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3.5.4 The Unit Map $u: X \rightarrow \operatorname{Sing}_{\bullet }(|X|)$

Our goal in this section is to prove the following result:

Theorem 3.5.4.1 (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.

Theorem 3.5.4.1 was proved by Milnor in [MR0084138]. It is closely related to the following earlier result of Giever ([MR0033002]):

Corollary 3.5.4.2. Let $X$ be a topological space. Then the counit map $v_{X}: | \operatorname{Sing}_{\bullet }(X) | \rightarrow X$ is a weak homotopy equivalence of topological spaces.

Proof. We must show that $\operatorname{Sing}_{\bullet }(v_ X): \operatorname{Sing}_{\bullet }(| \operatorname{Sing}_{\bullet }(X) | ) \rightarrow \operatorname{Sing}_{\bullet }(X)$ is a homotopy equivalence of Kan complexes. This is clear, since $\operatorname{Sing}_{\bullet }(v_ X)$ is left inverse to the unit map $u_{ \operatorname{Sing}_{\bullet }(X)}: \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(| \operatorname{Sing}_{\bullet }(X) | )$, which is a weak homotopy equivalence by virtue of Theorem 3.5.4.1 (and therefore a homotopy equivalence, since both $\operatorname{Sing}_{\bullet }(X)$ and $\operatorname{Sing}_{\bullet }( | \operatorname{Sing}_{\bullet }(X) | )$ are Kan complexes). $\square$

Proof of Theorem 3.5.4.1. Let $X$ be a simplicial set. By virtue of Remark 3.5.1.8, we can write $X$ as a filtered colimit of finite simplicial subsets $X' \subseteq X$. It follows from Proposition 3.5.1.9 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 3.2.7.3), 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 3.5.4.1 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 1.1.3.13, we can choose a pushout diagram

3.65
\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 3.4.2.6). 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 3.2.6.5), the unit map $u_{ \Delta ^{n} }$ is also a (weak) homotopy equivalence. Invoking Proposition 3.4.2.4, we see that $u_{X}$ is a homotopy equivalence if and only if the diagram of simplicial sets

3.66
\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

3.67
\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 3.5.3.3). It follows that the upper square and left square in diagram (3.67) are homotopy coCartesian (Proposition 3.4.2.5). Moreover, the outer rectangle on the right is homotopy coCartesian by virtue of Theorem 3.4.6.1. Applying Proposition 3.4.1.9, we deduce that the lower right square and bottom rectangle are also homotopy coCartesian. $\square$