Fan is to monoid as scheme is to ring: a generalization of the notion of a fan

Following DeMeyer, Ford & Miranda [DFM93], we define a topology on a fan ∆ in R by declaring the open sets to be its subfans. Then, like Kato [Kat94], we make our fans into monoided spaces by associating a sheaf of monoids to each ∆. (Our sheaf of monoids differs from Kato’s.) Observing that this new monoided space is locally isomorphic to the spectrum of some monoid in the same sort of way a scheme is locally isomorphic to the spectrum of some ring, we define any monoided space that is locally isomorphic to the spectra of monoids to be a (generalized) fan. The monoid algebra functor can then be used to associate a scheme to such a fan. If we use the monoid algebra functor over some base field k, the resultant scheme is a normal variety if and only if it is a toric variety.