# zbMATH — the first resource for mathematics

Homotopy theory of modules over operads in symmetric spectra. (English) Zbl 1235.55004
Algebr. Geom. Topol. 9, No. 3, 1637-1680 (2009); corrigendum ibid. 15, No. 2, 1229&ndash;1238 (2015).
Among the various formalizations of the homotopy theory of symmetric spectra are the flat stable model structures and the positive flat stable model structures. Results related to these model structures include the following:
- S. Schwede and B. Shipley [Proc. Lond. Math. Soc., III. Ser. 80, No. 2, 491–511 (2000; Zbl 1026.18004)], where it was shown that monoids in symmetric spectra inherit the model structures.
- B. Shipley [Contemporary Mathematics 346, 473–483 (2004; Zbl 1063.55006)] and M. A. Mandell, J. P. May, S. Schwede and B. Shipley [Proc. Lond. Math. Soc., III. Ser. 82, No. 2, 441–512 (2001; Zbl 1017.55004)], where it was shown that commutative monoids in symmetric spectra inherit the model structures.
- A. D. Elmendorf and M. A. Mandell [Adv. Math. 205, No. 1, 163–228 (2006; Zbl 1117.19001)], where it was shown that algebras over any operad $$\mathcal{O}$$ in simplicial sets inherit the model structure from the stable model structure on symmetric spectra. It was also shown that an object-wise weak equivalence map of operads induces a Quillen equivalence between the corresponding categories of algebras.
In this paper the author uses symmetric operads (certain algebraic structures that can be used to describe various structures, including monoids and commutative monoids) to uniformize and generalize the above results. The first main theorem is that for any operad $$\mathcal{O}$$ in symmetric spectra each of the categories of $$\mathcal{O}$$-algebras and left $$\mathcal{O}$$-modules inherits the (positive) flat stable model structure. The second main result is that an object-wise weak equivalence map of operads induces a Quillen equivalence between both the associated categories of algebras and the associated categories of left modules.
The article is written with great care for presentation, including a description of the model structures on symmetric spectra as well as a rather detailed account of operads. The article is scattered with many references to related works for further reading which serve to both place the article in its surrounding and to aid the reader with finding more information.
Reviewer: Ittay Weiss (Suva)

##### MSC:
 55P43 Spectra with additional structure ($$E_\infty$$, $$A_\infty$$, ring spectra, etc.) 55P48 Loop space machines and operads in algebraic topology 55U35 Abstract and axiomatic homotopy theory in algebraic topology
##### Keywords:
 [1] M Basterra, M A Mandell, Homology and cohomology of $$E_\infty$$ ring spectra, Math. Z. 249 (2005) 903 · Zbl 1071.55006 [2] C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805 · Zbl 1041.18011 [3] W Chachólski, J Scherer, Homotopy theory of diagrams, Mem. Amer. Math. Soc. 155 (2002) · Zbl 1006.18015 [4] W G Dwyer, J Spaliński, Homotopy theories and model categories (editor I M James), North-Holland (1995) 73 · Zbl 0869.55018 [5] A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys and Monogr. 47, Amer. Math. Soc. (1997) · Zbl 0894.55001 [6] A D Elmendorf, M A Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006) 163 · Zbl 1117.19001 [7] B Fresse, Lie theory of formal groups over an operad, J. Algebra 202 (1998) 455 · Zbl 1041.18009 [8] B Fresse, Koszul duality of operads and homology of partition posets (editors P G Goerss, S Priddy), Contemp. Math. 346, Amer. Math. Soc. (2004) 115 · Zbl 1077.18007 [9] B Fresse, Modules over operads and functors, Lecture Notes in Math. 1967, Springer (2009) · Zbl 1178.18007 [10] E Getzler, J D S Jones, Operads, homotopy algebra and iterated integrals for double