?? extra structure on categorified bialgebra arising from being categorified measures on categorified monoid where just _objects_ are invertible ... ???? ..... (?? contravariant functoriality of inverse ... ??? ....) ??? .... ???? non-toric analog ??? .... "triviality" of "affine" case ??? .... ???? .... ?? "serre duality" / "off-center center" / "twisting by line bundle" ?? .... ??? ....

?? "cartesian bimonoid(-al abstract category)" .... ??? mutltiplication and comultiplication as adjoint _to each other_ .... ???? ..... ??? adjointness and property-vs-structure(-vs-...) games ... ??? .... ?? generalizations .... span, span, span, ... ?? .....

?? doesn't this make it a lot more plausible that "unstable cartesian" might make sense ?? .... ??? ...

??? mystical idea that all higher coherence secretly arises from adjointness ?? ... ??

?? kock-zoeberlein here ??? .....level slips .... ??? .... ?? x,y having adjoints vs being adjoint to each other ..... ???? ..... bi-monoid and/or frobenius .... ???? ....

?? "spans forever" (... ?? ... ?? or at least up to a point ... cutoff subtleties ... ??? ...) as "walking object" _something_ ??? ... but what ??? .... ???? and what about span/co-span confusion (?? ...) here ??? .... bi-monoid and/or hopf vs frobenius .... ??? ..... ??? ambijunction vs certain sort of degenerate such ??? ....

??? microcosm .... ???? ..... "morphism becomes arity" ... ?? ...

?? no (?? ...) span-map manifestation evident between "toroic span bimodules" ??? ... ?? general idea of ... ?? .... well, like i almost said, retreating (?? ...) from suspecting x and y are adjoint to suspecting that they each (?? ...) have adjoints ...

?? automaticness of being able to cartesianize a smc by taking stable comonoids ... ?? counterpoint to idea of special "coincidence" .... ????? ......


???but ... "model diagonal" of topos .... ???? ...... ?? or of "distributive category" ??? .... .... ??? compatibility between topos structure and categorified bialgebra structure expressed in terms of this .... ??? ....

?? danger that bimonoid compatibility between "model diagonal" and "tensor product" will be vacuous ?? ... ??? .... ?? tensor product as .... ??? .... only significantly un-property-like structure / stuff ??? ... ???? ....

?? categorified bialgebra structure (...) on ordinary quasicoherent sheaves over toric variety .... ???? ....

?? arbitrary grothendieck topos has model diagonal ... ?? .... ??? ...

??? categorified bialgebra structure and that confusion about ... ??? ambiguity (?? ...) of "point of toric stack over comm monoid" ?? ..... ???? ..... ???? .....

?? trying to "untangle" / "straighten out" (?? ...) "mayhem" by ... ??? stuff about "what adjoint of f preserves in relation to what f preserves" .... ??? ... ??? limit-preservation vs colimit-preservation ... ??? ..... ????? ..... ?? some simple stuff here but maybe much trickier stuff too ?? ... "preserve" ... "commute with" ... ??? "beck-chevalley" ... ??? ... ?? adjoint functor preserving structure embodied by non-adjoint functor .... ??? .... ??? .... ??? ....