Koszul duality for algebras over infinity-operads

TL;DR

This paper extends Koszul duality to algebras and coalgebras over linear -infinity-operads and cooperads, introducing a new duality framework based on presheaves on tree categories, and defines their homology.

Contribution

It generalizes Koszul duality to -infinity-operads and develops a new duality approach using presheaves on trees, along with a homology theory for these algebras.

Findings

Extended Koszul duality to -infinity-operads and cooperads.

Established a duality between presheaves and copresheaves on tree categories.

Defined homology for -infinity-algebras and related it to tree category homology.

Abstract

In this paper, we introduce a new notion of algebra over a linear $\infty$-operad and a corresponding notion of coalgebra over an $\infty$-cooperad. We next extend the Koszul duality between linear $\infty$-operads and linear $\infty$-cooperads from our previous paper (arXiv:2105.11943) to their categories of algebras and coalgebras. This duality theorem specialises to the known duality in the case of algebras over classical (non-infinity) operads, but our proof is different. In fact, it is based on a much more general duality between presheaves and copresheaves on a category of trees. The latter duality is a priori independent of the (co)algebra structures, but we show that it can be lifted to (co)presheaves supporting such a structure. Based on this duality, we define the homology of an algebra over an $\infty$-operad, and prove that it can be described in terms of the homology of the same category of trees with coefficients in a presheaf.