discussion with alex today

we're trying to work out some of this stuff about homological algebra for the algebroid of short exact sequences...

trying to think of some stuff here as analog of "coherent cohomology" but for baby doctrine of finitely cocomplete algebroids...

something about coherent cohomology as derived functor of "sections" functor right adjoint to theory interpretation... ??something about measuring failure of "sections" functor to preserve colimits... ???something about recurring idea of "sheaves imitating spaces" or something like that... something about expressing variety as colimit, then interpreting this as expressing the structure sheaf as a colimit of pushforwarded structure sheaves... or something like that...

also something about vague idea that.... ???"restoration of exactness" and/or "serre duality" ... or something, and so forth or something... as relating to idea that "exactness of something fails, but secretly what's really happening is that homotopy-exactness succeeds" .... ?? though even if that philosophy is on the right track, is it still going to apply, just as easily, in this not-quite-abelian context that we're exploring??

anyway... we have a couple of theory interpretations from the walking object finitely cocomplete algebroid to the walking epi one: v |-> 0>->v->>v (which extracts the domain of an epi) and v |-> v>->v->>0 (which extracts the codomain)... and we can try to measure the failure of the right adjoints of these interpretations to preserve colimits. and the right adjoint of the "domain" interpretation seems to be "middle term of the ses", which _doesn't_ fail here. while the right adjoint of the "codomain" interpretation seems to be "first term of the ses", which _does_ seem to fail...

chris rogers pointed out fairly obvious fact that using injective resolution when "measuring failure to preserve colimits" goes along with idea that 0th derived functor being the original functor f follows from f preserving kernels... or something like that...