**3. Idealization Axiom**

From our viewpoint, nonstandard Analysis is nothing but a method of using an endofunctor, which satisfies the "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 ideas in this section have much in common with [4], although the functorial approach is not taken in internal set theory.

**Remark 2.** *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 Standard-* *ization (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 2.** *For any set X, X denotes the set of all finite subsets of X.* ˜

Idealization Axiom: Let *P* be an element of <sup>U</sup>(2*X*×*<sup>Y</sup>*). Then,

$$\forall\_{\mathbf{x'} \in \mathcal{R}} \exists\_{\mathbf{y} \in \mathcal{U}(\mathbf{y'})} \forall\_{\mathbf{x} \in \mathbf{x'}} P(\mathbf{x}, \mathbf{y}) \iff \exists\_{\mathbf{y} \in \mathcal{U}(\mathbf{y'})} \forall\_{\mathbf{x} \in \mathbf{x}} P(\mathbf{x}, \mathbf{y}) .$$

Or dually,

Idealization Axiom, dual form: Let *P* be an element of U(2*X*×*<sup>Y</sup>*) Then,

$$
\exists\_{\mathbf{x'} \in \mathcal{X}} \forall\_{\mathbf{y} \in \mathcal{U}(\mathbf{y'})} \exists\_{\mathbf{x} \in \mathcal{X}} P(\mathbf{x}, \mathbf{y}) \iff \forall\_{\mathbf{y} \in \mathcal{U}(\mathbf{y'})} \exists\_{\mathbf{x} \in \mathcal{X}} P(\mathbf{x}, \mathbf{y}) .
$$

When *X* is a directed set with an order ≤ and *P* ∈ U(2*X*×*<sup>Y</sup>*) satisfies the "filter condition", i.e.,

$$\forall\_{\mathbf{x}\_0 \in X} (P(\mathbf{x}\_0, y) \implies \forall\_{\mathbf{x} \in X} (\mathbf{x} \le \mathbf{x}\_0 \implies P(\mathbf{x}, y))),$$

or dually, the "cofilter condition", i.e.,

$$\forall\_{\mathbf{x}\_0 \in \mathcal{X}} (P(\mathbf{x}\_{0\prime} y) \implies \forall\_{\mathbf{x} \in \mathcal{X}} (\mathbf{x}\_0 \le \mathbf{x} \implies P(\mathbf{x}, y))),$$

then "idealization axiom" is simplified as the "commutation principle":

**Theorem 7** (Commutation Principle)**.** *If P* ∈ U(2*X*×*<sup>Y</sup>*) *satisfies the "filter condition" and "cofilter condition" above, respectively, and then*

$$\forall\_{\mathbf{x}\in\mathcal{X}}\exists\_{\mathbf{y}\in\mathcal{U}(\mathbf{y})}P(\mathbf{x},\mathbf{y}) \iff \exists\_{\mathbf{y}\in\mathcal{U}(\mathbf{y})}\forall\_{\mathbf{x}\in\mathcal{X}}P(\mathbf{x},\mathbf{y})$$

*and*

$$\exists\_{\mathbf{x}\in\mathcal{X}}\forall\_{\mathbf{y}\in\mathcal{U}(\mathbf{Y})}P(\mathbf{x},\mathbf{y}) \Longleftrightarrow \forall\_{\mathbf{y}\in\mathcal{U}(\mathbf{Y})}\exists\_{\mathbf{x}\in\mathcal{X}}P(\mathbf{x},\mathbf{y})$$

*holds, respectively.*

By the principle above, we can easily prove the existence of "unlimited numbers" in U(N), where all arithmetic operations and order structure on N are naturally extended.

**Theorem 8** (Existence of "unlimited numbers")**.** *There exists some ω* ∈ U(N) *such that n* ≤ *ω for any n* ∈ N*.*

**Proof.** 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 3.** *"Infinitesimals" do exist in* U(R)*. That is, there exists some r* ∈ U(R) *such that* |*r*| < *R for any positive R* ∈ R*.*

### **4. Topological Structure: Continuous Map and Uniform Continuous Map**

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—particularly, 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 2** (Infinitely close)**.** *Let* (*<sup>X</sup>*, *d*) *be a metric space. We call the relation* ≈ *on* U(*X*) *defined below as "infinitely close":*

$$\mathfrak{x} \approx \mathfrak{x}' \Longleftrightarrow \forall\_{\mathfrak{e} \in \mathbb{R}} d(\mathfrak{x}, \mathfrak{x}') < \mathfrak{e}.$$

That is, *d*(*<sup>x</sup>*, *x*) is infinitesimal. It is easy to see that ≈ is an equivalence relation on U(*X*).

