Proposition 1.2.3.4. For every simplicial set $S$, there exists a topological space $Y$ and a map $u: S \rightarrow \operatorname{Sing}_{\bullet }(Y)$ which exhibits $Y$ as a geometric realization of $S$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Proposition 1.2.3.4. Let $S = S_{\bullet }$ be a simplicial set; we wish to show that $S$ admits a geometric realization $|S|$. We first show that for each $n \geq -1$, the $n$-skeleton $\operatorname{sk}_{n}( S )$ admits a geometric realization. The proof proceeds by induction on $n$, the case $n = -1$ being trivial (since $\operatorname{sk}_{-1}( S )$ is empty). Let $C$ denote the collection of nondegenerate $n$-simplices of $C$. we note that Proposition 1.1.4.12 provides a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ \underset { \sigma \in C }{\coprod } \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \underset { \sigma \in C }{\coprod } \Delta ^{n} \ar [d] \\ \operatorname{sk}_{n-1}( S) \ar [r] & \operatorname{sk}_{n}( S). } \]
Combining our inductive hypothesis, Example 1.2.3.2, Example 1.2.3.11, and Lemma 1.2.3.6, we deduce that $\operatorname{sk}_{n}( S )$ admits a geometric realization $| \operatorname{sk}_{n} (S ) |$ which fits into a pushout diagram of topological spaces
\[ \xymatrix@R =50pt@C=50pt{ \underset { \sigma \in C }{\coprod } | \operatorname{\partial \Delta }^{n} | \ar [r] \ar [d] & \underset { \sigma \in C }{\coprod } |\Delta ^{n}| \ar [d] \\ |\operatorname{sk}_{n-1}( S )| \ar [r] & | \operatorname{sk}_{n}( S )|. } \]
Combining the equality $S = \bigcup _{n} \operatorname{sk}_{n} (S)$ of Remark 1.1.4.4 with Lemma 1.2.3.6, we deduce that the simplicial set $S$ also admits a geometric realization, given by the direct limit $\varinjlim _{n} | \operatorname{sk}_{n}( S ) |$. $\square$