Enhanced left triangulated categories

TL;DR

This paper investigates dg categories with homotopy kernels, showing their homotopy categories naturally form left triangulated structures and can be enhanced via bounded derived dg categories.

Contribution

It introduces a canonical dg enhancement for the stabilization of homotopy categories of dg categories with homotopy kernels.

Findings

Homotopy categories of dg categories with homotopy kernels admit a left triangulated structure.

The stabilization of such categories can be enhanced through bounded derived dg categories.

Prototypical examples include dg quotients of exact dg categories.

Abstract

In this short note, we study dg categories with homotopy kernels, whose homotopy categories are known to admit a natural left triangulated structure. Prototypical examples of such dg categories arise as dg quotients of exact dg categories. We demonstrate that the stablization of the homotopy category of such a dg category admits a canonical dg enhancement via its bounded derived dg category.