Koszul Duality in Higher Topoi

TL;DR

This paper establishes a Koszul duality framework in higher topoi, linking pointed, connective objects with group objects in truncated categories, extending classical algebraic models to sheaf contexts using bar and cobar constructions.

Contribution

It generalizes classical algebraic models for homotopy types to higher topoi and sheaves, introducing a Koszul duality via bar and cobar constructions in the $ty$-categorical setting.

Findings

Equivalence between pointed, connective objects and $_k$-group objects in truncated categories.

Extension of classical models to sheaves of homotopy types.

Identification of module and comodule categories via bar and cobar constructions.

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".