Abstract
We provide a very general approach to placing model structures and semi-model structures on algebras over symmetric colored operads. Our results require minimal hypotheses on the underlying model category \(\mathcal {M}\), and these hypotheses vary depending on what is known about the colored operads in question. We obtain results for the classes of colored operad which are cofibrant as a symmetric collection, entrywise cofibrant, or arbitrary. As the hypothesis on the operad is weakened, the hypotheses on \(\mathcal {M}\) must be strengthened. Via a careful development of the categorical algebra of colored operads we provide a unified framework which allows us to build (semi-)model structures for all three of these classes of colored operads. We then apply these results to provide conditions on \(\mathcal {M}\), on the colored operad O, and on a class \(\mathcal {C}\) of morphisms in \(\mathcal {M}\) so that the left Bousfield localization of \(\mathcal {M}\) with respect to \(\mathcal {C}\) preserves O-algebras. Even the strongest version of our hypotheses on \(\mathcal {M}\) is satisfied for model structures on simplicial sets, chain complexes over a field of characteristic zero, and symmetric spectra. We obtain results in these settings allowing us to place model structures on algebras over any colored operad, and to conclude that monoidal Bousfield localizations preserve such algebras.
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Barwick, C.: On left and right model categories and left and right Bousfield localizations. Homology, Homotopy Appl. 12(2), 245–320 (2010)
Batanin, M.: An operadic proof of the Baez-Dolan stabilisation hypothesis, In: Proceedings of the AMS. To appear. (2016)
Batanin, M., Berger, C.: Homotopy theory for algebras over polynomial monads. Theory Appl. Categ. 32, 148–253 (2017)
Batanin, M., White, D.: Bousfield Localization and Eilenberg-Moore Categories, preprint available electronically from arXiv:1606.01537 (2016)
Berger, C., Moerdijk, I.: The Boardman-Vogt resolution of operads in monoidal model categories. Topology 45, 807–849 (2006)
Berger, C., Moerdijk, I.: Resolution of coloured operads and rectification of homotopy algebras. Contemp. Math. 431, 31–58 (2007)
Berger, C., Moerdijk, I.: On the derived category of an algebra over an operad. Georgian Math. J. 16, 13–28 (2009)
Blumberg, A.J., Hill, M.A.: Operadic multiplications in equivariant spectra, norms, and transfers, preprint, arXiv:1309.1750 (2014)
Borceux, F.: Handbook of categorical algebra 2, categories and structures. Cambridge Univ. Press, Cambridge, UK.
Casacuberta, C., Gutiérrez, J. J., Moerdijk, I., Vogt, R.M.: Localization of algebras over coloured operads. Proc. Lond. Math Soc. (3) 101(1), 105–136 (2010)
Casacuberta, C., Raventos, O., Tonks, A.: Comparing Localizations Across Adjunctions, preprint, available electronically from arXiv:1404.7340(2014)
Elmendorf, A.D., Mandell, M.A.: Rings, modules, and algebras in infinite loop space theory. Adv. Math. 205, 163–228 (2006)
Elmendorf, A.D., Kriz, I., Mandell, M.A., May, J.P.: Rings, modules, and algebras in stable homotopy theory. Math Surveys and Monographs, vol. 47, Amer. Math. Soc., Providence, RI (1997)
Farjoun, E.D.: Cellular spaces, null spaces and homotopy localization. Lecture Notes in Math., vol. 1622. Springer-Verlag, Berlin (1996)
Frégier, Y., Markl, M., Yau, D.: The \(l_{\infty }\)-deformation complex of diagrams of algebras. New York J. Math. 15, 353–392 (2009)
Fresse, B.: Props in model categories and homotopy invariance of structures. Georgian Math. J. 17, 79–160 (2010)
Fresse, B.: Modules over operads, and functors. Lecture Notes in Math., vol. 1967. Springer-Verlag, Berlin (2009)
Gutiérrez, J. J., Röndigs, O., Spitzweck, M., Ostvær, P. A.: Motivic slices and colored operads. Journal of Topology 5, 727–755 (2012)
Harper, J.E.: Homotopy theory of modules over operads in symmetric spectra. Algebr. Geom. Topol. 9(3), 1637–1680 (2009)
Harper, J.E.: corrigendum 15, 1229–1238 (2015). 53
Harper, J.E.: Homotopy theory of modules over operads and non- Σ operads in monoidal model categories. J. Pure Appl. Algebra 214, 1407–1434 (2010)
