Sheaves of metric structures

TL;DR

This paper develops the theory of metric sheaves, constructing a generic model from sheaves of metric models and analyzing its semantics through forcing rules, advancing the understanding of metric structures in a sheaf-theoretic context.

Contribution

It introduces metric sheaves and constructs generic models, integrating metric structures with sheaf theory and forcing semantics for the first time.

Findings

Defined metric sheaves on topological spaces.

Constructed generic models via quotient spaces.

Provided a semantics framework controlled by forcing rules.

Abstract

We introduce and develop the theory of metric sheaves. A metric sheaf $\A$ is defined on a topological space $X$ such that each fiber is a metric model. We describe the construction of the generic model as the quotient space of the sheaf through an appropriate filter. Semantics in this model is completely controlled and understood by the forcing rules in the sheaf.