Carles Casacuberta (Universitat de Barcelona)
Homotopy reflectivity is equivalent to the weak Vopenka principle
Abstract: It is well known that the existence of homotopical localization with respect to every (possibly proper) class of maps between spaces or spectra is implied by suitable large-cardinal axioms. However, no concluding evidence had been given that the existence of such localizations could not be proved in ZFC. Using a recent result of Trevor Wilson, we prove that the existence of localizations with respect to classes of maps of spaces or spectra is equivalent to the weak Vopenka principle, stating that there is no full embedding of the opposite category of ordinals into any locally presentable category. In fact we prove that the weak Vopenka principle is equivalent to the claim that every colocalizing subcategory of the homotopy category of any stable locally presentable model category is reflective. This is joint work with Javier Gutiérrez.