Toen-Vaquié construct a category of schemes relative to some complete cocomplete closed symmetric monoidal category $C$. Affine schemes correspond by definition 1:1 to commutative monoid objects in $C$. For $C=\mathsf{Ab}$ we get the usual category of schemes, where affine schemes correspond to commutative rings. For $C=\mathsf{Set}$ one gets one of the various definitions of schemes over $\mathbb{F}_1$. One of the drawbacks of this quite general theory is that schemes are defined via their functors on commutative monoid objects, without any geometric incarnation.

**Question.** What happens when $C$ is the category of graded abelian groups (equipped either with the usual symmetry, or with the twisted symmetry)? Here affine schemes correspond to (graded) commutative rings. Is there any connection with the usual Proj construction? Is there any more geometric interpretation of these schemes? For example one might hope for a fully faithful functor into the category of locally ringed spaces. Is this category of schemes something new at all?

graded-ringedspaces. These are spaces with a sheaf of graded rings so that the stalks aregraded localin the sense that they have one homogeneous maximal ideal. – Dylan Wilson Mar 14 '13 at 12:34