Koszul duality for algebras over infinity-operads
Eric Hoffbeck, Ieke Moerdijk
TL;DR
The paper extends Koszul duality to algebras over linear $\infty$-operads by developing a bar-cobar theory on presheaves and copresheaves over a category of trees, and then lifting it to algebras/coalgebras. It constructs bar and cobar functors on the broader tree category $\mathbb R$, proves they are inverse up to quasi-isomorphism, and uses this to define dendroidal homology $DH_*(X)$ and $DH_*(M)$, connecting them to the Koszul dual cooperad in the classical case. A central achievement is a duality between conilpotent $\infty$-coalgebras over $\mathcal B X$ and $\infty$-algebras over $X$, along with a homological description of the bar complex in terms of category homology. The results provide a general framework for bar-cobar duality beyond classical operads, with implications for André-Quillen homology and indecomposables in the $\infty$-operadic setting, and relate dendroidal homology to traditional Koszul duality.
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.