Theorem (Milnor). Let $X$ be a simplicial set. Then the unit map $u_{X}: X \rightarrow \operatorname{Sing}_{\bullet }(|X|)$ is a weak homotopy equivalence of simplicial sets.

Proof of Theorem Let $X$ be a simplicial set. By virtue of Remark, we can write $X$ as a filtered colimit of finite simplicial subsets $X' \subseteq X$. It follows from Proposition that, for any compact topological space $K$, every continuous function $f: K \rightarrow |X|$ factors through $|X'| \subseteq |X|$ for some finite simplicial subset $X' \subseteq X$. Applying this observation in the case $K = | \Delta ^{n} |$, we conclude that the natural map $\varinjlim _{X' \subseteq X} \operatorname{Sing}_{\bullet }(|X'|) \rightarrow \operatorname{Sing}_{\bullet }(|X|)$ is an isomorphism of simplicial sets. It follows that the unit map $u_{X}: X \rightarrow \operatorname{Sing}_{\bullet }( |X| )$ can be realized as filtered colimit of unit maps $u_{X'}: X' \rightarrow \operatorname{Sing}_{\bullet }( |X'| )$. Since the collection of weak homotopy equivalences is closed under filtered colimits (Proposition, it will suffice to show that each of the morphisms $u_{X'}$ is a weak homotopy equivalence. Replacing $X$ by $X'$, we are reduced to proving Theorem under the additional assumption that the simplicial set $X$ is finite.

We now proceed by induction on the dimension of $X$. If $X$ is empty, then $u_{X}$ is an isomorphism and the result is obvious. Otherwise, let $n \geq 0$ be the dimension of $X$. We proceed by induction on the number of nondegenerate $n$-simplices of $X$. Using Proposition, we can choose a pushout diagram

\begin{equation} \label{equation:unit-of-realization-1} \begin{gathered} \xymatrix { \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \Delta ^ n \ar [d] \\ X' \ar [r] & X, }\end{gathered} \end{equation}

where $X'$ is a simplicial subset of $X$ with a smaller number of nondegenerate $n$-simplices. Since the inclusion $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^ n$ is a monomorphism, the diagram (3.65) is also a homotopy pushout square (Proposition By virtue of our inductive hypotheses, the unit morphisms $u_{X'}$ and $u_{ \operatorname{\partial \Delta }^{n} }$ are weak homotopy equivalences. Since the simplicial sets $\Delta ^{n}$ and $\operatorname{Sing}_{\bullet }( | \Delta ^{n} | )$ are contractible (Remark, the unit map $u_{ \Delta ^{n} }$ is also a (weak) homotopy equivalence. Invoking Proposition, we see that $u_{X}$ is a homotopy equivalence if and only if the diagram of simplicial sets

\begin{equation} \label{equation:unit-of-realization-2} \begin{gathered} \xymatrix { \operatorname{Sing}_{\bullet }(|\operatorname{\partial \Delta }^{n}|) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }(|\Delta ^ n|) \ar [d] \\ \operatorname{Sing}_{\bullet }(|X'|) \ar [r] & \operatorname{Sing}_{\bullet }(|X|), }\end{gathered} \end{equation}

is also homotopy pushout square.

Let $V = | \Delta ^ n | \setminus | \operatorname{\partial \Delta }^ n |$ be the interior of the topological $n$-simplex, and fix a point $v \in V$ having image $x \in |X|$. We then have a commutative diagram of simplicial sets

\begin{equation} \label{equation:unit-of-realization-3} \begin{gathered} \xymatrix { & \operatorname{Sing}_{\bullet }( V \setminus \{ v\} ) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }( V ) \ar [d] \\ \operatorname{Sing}_{\bullet }(|\operatorname{\partial \Delta }^{n}|) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }( | \Delta ^{n} | \setminus \{ v \} ) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }(|\Delta ^ n|) \ar [d] \\ \operatorname{Sing}_{\bullet }(|X'|) \ar [r] & \operatorname{Sing}_{\bullet }( |X| \setminus \{ x\} ) \ar [r] & \operatorname{Sing}_{\bullet }(|X|). }\end{gathered} \end{equation}

Note that the left horizontal maps and the upper vertical maps are homotopy equivalences, since they are obtained from homotopy equivalences of topological spaces

\[ |X'| \hookrightarrow |X| \setminus \{ x\} \quad \quad | \operatorname{\partial \Delta }^{n} | \hookrightarrow | \Delta ^{n} | \setminus \{ v\} \hookleftarrow V \setminus \{ v\} \quad \quad | \Delta ^ n | \hookleftarrow V \]

(see Example It follows that the upper square and left square in diagram (3.67) are homotopy coCartesian (Proposition Moreover, the outer rectangle on the right is homotopy coCartesian by virtue of Theorem Applying Proposition, we deduce that the lower right square and bottom rectangle are also homotopy coCartesian. $\square$