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 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 $x$:

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

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

For any category $C$, there exists a covariant functor $Aut : Core(C) \to Grp$ from the core to Grp, which maps each object $A$ to an automorphism group $Aut(A)$ and each morphism $f$ to a conjugate as follows.

$$\begin{array}{rl} \mathrm{Aut}(f) : \mathrm{Aut}(A) & \longrightarrow \mathrm{Aut}(B) \\ a & \mapsto f \circ a \circ f^{-1} \\ \end{array}$$

The contravariant form can be obtained by changing the order of the conjugate as $f^{-1} \circ a \circ f$.

Examples

• permutations are automorphisms in FinSet.

• automorphism group of a Lie algebra

• 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, ISBN:9780486816906]

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, Contemporary Mathematics 322 (2003) 217-244 [arXiv:math/0209053]

• Simon Henry (2017) [MO:a/262687]