Category Theory/Functors

This is the Functors chapter of Category Theory.

Definition

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

• The object function $T$, which assigns to each object $c$ in $𝒞$, an object $Tc$ in $ℬ$.
• The arrow function (also $T$), which assigns to each arrow $f:c\to c^{\prime }$ in $𝒞$, an arrow $Tf:Tc\to T{c}^{\prime }$ in $ℬ$, 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 $𝒫:\mathbf{\text{Set}}\to 𝐒𝐞𝐭$. Its object function assigns to every set $X$, its power set $𝒫X$ and its arrow function assigns to each map $f:X\to Y$, the map $𝒫f:𝒫X\to 𝒫Y$.
• The inclusion functor $ℐ:𝒮\to 𝒞$ sends every object in a subcategory $𝒮$ to itself (in $𝒞$).
• The general linear group ${\text{GL}}_n:𝐂𝐑𝐧𝐠\to 𝐆𝐫𝐩$ which sends a commutative ring $R$ to ${\text{GL}}_n(R)$.
• In homotopy, path components are a functor ${\pi }_0:𝐓𝐨𝐩\to 𝐒𝐞𝐭$, the fundamental group is a functor ${\pi }_1:𝐓𝐨𝐩\to 𝐆𝐫𝐩$, and higher homotopy is a functor ${\pi }_n:𝐓𝐨𝐩\to 𝐀𝐛$.
• 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 ${𝒞}_G$ denote this category. The group action functor $𝐀𝐜𝐭:{𝒞}_G\to 𝐒𝐞𝐭$ gives $𝐀𝐜𝐭(g)=X$ for some set $X$ and the set ${𝒞}_G(g,g)$ is sent to $𝐒𝐞𝐭(X,X)=\text{Aut}(X)$.

Types of functors

• A functor $T:𝒞\to ℬ$ is an isomorphism of categories if it is a bijection on both objects and arrows.
• A functor $T:𝒞\to ℬ$ is called full if, for every pair of objects $c,c^{\prime }$ in $𝒞$ and every arrow $g:Tc\to T{c}^{\prime }$ in $ℬ$, there exists an arrow $f:c\to c^{\prime }$ in $𝒞$ with $g=Tf$. In other words, $T$ is surjective on arrows given objects $c,c^{\prime }$.
• A functor $T:𝒞\to ℬ$ is called faithful if, for every pair of objects $c,c^{\prime }$ in $𝒞$ and every pair of parallel arrows $f_1,f_2:c\to c^{\prime }$ in $𝒞$, the equality $T{f}_1=T{f}_2:Tc\to T{c}^{\prime }$ implies that $f_1=f_2$. In other words, $T$ is injective on arrows given objects $c,c^{\prime }$. The inclusion functor is faithful.
• A functor $T:𝒞\to ℬ$ is called forgetful if it "forgets" some or all aspects of the structure of $𝒞$.
• A functor whose domain is a product category is called a bifunctor.

Types of subcategories

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

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