Kerodon

Variant 10.2.4.14. For every integer $n$, we let $\operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\leq n} = \operatorname{{\bf \Delta }}_{\operatorname{inj}} \cap \operatorname{{\bf \Delta }}^{\leq n}$ denote the category whose objects are linearly ordered sets $[m] = \{ 0 < 1 < \cdots < m \} $ for $0 \leq m \leq n$, and whose morphisms are strictly increasing functions. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, we say that a semisimplicial object $X_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ is $n$-coskeletal if it is right Kan extended from the full subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\leq n} )^{\operatorname{op}}$.