Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Theorem 3.4.6.6. Let $X$ be a topological space and let $\operatorname{\mathcal{U}}$ be a collection of subsets of $X$ satisfying $X = \bigcup _{U \in \operatorname{\mathcal{U}}} \mathring {U}$. Then the inclusion map $\operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \hookrightarrow \operatorname{Sing}_{\bullet }(X)$ is a weak homotopy equivalence.

Proof of Theorem 3.4.6.6. Let $X$ be a topological space and let $\operatorname{\mathcal{U}}$ be a collection of subsets of $X$ satisfying $X = \bigcup _{U \in \operatorname{\mathcal{U}}} \mathring {U}$. For each $k \geq 0$, let $Y(k) \subseteq \operatorname{Sing}_{\bullet }(X)$ denote the semisimplicial subset spanned by those singular $n$-simplices $f: | \Delta ^{n} | \rightarrow X$ having the property that, for every $m$-simplex $\sigma $ of the iterated subdivision $\operatorname{Sd}^{k}( \Delta ^ n )$, the composite map

\[ | \Delta ^{m} | \xrightarrow { | \sigma |} | \operatorname{Sd}^{k}( \Delta ^{n} ) | \simeq | \Delta ^{n} | \xrightarrow {f} X \]

is $\operatorname{\mathcal{U}}$-small; here the identification $| \operatorname{Sd}^{k}( \Delta ^ n ) | \simeq | \Delta ^ n |$ is given by iteratively applying the barycentric subdivision of Proposition 3.3.2.3. By construction, we have inclusions of semisimplicial sets

\[ \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) = Y(0) \subseteq Y(1) \subseteq Y(2) \subseteq \cdots \subseteq \operatorname{Sing}_{\bullet }(X). \]

We first claim that $\operatorname{Sing}_{\bullet }(X) = \bigcup _{k \geq 0} Y(k)$. Fix a continuous function $f: | \Delta ^{n} | \rightarrow X$, regarded as an $n$-simplex of $\operatorname{Sing}_{\bullet }(X)$; we wish to show that $f$ belongs to $Y(k)$ for $k \gg 0$. Let us identify the topological $n$-simplex $| \Delta ^{n} |$ with the subset of Euclidean space $V = \operatorname{\mathbf{R}}^{n+1}$ given by the convex hull of the standard basis vectors $\{ v_ i \} _{0 \leq i \leq n}$. Then the collection of inverse images $\{ f^{-1}(U) \} _{U \in \operatorname{\mathcal{U}}}$ can be refined to an open covering of $| \Delta ^{n} |$. It follows that there exists a positive real number $\epsilon $ with the property that, for every point $v \in | \Delta ^{n} |$, the open ball

\[ B_{\epsilon }(v) = \{ w \in | \Delta ^{n} |: | v-w|_{V} < \epsilon \} \]

is contained in $f^{-1}(U)$, for some $U \in \operatorname{\mathcal{U}}$. Choose an integer $k$ satisfying $( \frac{n}{n+1} )^{k} \mathrm{diam}( | \Delta ^{n} | ) < \epsilon $. It then follows from iterated application of Lemma 3.4.6.7 that the composite map

\[ | \operatorname{Sd}^{k}( \Delta ^{n} ) | \simeq | \Delta ^{n} | \xrightarrow {f} X \]

carries each simplex of $\operatorname{Sd}^{k}( \Delta ^ n )$ into a subset $U \subseteq X$ belonging to $\operatorname{\mathcal{U}}$, so that $f$ belongs to the semisimplicial subset $Y(k) \subseteq \operatorname{Sing}_{\bullet }(X)$.

