Enrichment and Representability for Triangulated Categories

Abstract

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