Localization theory for proarrow equipments and the inclusion ι: → DBimod

Develop a general theory of localizations for proarrow equipments and prove that the inclusion pseudo-functor ι: → DBimod, which embeds the bicategory of quasi-functors into the derived bicategory of dg bimodules, is a localization of the proarrow equipment dgCat → Bimod.

Background

Section 7 establishes a universal property for the inclusion of the bicategory of quasi-functors into the derived bicategory of dg bimodules, drawing an analogy with localizations of ordinary categories. While the paper proves factorization properties reminiscent of localization, it stops short of formulating and proving a full-fledged localization theory for proarrow equipments.

The authors explicitly pose as future work to formalize such a localization notion for proarrow equipments and to demonstrate that the inclusion ι: → DBimod realizes a localization of the classical equipment dgCat → Bimod given by dg functors and dg bimodules.