  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.

Category Theory;Algebraic Topology