?? (?? bi-stable) bi-monoids for lax interchange map of toric variety ... ?? (toric) geometric interpretation ... ?? ...

?? formal properties of co-/slice cat of cat of comm monoids ... ?? ?? hmm, maybe co- makes big difference here ?? .... ???? .... ??? paradox here ??? ... (?? obviously based on some stupid mistake i'm making ....) ?? .... "affine ..." ... hmmmm .... comm monoids under given such, vs comm rings under given such .... ???? vs ab gps under given such vs over given such .... ???? ..... ???? .... ??? .... "formal properties" of categories here .... ??? ...

??situation where taking vector space objects in x is right adjoint functor of x, vs situation where it's left adjoint ... and / or both .... ??? .....

??? doctrine morphism here ?? .... ???? .....


?? getting torus-equivariance involved here .... ?? ...