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I am reading the paper Chern-Weil map for principal bundles over groupoids. In page number $13$, authors say

let us recall the definition of fundamental group of a topological groupoid.

But, they do not give any reference (or I could not see where they have given reference) for original paper, where the notion of fundamental group of topological/Lie groupoid is introduced.

Can some one point me to some reference where this has been mentioned for the first time?

Side question : Do we have notion of fundamental group for ``other kinds" of algebraic/differential geometrical objects? Do we have fundamental group of, say, an algebraic variety, or a scheme or similar such thing?

There is another way to introduce the notion of fundamental group of Lie groupoid (Page $187$ in LMS Lecture notes series, titled Poisson geometry, deformation quantisation and group representations, Simone Hurt, John Rawnsley and Daniel Strenheimer (eds)). But, that is different from explanation in the above mentioned paper.

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    $\begingroup$ For your side question, there is a notion of fundamental group for pointed schemes. SGA1 is an early reference, and Deligne's "the fundamental group of the projective line minus 3 points" gives a big jump in the technology (see quomodocumque.wordpress.com/2009/07/24/… for a nice short comment) $\endgroup$
    – S. Carnahan ♦
    Commented Aug 26, 2019 at 12:25
  • $\begingroup$ @S.Carnahan Yes, I have seen that... :) Thanks for the link $\endgroup$
    – Praphulla Koushik
    Commented Aug 26, 2019 at 15:49
  • $\begingroup$ The notion of fundamental group(oid) of a topological groupoid has a long and complicated history. This would probably delve back into the literature on fundamental groups of orbifolds (here's an old reference I could see from something I had open at the moment: E. Salem, Riemannian foliations and pseudogroups of isometries, Appendix D in: Riemmanian Foliations by P. Molino, Progress in Mathematics 73, Birkhäuser, 1988, 265–296. but it's surely older than this. Wikipedia says the version described on the page on orbifolds was "adopted by Haefliger and known also to Thurston") $\endgroup$
    – David Roberts ♦
    Commented Sep 23, 2019 at 4:29
  • $\begingroup$ Here's an old reference, I can't guarantee it's the original source: Haefliger, A.; Quach Ngoc Du Appendice: une présentation du groupe fondamental d'une orbifold. (French) [Appendix: presentation of the fundamental group of an orbifold] Transversal structure of foliations (Toulouse, 1982). Astérisque No. 116 (1984), 98–107. $\endgroup$
    – David Roberts ♦
    Commented Sep 23, 2019 at 4:33
  • $\begingroup$ @DavidRoberts Thank you for those references. $\endgroup$
    – Praphulla Koushik
    Commented Sep 23, 2019 at 7:07

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