nLab
automorphism

Definition

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

Automorphism group

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

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

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

Related concepts

• automorphism group

  • inner automorphism group

  • outer automorphism group

  • automorphism Lie group

• automorphism 2-group

• automorphism ∞-group