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Homotopy reflectivity is equivalent to the weak Vopěnka principle

Carles Casacuberta, Javier J. Gutiérrez

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 Vopěnka's principle. In this article, we prove that each of the following statements is equivalent to an axiom of lower consistency strength than Vopěnka's principle, known as weak Vopěnka'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 Vopěnka's principle within the large-cardinal hierarchy.

Homotopy reflectivity is equivalent to the weak Vopěnka principle

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 Vopěnka's principle. In this article, we prove that each of the following statements is equivalent to an axiom of lower consistency strength than Vopěnka's principle, known as weak Vopěnka'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 -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 Vopěnka's principle within the large-cardinal hierarchy.

Paper Structure

This paper contains 18 sections, 19 theorems, 24 equations.

Table of Contents

  1. Preliminaries
  2. Large-cardinal principles
  3. Orthogonality
  4. Orthogonality in simplicial model categories
  5. Orthogonality in $\infty$-categories
  6. Comparison
  7. Underlying $\infty$-category
  8. Homotopy reflections
  9. Orthogonality in spectra
  10. From weak reflections to reflections
  11. Existence of weak reflections
  12. Spectra and simplicial sets
  13. Reflectivity in $\infty$-categories
  14. Reverse implications
  15. Triangulated categories
  16. ...and 3 more sections

Key Result

Proposition 1.1

Let ${\mathcal{M}}$ be a simplicial model category with cofibrant replacement $Q$ and fibrant replacement $R$. Given a class of morphisms ${\mathcal{S}}$ in ${\mathcal{M}}$, the class ${\mathcal{S}}^\perp({\mathcal{M}})$ is the closure of $(RQ{\mathcal{S}})^\perp({\mathcal{M}}^{\circ})$ under weak e

Theorems & Definitions (41)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Corollary 1.3
  • proof
  • Proposition 1.4
  • proof
  • Proposition 2.1
  • proof
  • ...and 31 more