Kerodon

Example 7.6.2.12. Let $Y$ be an essentially small Kan complex. Suppose we are given a morphism of simplicial sets $f: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, Y)$, which we identify with a morphism $\widetilde{f}: \underline{ \Delta ^0 }_{K} \rightarrow \underline{ Y }_{K}$ in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{S}})$. Then $f$ is a weak homotopy equivalence if and only if $\widetilde{f}$ exhibits $Y$ as a tensor product 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 7.6.2.3). We may therefore assume without loss of generality that $f$ factors through the homotopy equivalence $e: \operatorname{Fun}( \Delta ^0,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}(\Delta ^0,Y)$ of Remark 5.5.1.5, in which case the desired result follows from the criterion of Example 7.6.2.11 (applied in the case $X = \Delta ^0$). Taking $K = Y$ and $f = e$, we see that every Kan complex $Y$ can be viewed as a colimit of the constant diagram $Y \rightarrow \{ \Delta ^0 \} \hookrightarrow \operatorname{\mathcal{S}}$ (see Remark 7.6.2.6).