Category Theory/Functors

This is the Functors chapter of Category Theory.

Definition

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

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

Examples

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

Types of functors

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

Types of subcategories

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

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