Gray tensor products and lax functors of $(\infty,2)$-categories

Gagna, Andrea; Harpaz, Yonatan; Lanari, Edoardo

doi:10.48550/arxiv.2006.14495

Cited by 5 publications

(13 citation statements)

References 7 publications

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“…Remark 3.10. Since B b E commutes with colimits in both variables by corollary 3.9 and coincides with the usual Gray product (constructed for 8-categories in [GHL20]) on pθ n,m , θ s,t q, it follows that it coincides with the usual Gray tensor product for all pairs pB, Eq.…”

Section: Now the Results Follows From The Following Sequence Of Isomo...mentioning

confidence: 76%

“…In particular, giving a lax functor from a point to B is equivalent to giving a monad in B. The 8-categorical generalization of this concept can be found for example in [GHL20], [GR17] and [Kos21]. Our extension of this notion to categories with factorization systems is quite technical, but roughly speaking a lax functor from F to B contains all the data associated to the lax functor from the underlying categories V and H to B together with additional 2-morphisms γ h,v : F pvq ˝F phq Ñ F p r hq ˝F pr vq for every factorization v ˝h -r h ˝r v in F. We then define a distributive law to be a lax functor from a singleton category considered as a category with a factorization system.…”

Section: Introductionmentioning

confidence: 99%

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Factorization systems in $\infty$-categories

Kositsyn

2021

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We extend the notion of a factorization system in a category to the realm of 8-categories. To this end, we provide a description of the category of 8-categories with factorization systems as the category of presheaves of spaces on a certain category that satisfy a version of the Segal condition. Additionally, we study the notion of distributive laws between monads and, more generally, lax functors in the context of categories with factorization systems. In particular, we also characterize categories with factorization systems as distributive laws in the bicategory of spans.

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Section: Now the Results Follows From The Following Sequence Of Isomo...mentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Factorization systems in $\infty$-categories

Kositsyn

2021

Preprint

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“…The category rnsbrms is in fact the Gray tensor product of rns and rms viewed as 2-categories. The 8-catgorical version of this construction can be found in [GR17], [Hau20] and [GHL20b].…”

Section: Lax Functorsmentioning

confidence: 99%

Completeness for monads and theories

Kositsyn

2021

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We generalize the correspondence between theories and monads with arities of [BMW12] to 8categories. Additionally, we introduce the notion of complete theories that is unique to the 8-categorical case and provide a completion construction for a certain class of theories. Along the way we also develop the necessary technical material related to the flagged bicategory of correspondences and lax functor in the 8-categorical context. Contents 1. Introduction 1 2. Definition of Corr 3 3. Basic properties of Corr 11 4. Lax functors 21 5. Lax functors to Corr 34 6. Completeness for monads and theories 51 References 53

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“…As a result, there is now a long list of models of (∞, n)-categories, such as n-fold complete Segal spaces [Bar05], Θ n -spaces [Rez10a], Θ n -sets [Ara14], n-complicial sets [Ver08] and n-comical sets [DKLS20, CKM20] (see [Ber11] for a review of many of these models). Moreover, (∞, n)-categories have found interesting applications in derived algebraic geometry [GR17a,GR17b] (via the models of scaled simplicial sets [GHL20b] and comical sets) and topological field theories [Lur09c,CS19] (via the model of n-fold complete Segal spaces).…”

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confidence: 99%

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

Rasekh¹

2021

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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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Gray tensor products and lax functors of $(\infty,2)$-categories

Cited by 5 publications

References 7 publications

Factorization systems in $\infty$-categories

Factorization systems in $\infty$-categories

Completeness for monads and theories

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

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