**Theorem 9** (Characterization of continuity)**.** *Let* (*<sup>X</sup>*0, *d*0),(*<sup>X</sup>*1, *d*1) *be metric spaces and* <sup>≈</sup>0, <sup>≈</sup>1 *be infinitely close relations on them, respectively. A map f* : *X*0 −→ *X*1 *is continuous if and only if*

$$\forall\_{\mathbf{x}\in\mathcal{X}\_{0}}\forall\_{\mathbf{x'}\in\mathcal{U}(\mathcal{X}\_{0})} \left(\text{ }\mathbf{x}\Leftrightarrow\_{0}\text{ }\mathbf{x'}\Rightarrow f(\mathbf{x})\Leftrightarrow\_{1}f(\mathbf{x'})\text{ }\right)$$

*holds.*

**Proof.** We can translate the condition for *f* by using the usual logic, "commutation principle", and "transfer principle" as follows:

<sup>∀</sup>*x*∈*X*0∀*x*∈U(*<sup>X</sup>*0) ( *x* <sup>≈</sup>0 *x* ⇒ *f*(*x*) <sup>≈</sup>1 *f*(*x*) ) ⇐⇒ <sup>∀</sup>*x*∈*X*0∀*x*∈U(*<sup>X</sup>*0) ( ∀*δ*∈<sup>R</sup> *d*0(*<sup>x</sup>*, *x*) < *δ* ⇒ ∀∈<sup>R</sup> *d*1(*f*(*x*), *f*(*x*)) < ) ⇐⇒ <sup>∀</sup>*x*∈*X*0∀*x*∈U(*<sup>X</sup>*0)∀∈R∃*δ*∈<sup>R</sup> ( *d*0(*<sup>x</sup>*, *x*) < *δ* ⇒ *d*1(*f*(*x*), *f*(*x*)) < ) ⇐⇒ <sup>∀</sup>*x*∈*X*0∀∈R∀*x*∈U(*<sup>X</sup>*0)∃*δ*∈<sup>R</sup> ( *d*0(*<sup>x</sup>*, *x*) < *δ* ⇒ *d*1(*f*(*x*), *f*(*x*)) < ) ⇐⇒ <sup>∀</sup>*x*∈*X*0∀∈R∃*δ*∈R∀*x*∈U(*<sup>X</sup>*0) ( *d*0(*<sup>x</sup>*, *x*) < *δ* ⇒ *d*1(*f*(*x*), *f*(*x*)) < ) ⇐⇒ <sup>∀</sup>*x*∈*X*0∀∈R∃*δ*∈R∀*x*∈*X*0 ( *d*0(*<sup>x</sup>*, *x*) < *δ* ⇒ *d*1(*f*(*x*), *f*(*x*)) < ).

**Theorem 10** (Characterization of uniform continuity)**.** *Let* (*<sup>X</sup>*0, *d*0),(*<sup>X</sup>*1, *d*1) *be metric spaces and* <sup>≈</sup>0, <sup>≈</sup>1 *be infinitely close relations on them, respectively. A map f* : *X*0 −→ *X*1 *is uniformly continuous if and only if*

$$\forall\_{\mathbf{x}\in\mathcal{U}(\mathcal{X}\_0)} \forall\_{\mathbf{x'}\in\mathcal{U}(\mathcal{X}\_0)} \left(\mathbf{x}\approx\_0 \mathbf{x'} \Rightarrow f(\mathbf{x}) \approx\_1 f(\mathbf{x'})\right)$$

*holds.*

**Proof.** We can translate the condition for *f* by using usual logic, "commutation principle" and "transfer Principle" as follows:

