nuclear C*-algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Operator algebra
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A type of [[C*-algebra]].

Many equivalent definitions, not all of which were known to be equivalent when they were made. Usually credited to Takesaki (1964). An important characterization in terms of an appropriate "approximation property" was given by Lance (1972).

(...)

+-- {: .num_example }
###### Example

Every [[commutative C*-algebra]] is nuclear.

If $H$ is an infinite-dimensional Hilbert space then $B(H)$ is not nuclear.

=--

## Properties

### Relation to KK-theory

On K-nuclear $C^\ast$-algebras $C$, the [[KK-theory]] functor $KK(C,-)$ preserves [[short exact sequences]] in the middle (satisfies [[excision]]). ([Kasparov 80](#Kasparov80), [Skandalis 88](#Skandalis88)).

Hence restricted to nuclear $C*$-algebras the canonical functor $KK \to E$ from [[KK-theory]] to [[E-theory]] (see there) is a [[full and faithful functor]].

## References

The notion was studied as a condition for [[excision]] in [[KK-theory]] in section 7 of

* [[Gennady Kasparov]], _The operator $K$-functor and extensions of $C^{\ast}$-algebras_, Izv. Akad. Nauk SSSR Ser. Mat. __44__  (1980), no. 3, 571–636, 719, [MR81m:58075](http://www.ams.org/mathscinet-getitem?mr=582160), [Zbl](http://www.zentralblatt-math.org/zmath/search/?an=Zbl%200464.46054%7CZbl%200448.46051), [abstract](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1739&option_lang=eng), [english doi](http://dx.doi.org/10.1070%2FIM1981v016n03ABEH001320), free Russian original: [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=1739&volume=44&year=1980&issue=3&fpage=571&what=fullt&option_lang=eng)
 {#Kasparov80}

The generalization to K-nuclearity was introduced and discussed in

* [[Georges Skandalis]], _Une notion de nuclearité en K-theorie_, K-Theory 1 (1988) 549-574.
 {#Skandalis88}

* Wikipedia, _[Nuclear C-star-algebra](http://en.wikipedia.org/wiki/Nuclear_C*-algebra)_

[[!redirects nuclear C-star-algebra]]
[[!redirects nuclear C-star-algebras]]

[[!redirects nuclear C*-algebras]]

[[!redirects K-nuclear C-star-algebra]]
[[!redirects K-nuclear C-star-algebras]]

[[!redirects K-nuclear C*-algebra]]
[[!redirects K-nuclear C*-algebras]]


nuclear force

[[!redirects weak nuclear force]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Besides the [[forces]] of [[electromagnetism]] and [[gravity]] there are two more fundamental [[gauge fields]] in the [[standard model of particle physics]].

The _weak nuclear force_ is described by a [[gauge field]] with [[gauge group]] the [[special unitary group]] $SU(2)$ ([[W-boson]], [[Z-boson]]). It controls  aspects of the [[beta decay]].

The _strong nuclear force_ is described by a [[gauge field]] with [[gauge group]] the [[special unitary group]] $SU(3)$ ([[QCD]] ([[gluon]]). It governs the interaction between [[quarks]].

## Related concepts

* [[leptonic decay]]

* [[CKM matrix]]

* [[V_cb puzzle]]

* [[quarks]], [[confinement]]

* [[weakly interacting massive particle]]

## References

See also the references at:

* [[QCD]]

* [[electroweak field]], [[electroweak symmetry breaking]]

See also:

* Wikipedia, _[Weak interaction](http://en.wikipedia.org/wiki/Weak_interaction)_

* Wikipedia, _[Strong interaction](http://en.wikipedia.org/wiki/Strong_interaction)_

Textbook account on [[weak nuclear force]]-mediated [[decays]] with an eye towards in [[flavour physics]] and [[flavour anomalies]]:

* {#Buras20}  [[Andrzej Buras]], _Gauge Theory of Weak Decays_ (2020) ([doi:10.1017/9781139524100](https://doi.org/10.1017/9781139524100), [CERN review](https://cerncourier.com/a/the-hitchhikers-guide-to-weak-decays/) by [[Gino Isidori]])

[[!redirects nuclear force]]
[[!redirects nuclear forces]]

[[!redirects weak nuclear force]]
[[!redirects weak nuclear forces]]

[[!redirects weak interaction]]
[[!redirects weak interactions]]

[[!redirects strong nuclear force]]
[[!redirects strong interaction]]
[[!redirects strong interactions]]


nuclear magnetic resonance

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]], by *nuclear magnetic resonance* (NMR) one refers to resonant control of the [[spin]] of [[atomic nuclei]] through an external [[electromagnetic field]].

One way to realize [[qbits]] (in [[quantum information theory]] and [[quantum computing]]) is as spin-[[quantum states]] [[quantum gate|manipulated]] via spin resonance: [[spin resonance qbits]]. A prominent example of [[spin resonance qbit]] realizations uses a [[nitrogen-vacancy center in diamond]].

## Related concepts

* [[nitrogen-vacancy center in diamond]]

* [[spin resonance qbit]]

* [[quantum computing]]

* [[quantum sensing]]

## References

### General

See also

* Wikipedia, *[Nuclear magnetic resonance](https://en.wikipedia.org/wiki/Nuclear_magnetic_resonance)*

On [[Floquet theory]] in [[nuclear magnetic resonance]]:

* Konstantin L. Ivanov, Kaustubh R. Mote, Matthias Ernst, [[Asif Equbal]], Perunthiruthy K. Madhu, *Floquet theory in magnetic resonance: Formalism and applications*, Progress in Nuclear Magnetic Resonance Spectroscopy, **126**–**127** (2021) 17-58 [[doi:10.1016/j.pnmrs.2021.05.002](https://doi.org/10.1016/j.pnmrs.2021.05.002)]

Further developments:

* Navin Khaneja et al., *Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms*, Journal of Magnetic Resonance **172 2 (2005) 296-305 [[doi:10.1016/j.jmr.2004.11.004](https://doi.org/10.1016/j.jmr.2004.11.004)]

* [[Asif Equbal]] et al.,  *Role of electron spin dynamics and coupling network in designing dynamic nuclear polarization*, Progress in Nuclear Magnetic Resonance Spectroscopy **126**–**127** (2021) 1-16 [[doi:10.1016/j.pnmrs.2021.05.003](https://doi.org/10.1016/j.pnmrs.2021.05.003)]

[[!include spin resonance qbits -- references]]

[[!redirects nuclear magnetic resonances]]
[[!redirects NMR]]

[[!redirects spin resonance]]


nuclear matrix model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _nuclear matrix model_ ([Hashimoto-Iizuka-Yi 10](#HashimotoIizukaYi10), [Hashimoto-Matsuo-Morita 19](#HashimotoMatsuoMorita19)) is a [[matrix model]] for [[baryons]]/[[nucleons]] in [[nuclear physics]].

The model proceeds from the [[Witten-Sakai-Sugimoto model]] for [[QCD]] realized on [[intersecting D-brane model|intersecting D-branes]], where [[baryons]] are embodied by [[wrapped brane|wrapped]] [[D4-branes]] ("[Witten's baryon vertex](AdS-QCD+correspondence#Baryons)") on [[flavour brane|flavour]] [[D8-branes]] ([[D4-D8 brane bound states]]).

> (graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]])

Thus, encoding the [[D-brane]] dynamics of these [[D4-branes]] in a matrix model along the lines of the [[BFSS matrix model]]/[[BMN matrix model]] (for [[D0-branes]]) and the [[IKKT matrix model]] (for [[D(-1)-branes]]) leads to a matrix model for [[baryons]] and hence, potentially, for [[nucleons]].

## Related concepts

* [[holographic QCD]]

[[!include brane matrix models -- content]]

## References

Formulation of the model:

* {#HashimotoIizukaYi10} [[Koji Hashimoto]], [[Norihiro Iizuka]], [[Piljin Yi]], _A Matrix Model for Baryons and Nuclear Forces_, JHEP 1010:003, 2010 ([arXiv:1003.4988](https://arxiv.org/abs/1003.4988))

Review:

* {#AHI12} Sinya Aoki, [[Koji Hashimoto]], [[Norihiro Iizuka]], _Matrix Theory for Baryons: An Overview of Holographic QCD for Nuclear Physics_, Reports on Progress in Physics, Volume 76, Number 10 ([arxiv:1203.5386](https://arxiv.org/abs/1203.5386))

Further development:

* Si-wen Li, Tuo Jia, _Matrix model and Holographic Baryons in the D0-D4 background_, Phys. Rev. D 92, 046007 (2015) ([arXiv:1506.00068](https://arxiv.org/abs/1506.00068))

* {#HashimotoMatsuoMorita19} [[Koji Hashimoto]], [[Yoshinori Matsuo]], [[Takeshi Morita]], _Nuclear states and spectra in holographic QCD_, JHEP12 (2019) 001 ([arXiv:1902.07444](https://arxiv.org/abs/1902.07444))

* Yasuhiro Hayashi, Takahiro Ogino, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Stringy excited baryons in holographic QCD_, Prog Theor Exp Phys (2020) ([arXiv:2001.01461](https://arxiv.org/abs/2001.01461))

Computation of nuclear binding energies in the model:

* [[Koji Hashimoto]], [[Yoshinori Matsuo]], _Nuclear binding energy in holographic QCD_ ([arXiv:2103.03563](https://arxiv.org/abs/2103.03563))


nuclear object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **nuclear object** is a categorical generalization of the concept of a finite-dimensional vector space, or more generally, a finite-dimensional object.

## Definition

A _nuclear object_in a [[symmetric monoidal closed category]] $\mathcal{M}$ with unit $k$ is an object $A$ such that the canonical morphism $\phi _A:A\otimes A^*\to [ A, A]$ with $A^*:=[A,k]$ is an isomorphism.

A _morphism_ $f:A\to B$ is called _nuclear_ if there exists a morphism $p:k\to B\otimes A^*$ such that

$$\array{k & \overset{p}{\to} & B\otimes A^*\\
  n(f)\searrow && \swarrow\phi\\
  & [A,B] & }$$

is commutative, where $n(f)$, the name of $f$, obtains via adjointness from $k\otimes A\simeq A\overset{f}{\to} B$.

## Properties

* $A$ is nuclear precisely if $id_A$ is nuclear.

* If $A$ is nuclear then the canonical map to the double dual $\theta _A: A\to A^{**}$ is an isomorphism.

* The full subcategory $nuc(\mathcal{M})$ of nuclear objects is symmetric monoidal closed.

* $A$ is nuclear precisely if $A^*$ is nuclear.

* In a cartesian monoidal $\mathcal{M}$ only $1$ is nuclear.

* If $A$ is nuclear there exists a natural [[trace]] morphism $t_A:[A,A]\simeq A\otimes A^*\simeq A^*\otimes A \to k$.

## Examples

* Nuclear objects in the category of complete semilattices are precisely the [[completely distributive lattices]], or in absence of the [[axiom of choice]], more generally, the _constructively completely distributive lattices_ (Higgs&Rowe 1989, Rosebrugh&Wood 1994).

* Nuclear objects in the category of pointed sets are precisely the pointed sets with cardinality $\leq 2$.

* Nuclear objects in the category of [[Banach spaces]] with morphisms the bounded maps are the [[nuclear spaces]] (Rowe 1988).

## Related concepts

* [[nucleus of a profunctor]]

* [[completely distributive lattice]]

* [[nuclear space]]

* [[self-dual object]]

* [[dualizable object]]

* [[trace]]

## References

* D. A. Higgs, K. A. Rowe, _Nuclearity in the category of complete semilattices_ , JPAA **57** no.1 (1989) pp.67-78.

* R. Rosebrugh, R. J. Wood, _Constructive complete distributivity IV_ , App. Cat. Struc. **2** (1994) pp.119-144. ([preprint](http://www.mta.ca/~rrosebru/articles/ccd4.pdf))

* K. A. Rowe, _Nuclearity_ , Canad.Math.Bull. **31**  no.2 (1988) pp.227-235. ([pdf](http://cms.math.ca/cmb/v31/cmb1988v31.0227-0235.pdf))

* G. N. Raney, _Tight Galois Connections and Complete Distributivity_ , Trans.Amer.Math.Soc **97** (1960) pp.418-426. ([pdf](http://www.ams.org/journals/tran/1960-097-03/S0002-9947-1960-0120171-3/S0002-9947-1960-0120171-3.pdf))

[[!redirects nuclear objects]]


nuclear physics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[physics]] of [[atomic nuclei]].

