Enrichment and Representability for Triangulated Categories
Given a fixed tensor triangulated category $\mathsf S$ we consider triangulated categories $\mathsf T$ together with an $\mathsf S$-enrichment which is compatible with the triangulated structure of $\mathsf T$. It is shown that, in this setting, an enriched analogue of Brown representability holds when both $\mathsf S$ and $\mathsf T$ are compactly generated. A natural class of examples of such enriched triangulated categories are module categories over separable monoids in $\mathsf S$. In this context we prove a version of the Eilenberg--Watts theorem for exact coproduct and copower preserving $\mathsf S$-functors, i.e., we show that any such functor between the module categories of separable monoids in $\mathsf S$ is given by tensoring with a bimodule.
2010 Mathematics Subject Classification: Primary 16D90; Secondary 18E30, 55U35.
Keywords and Phrases: Tensor triangulated category, monoid, enriched category, representability.
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