisting of those T -algebras T X −→ X
that satisfy the equations x ◦ ηX = idX and x ◦ T x = x ◦ µX . The latter subcategory iscalled the Eilenberg-Moore category of the monad (T, η, µ).
The above observation makes it possible, in the following four steps, to express the“degree of algebraic character” of arbitrary concrete categories that have free objects:
Step 1: With every concrete category (A, U ) over X that has free objects (or, more genU

erally, with every adjoint functor A −−→ X) one can associate, in an essentially
unique way, an adjoint situation (η, ε) : F
U : A → X. (See Theorem 19.1
and Proposition 19.9.)

Step 2: With every adjoint situation (η, ε) : F
U : A → X one can associate amonad T = (T, η, µ) on X, where T = U ◦ F : X → X. (See Proposition 20.3.)

Step 3: With every monad T = (T, η, µ) on X one can associate a concrete subcategoryof Alg(T ) denoted by (X T , U T ) and called the category of T-algebras. (SeeDefinition 20.4.)
Step 4: With every concrete category (A, U ) over X that has free objects one canK

associate a distinguished concrete functor (A, U ) −−→ (X T , U T ) into the associated category of T-algebras called the comparison functor for (A, U ). (See
Proposition 20.37 and Definition 20.38.)
Concrete categories that are concretely isomorphic to a category of T-algebras for somemonad T have a distinct “algebraic flavor”. Such categories (A, U ) and their forgetful
functors U are called monadic. It turns out that a concrete category (A, U ) is monadic if
K

and only if it has free objects and its associated comparison functor (A, U ) −−→ (X T , U T )is an isomorphism. Thus, for concrete categories (A, U ) that have free objects, theassociated comparison functor can be considered as a means of measuring the “algebraic
character” of (A, U ); and the associated category of T-algebras can be considered to be
the “algebraic part” of (A, U ). In particular,(a) every finitary variety (see Definition 16.16) is monadic,316

Sec. 20]

Monads317(b) the category TopGrp, considered as a concrete category(1) over Top, is monadic,(2) over Set, is not monadic; the associated comparison functor is the forgetful
functor TopGrp → Grp, so that the construct Grp may be considered as the“algebraic part” of the construct TopGrp,(c) the construct Top is not monadic; the associated comparison functor is the forgetful
functor Top → Set itself, so that the construct Set may be considered as the“algebraic part” of the construct Top; hence the construct Top may be considered
as having a trivial “algebraic part”.
Among constructs, monadicity captures the idea of “algebraicness” rather well (as will
be demonstrated in §24). Unfortunately, however, the behavior of monadic categories ingeneral is far from satisfactory. Monadic functors can fail badly to reflect nice propertiesof the base category (e.g., the existence of colimits or of suitable factorization structures),and they are not closed under composition. Such deficiencies are shown near the end of
this section. Better behaved (and, moreover, simpler) concepts of “algebraicity” will be
developed in §23.

MONADS AND ALGEBRAS
20.1 DEFINITION
A monad on a category X is a triple T = (T, η, µ) consisting of a functor T : X → X
and natural transformations
η : idX → T

and µ : T ◦ T → Tsuch that the diagrams
T ◦T ◦T
µT

Tµ

/T ◦T
µ



T ◦T



µ

Tη

and

/ T ◦ T o ηT T
FF
yy
FF
yy
µ
y
F
y
id FF  yy id
" |y

T FF

/T

T

commute.
20.2 EXAMPLES(1) On every category X there is the trivial monad T = (T, η, µ) with T = idX andη = µ = idT .
(2) In Set the word-monad T = (T, η, µ) is defined as follows:(a) T : Set → Set assigns to each set X the set T X =
f

S

n∈

NX

n

of tuples orTf“words” over X and to each function X −→ Y the function T X −−→ T Y that
sends the empty word (∈ X 0 ) to the empty word and for n > 0 sends each word
(x1 , . . . , xn ) to (f (x1 ), . . . , f (xn )),9th July 2006

318

Adjoints and Monads[Chap. V(b) ηX : X → T X is defined by ηX (x) = (x), i.e., ηX interprets each member of X
as a one-letter word,(c) µX : T (T X) → T X is given by concatenation:µX ((x11 , . . . , x1n1 ), (x21 , . . . , x2n2 ), . . . , (xk1 , . . . , xknk )) =(x11 , . . . , x1n1 , x21 , . . . , x2n2 , . . . , xk1 , . . . , xknk ),i.e., µX interprets each word of words in the natural way, as a word.
(3) In Set the power-set monad T = (P, η, µ) is defined as follows:(a) P : Set → Set is the power-set functor,(b) ηX : X → PX is defined by ηX (x) = {x},(c) µX : P(PX) → PX is defined by µX (Z) =S

Z.

(4) Let X be a poset, considered as a category. A monad T on X consists of an orderpreserving function T : X → X with s ≤ T s (due to η) and T = T ◦ T (since(T ◦ T )s ≤ T s due to µ) — in other words, T is a closure operator on X.
20.3 PROPOSITION
Each adjoint situation (η, ε) : F(T, η, µ) on X, defined by

G : A → X gives rise to the associated monad

T =G◦F : X→Xandµ = GεF : T ◦ T → T.

Proof:(1). µ ◦ T µ = GεF ◦ GF GεF = G(ε ◦ F Gε)F = G(ε ◦ εF G)F = GεF ◦ GεF GF = µ ◦ µT ,since ε ◦ F Gε = ε ◦ εF G, in view of the fact that ε is a natural transformation.
(Cf. Exercise 19J.)(2). µ ◦ T η = GεF ◦ GF η = G(εF ◦ F η) = G(id) = id.
(3). µ ◦ ηT = GεF ◦ ηGF = (Gε ◦ ηG)F = (id)F = id.

20.4 DEFINITION
Let T = (T, η, µ) be a monad on X. The full concrete subcategory of Alg(T ) consistingxof all algebras T X −→ X that satisfy(1) x ◦ ηX = idX , and(2) x ◦ T x = x ◦ µX : T (T X) → X
is denoted by (XT , U T ) and is called the Eilenberg-Moore category of the monadT, or the category of T-algebras.
20.5 EXAMPLES(1) If T is the trivial monad on X, then (XT , U T ) is concretely isomorphic to (X, idX ).

9th July 2006

Sec. 20]

Monads319(2) If T is the word-monad on X = Set, then (XT , U T ) is concretely isomorphic to thexconstruct Mon of monoids. [If T X −→ X is a T-algebra, multiplication in X is
defined by a · b = x((a, b)), and if e is the value of x at the empty word, then (X, ·, e)
is a monoid.](3) If T is the power-set monad on X = Set, then (XT , U T ) is concretely isomorphicxto the construct JCPos. [If T X −→ X is a T-algebra, then a ≤ b if and only ifx({a, b}) = b defines a partial order on X with x(A) = sup A for each A ⊆ X; theT-morphisms are precisely the join-preserving maps.] Cf. Exercise 5N(a).
(4) If T = (T, η, µ) is a monad on a partially ordered set X, considered as a category,
and if E : Y → X is the embedding of the full subcategory Y of X, whose objects
are the fixed points of T , then (XT , U T ) is concretely isomorphic to (Y, E).
(5) The construct Vec is concretely isomorphic to the construct AbT for the following
monad T = (T, η, µ) on abelian groups: for each abelian group X, let T X = ⊗ X
be the tensor product, with morphisms ηX : X → ⊗ X given by x 7→ 1 ⊗ x, andµX :⊗ ⊗ X → ⊗ X given by r1 ⊗ r2 ⊗ x 7→ (r1 r2 ) ⊗ x.

R R

R

R

R

20.6 REMARKAs outlined in the introduction of this section, with each concrete category (A, U ) overU

X that has free objects (or, more generally, with every adjoint functor A −−→ X) one
can associate a category (XT , U T ) of T-algebras in three steps. Since the first of thesesteps does not yield a unique result, (XT , U T ) is not uniquely determined by (A, U ).
However, any two categories of algebras that are obtained from (A, U ) in this mannerare concretely isomorphic. (See Exercise 20A.) Thus, by “abuse of language”, we will
sometimes designate any category of algebras that is obtained from (A, U ) via the procedure outlined above as the category of algebras associated with (A, U ) (resp. withthe functor U ).
20.7 PROPOSITION
Every monad T = (T, η, µ) on X gives rise to an associated adjoint situation T(η, ε) : F T
U : XT → X, where(1) XT and U T are defined as in Definition 20.4,
FT

f

Tf

(2) X −−−→ XT is defined by F T (X −→ Y ) = (T X, µX ) −−→ (T Y, µY ),in particular, (T X, µX ) is a free object over X in (XT , U T ),ε(3) F T U T −→ idX T is defined by ε(X,x) = x.
Moreover, the monad associated with the above adjoint situation (20.3) is T itself.
Proof: Straightforward computations.

9th July 2006



320

Adjoints and Monads[Chap. V

MONADIC CATEGORIES AND FUNCTORS
20.8 DEFINITION(1) A concrete category over X is called monadic provided that it is concretely isomorphic to (XT , U T ) for some monad T on X.
U(2) A functor A −−→ X is called monadic provided that U is faithful and (A, U ) is
monadic.
20.9 REMARKIn Proposition 20.40 it will be shown that every monadic category is concretely isomorphic to its associated category of algebras.
20.10 EXAMPLES(1) Every isomorphism is monadic [cf. 20.5(1)].
(2) The construct Mon is monadic [cf. 20.5(2)]. So is every finitary variety (cf. 20.20).
(3) The construct JCPos is monadic [cf. 20.5(3)].
(4) An embedding of a full subcategory A of X into X is monadic if and only if A is
isomorphism-closed and reflective in X. (See Proposition 20.12, Theorem 20.17, andExercise 20F.) A concrete category (A, U ) over a partially ordered set X is monadic
U

if and only if, up to isomorphism, A −−→ X is the embedding of a full reflective
subcategory of X [cf. 20.5(4)]. In particular, a partially ordered set, considered as aconcrete category over 1, is monadic if and only if it contains precisely one element.
(5) None of the constructs Top, Rel, or Pos is monadic since the corresponding forgetful
functors don’t reflect isomorphisms (cf. 20.12).
20.11 LEMMA
The Eilenberg-Moore category of a monad T = (T, η, µ) is closed under the formation of
mono-sources in Alg(T ).
Proof: Let T = (T, η, µ) be a monad on X, and suppose that the diagram
TX

T mi

x





X

/ T Xi

mi

xi

/ Xi

m

i
commutes for each i ∈ I, where (X −−→
Xi )I is a mono-source in X, and each (Xi , xi )is a T-algebra. To show that (X, x) is a T-algebra, first notice that the commutativity

9th July 2006

Sec. 20]

Monads321of the diagram
mi

X
ηX

/ Xi




TX

T mi

/ T Xi

x





X

ηXi

mi

xi

/ Xi

and the fact that xi ◦ ηXi = idXi imply mi ◦ (x ◦ ηX ) = xi ◦ T mi ◦ ηX = xi ◦ ηXi ◦ mi =idXi ◦ mi = mi ◦ idX , and thus x ◦ ηX = idX. The commutativity of the five inner
quadrangles in the diagram
GG 2
GGT miGG
GG#

T 2 Xi

µ Xi

µX

/ TX
y
y
T mi yy
y
yy
|yy

Tx

T 2 XGT xi

: T Xivv
v
vv
vv
 vv T mi

TX/ T Xi


xi

xi

/ Xi

x

x

cFF
FF
FF
mi FFF

/X

implies that mi ◦ (x ◦ µX ) = mi ◦ (x ◦ T x) and thus x ◦ µX = x ◦ T x.



20.12 PROPOSITIONU
For monadic functors A −−→ X the following hold:(1) U is faithful,(2) (A, U ) is fibre-small,(3) U is adjoint, i.e., (A, U ) has free objects,(4) (A, U ) is uniquely transportable, hence amnestic,(5) U creates isomorphisms, hence reflects them,(6) U reflects epimorphisms and extremal epimorphisms,(7) U preserves and reflects mono-sources,(8) in (A, U ) mono-sources are initial,(9) U detects wellpoweredness, i.e., if X is wellpowered, then so is A,(10) U creates limits.
Proof: Since all of the properties mentioned above are invariant under concrete isomorphisms, we may assume that (A, U ) = (XT , U T ) for some monad T = (T, η, µ) on
X.

9th July 2006

322

Adjoints and Monads[Chap. V(1) and (2) follow immediately from the definition of monadicity.
(3) follows from Proposition 20.7.
(7) follows from (1) and (3).
(8) is immediate by straightforward computation.
(9) follows from (7) and (8).
`i(10). Let D : I → XT be a diagram and let L = (L −−→
Di )Ob(I) be a limit of U T ◦D. For

T`

x

i
i
each i ∈ Ob(I) let Di = (Xi , xi ). Then the source S = (T L −−−→T Xi −−→Xi )Ob(I)yis natural for U T ◦ D. Hence there exists a unique X-morphism T L −→ L with
S = L ◦ y, i.e., such that each diagram
TL

T `i

/ T Xi

y





L

`i

xi

/ Xi

commutes.
Thus it remains
to be shown that (L, y) is a T-algebra and that the`isource (L, y) −−→ Di
is a limit of D. The former follows from the fact that
Ob(I)L is a mono-source and from Lemma 20.11. The latter follows from (7) and (8)via Proposition 13.15.
(4) and (5) follow immediately from (10), by Proposition 13.36.
(6) follows immediately from (1), the preservation of monomorphisms (7) and thereflection of isomorphisms (5).

20.13 REMARKEven though monadic functors behave perfectly with respect to limits, their relationshipU

to colimits is rather complex. For example, if A −−→ X is a monadic functor with X
cocomplete and if A has coequalizers, then A is cocomplete (see Exercise 20D); but
A needn’t have coequalizers (see 20.47). In fact, monadic functors are characterized
by their (somewhat strange) behavior with respect to coequalizers, as will be seen in
Theorem 20.17.
20.14 DEFINITION
p
c /
// B
1. A fork A
C is called a congruence fork provided that (p, q) is aqcongruence relation of c, and c is a coequalizer of p and q.

9th July 2006

Sec. 20]

Monadsp

2. A fork A

q

//

B

c

323

/ C is called a split fork and c is called a split coequalizerof (p, q) provided that there exist morphisms s and t such that the diagram
c

B@

@@
@@t
@@


id

~~
~~q
~
 ~~

B

A

p

c

/B

/C
~
~
s ~
~~
~
~~
id
@@
@@
@
c @@

/C

commutes.
3. A colimit K of a diagram D : I → A is called an absolute colimit provided thatGfor each functor A −−→ B the sink GK is a colimit of G ◦ D.
In particular, c is called an absolute coequalizer of p and q in A provided thatGfor each functor A −−→ B, Gc is a coequalizer of Gp and Gq in B.
G

4. A functor A −−→ B is said to create absolute colimits provided that for eachD
diagram I −−→ A and each absolute colimit K of G ◦ D there exists a unique sinkci
C = (Di −−→ C)Ob(I) such that GC = K and, moreover, C is a (not necessarily
absolute) colimit of D.

A monadic functor creating colimits

20.15 EXAMPLES(1) If A is a category, f is an A-morphism, (p, q) is a congruence relation of f in A,

9th July 2006

324

Adjoints and Monads

and c is a coequalizer of p and q in A, then •pq[Chap. V// •c/ • is a congruence fork in

A. (Cf. Proposition 11.22.)p(2) A congruence fork •(3) If •

p
q

// •

(4) If (η, ε) : F

c

q

// •

c

/ • is a split fork if and only if c is a retraction.

/ • is a split fork, then c is an absolute coequalizer of p and q.


U : A → X is an adjoint situation, then for each A-object A
UFUFUAU F U εA//U εF U A

UFUA

U εA

/ UA

is a split fork in B since the diagramU εA

U F U AOOOOOOOηU F U A OOOO'U F U εAU pF U F U A

id



pp
ppp
p
p
wppp U εF U A

U εA

UFUA

/ UA
u
u
ηU A uuuuuzuu/ UFUA
id
II
II
II
I
U εA II$ / UAcommutes.
(5) If T = (T, η, µ) is a monad on X, then for each T-algebra (X, x)
Tx

T 2X

µX

//

x

TX

/X

is a split fork in X. Hence, in particular, for each X-object X
T µX

T 3X µTX// T 2 XµX/ TX

is a split fork in X.
20.16 PROPOSITION
Each monadic functor U creates absolute colimits.
Proof: Let T = (T, η, µ) be a monad on X and let (XT , U T ) be the associated categoryciof T-algebras. Let D : I → XT be a diagram and let K = (U T Di −−→C)Ob(I) be an
Tciabsolute colimit of U T ◦D. Then T K = (T U T Di −−−→T C)Ob(I) is a colimit of T ◦U T ◦D.
xi
ci
If Di = (Xi , xi ) for each i ∈ Ob(I), then S = (T Xi −−→ Xi −−→C)Ob(I) is a natural sink

9th July 2006

Sec. 20]

Monads325
cfor T ◦ U T ◦ D. Hence there exists a unique morphism T C −→ X with S = c ◦ T K, i.e.,such that for each i ∈ Ob(I) the diagram
T Xi
xi

T ci

/ TC
c



Xi



/C

ci

commutes. Thus it remains to be shown that (C, c) is a T-algebra and that the sinkci
(Di −−→(C, c))Ob(I) is a colimit of D. The former follows from the fact that K and T 2 K,
being colimits, are epi-sinks and from the equations(c ◦ ηC ) ◦ ci = c ◦ T ci ◦ ηXi = ci ◦ xi ◦ ηXi = ci ◦ idXi = idC ◦ ci
and(c ◦ T c) ◦ T 2 ci = c ◦ T ci ◦ T xi = ci ◦ xi ◦ T xi = ci ◦ xi ◦ µXi = c ◦ T ci ◦ µXi = (c ◦ µC ) ◦ T 2 ci .
fi

For the latter, consider a natural sink X = (Di −−→ (X, x))Ob(I) for D. Then U T X =fi(Xi −−→ X)Ob(I) is a natural sink for U T ◦ D. Thus there exists a unique X-morphismfC −→ X with fi = f ◦ ci for each i ∈ Ob(I). Hence T fi = T f ◦ T ci for each i ∈ Ob(I)and so x ◦ T f ◦ T ci = x ◦ T fi = fi ◦ xi = f ◦ ci ◦ xi = f ◦ c ◦ T ci . Since T K is an epi-sink,f

this implies that x ◦ T f = f ◦ c, i.e., (C, c) −→ (X, x) is the unique T -homomorphismwith X = f ◦ K.
20.17 CHARACTERIZATION THEOREM FOR MONADIC FUNCTORSU
For any functor A −−→ X the following conditions are equivalent:(1) U is monadic,(2) U is adjoint and creates absolute coequalizers,(3) U is adjoint and creates split coequalizers (as defined in 20.14).
Proof: (1) ⇒ (2) follows from Propositions 20.12 and 20.16.
(2) ⇒ (3) follows from Example 20.15(3).
(3) ⇒ (1). Since U is adjoint, there is an adjoint situation (η, ε) : F
U : A → X. Let
TT
T = (T, η, µ) be the associated monad and let (X , U ) be the associated category of TK

L

algebras. It suffices to construct functors A −−→ XT and XT −−→ A with U = U T ◦ K,
L ◦ K = idA , and K ◦ L = idXT ; since then U would be faithful, (A, U ) would be aK

concrete category, and (A, U ) −−→ (XT , U T ) would be a concrete isomorphism.
Construction of K: For each A-object A the pair (U A, U εA ) is a T-algebra [cf. 20.15(4)],f

Uf

and for each A-morphism A −→ B, (U A, U εA ) −−→ (U B, U εB ) is an T -homomorphism.

9th July 2006

326

Adjoints and Monads[Chap. V

K

Thus there exists a unique functor A −−→ XT with K(A) = (U A, U εA ) for each A-object
A.
Construction of L: For each XT -object (X, x) the fork
Tx

U (F x)X

U (εF X )x //X = U (F U F X)
T 2X µ / T X//

U (F X)x/X

x

is a split fork [cf. 20.15(5)]. Hence, by condition (3), U (F X) −→ X has a unique U-liftx
F X −→ L(X) and x is a coequalizer of εF X and F x in A. For each T -homomorphismf(X, x) −→ (Y, y) the following equalities hold:(y ◦ F f ) ◦ εF X = y ◦ εF Y ◦ F U F f = y ◦ F y ◦ F T f = (y ◦ F f ) ◦ F x.
Since x is a coequalizer of εF X and F x in A, there exists a unique A-morphism
L(f )L(X) −−−→ L(Y ) with y ◦ F f = L(f ) ◦ x. Since x is an epimorphism, the equalities U L(f ) ◦ x = U L(f ) ◦ U x = U y ◦ U F f = y ◦ T f = f ◦ x imply that U L(f ) = f ,
L(f )

f

f

i.e., U (L(X, x) −−−→ L(Y, y)) = X −→ Y = U T ((X, x) −→ (Y, y)). Thus L defines aLconcrete functor XT −−→ A. The equation L◦K = idA immediately follows (via concreteεUεness) from the fact that, for each A-object A, F U A −−A→ A U-lifts U F U A −−−A→ U A,
so that L ◦ K(A) = A. To show that K ◦ L = idXT , let (X, x) be an T-algebra. Then(K ◦ L)(X, x) = (U L(X, x), U εL(X,x) ). Since U L(X, x) = X, it remains (in view ofconcreteness) to be shown that x = εL(X,x) . By the universality of η, the equations
U x ◦ ηX = x ◦ ηX = idX = U εL(X,x) ◦ ηU L(X,x) = U εL(X,x) ◦ ηX imply that x = εL(X,x)
and hence that x = U x = U εL(X,x) .

20.18 PROPOSITION
Each construct of the form Alg(Ω) is monadic.
U

Proof: By Example 8.23(6) the forgetful functor Alg(Ω) −−→ Set is adjoint, so by theabove theorem it suffices to show that U creates absolute coequalizers. Consider a pairp(A, (αi )I )q// (B, (βi )I ) of Ω-homomorphisms and an absolute coequalizer B −→ Ccof p and q in Set. For each i ∈ I, the functor S ni : Set → Set [3.20(10)] preserves
cni

this coequalizer. Thus B ni −−−→ C ni is a coequalizer of pni and q ni in Set. Since
γi(c ◦ βi ) ◦ pni = c ◦ p ◦ αi = c ◦ q ◦ αi = (c ◦ βi ) ◦ q ni, there exists a unique function C ni −−→ Ccwith c ◦ βi = γi ◦ cni . Thus (B, (βi )I ) −→ (C, (γi )I ) is the unique U-lift of the costruccctured map (B, (βi )I ) −→ C. It follows easily that (B, (βi )I ) −→ (C, (γi )I ) is a coequalizerpof (A, (αi )I )q// (B, (βi )I ) in Alg(Ω). Thus U creates absolute coequalizers.



20.19 PROPOSITION
Let (A, U ) be a monadic category over X. Then each concrete full reflective subcategoryof A that is closed under the formation of regular quotients is also monadic over X.

9th July 2006

Sec. 20]

Monads327

Proof: Immediate from Theorem 20.17.



20.20 PROPOSITION
Each finitary variety is a monadic construct.
Proof: Immediate from Propositions 20.18 and 20.19.

E-MONADS AND E-MONADICCATEGORIES AND FUNCTORS
20.21 DEFINITION(1) A monad T = (T, η, µ) on X is called an E-monad provided that X is an (E, M)category for some M and T [E] ⊆ E.
RegEpi-monads in categories with regular factorizations are called regular monads.
U(2) A concrete category (A, U ) over X (or a faithful functor A −−→ X) is called Emonadic (resp. regularly monadic) provided that (A, U ) is concretely isomorphic
to (XT , U T ) for some E-monad (resp. regular monad) T on X.

20.22 PROPOSITION
Every monad on Set is regular.
Proof: This follows from the facts that in Set every regular epimorphism is a retraction
and that every functor preserves retractions.

20.23 DEFINITIONU
A functor A −−→ X lifts (E, M)-factorizations uniquely provided that for any source
S in A and any (E, M)-factorization U S = M◦e in X there exists a unique factorization
S = M̂ ◦ ê in A with U M̂ = M and U ê = e.
20.24 PROPOSITIONU
If X is an (E, M)-category and A −−→ X is E-monadic, then U lifts (E, M)-factorizations uniquely.
Proof: Assume that (A, U ) = (XT , U T ) for some E-monad T = (T, η, µ) on X. Let
fi

fi

e

m

i
S = ((X, x) −−→ (Xi , xi ))I be a source in XT and (X −−→ Xi )I = (X −→ Y −−→Xi )Ibe an (E, M)-factorization of U T S in X. Since T e ∈ E, there exists a unique diagonal

9th July 2006

328

Adjoints and Monads[Chap. Vy

T Y −→ Y in X that makes the diagram
T fi

T XE

EE
EE
E
T e EE"TY/ T Xix<x
xx
xxT mi
x
x

y

x

xi



(i ∈ I)< Y FF
FF mi
yy
FF
y
FF
y
y
F" 
 yy
/ Xi
X
e yy

fi

commute. That (Y, y) is a T-algebra follows from the fact that e and T 2 e are epimorphisms [15.5(2)] and from the equationsy ◦ ηY ◦ e = y ◦ T e ◦ ηX = e ◦ x ◦ ηX = e ◦ idX = idY ◦ e
and(y ◦ T y) ◦ T 2 e = y ◦ T (y ◦ T e) = y ◦ T (e ◦ x) = y ◦ T e ◦ T x = e ◦ x ◦ T x = e ◦ x ◦ µX
= y ◦ T e ◦ µX = (y ◦ µY ) ◦ T 2 e.



20.25 PROPOSITIONU
Let A be a full subcategory of an (E, M)-category B. Then the inclusion functor A −−→ B
is E-monadic if and only if the following conditions are satisfied:(1) A is reflective in B,
e

m

m◦e

(2) if A −→ B −−→ A0 is an (E, M)-factorization of an A-morphism A −−−→ A0 , thenB belongs to A.
Proof: If U is E-monadic, then by Propositions 20.12(3) and 20.24, conditions (1) and
(2) are satisfied. Conversely, let (1) and (2) be satisfied, and let (η, ε) : F
U :A→Bbe an adjoint situation. If T = (T, η, µ) is the associated monad, then (A, U ) can beshown to be concretely isomorphic to (BT , U T ). To see this consider the concrete functor−1
K : (A, U ) → (BT , U T ) given by KA = (A, ηA). K is an isomorphism since for eachT-algebra (B, b) we have b ◦ ηB = idB (where ηB is an A-reflection for B) and hence−1ηB ◦ b ◦ ηB = ηB implies that ηB ◦ b = id. Thus b = ηB
, and B ∈ Ob(A) since, by (2),
A is isomorphism-closed. It remains to be shown that T [E] ⊆ E. Consider a morphism
e

Te

ê

m̂

B −→ B 0 in E. Let T B −−→ T B 0 = T B −→ B 00 −−→ T B 0 be an (E, M)-factorization ofdT e. Then there exists a diagonal B −→ B 00 that makes the following diagram commute:
/ B0
y
y
d yy
ηB 0
ê◦ηB
yy
y
 |y

/ T B0
B 00

B

e

m̂

9th July 2006

Sec. 20]

Monads329dˆ
By the universal property of η there exists a morphism T B 0 −→ B 00 with d = dˆ ◦ ηB 0 .
Thus m̂ ◦ dˆ ◦ ηB 0 = m̂ ◦ d = ηB 0 implies that m̂ ◦ dˆ = id. Moreover, dˆ ◦ m̂ ◦ ê ◦ ηB =dˆ ◦ m̂ ◦ d ◦ e = dˆ ◦ ηB 0 ◦ e = d ◦ e = ê ◦ ηB implies that dˆ ◦ m̂ ◦ ê = ê. Thus, since ê isan epimorphism, dˆ ◦ m̂ = id. Hence m̂ is an isomorphism and, consequently, T e = m̂ ◦ ê
belongs to E.


20.26 COROLLARY
If A is an E-reflective subcategory of an (E, M)-category B, then the inclusion-functor
A → B is E-monadic.
Proof: Immediate from Theorem 16.8 and Proposition 20.25.



20.27 EXAMPLESAs the following examples show, there exist full subcategories A of (E, M)-categories B
such that the inclusion functor A → B is E-monadic, but such that A is neither closed
under the formation of M-subobjects nor under the formation of E-quotient objects:(1) Sgr is a (RegEpi, Mono-Source)-category, the full embedding Grp → Sgr is regularly monadic, and Grp is closed under the formation of regular quotients in Sgr.
But Grp is not closed under the formation of subobjects in Sgr.
(2) Top is an (Epi, ExtrMono-Source)-category and the full embedding of HComp into
Top is Epi-monadic. But HComp is neither closed under the formation of quotientsnor closed under the formation of extremal subobjects in Top.
20.28 PROPOSITION
If (A, U ) is an E-monadic category over an (E, M)-category X, then the following hold:(1) Every A-morphism f with U f ∈ E is final in (A, U ).
(2) A is an (U −1 [E], U −1 [M])-category.
Proof:(1). Assume that (A, U ) = (XT , U T ) for some E-monad T = (T, η, µ) on X. Leteef(X, x) −→ (Y, y) be a T -homomorphism with X −→ Y in E, and let Y −→ Z bef ◦ean X-morphism such that (X, x) −−−→ (Z, z) is a T -homomorphism. Since T e is an
epimorphism, the equalities(f ◦ y) ◦ T e = (f ◦ e) ◦ x = z ◦ T (f ◦ e) = (z ◦ T f ) ◦ T ef

imply that f ◦ y = z ◦ T f , i.e., that (Y, y) −→ (Z, z) is a T -homomorphism.
(2). By Proposition 20.24, A has (U −1 [E], U −1 [M])-factorizations. By (1) and the factthat U −1 [E] ⊆ Epi(A), A has the unique (U −1 [E], U −1 [M])-diagonalization property.


9th July 2006

330

Adjoints and Monads[Chap. V

20.29 COROLLARY
If (A, U ) is E-monadic over an E-co-wellpowered category, then A is U −1 [E]-co-wellpowered.

20.30 PROPOSITION
If (A, U ) is regularly monadic, then A has regular factorizations and U preserves andreflects regular and extremal epimorphisms.
Proof: By Proposition 20.28, A is an (E, Mono-Source)-category, where E is the classU −1 [RegEpi(X)]. Thus, by Proposition 15.8(1), E = ExtrEpi(A). It suffices to showethat in A every extremal epimorphism A −→ B is regular. Since U e is an extremal
epimorphism and hence a regular epimorphism in the base category X, it is a coequalizer
of some pair X

r1
r2// U A of X-morphisms. If X −→ U C is a universal arrow over X,ηr̂ithen for each i = 1, 2 there exists an A-morphism C −−→
A with ri = (U r̂i ) ◦ η. Since U e
is a coequalizer of r1 and r2 in X, the finality of e [cf. Proposition 20.28(1)] implies thate is a coequalizer of r̂1 and r̂2 in A(cf. Exercise 8O). Hence e is a regular epimorphismin A.


20.31 COROLLARY
Regularly monadic functors detect extremal co-wellpoweredness.



20.32 CHARACTERIZATION THEOREM FORREGULARLY MONADIC FUNCTORSU
A functor A −−→ X is regularly monadic if and only if the following conditions hold:(1) U is monadic,(2) X has regular factorizations,(3) U preserves regular epimorphisms.
Proof: By Proposition 20.30 every regularly monadic functor satisfies the above conditions. Conversely, assume that (A, U ) = (XT , U T ) for some monad T = (T, η, µ) on X,
and that conditions (2) and (3) are satisfied. Then F T and hence T = U T ◦ F T preserve
regular epimorphisms. Thus T is a regular monad and (A, U ) is regularly monadic. 
20.33 PROPOSITION
Regularly monadic functors detect colimits.
Proof: Let (A, U ) be a regularly monadic category over X, let D : I → A be a diagram,ci
and let (U Di −−→C)i∈Ob(I) be a colimit of U ◦ D. Consider the structured source
fj

fj ◦ci(C −−→ U Aj )j∈J ) consisting of all structured morphisms for which each U Di −−−−→ U Aj
η
is an A-morphism . If C −−C→ U A is a universal arrow over C, then for each j ∈ J

9th July 2006

Sec. 20]

Monads

331

gj

gj

there exists a unique A-morphism A −−→ Aj with fj = gj ◦ ηC . Let (A −−→ Aj )J =mje
(A −→ B −−→ Aj )J be a (RegEpi, Mono-Source)-factorization.
ηC
ci
e
/C
/ UB
/ UA
CC
TTTT
iiii
y
i
i
C
y
i
TTTT
C
gj yy
iiii mj
TTTT CCfCj
yy iiiiiii
TTTT C!
y|ytiiiiT*

U Di TTTU Aj

fj ◦ciSince mono-sources are initial and each U Di −−−−→ U Aj is an A-morphism, eache◦η ◦cie◦η ◦ci
U Di −−−C−−→ U B is an A-morphism. It follows easily that (Di −−−C−−→ B)i∈Ob(I)is a colimit of D.
MONADIC CONSTRUCTS
20.34 PROPOSITION
Monadic constructs are complete, cocomplete, wellpowered, extremally co-wellpowered,and have regular factorizations.
Proof: Let (A, U ) be a monadic construct. By Proposition 20.12, A is complete andwellpowered. By Proposition 20.22, U is regularly monadic. Hence Corollary 20.31
implies that A is extremally co-wellpowered, Proposition 20.30 implies that A has regularfactorizations, and Proposition 20.33 implies that A is cocomplete.

20.35 CHARACTERIZATION THEOREM FOR
MONADIC CONSTRUCTS
For constructs (A, U ) the following conditions are equivalent:(1) U is monadic,(2) U is regularly monadic,(3) U is adjoint and creates finite limits and coequalizers of congruence relations,(4) U is extremally co-wellpowered and creates limits and coequalizers of congruence
relations.
Proof: (1) ⇔ (2) by Proposition 20.22.
(1) ⇒ (3). It suffices to show that U creates coequalizers of congruence relations. LetpAq// B be a pair of A-morphisms and let U AUpUq// U Bc/ C be a congruence fork

in Set. By Example 20.15(2) it is a split fork. Thus the result follows from Theorem
20.17.

9th July 2006

332

Adjoints and Monads[Chap. V(3) ⇒ (1). By Theorem 20.17 it suffices to show that U creates coequalizers of splitpforks. Let Aq//

B be a pair of A-morphisms and let
c

U BF

FF
FFt
FF
F"

id

UA

xx
xx
x
x
 |xx U q

Up

c

UB

/C
{
{
s {{
{{
}{{

/ UB
id
CC
CC
C
c CCC
! 
/C

be a commutative diagram in Set. Let (c1 , c2 ) be a congruence relation of c in Set andlet (p1 , p2 ) be a congruence relation of p in A. Then (U p1 , U p2 ) is a congruence relationof U p in Set. Thus there exist unique functions u and v that make the diagrams
U p1

UP

/ UA

u

"

U p2

UA

Q

c2



Uq

c1

/ UB





and

c

"

c2

UP
U p2



/C

c

/ UB

v

Uq

/ UB



c1

Q

UB

t

U p1

/ UA




/ UA

t



Up

Up

/ UB

commute.
The equations ci ◦ (u ◦ v) = U q ◦ U pi ◦ v = U q ◦ t ◦ ci = ci = ci ◦ id for i = 1, 2 imply thatu ◦ v = id. Thus, if L

`1
`2

//

U P is a congruence relation of u, then u is a coequalizer of `1and `2 . A straightforward computation shows that the source (L, (`i )) can be considered
as a limit of the diagram
`1

/ UP
..
..
..
..
U (q◦p1 )`2..
..U (q◦p2 )..
U (q◦p1 )/ UB..
U P UUUUU
..
UUUU
UUUU
..
UUUU
UUUU .
U (q◦p2 )
U*

L

UB
`ˆ

i
Since U creates finite limits, there is a unique lift (L̂ −−→P ) for the structured source`i(L −−→U P ). Since U L̂

U `ˆ1
U `ˆ2

// U Pu/ Q is a congruence fork, U P −u→ Q has a unique

9th July 2006

Sec. 20]

Monads333ûlift P −→ Q̂ and û is a coequalizer of `ˆ1 and `ˆ2 in A. Thus the equations ci ◦ U û =ci ◦ u = U (q ◦ pi ) and the faithfulness of U imply that each ci can be lifted uniquely to anĉiA-morphism Q̂ −−→
B. Since U Q̂

U ĉ1
U ĉ2

// U Bcc/ C is a congruence fork, U B −→
C has

ĉ

a unique lift B −→ Ĉ and ĉ is a coequalizer of ĉ1 and ĉ2 in A. It remains to be shownthat ĉ is a coequalizer of p and q. Since ĉ ◦ p = ĉ ◦ q, this follows from the fact thatfwhenever Ĉ −→ D is an A-morphism with f ◦ p = f ◦ q, then f ◦ ĉ1 ◦ û = f ◦ q ◦ p1 =f ◦ p ◦ p1 = f ◦ p ◦ p2 = f ◦ q ◦ p2 = f ◦ ĉ2 ◦ û, and hence that f ◦ ĉ1 = f ◦ ĉ2 .
(3) ⇒ (4). In view of Proposition 18.11 this follows from (3) ⇒ (1) and Proposition 20.34.
(4) ⇒ (3). Since U creates limits and Set is strongly complete, so is A. Hence by
Theorem 17.11(1), U has (Generating,−)-factorizations, i.e., it is adjoint.

20.36 EXAMPLES(1) The construct HComp is monadic. Condition (3) of the above theorem is easily
seen to hold.
(2) Neither of the constructs Catof and (Ban,O) is monadic. The corresponding forgetful functors don’t preserve extremal epimorphisms [cf. 7.72(5)].

THE COMPARISON FUNCTOR
Next we will show that every monadic category is not only concretely isomorphic tosome category of algebras, but even to its associated category of algebras (cf. Remark
20.6). Moreover, for every concrete category (A, U ) that has free objects, there exists a
distinguished concrete functor into its associated category of algebras; and that functor
turns out to be an isomorphism if and only if (A, U ) is monadic. (See Proposition 20.40.)
20.37 PROPOSITIONU : A → X is an adjoint situation and (XT , U T ) is the associatedIf (η, ε) : F
K

category of algebras, then there exists a unique functor A −−→ XT such that the diagram
/A
|
K |||
U
FT
||
 }||

/X
XT

X

F

UT

commutes.
K

Proof: Existence: The functor A −−→ XT , defined byf

Uf

K(A −→ B) = (U A, U εA ) −−→ (U B, U εB ),has the desired properties.

9th July 2006

334

Adjoints and Monads[Chap. VK̂

Uniqueness: Let A −−→ XT be a functor satisfying U = U T ◦ K̂ and F T = K̂ ◦ F.
f

f

g

Consider an A-morphism A −→ B, and denote K̂(A −→ B) by (X, x) −→ (Y, y). ThenUfg

U = U T ◦ K̂ implies that U A −−→ U B = X −→ Y . Thus X = U A, Y = U B, andεg = U f . It remains to be shown that x = U εA . Since F U A −−A→ A is an A-morphism,
εUεK̂(F U A −−A→ A) = F T U A −−−A→ K̂A is a T -homomorphism. Thus the diagram
T U AKKK T η
KK U A
KK
KK
%
id

id

TTUA
! 

T U εA '/ TUAU εF U A

TUA

U εA



x

/ UA

commutes.
This implies that x = U εA , so that K̂ = K.

20.38 DEFINITIONK(1) For each adjoint situation (η, ε) : F
U : A → X, the unique functor A −−→ X Tof the above proposition is called its comparison functor.
U(2) For each adjoint functor A −−→ X (resp. each concrete category (A, U ) that has
free objects) the comparison functor of an associated adjoint situation is called acomparison functor for U (resp. for (A, U )).
20.39 REMARKS(1) In view of the essential uniqueness of comparison functors for U (cf. 20A) we will,by “abuse of language”, usually speak of the comparison functor for U .
(2) Let (η, ε) : F
U : A → X be an adjoint situation. Then Proposition 20.37 can T
be interpreted as saying that the adjoint situation (η T , εT ) : F T
U : XT → X
induced from the associated monad T is the “largest” one with the same associatedmonad. In Exercise 20B it is indicated that every monad also has a “smallest
realization”, called its Kleisli category.

20.40 PROPOSITION
An adjoint functor is monadic if and only if the associated comparison functor is aconcrete isomorphism.
Proof: If the comparison functor for an adjoint functor U is an isomorphism, then U isU
obviously monadic. Conversely, let A −−→ X be a functor, let T be a monad on X, and let
H T(A, U ) −−→ (XT , U T ) be a concrete isomorphism. Let (η, ε) : F T
U : XT → X be

9th July 2006

Sec. 20]

Monads335

the adjoint situation associated with T. Then (η, H −1 εH) : H −1 ◦ F T
is an adjoint situation whose associated monad is T. Since the diagram
H −1 ◦F T/

X
FT

{

H {{{

{{
 }{{



U : A→XAU

XT UT / X
commutes, H is the associated comparison functor for U . Thus the comparison functor
is an isomorphism.

20.41 EXAMPLES(1) The constructs Grp and TopGrp induce the same monads on Set. Thus the forgetful functor TopGrp → Grp is the comparison functor for the construct TopGrp.
(2) Since the constructs TfAb and Ab of torsion-free abelian groups and of abeliangroups induce the same monads in Set, the full embedding TfAb → Ab is thecomparison functor for TfAb.
(3) The construct Catof is not monadic. If Grph (= the category of oriented graphs)
is the full subconstruct of Alg(1, 1) that consists of all objects (X, c, d) that satisfy
the equations c ◦ c = d ◦ c = c and d ◦ d = c ◦ d = d, then the concrete functor
K

Catof −−→ Grph, defined by KC = (M or(C), c, d) (where M or(C)
f

c
d

//

M or(C)f

are given by d(A −→ B) = idA and c(A −→ B) = idB ), is the comparison functor
for Catof . K is not full.
(4) The full concrete embedding of the construct (Ban, O) into the construct TConv
of totally convex spaces is the comparison functor for (Ban, O).
(5) Since the constructs Top and Set induce the same monad on Set, the forgetful
functor Top → Set is the comparison functor for Top.
20.42 THEOREM
K
Let (A, U ) −−→ (XT , U T ) be a comparison functor. If A has coequalizers, then K is
adjoint.
Proof: Let (η, ε) : F
U : A → X be an adjoint situation and let T = (T, η, µ) becthe associated monad. Let (X, x) be a T-algebra and let F X −→ C be a coequalizer of
Fx

U c◦η

ε

X
X
F U F X −−→ F X and F U F X −−F−→ F X. Then (X, x) −−−−−→ KC is a K-universal
arrow as shown below:(a). Since (U c ◦ ηX ) ◦ x = U c ◦ U F x ◦ ηU F X = U c ◦ U εF X ◦ ηU F X = U c ◦ idU F X =U c ◦ U idF X = U c ◦ U (εF X ◦ F ηX ) = U (c ◦ εF X ) ◦ U F ηX = U (εC ◦ F U c) ◦ U F ηX =U c◦ηXU εC ◦ T (U c ◦ ηX ), it follows that (X, x) −−−−−→ (U C, U εC ) is an T -homomorphism.

9th July 2006

336

Adjoints and Monads[Chap. Vfˆf

Let (X, x) −→ KA be an arbitrary K-structured arrow. Then F X −→ A =εA ◦F fF X −−−−−→ A is the unique A-morphism with f = U fˆ ◦ ηX (cf. 19J). The equations
U (fˆ ◦ F x) ◦ ηU F X = U fˆ ◦ U F x ◦ ηU F X = U fˆ ◦ ηX ◦ x = f ◦ x = U εA ◦ U F f = U fˆ =U fˆ◦U εF X ◦ηU F X = U (fˆ◦εF X )◦ηU F X imply that fˆ◦F x = fˆ◦εF X . Since c is a coequalizer
(b).

˜

f
of F x and εF X in A, there exists an A-morphism C −→ A with fˆ = f˜ ◦ c. This impliesfU c◦ηXK f˜
that (X, x) −→ KA = (X, x) −−−−−→ KC −−−→ KA, since f = U fˆ ◦ ηX = U f˜ ◦ U c ◦ ηX
and K f˜ = U f˜. Uniqueness of f˜ follows from the fact that c is an epimorphism in A. 

20.43 THEOREM
Let (η, ε) : F
U : A → X be an adjoint situation with associated comparison functor
K

A −−→ XT . Then:(1) K is faithful if and only if U is faithful,(2) K is full and faithful if and only if ε is a RegEpi-transformation.
Proof:(1). Obvious.
(2). Let ε be a RegEpi-transformation. Then, by Theorem 19.14(1), U is faithful.
fThus K is faithful as well. Let A and B be objects in A and let KA −→ KB be
an XT -morphism.

If εA is a coequalizer of Cr/s

/ F U A , then the equations

U εB ◦ U F f ◦ U r = f ◦ U εA ◦ U r = f ◦ U εA ◦ U s = U εB ◦ U F f ◦ U s imply thatεB ◦ F f ◦ r = εB ◦ F f ◦ s by the faithfulness of U . Thus there exists an A-morphismfˆ
A −→ B with εB ◦ F f = fˆ ◦ εA . Since U εA as a retraction is an epimorphism, theequations K fˆ ◦ U εA = U fˆ ◦ U εA = U εB ◦ U F f = f ◦ U εA imply that K fˆ = f .
Thus K is full. Conversely, let K be full and faithful and let A be an A-object.
Then εA ◦ εF U A = εA ◦ F U εA . To see that εA is a coequalizer of εF U A andf

F U εA , let F U A −→ B be an A-morphism with f ◦ εF U A = f ◦ F U εA . The
equalities (U f ◦ ηU A ) ◦ U εA = U f ◦ U F U εA ◦ ηU F U A = U f ◦ U εF U A ◦ ηU F U A =U f = U f ◦ U εU F A ◦ U F ηU A = U εB ◦ U F U f ◦ U F ηU A = U εB ◦ T (U f ◦ ηU A )U f ◦η

UA
imply that KA −−−−−−→ KB is an T -homomorphism. By the fullness of K,fˆ
there exists an A-morphism A −→ B with K fˆ = U f ◦ ηU A . The equalities
U f ◦ ηU A = K fˆ = U fˆ = U fˆ ◦ U εA ◦ ηU A = U (fˆ ◦ εA ) ◦ ηU A imply that f = fˆ ◦ εA .

Uniqueness follows from the fact that (due to the faithfulness of K and hence of
U ) εA is an epimorphism [cf. Theorem 19.14(1)].

20.44 COROLLARYK
The comparison functor (A, U ) −−→ (XT , U T ) of a uniquely transportable concrete category, for which U reflects regular epimorphisms, is an isomorphism-closed full embedding.


9th July 2006

Sec. 20]

Monads337

DEFICIENCIES OF MONADIC FUNCTORS
20.45 EXAMPLE
A composite of two regularly monadic functors need not be monadic. Consider
U
/ Ab
II
II
II
V
V ◦U II$ 

TfAbI

Set

20.46 EXAMPLE
Concrete functors between monadic categories need not be adjoint. Consider
/ Grp
∞
II
II
II
V
U II$ 

Ab∞I

E

Set∞

where (in each case) A∞ is the full subcategory of A that consists of those objects of A
that have infinite underlying sets, where U and V are the forgetful functors, and E is
the inclusion functor.
20.47 EXAMPLE
Monadic functors need not detect colimits. The following yields a monadic category overPos that is not cocomplete.
Denote by P 0 : Set → Pos the power-set functor equipped with the ordering of PX suchthat ∅ is the least element, and PX−{∅} is discretely ordered. Denote by H : Pos → Set
the functor that assigns to each poset (X, ≤) the set
H(X, ≤) = { (x, y, z) ∈ X 3

x < y < z} ∪ {α}(where α 6∈ X 3 )and to each order-preserving function f : (X, ≤) → (Y, ≤), the map Hf with Hf (α) =α, and((f (x), f (y), f (z)), if f (x) 6= f (y) 6= f (z)Hf (x, y, z) =α,otherwise.
(a) The functor T = P 0 ◦ H : Pos → Pos is such that Alg(T ) has free objects. In fact,
for each poset A we have T A = T (A + T A) and, hence, the free T-algebra over A
ϕis (A + T A, ϕ), where T A −→ A + T A is the second coproduct injection, whereasηthe universal arrow A −→ A + T A is the first coproduct injection. Hence, Alg(T ) is
monadic over Pos.
(b) Alg(T ) does not have the coproduct of the following T-algebra (X, x) with itself:
X is the 3-chain a < b < c, and x : T X → X is the constant map to a.

9th July 2006

338

Adjoints and Monads[Chap. V

To prove this, for each ordinal i define T -homomorphisms fi , gi : (X, x) → (Yi , yi ) as
follows: Yi is the following poset
c1 @

c2

@@
@@
@@
@

b

. . . ggg j
pp 1
p
gggg
p
g
ggg g
pp
g
p
g
g
p
g
p ...
gg ...
ppp ggggg
pgpgpgggg...

0(j < i)aand for each j < i,j = yi ({ (a, b, k)k < j } ∪ { (a, b, ci )i = 1, 2 }),and otherwise, yi is constant to a. Finally, fi (a) = gi (a) = a, fi (b) = gi (b) = b, andfi (c) = c1 , gi (c) = c2 . It is easy to verify that fi and gi are homomorphisms such thatwhenever (X, x) + (X, x) exists, the factorizing homomorphism (X, x) + (X, x) → (Yi , yi )is surjective. This is impossible.
20.48 EXAMPLE
Monadic categories over co-wellpowered categories need not be extremally cowellpowered. See Exercise 16A(a).
20.49 EXAMPLE
Monadic categories over categories with regular factorizations need not have (Epi, MonoSource)-factorizations. See Exercise 16A(b).
20.50 EXAMPLE
Monadic functors need not preserve either extremal epimorphisms or regular epimorphisms. In fact they can map regular epimorphisms into non-extremal epimorphisms[see Exercise 16A(b)] and extremal epimorphisms into non-epimorphisms [see Exercise
16A(a)].
20.51 EXAMPLE
In monadic categories, regular epimorphisms need not be final. The comparison functor
K

Catof −−→ Grph for the construct Catof , described in Example 20.41(3), is monadic.
The functor F , described in Example 7.40(6), is a regular epimorphism in Catof that is
not final in (Catof , K).
20.52 EXAMPLE
Monadic functors need not reflect regular epimorphisms. As above, the comparison
K

G◦F

functor Catof −−→ Grph is monadic. The functor A −−−→ C, described in Remark
7.76(1), is not a regular epimorphism in Catof , but K(G ◦ F ) is a regular epimorphism
in Grph.

