**1. Introduction**

Nonstandard analysis and category theory are two of the grea<sup>t</sup> inventions in foundation (or organization) of mathematics . Both of these have provided productive viewpoints to organize many kinds of topics in mathematics or related fields [1,2]. On the other hand, a unification of the two theories is ye<sup>t</sup> to be developed, although there are some pioneering works, such as [3].

In the present paper, we propose a new axiomatic framework for nonstandard analysis in terms of category theory. Our framework is based on the idea of internal set theory [4], while we make use of an endofunctor U on a topos of sets S together with a natural transformation *υ*, instead of the terms as "standard", "internal", or "external".

The triple (S, U, *υ*) is supposed to satisfy two axioms. The first axiom ("elementarity axiom") introduced in Section 2 states that the endofunctor U should preserve all finite limits and finite coproducts. Then, the endofunctor U is viewed as some kind of extension of functions preserving all elementary logical properties. In Section 3, we introduce another axiom ("idealization axiom"), which is the translation of "the principle of idealization" in internal set theory and proves the appearance of useful entities, such as infinitesimals or relations, such as "infinitely close", in the spirit of Nelson's approach to nonstandard analysis [4].

Section 4 is devoted to provide a few examples of applications on topology (on metric spaces, for simplicity). Although the characterizations of continuous maps or uniform continuous maps in terms of nonstandard analysis are well known, we prove them from our framework for the reader's convenience. In Section 5, we characterize the notion of a bornologous map, which is a fundamental notion in coarse geometry [5].

In Section 6, we introduce the notion of U-space and U-morphism, which are the generalizations of examples in the previous two sections. We introduce the category U*Space* consisting of U-spaces and U-morphisms, which is shown to be Cartesian closed. This will give a unified viewpoint toward topological and coarse geometric structure, and will be useful to study symmetries/asymmetries of the systems with infinite degrees of freedom, such as quantum fields.

**Citation:** Saigo, H.; Nohmi, J. Categorical Nonstandard Analysis. *Symmetry* **2021**, *13*, 1573. https:// doi.org/10.3390/sym13091573

Academic Editor: Stefano Profumo

Received: 30 July 2021 Accepted: 23 August 2021 Published: 26 August 2021

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