Koszul duality in higher topoi

Abstract

We show that there is an equivalence in any $n$-topos $\mathcal{X}$ between the pointed and $k$-connective objects of $\mathcal{X}$ and the $\mathbb{E}_k$-group objects of $\mathcal{X}$. We also show that for any pointed and $k$-connective object $X$ of $\mathcal{X}$ there is an equivalence between the $\infty$-category of modules in $\mathcal{X}$ over the associative algebra $\Omega^k X$, and the $\infty$-category of comodules in $\mathcal{X}$ for the cocommutative coalgebra $\Omega^{k-1}X$. In doing so, we prove several results which may be of broader interest: we show that Lurie's straightening-unstraightening equivalence holds over an $(n-1)$-groupoid in any $n$-topos for $0\leq n\leq\infty$; we show that taking categories of $\mathcal{O}$-algebras, for $\mathcal{O}$ an $\infty$-operad, commutes with truncation; and we show that for a pointed and connected $(n-1)$-groupoid $Y$ there is an equivalence between $\Omega Y$-modules in $\mathcal{X}$ and $\mathcal{X}$-valued presheaves on $Y$.