9th July 2006

Sec. 20]

Monads339

VARIETORS AND FREE MONADS
We now explain the role that the categories of the form Alg(T ) play among monadic
categories. They are, up to isomorphism, precisely the Eilenberg-Moore categories offree monads (introduced below).
20.53 DEFINITION
A functor T : X → X is called a varietor provided that the concrete category Alg(T )has free objects.
20.54 EXAMPLES(1) S n : Set → Set is a varietor. Each set generates a free S n -algebra (i.e., an algebrawith one n-ary operation; cf. 8.23(6)).
(2) If X has countable colimits, then idX is a varietor. Each object X generates a free
idX -algebra whose underlying object is a coproduct of countably many copies of X.
More generally, every functor T : X → X that preserves colimits of ω-chains is avarietor (cf. Exercise 20P).
(3) The power-set functor P is not a varietor. If Alg(P) would have an initial object(X, x), then x : PX → X would be an isomorphism (cf. Exercise 20I). However,card PX > card X.
20.55 DEFINITION(1) Given monads T = (T, η, µ) and T0 = (T 0 , η 0 , µ0 ) over X, a natural transformation
τ : T → T 0 is called a monad morphism (denoted by τ : T → T0 ) provided thatη 0 = τ ◦ η and τ ◦ µ = µ0 ◦ τ T 0 ◦ T τ .
(2) A free monad generated by a functor T : X → X is a monad T# = (T # , η # , µ# )together with a natural transformation λ : T → T # that has the following universal
property: for every monad T0 = (T 0 , η 0 , µ0 ) and every natural transformation
τ : T → T 0 there exists a unique monad morphism τ # : T# → T0 with τ = τ # ◦ λ.
20.56 THEOREM
If T : X → X is a varietor, then Alg(T ) is monadic over X and the associated monadis a free monad generated by T .
Proof: Let U : Alg(T ) → X denote the forgetful functor, let F : X → Alg(T ) be thefree functor, let T # = U ◦ F , and let T# = (T # , η # , µ# ) be the associated monad.
(1). U is monadic because it creates absolute coequalizers (20.17). If (X, x)

f1
f2

are T -homomorphisms and X

f1
f2

//

Y

c

(Y, y)/ Z is an absolute coequalizer in X, thenT c is a coequalizer of T f1 and T f2 , and since(c ◦ y) ◦ T f1 = c ◦ f1 ◦ x = c ◦ f2 ◦ x = (c ◦ y) ◦ T f2 ,9th July 2006

//

340

Adjoints and Monads[Chap. Vzthere is a unique T Z −→ Z for which c : (Y, y) → (Z, z) is a T -homomorphism.
The fact that T c is a (regular) epimorphism makes it easy to see that(X, x)f1
f2// (Y, y)c/ (Z, z) is a coequalizer in Alg(T ).

(2). If F X = (X # , ϕX ), then the morphismsTηλX
T X −−−→ T #XϕX= T X −−−X→ T X # −−−→ X#

obviously satisfy the naturality condition; i.e., they form a natural transformation
λ : T → T # . To verify the universal property, let (T 0 , η 0 , µ0 ) be a monad, and let
τ : T → T 0 be a natural transformation. For each X-object X we have a T -algebra(τ T 0 )Xµ0X
T (T 0 X) −−−−−→ (T 0 )2 X −−−→ T 0 X.

#
: F X → (T 0 X, [µ0 ◦τ T 0 ]X )Furthermore, there exists a unique T -homomorphism τX#
0 . It is a straightforward computation to verify that the morphisms
with τX ◦ηX = ηX#
U τX form a natural transformation τ # : T # → T 0 that satisfies all the required
equalities.


20.57 COROLLARY#
If T : X → X is a varietor, then the category Alg(T ) is concretely isomorphic to XT
for a free monad T# .

20.58 REMARKThe following theorem provides a partial converse to the above (in the case where X isstrongly complete).
20.59 THEOREM
If X is a strongly complete category, then every functor T : X → X that generates afree monad is a varietor.
Proof: Let (T # , η # , µ# ) together with λ : T → T # be a free monad over T . To
prove that the forgetful functor U : Alg(T ) → X is adjoint, it is sufficient to verify
the solution-set condition of Theorem 18.12, since Alg(T ) is complete and U preserves
limits (13N). We will show that for each X-object X the T -algebra
λ #

µ#

X
T (T # X) −−T−−X→ (T # )2 X −−−→ T #X#
together with the X-morphism ηX: X → T # X forms a singleton solution set. In
other words, for each T -algebra (Y, y) and each X-morphism f : X → Y there is a
T -homomorphismg : (T # X, µ#X ◦ λT # X ) → (Y, y)#with f = g ◦ ηX.

9th July 2006

Sec. 20]

Monads341

Denote by (B, U 0 ) the full concrete subcategory of Alg(T ) consisting of all subalgebras
of products of the T -algebra (Y, y). Then B is closed under the formation of limits in
Alg(T ). Thus B is strongly complete and U 0 preserves limits. Moreover, (Y, y) is a
coseparator of B (10.38), and hence, U 0 is adjoint (18.17). Let F 0 : X → B denote thefree functor, F 0 X = (X ∗ , ϕX ), and let (T 0 , η 0 , µ0 ) be the associated monad. Then themorphismsTηϕX
T X −−−X→ T X ∗ −−−→ X∗

clearly form a natural transformation τ : T → T 0 . By the universal property of λ there
exists a unique natural transformation τ # : T # → T 0 with (a) τ = τ # ◦ λ,(b) η 0 = τ # ◦ η #, and (c) τ # ◦ µ# = µ0 ◦ τ # T 0 ◦ T # τ #.
If f : X → Y is a morphism, then since (Y, y) lies in B, there is a unique T -homomor0 . Then g = f ∗ ◦ τ # : T # X → Y is thephism f ∗ : T 0 X → (Y, y) with f = f ∗ ◦ ηX
X#
desired morphism. In fact, from (b) it follows that f = g ◦ ηX
, and from (a) and (c)and the properties of monads it is easy to verify that g is a T -homomorphism (i.e., thaty ◦ T y = g ◦ µ#X ◦ λT # x ).

Suggestions for Further Reading
Eilenberg, S., and J. C. Moore. Adjoint functors and triples.
(1965): 381–398.

Illinois J. Math.

9

Kleisli, H. Every standard construction is induced by a pair of adjoint functors. Proc.
Amer. Math. Soc. 16 (1965): 544–546.
Maranda, J. M. On fundamental constructions and adjoint functors. Canad. Math.
Bull. 9 (1966): 581–591.
Beck, J. Triples, algebras and cohomology. Ph. D. dissertation, Columbia University,
1967.
Manes, E. G. A triple miscellany: Some aspects of the theory of algebras over a triple.
Ph. D. dissertation, Wesleyan University, 1967.
Linton, F. E. J. Coequalizers in categories of algebras. Springer Lect. Notes Math. 80(1969): 75–90.
Duskin, J. Variations on Beck’s tripleability criterion. Springer Lect. Notes Math. 106(1969): 74–129.
Applegate, H., and M. Tierney. Iterated cotriples. Springer Lect. Notes Math. 137(1970): 56–99.
Pumplün, D. Eine Bemerkung über Monaden und adjungierte Funktoren. Math. Ann.
185 (1970): 329–337.
Gabriel, P., and F. Ulmer. Lokal präsentierbare Kategorien.
Math. 221 (1971): 1–200.

9th July 2006

Springer Lect. Notes

342

Adjoints and Monads[Chap. V

Paré, R. On absolute colimits. J. Algebra 19 (1971): 80–95.
Adámek, J. Colimits of algebras revisited. Bull. Austral. Math. Soc. 17 (1977): 433–
450.
Mac Donald, J. L., and A. Stone. The tower and regular decomposition.
Topol. Geom. Diff. 23 (1982): 197–213.

Cahiers

Pumplün, D., and H. Röhrl. Banach spaces and totally convex spaces I. Commun. Alg.
12 (1984): 953–1019.

EXERCISES
20A. Associated Categories of T-Algebras Are Essentially UniqueShow that(a) If (η, ε) : F
U : A → X and (η, ε) : F
U : A → X are adjoint situations
that are isomorphic in the sense of Proposition 19.9, then the associated monads
T = (T, η, µ) and T = (T , η, µ) are isomorphic in the following sense: there exists
σa natural isomorphism T −→ T that is a monad morphism (20.55).

(b) If monads T = (T, η, µ) and T = (T , η, µ) in X are isomorphic in the above sense,
then the associated categories of T-algebras and T-algebras are concretely isomorphic.
K(c) If (A, U ) is a concrete category over X with free objects and if (A, U ) −−→ (X T , U T )Kand (A, U ) −−→ (X T , U T ) are associated comparison functors, then there exists aH

concrete isomorphism (X T , U T ) −−→ (X T , U T ) with K = H ◦ K.
20B. The Kleisli Category of a Monad
The Kleisli category of a monad T = (T, η, µ) in X is the following concrete category(XT , UT ) over X: Ob(XT ) = Ob(X), homXT (X, Y ) = homX (X, T Y ), (idX )XT = ηX ,fg

and the composition of X −→ T Y with Y −→ T Z in XT is given by µZ ◦ T g ◦ f (in X).
fµ ◦T fFurthermore, UT (X −→ T Y ) = T X −−Y−−−→ T Y.
fη ◦f(a) Prove that UT has a co-adjoint FT with FT (X −→ Y ) = (X −−Y−−→ T Y ), and thatthe associated monad of this adjoint situation is T.
(b) Prove that for each adjoint situation F
G: A
X with the associated monad∗T, there exists a unique functor K : XT → A with F = K ∗ ◦ FT and UT = G ◦ K ∗.

(c) Describe the Kleisli category of the power-set monad as the category of “sets andrelations”.
(d) Show that UT need not detect finite limits. [Consider (c).]9th July 2006

Sec. 20]

Monads343* 20C. Algebraic Theories(a) Let T be a monad over Set. Verify that in the Kleisli category XT (20B) each object(set) X is a coproduct of X copies of 1 = {0} with coproduct injections kx : 1 → X
in XT corresponding to ηX (x) ∈ T X. Prove that for each T-algebra A the functor
FA : XT op → Set given byf

F f

FA (X −→ Y ) = hom((T X, µX ), A) −−A−→ hom((T Y, µY ), A)[where FA f (h) = h ◦ µX ◦ T f (in Set)] preserves products.
(b) Conversely, prove that every product-preserving functor F : (XT )op → Set is naturally isomorphic to FA for some T-algebra A. [Hint: Choose A to be (F (1), ϕ),where ϕ : T F (1) → F (1) assigns to each element of T F (1) represented by an XT morphism f : 1 → F (1), the element of F (1) that the function F f : F 2 (1) → F (1)assigns to id ∈ (F (1))F (1) .] Conclude that SetT is equivalent to the quasicategoryof all product-preserving functors from (XT )op to Set.
(c) An algebraic theory is a category L whose objects are precisely all sets and suchthat each object X is a coproduct of X copies of 1. A model of L is a productpreserving functor from Lop into Set. The quasicategory Mod(L) of models of
L is the full subcategory of [Lop , Set] that has models as objects. Prove that
Mod(L) is equivalent to a monadic construct. Moreover, show that the Kleisli
category of the corresponding monad is equivalent to L.
(d) Verify that the above transitions between monads and algebraic theories are essentially inverse to each other (i.e., inverse up to isomorphism of monads and up toequivalence of algebraic theories).
* 20D. Cocompleteness of Monadic Categories
Let (A, U ) be a monadic category over X and let A have coequalizers. Show that(a) A has colimits over each scheme over which colimits exist in X.
(b) If X is cocomplete then so is A.
* 20E. Concrete Functors Between Monadic CategoriesG

Let (A, U ) −−→ (B, V ) be a concrete functor between monadic categories. Show that(a) If A has coequalizers, then G is monadic.
(b) If (A, U ) and (B, V ) are regularly monadic, then G is regularly monadic.
20F. Idempotent Monadsµ

A monad T = (T, η, µ) is called idempotent provided that T 2 −→ T is a natural
isomorphism. Show that(a) For a monad T = (T, η, µ) the following conditions are equivalent:(1) T is idempotent,

9th July 2006

344

Adjoints and Monads[Chap. V(2) T η is an Epi-transformation,
UT(3) XT −−−→ X is full,UT(4) XT −−−→ X is an isomorphism-closed full reflective embedding.
U(b) If A −−→ X is an isomorphism-closed full reflective embedding, then the associatedmonad is idempotent.
(c) The monad associated with the forgetful functor Top → Set is idempotent.
* 20G. Monads With Rank
A monad T = (T, η, µ) is said to have rank k, where k is a regular cardinal, provided
that T preserves k-directed colimits, i.e., colimits of diagrams whose schemes are posets
in which every subset of cardinality less than k has an upper bound.
(a) Prove that for each variety of finitary algebras the corresponding monad in Set has
rank ℵ0.
(b) Let B be an isomorphism-closed, full reflective subcategory of a category A. Provethat the corresponding idempotent monad in A has rank k if and only if B is closed
under the formation of k-direct colimits in A.
(c) For each small category A prove that the forgetful functor [A, Set] → [Ob(A), Set]
(where the set Ob(A) is considered to be a discrete category) is monadic, and thatthe corresponding monad has rank ℵ0 .
* 20H. Locally Presentable Categories
An object A of a category A is called presentable provided that hom(A, −) : A → Set
preserves k-directed colimits for some regular cardinal k. A category A is called locallypresentable provided that it is cocomplete and has a dense (see 12D) subcategory
formed by presentable objects.
(a) Show that every object A is presentable in Set, Vec, Pos, and Aut. [Choose any
regular cardinal k larger than card A.] Show that in Top the only presentable objects
are the discrete spaces, and in HComp only the empty space is presentable.
(b) Show that Set, Vec, Pos, and Aut are locally presentable categories. Furthermore,show that [A, Set] is locally presentable for each small category A.
(c) Show that for each monad T with rank over a locally presentable category X thecategory XT is locally presentable. In particular, if Setm denotes the category
Setm for some cardinal number m, thenSetm, SetmT1 , (SetmT1 )T2 , . . .
are locally presentable categories for arbitrary monads T1 , T2 , . . . with rank.
(d) Conversely, prove that every locally presentable category A is equivalent to a category (SetmT1 )T2 for some monads T1 , T2 with rank; in fact, let B be a densesubcategory of A formed by presentable objects, then show that

9th July 2006

Sec. 20]

Monads345(1) the full and faithful functor E : A → [Bop , Set] of Exercise 12D(c) maps A ontoa reflective subcategory E[A], and the corresponding idempotent monad T2 on[Bop , Set] has rank,(2) [Bop , Set] is monadic with rank ℵ0 over Setn = Setn for n = card B [20G(c)].
(e) Let us call an object A strongly n-generated provided that hom(A, −) preserves
colimits of n-directed diagrams whose connecting morphisms are strong monomorphisms. Prove that then every quotient of A is also strongly n-generated.
In a locally n-presentable category X the strongly n-generated objects are precisely
the quotients of n-presentable objects. And given a monad T with rank n prove thata T-algebra is strongly n-presentable in XT iff it is a quotient of the free algebra
TX for some strongly n-generated object X.
(f) In every locally presentable category X prove that every object is strongly n-presentable for some n. Conclude that if X has, for every regular cardinal n, only a
set of strongly n-generated objects up to isomorphism, then X is co-wellpowered.
Moreover, XT is also co- wellpowered for every monad T with rank.
(g) Conclude that every locally presentable category is complete, wellpowered, and cowellpowered.
20I. Initial T -algebras
For an initial object (X, x) of Alg(T ) prove that x is an isomorphism. [Hint: Use theT -algebra (T X, T x).]* 20J. Monadic Towers
U
Let A be cocomplete and let A −−→ X be an adjoint functor. Call the associated monad
U1

1

1

T1 and the associated comparison functor A −−→ XT . Then U = U T ◦ U 1 . By
Theorem 20.42, U 1 is adjoint. Denote the associated monad by T2 and the associatedcomparison functor by U 2 . Then the diagram
2 U
XT o

2

A

||
2
||
|
U
UT
|
 }||

/X
T1
U1

X

UT

1

commutes. This process can be iterated over all ordinals (with limit-steps obtained byforming a limit in the quasicategory CAT).
2(a) Show that U T is an equivalence whenever U maps regular epimorphisms to epimorphisms.
(b) Let An (n a natural number) be the construct whose objects are pairs (X, (αi )i≤n ),where X is a set and αi is a partial endofunction of X defined on x ∈ X if and only if
αj (x) = x for all j < i. Morphisms from (X, (αi )i<n ) to (X 0 , (αi0 )i<n ) are functions
f : X → X 0 such that f (αi (x)) = αi0 (f (x)) whenever αi (x) is defined. Describe

9th July 2006

346

Adjoints and Monads[Chap. V

the free algebras, comparison functors, etc. in detail and prove that the iteration
described above takes precisely n steps. That is, if U0 is the forgetful functor of An
and Uk+1 is the comparison functor associated with the kth step (k = 0, 1, 2, . . .),then Un is an equivalence but Un−1 is not.
(c) Show that there exists a construct for which the above iteration does not stop.
20K. The Constructs Topop
0 and FramShow that (cf. 5L)(a) Fram is monadic.
(b) Topop
0 has free objects.
T(c) The concrete functor Topop−→ Fram described in Exercise 5L is a comparison
0 −functor.
20L. Regularly Monadic Functors Lift Regularity
Show that
U(a) If X is regular and A −−→ X is regularly monadic, then A is regular.
(b) Monadic constructs are regular.
20M. Extremally Monadic Functors
U

A functor A −−→ X (resp. a concrete category (A, U ) over X) is called extremallymonadic provided that X is an (ExtrEpi, Mono-Source)-category and U is ExtrEpimonadic. Show that a concrete category (A, U ) over X is regularly monadic if and only
if it is extremally monadic and X has regular factorizations.
20N. When Are Order Preserving Maps Monadic?
(a) Show that a morphism in Pos, considered as a functor between thin categories, ismonadic if and only if it is an embedding of a full reflective subcategory.
Pff(b) For any function A −→ B consider PA −−→ PB as a functor (cf. Exercise 6G).
Show that Pf is monadic if and only if f is surjective.
20O. Monadic Functors and Extremal Monomorphisms
Determine whether or not monadic functors preserve extremal monomorphisms.
* 20P. Varietors and Colimits of ω-chains
Let T : X → X be a functor with X having countable colimits. For each object X definea diagram D :→ X (whereis the thin category of natural numbers) byNN

D0 = X,
Dn+1 = X + T Dn ,D(0 → 1) = X → X + T X,

the first coproduct injection,
idX +T D(n→m)

D(n + 1 → m + 1) = X + T Dn −−−−−−−−−−−→ X + T Dm .

9th July 2006

Sec. 20]

Monads347(a) If T preserves the colimit of D, prove that X generates a free T -algebra. [Hint: If(Dn −−n→ C)N is a colimit of D, then there is a unique c : T C → C with c ◦ T dn =dd0dn+1 ◦ un , where un : T Dn → X + T Dn is the second injection. Then X −−→|(C, c)|is universal.](b) Conclude that each functor T : X → X that preserves colimits of ω-chains (i.e.,
diagrams with the scheme ) is a varietor.

N(c) Find a varietor that does not preserve colimits of ω-chains.

9th July 2006

349

Chapter VI

TOPOLOGICAL
CATEGORIESANDALGEBRAICAs demonstrated in §5, the vague concept of “structure” has a formalization in theconcept of a “concrete category” — the idea being that the forgetful or underlying
functor “forgets” the structure in question. Many structures can be decomposed into
more basic ones, which often can be classified as “topological” or “algebraic”. The natureof a structure is reflected not so much in properties of its abstract category, but rather
in properties of its underlying functor. For example, topological groups can be regarded:U(a) as topological structures via the forgetful functor TopGrp −−→ Grp, i.e., as topological structures over Grp, orV(b) as algebraic structures via the forgetful functor TopGrp −−→ Top, i.e., as algebraic
structures over Top, orW(c) as topologically algebraic structures via the forgetful functor TopGrp −−→ Set, i.e.,
as topologically algebraic structures over Set.
Since it is the underlying functor rather than the abstract category that determines
the character of a structure (= concrete category), in each case we study properties offunctors first. Surprisingly, each of the crucial properties under investigation (U being
topological, algebraic, or topologically algebraic ) implies that the functor U in question
is faithful, i.e., is the forgetful functor of a concrete category.
We choose terminology such that a concrete category (A, U ) has a certain property P
if and only if its forgetful functor U has the property P . A desirable characteristic ofsuch properties will be that they are closed under composition. Moreover, for “algebraic”
properties P , it will be desirable that concrete functors between concrete categories withproperty P will automatically have property P as well.
Most properties under investigation can be defined either in a rigid (uniquely transportable) version or in a more flexible (closed under equivalences) one. We have foundit somewhat more convenient and consistent with the current usage to choose the rigid
version in the topological and algebraic cases, and the flexible version in the topologicallyalgebraic case.

9th July 2006

21 Topological categories
The following phenomena are typical for topological categories (A, U ):fi(1) (A, U ) is initially complete, i.e., every structured source (X −−→ U Ai )I has a uniquefi

initial lift (A −−→ Ai )I ,fi(2) (A, U ) is finally complete, i.e., every structured sink (U Ai −−→ X)I has a uniquefi

final lift (Ai −−→ A)I ,(3) (A, U ) is fibre-complete, i.e., every fibre is a (possibly large) complete lattice (cf. Definition 5.7),(4) (A, U ) has discrete structures, i.e., every X-object has a discrete lift,(5) (A, U ) has indiscrete structures, i.e., every X-object has an indiscrete lift.
As we will see, the above conditions are not independent of each other. In fact, (1)and (2) are equivalent [Topological Duality Theorem (21.9)], and imply all the others.
Moreover, (1) and (2) have many other pleasant consequences. For example, they imply
that U lifts limits (and colimits) uniquely. The unique lifting of limits implies, together
with (5), all of the other conditions (21.18). However, forgetful functors of algebraic
categories also lift limits uniquely (see §23). Hence the very simple condition (5) insome sense may be considered to be at the heart of topology.

TOPOLOGICAL FUNCTORS
21.1 DEFINITIONG
A functor A −−→ B is called topological provided that every G-structured sourcefifi(B −−→ GAi )I has a unique G-initial lift (A −−→
Ai )I .

21.2 EXAMPLES(1) The forgetful functors of the constructs Top, Unif , PMet, Rel, and Prost are
topological, but those of the constructs Haus, Met, Vec, and Pos are not topological.
(2) For a thin category A, the unique functor to 1 is topological if and only if A is a(possibly large) complete lattice.
21.3 THEOREM
Topological functors are faithful.

350Sec. 21]Topological categories

Proof: Let A

351

r /
s

/ A0 be a pair of A-morphisms with Gr = Gs. Consider the sourcef

S = (GA −−h→ GAh )h∈M or(A) with (fh , Ah ) = (Gr, A0 ) for each h ∈ M or(A). LetfˆgŜ = (Â −−h→ A0 )h∈M or(A) be a G-initial lift of S. Define a source T = (A −−h→ A0 ) by(r, if fˆh ◦ h = s
gh =s, otherwise.
kThen GT = GŜ ◦ idGA . By G-initiality of Ŝ there exists a morphism A −→ Â with
T = Ŝ ◦ k, i.e., gh = fˆh ◦ k for each h ∈ M or(A). In particular, we obtain gk = fˆk ◦ k,which — by the definition of the gh ’s — is possible only for r = s.
Lifting of a source in a topological category

21.4 REMARK
We will see below that “to be topological” is a very strong and pleasant property. Muchof the strength lies in the fact that the G-structured sources are allowed to be large(as can be discerned from the preceding proof). For example, the natural “forgetful”
functor G from the category of modules to the category of rings is not faithful, hence not
topological, even though each small G-structured source has a G-initial lift (see Exercise
21D).
21.5 PROPOSITIONG
If A −−→ B is a functor such that every G-structured source has a G-initial lift, then thefollowing conditions are equivalent:

9th July 2006

352

Topological and Algebraic Categories[Chap. VI(1) G is topological,(2) (A, G) is uniquely transportable,(3) (A, G) is amnestic.
Proof: (1) ⇒ (2) is immediate from Proposition 8.14, once it is observed that G isfaithful. The latter follows as in the proof of Theorem 21.3.
(2) ⇒ (3) follows from Proposition 5.29.
f˜i

f

fi

i
(3) ⇒ (1). If (A −−→
Ai )I and (Ã −−→ Ai )I are initial lifts of (X −−→ GAi )I , then A ≤ Ãand Ã ≤ A; hence by (3), A = Ã.


21.6 PROPOSITIONGFF ◦G
If A −−→ B and B −−→ C are topological, then so is A −−−→ C.
Proof: Immediate by Proposition 21.5, since amnesticity and existence of initial liftings
are compositive properties.


TOPOLOGICAL CATEGORIES
21.7 DEFINITION
A concrete category (A, U ) is called topological provided that U is topological.
21.8 EXAMPLES(1) The constructs Top, Unif , PMet, Rel, and Prost are topological.
(2) All functor-structured categories Spa(T ) and all functor-costructured categories(Spa(T ))op are topological.
(3) A partially ordered set, considered as concrete category over 1, is topological if andonly if it is a complete lattice.
(4) TopGrp is topological if it is considered as a concrete category over Grp, but not
if it is considered as a concrete category over Top or over Set.
(5) The construct Top1 is not topological, even though for every structured sourcefi(X −−→ (Xi , τi ))I there exists a largest Top1 -structure (= the smallest T1 -topology)fiτ on X making each (X, τ ) −−→ (Xi , τi ) continuous.
The following theorem generalizes the well-known fact that each meet-complete poset isalso join-complete.
21.9 TOPOLOGICAL DUALITY THEOREM
If (A, U ) is topological over X, then (Aop , U op ) is topological over Xop (i.e., the existenceof unique U-initial lifts of U-structured sources implies the existence of unique U-final liftsof U-structured sinks).

9th July 2006

Sec. 21]Topological categories

353

Proof: Let (A, U ) be topological. It must be shown that in (A, U ) every structuredfisink S = (U Ai −−→ X)I has a final lift (since uniqueness follows from amnesticity).
gjConsider the structured source T = (X −−→ U Bj )J consisting of all structured arrows
gjfi(gj , Bj ) with the property that U Ai −−→ X −−→ U Bj is an A-morphism for each i ∈ I.
gjfi

If (A −−→ Bj )J is an initial lift of T , then (Ai −−→ A)I is a final lift of S.



21.10 REMARKThe above Topological Duality Theorem implies that (as was the case for abstract andfor concrete categories) there is also a Duality Principle available for topological categories. However, observe that, since the dual of a concrete (resp. topological) category
over X is a concrete (resp. topological) category over Xop , this does not imply a Duality
Principle for (topological) concrete categories over a fixed category X (unless Xop = X,
as for X = 1). In particular, there is not a Duality Principle available for topologicalconstructs.
21.11 PROPOSITION
Topological categories are fibre-complete. The smallest (resp. largest) member of eachfibre is discrete (resp. indiscrete).
Proof: Let (A, U ) be topological over X, let X be an X-object, and let (Ai )I be a
id

id

X
X
family of A-objects with U Ai = X. If (A −−−→ Ai )I is an initial lift of (X −−−→ U Ai )I ,then A = inf(Ai ) in the fibre of X. For I = ∅, we have that A is the largest element of
the fibre of X. It is an indiscrete object, since (A, ∅) is an initial source [cf. 10.42(1)].
By duality, the smallest element of the fibre of X must be discrete.


21.12 PROPOSITION
If (A, U ) is topological over X, thenF(1) U is an adjoint functor; its co-adjoint X −−→ A (the discrete functor) is a fullembedding, satisfying U ◦ F = idX .
G(2) U is a co-adjoint functor; its adjoint X −−→ A (the indiscrete functor) is a fullembedding, satisfying U ◦ G = idX .
Proof: Immediate from Proposition 21.11.



21.13 PROPOSITION
If (A, U ) is topological over X, then the following hold:(1) U preserves and reflects mono-sources and epi-sinks.
(2) An A-morphism is an extremal (resp. regular) monomorphism if and only if it isinitial and an extremal (resp. regular) X-monomorphism.
(3) An A-morphism is an extremal (resp. regular) epimorphism if and only if it is final
and an extremal (resp. regular) X-epimorphism.

9th July 2006

354

Topological and Algebraic Categories[Chap. VI

In particular, in topological constructs, the following hold:(4) embeddings = extremal monomorphisms = regular monomorphisms.
(5) quotient morphisms = extremal epimorphisms = regular epimorphisms.
Proof:(1). Since U is faithful and (co)adjoint, it reflects and preserves mono-sources (epi-sinks).
mm(2). Let A −−→ B be an extremal monomorphism in A. Then, by (1), U A −−→ U B ismefa monomorphism in X. Let U A −−→ U B = U A −→ X −→ U B be an (Epi,−)eefactorization in X. Then U A −→ X has a final lift A −→ C. Consequently, by (1),mfeA −−→ B = A −→ C −→ B is an (Epi,−)-factorization in A. Therefore, e is anmisomorphism in A and hence in X. Thus U A −−→ U B is an extremal monomorphismmmm
in X. To show initiality of A −−→ B, let A0 −−→ B be an initial lift of U A −−→ U B.
id
id
m
m
Then A −−→ B = A −−U−A→ A0 −−→ B is an (Epi,−)-factorization. Thus A −−U−A→ A0mm
is an isomorphism; hence by amnesticity A = A0 . So A −−→ B = A0 −−→ B is initial.
mm
For the converse, let A −−→ B be an initial morphism such that U A −−→ U B is anmextremal monomorphism in X. Then, by (1), A −−→ B is a monomorphism in A.
m

e

f

e

Let A −−→ B = A −→ C −→ B be an (Epi,−)-factorization in A. Then U A −→ U Cee
is an X-isomorphism and A −→ C is initial. Hence, by Proposition 8.14, A −→ C isman isomorphism. Thus A −−→ B is an extremal monomorphism in A.
m
Next, let A −−→ B be a regular monomorphism in A. Then it is extremal and henceminitial. By adjointness, U A −−→ U B is a regular monomorphism in X. Conversely,mm
let A −−→ B be an initial morphism such that U A −−→ U B is an equalizer ofUBrs// X in X. If Brs// C is a final lift, then by Proposition 13.15, m is anequalizer of r and s in A.
(3) follows by duality (21.9).
(4) and (5) are immediate from (2) and (3).



21.14 PROPOSITION
If (A, U ) is topological over an (E, M)-category X, then the following hold:(1) A is an (E, Minit )-category, where Minit consists of all initial sources in M.
(2) A is an (Ef in , M)-category, where Ef in consists of all final E-morphisms.



21.15 PROPOSITION
If (A, U ) is topological over X, then U uniquely lifts both limits (via initiality) andcolimits (via finality), and it preserves both limits and colimits.
Proof: The unique lifting follows immediately from Proposition 13.15 and its dual; thepreservation from (co)adjointness of U .


9th July 2006

Sec. 21]Topological categories

355

21.16 THEOREM
If (A, U ) is topological over X, then the following hold:(1) A is (co)complete if and only if X is (co)complete.
(2) A is (co-)wellpowered if and only if (A, U ) is fibre-small and X is (co-)wellpowered.
(3) A is extremally (co-)wellpowered if and only if X is extremally (co-)wellpowered.
(4) A is (Epi, Mono-Source)-factorizable if and only if X is (Epi, Mono-Source)-factorizable.
(5) A has regular factorizations if and only if X has regular factorizations.
(6) A has a (co)separator if and only if X has a (co)separator.
Proof: Let G : X → A be the indiscrete functor.
(1). If X is complete, then A is complete, since U lifts limits. Conversely, if A iscomplete, D : I → X is a small diagram, and L is a limit of G ◦ D : I → A, thenU L is a limit of U ◦ G ◦ D = D.
(2). Let (A, U ) be fibre-small and X be wellpowered. For any A-object A consider a setmi
M = { Xi −−→
U A | i ∈ I } of X-subobjects of U A such that every subobject of U A
miis isomorphic to some member of M. If M̂ consists of all possible lifts Ai −−→Aof members of M, then, by fibre-smallness, M̂ is a set. By transportability each
subobject of A is isomorphic to some member of M̂. Hence A is wellpowered.
Conversely, let A be wellpowered. For every X-object X, it is clear that the class
idX
A = { A ∈ Ob(A) | U A = X } is a set, since otherwise { A −−−→ GX | A ∈ A }would by amnesticity be a proper class of pairwise non-isomorphic subobjects ofmi
GX. Thus (A, U ) is fibre-small. If (Xi −−→X)I would be a proper class ofmipairwise non-isomorphic subobjects of X, then (GXi −−→
GX)I would be a properclass of pairwise non-isomorphic subobjects of GX in A. Hence X is wellpowered.
mi(3). Let X be extremally wellpowered. For any A-object A let M = { Xi −−→
U A }i∈I
be a set of extremal subobjects of U A such that every extremal subobject of U A
miis isomorphic to some member of M. For each i ∈ I let Ai −−→
A be the uniquemiinitial lift of Xi −−→
U A. Then every extremal subobject of A is isomorphic tomisome member of the set { Ai −−→
A | i ∈ I }. Hence A is extremally wellpowered.
mi
Conversely, let A be extremally wellpowered. If (Xi −−→X)I were a proper class ofpairwise non-isomorphic extremal subobjects of some X-object X, it would follow
mithat (GXi −−→
GX)I would be a proper class of pairwise non-isomorphic extremal
subobjects of GX in A. Hence X is extremally wellpowered.

fi

fi

e

m

i
(4). If (A −−→ Ai )I is a source in A, (U A −−→ U Ai ) = (U A −→ X −−→
U Ai ) ismian (Epi, Mono-Source)-factorization in X, and (B −−→ Ai )I is an initial lift of

fi

m

e

m

i
i
(X −−→U Ai )I , then (A −−→ Ai ) = (A −→ B −−→
Ai ) is an (Epi, Mono-Source)-fi

fifactorization in A. Conversely, if (X −−→ Xi )I is a source in X and GX −−→ GXi =
e

m

fi

i
GX −→ A −−→
GXi is an (Epi, Mono-Source)-factorization in A, then X −−→ Xi =mie
X −→ U A −−→ Xi is an (Epi, Mono-Source)-factorization in X.

9th July 2006

356

Topological and Algebraic Categories[Chap. VI(5). This follows as in (4) by means of Proposition 21.13(3).
(6). If A is an A-separator, then U A is an X-separator. If X is an X-separator, thenthe discrete object A with U A = X is an A-separator.
The “co”-parts follow by duality.



21.17 COROLLARY
Each topological construct(1) is complete and cocomplete,(2) is wellpowered (resp. co-wellpowered) if and only if it is fibre-small,(3) is an (Epi, Extremal Mono-Source)-category,(4) has regular factorizations,(5) has separators and coseparators.



21.18 INTERNAL TOPOLOGICAL CHARACTERIZATION THEOREM
A concrete category (A, U ) over X is topological if and only if it satisfies the following
conditions:(1) U lifts limits uniquely,(2) (A, U ) has indiscrete structures, i.e., every X-object has an indiscrete lift.
Proof: By Propositions 21.11 and 21.15 topological categories satisfy (1) and (2).
Conversely, let (A, U ) satisfy (1) and (2). Then (A, U ) is uniquely transportable. Hence,by Proposition 21.5 and by duality (21.9), it suffices to show that every structured sinkfi

(U Ai −−→ X)I has a final lift. Let B0 be the indiscrete object with U B0 = X andfilet (Bj )J be the family of all A-objects Bj such that U Bj = X and U Ai −−→ U Bj
id

X
is an A-morphism for each i ∈ I. The family (U Bj −−−→ U B0 )J has an intersection

id

id

X
X
X −−−
→ U B0 in X. By (1) this intersection can be lifted to an intersection B −−−
→ B0

fi

id

X
of the family (Bj −−−→ B0 )J . Consequently, S = (Ai −−→ B)I is a sink in A. To show

g

fi

g

that it is final, let U B −→ U C be an X-morphism such that each U Ai −−→ U B −→ U C
is an A-morphism. Let C0 be the indiscrete object with U C0 = U C. Then the pullback
in X
g
/ UC
X
idX





UB

g

idU C

/ U C0can be lifted to a pullback in A
P
idX

g





B

/C

g

idU C

/ C0

9th July 2006

Sec. 21]Topological categories

357

Since, for each i ∈ I, the diagram
Ai
fi

g◦fi





B

/C

g

idU C

/ C0
fi

commutes, by the pullback-property each U Ai −−→ U P is an A-morphism. Consequently, P is a member of (Bj )J . This implies that B ≤ P . Hence, by amnesticity,
g

g

fi

B = P . Thus B −→ C = P −→ C is an A-morphism. Consequently, (Ai −−→ B)I is afi

final lift of (U Ai −−→ X)I .



21.19 REMARKAs we will see later, the unique lifting of limits is a widespread property shared by manyreasonable forgetful functors not only in topology, but also in algebra. Hence the above
theorem shows that the existence of indiscrete structures is the crucial condition thatmakes (A, U ) topological.
21.20 EXAMPLE
For any construct of the form Alg(Σ) the forgetful functor Alg(Σ) → Set lifts limits
and colimits uniquely. But Alg(Σ) is topological only if Σ is the empty family, i.e., only
if Alg(Σ) is concretely isomorphic to the construct Set.
21.21 EXTERNAL TOPOLOGICAL CHARACTERIZATION THEOREM
Let CAT(X) be the quasicategory of all concrete categories and concrete functors overa fixed category X. If M is the conglomerate of all full functors in CAT(X), then foreach concrete category (A, U ) over X the following are equivalent:(1) (A, U ) is topological over X.
(2) (A, U ) is an M-injective object in CAT(X).
E

Proof: (1) ⇒ (2). Suppose that (A, U ) is topological over X, (B, V ) −−→ (C, W )G

belongs to M, and (B, V ) −−→ (A, U ) is a concrete functor over X. For each C-object
fi

C consider the E-structured source (C −−→ EBi )I consisting of all pairs (fi , Bi ) withfi

Bi ∈ Ob(B) and C −−→ EBi ∈ M or(C). Application of W yields the U-structuredfifisource (W C −−→ U GBi )I . By hypothesis it has a U-initial lift (AC −−→ GBi )I . Let
Ĝ(C, W ) −−→ (A, U ) be the unique concrete functor with ĜC = AC for each C-object C.
Then G = Ĝ ◦ E.
(2) ⇒ (1). Suppose that (A, U ) is M-injective in CAT(X). By Proposition 5.33 thereP

exists an amnestic (B, V ) and a functor (A, U ) −−→ (B, V ) that is surjective on objects
and belongs to M. By injectivity, P is a section, hence an isomorphism. Thus (A, U ) is

9th July 2006

358

Topological and Algebraic Categories[Chap. VIfiamnestic. It remains to be shown that every U-structured source S = (X −−→ U Ai )I has
a U-initial lift. Embed (A, U ) as a concrete and full subcategory of a concrete category(B, V ) by adding one object B with V B = X and adding the following morphisms (forA-objects A and X-morphisms f ):
fi ◦f

f

U A −→ V B ∈ M or(B) ⇐⇒ U A −−−→ U Ai ∈ M or(A)for each i ∈ I,f

V B −→ U A ∈ M or(B) ⇐⇒ there exists i ∈ I and an A-morphismgAi −→ A with f = g ◦ fi ,
fi ◦f

f

V B −→ V B ∈ M or(B) ⇐⇒ V B −−−→ U Ai ∈ M or(B)for each i ∈ I.

By M-injectivity, (A, U ) is a retract of (B, V ), i.e., the identity on A can be extended
G

fi

fi

to a concrete functor (B, V ) −−→ (A, U ). Then (GB −−→ Ai )I = (GB −−→ GAi )I is a
U-initial lift of S.

21.22 REMARKS(1) The above result remains true if only fibre-small or only amnestic concrete categories
are considered. In the amnestic case M consists precisely of the concrete full embeddings [cf. 5.10(4)]. Application of the fibre-small amnestic case to X = 1 yields
the result that the injectives in Pos are precisely the complete lattices [cf. 9.3(2)].
(2) From the above proof it is easily seen that topological categories have an alternative
external characterization as precisely the M-absolute retracts in CAT(X). For a
description of M-essential extensions and M-injective hulls in CAT(X) see Exercise
21J.

INITIALITY-PRESERVING CONCRETE FUNCTORSAn important property of abstract functors is the preservation of limits. Recall that every
adjoint functor preserves limits and that, under suitable assumptions, this preservation
property characterizes adjoints (see the adjoint functor theorems of §18). For concrete
categories a similar condition is that of the preservation of initial sources. (See Definition
10.47.) Among other things, we will show that for Galois correspondences (F, G) the
concrete functor G preserves initial sources, and that if G has a topological domain, thispreservation property characterizes those concrete functors G that are part of a Galois
correspondence.
21.23 PROPOSITION
Initiality-preserving concrete functors preserve indiscrete objects.
Proof: Immediate by Example 10.42(1).



21.24 GALOIS CORRESPONDENCE THEOREM
GFor concrete functors (A, U ) −−→ (B, V ) with topological domain (A, U ) the following
conditions are equivalent:

9th July 2006

Sec. 21]Topological categories

359(1) G preserves initial sources,
F(2) G is adjoint and has a concrete co-adjoint (B, V ) −−→ (A, U ),F(3) there exists a (unique) (B, V ) −−→ (A, U ) such that (F, G) is a Galois correspondence.
Proof:fi(1) ⇒ (2). For any B-object B consider the G-structured source SB = (B −−→ GAi )Ifi

consisting of all pairs (fi , Ai ) with Ai ∈ Ob(A) and B −−→ GAi ∈ M or(B). Applifi

fi

cation of V yields the U-structured source ŜB = (V B −−→ U Ai )I . Let (AB −−→ Ai )Ifi

be a U-initial lift of ŜB . By (1), (GAB −−→ GAi )I is V-initial.
fi

id

id

VB
VB
Since each V B −−−−→ V GAB −−→ V GAi is a B-morphism, V B −−−−→ V GAB must be a
B-morphism. Thus (idV B , AB ) is a G-universal arrow for B. Since it is identity-carried,
the associated co-adjoint F of G satisfies V = U ◦ F .

F(2) ⇒ (3). Let (B, V ) −−→ (A, U ) be a concrete co-adjoint of G, induced by G-universalηη
arrows B −−B→ GF B. By Theorem 19.1 each F B −−B→ F GF B is a section in A;ηB
hence V B −−→ U F B is a section in X. To show that it is an epimorphism in X, let/
U F B s / X be a pair of X-morphisms with r ◦ ηB = s ◦ ηB . If A is an indiscreterr // A is a pair of A-morphisms. Since
object in (A, U ) with U A = X, then F Bs(ηB , F B) is G-generating, the equality Gr ◦ ηB = Gs ◦ ηB implies that r = s. Thusη
η
V B −−B→ U F B is an X-isomorphism. Let AB −−B→ F B be a U-initial lift. Then,ηB
by Proposition 8.14, AB −−→ F B is an A-isomorphism. Hence, by Proposition 8.35,
id

G(ηB )−1

η

VB
B −−−
−→ GAB = B −−B→ GF B −−−−−−→ GAB is a G-universal arrow for B. If F̂ is theassociated co-adjoint of G, then (F̂ , G) is a Galois-correspondence. Uniqueness follows
from the amnesticity of U .

(3) ⇒ (1). Immediate from Proposition 10.49.



21.25 REMARK
If (A, U ) is not topological, then conditions (1) and (2) of the above theorem do not
imply (3). See Exercise 21E(b).
21.26 DEFINITIONUV
Let A −−→ X and B −−→ Y be functors. An adjoint situation (η̂, ε̂) : F̂Ĝ : Â → B̂is said to lift an adjoint situation (η, ε) : F
G : X → Y along U and V provided
that the following conditions are satisfied:(1) the diagrams
A
U



X

9th July 2006

Ĝ

/B


G

V

/Y

B
and

V

F̂



Y

/A


F

U

/X

360

Topological and Algebraic Categories[Chap. VI

commute.
(2) V η̂ = ηV,(3) U ε̂ = εU .
21.27 REMARKS(1) In the presence of (1), conditions (2) and (3) in the above definition imply eachother. They have both been included to make the symmetry apparent. If (1) and(2) hold, then (3) can be deduced from the equations GU ε̂A ◦ηGU A = V Ĝε̂A ◦ηV ĜA =V Ĝε̂A ◦ V η̂ĜA = V (Ĝε̂A ◦ η̂ĜA ) = V (idĜA ) = idV ĜA = idGU A = GεU A ◦ ηGU A , sinceηGA is G-generating.
(2) The following theorem, applied to G = idX , provides a new proof of the equivalenceof conditions (1) and (3) in Theorem 21.24.
21.28 TAUT LIFT THEOREM
Let (A, U ) be a topological category over the base category X and (B, V ) be a concreteĜG

category over the base category Y. If A −−→ B is a functor and X −−→ Y is an adjoint
functor with V ◦ Ĝ = G ◦ U, then the following conditions are equivalent:(1) Ĝ sends U-initial sources into V-initial sources,(2) every adjoint situation (η, ε) : F
G : X → Y can be lifted along U and V toan adjoint situation (η̂, ε̂) : F̂Ĝ : A → B.

Proof:fi(1) ⇒ (2). For each B-object B consider the Ĝ-structured source SB = (B −−→ ĜAi )fi

consisting of all pairs (fi , Ai ) with Ai ∈ Ob(A) and B −−→ ĜAi ∈ M or(B). For eachfˆii ∈ I let F V B −−→ U Ai be the unique X-morphism with
V fi

Gfˆi

η

V B −−−→ GU Ai = V B −−V−B→ GF V B −−−→ GU Ai .
fˆi

f˜i

Then the U-structured source (F V B −−→ U Ai )I has a U-initial lift TB = (AB −−→ Ai )I .
By (1), ĜTB is V-initial. Since V SB = V ĜTB ◦ ηV B
VB
V fi



V ĜAiηV B/ GF V B


Gfˆi

GU Ai

GU AB


GU f˜iGU Ai

V ĜABV Ĝf˜i

V ĜAiη̂

there exists a unique B-morphism B −−B→ ĜAB with V η̂B = ηV B and SB = ĜTB ◦ η̂B .
To show that the Ĝ-structured arrow (η̂B , AB ) is Ĝ-generating and hence Ĝ-universal,
let ABrs// A be a pair of A-morphisms with Ĝr ◦ η̂B = Ĝs ◦ η̂B . Then GU r ◦ ηV B =V Ĝr ◦ ηV B = V (Ĝr ◦ η̂B ) = V (Ĝs ◦ η̂B ) = V Ĝs ◦ ηV B = GU s ◦ ηV B implies that

9th July 2006

Sec. 21]Topological categories

361

Ĝ : A → B is theU r = U s. Hence r = s by the faithfulness of U . If (η̂, ε̂) : F̂
adjoint situation determined by Ĝ and the Ĝ-universal arrows (η̂B , AB ), then V η̂B = ηV B
and U ◦ F̂ (B) = U (AB ) = F ◦ V (B) for each B-object B. Thus for each B-morphismfB −→ B the equality GU F̂ f ◦ηV B = V (ĜF̂ f ◦η̂B ) = V (η̂B ◦f ) = ηV B ◦V f = GF V f ◦ηV B
implies that U F̂ f = F V f . Hence U ◦ F̂ = F ◦ V .
(2) ⇒ (1). Let (η, ε) : F

G : X → Y be an adjoint situation, let the adjoint situa-fiĜ : A → B be a lift along U and V, and let S = (A −−→ Ai )I be ation (η̂, ε̂) : F̂gi
U-initial source. To show that ĜS is V-initial, let T = (B −−→ ĜAi )I be a B-source andĝihV B −→ V ĜA be a Y-morphism with V T = V ĜS ◦ h. For each i ∈ I let F̂ B −−→ Aiĥ

be the unique A-morphism with gi = Ĝĝi ◦ η̂B . Furthermore, let F V B −→ U A be theunique X-morphism with h = Gĥ ◦ ηV B .
GU
F AJ

JJ
JJ
JJ
JJ
JJ GU f
JJ i
JJ
V ĜA
O JJ
JJ
JJV Ĝfi
JJ
JJ
JJ
h
JJ
JJ
J$
J$
V gi
Gĥ
/
GU
VB
V: ĜAi
: Ai
uu
tt

u
t

u
ηV B 
V η̂B tttuuuutt V Ĝĝi
u

t

u

uu
 V ĜF̂ B
uuGU ĝi

u

uu
uu

u

u

u
uu
 

GF V BGU F̂ B

Then G(U fi ◦ ĥ) ◦ ηV B = GU ĝi ◦ ηV B implies that U fi ◦ ĥ = U ĝi . Since S is U-initial,h̃

there exists a unique A-morphism F̂ B −→ A with U h̃ = ĥ. Thus h∗ = Ĝh̃ ◦ η̂B is a
B-morphism with V h∗ = V Ĝh̃ ◦ V η̂B = GU h̃ ◦ ηV B = Gĥ ◦ ηV B = h. Consequently,h∗h
V (B −−→ ĜA) = V B −→ V ĜA. Thus ĜS is V-initial.


TOPOLOGICAL SUBCATEGORIES
21.29 DEFINITION
A full concrete subcategory (A, U ) of a concrete category (B, V ) is called initiallyclosed in (B, V ) provided that every V-initial source whose codomain is a family ofA-objects has its domain in A.
Dual Notion: finally closed subcategory.
21.30 PROPOSITION
An initially closed subcategory of a topological category is topological.

9th July 2006



362

Topological and Algebraic Categories[Chap. VI

21.31 PROPOSITION
For any full concrete subcategory (A, U ) of a topological category (B, V ) the following
conditions are equivalent:(1) (A, U ) is initially closed in (B, V ),(2) (A, U ) is concretely reflective in (B, V ).
Proof: (1) ⇒ (2) follows from Theorem 21.24 applied to the inclusion functor G.
fi(2) ⇒ (1). Let S = (B −−→ Ai )I be a V-initial source with each Ai in A, and let
fi

id

VB
B −−−
−→ A be an identity-carried A-reflection arrow for B. Then each A −−→ Ai is a
idV B
morphism. Thus, by initiality of S, it follows that A −−−−→ B is a morphism. Hence, byamnesticity of (B, V ), A = B. Thus B belongs to A.


21.32 PROPOSITION
If a topological category (A, U ) is a finally dense full concrete subcategory of (B, V ),then (A, U ) is concretely reflective in (B, V ).
fi

Proof: For each B-object B consider the source (B −−→ Ai ), consisting of all morphismsfifi

B −−→ Ai with Ai in A, and let S = (A −−→ Ai )I be a U-initial lift of the U-structured
fi

id

VB
source (V B −−→ U Ai )I . By Proposition 10.71, S is V-initial. Thus V B −−−−→ V A is amorphism; hence it is an identity-carried A-reflection arrow for B.


21.33 THEOREM
For a full concrete subcategory (A, U ) of a topological category (B, V ) the following
conditions are equivalent:(1) (A, U ) is topological,(2) there exists a concretely reflective subcategory (C, W ) of (B, V ) such that (A, U ) is
concretely coreflective in (C, W ),(3) there exists a concretely coreflective subcategory (C, W ) of (B, V ) such that (A, U )is concretely reflective in (C, W ),R(4) there exists a concrete functor (B, V ) −−→ (A, U ) that leaves each A-object fixed.
Proof: (1) ⇒ (2). Let (C, W ) be the full concrete subcategory of (B, V ) that consists offi

all B-objects B for which there exists a V-initial source (B −−→ Ai )I with domain B andcodomain in A. Then (C, W ) is initially closed in (B, V ), hence concretely reflective in(B, V ). Moreover, (A, U ) is initially dense in (C, W ), so that by the dual of Proposition
21.32 it is concretely coreflective in (C, W ).
R

R

1
2
(2) ⇒ (4). If (B, V ) −−→(C, W ) is a concrete reflector and (C, W ) −−→(A, U ) is aconcrete coreflector, then R = R2 ◦ R1 has the desired properties.

9th July 2006

Sec. 21]Topological categories

363

fi

(4) ⇒ (1). Let S = (X −−→ U Ai )I be a U-structured source. Considered as a Vfi

fistructured source it has a V-initial lift T = (B −−→ Ai )I . Then RT = (RB −−→ Ai )I is
easily seen to be a U-initial lift of S.
(1) ⇔ (3) follows from (1) ⇔ (2) by duality (21.9).



