Lemma 7.3 (0NN4) · ScalingStacks

Lemma 7.3. Let ${\mathcal{C}}$ be an $\infty$ -category with finite colimits. Then for each $n$ , the forgetful functor

\Gap([n],{\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}(\Delta^{1,2,\ldots,n},{\mathcal{C}})

is an equivalence of $\infty$ -categories (and observe that $\Delta^{1,2,\ldots,n}\simeq\mathrm{N}([n-1])$ ).

0NN5

Proof. This follows from the fact that the space of colimits for a given diagram in an $\infty$ -category is contractible [52, 1.2.12.9, 1.2.13.5]. Alternatively, a constructive proof along the lines of [12, 2.9] (using a mapping cylinder argument) can be given using the comparison discussed in section 7.2 below. ∎