??so... 1-point compactification of N ... double negation topos of the localic topos coming from the stone space ... ???as ... ??sheaves over boolean locale correspodning to boolean frame of regular open sets, which is essentially the complete boolean algebra of arbitrary subsets of N (with infinity cleaving unto the closed complement of the regular open set ...) ... ???suggesting that the infinity model gets removed ???? .... ??this example actually happens to be coherent boolean locale corresponding to discrete N ??? .... ??or something ????

???also suggesting that ... ???closed point of sierpinski space should be thought of as _domain_ of model morphism ?????? ..... ???or something ????.... ???because domain of model morphism gets removed under double-negation topology?? or something ??? can we check this quasi-independently somehow, maybe ?? ... ??something about ... sheaves over sierpinski space .... closed point as non-open ... ??stalk over it as sections over its minimal open neigborhood which is the whole space ?? ???or something ??? ...restriction map from such global sections to sections over the open point .... ...???so in the "exponent category", which i think of as the "category of finitary models" (or something ...), the open point is the codomain of the non-trivial model morphism ... thus retained while the domain (=closed point) is removed ... seems to fit ... ???

??but ... ??what about ???... ???possible non-relationship (or something ...) here between ... [???"point in top space as in closure of some other points" .... ????or something ...] and ["model of geometric theory as filtered colimit of finitary models" .... ???or something ...] ?????? or something ????? ??weird puns on "filter" and "limit" and so forth here ???????? ..... ????? or something ?????.... ???so what _is_ going on here???????...... ????something about ... ????locale (or something ... ???) created by ... starting with point "0" ... then putting in point "1" with 0 in its closure ... then "2" with 1 in its closure ... and so forth .... ????something about point "infinity" materializing as filtered colimit of 0->1->2->... ????? ???or something??? ... ???does this make any sense ????...... ????direct colimit of locales here, vs of top spaces ???? .... ???or something ????? ..... ??????..... ??also vs direct limit of posets ???? ..... ???also vs direct limit of k-coherent locales for various k, or idealized limiting cases of such ... ???...

???is there something going on here about .... ????a model being an actual filtered colimit of models, vs being .... ????some sort of ultraproduct (????or something??????) of other models ??????????? ..... ????is an ultrafilter an ultraproduct or something of vanilla models????? ...... ????or something ?????? ..... ????.....

????something about elementary equivalences and / or some sort of "elementary equivalences of many variables" (or something .....????....) relating to ultraproduct situations ....... ???or something ????? ........

??something about ... "generalized birkhoff theorems for various doctrines" ??? .... ???or something ??? ... and so forth .... ???.....

??hmmm... ??i think that there is _something_ like this (??...) going on here ... ???something about ... single-environment (??"classical" ???....) model theory vs multi-environment here ... something about .... ??some sort of very straightforward operation producing for example "model parameterized by 2" from pair of "models parameterized by 1", but then also something about ... ??ultrafilter on x (or something ... ???and so forth ... ???might have some different doctrines here mixed up, but .... ???....) as giving operation from "model parameterized by x" to "model parameterized by 1" .... ??? or something ??? ... and so forth ... ???....

??something about ... ???categorified lawvere (or something ...???...) theory here?? ... ???something about "doctrine" and so forth ??? ....


??also something about ... ??our example of double negation topos of stone space given by 1-point compactification of N .... ???something about ... ??being cautious about relating this to "poset of forcing conditions", including possibility of shoe-horning in "non-standard analysis" as special case of this with discrete poset ... because in those cases the topos of which you take double-negation topos might be pretty different .... our example(s?...) of double negations toposes of stone spaces was just for fun and educational purposes, or something ... ??including attempt to possibly dispel some confusion about "stone space vs boolean locale" and so forth ??? ....

???what about something about boolean algebra given by something about ... ???regular open sets of unit interval ... ???.... "geometric realization of simplicial sets with orientation-switching ..." .... ???or something .... and so forth .... ????..... ????something about understanding boolean locale here ???????? ....... and so forth ..... ???????.......