nLab automorphism




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 endomorphism that is an isomorphism.

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.




Discussion of automorphism groups internal to sheaf toposes (“automorphism sheaves”):

Last revised on June 26, 2024 at 07:13:38. See the history of this page for a list of all contributions to it.