ScalingStacks

Remark 2.13. A small stable $\infty$-category corresponds to the notion of a pretriangulated spectral category, and the weak equivalences are given by exact functors which induce triangulated equivalences on passage to the homotopy category. We will make this correspondence precise in section 4, but for now observe that given a pretriangulated spectral category ${\mathcal{C}}$, the $\infty$-category $\mathrm{N}((\mathrm{Mod}({\mathcal{C}}))^{\cf})$ is stable. Recall that a stable model category is a pointed model category ${\mathcal{C}}$ for which the functors $\Sigma$ and $\Omega$ on $\Ho({\mathcal{C}})$ are inverse equivalences. Given a stable simplicial model category ${\mathcal{C}}$, the $\infty$-category $\mathrm{N}({\mathcal{C}}^{\cf})$ is stable. More generally, if ${\mathcal{C}}$ is a stable model category, $\mathrm{N}({\mathcal{C}}^{\mathrm{c}})[W^{-1}]$ is a stable $\infty$-category.