Homotopy reflectivity is equivalent to the weak Vop\v{e}nka principle

TL;DR

This paper demonstrates that certain localization properties in homotopy theory are equivalent to the weak Vopěnka principle, a set-theoretic axiom weaker than Vopěnka's principle, using recent large-cardinal results.

Contribution

It establishes the equivalence between weak Vopěnka's principle and various localization and reflectivity statements in homotopy theory and $mbda$-categories.

Findings

Localization exists in homotopy categories under weak Vopenka

Reflectivity of subcategories in triangulated categories under weak Vopenka

Use of Wilson's 2020 result to connect set theory and homotopy localization

Abstract

Homotopical localizations with respect to (possibly proper) classes of maps are known to exist assuming the validity of a large-cardinal axiom from set theory called Vop\v{e}nka's principle. In this article, we prove that each of the following statements is equivalent to an axiom of lower consistency strength than Vop\v{e}nka's principle, known as weak Vop\v{e}nka's principle: (a) Localization with respect to any class of maps exists in the homotopy category of simplicial sets; (b) Localization with respect to any class of maps exists in the homotopy category of spectra; (c) Localization with respect to any class of morphisms exists in any presentable $\infty$-category; (d) Every full subcategory closed under products and fibres in a triangulated category with locally presentable models is reflective. Our results are established using Wilson's 2020 solution to a long-standing open problem concerning the relative consistency of weak Vop\v{e}nka's principle within the large-cardinal hierarchy.