wowie the eilenberg-steenrod axioms for homology is literally just saying that the (โ,1)-category of homotopy types is the free co-complete (โ,1)-category generated by a point
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Conversation
and homology w/ coefficients in A is just the unique co-continuous thing that sends pt to (the eilenberg-maclane spectrum of) A, the latter living in the (โ,1)-category of spectra (which is co-complete)
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I guess this is just a fancy way of saying that sending X to Aโงฮฃ^โ(X) is left adjoint to the forgetful functor from A-module Spectra to โ-Grpds
Analogous to how X|->AโZ[X] is left adjoint to the forgetful functor from A-Mod to Sets
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classic QCposting, all universal properties, no proofs ๐
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dont need a proof if you work synthetically and that's your definition *taps forehead*
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