?? so consider some "anti-conservative" commutative monoid homomorphism ... "does nothing but invert stuff" .... ?? inclusion of n into z should do, i think ... ?? then we want to try to get two distinct geometric morphisms between accidental toposes corresponding to this ... ??

?? so first we get an essential geometric morphism, with lex left adjoint given by pulling back along the homomorphism .... ???

then we cannibalize this a bit ... since it's essential, there's another left adjoint invlved, namely its own left adjoint ... ??? and this is lex just in this special anti-conservative case, i think ... ?? ...

?? so maybe what's going on is something like .... the 2-category of accidental toposes, geometric morphisms, and natural transformations is equvalent to the poset-enriched category where an object is a toric variety, and a morphism is a toric-dense-openly defined toric map, and a 2-morphism is an "extension" relationship ... ?? with maybe some hopefully more or less obvious arrow reversal here ...

?? so this "extension relationship" stuff seems like somewhat of a baby (?? ...) version of idea of "extension of correspondences" ... ??? ....

(?? b(??? ...)-series flag geometry ??? .... ???? ....)

??? and ... we have these adjunctions (?? and asociated co-/monads) in this poset-enriched category .... ??? ..... ?? endo-map of toric variety with domain of definition given by some toric dense open, with the map equal to the identity on its domain of definition .... ??? unit / co-unit here ??? .......

(?? vague memories ... grothendieck and/or lawvere-tierney topology ... ?? analogy to kuratowski closure operator ... anti-kuratowski .... semilattice ... ?? distributive lattice ??? ..... partition of unity ???? ..... ??? qm "observable" ... ??? .... commutative vs noncommutative ... diagonal matrix vs matrix .... ???? ..... projection operator ... 0 / 1 .... ???? .... ???? ....)

?? in other direction get identity morphism ??? .... ??? localization as retract here ???? ..... ????? .....

??? hmmm .... ???? using extension relationship to get quotient (2-)category here ??? .... ??? "decategorificaiton" ??? ... ??? maybe some sort of "half-decategorification" ?? .... ???? model category structure here ????? ......