This surely rivals the Yoneda lemma to be the most important theorem of category theory - and it doesn't even have a namehttps://twitter.com/FunctorFact/status/1187400900291256325 …
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Well I didn't know about that adjoint, so the example still taught me something
you cannot get the van Kampen theorem this way; the adjunction only holds in the homotopy category but the homotopy category lacks most colimits, and in particular the functor from Top to the homotopy category doesn't preserve colimits. this is what homotopy colimits are for
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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
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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
End of conversation
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