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I show that any locally Cartesian left localisation of a presentable infinity-category admits a right proper model structure in which all morphisms are cofibrations, and obtain a Koszul duality classification of its fibrations. By a simple…

We show that the category of simplicial sets is a co-reflective subcategory of the category of cubical sets with connections, with the inclusion given by a version of the straightening functor. We show that using the co-reflector, one can…

We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model…

This is an exposition of homotopical results on the geometric realization of semi-simplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The…

We put a model structure on the category of categories internal to simplicial sets whose weak equivalences are reflected by the nerve functor to bisimplicial sets with Rezk's model structure. This model structure is shown to be Quillen…

We extend all known results about transferred model structures on algebraically cofibrant and fibrant objects by working with weak model categories. We show that for an accessible weak model category there are always Quillen equivalent…

We use Cisinski's machinery to construct and study model structures on the category of simplicial sets whose classes of fibrant objects generalize quasi-categories. We identify a lifting condition which captures the homotopical behavior of…

In this article, we construct a cofibrantly generated Quillen model structure on the category of small topological categories $\mathbf{Cat}_{\mathbf{Top}}$. It is Quillen equivalent to the Joyal model structure of $(\infty,1)$-categories…

We define a notion of groupoidal 2-quasi-categories and show that they are the fibrant objects of a model structure on the category of $\Theta_2$-sets. We show that this model category is Quillen equivalent to the Kan-Quillen model category…

We study $\infty$-categories in the synthetic simplicial type theory developed by Riehl and Shulman. In particular, we define cocartesian fibrations and prove their closure properties using a novel equivalence between LARI adjunctions and…

We construct on the category of diffeological spaces a Quillen model structure having smooth weak homotopy equivalences as the class of weak equivalences.

We show that the quasicategory of frames of a cofibration category, introduced by the second-named author, is equivalent to its simplicial localization.

We observe that the notion of a trivial Serre fibration, a Serre fibration, and being contractible, for finite CW complexes, can be defined in terms of the Quillen lifting property with respect to a single map M-->/\ of finite topological…

In this article, we develop a notion of Quillen bifibration which combines the two notions of Grothendieck bifibration and of Quillen model structure. In particular, given a bifibration $p:\mathcal E\to\mathcal B$, we describe when a family…

We provide a partial solution to the problem of defining a constructive version of Voevodsky's simplicial model of univalent foundations. For this, we prove constructive counterparts of the necessary results of simplicial homotopy theory,…

The main objective of this paper is to construct a symmetric monoidal closed model category of coherently commutative monoidal quasi-categories. We construct another model category structure whose fibrant objects are (essentially) those…

We describe a Quillen equivalence between quasi-categories and relative categories which is surprisingly similar to Thomason's Quillen equivalences between simplicial sets and categories.

In this article we introduce four variance flavours of cartesian 2-fibrations of $\infty$-bicategories with $\infty$-bicategorical fibres, in the framework of scaled simplicial sets. Given a map $p\colon \mathcal{E} \rightarrow\mathcal{B}$…

On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a…

In this note we show that a semisimplicial set with the weak Kan condition admits a simplicial structure, provided any object allows an idempotent self-equivalence. Moreover, any two choices of simplicial structures give rise to equivalent…