Monoidal model categories

Hovey, Mark

doi:10.1090/surv/063/04

Cited by 33 publications

(55 citation statements)

References 4 publications

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“…Unfortunately, it is often the case that one of the conditions for Kan's theorem cannot be checked fully, so that the resulting homotopical structure on the category of algebras is something less than a model category. This type of structure was first studied in [Hov98] and [Spi01], and later in published sources such as [Fre10] and [Fre09] (12.1).…”

Section: Semi-model Categoriesmentioning

confidence: 99%

Homotopical Adjoint Lifting Theorem

White

Yau

2019

Appl Categor Struct

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This paper provides a homotopical version of the adjoint lifting theorem in category theory, allowing for Quillen equivalences to be lifted from monoidal model categories to categories of algebras over colored operads. The generality of our approach allows us to simultaneously answer questions of rectification and of changing the base model category to a Quillen equivalent one. We work in the setting of colored operads, and we do not require them to be Σ-cofibrant. Special cases of our main theorem recover many known results regarding rectification and change of model category, as well as numerous new results. In particular, we recover a recent result of Richter-Shipley about a zig-zag of Quillen equivalences between commutative HQ-algebra spectra and commutative differential graded Q-algebras, but our version involves only three Quillen equivalences instead of six. We also work out the theory of how to lift Quillen equivalences to categories of colored operad algebras after a left Bousfield localization.

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Section: Semi-model Categoriesmentioning

confidence: 99%

Homotopical Adjoint Lifting Theorem

White

Yau

2019

Appl Categor Struct

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“…See the definition of monoidal model category given for example in [119] [188]. Our first axiom says that the unit object for the direct product is cofibrant, which is stronger than the unit axiom of [188].…”

Section: Lemma 996 the Class Of Trivial Fibrations Is Equal To Inj(i)mentioning

confidence: 99%

Homotopy Theory of Higher Categories

Simpson

2011

111

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This is the first draft of a book about higher categories approached by iterating Segal's method, as in Tamsamani's definition of n-nerve and Pelissier's thesis. If M is a tractable left proper cartesian model category, we construct a tractable left proper cartesian model structure on the category of M -precategories. The procedure can then be iterated, leading to model categories of (∞, n)-categories.

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“…Let S be a set. The category VCat(S) admits a model structure in which the weak equivalences and fibrations are defined pointwise [2,9]. (Ob (B)).…”

Section: 2mentioning

confidence: 99%