2021/07/29 by Jason Parker, Parker, Jason
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2107.13989
openalex publication_date 2021/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 GroupJ 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 J. In fact, we deduce such a characterization from a much more general result characterizing the categorical inner automorphisms in any category \mathbbTmodJ of presheaves of \mathbbT-models for a suitable first-order theory \mathbbT.