decategorified gabriel-ulmer duality and infra-sketching of meet-semilattices

alex and i have been talking a bit about decategorifying gabriel-ulmer duality instead of categorifying it, in part as a way of trying to ground what we're doing to some extent. and it seems to be somewhat interesting and useful to think about this stuff.

i wonder to what extent gabriel and ulmer might have explicitly thought about the decategorified version while developing their actual version. though i also wonder to what extent anyone anywhere has developed the decategorified version; i guess that it's very likely that for example some sort of "lattice theorists" have nailed it pretty far into the ground.

part of the attraction of the decategorified version is of course that some annoying "set-theoretical size" issues pretty much go away, allowing the duality (and some other things) to become more "perfect" in some ways, and allowing more definite answers (?more robust wrt change of set-theoretical foundations?) to some questions. this latter seems now to be helping to convince me that some of the tentative answers to questions that we've arrived at in the categorified case are on the right track, because they seem to resemble the more definite answers that we think we're seeing in the decategorified case.

let me try to give some sort of example here...

recently i've been suggesting that for example a doctrine whose syntactic (2,1)-category is essentially just the syntactic ordinary category of a finite limits theory can be sketched by giving a sketch of the finite limits theory, and then supplementing this by, for each generating object x in the original sketch, including in the new sketch as a homotopy-limit cone the "constant" diagram with value x on the diagram scheme given by "the cone over s^1". i probably wouldn't have stated it that way until today, though, because stating it that way is the result of alex and me thinking about the decategorified analog which goes something like this:

a finite limits theory whose syntactic category is essentially just the syntactic poset of a meet-semilattice can be sketched by giving a sketch (by which i mean here basically just a presentation) of the meet-semilattice, and then supplementing this by, for each generator x in the original sketch, including in the new sketch as a limit cone the "constant" diagram with value x on the diagram scheme given by "the cone over s^0".

and the pattern here is supposed to extend pretty straightforwardly to higher cases as well.