Grothendieck topos, gerbes and lifting actions of group objects

TL;DR

This paper investigates the problem of lifting group actions in a Grothendieck topos using non-commutative cohomology, extending previous results and applying to various geometric categories.

Contribution

It introduces a new cohomological framework for lifting group actions in topos theory, generalizing prior results and applying to manifolds, schemes, and algebraic varieties.

Findings

Established conditions for lifting actions via group object extensions

Applied the theory to topological and differentiable manifolds

Generalized existing results in the literature

Abstract

Let $C$ be a Grothendieck topos, $G$ and $H$ group objects of $C$. Let $p:P\rightarrow X$ be an $H$-torsor. Suppose that $X$ is endowed with an action of $G$. In this paper, we study the obstructions to lift the action of $G$ on $X$ to $P$ by using non commutative cohomology. Firstly, when a natural condition is satisfied, we associate to this problem an extension of groups objects in $C$ whose splittings correspond to the liftings of the action of $G$. We apply the results obtained to the categories of topological and differentiable manifolds, and to the category of schemes. For the categories of differentiable manifolds and affine varieties defined over a closed field, we use also another approach induced by the slice theorems of Koszul and Luna which enable to define Grothendieck topologies for $G$-invariant neighborhoods. This lifting problem has been studied in several categories by Brion, Hambleton, Hattori, Haussman, Lashof, May, Yoshida,... We recover and generalize some of their results