TITLE: Computing `$\pi_1 S^1$` using groupoids
QUESTION [9 upvotes]: I believe it is possible to compute $\pi_1 S^1$ by applying the groupoid version of the Seifert-Van Kampen Theorem (in the version presented in May's Concise Course) to a covering of the circle by three arcs.  Is there an account like this somewhere in the literature?  Ideally I'd like a discussion that a student familiar with May's book would be able to read.  (May doesn't take a 2-categorical approach to groupoids, and so he does not discuss the fact that a diagram of groupoids that is a point-wise equivalence induces an equivalence of colimits.  This is rather important for computations.)
Edit: this last statement is false in general!  I was thinking of homotopy colimits.  The relevant (correct) fact appears in Ronnie Brown's book: retracts of pushouts are pushouts.  This is the means by which one compares the Van Kampen theorem for the full fundamental groupoid - as in May's book - with the Van Kampen theorem for the fundamental groupoid on a set of basepoints.)

REPLY [8 votes]: A rather belated comment on these! I like the comparison between the circle $S^1$ as obtained from the unit interval $[0,1]$ by identifying $0$ and $1$ in the category of spaces, and the group of integers $\mathbb Z$ as obtained from the groupoid $\mathcal I$, which has objects $0$ and $1$ and exactly one arrow $\iota:0 \to 1$, by identifying $0$ and $1$ in the category of groupoids. 
I got hold of the idea in the 1960s from writing the first edition of this book that all of 1-dimensional homotopy theory was better expressed in terms of groupoids rather than groups This led to the question: are groupoids useful in higher homotopy theory? Is the 1-dimensional case a ``one-off''? or not? 
I liked the more exciting prospect, but it took 9 years to get with Philip Higgins in 1974  a good definition in dimension 2, namely the homotopy double groupoid of a pair of spaces, and a 2-dimensional van Kampen theorem. 
June 2, 2018  More background to this area is in the paper published this year:
Modelling and Computing Homotopy Types:I 
One of the points made is that the usual base point approach seems to assume all interesting spaces are path connected; yet the circle is a counterexample since it is nicely represented as a union of non connected spaces. The algebra of groupoids nicely copes with the more general case; one main point of my 1967 paper on the groupoid version of the van Kampen theorem was to obtain  by such a method the fundamental group of the circle! 
Note also that if the theorem is proved by verification of the universal property, then the proof for $\pi_1(X,C)$ is no more difficult than the proof for $\pi_1(X,c)$; the case of many sets is given in this paper.