Are all localizing subcategories of stable homotopy categories coreflective?

TL;DR

This paper demonstrates that under Vopenka's principle, all localizing subcategories in certain triangulated categories are coreflective, establishing a bijection with colocalizing subcategories, thus resolving an open problem in stable homotopy theory.

Contribution

It proves that assuming Vopenka's principle, localizing subcategories are coreflective and establish a bijection with colocalizing subcategories in triangulated categories with combinatorial models.

Findings

All localizing subcategories are coreflective under Vopenka's principle.

A bijective correspondence exists between localizing and colocalizing subcategories.

The results resolve open problems in stable homotopy category theory.

Abstract

We prove that, in a triangulated category with combinatorial models, every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large-cardinal axiom (Vopenka's principle) is assumed true. It follows that, under the same assumptions, orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories. The existence of such a bijection was left as an open problem by Hovey, Palmieri and Strickland in their axiomatic study of stable homotopy categories and also by Neeman in the context of well-generated triangulated categories.