# Category Theory/Functors

From Wikibooks, open books for an open world

This is the Functors chapter of Category Theory.

## Definition[edit]

A *functor* is a morphism between categories. Given categories and , a functor has domain and codomain , and consists of two suitably related functions:

- The
*object function*, which assigns to each object in , an object in . - The
*arrow function*(also ), which assigns to each arrow in , an arrow in , such that it satisfies and where is defined.

## Examples[edit]

- The
*power set functor*is a functor . Its object function assigns to every set , its power set and its arrow function assigns to each map , the map . - The
*inclusion functor*sends every object in a subcategory to itself (in ). - The general linear group which sends a commutative ring to .
- In homotopy, path components are a functor , the fundamental group is a functor , and higher homotopy is a functor .
- In group theory, a group can be thought of as a category with one object whose arrows are the elements of . Composition of arrows is the group operation. Let denote this category. The
*group action functor*gives for some set and the set is sent to .

## Types of functors[edit]

- A functor is an
*isomorphism of categories*if it is a bijection on both objects and arrows. - A functor is called
*full*if, for every pair of objects in and every arrow in , there exists an arrow in with . In other words, is surjective on arrows given objects . - A functor is called
*faithful*if, for every pair of objects in and every pair of parallel arrows in , the equality implies that . In other words, is injective on arrows given objects . The inclusion functor is faithful. - A functor is called
*forgetful*if it "forgets" some or all aspects of the structure of . - A functor whose domain is a product category is called a
*bifunctor*.

## Types of subcategories[edit]

is a *full subcategory* of if and only if the inclusion functor is full. In other words, if for every pair of objects in .

is a *lluf subcategory* of if and only if .