Homotopy localization with respect to proper classes of maps

Localizing with respect to sets of maps is a common technique in Mathematics. However, localizing with respect to proper classes of maps is more delicate, due to set-theoretical difficulties. In earlier joint work with Scevenels and Smith, we proved that, assuming the validity of a suitable large-cardinal axiom, homotopy localization exists with respect to any class of maps between simplicial sets, and any such localization is in fact determined by a single map. These results were extended to any combinatorial model category in joint work with Chorny.

We have recently proved with Gutierrez and Rosicky that, under a large-cardinal axiom, the following statements are true in the homotopy category of any combinatorial stable model category: every colocalizing subcategory is reflective, and every localizing subcategory is a coreflective cohomological Bousfield class. This gives a partial answer to a question asked by Hovey-Palmieri-Strickland. Counterexamples exist if the model category is not cofibrantly generated. (PDF file)