FIBRE-SMALL TOPOLOGICAL CATEGORIES
21.34 PROPOSITION
For fibre-small concrete categories (A, U ), the following conditions are equivalent:(1) (A, U ) is topological,
fi(2) every small structured source (X −−→ U Ai )I has a unique initial lift,fi(3) every small structured sink (U Ai −−→ X)I has a unique final lift.
Proof: (1) ⇒ (2) is obvious.
fi(2) ⇒ (1). Let (X −−→ U Ai )I be a structured source. For each i ∈ I there exists anfifi

initial lift Bi −−→ Ai of X −−→ U Ai . By fibre-smallness { Bithere exists a subset J of I with { Bjj ∈ J } = { Bii ∈ I } is a set. Thusfj

i ∈ I }. Let (A −−→ Aj )J be thefjinitial lift of (X −−→ U Aj )J . Then A ≤ Bj for each j ∈ J; hence A ≤ Bi for each i ∈ I.
fi

fi

Thus each A −−→ Ai is a morphism. Consequently, by Proposition 10.46, (A −−→ Ai )I is
initial. Uniqueness is immediate.
(1) ⇔ (3) follows from (1) ⇔ (2) by duality.



21.35 PROPOSITION
A fibre-small concrete category (A, U ) over a category X with products is topological if
and only if it satisfies the following conditions:(1) (A, U ) has concrete products,m(2) (A, U ) has initial subobjects, i.e., every structured X-monomorphism X −−→ U A
has a unique initial lift,(3) (A, U ) has indiscrete objects.
Proof: The conditions (1) – (3) are obviously necessary. To see that they are sufficient,fi

suppose that (X −−→ U Ai )I is a small structured source. Choose an element j0 withj0 6∈ I, use Aj0 to denote the indiscrete object with U Aj0 = X, let fj0 = idX , let
hfj i

pj

J = I ∪ {j0 }, and let (P −−→ Aj)J be a concrete product. Then X −−−→ U P is
hfj i

a structured X-section, and so has an initial lift A −−−→ P . By 10.53 and 10.45(1)9th July 2006

364

Topological and Algebraic Categories
fj[Chap. VIfi

the source (A −−→ Aj)J is initial. Since Aj0 is indiscrete, (A −−→ Ai )I is initial too.
Uniqueness follows from the uniqueness requirement in condition (2). Thus Proposition
21.34 implies that (A, U ) is topological.

21.36 PROPOSITIONfi
In a fibre-small topological category, a source (A −−→ Ai )I is initial if and only if there
fj

exists a subset J of I such that (A −−→ Aj)J is initial.
Proof: Immediate from the proof of (2) ⇒ (1) in Proposition 21.34.



21.37 PROPOSITION
Let (A, U ) be a full concrete subcategory of a fibre-small topological category (B, V ) overa category X with products. Then (A, U ) is concretely reflective in (B, V ) if and only if
it is closed under the formation of(a) products,(b) initial subobjects, and(c) indiscrete objects.
Proof: By Proposition 21.31, (A, U ) is concretely reflective in (B, V ) if and only if it isinitially closed in (B, V ). The latter is the case if and only if (A, U ) is closed under theformation of (a)–(c) in (B, V ). (Cf. the proof of Proposition 21.35.)
M-TOPOLOGICAL AND MONOTOPOLOGICAL CATEGORIES
Many familiar full subconstructs of Top, Unif , or Rel, defined by “separation axioms”,
e.g., Top0 , Top1 , Haus, Tych, Reg1 , HUnif, or Pos fail to be topological even though
they are “almost” so. This leads to a natural generalization of the concept of topologicalcategories over X that depends, however, on a distinguished factorization structure on
X. In particular, for constructs this leads to the concept of monotopological constructs,
since Set has only one interesting factorization structure:
21.38 DEFINITION
A concrete category (A, U ) over an (E, M)-category X is said to be M-topologicalprovided that every structured source in M has a unique initial lift. If M = MonoSources, the term monotopological is used.
21.39 EXAMPLES(1) The topological functors are precisely the Source-topological functors.
(2) The construct Pos is monotopological since it is closed under the formation of initialmono-sources in Rel.

9th July 2006

Sec. 21]Topological categories

365(3) The constructs Top0 , Top1 , Haus, Tych, and Reg1 (regular T1 topological spaces)
are monotopological (since they are closed under the formation of initial monosources in Top).
(4) The construct HUnif of separated uniform spaces is monotopological.
(5) Let PAlg(Ω) denote the construct of partial algebras of type Ω = (ni ), i.e., pairsof the form (X, {ωi }), where X is a set and the ωi : X ni → X are partial functions, i.e., functions ωi : Di → X, whose domains Di are subsets of X ni. Partialfalgebra homomorphisms (X, {ωi }) −→ (Y, {ωi0 }) are functions f : X → Y satisfying
f (ωi (g)) = ωi0 (f ni (g)) whenever ωi (g) is defined. Then PAlg(Ω) is monotopological.
(6) For concrete categories over 1, monotopological = topological.
(7) Unif , considered as a concrete category over Top is M-topological, where M is thecollection of initial mono-sources in Top.
21.40 M-TOPOLOGICAL CHARACTERIZATION THEOREM
A concrete category (A, U ) over an (E, M)-category X is M-topological if and only if(A, U ) is an E-reflective concrete subcategory of some topological category over X.
Proof: If (B, V ) is topological over X, then by Proposition 21.14, B is an (E, Minit )category. Thus if (A, U ) is an E-reflective subcategory of B, then by Theorem 16.8, A
is closed under Minit -sources in B. Hence (A, U ) is M -topological.
Conversely, let (A, U ) be M-topological. Define a concrete category (A∗ , U ∗ ) over
X as follows. A∗ -objects are triples (X, e, A) with X ∈ Ob(X), A ∈ Ob(A), ande
X −→ U A ∈ E. A∗ -morphisms from (X, e, A) to (X 0 , e0 , A0 ) are pairs (f, g) consistfg

ing of an X-morphism X −→ X 0 and an A-morphism A −→ A0 such that the diagram
X
e

f





UA

/ X0

g

e0

/ U A0commutes. A∗ -composition is defined coordinatewise. Let U ∗ : A∗ → X be defined by
(f,g)

f

U ∗ ((X, e, A) −−−→ (X 0 , e0 , A0 )) = X −→ X 0 .
Since E ⊆ Epi(X) (15.4), U ∗ is faithful. Hence (A∗ , U ∗ ) is a concrete category overfi

X. Moreover every U ∗ -structured source S = (X −−→ U ∗ (Xi , ei , Ai ))I has an initial lift
fi

e

e

m

i
i
that can be constructed as follows: Let X −−→ Xi −−→
U Ai = X −→ Y −−→
U Ai be anmi(E, M)-factorization and let (A −−→ Ai )I be a U-initial lift of the U-structured M-sourcemi(Y −−→U Ai )I . By the (E, M)-diagonalization-property, it is easily seen that the source(fi ,mi )((X, e, A) −−−−−→ (Xi , ei , Ai ))I is a U ∗ -initial lift of S. However, initial lifts need not
be uniquely determined. By Proposition 5.33 there exists an amnestic concrete category

9th July 2006

366

Topological and Algebraic Categories[Chap. VI

P(B, V ) and a surjective concrete equivalence (A∗ , U ∗ ) −−→ (B, V ). By Proposition 21.5,
E∗(B, V ) is topological. Now define a concrete full embedding (A, U ) −−→ (A∗ , U ∗ ) by
(f,f )f

E ∗ (A −→ B) = (U A, idA , A) −−−−→ (U B, idB , B).
Then E ∗ [A] is E-reflective in A∗ , since for each A∗ -object (X, e, A) the A∗ -morphism(e,idA )(X, e, A) −−−−−→ E ∗ A is an E ∗ [A]-reflection arrow. Thus
E∗

E

P

(A, U ) −−→ (B, V ) = (A, U ) −−→ (A∗ , U ∗ ) −−→ (B, V )is a full concrete embedding, whose image is an E-reflective subcategory of the topological category (B, V ).

21.41 REMARKS
E(1) Let (A, U ) −−→ (B, V ) be the concrete embedding constructed in the above proof.
Then the following hold, as can be seen easily:(a) E preserves M-initial sources. Thus E[A] is M-initially closed in (B, V ).
(b) If X is E-co-wellpowered and (A, U ) is fibre-small, then (B, V ) is fibre-small.
(c) If (A, U ) is topological, E needn’t be an isomorphism. For example, if Set isconsidered as a monotopological construct (A, U ) via the identity functor, then(B, V ) is the construct of equivalence relations.
(2) Whereas M-topological categories, i.e., E-reflective concrete subcategories of topological categories, still have a typical topological flavor, full reflective concrete subcategories of topological categories can have a completely different character. As wewill see in §25 the constructs Vec, Grp, and Rng are examples of this kind.
21.42 THEOREM
Let (A, U ) be a fibre-small concrete category over an E-co-wellpowered (E, M)-category
with products. Then (A, U ) is M-topological if and only if it satisfies the following
conditions:(1) (A, U ) has concrete products,m(2) (A, U ) has M-initial subobjects, i.e., every structured M-morphism X −−→ U A
has a unique initial lift.
Proof: If (A, U ) is M-topological, then it has concrete products, since X-products,
being extremal mono-sources, belong to M.
mi
Conversely, let (A, U ) satisfy the conditions (1) and (2). If S = (X −−→
U Ai )IQ
πj
is a small structured M-source and ( Ai −−→ Aj )I is a concrete product of (Aj )I ,

hmi i

hmi i

then the structured M-morphism X −−−→ U P has an initial lift A −−−→ P . Thus

9th July 2006

Sec. 21]m

Topological categories
hmi i

367

π

m

i
i
i
(A −−→Ai )I = (A −−−→ P −−→
Ai )I is an initial lift of S. If S = (X −−→
U Ai )I is alarge structured M-source, for each i ∈ I let

m

e

n

i
i
i
X −−→
U Ai = X −−→
Xi −−→
U Ai

n

n

i
i
be an (E, M)-factorization and let Bi −−→
Ai be an initial lift of Xi −−→
U Ai . Since X
ei
is E-co-wellpowered and (A, U ) is fibre-small, the structured source T = (X −−→
U Bi )I
is representable by a small source, i.e., there exists a subset J of I such that for eachi ∈ I there is a j(i) ∈ J and an isomorphism hi : Bj(i) → Bi with ei = hi ◦ ej(i) .
By Proposition 15.5(7), T belongs to M, and hence so does the small source T 0 =ejej(X −−→ U Bj )J . Thus T 0 has an initial lift (A −−→ Bj )J . It follows immediately thatthe source
ej(i)
mi
hi
ni
(A −−→Ai )I = (A −−−→ Bj(i) −−→
Bi −−→Ai )Iis an initial lift of S.



21.43 COROLLARY
A fibre-small construct is monotopological if and only if it has concrete products andinitial subobjects.

21.44 EXAMPLES(1) The constructs Vec and Grp have concrete products but fail to have initial subobjects.
(2) The constructs Met and Metc have initial subspaces but fail to have concrete
products.

Suggestions for Further ReadingHušek, M. S-categories. Comment. Math. Univ. Carolinae 5 (1964): 37–46.
Gray, J. W. Fibred and cofibred categories. Proceedings of the Conference on Categorical Algebra (La Jolla 1965), Springer, Berlin–Heidelberg–New York, 1966, 21–83.
Antoine, P. Étude élémentaire des catégories d’ensembles structurés. Bull. Soc. Math.
Belgique 18 (1966): 142–164 and 387–414.
Roberts, J. E. A characterization of initial functors. J. Algebra 8 (1968): 181–193.
Wyler, O. On the categories of general topology and topological algebra. Archiv Math.
22 (1971): 7–17.
Herrlich, H. Topological functors. Gen. Topol. Appl. 4 (1974): 125–142.
Hoffmann, R.-E. Topological functors and factorizations. Archiv Math.
1–7.
Herrlich, H. Initial completions. Math. Z. 150 (1976): 101–110.

9th July 2006

26 (1975):

368

Topological and Algebraic Categories[Chap. VI

Brümmer, G. C. L., and R.-E. Hoffmann. An external characterization of topological
functors. Springer Lect. Notes Math. 540 (1976): 136–151.
Hoffmann, R.-E. Topological functors admitting generalized Cauchy-completions.
Springer Lect. Notes Math. 540 (1976): 286–344.
Porst, H.-E. Characterizations of Mac Neille completions and topological functors.
Bull. Austral. Math. Soc. 18 (1978): 201–210.
Tholen, W. On Wyler’s taut lift theorem. Gen. Topol. Appl. 8 (1978): 197–206.
Herrlich, H. Categorical topology 1971-1981. General Topology and its Relations toModern Analysis and Algebra V. Proceedings of the Fifth Prague Topological Symposium 1981 (ed. J. Novak), Heldermann, Berlin, 1982, 279–383.
Brümmer, G. C. L. Topological categories. Topol. Appl. 18 (1984): 27–41.
Nakagawa, R. Categorical Topology. Topics in General Topology (eds. K. Morita and
J. Nagata), North Holland, Amsterdam, 1989, 563–623.

EXERCISES
21A. Existence of Indiscrete Structures
For transportable concrete categories, (A, U ), prove that the following conditions areequivalent:(1) (A, U ) has indiscrete structures,(2) U has an adjoint G with U ◦ G = idX ,(3) U has an adjoint G that is a full embedding,(4) there exists a Galois correspondence (U, G) : (X, idX ) → (A, U ).
21B. Fibre-Smallnessfi
Show that in fibre-small topological categories (resp. constructs) a sink (Ai −−→ A)I
is final (resp. is a final epi-sink) if and only if there exists a subset J of I such thatfj

(Aj −−→ A)J is final (resp. is a final epi-sink).
21C. Characterization of Topological Categories
Show that a concrete category (A, U ) over X is topological if and only if the following
conditions are satisfied:(a) (A, U ) is fibre-complete,ff(b) in (A, U ) each structured arrow X −→ U A has a unique initial lift B −→ A,ff(c) in (A, U ) each co-structured arrow U A −→ X has a unique final lift A −→ B.

9th July 2006

Sec. 21]Topological categories

369

21D. Non-topological Functors and Categories* (a) Consider the functor from Mod → Rng described as follows: Mod, the categoryof modules, has as objects all pairs (R, M ) consisting of a ring R and a left R(f,g)

module M , and as morphisms (R, M ) −−−→ (R0 , M 0 ) all pairs (f, g) consisting offga ring homomorphism R −→ R0 and an (abelian) group homomorphism M −→ M 0Gsatisfying the identity g(r · m) = f (r) · g(m). Let Mod −−→ Rng be the “forgetful”
functor, defined by
(f,g)

f

G((R, M ) −−−→ (R0 , M 0 )) = R −→ R0 .
Show that(1) Each small G-structured source has a G-initial lift.
F(2) There exists a functor Rng −−→ Mod that is simultaneously an adjoint and aco-adjoint for G.
(3) G is not faithful.
(4) G is not topological.
(b) Let A be the full subconstruct of Top consisting of all discrete and all indiscrete
spaces. Show thatfi(1) For each structured source (X −−→ (Xi , τi ))I there exists a largest A-structurefiτ on X making all (X, τ ) −−→ (Xi , τi ) continuous.
fi(2) For each structured sink ((Xi , τi ) −−→ X)I there exists a smallest A-structurefiτ on X making all (Xi , τi ) −−→ (X, τ ) continuous.
(3) A is fibre-complete.
(4) A has discrete and indiscrete structures.
(5) A is neither complete nor cocomplete.
(6) A is not topological.
/ • to • is not topological, even though(c) Show that the unique functor G from • o
G is faithful and each G-structured source has an initial lift.

21E. Concrete Co-adjoints and Galois Correspondences(a) Let X be a category consisting of a single object X and two morphisms idX and swith s2 = idX ; let A be the concrete category over X, consisting of two objects A0
and A1 and the following morphism sets:({idX }, if i = j
homA (Ai , Aj ) ={s},
if i 6= j;
and let G and F be concrete functors from A to A over X defined by GAi = Ai andF Ai = A1−i . Show that

9th July 2006

370

Topological and Algebraic Categories[Chap. VI(1) F is a co-adjoint for G.
(2) (F, G) is not a Galois correspondence between A and itself over X.
G* (b) Show that a concrete functor A −−→ B may have a concrete co-adjoint without being
part of a Galois correspondence (F, G). [Hint: Consider the concrete embedding
A −→ B constructed in Exercise 5E(d).] Cf. Theorem 21.24.
21F. Comma Categories(a) Let (A, U ) be topological over X, and let B be an A-object. Show that A ↓ B is
topological over X ↓ U B via the obvious forgetful functor induced by U .
(b) Consider
A↓B
T

W



X ↓ UB

/A


V

U

/X

where V, W, T are the obvious forgetful functors. Show that
(1) If U is topological, then so is T .
(2) W preserves initiality (i.e., sends T -initial sources into U-initial sources), but ingeneral does not preserve finality.
* 21G. Initial Completions
E

A full concrete embedding (A, U ) −−→ (B, V ) is called an initial completion of (A, U )provided that (B, V ) is topological and E[A] is initially dense in (B, V ). [Dual Notion:
E

1
final completion.] An initial completion (A, U ) −−→(B1 , U1 ) is said to be smallerE2than an initial completion (A, U ) −−→(B2 , U2 ) of (A, U ) — in symbols: E1 ≤ E2 ,E

provided that there exists a full concrete embedding (B1 , U1 ) −−→ (B2 , U2 ) with E2 =E ◦ E1 . Show that for initial completions E1 of (A, U ) the following hold:(a) The relation ≤ is reflexive and transitive.
(b) E1 ≤ E2 and E2 ≤ E1 hold simultaneously if and only if there exists a concreteH

isomorphism (B1 , U1 ) −−→ (B2 , U2 ) with E2 = H ◦ E1 .
(c) The following conditions are equivalent:(1) E1 ≤ E2 ,(2) there exists a (unique) initiality-preserving concrete functorG(B2 , U2 ) −−→ (B1 , U1 ) with E1 = G ◦ E2 ,E(3) there exist (unique) concrete functors (B1 , U1 ) −−→ (B2 , U2 ) andG(B2 , U2 ) −−→ (B1 , U1 ) with E2 = E ◦ E1 , E1 = G ◦ E2 and G ◦ E = id.

9th July 2006

Sec. 21]Topological categories

371* 21H. Mac Neille Completions
A completion of (A, U ) that is simultaneously an initial completion and a final completion is called a Mac Neille completion of (A, U ). Show that(a) [Uniqueness]

E

E

1
2
If (A, U ) −−→(B1 , V1 ) and (A, U ) −−→(B2 , V2 ) are Mac NeilleHcompletions, then there exists a concrete isomorphism (B1 , V1 ) −−→ (B2 , V2 ) with
E2 = H ◦ E1 .
E(b) [Minimality] If (A, U ) −−→ (B, V ) is a Mac Neille completion and if
E0(A, U ) −−→ (B0 , V 0 ) is a full concrete embedding of (A, U ) into a topological categoryH(B0 , V 0 ), then there exists a full concrete embedding (B, V ) −−→ (B0 , V 0 ) with V 0 =H ◦ V . In particular, Mac Neille completions of (A, U ) are the smallest initial
completions of (A, U ).
(c) [Construction] If (A, U ) is a full concrete subcategory of a topological category(B, V ), if (B0 , V 0 ) is the concretely reflective hull of A in (B, V ), and if (B00 , V 00 )is the concretely coreflective hull of A in (B0 , V 0 ), then (B00 , V 00 ) is a Mac Neillecompletion of (A, U ).
(d) [Existence] Let (A, U ) be an amnestic concrete category over X.
(1) If A is small, then (A, U ) has a Mac Neille completion. In particular, eachposet, considered as a concrete category over 1, has a Mac Neille completion (=its familiar Dedekind-Mac Neille completion).
(2) If (A, U ) is M-topological, then it has a Mac Neille completion.
(e) [Non-existence] Let C be a proper class. Order X = C × {0, 1} by(x, i) = (y, j)(x, i) ≤ (y, j) ⇔ or

i = 0 and j = 1 and x 6= y.
Then (X, ≤), considered as concrete category over 1, has no Mac Neille completion.
* 21I. Universal Initial Completions
E

An initial completion (A, U ) −−→ (B, V ) of (A, U ) is called a universal initial completion of (A, U ) provided that the following hold:(1) E preserves initiality,F(2) whenever (C, W ) is topological and (A, U ) −−→ (C, W ) is an initiality-preserving
concrete functor, then there exists a unique initiality-preserving concrete functor
G(B, V ) −−→ (C, W ) with F = G ◦ E.
Show thatEE12(a) [Uniqueness] If (A, U ) −−→(B1 , V1 ) and (A, U ) −−→(B2 , V2 ) are universal initialH

completions, then there exists a concrete isomorphism (B1 , V1 ) −−→ (B2 , V2 ) with
V2 = H ◦ V 1 .

9th July 2006

372

Topological and Algebraic Categories[Chap. VI(b) [Maximality] Universal initial completions of (A, U ) are the largest initiality-preserving initial completions of (A, U ).
(c) Every M-topological category has a universal initial completion.
(d) The construct Prox of (not necessarily T1 ) proximity spaces is a universal initial
completion (but not a Mac Neille completion) of HComp.
(e) If A is small and (A, U ) is amnestic, then (A, U ) has a universal initial completion.
In particular, if A is a finite set with n ≥ 2 elements, ordered by equality andconsidered as a concrete category over 1, then the universal initial completion of A
has 2n elements, whereas the Mac Neille completion of A has n + 2 elements.
(f) A proper class, ordered by equality and considered as concrete category over 1, has
a Mac Neille completion, but no universal initial completion.
* 21J. Essential Full Concrete Embeddings
Show that in the quasicategory CAT(X) of amnestic concrete categories and concreteEfunctors over X, a full concrete embedding (A, U ) −−→ (B, V ) is(a) essential [with respect to full concrete embeddings; cf. Definition 9.22(2)] if and only
if it is initially dense and finally dense,(b) a Mac Neille completion of (A, U ) if and only if it is an injective hull (with respect
to full concrete embeddings).
21K. M-Topological Functors
Let M be a conglomerate of sources in X. A functor G : A → X is called M-topologicalprovided that every G-structured source in M has a unique G-initial lift.
Prove that if X is an (E, M)-category, then every M-topological functor G : A → X isfaithful and uniquely transportable.
21L. Uniqueness of Forgetful Functors
Show that whenever (A, U ) and (A, V ) are topological constructs, then U and V are
naturally isomorphic.
21M. Topological Functors and Bimorphisms
U

Show that if A −−→ X is a topological functor, then E = U −1 [Iso(X)] satisfies:(1) E ⊆ Bimor(A) (= the class of bimorphisms in A),(2) if X is balanced, then E = Bimor(A),(3) A is an (E, −)-category and Aop is an (E, −)-category.
21N. Balanced Topological Categories
Show that a topological category (A, U ) over X is balanced if and only if X is balanced
and U is an isomorphism.

9th July 2006

22 Topological structure theorems
Universal algebra can be understood as the branch of mathematics in which “nice” subconstructs of the fundamental algebraic constructs Alg(Ω) of Ω-algebras are investigated.
Among these are the finitary varieties and quasivarieties. These “nice” subconstructs are
determined either internally (by means of suitable equations or implications) or externally (by means of certain closure properties). It turns out that an analogous situation
exists in the realm of topological categories over arbitrary (!) base categories. The roleof the fundamental topological categories is played by the functor-structured categories
Spa(T ) of T -spaces. “Nice” subcategories are determined either internally (by topological axioms) or externally (by suitable closure properties). In addition, duality provides
a dual structure theorem.

TOPOLOGICAL AXIOMS
Functor-structured categories Spa(T ) have been defined in Definition 5.40. By Example
21.8(2) they are fibre-small topological categories. Hence, by Proposition 21.30, each of
their initially closed (= concretely reflective) full concrete subcategories is a fibre-small
topological category as well. The main result of this section states that — conversely —every fibre-small topological category can be obtained in this way. Since, moreover, byTheorem 16.14 the isomorphism-closed E-reflective subcategories of an (E, M)-category
B are precisely the E-implicational subcategories of B, concretely reflective subcategories
of topological categories B are precisely the identity-carried implicational subcategories
of B. This justifies the following terminology (cf. Definition 16.12):
22.1 DEFINITION
Let (A, U ) be a concrete category over X.
p(1) Each identity-carried morphism P −→ P 0 is called a topological axiom in (A, U ).
An A-object A is said to satisfy the axiom p provided that A is {p}-injective; thatis, each A-morphism f : P → A is also an A-morphism f : P 0 → A.
(2) A full subcategory B of A is said to be definable by topological axioms in (A, U )provided that it is E-implicational in A, where E is a class of topological axioms in(A, U ); i.e., the objects in B are precisely those A-objects that satisfy each of theaxioms in E.
22.2 EXAMPLES(1) The construct Prost is definable by the following two topological axioms in Rel =Spa(S 2 ). Cf. Example 16.13(1).
(A1)

•

−→

•q

373

374

Topological and Algebraic Categories


•
(A2)

/•q


−→

•Q

[Chap. VI•@ /•q
@@
@@
 

•Q

(2) Let Bor denote the construct of bornological spaces, i.e., pairs (X, β), where X isa set and β is a bornology (i.e., a cover of X such that U ∈ β implies V ∈ β for all
V ⊆ U and U1 , U2 ∈ β implies U1 ∪ U2 ∈ β). Morphisms f : (X, β) → (X 0 , β 0 ) are
functions f : X → X 0 such that U ∈ β implies f [U ] ∈ β 0 . Bor is definable by thefollowing (proper) class of topological axioms in Spa(P):(A0 )(∅, ∅) → (∅, P(∅)),(A1 )({1}, ∅) → ({1}, P{1}),(AX,M,N )(X, {M, N }) → (X, P(M ∪ N ))for any set X and any M, N ⊆ X.

Bor cannot be defined by a set of topological axioms in Spa(P).
(3) Let Simp denote the construct of simplicial complexes, i.e., pairs (X, α), where X
is a set and α is a complex (i.e., a set of finite subsets of X that contains all subsetswith at most one member and has the property that U ∈ α implies V ∈ α for all
V ⊆ U ). Morphisms f : (X, α) → (X 0 , α0 ) are functions f : X → X 0 such thatU ∈ α implies f [U ] ∈ α0 . Let Pf in : Set → Set be the finite power-set functor thatassociates with each set X the set Pf in (X) of all finite subsets of X. Then Simp isdefinable by the following (proper) class of topological axioms in Spa(Pf in ):(A0 )(∅, ∅) → (∅, P(∅)),(A1 )({1}, ∅) → ({1}, P{1}),(AX,M )(X, {M }) → (X, P(M )) for every set X and every finite subset M of X.

(4) Let U : Set → Set be the ultrafilter functor defined as follows: For each set X,
U(X) is the set of all ultrafilters on X, and for each function f : X → Y , Uf (F) ={M ⊆ Yff −1 [M ] ∈ F }. Let F : Set → Set be defined by F (X −→ Y ) =Uf ×f

UX × X −−−−→ UY × Y . Then the construct PsTop of pseudotopological spaces isdefinable by one topological axiom in Spa(F ). An object (X, α) of Spa(F ) is given
by a set α ⊆ U(X) × X of pairs (F, x), where F is an ultrafilter and x is a point.
[Think of F → x if and only if (F, x) ∈ α.] Such an object is a pseudotopologicalspace if and only if for each point x, α contains the pair (ẋ, x) (where ẋ is the trivial
ultrafilter induced by x).
Thus PsTop is definable by the following topological axiom in Spa(F ):({1}, ∅) → ({1}, F {1}).
(5) Also Top is definable by topological axioms in Spa(F ). However, a proper class ofsuch axioms is needed.

9th July 2006

Sec. 22]Topological structure theorems

375

22.3 TOPOLOGICAL STRUCTURE THEOREM
For concrete categories (A, U ), the following conditions are equivalent:(1) (A, U ) is fibre-small and topological,(2) (A, U ) is concretely isomorphic to an initially closed full subcategory of a functorstructured category,(3) (A, U ) is concretely isomorphic to an isomorphism-closed concretely reflective subcategory of some functor-structured category,(4) (A, U ) is concretely isomorphic to a subcategory of a functor-structured category
Spa(T ) that is definable by topological axioms in Spa(T ).
Proof:(1) ⇒ (2). Let (A, U ) be a fibre-small topological category over X. Let T : X → Set be
the associated fibre-functor, defined on objects X by: T X = { A ∈ Ob(A) U A = X }ff

and on morphisms X −→ Y by: T f (A) = B ⇔ A −→ B is final. Then for each Aobject A the pair (U A , { B ∈ T U A B ≤ A }) is a T -space EA. The unique concrete
functor E : (A, U ) → Spa(T ), defined on objects as just described, is easily seen to
be a full embedding. Hence it remains to be shown that its image is initially closed infi

Spa(T ). To see this, consider an initial source ((X, α) −−→ EAi )I in Spa(T ), and letfifi(A −−→ Ai )I be the initial lift of the structured source (X −−→ U Ai )I in (A, U ). Thenby Example 10.42(4) the following conditions are equivalent for elements B of T X:(a) B ∈ α,(b) T fi (B) ≤ Ai for each i ∈ I,fi(c) B −−→ Ai is an A-morphism for each i ∈ I,(d) B ≤ A.
Thus (X, α) = EA.
(2) ⇒ (1) is immediate from Proposition 21.30 and the fact that each functor-structured
category is topological.
(2) ⇔ (3). Proposition 21.31.
(3) ⇔ (4) is immediate from Theorem 16.14, since each functor-structured category isan (Identity-Carried, Initial Source)-category.

22.4 COROLLARY
For a construct (A, U ) the following conditions are equivalent:(1) (A, U ) is fibre-small and topological,(2) (A, U ) can be concretely embedded in a functor-structured construct as a full subconstruct that is closed under the formation of:(a) products,

9th July 2006

376

Topological and Algebraic Categories[Chap. VI(b) initial subobjects, and(c) indiscrete objects.
Proof: This follows from Theorem 22.3 and Proposition 21.37.



22.5 REMARK
If (A, U ) is a topological category over X, then (Aop , U op ) is a topological category over
Xop . Thus, by the above structure theorem, there exists a functor T : Xop → Set, suchthat (Aop , U op ) is concretely isomorphic (over Xop ) to a full concrete subcategory of
Spa(T ) that is definable by topological axioms. Thus, by duality, (A, U ) is concretely
isomorphic over X to a full concrete subcategory of the functor-costructured category
Spa(T )op , definable by the dual of topological axioms, called topological co-axioms.
Such a representation of a topological category is sometimes more natural than the one
provided directly by the Topological Structure Theorem (22.3). Below we formulatethese results more explicitly and provide several examples.
22.6 DEFINITION
Let (A, U ) be a concrete category over X.
p(1) Each identity-carried morphism P 0 −→ P is called a topological co-axiom in(A, U ).71 An A-object A is said to satisfy the co-axiom p provided that A is {p}projective; that is, each A-morphism f : A → P is also an A-morphism f : A → P 0 .
(2) A full subcategory B of A is said to be definable by topological co-axioms in A
if there exists a class of topological co-axioms in A such that an A-object A satisfies
each of these co-axioms if and only if A ∈ Ob(B).
22.7 EXAMPLES(1) The construct Sym of symmetric relations is definable in Rel either(a) by the topological axiom
•→•

id

−−→

•↔•

or
(b) by the topological co-axiom


•o

71

/•q
O

•Q

−→


• @o / •O q
@
@@
@ 

•Q

Observe that f is a topological co-axiom if and only if it is a topological axiom. However, the concept“A satisfies the topological co-axiom f ” is dual to the concept “A satisfies the topological axiom f ”.

9th July 2006

Sec. 22]Topological structure theorems

377(2) The construct Top is definable by the following (proper) class of topological coaxioms in Spa(Q)op :(C1 ) ({0}, {{0}, ∅}) → ({0}, ∅),(C2 ) ({0, 1, 2, 3}, {{0, 1}, {0, 2}, {0}}) → ({0, 1, 2, 3}, {{0, 1}, {0, 2}}),SA ) (X, A ∪ { A}) → (X, A), for each set X and each family A of subsets of X.
(CX
22.8 THEOREM
For a concrete category (A, U ) the following are equivalent:(1) (A, U ) is fibre-small and topological,(2) (A, U ) is concretely isomorphic to a finally closed full subcategory of some functorcostructured category,(3) (A, U ) is concretely isomorphic to a full concretely coreflective subcategory of somefunctor-costructured category,(4) (A, U ) is concretely isomorphic to a subcategory of some functor-costructured category Spa(T )op that is definable by topological co-axioms in Spa(T )op .
D

M-TOPOLOGICAL STRUCTURE THEOREMS
22.9 M-TOPOLOGICAL STRUCTURE THEOREM
For concrete categories (A, U ) over an E-co-wellpowered (E, M)-category the following
conditions are equivalent:(1) (A, U ) is fibre-small and M-topological,(2) (A, U ) is concretely isomorphic to an M-initially closed full subcategory of a functorstructured category,(3) (A, U ) is concretely isomorphic to an isomorphism-closed E-reflective subcategoryof a functor-structured category,(4) (A, U ) is concretely isomorphic to an E-implicational subcategory of a functorstructured category.
Proof: (1) ⇒ (2). By Theorem 21.40 and Remark 21.41(1) there exists a full concreteEembedding (A, U ) −−→ (B, V ) of (A, U ) into a fibre-small topological category suchthat E[A] is M-initially closed in (B, V ). By Theorem 22.3 there exists a full concreteFembedding (B, V ) −−→ Spa(T ) of (B, V ) into some functor-structured category such
F ◦E

that F (B) is initially closed in Spa(T ). Thus (A, U ) −−−→ Spa(T ) is a full concrete
embedding such that (F ◦ E)[A] is M-initially closed in Spa(T ).
(2) ⇒ (1) is obvious.
(2) ⇔ (3) follows from Theorem 16.8 in view of Proposition 21.14(1).
(3) ⇔ (4) follows from Theorem 16.14.

9th July 2006



378

Topological and Algebraic Categories[Chap. VI

22.10 COROLLARY
For constructs (A, U ) the following conditions are equivalent:(1) (A, U ) is fibre-small and monotopological,(2) (A, U ) is concretely isomorphic to a full subconstruct of a functor-structured construct that is closed under the formation of products and initial subobjects,(3) (A, U ) is concretely isomorphic to an implicational subconstruct of a functor-structured construct.
Proof: This follows immediately from Theorem 22.9 and the observation that in a fibrefi

small construct, for each initial mono-source (A −−→ Ai )I , there exists a subset J of I
fjsuch that (A −−→ Aj )J is an initial mono-source (cf. Proposition 21.36).

Suggestions for Further Reading
Wyler, O. Top categories and categorical topology. Gen. Topol. Appl. 1 (1971): 17–28.
Menu, J., and A. Pultr. On categories determined by poset- and set-valued functors.
Comment. Math. Univ. Carolinae 15 (1974): 665–678.
Adámek, J., H. Herrlich, and G. E. Strecker. The structure of initial completions.
Cahiers Topol. Geom. Diff. 20 (1979): 333–352.
Herrlich, H Universal topology. Categorical Topology. Proceedings of the International
Conference, Toledo, Ohio, 1983 (ed. H. L. Bentley et. al.), Heldermann, Berlin,
1984, 223–281.
Adámek, J., and J. Reiterman. Topological categories presented by small sets of axioms. J. Pure Appl. Algebra 42 (1986): 1–14.

EXERCISES
22A. Characterization of Functor-Structured Categories
Show that a concrete category (A, U ) is concretely isomorphic to some functor-structured
category Spa(T ) if and only if the following conditions are satisfied:(1) (A, U ) is a topological category,(2) (A, U )-fibres are complete atomic Boolean algebras,f(3) if A −→ B is final in (A, U ), then the following hold:(a) if A is an atom (in its fibre), then so is B,(b) if B is discrete, then so is A.

9th July 2006

Sec. 22]Topological structure theorems

379

22B. Topological Theories
T

A topological theory in X is a functor X −−→ JCPos. For each topological theory
T in X denote by Top(T ) the concrete category over X whose objects are pairs (X, t)fwith X ∈ Ob(X) and t ∈ T (X), and whose morphisms (X, t) −→ (Y, s) are those Xf

morphisms X −→ Y that satisfy T f (t) ≤ s.
(a) Show that if T is a topological theory in X, then Top(T ) is a fibre-small topologicalcategory over X.
(b) Let (A, U ) be a topological category over X. Consider the so-called fibre-functor
T : X → JCPos that assigns to each X-object X the U-fibre of X, and to each
f

Tf

f

X-morphism X −→ Y the unique function T X −−→ T Y such that A −→ T f (A) isa final morphism in (A, U ) for each A ∈ T X. Prove that T is a topological theory
in X and that (A, U ) is concretely isomorphic to Top(T ).
(c) Show that if G : X → Set is a functor and P : Set → JCPos is the covariant
power-set functor, then P ◦ G is a topological theory in X and Top(P ◦ G) =Spa(G).
* 22C. Sets of Axioms
Prove that Top cannot be axiomatized by a set of axioms in a functor-structured category.
22D. Co-Axiomatization
Find a co-axiomatization of Prost in a functor-costructured category.

9th July 2006

23 Algebraic categories
In this section we introduce the concept of algebraic category, which is aimed at capturing
the intuitive concept of “algebras over a base category”. An attempt to describe algebraic
structures, via monads, has been presented in §20. For constructs this approach has beenquite successful. Unfortunately, however, besides being rather complicated, the conceptsof monadic functors and monadic categories have the severe deficiencies pointed out in§20.
Here we provide the concepts of essentially algebraic, algebraic, and regularly algebraic
functors and concrete categories. They are free from the above-mentioned deficiencies ofmonadic functors and categories. Moreover, as we will see in §24, the regularly algebraic
functors form the compositive hull of the regularly monadic functors.

ESSENTIALLY ALGEBRAIC FUNCTORS
Each of the following concepts, which under rather mild assumptions are equivalent (see17.13), captures the “essence of algebraicity” of a functor G:(1) G reflects isomorphisms,(2) G reflects limits,(3) G reflects equalizers,(4) G is faithful and reflects extremal epimorphisms,(5) mono-sources are G-initial.
Each of these concepts is too weak to have many striking consequences. However, when
any of them is combined with a suitably chosen factorization condition, such as (Generating, Mono-Source)-factorizability (which by itself has nothing to do with “algebraicity”),
a theory emerges that is surprisingly close to intuitive notions of what concrete categoriesof algebras should be. Just for convenience we will also require unique transportability.
This, however, is an inessential requirement.
23.1 DEFINITION
A functor is called essentially algebraic provided that it creates isomorphisms and is(Generating, Mono-Source)-factorizable.
23.2 PROPOSITION
For a uniquely transportable (Generating, Mono-Source)-factorizable functor
G : A → B, the following conditions are equivalent:(1) G is essentially algebraic,(2) G reflects isomorphisms,380

Sec. 23]

Algebraic categories

381(3) G reflects limits,(4) G reflects equalizers,(5) G is faithful and reflects extremal epimorphisms,(6) every mono-source in A is G-initial.
Proof: Immediate from Propositions 17.13 and 13.36.



23.3 PROPOSITION
Each essentially algebraic functor is a faithful adjoint functor.
Proof: Immediate from Propositions 23.2 and 18.3.
23.4 PROPOSITION
Essentially algebraic functors are closed under composition.



ESSENTIALLY ALGEBRAIC CATEGORIES
23.5 DEFINITION
A concrete category (A, U ) is called essentially algebraic provided that U is essentiallyalgebraic.
23.6 EXAMPLES(1) All monadic constructs, e.g., Alg(Ω), Vec, Grp, and HComp, are essentially algebraic. However, the monadic category over Pos exhibited in Example 20.47 is notessentially algebraic (since it is not cocomplete, cf. Theorem 23.11).
(2) The constructs BooSp, TfAb, Catof , and (Ban,O) are essentially algebraic, but
not monadic.
(3) Categories of topological algebras (e.g., topological groups, topological semigroups,etc.), considered as concrete categories over Top, are essentially algebraic.
(4) A topological category (A, U ) is essentially algebraic only if U is an isomorphism.
(5) A partially ordered class, considered as a concrete category over 1, is essentiallyalgebraic if and only if it has precisely one element.
23.7 PROPOSITION
In any essentially algebraic category, the embeddings are precisely the monomorphisms.
Proof: Immediate from Proposition 23.2.

9th July 2006



382

Topological and Algebraic Categories[Chap. VI

23.8 CHARACTERIZATION THEOREM FOR
ESSENTIALLY ALGEBRAIC CATEGORIESA concrete category (A, U ) is essentially algebraic if and only if the following conditions
are satisfied:(1) U creates isomorphisms,(2) U is adjoint,(3) A is (Epi, Mono-Source)-factorizable.
Proof:(a). If (A, U ) is essentially algebraic, then conditions (1) and (2) are clearly satisfiedfi(cf. 23.2 and 23.3). To show (3), let S = (A −−→ Ai )I be a source in A, and let
U fi

g

Um

i
U A −−−→ U Ai = U A −→ U B −−−→
U Ai

be a (Generating, Mono-Source)-factorization of the corresponding structuredgsource. Since by Proposition 23.2 each mono-source is U-initial, A −→ B is an
fi

g

m

i
Ai is an (Epi, Mono-Source)A-morphism. Hence A −−→ Ai = A −→ B −−→
factorization of S.

fi(b). Let conditions (1), (2), and (3) be satisfied, and let S = (X −−→ U Ai )I be au
structured source. If X −→ U A is a universal arrow, then for each i ∈ I there exists
gi
gi
mi
e
Ai is anan A-morphism A −−→ Ai with fi = (U gi ) ◦ u. If A −−→ Ai = A −→ B −−→fi(U e)◦uUmi
U Ai is(Epi, Mono-Source)-factorization, then X −−→ U Ai = X −−−−−→ U B −−−→a (Generating, Mono-Source)-factorization of S.


23.9 REMARKS(1) Condition (3) in the above theorem is surprising since it is a requirement on A andnot on U . In particular, the identity functor on a category A is essentially algebraic
if and only if A has (Epi, Mono-Source)-factorizations.
(2) Monadic categories satisfy conditions (1) and (2) of the above theorem, but may
fail to satisfy (3). Hence a monadic category (A, U ) is essentially algebraic if andonly if A has (Epi, Mono-Source)-factorizations. In particular, regularly monadic
categories are essentially algebraic.
23.10 COROLLARY
If (A, U ) is essentially algebraic, then A has coequalizers.
Proof: Immediate since (Epi, Mono-Source)-factorizable categories have coequalizers.
(Cf. Proposition 15.7 and Theorem 15.10.)

23.11 THEOREM
Every essentially algebraic functor

9th July 2006

Sec. 23]

Algebraic categories

383(1) detects colimits,(2) preserves and creates limits.
Proof: Let (A, U ) be an essentially algebraic category.
ci(1). Let D : I → A be a diagram, and let (U Di −−→K)I be a colimit of U ◦ D.

fj

Consider the structured source S = (K −−→ U Aj )J consisting of those structured
fj ◦ciarrows (fj , Aj ) with the property that U Di −−−−→ U Aj is an A-morphism for each
fj

U mj

g

i ∈ I. Let K −−→ U Aj = K −→ U A −−−−→ U Aj be a (Generating, Mono-Source)g◦ci
factorization of S. Since mono-sources in A are initial, each U Di −−−→ U A is ang◦ci
A-morphism. Hence (Di −−−→ A)I is a colimit of D.
ci

g

/ UA
y
FF
y
y
FF
fj yy
F
fj ◦ci FF#  |yyy U mj

U DiF

/K

U Aj

(2). As an adjoint functor, U preserves limits. To show creation, let D : I → A be a`idiagram and let L = (L −−→
U Di ) be a limit of the composite functor U ◦ D. If

g

`

Um

i
i
L −−→
U Di = L −→ U A −−−→U Di is a (Generating, Mono-Source)-factorization,

fi

m

i
then M = (A −−→
Di )I is a natural source for D. Let S = (B −−→ Di )I be an
arbitrary natural source for D. Then U S is a natural source for U ◦D. Consequently,fthere exists a morphism U B −→ L with U S = L ◦ f .
U BF

FF U f
FF j
FF
F#

`i
/ U Di
L
;
x
xx
x
g
xx
 xx U mi

f

UA

g◦f

Since M is a mono-source and, hence, is initial, U B −−−→ U A is an A-morphismwith S = M ◦ (g ◦ f ). Thus M is a limit of D. Since U preserves limits, U M is aglimit of U ◦ D. Hence L −→ U A is an isomorphism. Therefore, transportability of
U implies that U lifts limits. Hence, by Proposition 13.25, U creates limits.

23.12 PROPOSITION
If (A, U ) is essentially algebraic over X, then the following hold:(1) If X is (strongly) complete, then A is (strongly) complete.
(2) If X has coproducts, then A is cocomplete.

9th July 2006

384

Topological and Algebraic Categories[Chap. VI(3) If X is wellpowered, then A is wellpowered.
Proof: (1) follows from Theorem 23.11 and the fact that U , being adjoint, preservesmonomorphisms.
(2) follows from Theorem 23.11 and Corollary 23.10.
(3) follows from the fact that monomorphisms are initial and U preserves and reflectsthem.

23.13 COROLLARY
Every essentially algebraic construct is complete, cocomplete, and wellpowered.



23.14 PROPOSITION
For essentially algebraic categories, the following conditions are equivalent:(1) A is extremally co-wellpowered,(2) U is extremally co-wellpowered,(3) U is concretely co-wellpowered.
Proof: (1) ⇒ (2) follows from Proposition 18.11.
(2) ⇔ (3) is immediate by Proposition 23.7.
(3) ⇒ (1) is immediate by Remark 8.18(3).



23.15 THEOREM
If (A, U ) is a concretely co-wellpowered category over a strongly complete category X,
then the following conditions are equivalent:(1) (A, U ) is essentially algebraic,(2) U creates limits.
Proof: (1) ⇒ (2) follows from Theorem 23.11.
(2) ⇒ (1). By Theorem 13.19, A is strongly complete and U preserves strong limits.
Thus (1) follows from Theorem 17.11(2).

23.16 COROLLARY
Concretely co-wellpowered monadic categories over strongly complete categories are essentially algebraic.

23.17 THEOREM
Concrete functors between essentially algebraic categories are essentially algebraic.

9th July 2006

Sec. 23]

Algebraic categories

385

G

Proof: Let (A, U ) −−→ (B, V ) be a concrete functor between essentially algebraic categories over X. G reflects isomorphisms since V preserves and U reflects them. G
is faithful and amnestic since U is. G is transportable since U is transportable, V isamnestic, and B-monomorphisms are V-initial. Hence G creates isomorphisms. Letfi

S = (B −−→ GAi )I be a G-structured source. Consider a (Generating, Mono-Source)factorization
V fi
g
U mi
V B −−−→ U Ai = V B −→ U A −−−→U Ai
of the U-structured source (V B, (V fi , Ai )I ). Since U preserves and V reflects monoGmisources, (GA −−−→GAi )I is a mono-source, and hence is V-initial. Thus there exists aĝ

B-morphism B −→ GA with V ĝ = g and fi = Gmi ◦ ĝ for each i ∈ I. The G-structured
morphism (ĝ, A) is generating, since the U-structured morphism (V ĝ, A) is generating.
Hence
fi
ĝ
Gmi
B −−→ GAi = B −→ GA −−−→
GAi
is a (Generating, Mono-Source)-factorization of the G-structured source S.



23.18 COROLLARY
The forgetful functors Rng → Ab and Rng → Mon are essentially algebraic functors;
in particular, they are adjoint functors.


ALGEBRAIC FUNCTORS AND CATEGORIES
Since (Epi, Mono-Source)-factorizable categories are (Extremal Epi, Mono-Source) categories (15.10), there is a certain symmetry between mono-sources on the one hand andextremal epimorphisms on the other. Unfortunately, essentially algebraic functors, as
nice as they are otherwise, may not interact well with this symmetry. In particular,(1) they preserve mono-sources, but may fail to preserve extremal epimorphisms, e.g.,OU

Ban −−→ Set or Catof −−→ Set, cf. Example 7.72(5),(2) they detect wellpoweredness, but may fail to detect extremal co-wellpoweredness, cf.
Exercise 23I,
(3) mono-sources must be initial, but extremal epimorphisms may fail to be final, e.g,OU

Ban −−→ Set or Catof −−→ Set, cf. Example 8.11(4).
The following condition, which is quite often satisfied, restores a proper interaction withthis symmetry.
23.19 DEFINITION(1) A functor is called algebraic provided that it is essentially algebraic and preserves
extremal epimorphisms.
(2) A concrete category (A, U ) is called algebraic provided that U is algebraic.

9th July 2006

386

Topological and Algebraic Categories[Chap. VI

23.20 EXAMPLES(1) Each regularly monadic category and, in particular, each monadic construct is algebraic (cf. 20.30).
(2) Every full subconstruct of a monadic construct that is closed under the formation of
products and subobjects is algebraic. In particular, BooSp and TfAb are algebraic.
Neither, however, is monadic.
(3) The constructs Catof and (Ban, O) are essentially algebraic, but not algebraic[cf. 7.72(5)].
(4) Catof , considered as a concrete category over Alg(1, 1) via the functor that forgets
composition, is essentially algebraic and monadic but is not algebraic.
(5) TopGrp, considered as a concrete category over Top, is algebraic. The category of
Hausdorff semigroups, considered as a concrete category over Haus, is essentiallyalgebraic and monadic, but is not algebraic.
(6) The embedding Reg ,→ Top of the full subcategory of Top that consists of all regular spaces (the T1 -condition not being required) is essentially algebraic and monadic,
but is not algebraic.
23.21 PROPOSITION
Algebraic functors are closed under composition.



23.22 PROPOSITION
Each concrete functor between algebraic categories is algebraic.
Proof: This follows from Theorem 23.17 and the fact that (essentially) algebraic functors reflect extremal epimorphisms.

23.23 PROPOSITION
In algebraic categories every extremal epimorphism is final.
e

Proof: Suppose that A −→ B is an extremal epimorphism in an algebraic categoryfg(A, U ) over X. Let U B −→ U C be an X-morphism and A −→ C be an A-morphismwith U g = f ◦ U e. If
> BO
~
m
~
~
~~ e
/D
A@
@@
@@
n
g @@

e ~~

C

e

g

is an (ExtrEpi, Mono-Source)-factorization of the 2-source B ←− A −→ C (which exists
in view of Theorems 23.8 and 15.10), then U n ◦ U e = U g = f ◦ U e = f ◦ U m ◦ U e.
Since U is algebraic, U e is an (extremal) epimorphism in X, which implies that U n =9th July 2006

Sec. 23]

Algebraic categories

387

f ◦ U m. Since (U m, U n) is a mono-source, U m is a monomorphism in X, so that m is amonomorphism in A. Since e is an extremal epimorphism, m must be an isomorphism.
n◦m−1

Thus B −−−−−→ C is the unique A-morphism with U (n ◦ m−1 ) = f .

23.24 COROLLARY
Algebraic functors reflect regular epimorphisms.
Proof: This follows from Proposition 23.23, as in the proof of Proposition 20.30.



23.25 REMARK
Algebraic functors need not preserve regular epimorphisms. See Exercise 23J.
23.26 COROLLARY
In any algebraic category over a balanced category, the quotient morphisms are precisely
the extremal epimorphisms.

23.27 COROLLARY
If (A, U ) is algebraic over an extremally co-wellpowered category X, then A is extremallyco-wellpowered.

23.28 PROPOSITION
If (A, U ) is algebraic over X, then the following hold:(1) If extremal epimorphisms are closed under the formation of pullbacks (resp. products)
in X, then they are closed under the formation of pullbacks (resp. products) in A.
(2) If X has enough extremal projectives, then so does A.
Proof:(1). This follows from the fact that U preserves limits and preserves and reflects extremal
epimorphisms.
e(2). Let A be an A-object. Then in X there exists an extremal epimorphism P −→ U Auwith extremally projective domain P . If P −→ U P̂ is a universal arrow, then thereê
exists a unique A-morphism P̂ −→ A with e = U ê ◦ u. Since e is an extremal
epimorphism, so is U ê. Since U reflects extremal epimorphisms, ê is an extremalgepimorphism in A. It remains to show that P̂ is extremally projective. Let B −→ Cfbe an extremal epimorphism in A and P̂ −→ C be an arbitrary A-morphism. Sincek
U g is an extremal epimorphism in X, there exists some X-morphism P −→ U B
such that the square
u /
P
U P̂
k



UB
k̂



Ug

Uf

/ UC

commutes. If P̂ −→ B is the unique A-morphism with k = U k̂ ◦ u, then f = g ◦ k̂.