[[!include effective field theories of nuclear physics -- contents]]

## Related entries

* [[proton]], [[neutron]]

* [[atomic number]], [[periodic table of the elements]]

* [[meson]]-[[effective field theory]],

* [[chiral perturbation theory]]

* [[Skyrmion]]

* [[Witten-Sakai-Sugimoto model]], [[nuclear matrix model]]

* [[solid state physics]], [[high energy physics]]

## References

[[effective field theory]] for nuclear physics -- [[chiral perturbation theory]], [[Skyrme model]], [[quantum hadrodynamics]]:

* [[Maciej Nowak]], [[Mannque Rho]], [[Ismail Zahed]], _[[Chiral Nuclear Dynamics]]_,  World Scientific 1996 ([doi:10.1142/1681](https://doi.org/10.1142/1681))

* [[Mannque Rho]], _Lectures on Effective Field Theories for Nuclei, Nuclear Matter and Dense Matter_, 2002 ([arXiv:nucl-th/0202078](https://arxiv.org/abs/nucl-th/0202078), [cds:539674](https://cds.cern.ch/record/539674))

See also

* Wikipedia, _[Nuclear physics](https://en.wikipedia.org/wiki/Nuclear_physics)_


nuclear reaction

## Related concepts

* [[fundamental particle]]

* [[chemistry]]

## References

* Wikipedia, _[Nuclear reaction](https://en.wikipedia.org/wiki/Nuclear_reaction)_

[[!redirects nuclear reactions]]


nuclear space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A **nuclear vector space** is a [[locally convex topological vector space]] that is as far from being a [[normed vector space]] as possible.  Any map from a nuclear space into a [[normed vector space]] is compact, whence the only normed nuclear spaces are finite dimensional.

Nuclear spaces have very good properties with regard to [[topological tensor product]]s and [[dual vector space|duality]].

## Definition

To define a nuclear space we need to start with the concept of a nuclear map, first between [[Banach spaces]].

Let $E$ and $F$ be [[Banach spaces]].  Let $\mathcal{L}(E,F)$ be the Banach space of continuous linear maps $E \to F$.  Let $E^*$ denote the dual Banach space of $E$.  Let $E^* \widetilde{\otimes} F$ denote the completion of the [[projective tensor product]] of $E^*$ and $F$.  The bilinear map $E^* \times F \to \mathcal{L}(E,F)$ extends to a continuous linear map $E^* \widetilde{\otimes} F \to \mathcal{L}(E,F)$ (which might not be injective).

+-- {: .num_defn #nnmap}
###### Definition
Let $E$ and $F$ be Banach spaces.  A linear map $f \colon E \to F$ is **nuclear** if it lies in the image in $\mathcal{L}(E,F)$ of the completion of the projective tensor product $E^* \widetilde{\otimes} F$.
=--

From the notion of nuclear maps between Banach spaces we can define nuclear maps between arbitrary [[LCTVS]].  In essence, a linear map between arbitrary LCTVS is nuclear if it factors through a nuclear map of Banach spaces.

To make this precise, we need to recall how to associate Banach spaces to certain subsets of an LCTVS.  Let $E$ be an LCTVS and $U \subseteq E$ a [[convex set|convex]] [[circled set|circled]] $0$-neighbourhood.  Then we can define a Banach space $\widetilde{E}_U$ as follows: as $U$ is convex and circled, its [[Minkowski functional]] is a [[semi-norm]] on $E$.  The quotient $E_U \coloneqq E/\ker U$ is therefore a [[normed vector space]].  As $U$ is a $0$-neighbourhood, the quotient mapping defines a continuous linear function $E \to E_U$.  We define $\tilde{E}_U$ to be the Banach completion of $E_U$.

There is a dual notion.  Let $F$ be an LCTVS and $B \subseteq F$ a convex, circled, and [[bounded subset]] of $F$.  Let $F_B$ be the span of $B$ in $F$.  Then $B$ is [[absorbing]] in $F_B$ and so its Minkowski functional is defined.  If $F$ is [[Hausdorff]] then $B$ cannot contain a linear subspace and thus $F_B$ is a normed vector space.  We cannot complete $F_B$ to a Banach space but it might so happen that it is one.  As $B$ is bounded, the inclusion $F_B \to F$ is continuous.

(There is no danger of confusing the two notations since if $E$ admits a bounded $0$-neighbourhood then it is a normed vector space.)

Now we say that a continuous linear map $f \colon E \to F$ is **bounded** if for some $0$-neighbourhood $U$ of $E$ (which we may take to be [[circled set|circled]] and [[convex set|convex]]), $f(U)$ is bounded in $F$.  In which case, $f$ factors through a continuous map $f_{U,B} \colon E_U \to F_B$ where $B \subseteq F$ is bounded and contains $f(U)$.

+-- {: .num_defn #nmap}
###### Definition
Let $E$ and $F$ be LCTVS.  A linear map $f \colon E \to F$ is **nuclear** if there exists a [[convex set|convex]] [[circled set|circled]] $0$-neighbourhood, say $U$, in $E$ and a [[convex set|convex]] [[circled set|circled]] [[bounded set|bounded]], say $B$, in $F$ with $F_B$ complete such that $f(U) \subseteq B$ and the associated map $f_{U,B} \colon \tilde{E}_U \to F_B$ is nuclear.
=--

The following characterisation of nuclear maps is often helpful.

+-- {: .num_lemma #nchar}
###### Lemma
A linear map $f \colon E \to F$ is nuclear if and only if it is of the form:
$$
u(x) = \sum_{n=1}^\infty \lambda_n f_n(x) y_n
$$
where $\sum_{n=1}^\infty {|\lambda_n|} \lt \infty$, $\{f_n\}$ is an [[equicontinuous]] sequence in $E^*$, and $\{y_n\}$ is a sequence in $F$ contained in a convex, circled, bounded subset $B$ such that $F_B$ is complete.
=--

Now that we have the notion of a nuclear map, we can define a nuclear space.

+-- {: .num_defn #nsp}
###### Definition
A [[LCTVS]] $E$ is nuclear if it has a base $\mathcal{B}$ of convex circled $0$-neighbourhoods such that for $V \in \mathcal{B}$ the canonical mapping $E \to \tilde{E}_V$ is nuclear.
=--

## Properties

1. The following are equivalent:
    1. $E$ is nuclear,
    1. Every continuous linear map of $E$ into any Banach space is nuclear,
    1. Every convex, circled $0$-neighbourhood $U$ contains another, say $V$, such that the canonical map $\tilde{E}_V \to \tilde{E}_U$ is nuclear.
1. Every bounded subset of a nuclear space is precompact.
1. The completion of a nuclear space is a nuclear space.
1. A nuclear space is a projective limit of $\ell^p$ spaces (in particular, of [[Hilbert spaces]]).
1. Nuclearity is inherited by the following constructions: subspaces, separated quotients, arbitrary products, locally convex direct sum of a countable family, projective limits, countable inductive limits.
1. The [[projective tensor product]] (and its completion) of two nuclear spaces is nuclear.

## Examples

1. Any finite dimensional vector space is nuclear,
1. For a finite dimensional smooth manifold $M$, $C^\infty(M)$ is nuclear,
1. The space of rapidly decreasing functions on $\mathbb{R}^n$ is nuclear,
1. The product an arbitrary number of copies of $\mathbb{R}$ is nuclear,
1. The direct sum of a countable number of copies of $\mathbb{R}$ is nuclear,
1. The direct sum of $\mathbb{R}$-copies of $\mathbb{R}$ is *not* nuclear.

## References
* A. Grothendieck,  _Produits tensoriels topologiques et espaces nucléaires_, Memoirs of the Amer. Math. Soc. __16__, 190 pp. and 140 pp. (1955).
* [[Alexander Grothendieck]], _Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires_, Annales de l'institut Fourier __4__ (1952), p. 73-112, [numdam](http://www.numdam.org/item?id=AIF_1952__4__73_0)

category: functional analysis

[[!redirects Nuclear space]]
[[!redirects nuclear space]]
[[!redirects nuclear spaces]]
[[!redirects nuclear vector space]]
[[!redirects nuclear vector spaces]]
[[!redirects nuclear topological vector space]]
[[!redirects nuclear topological vector spaces]]


nuclear topological vector space > history

< [[nuclear topological vector space]]


nucleon

[[!redirects nucleons]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[proton|Protons]] and [[neutrons]] (both examples of [[baryons]], hence of [[hadrons]] are jointly called _nucleons_, since [[nucleus (physics)|atomic nuclei]] are [[bound states]] of these.

## Related concepts

* [[Walecka model]]

## References

### General

See also

* Wikipedia, _[Nucleon](https://en.wikipedia.org/wiki/Nucleon)_

### Via the Skyrme model

On nucleon [[interaction]] modeled via [[Skyrmions]]:

* [[Chris Halcrow]], [[Derek Harland]], _An attractive spin-orbit potential from the Skyrme model_ ([arXiv:2007.01304](https://arxiv.org/abs/2007.01304))

* [[Derek Harland]], [[Chris Halcrow]], _Nucleon-nucleon potential from skyrmion dipole interactions_ ([arXiv:2101.02633](https://arxiv.org/abs/2101.02633))

* [[Horatiu Nastase]], [[Jacob Sonnenschein]], _A $T \bar T$-like deformation of the Skyrme model and the Heisenberg model of nucleon-nucleon scattering_ ([arXiv:2101.08232](https://arxiv.org/abs/2101.08232))

[[!include quantum hadrodynamics -- references ]]

### Via holographic QCD

nucleon [[form factors]] via [[holographic QCD]]:

* Kiminad A. Mamo, [[Ismail Zahed]],  _Nucleon mass radii and distribution: Holographic QCD, Lattice QCD and GlueX data_ ([arXiv:2103.03186](https://arxiv.org/abs/2103.03186))

via a [[nuclear matrix model]]:

* [[Koji Hashimoto]], [[Norihiro Iizuka]], [[Piljin Yi]], _A Matrix Model for Baryons and Nuclear Forces_, JHEP 1010:003, 2010 ([arXiv:1003.4988](https://arxiv.org/abs/1003.4988))

* Si-wen Li, Tuo Jia, _Matrix model and Holographic Baryons in the D0-D4 background_, Phys. Rev. D 92, 046007 (2015) ([arXiv:1506.00068](https://arxiv.org/abs/1506.00068))

* [[Koji Hashimoto]], [[Yoshinori Matsuo]], [[Takeshi Morita]], _Nuclear states and spectra in holographic QCD_, JHEP12 (2019) 001 ([arXiv:1902.07444](https://arxiv.org/abs/1902.07444))

* Yasuhiro Hayashi, Takahiro Ogino, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Stringy excited baryons in holographic QCD_, Prog Theor Exp Phys (2020) ([arXiv:2001.01461](https://arxiv.org/abs/2001.01461))

nuclear binding energy

* Salvatore Baldino, Lorenzo Bartolini, [[Stefano Bolognesi]], Sven Bjarke Gudnason, _Holographic Nuclear Physics with Massive Quarks_ ([arXiv:2102.00680](https://arxiv.org/abs/2102.00680))

nuclear binding energy via the [[nuclear matrix model]]:

* [[Koji Hashimoto]], [[Yoshinori Matsuo]], _Nuclear binding energy in holographic QCD_ ([arXiv:2103.03563](https://arxiv.org/abs/2103.03563))


nucleosynthesis

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Generally, _nucleosynthesis_ refers to the formation of [[bound states]] of [[nucleons]] ([[protons]] and [[neutrons]]) to [[atomic nuclei]]. Specifically, _primordial nucleosynthesis_ (often meant by default) is this process taking place in the early [[observable universe]] some [[comoving time]] after the [[big bang]] (see also at _[[QCD cosmology]]_). Other natural processes of nucleosynthesis happened and happen in [[stars]] and in [[supernovae]].

## Related concepts

* [[baryogenesis]]

* [[leptogenesis]]

* [[confinement]]

## Literature

### General

See also:

* Wikipedia. _[Nucleosynthesis](https://en.wikipedia.org/wiki/Nucleosynthesis)_

### Primordial nucleosynthesis

* Dominik J. Schwarz, _The first second of the Universe_, Annalen Phys. 12:220-270, 2003 ([arXiv:astro-ph/0303574](https://arxiv.org/abs/astro-ph/0303574))

* [[Toshitaka Kajino]], *Cosmological Nucleosynthesis in the Big-Bang and Supernovae*,  Journal of Nuclear Science and Technology **39** sup2 ([doi:10.1080/00223131.2002.10875156](https://doi.org/10.1080/00223131.2002.10875156))

* Grant J. Mathews, Motohiko Kusakabe, [[Toshitaka Kajino]], *Introduction to Big Bang Nucleosynthesis and Modern Cosmology*,  International Journal of Modern Physics EVol. 26, No. 08, 1741001 (2017)  ([arXiv:1706.03138](https://arxiv.org/abs/1706.03138), [doi:10.1142/S0218301317410014](https://doi.org/10.1142/S0218301317410014))

* {#Alvey21} [[James Alvey]], *What can you measure with BBN in 2021?*, Cosmology Talks 2021 ([[Alvey_BBN2021.pdf:file]])

See also:

* Wikipedia, *[Big Bang nucleosynthesis](https://en.wikipedia.org/wiki/Big_Bang_nucleosynthesis)*

[[!redirects primordial nucleosynthesis]]


nucleus

> this entry is about the concept in [[topology]], especially in [[point-free topology]]. For the concept in [[physics]]/[[chemistry]] see [[nucleus (physics)]], or see [[nucleus (disambiguation)]].

