Inner horns for 2-quasi-categories

Maehara, Yuki

doi:10.1016/j.aim.2020.107003

Cited by 5 publications

(7 citation statements)

References 12 publications

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“…Presheaves on Θ 2 define Θ 2 -sets. Certain Θ 2 -sets, characterized by a right lifting property first introduced by Ara [A14] and later refined by Oury [O10] and Maehara [Ma20], define the 2-quasi-categories, a model of (∞, 2)-categories.…”

Section: Another ∞-Cosmos Of Bicategoriesmentioning

confidence: 99%

“…Remark 6.3. Maehara gives a combinatorially explicit description of the fibrations between fibrant objects in Ara's model structure as those maps with the right lifting property against the inner horizontal horn inclusions, inner vertical horn inclusions, vertical equivalence extensions, and horizontal equivalence extensions [Ma20]. In particular, a normal pseudofunctor between bicategories is an isofibration in the ∞-cosmos Bicat if and and only if its homotopy coherent cellular nerve lifts against these maps.…”

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On $\infty$-cosmoi of bicategories

Riehl¹,

Mira²

2021

Preprint

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An ∞-cosmos is a setting in which to develop the formal category theory of (∞, 1)-categories. In this paper, we explore a few atypical examples of ∞-cosmoi whose objects are 2-categories or bicategories rather than (∞, 1)categories and compare the formal category theory that is so-encoded with classical 2-category and bicategory theory. We hope this work will inform future explorations into formal (∞, 2)-category theory. Contents 1. Introduction 1 2. ∞-cosmoi of 2-categories and bicategories 2 3. Formal category theory in an ∞-cosmos 6 4. The formal theory of 2-categories, 2-functors, and 2-natural transformations 8 5. The formal theory of bicategories, normal pseudofunctors, and icons 12 6. Another ∞-cosmos of bicategories 18 References 21

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Section: Another ∞-Cosmos Of Bicategoriesmentioning

confidence: 99%

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confidence: 99%

On $\infty$-cosmoi of bicategories

Riehl¹,

Mira²

2021

Preprint

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“…Alternative characterisations of 2-quasi-categories in terms of lifting properties may be found in [Mae19].…”

Section: -Catmentioning

confidence: 99%

A homotopy coherent cellular nerve for bicategories

Campbell

2020

Advances in Mathematics

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The subject of this paper is a nerve construction for bicategories introduced by Leinster, which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over Joyal's category Θ2. We prove that the nerve of a bicategory is a 2-quasi-category (a model for (∞, 2)-categories due to Ara), and moreover that the nerve functor restricts to the right part of a Quillen equivalence between Lack's model structure for bicategories and a Bousfield localisation of Ara's model structure for 2-quasi-categories. We deduce that Lack's model structure for bicategories is Quillen equivalent to Rezk's model structure for (2, 2)-Θ-spaces on the category of simplicial presheaves over Θ2.To this end, we construct the homotopy bicategory of a 2-quasi-category, and prove that a morphism of 2-quasi-categories is an equivalence if and only if it is essentially surjective on objects and fully faithful. We also prove a Quillen equivalence between Ara's model structure for 2-quasi-categories and the Hirschowitz-Simpson-Pellissier model structure for quasi-categoryenriched Segal categories, from which we deduce a few more results about 2-quasi-categories, including a conjecture of Ara concerning weak equivalences of 2-categories.

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“…The following theorem characterises n-ary left Quillen functors out of Θ 2 . Its proof, which relies on the main result of [Mae20], is deferred to Appendix A. (i) each map in F (I, .…”

Section: Sendingmentioning

confidence: 99%

“…We review the necessary background material in this section. There is a significant overlap with [Mae20,§2].…”

Section: Introductionmentioning

confidence: 99%

The Gray tensor product for 2-quasi-categories

Maehara

2020

Preprint

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We construct an (∞, 2)-version of the (lax ) Gray tensor product. On the 1-categorical level, this is a binary (or more generally an n-ary) functor on the category of Θ 2 -sets, and it is shown to be left Quillen with respect to Ara's model structure. Moreover we prove that this tensor product forms part of a "homotopical" monoidal (closed) structure, or more precisely a normal lax monoidal structure that is associative up to homotopy.

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Inner horns for 2-quasi-categories

Cited by 5 publications

References 12 publications

On $\infty$-cosmoi of bicategories

On $\infty$-cosmoi of bicategories

A homotopy coherent cellular nerve for bicategories

The Gray tensor product for 2-quasi-categories

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