A model structure for weakly horizontally invariant double categories

Lyne, Moser,; Sarazola, Maru; Verdugo, Paula

doi:10.48550/arxiv.2007.00588

Cited by 1 publication

(12 citation statements)

References 9 publications

(48 reference statements)

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“…The existence of this model structure was independently noticed by Campbell [6]. As we show in [26], this new model structure is Quillen equivalent to the previous one through the identity functor on DblCat. The modification fixes the issue with respect to the vertical composition mentioned above, but the horizontal embedding H is not right Quillen anymore.…”

supporting

confidence: 61%

“…We then introduce notions of horizontal equivalences and weakly horizontally invertible squares in a double category, which allows us to define weakly horizontally invariant double categories. In Section 3, we recall the main features of Lack's model structure on 2Cat and of the model structure of [26] on DblCat. Then, in Section 4, we get to the ∞-setting and describe the model structures DblCat h ∞ and 2CSS for double (∞, 1)categories and 2-fold complete Segal spaces.…”

Section: Theorem B the Adjunctionmentioning

confidence: 99%

“…The functor H ≃ is not a left adjoint, since it does not preserve colimits; see [26,Remark 3.10]. However, it admits a left adjoint, which we describe below.…”

Section: Definition 212 An Equivalencementioning

confidence: 99%

“…However, as mentioned in the introduction, this model structure is not well-behaved with respect to composition of vertical morphisms in a double category. Therefore, in [26], we constructed another model structure obtained by adding the inclusion [0] ⊔ [0] → V [1] to the set of generating cofibrations and keeping the same weak equivalences. With this new model structure, the functor H is not right Quillen anymore, but the functor H ≃ fulfills this role.…”

Section: Model Structures On 2cat and Dblcatmentioning

confidence: 99%

“…To remedy this failure, the author, Sarazola, and Verdugo construct in [26] a model structure on DblCat with the same weak equivalences, i.e., the double biequivalences, and an additional cofibration [0] ⊔ [0] → V [1]. The existence of this model structure was independently noticed by Campbell [6].…”

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A double $(\infty,1)$-categorical nerve for double categories

Lyne¹

2020

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We construct a nerve from double categories into double (∞, 1)-categories and show that it gives a right Quillen and homotopically fully faithful functor between the model structure for weakly horizontally invariant double categories and the model structure on bisimplicial spaces for double (∞, 1)-categories seen as double Segal objects in spaces complete in the horizontal direction. We then restrict the nerve along a homotopical horizontal embedding of 2-categories into double categories, and show that it gives a right Quillen and homotopically fully faithful functor between Lack's model structure for 2-categories and the model structure for 2-fold complete Segal spaces. We further show that Lack's model structure is right-induced along this nerve from the model structure for 2-fold complete Segal spaces. Contents 1. Introduction 1.1. Outline Acknowledgments 2. Preliminaries on 2-dimensional categories 2.1. 2-categories, double categories, and their relations 2.2. Notions of equivalences in a double category 3. Model structures on 2Cat and DblCat 3.1. Lack's model structure for 2-categories 3.2. Model structure for weakly horizontally invariant double categories 4. Model structures for (∞, 2)-categories and double (∞, 1)-categories 4.1. Model structures for double (∞, 1)-categories 4.2. Model structure for 2-fold complete Segal spaces 5. Nerve of double categories 5.1. Definition of the nerve 5.2. The nerve N is right Quillen 5.3. The nerve N is homotopically fully faithful 5.4. Level of fibrancy of nerves of double categories 6. Nerve of 2-categories 6.1. The nerve NH ≃ is right Quillen and homotopically fully faithful 6.2. 2Cat is right-induced from 2CSS along NH ≃ 6.3. Comparison between the nerves NH and NH ≃ Appendix A. Weakly horizontally invertible squares A.1. Unique inverse lemma A.2. Weakly horizontally invertible square in HA, H ≃ A, and L ≃ A A.3. Horizontal pseudo-natural equivalences Appendix B. Explicit description of the nerves in lower dimensions B.1. Nerve of a double category B.2. Nerve of a 2-category B.3. Nerve of a horizontal double category References

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confidence: 61%

Section: Theorem B the Adjunctionmentioning

confidence: 99%

“…The functor H ≃ is not a left adjoint, since it does not preserve colimits; see [26,Remark 3.10]. However, it admits a left adjoint, which we describe below.…”

Section: Definition 212 An Equivalencementioning

confidence: 99%

Section: Model Structures On 2cat and Dblcatmentioning

confidence: 99%

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confidence: 99%

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