Inner Automorphisms of Presheaves of Groups

It has been proven by Schupp and Bergman that the inner automorphisms of groups can be characterized purely categorically as those group automorphisms that can be coherently extended along any outgoing homomorphism. One is thus motivated to define a notion of (categorical) inner automorphism in an arbitrary category, as an automorphism that can be coherently extended along any outgoing morphism, and the theory of such automorphisms forms part of the theory of covariant isotropy. In this paper, we prove that the categorical inner automorphisms in any category \(\textsf{Group}^\mathcal {J}\) of presheaves of groups can be characterized in terms of conjugation-theoretic inner automorphisms of the component groups, together with a natural automorphism of the identity functor on the index category \(\mathcal {J}\). In fact, we deduce such a characterization from a much more general result characterizing the categorical inner automorphisms in any category \(\mathbb {T}\textsf{mod}^\mathcal {J}\) of presheaves of \(\mathbb {T}\)-models for a suitable first-order theory \(\mathbb {T}\).