An automorphism of an object in a category is an isomorphism . In other words, an automorphism is an isomorphism that is an endomorphism.
Given an object , the automorphisms of form a group under composition, the automorphism group of , which is a submonoid of the endomorphism monoid of :
Aut_C(x) = End_C(x) \cap Iso(C) = Iso_C(x,x)
which may be written if the category is understood. Up to equivalence, every group is an automorphism group; see delooping.
Revised on November 10, 2010 12:13:09
by Urs Schreiber