2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74216It is known that the existence of localization with respect to an arbitrary (possibly proper) class of maps in the category of simplicial sets is implied by a large-cardinal axiom called Vopenka's principle.In this article we extend the validity of this result to any left proper, combinatorial, simplicial model category $\cat M$ and show that, under additional assumptions on $\cat M$, every homotopy idempotent functor is in fact a localization with respect to some set of maps. These results are valid for the homotopy category of spectra, among other applications.9 pagesAlgebraic TopologyCategory Theorytext