Kerodon

Example 7.1.2.10. Let $Y$ be an essentially small Kan complex. Suppose we are given a morphism of simplicial sets $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, Y)$. Then $e$ is a weak homotopy equivalence if and only if $\widetilde{f}$ exhibits $Y$ as a copower of $\Delta ^0$ by $K$ (in the $\infty $-category $\operatorname{\mathcal{S}}$). To prove this, we are free to modify the morphism $f$ by a homotopy (see Remark 7.1.2.3). We may therefore assume without loss of generality that $f$ factors through the homotopy equivalence $f: \operatorname{Fun}( \Delta ^0,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}(\Delta ^0,Y)$ of Remark 5.5.1.5, in which case the desired result follows from the criterion of Example 7.1.2.9 (applied in the case $X = \Delta ^0$). Taking $K = Y$ and $f = e$, we see that every essentially small Kan complex $Y$ can be viewed as a colimit of the constant diagram $Y \rightarrow \{ \Delta ^0 \} \hookrightarrow \operatorname{\mathcal{S}}$ (see Remark 7.1.2.6).