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Topological Quillen localization of structured ring spectra. (English) Zbl 1435.55006
Topological Quillen homology (or {TQ}-homology) is a homology theory for algebras over an operad. In this paper, the authors focus on operads in symmetric spectra (spectral operads) and construct a {TQ}-localization functor.
The method is to construct a semi-model structure on algebras over a spectral operad \(\mathcal{O}\) in terms of a left Bousfield localization. For the definition of a semi-model structure see [M. Spitzweck, Operads, algebras and modules in model categories and motives. Bonn: Univ. Bonn. Mathematisch-Naturwissenschaftliche Fakultät (Dissertation) (2001; Zbl 1103.18300)] or [P. G. Goerss and M. J. Hopkins, Contemp. Math. 265, 41–85 (2000; Zbl 0999.18009)]. With this model structure, the localisation functor is given by a fibrant replacement functor. Note that there are no connectivity assumptions required for this result.
The difficulty comes from the lack of left properness in the model structures. This is resolved by using a subcell lifting argument based on ideas in [P. G. Goerss and M. J. Hopkins, Moduli problems for structured ring spectra. (2005) available at http://hopf.math.purdue.edu]. Applications of the theory appear in [M. Ching and J. E. Harper, Tbil. Math. J. 11, No. 3, 69–79 (2018; Zbl 1440.55006)] and work of the second author [“Homotopy pro-nilpotent structured ring spectra and topological Quillen localization”, Preprint, arXiv:1902.03500].
MSC:
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
55P48 Loop space machines and operads in algebraic topology
55P60 Localization and completion in homotopy theory
18N40 Homotopical algebra, Quillen model categories, derivators
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References:
[1] M. Basterra. André-Quillen cohomology of commutative \(S\)-algebras. J. Pure Appl. Algebra, 144(2):111-143, 1999. · Zbl 0937.55006
[2] M. Basterra and M. A. Mandell. Homology and cohomology of \(E\sb \infty\) ring spectra. Math. Z., 249(4):903-944, 2005. · Zbl 1071.55006
[3] M. Basterra and M. A. Mandell. Homology of \(E_n\) ring spectra and iterated \(THH\). Algebr. Geom. Topol., 11(2):939-981, 2011. · Zbl 1219.55007
[4] K. Bauer, B. Johnson, and R. McCarthy. Cross effects and calculus in an unbased setting. Trans. Amer. Math. Soc., 367(9):6671-6718, 2015. With an appendix by R. Eldred. · Zbl 1332.55005
[5] A. K. Bousfield. The localization of spaces with respect to homology. Topology, 14:133-150, 1975. · Zbl 0309.55013
[6] A. K. Bousfield and D. M. Kan. Homotopy limits, completions and localizations. Lecture Notes in Mathematics, Vol. 304. Springer-Verlag, Berlin, 1972. · Zbl 0259.55004
[7] W. Chachólski and J. Scherer. Homotopy theory of diagrams. Mem. Amer. Math. Soc., 155(736):x+90, 2002. · Zbl 1006.18015
[8] M. Ching and J. E. Harper. Higher homotopy excision and Blakers-Massey theorems for structured ring spectra. Adv. Math., 298:654-692, 2016. · Zbl 1346.55009
[9] M. Ching and J. E. Harper. A nilpotent Whitehead theorem for TQ-homology of structured ring spectra. Tbilisi Math. J., 11:69-79, 2018.
[10] M. Ching and J. E. Harper. Derived Koszul duality and TQ-homology completion of structured ring spectra. Adv. Math., 341:118-187, 2019. · Zbl 1406.55003
[11] W. G. Dwyer. Localizations. In Axiomatic, enriched and motivic homotopy theory, volume 131 of NATO Sci. Ser. II Math. Phys. Chem., pages 3-28. Kluwer Acad. Publ., Dordrecht, 2004.
[12] W. G. Dwyer and J. Spaliński. Homotopy theories and model categories. In Handbook of algebraic topology, pages 73-126. North-Holland, Amsterdam, 1995. · Zbl 0869.55018
[13] E. Dror Farjoun. Cellular spaces, null spaces and homotopy localization, volume 1622 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1996. · Zbl 0842.55001
[14] J. Francis and D. Gaitsgory. Chiral Koszul duality. Selecta Math. (N.S.), 18(1):27-87, 2012. · Zbl 1248.81198
[15] B. Fresse. Lie theory of formal groups over an operad. J. Algebra, 202(2):455-511, 1998. · Zbl 1041.18009
[16] B. Fresse. Koszul duality of operads and homology of partition posets. In Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic \(K\)-theory, volume 346 of Contemp. Math., pages 115-215. Amer. Math. Soc., Providence, RI, 2004. · Zbl 1077.18007
[17] B. Fresse. Modules over operads and functors, volume 1967 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009. · Zbl 1178.18007
[18] S. Galatius, A. Kupers, and O. Randal-Williams. Cellular \({E}_k\)-algebras. Available at: arXiv:1805.07184 [math.AT], 2018.
