Suppose M is a model category and K a subset of morphisms. A left Bousfield localization is a left Quillen functor M → N sending elements of K to weak equivalences, universal for this property, and furthermore which induces an isomorphism of underlying categories and an isomorphism of classes of cofibrations. If it exists, it is unique up to isomorphism.

It is pretty well known that the left Bousfield localization exists whenever M is a left proper combinatorial model category. We refer to the references and particularly Hirschhorn [144] and Barwick [18] (see also Rosický and Tholen [221]) for this existence theorem and for some of the main characterizations, statements, details and proofs concerning this notion in general. Recall that the general definition of K-local objects and the localization functor depend on the notion of simplicial mapping spaces. Of course, simplicial mapping spaces are exactly the kind of thing we are looking at in the present book, but to start with these as basic building blocks would stretch the notion of “bootstrapping” pretty far. Therefore, in the present chapter, we consider a special case of left Bousfield localization in which everything is much more explicit.

Projection to a subcategory of local objects

Start with a left proper tractable model category (M, I, J), that is a left proper cofibrantly generated model category such that M is locally K-presentable for some regular cardinal K, and the domains of arrows in I and J are cofibrant.