Another basic Q. (sorry, I am comfortable with the Euler characteristic, and with filling holes in STL models, but not with homology):

Taking chains to be a rough analogue of functions, could we say that cycles on (X,A) serve as a kind of dual space for A? (because if we ignore intermediates, they're basically connecting A<->A?) So from the dual side, we seem to have the case that an appropriate quotient might be A/X?

(maybe an alternate view of what I'm trying to ask about: in Mathematics Made Difficult, we find many proofs of [something in A implies something else in A] which, instead of being limited in scope to A and its subfields as is customary, wander leisurely through X to ultimately prove the inclusion in A)