A generalization of Quillen's small object argument

We construct a cofibrantly generated Quillen model structure on the category of small differential graded categories. ----- Nous construisons une structure de categorie de modeles de Quillen a engendrement cofibrant sur la categorie des petites categories differentielles graduees.

We prove existence results for small presentations of model categories generalizing a theorem of D. Dugger from combinatorial model categories to more general model categories. Some of these results are shown under the assumption of Vop\v{e}nka's principle. Our main theorem applies in particular to cofibrantly generated model categories where the domains of the generating cofibrations satisfy a slightly stronger smallness condition. As a consequence, assuming Vop\v{e}nka's pr...

Let $D$ be a large category which is cocomplete. We construct a model structure (in the sense of Quillen) on the category of small functors from $D$ to simplicial sets. As an application we construct homotopy localization functors on the category of simplicial sets which satisfy a stronger universal property than the customary homotopy localization functors do.

In this paper we study a 2-dimensional version of Quillen's homotopy category construction. Given a category $\mathscr{A}$ and a class of morphisms $\Sigma \subset \mathscr{A}$ containing the identities, we construct a 2-category $\mathcal{H}o(\mathscr{A})$ obtained by the addition of 2-cells determined by homotopies. A salient feature here is the use of a novel notion of cylinder introduced in \cite{e.d.2}. The inclusion 2-functor $\mathscr{C} \longrightarrow \mathcal{H}o(\m...

We give a complete and careful proof of Quillen's theorem on the existence of the standard model category structure on the category of topological spaces. We do not assume any familiarity with model categories.

We extend the framework of combinatorial model categories, so that the category of small presheaves over large indexing categories and ind-categories would be embraced by the new machinery called class-combinatorial model categories. The definition of the new class of model categories is based on the corresponding extension of the theory of locally presentable and accessible categories developed in the companion paper [arXiv:1110.0605], where we introduced the concepts of l...

The definition of the homotopy limit of a diagram of left Quillen functors of model categories has been useful in a number of applications. In this paper we review its definition and summarize some of these applications. We conclude with a discussion of why we could work with right Quillen functors instead, but cannot work with a combination of the two.

In [Homotopical Algebra, Springer LNM 43] Quillen introduces the notion of a model category: a category $\mathcal{C}$ provided with three distinguished classes of maps $\{\mathcal{W},\, \mathcal{F},\, co\mathcal{F}\}$ (weak equivalences, fibrations, cofibrations), and gives a construction of the localization $\mathcal{C}[\mathcal{W}^{-1}]$ as the quotient of $\mathcal{C}$ by the congruence relation determined by the homotopies on the sets of arrows $\mathcal{C}(X,\,Y)$. We de...

We describe a method for constructing simplicial model structures on ind- and pro-categories. Our method is particularly useful for constructing "profinite" analogues of known model categories. Our construction quickly recovers Morel's model structure for pro-p spaces and Quick's model structure for profinite spaces, but we will show that it can also be applied to construct many interesting new model structures. In addition, we study some general properties of our method, suc...

In [BaSc2] the authors introduced a much weaker homotopical structure than a model category, called a "weak cofibration category". We further showed that a small weak cofibration category induces in a natural way a model category structure on its ind-category, provided the ind-category satisfies a certain two out of three property. The purpose of this paper is to serve as a companion to the papers above, proving results which say that if a certain property or structure exists...