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Grothendieck in his 1984 "Esquisse d'un programme" (Section 2) wrote (English translation):

" ..,people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups `a la van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points,.."

Question: Does anyone have any reference to relevant work on "descent theorems for fundamental groups"?

Relevant to this question is this mathoverflow discussion on several base points.

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    $\begingroup$ I think he is refering to all the statements that boils down to the fact that foundamental groupoids commute to homotopy colimit. Van kampen is the more simple case and which become a lot more powerfull when you allow several based point. $\endgroup$
    – Simon Henry
    Commented Oct 10, 2015 at 11:46

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I think I can now reasonably point to the book

Bourbaki, Topologie Algébrique: Chapitres 1 à 4 (Elements de Mathématique) 7 Apr 2016

which does use ideas of descent and the fundamental groupoid both for the van Kampen theorem and for discrete groups acting properly on spaces. They call the fundamental groupoid the "Poincaré groupoid", though I think its topological use goes back only to Reidemeister's 1932 book. However, despite the comment of Grothendieck, they do not use the fundamental groupoid on a set of base points.

Their Theorem 3 on p. 419 seems to overlap with the work on orbit groupoids of Chapter 11 of Topology and Groupoids, which gives the application to the symmetric square of a space.

Comments welcome!

Aug 21, 2016 I should mention the paper

"Van Kampen theorems for categories of covering morphisms in lextensive categories" R. Brown, G. Janelidze, J. Pure Applied Algebra I 19 (1997) 255-283. (pdf)

This considers the whole fundamental groupoid, not "many base points", but does use descent notions in general situations.

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