The enriched Grothendieck construction

Beardsley, Jonathan; Wong, Liang Ze

doi:10.1016/j.aim.2018.12.009

Cited by 22 publications

(3 citation statements)

References 8 publications

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“…This interesting subtlety concerning the transfer of monoidality from the target category to the very structure of the functor and vice versa could potentially bring new perspective into future variations of the Grothendieck construction. As an example, in [BW19] the authors work towards a 'fibrewise' enriched version of the correspondence between fibrations and indexed categories, hence future work could address the 'global' enriched Grothendieck construction.…”

Section: Introductionmentioning

confidence: 99%

Monoidal Grothendieck construction

Moeller,

Vasilakopoulou

2018

Preprint

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We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we investigate the relation between this 'global' monoidal structure where the total category is monoidal and the fibration strictly preserves the structure, and a 'fibrewise' one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, lax monoidal structures on the functor to the 2-category of categories correspond to lifts of the functor to the 2-category of monoidal categories. Finally, we give examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems. Contents 1. Introduction 1 2. Preliminaries 3 3. The Monoidal Grothendieck Construction 13 4. (Co)cartesian case: fibrewise and global monoidal structures 23 5. Examples

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Section: Introductionmentioning

confidence: 99%

Monoidal Grothendieck construction

Moeller,

Vasilakopoulou

2018

Preprint

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“…Remark 3.1. There is an enriched Grothendieck construction due to Beardsley and Wong [BW19], which holds for more general enrichments, however, only over free enriched categories, whereas we need the Grothendieck construction over arbitrary P(D)-enriched categories.…”

Section: Grothendieck Construction For D-simplicial Spacesmentioning

confidence: 99%

Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces

Rasekh¹

2021

Preprint

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For a small category D we define fibrations of simplicial presheaves on the category D × ∆, which we call localized D-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized D-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on D, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations and weak equivalences in this model structure and in particular obtain a Yoneda lemma.We apply this general framework to study Cartesian fibrations of (∞, n)-categories, for models of (∞, n)-categories that arise via simplicial presheaves, such as n-fold complete Segal spaces. This, in particular, results in the Yoneda lemma and Grothendieck construction for Cartesian fibrations of (∞, n)categories. CONTENTS 0. Introduction 1 1. A Deep Dive into Enrichments and Some Fibrations 8 2. From Left Fibrations to D-left Fibrations 29 3. Grothendieck Construction for D-Simplicial Spaces 39 4. Localized D-Left fibrations 57 5. (Segal) Cartesian Fibrations of (∞, n)-Categories 86 6. What needs to be done? 102 References 104

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“…For basic facts and homological properties on Grothendieck constructions of diagrams of categories, one can refer to e.g. [2,6,38,60].…”

Section: Introductionmentioning

confidence: 99%

Representations over diagrams of categories and abelian model structures

Di¹,

Li²,

Li³

et al. 2022

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In this paper we systematically consider representations over diagrams of abelian categories, which unify quite a few notions appearing widely in literature such as representations of categories, sheaves of modules over categories equipped with Grothendieck topologies, representations of species, etc. Since a diagram of abelian categories is a family of abelian categories glued by an index category, the central theme of our work is to determine whether local properties shared by each abelian category can be amalgamated to the corresponding global properties of the representation category. Specifically, we investigate the structure of the representation categories, describe important functors and adjunction relations between them, and construct cotorsion pairs in the representation category by local cotorsion pairs in each abelian category. As applications, we establish abelian model structures on the representation category for some particular index categories, and characterize special homological objects (such as projective, injective, flat, Gorenstein injective, and Gorenstein flat objects) in categories of presheaves of modules over some combinatorial categories. ContentsPart II. Functors, adjunctions, and their applications 5. The induction functor and its applications 5.1. The restriction functor and its left adjoint 5.2. Grothendieck structure and locally finitely presented property 5.3. Dual results for Rep-D 6. The lift and stalk functors 6.1. The lift functor and its left adoint 6.2. The stalk functor and its left adjoint 6.3. Dual results for Rep-D

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The enriched Grothendieck construction

Cited by 22 publications

References 8 publications

Monoidal Grothendieck construction

Monoidal Grothendieck construction

Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces

Representations over diagrams of categories and abelian model structures

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