Enrichment and Representability for Triangulated Categories

Abstract

Given a fixed tensor triangulated category mathsfS we consider triangulated categories mathsfT together with an mathsfS-enrichment which is compatible with the triangulated structure of mathsfT. It is shown that, in this setting, an enriched analogue of Brown representability holds when both mathsfS and mathsfT are compactly generated. A natural class of examples of such enriched triangulated categories are module categories over separable monoids in mathsfS. In this context we prove a version of the Eilenberg-Watts theorem for exact coproduct and copower preserving mathsfS-functors, i.e., we show that any such functor between the module categories of separable monoids in mathsfS is given by tensoring with a bimodule.