Scheme representation for first-order logic

Recall that every commutative ring R determines an affine scheme in algebraic geometry. This consists of two components: a topological space Spec(R) (the spectrum) and a sheaf of local rings OR (the structure sheaf). In this way, a scheme encodes both geometric and algebraic data. In this work, we present a construction of “logical schemes,” geometric entities which represent logical theories in much the same way that algebraic schemes represent rings. These also involve two components: a semantic spectral space and a syntactic structure sheaf. As in the algebraic case, we can recover a theory from its scheme representation (up to a conservative completion) and the structure sheaf is local in a certain logical sense. From these affine pieces we can build up a 2-category of logical schemes which share some of the nice properties of algebraic schemes. The “logical spectrum” of a first-order theory T is constructed from the semantics of T. Points in Spec(T) are models supplemented by certain variable assignments or labellings. Satisfaction induces a topology on these points, much as in the Stone space construction for propositional logic. As in the algebraic case, the spectrum is not a Hausdorff space. Instead the topology incorporates model theoretic information; notably, the closure of a point (model) M ∈ Spec(T) can be interpreted as the set of model homomorphisms into M . Every formula φ(x1, . . . , xn) determines a “definable sheaf” [[φ]] over the spectrum. Over each model M , the fiber of [[φ]] is the definable set