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category theory

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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.

Examples

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Last revised on January 8, 2023 at 04:50:18. See the history of this page for a list of all contributions to it.