# Nuclei
* table of contents
{: toc}

## Idea

Recall that [[frames]] are [[duality|dual]] to [[locales]], and locales are kinds of [[spaces]].  So, if you adopt locales as models for spaces, then your models for [[subspaces]] are [[sublocales]] or, dually, [[quotient object|quotient]] frames.  However, much as a [[quotient set]] can be described by an [[equivalence relation]] on the original [[set]], so a quotient frame may be described by an appropriate [[structure]] on the original frame.  That structure is a _nucleus_.

Thus, nuclei correspond to [[sublocales]].

## Definitions

Let $L$ be a [[frame]], that is a [[suplattice]] satisfying the infinite distributivity law.

A __nucleus__ on $L$ is a [[function]] $j\colon L \to L$ which is

1.  monotone, i.e. $j(a \wedge b) = j(a) \wedge j(b)$ for all $a,b\in L$,
2.  inflationary, $a \leq j(a)$ for all $a\in L$,
3.  satisfies $j(j(a)) \leq j(a)$ for all $a\in L$.

In other words, a nucleus on $L$ is a [[meet]]-preserving [[monad]] on $L$. If only the first two properties are satisfied, $j$ is sometimes called an __inflator__.

Note that the following properties of a nucleus might be included in the definition, but they follow from the above:

1.  $j(\top) = \top$,
2.  $j(a) \leq j(b)$ if $a \leq b$,
3.  (idempotency) $j(j(a)) = j(a)$.

Thus one sometimes defines a nucleus as a monotone, inflationary and idempotent (endo)function on $L$

## The subset of closed elements

Let $L$ be a frame.

As a nucleus $j$ on $L$ (being a monad on a [[poset]]) is a kind of [[Moore closure]], we say that an element $a$ of $L$ is __$j$-closed__ if $j(a) = a$.  (But note that this has nothing to do with the [[closed subspaces]] of the locale $L$.)

We may equivalently define a nucleus on $L$ to be a [[subset]] $J$ of $L$ that satisfies certain conditions, namely these identities:

1.  $\bigwedge A \in J$ whenever $A \subseteq J$ (using that $L$ is a [[complete lattice]]),
2.  $a \Rightarrow b \in J$ whenever $b \in J$ (using that $L$ is a [[Heyting algebra]]).

Then we recover $j\colon L \to L$ by

$$ j(a) \coloneqq \bigwedge \{ b\colon L \;|\; b \in J,\; a \leq b \} $$

and we have

$$ J = \{ a \colon L \;|\; j(a) = a \} .$$

+-- {: .query}
Check all this, and expand on it if necessary.
=--

This approach to nuclei is not appropriate in a [[predicative mathematics|predicative]] approach to topology, where we want to use [[large category|large]] (but [[accessible category|accessible]]) frames, which may not be meet-complete.

## Quotient frames and sublocales

Let $L$ be a frame, and let $j$ be a nucleus on $L$.

Let $L/j$ be the [[subset]] of $L$ consisting of the $j$-closed elements of $L$ (those elements $a$ such that $j(a) = a$).  Note that, by property (3) above, we may interpret $j$ as a function $j^*\colon L \to L/j$, which is a [[surjection|surjective]] frame [[homomorphism]].  Since [[Frm]] is an [[algebraic category]], this means that $L/j$ is a [[regular quotient]] of $L$.

Conversely, suppose that $M$ is any regular quotient of $L$; that is, we have a surjective frame homomorphism $k\colon L \to M$.  Since $k$ is a frame homomorphism, it has (by the [[adjoint functor theorem]]) a [[right adjoint]] $k_*\colon M \to L$.  Let $j\colon L \to L$ be the [[composite]] of $k$ followed by $k_*$.  Then $j$ is a nucleus, and $k_*$ is an [[embedding]] (in [[Pos]], not $Frm$) whose [[image]] is $L/j$.

In short, given a nucleus $j$, we have an [[adjunction]] $j^*\colon L \rightleftarrows L/j: j_*$, where $j^*$ is a surjective homomorphism and $j_*$ is the [[inclusion function]]; while, given a surjective homomorphism $k$, we have an adjunction $k\colon L \rightleftarrows M: k_*$, where $k_* \circ k$ is a nucleus and $k_*$ is an embedding.

If we think of $L$ as a [[locale]], then we define a __[[sublocale]]__ of $L$ to be a quotient frame of $L$, which corresponds to a nucleus on $L$ as above.

## For quantales

Quantales are a generalization of locales. The corresponding version of a nucleus is sometimes called [[quantic nucleus]].

## Categorification

If you [[categorify]] from locales to [[toposes]], then nuclei become [[Lawvere–Tierney topologies]], and the operation of the nucleus becomes [[sheafification]].

## References

* _[[Stone Spaces]]_, II.2
* [[André Joyal]], _A crash course on topos theory_, part 1/4, IHES 2006 [yt](https://www.youtube.com/watch?v=Ro8KoFFdtS4)

[[!redirects nucleus]]
[[!redirects nucleuses]]
[[!redirects nuclei]]


nucleus (disambiguation)

* [[nucleus]] in [[point-free topology]]

* [[atomic nucleus]] in [[nuclear physics]]

* [[nucleus of a profunctor]] in [[enriched category theory]]

category: disambiguation


nucleus (physics)

> this entry is about the concept in [[physics]]. For other meanings see [[nucleus (disambiguation)]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] or [[chemistry]] (or [[quantum chemistry]]) an _atomic nucleus_ --  or just  _nucleus_, for short --  is a [[bound state]] of [[protons]] and [[neutrons]]. If this in turn is surrounded by a shell of [[electrons]] the result is an [[atom (physics)|atom]], with the nucleus at its core.

The study of nuclei is the topic of [[nuclear physics]].

## Skyrmion models

Candidate models for [[baryons]] and even some aspects of [[atomic nuclei]]  are [[Skyrmions]] ([Riska 93](#Riska93), [Battye-Manton-Sutcliffe 10](#BattyeMantonSutcliffe10), [Manton 16](#Manton16), [Naya-Sutcliffe 18](#NayaSutcliffe18)).

For instance various resonances of thethe [[carbon]] [[nucleus]] are modeled well by a Skyrmion with baryon number 12 ([Lau-Manton 14](#LauMaonton14)):

\begin{center}

\end{center}

> graphics grabbed form [Lau-Manton 14](#LauMaonton14)

For Skyrmion models of nuclei to match well to [[experiment]], not just the [[pion field]] but also the heavier [[mesons]] need to be included in the construction. Including the [[rho meson]] gives good results for light nuclei ([Naya-Sutcliffe 18](#NayaSutcliffe18))

\begin{center}

\end{center}

> graphics grabbed form [Naya-Sutcliffe 18](#NayaSutcliffe18)

## Related concepts

* [[alpha particle]]

* [[confinement]]

* [[nuclear magnetic resonance]]

## References

On [[Skyrmion]] [[model (in theoretical physics)|models]] for nucleons:

* {#Riska93} D. O. Riska, _Baryons and nuclei as skyrmions_,  Czech J Phys (1993) 43: 449 ([doi:10.1007/BF01589856](https://doi.org/10.1007/BF01589856))

* {#BattyeMantonSutcliffe10} [[Richard A. Battye]], [[Nicholas Manton]], [[Paul Sutcliffe]], _Skyrmions and Nuclei_, pp. 3-39 (2010) ([doi:10.1142/9789814280709_0001](https://doi.org/10.1142/9789814280709_0001)) in:  [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

* {#LauManton14} P.H.C. Lau, [[Nicholas Manton]], _States of Carbon-12 in the Skyrme Model_,  Phys. Rev. Lett. 113, 232503 (2014) ([arXiv:1408.6680](https://arxiv.org/abs/1408.6680))

* {#Manton16} [[Nicholas Manton]], _Skyrmions and Nuclei_, talk at Brookhaven National Lab, November 2016 ([pdf](https://quark.phy.bnl.gov/~pisarski/talks/Manton_SkyBNL.pdf))

* {#NayaSutcliffe18} [[Carlos Naya]], [[Paul Sutcliffe]], _Skyrmions and clustering in light nuclei_, Phys. Rev. Lett. 121, 232002 (2018) ([arXiv:1811.02064](https://arxiv.org/abs/1811.02064))

* L. A. Ferreira, L. R. Livramento, *A False Vacuum Skyrme Model for Nuclear Matter* [[arXiv:2106.13335](https://arxiv.org/abs/2106.13335)]

* [[Nicholas Manton]], *Skyrmions as Models for Nuclei*, Ischia (2022) [[pdf](https://agenda.infn.it/event/19753/contributions/170544/attachments/90671/122280/SkyIschia.pdf)]

Textbook account:

* [[Nicholas Manton]], *Skyrmions – A Theory of Nuclei*, World Scientific (2022) [[doi:10.1142/q0368 ](https://doi.org/10.1142/q0368 )]

See also

* Wikipedia, _[Atomic nucleus](https://en.wikipedia.org/wiki/Atomic_nucleus)_
[[!redirects atomic nucleus]]

[[!redirects atomic nuclei]]


nucleus of a profunctor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition ##

If $C$ and $D$ are [[enriched categories]] over a [[cosmos]] $V$, that is, a complete and cocomplete symmetric [[closed monoidal category]], then a [[profunctor]] $F$, from $C$ to $D$, that is, a $V$-functor $D^{op} \otimes C\to V$, induces two adjoint $V$-functors: $F_{\ast}: (V^C)^{op} \to V^{D^{op}}$ and $F^{\ast}: V^{D^{op}} \to (V^C)^{op}$. Abstractly, one $V$-functor is given by currying to get a $V$-functor $C \to V^{D^{op}}$ and then using the [[universal property]] of the [[free completion]]; and the other $V$-functor is given by currying to get a $V$-functor $D^{op} \to V^C$, taking [[opposite category|opposites]] to get a $V$-functor $D \to (V^C)^{op}$, and then using the universal property of the [[free cocompletion]]. We then have an adjunction:

$$
(V^C)^{op}(F^{\ast} p, q) \cong V^{D^{op}}(p, F_{\ast} q).
$$

In the case of the $Hom$ profunctor, $Hom: A^{op} \otimes A \to V$, of a $V$-enriched category $A$, this adjunction is known as [[Isbell duality]] or Isbell conjugation. In analogy with this case, the general adjunction induced by a $V$-profunctor $F$ is called the **Isbell adjunction** induced by $F$.

The **nucleus** of $F$ is the [[center of an adjunction|center]] of this adjunction. In the case where $F = Hom$, the nucleus is called the [[reflexive completion]].

##Examples with $F = Hom$##

* $V = Ab$, the category of abelian groups. Let $k$ be a field, viewed as a one-object $Ab$-category. Both $[k^{op},Ab]$ and $[k,Ab]$ are the category of $k$-vector spaces, and both adjoints are the dual vector space construction. The nucleus of the profunctor, or reflexive completion $R(k)$ of $k$, is the category of $k$-vector spaces $V$ for which the canonical map $V \to V^{\ast \ast}$ is an isomorphism — in other words, the [[finite-dimensional vector spaces]].

##Examples with other profunctors##
* $V$ = truth values. Given two sets, $A$ and $B$, and a relation (truth-valued profunctor) between them, the adjunction is known as a [[Galois connection]], which restricts to the nucleus, a [[Galois correspondence]].
* $V = \overline{\mathbb{R}} = ([-\infty, \infty], \geq, +)$. Let a real vector space, $W$, be considered as a discrete $\overline{\mathbb{R}}$-category, and consider the $\overline{\mathbb{R}}$-profunctor corresponding to evaluation between an element of $W$ and an element of its dual. Then the nucleus is composed of $\overline{\mathbb{R}}$-valued functions on $W$, and the duality expresses the [[Legendre-Fenchel transform]]. (See Simon Willerton's [post](http://golem.ph.utexas.edu/category/2014/05/enrichment_and_the_legendrefen_1.html).)

##References

* [[Simon Willerton]], Tight spans, Isbell completions and semi-tropical modules, ([tac](http://www.tac.mta.ca/tac/volumes/28/22/28-22abs.html))
* [[Simon Willerton]], _The Nucleus of a Profunctor: Some Categorified Linear Algebra_, [blog post](http://golem.ph.utexas.edu/category/2013/08/the_nucleus_of_a_profunctor_so.html)
* [[Simon Willerton]], _Galois Correspondences and Enriched Adjunctions_, [blog post](http://golem.ph.utexas.edu/category/2014/02/galois_correspondences_and_enr.html)
* Shen, Lili, and Dexue Zhang. _Categories enriched over a quantaloid: Isbell adjunctions and Kan adjunctions_ (2013), ([TAC](http://www.tac.mta.ca/tac/volumes/28/20/28-20abs.html))


null homotopy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In the context of [[algebraic topology]], a [[continuous map]] $f \colon X \longrightarrow Y$ between two [[topological spaces]] $X,Y$, is said to be _null homotopic_ if it is [[homotopic]] to a [[constant map]] $g$. This is considered in particular in the context of [[pointed topological spaces]] with base point-preserving maps between them, hence for $g$ the map constant on the base point of $Y$ (which is the [[zero morphism]] in this context).

