nonstandard model of arithmetic

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### Context
#### Model theory
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[[!include model theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea {#sec:org4b65bd2}

In the same way that in [[nonstandard analysis]] one passes to [[elementary extension|elementary extensions]] of the field
$\mathbb{R}$ which realizes more [[type (in model theory)|types]] (in particular the types of infinite and
infinitesimal numbers), one can extend the standard model
$(\mathbb{N}, 0, 1, +, \times, S)$ of first-order [[Peano arithmetic]] to proper elementary extensions that have
infinite natural numbers.

These "points at infinity" that live in the nonstandard part tend to
embody the uniform behavior of the numbers from the standard part. (For
example, by [[compactness theorem|compactness]], the twin prime conjecture is true
if and only if in some nonstandard model of arithmetic there exists just
a single pair of nonstandard twin primes, and similarly Dirichlet's
theorem on arithmetic progressions can be reformulated as saying that
for every coprime positive pair of numbers $a$ and $b$ there exists just
one nonstandard prime congruent to $a$ modulo $b$.)

## Definition {#sec:org1621e14}

A **nonstandard model of arithmetic** is a proper
[[elementary extension]] of the *standard
model* $(\mathbb{N}, 0, 1, +, \times, S)$ of first-order
[[Peano arithmetic]].

## Examples {#sec:orgc18facb}

-   In the same way the [[hyperreal numbers]] are obtained by taking a countable
    ultrapower $\mathbb{R}^{\mathcal{U}}$ of $\mathbb{R}$, we can obtain
    the **hypernatural numbers** by taking a countable ultrapower
    $\mathbb{N}^{\mathcal{U}}$.

## Remarks {#sec:orgc4f9a0e}

This is all discussed in e.g. ([Gitman](#Gitman)).

-   While the standard model of arithmetic has no automorphisms, there
    exist countable models of arithmetic with
    [[continuum]]-many automorphisms.

-   A theorem due to Harvey Friedman states that every nonstandard model
    of arithemetic is isomorphic to some initial segment (in terms of
    the ordering discussed below) of itself.

-   A theorem due to Stanley Tennenbaum states that there is no
    countable nonstandard model of arithmetic for which there is an
    algorithm to compute addition or multiplication on the
    nonstandard part.

-   The usual ordering on the natural numbers is definable in
    [[Peano arithmetic|PA]] since
    $a \leq b \iff \exists c$ such that $a + c = b$.

## Denseness of ordering on nonstandard part {#sec:orgf41355d}

As remarked above, the usual ordering on $\mathbb{N}$ is definable in
$\mathsf{PA}$. Since the theory defines it, every nonstandard model of
arithmetic is linearly ordered. For every nonstandard number $\omega$,
one can generate using addition and subtraction a copy of $\mathbb{Z}$
around $\omega$, like so:

$$\left\{ \dots \omega - n, \dots, \omega - 1, \omega, \omega + 1, \dots, \omega + n, \dots\right\}$$

Since addition respects the ordering, $\omega + \omega$ is greater than
anything in this copy of $\mathbb{Z}$, and since multiplication respects
the ordering also, $\omega \times \omega$ is greater than
$n \times \omega$ for any finite $n$.

Similarly, if $\omega_1 < \omega_2$, then the copy of $\mathbb{Z}$
around $\omega_1$ is below the copy of $\mathbb{Z}$ around $\omega_2$ in
the ordering.

It turns out that if you quotient out these copies of $\mathbb{Z}$ in
the nonstandard part of (any) nonstandard model of arithmetic, the
resulting order on the nonstandard part is [[DLO|dense and
without endpoints]].

Let's prove this. We'll give two proofs; they're really the same proof,
but one uses the language of an
[[ultrapower]] (the "hypernatural numbers")
while the other doesn't.

+-- {: .num_prop}
###### Proposition

The ordering of the copies of $\mathbb{Z}$ in the nonstandard part of a
nonstandard model of arithmetic is dense and without endpoints.

=--

+-- {: .proof}
###### Proof

(First proof.)