9th July 2006

388

Topological and Algebraic Categories[Chap. VI

23.29 COROLLARY
Any algebraic construct is complete, cocomplete, wellpowered, and extremally co-wellpowered. It has enough extremal projectives, and its extremal epimorphisms are closed
under composition and the formation of pullbacks and products.
23.30 CHARACTERIZATION THEOREM FOR
ALGEBRAIC CATEGORIESA concrete category (A, U ) is algebraic if and only if the following conditions are satisfied:(1) A is (Epi, Mono-Source)-factorizable,(2) U is adjoint,(3) U is uniquely transportable,(4) U preserves and reflects extremal epimorphisms.
Proof: The necessity of these conditions has already been established. Conversely, itneed only be shown that for any concrete category (A, U ) with the above properties,
U must reflect isomorphisms. This follows immediately from the fact that U reflects
extremal epimorphisms and monomorphisms.

23.31 THEOREM
For a concrete category (A, U ) over an (Extremal Epi, Mono-Source)-category X thefollowing conditions are equivalent:(1) (A, U ) is algebraic,(2) U is an adjoint functor and lifts (Extremal Epi, Mono-Source)-factorizations
uniquely,(3) U is an adjoint functor and creates (Extremal Epi, Mono-Source)-factorizations, i.e., U lifts these factorizations uniquely and reflects extremal epimorphisms
and mono-sources.
Proof: (1) ⇒ (3). Since, by Theorem 23.30 and Theorem 15.10, A has (Extremal Epi,
Mono-Source)-factorizations and U preserves these and is transportable, U lifts suchfactorizations. Since, moreover, U reflects extremal epimorphisms and mono-sources
and is amnestic, the liftings are uniquely determined.
(3) ⇒ (2). Trivial.
(2) ⇒ (1). Since U , being faithful, reflects epimorphisms and mono-sources, A has (Epi,Mono-Source)-factorizations. Observe that U reflects identities, i.e., any A-morphismfA −→ B with U f = idX must be idA , since the (Extremal Epi, Mono-Source)Uf

id

id

f

id

X
X
B
factorization U A −−→ U B = U A −−−→ X −−−→ U B has the lifts A −→ B −−−
→ B
f
idA
and A −−−→ A −→ B. Next consider the class E of A-morphisms that are mapped bye
U into extremal epimorphisms. To show that any element A −→ B of E is final, let

9th July 2006

Sec. 23]

Algebraic categories

f

389

g

U B −→ U C be an X-morphism and let A −→ C be an A-morphism with U g = f ◦ U e.
Then the (Extremal Epi, Mono-Source)-factorization in X
B
? O

UB
x< O

e 

U e xxx

idU B
xx
xx U e
/ UB
U AF
FF
FF
f
F
U g FF" 

k

 e
/
A?
B
??
??
g ??? f 

has a lift

UC

in A.

C

f

Since U k = idU B , we conclude that k = idB ; in particular, B = B. Hence B −→ C is an
A-morphism with U f = f . This finality-property implies that A is an (E, M)-category,where M is the conglomerate of all A-sources that are mapped via U to mono-sources
in X. Since U is faithful and adjoint, M consists precisely of the mono-sources in A.
Thus, by Proposition 15.8(1), E consists precisely of the extremal epimorphisms in A.
Hence U preserves and reflects extremal epimorphisms. It remains to be shown that U ishtransportable. If X −→ U A is a structured isomorphism, then the (Extremal Epi, MonoU (idA )

h−1

h

f

g

Source)-factorization U A −−−−−→ U A = U A −−−→ X −→ U A has a lift A −→ B −→ A.
gHence B −→ A is a morphism with U g = h. Since U reflects extremal epimorphisms
and monomorphisms, g is both, i.e., an isomorphism.

23.32 COROLLARY
A full concrete subcategory (A, U ) of an algebraic category (B, V ) is algebraic if and only
if A is reflective in B and contains with every source its (Extremal Epi, Mono-Source)factorization in B.
Proof: By Propositions 23.21 and 23.22, (A, U ) is algebraic if and only if the inclusion
functor A → B is algebraic. Since B has (Epi, Mono-Source)-factorizations and hence isan (Extremal Epi, Mono-Source)-category, the desired result follows immediately from
Theorem 23.31.

23.33 COROLLARY
Each extremally epireflective concrete subcategory of an algebraic category is algebraic.
23.34 EXAMPLES(1) The constructs Grp and Sgr are algebraic, but Grp is not closed under the formation of subobjects in Sgr.
(2) HComp and Top, considered as concrete categories over Top, are algebraic, but
HComp is closed neither under the formation of subobjects nor under the formationof extremal quotient objects in Top.
(3) Epireflective full concrete subcategories of algebraic categories need not be algebraic.
See Exercise 16A.

9th July 2006

390

Topological and Algebraic Categories[Chap. VI

REGULARLY ALGEBRAIC FUNCTORS AND CATEGORIES
Algebraic functors whose codomains have regular factorizations are particularly wellbehaved. They are closely related to regularly monadic functors — in fact, they form
the compositive hull of the regularly monadic functors (see Theorem 24.2). This suggests
the following terminology:
23.35 DEFINITIONU(1) A functor A −−→ X is called regularly algebraic provided that U is algebraic and
X has regular factorizations.
(2) A concrete category (A, U ) is called regularly algebraic provided that U is regularly algebraic.
23.36 PROPOSITION
Regularly monadic functors are regularly algebraic.
Proof: Immediate from Proposition 20.24 and Theorem 23.31.



23.37 COROLLARY
Monadic constructs are regularly algebraic.



23.38 CHARACTERIZATION THEOREM FORREGULARLY ALGEBRAIC CATEGORIESA concrete category (A, U ) over X is regularly algebraic if and only if the following
conditions are satisfied:(1) A has coequalizers,(2) X has regular factorizations,(3) U is adjoint,(4) U is uniquely transportable,(5) U preserves and reflects extremal epimorphisms.
Proof: By Theorem 23.30 it suffices to show that conditions (1) – (5) imply that A isfi

(Epi, Mono-Source)-factorizable. Let (A −−→ Ai )I be a source in A and let
U fi

e

m

i
U A −−−→ U Ai = U A −→ X −−→
U Ai

be a (RegEpi, Mono-Source)-factorization in X. Then e is a coequalizer of some pair
p1

Y

p2

//

ηY

U A of morphisms in X. Let Y −−→ U B be a universal arrow for Y . Then therep̂jcexist A-morphisms B −−→ A with pj = (U p̂j ) ◦ ηY for j = 1, 2. Let A −→ C be acoequalizer of p̂1 and p̂2 in A. For each i ∈ I the equalities U (fi ◦ p̂1 ) ◦ ηY = U fi ◦ p1 =mi ◦ e ◦ p1 = mi ◦ e ◦ p2 = U fi ◦ p2 = U (fi ◦ p̂2 ) ◦ ηY imply that fi ◦ p̂1 = fi ◦ p̂2 .

9th July 2006

Sec. 23]

Algebraic categories

391
n

i
Thus for each i ∈ I there exists a unique A-morphism C −−→
Ai with fi = ni ◦ c.
ni
It remains to be shown that (C −−→ Ai )I is a mono-source in A. Since U c ◦ p1 =f

U (c ◦ p̂1 ) ◦ ηY = U (c ◦ p̂2 ) ◦ ηY = U c ◦ p2 , there exists a unique X-morphism X −→ U Cwith U c = f ◦ e. Since, by (5), U c is an extremal (and hence a regular) epimorphism
miin X and (X −−→
Ai )I is a mono-source in X, there exists a unique diagonal d in thefollowing diagram:
/ UC
x
x
x
dx
e
x
U ni
xx

 |xx
/ U Ai
X

UA

Uc

mi

Thus f ◦d◦U c = f ◦e = U c implies that f ◦d = idU C and d◦f ◦e = d◦U c = e implies thatnimidd ◦ f = idX . Hence d is an isomorphism and U (C −−→
Ai )I = (U C −→ X −−→
U Ai )Ini
is a mono-source. By faithfulness, (C −−→
Ai )I is a mono-source.

23.39 COROLLARY
If (A, U ) is a regularly algebraic category, then A has regular factorizations and U
preserves and reflects regular epimorphisms.
Proof: In the proof of Theorem 23.38 regular factorizations have been constructed.
Since in categories with regular factorizations every extremal epimorphism is regular(cf. 14.14), U preserves and reflects regular epimorphisms.

23.40 COROLLARY
Concrete functors between regularly algebraic categories are regularly algebraic.



23.41 THEOREM
A construct (A, U ) is monadic if and only if it is algebraic and U reflects congruence
relations.
Proof: (1). Let (A, U ) be a monadic construct. By Corollary 23.37, (A, U ) is algebraic. That U reflects congruence relations follows easily from the facts that U creates
coequalizers of congruence relations (20.35) and that every point-separating 2-source in
A is a mono-source and, hence, is initial.
(2). Let (A, U ) be an algebraic construct and let U reflect congruence relations. By
Theorem 20.35 it suffices to show that U creates coequalizers of congruence relations.
p

Let A

q

//

Up

B be a pair of A-morphisms and let U A

Uq// U B −→ C be a congruencecĉ

fork in Set. Let B −→ Ĉ be a coequalizer of p and q in A. Since U reflects congrup

ence relations, Proposition 11.22(1) implies that A

q

//

ĉ

B −→ Ĉ is a congruence fork

in A. Since U ĉ is an extremal (and hence a regular epimorphism) in Set and since

9th July 2006

392

Topological and Algebraic Categories
Up

U preserves pullbacks, Proposition 11.22(2) implies that U A

Uq[Chap. VI// U B −−→ U Ĉ is aU ĉhcongruence fork in Set. Thus there is an isomorphism C −→ U Ĉ with U ĉ = h ◦ c.
h̃

By transportability, h lifts to an isomorphism C̃ −→ Ĉ in A. Thus
c̃

h̃−1

ĉ

B −→ C̃ = B −→ Ĉ −−−→ C̃c0c

is a lift of U B −→ C to a coequalizer of p and q in A. Let B −−→ C 0 be an arbitraryclift of U B −→ C. Then faithfulness implies that c0 ◦ p = c0 ◦ q. Thus there exists ankA-morphism C̃ −→ C 0 with c0 = k ◦ c̃. Application of U yields c = U k ◦ c. Thuskid
U k = idC . Amnesticity yields C̃ −→ C 0 = C̃ 0 −−→ C̃. Therefore U creates coequalizers
of congruence relations.

23.42 REMARKThe following diagram exhibits some of the results on the “algebraic character” of funcU

tors A −−→ X:
regularly monadic

creates regular factorizations ?
regularly algebraic- preserves regular epis?
algebraic- reflects regular epis

creates(Extremal Epi, Mono-Source)- factorizations@@@
R detects stability properties@of extremal epis

detects extremal co-wellpoweredness%& - preserves extremal epis

extremal epis are final 

mono-sources are initial ?
essentially algebraic
domain category is(Epi, Mono-Source)-factorizable

detects wellpoweredness







faithfulA- reflects extremal epis

@ @
R
A
@ @
creates limits (cf. 13.38)
A
@
A
@
A
@
A
@
AU
@
R detects colimits@
adjoint

9th July 2006

Sec. 23]

Algebraic categories

393

An algebraic construct (A, U ) is monadic iff U reflects congruence relations

Suggestions for Further Reading
Dubuc, E. Adjoint triangles. Springer Lect. Notes Math. 61 (1968): 69–91.
Huq, S. A. An interpolation theorem for adjoint functors. Proc. Amer. Math. Soc. 25(1970): 880–883.
Herrlich, H. Regular categories and regular functors. Canad. J. Math.
709–720.

26 (1974):

Tholen, W. Adjungierte Dreiecke, Colimites und Kan-Erweiterungen. Math. Ann. 217(1975): 121–129.
Bargenda, H., and G. Richter. Varietal hulls of functors. Quaest. Math.
121–158.

4 (1980):

Richter, G. Algebraic categories of topological spaces. Springer Lect. Notes Math. 962(1982): 263–271.
Herrlich, H. Essentially algebraic categories. Quaest. Math. 9 (1986): 245–262.
Bargenda, H. Algebraische Hüllen rechtsadjungierter Funktoren. Ph. D. dissertation,
Bremen University, 1987.
Im, G. B., and G. M. Kelly. Adjoint triangle theorems for conservative functors. Bull.
Austral. Math. Soc. 36 (1987): 133–136.

9th July 2006

394

Topological and Algebraic Categories[Chap. VI

EXERCISES
23A. Weakly Algebraic Functors
A functor G : A → B is called weakly algebraic provided that it is an adjoint andmono-sources in A are G-initial. Show that(a) G is weakly algebraic if and only if G is an adjoint functor that reflects swell epimorphisms. (Cf. Exercise 15A).
(b) Weakly algebraic functors(1) reflect extremal epimorphisms, swell epimorphisms, epimorphisms, and isomorphisms,(2) are faithful,(3) preserve and reflect mono-sources,(4) preserve and reflect limits,(5) detect wellpoweredness.
(c)(1) Every monadic functor is weakly algebraic.
(2) Every essentially algebraic functor is weakly algebraic.
(3) Every full reflective embedding is weakly algebraic.
(4) Weakly algebraic embeddings need not be full. [A = pointed sets and all maps,
B = pSet.](5) A poset, considered as a concrete category over 1, is weakly algebraic if andonly if it contains exactly one element.
(d) G : A → B is essentially algebraic if and only if G is uniquely transportable andweakly algebraic and A is (Epi, Mono-Source)-factorizable.
(e) Weakly algebraic functors are closed under composition.
(f) Concrete functors between weakly algebraic categories need not be weakly algebraic.
(Cf. Example 20.46.)(g) If G : A → B is weakly algebraic and B has (Epi, Mono-Source)-factorizations,then each co-adjoint for G preserves extremal epimorphisms.
23B. Essentially Algebraic Embeddings(a) Prove that the embedding E : A → B of a full subcategory is essentially algebraic
if and only if the following two conditions hold:(1) E is monadic; i.e., A is reflective and isomorphism-closed in B.
(2) A is (Epi, Mono-Source) factorizable.

9th July 2006

Sec. 23]

Algebraic categories

395(b) Show that embeddings of extremally epireflective, full, isomorphism-closed subcategories into (Epi, Mono-Source)-factorizable categories are essentially algebraic.
(c) Prove that embeddings of epireflective, full, isomorphism-closed subcategories into
(Epi, Mono-Source)-factorizable categories need not be essentially algebraic. [SeeExercise 16A(b).]23C. Essentially Algebraic vs. Monadic Functors
Prove that a monadic functor with an extremally co-wellpowered domain and strongly
complete codomain is essentially algebraic. [Cf. Theorem 23.16 and Proposition 18.11.]
23D. Algebraic Categories(a) Let (A, U ) be a concrete category over an (E, M)-category and let U lift (E, M)factorizations uniquely. Show that(1) each A-morphism f with U f ∈ E is a final epimorphism,(2) A is an (U −1 [E], U −1 [M])-category.
(b) Prove that algebraic functors reflect regular epimorphisms.
23.23.][Hint: Proposition

23E. Regularly Algebraic Functors and Categories
Show that(a) If B has regular factorizations, then the following are equivalent for any functor
G : A → B:
(1) G is regularly algebraic,(2) G is essentially algebraic and preserves regular epimorphisms,(3) G is essentially algebraic and preserves swell epimorphisms,(4) G is adjoint and creates regular factorizations,(5) A has regular factorizations and G is adjoint, uniquely transportable, and preserves and reflects regular epimorphisms.
(b) Regularly algebraic functors are closed under composition.
(c) Regular epireflective full isomorphism-closed subcategories of regularly algebraic categories are regularly algebraic.
(d) If B has regular factorizations and G : A → B is an embedding of a full, reflective
subcategory, then the following are equivalent:(1) G is regularly algebraic,(2) A contains with any source its regular factorization in B,(3) A contains with any morphism its regular factorization in B,

9th July 2006

396

Topological and Algebraic Categories[Chap. VI(4) A contains any B-object that is simultaneously a subobject of some A-object
and a regular quotient object of some A-object,(5) A is isomorphism-closed in B, and G preserves regular epimorphisms.
(e) If B has regular factorizations, then the following conditions are equivalent for any
functor G : A → B:
(1) G is regularly monadic,(2) G is regularly algebraic and monadic.
* 23F. Varietal Functors and Varietal Categories
A functor G : A → B is called varietal provided that G is regularly algebraic, B has
pullbacks, and G reflects congruence relations. A concrete category (A, U ) is called
varietal provided that U is varietal. Show that(a) A concrete category (A, U ) over a category with pullbacks and regular factorizations
is varietal if and only if (A, U ) is algebraic and U creates coequalizers of congruence
relations.
(b) A regularly algebraic category (A, U ) over a category with pullbacks is monadic if
and only if U creates those coequalizers of congruence relations that are retractions.
(c) Every varietal functor is regularly monadic.
(d) A construct is varietal if and only if it is monadic.
(e) The full embeddings TfAb ,→ Ab and BooSp ,→ Top are regularly monadic, but
are not varietal.
(f) Varietal functors are closed under composition.
(g) Concrete functors between varietal categories are varietal.
G(h) If A −−→ B is varietal, then A has pullbacks and regular factorizations.
23G. Extremal Epimorphisms and Quotients
Prove that for algebraic constructs, quotients = extremal epimorphisms = swell epimorphisms = strict epimorphisms = regular epimorphisms.
23H. Extremal Projectives
G

Show that if A −−→ B is algebraic with co-adjoint F and if P is extremally projective in
B, then F P is extremally projective in A.
* 23I. Extremal Co-wellpoweredness
Show that the following (legitimate quasi-)construct (A, U ) is essentially algebraic but
is not extremally co-wellpowered:A-objects are pairs (X, (λβ )), consisting of a set X and a family (λβ ), indexed by all
ordinals, of maps λβ : Xβ → X, whereXβ = { x ∈ X | λγ (x) = x for each λ < β }.

9th July 2006

Sec. 23]

Algebraic categories
f

397
f

A-morphisms (X, (λβ )) −→ (X 0 , (λ0β )) are functions X −→ X 0 that satisfy the following
conditions:(a) f [Xβ ] ⊆ Xβ0 , for all ordinals β,(b) x ∈ Xβ implies that f (λβ (x)) = λ0β (f (x)).
23J. Algebraic Functors Need Not Preserve Regular Epimorphisms
Show that the full embedding of• OOO // • @OOO @@OOO @@OOO @@
OO'

into
•

◦
~? _@_@@@@@
~
~
@@@@@
~~
@@@
~
~
• OOO // • @◦OOO @@~~OOO @@~OOO @@ ~~~OO'  ~•is algebraic, but does not preserve regular epimorphisms.
* 23K. Algebraic Hulls
G

Let (A, U ) be a concrete category over X. A concrete functor (A, U ) −−→ (B, V ) is called
an algebraic hull of (A, U ) provided that (B, V ) is algebraic and for every concreteG0functor (A, U ) −−→ (B0 , V 0 ) with algebraic (B0 , V 0 ) there exists a unique concrete functor
F(B, V ) −−→ (B0 , V 0 ) with G0 = F ◦ G.
Let X have regular factorizations and enough RegEpi-projective objects, let (A, U ) beamnestic, and let U be adjoint. Show that:(a) (A, U ) has an essentially unique algebraic hull.
(b) The algebraic hull of (A, U ) is a full embedding if and only if U reflects regularepimorphisms.
(c) The algebraic hull of (A, U ) is a full reflective embedding if and only if A has
coequalizers and U reflects regular epimorphisms.
G(d) If X has coproducts, (A, U ) is finitary and (A, U ) −−→ (B, V ) is the algebraic hullof (A, U ), then (B, V ) is finitary.
23L. Full Reflective Restrictions of Algebraic Functors
UShow that a functor A −−→ X into a category X that has regular factorizations andenough RegEpi-projective objects is a restriction of an algebraic functor B → X to afull, reflective subcategory A if and only if A has coequalizers and U is faithful, amnestic,
and reflects regular epimorphisms.
23M. Regularly Algebraic Functors Lift Regularity
Show that

9th July 2006

398

Topological and Algebraic Categories[Chap. VI

U(a) If X is regular and A −−→ X is regularly algebraic, then A is regular.
(b) Algebraic constructs are regular.
23N. Extremally Monadic Functors Are Algebraic
Show that the title statement is true.

9th July 2006

24 Algebraic structure theorems
In the preceding sections the “algebraic nature” of concrete categories has been describedin various ways:(a) via operations and implications resp. equations in §16, which leads to the conceptsof finitary quasivarieties and finitary varieties,(b) via monads in §20, which leads to the concepts of monadic categories and regularly
monadic categories,(c) via axiomatic descriptions in §23, which leads to the concepts of algebraic categories
and regularly algebraic categories.
Here we analyze the relationships among the various descriptions of “algebraicness”.

REGULARLY ALGEBRAIC VIAREGULARLY MONADIC FUNCTORS
24.1 PROPOSITIONU
If A −−→ X is a regularly algebraic functor, then the associated monad T is regular andKthe associated comparison functor A −−→ XT is an embedding whose image is a regular
epireflective subcategory of XT .
Proof: Let F be a co-adjoint for U . Then F preserves regular epimorphisms (dual of
Proposition 18.9) as does U (Corollary 23.39) and, hence, so does T = U ◦ F . Thus T
is a regular monad and (XT , U T ) is a regularly monadic category over X. By Proposition 20.30, XT has regular factorizations. By Corollary 20.44, the comparison functor
K
A −−→ XT is an isomorphism-closed full embedding. Hence (by Theorem 16.8) it
remains to be shown that (X, x) belongs to K[A] for any K-structured mono-sourcemimi((X, x) −−→
KAi )I . Consider such a mono-source. Then (X −−→
U Ai )I is a monosource in X, and the diagram
TX

T mi

x





X

/ T U Ai

mi

U εAi

/ U Ai
εA

Fm

e

n

i
i
i
commutes for each i ∈ I. Let (F X −−−→F U Ai −−−→Ai )I = (F X −→ B −−→
Ai )I be a
(RegEpi, Mono-Source)-factorization in A. Then U e is a regular epimorphism in X andU ni ◦ U e = U εAi ◦ T mi = mi ◦ x. Thus there exists a diagonal d in X that makes thediagram

/ UB
x
x
d xx
x
U ni
xx
x

 {xx
/ U Ai
X

TX

Ue

mi

399

400

Topological and Algebraic Categories[Chap. VI

commute. Since x is a retraction, so is d. Since (U ni )I is a mono-source, d is amonomorphism and thus an isomorphism. Since (mi )I is a mono-source, the equations
mi ◦ (d ◦ U εB ) = U (ni ◦ εB ) = U (εAi ◦ F U ni ) = U εAi ◦ T mi ◦ T d = mi ◦ (x ◦ T d) implydd ◦ U εB = x ◦ T d, i.e., that K(B) −→ (X, x) is a morphism in XT . Since U T is knowndto reflect isomorphisms, K(B) −→ (X, x) is an isomorphism in XT . Finally, since K[A]is isomorphism-closed in XT , the object (X, x) must belong to K[A].

24.2 DECOMPOSITION THEOREM FORREGULARLY ALGEBRAIC FUNCTORSU
For a functor A −−→ X into a category X that has regular factorizations, the following
conditions are equivalent:(1) U is (regularly) algebraic,(2) U is the restriction of a regularly monadic functor to a regular epireflective
subcategory,(3) U is a composite of regularly monadic functors.
Proof: (1) ⇒ (2) follows from Proposition 24.1.
(2) ⇒ (3) follows from Corollary 20.26.
(3) ⇒ (1) follows from Proposition 23.36 and Proposition 23.21.



24.3 COROLLARY
Algebraic constructs are precisely the regular epireflective subconstructs of monadic constructs.


FINITARY (QUASI)VARIETIES
Finitary (quasi)varieties are characterized axiomatically below. The crucial concept that
describes “finiteness” in the realm of algebra is the following:
24.4 DEFINITIONU(1) A functor A −−→ X is called finitary provided that it maps directed colimits intoci
epi-sinks; i.e., whenever (Di −−→K)i∈I is a colimit of a diagram D : I → A with
Ucischeme an up-directed partially ordered set, then (U Di −−−→U K)i∈I is an epi-sink
in X.

(2) A concrete category (A, U ) is called finitary provided that U is finitary.
(3) A monad T = (T, η, µ) is called finitary provided that T is finitary.
24.5 EXAMPLES(1) Topological functors preserve all colimits, and hence are finitary.

9th July 2006

Sec. 24]

Algebraic structure theorems

401(2) Constructs of the form Alg(Ω) are finitary. In fact the associated forgetful functors
Alg(Ω) → Set preserve directed colimits [see Example 11.28(4)].
Since by Proposition 13.30 every U −1 [Epi]-reflective concrete subcategory of a finitary concrete category (A, U ) is obviously finitary, all finitary (quasi)varieties are
finitary. For finitary varieties (A, U ) the functor U even preserves directed colimits.
However, for finitary quasivarieties this need not be the case. (See Exercise 24A(b).)(3) Neither HComp nor JCPos is finitary.
24.6 PROPOSITION
A monad T in Set is finitary if and only if the Eilenberg-Moore category (SetT , U T ) is
finitary.
Proof: Let T = (T, η, µ) be a monad in Set, and let (A, U ) = (SetT , U T ) be theEilenberg-Moore category.
(1). Suppose that T is finitary. Let D : I → A be a diagram in A with directed scheme,ci(C, c))Ob(I) be a colimit
denote Di by (Xi , xi ) for each I-object i, let ((Xi , xi ) −−→
kiof D, and let (Xi −−→K)Ob(I) be a colimit of U ◦ D. Then there exists a unique

f

f

e

m

function K −→ C with ci = f ◦ki for each i ∈ Ob(I). Let K −→ C = K −→ X −−→ C
be an (Epi, Mono)-factorization in Set. Since e is a retraction, so is T e. Since T is
Tk

T e◦T kii
finitary, (T Xi −−−→T K)Ob(I) is an epi-sink in Set. Thus (T Xi −−−−−→T X)Ob(I)is an epi-sink in Set. Since m is a monomorphism in Set, there exists (by Exercise
15K) a diagonal map x that makes the diagram

T XiF

T ci

FF
FF
FF
F
T (e◦ki ) #

TX

xi



/ TC
y<
y
yy
yyT m
y
y
c

x

XE
x;
EE
EEm
x
EE
xx
EE
x
 x
" 
/C
Xi
ci
e◦ki xxx

commute. By Lemma 20.11, (X, x) is a T-algebra, and thus
c

e◦k

m

i
i
((Xi , xi ) −−→(C, c))Ob(I) = ((Xi , xi ) −−−→(X, x) −−→ (C, c))Ob(I)is a (−, Mono)-factorization of a colimit in A. Since each colimit is an extremal
epi-sink, m must be an isomorphism, and hence
c

k

e

m

i
i
(Xi −−→C)Ob(I) = (Xi −−→
K −→ X −−→ C)Ob(I)is an epi-sink in Set.

9th July 2006

402

Topological and Algebraic Categories[Chap. VI(2). Let U T be finitary. Then T = U T ◦ F T is finitary, since F T (being a co-adjoint)preserves all colimits.

24.7 CHARACTERIZATION THEOREM FOR FINITARY VARIETIESFor any construct (A, U ) the following conditions are equivalent:(1) (A, U ) is a finitary variety,(2) (A, U ) is finitary and monadic.
Proof: (1) ⇒ (2) follows from Proposition 20.20 and Example 24.5(2).
(2) ⇒ (1). By Proposition 24.6 we can assume that (A, U ) = (SetT , U T ) for somefinitary monad T = (T, η, µ) in Set. For notational convenience we assume that n ={0, 1, . . . , n − 1} for each n ∈ . DefineN

I = { (i, n)n∈N and i ∈ T n } andΩ = (n(i,n) )(i,n)∈I ,where n(i,n) = n.

x
For each T-algebra (X, x) and each (i, n) ∈ I define a map ω(i,n): X n → X byxω(i,n)(g) = (x ◦ T g)(i) (where g ∈ X n is considered as a map from n = {0, 1, . . . , n − 1}xto X). Then E(X, x) = (X, (ω(i,n))(i,n)∈I ) is an Ω-algebra. To show that this construcEtion determines a full concrete embedding (A, U ) −−→ Alg(Ω), it needs to be verifiedfthat for any function (X, x) −→ (Y, y) between A-objects the following conditions areequivalent:f(a) (X, x) −→ (Y, y) is an A-morphism (i.e., a T -homomorphism),f(b) E(X, x) −→ E(Y, y) is an Ω-homomorphism.
First, assume that (a) holds. Then f ◦ x = y ◦ T f . Hence for any (i, n) ∈ I and anyg ∈ X n the following equations hold:xx(f ◦ ω(i,n))(g) = f (ω(i,n)(g)) = f ((x ◦ T g)(i)) = (f ◦ x ◦ T g)(i) = (y ◦ T f ◦ T g)(i)yyy= (y ◦ T (f ◦ g))(i) = ω(i,n)(f ◦ g) = ω(i,n)(f n (g)) = (ω(i,n)◦ f n )(g).
yxThus f ◦ ω(i,n)= ω(i,n)◦ f n , i.e., (b) holds.

Next, assume that (b) holds. Then for each n ∈following equations:N, g ∈ X n, and i ∈ T n, we have theyx(f ◦ x ◦ T g)(i) = f ((x ◦ T g)(i)) = f (ω(i,n)(g)) = ω(i,n)(f n (g))
y= ω(i,n)(f ◦ g) = (y ◦ T (f ◦ g))(i) = (y ◦ T f ◦ T g)(i).

Therefore (f ◦ x) ◦ T g = (y ◦ T f ) ◦ T g. Since the set X is a colimit of the directed systemTgof its finite subsets and since T is finitary, it follows that the sink (T n −−→ T X)n∈N,g∈X n
is an epi-sink in Set. Thus f ◦ x = y ◦ T f , i.e., (a) holds.

9th July 2006

Sec. 24]

Algebraic structure theorems

403

E

Thus (A, U ) −−→ Alg(Ω) is a full concrete embedding. Hence, by Corollary 16.19, it
suffices to show that E[A] is closed under the formation of mono-sources and of homj
momorphic images in Alg(Ω). Let ((X, (ω(i,n) )(i,n)∈I ) −−→ E(Xj , xj ))j∈J be a monosource in Alg(Ω). For each n ∈and g ∈ X n define a map f(n,g) : T n → X byf(n,g) (i) = ω(i,n) (g). Then for each j ∈ J the following equations hold:xj
xj(mj ◦ f(n,g) )(i) = mj (ω(i,n) (g)) = ω(i,n)(mnj (g)) = ω(i,n)(mj ◦ g) = (xj ◦ T mj ◦ T g)(i).

N

Thus mj ◦ f(n,g) = xj ◦ T mj ◦ T g.

mj

Since (X −−→ Xj )j∈J is a mono-source andTg(T n −−→ T X)n∈N,g∈X n is an epi-sink (as seen earlier), this implies (see Exercise 15K)
that there exists a diagonal map x that makes the diagram
Tg

/ TX
z
z
x zz
f(n,g)
xj ◦T mj
z
z

 }zz
/ Xj
X

Tn

mj

N

commute (for every n ∈ and every g ∈ X n ). By Lemma 20.11, (X, x) is a T-algebra.
mj
mj
Thus both (E(X, x) −−→ E(Xj , xj ))j∈J and ((X, (ω(i,n) )(i,n)∈I ) −−→ E(Xj , xj ))j∈J are
mono-sources in Alg(Ω). Thus amnesticity and the initiality of mono-sources in Alg(Ω)
imply that E(X, x) = (X, (ω(i,n) )(i,n)∈I ). Hence E[A] is closed under the formationcof mono-sources in Alg(Ω). Finally let E(X, x) −→ (Y, (ω(i,n) )(i,n)∈I ) be a surjectiveΩ-homomorphism. Then c is a regular epimorphism and thus a coequalizer of its conp

gruence relation P// E(X, x) in Alg(Ω). Since pullbacks are mono-sources, the aboveq

implies that P belongs to E[A], i.e., there exists a T-algebra (Z, z) with E(Z, z) = P .
pSince U (Z, z)q// U (X, x) −→ Y is a congruence fork and hence a split fork in Set,ccand since, by Theorem 20.17, U = U T creates coequalizers of split forks, U (X, x) −→ Yccan be lifted via U to a coequalizer (X, x) −→ (Y, y) of (Z, z)cpq// (X, x) in A. Thusc

E(X, x) −→ E(Y, y) and E(X, x) −→ (Y, (ω(i,n) )(i,n)∈I ) are surjective morphisms in
Alg(Ω). Hence amnesticity and finality of surjective (= regular epi) morphisms in
Alg(Ω) imply that E(Y, y) = (Y, (ω(i,n) )(i,n)∈I .

24.8 REMARK
In view of the above theorem, the Characterization Theorem for Monadic Constructs(20.35), as well as Theorem 23.41, provides axiomatic descriptions of finitary varieties.
24.9 CHARACTERIZATION THEOREM FOR
FINITARY QUASIVARIETIESFor constructs (A, U ) the following conditions are equivalent:(1) (A, U ) is a finitary quasivariety,(2) (A, U ) is finitary and algebraic,

9th July 2006

404

Topological and Algebraic Categories[Chap. VI(3) (A, U ) is a regular epireflective subconstruct of some finitary variety,(4) (A, U ) is a regular epireflective subconstruct of some finitary monadic construct.
Proof: (1) ⇔ (3) is immediate by Corollary 16.19.
(3) ⇔ (4) follows from Theorem 24.7.
(2) ⇒ (4). Let (A, U ) be algebraic. If T is the associated monad, then by PropositionK

24.1 the associated comparison functor (A, U ) −−→ (SetT , U T ) is the embedding of aregular epireflective subcategory. If U is finitary, then so is T and hence, by Proposition
24.6, (SetT , U T ) is finitary as well.
(4) ⇒ (2). By Theorem 23.41 a regular epireflective subconstruct (A, U ) of a monadic
construct (B, V ) is algebraic. If (B, V ) is finitary, then so is (A, U ) in view of theconstruction of colimits in reflective subcategories (see Proposition 13.30) and the factthat U preserves regular epimorphisms.

24.10 REMARK
Because of the above theorem, Theorems 23.30, 23.31, and 23.38 provide axiomatic
descriptions of finitary quasivarieties.

VARIETIES AND QUASIVARIETIES
24.11 REMARKIn the characterization theorems 24.7 and 24.9 the concept finitary plays an importantrole. This, however, does not mean that the concepts algebraic and finitary are closely
related — in fact, they are not related at all. The correspondence is due to the fact thatin the definition of Alg(Ω) the operations were assumed to be finitary. If this restriction
is dropped (which can be done naturally in two steps), Theorems 24.7 and 24.9 can begeneralized to infinitary algebras. Whereas the first step is straightforward, the second
one involves tricky problems of size:
First Generalization: For any family Ω = (ni )i∈I of cardinal numbers ni , indexed by a set
I, one may define Alg(Ω) as in Example 3.3(2)(e). Bounded (quasi)varieties can be
defined as constructs that are concretely isomorphic to equational (resp. to implicational)
subconstructs of some Alg(Ω). Constructs (A, U ) can be called bounded provided that
there exists some infinite cardinal number k such that U maps k-directed colimits72
into epi-sinks. Then the following two theorems can be proved in a way completely
analogous to the proofs of Theorems 24.7 and 24.9:
Theorem A
Bounded varieties are precisely the bounded monadic constructs.
72

A partially ordered set is called k-directed provided that each subset of cardinality less than k has
an upper bound. Colimits of those diagrams whose schemes are k-directed partially ordered sets arecalled k-directed colimits.

9th July 2006

Sec. 24]

Algebraic structure theorems

405

Theorem B
Bounded quasivarieties are precisely the bounded algebraic constructs.
Second Generalization: For any family Ω = (ni )i∈I of cardinal numbers, indexed bysome class I, one may define Alg(Ω) as in Example 3.3(2)(e). However several problems
arise:(a) Alg(Ω) will generally be a quasiconstruct only — often an illegitimate one.
(b) In the illegitimate case, Alg(Ω) fails to have free objects, so that a careful redefinitionof “equation” is needed. However, this is possible under suitable assumptions on theset theory involved.
(c) Legitimate equational or implicational subquasiconstructs of some Alg(Ω) may failto have free objects, as the constructs CBoo, CLat, and Λ-JCPos demonstrate [cf.
Examples 8.23(7) and (8)].
However, the following result holds:
Theorem C
A construct is monadic (resp. algebraic) if and only if it is concretely isomorphic toa concretely co-wellpowered equational (resp. implicational) subquasiconstruct of some
Alg(Ω).
This justifies the following terminology:
24.12 DEFINITION
Monadic constructs are called varieties. Algebraic constructs are called quasivarieties.
24.13 EXAMPLES(1) TConv is a bounded variety. For every regular infinite cardinal number k, theconstruct that consists of those partially ordered sets for which each subset of cardinality less than k has a join (where morphisms are those functions between themthat preserve such joins) is a bounded variety.
(2) Fram, JCPos, and HComp are (unbounded) varieties.
(3) BooSp is an (unbounded) quasivariety.
(4) CBoo, CLat, and Λ-JCPos fail to be quasivarieties.
24.14 REMARK
Constructive descriptions of essentially algebraic constructs by means of monads orby means of partial operations and equations are possible, but are more complicated.
Roughly speaking, a co-wellpowered construct (A, U ) is essentially algebraic if and only
if U can be expressed, as described in Exercise 20J, as a (possibly infinite) composite ofmonadic functors. Precise formulations of such structure theorems are beyond the scopeof this book. The description of the essentially algebraic construct Catof by meansof partial operations and equations, given in Exercise 24D, may however provide someinsight.

9th July 2006

406

Topological and Algebraic Categories[Chap. VI

Suggestions for Further Reading
Birkhoff, G. On the structure of abstract algebras. Proc. Cambridge Phil. Soc. 31(1935): 433–454.
Lawvere, F. W. Functorial semantics of algebraic theories. Ph. D. dissertation, Columbia University, 1963.
Isbell, J. R. Subobjects, adequacy, completeness and categories of algebras. Rozprawy
Mat. 36 (1964): 1–33.
Linton, F. E. J. Some aspects of equational categories. Proceedings of the Conference
on Categorical Algebra (La Jolla, 1965), Springer, Berlin–Heidelberg–New York,
1966, 84–95.
Felscher, W. Kennzeichnung von primitiven und quasiprimitiven Kategorien von Algebren. Archiv Math. 19 (1968): 390–397.
Felscher, W. Birkhoffsche und kategorische Algebra. Math. Ann. 180 (1969): 1–25.
Porst, H. E. On underlying functors in general and topological algebra. Manuscr. Math.
20 (1977): 209–225.
Adámek, J., H. Herrlich, and J. Rosický. Essentially equational categories. Cahiers
Topol. Geom. Diff. 29 (1988): 175–192.
Herrlich, H. Remarks on categories of algebras defined by a proper class of operations.
Quaestiones Math. 13 (1990): 385–393.

EXERCISES
24A. Directed Colimits in Finitary Quasivarieties(a) Show that the forgetful functor of a finitary variety preserves directed colimits.
U(b) Let A be the epireflective hull of all finite abelian groups in Ab and let A −−→ Set
be the associated forgetful functor. Show that (A, U ) is a finitary quasivariety, butthat U does not preserve directed colimits. [Hint: Consider the directed diagram of
all finite subgroups of / .]QZ

* 24B. Finitary Quality of Extremally Epireflective Concrete Subcategories(a) Show that an extremally epireflective concrete subcategory of a finitary algebraic
category over X is finitary algebraic.
(b) Construct a non-finitary extremally epireflective concrete subconstruct of some finitary construct.
24C. Cat and Ban(a) Show that the construct Cat is finitary.
(b) Show that the construct (Ban, O) is not finitary.

9th July 2006

Sec. 24]

Algebraic structure theorems

407

24D. Categories and Banach Spaces as Equational Partial Algebras(a) Let I be the ordered set {1, 2, 3} and let Ω = (ni )i∈I be defined by n1 = n2 = 1 andn3 = 2. Let (B, V ) be the full subconstruct of PAlg(Ω) (21.39) consisting of thosepartial Ω-algebras (X, (ωi )i∈I ) that satisfy the conditions(1) domain ω1 = domain ω2 = X = { x ∈ X(2) domain ω3 = { (x, y) ∈ X 2

x = x },

ω1 (x) = ω2 (y) },(3) ω1 ◦ ω1 = ω2 ◦ ω1 = ω1 ,(4) ω2 ◦ ω2 = ω1 ◦ ω2 = ω2 ,(5) ω1 (ω3 (x, y)) = ω1 (y),(6) ω2 (ω3 (x, y)) = ω2 (x),(7) ω3 (x, ω3 (y, z)) = ω3 (ω3 (x, y), z),(8) ω3 (x, ω1 (x)) = x,(9) ω3 (ω2 (x), x) = x.
HShow that the concrete functor (B, V ) −−→ (Catof , U ), defined by H(X, (ωi )i∈I ) =(X, ω3 ), is a concrete isomorphism.
(b) Provide a description of (Ban, O) analogous to the one given in (a) for Catof .

9th July 2006

25 Topologically algebraic categories
In this section we investigate a common generalization of topological and algebraic functors, namely, functors that(a) forget some topological structure, e.g., Top → Set, or(b) forget some algebraic structure, e.g., Grp → Set, or(c) simultaneously forget some topological and some algebraic structure, e.g.,
TopGrp → Set.
Such functors share several pleasant properties. In particular they are adjoint functorsthat detect limits and colimits. Also, they often can be decomposed into topological andalgebraic functors, as the following diagram suggests:
/ Grp
KKK
KKK
KKK

% 
/ Set
Top

TopGrpK

Throughout this section we find it slightly more convenient to treat the more flexible
form of functors by requiring neither transportability nor amnesticity.

TOPOLOGICALLY ALGEBRAIC FUNCTORS AND CATEGORIES
25.1 DEFINITION(1) A functor is called topologically algebraic provided that it is (Generating, Initial
Source)-factorizable.
(2) A concrete category (A, U ) is called topologically algebraic provided that U is
topologically algebraic.
25.2 EXAMPLES(1) Topological (even M -topological) functors are topologically algebraic.
(2) Essentially algebraic functors are topologically algebraic.
(3) Full reflective embeddings are topologically algebraic. However, nonfull reflective
embeddings need not be topologically algebraic. Consider, e.g., the natural injectionof the preordered set
• _@

@@
@@
@@

•

?•
~~
~
~~
~~

• _@o
into the preordered set

both considered as categories. Cf. Exercise 13K.

408

/•
@@
~?
~
@@
~
@@ ~~~ ~•Sec. 25]

Topologically algebraic categories

409(4) Equivalences are topologically algebraic.
(5) The constructs TopGrp and HausVec (of Hausdorff topological vector spaces) are
topologically algebraic.
(6) A partially ordered set, considered as a concrete category over 1, is topologicallyalgebraic if and only if it is a complete lattice. Hence adjoint functors may fail badly
to be topologically algebraic.
(7) Monadic constructs are topologically algebraic. However, the monadic functor
Alg(T ) −→ Pos, exhibited in Example 20.47, fails to be topologically algebraic
since it does not detect colimits (see Theorems 25.11 and 25.14).
25.3 PROPOSITION
Every topologically algebraic functor is faithful and adjoint.
Proof: Immediate from Corollary 17.16 and Proposition 18.3.



25.4 DEFINITION
A structured arrow (e, A) in a concrete category (A, U ) is called strongly generating
provided that U has the ({(e, A)}, Initial Source)-diagonalization property.
25.5 PROPOSITION
Strongly generating structured arrows are generating.
eProof: Let X −→ U A be strongly generating and let A
U f1 ◦ e = U f2 ◦ e = f . The source B

m1
m2

//

f1
f2

//

B be A-morphisms with

B , defined by m1 = m2 = idB , is initial by

Proposition 10.59. Hence there exists a diagonal d/ UAxx
U d xx
U fi
f
xx

 |xx
/ UB
UB

X

e

U mi

which implies that f1 = m1 ◦ d = m2 ◦ d = f2 .



25.6 CHARACTERIZATION THEOREM FOR
TOPOLOGICALLY ALGEBRAIC CATEGORIESFor concrete categories (A, U ) the following conditions are equivalent:(1) (A, U ) is topologically algebraic,(2) U is a (Strongly Generating, Initial Source)-functor,(3) U is an (E, M)-functor for some E and M, such that every identity-carried structured morphism belongs to E,

9th July 2006

410

Topological and Algebraic Categories[Chap. VI(4) U is an (E, M)-functor for some E and M, such that every source in M is initial,(5) U is an adjoint functor and A is (Epi, Initial Source)-factorizable.
Proof: (1) ⇒ (2). It need only be shown that U is (Strongly Generating, Initial Source)fj

fifactorizable. Let S = (X −−→ U Ai )I be a structured source and let S = (X −−→ U Aj )J
be the smallest structured source with the following properties:(a) S contains S, i.e., I ⊆ J and (fi , Ai )I is the corresponding restriction of (fj , Aj )J ,g(b) if (f, A) belongs to S and A −→ B is an A-morphism, then (g ◦ f, B) belongs to S,fm(c) if (A −−−k→ Ak )K is an initial source and X −→ U A is a structured arrow such thateach (mk ◦ f, Ak ) belongs to S, then (f, A) belongs to S.
fj

U gj

e

Let X −−→ U Aj = X −→ U A −−−→ U Aj be a (Generating, Initial Source)-factorizationeof S. Then, by condition (c), the structured arrow X −→ U A belongs to S. This implies
gjethat X −→ U A is strongly generating. Also, the initiality of (A −−→ Aj )J immediatelygi
implies the initiality of the restriction (A −−→ Ai )I . Hence
fi

U gi

e

X −−→ U Ai = X −→ U A −−−→ U Ai

(i ∈ I)is the desired factorization of S.
idX(2) ⇒ (3) is immediate since every identity-carried structured arrow X −−−→ U A isstrongly generating.

(3) ⇒ (4). Let U be an (E, M)-functor such that every identity-carried structured arrowmibelongs to E. It suffices to show that every source (A −−→
Ai )I in M is initial. Let
mi ◦f

f

U B −→ U A be an X-morphism such that all B −−−−→ Ai are A-morphism. Since(idU B , B) belongs to E, there exists a diagonal d:
idU B

/ UB
x
x
d xx
x
f
mi ◦f
xx
x

 |x
/ U Ai
UA

UB

mi

f

d

Thus B −→ A = B −→ A is an A-morphism.
(4) ⇒ (5). Let U be an (E, M)-functor such that every source in M is initial. It follows
fifrom Proposition 18.3 that U is adjoint. Let S = (A −−→ Ai )I be a source in A, and let
U fi

Um

e

i
U A −−−→ U Ai = U A −→ U B −−−→U Ai be an (E, M)-factorization of U [S], consideredmie
as a structured source. Since (B −−→ Ai )I is initial, U A −→ U B is an A-morphism.

fi

e

m

i
Hence A −−→ Ai = A −→ B −−→
Ai is an (Epi, Initial Source)-factorization of S.

fiu(5) ⇒ (1). Let S = (X −−→ U Ai )I be a structured source, and let X −→ U A begi
a universal arrow. Then for each i ∈ I there exists a unique morphism A −−→ Ai

9th July 2006

Sec. 25]

Topologically algebraic categories
gi

411

m

e

i
with fi = U gi ◦ u. Let A −−→ Ai = A −→ B −−→
Ai be an (Epi, Initial Source)-(U e)◦ufi

Umi
factorization. Then X −−→ U Ai = X −−−−−→ U B −−−→U Ai is a (Generating, Initial
Source)-factorization of S.
SOLID FUNCTORS AND SOLID CATEGORIESTopologically algebraic functors have several pleasant properties, such as detecting limits and colimits. However, they fail to be closed under composition (see Exercise 25E
and Theorem 26.1). Those functors that can be expressed as composites of topologicallyalgebraic functors share such properties as detection of limits and colimits and, in addition, are closed under composition. Hence they form a more convenient family than do
the topologically algebraic functors. They are called solid and can be defined by meansof semifinal solutions:25.7 DEFINITION
fi
G
Let A −−→ B be a functor and let S = (GAi −−→ B)I be a G-structured sink. Thengiea pair (B −→ GA , (Ai −−→ A)I ) is called a semifinal solution for S provided thatgeiGgi = e ◦ fi for each i ∈ I, and whenever (B −→ GA , (Ai −−→A)I ) is a pair withg
Gg i = e ◦ fi for each i ∈ I, then there exists a unique A-morphism A −→ A withe = Gg ◦ e and g i = g ◦ gi for each i ∈ I.

GA
mm6|=

Ggi mmmm |||mm
mmm |||| e
m
m
m fi

/B
Gg
BB
QQQ
QQQ BBBeQQQ BBQQQB Gg i(

GAi QQGA

25.8 REMARKU
For faithful functors A −−→ X, resp. for concrete categories (A, U ) over X, the concept of
semifinal solutions can be expressed more succinctly as follows: a semifinal arrow for afie

U-structured sink S = (U Ai −−→ X)I is a structured arrow X −→ U A with the propertye◦fiethat each U Ai −−−→ U A is an A-morphism, and for each structured arrow X −→ U Ae◦fisuch that each U Ai −−−→ U A is an A-morphism, there exists a unique A-morphismgA −→ A with e = U g ◦ e. Then S has a semifinal solution if and only if it has a semifinalee◦fi

arrow. [The semifinal solution is formed by the pair (X −→ U A, (Ai −−−→ A)I ).] Noticethat semifinal arrows are generating. See Exercise 25C.
25.9 EXAMPLES
fi(1) Let S = (U Ai −−→ X)I be a structured sink in the construct

9th July 2006

412

Topological and Algebraic Categories
fi[Chap. VIid(a) Top. If (Ai −−→ A)I is the final lift of S, then X −−→ U A is a semifinal arrow for
S. Similarly, semifinal arrows can be constructed in all topological categories.
(b) Pos. Consider Pos as an epireflective subcategory of Prost. Let the sinkfie(Ai −−→ A)I be a final lift of S in Prost, and let A −→ B be a Pos-reflectionefor A. Then X −→ U B is a semifinal arrow for S. Similarly, semifinal arrows
can be constructed in all M -topological categories.
kj(c) TopGrp. Consider the structured source T = (X −−→ U Aj )J consisting of all
kj ◦fi(kj , Aj ) such that U Ai −−−−→ U Aj is an A-morphism for each i ∈ I. Let
kj

U mj

e

X −−→ U Aj = X −→ U A −−−−→ U Aj be a (Generating, Initial Source)e
factorization of T . Then X −→ U A is a semifinal arrow for S. Similarly,
semifinal arrows can be constructed in all topologically algebraic categories.
(2) If (A, U ) is a concrete category over X, thene(a) X −→ U B is a universal arrow over X if and only if it is a semifinal arrow for
the empty structured sink with codomain X,
id(b) (idU A , A) is a semifinal arrow for the structured 1-sink (U A −−→ U A).
25.10 DEFINITION(1) A functor G is called solid provided that each G-structured sink has a semifinal
solution.
(2) A concrete category (A, U ) is called solid provided that U is solid.
25.11 PROPOSITION
Topologically algebraic functors are solid.
G

Proof: Let A −−→ B be a topologically algebraic (and hence faithful) functor and letfi

S = (GAi −−→ B)I be a G-structured sink. Consider the G-structured source T =kj ◦fikj(B −−→ GAj )J consisting of all (kj , Aj ) such that GAi −−−−→ GAj is an A-morphism
kj

Gmj

e

for each i ∈ I. Let B −−→ GAj = B −→ GA −−−−→ GAj be a (Generating, Initiale
Source)-factorization of T . Then B −→ GA is a semifinal arrow for S.
25.12 PROPOSITION
Solid functors are faithful and adjoint.
G

Proof: Let A −−→ B be a solid functor.
(1). To show faithfulness, let Ars// B be a pair of A-morphisms with Gr = Gs = f .
fi

Consider the G-structured sink S = (GAi −−→ X)I with I = M or(A), X = GA, and9th July 2006

Sec. 25]

Topologically algebraic categories

413

gi

e

(Ai , fi ) = (A, idX ) for each i ∈ I. Let (X −→ GC , (Ai −−→ C)I ) be a semifinal solutiongeifor S. Consider (X −→ GC , (Ai −−→C)I ) with (e, C) = (f, B) and(r, if i ◦ gi = s
gi =s, otherwise.

g

Then Gg i = f ◦ fi for each i ∈ I. Hence there exists a unique A-morphism C −→ Cwith e = Gg ◦ e and g i = g ◦ gi for each i ∈ I. By choosing i = g we obtain g g = g ◦ gg ,which is possible only in the case that r = s.
(2). Adjointness follows immediately from Example 25.9(2)(a).



