An endomorphism of an object xx in a category CC is a morphism f:xxf : x \to x.

An endomorphism that is also an isomorphism is called an automorphism.


Given an object xx, the endomorphisms of xx form a monoid under composition, the endomorphism monoid of xx:

End C(x)=Hom C(x,x), End_C(x) = Hom_C(x,x) ,

which may be written End(x)End(x) if the category CC is understood. Up to equivalence, every monoid is an endomorphism monoid; see delooping.

An endomorphism monoid is a special case of a monoid structure on an end construction. Let d:DCd:D\to C be a diagram in CC, where CC is a monoidal category (in the case above the monoidal structure is the cartesian product and dd is a constant diagram from the initial category). One defines End(d)End(d) as an object in CC, equipped with a natural transformation a:End(d)dda: End(d) \otimes d \to d which is universal in the sense that for all objects ZCZ \in C, and any natural transformation f:Zddf: Z \otimes d \to d there exists a unique morphism g:ZEnd(d)g: Z \to End(d) such a(gd)=f:Zdda \circ (g \otimes d) = f: Z \otimes d \to d.


If the universal object (End(d),a)(End(d),a) exists then there is a unique structure of a monoid μ:End(d)End(d)End(d)\mu: End(d) \otimes End(d) \to End(d), such that the map a:End(d)dda: End(d) \otimes d \to d is an action.

Revised on March 30, 2017 08:31:43 by Urs Schreiber (