Homology of infinity-operads

TL;DR

This paper develops a homology theory for infinity-operads and dendroidal spaces, introduces bar and cobar constructions, and establishes a Koszul duality extending to presheaves on trees, using elementary methods.

Contribution

It introduces a novel homology theory for infinity-operads and dendroidal spaces, and establishes a general bar-cobar duality for presheaves on trees.

Findings

Homology theory extends classical homology of differential graded operads.

Bar and cobar constructions define an adjoint duality for infinity-operads and cooperads.

Duality applies broadly to presheaves and copresheaves on the category of trees.

Abstract

In a first part of this paper, we introduce a homology theory for infinity-operads and for dendroidal spaces which extends the usual homology of differential graded operads defined in terms of the bar construction, and we prove some of its basic properties. In a second part, we define general bar and cobar constructions. These constructions send infinity-operads to infinity-cooperads and vice versa, and define an adjoint bar-cobar (or "Koszul") duality. Somewhat surprisingly, this duality is shown to hold much more generally between arbitrary presheaves and copresheaves on the category of trees defining infinity-operads. We emphasize that our methods are completely elementary and explicit.