Harper, J.E., Hess, K.: Homotopy completion and topological Quillen homology of structured ring spectra. Geom. Topol. 17(3), 1325–1416 (2013)
Hill, M.A., Hopkins, M.J.: Equivariant localizations and commutative rings, preprint available electronically from arXiv:1303.4479 (2014)
Hill, M.A., Hopkins, M.J., Ravenel, D.C.: On the non-existence of elements of kervaire invariant one, version 4, preprint available electronically from arXiv:0908.3724 (2015)
Hirschhorn, P.S.: Model Categories and Their Localizations. Math. Surveys and Monographs, vol. 99. Amer. Math. Soc., Providence, RI (2003)
Hovey, M.: Model categories. Math. Surveys and Monographs, vol. 63. Amer. Math. Soc., Providence, RI (1999)
Hovey, M.: Monoidal model categories, preprint available electronically from arXiv:math/9803002
Hovey, M., Shipley, B., Smith, J.: Symmetric spectra. J. Amer. Math. Soc. 2000(1), 149–208. 1998.52
Johnson, M.W., Yau, D.: On homotopy invariance for algebras over colored PROPs. J. Homotopy and Related Structures 4, 275–315 (2009)
Kedziorek, M.: PhD Thesis, Sheffield University. Available electronically from http://etheses.whiterose.ac.uk/7699/ (2015)
Kelly, G.M.: On the operads of J.P.May. Theory Appl. Categ. 13, 1–13 (2005)
Lurie, J.: Higher topos theory. Princeton Univ. Press, Princeton, NJ (2009)
Mac Lane, S.: Categories for the working mathematician, 2nd edn, vol. 5. Springer-Verlag, New York (1998)
Mandell, M.A., May, J.P., Schwede, S., Shipley, B.: Model categories of diagram spectra. Proc. London Math. Soc. (3) 82(2), 441–512 (2001)
McClure, J.E., Smith, J.H.: A solution of Deligne’s Hochschild cohomology conjecture. In: Proceedings of the JAMI conference on Homotopy Theory. Contemp. Math. 293, 153–193 (2002)
Markl, M., Shnider, S., Stasheff, J.: Operads in algebra, topology and physics, math surveys and monographs, vol. 96. Amer. Math. Soc., Providence (2002)
May, J.P.: The geometry of iterated loop spaces, vol. 271. Springer-Verlag, New York (1972)
May, J.P.: Definitions: operads, algebras and modules. Contemporary Math. 202, 1–7 (1997)
Pavlov, D., Scholbach, J.: Admissibility and rectification of colored symmetric operads, preprint available electronically from arXiv:1410.5675 (2014)
Rezk, C.W.: Spaces of algebra structures and cohomology of operads, Ph.D. thesis MIT (1996)
Schwede, S., Shipley, B.: Algebras and modules in monoidal model categories. Proc. London Math. Soc. 80, 491–511 (2000)
Shipley, B.: A convenient model category for commutative ring spectra Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory, vol. 346, Contemp. Math., pp 473–483. Amer. Math. Soc., Providence, RI (2004)
Spitzweck, M.: Operads, algebras and modules in general model categories, preprint available electronically from arXiv:math/0101102 (2001)
White, D.: Model structures on commutative monoids in general model categories, accepted, Journal of Pure and Applied Algebra, available electronically from arXiv:1403.6759 (2017)
White, D.: Monoidal Bousfield localizations and algebras over operads, preprint available electronically from arXiv:1404.5197 (2014)
White, D.: Monoidal Bousfield localizations and algebras over operads. Thesis, (Ph.D.)-Wesleyan University (2014)
White, D., Yau, D.: Right Bousfield localization and operadic algebras, preprint available electronically from arXiv:1512.07570
White, D., Yau, D.: Homotopical Adjoint Lifting Theorem, preprint available electronically from arXiv:1606.01803
Yau, D., Johnson, M.W.: A Foundation for PROPs, Algebras, and Modules. Math. Surveys and Monographs, vol. 203. AMS, Providence, RI (2015)
Acknowledgments
The authors are indebted to John E. Harper for several helpful conversations and to Luis Pereira for pointing out a mistake in an earlier version of this paper. The authors would also like to thank the referee for several helpful comments and suggestions.
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White, D., Yau, D. Bousfield Localization and Algebras over Colored Operads. Appl Categor Struct 26, 153–203 (2018). https://doi.org/10.1007/s10485-017-9489-8
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DOI: https://doi.org/10.1007/s10485-017-9489-8