Related papers: An inductive model structure for strict $\infty$-c…
The familiar construction of categories of fractions, due to Gabriel and Zisman, allows one to invert a class W of arrows in a category in a universal way. Similarly, bicategories of fractions allow one to invert a collection of arrows in a…
We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or may not be required to be invertible. This is…
We explore the interlacing between model category structures attained to classes of modules of finite $\mathcal{X}$-dimension, for certain classes of modules $\mathcal{X}$. As an application we give a model structure approach to the…
In this note, we construct a closed model structure on the category of $\mathbb{Z}/2\mathbb{Z}$-graded complexes of projective systems of ind-Banach spaces. When the base field is the fraction field $F$ of a complete discrete valuation ring…
In the present article, we describe constructions of model structures on general bicomplete categories. We are motivated by the following question: given a category $\mathcal{C}$ with a subcategory $w\mathcal{C}$ closed under retracts, when…
We prove a generalization of Quillen's Theorem B to strict $\infty$-categories. More generally, we show that under similar hypothesis as for Theorem B, the comma construction for strict $\infty$-categories, that we introduced with…
We construct a flagged $\infty$-category ${\sf Corr}$ of $\infty$-categories and bimodules among them. We prove that ${\sf Corr}$ classifies exponentiable fibrations. This representability of exponentiable fibrations extends that…
In this paper we present a new way to construct the pro-category of a category. This new model is very convenient to work with in certain situations. We present a few applications of this new model, the most important of which solves an…
The purpose of this text is the study of the class of homotopy types which are modelized by strict \infty-groupoids. We show that the homotopy category of simply connected \infty-groupoids is equivalent to the derived category in…
We define admissible and weakly admissible subcategories in exact categories and prove that the former induce semi-orthogonal decompositions on the derived categories. We develop the theory of thin exact categories, an exact-category…
We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known model structures on simplicial, topological, dg- and…
We construct the covariant and the cocartesian model structures on the slice categories of cubical sets and marked cubical sets, respectively. As an application, we derive a version of the Bousfield-Kan formula for arbitrary cofibrantly…
We establish that a category of fibrant objects (in the sense of Brown) admits a Dwyer-Kan homotopical calculus of right fractions. This is done using a homotopical calculus of cocycles, which is an auxiliary structure that can be defined…
Given subsets $\mathcal{C},\mathcal{F}$ of a preorder $\mathcal{A}$, we give necessary and sufficient conditions for $\mathcal{A}$ to admit the structure of a model category whose cofibrant objects are $\mathcal{C}$ and whose fibrant…
We prove that categories enriched in the Thomason model structure admit a model structure that is Quillen equivalent to the Bergner model structure on simplicial categories, providing a new model for (infinity,1)-categories. Along the way,…
Associated to each small category $C$, there is a category of $C$-shaped diagrams of simplicial sets and an $\infty$-category of $NC$-shaped homotopy coherent diagrams of spaces. We present a functor which exhibits the latter as the…
Let $\mathcal{E}$ be a weakly idempotent complete exact category with enough injective and projective objects. Assume that $\mathcal{M} \subseteq \mathcal{E}$ is a rigid, contravariantly finite subcategory of $\mathcal{E}$ containing all…
We develop a model structure on presheaves of small simplicially enriched categories on a site $\mathscr{C}$, for which the weak equivalences are 'stalkwise' weak equivalences for the Bergner model structure. This model structure is right…
Let $R$ be a ring and Ch($R$) the category of chain complexes of $R$-modules. We put an abelian model structure on Ch($R$) whose homotopy category is equivalent to $K(Proj)$, the homotopy category of all complexes of projectives. However,…
We propose a simplified definition of Quillen's fibration sequences in a pointed model category that fully captures the theory, although it is completely independent of the concept of action. This advantage arises from the understanding…