nLab
endofunctor

Contents

Definition

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

Given any category CC, the functor category

End(C)=C CEnd(C) = C^C

is called the endofunctor category of CC. The objects of End(C)End(C) are endofunctors F:CCF: 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:

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

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

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

Monoids

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

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