Lemma 14.2 (05WH) · ScalingStacks

[05WH]

Lemma 14.2. If $C$ is a category, then $\widetilde{\discnerve C}$ is isomorphic to $NC$ . In particular, $NE\approx\widetilde{E}$ and $\widehat{E}$ are weakly equivalent to the terminal object in $s{\operatorname{\mathcal{S}}}$ .

[05WI]

Proof. The first statement is straightforward from the definitions. Since $E$ is equivalent to the terminal object in ${\operatorname{\mathcal{C}at}}$ , and $N$ takes equivalences to weak equivalences by (3.7), the second statement follows. ∎