<sup>∀</sup>*x*∈U(*<sup>X</sup>*0)∀*x*∈U(*<sup>X</sup>*0) ( *x* <sup>≈</sup>0 *x* ⇒ *f*(*x*) <sup>≈</sup>1 *f*(*x*) ) ⇐⇒ <sup>∀</sup>*x*∈U(*<sup>X</sup>*0)∀*x*∈U(*<sup>X</sup>*0) ( ∀*δ*∈<sup>R</sup> *d*0(*<sup>x</sup>*, *x*) < *δ* ⇒ ∀∈<sup>R</sup> *d*1(*f*(*x*), *f*(*x*)) < ) ⇐⇒ <sup>∀</sup>*x*∈U(*<sup>X</sup>*0)∀*x*∈U(*<sup>X</sup>*0)∀∈R∃*δ*∈<sup>R</sup> ( *d*0(*<sup>x</sup>*, *x*) < *δ* ⇒ *d*1(*f*(*x*), *f*(*x*)) < ) ⇐⇒ <sup>∀</sup>∈R∀*x*∈U(*<sup>X</sup>*0)∀*x*∈U(*<sup>X</sup>*0)∃*δ*∈<sup>R</sup> ( *d*0(*<sup>x</sup>*, *x*) < *δ* ⇒ *d*1(*f*(*x*), *f*(*x*)) < ) ⇐⇒ <sup>∀</sup>∈R∃*δ*∈R∀*x*∈U(*<sup>X</sup>*0)∀*x*∈U(*<sup>X</sup>*0) ( *d*0(*<sup>x</sup>*, *x*) < *δ* ⇒ *d*1(*f*(*x*), *f*(*x*)) < ) ⇐⇒ <sup>∀</sup>∈R∃*δ*∈R∀*x*∈*X*0∀*x*∈*X*0 ( *d*0(*<sup>x</sup>*, *x*) < *δ* ⇒ *d*1(*f*(*x*), *f*(*x*)) < ).

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.

### **5. 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 3** (Finitely remote)**.** *Let* (*<sup>X</sup>*, *d*) *be a metric space. We call the relation* ∼ *on* U(*X*) *defined below as "finitely remote":*

$$\mathbf{x} \sim \mathbf{x}' \Longleftrightarrow \exists\_{\mathcal{R} \in \mathcal{R}} d(\mathbf{x}, \mathbf{x}') < \mathcal{R}.$$

Note that we use ∃ instead of ∀, in contrast to "infinitely close". This kind of dual viewpoint will be proven to be useful in the geometric study of large scale structures, such as coarse geometry [5].

In fact, we can prove that a "bornologous map", a central notion of a morphism for coarse geometry, can be characterized as "a morphism with respect to ∼", similar to how (uniform) continuity can be viewed as "a morphism with respect to ≈".

**Definition 4** (Bornologous map)**.** *Let* (*<sup>X</sup>*0, *d*0) *and* (*<sup>X</sup>*1, *d*1) *be metric spaces. A map f* : *X*0 −→ *X*1 *is called a bornologous map when*

$$\forall\_{\mathcal{R}\in\mathbb{R}}\exists\_{\mathcal{S}\in\mathbb{R}}\forall\_{\mathbf{x}\in\mathcal{X}\_{0}}\forall\_{\mathbf{x'}\in\mathcal{X}\_{0}}(\operatorname{d}\_{0}(\mathbf{x},\mathbf{x'}) < R \Rightarrow \operatorname{d}\_{1}(f(\mathbf{x}),f(\mathbf{x'})) < \mathcal{S} \; )$$

*holds.*

**Theorem 11** (Characterization of bornologous map)**.** *Let* (*<sup>X</sup>*0, *d*0),(*<sup>X</sup>*1, *d*1) *be metric spaces and* ∼0, ∼1 *be finitely remote relations on them, respectively. A map f* : *X*0 −→ *X*1 *is bornologous if and only if*

$$\forall\_{\mathbf{x}\in\mathcal{U}(\mathcal{X}\_0)} \forall\_{\mathbf{x'}\in\mathcal{U}(\mathcal{X}\_0)} \left(\mathbf{x}\sim\_0 \mathbf{x'} \Rightarrow f(\mathbf{x}) \sim\_1 f(\mathbf{x'})\right),$$

*holds.*

**Proof.** We can translate the condition for *f* by using the usual logic, "commutation principle", and "transfer principle" as follows:

