= Automorphism
{title2=$\operatorname{Aut}_{\mathcal C}(X)$}
{wiki}
An <automorphism> is an <isomorphism> $f:X\to X$: a structure-preserving self-map that can be undone by another structure-preserving self-map. Equivalently, it is an invertible <endomorphism>. Its inverse satisfies
$$
f^{-1}\circ f=f\circ f^{-1}=1_X,
$$
where $1_X$ is the <identity morphism>. Both maps must be <morphisms> in the chosen <category>. For ordinary <groups>, <rings>, and <vector spaces> over a fixed <field>, a <bijective> <morphism> automatically has a compatible inverse. For <topological spaces>, an <automorphism> is a <homeomorphism> from the space to itself: a <continuous> <bijection> needs a <continuous> inverse as well. See https://emilyriehl.github.io/files/context.pdf[Riehl, Category Theory in Context, Definition 1.1.10 and Example 1.1.11].
For example, $T(x,y)=(x+y,y)$ is an <automorphism> of the <vector space> $\mathbb R^2$ over the <real numbers>, with inverse $T^{-1}(u,v)=(u-v,v)$. Both are <linear maps>. The <linear endomorphism> $P(x,y)=(x,0)$ from <endomorphism> preserves <addition> and <scalar multiplication> too, but discards a coordinate and cannot be undone. Preserving the structure and preserving enough information for reversal are separate requirements.
The <automorphisms> of a fixed object in a <locally small category> form a <group> under <composition in a category>: the <identity morphism> is the <identity element>, and the <inverse element> of a composite reverses the order of its factors. This makes reversible internal changes calculable. For a <group>, the resulting <automorphism group> records which relabelings preserve all products; the examples in <group automorphism> show how restrictive that condition can be.
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