We construct another model structure on the category DblCat of double categories and double functors, Quillen equivalent to the model structure on DblCat defined in a companion paper by the authors. The weak equivalences are still given by the double biequivalences; the trivial fibrations are now the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully faithful on squares; and the fibrant objects are the weakly horizontally invariant double categories.We show that the functor H ≃ : 2Cat → DblCat, a more homotopical version of the usual horizontal embedding H, is right Quillen and homotopically fully faithful when considering Lack's model structure on 2Cat. In particular, H ≃ exhibits a fibrant replacement of H. Moreover, Lack's model structure on 2Cat is right-induced along H ≃ from the model structure for weakly horizontally invariant double categories.We also show that this model structure is monoidal with respect to Böhm's Gray tensor product. Finally, we prove a Whitehead Theorem characterizing the double biequivalences between the fibrant objects as the double functors which admit a pseudo inverse up to horizontal pseudo natural equivalences.
In ordinary category theory, limits are known to be equivalent to terminal objects in the slice category of cones. In this paper, we prove that the 2-categorical analogues of this theorem relating 2-limits and 2-terminal objects in the various choices of slice 2-categories of 2-cones are false. Furthermore we show that, even when weakening the 2-cones to pseudo-or lax-natural transformations, or considering bi-type limits and bi-terminal objects, there is still no such correspondence.
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
In the enriched setting, the notions of injective and projective model structures on a category of enriched diagrams also make sense. In this paper, we prove the existence of these model structures on enriched diagram categories under local presentability, accessibility, and "acyclicity" conditions, using the methods of lifting model structures from an adjunction introduced
We introduce a functor V : DblCat h,nps → 2Cat h,nps extracting from a double category a 2-category whose objects and morphisms are the vertical morphisms and squares. We give a characterisation of bi-representations of a normal pseudo-functor F : C op → Cat in terms of double bi-initial objects in the double category Ð(F ) of elements of F , or equivalently as bi-initial objects of a special form in the 2-category V Ð(F ) of morphisms of F . Although not true in general, in the special case where the 2-category C has tensors by the category ¾ = {0 → 1} and F preserves those tensors, we show that a bi-representation of F is then precisely a bi-initial object in the 2-category el(F ) of elements of F . We give applications of this theory to bi-adjunctions and weighted bi-limits.
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