?? so is it a convincing argument that the diagonal of the accidental topos of the projective line (for example) is non-essential because "toric multiplication" is only partially defined on the projective line ?? ....

?? also, ... ??? possibility we just got it backwards as to which half (?? 1/4 ?? ... ???1/6 ??? ...) is missing ?? .... ??? aesthetic / philosophical / ??? preferences as to which way it should be ??? ... ??? "functions vs measures" ideas ??? .....

?? right adjoint of cocontinuous extension of tensor product acts as comultiplication of categorified bialgebra, with cartesian product as multiplication ?? .... ??? .....

???? hmmm, possibility we're really seeing here that we're _not_ (in any obvious way) getting a categorified bialgebra in the non-affine case .... ?? because sometimes (?? binary toric multiplication ?? ...) the "essentialness" is missing, and sometimes (?? co-nullary toric diagonal ?? ...) the "geometricness" is missing ... ??? .... ??? ....

?? and again this should have been pretty obvious all along ??? .... ???? ...

?? on the other hand can we salvage a compatibility relation here .... ???? ....

?? well, we already have the idea of getting a semi-monoidal topos here ... ?? do we get anything beyond that ?? ... still, might be worth making some things (compatibility conditions ...) explicit here ... ??? ....

?? "logic" .... "model" .... confusion .... ???? .... ?? confusion which i think i described somewhat incorrectly recently ... ?? ...

?? derived level .... ??? .... ??? "cohomology" ..... ?????? ......

?? "middle of the morphism" .... ???? ....

?? "mayhem" .... ???? ....

??? hmmm .... ??? cocontinuous extension of cartesian product, and biaction coming from action .... ?? as "main structure" ... (?? "middle of morphism" ... ?? ...) ?? all bialgebra compatibility as parsing here ??? .... ???? "logic" as not obviously really living here .... ?? backwards ??? .....

?? the missing nullary multiplication ..... ??? as .... ???? maybe existing in some "compensatorily shifted" form ??? ... ??? .... ??? ... derived level .... ???? .... ???? .... ???? .....

?? (?? which ?? .... ??? ....) adjoint to "global sections" in affine case ... ??? .... ???? .....