The theory of arithmetic proves that every other element
is divisible by $2$, so pick even representatives from every copy of
$\mathbb{Z}$ in the nonstandard part. Let $a$ represent a copy of
$\mathbb{Z}$ below a copy of $\mathbb{Z}$ represented by $b$. The theory
of arithmetic also proves that when $a$ and $b$ are even,

$$\models a < \frac{a+b}{2} < b$$

and since we have that $\models a + n < b$ for each finite $n$, we
can also see that

$$\models a + n < \frac{a + b}{2} < b$$

for each finite $n$ (resp. < $b - n$ for all finite $n$). This order
is dense and has no endpoints, since for any copy of $\mathbb{Z}$ in the
nonstandard part we can take an even representative and divide by $2$ to
obtain another nonstandard number which is below this copy of
$\mathbb{Z}$ (to see it has no upper bound, we can just look at
$\omega, \omega^2, \omega^3, \dots$.)

=--

+-- {: .proof}
###### Proof

(Second proof.)

Let $^*\mathbb{N}$ be any ultrapower nonstandard model of arithmetic. This is an uncountable and (at least) $\aleph_{1}$-saturated model of $\mathsf{PA}$, and its elements are equivalence classes ([[germs]]) of sequences $\overline{x}$ of natural numbers. The nonstandard part still consists of copies of $\mathbb{Z}$. Let $\overline{a}$ and $\overline{b}$ be nonstandard numbers such that $\overline{a}$ belongs to a lower copy of $\mathbb{Z}$ than $\overline{b}$ does. By the [[Los ultraproduct theorem]], $\overline{a}$ and $\overline{b}$ must therefore differ as sequences $\mathcal{U}$-often, with an unbounded sequence of differences. We can assume $\overline{a}$ and $\overline{b}$ are sequences of even numbers, since any perturbation of $\overline{a}$ and $\overline{b}$ by a sequence of $0$s and $1$s doesn't change what copy of $\mathbb{Z}$ $\overline{a}$ and $\overline{b}$ are in. Then the sequence $\overline{c}$ of their pointwise average will be between them, and will have unbounded $\mathcal{U}$-often difference from $\overline{a}$ and $\overline{b}$.

Again by the [[Los ultraproduct theorem]], the arithmetic operations in $^*\mathbb{N}$ are computed pointwise, so this argument will work in any elementary submodel of $^*\mathbb{N}$. In particular, any nonstandard model $M \models \mathsf{PA}$ embeds diagonally into its ultrapower $M^{\mathcal{U}}$.

=--

## Countable nonstandard models of arithmetic {#sec:org8d7c98e}

Peano arithmetic is the quintessential [[stability in model theory|unstable theory]], and so there are plenty of
countable nonstandard models of [[Peano arithmetic]] ($2^{\aleph_0}$-many.) One can
see this directly by [[type (in model theory)|omitting]] the types of nonstandard numbers whose
only finite prime divisors is an arbitrary set of finite primes.

By the discussion above, the nonstandard part is linearly ordered and
can be partitioned into copies of $\mathbb{Z}$, and in fact these copies
of $\mathbb{Z}$ are densely ordered without endpoints.

Since the [[DLO|theory of dense linear orders without endpoints]] has the unique countable model
$(\mathbb{Q}, <)$, this means that for any countable nonstandard
model of arithmetic, the order type of the nonstandard part is
$\mathbb{Q} \mathbb{Z}$, i.e. $\mathbb{Q}$, but every time there's a
rational number, you have a copy of the integers instead.

So in a way, the wild divergence among the countable models of
$\mathsf{PA}$ comes very concretely from the arithmetic part of
$\mathsf{PA}$, and not at all from the ordering. However, this niceness
("categoricity") of the ordering on nonstandard models of arithmetic is
also being controlled by the arithmetic part of $\mathsf{PA}$ (since
that was all we used in the proofs of denseness above): if we were looking at just
elementary extensions of $(\mathbb{N}, <)$, you would need to
stipulate $\omega$-saturation to force the order type on the
nonstandard part to be $\mathbb{Q} \mathbb{Z}$, because otherwise
there would be nothing wrong with just adjoining a single point at infinity and
generating a single copy of $\mathbb{Z}$ (possible with just the order because you can ask for the next or previous element in the ordering) around that for a countable
elementary extension of $(\mathbb{N}, <)$.