<sup>∀</sup>*x*∈U(*<sup>X</sup>*0)∀*x*∈U(*<sup>X</sup>*0) ( *x* ∼0 *x* ⇒ *f*(*x*) ∼1 *f*(*x*) ) ⇐⇒ <sup>∀</sup>*x*∈U(*<sup>X</sup>*0)∀*x*∈U(*<sup>X</sup>*0) ( ∃*R*∈<sup>R</sup> *d*0(*<sup>x</sup>*, *x*) < *R* ⇒ ∃*S*∈<sup>R</sup> *d*1(*f*(*x*), *f*(*x*)) < *S* ) ⇐⇒ <sup>∀</sup>*x*∈U(*<sup>X</sup>*0)∀*x*∈U(*<sup>X</sup>*0)∀*R*∈R∃*S*∈<sup>R</sup> ( *d*0(*<sup>x</sup>*, *x*) < *R* ⇒ *d*1(*f*(*x*), *f*(*x*)) < *S* ) ⇐⇒ <sup>∀</sup>*R*∈R∀*x*∈U(*<sup>X</sup>*0)∀*x*∈U(*<sup>X</sup>*0)∃*S*∈<sup>R</sup> ( *d*0(*<sup>x</sup>*, *x*) < *R* ⇒ *d*1(*f*(*x*), *f*(*x*)) < *S* ) ⇐⇒ <sup>∀</sup>*R*∈R∃*S*∈R∀*x*∈U(*<sup>X</sup>*0)∀*x*∈U(*<sup>X</sup>*0) ( *d*0(*<sup>x</sup>*, *x*) < *R* ⇒ *d*1(*f*(*x*), *f*(*x*)) < *S* ) ⇐⇒ <sup>∀</sup>*R*∈R∃*S*∈R∀*x*∈*X*0∀*x*∈*X*0 ( *d*0(*<sup>x</sup>*, *x*) < *R* ⇒ *d*1(*f*(*x*), *f*(*x*)) < *S* ).

### **6. The Notion of** *U***-Space and the Category** *USpace*

Based on the characterizations of topological and coarse geometrical structure, we introduce the notion of U-space.

**Definition 5** (U-space)**.** *A* U*-space is a triple* (*<sup>X</sup>*, *<sup>K</sup>*,) *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 (*<sup>X</sup>*, *<sup>K</sup>*,) 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 (*<sup>X</sup>*, U(*X*), ) where *x x* denotes the preorder "∀*O* ∈ *T x* ∈ U(*O*) =⇒ *x* ∈ U(*O*)". If (*<sup>X</sup>*, *T*) is a Hausdorff space, we can construct the symmetric U-space (*<sup>X</sup>*, *<sup>K</sup>*,), where *K* denotes

$$K = \{ \mathbf{x} \in \mathcal{U}(X) | \exists \mathbf{x}\_0 \in X \,\mathbf{x} \rightharpoonup \mathbf{x'} \}$$

and *x x* is defined as the relation "∃*<sup>x</sup>*0 ∈ *X x*0 *<sup>x</sup>*&*<sup>x</sup>*0 *<sup>x</sup>*." The transitivity of follows from the fact that if (*<sup>X</sup>*, *T*) is Hausdorff, *x*0 *x* and *x* 0 *x* imply *x*0 = *x* 0 for all *x*0, *x* 0 ∈ *X*. In fact, 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 6** (U-spatial morphism)**.** *Let* (*<sup>X</sup>*0, *<sup>K</sup>*0,0) *and* (*<sup>X</sup>*1, *<sup>K</sup>*1,1) *be* U*-spaces. A function f* : *X*0 → *X*1 *is called a* U*-spatial morphism from* (*<sup>X</sup>*0, *<sup>K</sup>*0,0) *to* (*<sup>X</sup>*1, *<sup>K</sup>*1,1) *when f*(*<sup>K</sup>*0) ⊂ *K*1 *and*

$$\mathbf{x} \sim\_0 \mathbf{x}' \implies f(\mathbf{x}) \sim\_1 f(\mathbf{x}')$$

*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 7** (Category U*Space*)**.** *The category* U*Space is a category whose objects are* U*-spaces and whose morphisms are* U*-spatial morphisms.*

