category theory

# Contents

## 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 transformations 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 $C$.

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