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.)

`$\pi_1 S^1$`

"? Are you referring to a combinatorial definition of the fundamental groupoid of a simplicial set? I'm familiar with Kan's combinatorial description of the homotopy groups of a Kan complex, and I guess you could mean something similar? Most importantly, how easy is it to identify this object with the fundamental groupoid of the geometric realization? Milnor's original proof that Kan's homotopy groups agree with the homotopy groups of the realization used the Van Kampen theorem. $\endgroup$ – Dan Ramras Nov 14 '10 at 6:01twointervals: just pick one basepoint in each component of the intersection. $\endgroup$ – Omar Antolín-Camarena Nov 14 '10 at 14:43