i'm going to try to sketch for todd here some ideas about non-toric analogs of some of the ideas that we've been trying to develop in the toric case...

so suppose that we take the topos x of "toric quasicoherent sheaves" on a "reasonable" toric variety, and then take its category c of models. then we conjecture that the comparison geometric morphism to (??or is it from??) the topos of filteredly cocontinuous set-valued functors on c is an equivalence.

then let's try working out a non-toric analog here...

the quasicoherent sheaves over a reasonable variety form an abelian category x... ??then the exact functors from x to modules over the base ring play the role of the mnodels here...

??except that here i should probably make up my mind for now as to whether i'll work with finite colimits or arbitrary ones... ?? ...