#### DOCUMENTA MATHEMATICA, Vol. 22 (2017), 1031-1062

Johan Steen and Greg Stevenson

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.

Full text: dvi.gz 65 k, dvi 289 k, ps.gz 362 k, pdf 295 k.