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Homotopy units in $ A$-infinity algebras


Author: Fernando Muro
Journal: Trans. Amer. Math. Soc. 368 (2016), 2145-2184
MSC (2010): Primary 18D50, 18G55
DOI: https://doi.org/10.1090/tran/6545
Published electronically: May 8, 2015
MathSciNet review: 3449236
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Abstract: We show that the canonical map from the associative operad to the unital associative operad is a homotopy epimorphism for a wide class of symmetric monoidal model categories. As a consequence, the space of unital associative algebra structures on a given object is up to homotopy a subset of connected components of the space of non-unital associative algebra structures.


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  • [1] Jiří Adámek and Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, vol. 189, Cambridge University Press, Cambridge, 1994. MR 1294136 (95j:18001)
  • [2] Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras, CRM Monograph Series, vol. 29, American Mathematical Society, Providence, RI, 2010. With forewords by Kenneth Brown and Stephen Chase and André Joyal. MR 2724388 (2012g:18009)
  • [3] Hans Joachim Baues, Algebraic homotopy, Cambridge Studies in Advanced Mathematics, vol. 15, Cambridge University Press, Cambridge, 1989. MR 985099 (90i:55016)
  • [4] Hans-Joachim Baues, Mamuka Jibladze, and Andy Tonks, Cohomology of monoids in monoidal categories, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math., vol. 202, Amer. Math. Soc., Providence, RI, 1997, pp. 137-165. MR 1436920 (98a:18007), https://doi.org/10.1090/conm/202/02597
  • [5] Francis Borceux, Handbook of categorical algebra. 1, Basic category theory, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. MR 1291599 (96g:18001a)
  • [6] Denis-Charles Cisinski, Catégories dérivables, Bull. Soc. Math. France 138 (2010), no. 3, 317-393 (French, with English and French summaries). MR 2729017 (2012b:18019)
  • [7] Benoit Fresse, Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads, Alpine perspectives on algebraic topology, Contemp. Math., vol. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 125-188. MR 2581912 (2011m:18016), https://doi.org/10.1090/conm/504/09879
  • [8] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009.
  • [9] -, Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009.
  • [10] Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR 1944041 (2003j:18018)
  • [11] Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134 (99h:55031)
  • [12] K. Lefèvre-Hasegawa, Sur les $ {A}_\infty $-catégories, Ph.D. thesis, Université Paris 7, 2003, arXiv:math/0310337v1 [math.CT].
  • [13] J. Lurie, Higher Algebra, available at the author's web page: http://www.math.harvard.
    edu/˜lurie, August 2012.
  • [14] V. Lyubashenko and O. Manzyuk, Unital $ {A}_\infty $-categories, arXiv:0802.2885v1 [math.CT], February 2008.
  • [15] Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872 (2001j:18001)
  • [16] Georges Maltsiniotis, La $ K$-théorie d'un dérivateur triangulé, Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, 2007, pp. 341-368 (French, with French summary). MR 2342836 (2008i:18008), https://doi.org/10.1090/conm/431/08280
  • [17] M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441-512. MR 1806878 (2001k:55025), https://doi.org/10.1112/S0024611501012692
  • [18] Martin Markl, Models for operads, Comm. Algebra 24 (1996), no. 4, 1471-1500. MR 1380606 (96m:18012), https://doi.org/10.1080/00927879608825647
  • [19] Sergei Merkulov and Bruno Vallette, Deformation theory of representations of prop(erad)s. I, J. Reine Angew. Math. 634 (2009), 51-106. MR 2560406 (2011d:18032), https://doi.org/10.1515/CRELLE.2009.069
  • [20] Sergei Merkulov and Bruno Vallette, Deformation theory of representations of prop(erad)s. II, J. Reine Angew. Math. 636 (2009), 123-174. MR 2572248 (2011d:18033), https://doi.org/10.1515/CRELLE.2009.084
  • [21] Fernando Muro, Homotopy theory of nonsymmetric operads, Algebr. Geom. Topol. 11 (2011), no. 3, 1541-1599. MR 2821434 (2012k:18015), https://doi.org/10.2140/agt.2011.11.1541
  • [22] Fernando Muro, Homotopy theory of non-symmetric operads, II: Change of base category and left properness, Algebr. Geom. Topol. 14 (2014), no. 1, 229-281. MR 3158759, https://doi.org/10.2140/agt.2014.14.229
  • [23] Fernando Muro, Moduli spaces of algebras over nonsymmetric operads, Algebr. Geom. Topol. 14 (2014), no. 3, 1489-1539. MR 3190602, https://doi.org/10.2140/agt.2014.14.1489
  • [24] Fernando Muro and Andrew Tonks, Unital associahedra, Forum Math. 26 (2014), no. 2, 593-620. MR 3176644, https://doi.org/10.1515/forum-2011-0130
  • [25] Charles W. Rezk, Spaces of algebra structures and cohomology of operads, ProQuest LLC, Ann Arbor, MI, 1996. Thesis (Ph.D.)-Massachusetts Institute of Technology. MR 2716655
  • [26] Stefan Schwede and Brooke E. Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80 (2000), no. 2, 491-511. MR 1734325 (2001c:18006), https://doi.org/10.1112/S002461150001220X
  • [27] Stefan Schwede and Brooke Shipley, Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003), 287-334 (electronic). MR 1997322 (2004i:55026), https://doi.org/10.2140/agt.2003.3.287
  • [28] James Dillon Stasheff, Homotopy associativity of $ H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293-312. MR 0158400 (28 #1623)
  • [29] Bertrand Toën, The homotopy theory of $ dg$-categories and derived Morita theory, Invent. Math. 167 (2007), no. 3, 615-667. MR 2276263 (2008a:18006), https://doi.org/10.1007/s00222-006-0025-y

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Additional Information

Fernando Muro
Affiliation: Facultad de Matemáticas, Departamento de Álgebra, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain
Email: fmuro@us.es

DOI: https://doi.org/10.1090/tran/6545
Keywords: Operad, $A$-infinity algebra, unit, model category, mapping space.
Received by editor(s): August 21, 2013
Received by editor(s) in revised form: February 17, 2014, April 22, 2014, and July 29, 2014
Published electronically: May 8, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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