Koszul duality for algebras over infinity-operads

Eric Hoffbeck; Ieke Moerdijk

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arXiv · 2602.08851

Koszul duality for algebras over infinity-operads

Eric Hoffbeck·Ieke Moerdijk

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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.

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Eric Hoffbeck, Ieke Moerdijk. 2026-02-09. Koszul duality for algebras over infinity-operads. https://arxiv.org/abs/2602.08851

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