# Categorical semantics of metric spaces and continuous logic

## TL;DR

This paper develops a categorical semantics for metric spaces that parallels classical logic, establishing an equivalence with continuous logic and introducing the concept of continuous subobject classifiers.

## Contribution

It introduces a metric analogue of categorical logic semantics, demonstrating the equivalence with continuous logic and defining continuous subobject classifiers in suitable categories.

## Key findings

- Real interval [0,1] acts as a continuous subobject classifier
- Categories of presheaves of metric spaces support continuous semantics
- Establishes a correspondence between predicate notions and quantification in continuous logic

## Abstract

Using the category of metric spaces as a template, we develop a metric analogue of the categorical semantics of classical/intuitionistic logic, and show that the natural notion of predicate in this "continuous semantics" is equivalent to the a priori separate notion of predicate in continuous logic, a logic which is independently well-studied by model theorists and which finds various applications. We show this equivalence by exhibiting the real interval [0,1] in the category of metric spaces as a "continuous subobject classifier" giving a correspondence not only between the two notions of predicate, but also between the natural notion of quantification in the continuous semantics and the existing notion of quantification in continuous logic. Along the way, we formulate what it means for a given category to behave like the category of metric spaces, and afterwards show that any such category supports the aforementioned continuous semantics. As an application, we show that categories of presheaves of metric spaces are examples of such, and in fact even possess continuous subobject classifiers.