Lemma 10.7 (0NR5) · ScalingStacks

Lemma 10.7. The functors

TR^{n},TC^{n},TC\colon\Cat_{\mathcal{S}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{S}

induces functors of $\infty$ -categories

TR^{n},TC^{n},TC\colon\Cat_{\infty}^{\ex}\simeq\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}((\mathcal{S})^{\mathrm{c}})[W^{-1}]\simeq{\mathcal{S}}_{\infty}.

0NR6

Proof. By [10, 3.9], maps which induce equivalences on $THH$ induce equivalences on $TR^{n}$ , $TC^{n}$ , and $TC$ . As a consequence, the result follows from lemma 10.1. ∎