loop spaces [11] V Ginzburg, M Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994) 203 · Zbl 0855.18006 [12] P G Goerss, M J Hopkins, André-Quillen (co)-homology for simplicial algebras over simplicial operads (editors D Arlettaz, K Hess), Contemp. Math. 265, Amer. Math. Soc. (2000) 41 · Zbl 0999.18009 [13] P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra (editors A Baker, B Richter), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151 · Zbl 1086.55006 [14] P G Goerss, M J Hopkins, Moduli problems for structured ring spectra (2005) · Zbl 1086.55006 [15] P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Math. 174, Birkhäuser Verlag (1999) · Zbl 0949.55001 [16] J E Harper, Homotopy theory of modules over operads and non-$$\Sigma$$ operads in monoidal model categories · Zbl 1231.55011 [17] V Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (1997) 3291 · Zbl 0894.18008 [18] V Hinich, V Schechtman, Homotopy Lie algebras (editors S Gel$$^{\prime}$$fand, S Gindikin), Adv. Soviet Math. 16, Amer. Math. Soc. (1993) 1 · Zbl 0823.18004 [19] P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monogr. 99, Amer. Math. Soc. (2003) · Zbl 1017.55001 [20] M Hovey, Model categories, Math. Surveys and Monogr. 63, Amer. Math. Soc. (1999) · Zbl 0909.55001 [21] M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149 · Zbl 0931.55006 [22] M Kapranov, Y Manin, Modules and Morita theorem for operads, Amer. J. Math. 123 (2001) 811 · Zbl 1001.18004 [23] G M Kelly, On the operads of J P May, Repr. Theory Appl. Categ. (2005) 1 · Zbl 1082.18009 [24] I K\vz, J P May, Operads, algebras, modules and motives, Astérisque (1995) · Zbl 0840.18001 [25] L G Lewis Jr., M A Mandell, Modules in monoidal model categories, J. Pure Appl. Algebra 210 (2007) 395 · Zbl 1123.18010 [26] S Mac Lane, Categories for the working mathematician, Graduate Texts in Math. 5, Springer (1998) · Zbl 0906.18001 [27] M A Mandell, $$E_\infty$$ algebras and $$p$$-adic homotopy theory, Topology 40 (2001) 43 · Zbl 0974.55004 [28] M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. $$(3)$$ 82 (2001) 441 · Zbl 1017.55004 [29] M Markl, S Shnider, J Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monogr. 96, Amer. Math. Soc. (2002) · Zbl 1017.18001 [30] J P May, The geometry of iterated loop spaces, Lectures Notes in Math. 271, Springer (1972) · Zbl 0244.55009 [31] J E McClure, J H Smith, A solution of Deligne’s Hochschild cohomology conjecture (editors D M Davis, J Morava, G Nishida, W S Wilson, N Yagita), Contemp. Math. 293, Amer. Math. Soc. (2002) 153 · Zbl 1009.18009 [32] J E McClure, J H Smith, Operads and cosimplicial objects: an introduction (editor J P C Greenless), NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer Acad. Publ. (2004) 133 · Zbl 1080.55010 [33] D G Quillen, Homotopical algebra, Lecture Notes in Math. 43, Springer (1967) · Zbl 0168.20903 [34] D G Quillen, Rational homotopy theory, Ann. of Math. $$(2)$$ 90 (1969) 205 · Zbl 0191.53702 [35] C Rezk, Spaces of algebra structures and cohomology of operads, PhD thesis, Massachusetts Institute of Technology (1996) [36] S Schwede, $$S$$-modules and symmetric spectra, Math. Ann. 319 (2001) 517 · Zbl 0972.55005 [37] S Schwede, An untitled book project about symmetric spectra (2007) [38] S Schwede, B Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. $$(3)$$ 80 (2000) 491 · Zbl 1026.18004 [39] B Shipley, A convenient model category for commutative ring spectra (editors P G Goerss, S Priddy), Contemp. Math. 346, Amer. Math. Soc. (2004) 473 · Zbl 1063.55006 [40] V A Smirnov, Homotopy theory of coalgebras, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985) 1302, 1343 · Zbl 0595.55008 [41] M Spitzweck, Operads, algebras and modules in general model categories · Zbl 1103.18300