Variant 10.2.1.2 (04QS)

Variant 10.2.1.2. Let $S = S_{\bullet }$ be a simplicial set. Then the Kan complex $\operatorname{Sing}_{\bullet }( | S | )$ is a colimit of the diagram

\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { S } \operatorname{N}_{\bullet }( \operatorname{Set}) \subset \operatorname{\mathcal{S}}. \]

Proof of Variant 10.2.1.2. Let $\operatorname{Kan}$ denote the ordinary category of Kan complexes, and let $\mathscr {F}: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Kan}$ be the functor which carries each object $[n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the set of $n$-simplices $S_{n}$ (regarded as a constant simplicial set). Let $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ denote the homotopy colimit of the diagram $\mathscr {F}$ (Construction 5.3.2.1). By virtue of Proposition 7.5.7.1 (and Example 1.4.2.5), it will suffice to show that there is a weak homotopy equivalence of simplicial sets $v: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )^{\operatorname{op}} \rightarrow \operatorname{Sing}_{\bullet }( | S | )$. Using Example 5.3.2.5 (and Example 5.2.6.4), we can identify $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )^{\operatorname{op}}$ with the nerve of the category of simplices $\operatorname{{\bf \Delta }}_{S}$ (see Construction 1.1.3.9). We complete the proof by taking $v$ to be the composition

\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \xrightarrow {\psi } S \xrightarrow {u} \operatorname{Sing}_{\bullet }( |S| ), \]

where $u$ is the weak homotopy equivalence of Theorem 3.6.4.1 and $\psi : \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \rightarrow S$ is the comparison map of Construction 3.3.3.10. By virtue of Variant 6.3.7.4, the morphism $\psi $ is universally localizing, and is therefore also a weak homotopy equivalence (Remark 6.3.6.5). $\square$