TITLE: The fundamental groupoid and a pushout in the category of groupoids.
QUESTION [8 upvotes]: 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?

REPLY [4 votes]: I think the relevant formula is 8.4.1 in T&G. This is applied in section 9.2 to the Phragmen-Brouwer property and the Jordan Curve Theorem.
My original motivation for the investigation was to avoid a detour to compute the fundamental group of the circle: a basic theorem should compute THE basic example! I like the view of the integers (an infinite set)  as an identification of a groupoid $\mathbf I $ with 4 arrows, identifying 0 and 1.
Also I tend to see covering spaces in terms of covering morphisms of groupoids, since then a covering map is algebraically modelled by a covering morphism, whereas an action is one step further.
In the new book `Nonabelian algebraic topology', published by the EMS, the van Kampen style arguments are used to compute relative homotopy groups as modules, and second relative homotopy groups as crossed modules, using colimit calculations.
January 25,2016 There is a small correction to the Phragmen-Brouwer proof.
October29, 2020
It may be useful to think about forming the coequaliser of two morphisms $a,b: G \to H$ of groupoids. If $a,b$ are the identity on objects, the formula is just as you would expect: factorise by the normal subgroupoid of $H$  generated by ....If they are not the identity on objects, you first have to coequalise the objects. This gives a function $f: Ob(G) \to Ob(H)$. So you now need Higgins' "Universal morphism" say $U_f : G \to f_*(G)$. The construction of this generalises free groups, free products of groups, free groupoids ... and is well explained in Higgins' book. "Categories and Groupoids" available as a TAC reprint.
Nov 3, 2020  In the NAT book, Appendix B, this is put in the context of the functor $Ob: Gpds \to Sets$ being a bifibration of categories.