??what about something about interval object which is not only a flat co-presheaf but also "connected" ??? or something ... and so forth ... ???something about doctrine incorporating something about exponentials?? ??or something?? .. and so forth ...

??hmm, is ".->.<-." an interval object in _simplicial set_ ????... ???is the "concatenation" of interval objects an interval object ??? .... ???? .... ??and so forth ... ???... 2 3 4 5 6 .... 4 6 8 10 12 ... 3 5 7 9 11 ... ??... ??even if this does work (still not clear to me...), staring at the interval squared makes it seem like this is giving something other than barycentric subdivision ... ???or something ?? ... ??what about something about edges of simplex with "rectilinear backbone with all steps the same size" as having edges of lengths sqrt(1),sqrt(2),...,sqrt(n) ?? ... ???... ??hmm, so maybe this really is an interval object, and maybe the left exact geometric realization process here is a sort of "modified version of barycentric subdivision where the barycenter of a simplex is pushed over to the barycenter of its longest edge" ??? ... ??or something ???... ??what about something about .... ??as maybe implicit in this, some way of encoding subset (??or something??) of {backbone edges} as total order on {vertexes} ...??and/or of binning the total orders into the subsets, or something ?? ???hmm, maybe something about "only keeping track of comparisons between immediate neighbors wrt the original total order" ??? ... or something ... ???sounds vaguely familiar for some reason?? ??what about whether this alleged modified barycentric subdivision is "good enough for purposes to which ordinary barycentric subdivision is put" ... ??whatever they are ... ?? ??or something ... ???.... ??what about if you do this to the nerve of a category?? ???is it then again the nerve of a category, and if so then which one ???.... ???hmm, so _is_ this just the twisted arrow category of the original category ?? ???or something ???... ??what about something about extra structure on twisted arrow category allowing you to recover the original category from it ?? .. and so forth ... ???... ??maybe _pair_ of factorization systems ?? ...???or something ???... ??hmm, so what about something about ... ???untwisted arrow category as corresponding to _another_ kind of "subdivision" ??? ??or something ?? ... and so forth ... ???.... ??something about dependence on "orientation" here ...??or something ???... ??something about sort of (again, somewhat orientation-dependent, or something ...??...) "ternary subdivision" ... ??flatly cosimplicial [simplicial set] ".->.->.->." ... ???as corresponding to something about.... ???internal homming from ".->.->." ???? or something ????...... ??????.... ???some sort of "poincare duality" here????? ???or something???? ....

??what about something about "segal category" here, or something ??? .... ... and so forth ... ???....

??... hmm, so what _about_ something about "flatly cosimplicial [simplicial set]" here?? ...?? or something??? ... ??somehow i didn't even think about this aspect of it??? ????something about ... ???geometric morphism from topos of interval objects to topos of interval objects ... ??something about ... ??relationship to some vague sort of "poincare duality" (or something...) that i thought i vaguely perceived showing up here .... ???or something ????....

??interval object in _simplicial set_ ... ??corresponding to geometric endomorphism of topos of interval objects ... ???or something ...

??i forget, does "interval object" include something about the extreme points being distinct ??? ... ??or something ?? ...

???internal homming from "." ... as identity geometric endomorphism ... ???...

???internal homming from ".->." as geometric endomorphism corresponding to interval object ".->.->." ... ???...

???so _is_ "." an interval object, and if so then what geometric endomorphism does it correspond to ??? ....

???internal homming from walking 2-simplex as geometric endomorphism corresponding to interval object ".->.->.->." ?????.... ??or something ????

???internal homming from representable presheaf as ... ????left exact left adjoint ??????? ..... ???left exact because of "connected projectiveness" of representable ??? (something about splitting idempotent here ... ??...) ... ???left adjoint because right adjoint is .... ???hmm, or did i get it backwards here ??? ... ??or something??? ... ???is there something here about "essential geometric endomorphism" ?? ???or something ???....

???hmmm, is there something here about "tiny object" or something ??? ... and so forth ... joyal ... differential forms ... ??or something ??? ....


??hmmm, so far we're saying what the interval objects are that correspond to these geometric endomorphisms, but we're not really saying much yet about what the geometric endomorphisms themselves really are ... as constructions on models of the topos ... ?? or something ... and so forth ... ???....


??is (left exact left adjoint part of ...) [geometric endomorphism corresponding to interval object ".->.<-." ] equivalent to internal homming from something ??? .... ??? .... ??or something ??? ....

??something about ...???maybe some "poincare duality" stuff here as wrapped up with something about ... ???using left exact left adjoint f evaluated at walking 1-simplex as shorthand for f ... ???or something ?? ... and so forth ... ?? ...

???are representable presheaves always "internally" projective ???? ??? or something??? ... and so forth ... ???....

??not at all clear to me how to use arbitrary "compact model" (??or something??...) to get geometric endomorphism ... i mean at intuitive "construction on models of topos" level ... ???or something ??? ....


??so what _about_ something about concatenation of interval objects??? ??as non-symmetric tensor product, or something ?? ... and what _about_ whether there's a unit object for it??? ... that is, again something about whether "interval object" is automatically (...) "strict" ... ??or something ... ?? ...

??and what _about_ relationship of some stuff here to something about ... ???whole bunch of batanin / joyal / berger / [and so forth] stuff ??? ??? or something ... ???....

???hmmm... ???so let's try evaluating "internal homming from walking 2-simplex" (for example...) at walking 1-simplex ... ???or something ... ???hmmm, well, maybe right _there_'s your poincare duality; something about "(??internal?? ... ??...) homming into the walking 1-simplex" ...

2-simplexes in walking 1-simplex ... ??... ??something about 3-simplex vs backbone of 3-simplex here ???? .... ???or something ???? ....

??we still seem to sort of have an argument saying that "self-model of topos" is roughly _equivalent_ (??or something??) to model of topos ...??or do we?? ... but in any case, this sounds insane, right?? ??"in both directions", or something ??? ... ??so where's the resolution of the confusion??? ... ???something about "projective vs internally projective" or something ???? .... or what ??? ... and so forth ... ???...

??hmm, a lot of this could probably be straightened out fairly easily ... ??...

??so what _about_ possibility of relationship (??or something... ??maybe with level slip in between, or something ... ??...) between "internally projective" and "exponentiable" or "powerable" or "copowerable" or something ... ?? ...

??maybe walking j-simplex and it's "backbone" (or something ...) are _both_ interval objects ... ??? ??or something ???...