In the context of [[homological algebra]], a _null homotopy_ is a [[chain homotopy]] from (or to) the [[zero map]].

Generally in abstract [[homotopy theory]], a null homotopy in an [[pointed (infinity,1)-category]] is a [[2-morphism]] to (or from) a [[zero morphism]].

This general concept subsumes the previous two cases via the [[presentable (infinity,1)-category|(infinity,1)-categories presented by]] the [[classical model structure on pointed topological spaces]] or a [[model structure on chain complexes]], respectively.

## Related concepts

* [[homotopy category of chain complexes]]

* [[contractible chain complex]]

## References

Textbook accounts:

* [[James Munkres]], Section 51 of: _[[Introduction to Topology]]_

* [[Charles Weibel]], Section 1.4 of: _[[An Introduction to Homological Algebra]]_

[[!redirects null homotopies]]

[[!redirects contracting homotopy]]
[[!redirects contracting homotopies]]


null hypothesis

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related entries

* [[hypothesis testing]]

* [[statistical significance]]

* [[null result]]

## References

See also

* Wikipedia, _[Null hypothesis](https://en.wikipedia.org/wiki/Null_hypothesis)_

[[!redirects null hypotheses]]


null result

#Contents#
* table of contents
{:toc}

## Ideaf

An [[experiment]]/[[measurement]] is said to produce a _null result_ if it supports the [[null hypothesis]], typically in that the result has low [[statistical significance]].

## Examples

### Null results in physics

Historically important null results in [[physics]]:

* [[Michelson-Morley experiment]] led to development of [[special relativity]], thus of [[general relativity]] and the modern [[theory (physics)|theory]] of [[gravity]].

* absence of the [[ultraviolet catastrophe]] predicted by $H_0 = $ [[classical field theory|classical]] [[electromagnetism]]] led to the hypothesis of [[energy]]/[[frequency]]-quantization (now expressed by [[Planck's constant]]) and hence to the development of [[quantum physics]]

## References

See also

* Wikipedia, _[Null result](https://en.wikipedia.org/wiki/Null_result)_

[[!redirects null results]]


null sequence

Let $X$ be a topological vector space. A null sequence in $X$ is a convergent sequence whose limit is $0 \in X$.


null subset

> ([[null set|Null set]] redirects here; for the notion in set theory, see [[empty set]].)

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

# Null and full sets
* table of contents
{: toc}

## Idea

In [[measure theory]], a _null set_ is a [[subset]] of a [[measure space]] (or [[measurable space]]) that is so small that it can be neglected: it might as well be the [[empty subset]]; its measure is [[zero]].  Similarly, a _full set_ is a subset that is so large that it might as well be the [[improper subset]] (the entire space).  One also says that a null set has _null measure_ and a full set has _full measure_.

Traditionally, full sets are not usually referred to explicitly; in [[classical mathematics]], they are simply the [[complements]] of null sets.  However, they are often referred to implicitly through such terminology as 'almost everywhere'.  Also, in [[constructive mathematics]], full sets are more fundamental than null sets; they are not simply the complements of the latter.

## Definitions

The definitions depend on the context.

### In a measure space

In a traditional [[measure space]], we have an abstract [[set]] $X$, a $\sigma$-[[sigma-algebra|algebra]] (or similar structure) $\mathcal{M}$ consisting of the [[measurable subsets]] of $X$, and a [[measure]] $\mu$ mapping each measurable set $A$ to a [[real number]] (or similar quantity) $\mu(A)$, the measure of $A$.

A measurable subset $B$ of $X$ is __full__ if, given any measurable set $A$, $\mu(A \cap B) = \mu(A)$; an arbitrary subset of $X$ is __full__ if it\'s a [[superset]] of a full measurable set.  [[de Morgan duality|Dually]], a measurable set $B$ is __null__ if, given any measurable set $A$, $\mu(A \cup B) = \mu(A)$; an arbitrary subset of $X$ is __null__ if it\'s a [[subset]] of a null measurable set.

Some equivalent characterisations ([[constructive mathematics|constructively]] valid for measures on [[Cheng spaces]] except as stated):

*  A measurable set $B$ is null iff $\mu(C) = 0$ for every measurable subset of $B$.
*  If $\mu$ is a [[positive measure]], then a measurable set $B$ is null iff $\mu(B) = 0$.
*  If $\mu$ is a [[finite measure]] with total measure $I$, then a measurable set $B$ is full iff $\mu(C) = I$ for every measurable superset of $B$.
*  If $\mu$ is both positive and finite (so a [[probability measure]] up to rescaling), then a measurable set $B$ is full iff $\mu(B) = I$.
*  If $\mu$ is [[complete measure|complete]], then every null set is measurable and every full set is measurable (which is basically the definition of 'complete') and consequently the preceding properties continue to hold when the adjective 'measurable' is removed.
*  Using [[excluded middle]], a set is null iff its complement is full, and a set is full iff its complement is null.  (Even constructively, if a set is null, then its complement is full.)
*  Even constructively, a measurable set is null iff its measurable complement (the complement in the algebraic structure of complemented pairs in a Cheng measurable space) is full, and a measurable set is full iff its measurable complement is null.

### In untraditional measurable spaces

Traditionally, a [[measurable space]] is simply an abstract [[set]] $X$ and a $\sigma$-[[sigma-algebra|algebra]] (or similar structure) $\mathcal{M}$ consisting of the [[measurable subsets]] of $X$.  There is no notion of null or full subsets of such a space.  However, there are two (essentially equivalent) variations of this concept in which null and full subsets do make sense.

One variation is to simply equip the space with a $\delta$-[[delta-filter|filter]] of measurable subsets, which are declared to be the full measurable subsets.  Then a general __full subset__ is a superset of a measurable full subset, and a __null subset__ is any set whose complement is full.  (Alternatively, equip the space with a $\sigma$-[[sigma-ideal|ideal]] of measurable subsets, which are declared to be the null measurable subsets.)  In particular, a [[localizable measurable space]] is a measurable space so equipped such that the [[Boolean algebra]] of measurable sets modulo null sets (or modulo full sets if this is done by identifying the full sets with $X$) is [[complete boolean algebra|complete]].

Another variation, used especially in [[constructive mathematics]], is a [[Cheng measurable space]].  This consists of a set $X$ equipped with a $\sigma$-semialgebra of [[disjoint subsets|disjoint]] *pairs* of subsets of $X$, declared to be the _complemented pairs_.  A set is measurable iff it appears as one component of a complemented pair.  A measurable subset is _full_ if it appears as one component of a complemented pair whose other component is [[empty subset|empty]], or equivalently (given the structure of the algebra of complemented pairs) if it is the [[union]] of the two components of any complemented pair.  Then a general __full subset__ is a superset of a measurable full subset, and a __null subset__ is any set whose complement is full.

These are actually equivalent concepts.  Given a measurable space equipped with a $\delta$-filter of measurable full subsets, define a complemented pair to be a pair of disjoint measurable subsets whose union is full.  Conversely, given a Cheng measurable space, the measurable subsets and measurable full subsets as defined above comprise a $\sigma$-algebra and a $\delta$-filter in it.  (But constructively, the algebra of measurable subsets, while closed under the appropriate operations, will generally not be a [[boolean algebra]].)

### In smooth manifolds

A subset $A$ of an $n$-dimensional [[smooth manifold]] $X$ is __null__ or __full__ (respectively) if its [[preimage]] under every [[coordinate chart]] is a null or full subset (respectively) of the chart\'s domain (which is an [[open subset]] of the [[Cartesian space]] $\mathbb{R}^n$) under [[Lebesgue measure]].

This is actually better behaved than it may at first seem.  If $A$ is [[open cover|covered]] by an [[atlas]] $(\phi_i\colon U_i \to X)_i$, then $A$ is null or full as soon as $\phi_i^*(A)$ is null/full in $U_i$ for every index $i$.  In particular, if $A$ is contained in a single coordinate chart (which is not very likely for a full set but fairly common for null sets), then it is sufficient to check its preimage under that one.  This fact depends on the smoothness and fails for [[topological manifolds]].

As we can define a [[measurable subset]] of a smooth manifold similarly, this means that every smooth manifold gives rise to a measurable space equipped with a $\delta$-filter of full subsets (and hence to a Cheng measurable space); this space is always localizable.

(Details?  Is $C^1$ sufficient?  Conversely, is paracompactness necessary to keep the covers manageable?)

## Logic of full/null sets

A property of elements of $X$ (given by a [[subset]] $S$ of $X$) can be considered modulo null sets.  We say that the property $\phi$ is true __almost everywhere__ or __almost always__ if it is true on some full set, that is if $\{X | \phi\}$ is full.  Dually, we say that $\phi$ is true __almost nowhere__ or __almost never__ if $\{X | \phi\}$ is null.  It is better to use the [[negation]] of 'almost nowhere', although the terminology for this is not really standard; say that $\phi$ is true __somewhere significant__ if $\{X | \phi\}$ is non-null.

Note that being true almost everywhere is a weakening of being true everywhere (given by the [[universal quantifier]] $\forall$), while being true somewhere significant is a strengthening of being true somewhere (given by the [[particular quantifier]] $\exists$).  Indeed we can build a logic out of these.  Use $\ess\forall i, \phi[i]$ or $\ess\forall \phi$ to mean that a [[predicate]] $\phi$ on $X$ is true almost everywhere, and use $\ess\exists i, \phi[i]$ or $\ess\exists \phi$ to mean that $\phi$ is true somewhere significant.  Then we have:
$$\forall \phi \;\Rightarrow\; \ess\forall \phi$$
$$\ess\exists \phi \;\Rightarrow\; \exists \phi$$
$$\ess\forall (\phi \wedge \psi) \;\Leftrightarrow\; \ess\forall \phi \wedge \ess\forall \psi$$
$$\ess\exists (\phi \wedge \psi) \;\Rightarrow\; \ess\exists \phi \wedge \ess\exists \psi$$
$$\ess\forall (\phi \vee \psi) \;\Leftarrow\; \ess\forall \phi \wedge \ess\forall \psi$$
$$\ess\exists (\phi \vee \psi) \;\Leftrightarrow\; \ess\exists \phi \vee \ess\exists \psi$$
$$\ess\forall \neg{\phi} \;\Leftrightarrow\; \neg\ess\exists \phi$$
and other analogues of theorems from [[predicate logic]].  Note that the last item listed requires [[excluded middle]] even though its analogue from ordinary predicate logic does not.

A similar logic is satisfied by '[[eventually]]' and its dual ('frequently') in [[filters]] and [[nets]].

## Related concepts

* [[almost surely]]

## References

See also

* Wikipedia, *[Null set](https://en.wikipedia.org/wiki/Null_set)*

[[!redirects negligible set]]
[[!redirects negligible sets]]
[[!redirects negligible subset]]
[[!redirects negligible subsets]]

[[!redirects null set]]
[[!redirects null sets]]
[[!redirects null subset]]
[[!redirects null subsets]]
[[!redirects null measure]]
[[!redirects set of null measure]]
[[!redirects sets of null measure]]
[[!redirects subset of null measure]]
[[!redirects subsets of null measure]]

[[!redirects full set]]
[[!redirects full sets]]
[[!redirects full subset]]
[[!redirects full subsets]]
[[!redirects full measure]]
[[!redirects set of full measure]]
[[!redirects sets of full measure]]
[[!redirects subset of full measure]]
[[!redirects subsets of full measure]]

[[!redirects almost everywhere]]
[[!redirects almost nowhere]]
[[!redirects somewhere significant]]


null system

#Idea#

A _null system_ in a [[triangulated category]] is a triangulated [[subcategory]] whose objects may consistently be regarded as being equivalent to the [[zero object]]. Null systems give a convenient means for encoding and computing [[localization]] of [[triangulated category|triangulated categories]].

#Definition#

A **null system** of a [[triangulated category]] $C$ is a full [[subcategory]] $N \subset C$ such that

* $N$ is _saturated_: every object $X$ in $C$ which is [[isomorphism|isomorphic]] in $C$ to an object in $N$ is in $N$;

* the [[zero object]] is in $N$;

* $X$ is in $N$ precisely if $T X$ is in $N$;

* if $X \to Y \to Z \to T X$ is a distinguished triangle in $C$ with $X, Z \in N$, then also $Y \in N$.

#Properties#

The point about null systems is the following:

for $N$ a null system, let $N Q$ be the collection of all morphisms in $C$ whose "[[mapping cone]]" is in $N$, precisely: set

$$
  N Q := \{ X \stackrel{f}{\to} Y | \exists dist. tri. X \to Y \to Z \: in \: C \: with \: Z \in N\}
  \,.
$$

Then
$N Q$ admits a left and right [[calculus of fractions]] in $C$.

