Derived categories for Grothendieck categories of enriched functors

TL;DR

This paper investigates the derived categories of enriched functor categories over Grothendieck categories, establishing conditions under which these categories are compactly generated and providing explicit descriptions of their generators.

Contribution

It extends the understanding of derived categories for enriched functor categories, showing they are compactly generated under certain conditions and giving explicit generators.

Findings

Derived category $D[C,V]$ is compactly generated under specified conditions.

Explicit descriptions of generators for $D[C,V]$ are provided.

Conditions on $V$ ensure the transfer of compact generation to $D[C,V]$.

Abstract

The derived category $D[C,V]$ of the Grothendieck category of enriched functors $[C,V]$, where $V$ is a closed symmetric monoidal Grothendieck category and $C$ is a small $V$-category, is studied. We prove that if the derived category $D(V)$ of $V$ is a compactly generated triangulated category with certain reasonable assumptions on compact generators or $K$-injective resolutions, then the derived category $D[C,V]$ is also compactly generated triangulated. Moreover, an explicit description of these generators is given.