Proposition 4.28 (0NLF) · ScalingStacks

0NLF

Proposition 4.28. Let $I$ be a small category. Given a diagram ${\mathcal{D}}$ of small stable $\infty$ -categories indexed by $\mathrm{N}(I)$ , there exists an $I$ -diagram of pretriangulated spectral categories $\widetilde{{\mathcal{D}}}$ lifting ${\mathcal{D}}$ .

0NLG

Proof. This is a consequence of [52, 4.2.4.4]. Given a diagram of small stable $\infty$ -categories, the equivalence in theorem 4.22 gives rise to a diagram in the localization of $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ . Including the localization into $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ , we now obtain a diagram in $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\simeq\mathrm{N}((\Cat_{\mathcal{S}})^{\cf})$ and we can use [52, 4.2.4.4] to lift this to a rigid diagram in $\Cat_{\mathcal{S}}$ . ∎