TITLE: Algebraic geometry for cocommutative corings with counit.
QUESTION [8 upvotes]: Is there a notion of algebraic geometry for these objects? If we take the dual category of the category of cocommutative corings with counit, is there geometry in it in a sense dual to affine schemes? Can we look at the set of coideals of a coring, put a space structure on it and sheaves (maybe cosheaves) of sections?

REPLY [5 votes]: One can do (relative) algebraic geometry with respect to any symmetric monoidal category, with affine schemes corresponding to the opposite category of (commutative) monoid objects. This viewpoint is developed in the paper Au-dessous de Spec Z (in French) by Toen and Vaquie. By taking the category of $\mathbb{Z}$--modules with tensor products over $\mathbb{Z}$ one recovers usual algebraic geometry.
For (cocommutative) corings, you just have to do the same in the opposite category $(\mathbb{Z}-Mod)^{op}$. The big question is: what do you take as your monoidal structure? If you just dualize diagrams, your new coproduct would be the old product, i.e. direct product of the underlying abelian groups. Or instead of this you take usual tensor products. Each of these approaches will produce a different geometry.
I remember a course given by Lieven Le Bruyn on Kontsevich-Soibelman (the noncommutative coalgebra thing) and he mentioning that by the "discrete" nature of coalgebras one could never expect them to describe anything but the etale topology of a scheme, which I think is precisely what Leonid observed in his answer.

REPLY [4 votes]: Leonid speaks with great authority.  I'd like to point out that you get a different answer in the category of graded coalgebras.  If $A$ is a graded vector space, it has a graded dual $A'$ which is smaller than its full vector space dual $A^*$.  Indeed, if the grading is locally finite, then this $A'$ has the same dimension sequence as $A$ and $A'' = A$.  If $A$ is a locally finite, graded coalgebra, then its ring structure is given by an infinite sequence of finite tensors, and $A'$ is equivalently a graded algebra.  Graded coalgebra homomorphisms also transpose to graded algebra homomorphisms.  Recall that if $A'$ is also finitely generated (and scalar in degree $0$), then it corresponds to a projective variety with a choice of an ample line bundle.  The morphisms between these are a perfectly good non-full subcategory of the projective schemes.
The full dual $A^*$ is also the graded completion of $A'$.  So what Leonid is saying in this case is that if you take the graded completion of a finitely generated, graded algera, it localizes the projective variety to the apex of its affine cone.  This is similar to what Leonid describes for $\text{Spec}(A^*)$ in general.
I said in a previous version of this answer that $\text{Spec}(A^*)$ atomizes the variety $\text{Proj}(A')$ to a "Cantor-set-like" structure.  As Leonid points out in a comment, this is totally wrong in context.  However, it is true that if $A$ is a finitely generated algebra, it has a completion $\hat{A}$ related to coalgebras such that $\text{Spec}(\hat{A})$ is an atomized form of $\text{Spec}(A)$.  Namely, let $A'$ be the vector space of those dual vectors on $A$ that factor through finite-dimensional algebra quotients of $A$.  Then $A'$ is a coalgebra, and $\hat{A} = (A')^*$ is an atomization in the sense that $\text{Spec}(\hat{A})$ is a 0-dimensional scheme whose points are the closed points of $\text{Spec}(A)$.  (It isn't a Cantor set though.)