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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].
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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