functor

Definition

Let $\mathcal{C}$, $\mathcal{D}$ be categories. A functor $F:\mathcal{C}\to \mathcal{D}$ consists of:

A function on objects:

$$\begin{array}{rl}F:=\text{Ob}F:\text{Ob}(\mathcal{C})& ⟶\text{Ob}(\mathcal{D})\\ X& ⟼F(X)\end{array}$$

and a function on morphisms:

$$\begin{array}{rl}F:=F_{X,Y}:{\text{Hom}}_{\mathcal{C}}(X,Y)& ⟶{\text{Hom}}_{\mathcal{D}}(F(X),F(Y))\\ f& ⟼F(f)\end{array}$$

that satisfies

$$F(f\circ \tilde{f})=F(f)\circ F(\tilde{f})\text{for}X\stackrel{\tilde{f}}{\to }Y\stackrel{f}{\to }Z\text{in }\mathcal{C}$$

and

$$F({\text{id}}_X)={\text{id}}_{F(X)}$$

Usefulness

Functors are extremely important and useful. Many of the most important machinery in mathematics end up being functors since we can use (hopefully better understood) constructs to help get insight on another construction by using a map between them that preserves their structures in some way.

Examples

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