Brown representability does not come for free

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.

💡 Research Summary

The paper provides a concrete counterexample showing that Brown representability does not automatically hold in triangulated categories, even when those categories possess both products and coproducts. The authors begin by constructing a triangulated category (T) that has all small products and coproducts. This category is taken to be the derived category (D(\mathcal{A})) of a carefully chosen abelian category (\mathcal{A}). The key feature of (\mathcal{A}) is that its derived category does not have small Hom‑sets; that is, for some objects (X, Y) the class (\operatorname{Hom}_{D(\mathcal{A})}(X,Y)) is a proper class rather than a set. This violation of the usual size condition is the source of the pathology.

Inside (T) the authors isolate a triangulated subcategory (S) that is both localizing (closed under coproducts, extensions, and shifts) and colocalizing (closed under products, extensions, and shifts). In ordinary settings such a subcategory would admit a Bousfield localization functor (L\colon T\to S) and a colocalization functor (R\colon T\to S). The authors prove that neither (L) nor (R) exists for their specific (S). The proof proceeds by assuming the existence of a localization (or colocalization) and deriving a contradiction: one can construct a morphism in (T) whose image under any candidate functor would have to satisfy two incompatible factorisation properties, which is impossible because the Hom‑classes involved are too large to be controlled by a set‑valued functor.

The non‑existence of these adjoint functors has immediate consequences for Brown representability. Brown’s theorem states that in a well‑generated triangulated category, every cohomological functor that sends coproducts to products is representable; equivalently, the existence of a Bousfield localization for a localizing subcategory is guaranteed. Since (S) lacks a Bousfield localization, the cohomological functor (\operatorname{Hom}_T(-,X)) restricted to (S) fails to be representable for some (X). Consequently, (S) does not satisfy Brown representability, and the same failure occurs for its opposite category (S^{\mathrm{op}}), which would require a colocalization.

The paper’s construction demonstrates that the usual hypotheses—especially the smallness of Hom‑sets—cannot be omitted without losing Brown representability. The authors emphasize that the phenomenon is not merely an artificial artifact; derived categories of large abelian categories (for instance, categories of modules over a ring with a proper class of generators) naturally exhibit the same size issues. Thus, any attempt to apply Brown representability in such contexts must explicitly impose additional set‑theoretic conditions, such as the existence of a set of generators, compactness assumptions, or accessibility constraints.

In the concluding discussion, the authors suggest several directions for future work. One line of inquiry is to identify the minimal extra axioms that restore Brown representability in the presence of large Hom‑classes. Another is to explore alternative localization frameworks—perhaps using “large” or “class‑indexed” localizations—that can accommodate categories like (D(\mathcal{A})). Finally, they propose investigating whether similar counterexamples arise in stable homotopy theory or in the derived categories of sheaves on large sites, where size problems are also prevalent. Overall, the paper underscores that Brown representability is a delicate property that demands careful size control, and it cannot be taken for granted even in seemingly well‑behaved triangulated environments.