# Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics

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## Abstract

**:**

## 1. Introduction

## 2. Results

**Lemma**

**1.**

**Proof.**

- The terminal object 1 in Set is an arbitrary singleton since there exists a unique arrow (function) $X\to 1$ for every set X.
- There exist exponentiable set ${X}^{Y}$ of functions for every pair $(X,Y)$ of sets;
- The fibered products: Given $f:B\to A$ and $g:C\to A$ their fibered product is the set $B{\times}_{A}C=\{(b,c)\in B\times C:f\left(b\right)=g\left(c\right)\}$.
- The sub-object classifier and truth object: $\Omega =\{0,1\}$ determines subsets as the codomain of the characteristic functions. The truth arrow reads $1\stackrel{\top}{\to}\Omega $.

**Lemma**

**2**

**.**The embedding of the category $\mathbb{M}$ into $\mathcal{B}$, $s:\mathbb{M}\hookrightarrow \mathcal{B}$, is full and faithful.

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Proposition**

**1.**

**Remark**

**4.**

**Remark**

**5.**

**Lemma**

**3.**

- $s:\mathbb{M}\to \mathcal{B}$ does not preserve compactness, e.g., $[0,1]\subset R=s\left(\mathbb{R}\right)$ is not compact;
- s does not preserve open covers;
- s does not preserve partitions of unity subordinated to open covers;
- the ring R is not any local ring.

#### 2.1. Local Ext/Int Symmetry on Smooth Manifolds

**Remark**

**6.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Remark**

**7.**

**Definition**

**1.**

**Remark**

**8.**

**Proposition**

**2.**

**Proof.**

**Corollary**

**1.**

#### 2.2. Is Any Exotic ${\mathbb{R}}^{4}$ Equivalent to a $\phantom{\rule{3.33333pt}{0ex}}\mathcal{B}$-Invariant Structure?

#### 2.2.1. First Comparison of Distributions and Exotic Smoothness Structures on ${\mathbb{R}}^{4}$

**Remark**

**9.**

**Remark**

**10.**

**Theorem**

**1**

**.**Suppose $h\in {\mathcal{D}}^{\prime}(\Omega )$, and K is a compact subset of Ω. Then there is a continuous function f in Ω and there is a multi-index α such that

**Theorem**

**2**

**.**Suppose $h\in {\mathcal{D}}^{\prime}(\Omega )$. There exist continuous functions ${g}_{\alpha}$ in Ω, one for each multi-index α, such that

- a.
- each compact $K\subset \Omega $ intersects the supports of only finitely many ${g}_{\alpha}$, and
- b.
- $h={\sum}_{\alpha}{D}^{\alpha}{g}_{\alpha}$

#### 2.2.2. Distributions in $\mathcal{B}$

**Remark**

**11.**

- $s:\mathbb{M}\to \mathcal{B}$ preserve compactness, e.g., $[0,1]\subset R=s\left(\mathbb{R}\right)$ is s-compact (smooth compact, i.e., with respect to N);
- s preserves open covers with respect to N;
- s preserves partitions of unity subordinated to open covers (with respect to N);
- the ring R is a local ring;

**Remark**

**12.**

**Remark**

**13.**

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**3**

**.**The global sections functor $\Gamma :\mathcal{B}\to \mathrm{Sets}$ induces a bijection between distributions ${F}_{n}\stackrel{\nu}{\to}R$ in $\mathcal{B}$ and external distributions $\Gamma \nu :{C}_{c}^{\infty}\left({\mathbb{R}}^{n}\right)\to \mathbb{R}$, and between distributions with compact support, i.e., R-linear maps ${R}^{{R}^{n}}\stackrel{\nu}{\to}R$ and external distribution with compact support ${C}^{\infty}\left({\mathbb{R}}^{n}\right)\stackrel{\Gamma \nu}{\to}\mathbb{R}$.

**Theorem**

**4**

**.**For every distribution μ on ${R}^{n}$ there exists a predistribution ${\mu}_{0}:{R}_{acc}^{n}\to R$ such that for all $f\in {F}^{n}$

**Remark**

**14.**

#### 2.2.3. The Construction of $\phantom{\rule{3.33333pt}{0ex}}\mathcal{B}$-Invariant Functions as Smooth Exotic

**Definition**

**4.**

**Remark**

**15.**

**Theorem**

**5.**

**Proposition**

**3.**

**Proof.**

**Proof**

**of**

**Theorem**

**5.**

**Remark**

**16.**

**Remark**

**17.**

#### 2.3. Local $\phantom{\rule{3.33333pt}{0ex}}\mathcal{B}$-Invariance and General Tovariance

**Remark**

**18.**

**Remark**

**19.**

**Proposition**

**4**

**.**The inverse image part ${g}^{*}$ of a geometric morphism $g:{\mathcal{T}}_{1}\to {\mathcal{T}}_{2}$ preserves any geometric theory.

**Proposition**

**5**

**.**For any geometric morphism $g=({g}_{*},{g}^{*})$ between toposes ${\mathcal{T}}_{1},{\mathcal{T}}_{2}$ the inverse image functor ${g}^{*}$ preserves NNO, i.e., if ${\mathbb{N}}_{2}$ is a NNO in ${\mathcal{T}}_{2}$ then ${g}^{*}{\mathbb{N}}_{2}$ is the NNO in ${\mathcal{T}}_{1}$.

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Remark**

**20.**

Special tovariance principle. When a physical theory is formulated in the geometric language one cannot experimentally distinguish between their realizations in different toposes with NNO when the change of the toposes is via geometric morphisms.

- We follow ‘global-to-local’ pattern known from gauge theories. This means that given local $\mathcal{B}$-structure on a smooth manifold $M\in \mathbb{M}$ one switches between Set and $\mathcal{B}$ frames without possibility to leave entirely any of them (Definition 1). The Definition 1 can thus serve as an obstruction to the global choice of a topos on M, i.e., Set or $\mathcal{B}$. It can be restated as the property of non-existence of any global Set or $\mathcal{B}$-sections on M.
- Any $\mathcal{B}$-invariant structure on M would rely on generalized equivalence between the Set construction with respect to $\mathbb{N}$ and $\mathcal{B}$ construction with respect to N. The constructions are not equivalent by geometric morphisms since N is not preserved - it does not exist in general (Corollaries 2 and 3).

**Conjecture**

**1.**

## 3. Discussion

- The global section functor $\Gamma :E\to \mathrm{Set}$ is faithful—the terminal object is a generator,
- 1 is not an initial object (E is nondegenerate).

**Theorem**

**6**

**.**Set is, up to equivalence, the unique locally small and cocomplete well-pointed topos (or the unique locally small and complete well-pointed topos).

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**$\mathcal{B}$-smoothing of the continuous function f. Differentiating of f in Set gives rise to the distribution which internally is certain unique $\mathcal{B}$-distribution ${\mu}_{\widehat{f}}$. This last is internally a regular distribution (function) ${\mu}_{\widehat{f}}^{0}$ due to the replacement of $\mathbb{N}$ by N.

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**MDPI and ACS Style**

Asselmeyer-Maluga, T.; Król, J.
Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics. *Symmetry* **2019**, *11*, 1429.
https://doi.org/10.3390/sym11121429

**AMA Style**

Asselmeyer-Maluga T, Król J.
Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics. *Symmetry*. 2019; 11(12):1429.
https://doi.org/10.3390/sym11121429

**Chicago/Turabian Style**

Asselmeyer-Maluga, Torsten, and Jerzy Król.
2019. "Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics" *Symmetry* 11, no. 12: 1429.
https://doi.org/10.3390/sym11121429