[19] P. G. Goerss. On the André-Quillen cohomology of commutative \({\bf F}\sb 2\)-algebras. Astérisque, (186):169, 1990. · Zbl 0742.13008
[20] P. G. Goerss and M. J. Hopkins. André-Quillen (co)-homology for simplicial algebras over simplicial operads. In Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999), volume 265 of Contemp. Math., pages 41-85. Amer. Math. Soc., Providence, RI, 2000. · Zbl 0999.18009
[21] P. G. Goerss and M. J. Hopkins. Moduli spaces of commutative ring spectra. In Structured ring spectra, volume 315 of London Math. Soc. Lecture Note Ser., pages 151-200. Cambridge Univ. Press, Cambridge, 2004. · Zbl 1086.55006
[22] P. G. Goerss and M. J. Hopkins. Moduli problems for structured ring spectra. 2005. Available at http://hopf.math.purdue.edu.
[23] P. G. Goerss and J. F. Jardine. Simplicial homotopy theory, volume 174 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1999. · Zbl 0949.55001
[24] J. E. Harper. Bar constructions and Quillen homology of modules over operads. Algebr. Geom. Topol., 10(1):87-136, 2010. · Zbl 1197.18002
[25] J. E. Harper and K. Hess. Homotopy completion and topological Quillen homology of structured ring spectra. Geom. Topol., 17(3):1325-1416, 2013. · Zbl 1270.18025
[26] G. Heuts. Lie algebras and \(\nu_n\)-periodic spaces. Available at: arXiv:1803.06325 [math.AT], 2018.
[27] P. Hilton, G. Mislin, and J. Roitberg. Localization of nilpotent groups and spaces. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematics Studies, No. 15, Notas de Matemática, No. 55. [Notes on Mathematics, No. 55]. · Zbl 0323.55016
[28] P. S. Hirschhorn. Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003. · Zbl 1017.55001
[29] M. Hovey, B. Shipley, and J. H. Smith. Symmetric spectra. J. Amer. Math. Soc., 13(1):149-208, 2000. · Zbl 0931.55006
[30] J. F. Jardine. Local homotopy theory. Springer Monographs in Mathematics. Springer, New York, 2015. · Zbl 1320.18001
[31] N. J. Kuhn. Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces. Adv. Math., 201(2):318-378, 2006. · Zbl 1103.55007
[32] N. J. Kuhn. Adams filtration and generalized Hurewicz maps for infinite loopspaces. Invent. Math., 214(2):957-998, 2018. · Zbl 1403.55007
[33] N. J. Kuhn and L. A. Pereira. Operad bimodules and composition products on André-Quillen filtrations of algebras. Algebr. Geom. Topol., 17(2):1105-1130, 2017. · Zbl 1362.55008
[34] M. A. Mandell. \(E\sb \infty\) algebras and \(p\)-adic homotopy theory. Topology, 40(1):43-94, 2001. · Zbl 0974.55004
[35] J. P. May and K. Ponto. More concise algebraic topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2012. Localization, completion, and model categories. · Zbl 1249.55001
[36] H. R. Miller. The Sullivan conjecture on maps from classifying spaces. Ann. of Math. (2), 120(1):39-87, 1984. Correction: Ann. of Math. (2), 121(3):605-609, 1985. · Zbl 0575.55011
[37] L. A. Pereira. Goodwillie calculus in the category of algebras over a spectral operad. 2013. Available at: http://math.mit.edu/~luisalex/.
[38] D. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin, 1967. · Zbl 0168.20903
[39] C. Rezk. Spaces of Algebra Structures and Cohomology of Operads. PhD thesis, MIT, 1996. Available at http://www.math.uiuc.edu/~rezk/.
[40] S. Schwede. An untitled book project about symmetric spectra. 2007,2009. Available at: http://www.math.uni-bonn.de/people/schwede/.
[41] S. Schwede. On the homotopy groups of symmetric spectra. Geom. Topol., 12(3):1313-1344, 2008. · Zbl 1146.55005
[42] B. Shipley. A convenient model category for commutative ring spectra. In Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic \(K\)-theory, volume 346 of Contemp. Math., pages 473-483. Amer. Math. Soc., Providence, RI, 2004. · Zbl 1063.55006
[43] M. Spitzweck. Operads, algebras and modules in model categories and motives. PhD thesis, Rheinischen Friedrich-Wilhelms-Universitat Bonn, 2001. Available at: http://hss.ulb.uni-bonn.de/2001/0241/0241.pdf. · Zbl 1103.18300
[44] D. White. Model structures on commutative monoids in general model categories. J. Pure Appl. Algebra, 221(12):3124-3168, 2017. · Zbl 1387.18016
[45] D. White and D. Yau. Bousfield localization and algebras over colored operads. Appl. Categ. Structures, 26(1):153-203, 2018. · Zbl 1397.18023
[46] Y. Zhang. Homotopy pro-nilpotent structured ring spectra and topological Quillen localization. arXiv:1902.03500 [math.AT], 2019.
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