#Examples#

* For $K(C)$ the [[category of chain complexes]] modulo [[chain homotopy]] of an [[abelian category]], the full [[subcategory]] of $K(C)$ of chain complexes $V$ whose [[homology]] vanishes, $H(V) \simeq 0$ is a null system.
Then$D(C) := K(C)/N$ is the [[derived category]] of $C$.

+-- {: .query}
[[David Roberts]]: Would [[Serre class]]es fit in here? Perhaps that's one step back.
=--

#References#

For instance section 10.2 of

* Kashiwara--Schapira, [[Categories and Sheaves]]


Nullstellensatz

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Nullstelle_ means [[zero locus]]. Hilbert's _Nullstellensatz_ (theorem about zero loci) characterizes the joint zero loci of an [[ideal]] of functions in a [[polynomial ring]]. This is a foundational result in [[algebraic geometry]] at the heart of the [[Isbell duality|duality]] by which [[rings]] are dually interpreted as [[spaces]] ([[varieties]]).

## Classical formulation in algebraic geometry

In classical algebraic geometry over an [[algebraically closed field]] $k$, the Nullstellensatz concerns the [[fixed points]] of a standard [[Galois connection]] between [[ideals]] $I$ of the [[polynomial ring]] $k[x_1, \ldots, x_n]$ and [[subsets]] $V \subseteq k^n$. The Galois connection is induced by a [[relation]] $I \perp V$ iff $f(x) = 0$ for all $f \in I, x \in V$. Accordingly, letting $Idl(k[x_1, \ldots, x_n])$ be the set of ideals ordered by inclusion and $P(k^n)$ the set of subsets of $k^n$ ordered by inclusion, there are contravariant maps $Ideal: P(k^n) \to Idl(k[x_1, \ldots, x_n])$ and $Var: Idl(k[x_1, \ldots, x_n]) \to P(k^n)$ defined by

$$Ideal(S) = \{f \in k[x_1, \ldots, x_n]: \forall_x x \in S \Rightarrow f(x) = 0\}, \qquad Var(I) = \{x \in k^n: \forall_f f \in I \Rightarrow f(x) = 0\}$$

which defines the Galois connection.

The fixed points of $Var \circ Ideal: P(k^n) \to P(k^n)$ are, by definition, the closed sets of the [[Zariski topology]] on the [[affine space]] $k^n$ ("affine space" here in the sense of algebraic geometry). These are the closed (not necessarily reduced) [[subvarieties]] of $k^n$.

On the other side, the fixed points of $Ideal \circ Var$ is what the classical Nullstellensatz is about:

+-- {: .num_theorem}
###### Theorem
**(Strong Nullstellensatz)** The fixed points of $Ideal \circ Var$ are precisely the [[radical]] ideals $I$ of $k[x_1, \ldots, x_n]$, i.e., ideals $I$ such that for any $r \in \mathbb{N}$, the condition $f^r \in I$ implies $f \in I$.
=--

Thus there is a [[Galois correspondence]] between closed subvarieties and radical ideals.

Typically the way that the strong Nullstellensatz is proved is by reduction to the so-called "weak Nullstellensatz" by means of the "Rabinowitsch trick". (The "weak" here may be misleading, as the weak Nullstellensatz may be considered the core result and the strong Nullstellensatz a corollary.) We now turn to these.

### A weak Nullstellensatz: Existence of points
 {#ExistenceOfPoints}

One weak version of the Nullstellensatz says the following (e.g. Theorem 3.99.1 here: [[Frank-ALGANT-2010-7-12.pdf:file]])

For $k$ an [[algebraically closed field]] and $I$ a proper [[ideal]] in the [[polynomial ring]] $k[X_1, \cdots, X_n]$, the set $V(I)$ (of $n$-[[tuples]] $x = (x_i) \in k^n$ such that all polynomials in $I$ vanish when evaluated on these $x$) is an [[inhabited set]].

Before giving a proof, we remark (in view of more abstract formulations to come later) that an element of $V(I)$ is just a $k$-[[associative algebra|algebra]] [[homomorphism]] of the form

$$
  k \longleftarrow  k[X_1, \cdots, X_n]/I
  \,.
$$

[[Isbell duality|Dually]] this is a morphism of [[affine schemes]] ([[spectrum of a commutative ring|ring spectra]]) of the form

$$
  Spec(k) \longrightarrow Spec(k[X_1, \cdots, X_n]/I)
  \,.
$$

Moreover, since $Spec(k)$ is the [[terminal object]] in this context, such a map is the same as a "point", a [[global element]] of $Spec(k[X_1, \cdots, X_n]/I)$.

Hence in this form the Nullstellensatz simply says that (for $k$ algebraically closed) affine schemes have points. This formulation of the Nullstellensatz leads one to a [more general abstract formulation](#GeneralAbstract).

We turn now to the proof of the weak Nullstellensatz.

+-- {: .num_lemma #uncountable}
###### Lemma
If $k$ is an **uncountable** algebraically closed field, then for any proper ideal $I \subset k[X_1, \ldots, X_n]$ there is $(a_1, \ldots, a_n) \in k^n$ such that $f(a_1, \ldots, a_n) = 0$ for all $f \in I$.
=--

+-- {: .proof}
###### Proof
By the [[axiom of choice]], $I$ is contained in a [[maximal ideal]] $M$. So it suffices to prove it in the case where $I = M$ is maximal, where we prove there is an isomorphism $\phi: k[X_1, \ldots, X_n]/M \cong k$. Then the composite

$$k[X_1, \ldots, X_n] \stackrel{quot}{\twoheadrightarrow} k[X_1, \ldots, X_n]/M \stackrel{\phi}{\to} k$$

sends each $X_i$ to a value $a_i$, so we have an inclusion of maximal ideals

$$(X_1 - a_1, \ldots, X_n - a_n) \subseteq M$$

which must be an equality. In that case $(a_1, \ldots, a_n)$ is the desired point.

Since $k$ is algebraically closed, it suffices to prove that the extension field $F = k[X_1, \ldots, X_n]/M$ over $k$ is algebraic over $k$. Clearly $F$ as a [[vector space]] over $k$ has countable [[dimension]]. Supposing to the contrary there is an element $t \in F$ that is transcendental over $k$, we have a [[subquotient]] vector space of $F$,

$$\frac{k(t)}{k[t]} \cong \bigoplus_{a \in k} \frac{k[\frac1{t - a}]}{k},$$

where the isomorphism is by partial fraction decomposition (ultimately by the [[Chinese remainder theorem]]), but the vector space on the right clearly has uncountable dimension because there are uncountably many $a \in k$, and this leads to a contradiction.
=--

Now we reduce to the case where $k$ is uncountable by employing a [[model theory|model-theoretic]] argument:

+-- {: .num_theorem #weak}
###### Theorem
**(Weak Nullstellensatz)**
If $k$ is an algebraically closed field, then for any proper ideal $I \subset k[X_1, \ldots, X_n]$ there is $(a_1, \ldots, a_n) \in k^n$ such that $f(a_1, \ldots, a_n) = 0$ for all $f \in I$.
=--

+-- {: .proof}
###### Proof
Let $U$ be a non-principal [[ultrafilter]] on $\mathbb{N}$, and let $K$ be the [[ultraproduct|ultrapower]] $k^\mathbb{N}/U$. By [[Łoś's ultraproduct theorem]], $K$ is an algebraically closed field. Also $K$ is uncountable (Lemma \ref{ultrapower} below).

First we claim $K \otimes_k I$ is a proper ideal of $K \otimes_k k[X_1, \ldots, X_n] \cong K[X_1,\ldots, X_n]$. For suppose we have $1 = \sum_{i = 1}^s f_i p_i$ for $p_1, \ldots, p_s \in I$ and $f_i \in K[X_1, \ldots, X_n]$. Writing the $f_i$ as $K$-linear combinations of monomials $X^\alpha$, there are finitely many $q_1, \ldots, q_t \in I$, each of the form $X^\alpha p_i$, so as to enable us to write $1 = \sum_{i=1}^t a_i q_i$ for $a_i = [(a_{i, m})] \in K$ and $q_i \in I$. By Łoś's theorem,

$$\{m \in \mathbb{N}: 1 = \sum_{i=1}^t a_{i, m} q_i\; in\; k[X_1, \ldots, X_n]\} \in U;$$

this set is nonempty, and so there is at least one $m$ rendering the equation true. This contradicts $1 \notin I$.

By Lemma \ref{uncountable}, there exists $(a_1, \ldots, a_n) \in K^n$ belonging to $Var(K \otimes_k I)$. By [[noetherian ring|Hilbert's basis theorem]], $I \subset k[X_1, \ldots, X_n]$ is generated by finitely many polynomials $p_1, \ldots, p_s$, so $Var(K \otimes_k I) \subseteq K^n$ is defined by finitely many formulas $p_1(X) = 0, \ldots, p_s(X) = 0$ and thus by Łoś's theorem again,

$$\{m \in \mathbb{N}: p_i(a_{1, m}, a_{2, m}, \ldots, a_{n, m}) = 0\; for\; i = 1, \ldots, s\} \in U$$

and so this set is nonempty and therefore there is a point $(a_{1, m}, \ldots, a_{n, m}) \in k^n$ that belongs to $Var(I)$.
=--

To prove $K$ uncountable, we use the fact that an algebraically closed field is at least infinite. Then the missing piece is supplied by the following result.

+-- {: .num_lemma #ultrapower}
###### Lemma
If $X$ is an infinite set and $U$ is a non-principal ultrafilter on $\mathbb{N}$, then the ultrapower $X^\mathbb{N}/U$ has cardinality ${|X|}^{\aleph_0}$ (so at least $2^{\aleph_0}$).
=--

+-- {: .proof}
###### Proof
For $j \in \mathbb{N}$, put $[j] = \{i \in \mathbb{N}: i \lt j\}$. If $X$ is infinite, then there is a mono $i_j: X^{[j]} \to X$ for each $j \geq 0$. For any $h: \mathbb{N} \to X$, let $h|_{[j]}$ denote the composite $[j] \hookrightarrow \mathbb{N} \stackrel{h}{\to} X$, and define $h': \mathbb{N} \to X$ by $h'(j) = i_j(h|_{[j]})$. Let $[h']_U$ denote the value of $h'$ under the quotient map $X^\mathbb{N} \to X^\mathbb{N}/U$.

Claim: The map $X^\mathbb{N} \to X^\mathbb{N}/U$ that takes $h$ to $[h']_U$ is injective. For, if $g, h: \mathbb{N} \to X$ are distinct, then they differ at some $i \in \mathbb{N}$. Then $g|_{[j]} \neq h|_{[j]}$ whenever $i \lt j$, which in turn implies $i_j(g|_{[j]}) \neq i_j(h|_{[j]})$ i.e. $g'(j) \neq h'(j)$ for all $j$ such that $j \gt i$. (That is, "for almost all $j$", belonging to a cofinite set that belongs to $U$.) Thus $[g']_U \neq [h']_U$, which establishes the claim. The lemma follows immediately from the claim.
=--

### Rabinowitsch's "trick"

Rabinowitsch, who later renamed himself Rainich (see the story [here](http://mathoverflow.net/a/45195/2926)), showed how to deduce the Strong Nullstellensatz from the Weak Nullstellensatz. Often this is described as Rabinowitsch's "trick" (i.e., something involving a stroke of cleverness), but see the remark below which places the trick conceptually within the general method of localization.

Let $k$ be algebraically closed and let $I \subseteq k[x_1, \ldots, x_n]$ be an ideal. To show that $Ideal(Var(I)) = \sqrt{I}$, suppose that $f \in k[x_1, \ldots, x_n]$ vanishes at any point where all of a given set of polynomials $f_1, \ldots, f_s \in k[x_1, \ldots, x_n]$ vanish. It suffices to show that for some $r$ we have that $f^r$ belongs to the ideal generated by the $f_i$.

Introduce a fresh variable $x_0$, and observe that there is no point $(a_0, a_1, \ldots, a_n) \in k^{n+1}$ where the polynomials $1 - x_0 f, f_1, \ldots, f_s$ simultaneously vanish. By the Weak Nullstellensatz (Theorem \ref{weak}), these polynomials must generate the unit ideal, say

$$1 = p_0(x)(1 - x_0 f) + p_1(x) f_1 + \ldots + p_s(x) f_s$$

where $x = (x_0, x_1, \ldots, x_n)$. There is a $k$-algebra map $k[x_0, x_1, \ldots, x_n] \to k(x_1, \ldots, x_n)$ sending $x_0 \mapsto 1/f$ and $x_i \mapsto x_i$ for $1 \leq i \leq n$, where the above identity yields

$$1 = \sum_{i=1}^s p_i(1/f, x_1, \ldots, x_n) f_i$$

which, if $r$ is the highest degree of $x_0$ appearing in a $p_i$, may be rewritten in the form $f^r = \sum_{i=1}^s g_i f_i$ in $k[x_1, \ldots, x_n]$ by clearing denominators.

