there's lots of interesting digressions that ideally i'd like to explore here... not sure whether i should instead try to stick to the main path, towards the relationship between homotopy limits and homotopy colimits in stable homotopy theory...
not sure how much i should bring up the "abelian analog of diaconescu's theorem" idea here... (and other ideas about "flat modules" ... algebraico-geometric interpretation of flatness...) perhaps try looking up that n-category cafe thread about flat presheaves (or something...) to see how explicitly the analogy is spelled out there...
perhaps i should try to get pretty quickly to the adjunction between cokernel and kernel in an abelian category, and how this is a prelude to homotopy cokernel and homotopy kernel being inverse in stable homotopy theory... or something like that... (digress about relationship to snake lemma??)
then introduce the issue of "weights" for diagram schemes... motivated in part by idea that homotopy cokernel and homotopy kernel aren't just inverse to each other, but are "the same" up to a specification of weight and co-weight... or something like that... develop the analogy between the colimit of a weighted diagram and the expectation of a random variable... (digress about "calculus of co-ends" and/or "calculus of weighted colimits" ??...)
develop interpretation of weighted homotopy limits in chain complex world as weak homming from a dg module of a dg algebroid, and of weighted homotopy colimits as weak tensoring with a dg op-module of a dg algberoid... then develop equivalence between weighted homotopy limits and weighted homotopy colimits in terms of appropriate kind of "weak duality" between a dg module and a dg op-module...
...try to work out many examples... non-existence of duals in set-based or vector-space-based case, vs existence of weak duals in chain-complex-based case...
...try to relate this "duality" stuff to "absolute colimits" and so forth... and to algebraic geometry...
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