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Daniel Plácido (Feb 01 2021 at 17:45):Does the singular complex functor admit left or right adjoint?
Notice that the standard adjunctions image.png don't compose "the right way".
Reid Barton (Feb 01 2021 at 18:00):No, though it is a left adjoint in a suitable derived sense (e.g. on the homotopy categories), since then the geometric realization-Sing part is an equivalence.
Reid Barton (Feb 01 2021 at 18:01):Imagine you cover a topological space by, say, two open subsets; then the whole space is presented as a pushout of the two opens along their intersection. If you take a chain consisting of a "large" singular simplex (one which isn't contained in either open), then it won't be equal to any chain in the corresponding pushout of chain complexes; only homologous.
Joe Moeller (Feb 01 2021 at 18:03):What do you mean by "don't compose in "the right way"."?
Reid Barton (Feb 01 2021 at 18:08):I assume that it's a composition of a left adjoint and a right adjoint
Joe Moeller (Feb 01 2021 at 18:09):Oh I see, yeah.
John Baez (Feb 01 2021 at 21:21):Hmm, so this is a very nice, very fundamental advertisement for interpreting these functors "in a suitable derived sense." One certainly wants all the time to take a chain complex of abelian groups and view it as a space.