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