Kerodon

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

Theorem 3.4.7.2 (Seifert-van Kampen, Groupoid Version). 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 fundamental groupoids

\[ \xymatrix { \pi _{\leq 1}(U \cap V) \ar [r] \ar [d] & \pi _{\leq 1}(U) \ar [d] \\ \pi _{\leq 1}(V) \ar [r] & \pi _{\leq 1}(X) } \]

is a pushout square in the (ordinary) category $\operatorname{Cat}$.

Proof. Let $\operatorname{\mathcal{C}}$ be a category; we wish to show that the diagram of sets $\sigma :$

\[ \xymatrix { \operatorname{Hom}_{\operatorname{Cat}}( \pi _{\leq 1}(U \cap V), \operatorname{\mathcal{C}}) & \operatorname{Hom}_{\operatorname{Cat}}( \pi _{\leq 1}(U), \operatorname{\mathcal{C}}) \ar [l] \\ \operatorname{Hom}_{\operatorname{Cat}}( \pi _{\leq 1}(V), \operatorname{\mathcal{C}}) \ar [u] & \operatorname{Hom}_{\operatorname{Cat}}( \pi _{\leq 1}(X), \operatorname{\mathcal{C}}) \ar [u] \ar [l] } \]

is a pullback square. Replacing $\operatorname{\mathcal{C}}$ by its core $\operatorname{\mathcal{C}}^{\simeq }$ (Construction 1.2.4.4), we may assume without loss of generality that $\operatorname{\mathcal{C}}$ is a groupoid. Let $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ denote the nerve of $\operatorname{\mathcal{C}}$, so that we can identify $\sigma $ with the diagram

\[ \xymatrix { \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }(U \cap V), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }(U), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \ar [l] \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }(V), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \ar [u] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }(X), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ). \ar [u] \ar [l] } \]

Let $K$ denote the pushout $\operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }(U \cap V) } \operatorname{Sing}_{\bullet }(V)$, which we regard as a simplicial subset of $\operatorname{Sing}_{\bullet }(X)$. Unwinding the definitions, we must show that every morphism of simplicial sets $f: K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ extends uniquely to a map $\overline{f}: \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Note that the inclusion $K \hookrightarrow \operatorname{Sing}_{\bullet }(X)$ is a weak homotopy equivalence (Theorem 3.4.6.1) and therefore anodyne (Corollary 3.3.7.5), so the existence of $\overline{f}$ follows from the observation that $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a Kan complex (Proposition 1.2.4.2). To prove uniqueness, suppose that we are given a pair of maps $\overline{f}, \overline{f}': \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ satisfying $\overline{f}|_{K} = f = \overline{f}'|_{K}$. It follows that there exists a homotopy $h: \Delta ^1 \times \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which is constant when restricted to $\Delta ^1 \times K$. Note that $\overline{f}$ and $\overline{f}'$ can be identified with functors $F,F': \pi _{\leq 1}(X) \rightarrow \operatorname{\mathcal{C}}$, and $h$ with a natural transformation of functors $H: F \rightarrow F'$. Since every vertex of $\operatorname{Sing}_{\bullet }(X)$ is contained in $K$, this natural transformation carries each point $x \in X$ to the identity morphism $\operatorname{id}_{ \overline{f}(x) }: F(x) \rightarrow F(x) = F'(x)$. It follows that the functors $F$ and $F'$ are identical, so that the morphisms $\overline{f}$ and $\overline{f}'$ are the same. $\square$