Brown representability does not come for free

TL;DR

The paper presents a specific triangulated category example demonstrating that Brown representability does not automatically hold, especially when certain localizing and colocalizing subcategories lack Bousfield localizations.

Contribution

It constructs a triangulated category with particular properties showing the failure of Brown representability in certain subcategories, highlighting limitations of existing theoretical assumptions.

Findings

Existence of a triangulated category with both products and coproducts

Identification of a subcategory lacking Bousfield localization

Demonstration that Brown representability fails in this context

Abstract

We exhibit a triangulated category T having both products and coproducts, and a triangulated subcategory S of T which is both localizing and colocalizing, for which neither a Bousfield localization nor a colocalization exists. It follows that neither the category S nor its dual satisfy Brown representability. Our example involves an abelian category whose derived category does not have small Hom-sets.