# Category Theory

## Table of Contents

MEGA resource:

## Notation

## Definition

A **category** consists of the three mathematical entities:

- A
*class*, whose elements are the*objects* - A
*class*, whose elements are called*morphisms*or*maps*or*arrows*. Each morphism has a*source object*and*target object* A

*binary operation*, called*composition of morphisms*, such that for any three objects , , and , we haveThe composition of and is governed by two axioms:

*Associativity*:*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