About Quillen's homotopy theorie of derivatives
This article is concerned with the 2-category of combinatorial left proper Quillen model categories, where the morphisms are Quillen pairs and the \(2\)-morphisms are natural transformations between left adjoints. A Quillen homotopy is such a natural transformation which is a weak equivalence on any cofibrant object. The author constructs path and cylinder model categories and shows that, by first inverting Quillen homotopies, one obtains a 2-category where Quillen equivalences admit a calculus of fractions as studied by \textit{D. A. Pronk} [Compos. Math. 102, No. 3, 243--303 (1996; Zbl 0871.18003)]. The corresponding bilocalization is the 2-category of combinatorial left proper Quillen homotopy theories. Next one can construct from a model category a prederivator. It associates to any small category \({\mathcal A}\) the homotopy category of diagrams of shape~\({\mathcal A}\). The good behavior between pull-back morphisms along functors \({\mathcal A}\to{\mathcal B}\) and their Kan adjoints make it actually into a derivator. The main result of this article is that the pseudo-functor sending combinatorial left proper Quillen homotopy theories to their associated derivators is a local equivalence, i.e., the categories of morphisms between model categories and between their associated derivators are equivalent. The proof is based on a representation result of \textit{D.-C. Cisinski} [Theory Appl. Categ. 20, 605--649 (2008; Zbl 1188.18009)] and the fact that combinatorial model categories have small presentations in the sense of \textit{D. Dugger} [Adv. Math. 164, No. 1, 177--201 (2001; Zbl 1001.18001)].
scientific article; zbMATH DE number 5566281