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 the $(n-k-1)$-truncation of $\mathcal{X}$. This recovers, up to equivalence of $\infty$-categories, some classical results regarding algebraic models for $k$-connective, $(n-1)$-coconnective homotopy types. Further, it extends those results to the case of sheaves of such homotopy types. 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 $\Omega^{k-1}X$. All of these equivalences are given by truncations of Lurie's $\infty$-categorical bar and cobar constructions, hence the terminology "Koszul duality".