Kerodon

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Theorem 3.4.6.1 (Excision). Let $X$ be a topological space, and let $U,V \subseteq X$ be subsets whose interiors $\mathring {U} \subseteq U$ and $\mathring {V} \subseteq V$ comprise an open covering of $X$. Then the diagram of singular simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{Sing}_{\bullet }(U \cap V) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }(U) \ar [d] \\ \operatorname{Sing}_{\bullet }(V) \ar [r] & \operatorname{Sing}_{\bullet }(X) }$

is a homotopy pushout square (Definition 3.4.2.1).

Proof of Theorem 3.4.6.1 from Theorem 3.4.6.6. Let $X$ be a topological space and let $\operatorname{\mathcal{U}}= \{ U, V \}$ be a pair of subsets of $X$. Then $\operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X)$ can be identified with the pushout

$\operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }( U \cap V) } \operatorname{Sing}_{\bullet }(V),$

formed in the category of simplicial sets. Theorem 3.4.6.6 then asserts that if $X = \mathring {U} \cup \mathring {V}$, then the inclusion

$\operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }( U \cap V) } \operatorname{Sing}_{\bullet }(V) \hookrightarrow \operatorname{Sing}_{\bullet }(X)$

is a weak homotopy equivalence. By virtue of Remark 3.4.6.2, this is equivalent to Theorem 3.4.6.1. $\square$