nLab inner automorphism

Definition

We give a definition of an inner automorphism in any category, originally due to Bergman.

Suppose $C$ is a category and $G$ is an object of $C$. An inner automorphism of $G$ is a family of automorphisms $p_f\colon H\to H$, one for every morphism $f\colon G\to H$ in $C$, for which the following square commutes for every morphism $g\colon H\to K$:

For example, if $C$ is the category of groups, then every such family $p$ can be shown to satisfy $p_f(h)=f(g) h f(g)^{-1}$ for an element $g\in G$.

Bergman computes inner automorphisms for many algebraic structures, which tend to be given by a conjugation of some sort.

Groups

An inner automorphism $\phi:G\to G$ of a group $G$ is any automorphism $\phi_g$ of the form $h\mapsto g h g^{-1}$. The inner automorphisms form a subgroup $Inn(G)$, called the inner automorphism group of $G$, of the entire automorphism group $Aut(G)$; it is the image of the natural map $G\to Aut(G)$ given by $g\mapsto\phi_g$. The center of a group $G$ is precisely the kernel of this natural map. Similarly, the monoidal center due to Drinfel’d and Majid, in the case when the monoidal category is Picard, is a $2$-category-theoretic kernel (an observation due to L. Breen).

Higher analogues of the inner automorphism group were studied by Roberts and Schreiber.

Related concepts

• automorphism

References

• George Bergman, An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange, Publicacions Matemàtiques 56 (2012), 91-126. DOI.