+-- {: .num_remark}
###### Remark
Proving that $f \in \sqrt{I}$ means we prove that $f$ is nilpotent in $k[x_1, \ldots, x_n]/I$, in other words that the localization of $k[x_1, \ldots, x_n]/I$ with respect to the (image of the) multiplicative system $\{1, f, \ldots, f^n, \ldots\}$ vanishes. Clearly we localize by inverting (the image of) $f$, by constructing
$$\left(k[x_1, \ldots, x_n]/I\right)[f^{-1}] = k[x_0, x_1, \ldots, x_n]/(I, 1 - x_0 f).$$
Clearly $Var(I, 1 - x_0 f) = \emptyset$, and so by the weak Nullstellensatz the ideal $(I, 1 - x_0 f)$ is improper, i.e., the localization vanishes.
=--

## General abstract formulations
 {#GeneralAbstract}

By the discussion above at _[Existence of points](#ExistenceOfPoints)_ it follows that phrased in terms of the [[sheaf topos]] over the [[Zariski site]], the weak Nullstellensatz says that "objects have global points".

Motivated by this, in ([Lawvere 07, def. 2](#Lawvere07)) is a proposal for a [[general abstract]] formulation of what constitutes a _Nullstellensatz_, formalized in the context of [[cohesion]].

A [[cohesive topos]] $\mathbf{H}$ over some [[base topos]] $\mathbf{S}$ is a relative [[topos]] such that the terminal [[geometric morphism]] extended to an [[adjoint quadruple]] $(\Pi \dashv Disc \dashv \Gamma \dashv coDisc)$ with $Disc, coDisc \colon \mathbf{S}\hookrightarrow \mathbf{H}$ [[full and faithful functor|full and faithful]] and $\Pi$ preserving [[finite products]]. The top [[adjoint triple]] here induces a canonical [[natural transformation]]

$$
  ptp \;\colon\; \Gamma \longrightarrow \Pi
$$

which deserves to be called the _[points-to-pieces transformation](cohesive+topos#CanonicalComparison)_.

Consider the condition that $ptp$ is on every object $X$ of $\mathbf{H}$ an [[epimorphism]]. In terms of the geometric interpretation of [[cohesion]] this means that "all pieces of $X$ have at least one point". (See at _[cohesive topos -- Pieces have points](cohesive+topos#PiecesHavePoints)_)
This statement is what in [Lawvere 07, def. 2 c)](#Lawvere07) is also referred to as **Nullstellensatz**. More comments on this are in ([Lawvere 11](#Lawvere11)). (In ([Johnstone 11](#Johnstone11)) this condition is called "punctual local connectedness".)

For a detailed analysis of how this general abstract concept indeed captures the traditional meaning of _Nullstellensatz_, see ([Lawvere 15](#Lawvere15), [Menni](#Menni)), also ([Tholen](#Tholen)).

## References

* Wikipedia, _[Hilbert's Nullstellensatz](http://en.wikipedia.org/wiki/Hilbert's_Nullstellensatz)_

* David Hilbert. Ueber die vollen Invariantensysteme. Mathematische Annalen **42(3)**, pp. 313–373. 1893.

### The model theoretic perspective

* [[Marta Bunge|M. Bunge]], _On the transfer of an abstract Nullstellensatz_ , Comm. Alg. **10** (1982) pp.1891-1906.

* W. A. MacCaull, _Hilbert's Nullstellensatz revisited_ , JPAA **54** (1988) pp.289-297.

* V. Weispfenning, _Nullstellensätze - a model theoretic framework_ , Z. Math. Logik Grundl. Math. **23** (1977) pp. 539-545.

### In terms of cohesion

Discussion of the Nullstellensatz in terms of [[cohesion]]:

* {#Lawvere07} [[William Lawvere]], _Axiomatic cohesion_ Theory and Applications of Categories, Vol. 19, No. 3 (2007) pp. 41–49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))

* {#Lawvere11} [[William Lawvere]], _[MO comment on the general abstract Nullstellensatz](http://mathoverflow.net/a/52894/381)_ (2011)

* {#Lawvere15}[[F. William Lawvere]], _Birkhoff's Theorem from a geometric
perspective: A simple example_ , Categories and General Algebraic Structures with Applications, Vol. 4, No. 1 (2015) pp. 1–7. ([pdf](http://www.cgasa.ir/pdf_12425_5ddf468d82aad53ce2e8ee88dd3bf84d.html))

* {#Johnstone11} [[Peter Johnstone]], _Remarks on punctual local connectedness_, Theory and Applications of Categories, Vol. 25, 2011, No. 3, pp 51-63 ([TAC](http://tac.mta.ca/tac/volumes/25/3/25-03abs.html))

* {#RRZ04} [[Marie La Palme Reyes]], [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _Generic Figures and their Glueings_ , Polimetrica Milano 2004. (pp.206-224)

* {#Menni} [[Matias Menni]], _The manifestation of Hilbert's Nullstellensatz in Lawvere's Axiomatic Cohesion_ ([pdf slides](http://www.mat.uc.pt/~workct/slides/Menni.pdf) [pdf abstract](http://www.mat.uc.pt/~workct/abstracts/Menni.pdf))

* {#Menni14a} [[Matías Menni]], _Sufficient Cohesion over Atomic Toposes_ , Cah.Top.Géom.Diff.Cat. **LV** (2014). ([preprint](https://sites.google.com/site/matiasmenni/SufCohesion12.pdf?attredirects=0))

* {#Menni14b} [[Matías Menni]], _Continuous Cohesion over Sets_ , TAC **29** no.20 (2014) pp.542-568. ([pdf](http://www.tac.mta.ca/tac/volumes/29/20/29-20.pdf))

* {#Tholen} [[Walter Tholen]], _Nullstellen and Subdirect Representation_ ([pdf](http://www.math.yorku.ca/~tholen/nullstMarch2013bis.pdf))

### In spectral algebraic geometry

For a version of the Nullstellensatz in [[spectral algebraic geometry]] see

* Robert Burklund, Tomer M. Schlank, Allen Yuan, _The Chromatic Nullstellensatz_ ([arXiv:2207.09929](https://arxiv.org/abs/2207.09929))

[[!redirects Nullstellensätze]]

[[!redirects Hilbert's Nullstellensatz]]

[[!redirects nullstellensatz]]


number

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Arithmetic
+-- {: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Throughout [[mathematics]] there are various entities referred to as "numbers"; in modern mathematics it would be more accurate to refer to any one of various [[types]] called "number systems", and simply define a *number* to be a [[term]] of such a type.

A *[[numeral]]* on the other hand is a [[syntax|syntactic]] representation of a number, part of a system of numeration.

It is interesting to try to describe the general conditions under which something comes to be designated as a "number". After all, such number systems tend to form [[commutative rings]] or at least commutative [[rigs]], but not all commutative rigs are considered to consist of "numbers", or at least that is not how the language is used in practice.

The root notion, known to the great ancient civilizations and particularly ancient Hellenic civilization, is that of (the system of) [[natural numbers]], and in some way or other each of the various number systems are extensions of that one. If we may attempt a broad classification, qualitatively there are two broad types of number systems:

* systems of [[transfinite arithmetic|transfinite numbers]] (e.g., cardinal numbers, ordinal numbers, surreal numbers)

Roughly speaking, transfinite numbers tend to be quantities that measure sizes of elements in ZF-type structures (as in [[algebraic set theory]], or in the study of games in the sense of Conway), sometimes arising through a process of [[decategorification]].

* structures arising in [[algebraic number theory]]

In algebraic number theory, there seem to be two distinct general trends in the creation of new number systems from old (starting from the natural numbers):

* * Passing to a [[completion]] in some sense

* * Passing to a *finite* extension of some sort

For example, one passes from the rig of natural numbers to the ring of integers through additive [[group completion]], or from the [[integral domain]] of integers to the [[field]] of rationals through a [[field of fractions]] completion. Or, one constructs number fields as finite extensions of the field of rational numbers (similarly, algebraic integers $\alpha$ inside such a number field $K/\mathbb{Q}$ as those elements such that $\mathbb{Z}[\alpha]$ is a finite rank extension of $\mathbb{Z}$). Or, one constructs various [[local field]] completions of number fields, based on the [[valuations]] one can define on them; these include the real numbers and complex numbers and $p$-adic numbers. Or, one has various finite-dimensional algebra extensions such systems, including for example the [[quaternions]] (aka Hamiltonian numbers) and [[octonions]] (aka Cayley numbers), and various $p$-adic relatives of these.

Sometimes one singles out subtypes of such types; for example, the field of all algebraic numbers is the union of all number fields considered within the type of complex numbers. Sometimes numbers are designated according to what they are *not*, for example one speaks of irrational (= non-rational) numbers and of transcendental (= non-algebraic complex) numbers.

## In dependent type theory
 {#InDependentTypeTheory}

Assuming the [[type of natural numbers]], we display the construction of, consecutively, the [[rings]] of

* [[integer numbers]]

* [[rational numbers]]

* [[real numbers]]

* [[complex numbers]]

in [[dependent type theory]] with [[quotient types]] (such as [[homotopy type theory]] with the [[univalence axiom]]), as [[terms]] of [[ring]] [[structure]] (see [there](structure#RingDataStructure)):

Here we use the notation for [[dependent functions]], [[dependent pairs]] and their [[type telescopes]] from *[[dependent functions and dependent pairs -- table]]*.

Throughout we omit displaying the certificates of [[identification type]] involved ([[associativity]] etc.). Mostly their construction is immediate, a little thought is required only for the [[convergence of a sequence|convergence]] certificates of the operations on the [[real numbers]]: We show the "regular Cauchy real number type" due to [Bishop (1967)](#Bishop67); the ideas for obtaining the required identification certificates may be found in [Bishop (1967, pp. 15)](#Bishop67) and [Bishop & Bridges (1985, pp. 18)](#BishopBridges85), full formalization in [[Agda]] is in [Lundfall (2015)](#Lundfall15),  [Murray (2022)](#Murray22).

For $X \,\colon\, Set$ and $P \colon X \to Prop$ an [[equivalence relation]] with [corresponding](dependent+type#TypeBundlesReflectedInTypeUniverse) $R \,\coloneqq\, (x \colon X) \times P(x)$, we write

$$
  X / R \,\colon\, Set
$$

for the corresponding [[quotient type]].

We display functions on such quotient sets as operations on the quotien*ed* set $X$, leaving again implicit the certificates that these functions do respect the relations and hence do restrict to the quotient.

### The natural numbers

Here we assume the [[type of natural numbers]], for detailed discussion see there.

Just briefly:

$$
  \mathbb{N}
  \;\colon\;
  Type
$$

denotes the [[inductive type]] generated from

$$
  0 \,\colon\, \mathbb{N}
  ,\;\;\;\;
  succ \,\colon\, \mathbb{N} \to \mathbb{N}
  \,.
$$

By [[recursion]] this induces the usual operations of
[[addition]] and [[multiplication]], which we denote as usual:

$$
  \begin{array}{ccl}
  + &\colon& \mathbb{N} \times \mathbb{N} \to \mathbb{N}
  \\
  \cdot &\colon& \mathbb{N} \times \mathbb{N} \to \mathbb{N}
  \,.
  \end{array}
$$

Not to clutter the notation, we will consecutively *overload* this notation in the following. For example, the addition of integer numbers in the following is defined on representatives as

$$
  (n_1 ,\, m_1)
  \;+\;
  (n_2 ,\, m_2)
  \;\;
  =
  \;\;
  \big(
    n_1 + n_2
    ,\,
    m_1 + m_2
  \big)
$$

and it is tacitly understood that the addition symbol operation on the left goes $\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$, while that on the right is the one from above: $\mathbb{N} \times \mathbb{N} \to \mathbb{N}$.