## Related entries {#sec:orgb56552f}

-   [[natural number]]

-   [[nonstandard analysis]]

-   [[ultraproduct]]

-   [[Peano arithmetic]]

## References {#sec:org926dcee}

-   {#Gitman}Victoria Gitman, [An introduction to nonstandard models
    of arithmetic](https://victoriagitman.github.io/talks/2015/04/22/an-introduction-to-nonstandard-model-of-arithmetic.html)

-   R. Kossak and J. H. Schmerl, The structure of models of Peano arithmetic, The Clarendon Press, Oxford University Press, Oxford, 2006, vol. 50.

[[!redirects nonstandard model of arithmetic]]
[[!redirects nonstandard models of arithmetic]]
[[!redirects nonstandard model of Peano arithmetic]]
[[!redirects nonstandard models of Peano arithmetic]]
[[!redirects nonstandard arithmetic]]

[[!redirects hypernatural number]]
[[!redirects hypernatural numbers]]
[[!redirects nonstandard natural number]]
[[!redirects nonstandard natural numbers]]


nonstationary p-divisible group

# Contents
* toc
{: toc}

## Idea

We wish to more generally discuss the criterion in the [[Landweber exact functor theorem]]. Rather than looking at formal groups classified by maps from $Spec(R) \to M_{fg}$ that are **etale**, we look at p-divisible groups classified by maps from $Spec(R) \to M_{p-div}$ which are **unramified**.

Lurie defines the condition for a [[p-divisible group]] $G$ over $R$ to be nonstationary. Heuristically, we think of $G$ as a family of p-divisible groups parameterized by $Spec(R)$. The concept of $G$ being nonstationary is that this family is _nonconstant along every tangent vector_ in $Spec(R)$.
This is equivalent to the map classifying $G$ from $Spec(R) \to M_{p-div}$ being unramified.

## Definition

Let $R$ be a commutative ring and let $G$ be a p-divisible group over $R$. Let $\kappa(x)$ be the residue field of $R$ at $x \in |Spec(R)|$.

Let $G$ be a p-divisible group defined over a commutative ring $R$. Suppose that we are given a point $x \in |Spec(R)|$ and a derivation $d: R \to \kappa(x)$, then the canonical map $\beta_0: R \to \kappa(x)$ lifts to a ring homomorphism $\beta: R \to \kappa(x)[\epsilon]/(\epsilon^2)$, given by the formula $\beta(t) = \beta_0(t) + \epsilon dt$.

Let $G_d$ denote the p-divisible group over $\kappa(x)[\epsilon]/(\epsilon^2)$ obtained from $G$ by extending scalars along $\beta$.

We say $G$ is **nonstationary** if:

* For every point $x \in |Spec(R)|$ and every nonzero derivation $d: R \to \kappa(x)$. the p-divisible group $G_d$ is a nontrivial first order deformation $G_{\kappa(x)}$.

(If the map $d = 0$, then $G_d$ is a trivial first order deformation of $G_{\kappa(x)}$).


nonunital algebra

#Contents#
* table of contents
{:toc}

## Idea

A _nonunital algebra_ is like an [[associative algebra]] but without specified [[identity]] element. The associative algebra version of _[[nonunital ring]]_, see there for more.

## Related concepts

* [[unitisation of C*-algebras]].

* [[noncommutative algebra]], [[nonassociative algebra]]

## References

A definition of [[algebraic K-theory]] for [[nonunital rings]] is due to

* {#Quillen} [[Daniel Quillen]], _$K_0$ for nonunital rings and Morita invariance_, J. Reine Angew. Math., 472:197-217, 1996.

with further developments (in [[KK-theory]]) including

* [[Snigdhayan Mahanta]], _Higher nonunital Quillen K'-theory, KK-dualities and applications to topological T-dualities_,  J. Geom. Phys., 61 (5), 875-889, 2011. ([pdf](http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf))

[[!redirects nonunital algebras]]

[[!redirects non-unital algebra]]
[[!redirects non-unital algebras]]


nonunital Ek-algebra

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###Context###
#### Higher algebra
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[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The nonunital variant of _[[Ek-algebra]]_, hence the generalization of the concept of [[nonunital ring]] from [[algebra]] to [[higher algebra]].

## Properties

### Relation to augmented $\mathbb{E}_k$-algebra
 {#RelationToAugmentedEkAlgebras}

Nonunital $\mathbb{E}_k$-algebras are equivalent to unital $\mathbb{E}_k$-algebras sliced over the free $\mathbb{E}_k$-algebra on a single generator.

