no. 2
Homotopy theory of homotopy algebras
[Théorie homotopique des algèbres à homotopie près]
Vallette, Bruno1
1 Laboratoire Analyse, Géométrie et Applications (UMR 7539) CNRS, Université Paris 13 - Sorbonne Paris Cité Université Paris 8 93430 Villetaneuse (France)
Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 683-738.

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.

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.

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Révisé le :
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DOI : 10.5802/aif.3322
Classification : 18G55, 18D50
Keywords: Homotopical algebra, model category, coalgebras, operads
Mot clés : Algèbre homotopique, catégorie de modèles, cogèbres, opérades
Vallette, Bruno 1

1 Laboratoire Analyse, Géométrie et Applications (UMR 7539) CNRS, Université Paris 13 - Sorbonne Paris Cité Université Paris 8 93430 Villetaneuse (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Vallette, Bruno. Homotopy theory of homotopy algebras. Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 683-738. doi : 10.5802/aif.3322. https://aif.centre-mersenne.org/articles/10.5802/aif.3322/

[1] Aubry, Marc; Chataur, David Cooperads and coalgebras as closed model categories, J. Pure Appl. Algebra, Volume 180 (2003) no. 1-2, pp. 1-23 | DOI | MR | Zbl

[2] Brown, Kenneth S. Abstract homotopy theory and generalized sheaf cohomology, Trans. Am. Math. Soc., Volume 186 (1973), pp. 419-458 | DOI | MR | Zbl

[3] Buijs, Urtzi; Murillo, Aniceto Algebraic models of non-connected spaces and homotopy theory of L ∞ -algebras, Adv. Math., Volume 236 (2013), pp. 60-91 | DOI | MR | Zbl

[4] Cheng, Xue Zhi; Getzler, Ezra Transferring homotopy commutative algebraic structures, J. Pure Appl. Algebra, Volume 212 (2008) no. 11, pp. 2535-2542 | DOI | MR | Zbl

[5] Dolgushev, Vasily A.; Hoffnung, Alexander E.; Rogers, Christopher L. What do homotopy algebras form?, Adv. Math., Volume 274 (2015), pp. 562-605 | DOI | MR | Zbl

[6] Dotsenko, Vladimir; Poncin, Norbert A tale of three homotopies, Appl. Categ. Struct., Volume 24 (2016) no. 6, pp. 845-873 | DOI | MR | Zbl

[7] Dotsenko, Vladimir; Shadrin, Sergey; Vallette, Bruno Pre-Lie deformation theory, Mosc. Math. J., Volume 16 (2016) no. 3, pp. 505-543 | DOI | MR | Zbl

[8] Drummond-Cole, Gabriel C.; Hirsh, Joseph Model structures for coalgebras, Proc. Am. Math. Soc., Volume 144 (2016) no. 4, pp. 1467-1481 | DOI | MR | Zbl

[9] Dupont, Johan L. Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology, Volume 15 (1976) no. 3, pp. 233-245 | DOI | MR | Zbl

[10] Dwyer, William G.; Kan, Daniel M. Function complexes in homotopical algebra, Topology, Volume 19 (1980) no. 4, pp. 427-440 | DOI | MR | Zbl

[11] Gálvez-Carrillo, Imma; Tonks, Andrew; Vallette, Bruno Homotopy Batalin–Vilkovisky algebras, J. Noncommut. Geom., Volume 6 (2012) no. 3, pp. 539-602 | DOI | MR | Zbl

[12] Getzler, Ezra Lie theory for nilpotent L ∞ -algebras, Ann. Math., Volume 170 (2009) no. 1, pp. 271-301 | DOI | MR | Zbl

[13] Getzler, Ezra; Goerss, Paul A model category structure for differential graded coalgebras, 1999 (http://www.math.northwestern.edu/~pgoerss/papers/model.ps)

[14] Getzler, Ezra; Jones, J. D. S. Operads, homotopy algebra and iterated integrals for double loop spaces (1994) (https://arxiv.org/abs/hep-th/9403055)

[15] Ginzburg, Victor; Kapranov, Mikhail Koszul duality for operads, Duke Math. J., Volume 76 (1994) no. 1, pp. 203-272 | DOI | MR | Zbl

[16] Grothendieck, Alexander Sur quelques points d’algèbre homologique, Tôhoku Math. J., Volume 9 (1957), pp. 119-221 | Zbl

[17] Hess, Kathryn; Shipley, Brooke The homotopy theory of coalgebras over a comonad, Proc. Lond. Math. Soc., Volume 108 (2014) no. 2, pp. 484-516 | DOI | MR | Zbl

[18] Hinich, Vladimir Descent of Deligne groupoids, Int. Math. Res. Not. (1997) no. 5, pp. 223-239 | DOI | MR | Zbl

[19] Hinich, Vladimir DG coalgebras as formal stacks, J. Pure Appl. Algebra, Volume 162 (2001) no. 2-3, pp. 209-250 | DOI | MR | Zbl

[20] Hirschhorn, Philip S. Model categories and their localizations, Mathematical Surveys and Monographs, 99, American Mathematical Society, 2003, xvi+457 pages | MR | Zbl

[21] Hovey, Mark Model categories, Mathematical Surveys and Monographs, 63, American Mathematical Society, 1999, xii+209 pages | MR | Zbl

[22] Lazarev, Andrey; Markl, Martin Disconnected rational homotopy theory, Adv. Math., Volume 283 (2015), pp. 303-361 | DOI | MR | Zbl

[23] Lefevre-Hasegawa, Kenji Sur les A-infini catégories (2003) (https://arxiv.org/abs/math/0310337)

[24] Loday, Jean-Louis; Vallette, Bruno Algebraic operads, Grundlehren der Mathematischen Wissenschaften, 346, Springer, 2012, xviii+512 pages | MR | Zbl

[25] Mac Lane, Saunders Homology, Classics in Mathematics, Springer, 1995, x+422 pages (Reprint of the 1975 edition) | Zbl

[26] Munkholm, Hans J. DGA algebras as a Quillen model category. Relations to shm maps, J. Pure Appl. Algebra, Volume 13 (1978) no. 3, pp. 221-232 | DOI | MR | Zbl

[27] Quillen, Daniel G. Homotopical algebra, Lecture Notes in Mathematics, 43, Springer, 1967 | MR | Zbl

[28] Quillen, Daniel G. Rational homotopy theory, Ann. Math., Volume 90 (1969), pp. 205-295 | DOI | MR | Zbl

[29] Robert-Nicoud, Daniel Deformation theory with homotopy algebra structures on tensor products, Doc. Math., Volume 23 (2018), pp. 189-240 | MR | Zbl

[30] Smith, Justin R. Model-categories of coalgebras over operads, Theory Appl. Categ., Volume 25 (2011), pp. 189-246 | MR | Zbl

[31] Sullivan, Dennis Infinitesimal computations in topology, Publ. Math., Inst. Hautes Étud. Sci. (1977) no. 47, pp. 269-331 | DOI | Numdam | MR | Zbl

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