Approximation of structures:local and global

TL;DR

This paper introduces a rigorous definition of local approximation for structures, demonstrating that compact simple Lie groups can be locally approximated by finite groups using model-theoretic ultraproduct techniques.

Contribution

It provides a new formal framework for local approximation of structures, with applications to Lie groups, inspired by physics and developed through model theory.

Findings

Any compact simple Lie group is locally approximated by finite groups.

Introduces ultraproducts of metric structures to generalize ultraproducts in metric model theory.

Defines local approximation in a mathematically rigorous way.

Abstract

We provide a mathematically rigorous definition of local approximation and demonstrate its applicability to some interesting classes of structures. In particular, we prove that any compact simple Lie group is locally approximated by finite groups. The definition and main examples are motivated by physics but the techniques are of model theory. Namely, we introduce the ultraproduct of emerging metric structures, which generalises the ultraproduct in metric model theory.