Seifert-Van Kampen theorem: Difference between revisions

Latest revision as of 18:54, 14 May 2026

Statement

Let $X$ be a topological space and let $𝒰 = {U_{i}}_{i \in I}$ be an open cover of $X$ .

Let $ℱ$ be the set of all finite non-empty intersections of members of $𝒰$ : $ℱ = {⋂_{i \in J} U_{i} | \emptyset \neq J \subseteq I, | J | < \infty}$ .

Regard $ℱ$ as a category whose objects are the elements of $ℱ$ and in which there is a unique morphism $V \to W$ whenever $V \subseteq W$ .

Let $Π_{1} : ℱ \to 𝖦 𝗉 𝖽$ be the functor that sends each $V \in ℱ$ to its fundamental groupoid $Π_{1} (V)$ and each inclusion $V ↪ W$ to the induced morphism of groupoids.

Then the canonical morphism ${colim}_{V \in ℱ} Π_{1} (V) \to Π_{1} (X)$ 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|\emptyset\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">\operatorname{colim}_{V\in \mathcal{F}}\Pi_1(V)\to \Pi_1(X)</math>
+is an isomorphism of groupoids.