Algebraic & Geometric Topology

Classifying spaces of algebras over a prop

Sinan Yalin

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Abstract

We prove that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of algebras. This statement generalizes to the prop setting a homotopy invariance result which is well known in the case of algebras over operads. The absence of model category structure on algebras over a prop creates difficulties and we introduce new methods to overcome them. We also explain how our result can be extended to algebras over colored props in any symmetric monoidal model category tensored over chain complexes.

Article information

Source
Algebr. Geom. Topol., Volume 14, Number 5 (2014), 2561-2593.

Dates
Received: 9 October 2012
Revised: 19 December 2013
Accepted: 24 February 2014
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715996

Digital Object Identifier
doi:10.2140/agt.2014.14.2561

Mathematical Reviews number (MathSciNet)
MR3276841

Zentralblatt MATH identifier
1315.18025

Subjects
Primary: 18G55: Homotopical algebra
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 18D50: Operads [See also 55P48]

Keywords
props classifying spaces moduli spaces bialgebras category homotopical algebra homotopy invariance

Citation

Yalin, Sinan. Classifying spaces of algebras over a prop. Algebr. Geom. Topol. 14 (2014), no. 5, 2561--2593. doi:10.2140/agt.2014.14.2561. https://projecteuclid.org/euclid.agt/1513715996


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References

  • J M Boardman, R M Vogt, Homotopy-everything $H\!$–spaces, Bull. Amer. Math. Soc. 74 (1968) 1117–1122
  • J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics 347, Springer (1973)
  • W G Dwyer, D M Kan, Function complexes in homotopical algebra, Topology 19 (1980) 427–440
  • W G Dwyer, J Spaliński, Homotopy theories and model categories, from: “Handbook of algebraic topology”, North-Holland (1995) 73–126
  • B Enriquez, P Etingof, On the invertibility of quantization functors, J. Algebra 289 (2005) 321–345
  • B Fresse, Modules over operads and functors, Lecture Notes in Mathematics 1967, Springer (2009)
  • B Fresse, Props in model categories and homotopy invariance of structures, Georgian Math. J. 17 (2010) 79–160
  • P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc. (2003)
  • M Hovey, Model categories, Mathematical Surveys and Monographs 63, Amer. Math. Soc. (1999)
  • M W Johnson, D Yau, On homotopy invariance for algebras over colored PROPs, J. Homotopy Relat. Struct. 4 (2009) 275–315
  • S Mac Lane, Categorical algebra, Bull. Amer. Math. Soc. 71 (1965) 40–106
  • M Markl, Homotopy algebras are homotopy algebras, Forum Math. 16 (2004) 129–160
  • J P May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies 11, D Van Nostrand (1967)
  • J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer (1972)
  • C W Rezk, Spaces of algebra structures and cohomology of operads, PhD thesis, Massachusetts Institute of Technology (1996) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/304311860 {\unhbox0