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SYMMETRIC OPERADS IN ABSTRACT SYMMETRIC SPECTRA

Part of: Homological algebra Categories with structure Applied homological algebra and category theory (Co)homology theory Homotopy theory Algebraic geometry: Foundations

Published online by Cambridge University Press:  25 May 2018

Dmitri Pavlov  and
Jakob Scholbach Open the ORCID record for Jakob Scholbach [Opens in a new window]
Dmitri Pavlov
Affiliation:
Faculty of Mathematics, University of Regensburg Department of Mathematics and Statistics, Texas Tech Universityhttps://dmitripavlov.org/
Jakob Scholbach
Affiliation:
Mathematical Institute, University of Münster, Germany https://wwwmath.uni-muenster.de/u/jakob.scholbach/
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Abstract

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.

Keywords

algebraic topologycategory theoryhomological algebrahomotopy theoryoperads

MSC classification

Primary: 55P43: Spectra with additional structure ($E_infty$, $A_infty$, ring spectra, etc.) 55P48: Loop space machines, operads 18D50: Operads
Secondary: 55U40: Topological categories, foundations of homotopy theory 55P42: Stable homotopy theory, spectra 55U35: Abstract and axiomatic homotopy theory 18G55: Homotopical algebra 18D20: Enriched categories (over closed or monoidal categories) 14F42: Motivic cohomology; motivic homotopy theory 14F35: Homotopy theory; fundamental groups 14A20: Generalizations (algebraic spaces, stacks) 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 4 , July 2019 , pp. 707 - 758
Creative Commons
This is a work of the U.S. Government and is not subject to copyright protection in the United States.
Copyright
© Cambridge University Press 2018

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Footnotes

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