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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, 1229–1238 (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
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