Seifert-Van Kampen theorem: Difference between revisions
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Let <math>\mathcal{F}</math> be the set of all finite non-empty intersections of members of <math>\mathcal{U}</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|\ | <math display="block">\mathcal{F}=\left\{\left.\bigcap_{i\in J}U_i\right|\varnothing\neq J\subseteq I, |J|<\infty\right\}</math>. | ||
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>. | 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>. | ||
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Then the canonical morphism | Then the canonical morphism | ||
<math display="block">\operatorname{colim} | <math display="block">\mathop{\operatorname{colim}}\limits_{V\in\mathcal{F}}\Pi_1(V)\to \Pi_1(X)</math> | ||
is an isomorphism of groupoids. | is an isomorphism of groupoids. | ||
Latest revision as of 11:25, 10 July 2026
Statement
Let 
be a topological space and let 
be an open cover of .
Let 
be the set of all finite non-empty intersections of members of 
:
.
Regard as a category whose objects are the elements of and in which there is a unique morphism 
whenever 
.
Let 
be the functor that sends each 
to its fundamental groupoid 
and each inclusion 
to the induced morphism of groupoids.
Then the canonical morphism
is an isomorphism of groupoids.