Koszul duality between $E_n$-algebras and coalgebras in a filtered category

TL;DR

This paper explores the duality between $E_n$-algebras and coalgebras within a filtered symmetric monoidal stable infinity category, establishing an equivalence under certain positivity and completeness conditions.

Contribution

It extends Koszul duality to filtered $E_n$-algebras and coalgebras, showing an equivalence and functoriality in generalized Morita categories.

Findings

Koszul duality restricts to an equivalence under positivity and completeness.

Duality is functorial in generalized Morita categories.

Establishes a framework for duality in filtered symmetric monoidal categories.

Abstract

We study the Koszul duality between augmented $E_n$-algebras and augmented $E_n$-coalgebras in a symmetric monoidal stable infinity $1$-category equipped with a filtration in a suitable sense. We obtain that the Koszul duality constructions restrict to an equivalence between augmented algebras and coalgebras which have some positivity and completeness with respect to the filtration. We also obtain that the Koszul duality construction is functorial between carefully constructed generalized Morita categories consisting of those algebras/coalgebras in each dimension.