Seifert-Van Kampen theorem

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.