Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T08:35:08.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Simplicial localisation of homotopy algebras over a prop

Published online by Cambridge University Press:  13 October 2014

SINAN YALIN
SINAN YALIN*
Affiliation:
Mathematics Research Unit, Luxembourg University, 6 Rue Richard Coudenhove–Kalergi, L-1359 Luxembourg.
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer–Kan equivalence between the simplicial localisations of the associated categories of algebras. This homotopy invariance under base change implies that the homotopy category of homotopy algebras over a prop P does not depend on the choice of a cofibrant resolution of P, and gives thus a coherence to the notion of algebra up to homotopy in this setting. The result is established more generally for algebras in combinatorial monoidal dg categories.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 3 , November 2014 , pp. 457 - 468
Copyright
Copyright © Cambridge Philosophical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Barwick, C. and Kan, D. M.Relative categories:another model for the homotopy theory of homotopy theories. Indag. Math. 23 (2012), 42–68.CrossRefGoogle Scholar
[2]Barwick, C. and Kan, D. M.A characterisation of simplicial localisation functors and a discussion of DK equivalences. Indag. Math. 23 (2012), 69–79.CrossRefGoogle Scholar
[3]Berger, C. and Moerdijk, I.Axiomatic homotopy theory for operads. Comment. Math. Helv. 78 (2003), 805–831.CrossRefGoogle Scholar
[4]Berger, C. and Moerdijk, I.Resolution of coloured operads and rectification of homotopy algebras. Categories in algebra, geometry and mathematical physics. Contemp. Math. 431 (2007), 31–58.CrossRefGoogle Scholar
[5]Bergner, J.A model category structure on the category of simplicial categories. Trans. Amer. Math. Soc. 359 (2007), 2043–2058.CrossRefGoogle Scholar
[6]Bergner, J.A survey of (∞,1)-categories. In Baez, J. and May, J. P., Towards Higher Categories, IMA Volumes in Mathematics and Its Applications (Springer, 2010), 69–83.CrossRefGoogle Scholar
[7]Boardman, J. M. and Vogt, R. M.Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Math. 347 (Springer-Verlag, 1973).CrossRefGoogle Scholar
[8]Dugger, D.Combinatorial model categories have presentations. Adv. Math. 164 (2001), no. 1, 177–201.CrossRefGoogle Scholar
[9]Dwyer, W. G. and Kan, D. M.Simplicial localization of categories. J. Pure Appl. Alg. 17 (1980), 267–284.CrossRefGoogle Scholar
[10]Dwyer, W. G. and Kan, D. M.Calculating simplicial localizations. J. Pure Appl. Alg. 18 (1980), 17–35.CrossRefGoogle Scholar
[11]Dwyer, W. G. and Kan, D. M.Function complexes in homotopical algebra. Topology 19 (1980), 427–440.CrossRefGoogle Scholar
[12]Enriquez, B. and Etingof, P.On the invertibility of quantization functors. J. Algebra 289 (2005), 321–345.CrossRefGoogle Scholar
[13]Fregier, Y., Markl, M. and Yau, D.The L∞-deformation complex of diagrams of algebras. New York J. Math. 15 (2009), 353–392.Google Scholar
[14]Fresse, B.Props in model categories and homotopy invariance of structures. Georgian Math. J. 17 (2010), 79–160.CrossRefGoogle Scholar
[15]Getzler, E. and Jones, J. D. S. Operads, homotopy algebra and iterated integrals for double loop spaces. Preprint arXiv:hep-th/9403055 (1994).Google Scholar
[16]Hirschhorn, P. S.Model categories and their localizations. Mathematical Surveys Monographs 99 (AMS, 2003).Google Scholar
[17]Hovey, M.Model categories. Math. Surveys Monogr. 63 (AMS, 1999).Google Scholar
[18]Johnson, M. W. and Yau, D.On homotopy invariance for algebras over colored PROPs. J. Homotopy and Related Structures 4 (2009), 275–315.Google Scholar
[19]Loday, J-L.Generalized bialgebras and triples of operads. Astérisque 320 (SMF, 2008).Google Scholar
[20]Loday, J-L. and Vallette, B.Algebraic Operads. Grundlehren der mathematischen Wissenschaften 346 (Springer-Verlag, 2012).CrossRefGoogle Scholar
[21]Lurie, J.Higher Topos Theory. Ann. Math. Stud. 170 (Princeton University Press, 2009).CrossRefGoogle Scholar
[22]Maclane, S.Categorical algebra. Bull. Amer. Math. Soc. 71, Number 1 (1965), 40–106.CrossRefGoogle Scholar
[23]Maclane, S.Categories for the working mathematician. Graduate Texts in Math. second edition (Springer-Verlag, 1998).Google Scholar
[24]Markl, M.Homotopy algebras are homotopy algebras. Forum Math. 16 (2004), 129–160.CrossRefGoogle Scholar
[25]Rezk, C. Spaces of algebra structures and cohomology of operads. PhD. thesis. Massachusetts Institute of Technology (1996).Google Scholar
[26]Rezk, C.A model for the homotopy theory of homotopy theory. Trans. Amer. Math. Soc. 353 (2001), 973–1007.CrossRefGoogle Scholar
[27]Yalin, S. Classifying spaces of algebras over a prop. To appear in Algebr. Geom. Topol. See arxiv.org/abs/1207.2964Google Scholar