Theorem 3.6.4.1 (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.
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3.6.4 The Unit Map $u: X \rightarrow \operatorname{Sing}_{\bullet }(|X|)$
Our goal in this section is to prove the following result:
Theorem 3.6.4.1 was proved by Milnor in [MR0084138]. It is closely related to the following earlier result of Giever ([MR0033002]):
Corollary 3.6.4.2. 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 3.6.4.1 (and therefore a homotopy equivalence, since both $\operatorname{Sing}_{\bullet }(X)$ and $\operatorname{Sing}_{\bullet }( | \operatorname{Sing}_{\bullet }(X) | )$ are Kan complexes). $\square$
Corollary 3.6.4.3. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. The following conditions are equivalent:
The morphism $f$ is a weak homotopy equivalence, in the sense of Definition 3.1.6.12.
The induced map of topological spaces $|X| \rightarrow |Y|$ is a weak homotopy equivalence, in the sense of Definition 3.6.3.1.
The induced map of topological spaces $|X| \rightarrow |Y|$ is a homotopy equivalence.
Proof. We have a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d]^{u_{X}} & Y \ar [d]^{u_{Y}} \\ \operatorname{Sing}_{\bullet }( |X| ) \ar [r]^-{ \operatorname{Sing}_{\bullet }(|f| ) } & \operatorname{Sing}_{\bullet }( |Y| ), } \]
where the vertical maps are weak homotopy equivalences by virtue of Theorem 3.6.4.1. The equivalence $(1) \Leftrightarrow (2)$ now follows from Remark 3.1.6.16. The implication $(3) \Rightarrow (2)$ follows from Example 3.6.3.3, and the reverse implication is a special case of Proposition 3.6.3.8 (since the topological spaces $|X|$ and $|Y|$ are CW complexes; see Remark 1.2.3.12). $\square$
Example 3.6.4.4. A simplicial set $X$ is weakly contractible if and only if the geometric realization $|X|$ is a contractible topological space.
Proof of Theorem 3.6.4.1. Let $X$ be a simplicial set. By virtue of Remark 3.6.1.8, we can write $X$ as a filtered colimit of finite simplicial subsets $X' \subseteq X$. It follows from Proposition 3.6.1.11 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 3.2.8.3), 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 3.6.4.1 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 1.1.4.12, we can choose a pushout diagram
3.81
\begin{equation} \label{equation:unit-of-realization-1} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \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 (3.81) is also a homotopy pushout square (Proposition 3.4.2.11). 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 3.2.4.17), the unit map $u_{ \Delta ^{n} }$ is also a (weak) homotopy equivalence. Invoking Proposition 3.4.2.9, we see that $u_{X}$ is a homotopy equivalence if and only if the diagram of simplicial sets
3.82
\begin{equation} \label{equation:unit-of-realization-2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \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
3.83
\begin{equation} \label{equation:unit-of-realization-3} \begin{gathered} \xymatrix@R =50pt@C=50pt{ & \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 3.6.3.3). It follows that the upper square and left square in diagram (3.83) are homotopy pushout squares (Proposition 3.4.2.10). Moreover, the outer rectangle on the right is a homotopy pushout square by virtue of Theorem 3.4.6.1. Applying Proposition 3.4.1.11, we deduce that the lower right square and bottom rectangle are also homotopy pushout squares. $\square$