coalgebras - Algebraic geometry for cocommutative corings with counit.

I like to think of algebraic geometry as being born out of the fact that Ring behaves a lot like Set^op. For instance, in Ring, A ∐ (B × C) = (A ∐ B) × (A ∐ C), where ∐ is the coproduct in Ring, which is just the tensor product. This formula is also true in Set after we swap ∐ and ×. This suggests that we can use our intuition about Set to think about Ring if we replace Ring by its opposite, the category of affine schemes.

Cocommutative corings with counit are monoid objects in the opposite category of (Ab, ⊗). However, to get the morphisms to point in the right direction, we need to take the opposite category again: so Coring is (CAlg((Ab, ⊗)^op))^op. It follows that products in Coring are computed by tensor products in Ab, and colimits are formed by taking colimits of underlying abelian groups; and in fact Coring is a closed cartesian category, even more like Set than Ring^op is. In particular, we don't want to take the opposite category of Coring. Maybe this isn't surprising, since every set is already a cocommutative comonoid (w.r.t. ×) in a unique way, and we have a functor Set → Coring taking a set to the free abelian group on it.

These are purely formal observations, and I don't know whether anyone has built a more concrete geometric theory, with say a functor from Coring to some kind of topological spaces with extra structure.