An automorphism of an object xx in a category CC is an isomorphism f:xxf : x \to x. In other words, an automorphism is an isomorphism that is an endomorphism.

Automorphism group

Given an object xx, the automorphisms of xx form a group under composition, the automorphism group of xx, which is a submonoid of the endomorphism monoid of xx:

Aut C(x)=End C(x)Iso(C)=Iso C(x,x), Aut_C(x) = End_C(x) \cap Iso(C) = Iso_C(x,x) ,

which may be written Aut(x)Aut(x) if the category CC is understood. Up to equivalence, every group is an automorphism group; see delooping.

Revised on March 7, 2017 05:14:03 by Urs Schreiber (