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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 endomorphism that is an isomorphism.

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 $x$:

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

Examples

(…)

• Permutations are automorphisms in FinSet.

• automorphism of a vertex operator algebra

Related concepts

• automorphism group

  • inner automorphism group

  • outer automorphism group

  • automorphism Lie group

• automorphism 2-group

• automorphism ∞-group

References

• Marshall Hall, §6 in: The Theory of Groups, Macmillan (1959), AMS Chelsea (1976), Dover (2018) [[ISBN:978-0-8218-1967-8](https://bookstore.ams.org/view?ProductCode=CHEL/288), ISBN:9780486816906]

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

• Simon Robert Henry Friedman, MO:a/262687 John W. Morgan, §2.1 in: Automorphism sheaves, spectral covers, and the Kostant and Steinberg sections, Contemporary Mathematics 322 (2003) 217-244 [[arXiv:math/0209053](https://arxiv.org/abs/math/0209053)]

• Robert Friedman, John W. Morgan, §2.1 in: Simon HenryAutomorphism sheaves, spectral covers, and the Kostant and Steinberg sections (2017) [[MO:a/262687](https://mathoverflow.net/a/262687/381)] [[arXiv:math/0209053](https://arxiv.org/abs/math/0209053)]