The Gray tensor product for 2-quasi-categories

Maehara, Yuki

doi:10.1016/j.aim.2020.107461

Cited by 7 publications

(4 citation statements)

References 9 publications

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“…3.2.6 and 3.2.9], given there without a proof. We should note however that these assertions were previously proven in [Ver08b] and [Mae20] in the contexts of complicial sets and 2-quasicategories respectively.…”

Section: Introductionsupporting

confidence: 55%

A cubical model for $(\infty, n)$-categories

Campion,

Kapulkin,

Maehara

2020

Preprint

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We propose a new model for the theory of (∞, n)-categories (including the case n = ∞) in the category of marked cubical sets with connections, similar in flavor to complicial sets of Verity. The model structure characterizing our model is shown to be monoidal with respect to suitably defined (lax and pseudo) Gray tensor products; in particular, these tensor products are both associative and biclosed. Furthermore, we show that the triangulation functor to pre-complicial sets is a left Quillen functor and is strong monoidal with respect to both Gray tensor products.

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Section: Introductionsupporting

confidence: 55%

A cubical model for $(\infty, n)$-categories

Campion,

Kapulkin,

Maehara

2020

Preprint

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“…In this paper, we prove that this version of the Gray tensor is equivalent to ours, via the equivalence established in [3]. Another approach has been recently provided by Maehara [12] via the combinatorics of Θ 2sets.…”

Section: Introductionmentioning

confidence: 66%

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

Gagna¹,

Harpaz²,

Lanari³

2020

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We give a definition of the Gray tensor product in the setting of scaled simplicial sets which is associative and forms a left Quillen bifunctor with respect to the bicategorical model category of Lurie. We then introduce a notion of oplax functor in this setting, and use it in order to characterize the Gray tensor product by means of a universal property. A similar characterization was used by Gaitsgory and Rozenblyum in their definition of the Gray product, thus giving a promising lead for comparing the two settings. AcknowledgementsThe first author is supported by GA ČR EXPRO 19-28628X. The third author gratefully acknowledges the support of Praemium Academiae of M. Markl and RVO:67985840, as well as fruitful conversations with Nick Rozenblyum during his stay at MSRI.

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“…That this model structure is monoidal with respect to ⊗ follows from [Mae21,Proposition A.4] and (4-6).…”

Section: Introductionmentioning

confidence: 97%

Equivalence of cubical and simplicial approaches to $(\infty,n)$-categories

Doherty¹,

Kapulkin²,

Maehara³

2021

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We prove that the marked triangulation functor from the category of marked cubical sets equipped with a model structure for (n-trivial, saturated) comical sets to the category of marked simplicial set equipped with a model structure for (n-trivial, saturated) complicial sets is a Quillen equivalence. Our proof is based on the theory of cones, previously developed by the first two authors together with Lindsey and Sattler.Proof. We will apply [Ols09, Theorem 2.2.5] (with L taken to be all monomorphisms). Our functorial cylinder is given byNote that (1), ( 2) and (4) imply that X ⊗ (∂ 0 , ∂ 1 ) is a monomorphism for each X since it can be written as i X ⊗(∂ 0 , ∂ 1 ) where i X ∶ 0 → X is the unique map. Similarly, since any map from a terminal object (and in particular ∂ 0 , ∂ 1 ) is a monomorphism, we can deduce that X ⊗ ∂ 0 and X ⊗ ∂ 1 are always

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The Gray tensor product for 2-quasi-categories

Cited by 7 publications

References 9 publications

A cubical model for $(\infty, n)$-categories

A cubical model for $(\infty, n)$-categories

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

Equivalence of cubical and simplicial approaches to $(\infty,n)$-categories

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