Is there a surprising corollary, or one that is much easier to prove using this fact?
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what is true in a higher categorical setting is that the fundamental groupoid functor, from the oo-category of homotopy types to groupoids, is left adjoint to the inclusion of groupoids (as homotopy 1-types) into all homotopy types
this tells you that Pi_1 preserves homotopy colimits, but this *still* doesn't get you van Kampen; van Kampen is a statement about how certain particularly nice pushouts in Top are actually homotopy pushouts
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this line of reasoning does correctly suggest a stronger version of the van Kampen theorem concerning more general homotopy colimits which is e.g. capable of computing the fundamental group of the circle
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