A formal categorical approach to the homotopy theory of dg categories

We introduce a bicategory which 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 DBimod whose Hom categories are the derived categories of dg bimodules and then define the desired bicategory as the sub-bicategory DBimodʳqr consisting only 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 DBimodʳqr DBimod is a proarrow equipment in the sense of Richard J. Wood, which is a framework for formal category theory and makes it possible to talk about (weighted) (co) limits in an abstract way. Thus we obtain the notion of homotopical (co) limits in a dg category, including homotopical shifts and cones, by which we obtain a formal categorical characterization of pretriangulated dg categories. As an immediate application we give a conceptual proof of the fact that the pretriangulatedness is preserved under the gluing procedure.