In particular nonunital [[E-∞ algebras]] are [[equivalence of (∞,1)-categories|equivalent]] to unital $E_\infty$-algebras sliced over the [[sphere spectrum]]:

$$
  E_\infty Ring^{nu} \simeq E_\infty Ring_{/\mathbb{S}}
  \,,
$$

where the map from right to left is given by $\mathbb{E}_k$-[[unitalization]] and the converse by forming [[augmentation ideals]].

([Lurie, prop. 5.4.4.10](#Lurie))

## References

* {#Lurie} [[Jacob Lurie]], section 5.4.4 of _[[Higher Algebra]]_

[[!redirects nonunital Ek-algebras]]

[[!redirects non-unital Ek-algebra]]
[[!redirects non-unital Ek-algebras]]

[[!redirects nonunital E-∞ algebra]]
[[!redirects nonunital E-∞ algebras]]

[[!redirects nonunital E? algebra]]
[[!redirects nonunital E? algebras]]

[[!redirects nonunital E-infinity algebra]]
[[!redirects nonunital E-infinity algebras]]

[[!redirects nonunital E-∞ ring]]
[[!redirects nonunital E-∞ rings]]

[[!redirects nonunital E? ring]]
[[!redirects nonunital E? rings]]

[[!redirects nonunital E-infinity ring]]
[[!redirects nonunital E-infinity rings]]


nonunital ring

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

# Contents
* table of contents
{: toc}

## Idea

The concept of _nonunital ring_ is like that of [[ring]] but without the requirement of the existence of an [[identity]] element ("unit" element).

Nonunital rings with [[homomorphisms]] between them form the [[category]] [[Rng]].

Historically, this was in fact the original meaning of "ring", and while mostly "ring" has come to mean by default the version with identity element, nonunital rings still play a role (see e.g. the review in [Anderson 06](#Anderson06)) and in some areas of mathematics "nonunital ring" is still the default meaning of "ring". In particular, non-unital rings may naturally be identified with the [[augmentation ideals]] of $\mathbb{Z}$-[[augmented algebra|augmented]] unital rings, see the discussion [below](#AsSlicesOfRings).

+-- {: .num_remark}
###### Remark on terminology

The term "non-unital ring" may be regarded as an example of the "[[red herring principle]]", as a non-unital ring is not in general a ring in the modern sense of the word.

In [[Bourbaki|Bourbaki 6, chapter 1]] the term _pseudo-ring_ is used, but that convention has not become established.

Another terminology that has been suggested for "nonunital ring", and which is in use in part of the literature (e.g. [Anderson 06](#Anderson06)) is "rng", where dropping the "i" in "ring" is meant to be alluding to the absence of _i_dentity elements. This terminology appears in print first in ([Jacobson](#Jacobson)), where it is attributed to Louis Rowen.

Similarly there is, for whatever it's worth, the suggestion that a ring without negatives, hence a [[semiring]], should be called a _[[rig]]_ (although here one may make a technical distinction about the _additive_ identity).

=--

## Definitions

### For rings

Specifically:

+-- {: .num_defn}
###### Definition

A __nonunital ring__ or _rng_ is a [[set]] $R$ with operations of addition and multiplication, such that:

*  $R$ is a [[semigroup]] under multiplication;
*  $R$ is an [[abelian group]] under addition;
*  multiplication distributes over addition.

=--

More sophisticatedly, we can say that, just as a ring is a [[monoid object]] in [[Ab]], so

+-- {: .num_defn}
###### Definition

A _nonunital ring_ or _rng_ is a [[semigroup]] object in [[Ab]].

=--

+-- {: .num_remark}
###### Remark

A non-unital ring may well contain an element that behaves as the [[identity]] element, i.e. it may be in the image of the [[forgetful functor]] from unital rings to nonunital rings. But if so, then this element is still not part of the defining data and in particular a [[homomorphism]] of non-unital rings need not to preserve whatever identity elements may happen to be present.

=--

### For $\mathbb{E}_k$-algebras

[[nonunital Ek-algebras]] are discussed in ([Lurie, section 5.2.3](#Lurie)).

