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.
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.
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Permutations are automorphisms in FinSet.
automorphism group
Discussion of automorphism groups internal to sheaf toposes (“automorphism sheaves”):
Robert Friedman, John W. Morgan, §2.1 in: Automorphism sheaves, spectral covers, and the Kostant and Steinberg sections [arXiv:math/0209053]
Last revised on September 23, 2023 at 23:42:58. See the history of this page for a list of all contributions to it.