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 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.
Highlights
U Space consisting of U -spaces and U -morphisms, which is shown to be Cartesian closed
Internal set theory (IST) is a syntactical approach to nonstandard analysis consisting of the “principle of Idealization (I)” and the two more basic principles, called “principle of Standardization (S)” and “Transfer principle (T)”
We will take an example of basic applications of nonstandard analysis within our framework, i.e., the characterization of continuity and uniform continuity in terms of a relation ≈ (“infinitely close”) on U ( X ), which is based on essentially the same arguments that are well-known in nonstandard analysis—internal set theory [4]
Summary
Nonstandard analysis and category theory are two of the great 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]. The first axiom (“elementarity axiom”) introduced in Section 2 states that the endofunctor U should preserve all finite limits and finite coproducts. 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
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