## Properties

### Unitization

+-- {: .num_defn #Unitisation}
###### Definition

Given a non-unital commutative ring $A$, then its _[[unitisation]]_ is the [[commutative ring]] $F(A)$ obtained by [[free construction|freely]] adjoining an [[identity]] element, hence the ring whose underlying [[abelian group]] is the [[direct sum]] $\mathbb{Z} \oplus A$ of $A$ with the [[integers]], and whose product operation is defined by

$$
  (n_1, a_1) (n_2, a_2) \coloneqq (n_1 n_2, n_1 a_2 + n_2 a_1 + a_1 a_2)
  \,,
$$

where for $n \in \mathbb{Z}$ and $a \in A$ we set $n a \coloneqq \underset{n\;summands}{\underbrace{a + a + \cdots + a}}$.

=--

+-- {: .num_remark}
###### Remark

In the unitization $\mathbb{Z} \oplus A$ we have $(n,0) + (0,a) = (n,a)$ and hence it makes sense to abbreviate $(n,a)$ simply to $n+a$.  The product in the [[unitisation]] is then fixed by the defining requirement that $1 \cdot a = a$ and by the [[distributivity law]].

=--

+-- {: .num_remark}

###### Remark

One can also think of the unitisation as the quotient of the polynomial ring "$A[x]$" ($A$ "with a generic element adjoined", i.e. a term algebra) quotiented by the relations $ax - a = 0, xa - a = 0$, so that this $x$ must be a right and left identity for multiplication $a$. Cosets must be represented by expressions of the form $a + nx$; this provides an obvious isomorphism to the above definition, and motivates the multiplication for $\mathbb{Z} \oplus A$ above.

=--

+-- {: .num_remark}

###### Remark

Since $A$ embeds into its unitisation $F(A)$, every [[rng]] lives as an ideal in some [[unital ring|ring]]. We can consider $A$-linear actions $A \curvearrowright M$ on abelian groups $M$. Since the endomorphism rings of abelian groups are always unital, the [[universal property]] of the unitisation-forgetful adjunction (see below) ensures that there is a unique extension of the action map $A \to \operatorname{End}_{\mathbf{Ab}}(M)$ along $A \hookrightarrow \operatorname{for} \circ F(A)$ to an action map $F(A) \to \operatorname{End}_{\mathbf{Ab}}(M)$. This induces an equivalence $A\text{-}\mathbf{Mod} \simeq F(A)\text{-}\mathbf{Mod},$ so that one may as well study $F(A)$ if one wanted to study $A$ through its module category.

=--

+-- {: .num_remark}
###### Remark

Similar [[unitisation]] prescriptions work for non-commutative rings and for [[nonunital algebras]] over a fixed base ring, see also at

* _[[unitisation of C*-algebras]]_.

=--

+-- {: .num_prop}
###### Proposition

Unitisation in def. \ref{Unitisation} extends to a [[functor]] from $CRng$ to [[CRing]] which is [[left adjoint]] to the [[forgetful functor]] from [[commutative rings]] to non-unital commutative rings.

$$
  F \colon CRng \leftrightarrow CRing \colon U
  \,.
$$

=--

+-- {: .proof}
###### Proof

This is because the definition of any ring [[homomorphism]] out of $F(A)= (\mathbb{Z} \oplus A, \cdot)$ is uniquely fixed on the $\mathbb{Z}$-summand.

=--

### Nonunital rings as slices of rings
 {#AsSlicesOfRings}

+-- {: .num_defn #AugmentationIdealFunctor}
###### Definition

Write $CRing_{/\mathbb{Z}}$ for the [[slice category]] of [[CRing]] over the ring of [[integers]] ([[augmented algebra|augmented commutative rings]]). Write

$$
  CRing_{/\mathbb{Z}} \longrightarrow CRng
$$

for the [[functor]] to commutative non-unital rings which sends any $(R \stackrel{\phi}{\to} \mathbb{Z})$ to its _[[augmentation ideal]]_, hence to the [[kernel]] of $\phi$

$$
  (R \stackrel{\phi}{\to} \mathbb{Z}) \mapsto ker(\phi)
  \,.
$$

=--

+-- {: .num_defn #EquivalenceOfNonunitalAndSliced}
###### Proposition

The [[augmentation ideal]]-functor in def. \ref{AugmentationIdealFunctor} is an [[equivalence of categories]] whose inverse is given by [[unitisation]], def. \ref{Unitisation}, remembering the [[projection]] $(\mathbb{Z} \oplus A) \to \mathbb{Z}$:

$$
  CRng \simeq CRing_{/\mathbb{Z}}
  \,.
$$

=--

+-- {: .proof}
###### Proof

That the functor is [[fully faithful]] is to observe that for a ring $R \stackrel{\phi}{\to} \mathbb{Z}$ the [[fiber]] $R_{n}$ over $n \in \mathbb{Z}$ is a [[torsor]] over the additive group underlying the [[augmentation ideal]] $A = R_0 = ker(\phi)$, and moreover it is a pointed torsor, the point being $n$ itself, hence is canonically equivalent to the [[augmentation ideal]] $A$, the equivalence being addition by $n$ in $R$. Hence any homomrphism of rings with identity over $\mathbb{Z}$

$$
  \array{
    R_1 && \stackrel{f}{\longrightarrow} && R_2
    \\
    & {}_{\mathllap{\phi_1}}\searrow && \swarrow_{\mathrlap{\phi_2}}
    \\
    && \mathbb{Z}
  }
$$

is uniquely fixed by its restriction to the augmentation ideal $ker(\phi_1)$, whose image, moreover, has to be in the augmentation ideal $ker(\phi_2)$.

=--

The identification of non-unital algebras as augmentation ideals of augmented unital algebras is used for instance in ([Fresse 06](#Fresse06)).

+-- {: .num_remark}
###### Remark

In terms of [[arithmetic geometry]], the [[Isbell duality|formally dual]]  statement of prop. \ref{AugmentationIdealFunctor} is that arithmetic geometry induced by non-unital rings is equivalently ordinary arithmetic geometry _under_ [[Spec(Z)]].

=--

+-- {: .num_remark}
###### Remark

The generalization of prop. \ref{EquivalenceOfNonunitalAndSliced} to [[nonunital Ek-algebras]] is ([Lurie, prop. 5.2.3.15](#Lurie)).

=--

## Related concepts

* [[unitisation of C*-algebras]]

* [[noncommutative ring]], [[nonassociative ring]]

* [[rig]]

* [[nonunital Ek-algebra]]

* [[paraunital ring]]

## References
 {#References}

### Nonunital ring theory
 {#ReferencesTheory}

A survey of various notions between unital rings and nonunital rings:

* Patrik Nystedt, _A survey of s-unital and locally unital rings_, Revista Integración, temas de matemáticas
Escuela de Matemáticas, Universidad Industrial de Santander, Vol. 37, N◦ 2, 2019, pág. 251–260.  [doi](http://dx.doi.org/10.18273/revint.v37n2-2019003), [arXiv](https://arxiv.org/abs/1809.02117).

The terminology "rng" originates in

* {#Jacobson} Nathan Jacobson _Basic Algebra_.

A survey of commutative rng theory is in

* {#Anderson06} D. Anderson, _Commutative rngs_, in J. Brewer et al. (eds.) _Multiplicative ideal theory in Commutative Algebra_, 2006

Discussion of [[module]] theory over rngs (with close relation to [[torsion modules]]) is in

* {#Quillen96} [[Daniel Quillen]], _Module theory over nonunital rings_, August 1996 ([[QuillenModulesOverRngs.pdf:file]])

Discussion of non-unital rings as [[augmentation ideals]] of augmented unital rings includes

* {#Fresse06} [[Benoit Fresse]], _The Bar Complex of an E-infinity Algebra_, Adv. Math.  223 (2010), pages 2049-2096 ([arXiv:math/0601085](http://arxiv.org/abs/math/0601085))

A definition of [[algebraic K-theory]] for nonunital rings is due to

* {#Quillen} [[Daniel Quillen]], _$K_0$ for nonunital rings and Morita invariance_, J. Reine Angew. Math., 472:197-217, 1996.

with further developments (in [[KK-theory]]) including

* [[Snigdhayan Mahanta]], _Higher nonunital Quillen K'-theory, KK-dualities and applications to topological T-dualities_,  J. Geom. Phys., 61 (5), 875-889, 2011. ([pdf](http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf))

Discussion in the context of [[higher algebra]] ([[nonunital Ek-algebras]]) is in

