Homotopy theory of homotopy algebras
Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 683-738.

This paper studies the homotopy theory of algebras and homotopy algebras over an operad. It provides an exhaustive description of their higher homotopical properties using the more general notion of morphism called infinity-morphism. The method consists in using the operadic calculus to endow the category of coalgebras over the Koszul dual cooperad or the bar construction with a new type of model category structure, Quillen equivalent to that of algebras. We provide an explicit homotopy equivalence for infinity-morphisms, which gives a simple description of the homotopy category, and we endow the category of homotopy algebras with an infinity-category structure.

Cet article porte sur la théorie homotopique des algèbres et des algèbres à homotopie près sur une opérade. Il fournit une description exhaustive de leurs propriétés homotopiques supérieures en utilisant la notion générale de morphisme appelé infini-morphisme. La méthode consiste à utiliser le calcul opéradique pour munir la catégorie des cogèbres sur la coopérade duale de Koszul ou sur la construction bar d’un nouveau type de structure de modèles, équivalente au sens de Quillen de celle des algèbres. Nous introduisons une notion d’équivalence homotopique explicite pour les infinis-morphismes, qui induit une description simple de la catégorie homotopique, et nous munissons la catégorie des algèbres à homotopie près d’une structure d’infinie-catégorie.

Received: 2016-02-07
Revised: 2018-02-12
Accepted: 2019-03-12
Published online: 2020-05-28
DOI: https://doi.org/10.5802/aif.3322
Classification: 18G55,  18D50
Keywords: Homotopical algebra, model category, coalgebras, operads
@article{AIF_2020__70_2_683_0,
     author = {Vallette, Bruno},
     title = {Homotopy theory of homotopy algebras},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {2},
     year = {2020},
     pages = {683-738},
     doi = {10.5802/aif.3322},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_2_683_0/}
}
Vallette, Bruno. Homotopy theory of homotopy algebras. Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 683-738. doi : 10.5802/aif.3322. https://aif.centre-mersenne.org/item/AIF_2020__70_2_683_0/

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