Category Theory/Functors

This is the Functors chapter of Category Theory.

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.

• 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 ${\displaystyle \mathbf {Set} (X,X)={\text{Aut}}(X)}$.

• 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.

${\displaystyle {\mathcal {S}}}$ 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 ${\displaystyle {\text{ob}}({\mathcal {S}})={\text{ob}}({\mathcal {C}})}$.