Categorical Nonstandard Analysis

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.


Introduction
Nonstandard analysis and category theory are two of the great inventions in foundation (or organization) of mathematics . Both of them have provided productive viewpoints to organize many kinds of topics in mathematics or related fields [5,3]. On the other hand, unification of two theories still seems to be developed, although there are some pioneering works such as [1].
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 endofunctor an 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 give just 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, but we prove them from our framework for the reader's convenience. In section 5, we characterize the notion of bornologous map, which is a fundamental notion in coarse geometry [6].
In section 6, we introduce the notion of U-space and U-morphism, which are the generalization of examples in the previous two sections. We introduce the category USpace consisting of U-spaces and U-morphisms, which is shown to be cartesian closed. It 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.
From the discussion above, a set X in S is to be considered as a canonical subset of U(X) through υ X : X −→ U(X). Hence, U(f ) : U(X 0 ) −→ U(X 1 ) can be considered as "the function induced from f : X 0 −→ X 1 through υ." where 1 X ×ĝ denotes the function satisfying We define a family of functions κ A,B : The theorem below means that κ A,B • υ B A represents "inducing U(f ) from f through υ" in terms of exponentials.
By naturality of υ and functorial properties of U, it is calculated as follows: Notation 8. From now on, we omit υ and κ. U(f ) : U(X) −→ U(Y ) will be often identified with f : X −→ Y and denoted simply as f instead of U(f ).
Theorem 9. Let P : X −→ 2 be any proposition (function in S). Then Dually, we obtain the following: The two theorems above are considered as the simplest versions of "transfer principle". To treat with free variables and quantification, the theorem below is important. (The author thanks Professor Anders Kock for indicating this crucial point.) Theorem 11. U preserves images.
Proof. As U preserves all finite limits, it preserves monics. On the other hand, it also preserves epic since every functor preserves split epics and every epic in S is split epic (axiom of choice). Hence the image, which is nothing but the epi-mono factorization, is preserved.

Idealization Axiom
From our viewpoint, nonstandard Analysis is nothing but a method of using an endofunctor which satisfies "elementarity axiom" and the following "idealization axiom". The name is after "the principle of idealization" in Nelson's internal set theory [4]. Most of the basic idea in this section is much in common with [4], although the functorial approach is not taken in internal set theory.

Remark 12. 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)". In our framework, the role of (S) is played by the axiom of choice for S and (T) corresponds to the contents of section 2.
Notation 13. For any set X,X denotes the set of all finite subsets of X.
Idealization Axiom: Let P be an element of U(2 X×Y ). Then Or dually, Idealization Axiom, dual form: Let P be an element of U(2 X×Y ) Then When X is a directed set with an order ≤ and P ∈ U(2 X×Y ) satisfies the "filter condition", i.e., or dually, the "cofilter condition", i.e., then idealization axiom is simplified as "commutation principle": Theorem 14 (Commutation Principle). If P ∈ U(2 X×Y ) satisfies the "filter condition" and "cofilter condition" above, respectively, then holds, respectively.
By the principle above, we can easily prove the existence of "unlimited numbers" in U(N), where all arithmetic operation and order structure on N is naturally extended.
Theorem 15 (Existence of "unlimited numbers"). There exists some ω ∈ U(N) such that n ≤ ω for any n ∈ N.
Proof. Because it is obvious that for any n ∈ N there exists some ω ∈ N ⊂ U(N) such that n < ω.
As in S we can construct rational numbers and the completion of them as usual, we have the object R, the set of real numbers. Then we obtain the following: Corollary 16. "Infinitesimals" do exist in U(R). That is, there exists some r ∈ U(R) such that |r| < R for any positive R ∈ R.

