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?



Algebraic geometry for cocommutative corings with counit.
Let's consider coalgebras over a field rather than corings. There is a theorem that every (coassociative) coalgebra over a field is the union of its finite-dimensional subcoalgebras. So the category of coalgebras over a field k is the category of ind-objects in the category of finite-dimensional coalgebras, while the latter is the opposite category to the category of finite-dimensional algebras. Now if we restrict ourselves to the commutative case, then the category of finite-dimensional commutative algebras with unit over k is the opposite category to the category of 0-dimensional schemes of finite type over k, or just schemes finite over Spec k. Combining it all, the category of cocommutative coalgebras with counit over k is equivalent to the category of ind-0-dimensional ind-schemes of ind-finite type over k, or just ind-finite ind-schemes over Spec k. This explains, in particular, that the "underlying topological space" functor maps cocommutative coalgebras to discrete sets rather than anything else, and the coalgebra itself is simply the infinite direct sum of the coalgebras sitting at the points of this set.


