Model structures on commutative monoids in general model categories

White, David

doi:10.1016/j.jpaa.2017.03.001

Cited by 63 publications

(85 citation statements)

References 40 publications

(169 reference statements)

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“…4.13] from single maps to compositions of maps. A similar result, with modified hypotheses and conclusions but sharing some of the key ideas in the proof, was proven independently by David White in [17].…”

Section: Lax σ N -Cofibrancy Of N-fold Pushout Productssupporting

confidence: 63%

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Cofibrancy of operadic constructions in positive symmetric spectra

Pereira¹

2016

Homology, Homotopy and Applications

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We show that when using the underlying positive model structure on symmetric spectra one obtains cofibrancy conditions for operadic constructions under much milder hypothesis than one would need for general categories. Our main result provides such an analysis for a key operation, the "relative composition product" ○ O between right and left O-modules over a spectral operad O, and as a consequence we recover (and usually strengthen) previous results establishing the Quillen invariance of model structures on categories of algebras via weak equivalences of operads, compatibility of forgetful functors with cofibrations and Reedy cofibrancy of bar constructions.Key to the results above are novel cofibrancy results for n-fold smash powers of positive cofibrant spectra (and the relative statement for maps). Roughly speaking, we show that such n-fold powers satisfy a (new) type of Σn-cofibrancy which can be viewed as "lax Σn-free/projective cofibrancy" in that it determines a larger class of cofibrations still satisfying key technical properties of "true Σn-free/projective cofibrancy".

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Section: Lax σ N -Cofibrancy Of N-fold Pushout Productssupporting

confidence: 63%

“…is well defined since πσ is tidy only for σ ∈ Σ a ⊂ Σ w and such shuffles do not change (17). It follows that F M is well defined on all arrows.…”

Section: Remark 52mentioning

confidence: 90%

Cofibrancy of operadic constructions in positive symmetric spectra

Pereira¹

2016

Homology, Homotopy and Applications

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“…This was later generalized by Lurie (see [18]) to model categories which are free powered. Lurie also showed that in this case the resulting model category CAlg(M) models the ∞-category of E ∞ -algebra objects in M. If one is only interested in the existence of the transferred model structure, weaker conditions were established by White [29]. However, one should be mindful that under the assumptions of [29] the comparison between the resulting model structure and its ∞-categorical analogue may fail in general, see Example 6.4.6 below.…”

Section: Modules Over Commutative Algebrasmentioning

confidence: 99%

“…The main assumption used in [29] is the following: Example 6.4.5. Let M be the category of symmetric spectra endowed with the positive flat stable model structure (see [25]).…”

Section: Modules Over Commutative Algebrasmentioning

confidence: 99%

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The Grothendieck construction for model categories

Harpaz

Prasma²

2015

Advances in Mathematics

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The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing category. In this paper we consider the analogous correspondence in the setting of model categories. As a main result, we establish an equivalence between suitable diagrams of model categories indexed by M and a new notion of model fibrations over M. When M is a model category, our construction endows the Grothendieck construction with a model structure which gives a presentation of Lurie's ∞-categorical Grothendieck construction and enjoys several good formal properties. We apply our construction to various examples, yielding model structures on strict and weak group actions and on modules over algebra objects in suitable monoidal model categories.

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Anna Gavalda

Lenz¹

2016

Kindler Kompakt Französische Literatur 20. Jahrhundert

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We develop an analogue of the theory of * -modules in the world of simplicial sets, based on actions of a certain simplicial monoid EM originally appearing in the construction of global algebraic K-theory.As our main results, we show that strictly commutative monoids with respect to a certain box product on these simplicial * -modules yield models of equivariantly and globally coherently commutative monoids, and we give a characterization of simplicial * -modules in terms of a certain mildness condition on the EM-action, relaxing the notion of tameness previously investigated by Sagave-Schwede and the first author.

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Model structures on commutative monoids in general model categories

Cited by 63 publications

References 40 publications

Cofibrancy of operadic constructions in positive symmetric spectra

Cofibrancy of operadic constructions in positive symmetric spectra

The Grothendieck construction for model categories

Anna Gavalda

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