25.13 PROPOSITION
Solid functors are closed under composition.
G

F

Proof: If A −−→ B and B −−→ C are solid, then G, F , and hence F ◦ G are faithful. Letfie

S = (F GAi −−→ C)I be an (F ◦ G)-structured sink. Let C −→ F B be a semifinal arrowe◦fi

for S, considered as an F -structured sink. Then (F GAi −−−→ F B)I can be considerede◦fie

as a sink (GAi −−−→ B)I in B, hence as a G-structured sink. If B −→ GA is a semifinalF e◦earrow for this sink, then C −−−−→ F GA is a semifinal arrow for S.



25.14 THEOREM
Solid functors detect colimits and preserve and detect limits.
D

Proof: Let (A, U ) be a solid category over X and let I −−→ A be a diagram.
cei(1). Let S = (U Di −−→K)I be a colimit of U ◦ D. If K −→ U A is a semifinal arrow fore◦ci
the structured sink S, then (Di −−−→A)I is a colimit of D.

`i(2). Preservation of limits follows from adjointness. Let (L −−→
U Di )I be a limit offj

U ◦ D. Consider the structured sink S = (U Aj −−→ L)J consisting of all costructured`i ◦fjearrows (Aj , fj ) such that U Aj −−−−→ U Di is a morphism for each i ∈ I. Let L −→ U A befia semifinal arrow for S. Then, for each i ∈ I, there exists a unique morphism A −−→ Difiwith `i = U fi ◦ e. The source (A −−→ Di )I is easily seen to be a limit of D.



25.15 COROLLARY
Every solid category over a complete (resp. strongly complete, resp. cocomplete) category
is complete (resp. strongly complete, resp. cocomplete).

25.16 COROLLARY
Solid constructs are strongly complete and cocomplete.

9th July 2006



414

Topological and Algebraic Categories[Chap. VI

25.17 REMARK(1) A solid construct need be neither topologically algebraic nor strongly cocomplete.
The construct Λ-CCPos is neither strongly complete nor topologically algebraic,esince the source (B0 −−α→ Bα ) described in Exercise 15D(d), has no cointersection,eand the structured source (|B0 | −−α→ Aα ) described in Exercise 15D(b) has no (Generating, Initial Source)-factorization. However, Λ-CCPos is solid. The latter follows
from the observation that Λ-CCPos is a full reflective concrete subcategory of the
topologically algebraic construct that is defined in the same way as Λ-CCPos exceptthat λ is required only to be a partial unary operation.
(2) Although there exist solid categories that are not topologically algebraic, the concepts of solid and topologically algebraic categories coincide under rather natural
assumptions, as the following results and Exercise 25D demonstrate.
25.18 THEOREM
Let (A, U ) be a concrete category over a strongly complete category X. If U is concretely
co-wellpowered, then the following conditions are equivalent:(1) (A, U ) is topologically algebraic,(2) (A, U ) is solid,(3) U detects and preserves limits.
Proof: (1) ⇒ (2) ⇒ (3) has already been established.
(3) ⇒ (1) is an immediate consequence of Theorem 17.11.



25.19 CHARACTERIZATION THEOREM FOR SOLID CATEGORIESLet (A, U ) be a concrete category. If A is an (Epi,−)-category (in particular, if A isstrongly cocomplete), then the following conditions are equivalent:(1) (A, U ) is topologically algebraic,(2) (A, U ) is solid,(3) U is adjoint.
Proof: (1) ⇒ (2) ⇒ (3) has already been established.
(3) ⇒ (1). Let A be an (Epi, M)-category. By adjointness it follows that U is a
(Generating, M)-functor. By faithfulness, every identity-carried structured morphism isgenerating. Hence, by Theorem 25.6, (A, U ) is topologically algebraic.

25.20 REMARKS(1) A partially ordered set with a smallest element, considered as a concrete category(A, U ) over 1, may fail badly to be cocomplete and hence solid, even though A is
co-wellpowered and U is adjoint.
(2) Monadic categories over Pos may fail to be cocomplete and hence to be solid (cf.
Example 20.47).

9th July 2006

Sec. 25]

Topologically algebraic categories

415

25.21 PROPOSITION
Solid functors preserve extremal monomorphisms.
m

Proof: Let (A, U ) be a solid category over X. If A −−→ B is an extremal monomorphismmin A, then U A −−→ U B is a monomorphism in X. To show that it is extremal, let
m

f

e

e

U A −−→ U B = U A −→ X −→ U B be an (Epi,−)-factorization in X. If X −→ U Ce◦ee
is a semifinal arrow for the structured 1-sink (U A −→ X), then U A −−−→ U C is anfA-morphism and there exists a unique A-morphism C −→ B with f = f ◦ e. Thus
m

f

e◦e

e◦e

A −−→ B = A −−−→ C −→ B is a factorization in A. Since A −−−→ C is (easily seen tobe) an epimorphism in A, it is an isomorphism in A and hence in X. Thus e is a sectionin X and, consequently, an isomorphism in X.

25.22 REMARKThe following diagram exhibits several results of this section in diagrammatic form.
essentiallyalgebraic (cf. 23.42)

M-topological
PP
PP
PP

full reflective
embeddingequivalence





)


@
PP
q
P

@
R
@

topologically algebraic

?














9

adjoint

detects limits

solid
?

X
H XXXX
HH
XXX
XXX
H
HH
XXX
j
z

preserves
extremal monosdetects colimits

faithful

Suggestions for Further Reading
Hong, S. S. Categories in which every mono-source is initial. Kyungpook Math. J. 15(1975): 133–139.
Trnková, V. Automata and categories. Springer Lect. Notes Comp. Sci. 32 (1975):
132–152.
Hoffmann, R.-E. Semi-identifying lifts and a generalization of the duality theorem for
topological functors. Math. Nachr. 74 (1976): 295–307.
Fay, T. H. An axiomatic approach to categories of topological algebras. Quaest. Math.
2 (1977): 113–137.

9th July 2006

416

Topological and Algebraic Categories[Chap. VI

Tholen, W. Semi-topological functors I. J. Pure Appl. Algebra 15 (1979): 53–73.
Börger, R., and W. Tholen. Remarks on topologically algebraic functors. Cahiers Top.
Geom. Diff. 20 (1979): 155–177.
Herrlich, H., R. Nakagawa, G. E. Strecker, and T. Titcomb. Equivalence of topologically algebraic and semi-topological functors. Canad. J. Math. 32 (1980): 34–39.
Porst, H.-E. T -regular functors. Categorical Topology. Proceedings of the International
Conference, Toledo, Ohio, 1983 (ed. H. L. Bentley et. al.), Heldermann, Berlin,
1984, 425–440.

EXERCISES
25A.
Prove that(a) Every faithful (Generating, −)-functor is topologically algebraic.
(b) Set → 1 is a (Generating, Source)-functor that is not faithful.
25B. M -Topological =⇒ Topologically Algebraic
UShow that if X is an (E, M)-category and A −−→ X is M-topological, then U is topologically algebraic.
25C. Semifinal Solutions and Semifinal Arrows
Show that(a) Semifinal arrows are concretely generating.
gie(b) If (B −→ GA, (Ai −−→ A)I ) is a semifinal solution for a G-structured sink, theneB −→ GA need not be G-generating. [Hint: Consider the nonfaithful functorG

A −−→ B indicated by
A1 3

f

333
3333
3333
r
3333sf
rf 333
333
3333
 

/ A2
G

−−→

s

A3

Gf

/ GA2
w;
55GGG kw
e ww
55 GGGw
55 GG# wwww
55
Gr=Gs
G(rf )=G(sf ) 55 B
55
55 `
  

GA51G

GA3

e

f

and the semifinal solution (B −→ GA2 , (A1 −→ A2 )) of the G-structured 1-sinkk(GA1 −→ B).]
* 25D. Solid = Topologically Algebraic
GShow that in any of the following situations, A −−→ B is solid if and only if G is
topologically algebraic:

9th July 2006

Sec. 25]

Topologically algebraic categories

417(a) A is finite,(b) A is strongly complete and co-wellpowered,(c) B = 1,(d) G is full.
* 25E. Solid 6= Topologically Algebraic(a) Let B be the following subcategory of Set:
BO 1 NN

NNN
NNgNn
hn
NNN
NN&
t
s r
p8 B3
p
p
ppp
pppkn
p
p
 pp

B2

with B1 = {1}, B2 = {2}, B3 = N, r(1) = 2, s(2) = 1, gn (1) = kn (2) = n, andhn (m) = n + m. Let A be the subcategory of B obtained by removing r and g0 .
GShow that the inclusion functor A −−→ B is solid but is not topologically algebraic.
(b) Denote by ∆Pos the category of ∆-complete partially ordered sets (i.e., partiallyordered sets in which each nonempty up-directed subset has a join) and ∆-continuousfunctions (i.e., functions preserving up-directed joins).
(1) Verify that ∆Pos is a solid construct (cf. Theorem 25.18).
(2) If T : ∆Pos → ∆Pos is the “discretization” concrete functor that sends eachpartially ordered set to the discrete partially ordered set on its underlying set,
verify that Alg(T ) is a solid concrete category over ∆Pos. [Hint: Alg(T ) is areflective subcategory of the category of partial T -algebras, defined analogously
to Alg(T ) except that objects (X, x) are given by partial maps x : T X → X.
The latter category is topologically algebraic over ∆Pos.](3) If U is the composite of the forgetful functors Alg(T ) → ∆Pos → Set, conclude
that the construct (Alg(T ), U ) is solid. Show, however, that it is not topologi-Nfically algebraic since the source (( , x) −−→ (Yi , yi ))O defined below has no (Epi,
Initial Source)-factorization:
O is the class of all infinite ordinals;

N has the discrete ordering, with x(n) = n + 1;
Yi is the set of all ordinals smaller than i + ω with the natural linear orderon i (= {j j < i}) and with i + n incompatible to any other element for
all n ∈ , yi (j) = j + 1;Nfi (n) = n.

9th July 2006

418

Topological and Algebraic Categories[Chap. VI(4) Verify that Alg(T ) is not strongly cocomplete by showing that the source in (3)consists of epimorphisms and has no cointersection.
* 25F. Solid vs. Monadic Functors
A solid functor whose associated monadic functor is not solid:(a) For the non-cocomplete monadic category Alg(T ) over Pos of Example 20.47 denote
by A the full subcategory of all T -algebras (X, x) such that x(∅) lies in no 3-chainof X [more precisely, if (p1 , p2 , p3 ) ∈ H(X), then x(∅) 6= pi ]. Verify that A contains
all free T -algebras, and conclude that its underlying functor is an adjoint functorU
A −−→ Pos whose associated monadic functor is not solid [it being just the forgetful
functor of Alg(T )].
(b) Prove that A is closed under the formation of limits in Alg(T ), and conclude that
A is strongly complete.
(c) Prove that U is co-wellpowered, and hence is solid (cf. Theorem 25.18).
* 25G. Solid Categories and Mac Neille CompletionsShow that the following conditions are equivalent for amnestic concrete categories (A, U ):(1) U is solid,E(2) (A, U ) has a Mac Neille completion (A, U ) −−→ (B, V ) and E[A] is reflective in B,
E(3) If (A, U ) −−→ (B, V ) is an initially dense and finally dense full concrete embedding,then E[A] is reflective in B.
* 25H. Topologically Algebraic Categories and Universal Initial CompletionsShow that the following conditions are equivalent for amnestic concrete categories (A, U ):(1) (A, U ) is topologically algebraic,E(2) (A, U ) has a universal initial completion (A, U ) −−→ (B, V ) and E[A] is reflectivein B(cf. Exercise 21I),E(3) if (A, U ) −−→ (B, V ) is an initiality-preserving and initially dense full concrete embedding, then E[A] is reflective in B.
25I. A Characterization for Solid Categories
Let (A, U ) be a concrete category over a cocomplete category X. Prove that if A is
co-wellpowered, then the following conditions are equivalent:(1) (A, U ) is topologically algebraic,(2) (A, U ) is solid,(3) A is cocomplete and U is adjoint.

9th July 2006

Sec. 25]

Topologically algebraic categories

419

25J. Solid vs. Topological
Prove that a solid category is topological if and only if it is uniquely transportable andhas indiscrete structures.
25K. Alternative Proof of Theorem 12.13
Use results of this section to obtain a short proof that every strongly cocomplete category A with a separating set S is strongly complete (cf. Exercise 12N). [Define a faithfulUfunctor A −−→ [S op , Set] by restricting the contravariant hom-functors and the corresponding natural transformations. Use cocompleteness to show that U is adjoint andTheorem 25.19 to show that it is solid. Apply Corollary 25.15.]
25L. Topologically Algebraic Functors and Lifting Adjoints
Let G, Ĝ, U , and V be functors with V ◦ Ĝ = G ◦ U . Let U be topologically algebraic,
V faithful, and let Ĝ send U -initial sources into V -initial sources. Show that adjointnessof G implies adjointness of Ĝ. [Cf. Theorem 21.28.]

9th July 2006

26 Topologically algebraic structure theorems
26.1 THEOREM
For a functor G the following conditions are equivalent:(1) G is solid,(2) G can be expressed as the composite G = F ◦K of two topologically algebraic functors,(3) G can be expressed as the composite G = T ◦ H ◦ E of a topological functor T , anequivalence H, and a full reflective embedding E,(4) G belongs to the smallest conglomerate of functors that contains all topologicallyalgebraic functors and is closed under composition,(5) G belongs to the smallest conglomerate of functors that contains all topological functors, all equivalences, all full reflective embeddings, and is closed under composition.
G

Proof: (1) ⇒ (3). If A −−→ X is a solid functor, consider (A, G) as a concrete category
over X, and let (B, U ) be the following concrete category over X: objects are pairse(X, (e, A)), consisting of an X-object X and a semifinal G-structured arrow X −→ GA;ffmorphisms (X, (e, A)) −→ (X 0 , (e0 , A0 )) are those X-morphisms X −→ X 0 for which theregexists a (necessarily unique) A-morphism A −→ A0 with e0 ◦ f = Gg ◦ e. The concreteEfunctor (A, G) −−→ (B, U ) defined by E(A) = (GA, (idGA , A)) is a full embedding.
eMoreover, for each B-object (X, (e, A)), the E-structured arrow (X, (e, A)) −→ EA is
E

E-universal. Hence A −−→ B is a full reflective embedding. In (B, U ) every structuredfisource S = (X −−→ U (Xi , (ei , Ai )))I has a (not necessarily unique) initial lift. This cangbe seen as follows: in (A, G) consider the structured sink T = (GAk −−k→ X)K thatei ◦fi ◦gconsists of all pairs (Ak , gk ) such that GAk −−−−−−k→ GAi is an A-morphism for eachei ∈ I. Let the G-structured arrow X −→ GA be a semifinal arrow for T . It follows
fithat ((X, (e, A)) −−→ (Xi , (ei , Ai )))I is a U-initial lift of S. Hence (B, U ) may fail to be
topological only by not being amnestic. By Proposition 5.33 there exists an amnesticH
concrete category (C, T ) over X and a surjective concrete equivalence (B, U ) −−→ (C, T ).
Obviously, T is topological. Hence G = T ◦ H ◦ E is the desired factorization.
(3) ⇒ (5) is obvious.
(5) ⇒ (4) is immediate from the fact that topological functors, equivalences, and fullreflective embeddings are topologically algebraic.
(4) ⇒ (1) is immediate from the fact that topologically algebraic functors are solid andsolid functors are closed under composition.
(3) ⇒ (2) and (2) ⇒ (4) are obvious.

420Sec. 26]

Topologically algebraic structure theorems

421

26.2 COROLLARY
A concrete category is solid and amnestic if and only if it is a full reflective concrete
subcategory of a topological category.
H◦E

Proof: In the proof of Theorem 26.1 amnesticity of G implies that A −−−−→ C is a fullreflective embedding.

26.3 DECOMPOSITION THEOREM FOR SOLID FUNCTORS
For any functor G between (Epi, Mono-Source)-factorizable categories, the following
conditions are equivalent:(1) G is solid and uniquely transportable,(2) G can be expressed as a composite G = T ◦ A of a topological functor T and an
essentially algebraic functor A,(3) G belongs to the smallest conglomerate of functors that contains all topological andall essentially algebraic functors and is closed under composition.
Proof: (1) ⇒ (2). By Theorem 26.1 G can be expressed as a composite G = T ◦H ◦E of
a topological functor T , an equivalence H, and a full reflective embedding E. In the casethat G is uniquely transportable, the above constructed A = H ◦ E is an isomorphismclosed full reflective embedding. Hence, by Theorem 23.8, it is an essentially algebraic
functor.
(2) ⇒ (3) ⇒ (1) is obvious.



26.4 THEOREM
G
For any functor A −−→ X into a category X that has regular factorizations the following
conditions are equivalent:(1) G is solid, uniquely transportable, and preserves regular epimorphisms,(2) G can be expressed as a composite G = T ◦ R of a topological functor T and a
regularly monadic functor R,(3) G belongs to the smallest conglomerate of functors that contains all topological andall regularly monadic functors and is closed under composition.
Proof: (1) ⇒ (2). In view of Theorem 20.32 it suffices to show that the full reflecEtive embedding A −−→ B constructed in the proof of Theorem 26.1 preserves regularcepimorphisms. Let B −→ C be a regular epimorphism in A, and thus a coequalizerpof some pair A

q

//

Gc

B of A-morphisms. Then GB −−→ GC is a regular epimorphismin X, and so is a coequalizer of some pair Xr /s/ GB of X-morphisms. Consider thefi

G-structured sink S = (GAi −−→ X)I that consists of those G-costructured arrows
fi

r◦fi

s◦fi

GAi −−→ X, for which GAi −−−→ GB and GAi −−−→ GB are A-morphisms, and9th July 2006

422

Topological and Algebraic Categories[Chap. VIre

let X −→ GA be a semifinal arrow for S. Then there exist A-morphisms A −→ Brs

and A −→ B with r = Gr ◦ e and s = Gs ◦ e. Thus (X, (e, A))sc//

E(B) is a pairGcof B-morphisms. It suffices to show that E(B −→ C) = E(B) −−→ E(C) is a cof

equalizer of this pair in B. Obviously, Gc ◦ r = Gc ◦ s. Let E(B) −→ (X, (e, B))
be a B-morphism with f ◦ r = f ◦ s. Since Gc is a coequalizer of r and s in X,fthere exists a unique X-morphism GC −→ X with f = f ◦ Gc. Hence it suffices toffshow that E(C) −→ (X, (e, B)) is a B-morphism. Since E(B) −→ (X, (e, B)) is ag
B-morphism, there exists an A-morphism B −→ B with e ◦ f = Gg ◦ idGB . ThusG(g ◦ p) = e ◦ f ◦ Gp = e ◦ f ◦ Gc ◦ Gp = e ◦ f ◦ Gc ◦ Gq = e ◦ f ◦ Gq = G(g ◦ q).
Hence g ◦ p = g ◦ q. Since c is a coequalizer of p and q in A, there exists an A-morphismgC −→ B with g = g ◦c. Thus the equations Gg ◦Gc = Gg = e◦f = e◦f ◦Gc and the fact
fthat Gc is an epimorphism in X imply that Gg = e ◦ f , i.e., that E(C) −→ (X, (e, B))
is a B-morphism.
fi

GAi

/X
r



e
s

GB
Gc
f

X



p GC

ppp
ppp
p
p
wpppp f

/ GA
Gr

id

id

e

Gs


Gp
/ GB oo
GA
>>
>> Gq
Gc >>
>>

>>Gg
/ GC
NNN >>>
NNN >>
NNN >>
Gg NNN>
'/ GB(2) ⇒ (3) ⇒ (1) obvious.



Next we wish to characterize those concrete categories that can be embedded as fullreflective concrete subcategories of fibre-small topological categories. By Theorem 22.3
this will provide a characterization of full reflective subcategories of functor-structured
categories. By the above theorem they must be solid and, obviously, fibre-small. But
fibre-smallness is not enough. We now introduce a slightly stronger condition than fibresmallness, and show that for concrete categories that have this property, solidness isequivalent to being a reflective subcategory of some functor-structured category.
26.5 DEFINITION
f1
f2(1) Structured arrows X −−→ U A1 and X −−→ U A2 in a concrete category (A, U ) aregsaid to be equivalent if for each costructured arrow U B −→ X the following holds:
f1 ◦g

f2 ◦g

B −−−→ A1 is an A-morphism if and only if B −−−→ A2 is an A-morphism.
(2) A concrete category over X is said to be strongly fibre-small provided that foreach X-object X there exists no proper class of pairwise non-equivalent structured
arrows with domain X.

9th July 2006

Sec. 26]

Topologically algebraic structure theorems

423

26.6 EXAMPLES(1) Each strongly fibre-small amnestic concrete category is fibre-small.
(2) A topological category is strongly fibre-small if and only if it is fibre-small. Thisffollows immediately from the fact that each structured arrow X −→ U A is equivalent
idX(via its initial lift) to a structured arrow of the form X −−−→ U A0 .
(3) Each fibre-small hereditary (i.e., monomorphisms have initial lifts) construct (e.g.,each of Pos, Top, Haus, and Met) is strongly fibre-small: for each structured
f0

farrow X −→ U A consider the factorization of f as a surjective function X −−→ U A0
fi

followed by an initial monomorphism A0 → A. Then two such arrows X −−→ U Ai ,fi0

i = 1, 2, are equivalent if and only if X −−→ U A0i are equivalent.
(4) Each fibre-small uniquely transportable construct that has (surjective, injective)factorizations is strongly fibre-small. Thus Vec, Grp, HComp, Ban, Boo, Σ-Seq,
etc. are strongly fibre-small constructs.
(5) For concrete categories over X = 1, smallness, fibre-smallness, and strong fibresmallness are equivalent.
(6) Let A be a large discrete category and let U : A → Set be an injective functor withcard(U A) = 1 for all A-objects A. Then (A, U ) is fibre-small, but not strongly
fibre-small.
26.7 STRUCTURE THEOREM FOR SOLID CATEGORIESFor a concrete category (A, U ) the following conditions are equivalent:(1) (A, U ) is amnestic, solid, and strongly fibre-small,(2) (A, U ) is a full reflective concrete subcategory of a fibre-small topological category,(3) (A, U ) is concretely isomorphic to a full reflective subcategory of some functorstructured category.
E

Proof: (1) ⇒ (2). Let (A, G) be a solid category and let (A, G) −−→ (B, U ) andH(B, U ) −−→ (C, T ) be the concrete functors constructed in the proof of Theorem 26.1.
If G is amnestic, then H ◦ E is a full reflective embedding. Thus it suffices to showthat strong fibre-smallness of (A, G) implies fibre-smallness of (C, T ). For this it issufficient to show that for any two B-objects (X, (e, A)) and (X, (e0 , A0 )) that are equivee0

alent when considered as G-structured arrows X −→ GA and X −−→ GA0 , the equality
H(X, (e, A)) = H(X, (e0 , A0 )) holds. This follows immediately from the fact that for any
idX
such pair (X, (e, A)) −−−→ (X, (e0 , A0 )) is a B-isomorphism.

(2) ⇒ (3) follows from Theorem 22.3.
(3) ⇒ (1) follows immediately from Corollary 26.2, Example 26.6(2), and the fact thatevery full concrete subcategory of a strongly fibre-small category is strongly fibre-small.

9th July 2006

424

Topological and Algebraic Categories[Chap. VI

26.8 COROLLARY
For a co-wellpowered concrete category (A, U ) over a cocomplete base category, the following are equivalent:(1) (A, U ) is cocomplete, amnestic, and has free objects,(2) (A, U ) is concretely isomorphic to a full reflective subcategory of a functor-structured
category.
Proof: (1) ⇒ (2). By Proposition 18.11, U is co-wellpowered. By Corollary 15.17, A
is an (Epi, ExtrMono-Source)-category, so that by Theorem 25.19, U is topologicallyalgebraic, and thus has (Generating, InitialSource)-factorizations. These two facts immediately imply that (A, U ) is strongly fibre-small. Thus (2) follows from Theorem
26.7.
(2) ⇒ (1) is immediate from the fact that full reflective embeddings detect colimits(cf. 13.32).
Suggestions for Further Reading
Hoffmann, R.-E. Note on semi-topological functors. Math. Z. 160 (1978): 69–74.
Rosický, J. Concrete categories and infinitary languages. J. Pure Appl. Algebra 22(1981): 309–338.
Rosický, J. Semi-initial completions. J. Pure Appl. Algebra 40 (1986): 177–183.
Herrlich, H., Mossakowski, T., and G. E. Strecker. Algebra ∪ Topology. Category
Theory at Work (eds. H. Herrlich and H.-E. Porst), Heldermann Verlag 1991,
137–148.

EXERCISES* 26A. Decomposition of Functors and Factorization StructuresG

Let A −−→ B be a uniquely transportable, solid functor, and let B have regular factorizations. Show that:(a) A need not have (Epi, Mono-Source)-factorizations. [Cf. Exercise 16A(b).](b) There need not exist a decomposition G = T ◦A with T topological and A essentiallyalgebraic.
26B. Composites are Topologically Algebraic
Show that each composite G = A ◦ T of a monotopological functor T and an essentiallyalgebraic functor A is topologically algebraic.

9th July 2006

Sec. 26]

Topologically algebraic structure theorems

425

26C. Topological-Algebraic via Algebraic-Topological Decompositions(a) Consider the conditions:(1) there exist a topological functor T and an essentially algebraic functor A with
G = A ◦ T,(2) there exist a topological functor T and an essentially algebraic functor A with
G = T ◦ A.
Show that:(i) (1) implies (2).
(ii) If (A, U ) is a non-topological, monotopological construct and there exists an
A-object A such that U A has at least 2 points, then U satisfies (2) but not(1).
(iii) The forgetful functor for Λ-CCPos satisfies (2) but not (1). Cf. Exercise 26B.
(b) Let (A, U ) be a concrete category over a category X with regular factorizations. Let
U be uniquely transportable, solid, and preserve regular epimorphisms. Let E be theclass of those A-morphisms whose underlying morphisms are regular epimorphisms
in X. Show that the following conditions are equivalent:(1) A is an (E, InitMono-Source)-category,(2) U can be expressed as a composite U = A ◦ T of a monotopological functor T
and a regularly monadic functor A,(3) the comparison functor for U is monotopological.
26D. Strong Fibre-Smallness(a) Verify that a construct is strongly fibre-small if and only if it can be fully embedded
into a fibre-small topological construct. Under what assumptions on X does thesame result hold for concrete categories over X?
(b) Show that in Corollary 26.8 it is not possible to delete the (co-)wellpoweredness
hypothesis. [Hint: Consider Ord as a concrete category over itself.]

9th July 2006

427

Chapter VII

CARTESIAN
CLOSEDNESSPARTIAL MORPHISMS

9th July 2006

AND

27 Cartesian closed categories
In some constructs, we can form objects B A by structuring the set of all morphisms from
A to B in such a way that the evaluation map
ev : A × B A → B

given by: (x, f ) 7→ f (x)becomes a co-universal arrow. For example, in Set we have the set B A of all functions
from A to B, in Pos we have the poset B A of all order-preserving functions from A to
B (ordered pointwise), etc. We will study this phenomenon first for abstract categories,
then for concrete ones, and finally for constructs.
If a category A has finite products, then for each A-object A we can define a functor
id ×f

f

(A × −) : A → A by (B −→ B 0 ) 7→ (A × B −−A−−→ A × B 0 ). (Since products are
not unique, such a rule usually gives a large collection of pairwise naturally isomorphic
endofunctors of A. By an “abuse of language” we will, however, speak about the functor(A × −), meaning that a certain product A × B has been chosen for each B.)
27.1 DEFINITION
A category A is called cartesian closed provided that it has finite products and foreach A-object A the functor (A × −) : A → A is co-adjoint.
27.2 NOTATIONThe essential uniqueness of products and of co-universal arrows allows us to introduce
the following standard notation for cartesian closed categories: “The” adjoint functor
for (A × −) is denoted on objects by B 7→ B A , and “the” associated co-universal arrows
are denoted by
ev : A × B A → B.
Thus, a category with finite products is cartesian closed if and only if for each pair (A, B)
of objects there exists an object B A and a morphism ev : A × B A → B with the following
universal property: for each morphism f : A × C → B there exists a unique morphismfˆ: C → B A such thatA × CG

GG f
GG
GG
GG

#
/B
A
A×B

idA ×fˆev

commutes. We call the objects B A power objects, the morphisms ev : A × B A → A
evaluation morphisms, and the fˆ, associated with f , the exponential morphism
for f .

428

Sec. 27]

Cartesian closed categories

429

A power object in a cartesian closed category

27.3 EXAMPLES(1) The following categories are cartesian closed:(a) Set: B A is the set of all functions from A to B; ev is the usual evaluation map;(fˆ(c))(a) = f (a, c).
(b) Rel: (B, σ)(A,ρ) is (B A , τ ) with f τ g ⇔ [aρa0 ⇒ f (a)σg(a0 )]; ev is the usual
evaluation map; (fˆ(c))(a) = f (a, c).
(c) Pos: (B, ≤)(A,) is the set of all order-preserving functions from A to B, ordered
as follows: f ≤ g ⇔ f (a) ≤ g(a) for all a ∈ A; ev is the usual evaluation map;(fˆ(c))(a) = f (a, c).
(d) Alg(1) (algebras with one unary operation): B A is the set of all homomorphisms
from A ×
to B [where
denotes the algebra of all natural numbers withthe successor operation], together with the unary operation that sends eachh : A× → B to h0 : A× → B, where h0 (a, n) = h(a, n+1); ev(a, f ) = f (a, 0);and for each homomorphism f : A × C → B, fˆ(c) sends (a, n) to f (a, γ n (c)),
where γ is the unary operation for C.

N

N

N

N

(e) Cat: BA is the functor category [A, B]; ev : A × BA → B is defined on objectshby ev(A, F ) = F A and on morphisms by ev(h, τ ) = τA0 ◦ F h [where A −→ A0 ];for F : A × C → B we have F̂ C : A → B defined byhF (h,id)F̂ C(A −→ A0 ) = F (A, C) −−−−−→ F (A0 , C).

9th July 2006

430

Cartesian Closedness and Partial Morphisms[Chap. VII

Q(2) Top is not cartesian closed (since ( ×−) : Top → Top does not preserve quotients,and hence does not preserve coequalizers). However, Top has cartesian closed supercategories [e.g., Conv and PsTop, where the power objects carry the structureof “continuous convergence” (see Exercise 27G)] as well as cartesian closed subcategories [e.g., the category kTop of (compact Hausdorff)-generated topological spaces,
where the power objects are formed by the kTop-coreflections of the compact-open
topologies].
(3) A poset A, considered as a category, is cartesian closed if and only if A has finite
meets and for each pair (a, b) of elements, the set { x ∈ A a ∧ x ≤ b } has a largest
member. In particular, a W
complete
W lattice is cartesian closed if and only if it satisfies
the distributive law: a ∧ bi = (a ∧ bi ); i.e., if and only if it is a frame.
27.4 CHARACTERIZATION THEOREM FOR
CARTESIAN CLOSED CATEGORIESLet A be a cocomplete and co-wellpowered category that has a separator. Then A iscartesian closed if and only if it has finite products, and for each A-object A the functor(A × −) preserves colimits.
Proof: Immediate from the Special Adjoint Functor Theorem (18.17).



27.5 DEFINITION
Let A be a cartesian closed category. For each object C(1) “the” covariant exponential functor for C, denoted by (−)C : A → A, is “the”fadjoint functor for (C × −) and is defined (for an A-morphism A −→ B) by:
fC

f(−)C (A −→ B) = AC −−→ B C ,where f C is the unique A-morphism that makes the diagram
C × AC
id×f C

ev

/A
f



C × B C ev

/Bcommute;(2) “the” contravariant exponential functor for C, denoted by C (−) : Aop → A,f

is defined (for an A-morphism A −→ B) by
Cff

C (−) (A −→ B) = C B −−→ C A ,where C f is the unique A-morphism that makes the diagram
A × CB
id×C f

f ×id

ev



A × CA/ B × CBev

/C

9th July 2006

Sec. 27]

Cartesian closed categories

431

commute.
27.6 REMARK
Observe that exponential functors are not determined uniquely, but that any two covariant (resp. contravariant) exponential functors for an A-object C are naturally isomorphic.
27.7 PROPOSITION
In a cartesian closed category A, every contravariant exponential functor C (−) is anadjoint functor and has its own dual (C (−) )op as a co-adjoint.
Proof: Let A be an A-object and let s : C A × A → A × C A be the isomorphism thatis determined by interchanging the projections. Then the unique A-morphismAuA : A → C (C ) that makes the diagram

id×uA/ A × CAs

CA × A

ev





/C

A

C A × C (C ) ev

commute is a C (−) -universal arrow for A. To see this let f : A → C B be a C (−) -structured
arrow, let s : A × B → B × A be the isomorphism determined by interchanging the
projections, and let f : B → C A be the unique A-morphism that makes the diagram
A×B
id×f

(id×f )◦s

/ B × CBev

A × CAev/C

commute. Then id×f = s◦(f ×id)◦s implies that in the following diagram 1 commutes:
B×A
??
||
??
|
|
f ×id ??
|
|
??
|

??
||
|
A
??
|
C
×
A
|
H
??
HH
}
||
}
|
?
H
s
}
H
|
HH 1 ???
4 }}}
||
H
|
??
H$
|
}}
??id×fid×uA |||
}}
A
?
}
2
|
A
×
C
}
|
@@ ??
} id×uA|}|@}|@@ev ???
}}
||
@@ ??
}
|
}
|
??
@
~
}
|
?
ev
||
A)
/
A
(C
|
C eJJ ???
|
C6 × C|Jm|
JJ ???
|| mmmmmm
JJ ?
|
J
3
|| mmmev JJ???
J}|| mmm f ×id
A)/(C
B × CB
B×C
id×C f

9th July 2006

432

Cartesian Closedness and Partial Morphisms[Chap. VII

2 commutes by the definition of u, 3 commutes by the definition of C (−), and 4commutes trivially. Thus f = C (f ◦uA ) . That f is determined by this equality follows
immediately by retracing the steps from C f via f to f in the above diagram. ThusAuA : A → C (C ) is a C (−) -universal arrow. In order to see that (C (−) )op : A → Aop is aco-adjoint for C (−) it need only be shown that for any A-morphism f : A → B the equal(−) op
ity uB ◦ f = C (C ) (f ) ◦ uA holds. This follows immediately from the commutativity of
the following diagram, where s, s, and s̃ are isomorphisms determined by interchanging
the projections

CB × BJ
id×uB

C B × FA XXXXX
XXXXX
gggg xxg
FF
ggg
XXXXid×u
id×fgggg
x
FF
x
XXXA
g
g
x
XXXXX
g
F
xs
ggg
x
g
f ×id FF
XXXXX
g
g
x
C
g
g
x
"
|
X+
g
g
sg

JJ
JJ s
JJ
JJ
J$
B

A × CFBxf ×id xxx
x
x
x
|xx

C B × C (C ) WWWB × C B SS

FF
FFid×C f
FF
FF"

CA × A
I

y
s yyy
y
y
y
|yy

II
IIid×uA
II
II
$

A

A

C B × C (C )q
C f ×idqqqq
q
q
xqqq

(C A )

A

A×C
C
C ×C
ffff
WWWWW
S
jj
S
S
fffff
WWWWW SSev
evjjjjj
f
f
f
f
S
j
ev
ff
WWWWW SSSSev
jjjj ffffff ev
WWWWWSSSWWWSWS)+  rfujjfjfjfjfjffffff

id×C (C )B × C (C B )C
27.8 PROPOSITION
In a cartesian closed category the following hold:(1) First Exponential Law: AB×C ∼= (AB )C ,Q
Q
(2) Second Exponential Law: ( Ai )B ∼
= (AB
i ),
Q
(3) Third Exponential Law: AqBi ∼= (ABi ),``(4) Distributive Law: A × Bi ∼= (A × Bi ),(5) Finite products of epimorphisms are epimorphisms.
Proof:(1). This follows from the definition of covariant exponential functors (27.5) and thefact that, up to natural equivalence, the functor ((B × C) × −) is the composite of(B × −) with (C × −).
(2). (−)B preserves products since it is an adjoint functor.
(3). A(−) : Aop → A preserves products since it is an adjoint functor.
(4). (A × −) preserves coproducts since it is a co-adjoint functor.
(5). This follows from the fact that the functors (A × −), as co-adjoints, preserve epimorphisms and the fact thatf1 ×f2

f1 ×idid×f2

A1 × A2 −−−−→ B1 × B2 = A1 × A2 −−−−→ B1 × A2 −−−−→ B1 × B2 .



9th July 2006

Sec. 27]

Cartesian closed categories

433

CARTESIAN CLOSED SUBCATEGORIES
27.9 PROPOSITION
Let A be an isomorphism-closed full subcategory of a cartesian closed category B.
(1) If A is reflective in B and the A-reflector preserves finite products, then A is closed
under the formation of finite products and powers in B, and hence is cartesianclosed.
(2) If A is coreflective in B and is closed under the formation of finite products in B,
then A is cartesian closed.
Proof:(1). Let A and B be A-objects and let r : B A → A be an A-reflection arrow for B A .
Then by assumption id × r : A × B A → A × A is an A-reflection arrow for A × B A .
Thus there exists a unique morphism e : A × A → B with ev = e ◦ (idA × r).
Furthermore, there exists a unique morphism s : A → B A with e = ev ◦ (idA × s).
A ×O AGGG
GGe
idA ×r
GG
GG
#
/B
A
A×B
ev

A × AG

GG
GGe
GG
GG

#
/B
A
A×B

idA ×sev

This immediately implies that s ◦ r = idB A . Hence r ◦ s ◦ r = idA ◦ r, so thatr ◦ s = idA . Consequently, r : B A → A is an isomorphism; i.e., B A belongs to A.
(2). If A and B are A-objects, ev : A × B A → B is an evaluation morphism in B, andc : A → B A is an A-coreflection arrow for B A , then ev ◦ (idA × c) : A × A → B iseasily seen to be an evaluation morphism in A.

27.10 EXAMPLES(1) The full subcategory Sym of Rel is cartesian closed (and is closed under the formation of power objects in Rel) since it is a reflective subcategory of Rel andthe reflector preserves products. In contrast, Pos, although it is cartesian closed,
is not closed under the formation of power objects in Rel. Also, the full (reflective) subcategory that consists of all transitive relations is not cartesian closed: for
A = ({0, 1}, {0, 1}) the functor (A × −) does not preserve the quotient of A + A that
identifies 1 of the first copy of A with 0 of the second copy.
(2) Top is not cartesian closed even though it is a reflective subcategory of Conv, whichis cartesian closed.

CARTESIAN CLOSED CONCRETE CATEGORIES
27.11 DEFINITION
A concrete category (A, U ) over X is called concretely cartesian closed provided
that the following hold:

9th July 2006

434

Cartesian Closedness and Partial Morphisms[Chap. VII(1) A and X are cartesian closed,(2) U preserves finite products, power objects, and evaluation; in particular, wheneverev
A × B A −−→ B is an evaluation in A, thenevev

U (A × B A −−→ B) = U A × U B U A −−→ U B
is an evaluation in X.
27.12 EXAMPLES(1) Rel is a concretely cartesian closed construct. Cf. Example 27.3(1)(b).
(2) Pos and Alg(1) are constructs that are cartesian closed but not concretely so. Cf.
Examples 27.3(1)(c) and (d).
(3) A partially ordered set, considered as concrete category over 1, is cartesian closed if
and only if it is concretely so.
(4) Spa(P) is concretely cartesian closed. More generally, a functor-structured category
is cartesian closed if and only if it is concretely so (cf. Exercise 27I).
27.13 REMARKFor topological categories, concrete cartesian closedness is well understood, as the following results indicate.
27.14 PROPOSITION
If (A, U ) is a topological category over X and if A is cartesian closed, then so is X.
Proof: X is isomorphic to the full subcategory of A consisting of all indiscrete objects,
which by Proposition 27.9(1) is cartesian closed.

27.15 CHARACTERIZATION THEOREM FOR CONCRETELYCARTESIAN CLOSED TOPOLOGICAL CATEGORIESFor a topological category (A, U ) over a cartesian closed category X the following are
equivalent:(1) (A, U ) is concretely cartesian closed,(2) A is cartesian closed and every A-morphism with a discrete codomain has a discrete
domain,(3) for each A-object A the functor (A × −) preserves final sinks.
fi

Proof: (1) ⇒ (3). Let S = (Bi −−→ B)I be a final sink. To show that A × S =id ×fifA(A × Bi −−−−−→ A × B)I is final, let U (A × B) −→ U C be an X-morphism such thatf ◦(idA ×fi )ev

all A × Bi −−−−−−−→ C are A-morphisms. Let A × C A −−→ C be an evaluation in A.
ev
Then by (1), U A × U C U A −−→ U C is an evaluation in X. Thus in A, for each i ∈ I,
gi
there exists a unique morphism Bi −−→ C A with f ◦ (idA × fi ) = ev ◦ (idA × gi ), and9th July 2006

Sec. 27]

Cartesian closed categories

435g

in X there exists a unique morphism U B −→ U C U A with f = ev ◦ (idU A × g). Henceev ◦ (idA × (g ◦ fi )) = ev ◦ ((idA × g) ◦ (idA × fi )) = f ◦ (idA × fi ) = ev ◦ (idA × gi ) impliesgthat g ◦fi = gi , so that since S is final, U B −→ U (C A ) is an A-morphism. Consequently,
id ×g

f

ev

U (A × B) −→ U C = U (A × B) −−A−−→ U (A × C A ) −−→ U C is an A-morphism as well.
ev(3) ⇒ (1). Let A and B be A-objects and let U A × U B U A −−→ U B be an evaluationfimorphism in X. Consider the structured sink S = (U Ai −−→ U B U A )I consisting of all Xev◦(idA ×fi )fimorphisms fi such that U (A×Ai ) −−−−−−−−→ U B is an A-morphism. Let (Ai −−→ B A )I
id ×fiAbe a final lift of S. Then, by (3), the sink (A × Ai −−−−−→ A × B A )I is final too. Thusev
U (A × B A ) −−→ U B is an A-morphism. It is easily seen to be the desired evaluation inA.

f(1) ⇒ (2). Let A −→ D be an A-morphism with discrete codomain D. To show that A
gis discrete, consider an X-morphism U A −→ U B. Let (A × D , (πA , πD )) be a product.
Since an object C is discrete if and only if the empty sink with codomain C is final,g◦πA→ U B is an A-morphism.
(3) implies that A × D is discrete. Hence U (A × D) −−−−
hidA ,f i

g

g◦π

A
→ U B is an A-morphism.
Consequently, U A −→ U B = U A −−−−−→ U (A × D) −−−−ev(2) ⇒ (1). Let A × B A −−→ B be an evaluation in A. To establish that its imagefev

U A × U (B A ) −−→ U B is an evaluation in X, let U A × X −→ U B be an X-morphism andlet D be the discrete object with U D = X. By condition (2), A × D is discrete since theπfprojection A×D −−D→ D is an A-morphism. Thus U (A×D) −→ U B is an A-morphism.
g
Hence there exists a unique A-morphism D −→ B A with f = e ◦ (idA × g). Thus, bygdiscreteness of D, X −→ U (B A ) is the unique X-morphism with f = e ◦ (idU A × g). 

CARTESIAN CLOSED CONSTRUCTS
Relatively few of the familiar cartesian closed constructs are concretely cartesian closed.
More frequently, the underlying sets U (B A ) of powers in cartesian closed constructs(A, U ) are not of the form U B U A, but rather of the form homA (A, B). As we will see,this phenomenon is closely related to the fact that the terminal objects are often discrete,or, equivalently, that all constant maps between A-objects are A-morphisms.
27.16 PROPOSITION
Every concretely cartesian closed amnestic construct with discrete terminal object is, up
to concrete isomorphism, a full subconstruct of Set.
Proof: In Theorem 27.15 the implication (1) ⇒ (2) holds without the assumption that
(A, U ) is topological. Thus the discreteness of the terminal object implies that every
object is discrete.

27.17 DEFINITION
A construct (A, U ) is said to have function spaces provided that the following hold:

9th July 2006

436

Cartesian Closedness and Partial Morphisms[Chap. VII(1) (A, U ) has finite concrete products,ev(2) A is cartesian closed and the evaluation morphisms A × B A −−→ B can be chosen
in such a way that U (B A ) = homA (A, B) and ev is the restriction of the canonicalevaluation map in Set.
27.18 PROPOSITION
Let (A, U ) be a construct with finite concrete products. If A is cartesian closed, then thefollowing conditions are equivalent:(1) (A, U ) has function spaces,(2) terminal A-objects are discrete,(3) each constant function73 between A-objects is an A-morphism.
f

Proof: (1) ⇒ (2). Let T be a terminal object in A and let U T −→ U A be a function.
Then f is a constant function, whose value will be denoted by a. Consider the evaluationev
π
T × AT −−→ A as specified by Definition 27.17. Since the projection T × A −→ A is aπ̂morphism, A −→ AT is a morphism. Hence π̂(a) ∈ hom(T, A). Since π̂(a) agrees withff , it follows that U T −→ U A is a morphism. Thus T is discrete.
ev(2) ⇒ (1). Let A × B A −−→ B be an evaluation morphism in A and let T be a terminal
A-object. Since empty products are concrete, U T is a singleton set, say, {t}. Since T
is terminal, it is clear that the function h : homA (A, B) → homA (A × T, B) definedfπfby h(A −→ B) = A × T −−A→ A −→ B is a bijection. Also, we have the bijectionfk : homA (A × T, B) → homA (T, B A ) that associates with any morphism A × T −→ Bk(f )the unique morphism T −−−→ B A with f = ev ◦ (idA × k(f )). The function` : homA (T, B A ) → U (B A ) defined by `(f ) = f (t) is a bijection since T is terminal, hence discrete by (2).
`◦k◦h

`◦k◦h

Let C −−−−→ B A be an initial lift of the bijection homA (A, B) −−−−→ U (B A ) and set
ev
ev = ev ◦ (idA × (` ◦ k ◦ h)). Then A × C −−→ B is an evaluation with U C = homA (A, B).
A simple calculation shows that ev(a, f ) = f (a) for a ∈ U A and f ∈ homA (A, B). Thus
(A, U ) has function spaces.
(2) ⇔ (3) is obvious.



27.19 REMARK
In topological constructs, terminal objects are discrete if and only if the fibres of eachone-element set contain exactly one element. To avoid uninteresting pathologies we will
require a little more:
27.20 DEFINITION
A construct is called well-fibred provided that it is fibre-small and for each set with at
most one element, the corresponding fibre has exactly one element.
73

A function is called constant provided that it factors through a one-element set.

9th July 2006

Sec. 27]

Cartesian closed categories

437

27.21 EXAMPLES
The following topological constructs are well-fibred and have function spaces: Conv of
convergence spaces, PsTop of pseudotopological spaces, sTop of sequential topologicalspaces, kTop of (compact Hausdorff)-generated topological spaces, Bor of bornologicalspaces, Simp of simplicial complexes, Prost of preordered sets, and Rere of reflexive
relations.
27.22 THEOREM
For well-fibred topological constructs the following conditions are equivalent:(1) A is cartesian closed,(2) (A, U ) has function spaces,(3) for each A-object A the functor (A × −) preserves final epi-sinks,(4) for each A-object A the functor (A × −) preserves colimits,(5) for each A-object A the functor (A × −) preserves (a) coproducts and (b) quotients.
Proof:(1) ⇔ (2). Immediate from Proposition 27.18.
(1) ⇔ (4). Theorem 27.4.
(4) ⇒ (5) follows from the fact that quotient maps are regular epimorphisms. (Cf.
Proposition 21.13.)fi(5) ⇒ (3) follows from the facts that: (a) a sink (Bi −−→ B)I is a final epi-sink if and only
fj

if there exists a subset J of I such that the small sink (Bj −−→ B)J is a final epi-sink,
`
[fj ]
fj
and (b) a small sink (Bj −−→ B)J is a final epi-sink if and only if J Bj −−−→ B is aquotient map.
(3) ⇒ (4) follows from concreteness of both colimits and finite products in (A, U ) andthe fact that the functor (U A × −) : Set → Set preserves colimits.

27.23 EXAMPLES(1) Top is a well-fibred topological construct that satisfies the above condition (5)(a),but not (5)(b).
(2) Unif is a well-fibred topological construct that satisfies the above condition (5)(b),but not (5)(a).
(3) Alg(1) and Cat [considered as a construct via the functor U : Cat → Set, definedFF

M
by U (A −−→ B) = M or(A) −−−→ M or(B)] are cartesian closed well-fibred nontopological constructs that have finite concrete products, but don’t have function
spaces.

27.24 PROPOSITION
For cartesian closed, well-fibred topological constructs the following hold:

9th July 2006

438

Cartesian Closedness and Partial Morphisms[Chap. VII(1) products with discrete factors A are coproducts:
a
A×B ∼
B,
= |A| B =x∈|A|(2) power objects with discrete exponents A are powers:
Y
BA ∼
B.
= B |A| =x∈|A|`
Proof: This follows immediately from A ∼= x∈X T , where T is a terminal object, andfrom Proposition 27.8(3) and (4).


Suggestions for Further Reading
Brown, R. Ten topologies for X × Y . Quart. J. Math. 2(14), 1963: 303–319.
Eilenberg, S., and G. M. Kelly. Closed categories. Proceedings of the Conference on
Categorical Algebra (La Jolla, 1965), Springer, Berlin–Heidelberg–New York, 1966,
421–562.
Day, B. J., and G. M. Kelly. On topological quotient maps preserved by pullbacks or
products. Proc. Cambridge Phil. Soc. 67 (1970): 553–558.
Day, B. A reflection theorem for closed categories. J. Pure Appl. Algebra 2 (1972):
1–11.
Wyler, O. Convenient categories for topology. Gen. Topol. Appl. 3 (1973): 225–242.
Herrlich, H. Cartesian closed topological categories. Math. Coll. Univ. Cape Town 9(1974): 1–16.
Nel, L. D. Initially structured categories and cartesian closedness. Canad. J. Math. 27(1975): 1361–1377.
Isbell, J. R. Function spaces and adjoints. Math. Scand. 36 (1975): 317–339.
Herrlich, H., and L. D. Nel. Cartesian closed topological hulls. Proc. Amer. Math. Soc.
62 (1977): 215–222.
Wyler, O. Function spaces in topological categories. Springer Lect. Notes Math. 719(1979): 411–420.
Frölicher, A. Catégories cartésienne fermées engendrées par des monoids.
Topol. Geom. Diff. 21 (1980): 367–375.

