Lemma 2.11 (0KEG) · ScalingStacks

Lemma 2.11. Let $d\geq-1$ and let $\mathcal{C}$ be an $\infty$ -category.

(1)

The simplicial set $h_{d}\mathcal{C}$ is a $d$ -category.
(2)

The canonical map $\mathcal{C}\to h_{d}\mathcal{C}$ is an isomorphism if and only if $\mathcal{C}$ is a $d$ -category.

(3)

For every $d$ -category $\mathcal{D}$ , composition with the canonical map $\mathcal{C}\to h_{d}\mathcal{C}$ induces an isomorphism of simplicial sets

\operatorname{Fun}\left(h_{d}\mathcal{C},\mathcal{D}\right)\overset{\sim}{\longrightarrow}\operatorname{Fun}\left(\mathcal{C},\mathcal{D}\right).

[0KEH]

Proof. For $d\geq 1$ this is the content of T.2.3.4.12. For $d=-1$ this is trivial. For $d=0$ , (1) and (2) are obvious from the definition. For (3) observe that we have a factorization of the map in question:

\operatorname{Fun}\left(h_{0}\mathcal{C},\mathcal{D}\right)\to\operatorname{Fun}\left(h_{1}\mathcal{C},\mathcal{D}\right)\overset{\sim}{\longrightarrow}\operatorname{Fun}\left(\mathcal{C},\mathcal{D}\right),

where the second map is an isomorphism (from the claim for $d=1$ ). Therefore, we can assume that $\mathcal{C}$ is an ordinary category and $\mathcal{D}$ is a poset and hence both simplicial sets are discrete. The result now follows from the observation that every functor $\mathcal{C}\to\mathcal{D}$ factors uniquely through $h_{0}\mathcal{C}$ . ∎