# Category Theory

MEGA resource:

## Definition

A category consists of the three mathematical entities:

1. A class , whose elements are the objects
2. A class , whose elements are called morphisms or maps or arrows. Each morphism has a source object and target object
3. A binary operation , called composition of morphisms, such that for any three objects , , and , we have

The composition of and is governed by two axioms:

1. Associativity:

2. Identity map:

## Category of Sets

The category of for any partially ordered set , where

• a is a function from a set to
• a morphism of and is a function

Let , i.e. be the singleton set, then, for any set there is an isomorphism of sets

Just to make the point clear here, for the set , where is defined by the mapping:

### Products

Let and be sets.

The product of and , denoted , is defined as the set of ordered pairs where and . That is,

There are two natural projections and .

Let and be sets.

For any set and functions and there exist a unique function such that the following diagram commutes:

We might write the unique function as

Suppose we are given as above. To provide a function is equivalent of providing an element for each .

We need such a function for which

An element of is an ordered pair , and so we can use

Hence, it's necessary and sufficient to define

### Coproducts

Let and be sets.

The coproduct of and , denoted is defined as the disjoint union of and , i.e. the set for which an element is either an element of or an element of .

If something is an element of both and then we include both copies, and distinguish between them, in .

There are two natural inclusion functions

### Finite limits in Set

#### Pullbacks

Suppose the given diagram of sets and functions below

Its fiber product is the set

There are obvious projections

Note that if

then the diagram commutes.

Given the setup in the diagram above, we define the pullback of and over to be any set for which we have an isomorphism

The corner symbol denotes that is the pullback.

Sometimes you'll see the fiber product denoted .

Suppose given the diagram of sets and functions as below.

For any set and commutative solid arrow diagram as below (i.e. functions and such that ),

there exists a unique arrow making everything commmute,

## Category of Fuzzy Sets

The category of fuzzy subsets is denoted

The objects of are all pairs where

• is a set
• is a function from to the unit interval

The maps of are defined by

where denotes the category of sets.

• With composition simply being the composition of functions

## Course

### Notation

• morphism is a structure-preserving map

### Lecture 1

A category consists of

1. A collection of objects
2. A collection of morphisms
3. Two operations and from to : we write for " and and ".
4. An operation from to , s.t.
5. A partial binary operation on , defined iff , and satisfying , .
6. Satisfying

7. whenever it makes sense (i.e. domain- and co-domain-conditions in (3) are satisfied)
1. We don't require and to be sets
• If they are sets, we call small
2. Could formualte the definition with "morphism" as the only primitive notion, identifying "objects" with identity morphisms.
• However, in practice the objects are often logically prior to the morphisms

#### Examples

1. The category has all sets as objects, and all functions between them as morphisms.
• Formally, morphisms are pairs where is a set-theoretic function and
2. Algebraic ones:
• is the category of groups and group homomorphisms
• is the category of rings and ring homomorphisms
• is the category of R-modules and R-module homomorphisms, for a particular ring
3. Topological:
• is the category of topological spaces with continuous maps as morphisms
• is the category of smooth manifolds and maps as morphisms
4. The category has the same objects as , but morphisms are homotopy classes of continuous maps.
• More generally, given an equivalence relation on s.t.

1. implies and
2. implies and whenever the composites are defined

we have a new category with the same objects as but equivalence classes as morphisms

5. The category of relations, , has the same objects as , but morphisms are relations with composition of defined to be

• The category also has sets as objects, but morphisms are partial functions, i.e. functions for some
6. For any category , the opposite category has the same objects and morphisms as , but the and are interchanged, and thus the order of composition has to be interchanged, i.e. in is in .
• Hence we have the duality principle: if is a true statement about categories, then so is the statement I get by reversing all the arrows, denoted .
7. A category with a single object has for all morphisms , and so composition is defined everywhere. Hence a samll category with one object may be "identified" with a monoid (i.e. a semigroup with 1)
• In particular, a group is a small category with one object in which all morphisms are isomorphisms.
8. A (Brandt) groupoid is a category in which all morphisms as isomorphisms.
• E.g. for any category , has the same objects as , but only the isomorphisms of as morphisms
• Also, for any topological space , the fundamental groupoid has points of as objects and morephisms are (homotopy classes of) paths with and .