# Category Theory/Functors

This is the Functors chapter of Category Theory.

## Definition

A functor is a morphism between categories. Given categories ${\displaystyle {\mathcal {B}}}$  and ${\displaystyle {\mathcal {C}}}$ , a functor ${\displaystyle T:{\mathcal {C}}\to {\mathcal {B}}}$  has domain ${\displaystyle {\mathcal {C}}}$  and codomain ${\displaystyle {\mathcal {B}}}$ , and consists of two suitably related functions:

• The object function ${\displaystyle T}$ , which assigns to each object ${\displaystyle c}$  in ${\displaystyle {\mathcal {C}}}$ , an object ${\displaystyle Tc}$  in ${\displaystyle {\mathcal {B}}}$ .
• The arrow function (also ${\displaystyle T}$ ), which assigns to each arrow ${\displaystyle f:c\to c'}$  in ${\displaystyle {\mathcal {C}}}$ , an arrow ${\displaystyle Tf:Tc\to Tc'}$  in ${\displaystyle {\mathcal {B}}}$ , such that it satisfies ${\displaystyle T(1_{c})=1_{Tc}}$  and ${\displaystyle T(g\circ f)=Tg\circ Tf}$  where ${\displaystyle g\circ f}$  is defined.

## Examples

• The power set functor is a functor ${\displaystyle {\mathcal {P}}:{\textbf {Set}}\to \mathbf {Set} }$ . Its object function assigns to every set ${\displaystyle X}$ , its power set ${\displaystyle {\mathcal {P}}X}$  and its arrow function assigns to each map ${\displaystyle f:X\to Y}$ , the map ${\displaystyle {\mathcal {P}}f:{\mathcal {P}}X\to {\mathcal {P}}Y}$ .
• The inclusion functor ${\displaystyle {\mathcal {I}}:{\mathcal {S}}\to {\mathcal {C}}}$  sends every object in a subcategory ${\displaystyle {\mathcal {S}}}$  to itself (in ${\displaystyle {\mathcal {C}}}$ ).
• The general linear group ${\displaystyle {\text{GL}}_{n}:\mathbf {CRng} \to \mathbf {Grp} }$  which sends a commutative ring ${\displaystyle R}$  to ${\displaystyle {\text{GL}}_{n}(R)}$ .
• In homotopy, path components are a functor ${\displaystyle \pi _{0}:\mathbf {Top} \to \mathbf {Set} }$ , the fundamental group is a functor ${\displaystyle \pi _{1}:\mathbf {Top} \to \mathbf {Grp} }$ , and higher homotopy is a functor ${\displaystyle \pi _{n}:\mathbf {Top} \to \mathbf {Ab} }$ .
• In group theory, a group ${\displaystyle G}$  can be thought of as a category with one object ${\displaystyle g}$  whose arrows are the elements of ${\displaystyle G}$ . Composition of arrows is the group operation. Let ${\displaystyle {\mathcal {C}}_{G}}$  denote this category. The group action functor ${\displaystyle \mathbf {Act} :{\mathcal {C}}_{G}\to \mathbf {Set} }$  gives ${\displaystyle \mathbf {Act} (g)=X}$  for some set ${\displaystyle X}$  and the set ${\displaystyle {\mathcal {C}}_{G}(g,g)}$  is sent to ${\displaystyle \mathbf {Set} (X,X)={\text{Aut}}(X)}$ .

## Types of functors

• A functor ${\displaystyle T:{\mathcal {C}}\to {\mathcal {B}}}$  is an isomorphism of categories if it is a bijection on both objects and arrows.
• A functor ${\displaystyle T:{\mathcal {C}}\to {\mathcal {B}}}$  is called full if, for every pair of objects ${\displaystyle c,c'}$  in ${\displaystyle {\mathcal {C}}}$  and every arrow ${\displaystyle g:Tc\to Tc'}$  in ${\displaystyle {\mathcal {B}}}$ , there exists an arrow ${\displaystyle f:c\to c'}$  in ${\displaystyle {\mathcal {C}}}$  with ${\displaystyle g=Tf}$ . In other words, ${\displaystyle T}$  is surjective on arrows given objects ${\displaystyle c,c'}$ .
• A functor ${\displaystyle T:{\mathcal {C}}\to {\mathcal {B}}}$  is called faithful if, for every pair of objects ${\displaystyle c,c'}$  in ${\displaystyle {\mathcal {C}}}$  and every pair of parallel arrows ${\displaystyle f_{1},f_{2}:c\to c'}$  in ${\displaystyle {\mathcal {C}}}$ , the equality ${\displaystyle Tf_{1}=Tf_{2}:Tc\to Tc'}$  implies that ${\displaystyle f_{1}=f_{2}}$ . In other words, ${\displaystyle T}$  is injective on arrows given objects ${\displaystyle c,c'}$ . The inclusion functor is faithful.
• A functor ${\displaystyle T:{\mathcal {C}}\to {\mathcal {B}}}$  is called forgetful if it "forgets" some or all aspects of the structure of ${\displaystyle {\mathcal {C}}}$ .
• A functor whose domain is a product category is called a bifunctor.

## Types of subcategories

${\displaystyle {\mathcal {S}}}$  is a full subcategory of ${\displaystyle {\mathcal {C}}}$  if and only if the inclusion functor ${\displaystyle {\mathcal {I}}:{\mathcal {S}}\to {\mathcal {C}}}$  is full. In other words, if ${\displaystyle {\mathcal {S}}(X,Y)={\mathcal {C}}(X,Y)}$  for every pair of objects ${\displaystyle X,Y}$  in ${\displaystyle {\mathcal {S}}}$ .

${\displaystyle {\mathcal {S}}}$  is a lluf subcategory of ${\displaystyle {\mathcal {C}}}$  if and only if ${\displaystyle {\text{ob}}({\mathcal {S}})={\text{ob}}({\mathcal {C}})}$ .