nLab endofunctor

Definition

A functor from a category to itself is called an endofunctor.

Given any category $C$, the functor category

$$End(C) = C^C$$

is called the endofunctor category of $C$. The objects of $End(C)$ are endofunctors $F: C \to C$, and the morphisms are natural transformation between such endofunctors.

Properties

Monoidal structure

The endofunctor category is a strict monoidal category, thanks to our ability to compose endofunctors:

$$\circ : End(C) \times End(C) \to End(C)$$

The unit object of this monoidal category is the identity functor from $C$ to itself:

$$1_C \in End(C)$$

Monoids

A monoid in this endofunctor category is called a monad on $C$.

Related concepts

• algebra over an endofunctor

• coalgebra over an endofunctor

• pointed endofunctor