Koszul duality for algebras over infinity-operads
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912891204009984 |
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| author | Hoffbeck, Eric Moerdijk, Ieke |
| author_facet | Hoffbeck, Eric Moerdijk, Ieke |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08851 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Koszul duality for algebras over infinity-operads Hoffbeck, Eric Moerdijk, Ieke Category Theory Algebraic Topology 18N70, 55N35, 18M70 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. |
| title | Koszul duality for algebras over infinity-operads |
| topic | Category Theory Algebraic Topology 18N70, 55N35, 18M70 |
| url | https://arxiv.org/abs/2602.08851 |