in this paper we present some answers to the question: what happens to the tannakian philosophy of algebraic geometry (roughly, "to know a variety is to know the tensor category of quasicoherent sheaves over it") when the varieties that you're studying are toric varieties?
one answer is: an extra "toric convolution" tensor product of quasicoherent sheaves appears. (for example in the case of an affine toric variety, its coordinate algebra is a monoid algebra and thus a bialgebra; toric convolution is then the tensor product associated to the bialgebra comultiplication.)
(an extra "convolution" tensor product of quasicoherent sheaves also appears in the case of for example an abelian variety, though with somewhat different formal properties.)
another answer, more oriented towards the study of toric varieties as objects in themselves than as ordinary varieties with extra structure, is: instead of studying quasicoherent sheaves of modules over a structure sheaf of commutative rings, we study quasicoherent sheaves of actions of a structure sheaf of commutative monoids ("toric quasicoherent sheaves"), and the "tensor category" of these has an underlying grothendieck topos instead of an underlying abelian category.
the most direct relationship between these two answers is that the ordinary k-based quasicoherent sheaves over a toric variety x appear as the k-module objects in the topos t(x) of toric quasicoherent sheaves over x. these k-module objects can be "tensored" in two different ways: either making use of the tensor product on t(x), or in the way that k-module objects in any topos can be tensored. the former amounts to the usual tensor product of quasicoherent sheaves, while the latter is the extra "toric convolution" product.
the topos t(x), when x is non-affine, is among the simplest sort of example of a grothendieck topos that is not a presheaf topos and not "totally distributive". it is glued together from toposes of presheaves over single-object symmetric monoidal categories (aka "toposes of actions of commutative monoids"), but the glueing is along non-essential "localization" inclusions, which results in the lack of total distributivity. this may make t(x) interesting even from the standpoint of pure topos theory.
more generally, although the relationship between algebraic geometry and topos theory discussed here is related to more oft-mentioned such relationships in a somewhat peculiar way, this work does seek to revive a particular form of interaction between algebraic geometry and category theory that has not been developed to its full potential, the idea that the spaces (or more generally "stacks") that algebraic geometers study should be seen as "classifying stacks" for "models" of some kind of "theories" in lawvere's sense (that a "theory" is a category equipped with some sort of extra algebraic structure).
the specific example of this philosophy explored in this paper, focusing on toric varieties, is intended as a toy example, in the same way that toric varieties give toy examples of many phenomena.
(in order to get a fresh viewpoint in this preliminary version of the paper we have to some extent avoided consulting the literature on toric varieties; thus some later rewriting to account for this is to be expected.)
1 construction of the topos t(x) from the fan of a toric variety x
the cones of the fan of a toric variety x form a finite poset under inclusion, or equivalently a finite t0-space, with the open subsets being the downward-closed ones. this finite t0-space is a toric analog of the zariski t0-space of a scheme. being finite it is covered by the minimal neighborhoods of its points, which correspond to the affine open toric subvarieties which glue together to give x.
in this context, the concept of "quasicoherent sheaf of actions of the structure sheaf of commutative monoids of the toric variety x" goes through straightforwardly, in imitation of the usual concept of "quasicoherent sheaf of modules of the structure sheaf of commutative rings of x" for a scheme x. the main difference is that instead of forming a tensored abelian category they form a tensored grothendieck topos.
definition: the "toric quasicoherent sheaves" over x are the tensored grothendieck topos t(x) described above.
alternatively, t(x) can be defined as the filteredly cocontinuous set-valued functors on the cocone category x# of x, which is the category where an object is the cocone dual to a cone in the fan of x and a morphism is a translation map. x# can be recovered as the category of models of the topos t(x).
theorem 1.1: the two alternative definitions of the topos t(x) are equivalent.
2 geometric morphisms from t(x) to t(y)
a geometric morphism from a grothendieck topos t1 to another such t2 is a left-exact left adjoint functor from t2 to t1. as in the ordinary non-toric case, an arbitrary left adjoint functor between categories of toric quasicoherent sheaves has a geometric interpretation as a sort of "correspondence", so a geometric morphism from t(x) to t(y) will have a geometric interpretation as a particular kind of toric correspondence. even if there's no obvious fundamental geometric significance to this particular kind of correspondence, it will be useful to know what it is.
theorem 2.1: consider the (weak) 2-category where an object is a toric variety, a morphism from x to y is a geometric morphism from t(x) to t(y), and a 2-morphism from f:x->y to g:x->y is a natural transformation from the left-exact left adjoint of g to that of f. this 2-category can equivalently be described in the following two ways:
1) a morphism from x to y is a filteredly cocontinuous functor from x# to y#, and a 2-morphism from f:x->y to g:x->y is a natural transformation from f to g.
2) a morphism from x to y is a toric map m from a dense toric open subvariety o of y to x, such that inverse image under f preserves affine toric varieties, equipped with a toric line bundle i over o.
