??wait a minute; is there a problem here?:

on the one hand, we sort of think that there's this category of "affine toric varieties" that's equivalent to .... ???hmmm, i think that i was going to say something like "to the category of commutative monoids" or something like that... which is maybe not that far from the/a truth .... depending in part on what "affine toric variety" should mean exactly... might also try to develop concept of "affine toric scheme" and so forth ... anyway, also possibilities like considering just those commutative monoids that are finitely generated submonoids of free abelian groups... or something .... ??or maybe of abelian groups in general??? .... maybe lots of possibilities...

anyway... meanwhile ... on some other hand, we also sort of think that ... this same category of affine toric varieties (whatever it is...???....) should also be equivalent to ... some ("essentially (1,1)" ... or something...) (2,1)-category of certain toposes, in turn equivalent to (2,1)-category of certain filteredly-cocomplete small categories ... ??which are free filteredly-cocomplete on those commutative monoids (construed as 1-object categories) that we mentioned ??? ... ???or something???

??anyway, the potential problem bugging me here is something like ... mismatch in general between the homomorphisms between the commutative monoids, and the filtered-colimit-preserving functors between the corresponding free filteredly-cocomplete categories on them ... ???... ???something about "kleisli morphism" here ... ???....

???so what's going on here???

??shouldn't it be not that difficult to straighten this out?? ???hopefully ??... ??... ??something about ... morphisms of toric varieties (in some sense...) between affine line and punctured affine line ?????? ...... and so forth .... ????? ......

??so what _about_ how geometric morphisms here get along with day convolution wrt the (...??...) symmetric monoidal structure ??? .... and so forth ....

???hmm, so what about something about preservation of unit object here ????....

??something about ... ???maybe running into subtle questions about "property-like" here ?? .... ???and so forth ??? ....

or maybe not that subtle ... ??something about ... geometric morphisms one way or other between accidental topos and "toric small zariski topos" ... ??? ... and so forth ...