Example 6.2.2.7. Let $f: X \rightarrow Y$ be a map of Kan complexes, which we regard as a morphism in the $\infty $-category of spaces $\operatorname{\mathcal{S}}$. For every integer $n$, the following conditions are equivalent:
- $(1_ n)$
The morphism $f$ exhibits $Y$ as an $n$-truncation of $X$: that is, $Y$ is $n$-truncated and $f$ is $(n+1)$-connective (see Definition 3.5.7.19)
- $(2_ n)$
The morphism $f$ exhibits $Y$ as a $\operatorname{\mathcal{S}}_{\leq n}$-reflection of $X$, where $\operatorname{\mathcal{S}}_{\leq n} \subset \operatorname{\mathcal{S}}$ is the full subcategory spanned by the $n$-truncated Kan complexes.
This follows from the characterization of $n$-truncations supplied by Proposition 3.5.7.29 (together with Remark 5.5.1.5). For example, if $n \geq 0$, we can regard the fundamental $n$-groupoid $\pi _{\leq n}(X)$ as a $\operatorname{\mathcal{S}}_{\leq n}$-reflection of $X$ (Example 3.5.7.25). It follows that $\operatorname{\mathcal{S}}_{\leq n}$ is a reflective subcategory of $\operatorname{\mathcal{C}}$.