The Grothendieck construction for model categories

Harpaz, Yonatan; Prasma, Matan

doi:10.1016/j.aim.2015.03.031

Cited by 45 publications

(55 citation statements)

References 24 publications

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“…Roig's construction has also recently been significantly generalized by Harpaz and Prasma [14]. We include a proof for completeness.…”

Section: Remark 427mentioning

confidence: 99%

Rectification of enriched ∞–categories

Haugseng

2015

Algebr. Geom. Topol.

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Rectification of enriched ∞-categories RUNE HAUGSENGWe prove a rectification theorem for enriched ∞-categories: If V is a nice monoidal model category, we show that the homotopy theory of ∞-categories enriched in V is equivalent to the familiar homotopy theory of categories strictly enriched in V. It follows, for example, that ∞-categories enriched in spectra or chain complexes are equivalent to spectral categories and dg-categories. A similar method gives a comparison result for enriched Segal categories, which implies that the homotopy theories of n-categories and (∞, n)-categories defined by iterated ∞-categorical enrichment are equivalent to those of more familiar versions of these objects. In the latter case we also include a direct comparison with complete n-fold Segal spaces. Along the way we prove a comparison result for fibrewise simplicial localizations potentially of independent use. 18D2, 55U35; 18D50, 55P48 IntroductionIn [13], David Gepner and I set up a general theory of "weakly enriched categories" -more precisely, we introduced a notion of ∞-categories enriched in a monoidal ∞-category, and constructed an ∞-category of these objects where the equivalences are the natural analogue of fully faithful and essentially surjective functors in this context. In this paper we are interested in the situation where the monoidal ∞-category we enrich in can be described by a monoidal model category -this applies to many, if not most, interesting examples of monoidal ∞-categories. If V is a model category, then inverting the weak equivalences W gives an ∞-category V[W −1 ]; if V is a monoidal model category, then V[W −1 ] inherits a monoidal structure, so our theory produces an ∞-category of V[W −1 ]-enriched ∞-categories. On the other hand, there is also often a model structure on ordinary V-enriched categories (cf. [19,6,32,24]) where the weak equivalences are the so-called DK-equivalences, namely the functors that are weakly fully faithful (i.e. given by weak equivalences in V on morphism objects), and essentially surjective (up to homotopy). Our main goal in this paper is to prove a rectification theorem in this setting: In particular, V[W −1 ]-enriched ∞-categories can be rectified to V-categories: every V[W −1 ]-enriched ∞-category is equivalent to one coming from a category enriched in V. We will state and prove a precise version of this result in §5. The precise meaning of "nice" required applies, for example, to the category of chain complexes over a ring with the usual projective model structure, and certain model structures on symmetric spectra. We can therefore conclude that the ∞-category of spectral categories is equivalent to that of spectral ∞-categories, and the ∞-category of dg-categories to that of ∞-categories enriched in the derived ∞-category of abelian groups. We will prove a precise version of this theorem in §6. From this we can conclude that the homotopy theories of n-categories and (∞, n)-categories constructed in [13, §6.1] using iterated enrichment are equivalent to those construct...

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“…Roig's construction has also recently been significantly generalized by Harpaz and Prasma [14]. We include a proof for completeness.…”

Section: Remark 427mentioning

confidence: 99%

Rectification of enriched ∞–categories

Haugseng

2015

Algebr. Geom. Topol.

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“…. Harpaz and Prasma [HP15] have identified conditions under which a model structure on the base category and model structures on the values of a pseudofunctor assemble to a so-called integral model structure on the Grothendieck construction, and we verify these conditions in the case at hand. Theorem 1.3.…”

Section: Introductionmentioning

confidence: 67%

“…It is shown in [HP15, Theorem 3.0.12] that these classes of maps form a model structure when F is a proper relative pseudofunctor (in the language of [HP15,§3]). This amounts to requiring that the conditions of Lemma 2.16 hold for F .…”

Section: 4mentioning

confidence: 99%

“…bicartesian fibrations where the right equivalence follows from the unstraightening equivalence [Lur09, Theorem 2.2.1.2], and the left is immediate from the main theorem of[HP15] together with the compatibility between diagram categories of model and ∞-categories.Next we give an ∞-categorical interpretation of the category of symmetric spectra in retractive spaces Sp Σ R . To do so let us denoteT S ∞ = Grothendieck construction of Fun(−, Sp ∞ ) : S ∞ → Cat ∞ ,which is a model for the tangent bundle of the category S ∞ as in [Lur16, Section 7.3.1].…”

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confidence: 99%

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Multiplicative Parametrized Homotopy Theory via Symmetric Spectra in Retractive Spaces

Hebestreit

Sagave

Schlichtkrull

2020

Forum of Mathematics, Sigma

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In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding ∞-categorical product and allows for convenient constructions of commutative parametrized ring spectra. As an immediate application, we compare various models for generalized Thom spectra. In a companion paper, this approach is used to compare homotopical and operator algebraic models for twisted K-theory.

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“…(cf. [HP,Definition 2.16]) We denote by ModCat the (2, 1)category whose objects are the model categories and whose morphisms the Quillen pairs (composition is done in the direction of the left Quillen functor). The 2-morphisms are given by the pseudo-natural isomorphisms of (Quillen) adjunctions.…”

Section: Model Categoriesmentioning

confidence: 99%

An integral model structure and truncation theory for coherent group actions

Harpaz

Prasma

2017

Isr. J. Math.

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In this work we study the homotopy theory of coherent group actions from a global point of view, where we allow both the group and the space acted upon to vary. Using the model of Segal group actions and the model categorical Grothendieck construction we construct a model category encompassing all Segal group actions simultaneously. We then prove a global rectification result in this setting. We proceed to develop a general truncation theory for the model-categorical Grothendieck construction and apply it to the case of Segal group actions. We give a simple characterization of n-truncated Segal group actions and show that every Segal group action admits a convergent Postnikov tower.

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The Grothendieck construction for model categories

Cited by 45 publications

References 24 publications

Rectification of enriched ∞–categories

Rectification of enriched ∞–categories

Multiplicative Parametrized Homotopy Theory via Symmetric Spectra in Retractive Spaces

An integral model structure and truncation theory for coherent group actions

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