Kerodon

Proof of Proposition 4.8.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. For every integer $m \geq 0$, we let $\theta _{m}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, \operatorname{\mathcal{C}})$ denote the restriction map. By virtue of Corollary 4.8.3.11, the map $\theta _{m}$ is surjective for $m \geq n+2$ if and only if $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated (and nonempty if $n \leq -2$). Assume that these equivalent conditions are satisfied. Then:

$(1)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is $(n+1)$-coskeletal if and only if $\theta _{m}$ is injective for $m \geq n+2$. By virtue of Proposition 4.8.3.12 (and Remark 4.8.3.13), this is equivalent to the requirement that $\operatorname{\mathcal{C}}$ is minimal in dimensions $\geq n+2$.

$(2)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is weakly $n$-coskeletal if and only if $\theta _{m}$ is injective for $m \geq n+1$. By virtue of Proposition 4.8.3.12 (and Remark 4.8.3.13), this is equivalent to the requirement that $\operatorname{\mathcal{C}}$ is minimal in dimensions $\geq n+1$.

$(3)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if and only if $\operatorname{\mathcal{C}}$ is minimal in dimensions $\geq n$. This follows immediately from $(2)$ (see Definition 4.8.1.8).