Endofunctor

category-theory

Definition

Endofunctor

An endofunctor is a functor $F:\mathcal{C}\to \mathcal{C}$ from a category $\mathcal{C}$ to itself.

It maps every object $X\in \mathcal{C}$ to an object $F(X)\in \mathcal{C}$ and every morphism $f:X\to Y$ to a morphism $F(f):F(X)\to F(Y)$, preserving identity and composition:

$$F({\mathrm{id}}_X)={\mathrm{id}}_{F(X)}F(g\circ f)=F(g)\circ F(f)$$

Examples

Identity endofunctor

The identity functor ${\mathrm{Id}}_{\mathcal{C}}:\mathcal{C}\to \mathcal{C}$ is the simplest endofunctor, sending every object and morphism to itself.

List functor

On the category of sets, the list endofunctor sends a set $X$ to the set of finite lists over $X$, and a function $f:X\to Y$ to the function that applies $f$ element-wise.

Power set functor

The power set endofunctor on sets sends a set $X$ to its power set $\mathcal{P}(X)$, and a function $f:X\to Y$ to the direct image map $\mathcal{P}(X)\to \mathcal{P}(Y)$.