Kerodon

Warning 10.2.1.4. Let $S = S_{\bullet }$ be a simplicial set. Proposition 10.2.1.1 asserts that the topological space $| S |$ introduced in §1.2.3 is a geometric realization of $S$ (in the sense of Definition 10.2.1.3), provided that we regard $S$ as a simplicial object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top})$ (by equipping each of the sets $S_{n}$ with the discrete topology). Beware that $| S_{\bullet } |$ is usually not a geometric realization of $S$ (in the sense of Definition 10.2.1.3) if we regard $S$ as a simplicial object of the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{Set})$. The latter is a colimit of the diagram $\operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { S } \operatorname{Set}$, which identifies with the set of connected components $\pi _0( S )$ (see Remark 1.2.1.20), or equivalently with the set of path components of the topological space $|S|$ (see Corollary 1.2.3.19).