Sheaves of Metric Structures

Abstract

We introduce sheaves of metric structures and develop their basic model theory. The metric sheaves defined here provide a way to construct new metric models on sheaves (a strong generalization of the ultraproduct construction), with the additional property of having the theory of the resulting model controlled by the topology of a given space. More specifically, a metric sheaf $\mathfrak {A}$ is defined on a topological space X such that each fiber is a metric model. A new model, the generic metric model, is obtained as the quotient space of the sheaf through an appropriate filter of open sets. Semantics in the generic model is completely controlled and understood by the forcing rules in the sheaf and Theorem 3. This work extends early constructions due to Comer [5] and Macintyre [12] and later developments due to Caicedo [3], to the context of continuous logic. We illustrate these concepts by studying the metric sheaf of the continuous cyclic flow on tori.

A. Villaveces—The second author was partially supported by Colciencias (Departamento Administrativo de Ciencia, Tecnología e Innovación) for the research presented here.