A formal category theoretic approach to the homotopy theory of dg categories

TL;DR

This paper develops a category-theoretic framework for dg categories using bicategories and formal category theory, providing new insights into homotopy limits, pretriangulated structures, and gluing procedures.

Contribution

It introduces a bicategory of dg categories with derived bimodules, formalizes homotopical (co)limits, and characterizes pretriangulated dg categories within this framework.

Findings

Homotopical (co)limits are defined via formal category theory.

Pretriangulated dg categories are characterized formally.

Gluing preserves pretriangulatedness and reflection results are established.

Abstract

We introduce a bicategory that refines the localization of the category of dg categories with respect to quasi-equivalences and investigate its properties via formal category theory. Concretely, we first introduce the bicategory of dg categories $\mathsf{DBimod}$, whose Hom categories are given by the derived categories of dg bimodules, and then define the desired bicategory as the sub-bicategory $\mathsf{DBimod}^\text{rqr}$ consisting of right quasi-representable dg bimodules. The first half of the paper is devoted to the study of adjunctions and equivalences in these bicategories. We then show that the embedding $\mathsf{DBimod}^\text{rqr} \hookrightarrow \mathsf{DBimod}$ forms a proarrow equipment in the sense of Richard J. Wood, which provides a framework for formal category theory and enables us to define (weighted) (co)limits in an abstract setting. From this proarrow equipment, we derive the notion of homotopical (co)limits in dg categories, including homotopical shifts and cones, which in turn allows us to give a formal characterization of pretriangulated dg categories. As an application, we provide a conceptual proof of the fact that pretriangulatedness is preserved under the gluing procedure, and we establish reflection results concerning adjoints and colimits.