  There are many examples of dualities between topological spaces and algebras
in the literature. Particularly, many of those examples come from the algebraic
counterpart of a logical system, e.g, boolean and heyting algebras,
MV-algebras, monoidal categories and linear logic etc. This process can be
generalized to a representation of an algebraic structure by a sheaf in a
suitable topological space. Recently, Awodey and his contributors constructed a
duality between first order logic and topological grupoids from which they have
proven some results of completeness. Furthermore, they built a notion of scheme
of a first order theory. The attempt of this project is to use the construction
of abstract schemes invented by Toen to give an unified approach from which we
can generalize the aforementioned results.
