Kerodon

Example 9.2.4.5. Fix an integer $n \geq 0$. Let $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ be the $\infty $-category of spaces and let $W$ be the collection of morphisms of $\operatorname{\mathcal{C}}$ given by the inclusion maps $\{ \operatorname{Ex}^{\infty }( \operatorname{\partial \Delta }^{m} ) \hookrightarrow \operatorname{Ex}^{\infty }( \Delta ^ m ) \} _{0 \leq m \leq n}$. Then an object $X \in \operatorname{\mathcal{C}}$ weakly $W$-local if and only if it is $n$-connective (see Definition 3.5.1.1). In this case, Theorem 9.2.4.3 asserts that every Kan complex $X$ admits a morphism $f: X \rightarrow Y$, where $Y$ is an $n$-connective Kan complex which can be obtained from $X$ by attaching cells of dimension $\leq n$. Beware that, if $n > 0$, then $Y$ cannot be chosen to depend functorially on $X$.