i still have this idea that certain ways of getting a "ringed topos" from a symmetric monoidal finitely cocomplete algebroid should correspond to interpretations of doctrines, but i still haven't worked out the details enough to know whether it really works; the lack of someone to discuss it with is really slowing me down... (a couple of days after i wrote that i started discussing some of this stuff with todd though...)

i've vaguely thought a bit about trying to develop some sort of decategorified analog of this, but at least one attempt to do this seems problematic enough to make me wonder whether it's telling me that it's the wrong idea in the categorified case as well...

consider obtaining a distributive lattice (or "frame" or something...) from a commutative ring in the more or less usual "zariski" way... or something ilke that...

one question is whether this corresponds to something like an interpretation of lex theories... ????....

recently learning a bit about "malcev varieties" from the viewpoint that it has something to do with lattices of congruences being modular... ??thought of as a step on the way to distributiveness?? ... has me wondering about thinking of the decategorified zariski construction this way... ??and about how this compares to the doctrine interpretation idea... ??and also about the possibility of categorifying the malcev variety idea or its "distributive analog" ....

??what about... ??something about general idea of... ??relationships between properties of categories and of subobject lattices here?? ... like "distributiveness" ... ???and maybe "modularness" in some way???... hmmm...??...
??sort of de/categorification here?? ... something about locale vs topos, and so forth... ??...