Cahiers

Schwarz, F. Powers and exponential objects in initially structured categories and applications to categories of limit spaces. Quaest. Math. 6 (1983): 227–254.
Adámek, J., and H. Herrlich. Cartesian closed categories, quasitopoi and topologicaluniverses. Comment. Math. Univ. Carolinae 27 (1986): 235–257.

9th July 2006

Sec. 27]

Cartesian closed categories

439

Herrlich, H., and G. E. Strecker. Cartesian closed topological hulls as injective hulls.
Quaest. Math. 9 (1986): 263–280.
Isbell, J. R. General function spaces, products and continuous lattices. Math. Proc.
Cambridge Phil. Soc. 100 (1986): 193–205.

EXERCISES
27A. Cartesian Closed Categories With Zero-Objects
Let A be cartesian closed. Show that(a) If I is an initial object, then A × I ∼= I for each A-object A.
(b) The following are equivalent:(1) A is pointed,(2) A has a zero-object,(3) A is equivalent to 1.
27B. ∆Pos is Cartesian Closed
Show that the category ∆Pos is cartesian closed. [Analogously to the situation in Pos,B A is the pointwise-ordered set of all ∆-continuous functions from A to B.]
27C. Coreflective Hulls and Cartesian Closedness
Let (A, U ) be a cartesian closed topological construct and let B be a full subcategoryof A that is closed under the formation of finite products. Show that the bicoreflective
hull of B in A is cartesian closed.
27D. Cartesian Closed Constructs
Let T be a terminal object in a cartesian closed topological construct (A, U ), and foreach A-object A let A∗ denote the discrete object with U A∗ = U A. Show thatev(a) Power objects B A and evaluation morphisms A × B A −−→ B can be chosen such thatfπf

U (B A ) = {U A −→ U B A × T ∗ −−A→ A −→ B

is an A-morphism}

ev(a, f ) = f (a).
π(b) (A, U ) has function spaces if and only if A × T ∗ −−A→ A is an isomorphism for eachA-object A.
(c) (A, U ) is concretely cartesian closed if and only if A × T ∗ = (A × T )∗ for eachA-object A.
(d) (A, U ) is concretely cartesian closed and has function spaces if and only if U is an
isomorphism.

9th July 2006

440

Cartesian Closedness and Partial Morphisms[Chap. VII

27E. Composition as a Morphism
In cartesian closed constructs describe explicitly the unique morphismcomp : B A × C B → C A that makes the diagram
A × (B A × C B )idA ×compev×idC Bev



A × CA/ B × CBev

/Ccommute.
* 27F. (Concretely) Cartesian Closed Topological Categories
as Injective Objects
Let X be a cartesian closed category and let CATp (X) be the quasicategory whose
objects are the amnestic concrete categories over X with finite concrete products, andwhose morphisms are the concrete functors over X that preserve finite products.
(a) Show that the injective objects in CATp (X) (with respect to full embeddings) are
precisely the concretely cartesian closed topological categories over X.
Let CONSTp be the quasicategory whose objects are the amnestic well-fibred constructs with finite concrete products, and whose morphisms are the concrete functorsthat preserve finite products.
(b) Show that the injective objects in CONSTp (with respect to full embeddings) are
precisely the well-fibred topological constructs that have function spaces.
* 27G. (Concretely) Cartesian Closed Topological Hulls
E

Let X be a cartesian closed category. A morphism (A, U ) −−→ (B, V ) in CATp (X) is
called a concretely cartesian closed topological hull (shortly: a CCCT hull) of(A, U ) provided that the following conditions are satisfied:(1) E is a full embedding,(2) E[A] is finally dense in (B, V ),(3) { EAEA

A, A ∈ Ob(A) } is initially dense in (B, V ),(4) (B, V ) is a concretely cartesian closed topological category.
(a) Show that the injective hulls in CATp (X) are precisely the CCCT hulls.
E

A morphism (A, U ) −−→ (B, V ) in CONSTp is called a cartesian closed topological hull (shortly: a CCT hull) of (A, U ) provided the above conditions (1),(2), (3), and the following (4∗ ) hold:(4∗ ) (B, V ) is a cartesian closed topological category.
Show that(b) The injective hulls in CONSTp are precisely the CCT hulls.

9th July 2006

Sec. 27]

Cartesian closed categories

441(c) The concrete embedding PrTop ,→ PsTop is a cartesian closed topological hull of
PrTop.
27H. Well-Fibred Topological Constructs
Show that(a) In well-fibred topological constructs extremal subobjects of discrete objects are discrete. [Cf. Exercise 8M(a).](b) A functor-structured construct Spa(T ) is well-fibred only if T is the constant functor,fid∅defined by T (X −→ Y ) = ∅ −−→ ∅, i.e., only if Spa(T ) is concretely isomorphic tothe construct Set.
27I. Cartesian Closed Functor-Structured Categories(a) Prove that if Spa(T ) is cartesian closed, then it is concretely cartesian closed.
(b) Prove that Spa(T ) is (concretely) cartesian closed whenever X is cartesian closedfg

and T weakly preserves pullbacks, i.e., for each 2-sink • −→ • ←− • the factorizing morphism of the T -image of the pullback of (f,g) through the pullback of(T f, T g) is a retraction.
(c) Verify that the Set-functors S n and P weakly preserve pullbacks.
27J. Re: Theorem 27.15
Prove that the equivalence of 27.15(1) and 27.15(3) follows from the dual of the Galois
Correspondence Theorem (21.24).

9th July 2006

28 Partial morphisms, quasitopoi, andtopological universes
A partial morphism from A to B is a morphism from a subobject of A to B. Sometimes it
is possible to “represent” partial morphisms into B by ordinary morphisms into a suitable
extension B ∗ of B in such a way that partial morphisms from A into B are precisely
the pullbacks of ordinary morphisms from A to B ∗ along a distinguished embeddingmB
B −−−→ B ∗ . Since there is no satisfactory categorical concept of subobject or of embedding, partial morphisms will be defined with respect to an arbitrary class M ofmorphisms, acting as “embeddings of subobjects”. The most interesting cases will bethose where M consists of all extremal monomorphisms resp. of all monomorphisms. Asin §27 we first will consider abstract categories, then topological categories, and, finally,
topological constructs.

REPRESENTATIONS OF PARTIAL MORPHISMS
28.1 DEFINITION
Let M be a class of morphisms in A.
fm(1) A 2-source (A ←−− • −→ B) with m ∈ M is called an M -partial morphismfrom A into B. (Extremal) Mono-partial morphisms are called (extremal) partial
morphisms.
mB(2) An M -morphism B −−−→ B ∗ is said to represent M -partial morphisms into B
provided that the following two conditions are satisfied:f(a) for every morphism • −→ B ∗ there exists a pullback
◦
f





B

/•

m

mB

f

/ B∗

and every such m belongs to M ,fm(b) for every M -partial morphism (• ←−− ◦ −→ B) there exists a unique morphismf∗• −−→ B ∗ such that
◦
f

442

/•




B
is a pullback.

m

mB

f∗

/ B∗

Sec. 28] Partial morphisms, quasitopoi, and topological universes

443(3) A is said to have representable M -partial morphisms provided that for eachmB
A-object B there exists some B −−−→ B ∗ that represents M -partial morphisms into
B.
28.2 EXAMPLES(1) Set has representable (extremal) partial morphisms. For each set B the one-point
extension B ,→ B ] {∞} represents both types of partial morphisms into B: givenf

A0 ⊆ A and A0 −→ B, define f ∗ : A → B ] {∞} by(f (x), for x ∈ A0f ∗ (x) =∞,otherwise.
(2) The category Alg(1) of algebras with one unary operation has representable (extremal) partial morphisms. For each object (B, β) the extensionm(B, β) −−→ ((B × ) ∪ {∞} , β ∗ ), defined by m(b) = (b, 0) and
if x = ∞∞,∗β (x) = (b, n),if x = (b, n + 1)(β(b), 0), if x = (b, 0),N

represents (extremal) partial morphisms into (B, β).
(3) Rel has representable extremal partial morphisms. For each object (X, ρ) the onepoint extension(X, ρ) ,→ (X ] {∞} , ρ ∪ (X × {∞}) ∪ ({∞} × X) ∪ {(∞, ∞)})
represents extremal partial morphisms into (X, ρ). However, Rel does not haverepresentable partial morphisms.
(4) Top has neither representable extremal partial morphisms nor representable partial
morphisms. However, for the class M of all closed [resp. open] embeddings, it does
have representable M -partial morphisms. The one-point extension(X, τ ) ,→ (X ] {∞} , {∅} ∪ { A ] {∞} A ∈ τ })[resp. (X, τ ) ,→ (X ] {∞} , τ ∪ {X ] {∞}})]
represents M -partial morphisms into (X, τ ).
(5) Every category has representable Iso-partial morphisms, since these are essentiallyjust morphisms. In particular, every partially ordered set, considered as a category, has representable extremal partial morphisms. However, such a category has
representable partial morphisms only if the order relation is equality.
28.3 PROPOSITION
If A has representable M -partial morphisms, then the following hold:

9th July 2006

444

Cartesian Closedness and Partial Morphisms[Chap. VII(1) Iso(A) ⊆ M ⊆ RegMono(A).
(2) Pullbacks of M -morphisms exist and belong to M .
Proof:
fmB(1). Let A −→ B be a morphism and let B −−−→ B ∗ represent M -partial morphismsginto B. If f belongs to M , then there exists a morphism B −→ B ∗ such that
f

A
f

/B




B

mB

g

/ B∗

is a pullback. This implies that f is an equalizer of g and mB , and hence is a regularmonomorphism.
If f is an isomorphism, then the square
A
f

f





B

/B

mB

mB

/ B∗

is a pullback since mB , by the above, is a monomorphism. Thus f ∈ M .
m

f

m

A
(2). Let (A −−→ B ←− C) be a 2-sink with m ∈ M , and let A −−−→ A∗ represent
M -partial morphisms into A. Then there exist pullbacks

A
idA

m





A

/B

mA

•

g

and

f

/ A∗

/C

m





A

mA

g◦f

/ A∗

k

with m ∈ M . Thus there exists a unique morphism • −→ A such that
m
/C
>
// >
// >>>kf// >> ∗///Bmf // A//g// idA


/ A∗
A

•/>

mA

commutes. By Proposition 11.10(2) the square designated by ∗ is a pullback.



28.4 REMARK
If A has representable M -partial morphisms, then M need not be closed under composition, as the following example shows: A is the category of sets and injective maps, andM consists of all injective maps f : A → B such that B \ f [A] has at most one element.

9th July 2006

Sec. 28] Partial morphisms, quasitopoi, and topological universes

445

28.5 PROPOSITION
If A has finite products and representable M -partial morphisms, where M is a family
that contains all sections, then the following hold:(1) A is finitely complete.
(2) M = RegMono(A).
Proof:(1). In view of Theorem 12.4 it suffices to show that any pair of A-morphisms A

f
g

// B

has an equalizer. Since hidA , f i : A → A × B is a section, it belongs to M . Henceby Proposition 28.3 there exists a pullback
•

/A

m

hidA ,gi

×B
A hid ,f/ A
iA

with m ∈ M . By Proposition 11.14, m is an equalizer of f and g.
(2). By Proposition 28.3(1) it suffices to show that every regular monomorphism belongse

to M . Let A −→ B be an equalizer of B
A
e



B

f

//

g

C . Then/BehidB ,gi/B×C

hidB ,f i

is a pullback. Since hidB , f i is a section, and so belongs to M , by Proposition
28.3(2), e belongs to M .
28.6 COROLLARY
If A has finite products and representable (extremal) partial morphisms, then the following hold:(1) A is finitely complete.
(2) (Extr)Mono(A) = RegMono(A).
(3) In A regular monomorphisms are closed under composition.
Proof: (1) and (2) follow from Proposition 28.5, and (3) follows from (2) and Proposition
7.62(1).


9th July 2006

446

Cartesian Closedness and Partial Morphisms[Chap. VII

QUASITOPOI
28.7 DEFINITION
Let M be a class of morphisms in a category A. Then A is called an M -topos provided
that it:(1) has representable M -partial morphisms,(2) is cartesian closed, and(3) is finitely cocomplete.
Mono-topoi are called topoi, and ExtrMono-topoi are called quasitopoi.
28.8 PROPOSITION
Topoi are precisely the balanced quasitopoi.
Proof: Immediate from Proposition 28.5(2) and Proposition 7.67.



28.9 EXAMPLES(1) Set, Alg(1), the category of finite sets, and all functor categories of the form
[A, Set], where A is small, are topoi.
(2) Rel, Conv, and PsTop are quasitopoi, but are not topoi.
(3) A linearly ordered set, considered as a category, is(a) cartesian closed if and only if it has a largest element,(b) a quasitopos if and only if it has a largest element and a smallest element,(c) a topos if and only if it has precisely one element.
(4) Cat and kTop are cartesian closed and cocomplete, hence Iso-topoi; but they are
not quasitopoi.
(5) pSet, and PrTop (equivalently: Clos, see 5N) are complete and cocomplete andtheir extremal partial morphisms are representable, but they fail to be cartesianclosed.
28.10 PROPOSITION
Each quasitopos is (Epi, RegMono)-structured.
Proof: Immediate from Proposition 28.6(3) and the dual of Proposition 14.22.



28.11 REMARKS
Quasitopoi have a rich inner structure:

9th July 2006

Sec. 28] Partial morphisms, quasitopoi, and topological universes

447(1) There is an “extremal-subobject classifier”, i.e., a morphism T → Ω that represents
extremal partial morphisms into a terminal object T .
Cartesian closed categories with subobject classifiers are automatically topoi. But
cartesian closed categories with extremal-subobject classifiers may fail badly to bequasitopoi. For example,(a) Prost, Cat, and kTop are cartesian closed, cocomplete, and have extremalsub-object classifiers, but fail to be quasitopoi,(b) the ordered sets

•

1

•

1

•

2

•

2

3

•
..
.

3

•
..
.

n

•
..
.

n

•
..
.

and

•

•
@
@

•

considered as categories, are cartesian closed and have representable extremal
partial morphisms, but fail to be finitely cocomplete.
(2) For each object A there is an “elementhood-morphism” εA , defined by the pullback•εA/ A × ΩA




ev

/Ω

T

(3) For each object A there is a “singleton-morphism” σA : A → ΩA , defined by the
diagramAhidA ,idA ipullback

T

idA ×σA/ A × ΩA
n
n
n
nnn
nnnev
n
n
 nn
/ vn

/A×A

Ω

f

(4) For each morphism A −→ B there is a “preimage-morphism” f ∗ : B → ΩA , definedby the diagramA

T

9th July 2006

hidA ,f ipullback/A×B

idA ×f ∗/ A × ΩA
n
n
n
nnn
nnnev
n
n
 vnnn/Ω448

Cartesian Closedness and Partial Morphismsf[Chap. VIIΩf(5) For each morphism A −→ B there is a “power-set-morphism” ΩB −−→ ΩA , definedby the diagramA × ΩB
idA ×Ωff ×id/ B × ΩBev



A × ΩA



/Ω

ev

f

(6) For each morphism A −→ B there is a “pullback along f ” functor(A ↓ B) → (A ↓ A) between the corresponding comma categories that is simultaneously an adjoint and a co-adjoint functor.

CONCRETE QUASITOPOI
If a topological category (A, U ) has representable partial morphisms, then, by Proposition 28.3(1), A is balanced, so that U is an isomorphism (cf. Exercise 21N). Thus
for topological categories the concepts of representability of partial morphisms, and inparticular of topoi, are not interesting. However, the concepts of representability of
extremal partial morphisms (and in particular of quasitopoi) are fruitful.
28.12 PROPOSITION
Let (A, U ) be a topological category over X, let M be a class of morphisms in X, and let
Minit be the class of all initial A-morphisms m with U m ∈ M . If A has representable
Minit -partial morphisms, then X has representable M -partial morphisms. U preserves
these representations if and only if Minit -subobjects of discrete objects are discrete.
Proof:(1). To show that X has representable M -partial morphisms, consider the full subcategory B of A that consists of all indiscrete objects. Since U restricts to an
isomorphism from B to X, and since B is closed under the formation of pullbacksmB
in A, it suffices to show that B is closed under representations B −−−→ B ∗ of Minit partial morphisms in A, i.e., that B ∈ B implies that B ∗ ∈ B. For this purposeflet U A −→ U B ∗ be an X-morphism. Let D be the discrete object with U D = U A,
and let
P
fˆ

m̂





B

/D

mB

f

/ B∗
m

B
be a pullback in A. Then m̂ ∈ Minit . Since B −−−→ B ∗ represents Minit -partialfmorphisms into B and since D is discrete, U D −→ U B ∗ is the only X-morphism g

9th July 2006

Sec. 28] Partial morphisms, quasitopoi, and topological universes

449

that makes
UP
fˆ

m̂

mB

(∗)

g





UB

/ UD
/ U B∗m̂m̂

m̂

a pullback in X. Let C −−→ A be an initial lift of U P −−→ U A. Then C −−→ Afˆ

belongs to Minit . Moreover, U C −→ U B is an A-morphism, since B is indiscrete.
m̂fˆHence A ←−− C −→ B is an Minit -partial morphism from A to B. Thus there existsgan A-morphism A −→ B ∗ that makes
C
fˆ

m̂

g





B

/A

mB

/ B∗

a pullback in A. Hence
UC
fˆ

m̂





UB

/ UA

mB

g

/ U B∗
fa pullback in X. By (∗) this implies that g = f . Hence U A −→ U B ∗ is an Amorphism.
(2). That U preserves representations if and only if Minit -subobjects of discrete objects
are discrete follows by a straightforward calculation.

28.13 DEFINITION
fim(1) Let B −−→ A be a morphism and let S = (Ai −−→ A) be a sink. If for each i ∈ I thediagram
Bi
fˆi

mi



B

/ Ai


m

fi

/A

fˆi

is a pullback, then the sink (Bi −−→ B)I is called a pullback of S along m.
(2) Let M be a class of morphisms and let C be a conglomerate of sinks in a category
A. C is called stable under pullbacks along M provided that every pullback ofa sink in C along a M -morphism is a member of C.
In particular, C is called pullback-stable provided that C is stable under pullbacks
along M or(A). In the case that M is a class of monomorphisms, C is called reducible provided that it is stable under pullbacks along M . When M is the classof all extremal monomorphisms, we say that C is extremally reducible.

9th July 2006

450

Cartesian Closedness and Partial Morphisms[Chap. VII

28.14 REMARK
In view of the fact that diagrams of the form
A×B
idA ×f

πB

/B



f



/C

A × C πC

are pullbacks (see Exercise 11D), Theorem 27.15 can be restated as follows:
If X is cartesian closed, then for each topological category (A, U ) over X the following
conditions are equivalent:(1) (A, U ) is concretely cartesian closed,(2) final sinks in (A, U ) are stable under pullbacks along projections.
28.15 THEOREM
If X has representable M -partial morphisms, then for each topological category (A, U )over X the following conditions are equivalent:(1) A has representable Minit -partial morphisms and U preserves these representations,(2) final sinks in (A, U ) are Minit -reducible.
fim

Proof: (1) ⇒ (2). Let S = (Ai −−→ A)I be a final sink and let B −−→ A belong to Minit .
Then for each i ∈ I there exists a pullback
Bi
fˆi

mi

/ Ai

B

fi





/A

m

fˆi

g

with mi ∈ Minit . To show that the sink T = (Bi −−→ B)I is final, let U B −→ U C be an
g◦fˆi

m

C
→ C ∗ represent
X-morphism such that all U Bi −−−→ U C are A-morphisms. Let C −−−mC∗
Minit -partial morphisms into C. Then, by (1), U C −−−→ U C represents M -partialĝ

morphisms into U C. Hence there exists a unique X-morphism U A −→ U C ∗ such that
/ UA

m

UB
g

ĝ





UC

mC

/ U C∗
gi

is a pullback in X, and for each i ∈ I there is a unique A-morphism Ai −−→ C ∗ such
that
mi
/ Ai
Bi
g◦fˆi





C

mC

gi

/ C∗

9th July 2006

Sec. 28] Partial morphisms, quasitopoi, and topological universes

451

is a pullback in A. Thus, for each i ∈ I,
U Bi
g◦fˆi

mi





UC

/ U Ai

mC

U Bi
and

gi

g◦fˆi

/ U C∗fi/ U Ai




UC

gi

mi

mC

ĝ◦fi

/ U C∗ĝare pullbacks in X. Hence U Ai −−→ U C ∗ = U Ai −−→ U A −→ U C ∗. Since S is final,
ĝ

g

m

ĝ

m

C
U A −→ U C ∗ is an A-morphism. Thus U B −→ U C −−−→ U C ∗ = U B −−→ U A −→ U C ∗g
mC∗
is an A-morphism. Since C −−−→ C is initial, U B −→ U C is an A-morphism.

mB(2) ⇒ (1). Let B be a B-object and let U B −−−→ X represent M -partial morphisms

fi

m

i
into U B. Consider the family (Ai ←−−Bi −−→ B)I of all Minit -partial morphisms intogi
B. Then for each i ∈ I there exists a unique X-morphism U Ai −−→ X such that

U Bi
fi

mi



UB

/ U Ai
gi



mB

/X
gi

gi

is a pullback in X. Let (Ai −−→ B ∗ )I be a final lift of the structured sink (U Ai −−→ X)I .
mB
In order to prove that B −−−→ B ∗ represents Minit -partial morphisms into B it sufficesmB∗
to show that U B −−−→ U B belongs to Minit . Since
UB
idU B

idB



UB

/ UB


mB

mB

/X
m

B
is a pullback, there exists an i ∈ I with (Ai , gi ) = (B, mB ). Thus U B −−−→ U B∗ =gi
mm
B
B
U Ai −−→ U B ∗ is an A-morphism. If A −−−→ B ∗ is an initial lift of U B −−−→ U B ∗ , thenB ≤ A (cf. 5.4). So it remains to be shown that A ≤ B. For each i ∈ I the diagram

Bi
fi

mi





A

/ Ai

mB

gi

/ B∗
m

i
is a pullback in A since it is a pullback in X and Bi −−→
Ai is initial. Hence, by (2),

fi

fi

fi

id

the sink (Bi −−→ A)I is final. Thus U Bi −−→ U B = U Bi −−→ U A −−U−A→ U B implies
idthat U A −−U−A→ U B is an A-morphism, i.e., that A ≤ B.

28.16 DEFINITION
A concrete category is called universally topological provided that it is topologicaland final sinks are pullback-stable.

9th July 2006

452

Cartesian Closedness and Partial Morphisms[Chap. VII

28.17 EXAMPLES(1) Rel is a universally topological construct.
(2) If a functor T : X → Set weakly preserves pullbacks [cf. Exercise 27I], then Spa(T )is universally topological over X. In particular, Spa(P) is universally topological.
(3) Frames are (cartesian closed and hence) universally topological over 1.
(4) The construct of compactly generated topological spaces is not universally topological (although it is cartesian closed and topological). In fact, a “non-concretely”
cartesian closed construct cannot be universally topological:
28.18 THEOREM
For topological categories (A, U ) over a quasitopos X the following conditions are equivalent:(1) (A, U ) is universally topological,(2) (A, U ) is a concrete quasitopos, i.e., A is a quasitopos and U preserves evaluation,
power objects, and representations of extremal partial morphisms.
Proof: (1) ⇒ (2). Immediate from Remark 28.14, Theorem 28.15, and Proposition
28.12.
(2) ⇒ (1). By the above-mentioned results, (2) implies that in (A, U ) final sinks arestable under pullbacks along projections and along extremal monomorphisms. Since eachffhidA ,f iπA-morphism A −→ B can be expressed as A −→ B = A −−−−−→ A × B −−B→ B, sincehidA , f i is a section and hence is an extremal monomorphism, and since pullbacks can becomposed [cf. Proposition 11.10(1)], this implies that (A, U ) is universally topological.

TOPOLOGICAL UNIVERSES
28.19 THEOREM
For well-fibred topological constructs, the following conditions are equivalent:(1) extremal partial morphisms are representable,(2) final sinks are extremally reducible,(3) final epi-sinks are extremally reducible,(4) coproducts and quotients are extremally reducible.
Proof: (1) ⇔ (2) is immediate from Theorem 28.15. Cf. Exercise 28B(a).
(2) ⇒ (3) is obvious.

9th July 2006

Sec. 28] Partial morphisms, quasitopoi, and topological universes

453(3) ⇒ (2). Let O (resp. T ) be the unique object with underlying set ∅ (resp. {0}). Iffjfi

S = (Ai −−→ A)I is a final sink, enlarge S to a final epi-sink T = (Aj −−→ A)J by addingm
all morphisms from T to A. Let B −−→ A be an extremal monomorphism and let
Bj
fˆj

m̂j





B

/ Aj

m

fj

/A

fˆj

be a pullback for each j ∈ J. Then (Bj −−→ B)J is a final epi-sink. Since for eachj ∈ J \ I the morphism fˆj has a discrete domain (namely, O or an object isomorphic tofˆiT ), the restricted sink (Bi −−→ B)I is final as well.
(3) ⇔ (4) as in Theorem 27.22.



28.20 EXAMPLES(1) Extremal partial morphisms are representable in each of the following well-fibred
topological constructs: Conv, PsTop, PrTop (= pretopological spaces = closure
spaces), Bor, Simp, and Rere.
(2) Extremal partial morphisms are not representable for any of the following well-fibred
topological constructs: Top, Unif , PMet, sTop, kTop, and Prost.
28.21 DEFINITION
A well-fibred topological construct (A, U ) for which A is a quasitopos is called a topological universe.

9th July 2006

A topological universe

454

Cartesian Closedness and Partial Morphisms[Chap. VII

28.22 THEOREM
For a well-fibred topological construct (A, U ) the following conditions are equivalent:(1) (A, U ) is a topological universe,(2) (A, U ) has function-spaces and representable extremal partial morphisms,(3) in (A, U ) final epi-sinks are pullback-stable.
Proof: (1) ⇔ (2) is immediate from Theorem 27.22.
(2) ⇔ (3) can be established as in the proof of Theorem 28.18.



28.23 EXAMPLES
Conv, PsTop, Simp, Bor, and Rere are topological universes.

Suggestions for Further Reading
Lawvere, F. W. Introduction. Springer Lect. Notes Math. 274 (1972): 1–12.
Freyd, P. Aspects of topoi. Bull. Austral. Math. Soc. 7 (1972): 1–76 and 467–480.
Paré, R. Co-limits in topoi. Bull. Amer. Math. Soc. 80 (1974): 556–561.
Wyler, O. Are there topoi in topology? Springer Lect. Notes Math. 540 (1976): 700–
719.
Penon, J. Sur les quasi-topos. Cahiers Topol. Geom. Diff. 18 (1977): 181–218.
Booth, P. I., and R. Brown. Spaces of partial maps, fibred mapping spaces and thecompact-open topology. Gen. Topol. Appl. 8 (1978): 181–195.
Dubuc, E. J. Concrete quasitopoi. Springer Lect. Notes Math. 753 (1979): 239–254.
Lambek, J., and P. Scott. Algebraic aspects of topos theory. Cahiers Topol. Geom.
Diff. 22 (1981): 129–140.
Nel, L. D. Topological universes and smooth Gelfand-Naimark duality. Contemp. Math.
30 (1984): 244–276.
Dyckhoff, R., and W. Tholen. Exponentiable morphisms, partial products and pullback complements. J. Pure Appl. Algebra 49 (1987): 103–116.
Herrlich, H. On the representability of partial morphisms in Top and in related constructs. Springer Lect. Notes Math. 1348 (1988): 143–153.
Schwarz, F. Description of the topological universe hull. Springer Lect. Notes Comp.
Sci. 393 (1989): 325–332.

9th July 2006

Sec. 28] Partial morphisms, quasitopoi, and topological universes

455

EXERCISES
28A. Injectivity in Topological Constructs
Let (A, U ) be a well-fibred topological construct with representable extremal partial
morphisms. Show that in (A, U )m(a) An object B is injective if and only if there exists some A −−→ B that represents
extremal partial morphisms into A.
m(b) A −−→ B is an injective hull if and only if either(1) A is injective and m is an isomorphism, orm(2) A is not injective and A −−→ B represents extremal partial morphisms into A.
28B. Extensional Topological Constructs
A well-fibred topological construct (A, U ) is called extensional provided that it has
representable extremal partial morphisms.
(a) Show that if (A, U ) is extensional then U preserves representations of extremal
partial morphisms. [Cf. Proposition 28.12 and Exercise 27H(a).](b) For each A-object B denote by B ∗ the final lift of the structured sink consisting ofaall A −→ |B| ] ∞ whose restriction a0 to the initial subobject A0 of A on the seta0

a−1 [|B|] is an A-morphism A0 −−→ B. If the inclusion map B ,→ B ∗ is an initial(= extremal) monomorphism, we call B ∗ the canonical one-point extension of
B. Prove that a well-fibred topological construct is extensional if and only if every
A-object has a canonical one-point extension.
* (c) Let CONSTs be the quasicategory, whose objects are the well-fibred constructs
in which every injective structured arrow has a unique initial lift, and whose morphisms are the concrete functors that preserve embeddings. Show that the injective
objects in CONSTs (with respect to full embeddings) are precisely the extensional
topological constructs.
* 28C. Extensional Topological Hulls
E

A morphism (A, U ) −−→ (B, V ) in CONSTs is called an extensional topological hullof (A, U ) provided that the following conditions are satisfied:(1) E is a full embedding,(2) E[A] is finally dense in (B, V ),(3) the image under E of
{ BA

m

A
A ∈ Ob(A) and A −−−→ BA represents extremal partial morphisms into A }is initially dense in (B, V ),(4) (B, V ) is an extensional topological construct.

9th July 2006

456

Cartesian Closedness and Partial Morphisms[Chap. VII

Show that(a) Injective hulls in CONSTs are precisely the extensional topological hulls.
(b) Every object in (A, U ) has an injective hull.
(c) The concrete embedding Top ,→ PrTop is an extensional topological hull of Top.
28D. Posets as Quasitopoi
Let A be a poset, considered as a category. Show that:(a) A is a topos if and only if A contains precisely one element.
(b) If A is complete, then A is a quasitopos if and only if A is a frame.
28E. A Characterization of Topoi(a) Show that A is a topos if and only if A is cartesian closed and has representablepartial morphisms.
(b) Show that the category A of nonempty sets has the following properties:(1) A is cartesian closed,mB(2) for each A-object B there exists a monomorphism B −−−→ B ∗ such that for

f∗

f

m

every partial morphism • ←−− ◦ −→ B there exists a unique morphism • −−→ B
such that◦ m /•
f





B

mB

f∗

/ B∗

is a pullback,(3) in A partial morphisms are not representable.
28F. Regular Monomorphisms in (Quasi)Topoi
Show that:(a) In a quasitopos pushouts of regular monomorphisms are monomorphisms.
(b) In a topos monomorphisms and regular monomorphisms are pushout-stable.
(c) In a (quasi)topos every (extremal) monomorphism is regular.
* 28G. Universally Topological Categories as Injective Objects
Let X be a quasitopos and let CATps be the quasicategory whose objects are the concrete categories over X with finite concrete products and unique initial lifts of regularmonomorphic structured arrows, and whose morphisms are the concrete functors that
preserve finite products and those embeddings that are carried by regular monomorphisms.

9th July 2006

Sec. 28] Partial morphisms, quasitopoi, and topological universes

457(a) Show that the injective objects in CATps (with respect to full embeddings) are
precisely the universally topological categories.
(b) Describe injective hulls in CATps (X).
(c) Show that the concrete embedding Top ,→ PsTop is an injective hull in the quasicategory CATps (Set).
28H. Relational Categories
For every small concrete category (A, U ) over X the concrete category Rel(A, U ) of
(A, U )-ary relations over X is Spa(F(A,U ) ), where F(A,U ) : X → Set is the functor,
defined byaF(A,U ) (X) = { (A, a) | A ∈ Ob(A) and U A −→ X ∈ M or(X) }and F(A,U ) (f )(A, a) = (A, f ◦ a). A concrete category that is concretely isomorphic tosome Rel(A, U ) is called relational. Show that(a) Rel is a relational construct.
(b) Every relational category is functor-structured.
(c) If a functor-costructured construct (A, U ) is relational, then U is an isomorphism.
(d) Top is not concretely isomorphic to a full subconstruct of some relational construct.
(e) Relational categories are universally topological.
28I. Functor-Costructured Constructs
Prove that for functor-costructured constructs (A, U ), the following conditions are equivalent:(1) (A, U ) is relational,(2) (A, U ) is functor-structured,(3) (A, U ) is universally topological,(4) (A, U ) is cartesian closed and there exists only one A-object A with U A = ∅,(5) (A, U ) is well-fibred,(6) U is an isomorphism.
28J. Products of Final Morphisms
Prove that(a) In relational categories, products of final morphisms are final.
(b) In universally topological categories, finite products of final morphisms are final.
(c) If T : Set → Set is defined by T (X) is the set of all finite subsets of X, and T f (Y ) =f [Y ], then Spa(T ) is universally topological, but in Spa(T ) countable products offinal morphisms can fail to be final.

9th July 2006

458

Cartesian Closedness and Partial Morphisms[Chap. VII(d) In concretely cartesian closed topological categories, finite products of final maps
are final.
(e) In a well-fibred topological construct (A, U ), finite products of final maps are final
if and only if U is an isomorphism.
(f) In a topological construct with function spaces, finite products of quotient maps are
quotient maps.
(g) A topological full subconstruct (A, U ) of Top has function spaces if and only if in(A, U ) finite products of quotient maps are quotient maps.
28K. Functor-Structured Categories
Prove that Spa(T ) is universally topological if and only if the base category is a quasitopos and T weakly preserves pullbacks (cf. Exercise 27I).
28L. Re: Theorems 27.22 and 28.19
Let A (resp. B) be the full subconstruct of Rel (resp. Rere) that consists of all objects(X, ρ) that have no proper cycles; i.e., x1 ρx2 , x2 ρx3 , · · · , xn−1 ρxn and x1 = xn imply
that x1 = x2 = · · · = xn . Show that the following hold:(a) A (resp. B) is closed under the formation of mono-sources in Rel (resp. Rere),(b) A and B are monotopological constructs,(c) A and B are wellpowered and co-wellpowered (epimorphisms are surjective),(d) A and B are locally presentable,(e) B is well-fibred; A is not,(f) in A final sinks are pullback stable; in B final epi-sinks are pullback stable,(g) in B all functors of the form (B × −) preserve colimits; in A some functors of theform (A × −) don’t preserve coequalizers,(h) B is cartesian closed; A is not,(i) neither A nor B has representable extremal partial morphisms; neither even has anextremal-subobject classifier,(j) neither A nor B is a quasitopos.
28M. Re: Theorem 28.19
Show that for a topological construct A the following are equivalent:(1) in A final sinks are extremally reducible,(2) in A(i) final epi-sinks are extremally reducible, and(ii) subobjects of discrete objects are discrete.
Observe that condition (2)(ii) is automatically satisfied whenever the fibre of the empty
set has exactly one element.

9th July 2006

BIBLIOGRAPHY
Adámek, J. Theory of Mathematical Structures. Reidel, Dordrecht–Boston–Lancaster,1983.
Adámek, J., and J. Rosicky. Locally Presentable and Accessible Categories. Cambridge
University Press, 1993.
Corrections under http://www.iti.cs.tu-bs.de/~adamek/corrections.ps.
Adámek, J., and V. Trnková. Automata and Algebras in Categories.
drecht, 1989.
Arbib, M. A. Theories of Abstract Automata.
1969.

Kluwer, Dor-

Prentice-Hall, Englewood Cliffs, NJ,

Arbib, M. A., and E. G. Manes. Arrows, Structures and Functors. The Categorical
Imperative. Academic Press, New York, 1975.
Barr, M. ∗-Autonomous Categories. Lecture Notes in Mathematics, Vol. 752. Springer,
Berlin–Heidelberg–New York, 1979.
Barr, M., P. A. Grillet, and D. H. van Osdol. Exact Categories and Categories ofSheaves. Lecture Notes in Mathematics, Vol. 236. Springer, Berlin–Heidelberg–
New York, 1971.
Barr, M., and C. Wells. Category Theory for Computer Science. Third edition, CRM
Montreal, 1999.
Toposes, Triples and Theories. Springer, Berlin–Heidelberg–New York, 1985. Revised and corrected version:
http://www.cwru.edu/artsci/math/wells/pub/ttt.html
Bell, J. L. Toposes and Local Set Theories — An Introduction. Oxford Logic Guides,
Vol. 14. Clarendon Press, New York–Oxford, 1988.
Bénabou, J. Introduction to Bicategories. Lecture Notes in Mathematics, Vol. 47.
Springer, Berlin–Heidelberg–New York, 1967.
Structures Algébriques dans les Catégories. Cahiers Topol. Géom. Diff. 10 (1968):
1–126.
Blyth, T. S. Categories. Wiley, New York, 1986.
Borceaux, F. Handbook of Categorical Algebra. Three volumes, Cambridge University
Press, 1994.
Borceaux, F., and G. Janelidze. Galois Theories. Cambridge University Press, 2001.
Brinkmann, H.-B., and D. Puppe. Kategorien und Funktoren. Lecture Notes in Mathematics, Vol. 18. Springer, Berlin–Heidelberg–New York, 1966.
Abelsche und exakte Kategorien, Korrespondenzen. Lecture Notes in Mathematics,
Vol. 96. Springer, Berlin–Heidelberg–New York, 1969.

459

460

BIBLIOGRAPHY

Bucur, I., and A. Deleanu (with the collaboration of P. J. Hilton and N. Popescu).
Introduction to the Theory of Categories and Functors. Wiley, New York, 1968.
Budach, L. Quotientenfunktoren und Erweiterungstheorie. Mathematische Forschungsberichte, Vol. 22. Deutscher Verlag der Wissenschaften, Berlin, 1967.
Budach, L., and H.-J. Hoehnke. Automaten und Funktoren. Akademie Verlag, Berlin,
1975.
Budach, L., and H.-P. Holzapfel. Localizations and Grothendieck Categories, Deutscher
Verlag der Wissenschaften, Berlin, 1975.
Burmeister, P., and H. Reichel. Introduction to Theory and Application of Partial Algebras. Akademie Verlag, Berlin, 1986.
Castellini, G. Categorical Closure Operators. Birkhäuser, Boston–Basel–Berlin, 2003.
Diers, Y. Categories of Commutative Algebras. Oxford University Press, Oxford, 1992.
Dubuc, E. Kan Extensions in Enriched Category Theory. Lecture Notes in Mathematics, Vol. 145. Springer, Berlin–Heidelberg–New York, 1970.
Ehresmann, C. Catégories et Structures. Dunod, Paris, 1965.
Prolongements Universels d’un Foncteur par Adjonction de Limites. Diss. Math.
Rozprawy Mat., Vol. 64, 1969.
Oevres Complétes et Commentées I-IV (ed. by A. C. Ehresmann). Cahiers Topol.
Géom. Diff., Vols. 20-24, (1979–83): Supplements.
Ehrig, H., K. D. Kiermeier, H.-J. Kreowski, and W. Kühnel. Universal
Automata. Teubner, Stuttgart–Leipzig, 1974.

TheoryofEilenberg, S. Automata, Languages and Machines Vol.I. Academic Press, New York,
1974.
Eilenberg, S., and S. Mac Lane. Collected Works. Academic Press, New York, 1986.
Freyd, P. Abelian Categories. An Introduction to the Theory of Functors. Harper &Row, New York, 1964.
Freyd, P., and A. Scedrov. Categories, Allegories. North-Holland, Amsterdam, 1990.
Frölicher A., and A. Kriegl. Linear Spaces and Differentiation Theory. Wiley, New
York, 1988.
Gabriel, P., and F. Ulmer. Lokal präsentierbare Kategorien. Lecture Notes in Mathematics, Vol. 221. Springer, Berlin–Heidelberg–New York, 1971.
Goldblatt, R. Topoi. North Holland, Amsterdam, 1984.
Gorey, E. Cate gor y. Adama Books, 1973.
Grätzer, G. Universal Algebra. Van Nostrand, New York, 1968.

9th July 2006

BIBLIOGRAPHY

461

Grothendieck, A. Catégories Cofibrées Additives et Complexe Cotangent Relatif. Lecture Notes in Mathematics, Vol. 79. Springer, Berlin–Heidelberg–New York, 1968.
Hasse, M., and L. Michler. Theorie der Kategorien. Mathematische Monographien,Vol. 7. Deutscher Verlag der Wissenschaften, Berlin, 1966.
Henkin, L., J. D. Monk, and A. Tarski. Cylindric Algebras Part II. North Holland,
Amsterdam, 1985.
Herrlich, H. Topologische Reflexionen und Coreflexionen. Lecture Notes in Mathematics, Vol. 78. Springer, Berlin–Heidelberg–New York, 1968.
Herrlich, H., and G. E. Strecker. Category Theory. Allyn Bacon, Boston, 1973, 2nded. Heldermann, Berlin, 1979.
Isbell, J. R. Subobjects, Adequacy, Completeness and Categories of Algebras.
Math. Rozprawy Mat., Vol. 36, 1964.

Diss.

Johnstone, P. T. Topos Theory. Academic Press, New York, 1977.
Stone Spaces. Cambridge Univ. Press, Cambridge–New York–Melbourne, 1982.
Sketches of an Elephant. A Topos Theory Compendium. Two volumes, Oxford
University Press, Oxford, 2002.
Kelly, G. M. Basic Concepts of Enriched Category Theory. Cambridge Univ. Press,
Cambridge–New York–Melbourne, 1982.
Kuroš, A. G., A. H. Livšic, E. G. Šul’geifer, and M. S. Calenko. Zur Theorie derKategorien, Mathematische Forschungsberichte, Vol. 15. Deutscher Verlag derWissenschaften, Berlin, 1963.
Lambek, J. Completions of Categories. Lecture Notes in Mathematics, Vol. 24.
Springer, Berlin–Heidelberg–New York, 1966. tablecat.tex
Lambek, J., and P. J. Scott. Introduction to Higher-Order Categorical Logic. Cambridge Univ. Press, Cambridge–New York–Melbourne, 1986.
Leinster, Tom. Higher Operads, Higher Categories, Cambridge University Press, 2003.
Available from http://www.arxiv.org/abs/math.CT/0305049
Mac Lane, S. Categories for the Working Mathematician. Springer, Berlin–Heidelberg–
New York, 2nd ed., 1997.
Mac Lane, S., and I. Moerdijk. Sheaves in Geometry and Logic. Springer, Berlin–Heidelberg–New York, 1992.
Mal’cev, A. I. Algebraic Systems. Die Grundlehren der mathematischen Wissenschaften,
Vol. 192. Springer, Berlin–Heidelberg–New York, 1972.
McLarty, C. Elementary Categories, Elementary Toposes. Oxford University Press, Oxford, 1992.
Makkai, M., and R. Paré. Accessible Categories: The Foundations of Categorical
Model Theory. American Mathematical Society, 1989.

9th July 2006

462

BIBLIOGRAPHY

Manes, E. G. Algebraic Theories. Springer, Berlin–Heidelberg–New York, 1976.
Michor, P. W. Functors and Categories of Banach Spaces. Lecture Notes in Mathematics, Vol. 651. Springer, Berlin–Heidelberg–New York, 1978.
Mitchell, B. Theory of Categories. Pure and Applied Mathematics 17. Academic Press,
New York, 1965.
Morita, K., and J. Nagata (editors). Topics in General Topology. North Holland, Amsterdam, 1989.
Pareigis, B. Categories and Functors. Academic Press, New York, 1970.
Popescu, N. Abelian Categories with Applications to Rings and Modules. Academic
Press, New York, 1973.
Preuss, G. Theory of Topological Structures.
1988.

Reidel, Dordrecht–Boston–Lancaster,

Pultr, A., and V. Trnková. Combinatorial, Algebraic and Topological Representationsof Groups, Semigroups and Categories. North Holland, Amsterdam, 1980.
Richter, G. Kategorielle Algebra. Akademie Verlag, Berlin, 1979.
Rivano, N. S. Catégories Tannakiennes. Lecture Notes in Mathematics, Vol. 265.
Springer, Berlin–Heidelberg–New York, 1972.
Rosenthal, K. I. The Theory of Quantaloids. Pitman Research Notes in Mathematics
348, Longman, Harlow, 1996.
Salicrup, G. Categorical Topology – The Complete Work of Graciela Salicrup (ed. by
H. Herrlich and C. Prieto). Aportaciones Matemática, Vol. 2, 1988.
Schubert, H. Categories. Springer, Berlin–Heidelberg–New York, 1972.
Semadeni, Z. Projectivity, Injectivity, and Duality. Diss. Math. Rozprawy Mat., Vol.
35, 1963.
Banach Spaces of Continuous Functions. Polish Scientific Publishers, Warsaw,
1971.
Semadeni, Z., and A. Wiweger. Einführung in die Theorie der Kategorien und Funktoren. Teubner, Stuttgart–Leipzig, 1979.
Slominsky, J. The Theory of Abstract Algebras with Infinitary Operations. Diss. Math.
Rozprawy Mat., Vol. 18, 1959.
Taylor, P. Practical Foundations of Mathematics. Cambridge University Press, 1999.
Wyler, O. Lecture Notes on Topoi and Quasitopoi. World Scientific, Singapore-New
Jersey-London-Hong Kong, 1991.

