We give an example of a morphism of simplicial sets which is a monomorphism, bijective on 0-simplices, and a weak categorical equivalence, but which is not inner anodyne.This answers an open question of Joyal. Furthermore, we use this morphism to refute a plausible description of the class of fibrations in Joyal's model structure for quasi-categories.
Abstract. This paper introduces a skew variant of the notion of enriched category, suitable for enrichment over a skew-monoidal category, the main novelty of which is that the elements of the enriched hom-objects need not be in bijection with the morphisms of the underlying category. This is the natural setting in which to introduce the notion of locally weak comonad, which is fundamental to the theory of enriched algebraic weak factorisation systems. The equivalence, for a monoidal closed category V, between tensored V-categories and hommed V-actegories is extended to the skew setting and easily proved by recognising both skew V-categories and skew V-actegories as equivalent to special kinds of skew V-proactegory.
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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