so consider the walking epi finitely cocomplete algebroid... which by gabriel-ulmer duality we can describe pretty explicitly...

and then consider chain complexes of such things... ??

but we have to work out the details of what we really mean by this... i mean, we want to create something like a derived category (or more saliently its (infinity,1)-category precursor, or something like that...), but we don't actually have an abelian category to start with so we probably have to tread lightly...

but assuming that we can actually figure out what we want here, then...

??the next thing that we want to try here is to consider homotopy limits and homotopy colimits in this context ... or something... ???...

see whether they "behave as expected" or something like that...

??which might actually come down to checking whether the "derived category" that we're getting is essentially just the derived category of some certain actual abelian category?? or something??

so far there does seem to be some danger here of cheating out of confusion... or something... ??...