Homotopy theory of symmetric powers

Pavlov, Dmitri; Scholbach, Jakob

doi:10.4310/hha.2018.v20.n1.a20

Cited by 15 publications

(59 citation statements)

References 27 publications

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“…It does not apply to chain complexes of abelian groups, and in fact the commutative operad is provably not admissible in this category. Moreover, as was shown in , symmetric h‐monoidality (and similarly with symmetroidality and symmetric flatness) are stable under transfer and monoidal left Bousfield localizations, which allows to easily promote these properties from basic model categories to more advanced model categories, such as spectra. The latter are shown in to be symmetric h‐monoidal, symmetroidal, and symmetric flat.…”

Section: Introductionmentioning

confidence: 83%

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Admissibility and rectification of colored symmetric operads

Pavlov

Scholbach

2018

Journal of Topology

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We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, that is, the category of algebras over any operad admits a model structure transferred from the original model category. We also give a necessary and sufficient criterion that ensures that a given weak equivalence of admissible operads admits rectification, that is, the corresponding Quillen adjunction between the categories of algebras is a Quillen equivalence. In addition, we show that Quillen equivalences of underlying symmetric monoidal model categories yield Quillen equivalences of model categories of algebras over operads. Applications of these results include enriched categories, colored operads, prefactorization algebras, and commutative symmetric ring spectra. Contents

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Section: Introductionmentioning

confidence: 83%

“…In § 2, we recall the symmetricity properties introduced in : symmetric h‐monoidality, symmetroidality, and symmetric flatness, and a few other basic notions on model categories. As was shown in [, 5.7, 5.8, 6.4, 6.5], these properties are stable transfer and monoidal Bousfield localizations.…”

Section: Introductionmentioning

confidence: 99%

Admissibility and rectification of colored symmetric operads

Pavlov

Scholbach

2018

Journal of Topology

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“…In contrast, our result will place no hypothesis on the maps in C at all, beyond our standing hypothesis that these maps are cofibrations. We additionally remark that the preprint [PS15] has independently considered the question of when Bousfield localization preserves the monoid axiom, towards the goal of rectification results in general categories of spectra.…”

Section: Bousfield Localization and The Monoid Axiommentioning

confidence: 99%

Bousfield Localization and Algebras over Colored Operads

White

Yau

2017

Appl Categor Struct

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ABSTRACT. We provide a very general approach to placing model structures and semi-model structures on algebras over symmetric colored operads. Our results require minimal hypotheses on the underlying model category M, and these hypotheses vary depending on what is known about the colored operads in question. We obtain results for the classes of colored operad which are cofibrant as a symmetric collection, entrywise cofibrant, or arbitrary. As the hypothesis on the operad is weakened, the hypotheses on M must be strengthened. Via a careful development of the categorical algebra of colored operads we provide a unified framework which allows us to build (semi-)model structures for all three of these classes of colored operads. We then apply these results to provide conditions on M, on the colored operad O, and on a class C of morphisms in M so that the left Bousfield localization of M with respect to C preserves O-algebras. Even the strongest version of our hypotheses on M is satisfied for model structures on simplicial sets, chain complexes over a field of characteristic zero, and symmetric spectra. We obtain results in these settings allowing us to place model structures on algebras over any colored operad, and to conclude that monoidal Bousfield localizations preserve such algebras.

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“…The following basic assumption ensures, in particular, that the forgetful functor from O-algebras to the underlying category of R-modules preserves cofibrant objects; see, for instance, [26,28,33]. Recall from [18] that associated to the operad O is the tower of operads…”

Section: Nilpotent Structured Ring Spectramentioning

confidence: 99%

A nilpotent Whitehead theorem for $\mathsf{TQ}$-homology of structured ring spectra

Ching¹,

Harper²

2018

Tbilisi Math. J.

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The aim of this short paper is to prove a TQ-Whitehead theorem for nilpotent structured ring spectra. We work in the framework of symmetric spectra and algebras over operads in modules over a commutative ring spectrum. Our main result can be thought of as a TQ-homology analog for structured ring spectra of Dror's generalized Whitehead theorem for topological spaces; here TQ-homology is short for topological Quillen homology. We also prove retract theorems for the TQ-completion and homotopy completion of nilpotent structured ring spectra.

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Homotopy theory of symmetric powers

Cited by 15 publications

References 27 publications

Admissibility and rectification of colored symmetric operads

Admissibility and rectification of colored symmetric operads

Bousfield Localization and Algebras over Colored Operads

A nilpotent Whitehead theorem for $\mathsf{TQ}$-homology of structured ring spectra

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