Seifert-Van Kampen theorem: Difference between revisions

Latest revision as of 11:25, 10 July 2026

Statement

Let $\text{[math]}$ be a topological space and let $\text{[math]}$ be an open cover of $\text{[math]}$ .

Let $\text{[math]}$ be the set of all finite non-empty intersections of members of $\text{[math]}$ :

$\text{[math]}$

.

Regard $\text{[math]}$ as a category whose objects are the elements of $\text{[math]}$ and in which there is a unique morphism $\text{[math]}$ whenever $\text{[math]}$ .

Let $\text{[math]}$ be the functor that sends each $\text{[math]}$ to its fundamental groupoid $\text{[math]}$ and each inclusion $\text{[math]}$ to the induced morphism of groupoids.

Then the canonical morphism

$\text{[math]}$

is an isomorphism of groupoids.

@@ Line 1: / Line 1: @@
 == Statement ==
-Let <math>X</math> be a topological space, and <math>U, V\subset X</math> be open sets such that <math>X = U\cup V</math>, and <math>U</math>, <math>V</math> and <math>U\cap V</math> are path-connected. Take a basepoint <math>x_0\in U\cap V</math> with inclusion maps:
+Let <math>X</math> be a topological space and let <math>\mathcal{U}=\{U_i\}_{i\in I}</math> be an open cover of <math>X</math>.
-<math display="block">i\colon U\cap V\hookrightarrow U,\quad j\colon U\cap V\hookrightarrow V,\quad k\colon U\hookrightarrow X,\quad l\colon V\hookrightarrow X,</math>
+Let <math>\mathcal{F}</math> be the set of all finite non-empty intersections of members of <math>\mathcal{U}</math>:
+<math display="block">\mathcal{F}=\left\{\left.\bigcap_{i\in J}U_i\right|\varnothing\neq J\subseteq I, |J|<\infty\right\}</math>.
-then the following diagram is a pushout:
+Regard <math>\mathcal{F}</math> as a category whose objects are the elements of <math>\mathcal{F}</math> and in which there is a unique morphism <math>V\to W</math> whenever <math>V\subseteq W</math>.
-<div style="display: flex; justify-content: center; gap: 40px;">
-<kroki lang="tikz">
-\documentclass[tikz]{standalone}
-\usepackage{quiver}
-\begin{document}
-\begin{tikzpicture}[baseline=(current bounding box.center)]
-\node[scale=1.5] {
-\begin{tikzcd}
-	{\pi_1(U\cap V,x_0)} & {\pi_1(U,x_0)} \\
-	{\pi_1(V,x_0)} & {\pi_1(X,x_0)}
-	\arrow["{i_*}", from=1-1, to=1-2]
-	\arrow["{j_*}"', from=1-1, to=2-1]
-	\arrow["{k_*}", from=1-2, to=2-2]
-	\arrow["{l_*}"', from=2-1, to=2-2]
-\end{tikzcd}
-  };
-\end{tikzpicture}
-\end{document}
+Let <math>\Pi_1\colon \mathcal{F}\to \mathsf{Gpd}</math> be the functor that sends each <math>V\in \mathcal{F}</math> to its fundamental groupoid <math>\Pi_1(V)</math> and each inclusion <math>V\hookrightarrow W</math> to the induced morphism of groupoids.
-</kroki>
-</div>
+Then the canonical morphism
+<math display="block">\mathop{\operatorname{colim}}\limits_{V\in\mathcal{F}}\Pi_1(V)\to \Pi_1(X)</math>
+is an isomorphism of groupoids.