**Definition 8.** *Let* (*<sup>X</sup>*0, *<sup>K</sup>*0,0) *and* (*<sup>X</sup>*1, *<sup>K</sup>*1,1) *be* U*-spaces. The* U*-space* (*<sup>X</sup>*0 × *X*1, *K*0 × *<sup>K</sup>*1,)*, where the preorder is defined as*

$$(\mathfrak{x}\_{0\prime}\mathfrak{x}\_1) \leadsto (\mathfrak{x}\_{0\prime}'\mathfrak{x}\_1') \Longleftrightarrow \mathfrak{x}\_0 \leadsto\_0 \mathfrak{x}\_0' \& \ \mathfrak{x}\_1 \leadsto\_1 \mathfrak{x}\_{1\prime}'$$

*is called the product* U*-space of* (*<sup>X</sup>*0, *<sup>K</sup>*0,0) *and* (*<sup>X</sup>*1, *<sup>K</sup>*1,1)*.*

**Theorem 12.** *The projections become* U*-spatial morphisms. The diagram consisting of two* U*spaces, the product space of them, and projections becomes a product in* U*Space.*

**Proof.** Easy.

**Definition 9** (Exponential U-space)**.** *Let* (*<sup>X</sup>*0, *<sup>K</sup>*0,0) *and* (*<sup>X</sup>*1, *<sup>K</sup>*1,1) *be* U*-spaces. We denote the set of all* U*-spatial morphisms from X*0 *to X*1 *as* [*XX*<sup>0</sup> 1 ]*, which is the subset of XX*<sup>0</sup> 1 *. The restriction of evX*0,*X*1 : *X*0 × *XX*<sup>0</sup> 1 −→ *X*1 *onto X*0 × [*XX*<sup>0</sup> 1 ] −→ *X*1 *is denoted as* [*evX*0,*X*1 ]*. The* U*-space* ([ *XX*<sup>0</sup> 1], *<sup>K</sup>*,)*, where K is defined as the subset of* U([ *XX*<sup>0</sup> 1])*,*

$$K = \{ f \mid \forall \mathbf{x} \in \mathcal{K}\_0[\varepsilon \upsilon\_{X\_0, X\_1}](\mathbf{x}, f) \in \mathcal{K}\_1 \text{ and } \mathbf{x} \prec\_0 \mathbf{x}' \implies [\varepsilon \upsilon\_{X\_0, X\_1}](\mathbf{x}, f) \leadsto\_1 [\varepsilon \upsilon\_{X\_0, X\_1}](\mathbf{x}, f') \} $$

*and is defined as*

$$f \leadsto f' \iff \forall \mathbf{x} \in \mathsf{K}\_0 \; [\mathsf{ev}\_{X\_0, X\_1}](\mathbf{x}, f) \leadsto\_{\mathbf{1}} [\mathsf{ev}\_{X\_0, X\_1}](\mathbf{x}, f'),$$

*is called the exponential* U*-space from* (*<sup>X</sup>*0, *<sup>K</sup>*0,0) *to* (*<sup>X</sup>*1, *<sup>K</sup>*1,1)*.*

**Theorem 13.** *Let* (*<sup>X</sup>*0, *<sup>K</sup>*0,0) *and* (*<sup>X</sup>*1, *<sup>K</sup>*1,1) *be* U*-spaces and* ([ *XX*<sup>0</sup> 1 ], *<sup>K</sup>*,) *be the exponential* U*-space from* (*<sup>X</sup>*0, *<sup>K</sup>*0,0) *to* (*<sup>X</sup>*1, *<sup>K</sup>*1,1)*. The morphism* [*evX*0,*X*1 ] : *X*0 × [*XX*<sup>0</sup> 1 ] −→ *X*1*, the restriction of evX*0,*X*1 *, is a* U*-spatial morphism. Moreover, it becomes an evaluation in* U*Space and* [*XX*<sup>0</sup> 1] *is an exponential in* U*Space.*

**Proof.** First, we prove that [*evX*0,*X*1 ] is a U-spatial morphism: For any (*<sup>x</sup>*, *f*) ∈ *K*0 × *K*, [*evX*0,*X*1 ](*<sup>x</sup>*, *f*) is in *K*1 since *x* ∈ *K*0 and *f* ∈ *K*. Suppose that (*<sup>x</sup>*, *f*),(*x*, *f* ) ∈ *K*0 × *K* and (*<sup>x</sup>*, *f*) (*x*, *f* ), that is, *x*, *x* ∈ *K*0, *f* , *f* ∈ *K*, *x x* and *f f* . Then, we have

