?? universal [cocontinuous (by which i mean, in this context, between cocomplete categories ... ???)and universalizing some given cocones] as ... ???reducible to universal [cocontinuous and inverting some given morphisms] ??? ....

??by ... ???given co-yoneda morphism on cocomplete category, representabilizing its source presheaf ... ????? ....

(?? situation where might be reducible further to universal inverting some given morphisms ?? .... ... ?? ...)

??start with cocone ... get co-yoneda morphism from it ... get co-yoneda monomorphism from that as image inclusion ... convert back to cocone ... ??? .... for example some tendency to convert discrete cobase cocone to one with cobase "revealing overlaps" ??? .... ???? .... ???try example ??? say, co-span in _top sp_ ... (1,3) and (2,4) included into (0,5), say ... ???.... ??hmm, so maybe we should look for a nice presentation of the subrepresentable presheaf here ... though maybe think just a bit about what the canonical presentation looks like ... i guess, include a generator for any map into one of the "pieces", and include lots of relators ... anyway, seems like a relatively nice presentation would be ... about what you'd probably expect ... "pushout input" ....

(??something here vaguely reminding me of .... ???filter (??? ...) as "plateau" ??? ... plateau/s achieved by a net ... ???? .... ?????... ?? and / or ... ??? grothendieck topology where any open set of diameter > d gets covered ... limit as d approaches 0 ... ??? .... ??? .... ??? .... )



(?? "dense" (??) morphism of diagram schemes ... ??? ...)


(??general sketches in co-yoneda + co-co-yoneda form ... ???vague feeling about "isbell conjugation" ideas maybe showing up here ??? .... ??or am i making some level slip in trying to remember what isbell conjugation is like ??? ... ??or maybe not ??? ...)

???sheaves on 1-point space .... ????... site of open subspaces ... ?? "empty cover of empty subspace" ...

???more generally ....

?? is it true that ... ???if you take, say, a (nice?? ??or maybe completely arbitrary ??? ....) locale, and take the "co-yoneda sketch" where the site category is the syntactic poset of the locale and the specified co-yoneda morphisms are all the universal ones, then the syntactic category of the resulting colimits theory is just the syntactic poset again, while if the specified co-yoneda morphisms are just the monic universal ones (???and/or ... ???image inclusions of the universal ones .... ???? ....), then ... ???instead the syntactic category is the sheaves over the locale ???? .... ???

(???co-monad here ???? .....???? on _cocomplete category_ ??? ... ???? ....)

??does this actually make sense ??? .... if so then what's going on here??? ??relationship to mystical stuff about choosing grothendieck topologies where coverings are "covering-like" in some vague geometric sense ??? ... ??relationship to quasitopos ideas ??? ...

??? generalizations .... ???? .... ??? getting higher moore-postnikov factorizations involved ... ??? ....

??issue of ... ?? limits as well as colimits here ??? ...

??hmm, so maybe 1-point space really is good place to test the conjecture, to start with ... ??? ....

??total subspace as own co-square ... ???.... ??hmm, back to example theory "object with invertible codiagonal" i mentioned in other post ... ??...

???hmmm ... ??? so what happens when you apply this idea of "taking all the universal co-yoneda morphisms vs just the monic universal ones (??and/or image inclusion of the universal ones ... ???...)" to just a plain old semi-lattice instead of a locale ?? .... ??does this give something interesting, and/or secretly familiar in disguise ??? ....

??perhaps get quantales involved here ??? ...

i forgot to ask ... is image inclusion of universal co-yoneda morphism also universal ??? ... ?? and if not always then can we find some nice simple counter-example ?? ... ... ??and similar questions?? ... ??? ....

???maybe ... it's at least true for posets ???? .... ?????....

??over a cocomplete poset, a co-yoneda morphism as universal precisely in case it's surjective ??? .... ???? .... ???if correct, then generalizations ?? ....

did that make any sense at all ??? ... ??then a monic co-yoneda morphism is ... ????.... ???? .......

??maybe what i meant to say was that .... ???in poset case, co-yoneda morphism as universal precisely in case its image inclusion is universal ... ????.....

???the basic non-distributive lattice ??? ... ???? .... ???"triple of sets, equipped with isomorphisms between the product of any two .... ??? ...." .... ????? ..... ???? ...

??vaguely reminds me of ... ?? "latin square" ??? ... ???? ....

??so is category of latin squares "modular" / "malchev" ??? ... ???...

(??any interesting sense in which "distributive" category is "modular" ?? ... ???confusion ?? ...???...)

??and what about the primeval non-modular lattice ??? ...

?? actually, if it goes the way that i'm imagining at the moment ... ?? "sheaves over a modular lattice" ??? .... ??doesn't it seems likely that this (...??...) stuff is already well-known ???...

??? "sheafification" of "pre-latin square" subobject classifier ????....


??so... let's consider the subobject classifier for pre-sheaves on the primeval non-modular lattice ... ???

?? objects a,b,c,d,e .... morphisms a->b->c->d and a->e->d ... all diagrams commute ...

walking a-elt ... 10000

walking b-elt ... 11000

walking c-elt ... 11100

walking d-elt ... 11111

walking e-elt ... 10001

??...

?????"representabilization" ... ????

???only subobject of d that gets promoted to true is true ???? ???or wait ... ??not true???... ... ???? .....

??empty subobject of a gets promoted to true ... ??so a is basically out of the game ...

?? now it does seem to come down to which subobjects of d are dense, in fact ... ???...

??the important one seems to be that 11001 is dense ... ??? is it true that 11101 doesn't matter that much ?? ... ???? ....

???maybe .... ??? "either restriction from c to b is invertible, or e is empty" ... ????.... ??something about this as maybe seeming vaguely familiar ??? ....

??? vague memory about "constant" ??? .... ???? ....

???"each element of e gives a bijection between b and c, but e might be empty so there might be no bijection" ... ???? but i'm confused ... ??is it always the same bijection that's given ???..... ??maybe yes??? .... ???maybe they're all 2-sided inverses for the restriction from c to b ... ????....

?? don't forget could be of limits doctrine rather than just products ... ?? ....


?? how close this comes to giving a grothendieck topology ???...

???canonical grothendieck topology as geometrically smallest for which representables are sheaves ????? ...... ??????..... ???vs ... "representabilization" ??? ....???? ...

??trying to understand canonical and / or subcanonical ?? ...) grothendieck topology in terms of _models_ ... of .... theory ... in some doctrine ... ???....