no. 3
Homology of infinity-operads
Hoffbeck, Eric1; Moerdijk, Ieke2
1 Université Sorbonne Paris Nord, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse (France)
2 Department of Mathematics, Utrecht University, PO BOX 80.010, 3508 TA Utrecht (The Netherlands)
Annales de l'Institut Fourier, Volume 75 (2025) no. 3, pp. 929-965.

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 a 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.

Dans une première partie, nous introduisons une théorie de l’homologie pour les infini-opérades et les espaces dendroïdaux, qui étend l’homologie usuelle des opérades différentielles graduées définie au moyen de la construction bar, et nous prouvons certaines de ses propriétés de base. Dans une seconde partie, nous définissons des constructions bar et cobar générales. Ces constructions envoient les infini-opérades sur des infini-coopérades et vice versa, et définissent une dualité bar-cobar (ou «  de Koszul  »). De façon assez surprenante, cette dualité est encore vraie pour des préfaisceaux et copréfaisceaux quelconques sur la catégorie des arbres définissant les infini-opérades. Nous insistons sur le caractère élémentaire et explicite de nos méthodes.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3653
Classification: 18N70, 55N35, 18M70
Keywords: Koszul duality, infinity-operads, dendroidal sets.
Mots-clés : Dualité de Koszul, infini-opérades, ensembles dendroidaux.

Hoffbeck, Eric 1; Moerdijk, Ieke 2

1 Université Sorbonne Paris Nord, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse (France)
2 Department of Mathematics, Utrecht University, PO BOX 80.010, 3508 TA Utrecht (The Netherlands)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Hoffbeck, Eric; Moerdijk, Ieke. Homology of infinity-operads. Annales de l'Institut Fourier, Volume 75 (2025) no. 3, pp. 929-965. doi : 10.5802/aif.3653. https://aif.centre-mersenne.org/articles/10.5802/aif.3653/

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