Fra\"iss\'e limits of metric structures

TL;DR

This paper extends Fra"iss"e theory to metric structures, establishing a correspondence between Fra"iss"e classes and separable approximately homogeneous structures using approximate isometries.

Contribution

It introduces a new approach to Fra"iss"e theory in metric structures by employing approximate isometries to encode finite maps up to errors.

Findings

Characterizes Fra"iss"e classes as ages of separable approximately homogeneous structures.

Proves the uniqueness and universality of the Fra"iss"e limit in this context.

Develops a novel method using approximate isometries for the theory.

Abstract

We develop \emph{Fra\"iss\'e theory}, namely the theory of \emph{Fra\"iss\'e classes} and \emph{Fra\"iss\'e limits}, in the context of metric structures. We show that a class of finitely generated structures is Fra\"iss\'e if and only if it is the age of a separable approximately homogeneous structure, and conversely, that this structure is necessarily the unique limit of the class, and is universal for it. We do this in a somewhat new approach, in which ''finite maps up to errors'' are coded by \emph{approximate isometries}.