Hi,
Recently I've been looking at the more general version of Van Kampen's theorem, or R. Browns version of it, for the fundamental groupoid. It mentions that if a space X is the union of the interiors of $X_1$ and $X_2$ , then:
is a pushout in the category of groupoids. In the category of groups, we have the concrete description that the pushout is just the free product with amalgation. Does something similar hold here? Is there any explicit description, like free product with amalgation?
Best Answer
I agree with the comments above: being a pushout is a categorical property. What is useful is to be able to compute explictly such pushouts and, as you say, free/amalgamated products do so in the category of groups.
In his paper Le théorème de Van Kampen (Cahiers de Topologie et Géométrie Différentielle Catégoriques, 33 no. 3 (1992), p. 237-251. Available on Numdam, http://www.numdam.org/item?id=CTGDC_1992__33_3_237_0), André Gramain gives (part of) an explicit recipe to compute the isotropy groups of a coequalizer of a pair $(\phi,\psi)$ of morphisms of groupoids. This recipe applies to your case by considering (as in van Kampen's theory) the disjoint sum of the groupoids $\pi_1(X_1,A)$ and $\pi_1(X_2,A)$ and the two morphisms from $\pi_1(X_0,A)$ to this disjoint union.
In SGA 1 (Revêtements étales et groupe fondamental, Exposé IX, §5), Grothendieck had given the same recipe for the fundamental group of schemes. However, his proof is more categorical and based on the correspondence between coverings and sets with action of the fundamental groups, and on descent theory for coverings.