first let me try to give a reasonable definition of "bistable bimonoid in a symmetric monoidal 2-category". i'd guess that this is fairly standard (because it's a straightforward categorification of the concept of "bicommutative bimonoid in a symmetric monoidal 1-category") ...
define wbb ("walking bistable bimonoid") as the symmetric monoidal 2-category where a 0-cell is a finite set, a 1-cell is a finite span, a 2-cell is an isomorphism of parallel spans, and the rest of the structure is hopefully straightforward. a bistable bimonoid in a symmetric monoidal 2-category x is defined to be a symmetric monoidal 2-functor wbb -> x.
define a "bistable categorified bialgebra" to be a bistable bimonoid in the symmetric monoidal 2-category of small-cocomplete categories (with small-cocontinuous functors as 1-cells and natural transformations as 2-cells). define such a bistable categorified bialgebra to be "special" just in case its underlying small-cocomplete category is a presheaf category (so that small-cocontinuous functors are given by bi-presheaves, and natural transformations between them by bi-presheaf morphisms).
given a small symmetric monoidal category m (we'll be especially interested in the case where m has only a single object), the category of presheaves on it is a special bistable categorified bialgebra in a straightforward way (actually in two "opposite" straightforward ways). the corresponding symmetric monoidal 2-functor (finite set, finite span, span isomorphism) -> (small cat, bi-presheaf, bi-presheaf morphism) takes finite set s to the small category m^s, and takes a finite span s <- a -> t to the bi-presheaf given as follows ...
[to be continued]
[note: delineation of the categorified bialgebra structure may seem like conceptual overkill in the case of an affine toric variety (that is, the case of presheaves on a single-object symmetric monoidal category), but my suspicion is that it will turn out to be the right concept in the case of non-affine toric varieties.]
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