9th July 2006

TABLESFUNCTORS AND MORPHISMS: PRESERVATION PROPERTIES

topologicalmonadic
reg. monadic
reg. algebraic
algebraicess. algebraic
solidfull refl. emb.
adjointfaithful

monos
+
+
+
+
+
+
+
+
+
−

extr. monos
+
?
+
+
+
+
+
+
−
−

reg. monos
+
+
+
+
+
+
+
+
+
−

epis
+
−
−
−
−
−
−
−
−
−

extr. epis
+
−
+
+
+
−
−
−
−
−

reg. epis
+
−
+
+
−
−
−
−
−
−

FUNCTORS AND MORPHISMS: REFLECTION PROPERTIES

topologicalmonadic
reg. monadic
reg. algebraic
algebraicess. algebraic
solidfull refl. emb.
adjointfaithful

monos
+
+
+
+
+
+
+
+
−
+

extr. monos
−
−
−
−
−
−
−
−
−
−

reg. monos
−
−
−
−
−
−
−
−
−
−

epis
+
+
+
+
+
+
+
+
−
+

extr. epis
−
+
+
+
+
+
−
+
−
−

reg. epis
−
−
+
+
+
−
−
+
−
−

isos
−
+
+
+
+
+
−
+
−
−

463

464

TABLESFUNCTORS AND LIMITS

topologicalmonadic
ess. algebraic
solid, uniquely transp.
adjoint

creates−++−−

lifts uniquely
+
+
+
+
−

preserves
+
+
+
+
+

reflects−++−−

FUNCTORS AND COLIMITS

topologicalmonadic
solid

lifts uniquely+−−

preserves+−−detects+−+

STABILITY PROPERTIES OF SPECIAL EPIMORPHISMS

isomorphisms
retractions
reg. epis
strict epis
swell epis
strong epis
extr. epis
epis

composition
+
+
−
−
+
+
−
+

pushouts
+
−
+
+
+
+
−
+

cointersections
+
−
−
+
+
+
−
+

9th July 2006

TABLE OF CATEGORIES(A ↓ K), 3K (objects over K)
Ab, 3.26 (abelian groups) 3.39, 4.17, 7.18, 7.69, 8.23, 9.3, 9.13, 9.17, 9.24, 9.28, 9A,
10G, 11O, 12.11, 14F, 16.13, 24A
AbTop, 16.13 (abelian topological groups)
AbTor, 10.20 (abelian torsion groups) 10.55, 13.2
Alg(1), 27.3 (algebras with one unary operator) 4K, 5O, 9.3, 27.12, 28.2, 28.9
Alg(S 2 ), 5.38 (binary algebras)
Alg(T ), 5.37 (T -algebras) 5G, 10U, 13N, 20.56, 20I
Alg(Σ), 3.26 (Σ-algebras) 8.41, 18.2, 21.20
Alg(Ω), 3.3 (Ω-algebras) 5O, 6I, 7.2, 7.40, 7.69, 7.72, 7O, 8.23, 8H, 9.28, 10.67, 10U,
16.19, 20.18, 24.5
Aut, 3.3 (automata) 4K, 5.2, 7.15, 13.13, 15.3, 20H
Autr , 8.31 (reachable automata) 18.2, 19.12
Ban, 3.3 (Banach spaces with linear contractions) 7.9, 7.18, 7.40, 7.50, 7.58, 9.3, 10.20,
10.31, 10J, 26.6
Banb , 3.3 (Banach spaces with bounded linear maps) 5.6, 7.9, 7.18, 7.40, 7.50, 7.58, 8.8,
8.23, 10.31, 10J, 13.18(Ban, O), 5.2 (Banach spaces as a construct with unit ball functor) 6.11, 6E, 7.72, 8.8,
8.11, 8.23, 8K, 10.55, 10J, 20.36, 20.41, 23.20, 24C(Ban, U ), 5.2 (Banach spaces as a construct with usual forgetful functor) 6.11, 6E, 8.8,
8.23, 8K 10.55, 10J, 13.2, 18.2
Beh, 8.31 (behaviors) 18.2, 19.12
BiComp, 4F (bicompact spaces) 16.10
BiTop, 4F (bitopological spaces) 16.10
Boo, 3.20 (boolean algebras) 3.26, 3.39, 7.2, 7.18, 9.3, 9.13, 9A, 26.6
BooRng, 3.26 (boolean rings)
BooSpa, 3.39 (boolean spaces) 9.3, 9.18, 9.28, 10C, 12M, 23.6, 24.13
Bor, 22.2 (bornological spaces) 27.21, 28.20, 28.23
Cat, 3.47 (small categories); 5I, 7.40, 7.72, 7.76, 7S, 8.11, 8.23, 8C, 11J, 14.2, 14.23,
15.3, 15M, 24C, 27.3, 27.23, 28.9, 28.11

465

466

TABLE OF CATEGORIES

CAT, 3.50 (categories) 7.18, 7.33
CAT(X), 5.15 (concrete categories over X) 7.5, 9.24, 21.21, 21.22, 21J
Catof , 3.55 (object-free small categories); 5I, 20.36, 20.41 20.51, 20.52, 23.6, 23.20, 24D
CATof , 3.55 (object-free categories) 13.18
CBoo, 8E (complete boolean algebras) 8F, 8.23, 10S, 12.6, 12I, 18.13, 18G
Class, 15.3 (classes)
CLat, 4.3 (complete lattices with join- and meet-preserving maps) 8.23, 8G, 10S, 12.6,
12I, 18.13
Clos 5N (closure spaces)
Comp, 5A (compact spaces)
CONST, 5.15 (constructs)
Conv, 5N (convergence spaces) 27.3, 27.21, 28.9, 28.20, 28.23∆Pos, 25E (∆-complete partially ordered sets) 27B
DivAb, 7.33(5) (divisible abelian groups)
DLat, 9.3 (distributive lattices)
DRail, 8.8 (derailed spaces) 11H(F ↓ G), 3K (comma category F over G)
FHaus, 7.65 (functionally Hausdorff spaces) 7I, 7J, 8.8
Field, 9.8 (fields) 9.17, 10I, 12.6, 16.13
Fram, 5L (frames) 8G, 20K, 24.13
Grp, 3.3 (groups) 4.17, 7.18, 7.58, 7C, 7H, 8.8, 9.3, 10.67, 14F, 16N, 17.5, 20.27, 23.34,
26.6
Grph, 20.41 (oriented graphs) 20.51, 20.52
Haus, 4.3 (Hausdorff spaces) 5O, 7.18, 7.40, 7.58, 8.8, 9.3, 9E, 11J, 13.2, 21.39
HausVec, 25.2 (Hausdorff topological vector spaces)
HComp, 3.39 (compact Hausdorff spaces) 4.17, 4D, 7.58, 9.3, 9.18, 9.28, 9A, 9D, 10.18,
10.67, 10C, 12K, 20.27, 20.36, 21I, 23.13, 23.34, 24.5, 26.6
HCompop , 5M (as a construct)hTop, 5J(d) (topological spaces and homotopy equivalence classes of maps) 7.33, 7.40
HUnif, 21.39 (separated uniform spaces)
JCPos, 4.3(3) (complete lattices and join-preserving maps) 4.17, 5N, 8.23, 8G, 9.3, 20.5,
22B, 24.5, 24.13

9th July 2006

TABLE OF CATEGORIES

467

JPos, 4.17(10) (posets and join-preserving maps) 8G, 12.6
(K ↓ A), 3K (objects under K)
kTop, 27.3 ((compact Hausdorff)-generated topological spaces) 7.52, 27.21, 28.9, 28.11,
28.20
Λ-CCPos, 15D (chain complete λ-posets and chain join-preserving λ-homomorphisms)
15.26, 15J, 18.16, 18B, 25.17, 26C
Lat, 4.3 (lattices) 7.40, 7.72, 9.13, 9H
Loc 5L (locales)
M -Act, 3.26 (M -actions)
Mat, 3.3 (matrices (as morphisms)) 3.35
MCPos, 5N (complete lattices and meet-preserving maps)
Met, 3.3 (metric spaces and non-expansive maps) 8.8, 9.3, 9.13, 9.17, 9.24, 10.31, 10.70
Metc , 3.3 (metric spaces and continuous maps) 3.35, 5.6, 5.34, 10.31
Metu , 3.3 (metric spaces and uniformly continuous maps) 10.31
Mod, 21D (modules)
Mod-R, 3.26 (right R-modules)
Mon, 4.3 (monoids) 4.17, 8.23, 14F, 20.5
Neigh, 5N (neighborhood spaces)
1, 3.3 (terminal category (with only one morphism))
Ord, 7.83 (ordinal numbers) 7.90, 12.6, 15.8, 18.13, 18.16
PAlg(Ω), 21.39 (partial algebras of type Ω)
pGrp, 3B (pointed groups)
PMet, 8.2 (pseudometric spaces) 8.11, 10.31, 10.42, 21.8
Pos, 4.3 (partially ordered sets = posets) 4.17, 6I, 7.18, 8.8, 8.23, 9.3, 9.13, 9.17, 10.39,
10.55, 10.64, 10.65, 10.66, 10.67, 10.68, 10.70, 11.12, 12.6 12.11, 13.2, 16.13, 20H,
21.22, 21.39, 25.9, 27.3, 27.10, 27.12
Prost, 4.3 (preordered sets) 4.17, 21.8, 22.2, 27.21, 28.11, 28.20
Prox, 21I (proximity spaces)
PrTop, 5N (pretopological spaces) 27G, 28.9, 28.20, 28C
pSet, 3B (pointed sets) 14F, 16.13, 23A, 28.9
PsTop, 4F (pseudotopological spaces) 22.2, 27.3, 27.21, 27G, 28.9, 28.20, 28.23, 28G

9th July 2006

468

TABLE OF CATEGORIES

pTop, 3B (pointed spaces)
Reg, 23.20 (regular topological spaces)
Reg1 21.39 (regular T1 topological spaces)
Rel, 3.3 (relations) 4.7, 4E, 5K, 5.12, 7.69, 8.8, 8.11, 13I, 18.2, 21.8, 21.37, 27.3, 28.2,
28.9, 28L
Rere, 27.21 (reflexive relations) 15L, 27.21, 28.20, 28.23, 28L
R-Mod, 3.26 (left R-modules)
Rng, 5.11 (rings) 7.2, 7.18, 7.40, 7.58, 8.8, 8.23, 21D, 23.18
Set, 3.3 (sets) 3.39, 4D, 4E, 6F, 6I, 7.18, 7.40, 7.48, 7.69, 9.3, 9.13, 9.17, 9A, 10.6, 10.20,
10.28, 10.64, 10.67, 11.4, 11.28, 11K, 11P, 12.11, 12E, 13.2, 13B, 13I, 14.2, 14B,
15.3, 15K, 15J, 16.5, 18.2, 18A, 18E, 19G, 20.22, 20H, 27.3, 28.2
Setop , 5.2(4) (as a construct)
Set2 , (arrow category of Set, i.e., functions) 9E, 15N, 16I
Sgr, 4.3 (semigroups) 4.7, 6I, 7.18, 7.40, 7.58, 7.60, 8.23, 8B, 14.23, 14I, 16N
Σ-Seq, 3.3 (Σ-acceptors) 3.35, 4.17, 4.26, 4K, 7.2, 7.33, 7.58, 7.69, 8.8, 8.23, 8P, 9I,
10.20, 10.64, 10.70, 16.13, 26.6
Σ-Seq0 , 10.18 (observable Σ-acceptors with no initial state)
Simp, 22.2 (simplicial complexes) 27.21, 28.20, 28.23
SLat, 4.3(3) (semi-lattices)
Spa(T ), 5.40 (T -spaces) 5H, 6C, 6I, 7.2, 7R, 8.41, 10.42, 10.68, 13.13, 14.23, 18.2, 21.8,
22A, 22B, 27H, 27I, 28.17, 28H, 28J, 28K
sTop, 27.21 (sequential topological spaces) 4.26, 7.52, 27.21, 28.20
Sym, 4.26 (symmetric relations) 4E, 22.7, 27.10
TConv, 20.41 (totally convex spaces) 24.13
TfAb, 4.17 (torsion-free abelian groups) 7.40, 16N, 20.41, 20.45, 23.6
Top, 3.3 (topological spaces) 4E, 5.12, 5D, 7.18, 7.58, 7.60, 7.69, 7L, 8.8, 8.11, 8.23,
8.41, 9.3, 9.24, 9.26, 9E, 10.18, 10.42, 10.64, 10.66, 10.67, 10.70, 12.11, 13I, 13J,
14.2, 15.22, 15L, 16.25, 16D, 16L, 17.5, 18.2, 21.8, 21D, 22.7, 22C, 27.3, 27.23, 28.2
Topop , 3.20
Top0 , 4.17 (T0 topological spaces) 5L, 7.18, 9.3, 9.24, 21.39
Top0 op , 5L (as a construct) 20K
Top1 , 7.73 (T1 topological spaces) 8.5, 9.3, 9F, 16.13, 21.8, 21.39

9th July 2006

TABLE OF CATEGORIES

469

Topm , 3.35 (metrizable topological spaces) 5.34
TopGrp, 5.2 (topological groups) 5A, 8.8, 18.2, 20.41, 21.8, 23.6, 23.20, 25.2, 25.9
TopVec, 5.2 (topological vector spaces)
Tych, 4.3 (Tychonoff spaces) 7.18, 9.3, 9.13, 9.21, 10.18, 10C, 14.2, 16F, 21.39
Unif, 9.3 (uniform spaces) 21.8, 21.39, 27.23
Vec, 3.3 (vector spaces) 3.20, 4H, 6I, 7.40, 7.58, 8.8, 8.23, 9.3, 9.13, 9.17, 9.28, 10.67,
10.70, 12.11, 13I, 14F, 16.5, 20.5, 20H, 26.6

9th July 2006

TABLE OF SYMBOLS{x ∈ X | P (x)}, 2.1
P(X), 2.1 (power set)
{X, Y }, 2.1(X, Y ), 2.1 (ordered pair)X ∪ Y , 2.1 (binary union)
X ∩ Y , 2.1 (binary intersection)
X × Y , 2.1 (binary cartesian product)
X \ Y , 2.1 (relative complement)
Y X , 2.1
{Xi | i ∈ I}, 2.1
Si∈I Xi , 2.1 (union)
Ti∈I Xi , 2.1 (intersection)
Qi∈I Xi , 2.1 (cartesian product)
Ui∈I Xi , 2.1 (disjoint union)

N, 2.1 (natural numbers)
Z, 2.1 (integers)Q, 2.1 (rational numbers)
R, 2.1 (real numbers)
C, 2.1 (complex numbers)hom(A, B), 3.1 (hom-set)
id, 3.1 (identity)◦, 3.1 (composition)fA −→ B, 3.1 (morphism)
Ob(A), 3.2 (class of A-objects)
M or(A), 3.2 (class of A-morphisms)
dom(f ), 3.2 (domain of f )cod(f ), 3.2 (codomain of f )470

TABLE OF SYMBOLS
1, 3.3(4) (terminal category)A × B, 3.3(4) (product of categories)
Aop , 3.5 (dual category)(end of proof)
D , 3.7 (proof by duality)
A , 8.25 (footnote) (proof by analogy)

f −1 , 3.12 (inverse of f )P, 3.20(8) (covariant power-set functor)
Q, 3.20(9) (contravariant power-set functor)
S n , 3.20(10) (nth power functor)
F n , 3.23 (nth composite of F )F op , 3.41 (dual functor)(F ↓ G), 3K (comma category F over G)(A ↓ K), 3K (category of objects over K)(K ↓ A), 3K (category of objects under K)
A2 3K (arrow category of A)| |, 5.3 (underlying functor)A ≤ B, 5.4 (order in a fibre)
F ≤ G, 5.18 (order of concrete functors)
τ op , 6.3 (dual natural transformation)[A, B], 6.15 (functor quasicategory)(F̂ , Ĝ) ◦ (F, G), 6.27 (composition of Galois correspondences)(A, m) ≤ (A0 , m0 ), 7.79 (order of subobjects)(e, B) ≥ (e0 , B 0 ), 7.85 (order of quotient objects)f

X −→ |A| = (f, A), 8.15 (structured arrow)fi(A −−→ Ai )I = (A, fi )I = (A, fi ) = (A, (fi )i∈I ), 10.1, 10.2 (source)Q( Ai , πj )j∈I , 10.23 (product of objects)
hfi i, 10.23 (morphism induced by product)
A × B, 10.23 (binary product)
Πfi , f × g 10.34 (product of morphisms)

9th July 2006

471

472

TABLE OF SYMBOLS

AI , 10.37 (I-th power of A)`(µj , Ai )j∈I , 10.63 (coproduct of objects)[fi ], 10.63 (morphism induced by coproduct)qfi , 10.63 (coproduct of morphisms)
A + B, f + g, 10.63 (binary coproduct)
IA, 10.63 (I-th copower of A)∆A , 10V (diagonal morphism of A)fi(X −−→ GAi )i∈I , 17.2 (structured source)
ev : A × B A → B, 18.2, 27.2 (evaluation)(η, ε) : F



G : A → B, 19.3 (adjoint situation)

T = (T, η, µ), 20.1 (monad)(XT , U T ), 20.4 (category of T-algebras)
ConGen = ConGen(A) = the class of concretely generating structured arrows (in A)
Emb = Emb(A) = the class of embeddings (in A)
Epi = Epi(A) = the class of epimorphisms (in A)
ExtrEpi = ExtrEpi(A) = the class of extremal epimorphisms (in A)
ExtrGen = ExtrGen(A) = the class of extremally generating structured arrows (in A)
ExtrMono = ExtrMono(A) = the class of extremal monomorphisms (in A)
ExtrMono-Source = ExtrMono-Source(A) = the conglomerate of extremal
mono-sources (in A)
Final = Final(A) = the class of final morphisms (in A)
Gen = Gen(A) = the class of generating structured arrows (in A)
Init = Init(A) = the class of initial morphisms (in A)
Iso = Iso(A) = the class of isomorphisms (in A)
Mono = Mono(A) = the class of monomorphisms (in A)
Mono-Source = Mono-Source(A) = the conglomerate of mono-sources (in A)
Mor = Mor(A) = the class of morphisms (in A)
NormalEpi = NormalEpi(A) = the class of normal epimorphisms (in A)
NormalMono = NormalMono(A) = the class of normal monomorphisms (in A)
Quot = Quot(A) = the class of quotient morphisms (in A)

9th July 2006

TABLE OF SYMBOLS
RegEpi = RegEpi(A) = the class of regular epimorphisms (in A)
RegMono = RegMono(A) = the class of regular monomorphisms (in A)
Retr = Retr(A) = the class of retractions (in A)
Sect = Sect(A) = the class of sections (in A)
Source = Source(A) = the conglomerate of sources (in A)
StableEpi = StableEpi(A) = the class of stable epimorphisms (in A)
StrictMono = StrictMono(A) = the class of strict monomorphisms (in A)
StrongMono = StrongMono(A) = the class of strong monomorphisms (in A)
Surj = Surj(A) = the class of surjective morphisms (in A)
SwellEpi = SwellEpi(A) = the class of swell morphisms (in A)

9th July 2006

473

INDEX
Absolute coequalizer, 20.14
Absolute colimit, 20.14
Absolute retract, 9.6
vs. enough injectives, 9.10
Adjoint, for a functor,19.10
Adjoint functor, 18.1 ff
characterizations, 18.3
characterization theorems, 18.12,
18.14, 18.17, 18.19
comparison functor for, 20.38, 20.42
composition of, 18.5
monadic functor is, 20.12
between posets, 18H
preserves mono-sources and limits,
18.6, 18.9
smallness conditions for, 18B
theorem, concrete, 18.19
theorem, first, 18.12
theorem, special, 18.17
vs. adjoint situation, 19.1, 19.4, 19.8
vs. algebraic category, 23.31
vs. co-adjoint functor, 18A, 19.1
vs. colimit, 18D
vs. co-wellpoweredness, 18.11, 18.14,
18.19
vs. completeness, 18.12, 18.14, 18.17,
18.19
vs. essentially algebraic functor, 23.8
vs. exponential functor, 27.7
vs. extremal monomorphism, 18J
vs. free object, 18.19
vs. full, faithful functor, 19I
vs. (Generating, M)-functor, 18.4
vs. monadic category, 20.46
vs. monadic functor, 20.12, 20.17
vs. reflection of epimorphisms, 18I,
19B
vs. regularly algebraic category, 23.38
vs. representable functor, 18C
vs. solid functor, 25.12, 25.19

474

vs. topologically algebraic functor,
25.3, 25.6, 25.19
vs. wellpoweredness, 18.19
Adjoint sequence, 19F
Adjoint situation, 19.3 ff
alternative description, 19A
associated with a monad, 20.7
composition of, 19.13
consequences of, 19.14
duality for, 19.6
gives rise to a monad, 20.3
induced by (co-)adjoint functor, 19.7
lifting of, 21.26, 21.28
uniqueness, 19.9
vs. adjoint functor, 19.1, 19.4, 19.8
vs. equivalence functor, 19.8, 19H
vs. free object, 19.4
vs. Galois correspondence, 19.8
vs. indiscrete structure, 21A
vs. monad, 20A
vs. reflective subcategory, 19.4
vs. universal arrow, 19.7
Algebra, partial binary, 3.52
Algebraic category, 23.19 ff, 23D
characterization theorem for, 23.30,
23.31
concrete functor between, 23.22
implies extremal epimorphisms are
final, 23.23
vs. adjoint functor, 23.31
vs. (Epi, Mono-Source)-factorizable
category, 23.30, 23.31
vs. extremal co-wellpoweredness,
23.27
vs. extremal epimorphism, 23.30
vs. unique lift of (ExtrEpi,
Mono-Source)-factorizations, 23.31
Algebraic construct, = regular
epireflective subconstruct of monadic

INDEX
construct, 24.3
bounded, vs. bounded quasivariety,24.11
vs. monadic construct, 23.41

475
structured, = structured arrow, 8.15
universal, = universal structured
arrow, 8.22

Arrow category, 3K, 6.17

Algebraic functor, 23.19 ff
closed under composition, 23.21
need not preserve regularepimorphisms, 23.25, 23J
reflects regular epimorphisms, 23.24
restrictions of, 23L
vs. composite of regular monadic
functors, 24.2
vs. regular epimorphism, 23Jvs. regular factorization, 24.2
vs. uniquely transportable functor,
23.30

Associativity, composition of morphisms
has, 3.1, 3.53, 3C

Algebraic hull of a concrete category, 23K

Bicoreflective subcategory,vs. coreflective subcategory, 16.4
vs. separator, 16.4

Algebraic subcategory, conditions for,23.32, 23.33
vs. extremally epireflective
subcategory, 23.33
Algebraic theory, 20C
Algebraic-topological decompositions,
26C

Axiom, topological, 22.6
Axiom of choice, 2.3, 9A
Axiom of replacement, 2.2
Balanced category, 7.49 ff
topological, scarcity of, 21N
vs. extremal monomorphism, 7.67
vs. mono-source, 10.12
Base category, 5.1 ff

Bimorphism, 7.49
vs. topological functor, 21M
Bireflective subcategory, 16.1
monoreflective subcategory is, 16.3
Birkhoff-theorem, 16G

Algebraic-type functors, relationships
among, 23.42

Boolean ring, 3.26, footnote

Amnesticity, vs. lifting of limits, 13.21

Cartesian closed category, 27.1 ff
characterization theorem, 27.4
vs. coreflective hull, 27C
vs. products of epimorphisms, 27.8
vs. zero object, 27A

Amnestic concrete category, 5.4 ff
vs. Galois correspondence, 6.29–6.36
vs. topological functor, 21.5
vs. transportable concrete category,
5.29, 5.30
Amnestic functor, 3.27, 5.6, 13.21, 13.25,
13.28

Cartesian closed construct, 27.16 ff, 27D
vs. well-fibred topological construct,
27.22

Amnestic modification, of a concrete
category, 5.6, 5.33, 5.34, 5F

Cartesian closed subcategory, 27.9
vs. (co)reflective subcategory, 27.9

Arrow,
costructured, = costructured arrow,
8.40
co-universal, = co-universal
costructured arrow, 8.40
reflection, = reflection arrow, 4.16

Cartesian closed topological category,
implies base category is cartesianclosed, 27.14
as injective object, 27F

9th July 2006

Cartesian closed topological construct,vs. powers with discrete exponents

476

INDEX

(resp. factors), 27.24
Cartesian closed topological hull, 27G
Cartesian product of sets, 2.1
Categorical statement involving functors,
dual of, 3.40, 3.42
Category, 3.1 ff
algebraic, = algebraic category,
23.19 ff
of all categories, can’t be formed
because of set-theoretical
restrictions, 3.48
with all products must be thin, 10.32
alternative definition, 3C
arrow, 3K
base, = base category, 5.1 ff
cartesian closed, = cartesian closed
category, 27.1(co)complete, = (co)complete
category, 12.2
comma, 3K, 5.38, 21F
compact, 18Kof concrete categories, 5.15, 5C
concretely cartesian closed, =concretely cartesian closed category,
27.11
concretizable, 5J
co-wellpowered, = co-wellpowered
category, 7.87
discrete, 3.26
dual, = opposite category, 3.5
Eilenberg-Moore, 20.4
(E, M), = (E, M)-category, 15.1 ff
empty, 3.3(4)essentially algebraic, = essentiallyalgebraic category, 23.5 ff(E, −)-structured, 14H
exact, 14F
extremally co-wellpowered, =extremally co-wellpowered category,
7.87
extremally wellpowered, = extremallywellpowered category, 7.82
finitary, 24.4, 24.7

finitely (co)complete, 12.2
free, 3A
functor-costructured, 5.43
functor-structured, =functor-structured category, 5.40
graph of, 3A
isomorphic, 3.24
Kleisli, 20.39, 20B
large, 3.44
locally presentable, 20H
monadic, = monadic category, 20.8 ff
non-co-wellpowered, 7L
object-free, = object-free category,
3.53
of objects over (resp. under) an
object, 3K
opposite, = opposite category, 3.5
pointed, 3B, 7B
product, 3.3(4)is a quasicategory, 3.51
regular co-wellpowered, 7.87–7.89
regular wellpowered, 7.82, 7.88, 7.89
regularly algebraic, = regularly
algebraic category, 23.35 ff
skeleton of a, 4.12–4.15, 4I
small, 3.44–3.45
of small categories, 3.47
— is a large category, 3.48
— vs. concrete category, 5I
solid, = solid category, 25.10 ff
strongly (co)complete, = strongly(co)com- plete category, 12.2
sub-, see Subcategory
terminal, 3.3(4)thin, 3.26, 3.29, 3G
topological, = topological category,
21.7 ff
topologically algebraic, =topologically algebraic category,
25.1 ff
total, 6Iof type 2, 3C
universal, 4J
universally topological, 28.16, 28.18,

9th July 2006

INDEX
28G
wellpowered, = wellpowered category,
7.82
Category theory, “object-free version”,
3.55
Choice, axiom of, 2.3, 9A
Class, 2.2
of all sets, = universe, 2.2
as a category, (is not a construct),3.3(4)
large, = proper class, 2.2
preordered, 3.3(4)proper, 2.2
small, = set, 2.2
Closed under the formation ofintersections, 11.26
Closed under the formation of M-sources,
16.7
and E-quotients, vs. E-equational
subcategory, 16.17
vs. E-reflective, 16.8
Closed under the formation of products,
and extremal subobjects, vs.
(epi)reflective subcategory, 16.9, 16.10
vs. E-equational subcategory, 16.17
Closed under the formation of pullbacks,
11.17
Closure space, 5N
Co-, see Dual concept
Co-adjoint, for a functor, 19.10
Co-adjoint functor, 18.1 ff
vs. adjoint functor, 18A, 19.1, 19.2,
19.11
vs. colimit, 18D
vs. contravariant exponential functor,
27.7
Coarser than, (preorder) relation on
concrete functors, 5.18
Co-axiom, topological, 22.6
Cocomplete category, 12.2 ff
almost implies complete, 12.7 ff

9th July 2006

477
with a small colimit-densesubcategory is complete, 12.12
vs. cartesian closed category, 27.4
vs. colimit-dense subcategory, 12.12
vs. complete category, 12.13, 25K
vs. copower, 12H
vs. essentially algebraic category,
23.12, 23.13
vs. solid category, 25.15, 25.16
vs. topological category, 21.16, 21.17

Cocomplete subcategory, vs. reflective
subcategory, 12K
Cocone, = natural sink, 11.27
Codomain,of a function, 2.1
of a morphism, 3.2
of a source, 10.1
of a sink, 10.62
of a structured source, 17.1
Coequalizer, 7.68 ff
absolute, 20.14
as a colimit, 11.28
existence, 12.1, 15Fpreservation, 13.1
split, 20.14
uniqueness, 7.70
vs. completeness, 12J
vs. epimorphism, 7M
vs. forgetful functor, 7.73
vs. isomorphism, 7.70
vs. monomorphism, 7.70
vs. pushout square, 11.33
Coessential morphism, w.r.t. a class ofmorphisms, 9.27
Coessential quotient map, 9.27
Cointersection, 11.28
existence, 12.1
vs. (E, M)-category, 15.14, 15.15,
15.16
Colimit, 11.27 ff
absolute, 20.14
concrete, 13.12
creation, 13.17

478

INDEX

detected by regularly monadic
functor, 20.33
detection, = detection of colimit,
13.22
directed, = directed colimit, 11.28
domain of, 11.27
lifting of, 13.17— vs. topological functor, 21.15
preservation, 13.1
— vs. topological functor, 21.15
reflection, 13.22
uniqueness, 11.29
vs. (co-)adjoint functor, 18D
vs. coproduct, 11.28
vs. monadic functor, 20.13
vs. reflection arrow, 13.30
Colimit-closed subcategory, vs. reflective
subcategory, 13.29
Colimit-dense full embedding, preserves
limits, 13L
Colimit-dense subcategory, 12.10
embedding of preserves limits, 13.11
vs. (co)completeness, 12.12
Comma category, 5.38, 3K, 21F
Commuting triangle and square, 3.4
Compact category, 18K
Comparison functor, 20.37 ff
vs. adjoint functor, 20.42
vs. faithful functor, 20.43
vs. full functor, 20.43, 20.44
Complete category, 12.2 ff
characterization, 12.3
and wellpowered category, is strongly
complete, 12.5
vs. adjoint functor, 18.12, 18.14,
18.17, 18.19
vs. cocomplete category, 12.13, 25K
vs. coequalizer, 12J
vs. colimit-dense subcategory, 12.12
vs. essentially algebraic category,
23.12, 23.13
vs. solid category, 25.15
vs. strongly complete category, 12.5,

12I
vs. topological category, 21.16, 21.17
Complete lattice, vs. has products, 10.32
Completion,of abstract categories, 12L
regarded as reflection, 4.17
universal, 12M
Complex numbers, 2.1
Composite,of (concrete) functors, 3.23, 5.14
— vs. natural transformation, 6.3
of epimorphisms, is epimorphism, 7.41
of essential embeddings, 9.14
of functions, 2.1
of isomorphisms, is isomorphism, 3.14
of monomorphisms, ismonomorphism, 7.34
of morphisms, 3.1
of natural transformations, 6.13
of sources, 10.3
— vs. (E, M)-category, 15.5
Composition, as a morphism, 27E
Composition of morphisms, 3.1 ff
is associative, 3.1
in a quasicategory, 3.49
vs. (E, M)-category, 15.14, 15.15
vs. (E, M )-structured category, 14.6
Concept, (self-)dual, 3.7
Concrete adjoint functor, vs. concrete coadjoint functor, 21.24
Concrete category, 5.1 ff
amnestic, = amnestic concrete
category, 5.4 ff
amnestic modification, 5.6, 5.33, 5.34
(A, U ) has property P means A (orU ) has property P , 5.3“concretely isomorphic” is stronger
than “isomorphic”, 5.12
duality principle for, 5.20
fibre-complete, 5.7, 5.42
fibre-discrete, 5.7, 5.8, 5.39
monadic, 20.8 ff

9th July 2006

INDEX
over 1, vs. concrete functors, 5.11
over Set, = construct, 3.3(2), 5.1 ff
solid, = solid concrete category,
25.10 ff
strongly fibre-small, 26.5
topological, = topological concrete
category, 21.7 ff(uniquely) transportable, 5.28, 5.35,
5.36
vs. Cat, 5I
Concrete co-adjoint functor,
vs. concrete adjoint functor, 21.24
vs. Galois correspondence, 21E
Concrete colimit, 13.12
Concrete coproduct, 13.12
Concrete coreflector, i.e., coreflector
induced by identity-carriedcoreflection arrows, 5.22

479
vs. universal arrow, 8.24

Concrete isomorphism, 5.12, 5N
Concrete limit, 13.12
characterized, 13.15
reflection of, 13C
two step construction, 13.16
vs. concrete colimit, 13.10
Concrete limit dense, 12M
Concretely cartesian closed category,
27.11
vs. discrete objects, 27.15
vs. topological category, 27.15
Concretely cartesian closed construct, vs.
subconstruct of Set, 27.16
Concretely complete, 12M
Concretely coreflective subcategory, 5.22
vs. topological subcategory, 21.33

Concrete co-wellpoweredness, 8.19
vs. extremal co-wellpoweredness, 8E

Concretely co-wellpowered concrete
category, 8.19

Concrete embedding, essential, 21J

Concretely co-wellpowered functor, 8.37
vs. (ConGen, Initial Mono-Source)
functor, 17.11

Concrete equivalence, 5.13
Concrete functor, 5.9 ff
between algebraic categories isalgebraic, 23.22
composite, 5.14
between constructs, 5D
between essentially algebraic
categories is essentially algebraic,
23.17
existence of a concrete naturaltransformation between, 6.24
between regularly algebraic categories
is regularly algebraic, 23.40
specified by its values on objects,
5.10, 5.11
vs. embedding functor and equivalent
objects, 5.10
vs. monadic category, 20E
Concrete generation of an object, 8.15
vs. extremal epimorphism, 8.18
vs. (extremal) generation, 8.16

9th July 2006

Concretely equivalent, is not symmetric,
5.13
Concretely generating structured arrow,see Concrete generation of an object
Concretely reflective concrete
subcategory, 5.22of amnestic category is full, 5.24
of non-amnestic concrete category
need not be full, 5.25, 4.21
vs. finally dense subcategory, 21.32
vs. initial source, 10.50
vs. initially closed subcategory, 21.31
vs. reflector, 5.26, 5.27, 5.31, 5.32
vs. topological (sub)category, 21.32,
21.33
Concrete natural transformation, 6.23
vs. concrete functors, 6.24
Concrete product, 10.52
as a concrete limit, 13.12

480

INDEX

preserved by composition, 10.56
vs. has M-initial subobjects, 21.42
vs. initial source, 10.53
vs. monotopological construct, 21.42
Concrete quasitopos, vs. universally
topological category, 28.18
Concrete reflector, = reflector induced by
identity-carried reflection arrows,
5.22, 5E
vs. concrete functors that arereflectors, 5.22
Concrete subcategory, 5.21 ff
concretely (co)reflective, = concretely(co)reflective subcategory, 5.22
Concretizable category, 5J, 10L
vs. separating set, 7QCone, = natural source, 11.3
(ConGen, Initial
Mono-Source)-factorizable category,
17J(ConGen, Initial Mono-Source)-functor,
vs. concretely co-wellpowered, 17.11
Conglomerate, 2.3
of all classes, 2.3
codability by, 2.3(il)legitimate, 2.3
of morphisms between two objects ina quasicategory, 3.49
of objects in a quasicategory, 3.49
small, 2.3
Congruence fork, 20.14
Congruence relation, 11.20
coequalizer of, vs. monadic construct,
20.35
vs. equalizer, 11.20
vs. monomorphism, 11.20, 11R
vs. pulation square, 11.33
Constant functions are morphisms, vs.
has function spaces, 27.18
Constant functor, 3.20(2)Constant morphism, 7A, 10W

Construct, 3.3(2), 5.1 ffbounded, 24.11
concretely co-wellpowered, 8D
monadic, = monadic construct,20.34 ffmust be regular wellpowered andregular co-wellpowered, 7.88
not determinated by object class,3.3(2)
Contraction, 3.3(3)Contravariant exponential functor, 27.5
vs. (co-)adjoint functor, 27.7
Contravariant hom-functor, 3.20(5)Contravariant power-set functor, 3.20(9)Copower, of an object, 10.63
non-existence, 10S
vs. cocompleteness, 12H
vs. free object, 10R
Coproduct, 10.63 ff
concrete, 13.12
existence, 10I, 12.1, 12G
of functors, 10U
preservation of, 13.1
vs. colimit, 11.28
Coreflection, Galois, 6.26
Coreflection arrow, 4.25 ff
Coreflective hull, vs. cartesian-closed
category, 27C
Coreflective modification, of a concrete
category, 5.22, 5K
Coreflective subcategory, 4.27, 16.1 ff
vs. bicoreflective subcategory, 16.4
vs. cartesian closed subcategory, 27.9
Coreflector for a coreflective subcategory,
4.27
Coreflector, concrete, 5.22
Coseparating set, 18L
Coseparator, 7.16
category with, is regular wellpowered
and regular co-wellpowered, 7.89
extremal, 10.17

9th July 2006

INDEX
is extremal in balanced category,
10.18
vs. faithful hom-functor, 7.17
vs. power of an object, 10.38
vs. topological category, 21.16, 21.17
Costructured arrow, 8.40
Costructured sink, 17.4
Co-unit of an adjunction, 19.3, 19J
Co-universal arrow, 8.40
vs. universal arrow, 19.1, 19.2
Covariant exponential functor, 27.5
Covariant hom-functor, 3.20(4)Covariant power-set functor, 3.20(8)Cover, projective, 9.27
Co-wellpowered category, 7.87 ff
concrete, 8.19
vs. adjoint functor, 18.11, 18.14, 18.19
vs. cartesian closed category, 27.4
vs. (E, M)-category, 15.25, 15.26
vs. epireflective hulls, 16C
vs. essentially algebraic category,
23.14
vs. extensions of factorization
structures, 15.20, 15.21
vs. monadic category, 20.29
vs. topological category, 21.16, 21.17
vs. wellpoweredness, 12.13
Co-wellpowered functor, 8.37
concrete, 8.37
extremal, 8.37
implies domain is co-wellpowered, 8.38
vs. adjoint functor, 18.11, 18.14
vs. essentially algebraic category,
23.14
Creation of absolute coequalizers, vs.
monadic functor, 20.17
Creation of absolute colimits, 20.14
monadic functor does, 20.16

481

Creation of finite limits, vs. monadic
construct, 20.35
Creation of isomorphisms, 13.35, 13M
vs. creation of limits, 13.36
vs. essentially algebraic functor, 23.8
vs. monadic functor, 20.12
vs. reflection of identities &isomorphisms, 13.36
Creation of limits, 13.17, 13N
monadic functor does, 20.12
vs. creation of isomorphisms, 13.36
vs. essentially algebraic functor, 23.15
vs. lifts limits uniquely and reflects
limits, 13.20, 13.25
vs. reflection of isomorphisms, 13.25
Creation of split coequalizers, vs.
monadic functor, 20.17
Decomposition of functors, 3N
vs. factorization structure, 26A
Decomposition of Galois correspondence,
6.35
Decomposition theorems for solidfunctors, 26.3, 26.4
Dense subcategory, 12D
Detection of colimits, 13.22
vs. reflective subcategory, 13.32
vs. solid functor, 25.14
Detection of limits, 13.22
vs. lifting of limits, 13.34
vs. solid functor, 25.14
Detection and preservation of limits,
vs. solid functor, 25.18, 25.19
vs. topologically algebraic functor,
25.18, 25.19
Detection of wellpoweredness, vs.
monadic functor, 20.12
Diagonal, 14.1, 14B, 15.1, 15K, 17.3, 17D

Creation of colimits, 13.17

Diagonal morphism, 10V, 10W

Creation of (Extremal Epi,
Mono-Source)- factorizations, 23.31

Diagonalization property,
causes E and M to determine each

9th July 2006

482

INDEX

other, 15.5
(E, M), = (E, M)-diagonalization
property, 15.1
w.r.t. a functor, 17.3

Duality principle,
for categories, 3.7, 3E
for concrete categories, 5.20
for topological category, 21.10

Diagram, = functor, 11.1 ff
limit of, 11.3

Duality theories, vs. representability, 10N

Directed colimit, 11.28, 11O
vs. finitary quasivariety, 24A

Dual property, 3.7

Dually equivalent categories, 3.37, 3.38

Discrete category, 3.26

Dual statement, of a “categoricalstatement”, 3.7

Discrete functor, vs. topological functor,
21.12(E, −)-category, 15B, 15D

Discrete object, 8.1, 8M
must be smallest element in the fibre,
8.4
vs. concretely cartesian closed
subcategory, 27.15
vs. topological category, 21.11

E-equation, 16.16

Discrete quasicategory, 3.51

(E, −)-functor, 17.4

Discrete space functor, is a fullembedding, 3.29

Eilenberg-Moore category, 20.4

Discrete terminal object, vs. has function
spaces, 27.18
Disjoint union, of a family of sets, 2.1
Dispersed factorization structure, 15L
Distributive law for cartesian closed
category, 27.8
Domain,of a diagram, = scheme, 11.1
of a function, 2.1
of a morphism, 3.2
of a sink, 10.62
of a source, 10.1
of a structured source, 17.1
Dominion, 14J
Down-directed poset, 11.4
Dual of a categorical statement involving
functors, 3.40, 3.42
Dual category, = opposite category, 3.5
Dual concept, of a “categorical concept”,
3.7
Dual functor, = opposite functor, 3.41

E-equational subcategory, 16.16
vs. closed under the formation of Msources and E-quotients, 16.17
vs. closed under the formation of
products, 16.17

E-implicational subcategory, 16.12
vs. E-reflective subcategory, 16.14
Elementhood-morphism, 28.11
Embeddable, fully, 4.6
Embedding functor, 3.27
closed under composition and first
factor, 3.30
finally dense, 10.72
is (up to isomorphism) inclusion of
subcategory, 4.5
vs. faithful functor, 3.28, 4.5
vs. full functor, 3.29
vs. preservation of limit, 13.11
Embedding morphism, 8.6 ff
in Cat, 8C
composition, 8.9
essential, 9.12
first factor of, 8.9
vs. monomorphism, 8.7
vs. regular monomorphism, 8.7, 8A
vs. section, 8.7
Embedding, Yoneda, 6.19, 6J

9th July 2006

INDEX(E, M)-category, 15.1
conditions on E, 15.14, 15.15
consequences of, 15.5
implies E ⊆ Epi, 15.4
values for E and M, 15.8
vs. co-wellpowered category, 15.25,
15.26
vs. (E, M)-factorization, 15.10
vs. (E, M)-functor, 17.4
vs. extremally co-wellpowered
category, 15.25, 15.26
vs. mono-source, 15.6–15.9
vs. regular epimorphism, 15.7
vs. strongly complete category, 15.25,
15.26
vs. topological category, 21.14(E, M )-diagonalization property, 14.1
(E, M)-diagonalization property, 15.1
(E, M )-factorization, 14.1
vs. (E, M )-structured category, 14.7
(E, M)-factorization, 15.1
uniqueness, 15.5
vs. (E, M)-category, 15.10
vs. faithful functor, 17.15, 17.16
vs. topological category, 21.16, 21.17(E, M)-functor, = adjoint functor, 17.3,
18.3, 18.4
implies E ⊆ Gen, 17.6
implies factorizations are essentiallyunique, 17.7
implies M determines E, 17.7
M need not be closed undercomposition nor determined by E,
17.8
not every functor is, 17.4
properties of, 17E
vs. (E, M)-category, 17.4
vs. has (E, M)-factorizations, 17.10
vs. topologically algebraic functor,
25.6

483
20.24
vs. reflective subcategory, 20.25(E, Mono-Source)-category, vs. (Epi,M)-category, 15.11(E, Mono-Source)-functor, implies E ⊆ExtrGen, 17.9
(E, Mono)-structured category, vs.
regular epimorphism, 14.14
Empty category, 3.3(4)Empty source, 10.2
vs. product, 10.20(E, M )-structured category, 14.1 ff
consequences of, 14.6, 14.9, 14.11
duality for, 14.3
relationship to limits, 14.15 ff
relationship to special morphisms,
14.10 ff
uniqueness of factorizations, 14.4
vs. composition of morphisms, 14.6
vs. (E, M )-factorization property, 14.7
vs. (extremal) epimorphisms,
14.10–14.14
vs. extremal monomorphisms, 14.10
vs. isomorphisms, 14.5
Enough injectives, 9.9
vs. absolute retract, 9.10
vs. injective hull, 9D(Epi, ExtrMono-Source)-category, vs.
(ExtrEpi, Mono-Source)-category,
15C
characterization, 15.16
strongly complete category is, 15.17(Epi, ExtrMono)-structured category,vs. equalizer, 14.19
vs. intersection, 14.19(Epi, Initial Source)-factorizable category,vs. topologically algebraic category,
25.6

E-monad, 20.21 ff(Epi, M)-category, characterization, 15.16
vs. (E, Mono-Source)-category, 15.11

E-monadic functor, 20.21 ff
lifts (E, M)-factorizations uniquely,(Epi, Mono-Source)-factorizations, vs.
monadic category, 20.49

9th July 2006

484

INDEX

(Epi, Mono-Source)-factorizable category,vs. algebraic category, 23.30, 23.31
vs. essentially algebraic category, 23.8,
23.9(Epi, Mono-Source)-factorizations, imply(ExtrEpi, Mono-Source)-category,
15.10
Epimorphism, 7.38 ff
closed under composition, 7.41
equals implication, 16.12
extremal, = extremal epimorphism,
7.74
as extremal monomorphism isisomorphism, 7.66
for groups, 7H
preserved and reflected by equivalence
functor, 7.47
products of, 10D
reflected by faithful functor, 7.44
regular, = regular epimorphism, 7.71
as section is isomorphism, 7.43
split, = retraction, 7.24
stable, 11J
swell, 7.76, 15A
types, 7.76
vs. coequalizer, 7M
vs. (E, M)-category, 15.8, 15.14, 15.15
vs. (E, M )-structured category, 14.10
vs. equalizer, 7.54
vs. generating structured arrows, 8.16,
8.18, 8.36
vs. hom-functor, 19C
vs. quotient morphism, 8.12
vs. regular epimorphism, 7O
vs. retraction, 7.42
Epireflective hulls, vs.
co-wellpoweredness, 16C
Epireflective subcategory, 16.1 ff
with bad behavior, 16A
regular, 16I
vs. closure under the formation of
products and extremal subobjects,

16.9, 16.10
vs. extremal mono-source, 16H
vs. strongly limit-closed subcategory,
16L
Epi-sink, 10.63
vs. pullback, 11I
Epi-transformation, 6.5, 7R
vs. adjoint situation, 19.14
Equalizer, 7.51 ff
existence, 12.1
preservation of, 13.1, 13.3
and product, vs. pullback square,
11.11, 11.14
reflection of, 17.13
uniqueness, 7.53
vs. congruence relation, 11.20
vs. (Epi, ExtrMono)-structured
category, 14.19
vs. epimorphism and isomorphism,
7.54
vs. product and pullback square,
10.36, 11.14, 11S
vs. regular monomorphism, 13.6
Equation, = regular implication with freedomain, 16.16
Equational subcategory, 16.16
of Alg(Ω), char. 16.19
Equational subconstruct, vs. epireflective
subconstruct, 16.18
Equivalence,natural, = natural isomorphism, 6.5of quasicategories, 3.51
of “standard” and “object-free”
versions of category theory, 3.55
Equivalence functor, = full, faithful, andiso-morphism-dense functor, 3.33, 3H
concrete, 5.13
is topologically algebraic, 25.2
preserves and reflects special
morphisms, 7.47
properties of, 3.36
vs. adjoint situation, 19.8, 19H
vs. natural isomorphism, 6.8

9th July 2006

INDEX
Equivalence relation on a conglomerate,
2.3
Equivalent categories, 3.33
dually, 3.37, 3.38
vs. isomorphic categories, 3.35
Equivalent objects, in a concrete
category, 5.4
E-reflective hull, = smallest E-reflective
subcategory, 16.21
characterization of members, 16.22

485
vs. solid functor, 26.3

Essential morphism, w.r.t. a class of morphisms, 9.22
Essential uniqueness, of universal arrow,
8.25, 8.35
(E, −)-structured category, 14H
Evaluation morphism, 27.2
Exact category, 14F
Exponential functors, 27.5

E-reflective subcategory, 16.1
intersection of, 16.20
vs. closure under the formation of
M-sources, 16.8
vs. E-implicational subcategory, 16.14
vs. E-monadic functor, 20.25

Exponential law for cartesian closed
category, 27.8

Essential embedding, 9.12
properties of, 9.14

Extension of a factorization structure,
15.19 ff
equivalent conditions for, 15.20
vs. has products, 15.19, 15.21

Essential extension, 9.11, 9H
types of, 9.20
vs. injective object, 9.15
Essentially algebraic category, 23.5 ff
characterization, 23.8
concrete functor between, 23.17
has coequalizers, 23.10
implies embeddings =monomorphisms, 23.7
vs. completeness & cocompleteness,
23.12, 23.13
vs. co-wellpoweredness, 23.14
vs. monadic category, 23.16
vs. wellpoweredness, 23.12, 23.13
Essentially algebraic embedding, 23B
Essentially algebraic functor, 23.1 ff
closed under composition, 23.4
detects colimits, 23.11
equivalent conditions, 23.2
is faithful adjoint functor, 23.3
preserves and creates limits, 23.11
is topologically algebraic, 25.2
vs. creates limits, 23.15
vs. monadic functor, 23C

9th July 2006

Exponential morphism, 27.2
Extension, injective, 9.11
Extensional topological construct & hull,
28B, 28C

Extension of an object, 8.6
Extremal coseparator, 10.17
characterization, 10B, 10C
Extremal co-wellpoweredness, 7.87 ff, 23I
detected by regularly monadic
functor, 20.31
vs. concrete co-wellpoweredness, 8E
vs. monadic category, 20.48
Extremal epimorphism, 7.74 ff
is final morphism in algebraic
category, 23.23
preservation & reflection of, vs.
regularly monadic category, 20.30
preservation & reflection of, vs.
regularly algebraic category, 23.38
reflection of, 17.13
vs. concrete generation, 8.18
vs. (E, M)-category, 15.8
vs. (E, M )-structured category, 14.12,
14.13, 14.14
vs. extremally generating structured
arrows, 8.16, 8.18, 8.36

486

INDEX

vs. final morphism, 8.11(5)vs. pullbacks and products, 23.28,
23.29
vs. quotient morphism in algebraic
category, 23.26, 23G
vs. regular epimorphism, 7.75, 21.13
vs. topological category, 21.13
Extremal epi-sink, 10.63
Extremal generation of an object, 8.15
vs. concrete generation, 8.16
vs. extremal epimorphism, 8.16, 8.18
Extremally co-wellpowered category, 7.87
vs. algebraic category, 23.27
vs. (E, M)-category, 15.25, 15.26
Extremally co-wellpowered functor, 8.37

vs. extremal mono-source, 10.26
vs. monadic functor, 20O
vs. regular monomorphism, 7.62, 7.63,
12B, 14.20, 14I, 21.13
vs. topological category, 21.13
Extremal mono-source, 10.11 ff
characterization, 10A
not preserved by composition, 10.14,
7N
preserved by first factor, 10.13
vs. (E, M)-category, 15.8
vs. epireflective subcategory, 16H
vs. extremal monomorphism, 10.26
vs. product, 10.21
vs. pullback square, 11.9
vs. subsource, 10.15

Extremally epireflective subcategory,
16.1 ff
characterization, 16M
vs. algebraic subcategory, 23.33

Extremal partial morphism, 28.1

Extremally generating structured arrow,
8.15
with respect to a functor, 8.30

Extremal separator, 10.63
characterization, 10B, 10C

Extremally monadic construct, is
topologically algebraic, 25.2

Extremal-subobject classifier, 28.11

Extremally monadic functor, 20M, 23N
Extremally projective object, 23H
Extremally reducible conglomerate ofsinks, 28.13
Extremally wellpowered category, 7.82 ff
vs. wellpowered category, 7.83
Extremal monomorphism, 7.61 ff
closure under composition andintersections, 14.18
composite need not be extremal, 7N
as epimorphism is isomorphism, 7.66
preserved by solid functors, 25.21
products of, 10D
vs. adjoint functor, 18J
vs. balanced category, 7.67
vs. (E, M )-diagonalization property,
14.18
vs. (E, M )-structured category, 14.10

Extremal quotient object, 7.84
Extremal reducibility of final sinks, 28.19

Extremal subobject, 7.77
(ExtrEpi, Mono-Source)-category, vs.
(Epi, ExtrMono-Source)-category,
15C(ExtrGen, Mono)-factorization, 17I(ExtrGen, Mono-Source)-functor, 17K
vs. preservation of strong limits,
17.11, 17H
Factorization lemma, 14.16
Factorization structure for morphisms,
14.1 ff, 14A
extension of, to a factorization
structure for sources, 15.24
Factorization structure, 15.1 ff
dispersed, 15J
for empty sources, 15.2, 15G
extensions of, 15.19 ff
w.r.t. a functor, 17.3
— existence, 17C

9th July 2006

INDEX487— two methods, 17.5
inheritance, 16B
for sinks, 15.2
for small sources, 15J
for 2-sources, 15I
vs. decomposition of functors, 26A
vs. monadic category, 20.28

Fibre-discrete concrete category, means
fibres are ordered by equality, 5.7
categories of T -algebras are, 5.39
concrete category is, iff forgetful
functor reflects identities, 5.8

Faithful functor, 3.27 ff, 7G
characterization, 8N
closed under composition, 3.30
is isomorphism iff full and bijective on
objects, 3.28
means epimorphisms are generating,
8.32
monadic functor is, 20.12
reflects epimorphisms, 7.44
reflects monomorphisms, 7.37
reflects mono-sources, 10.7
topological functor is, 21.3
vs. adjoint situation, 19.14
vs. comparison functor, 20.43
vs. concrete functor, 5.10
vs. embedding functor, 3.28, 3.29, 4.5
vs. (E, M)-factorization, 17.15, 17.16
vs. essentially algebraic functor, 23.2
vs. inclusions of subcategories, 4.5
vs. initial source, 10.59
vs. reflection of equalizers, 13.24,
17.13, 17.14
vs. reflection of (extremal)
epimorphisms, 19.14
vs. reflection of products, 10.60
vs. reflection of special morphisms,
17G
vs. solid functor, 25.12
vs. thin category, 3.29, 3G
vs. topologically algebraic functor,
25.3

Fibre-small topological category, 21.34 ff,
21B

Fibre of a base object, 5.4, 5A
largest element need not be discrete,
8.5
order on, 5.4

Finitary category, 24.4
= finitary variety, if monadic, 24.7

Fibre-complete concrete category, 5.7
functor-structured categories are, 5.42

Finitary quasivariety, 16.12

9th July 2006

Fibre-small concrete category, 5.4, 5.6
monadic category is, 20.12

Final lift, vs. topological category, 21.34
Finally closed subcategory, 21.29
vs. concretely coreflective
subcategory, 22.8
Finally dense embedding, 10.72
Finally dense subcategory, 10.69
vs. concretely reflective subcategory,
21.32
vs. preserves initial sources, 10.71
vs. topological category, 21.32
Final morphism, 8.10 ff
composition of, 8.13
is extremal epimorphism in algebraic
category, 23.23
first factor of, 8.13
products of, 28J
vs. extremal epimorphism, 8.11(5)vs. isomorphism, 8.14
vs. monadic category, 20.28
vs. regular epimorphism, 8.11(4), 8O,
20.51
Final quotient object, 8.10
Final sink, 10.63 ff
reducibility of, vs. representablepartial morphisms, 28.15
Finer than, (preorder) relation on
concrete functors, 5.18

Finitary functor, 24.4
Finitary monad, 24.4

488

INDEX

axiomatic descriptions, 23.30, 23.31,
23.38, 24.9, 24.10,
characterization theorem, 24.9
= finitary and algebraic category, 24.9
= regular epireflective subconstruct ofsome finitary monadic construct,
24.9
vs. directed colimit, 24A
vs. finitary variety, 16K, 16N, 24.9
Finitary variety, 16.16
axiomatic descriptions, 20.35, 23.40,
24.7, 24.8
characterization theorem for, 24.7
is monadic, 20.20
vs. finitary quasivariety, 16K, 16N,
24.9
vs. monadic category, 24.7
Finitely (co)complete category, 12.2
characterization, 12.4
Finitely complete category, vs. has finite
products and representable M -partial
morphisms, 28.5, 28.6
Finite product, 10.29
vs. product and projective limit, 11B
Forgetful functor, 3.20(3), 5.1 ff
vs. coequalizer, 7.73
vs. regular epimorphism, 7.73
vs. topological construct, 21L
Fork, congruence, 20.14
split, 20.14
Frame, 5L, 8G
Free automata, 8P
Free category, 3A
Free functor, 19.4
Free monad, 20.55
vs. varietor, 20.56, 20.57, 20.58
Free object, 8.22, 8F–8L
existence, 18.15
retract of, 9.29
vs. adjoint functor, 18.19
vs. adjoint situation, 19.4
vs. copower, 10R

vs. initial object, 8.23
vs. monomorphism, 8.28, 8.29
vs. preservation and reflection of
mono-sources, 18.10
vs. projective object, 9.29, 9.30
vs. representable forgetful functor,
8.23
vs. wellpoweredness, 18.10
Full embeddability of familiar constructs,
4.7, 4K
Full embedding functor, 4.6
vs. comparison functor, 20.44
vs. inclusion of a full subcategory, 4.5,
4.6
Full faithful functor,
is isomorphism iff bijective on objects,
3.28
properties of, 3.31
reflects isomorphisms, 3.32
vs. adjoint situation, 19.14
Full functor, 3.27 ff
closed under composition, 3.30
second factor of is not always full, 3.30
vs. comparison functor, 20.43
Full reflective embedding,is topologically algebraic, 25.2
vs. solid functor, 26.1
Full reflective subcategory,characterization, 4.20
Full subcategory, 4.1 ff, 4C
isomorphism-closed, 4.9, 4B
isomorphism-dense, 4.9
vs. concretely reflective subcategoryof amnestic category, 5.24
Function,
between classes, 2.2
between conglomerates, 2.3
between sets, 2.1
Functor, 3.17 ff
adjoint, = adjoint functor, 18.1 ff
algebraic, = algebraic functor, 23.19 ff
co-adjoint, = co-adjoint functor,