$$[ev\_{\mathcal{X}\_0, \mathcal{X}\_1}](\mathfrak{x}\_\prime f) \leadsto [ev\_{\mathcal{X}\_0, \mathcal{X}\_1}](\mathfrak{x}\_\prime f^\prime)$$

since *f f* . We also have

$$[\operatorname{ev}\_{\mathcal{X}\_0, \mathcal{X}\_1}](\mathfrak{x}, f') \leadsto [\operatorname{ev}\_{\mathcal{X}\_0, \mathcal{X}\_1}](\mathfrak{x'}, f')$$

since *f* ∈ *K*. Hence, [*evX*0,*X*1 ](*<sup>x</sup>*, *f*) [*evX*0,*X*1 ](*x*, *f* ).

Next, we prove that [*evX*0,*X*1 ] : *X*0 × [*XX*<sup>0</sup> 1 ] −→ *X*1 becomes an evaluation in U*Space*, and [*XX*<sup>0</sup> 1 ] is an exponential in U*Space*: Let (*<sup>X</sup>*2, *<sup>K</sup>*2,2) be any U-space and *f* : *X*0 × *X*2 −→ *X*1 be any U-spatial morphism. Consider the lambda conversion ˆ *f* : *X*2 −→ *XX*<sup>0</sup> 1 . By assumption that *f* is U-spatial,

$$f(\mathbf{x}, \mathbf{c}), (\mathbf{x'}, \mathbf{c'}) \in \mathcal{K}\_0 \times \mathcal{K}\_2 \text{ \& (x, c) \leadsto (x', c')} \implies f(\mathbf{x}, \mathbf{c}), f(\mathbf{x'}, \mathbf{c'}) \in \mathcal{K}\_1 \text{ \& } f(\mathbf{x}, \mathbf{c}) \leadsto\_1 f(\mathbf{x'}, \mathbf{c'})$$

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* implies that

$$\text{ax}, \text{x}' \in \mathsf{K}\_0 \& \text{ax} \leadsto\_0 \text{x}' \implies \hat{f}(\mathfrak{c})(\texttt{x}), \hat{f}(\mathfrak{c}')(\texttt{x}') \in \mathsf{K}\_1 \& \hat{f}(\mathfrak{c})(\texttt{x}) \leadsto\_1 \hat{f}(\mathfrak{c}')(\texttt{x}').$$

Applying the implication above for the case *c* = *c* ∈ *X*2, we have ˆ*f*(*c*) ∈ [*XX*<sup>0</sup> 1 ]. Hence, we can replace ˆ *f* : *X*2 −→ *XX*<sup>0</sup> 1 with [ ˆ*f* ] : *X*2 −→ [*XX*<sup>0</sup> 1 ] by restricting the codomain to the image of ˆ *f* . Moreover, we can also prove that [ ˆ*f* ] is U-spatial from the implication: By the implication above, we have [ ˆ*f* ](*c*), [ ˆ*f* ](*c*) ∈ *K* and [ ˆ*f* ](*c*) [ ˆ*f* ](*c*) when *c*, *c* ∈ *K*2 and *c* 2 *<sup>c</sup>*. This means that [ ˆ*f* ] is U-spatial.

It is easy to show that this [ *f* ] is the unique U-spatial morphism from *X*2 to [*XX*<sup>0</sup> 1 ] satisfying [*evX*0,*X*<sup>1</sup>] ◦ (<sup>1</sup>*X*0 × [ ˆ*f* ]) = *f* . This completes the proof.

Combining the two theorems above, we have:

**Theorem 14.** *The category* U*Space is a Cartesian closed category.*

ˆ

**Author Contributions:** Conceptualization, H.S. and J.N.; Investigation, H.S. and J.N.; Methodology, H.S.; Writing—original draft, H.S.; Writing—review & editing, J.N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partially supported by Research Origin for Dressed Photon, JSPS KAKENHI (grant number 19K03608 and 20H00001) and JST CREST (JPMJCR17N2).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** The authors would like to express their sincere thanks to Anders Kock and Edward Nelson for their encouragements. They are grateful to Hiroshi Ando, Izumi Ojima, Kazuya Okamura, Misa Saigo, Hiroki Sako, and Ryokichi Tanaka for the fruitful discussions and comments.

**Conflicts of Interest:** The authors declare no conflict of interest.
