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Symmetric operads in abstract symmetric spectra. (English) Zbl 1503.55006

J. Inst. Math. Jussieu 18, No. 4, 707-758 (2019); erratum ibid. 18, No. 5, 1113 (2019).
Summary: This paper sets up the foundations for derived algebraic geometry, Goerss-Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën-Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith’s stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and \(\text{E}_{\infty }\)-ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of \(\text{E}_{\infty }\)-rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent. This paper is also available at arXiv:1410.5699v2.

MSC:

55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55P48 Loop space machines and operads in algebraic topology
18M60 Operads (general)
55U40 Topological categories, foundations of homotopy theory
55P42 Stable homotopy theory, spectra
55U35 Abstract and axiomatic homotopy theory in algebraic topology
18N99 Higher categories and homotopical algebra
18D20 Enriched categories (over closed or monoidal categories)
14F42 Motivic cohomology; motivic homotopy theory
14F35 Homotopy theory and fundamental groups in algebraic geometry
14A20 Generalizations (algebraic spaces, stacks)
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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References:

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