??so what about the geometric theory of "dense" intervals ?? or something...

??hmmm, not just putting lawvere-tierney topology on topos of simplicial sets ... ?? ...

??consider classifying topos for inverse pair of real vector spaces ... ???and so forth ... ???

??vs classifying ringed topos for inverse pair of modules ... ???and so forth ... ???

??what about "classifying topos for invers pair of real vector spaces, where real vector spaces are interpreted in terms of usual topology ..." ... ????

??... classifying topos for truth value ... ???....

??so what about interval objects in _space_ (or soemthing...), and projective plane objects in _space_, and so forth ... ????....

???so.... consider _set_^[ne fin toset, with injective order-preserving maps] ... ??? or something... ???.....

??so what about ... "compact object" vs "finite colimit of compact objects" ???... or something... ???and so forth ... ???....

???something about models of boolean (or something ...) toposes, vs models in boolean toposes ... ??? and so forth ... ???...

??something about "cantor orders" ... ??or something ...

morphism from "strictly ascending j-tuples" to "strictly ascending k-tuples" ... ???as given by order-preserving injection k -> j ... ????....

???so seems like we want _set_^_"decidable toset"_ ... ???but then just the projective objects, or something ??? ........

1 1 1 1 1 1 1 1 1 1 1 ...

0 1 2 3 4 5 6 7 8 9 10 ...

0 0 1 3 6 10 15 21 28 36 45 ...

0 0 0 1 4 10 20 ...

.
.
.




0 1 4 9 16 25 36 ... =

0 1 2 3 4 5 6 ... +

0 0 1 3 6 10 15 ... +

0 0 1 3 6 10 15 ...


???....

???hmm, so we visualize the building blocks here as simplexes, but with only the degeneracy maps available ??? ... ???or something ??? ....

??1-simplex squared as 2 2-simplexes plus the diagonal 1-simplex ??...

1 * 1 -> 1 + 2 + 2

1 <- 1+2+2 -> 1


j^k = ????




???presheaf topos as always coherent??? ???os??? with the compact (??or something...) objects being ... ??? or something??? ... and so forth ... ???...

??so what about ... ???a chain complex as a nice way of getting a vector space object from an interval object ... ???or something ???....

??so what about diaconescu's theorem in the simplicial set case ??? ??what's the flat thing corresponding to an interval object ?? or something?? ... ??is this something that we've thought about before ?? ... ??well for one thing, it's the "good cosimplicial object" ... with geometric realization preserving products ... or something ... and so forth ... ???...

??so what about the underlying "degeneracy set" of a simplicial set?? ... ??...

??degeneracy site maps to simplex site, or something?? ???so expect to be able to extract model of simplicial topos from model of degeneracy topos ... so expect geometric morphism from degeneracy to simplicial ... corresponding to algebraic morphism from simplicial to degeneracy ... ??? ...

??what about something about euler characteristic here???? ??or something ... ??? .... ???schanuel ... ????....

??so what about something about "boolean" boolean algebras ?? ... ??or something .... ??something about surjections between finite sets??? .... ???injective homomorphisms between finite boolean algebras ... ????...

??so what about for example underlying degeneracy set of nerve of Z/2 ?? ...

?? hmmm ... ???...

1 * (1 1 1 1 1 1 ...) +
(g-1) * (0 1 2 3 4 5 ...) +
(g-1)^2 * (0 0 1 3 6 10 ...) +
.
.
.

??so what about subobject classifier for degeneracy sets ??? ....

??hmm, maybe subobject classifier here is very big ???? ???? or something ???...

???hmmm, so what about something about ... ???degeneracy set consisting of ... ???two 0-simplexes whose entire degeneracy parts (or something ...) agree ??? .... hmmm .... ???...

???so what about something about ... simplicial set vs augmented simplicial set in some of above discussion ... ???...

???and so then what _about_ some sort of grothendieck topology maybe amounting to denseness or something ??? ....

???and what _about_ "booleanness" (or something ...) here ?? ....

???what about idea that ... ???grothendieck topology here might have suspiciously similar to ... ???something about ... ???what happens automatically as result of coming from augmented simplicial set ... ??? ... ???or something ??? ....

??hmm, so what about something about ... ?? "promote anything not absolutely false to true" ????? ?? or something ?? ...

??hmm, so what about idea that ... ??maybe it makes some sense that adding denseness axiom to theory of "decidable toset" causes booleanness (or something ...) as side effect ?? .... ????...

???what about relationship between "completeness" and "booleanness" of theories ... ??? or something??? .... ????....

??so what _about_ vector space objects in this topos ??? .... or something .... ????....

??so what about double negation topology here?? ... ???and what about something about "coherence" ??... ??extent to which "coherence" lives at truth-value level ??? ... or something ...

??so what _about_ lawvere-tierney topology associated with model ??? .... or something ... and so forth ... ??...

??relationship to "structure /semantics adjunction" ??? ....

??hmmm, so what about double-negation topology for "decidable set" ?? ... and so forth ... ??? ... what about "tannakian" take here ??? ....

??? ....