Combinatorial and accessible weak model categories

In a previous work, we have introduced a weakening of Quillen model categories called weak model categories. They still allow all the usual constructions of model category theory, but are easier to construct and are in some sense better behaved. In this paper we continue to develop their general theory by introducing combinatorial and accessible weak model categories. We give simple necessary and sufficient conditions under which such a weak model category can be extended into a left and/or right semi-model category. As an application, we recover Cisinski-Olschok theory and generalize it to weak and semi-model categories. We also provide general existence theorems for both left and right Bousfield localization of combinatorial and accessible weak model structures, which combined with the results above gives existence results for left and right Bousfield localization of combinatorial and accessible left and right semi-model categories, generalizing previous results of Barwick. Surprisingly, we show that any left or right Bousfield localization of an accessible or combinatorial Quillen model category always exists, without properness assumptions, and is simultaneously both a left and a right semi-model category, without necessarily being a Quillen model category itself.     

Chains of Group Localizations

We construct long sequences of localization functors L _α in the category of abelian groups such that L _α ≥ L _β for infinite cardinals α < β less than some κ. For sufficiently large free abelian groups F and α < β we have proper inclusions L _α F ? L _β F.     

Large localizations of finite groups

We construct examples of localizations in the category of groups which take the Mathieu group M₁₁ to groups of arbitrarily large cardinality which are “abelian up to finitely many generators.” The paper is part of a broader study on the group theoretic properties which are or are not preserved by localizations.