# Coherence Spaces and Uniform Continuity

## TL;DR

This paper introduces a model of classical linear logic based on coherence spaces with totality, showing that uniform spaces can be represented and that uniformly continuous functions correspond to total linear maps within this framework.

## Contribution

It establishes a novel connection between coherence spaces with totality and uniform spaces, providing a new categorical model where uniform continuity is characterized by linear maps.

## Key findings

- Uniform spaces can be represented by coherence spaces with totality.
- Uniformly continuous functions correspond to total linear maps in the model.
- The model applies to separable, metrizable uniform spaces like the real line.

## Abstract

In this paper, we consider a model of classical linear logic based on coherence spaces endowed with a notion of totality. If we restrict ourselves to total objects, each coherence space can be regarded as a uniform space and each linear map as a uniformly continuous function. The linear exponential comonad then assigns to each uniform space X the finest uniform space !X compatible with X. By a standard realizability construction, it is possible to consider a theory of representations in our model. Each (separable, metrizable) uniform space, such as the real line, can then be represented by (a partial surjecive map from) a coherence space with totality. The following holds under certain mild conditions: a function between uniform spaces X and Y is uniformly continuous if and only if it is realized by a total linear map between the coherence spaces representing X and Y.