category theory

# Contents

## Definition

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

Given any category $C$, the functor category

$\mathrm{End}\left(C\right)={C}^{C}$End(C) = C^C

is called the endofunctor category of $C$. The objects of $\mathrm{End}\left(C\right)$ are endofunctors $F:C\to C$, and the morphisms are natural transformations between such endofunctors.

## Properties

### Monoidal structure

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

$\circ :\mathrm{End}\left(C\right)×\mathrm{End}\left(C\right)\to \mathrm{End}\left(C\right)$\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 \mathrm{End}\left(C\right)$1_C \in End(C)

### Monoids

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

Revised on December 6, 2012 12:37:36 by Urs Schreiber (82.169.65.155)