Cauchy completeness and causal spaces

TL;DR

This paper explores the use of enriched category theory to model relativistic causal spaces, establishing conditions under which these spaces are Cauchy complete and linking this completeness to the splitting of idempotent arrows.

Contribution

It introduces a novel base of enrichment for describing causal spaces and characterizes their Cauchy completeness in terms of idempotent splitting.

Findings

All causal spaces considered are Cauchy complete.

Cauchy completeness is equivalent to splitting of idempotent arrows under certain conditions.

Provides a new categorical framework for relativistic causal structures.

Abstract

Following Lawvere's description of metric spaces using enriched category theory, we introduce a change in the base of enrichment that allows description of some aspects of (relativistic) causal spaces. All such spaces are Cauchy complete, in the sense of enriched category theory. Furthermore, we give sufficient conditions on a base monoidal category for which enriched categories are Cauchy complete if and only if their underlying categories are (their idempotent arrows split).