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Class-combinatorial model categories. (English) Zbl 1263.18011

Combinatorial model categories, introduced in the 1990s, have become a standard tool of homotopical algebra. A model category is combinatorial if it is locally presentable as a category and its model category structure is cofibrantly generated. Although of great utility, there are examples of model categories that are extremely useful for applications, but are not combinatorial. (Isaksen gave important non-cofibrantly generated model structures on categories of pro-spaces and ind-spaces, for instance, and studies of the Goodwillie calculus in terms of model categories requires study of small functors defined on large categories and that again is not cofibrantly generated.)
In this paper, the authors propose a framework extending that of combinatorial model categories, so that categories of small simplicial presheaves on large categories and ind-categories of model categories would become examples of these newly defined class-combinatorial model categories. (A pre-requisite for understanding this paper is a companion paper by the same authors on class-locally presentable and class-accessible categories.) A class-combinatorial model category is class-locally presentable and class-cofibrantly generated. These generalizations are effectuated by keeping close control of the ‘size’ of the objects and morphisms involved in the constructions. One of the main results is that the left Bousfield localization of a class-combinatorial model category with respect to a strongly class-accessible localization functor is once again class-combinatorial.

MSC:

18G55 Nonabelian homotopical algebra (MSC2010)
55P60 Localization and completion in homotopy theory
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