Algebraic & Geometric Topology

Relative left properness of colored operads

Philip Hackney, Marcy Robertson, and Donald Yau

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Abstract

The category of –colored symmetric operads admits a cofibrantly generated model category structure. In this paper, we show that this model structure satisfies a relative left properness condition, ie that the class of weak equivalences between Σ–cofibrant operads is closed under cobase change along cofibrations. We also provide an example of Dwyer which shows that the model structure on –colored symmetric operads is not left proper.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 5 (2016), 2691-2714.

Dates
Received: 1 December 2014
Revised: 9 December 2015
Accepted: 1 February 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841225

Digital Object Identifier
doi:10.2140/agt.2016.16.2691

Mathematical Reviews number (MathSciNet)
MR3572345

Zentralblatt MATH identifier
1350.18016

Subjects
Primary: 18D50: Operads [See also 55P48] 55U35: Abstract and axiomatic homotopy theory
Secondary: 18G55: Homotopical algebra 55P48: Loop space machines, operads [See also 18D50] 18D20: Enriched categories (over closed or monoidal categories)

Keywords
operads model categories left proper

Citation

Hackney, Philip; Robertson, Marcy; Yau, Donald. Relative left properness of colored operads. Algebr. Geom. Topol. 16 (2016), no. 5, 2691--2714. doi:10.2140/agt.2016.16.2691. https://projecteuclid.org/euclid.agt/1510841225


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References

  • M Batanin, C Berger, Homotopy theory for algebras over polynomial monads, preprint (2013)
  • C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805–831
  • C Berger, I Moerdijk, The Boardman–Vogt resolution of operads in monoidal model categories, Topology 45 (2006) 807–849
  • C Berger, I Moerdijk, Resolution of coloured operads and rectification of homotopy algebras, from “Categories in algebra, geometry and mathematical physics” (A Davydov, M Batanin, M Johnson, S Lack, A Neeman, editors), Contemp. Math. 431, Amer. Math. Soc., Providence, RI (2007) 31–58
  • J,M Boardman, R,M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics 347, Springer, Berlin (1973)
  • G Caviglia, A model structure for enriched coloured operads, preprint (2014)
  • D-C Cisinski, I Moerdijk, Dendroidal sets and simplicial operads, J. Topol. 6 (2013) 705–756
  • W Dwyer, K Hess, Long knots and maps between operads, Geom. Topol. 16 (2012) 919–955
  • A,D Elmendorf, I Kriz, M,A Mandell, J,P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, Amer. Math. Soc., Providence, RI (1997)
  • B Fresse, Koszul duality of operads and homology of partition posets, from “Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$–theory” (P Goerss, S Priddy, editors), Contemp. Math. 346, Amer. Math. Soc., Providence, RI (2004) 115–215
  • B Fresse, Props in model categories and homotopy invariance of structures, Georgian Math. J. 17 (2010) 79–160
  • V Ginzburg, M Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994) 203–272
  • P Hackney, M Robertson, D Yau, Shrinkability, relative left properness, and derived base change, preprint (2015)
  • J,E Harper, Homotopy theory of modules over operads and non–$\Sigma$ operads in monoidal model categories, J. Pure Appl. Algebra 214 (2010) 1407–1434
  • P,S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc., Providence, RI (2003)
  • M Hovey, Model categories, Mathematical Surveys and Monographs 63, Amer. Math. Soc., Providence, RI (1999)
  • M,W Johnson, D Yau, On homotopy invariance for algebras over colored PROPs, J. Homotopy Relat. Struct. 4 (2009) 275–315
  • G,M Kelly, On the operads of J P May, Repr. Theory Appl. Categ. (2005) 1–13
  • M Kontsevich, Y Soibelman, Deformations of algebras over operads and the Deligne conjecture, from “Conférence Moshé Flato 1999, I” (G Dito, D Sternheimer, editors), Math. Phys. Stud. 21, Kluwer Acad. Publ., Dordrecht (2000) 255–307
  • S Mac Lane, Categories for the working mathematician, 2nd edition, Graduate Texts in Mathematics 5, Springer, New York (1998)
  • M Markl, S Shnider, J Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs 96, Amer. Math. Soc., Providence, RI (2002)
  • J,P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer, Berlin (1972)
  • J,P May, Definitions: operads, algebras and modules, from “Operads: Proceedings of Renaissance Conferences” (J-L Loday, J,D Stasheff, A,A Voronov, editors), Contemp. Math. 202, Amer. Math. Soc., Providence, RI (1997) 1–7
  • F Muro, Homotopy theory of non-symmetric operads, II: Change of base category and left properness, Algebr. Geom. Topol. 14 (2014) 229–281
  • C,W Rezk, Spaces of algebra structures and cohomology of operads, PhD thesis, Massachusetts Institute of Technology (1996)
  • M Robertson, The homotopy theory of simplicially enriched multicategories, preprint (2011)
  • S Schwede, B,E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491–511
  • M Spitzweck, Operads, algebras and modules in general model categories, preprint (2001)
  • D White, D Yau, Bousfield localization and algebras over colored operads, preprint (2015)
  • D Yau, Colored operads, Graduate Studies in Mathematics 170, Amer. Math. Soc., Providence, RI (2016)
  • D Yau, M,W Johnson, A foundation for PROPs, algebras, and modules, Mathematical Surveys and Monographs 203, Amer. Math. Soc., Providence, RI (2015)