(a "toric line bundle" over toric variety x is an object in t(x) invertible under tensor product. after "toric convolution" is defined, "toric line bundle" can alternatively but equivalently be defined as an ordinary line bundle with a cocommutative comonoid structure under toric convolution.)
a 2-morphism from (o,m,i):x->y to (o',m',i'):x->y requires o contain o' and m' = m restricted to o', and consists then of a morphism from i' to the pullback of i to o'.
composition of 1-morphisms and 2-morphisms is fairly straightforward; thus the composite of (o,m,i):x->y and (o',m',i'):y->z is defined on the intersection of o' and m'^*(o), and the line bundles are tensored after both being pulled back to this intersection.
for many of the ideas in this paper we can ask whether an analog is known or exists in the case of ordinary non-toric varieties; for theorem 2.1 perhaps the right analog would give the geometric interpretation of left-exact left adjoints between categories of ordinary quasicoherent sheaves.
the subtopos of t(y) given as the "image" of a geometric morphism (o,m,i) from t(x) to t(y) is again of the form t(z) for some toric variety z; in fact z=o. (thus (o,m,i) is surjective in the topos-theory sense iff m is totally defined from y to x in the toric variety sense; that is, iff o=y.) in fact, every nonempty subtopos of t(y) is of the form t(z) for some dense open toric subvariety z of y. a prominent example is the "double negation" subtopos of t(y), in which case the corresponding dense open toric subvariety is the dense torus in y; thus the double negation topology is responsible for the "toric" nature of toric varieties.
3 construction of the tensor product on t(x)
in the affine case, the tensor product of toric quasicoherent sheaves is simply the tensor product of actions of the corresponding commutative monoid (that is, day convolution for the commutative monoid viewed as single-object symmetric monoidal category). in the non-affine case, the usual construction of the global tensor product by glueing together the local tensor products goes through straightforwardly.
alternatively, we can understand the tensor product with the help of theorem 2.1.
given a toric variety x, consider the following four partial toric maps:
1) "binary multiplication" xXx -> x. this is total when x is affine but in general only partial; inverse image preserves affineness.
2) "nullary multiplication" 1 -> x. total, and inverse image preserves affineness.
3) "co-binary diagonal" x -> xXx. total, and inverse image preserves affineness.
4) "co-nullary diagonal" x -> 1. total, but inverse image preserves affineness only when x is affine.
applying theorem 2.1 to these four partial maps, we get geometric morphisms from the first three, but from the last one as well only when x is affine.
the partialness of 1) corresponds under theorem 2.1 to the non-essentialness of the co-binary diagonal operation of t(x); this shows that t(x) is totally distributive iff x is affine.
the totalness of 3) gives by 2.1 an essential geometric morphism, making t(x) into a symmetric semi-monoidal topos (which by the failure of 4 to give a geometric morphism is not fully monoidal). the extra left adjoint due to essentialness gives the tensor product of toric quasicoherent sheaves.
the filteredly cocontinuous functor corresponding to this geometric morphism can be thought of as the tensor product of categories enriched over the discrete symmetric closed monoidal category given by the dual lattice of the torus of the toric variety x, interpreting the objects of x# as such enriched categories. this makes x# into a symmetric semi-monoidal filteredly cocomplete category, as the unit object for this tensor product exists as an enriched category but not as an object of x#.
3 relationship of toric quasicoherent sheaves to ordinary quasicoherent sheaves on a toric variety
theorem 3.1: for a commutative ring k, the k-module objects in the topos t(x) are essentially the k-based ordinary quasicoherent sheaves over x.
thus since the ordinary quasicoherent sheaves over x are the k-module objects in a grothendieck topos, they can be tensored together like k-module objects in any such topos. this is not the ordinary tensor product of quasicoherent sheaves, however.
definition: "toric convolution" of k-based ordinary quasicoherent sheaves on a toric variety x is the tensor product arising here.
alternatively, toric convolution can be defined by patching together the "tensor product" functors corresponding to the bialgebra comultiplications on each affine toric open.
taking the free k-module object on a toric quasicoherent and then applying the equivalence of theorem 3.1 gives a quasicoherent sheaf which is cocommutative comonoidal wrt toric convolution. this is a full embedding from the topos of toric quasicoherent sheaves to the cartesian closed category of toric convolution cocommutative comonoids. it would be nice to have a good characterization of the image of this embedding so as to recover the topos t(x) from the k-module-enriched category of k-based ordinary quasicoherent sheaves over x equipped with its two tensor products (ordinary tensor product and toric convolution).
4 toric "proj" construction
let m be a commutative monoid "graded by an abelian group g"; in other words with a homomorphism h to g. this is equivalent to a strict symmetric monoidal category c(h) where all of the objects are strictly invertible and all of the self-braiding morphisms are identity morphisms. (the objects are the elements of g and the hom-set [g1,g2] is the fiber of h over g2-g1.) let t(h) be the symmetric monoidal object in the 2-category of toposes obtained as the presheaf category over c(h).
specialize to the case where g = the integers and grade 1 is finite and generates m, and consider the grothendieck pretopology on c(h) containing for each object n the cover of it by the morphisms from n-1. the sheaf topos for this topology is the "toric proj" construction proj(h). when h is the grading of the homogeneous coordinate monoid of a projective toric variety x, proj(h) is naturally equivalent to t(x) as a symmetric semi-monoidal topos.
addendum 2012-1-11: there should be a lot of material here from the "categorified bialgebra" viewpoint. i neglected this viewpoint originally because it didn't fit with a certain big picture that i was trying to develop, but after further consideration this viewpoint seems too central to the geometry of toric varieties to neglect.
addendum 2012-2-22: the situation isn't as simple as i thought when i wrote the previous addendum ...