Related papers: Fibration Categories are Fibrant Relative Categori…

We lift Charles Rezk's complete Segal space model structure on the category of simplicial spaces to a Quillen equivalent one on the category of relative categories.

We show that the category of N-complexes has a Str\om model structure, meaning the weak equivalences are the chain homotopy equivalences. This generalizes the analogous result for the category of chain complexes (N = 2). The trivial objects…

We develop further the theory of weak factorization systems and algebraic weak factorization systems. In particular, we give a method for constructing (algebraic) weak factorization systems whose right maps can be thought of as (uniform)…

We develop new techniques for constructing model structures from a given class of cofibrations, together with a class of fibrant objects and a choice of weak equivalences between them. As a special case, we obtain a more flexible version of…

In the first part of this paper we study fibrations of $(\infty,2)$-categories. We give a simple characterization of such fibrations in terms of a certain square being a pullback, and apply this to show that in some cases…

We prove that the category of trees $\Omega$ is a test category in the sense of Grothendieck. This implies that the category of dendroidal sets is endowed with the structure of a model category Quillen-equivalent to spaces. We show that…

For each $n \geq -1$, a quasi-category is said to be $n$-truncated if its hom-spaces are $(n-1)$-types. In this paper we study the model structure for $n$-truncated quasi-categories, which we prove can be constructed as the Bousfield…

We define groupoidal and $(n+k)$-truncated $n$-quasi-categories, which are the translation to the world of $n$-quasi-categories of groupoidal and truncated $(\infty, n)$-$\Theta$-spaces defined by Rezk. We show that these objects are the…

There are various adjunctions between model (co-)slice and slice categories. We characterize when these adjunctions are Quillen equivalences. As an application, a triangle equivalence between the stable category of a Frobenius category and…

We extend the Quillen Theorem Bn for homotopy fibers of Dwyer, et al. to similar results for homotopy pullbacks and note that these results imply similar results for zigzags in the categories of relative categories and k-relative…

We construct a model structure on the category $\mathrm{DblCat}$ of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully…

We construct a Quillen model structure on the category of spectral categories, where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories.

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 formulate a model-independent theory of co/cartesian morphisms and co/cartesian fibrations: that is, one which resides entirely *within the $\infty$-category of $\infty$-categories*. We prove this is suitably compatible with the…

We prove that each of the model structures for ($n$-trivial, saturated) comical sets on the category of marked cubical sets having only faces and degeneracies (without connections) is Quillen equivalent to the corresponding model structure…

Various models of $(\infty,1)$-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an $\infty$-cosmos. In a generic $\infty$-cosmos, whose…

For each pair of simplicial sets $A$ and $B$, the category $\mathbf{Cyl}(A,B)$ of cylinders (also called correspondences) from $A$ to $B$ admits a model structure induced from Joyal's model structure for quasi-categories. In this paper, we…

We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation,…

We exhibit a Quillen equivalence between two model categories encoding the homotopy theory of stratified spaces : the model category of filtered simplicial sets, and that of filtered spaces. Additionally, we introduce a new class of…

In this note we show that in the simplicial setting, the classifying space construction converts short exact sequences of groups not just to homotopy fibrations, but in fact to fibre bundles.