?? 3/4 ... herring (red) .... derived .... claws (hanging on by ...) ......... boolean ....

?? presheaves on single-object symmetric monoidal category x as same as for x^op ... ??? not so in multi-object case ??????? ..... ????? .... ????? ..... i guess : single-object smc as self-opposite; not so in multi-object case ?? ... ???? ... ??? identity fr as "star-structure" ??? ..... ????? ...... ???? ..... ??? "serre duality" and inverse line bundle ??? .... ????? .....


?? so i'm hopeful that a bunch of stuff (about the algebraic structure on the category tqs(x) of "toric quasicoherent sheaves" over a toric variety x, roughly) is fitting together reasonably well now ... i'll try to describe it here ... it seems to be fitting together well enough to give me that feeling of "why didn't i see it before when it seems so clear now?" (and associated feeling of "everyone (or at least someone) probably knows this already") even though i think that i actually understand some of the reasons that i didn't see it all so clearly before ...

it seems useful to organize the ideas here by first considering the ("natural" in some sense ... though there's some trickiness involved here ...) algebraic structure on tqs(x) when x is affine, and then picking out from among that considerable amount of structure the part that survives to the non-affine case ...

in the affine case, tqs(x) is a symmetric monoidal object in the (cartesian symmetric monoidal) 2-category where the objects are the totally distributive toposes and the morphisms are the essential geometric morphisms ...

(more generally, the category of presheaves on any symmetric monoidal small category is such a symmetric monoidal object ... for pretty straightforward reasons ...)

in the non-affine case, tqs(x) is a symmetric monoidal object in the (symmetric monoidal) 2-category where the objects are the small-cocomplete categories and the morphisms are the left-adjoint functors, and it's simultaneously also a grothendieck topos whose diagonal geometric morphism is essential, with the extra left adjoint of the diagonal acting as a categorified comultiplication engaged in categorified bialgebra compatibility with the tensor product acting as categorified multiplication.

it seems reasonable to take more or less the above description as an axiomatic definition of some sort of "generalized toric category" ... more, in that for example we might want for some (not all ...) purposes to impose the further condition that the double negation topos is a categorified hopf algebra ... less, in that for example, we might want to separate out the categorified bialgebra and the topos as separate categories, conceptually "dual" to each other in some sense ... not sure how that idea might play out ...

there's a bit more of the structure in the affine case that survives to the non-affine case, but it seems not so fundamental offhand ... the totally distributive topos with essential symmetric monoidal structure hangs on just barely as a non-totally-distributive topos with non-essential symmetric semi-monoidal structure ... one of the reasons that this seems less fundamental is because of the (contestable ...) philosophy that "semi-" concepts are less fundamental ... still, you could try incorporating this extra structure in the axiomatic definition ...

?? hmmm .... ?? 3/4 vs ?/6 ??? .... ???? .... essential geometric morphism as made of three parts ... ??? ....

?? stuff about cartesianness of categorified bialgebra as red herring ???? ... ???? .... still ..... ???? .... ???? ...... ??? trying to study cateogrified prop of natural operations here .... ??? "distributive category" ... ?? ....

?? idea that at derived level, there really is more surviving ... ???? 4/4 ??? .... ??? symmetric monoidal topos ???? ..... ???? ..... .....

?? extra left adjoint of essential diagonal acting as categorified comultiplication with tensor product acting as categorified multiplication .... ??? vs ... ??? "maxwell" ... ??? .... [right adjoint of cocontinuous extension of tensor product] being also a left adjoint of something, so that it can act as a categorified comultiplication with cartesian product acting as categorified multiplication .... ?? see above bit about derived level ... ?? where distinction between left and right adjoint maybe more or less vanishes, thus restoring full 4/4 or 6/6 or whatever .... ??? ... ... ??? ....

?? 6/6 ... ??? grothendieck's 6 operations ??? ... ??? .... perhaps not ... ?? ....

?? checking notes from december for fit with current picture ... ?? "generalized day convolution" .... ??? tensor product as "essentialness" of something ..... ????? ....

?? getting more explicit about how the structure described in the non-affine case fits as part of the structure described in the affine case ... the "left-adjoint functors" as the _"essentialness"_ of the corresponding "essential geometric morphisms" ..... ????? .....

?????? extra left adjoint of cartesian product vs extra _second_ right adjoint of tensor product ..... ????? extra right adjoint of .... ??? candidate for certain categorified coproduct ... ?? ....

???? maybe 5/6 ??? ..... ????? .....