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

A differential graded category (or dg category) is an enriched category over the symmetric monoidal closed category of cochain complexes of modules (over a field, for simplicity). It is widely used in algebraic geometry and representation theory as an enhancement of triangulated categories ([1, 6]), providing a richer underlying structure. The homotopy theory of complexes up to quasi-isomorphism induces a natural homotopical structure on dg categories ([5]), whose weak equivalences are called quasi-equivalences. Accordingly, one can construct the localization $\mathrm{𝖧𝗈}(\mathrm{𝖽𝗀𝖢𝖺𝗍})$ of the category of dg categories with respect to quasi-equivalences.

In this talk, we will present an approach to the homotopy theory of dg categories from the perspective of formal category theory. We introduce a bicategory that serves as a natural refinement of $\mathrm{𝖧𝗈}(\mathrm{𝖽𝗀𝖢𝖺𝗍})$, and show that this bicategory carries the structure of a proarrow equipment in the sense of Richard J. Wood [3, 4]. Proarrow equipments provide a general framework for formal category theory and allow one to define notions of (co)limits in an abstract setting. Applying this framework to our proarrow equipment, we derive a notion of homotopical (co)limits in dg categories that is respected by quasi-equivalences. We show that these homotopical limits include homotopical shifts and cones, yielding a formal characterization of pretriangulated dg categories. As an application, we also establish reflection results concerning adjoints and colimits.

This talk is based on the paper [2].