?? lex geometric realization of (??say for example "level 2" ??? .... "2-stage" ...) globoplexes ... ???

?? trying to really understand joyal(?? ...)'s ideas about ... whatever's going on here ... ???...

???model of (??nice???) geometric theory in something not quite a topos ... as with standard interval object in more or less usual not-quitr-topos category of "spaces" ... ????..... ???? .... presumably happening here as well ... ???....

??for simplexes, the key object to realize as "the 1-simplex" ... "generates" everything else ... ??so analogously seems like we should be particularly interested in the top-dimensional globe ...

???meanwhile general model as filtered colimit of objects of site^op ... ???

??which, in part, means that ... ??? to ["interpret as model in form of concretely structured set ..."] such an object m of site^op, we should ... ???? look at the globoplex morphisms from the walking top-dimensional globe to the walking

???so _is_ it really true that ... ???to interpret a simplex x as a model of the theory, you should ... look at the simplex morphisms from the 1-simplex to x?? ... ??or is it the other way around?? ... ?? ??slightly annoying to try to straighten out... at least if you try to do it by just guessing ... ???....

(???"sesquisimplicial set" ... ??? .... ?? "barycentric subdivision" (?? ...) .... ???? ... variants and relatives thereof ...)

??well, so... ???the tame (??... ?? "mundane") models are supposed to be ... site^op objects ... ???....

??and we're under the vague impression that "interval object" is supposed to be a reasonable name for this ....

???and i think that i'm expecting that a 1-point set would probably qualify as an "interval object", this being the case where "arrow between vertexes gets interpreted as equation" ... ????... ??but who knows?? ...

(??hmm, lex geometric realization really does require that "vertex" gets realized as single point ... ???...)

??well so anyway, let's try as a guess the category of "finite tosets with top and bottom" ... ???? ..... well, at first i was thinking "with" here as property, which made it seem like just plain "non-empty finite tosets" which annoyingly seems to be site rather than site^op ... but then i tried "with" as structure, and nicer things started happening ... like for example the decision about whether to include the so-called "[-1]-simplex" now seems to be the question of whether to allow case bottom = top, which moreover seems to correspond very nicely to that case where "arrows get implemented as equations", which threatens to fit in very nicely with mystical philosophy of "bar resolution" .... ?? the things being resolved as the case where arrows get implemented as equations ... ????....

(?? ... ??? globoplexes and "resolution" ... ??? .... ??? ... ??? "degenerate globoplex" ... ????...)

???so ... still using a lot of guessing, though in principle we ought to be able to work it out pretty much more systematically ... seems like ...

0-simplex as site^op object gives 2-point interval, just bottom and top ... so that seems like it's got to be [0-simplex,1-simplex]_site ... in particular rather than [1-simplex,0-simplex]_site = [0-simplex,1-simplex]_[site^op] which would be only one point ... so in general we turn a site^op object into a "model viewed concretely in the 1-simplex picture" by homming into the 1-simplex as a site object, or out of it as a site^op object ...

??? so in particular, modulo parity of number of sign mistakes here ... ????... ?? seems like i'm getting that ... ?? to see globoplex x as "model viewed concretely in top-dim globe picture", we should ... ??? hom the globoplex x into the top-dimensional globe ... ????....


???mental picture of ... ?? "generic (!!??....) random dynamically growing (via monos of site^op objects .... no non-monos ... ???...) converging to filtered colimit ..." ... ?? ... in fact not just converging to but reaching the (co...)limit ... model of double-negation subtopos ... ???... ??countable and/or continuum-sized ... ???? ..... ????....

??globoplexes corresponding to notable relations .... ??? ???hmmm , particularly ... ??? "n-stage trees with one excess ..." ... ??well, not sure exactly, but ...??? interesting possibilities ??? .... ??? ....

?? "horizontal vs vertical ... " ... ??...

??globe globoplexes of all dimensions ?? ... ??? ....