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 $\mathcal{C}$).

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 if "forgets" some or all aspects of the structure of $\mathcal{C}$.