### The integer numbers

The [[ring]] of [[integer numbers]]:

\[
  \label{TypeOfIntegers}
  \begin{array}{l}
    \mathllap{
      \vdash
      \;\;\;\;\;\;
    }
    \big(
      \mathbb{Z}
      ,\,
      0
      ,\,
      +
      ,\,
      -
      ,\,
      1
      ,\,
      \cdot
    \big)
    \;\colon\;
    Ring
    \\
    \text{where}
    \\
    \begin{array}{ccl}
      \mathbb{Z}
      &\coloneqq&
      \mathbb{N} \times \mathbb{N}
      \;\bigg/\;
      \big(
        (n_1,\,m_1)
        ,\,
        (n_2,\,m_2)
        \;\colon\;
        \mathbb{N} \times \mathbb{N}
      \big)
      \times
      Id_{\mathbb{N}}
      \big(
        n_1 + m_2
        ,\;
        m_1 + n_2
      \big)
      \\
      0
      &\coloneqq&
      (0,0)
      \\
      +
      &\colon&
      \big(
        (n_1,\,m_1)
        ,\,
        (n_2,\,m_2)
      \big)
      \;\mapsto\;
      \big(
        n_1 + n_2
        ,\,
        m_1 + m_2
      \big)
      \\
      -
      &\colon&
      (n,\, m)
      \;\mapsto\;
      (m,\, n)
      \\
      1
      &\coloneqq&
      \big(
        \mathrm{succ}(0)
        ,\,
        0
      \big)
      \\
      \cdot
      &\colon&
      \big(
        (n_1,\,m_1)
        ,\,
        (n_1,\,m_2)
      \big)
      \;\mapsto\;
      \big(
        n_1 \cdot n_2
        +
        m_1 \cdot m_2
       ,\;\;
       m_1 \cdot n_2
       +
       n_1 \cdot m_2
      \big)
    \end{array}
  \end{array}
\]

The inclusion of [[natural numbers]] into [[integer numbers]]:

\[
  \begin{array}{rcl}
    \iota \;\colon &
    \mathbb{N}
    &\longrightarrow&
    \mathbb{Z}
    \\
    &
    n &\mapsto& (n,\,0)
  \end{array}
\]

The [[ordering]] of [[integer numbers]]:

\[
  \label{OrderRelationOnIntegers}
  \begin{array}{rrcl}
    \leq \;\colon
    &
    \mathbb{Z} \times \mathbb{Z}
    &\longrightarrow&
    Prop
    \\
    &
    (n_1 ,\, n_2)
      &\mapsto&
    (k \colon \mathbb{N})
     \times
    Id_{\mathbb{Z}}
    \big(
      n_2 ,\;  n_1 + \iota(k)
    \big)
  \end{array}
\]

The [[positive number|positive]] [[integer numbers]]:

\[
  \label{PositiveNaturalNumbers}
    \mathbb{Z}_+
    \;\coloneqq\;
    (n \,\colon\, \mathbb{Z})
    \times
    (1 \leq n)
\]

### The [[rational numbers]]

The [[ring]] of [[rational numbers]]:

\[
  \label{DataStructureOfRationalNumbers}
  \begin{array}{l}
    \mathllap{
      \vdash
      \;\;\;\;\;\;
    }
    \big(
      \mathbb{Q}
      ,\,
      0
      ,\,
      +
      ,\,
      -
      ,\,
      1
      ,\,
      \cdot
    \big)
    \;\colon\;
    Ring
    \\
    \text{where}
    \\
    \begin{array}{ccl}
      \mathbb{Q}
      &\coloneqq&
     \mathbb{Z}
       \times
     \mathbb{Z}_+
     \;\bigg/\;
     \big(
       (p_1 ,\, q_1)
       ,\,
       (p_1 ,\, q_2)
       \,:\,
       \mathbb{Z}
       \times
       \mathbb{Z}_+
     \big)
     \times
     \Id_{\mathbb{Z}}
     \big(
       p_1 \cdot q_2
       ,\;
       q_1 \cdot p_2
     \big)
      \\
      0 &\coloneqq& (0,1)
      \\
      +
      &:&
      \big(
        (p_1 ,\, q_1)
        ,\,
        (p_2 ,\,q_2)
      \big)
      \;\;\mapsto\;\;
      \big(
        p_1 \cdot q_2
        \,+\,
        p_2 \cdot q_1
        ,\;\;
        q_1 \cdot q_2
      \big)
      \\
      -
      &\colon&
      (p,\,q)
        \;\mapsto\;
      (-p,\, q)
      \\
      1
      &\coloneqq&
      (1,\, 1)
      \\
      \cdot
      &\colon&
      \big(
        (p_1,\, q_1)
        ,\,
        (p_,\,q_2)
      \big)
      \;\mapsto\;
      \big(
        p_1 \cdot p_2
        ,\;
        q_1 \cdot q_2
      \big)
    \end{array}
  \end{array}
\]

[[ordering]] of the rational numbers induced from the ordering (eq:OrderRelationOnIntegers) of the integer numbers:

\[
  \label{OrderRelationOnRationals}
  \begin{array}{rrcl}
    \leq \;\colon
    &
    \mathbb{Q} \times \mathbb{Q}
    &\longrightarrow&
    Prop
    \\
    &
    \big(
      (q_1 ,\, p_1)
      ,\,
      (q_2 ,\, p_2)
    \big)
      &\mapsto&
    \big(
      q_1 \cdot p_2
      \,\leq\,
       q_2 \cdot p_1
    \big)
  \end{array}
\]

When constructing the real numbers in (eq:DataStructureOfRealNumbers) below, we use the following common abbreviations:

The [[fraction|multiplicative inverse]] of a [[positive number|positive]] [[natural number]]:

\[
  \label{MultiplicativeInverseOfPositiveNaturalNumber}
  \begin{array}{rrcl}
    \frac{1}{(-)}
    \;\colon
    &
    \mathbb{N}_+
    &\longrightarrow&
    \mathbb{Q}
    \\
    &
    n &\mapsto& (1,\,n)
  \end{array}
\]

The [[monomial|square]] of a [[rational number]]:

\[
  \label{SquareOfARationalNumber}
  \begin{array}{rrcl}
    (-)^2
    \;\colon
    &
    \mathbb{Q}
    &\longrightarrow&
    \mathbb{Q}
    \\
    &
    r &\mapsto& r \cdot r
  \end{array}
\]

### The real numbers

The [[real numbers]]:

\[
  \label{DataStructureOfRealNumbers}
  \begin{array}{l}
    \mathllap{
      \vdash
      \;\;\;\;\;\;
    }
    \big(
      \mathbb{R}
      ,\,
      0
      ,\,
      +
      ,\,
      -
      ,\,
      1
      ,\,
      \cdot
    \big)
    \;\colon\;
    Ring
    \\
    \text{where}
    \\
    \begin{array}{ccl}
    \mathbb{R}
    &\coloneqq&
    \Big(
      x_{(-)}
      :
      \mathbb{Z}_+
      \to
      \mathbb{Q}
    \Big)
    \times
    \bigg(
      (
        n,\,m \,\colon\, \mathbb{Z}_+
      )
      \to
      \Big(
        (x_n - x_m)^2
        \,\leq\,
        \big(
          \frac{1}{n} + \frac{1}{m}
        \big)^2
      \Big)
    \bigg)
    \\
    &&
    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
    \bigg/
    \Big(
      x_{(-)} ,\, y_{(-)}
    \Big)
    \times
    \bigg(
      (n \colon \mathbb{Z}_{+})
      \to
      \Big(
        (x_n - y_b)^2
        \,\leq\,
        \big(\frac{2}{n}\big)^2
      \Big)
    \bigg)
    \\
    0 &\colon& n \mapsto 0
    \\
    +
    &\colon&
    \big(
      x_{(-)}
      ,\,
      y_{(-)}
    \big)
    \;\mapsto\;
    \big(
      n \,\mapsto\, x_n + y_n
    \big)
    \\
    -
    &\colon&
    x_{(-)}
    \,\mapsto\,
    \big(
      n \,\mapsto\, - x_n
    \big)
    \\
    1
    &\colon&
    n \,\mapsto\, 1
    \\
    \cdot
    &\colon&
    \big(
      x_{(-)}
      ,\,
      y_{(-)}
    \big)
    \;\mapsto\;
    \big(
      n
      \,\mapsto\,
      x_n
      \cdot
      y_n
    \big)
    \end{array}
  \end{array}
\]

### The complex numbers

The [[complex numbers]]:

$$
  \begin{array}{l}
    \mathllap{
      \vdash
      \;\;\;\;\;\;
    }
    \big(
      \mathbb{C}
      ,\,
      0
      ,\,
      +
      ,\,
      -
      ,\,
      1
      ,\,
      \cdot
    \big)
    \;\colon\;
    Ring
    \\
    \text{where}
    \\
    \begin{array}{rcl}
    \big(
      z \colon \mathbb{C}
    \big)
    &\coloneqq&
    \big(
      \mathrm{Re}(z)
      ,\,
      \mathrm{Im}(z)
    \big)
    \,:\,
    \mathbb{R} \times \mathbb{R}
    \\
    0
    &\coloneqq&
    (0,\,0)
    \\
    +
    &\colon&
    \Big(
    \big(
      \mathrm{Re}(z_1)
      ,\,
      \mathrm{Im}(z_1)
    \big)
    ,\,
    \big(
      \mathrm{Re}(z_2)
      ,\,
      \mathrm{Im}(z_2)
    \big)
    \Big)
    \;\;\mapsto\;\;
    \big(
      \mathrm{Re}(z_1) + \mathrm{Re}(z_2)
      ,\,
      \mathrm{Re}(z_1) + \mathrm{Re}(z_2)
    \big)
    \\
    -
    &\colon&
    \big(
      \mathrm{Re}(z)
      ,\,
      \mathrm{Im}(z)
    \big)
    \;\mapsto\;
    \big(
      - \mathrm{Re}(z)
      ,\,
      - \mathrm{Im}(z)
    \big)
    \\
    1
    &\coloneqq&
    (1,\, 0)
    \\
    \cdot
    &\colon&
    \Big(
      \big(
        \mathrm{Re}(z_1)
        ,\,
        \mathrm{Im}(z_1)
      \big)
      ,\,
      \big(
        \mathrm{Re}(z_2)
        ,\,
        \mathrm{Im}(z_2)
      \big)
    \Big)
    \\
    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
    &&
    \;\;\mapsto\;\;
    \Big(
    \mathrm{Re}(z_1)
    \cdot
    \mathrm{Re}(z_2)
    -
    \mathrm{Im}(z_1)
    \cdot
    \mathrm{Im}(z_2)
    ,\;\;
    \mathrm{Re}(z_1) \cdot \mathrm{Im}(z_2)
    +
    \mathrm{Im}(z_1) \cdot \mathrm{Re}(z_2)
    \Big)
    \end{array}
  \end{array}
$$

\linebreak

\linebreak

## Related pages on numbers

* [[natural number]]

  * [[even number]], [[odd number]]

* [[integer]]

  * [[positive number]], [[negative number]]

* [[real number]]

  * [[rational number]], [[irrational number]]

  * [[algebraic number]], [[transcendental number]]

* [[complex number]]

  * [[imaginary number]]

* [[hypercomplex number]]

  * [[dual number]], [[quaternion]], [[octonion]]

* [[adic number]]

* [[infinite number]], [[infinitesimal]]

  * [[extended real number]]

  * [[cardinal number]]

  * [[ordinal number]]

  * [[nonstandard natural number]]

  * [[hyperreal number]]

  * [[surreal number]]

## Related entries

* [[number of supersymmetries]]

(...)

## References

History:

* [[Leo Corry]], *A Brief History of Numbers*, Oxford University Press (2015) [[ISBN:9780198702597](https://global.oup.com/academic/product/a-brief-history-of-numbers-9780198702597)]

Construction of the number systems up to the [[real numbers]] in [[constructive analysis]]:

* {#Bishop67} [[Errett Bishop]], *[[Foundations of Constructive Analysis]]*, McGraw-Hill (1967)

* {#BishopBridges85} [[Errett Bishop]], [[Douglas Bridges]] *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

Formalization in [[Agda]]:

* {#Lundfall15} [[Martin Lundfall]], *Formalizing real numbers in Agda* (2015) [pdf, [[Lundfall-RealNumbersInAgda.pdf:file]], [github](https://github.com/MrChico/Reals-in-agda)]

* {#Murray22} [[Zachary Murray]], *Constructive Analysis in the Agda Proof Assistant* [[arXiv:2205.08354](https://arxiv.org/abs/2205.08354), [github](https://github.com/z-murray/honours-project-constructive-analysis-in-agda)]

[[!redirects number]]
[[!redirects numbers]]


number field

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A **number field** is a [[finite number|finite]] [[field extension]] of the [[field]] of [[rational numbers]], $\mathbb{Q}$.

In other words, a number field is a field $k$ of [[characteristic zero]] such that under the field [[homomorphism]] $i: \mathbb{Q} \hookrightarrow k$, the field $k$ is a [[finite number|finite]]-[[dimension|dimensional]] [[vector space]] over $\mathbb{Q}$ with respect to the scalar multiplication [[action]] of $\mathbb{Q}$

$$\mathbb{Q} \otimes k \stackrel{i \otimes 1}{\to} k \otimes k \stackrel{mult}{\to} k$$

on the underlying additive [[group]] of $k$.

## Examples

* the [[rational numbers]] $\mathbb{Q}$;

* the [[Gaussian numbers]] $\mathbb{Q}(i)$;

* for certain $d$ the [[quadratic field]] $\mathbb{Q}(\sqrt{d})$;

* the [[cyclotomic fields]] $\mathbb{Q}(\zeta_n)$

Counterexamples:

* the [[real numbers]] $\mathbb{R}$ and [[complex numbers]] $\mathbb{C}$ are **not** number fields, since, while being [[vector spaces]] over $\mathbb{Q}$, they are not [[finite number|finite]] [[dimension|dimensional]] as $\mathbb{Q}$-vector spaces.

## Applications

Number fields are the basic objects of study in [[algebraic number theory]]. For example, one is typically interested in the arithmetic structure of $k$, including for example the structure of the ring of [[algebraic integer]]s $\mathcal{O}_k$ in $k$, the decomposition of primes in $\mathbb{Z}$ in terms of prime ideals in $\mathcal{O}_k$, the structure of the unit group of $\mathcal{O}_k$, the structure of the [[ideal class group]], the detailed study of the [[zeta function]] of $k$, and much more.

