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Sam Tenka (Feb 08 2023 at 22:54):There is a van kampen theorem for the fundamental 1 groupoid. And Ronnie Brown has a paper giving the same theorem ("taking fundamental groupoids preserves such and such kinds of colimit") for some notion of fundamental 2groupoid. Ronnie Brown's book on "nonabelian algebraic topology" seems to say that the same approach extends to higher ns BUT do not give full information, e.g. don't reveal whitehead products.
My question is: is there at least a vankampenish theorem for fundamental 3groupoids that allows one in principle to compute whitehead products? Or, is there some result that suggests why there ought not to be such a theorem (maybe van kampenish things don't jobe with unstable phenomena?) I'll settle for 3 since infinity is too big for me :smile: I'd think the main challenge here is in defining what "fundamental 3groupoid" ought to mean here. Already with RB's 2groupoids, one has to be careful about exactly how strict each aspect should be to get the theorem to work.
Thanks!