Kerodon

Proof of Proposition 10.2.1.1. Let $\operatorname{\mathcal{T}}$ be the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top})$, and let $\operatorname{\mathcal{T}}_{0} \subseteq \operatorname{\mathcal{T}}$ be the full subcategory spanned by those topological spaces which have the homotopy type of a CW complex. It follows from Example 6.2.2.9 that $\operatorname{\mathcal{T}}_0$ is a coreflective subcategory of $\operatorname{\mathcal{T}}$; in particular, the inclusion map $\operatorname{\mathcal{T}}_0 \hookrightarrow \operatorname{\mathcal{T}}$ preserves colimits (Variant 7.1.4.25). It will therefore suffice to show that for every simplicial set $S$, the geometric realization $| S |$ is a colimit of the diagram $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { S } \operatorname{N}_{\bullet }( \operatorname{Set}) \subset \operatorname{\mathcal{T}}_{0}$. This is a reformulation of Variant 10.2.1.2, since the functor $X \mapsto \operatorname{Sing}_{\bullet }(X)$ determines an equivalence of $\infty $-categories $\operatorname{\mathcal{T}}_{0} \rightarrow \operatorname{\mathcal{S}}$ (Remark 5.5.1.9). $\square$