## Properties

### As global fields

Number fields $k$ are examples of [[global field]]s, in fact they are the global fields of [[characteristic zero]]. They are often studied in terms of how they embed in their rings of [[adele ring|adeles]] $\mathbb{A}_k$, which are built from the [[local field|local completions]] of $k$.

### Class number formula

Given a number field $K$, the [[Dedekind zeta function]] $\zeta_K$ of $K$ has a [[simple pole]] at $s = 1$. The _class number formula_ says that its [[residue]] there is proportional the product of the [[regulator of a number field|regulator]] with the [[class number]] of $K$

$$
  \underset{s\to 1}{\lim} (s-1) \zeta_K(s)
  \propto
  ClassNumber_K \cdot Regulator_K
  \,.
$$

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[algebraic curve]]

* [[function field]]

## Literature

* {#NSW08} [[Jürgen Neukirch]], Alexander Schmidt, Kay Wingberg, *Cohomology of Number Fields*, Springer Grundlehren der mathematischen Wissenschaften **323**, Springer 2008 ([doi:10.1007/978-3-540-37889-1](https://link.springer.com/book/10.1007/978-3-540-37889-1), [webpage](https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/))

[[!redirects number fields]]

[[!redirects algebraic number field]]
[[!redirects algebraic number fields]]


number line rig

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In this article we [[axioms|axiomatize]] the [[algebra|algebraic]], [[order theory|order theoretic]], and [[metric space|metric]] properties which are common to the basic [[types]] of [[numbers]] found in [[mathematics]]: the [[natural numbers]], [[integers]], [[rational numbers]], and [[Dedekind real numbers]], and which remain valid in [[constructive mathematics|constructive]] and [[predicative mathematics]].

## Definition

A **number line rig** $S$ is

* A [[commutative rig]] $(S, 0, 1, +, \cdot)$. It follows that there is a [[rig homomorphism]] $h:\mathbb{N} \to S$, since the natural numbers are the [[initial object|initial]] [[commutative rig]]

* with a [[strict total order]] $(S, \lt)$ such that
  * $0 \lt 1$
  * $0 \lt a$ and $0 \lt b$ imply that $0 \lt a + b$
  * $0 \lt a$ and $0 \lt b$ imply that $0 \lt a \cdot b$
* with a [[tight apartness relation]] $(S, \#)$ such that $a \# b$ if and only if $a \lt b$ or $b \lt a$
* with a [[pseudolattice]] structure $(S, \leq, \min, \max)$ such that
  * $a \leq b$ if and only if $\neg (b \lt a)$
  * $a \leq b$ if and only if $a + c \leq b + c$
* with a [[metric]] $\rho:S \times S \to S$ such that $\rho(a, b) + \min(a, b) = \max(a, b)$. Every element in the image of $\rho$ is always not less than zero, because by definition of a pseudolattice, $\min(a, b) \leq \max(a, b)$. This can be shown to be a translation-invariant metric, where it follows that addition is a [[cancellative monoid]]:
  * $a = b$ if and only if $\rho(a, b) = 0$
  * $\rho(a, b) = \rho(b, a)$
  * $\rho(a, b) \leq \rho(a, c) + \rho(c, b)$
  * $\rho(a + c, b + c) = \rho(a, b)$.
* The [[multiplicative subset of cancellative elements]] is the set of elements apart from zero
* and $S$ satisfies the [[Archimedean property]]: for every [[positive number|positive]] element $0 \lt a$, there is a positive natural number $n$ such that $a \lt h(n)$, and there is a positive natural number $m$ such that $1 \lt h(m) \cdot a$.

## Group completion, rig of fractions, and completions

Given any number line rig, one could construct the [[group completion]], which results in a [[Archimedean integral domain]]. One could also [[localization of a ring|localize]] at the subset of [[regular elements]], which results in a rig of fractions. Doing both results in an [[Archimedean field]]. One could also complete the resulting field by [[Dedekind cuts]], resulting in the [[Dedekind real numbers]], or complete it by [[Cauchy sequences]], resulting in a [[sequentially Cauchy complete]] [[Archimedean field]] such as the [[HoTT book real numbers]].

## See also

* [[natural numbers]]
* [[integers]]
* [[rational numbers]]
* [[real numbers]]
* [[streak]]


number of supersymmetries

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In discussion of [[supersymmetry]]: the [[number]] of [[generators and relations|generators]] of odd degree, suitably conceived.

## Definition

A [[supersymmetry]] [[super Lie algebra]] or [[super Lie group]] is determined by the underlying bosonic algebra/group ([[body]]) and a [[real spin representation]] $\mathbf{N}$.

One says that the corresponding _number of supersymmetries_ is either the [[dimension]]

$$
  N \;\coloneqq\; dim_{\mathbb{R}}\big( \mathbf{N} \big)
$$

of the [[real spin representation]] $\mathbf{N}$, and as such denoted by a roman "$N$",

or, alternatively, the multiplicity

$$
  \mathcal{N}
  \;\coloneqq\;
  (n_i)_{i \in I}
$$

of the [[irreducible representation|irreducible]] [[real spin representations]] $\mathbf{N}^{irr}_{i}$ in a [[direct sum]] decomposition

$$
  \mathbf{N}
  \;\simeq\;
  \underset{i \in I}{\oplus}
  n_i \mathbf{N}^{irr}_i
$$

of this [[real spin representation]] $\mathbf{N}$, and as such denoted by a list of calligraphic "$\mathcal{N}$"s.

Typically there is either a single [[irrep]] or precisely two, in which case these multiplicities are either a single [[natural number]]

$$
  \mathcal{N}
  \;\in\;
  \mathbb{N}
$$

or a [[pair]] of them

$$
  \mathcal{N}
  \;=\;
  (\mathcal{N}_+, \mathcal{N}_-)
  \,,
$$

respectively.

## References

See the references at _[[supersymmetry]]_, for instance

* [[Urs Schreiber]],  _[[geometry of physics --  supersymmetry]]_

[[!redirects numbers of supersymmetry]]


number realizability

#Contents#
* table of contents
{:toc}

## Idea

The version of [[realizability]] where truth/proof is realized by [[numbers]].

As opposed to _[[function realizability]]_.

## Related concepts

[[!include computable mathematics -- table]]


number theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

**Number theory** (or **[[arithmetic]]**) studies [[numbers]], especially the [[natural numbers]] and the [[integers]]. Typical questions are the distribution of [[prime numbers]] and the study of integer (or rational) [[solutions]] of [[algebraic equations]] with [[integer]] [[coefficients]], also called **[[Diophantine equations]]**.

(Notice that by a theorem of [[Matiyasevich]] (spelled also Matijasevich), for every statement in [[mathematics]] (say ZF [[set theory]]) there is [[Diophantine equation]] whose [[solution|solvability]] is equivalent to the validity of the statement. Of course, that does not mean that addressing a problem as a Diophantine equation helps in solving it. )

The analytic study of the asymptotic behaviour of the distribution of prime numbers on the positive integer line is the main subject of [[analytic number theory]]; it also studies the distribution of rational numbers with small denominators.

A root of an algebraic equation with integral (or equivalently, with rational) coefficients is called an [[algebraic number]]. Extending the field $\mathbb{Q}$ of rational numbers by abstract solutions of such an algebraic equation in a minimal way leads to an algebraic extensions of rationals, a so called [[number field]]. There are a number of algebraic structures related to the study of algebraic integers and number fields in particular; these questions comprise [[algebraic number theory]] including its central part, [[class field theory]]. [[Galois theory]] is one of the principal ways of studying such questions. [[algebraic geometry|Algebraic geometry]] is very effective in the investigation of more elaborate questions in this area, and through its historical development it has given rise to many major concepts, such as [[group]], [[ideal]], and Grothendieck [[motive]].

It is one of the oldest branches of mathematics and very popular as many difficult problems can be stated in a rather elementary and simple manner, e.g. Fermat's Last Theorem and the [[Riemann hypothesis]].

## Related concepts

* [[analytic number theory]]

* [[objective number theory]]

* [[elementary function arithmetic]]

* [[second order arithmetic]]

* [[arithmetic geometry]]

* [[Weil conjecture]], [[Fermat's last theorem]], [[abc conjecture]], [[Mordell conjecture]]

* [[exact real computer arithmetic]]

* [[number theory and physics]]

* [[arithmetic pretopos]]

## References

Historical origins:

* [[Carl Gauss]], _[[Disquisitiones Arithmeticae]]_, 1801

Textbook accounts:

* {#FroehlichCassels67} [[Albrecht Fröhlich]], [[J. W. S. Cassels]] (eds.), _Algebraic number theory_, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, [pdf](https://www.math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf), [errata pdf](https://www.ma.imperial.ac.uk/~buzzard/errata.pdf) by [[Kevin Buzzard]])

* [[Albrecht Fröhlich]], Martin J. Taylor, _Algebraic number theory_, Cambridge Studies in Advanced Mathematics **27**, 1993

In a context of [[automorphic forms]] and the [[Langlands correspondence]]:

* {#Tate77} [[John Tate]], *Number theoretic background*, in: *Automorphic forms, representations and L-functions*, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. (1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26. Amer. Math. Soc., Providence, RI ([ISBN:978-0-8218-3371-1](https://bookstore.ams.org/pspum-33-2), [pdf](http://www.math.ucsd.edu/~csorense/teaching/math205/Tate_Weil.pdf), [[TateNumberTheory.pdf:file]])

Lecture notes:

* [[Andrew Sutherland]], _[Number theory I](https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2019/)_, 2019

Further collection of lecture notes is [here](http://www.math.caltech.edu/~jimlb/coursenotes.html).

The following survey of Connes-Marcolli work has an accessible quick introduction to algebraic number theory

* P. Almeida, _Noncommutative geometry and arithmetics_, Russian Journal of Mathematical Physics __16__, No. 3, 2009, pp. 350–362, [doi](http://dx.doi.org/10.1134/S1061920809030030), see also $n$Lab:[[arithmetic and noncommutative geometry]]

See also:

* Wikipedia en:[number theory](http://en.wikipedia.org/wiki/Number_theory), fr:l'[arithmétique modulaire](http://fr.wikipedia.org/wiki/Arithm%C3%A9tique_modulaire)

Some relation to [[modular tensor categories]]:

* {#DavidovichHaggeWang2013} [[Orit Davidovich]], [[Tobias Hagge]], [[Zhenghan Wang]], _On Arithmetic Modular Categories_, arXiv preprit, 2013 [[arXiv:1305.2229](https://arxiv.org/abs/1305.2229)]

[[!redirects number theory]]


number theory and physics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[number theory]] and [[physics]]

## Examples

* [[Riemann hypothesis and physics]]

* [[Langlands duality]]

* [[p-adic physics]]

  * [[p-adic string theory]]

  * [[p-adic AdS-CFT]]

## References

The relation between [[string theory]] with its [[scattering amplitudes]] on [[Riemann surfaces]] and the theory of [[automorphic forms]] was maybe first highlighted in

* {#Witten88} [[Edward Witten]], _Quantum field theory, Grassmannians, and algebraic curves_, Comm. Math. Phys. Volume 113, Number 4 (1988), 529-700 ([Euclid](http://projecteuclid.org/euclid.cmp/1104160350))

Reviews include

* [[Matthew Watkins]], _[Number theory and physics archive](http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm)_

* [[Michael Atiyah]], [[Robbert Dijkgraaf]], [[Nigel Hitchin]] _Geometry and physics_, Phil. Trans. R. Soc. A 2010 368, doi: 10.1098/rsta.2009.0227 ([pdf](http://rsta.royalsocietypublishing.org/content/368/1914/913.full.pdf))

  (despite the title, this is really mostly about number theory/enumerative geometry and physics)


numerable fiber bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For the theory of [[fiber bundles]] to be well-behaved one typically needs to restrict to those that have a _[[local trivialization]]_ not just over any [[open cover]] but over a [[numerable open cover]] -- these are the numerable bundles. Since this condition is automatic if the base space is [[paracompact topological space|paracompact]] the condition is often not made explicit.

## Definition

\begin{definition}\label{NumerableFiberBundle}
**(numerable fiber bundle)**

A [[fiber bundle]] over a [[topological space]] is called _numerable_ if it admits a [[local trivialization]] over a [[numerable open cover]].

\end{definition}

## Properties

\begin{remark}
Various results for [[classifying spaces]] that classify arbitrary [[fiber bundles]] over [[paracompact topological spaces]] generalizes (only) to a classification of numerable bundles over general topological spaces.

\end{remark}

\begin{remark}
 The conclusion that the bundle projection of a topological [[fiber bundle]] is a [[Hurewicz fibration]] follows generally only for numberable bundles (by [this Prop.](Hurewicz+fibration#MapIsHurewiczFibrationIfPullbackToNumerableCoverIsSo))
\end{remark}

[[!redirects numerable fiber bundles]]

[[!redirects numerable bundle]]
[[!redirects numerable bundles]]