Extensions of $\infty$-group sheaves

TL;DR

This paper explores the structure of $ abla$-group sheaves within $ abla$-toposes, establishing a universal fiber bundle and classifying extensions and semidirect products of group objects in this context.

Contribution

It introduces a universal fiber bundle for $ abla$-group sheaves and classifies extensions and semidirect products in $ abla$-toposes.

Findings

Existence of a universal $ extbf{BA}$-fiber bundle for $ abla$-group sheaves.

Classification of semidirect products of group objects by $A$.

Identification of universal extensions via looping-delooping equivalence.

Abstract

Let $\mathscr X$ be an $\infty$-topos, for example the $\infty$-category of simplicial sheaves on a Grothendieck site. Then $\infty$-group sheaves are group objects in $\mathscr X$. Let $A\in\mathrm{Grp}\mathscr X$ be such a group object. Then as $\mathscr X$ is an $\infty$-topos, there exists a universal $\mathbf BA$-fiber bundle $\mathbf BA//\mathbf{Aut} A\xrightarrow q\mathbf B\mathbf{Aut} A$. We make $q$ pointed, and show that as a pointed map, via the looping-delooping equivalence, it is a universal extension of group objects by $A$. In particular, semidirect products of group objects by $A$ are classified by $\mathbf BA//\mathbf{Aut} A$.