9. Category theory

9. Category theory

Of the branches of mathematics,

category theory

is one which perhaps fits the least comfortably in set theoretic

foundations

.
One problem is that most of category theory is

invariant

under weaker notions of “sameness” than equality, such as

isomorphism

in a

category

or

equivalence of categories

, in a way which

set theory

fails to capture.
But this is the same sort of problem that the univalence axiom solves for types, by identifying equality with

equivalence

.
Thus, in

univalent foundations

it makes sense to consider a notion of “category” in which equality of objects is identified with isomorphism in a

similar

way.

Ignoring size issues, in set-based mathematics a category consists of a

set

A

0

of objects and, for each

x

,

y

∈

A

0

, a

set

hom

A

⁡

(

x

,

y

)

of

morphisms

.
Under univalent foundations, a “naive”

definition

of category would simply mimic this with a

type

of objects and

types

of morphisms.
If we allowed these types to contain arbitrary higher homotopy, then we ought to impose higher coherence conditions, leading to some notion of

(

∞

,

1

)

-category,
but at present our goal is more modest.
We consider only 1-categories, and therefore we restrict the types

hom

A

⁡

(

x

,

y

)

to be sets, i.e. 0-types.
If we impose no further conditions, we will call this notion a

precategory

.

If we add the requirement that the type

A

0

of objects is a set, then we end up with a definition that behaves much like the traditional set-theoretic one.
Following Toby Bartels, we call this notion a

strict category

.
But we can also require a generalized version of the univalence axiom, identifying

(

x

=

A

0

y

)

with the type

𝗂𝗌𝗈

⁢

(

x

,

y

)

of isomorphisms from

x

to

y

.
Since we regard this as usually the “correct” definition, we will call it simply a

category

.

A good example of the

difference

between the three notions of category is provided by the statement “every fully

faithful

and essentially surjective functor is an equivalence of categories”, which in classical set-based category theory is

equivalent

to the

axiom of choice

.

1.

For strict categories, this is still equivalent to the axiom of choice.

2.

For precategories, there is no

consistent

axiom of choice which can make it true.

3.

For categories, it is provable

without

any axiom of choice.

We will prove the latter statement in this chapter, as well as other pleasant

properties

of categories, e.g. that equivalent categories are equal (as elements of the type of categories).
We will also describe a

universal

way of “saturating” a precategory

A

into a category

A

^

, which we call its

Rezk completion

,
although it could also reasonably be called the

stack completion

(see the Notes).

The Rezk completion also sheds further light on the notion of equivalence of categories.
For

instance

, the

functor

A

→

A

^

is always fully faithful and

essentially surjective

, hence a “

weak equivalence

”.
It follows that a precategory is a category exactly when it “sees” all fully faithful and essentially surjective functors as equivalences; thus our notion of “category” is already inherent in the notion of “fully faithful and essentially surjective functor”.

We assume the reader has some basic familiarity with classical category theory.
Recall that whenever we write

𝒰

it denotes some

universe

of types, but perhaps a different one at different times; everything we say remains true for any consistent choice of universe levels.
We will use the basic notions of

homotopy type theory

from

\autoref

cha:typetheory,

\autoref

cha:basics and the propositional

truncation

from

\autoref

cha:logic, but not much else from

\autoref

part:foundations, except that our second construction of the Rezk completion will use a

higher inductive type

.

Title

9. Category theory

\metatable

Generated on Thu Feb 8 19:43:16 2018 by

LaTeXML