* {#Lurie} [[Jacob Lurie]], section 5.2.3 of _[[Higher Algebra]]_

[[!redirects rng]]
[[!redirects rngs]]

[[!redirects nonunital ring]]
[[!redirects nonunital rings]]
[[!redirects non-unital ring]]
[[!redirects non-unital rings]]

[[!redirects unitization of a ring]]
[[!redirects unitization of rings]]
[[!redirects unitizations of rings]]
[[!redirects unitisation of a ring]]
[[!redirects unitisation of rings]]
[[!redirects unitisations of rings]]

[[!redirects unitization of a commutative ring]]
[[!redirects unitization of commutative rings]]
[[!redirects unitizations of commutative rings]]
[[!redirects unitisation of a commutative ring]]
[[!redirects unitisation of commutative rings]]
[[!redirects unitisations of commutative rings]]


Noon Silk

## Noon Silk ##

[Website](https://sites.google.com/site/noonsilk/)


Nora Ganter

* [Homepage](https://researchers.ms.unimelb.edu.au/~nganter@unimelb/)

* [School staff page](http://www.ms.unimelb.edu.au/Personnel/profile.php?PC_id=662)

## Research program

* (with [[Matthew Ando]]), _Subtle symmetries and the refined Monster_, ([pdf](https://researchers.ms.unimelb.edu.au/~nganter@unimelb/research_program_ganter.pdf))

## Selected writings

On [[Tate K-theory]] and [[equivariant elliptic cohomology]]:

* [[Nora Ganter]], _Stringy power operations in Tate K-theory_, [math.AT/0701565](http://arxiv.org/abs/math/0701565)

* {#Ganter13} [[Nora Ganter]], _Power operations in orbifold Tate K-theory_, Homology Homotopy Appl. Volume 15, Number 1 (2013), 313-342. ([arXiv:1301.2754](https://arxiv.org/abs/1301.2754), [euclid:hha/1383943680](https://projecteuclid.org/euclid.hha/1383943680))

* [[Nora Ganter]], _Orbifold genera, product formulas and power operations_, Adv. Math. __205__ (2006), no. 1, 84-133 (revised Ph. D. Thesis) [math.AT/0407021](http://arxiv.org/abs/math/0407021)

* [[Nora Ganter]], _Hecke operators in equivariant elliptic cohomology and generalized moonshine_, [arxiv/0706.2898](http://arxiv.org/abs/0706.2898)

* [[Nora Ganter]], [[Matthew Ando]], C. French, _The Jacobi orientation adn the two-variable elliptic genus_, Algebraic and Geometric Topology __8__, 493-539, 2008, [math.AG/0605554](http://arxiv.org/abs/math/0605554)

On [[2-representations]]:

* _Inner products of 2-representations_, [http://arxiv.org/abs/1110.1711](http://arxiv.org/abs/1110.1711)

* [[Nora Ganter]], [[Mikhail Kapranov]], _Representation and character theory in 2-categories_, Adv. Math. __217__ (2008), no. 5, 2268-2300, [math.AT/0602510](http://arxiv.org/abs/math/0602510); _Symmetric and exterior powers of categories_, [arxiv/1110.4753](http://arxiv.org/abs/1110.4753)

On [[2-group]]-[[higher central extensions|extensions]] by the [[circle 2-group]]:

of [[tori]] (see also at [[T-duality 2-group]]):

* [[Nora Ganter]], _Categorical Tori_, SIGMA 14 (2018), 014, 18 ([arXiv:1406.7046](https://arxiv.org/abs/1406.7046))

of [[finite subgroups of SU(2)]] (to [[Platonic 2-groups]]):

* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, Higher Structures 1(1):122-146, 2017 ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192), [hs:30](https://journals.mq.edu.au/index.php/higher_structures/article/view/30))

See also:

* _Smash products of E(1)-local spectra at an odd prime_, Cah. Topol. Geom. Differ. Categ. __48__ (2007), no.1, 3-54, [math.CT/0410589](http://arxiv.org/abs/math/0410589)
category: people


Norbert Dragon

* [webpage](https://www.itp.uni-hannover.de/~dragon/)

## related $n$Lb entries

* [[Wigner classification]]

category: people


Norbert Kaiser

* [webpage](http://www.t39.ph.tum.de/T39_files/T39_people_files/kaiser.php)

