How to read this book

How to read this book

This book is divided into two parts.

http://planetmath.org/node/87878

Part I, “

Foundations

”, develops the fundamental

concepts

of

homotopy type theory

.
This is the

mathematical foundation

on which the

development

of specific subjects is built, and which is required for the understanding of the

univalent foundations

approach. To a programmer, this is “library code”.
Since univalent foundations is a new and different kind of mathematics, its basic notions take some getting used to; thus

http://planetmath.org/node/87878

Part I is fairly extensive.

http://planetmath.org/node/87879

Part II, “Mathematics”, consists of four chapters that build on the basic notions of

http://planetmath.org/node/87878

Part I to exhibit some of the new things we can do with univalent foundations in four different areas of mathematics:

homotopy theory

(

http://planetmath.org/node/87582

Chapter 8),

category theory

(

http://planetmath.org/node/87583

Chapter 9),

set theory

(

http://planetmath.org/node/87584

Chapter 10), and real analysis (

http://planetmath.org/node/87585

Chapter 11).
The chapters in

http://planetmath.org/node/87879

Part II are more or less independent of each other, although occasionally one will use a lemma proven in another.

A reader who wants to seriously understand univalent foundations, and be able to work in it, will eventually have to read and understand most of

http://planetmath.org/node/87878

Part I.
However, a reader who just wants to get a taste of univalent foundations and what it can do may understandably balk at having to work through over 200 pages before getting to the “meat” in

http://planetmath.org/node/87879

Part II.
Fortunately, not all of

http://planetmath.org/node/87878

Part I is necessary in order to read the chapters in

http://planetmath.org/node/87879

Part II.
Each chapter in

http://planetmath.org/node/87879

Part II begins with a brief overview of its subject, what univalent foundations has to contribute to it, and the necessary background from

http://planetmath.org/node/87878

Part I, so the courageous reader can turn immediately to the appropriate chapter for their favorite subject.
For those who want to understand one or more chapters in

http://planetmath.org/node/87879

Part II more deeply than this, but are not ready to read all of

http://planetmath.org/node/87878

Part I, we provide here a brief summary of

http://planetmath.org/node/87878

Part I, with remarks about which parts are necessary for which chapters in

http://planetmath.org/node/87879

Part II.

http://planetmath.org/node/87533

Chapter 1 is about the basic notions of type theory, prior to any homotopical

interpretation

.
A reader who is familiar with Martin-Löf

type theory

can quickly skim it to pick up the particulars of the theory we are using.
However, readers without experience in type theory will need to read

http://planetmath.org/node/87533

Chapter 1, as there are many subtle

differences

between type theory and other foundations such as set theory.

http://planetmath.org/node/87569

Chapter 2 introduces the homotopical viewpoint on type theory, along with the basic notions supporting this view, and describes the homotopical

behavior

of each component of the type theory from

http://planetmath.org/node/87533

Chapter 1.
It also introduces the

univalence axiom

(§2.10 (

http://planetmath.org/210universesandtheunivalenceaxiom

)) — the first of the two basic innovations of homotopy type theory.
Thus, it is quite basic and we encourage everyone to read it, especially §2.1 (

http://planetmath.org/21typesarehighergroupoids

) to §2.4 (

http://planetmath.org/24homotopiesandequivalences

).

http://planetmath.org/node/87576

Chapter 3 describes how we

represent

logic in homotopy type theory, and its

connection

to

classical logic

as well as to constructive and

intuitionistic logic

.
Here we define the

law of excluded middle

, the

axiom of choice

, and the axiom of propositional resizing (although, for the most part, we do not need to assume any of these in the rest of the book), as well as the

propositional

truncation

which is essential for representing traditional logic.
This chapter is essential background for

http://planetmath.org/node/87584

Chapter 10,

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Chapter 11, less important for

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Chapter 9, and not so necessary for

http://planetmath.org/node/87582

Chapter 8.

