??hmm.... danger here (...) of categorifying certain mistake i sometimes make with cocomplete posets ... ?? or _is_ it a mistake ??? .... "subsemilattice" ... as automatically corresponding to "closure operation" ... vs ... ???

i knw that we had some confusions here, but i don't completely remember yet what they were... which perhaps is a bit like still being confused ... ??...

"kuratowski vs anti-kuratowski" ... ???...

?? various 3-elt (??) quantales here ?? ...

modules of quantales ... ????....

?? "semilattice equipped with closure operation" vs ... ??module of certain 3-elt quantale ??? .... ??? what _is_ the difference here, if any ??? ...

???whether closure operation is itself cocontinuous ??? .... ???and/or continuous ... ????? ......

?? different "gauge-fixing" conventions here ... ??...

(?? "gauge-fixing" and "wlog"/"mawa" ?? ...????)

?? might as well assume ... (sup) semi-lattice with endomorphism ... ???... which could then also be monadic or comonadic ... ??? .... ??vs some other convention where ... ???? might as well assume monad on (sup) semi-lattice ...


???anyway, more or less seems like ... ??it was a mistake to think that i might be categorifying that mistake here ??? .... ?? full (?? ... ??consider decategorified case again ... ?? ...) subcategory being closed under limits as more or less enough to get reflector left adjoint to full inclusion ... ???? ...




?? speaking of quantales .... ?? decategorified version of "combined doctrine" phenomenon here ?? .... ???relationship to "boolean hecke algebra" and "hecke quantale" ???? .... ... ????any direct connection between hecke quantale and toric varieties ???? ..... ????? .....

??maybe sort of in line with something todd was suggesting .... ??? ....


??vague memories of ... ???at one point feeling like we'd pretty nicely categorified a bunch of stuff about sub-semilattices and so forth ... ??riding to box mountain (???) park ... ???....


???categorified version of "kuratowski / anti-kuratowski situation as module of 3-elt quantale" ????? ..... hmmmm .... lawvere-tierney topology ... ??? ....


?? hmm ... having remembered a bit more, and/or thought things through a bit more, it seems like if we avoided categorifying those mistakes then it was probably only through some sort of compensation oferrors ... ???...

??? ... confusion between reflector and monad carrier ... ?? monad carrier as composite of right adjoint full inclusion and left adjoint anti-[cocone-reflecting] reflector (?? ...) has nice properties of neither ... ??... so not even cocontinuous, let alone also continuous ... ??? .....

(??so one thing to straighten out here a bit further is that stuff about 3-elt quantales ... the nature of the confusion about them, and possibly categorifying all that (including the confusion ...) ... ????... ??and it should be fairly easy to straighten out, assuming that we actually get the time and motivation to do so ...)

?? but still seems to be true that, morally, (?? full and) closed under limits = has left adjoint reflector .... ??? ....

??so for example, consider limits-closed full subcat of _graph_ generated by some edgeless graph ... ??evidently the reflector takes any non-edgeless graph to the terminal graph ... ??? ....