Category Theory/Functors

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  .