http://planetmath.org/node/87577

Chapter 4,

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Chapter 5 study two special topics in detail:

equivalences

(and related notions) and generalized inductive definitions.
While these are important subjects in their own rights and provide a deeper understanding of homotopy type theory, for the most part they are not necessary for

http://planetmath.org/node/87879

Part II.
Only a few lemmas from

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Chapter 4 are used here and there, while the general discussions in §5.1 (

http://planetmath.org/51introductiontoinductivetypes

),§5.6 (

http://planetmath.org/56thegeneralsyntaxofinductivedefinitions

),§5.7 (

http://planetmath.org/57generalizationsofinductivetypes

) are helpful for providing the intuition required for

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Chapter 6.
The generalized sorts of inductive definition discussed in §5.7 (

http://planetmath.org/57generalizationsofinductivetypes

) are also used in a few places in

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Chapter 10,

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Chapter 11.

http://planetmath.org/node/87579

Chapter 6 introduces the second basic innovation of homotopy type theory —

higher inductive types

— with many examples.
Higher inductive types are the

primary

object of study in

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Chapter 8, and some particular ones play important roles in

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Chapter 10,

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Chapter 11.
They are not so necessary for

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Chapter 9, although one example is used in §9.9 (

http://planetmath.org/99therezkcompletion

).

Finally,

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Chapter 7 discusses

homotopy

n

-types and related notions such as

n

-

connected

types.
These notions are important for

http://planetmath.org/node/87582

Chapter 8, but not so important in the rest of

http://planetmath.org/node/87879

Part II, although the case

n

=

-

1

of some of the lemmas are used in §10.1 (

http://planetmath.org/101thecategoryofsets

).

This

completes

http://planetmath.org/node/87878

Part I.
As mentioned above,

http://planetmath.org/node/87879

Part II consists of four largely unrelated chapters, each describing what univalent foundations has to offer to a particular subject.

Of the chapters in

http://planetmath.org/node/87879

Part II,

http://planetmath.org/node/87582

Chapter 8 (Homotopy theory) is perhaps the most

radical

.
Univalent foundations has a very different “synthetic” approach to homotopy theory in which

homotopy types

are the basic objects (namely, the types) rather than being constructed using

topological spaces

or some other set-theoretic model.
This enables new styles of proof for classical

theorems

in

algebraic topology

, of which we present a sampling, from

π

1

⁢

(

𝕊

1

)

=

ℤ

to the

Freudenthal suspension theorem

.

In

http://planetmath.org/node/87583

Chapter 9 (Category theory), we develop some basic (1-)category theory, adhering to the principle of the univalence axiom that

equality is

isomorphism

.
This has the pleasant effect of ensuring that all

definitions

and constructions are automatically

invariant

under

equivalence of categories

: indeed,

equivalent

categories

are equal just as equivalent types are equal.
(It also has connections to higher category theory and higher topos theory.)

http://planetmath.org/node/87584

Chapter 10 (Set theory) studies

sets in univalent foundations

.
The

category of sets

has its usual

properties

, hence provides a foundation for any mathematics that doesn’t need homotopical or higher-categorical

structures

.
We also observe that univalence makes

cardinal

and

ordinal numbers

a bit more pleasant, and that higher inductive types yield a

cumulative hierarchy

satisfying the usual axioms of Zermelo–Fraenkel set theory.

In

http://planetmath.org/node/87585

Chapter 11 (

Real numbers

), we summarize the construction of Dedekind real numbers, and then observe that higher inductive types allow a definition of Cauchy real numbers that avoids some associated problems in constructive mathematics.
Then we sketch a similar approach to Conway’s

surreal numbers

.

Each chapter in this book ends with a Notes

section

, which collects historical comments,

references

to the literature, and attributions of results, to the extent possible.
We have also included Exercises at the end of each chapter, to assist the reader in gaining familiarity with doing mathematics in univalent foundations.

Finally, recall that this book was written as a massively collaborative effort by a large number of people.
We have done our best to achieve

consistency

in terminology and notation, and to put the mathematics in a linear

sequence

that flows logically, but it is very likely that some imperfections remain.
We ask the reader’s forgiveness for any such infelicities, and welcome suggestions for improvement of the next edition.

References

1

Title

How to read this book

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