Proposition 4.6 (0NKE) · ScalingStacks

0NKE

Proposition 4.6. The $\infty$ -category obtained by applying the simplicial nerve to (a fibrant replacement of) $\Cat_{\mathcal{T}}^{\ex}$ is equivalent to the $\infty$ -category $\Cat_{\infty}^{\ex}$ . That is, the equivalence (induced by the simplicial nerve [52, 2.2.0.1])

\mathrm{N}((L^{H}(\Cat_{\mathcal{T}})^{\mathrm{fib}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}

restricts to an equivalence

\mathrm{N}((\Cat_{\mathcal{T}}^{\ex})^{\mathrm{fib}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\ex}.

0NKF

Proof. It suffices to show that the mapping spaces in $\Cat_{\mathcal{T}}^{\ex}$ have the correct homotopy type, and this follows from the comparison between the mapping spaces of $\Cat_{\mathcal{T}}$ and $\Cat_{\infty}$ [52, 2.2.0.1] and the fact that on both sides we define the mapping spaces by the same restriction of vertices. ∎