Theorem 3.4.7.1 (Seifert-van Kampen). Let $X$ be a topological space containing a pair of subsets $U,V \subseteq X$ which satisfy the following conditions:
- $(1)$
The topological spaces $U$, $V$, and $U \cap V$ are path connected.
- $(2)$
The interiors $\mathring {U} \subseteq U$ and $\mathring {V} \subseteq V$ comprise an open covering of $X$.
Then, for every point $x \in U \cap V$, the diagram
\[ \xymatrix@R =50pt@C=50pt{ \pi _{1}( U \cap V, x) \ar [r] \ar [d] & \pi _{1}(U,x) \ar [d] \\ \pi _{1}(V,x) \ar [r] & \pi _{1}(X,x) } \]
is a pushout square in the category of groups.
Proof of Theorem 3.4.7.1.
For every group $G$, let us write $BG$ for the groupoid having a single object with automorphism group $G$ (Remark 1.3.2.4). Fix a point $x \in U \cap V$. To show that the diagram
\[ \xymatrix@R =50pt@C=50pt{ \pi _{1}( U \cap V, x) \ar [r] \ar [d] & \pi _{1}(U,x) \ar [d] \\ \pi _{1}(V,x) \ar [r] & \pi _{1}(X,x) } \]
is a pushout square in the category of groups, it will suffice to show that the diagram $\sigma _0$:
\[ \xymatrix@R =50pt@C=50pt{ B \pi _{1}( U \cap V, x) \ar [r] \ar [d] & B \pi _{1}(U,x) \ar [d] \\ B\pi _{1}(V,x) \ar [r] & B \pi _{1}(X,x) } \]
is a pushout square in the (ordinary) category $\operatorname{Cat}$.
For each point $y \in X$, choose a continuous path $p_{y}: [0,1] \rightarrow X$ satisfying $p(0) = x$ and $p(1) = y$. By virtue of our assumption that $U$, $V$, and $U \cap V$ are path connected, we can arrange that these paths satisfy the following requirements:
If $y = x$, then $p_ y: [0,1] \rightarrow X$ is the constant map taking the value $x$.
If $y$ is contained in the intersection $U \cap V$, then the path $p_{y}$ factors through $U \cap V$.
If $y$ is contained in $U$, then the path $p_{y}$ factors through $U$.
If $y$ is contained in $V$, then the path $p_{y}$ factors through $V$.
Note that, for $W \in \{ X, U, V, U \cap V \} $, we can identify $B \pi _{1}(W,x)$ with the full subcategory of $\pi _{\leq 1}(W)$ spanned by the point $x$. Let $r_{W}: \pi _{\leq 1}(W) \rightarrow B \pi _{1}(W,x)$ be the functor which carries each point of $W$ to the point $x$, and each morphism $\alpha \in \operatorname{Hom}_{\pi _{\leq 1}(W)}( y, z )$ to the composition $[p_{z}]^{-1} \circ \alpha \circ [p_{y}]$ (where $[p_ y]$ and $[p_ z]$ denote the homotopy classes of the paths $p_ y$ and $p_ z$, regarded as morphisms in the fundamental groupoid $\pi _{\leq 1}(W)$). The functors $r_{W}$ restrict to the identity on $B \pi _{1}(W,x)$ and are compatible as $W$ varies, and therefore exhibit $\sigma _0$ as a retract of the diagram $\sigma :$
\[ \xymatrix@R =50pt@C=50pt{ \pi _{\leq 1}(U \cap V) \ar [r] \ar [d] & \pi _{\leq 1}(U) \ar [d] \\ \pi _{\leq 1}(V) \ar [r] & \pi _{\leq 1}(X) } \]
in the category $\operatorname{Fun}( [1] \times [1], \operatorname{Cat})$. Since $\sigma $ is a pushout square (by virtue of Theorem 3.4.7.2), it follows that $\sigma _0$ is also a pushout square.
$\square$