In case \(\mathbb{K}= Hk\) for a field \(k\) of characteristic zero, it follows from [Coh16] that \(\mathrm{st}_k\coloneqq \mathrm{st}_{Hk}\) is the localization of the ordinary \(1\)-category \(\mathrm{dgCat}^{\mathrm{idem}, \mathrm{pretriang}}_k\) of small idempotent-complete pretriangulated dg-categories at the quasi-equivalences, i.e. those dg-functors which induces triangulated equivalences on homotopy categories. Hence, the reader may consider \(\mathrm{st}_k\) as our \(\infty\)-categorical stand-in for the theory of dg-categories. In practice, the localization functor \(\mathrm{dgCat}^{\mathrm{idem}, \mathrm{pretriang}}_k \rightarrow\mathrm{st}_k\) provides an easy way to construct objects and morphisms of \(\mathrm{st}_k\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2