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A homotopy coherent cellular nerve for bicategories. (English) Zbl 1451.18038

When generalizing \(n\)-categories to \((\infty,n)\)-categories we often use presheaf categories. As a matter of fact, quasi-categories, the most common model of \((\infty,1)\)-categories, are just certain simplicial sets. Thus in order to understand the relationship between strict categories and weak categories, we need to understand how to construct presheaves out of categories. Such constructions are generally known as nerves.
In the context of \((\infty,1)\)-categories the nerve construction takes a category to a simplicial set. This construction was already known to Grothendieck and has many desirable properties. In particular, it is fully faithful and in fact a right Quillen functor from the canonical model structure for categories to the model structure for quasi-categories. Moreover, we can precisely characterize the simplicial sets that lie in the essential image, via the Segal condition.
Quasi-categories have been generalized by Ara to \(2\)-quasi-categories, which is a model of \((\infty,2)\)-categories, and are certain presheaves on the category \(\Theta_2\) [D. Ara, J. \(K\)-Theory 14, No. 3, 701–749 (2014; Zbl 1322.18002)]. Hence, in order to understand the relationship between \(2\)-categories and \((\infty,2\))-categories, we need a nerve construction that takes a bicategory to a \(2\)-quasi-category and that has similar properties to the nerve construction for \(1\)-categories. However, as of now the appropriate nerve construction has eluded us. In particular, Ara himself suggested a nerve construction for bicategories, the strict nerve, that turned out not to actually take value in \(2\)-quasi-categories.
This paper successfully resolves this problem. The author proves there is a coherent nerve functor from bicategories to \(\Theta_2\)-sets, motivated by work of T. Leinster [Theory Appl. Categ. 10, 1–70 (2002; Zbl 0987.18007)], with the following properties:
It is fully faithful (Theorem 3.18).
It is a right Quillen functor from Lack’s model structure for bicategories [S. Lack, \(K\)-Theory 33, No. 3, 185–197 (2004; Zbl 1069.18008)] to Ara’s model structure for \(2\)-quasi-categories (Theorem 5.10). Hence it takes bicategories to \(2\)-quasi-categories.
The coherent nerve as a functor from bicategories to \(2\)-quasi-categories has a left adjoint, the homotopy bicategory (Theorem 6.29).
The essential image of the coherent nerve consist of \(2\)-truncated \(2\)-quasi-categories (Theorem 7.28). In fact, there is a Bousfield localization of the model structure for \(2\)-quasi-categories with fibrant objects \(2\)-truncated \(2\)-quasi-categories, that is Quillen equivalent to Lack’s model structure for bicategories (Theorem 8.7) and Rezk’s model structure for \((2,2)\)-\(\Theta\)-spaces [C. Rezk, Geom. Topol. 14, No. 1, 521–571 (2010; Zbl 1203.18015)] (Proposition 8.11).
More generally, there is an equivalence of tricategories between bicategories and \(2\)-truncated \(2\)-quasi-categories (Theorem 9.12).
The coherent nerve is the fibrant replacement of the strict nerve (Theorem 10.10).
As an application of the previous result the author also resolves Ara’s conjecture: A functor of bicategories is an equivalence if and only if the strict nerve is an equivalence in the model structure for \(2\)-quasi-categories (Theorem 10.11).
In order to prove these results about the coherent nerve the author also proves many interesting results about \(2\)-quasi-categories that are of independent interested:
There is a computationally feasible recognition principle for left Quillen functors out of the model structure for \(2\)-quasi-categories (Proposition 4.13).
There is a mapping quasi-category for a \(2\)-quasi-category and any choice of \(2\) objects (Proposition 6.5).
A map of \(2\)-quasi-categories is an equivalence if and only if it is an equivalence of mapping quasi-categories and essentially surjective (Theorem 7.25).
The underlying bisimplicial set of a \(2\)-quasi-category is a quasi-category-enriched Segal category (Theorem 6.14).
More generally, there is a Quillen equivalence between \(2\)-quasi-categories and quasi-category-enriched Segal categories (Theorem 10.7).
There is a Quillen equivalence between the model structure for quasi-categories and locally Kan \(2\)-quasi-categories (Theorem 11.14).

MSC:

18N10 2-categories, bicategories, double categories
18N20 Tricategories, weak \(n\)-categories, coherence, semi-strictification
18N40 Homotopical algebra, Quillen model categories, derivators
18N55 Localizations (e.g., simplicial localization, Bousfield localization)
18N60 \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories
18N65 \((\infty, n)\)-categories and \((\infty,\infty)\)-categories
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References:

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