?? moore-postnikov factorization of filteredly cocontinuous functor between accidental toposes .... ??? corresponding to what factorization of corresponding partial toric map ?? ....

?? might as well throw in moore-postnikov factorization of (?? left adjoint part of ???) topos-geometric morphism here too ??? ....

?? "injective" (topos-)geometric morphism ... ?? ... lawvere-tierney topology .... ?? maybe corresponding to full-and-faithful on model level ?? ... ?? precisely, or more or less ... ??? ... ??? idempotent monadic on formula level ??? .... ???? .....

?? surjective/injective factorization of geometric morphisms as corresponding to

(?? hyperconnected ??? .... ??? ...)

?? hmmm .... you don't get to independently specify domain of definition; it happens automatically ??? .... ??? part of cockett's point ??? ...... ????? ....

???? co-span ???? .... ??? ...

??? map torus of domain to co-domain .... ?? determine domain of definition from that ?? ... ???? ....

?? "toric birational geometry" .... ????....

??? topology on commutative ring as generalized ideal ?????? .... ???? ..... ?? ...

?? topology on commutative monoid ??? .... ???? ..... ??? .... "toric ideal" ... ??? .... ????? .....

?? essentially surjective filteredly cocontinuous functor followed by full-and-faithful filteredly cocontinuous functor .... (?? _is_ "image" here guaranteed to be filteredly cocomplete ??? ....) ??? toric map from toric variety to some (????) completion of another toric variety .... followed by "openly defined isomorphism" ... ???? ..... ???? ..... ??? blow-up/down here somewhere maybe ?? ....

??? (?? weak ?? ...) pullback of co-span here ??? .... ... ?? ...

?? "maximally extended map" .... ??? .... ??? zorn ... cohomology ... ??? .... ?? "obstruction" ... ??? ....