*Article* **Categorical Nonstandard Analysis**

**Hayato Saigo 1,\* and Juzo Nohmi 2**

	- **\***Correspondence: h\_saigoh@nagahama-i-bio.ac.jp

**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 U on a topos of sets *S* together with a natural transformation *υ*, instead of the terms as "standard", "internal", or "external". Moreover, we propose a general notion of a space called U-space, and the category U*Space* whose objects are U-spaces and morphisms are functions called U-spatial morphisms. The category U*Space*, which is shown to be Cartesian closed, gives a unified viewpoint toward topological and coarse geometric structure. It will also be useful to further study symmetries/asymmetries of the systems with infinite degrees of freedom, such as quantum fields.

**Keywords:** category theory; nonstandard analysis; coarse geometry
