Mathematical Research Letters
Volume 16 (2009)
Brown representability does not come for free
Pages: 1 – 5
We exhibit a triangulated category $\ct$ having both products and coproducts, and a triangulated subcategory $\cs\subset \ct$ which is both localizing and colocalizing, for which neither a Bousfield localization nor a colocalization exists. It follows that neither the~category $\cs$ nor its dual satisfy Brown representability. Our example involves an abelian category whose derived category does not have small Hom-sets.