Enrichment and representability for triangulated categories

TL;DR

This paper explores enriched triangulated categories over a fixed tensor triangulated category, establishing an enriched Brown representability theorem and a version of the Eilenberg-Watts theorem for module categories over separable monoids.

Contribution

It introduces an enriched analogue of Brown representability and proves a version of the Eilenberg-Watts theorem within the context of S-enriched triangulated categories.

Findings

Enriched Brown representability holds for compactly generated S- and T.

Functor categories between modules over separable monoids are given by tensoring with bimodules.

Provides a framework for understanding enriched structures in 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 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.