Covariant Isotropy of Grothendieck Toposes

TL;DR

This paper characterizes the covariant isotropy group and the center of any Grothendieck topos, providing explicit descriptions of these automorphism groups in the context of sheaf theory and category theory.

Contribution

It offers a new explicit characterization of the covariant isotropy group and the center of Grothendieck toposes, advancing understanding of their automorphism structures.

Findings

Explicit description of the covariant isotropy group for any Grothendieck topos

Explicit characterization of the center of a Grothendieck topos

Provides tools for analyzing automorphisms in sheaf categories

Abstract

We provide an explicit characterization of the covariant isotropy group of any Grothendieck topos, i.e. the group of (extended) inner automorphisms of any sheaf over a small site. As a consequence, we obtain an explicit characterization of the centre of a Grothendieck topos, i.e. the automorphism group of its identity functor.