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Article

Categorical Nonstandard Analysis

by 1,* and 2
1
Nagahama Institute of Bio-Science and Technology, Shiga 526-0829, Japan
2
Independent Reseacher, Tokyo 103-0008, Japan
*
Author to whom correspondence should be addressed.
Academic Editor: Stefano Profumo
Symmetry 2021, 13(9), 1573; https://doi.org/10.3390/sym13091573
Received: 30 July 2021 / Revised: 21 August 2021 / Accepted: 23 August 2021 / Published: 26 August 2021
(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)
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 USpace whose objects are U-spaces and morphisms are functions called U-spatial morphisms. The category USpace, 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. View Full-Text
Keywords: category theory; nonstandard analysis; coarse geometry category theory; nonstandard analysis; coarse geometry
MDPI and ACS Style

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

AMA Style

Saigo H, Nohmi J. Categorical Nonstandard Analysis. Symmetry. 2021; 13(9):1573. https://doi.org/10.3390/sym13091573

Chicago/Turabian Style

Saigo, Hayato, and Juzo Nohmi. 2021. "Categorical Nonstandard Analysis" Symmetry 13, no. 9: 1573. https://doi.org/10.3390/sym13091573

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