Topological Structure: Continuous Map and Uniform Continuous Map
We will take an example of basic applications of nonstandard analysis within our framework, i.e., characterization of continuity and uniform continuity in terms of a relation ≈ ("infinitely close") on U(X), which is based on essentially the same arguments well-known in nonstandard analysis, especially, internal set theory [4]. For simplicity, we will discuss only for metric spaces here. (For more general topological spaces, we can define ≈ in terms of the system of open sets. See [4] for example.) Definition 17 (Infinitely close). Let (X, d) be a metric space. We call the relation ≈ on U(X) defined below as "infinitely close": That is, d(x, x ′ ) is infinitesimal. It is easy to see that ≈ is an equivalence relation on U(X).

holds.
Proof. We can translate the condition for f by using usual logic, "commutation principle" and "transfer principle" as follows: Theorem 19 (Characterization of uniform continuity). Let (X 0 , d 0 ), (X 1 , d 1 ) be metric spaces and ≈ 0 , ≈ 1 be infinitely close relations on them, respectively. A map f : X 0 −→ X 1 is uniformly continuous if and only if holds.
Proof. We can translate the condition for f by using usual logic, "commutation principle" and "transfer Principle" as follows: As we have seen, a morphism between metric spaces is characterized as "a morphism with respect to ≈". This suggests the possibility for considering other kinds of "equivalence relations on (some subset of) U(X)" as generalized spatial structures on X. In the next section, we will take one example related to large scale geometric structure.

Coarse Structure: Bornologous Map
Let us consider another kind of equivalence relation ∼ ("finitely remote") defined below. For simplicity, we will discuss only for metric spaces here.
Definition 20 (Finitely remote). Let (X, d) be a metric space. We call the relation ∼ on U(X) defined below as "finitely remote": Note that we use ∃ instead of ∀, in contrast to "infinitely close". This kind of dual viewpoint will be proved to be useful in the geometric study of large scale structure such as coarse geometry [6].
In fact, we can prove that "bornologous map", a central notion of a morphism for coarse geometry, can be characterized as "a morphism with respect to ∼", just like (uniform) continuity can be viewed as "a morphism with respect to ≈".
Proof. We can translate the condition for f by using usual logic, "commutation principle" and "transfer principle" as follows: 6 The notion of U -space and the Category U Space Based on the characterizations of topological and coarse geometrical structure, we introduce the notion of U-space.
Definition 23 (U-space). A U-space is a triple (X, K, ) consisting of a set X, a subset K of U(X) which includes X as a subset and a preorder defined on K.
When the preorder is an equivalence relation, i.e., a preorder satisfying symmetry, we call the U-space symmetric. A symmetric U-space (X, K, ) is called uniform if K = U(X). The "infinitely close" relation and the "finitely remote" relation provide the simplest examples of uniform U-space structure.
Actually, any topological space X with the set of open sets T can be viewed as U-space (X, U(X), ⇀) where x ⇀ x ′ denotes the preorder "∀O ∈ T x ∈ U(O) =⇒ x ′ ∈ U(O)". If (X, T ) is a Hausdorff space, we can construct the symmetric U-space (X, K, ), where K denotes x ′ is defined as the relation "∃x 0 ∈ X x 0 ⇀ x&x 0 ⇀ x ′ ." The transitivity of ⇀ follows from the fact that if (X, T ) is Hausdorff, x 0 ⇀ x and Actually, the preorder becomes an equivalence relation.
The concept of U-space will provide a general framework to unify various spatial structure such as topological structure and coarse structure. The notion of morphism between U-spaces is defined as follows: Definition 24 (U-spatial morphism). Let (X 0 , K 0 , 0 ) and (X 1 , K 1 , 1 ) be U-spaces. A function f : X 0 → X 1 is called a U-spatial morphism from (X 0 , K 0 , 0 ) to (X 1 , K 1 , 1 ) when f (K 0 ) ⊂ K 1 and holds for any x, x ′ ∈ K 0 .
The uniform continuous maps and bornologous maps between metric spaces are nothing but U-spatial morphisms between corresponding uniform U-spaces. The notion of continuous maps between Hausdorff spaces can be characterized as U-spatial morphisms between the corresponding symmetric U-spaces.
Definition 25 (Category USpace). The category USpace is a category whose objects are U-spaces and whose morphisms are U-spatial morphisms.
where the preorder is defined as is called the product U-space of (X 0 , K 0 , 0 ) and (X 1 , K 1 , 1 ).
Theorem 27. The projections become U-spatial morphisms. The diagram consisting of two U-spaces, the product space of them and projections become a product in USpace.
By assumption that f is U-spatial, holds, where denote the preorder on K 0 ×K 2 . It is equivalent to the statement that c, c ′ ∈ K 2 and c 2 c ′ imply Applying the implication above for the case c = c ′ ∈ X 2 , we havef (c) ∈