Categorical Nonstandard Analysis

TL;DR

This paper introduces a new axiomatic framework for nonstandard analysis using category theory, defining $$-spaces and their morphisms to unify topological and geometric structures.

Contribution

It proposes a novel categorical axiomatization of nonstandard analysis and introduces $$-spaces, providing a unified approach to topology, coarse geometry, and systems with infinite degrees of freedom.

Findings

Category $$Space is cartesian closed.

Unified framework for topological and coarse geometric structures.

Potential applications to quantum field symmetries.

Abstract

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while we make use of an endofunctor $\mathcal{U}$ on a topos of sets $S$ together with a natural transformation $\upsilon$, instead of the terms as "standard", "internal" or "external". Moreover, we propose a general notion of a space called $\mathcal{U}$-space, and the category $\mathcal{U}space$ whose objects are $\mathcal{U}$-spaces and morphisms are functions called $\mathcal{U}$-spatial morphisms. The category $\mathcal{U}Space$, which is shown to be cartesian closed, will give a unified viewpoint toward topological and coarse geometric structure. It will also useful to study symmetries/asymmetries of the systems with infinite degrees of freedom such as quantum fields.