The derivator of a dg-category

TL;DR

This paper constructs a stable derivator for dg-categories using explicit homotopy Kan extensions, providing new insights into their structure and applications to Frobenius categories and Gorenstein projective diagrams.

Contribution

It explicitly defines the derivator for dg-categories via weighted homotopy limits and colimits, extending to pretriangulated categories and linking to Gorenstein projective diagrams.

Findings

Explicit construction of the derivator for dg-categories.

Extension of the derivator to pretriangulated dg-categories.

Characterization of the derivator for Frobenius categories in terms of Gorenstein diagrams.

Abstract

In this work, we construct the stable derivator associated to a homotopically complete and cocomplete dg-category by explicitly defining homotopy Kan extensions via suitable weighted homotopy limits and colimits in dg-categories. By restricting the domain of the derivator to finite direct categories, we obtain a well-defined derivator even for pretriangulated dg-categories. This definition enables an explicit description of the derivator associated to a weakly idempotent complete Frobenius exact category, leading to a more direct characterization in terms of Gorenstein projective (equivalently, Gorenstein injective) diagrams.