9th July 2006

INDEX
18.1 ff
comparison, = comparison functor,
20.37 ff
composite of, 3.23
concretely co-wellpowered, 8.37
constant functor, 3.20(2)contravariant hom-functor, 3.20(5)contravariant power-set functor,
3.20(9)coproduct of, 10U
coreflector for a coreflective
subcategory, 4.27
covariant hom-functor, 3.20(4)covariant power-set functor, 3.20(8)co-wellpowered, implies domain
co-well- powered, 8.37, 8.38
decomposition of, 3N
discrete space, 3.29
dual, = opposite functor, 3.41
duality functor for vector spaces,
3.20(12)(E, −), = (E, −)-functor, 17.4
embedding, = embedding functor,
3.27
is embedding iff faithful and injective
on objects, 3.28
(E, M), = (E, M)-functor, 17.3 ff
E-monadic, = E-monadic functor,
20.21 ff
equivalence, = equivalence functor,
3.33
essentially algebraic, = essentiallyalgebraic functor, 23.1 ff
exponential, 27.5
extremally co-wellpowered, 8.37
faithful, = faithful functor, 3.27
as family of functions between
morphism classes, 3.19
finitary, 24.4
forgetful functor, = underlying orforgetful functor, 3.20(3), 5.1
free, 19.4
full, = full functor, 3.27
fundamental group, 3.22

9th July 2006

489
identity, 3.20(1)inclusion, 4.4
indiscrete space, 3.29, 21.12
inverse, 3.25
isomorphism, 3.24
is isomorphism iff full, faithful, andbijective on objects, 3.28
isomorphism-dense, 3.33
monadic, = monadic functor, 20.8 ff
M-topological, 21K
naturally isomorphic to idA , 6D
need not reflect isomorphisms, 3.22
notation, 3.18
nth power functor, 3.20(10)object-free, 3.55
object-part determined by themorphism-parts, 3.19
opposite, = opposite functor, 3.41
preserves isomorphisms, 3.21
between quasicategories, 3.51
regularly algebraic, = regularly
algebraic functor, 23.35 ff
regularly monadic, = regularly
monadic functor, 20.21 ff
representable, = representablefunctor, 6.9
solid, = solid functor, 25.10 ff
Stone-functor, 3.20(11)topologically algebraic, =topologically algebraic functor,
25.1 ff
topological, = topological functor,
21.1 ff
underlying, = forgetful functor,
3.20(3), 5.1
vs. reflection of identities, 3D

Functor-costructured category, 5.43
Functor quasicategory, 6.15, 6H
full embedding of any category into,
6.20
Functors, relationships among, 25.22
Functor-structured category, 5.40, 6C,
28K

490

INDEX

cartesian closed, 27I
characterization, 22A
reflective subcategory of, vs. solid
strongly fibre-small category, 26.7
Galois adjoint, = residual functor, 6.25
Galois adjunction, 19D

extremally, 8.15
strongly generating structured arrowis, 25.5
w.r.t. a functor, 8.30
Generation,
vs. concrete generation, 8.16
vs. epimorphism, 8.16, 8.18

Galois co-adjoint, = residuated functor,
6.25

Graph of a category, 3A

Galois connection, 6.26, 6G
composition of, 6.27(1)dual of, 6.27(2)Has a separator, vs. cartesian closed
category, 27.4

Galois coreflection, 6.26, 6.35

Has coequalizers, 12.1
vs. regularly algebraic category, 23.38

Galois correspondence, 6.25, 19E
between amnestic concrete categories,
6.29–6.36
decomposition of, 6.35
equivalently described, 6.28
for constructs, 6.26
theorem, 21.24
vs. adjoint situation, 19.8
vs. concrete co-adjoint, 21E
vs. initial source preservation, 10.49,
21.24

Has concrete (co)limits, 13.12

Galois isomorphism, 6.26, 6.35

Has (finite) (co)intersections, 12.1

Galois reflection, 6.26
characterization, 6.34
vs. decomposition of Galois
correspondence, 6.35

Has (finite) (co)products, 10.29, 12.1

(Generating, −)-factorizable functor,
composites of, 17B(Generating, −)-factorization, for
2-sources implies preservation of
mono-sources, 17.12(Generating, M)-functor, vs. adjoint
functor, 18.4
(Generating, Mono-Source)-factorizationsof 2-sources, vs. reflection of
isomorphisms, 17.13
Generating structured arrow, 8.15 ff
characterization, 17A
concretely, 8.15

Has concrete products, 10.54
Has (E, M)-factorizations, vs.
(E, M)-functor, 17.10
w.r.t. a functor, 17.3
vs. adjoint functor, 18.3
Has enough injectives, 9.9, 9.22
Has equalizers, 12.1
and (finite) products, means (finitely)complete, 12.3, 12.4

Has finite intersections,and finite products, means finitely
complete, 12.4
and products, means complete, 12.3
Has finite products, and representable
M -partial morphisms, vs. finitely
complete, 28.5, 28.6
Has free objects, 8.26
Has function spaces, 27.17
vs. constant functions are morphisms,
27.18
vs. discrete terminal objects, 27.18
vs. topological category, 27.22
Has limits, vs. (E, M )-structured, 14.17
Has M-initial subobjects, vs. concrete
products, 21.42

9th July 2006

INDEX
Has products, 10.29
of all sizes, implies thin category,
10.32
and equalizers (resp. intersections),means complete, 12.3
vs. extension of factorization
structure, 15.19, 15.21
Has pullbacks, 12.1
and a terminal object, means finitely
complete, 12.4
Has pushouts, 12.1
Has regular factorizations, 15.12
vs. regularly monadic functor, 20.32
Has representable M -partial morphisms,
28.1
consequences of, 28.3
Has small concrete colimits,characterized, 13.14
Hewitt, E. see Duality principle
Hom-functor, 3.20(4), 3.20(5)preserves limits, 13.7
vs. (co)separator, 7.12, 7.17
vs. epimorphism, 19C
vs. limit, 13H
vs. product, 10E, 10F
vs. quasicategory, 3.51
vs. set-valued functor, 6.18
Hull,algebraic, 23K
E-reflective, = E-reflective hull, 16.21
injective, 9.16

491
reflection, 3D
vs. isomorphism, 3.13
vs. object, 3.19
vs. subcategory, 4A

Identity natural transformation, 6.6
Illegitimate conglomerate, 2.3
Illegitimate quasicategory, 6.16
Image of the indexing function, 2.1
Image, inverse 11.19
Implication, = epimorphism, 16.12
satisfaction of, 16.12
Implicational subcategory, 16.12
Inclusion functor, 4.4
Indiscrete object, 8.3
must be largest element in the fibre,
8.4
preservation, 21.23
vs. initial source, 10.42
vs. initial subobjects, 21.35
vs. topological category, 21.11, 21.35
Indiscrete space functor,
is a full embedding, 3.29
vs. topological functor, 21.12
Indiscrete structure, vs. adjoint situation,
21A
vs. topological functor, 21.18, 21.19,
21.20
Inductive limit, 11.28
Initial completion, 21G
Initiality-preserving concrete functor,
preserves indiscrete objects, 21.23

Idempotent monad, 20F

Initial lift, vs. topological category, 21.34

Identity-carried A-morphism, 5.3

Initially closed subcategory, 21.29
of a topological category is
topological, 21.30
vs. concretely reflective subcategory,
21.31, 22.3, 22.4

Identity-carried reflection arrow, seeConcretely reflective concrete
subcategory
Identity functor, 3.20(1), 5.14
Identity morphism, 3.1 ff
in a category, as unit in thecorresponding object-free category,
3.54

9th July 2006

Initially dense subcategory, 10.69
Initial morphism, 8.6
composition of, 8.9
first factor of, 8.9

492

INDEX

universally, 10P
vs. injective morphism, 8B
vs. isomorphism, 8.14
Initial object, 7.1
uniqueness, 7.3
vs. free object, 8.23
vs. limit, 11A
Initial source, 10.41
composition of, 10O
domain of, 10.43
preservation, 10.47— vs. Galois correspondence, 21.24
preserved by composition, 10.45
preserved by first factor, 10.45
vs. concrete product, 10.53
vs. concretely reflective subcategory,
10.50
vs. faithful functor, 10.59
vs. finally dense subcategory, 10.69
vs. indiscrete object, 10.42
vs. initial subsource, in a fibre-small
topological category, 21.36
vs. mono-source, 17.13
vs. subsource, 10.46
w.r.t. a functor, 10.57
Initial subobject, 8.6
vs. indiscrete object, 21.35
vs. monotopological construct, 21.42
vs. topological category, 21.35
Initial T -algebra, 20I
Injection morphism, 10.63
Injective, enough, 9.9, 15H
Injective automata, 9I
Injective extension, 9.11
types of, 9.20
Injective hull, 9.16
w.r.t. a class of morphisms, 9.22
existence, 9C
for finitary varieties, 16J
uniqueness, 9.19
vs. enough injectives, 9D
Injective morphism, vs. initial morphism,

8B
Injective object, 9.1
is an absolute retract, 9.7
in CAT, vs. topological category,
21.21
characterization, 9B
w.r.t. a class of morphisms, 9.22
preserved by product, 10.40
terminal object is, 9.4
in topological construct, 28A
vs. essential extension, 9.15
vs. maximal essential extension, 9F
vs. reflective subcategory, 9.25
vs. retract, 9.5
Injectivity class, 9.26
Insertion of generators, 19.4
Intersection, of a family of sets, 2.1
Intersection of subobjects, 11.23
closure under the formation of, 11.26
existence, 12.1
as pullback, 11F
vs. (E, M )-structured category, 14.15
vs. (Epi, ExtrMono)-structured
category, 14.19
vs. isomorphism, 11.25
vs. limit, 11.25
vs. regular subobject & section, 11K
Inverse functor, unique determination,
3.25
Inverse image, of a subject, 11.19
Inverse of an isomorphism, notation, 3.12
Inverse limit, 11.4
Inverse of a morphism, 3.8
uniqueness, 3.11
“is equivalent to”, is equivalence relation
on categories, 3.34
“is isomorphic to” is an equivalence
relation on categories, 3.25
Isomorphic, “is isomorphic to” is anequivalence relation on categories,
3.25
concretely, 5.12

9th July 2006

INDEX
naturally, 6.5493
limit-closed, 13.27

Isomorphic categories, 3.24, 3.25
vs. equivalent categories, 3.35

Isomorphism-dense full subcategory, 4.9,
4.10, 4.12

Isomorphic objects, 3.15
yields equivalence relation on theobject class, 3.16

Isomorphism-dense functor, 3.33

Isomorphic structured arrows, 8.19, 8.34
Isomorphism, 3.8 ff, 3F
in a category, vs. isomorphic objects,
3.15
composite of, is an isomorphism, 3.14
concrete, 5N
creation, 13.35, 13M
functor is isomorphism iff full,
faithful, and bijective on objects,
3.28
functors need not reflect, 3.22
Galois, 6.26
inverse is an isomorphism, 3.14
natural, 6.5
non-concrete, 5B
is 1-product, 10.20
preserved and reflected by equivalence
functor, 7.47
preserved by functors, 3.21
preserved by products, 10.35
reflection of, 13.35, 13F
vs. coequalizer, 7.70
vs. (E, M)-category, 15.5
vs. epimorphism, 7.43, 7.66
vs. equalizer, 7.54
vs. extremal monomorphism, 7.66
vs. final morphism, 8.14
vs. identity morphism, 3.13
vs. initial morphism, 8.14
vs. intersection, 11.25
vs. monomorphism, 7.36
vs. product source, 10.26
vs. retraction, 7.36
vs. section, 7.43

Isomorphism functor, 3.24
closed under composition, 3.30
is isomorphism in CAT, 3.51
Iso-transformation, 6.5
Kernel, 7C
vs. pullback, 11E
Kleisli category, 20.39, 20B
Large category, 3.44
Large class, 2.2
Largest essential extension, 9.20
Left adjoint, = co-adjoint, 19.11
Left inverse of a morphism, vs. rightinverse, 3.10
Legitimate conglomerate, 2.3
Legitimate quasicategory, 6.16
Lifting of an adjoint situation, 21.26
vs. topological category, 21.28
Lifting of colimits, 13.17
Lifting of (E, M)-factorizations, uniquely,
20.23
vs. E-monadic functor, 20.24
Lifting of (Extremal Epi,
Mono-Source)-factorizations uniquely,
vs. algebraic category, 23.31

Isomorphism-closed (full) subcategory,
4.9, 4B

Lifting of limits, 13.17
uniquely, and reflect limits, vs. create
limits, 13.25
vs. amnesticity, 13.21
vs. create limits, 13.20
vs. detect limits, 13.34
vs. preserve small limits, 13.19
vs. reflect limits, 13.23
vs. transportability, 13E

Isomorphism-closed full subcategory, vs.

Limit-closed subcategory, 13.26, 13I, 13J

9th July 2006

494

INDEX

vs. isomorphism-closed subcategory,
13.27
Limit-dense subcategory, rarity, 12E
Limit (of a diagram), 11.3 ff
concrete, = concrete limit, 13.12
construction via large colimits, 12.7
creation, 13.17
detection, = detection of limit, 13.22
is extremal mono-source, 11.6
inductive, 11.28
inverse, 11.4
lifting of, 13.17, 13D, 13E— vs. topological functor, 21.15
preservation of, = preservation oflimit, 13.1 ff, 13B— vs. preservation of monos, 13.5
— vs. preservation of strong limits,
13.1— vs. topological functor, 21.15
projective, 11.4, 11B
reflection of, 13.22, 13G, 17.13
uniqueness, 11.7
vs. factorization, 14.16
vs. hom-functors, 13H
vs. initial object, 11A
vs. intersection, 11.25
vs. pullback square, 11.9
vs. reflective subcategory, 13.28
Locally presentable category, 20H
Mac Neille completion, 21H
vs. solid category, 25G
M -action, category of all M -actions andaction homomorphisms, 3.26
Map, quotient, = quotient map, 9.27
Matching condition, see Object-freecategory
Maximal essential extension, 9.20
vs. injective object, 9F
Minimal injective extension, 9.20
vs. smallest injective extension, 9G
M-initially closed subcategory, vs.

concretely E-reflective subcategory,
22.9
Modification, amnestic, = amnestic
modification, 5.6, 5.33, 5.34, 5F
Modifications of structure, vs.
(co)reflective subcategory, 4.17, 4.26
Monad, 20.1 ff
associated with an adjoint situation,
20.3, 20A
finitary, 24.4
gives rise to an adjoint situation, 20.7
idempotent, 20F
on Set, is finitary if and only if theassociated construct is finitary, 24.6
— is regular, 20.22
regular, 20.21
trivial, 20.2
with rank, 20G
Monadic concrete category, 20.8 ff
cocompleteness of, 20D∼= its associated category of algebras,
20.40
properties of, 20.12
vs. adjoint functor, 20.46
vs. concrete functor, 20E
vs. co-wellpoweredness, 20.29
vs. (Epi,Mono-Source)-factorization,
20.49
vs. essentially algebraic category,
23.16
vs. extremal co-wellpoweredness,
20.48
vs. factorization structure, 20.28
vs. final morphism, 20.28
vs. finitary variety, 24.7
vs. monadic subcategory, 20.19
vs. reflective subcategory, 20.18
vs. varietor, 20.56, 20.57, 20.58
Monadic construct, 20.34 ff
bounded, vs. bounded variety, 24.11
characterization theorem for, 20.35
is complete, cocomplete, wellpowered,co-wellpowered, and has regular

9th July 2006

INDEX
factorizations, 20.34
is regularly algebraic, 23.37
vs. algebraic construct, 23.41
vs. coequalizer of congruence relation,
20.35
vs. creation of finite limits, 20.35
Monadic functor, 20.8 ff
characterization theorem for, 20.17
composition of, need not be monadic,
20.45
creates absolute colimits, 20.16
deficiencies of, 20.45 ff
need not detect colimits, 20.47
need not reflect regular epimorphisms,
20.52
properties of, 20.12
regularly, = regularly monadic
functor, 20.21 ff
vs. colimit, 20.13
vs. essentially algebraic functor, 23C
vs. extremal monomorphism, 20O
vs. order preserving map, 20N
vs. preservation of extremal
epimorphisms, 20.50
vs. preservation of regularepimorphisms, 20.50
vs. regularly monadic functor, 20.32,
20.35
vs. solid functor, 25F
Monadic subcategory, vs. monadic
category, 20.18
Monadic towers, 20J
Monad morphism, 20.55
Monoid (as a category), 3.3(4)M -actions as functors from a monoid
M to Set, 3.20(13)Monomorphism, 7.32 ff
closed under composition, 7.34
extremal, = extremal monomorphism,
7.61 ff
normal, 7C
preserved by products, 10.35
preserved and reflected by equivalence

9th July 2006

495
functor, 7.47
preserved by representable functor,
7.37
is pullback stable, 11.18
reflected by faithful functor, 7.37
regular, = regular monomorphism,
7.56 ff
as retraction means isomorphism, 7.36
split, = section, 7.19 ff
strict, = strict monomorphism, 7.76,
7D
types of, 7.76
vs. coequalizer, 7.70
vs. congruence relation, 11.20, 11R
vs. embedding, 8.7
vs. (E, M )-structured category, 14.9
vs. free object, 8.28, 8.29
vs. mono-source, 10.26
vs. preservation of limits, 13.5
vs. pullback square, 11.15, 11.16
vs. regular monomorphism, 7O
vs. section, 7.35, 7P

Monoreflective subcategory, 16.1
is bireflective, 16.3
Mono-source, 10.5 ff
characterization, 10A
extremal, = extremal mono-source,
10.11
is extremal in balanced category,
10.12
is initial, vs. essentially algebraic
functor, 23.2
— vs. monadic category, 20.12— vs. reflection of limits, 17.13, 17.14
preservation, by representablefunctors, 10.7
— implied by (Generating,−)-factoriza- tion for 2-sources,
17.12
preserved by composition, 10.9
preserved by first factor, 10.9
reflection, by faithful functors, 10.7
vs. balanced category, 10.12
vs. (E, M)-category, 15.6–15.9

496

INDEX

vs. initial source, 17.13
vs. monomorphism, 10.26
vs. point-separating source, 10.8, 10T
vs. preservation of limits, 13.5
vs. pushout, 11P
vs. subsource, 10.10
Monotopological category, 21.38 ff

M-topological category, 21.38 ff
characterization theorem, 21.40
is E-reflective in some topologicalcategory, 21.40
fibre-small, characterization, 22.9
vs. subcategory of afunctor-structured category, 22.9

Monotopological construct,characterization of fibre-small, 22.10
vs. concrete products, 21.42
vs. initial subobjects, 21.42
vs. subconstruct of afunctor-structured construct, 22.10

M-topological functor, 21K

Mono-transformation, 6.5, 7R

Multiple pullback, 11L–11N, 12C
preservation, 13A

Morphism, 3.1 ff(co)domain of, 3.2
coessential, 9.27
composite of, see Composition ofmorphisms
constant, 7A, 10W
diagonal, 10V, 10W
epi-, see Epimorphism
evaluation, 27.2
exponential, 27.2
final, = final morphism, 8.10 ff
initial, = initial morphism, 8.6 ff
isomorphism, 3.8
mono-, see Monomorphism
in an object-free category, 3.53
partial, 28.1
product of, 10.34
quotient, = quotient morphism, 8.10
retraction, 7.24 ff
section, 7.19 ff
zero, 7A
Morphism class of a category, 3.2
as corresponding object-free category,
3.54
of small category is a set, 3.45
M -reducible, 28.13
M-sources, closed under the formation of,
16.7
M-subobject, = singleton M-source, 16.8

M-topological structure theorem, 22.9
M -topos, 28.7
M -transformation, 6.5
Multiple equalizer, 11.4

Natural equivalence, = natural
isomorphism, 6.5
Natural isomorphism, 6.5
Natural numbers, 2.1
Naturality condition, 6.1
Naturally isomorphic, 6.5
Natural sink, 11.27
Natural source, 11.3
is natural transformation, 11.5
Natural transformation, 6.1 ff
cardinality of, 6B
composition, 6.13, 6A
concrete, 6.23, 6.24
identity-carried, = concrete naturaltransformation, 6.23
M -transformation, 6.5
vs. composition of functors, 6.3
vs. opposite functor, 6.3
vs. Set-valued functor, 6.18
Neighborhood space, 5N
Normal monomorphism, 7C
nth power functor, 3.20(10)n-tuple of sets or classes, 2.1, 2.2
Object, 3.1 ff

9th July 2006

INDEX
copower of, 10.63
discrete, = discrete object, 8.1
equivalent to another, in a concrete
category, 5.4
free, = free object, 8.22
indiscrete, = indiscrete object, 8.3
initial, = initial object, 7.1
injective, = injective object, 9.1
isomorphic, 3.15, 3.16
power, 27.2
power of, 10.37
projective, = projective object, 9.27
quotient, = quotient object, 7.84 ff
terminal, = terminal object, 7.4
zero, 7.7, 27A

497

Partial binary algebra, 3.52
Partial binary operation, 3.52
Partial morphism, 28.1
Pointed category, 3B, 7B
Point-separating source, 10.5
vs. mono-source, 10.8, 10T
Posets, category of, 4.3
down-directed & up-directed, 11.4
Power object, 27.2
Power-set, = set of all subsets, 2.1
Power-set functor, 3.20(8), 3.20(9)representability of, 6F
vs. adjoint functor, 18E

Object class of a category, 3.1, 3.2
vs. identity morphisms, 3.19

Power-set monad, 20.2

Object-free category, 3.53
corresponding to a category, 3.54, 3.55

Power (of an object), 10.37
with discrete exponent is product, vs.
cartesian closed topologicalconstruct, 27.24
comparison of, 10K
vs. coseparator, 10.38

Object-free functor, 3.55
Opposite category, 3.5
dually equivalent means opposite
category is equivalent, 3.38
vs. contravariant hom-functor, 3.20(5)vs. contravariant power-set functor,
3.20(9)vs. dual functor for vector spaces,
3.20(12)
vs. Stone-functor, 3.20(11)Opposite functor, 3.41
vs. Galois connection, 6.27 (2)vs. natural transformation, 6.3
Ordered pair of sets, 2.1
Order on concrete functors, 5.18
vs. concretely reflective subcategory,
5.26
Order on objects, in a concrete category,
5.4, 5.5
Order preserving map, = morphism inPos, 4.3
vs. monadic functor, 20N

9th July 2006

Power-set-morphism, 28.11

Preimage-morphism, 28.11
Preordered class, as a category, 3.3(4)Preorder relation on the fibres, in a
concrete category, 5.5
Preservation of coequalizers, 13.1
Preservation of colimits, 13.1
Preservation of coproducts, 13.1
Preservation of equalizers, 13.1, 13.3
Preservation of extremal epimorphisms,
vs. monadic functor, 20.50
Preservation of initial sources, 10.47
vs. finally dense subcategory, 10.71
Preservation of limits & colimits, §13
Preservation of limits,adjoint functor does, 18.9
vs. adjoint functor, 18.12, 18.14,
18.17, 18.19
vs. embedding, 13.11
vs. hom-functor, 13.7

498

INDEX

vs. monomorphism & mono-source,
13.5
vs. representable functor, 13.9
vs. solid functor, 25.14
Preservation of mono-sources, implied by(Generating, −)-factorization for
2-sources, 17.12
Preservation of products, 13.1, 13.3
Preservation of pullbacks, 13.3
Preservation and reflection of
mono-sources, vs. free object, 18.10
vs. monadic functor, 20.12
Preservation of regular epimorphisms, vs.
(regularly) monadic functor, 20.32,
20.50
Preservation of small limits, vs. lift
limits, 13.19
Preservation of strong limits, vs.
(ExtrGen, Mono-Source) functor,
17.11, 17H
Preservation of terminal object, 13.3
Pretopological space, 5N
Product (of morphisms), 10.34
of epimorphisms, vs. cartesian closed
category, 27.8
Product (of objects), 10.19 ff
in abelian groups, 10G
for Banach spaces, 10J
characterization, 10Q
composition of, 10.25
concrete, = concrete product, 10.52,
13.12
with discrete factors are coproducts,
vs. cartesian closed topologicalconstruct, 27.24
existence, 10.29, 10I, 12.1
is extremal mono-source, 10.21
notation for, 10.23
of pairs, 10.30
preservation of, 13.1
preserved by first mono-factor, 10.56
and pullback square, vs. equalizer,

11.11, 11.14
uniqueness, 10.22
vs. empty source, 10.20
vs. (E, M )-structured category, 14.15
vs. equalizer, 10.36
vs. finite product, 11B
vs. first factor, 10.25
vs. hom-functor, 10E, 10F
vs. isomorphism, 10.20
vs. projective limit, 11B
vs. pullback square, 11.13, 11C, 11D
vs. retraction, 10.28
vs. terminal object, 10.20, 10.30, 10H
Product category, 3.3(4)Product source, vs. isomorphism, 10.26
Projection morphism, 10.23
vs. retraction, 10.27
Projective cover, 9.27
Projective hull, w.r.t. a class ofmorphisms, 9.27
Projective limit, 11.4
vs. (finite) product, 11B
Projective object, 9.27
w.r.t. a class of morphisms, 9.27
extremal, in algebraic category, 23.28,
23.29
regular, 9E
vs. free object, 9.29, 9.30
Proper class, 2.2 See also Large category
Proper quasicategory, 3.50, 3.51
quasicategory of all categories is, 3.51
Property, dual, 3.7
universal, 4.16, 4.25
Property of objects, as(isomorphism-closed) full subcategory,
4.8
Pulation square, 11.32, 11Q
vs. congruence relation, 11.33
Pullback, 11.8 ff. See also Pullbacksquareclosure under the formation of, 11.17
existence, 12.1

9th July 2006

INDEXof a morphism, 11.8
multiple, 11L–11N
of a sink along a morphism, 28.13
of a 2-sink, 11.8
vs. (E, M )-structured category, 14.15
vs. epi-sink, 11I
vs. equalizer, 11Svs. kernel, 11E
vs. product, 11C, 11D
vs. section, 11H
vs. strict monomorphism, 11H
Pullback square, 11.8 ff. See also
Pullback
cancellation of, 11.10, 11.15
composition of, 11.10
and product, vs. equalizer, 11.14
vs. equalizer and product, 11.11
vs. extremal mono-source, 11.9
vs. limit, 11.9
vs. monomorphism, 11.15, 11.16
vs. product, 11.13
vs. terminal object, 11.13

499
proper, 3.50, 3.51
vs. hom-functor, 3.51

Quasiconstruct of quasitopologicalspaces, 5.6
Quasitopos, 28.7
is (Epi, RegMono)-structured, 28.10
Quasivariety, = algebraic construct, 24.12
finitary, = finitary quasivariety, 16.12
Quotient map, 9.27
coessential, 9.27
Quotient morphism, 8.10
composition & first factor of, 8.13
vs. extremal epimorphism, 23G
vs. (regular) epimorphism &retraction, 8.12
Quotient object, 7.84 ff
extremal, = extremal quotient object,
7.84
order on, 7.85
regular, = regular quotient object,
7.84

Pullback stable, 11.17, 28.13
Pushout, 11.30 ff
existence, 12.1
of a 2-source, 11.30
vs. (E, M)-category, 15.14, 15.15,
15.16
vs. mono-source, 11P

Rational & real numbers, 2.1

Pushout square, 11.30 ff
vs. coequalizer, 11.33

Reflection, See also Reflector
concrete, 5.22, 5E
Galois, 6.26

Quasicategory, 3.49 ff
of all categories, is not a category, 3.5,
3.51
of all concrete categories over a given
base category, 5.14, 5.15
of all object-free categories, 3.55
of all quasicategories, yields
Russell-like paradox, 3.51, 3L
category is, 3.51
functor quasicategory, 6.15, 6H(il)legitimate, 6.16

Reflection of colimits, 13.22

9th July 2006

Realization = full, concrete embedding,
5O
Reflection arrow, 4.16 ff
uniqueness, 4.19
vs. colimit, 13.30

Reflection of equalizers, 17.13
vs. essentially algebraic functor, 23.2
vs. faithful functor, 13.24, 17.13, 17.14
vs. reflection of limits, 17.13, 17.14
Reflection of (extremal) epimorphisms,
17.13
vs. adjoint situation, 19.14
vs. essentially algebraic functor, 23.2
vs. faithful functor, 19.14
vs. monadic functor, 20.12

500

INDEX

Reflection of identities, vs. creation of
isomorphisms, 13.36
Reflection of isomorphisms, 13.35
functors need not, 3.22
vs. creation of isomorphisms, 13.36
vs. creation of limits, 13.25
vs. essentially algebraic functor, 23.2
vs. (Generating,Mono-Source)-factoriza- tions of
2-sources, 17.13
vs. reflection of limits, 17.13, 17.14
Reflection of limits, 13.22, 13G, 17.13
vs. essentially algebraic functor, 23.2
vs. lifting of limits, 13.23
vs. mono-sources are initial, 17.13,
17.14
vs. reflection of equalizers, 17.13,
17.14
vs. reflection of isomorphisms, 17.13,
17.14
Reflection of products, vs. faithful
functor, 10.60
Reflection of regular epimorphisms, vs.
initiality of mono-sources, 19.14
Reflective embedding, misbehaved, 13K
Reflective modification, of a concrete
category, 5.22, 5K
Reflective subcategory, 4.16, 16D, 16E
characterization of fullness, 4.20
embedding of, preserves and reflectsmono-sources, 18.7
full, 4.20, 7F
of a functor-structured category, vs.
cocomplete category that has freeobjects, 26.8
intersections, 4F
nonfull, with reflection arrows
isomorphisms, example, 4.21, 13K
reflectors are naturally isomorphic, 6.7
of special categories, 4D
vs. adjoint situation, 19.4
vs. cartesian closed subcategory, 27.9
vs. cocomplete subcategory, 12K

vs. colimit-closed subcategory, 13.29
vs. detection of colimits, 13.32
vs. E-monadic subcategory, 20.25
vs. injective objects, 9.25
vs. limits, 13.28
vs. monadic category, 20.18
Reflector (for a reflective subcategory),
4.23, 4H
as composite of epireflectors, 16.24
concrete, 5.22, 5E
existence, 4.22
naturally isomorphic to others, 6.7
uniqueness 4.24, 6.7
vs. concretely reflective subcategory,
5.26, 5.27, 5.31, 5.32
(RegEpi, Mono-Source)-category,characterization, 15.25
vs. regular factorization, 15.13(RegEpi, Mono)-structured category,
14.22, 14D
Regular category, 14E
Regular co-wellpowered category, 7.87
vs. category with separator orcoseparator, 7.89
vs. construct, 7.88
Regular epimorphism, 7.71 ff
closed under composition, vs.
(RegEpi, Mono-Source)-category,
15.25
composition of, 14.22
is extremal epimorphism, 7.75
preservation and reflection, vs.
regularly algebraic category, 23.39
vs. (E, M)-category, 15.7, 15.8
vs. (E, Mono)-structured category,
14.14
vs. (extremal) epimorphism, 7O, 21.13
vs. final morphism, 8.11(4), 8O, 20.51
vs. forgetful functor, 7.72(5), 7.73
vs. monadic functor, 20.51, 20.52
vs. quotient morphism, 8.12
vs. retraction, 7.75
Regular epireflective subcategory, 16.1

9th July 2006

INDEX
Regular epireflective subconstruct, vs.
(regular) equational subconstruct,
16.18
Regular equation, 16.16
Regular equational subconstruct, vs.
regular epireflective subconstruct,
16.18
Regular factorization, 15.12, 20.32
vs. algebraic functor, 24.2
vs. (RegEpi,Mono-Source)-category,
15.13
vs. regularly algebraic category, 23.38,
23.39
vs. regularly monadic category, 20.30
Regular monomorphism, 7.56 ff
composition of, 7J, 10M, 14I, 28.6
is (extremal) monomorphism, 7.59,
7.63
first factor, 14I
in functionally Hausdorff spaces, 7J
preserved by products, 10.35
is pullback stable, 11.18
in semigroups, 14I
vs. embedding, 8.7, 8A
vs. equalizer, 13.6
vs. extremal monomorphism, 7.62,
7.63, 7.65, 7J, 12B, 14.20, 14I, 21.13
vs. monomorphism, 7O
vs. section, 7.59
vs. strict monomorphism, 12A
vs. topos, 28F
Regular projective object, 9E
Regular quotient object, 7.84
Regular subobject, 7.77
vs. intersection, 11K
Regular wellpowered category, 7.82
vs. category with separator orco-separator, 7.89
vs. construct, 7.88
vs. wellpowered category, 7.83
Regularly algebraic category, 23.35 ff
characterization theorem for, 23.38

9th July 2006

501
vs. adjoint functor, 23.38
vs. regular factorization, 23.39
vs. uniquely transportable functor,
23.38

Regularly algebraic functor, 23.35 ff, 23E,
23M
decomposition theorem for, 24.2
implies associated monad is regular,
24.1
implies comparison functor is aregular epireflective embedding, 24.1
Regularly monadic category, vs. regularfactorization, 20.30
Regularly monadic functor, 20.21 ff, 20L
characterization theorem for, 20.32
composition of, 24.2
detects colimits, 20.33
detects extremal co-wellpoweredness,
20.31
is regularly algebraic, 23.36
vs. monadic functor, 20.35
vs. solid functor, 26.4
Relation, congruence, = congruence
relation, 11.20
Relational category, 28H
Representable extremal partial
morphisms, 28.19
Representable functor, 6.9, 6K
preserves limits, 13.9
preserves monomorphisms, 7.37
preserves mono-sources, 10.7
vs. adjoint functor, 18C
vs. duality, 10N
vs. free object, 8.23
Representable M -partial morphisms, vs.
topological category, 28.12
Representable partial morphisms,
vs. final sinks are reducible, 28.15
vs. topological category, 28.15
Represent M -partial morphisms, 28.1
Retract, 7.24
absolute, 9.6

502

INDEX

in an isomorphism-closed fullreflective subcategory, 7.31
vs. injective object, 9.5
Retraction, 7.24 ff
is epimorphism, 7.42
as monomorphism means
isomorphism, 7.36
preserved by all functors, 7.28
preserved by products, 10.35
preserved and reflected by equivalence
functor, 7.47
is pullback stable, 11.18
reflected by full, faithful functor, 7.29
is regular epimorphism, 7.75
vs. (E, M )-structured category, 14.9
vs. product, 10.28
vs. projection morphism, 10.27
vs. quotient morphism, 8.12
Right adjoint, = adjoint, 19.11
Right inverse of a morphism, vs. left
inverse, 3.10
Russell’s paradox, 2.2, 2.3, 3.51, 3L
Satisfaction of an implication, =e-injective, 16.11, 16.12

Self-dual properties for functors, 3.43
Semifinal arrow, for a structured sink,
25.8, 25C
Semifinal solution, for a structured sink,
25.7, 25C
Separating set, 7.14
vs. concretizable category, 7Q
Separator, 7.10 ff
category with, is regular(co-)wellpowered, 7.89
extremal, 10.63
vs. bicoreflective subcategory, 16.4
vs. (co)wellpoweredness and(co)complete- ness, 12.13
vs. faithful hom-functor, 7.12
vs. topological category, 21.16, 21.17
Set,
constructions that can be performed
with sets, 2.1
morphism class of small category is aset, 3.45
separating, = separating set, 7.14
is a (small) class, 2.2
underlying, = underlying set, see
Forgetful functor

Satisfaction of topological (co-)axiom,
22.1, 22.6

Set-indexed family of sets, 2.2

Scheme, = domain of a diagram, 11.1

Set-valued functor, vs. hom-functor, 6.18
vs. natural transformation, 6.18

Section, 7.19 ff, 7E
as epimorphism is isomorphism, 7.43
preserved by all functors, 7.22
preserved by products, 10.35
reflected by full, faithful functor, 7.23
is (regular) monomorphism, 7.35, 7.59
vs. embedding, 8.7
vs. equivalence functor, 7.47
vs. intersection, 11K
vs. monomorphism, 7P
vs. pullback, 11H
Self-adjoint endofunctor, 19G
Self-dual concept, 3.7 See also PulationsquareSet-indexed source, 10.2

Singleton-morphism, 28.11
Sink, 10.62 ff
costructured, 17.4
final, 10.63, 28.15
natural, 11.27
Situation, adjoint, = adjoint situation,
19.3 ff
Skeleton of a category, 4.12–4.15, 4I
Small category, 3.44. See also
Fibre-small
is a set, 3.45
Small class, 2.2

9th July 2006

INDEX
Small conglomerate, 2.3
Smallest containing E-reflective
subcategory, = E-reflective hull,
16.20, 16.21
Smallest injective extension, 9.20
vs. minimal injective extension, 9G
Smallness of hom-class condition, see
Object-free category
Small object-free category, 3.55
Small source, 10.2
Solid concrete category, 25.10 ff
is almost topologically algebraic,
25.18, 25.19
characterization theorems, 25.19, 25I
structure theorem, 26.7
vs. (co)complete category, 25.15, 25.16
vs. Mac Neille completion, 25G
vs. subcategory of topologicalcategory, 26.2
vs. topological category, 25J
Solid construct, need not be topologicallyalgebraic nor strongly cocomplete,
25.17
Solid functor, 25.10 ff
is almost topologically algebraic,
25.18, 25.19
composite of topological andessentially algebraic functors, 26.3
composite of topologically algebraic
functors, 26.1
composite of topological and regularly
monadic functors, 26.3
composition of, 25.13
detects colimits and preserves anddetects limits, 25.14
is faithful and adjoint, 25.12
preserves extremal monomorphisms,
25.21
topologically algebraic functor is,
25.11
vs. adjoint functor, 25.12, 25.19
vs. detection and preservation oflimits, 25.18, 25.19

9th July 2006

503
vs. essentially algebraic functor, 26.3
vs. faithful functor, 25.12
vs. full reflective embedding, 26.1
vs. monadic functor, 25F
vs. regularly monadic functor, 26.4
vs. topologically algebraic functor,
25D, 25E, 26.1
vs. topological functor, 26.1, 26.3, 26.4

Solution set condition, 18.12
Source, 10.1 ff
codomain of, 10.1
composite of, 10.3
domain of, 10.1
empty, 10.2
initial, = initial source, 10.41, 10.57
natural, 11.3
notation for, 10.2, 10.4
set-indexed, 10.2
small, 10.2
structured, w.r.t. a functor, 17.1
Split coequalizer, 20.14
Split epimorphism, = retraction, 7.24 ff
Split fork, 20.14
Split monomorphism, = section, 7.19 ff
Square,commutes, 3.4
pulation, 11.32, 11.33, 11Q
pullback, = pullback square, 11.8 ff
Stability under pullbacks, 11.17, 28.13
Stable epimorphism, 11J
vs. adjoint functor, 18I
Statement, dual, = dual statement, 3.7
Stone-functor, 3.20(11)Strict monomorphism, 7D
vs. other types of monomorphisms,
7.76
vs. pullback, 11H
vs. regular monomorphism, 12A
Strong fibre-smallness, 26D
Strongly cocomplete category, 12.2
is (Epi,ExtrMono-Source)-category,

504

INDEX

15.17
vs. (co)complete category, 12.5, 12I,
12N, 12O
Strongly complete category, 12.2
is (E, M )-structured, 14.21
vs. (co)complete category, 12.5, 12I,
12N, 12O
vs. (E, M)-category, 15.25, 15.26
vs. free monad & varietor, 20.59
Strongly fibre-small concrete category,
26.5(Strongly Generating, Initial
Source)-functor, is topologicallyalgebraic functor, 25.6
Strongly generating structured arrow,
25.4
is generating, 25.5
Strongly n-generating object, 20H
Strongly limit-closed subcategory, vs.
epireflective subcategory, 16L
Strong monomorphism, 7.76, 14C
Structure, factorization:— for morphisms, 14.1 ff— for sources, 15.1 ff— w.r.t. a functor, 17.3 ff
Structured arrow, 8.15
equivalence for, 26.5
(extremally) generating, vs.
(extremal) epimorphism, 8.36
generating, = generating structured
arrow, 8.15
isomorphic, 8.19, 8.34
strongly generating, 25.4, 25.5
universal, 8.30
w.r.t. a functor, 8.30
Structured source,
w.r.t. a functor, 17.1
notation for, 17.2
self-indexed, 17.2
Structured 2-source, factorizations of, 17F
Subcategory, 4.1 ff
algebraic, = algebraic subcategory,

23.32
bireflective, 16.1, 16.3
colimit-closed, 13I
colimit-dense, = colimit-densesubcategory, 12.10
concrete, 5.21 ff
coreflective, = coreflective
subcategory, 4.25, 16.1 ff
definable by topological (co-)axioms,
22.1, 22.3, 22.4, 22.6
dense, 12D
E-equational, 16.16
epireflective, 16.1
E-reflective, 16.1
extremally epireflective, 16.1
finally closed, 21.29
finally dense, 10.69, 10.69
full, 4.1(2) ff
of a functor-costructured category, vs.
topological category, 22.8
of a functor-structured category, vs.
(M-) topological category, 22.3,
22.4, 22.9
implicational, 16.12
initially closed, 21.29
initially dense, 10.69
isomorphism-dense, 4.9
limit-closed, 13.26, 13.27, 13I, 13J
limit-dense, 12E
M-initially closed, vs. concretely
E-reflec- tive subcategory, 22.9
monadic, conditions for, 20.19
monoreflective, 16.1
nonfull, 4.3(3)reflective, = reflective subcategory,
4.16, 16.1 ff
regular epireflective, 16.1
simultaneously reflective andcoreflective, 4E
of subcategories, 4G
topological, = topologicalsubcategory, 21.29 ff
vs. identities, 4A
Subconstruct, 5.21

9th July 2006

INDEXof a functor-structured construct, vs.
monotopological construct, 22.10
of Set vs. concretely cartesian closed
construct, 27.16
Subobject, 7.77 ff
extremal, = extremal subobject, 7.77
initial, = initial subobject, 8.6
intersection of, 11.23
non-isomorphic, 7K
order on, 7.79
regular, = regular subobject, 7.77
vs. M-subobject, 16.8
Subset, 2.1. See also Power set
Swell epimorphism, 7.76, 15A
Swell separator, 19K
T -algebra, 5.37
T-algebra, category of, 20.4, 5G— closed under the formation of
mono-sources, 20.11— three-step construction, 20.6
Taut lift theorem, 21.28
Terminal, weakly terminal set of objects,
12F
Terminal category, 3.3(4)Terminal object, 7.4
uniqueness, 7.6
vs. product, 10.20, 10.30, 10H
vs. pullback square, 11.13
vs. weak terminal object, 12.9
Thin category, 3.26
vs. faithful functor, 3.29, 3G
Thin quasicategory, 3.51
T -homomorphism, 5.37
Topological axiom, 22.1
Topological characterization theorems,
external, 21.21
internal, 21.18
Topological (co-)axiom, 22.6
Topological concrete category, 21.7 ff
characterization, 21C

9th July 2006

505
duality for, 21.9
is fibre-complete, 21.11
fibre-small, 21.34 ff— characterization of, 22.3, 22.4, 22.8
— vs. closure under the formation ofindiscrete objects, initial subobjects,
and products, 21.37
fibre-small reflective subcategory, vs.
solid strongly fibre-small category,
26.7
subcategory of, vs. solid category, 26.2
vs. (co)completeness, 21.16, 21.17
vs. concretely cartesian closed
subcategory, 27.15
vs. concretely reflective subcategory,
21.32
vs. coseparator, 21.16, 21.17
vs. co-wellpoweredness, 21.16, 21.17
vs. discrete object, 21.11
vs. (E, M)-category, 21.14
vs. (E, M)-factorization, 21.16, 21.17
vs. extremal morphism, 21.13
vs. final lift, 21.34
vs. finally dense subcategory, 21.32
vs. has function spaces, 27.22
vs. indiscrete object, 21.11, 21.35
vs. initial lift, 21.34
vs. initially closed subcategory, 21.30,
21.31
vs. initial subobjects, 21.35
vs. injective object, 21.21
vs. lift of an adjoint situation, 21.28
vs. representable (M -)partial
morphisms, 28.12, 28.15
vs. separator, 21.16, 21.17
vs. solid category, 25J
vs. subcategory of afunctor-(co)structured category,
22.3, 22.4, 22.8
vs. wellpoweredness, 21.16, 21.17

Topological construct, fibre-small,characterization of, 22.4
vs. forgetful functor, 21L
Topological functor, 21.1 ff

506

INDEX

is (co-)adjoint, 21.12
composition of, 21.6
is faithful, 21.3
lifts limits and colimits, 21.15
preserves limits and colimits, 21.15
preserves and reflects mono-sourcesand epi-sinks, 21.13
is topologically algebraic, 25.2
vs. amnestic concrete category, 21.5
vs. bimorphism, 21M
vs. discrete and indiscrete functors,
21.12
vs. indiscrete structures, 21.18, 21.19,
21.20
vs. solid functor, 26.1, 26.3, 26.4
vs. topologically algebraic functor,
25Bvs. unique lifting of limits, 21.18,
21.20
vs. uniquely transportable concrete
category, 21.5

Topos, 28.7
is balanced quasitopos, 28.8
characterization, 28E
vs. regular monomorphism, 28F

Topologically algebraic category, 25.1 ff
characterization theorem, 25.6
vs. universal initial completion, 25H

Underlying function, see Forgetful
functorTopologically algebraic functor, 25.1 ff
almost is solid, 25.18, 25.19
is composite of other types, 26B
composites yield solid functor, 26.1
is faithful and adjoint, 25.3
is solid, 25.11
vs. adjoint functor, 25.3, 25.6, 25.19
vs. detection and preservation oflimits, 25.18, 25.19
vs. solid functor, 25D, 25E
vs. topological functor, 25B
Topological structure theorem, 22.3
Topological subcategory, 21.29 ff
vs. concretely (co)reflective
subcategory, 21.33
Topological theory, 22B
Topological universe, 28.21
equivalent conditions, 28.22
Topology, three approaches, 5N

Total category, 6I
Transformation, natural, = naturaltransformation, 6.1 ff
Transportable concrete category, 5.28
vs. amnestic concrete category, 5.29,
5.30
vs. concrete category, 5.35, 5.36
Transportable functor, monadic functor
is, 20.12
vs. lifting of limits, 13E
Triangle, commutes, 3.4
Trivial factorization structure, 15.3
Trivial monad, 20.2
T -space, 5.40, 5H

Underlying functor, = forgetful functor,
3.20(3), 5.1 ff
Union, of a family of sets, 2.1
Unique diagonalization property, 14.1,
15.1
Unique lifting of limits, vs. topological
functor, 21.18, 21.20
Uniquely transportable concrete category,
5.28
vs. topological functor, 21.5
Uniquely transportable functor,
vs. algebraic functor, 23.30
vs. regularly algebraic category, 23.38
Unit of an adjunction, 19.3
Unit ball functor, 5.2
Unit existence condition, see Object-freecategory
Unit of a partial binary algebra, 3.52, 19J
Universal arrow, 8.22

9th July 2006

INDEX
must be extremally generating, 8.33
uniqueness, 8.25, 8.35
vs. concrete generation, 8.24
vs. co-universal arrow, 19.1, 19.2

507
vs. equational subquasiconstruct ofsome Alg(Ω), 24.11

Weakly algebraic functor, 23A

Universal category, 4J

Weakly terminal set of objects, 12F

Universal completion, 12M

Weak terminal object, vs. terminal
object, 12.9

Universal initial completion, 21I
vs. topologically algebraic category,
25H
Universally initial morphism, 10P
Universally topological category, 28.16
as injective object, 28G
vs. concrete quasitopos, 28.18
Universal property, 4.16, 4.25, 8.22, 8.30
Universal structured arrow, 8.30
Universe, 2.2
Up-directed poset, 11.4
Upper semicontinuity, as adjointness, 18F
Varietal category & functor, 23F
Varietor, 20.53
vs. colimits of ω-chains, 20P
vs. free monad & monadic category,
20.56–20.58
Variety, = monadic construct, 24.12
bounded, vs. bounded monadic
construct, 24.11
finitary, = finitary variety, 16.16,
20.20

Well-fibred construct, 27.20
Well-fibred topological construct, vs.
cartesian closed construct, 27.22
Wellpowered category, 7.82 ff
and complete category, is strongly
complete, 12.5
is regular wellpowered and extremallywellpowered, 7.83
vs. adjoint functor, 18.19
vs. co-wellpoweredness, 12.13
vs. essentially algebraic category,
23.12, 23.13
vs. free object, 18.10
vs. topological category, 21.16, 21.17
Word-monad, 20.2
Yoneda embedding, 6J
Yoneda Lemma, 6.19
Zero morphism, 7A
Zero object, 7.7
vs. Cartesian closed category, 27A

This article may be accessed via WWW at http://www.tac.mta.ca/tac/reprints

9th July 2006

REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES will disseminate articles from thebody of important literature in Category Theory and closely related subjects which have never been
published in journal form, or which have been published in journals whose narrow circulation makes
access very difficult. Publication in ‘Reprints in Theory and Applications of Categories’ will permit
free and full dissemination of such documents over the Internet. Articles appearing have been criticallyreviewed by the Editorial Board of Theory and Applications of Categories. Only articles of lasting
significance are considered for publication. Distribution is via the Internet tools WWW/ftp.

Subscription information
Individual subscribers receive (by e-mail) abstracts of articles as they are published. To subscribe, sende-mail to tac@mta.ca including a full name and postal address. For institutional subscription, sendenquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Selection of reprints
After obtaining written permission from any copyright holder, any three TAC Editors may propose awork for TAC Reprints to the Managing Editor. The proposal will be reported to all Editors. The
Managing Editor may either accept the proposal or require that the Editors vote on it. Before such avote, the author, title and original publication data will be circulated to Editors. If a 2/3 majority of
those TAC Editors responding within one month agrees, the work will be accepted for TAC Reprints.
After a work is accepted, the author or proposer must provide to TAC either a usable TeX source or a
PDF document acceptable to the Managing Editor that reproduces a typeset version. Up to five pagesof corrections, commentary and forward pointers may be appended by the author. When submittingcommentary, authors should read and follow the Format for submission of Theory and Applications of
Categories at http://www.tac.mta.ca/tac/.

Managing editor
Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor
Michael Barr, McGill University: mbarr@barrs.org

Transmitting editors
Richard Blute, Université d’Ottawa: rblute@mathstat.uottawa.ca
Lawrence Breen, Paris 13: breen@math.univ-paris13.fr
Ronald Brown, University of North Wales: r.brown@bangor.ac.uk
Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it
Valeria de Paiva, Xerox Palo Alto Research Center: paiva@parc.xerox.com
Ezra Getzler, Northwestern University: getzler(at)math(dot)northwestern(dot)edu
Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk
P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk
G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au
Anders Kock, University of Aarhus: kock@imf.au.dk
Stephen Lack, University of Western Sydney: s.lack@uws.edu.au
F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu
Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr
Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl
Susan Niefield, Union College: niefiels@union.edu

INDEX
Robert Paré, Dalhousie University: pare@mathstat.dal.ca
Jiri Rosicky, Masaryk University: rosicky@math.muni.cz
Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu
James Stasheff, University of North Carolina: jds@math.unc.edu
Ross Street, Macquarie University: street@math.mq.edu.au
Walter Tholen, York University: tholen@mathstat.yorku.ca
Myles Tierney, Rutgers University: tierney@math.rutgers.edu
Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it
R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca

9th July 2006

509