Note that the inclusion $\iota : \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \hookrightarrow \operatorname{Sing}_{\bullet }(X)$ is a weak homotopy equivalence of simplicial sets if and only if it is a weak homotopy equivalence when regarded as a morphism of semisimplicial sets (Corollary 3.4.5.5). It follows from the preceding argument that, as a morphism of semisimplicial sets, $\iota $ can be realized as a filtered colimit of the inclusion maps $\iota (k): \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) = Y(0) \hookrightarrow Y(k)$. Since the collection of weak homotopy equivalences is closed under filtered colimits (Remark 3.4.5.2), it will suffice to show that each $\iota (k)$ is a weak homotopy equivalence. Proceeding by induction on $k$, we are reduced to showing that each of the inclusion maps $Y(k) \hookrightarrow Y(k+1)$ is a weak homotopy equivalence. Note that the semisimplicial isomorphism $\varphi : \operatorname{Sing}_{\bullet }(X) \simeq \operatorname{Ex}( \operatorname{Sing}_{\bullet }(X) )$ of Example 3.3.2.9 restricts to a map $\varphi ^{\operatorname{\mathcal{U}}}: \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \rightarrow \operatorname{Ex}( \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) )$ (which is generally not an isomorphism). Unwinding the definitions, we see that the inclusion $Y(k) \hookrightarrow Y(k+1)$ can be identified with the map $\operatorname{Ex}^{k}( \varphi ^{\operatorname{\mathcal{U}}} ): \operatorname{Ex}^{k}( \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) ) \rightarrow \operatorname{Ex}^{k+1}( \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) )$ (see Variant 3.3.2.10). By virtue of Corollary 3.4.5.8, it will suffice to show that $\varphi ^{\operatorname{\mathcal{U}}}$ is a weak homotopy equivalence.

Fix an integer $n \geq 0$ as above, let $\operatorname{Chain}[n]$ denote the collection of all nonempty subsets of $[n] = \{ 0 < 1 < \cdots < n \} $. Let $\sigma $ be an $n$-simplex of the simplicial set $\Delta ^{1} \times \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X)$, which we identify with a pair $(\epsilon , f)$ where $\epsilon : [n] \rightarrow [1]$ is a nondecreasing function and $f: | \Delta ^{n} | \rightarrow X$ is a continuous map of topological spaces. Define a map of sets $g_{\epsilon }: \operatorname{Chain}[n] \rightarrow | \Delta ^{n} |$ by the formula

\[ g_{\epsilon }(S) = \begin{cases} \frac{ \sum _{i \in S} v_{i} }{|S|} & \text{ if } \epsilon |_{S} = 0 \\ v_{ \mathrm{Max}(S)} & \text{ otherwise. } \end{cases} \]

Then $g_{\epsilon }$ extends to a continuous map

\[ \overline{g}_{\epsilon }: | \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \rightarrow | \Delta ^ n | \]

which is affine when restricted to each simplex of $| \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \simeq | \operatorname{Sd}(\Delta ^ n) |$. The composite map

\[ | \operatorname{Sd}( \Delta ^ n) | \xrightarrow { \overline{g}_{\epsilon } } | \Delta ^ n | \xrightarrow {f} X \]

can be identified with an $n$-simplex of $\operatorname{Ex}( \operatorname{Sing}^{\operatorname{\mathcal{U}}}_{\bullet }(X) )$, which we will denote by $h( \sigma )$. It is not difficult to see that the construction $\sigma \mapsto h(\sigma )$ is compatible with face operators, and therefore determines a morphism of semisimplicial sets $h: \Delta ^{1} \times \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \rightarrow \operatorname{Ex}( \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) )$. By construction, this morphism fits into a commutative diagram of semisimplicial sets

\[ \xymatrix@C =50pt@R=50pt{ \Delta ^{1} \times \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \ar [dr]^{h} & \{ 0\} \times \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \ar [d]^{ \varphi ^{\operatorname{\mathcal{U}}} } \ar [l]_-{i_0} \\ \{ 1\} \times \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \ar [u]^{i_1} \ar [r]^-{ \rho } & \operatorname{Ex}( \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) ), } \]

where $i_0$ and $i_1$ are the inclusion maps and $\rho = \rho _{ \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) }$ is the comparison map of Construction 3.3.4.3. Note that the morphisms $i_0$, $i_1$, and $\rho $ are weak homotopy equivalences of simplicial sets (Theorem 3.3.5.1), and therefore also weak homotopy equivalences of semisimplicial sets (Corollary 3.4.5.5). Invoking the two-out-of-three property (Remark 3.4.5.3), we conclude that $h$ and $\varphi ^{\operatorname{\mathcal{U}}}$ are also weak homotopy equivalences of semisimplicial sets. $\square$