Extensions of $\infty$-group sheaves

Abstract

Let $\mathcal{X}$ be an $\infty$-topos, for example the $\infty$-category of simplicial sheaves on a Grothendieck site. Then $\infty$-group sheaves are group objects in $\mathcal{X}$. Let $A\in \mathrm{Grp}\mathcal{X}$ be such a group object. Then as $\mathcal{X}$ is an $\infty$-topos, there exists a universal $\mathbf{B}A$-fiber bundle $\mathbf{B}A//\mathbf{Aut}A\stackrel{q}{\to }\mathbf{B}\mathbf{Aut}A$. 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 $\mathbf{B}A//\mathbf{Aut}A$.

10 pages