A functor from a category to itself is called an endofunctor.
Given any category , the functor category
End(C) = C^C
is called the endofunctor category of . The objects of are endofunctors , and the morphisms are natural transformations between such endofunctors.
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 to itself:
1_C \in End(C)
A monoid in this endofunctor category is called a monad on .
Revised on December 6, 2012 12:37:36
by Urs Schreiber