??so what _about_ property (or something .... ???...) of grothendieck topos of inverse image functor from _set_ being logical?? ...

??includes "2-valued" or something ??? ... (??or _is_ that correct ??? ...) ???anything else ??? .... ???....

(1+....+1)^(1+...+1) ... ???... ???something about 1^1 ... ???....

?? _set_^(1+1) ... ??...


??so what about ... ???doctrine (or something ...) of "geometric theories over" an elementary topos ?? ... or something ... ??and so forth ?? ... also ag theories over elementary topos, or something ... ???...

??something about subcanonicalness (or something ...) of double negation topology on presheaf topos and monicness of all morphisms in exponent ?? .... or something ...

??what about other correspondences (or something ...) here ?? ??something about ... arbitrary double negation sheaf as sum of representables ... ??or something ... ???...

??"boolean-valued model ..." ... ???...

something about "every morphism out of a double-negation model is an "elementary equivalence"" ??? ... or something ... ??? ??and characterizing double-negation models that way, or something ... ??....




"elementary topos as set theory... ???equipped with generic model of ... " ... ???or something ??? (??does that mean that ... ??a logical morphism (or something...) between elementary toposes is something like a morphism of set theory models, picking out in the codomain model a model of a first-order theory correposnding tothe domain ... ???or something??? ...) ...and so forth??? ... ??so what _about_ _elementary_ topos theory as all about "forcing" ??? .... ... or something .... ???....

??so what _about_ factoring geometric morphisms into logical and "anti-logical" part ??? .... or something ... ???....