## Selected writings

On [[nucleons]] as [[Skyrmions]] in [[quantum hadrodynamics]] with [[pions]], [[omega-mesons]] and [[rho-mesons]] ($\pi$-$\rho$-$\omega$-model):

* [[Ulf-G. Meissner]], [[Norbert Kaiser]], [[Wolfram Weise]], _Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties_, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (doi:10.1016/0375-9474(87)90463-5)

* [[Ulf-G. Meissner]], [[Norbert Kaiser]], Andreas Wirzba, [[Wolfram Weise]], _Skyrmions with $\rho$ and $\omega$ Mesons as Dynamical Gauge Bosons_, Phys. Rev. Lett. 57, 1676 (1986) ([doi:10.1103/PhysRevLett.57.1676](https://doi.org/10.1103/PhysRevLett.57.1676))

category: people


Norbert Poncin

* [Institute page](https://wwwfr.uni.lu/recherche/fstm/dmath/people/norbert_poncin)

* [research group page](https://wwwde.uni.lu/recherche/fstm/dmath/research_areas/geometry_and_the_mathematical_theory_of_quantisation/team_norbert_poncin)

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=191593)

## Selected writings

On [[homotopy limits]], [[homotopy pullbacks]] and their [[pasting law for pullbacks|pasting law]] in/via [[model categories]]:

* [[Alisa Govzmann]], [[Damjan Pištalo]], [[Norbert Poncin]], *Indeterminacies and models of homotopy limits* [[arXiv:2109.12395](https://arxiv.org/abs/2109.12395)]

category: people


Norbert Schappacher

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Norbert_Schappacher)

## Selected writings

On [[Deligne cohomology]] and [[Beilinson's conjectures]] on [[special values of L-functions]]:

* [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_  Perspectives in Mathematics, Volume 4, Academic Press, Inc. 1988 (ISBN:978-0-12-581120-0)

category: people


Norbert Straumann

* [Wikipedia entry (German)](http://de.wikipedia.org/wiki/Norbert_Straumann)

* [publications](http://www.itp.phys.ethz.ch/education/articles)

category: people


Norbert Wiener

* [Wikipedia entry](https://en.wikipedia.org/wiki/Norbert_Wiener)

* inControl podcast, *[Norbert Wiener and Cybernetics](https://youtu.be/ggKVMvxBFGw)*

## Selected writings

Description of [[Brownian motion]] as a [[stochastic process]] that came to be known as the [[Wiener process]]:

* [[Norbert Wiener]], *The Average of an Analytic Functional and the Brownian Movement*, Proceedings of the National Academy of Sciences of the United States of America, **7** 10 (1921) 294-298 [[jstor:84434](https://www.jstor.org/stable/84434)]

category: people


Norberto Scoccola

* [webpage](http://www.tandar.cnea.gov.ar/~scoccola/)

## Selected writings

On [[Skyrmions]] with [[heavy mesons]]:

* [[Norberto Scoccola]], D. P. Min, H. Nadeau, [[Mannque Rho]], _The strangeness problem: An $SU(3)$ skyrmion with vector mesons_, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (doi:10.1016/0375-9474(89)90029-8)

* [[Mannque Rho]], D. O. Riska, [[Norberto Scoccola]], _Charmed baryons as soliton - D meson bound states_, Phys. Lett.B 251 (1990) 597-602 ([spire:297771](https://inspirehep.net/literature/297771), doi:10.1016/0370-2693(90)90802-D)

* [[Mannque Rho]], D. O. Riska, [[Norberto Scoccola]], _The energy levels of the heavy flavour baryons in the topological soliton model_, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages 343–352 (1992) ([doi:10.1007/BF01283544](https://doi.org/10.1007/BF01283544))

* Yongseok Oh, Dong-Pil Min, [[Mannque Rho]], [[Norberto Scoccola]], _Massive-quark baryons as skyrmions: Magnetic moments_, Nuclear Physics A
Volume 534, Issues 3–4 (1991) Pages 493-512 (doi:10.1016/0375-9474(91)90458-I)

* [[Norberto Scoccola]], _Heavy quark skyrmions_, ([arXiv:0905.2722](https://arxiv.org/abs/0905.2722), [doi:10.1142/9789814280709_0004](https://doi.org/10.1142/9789814280709_0004)), Chapter 4 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

category: people


Nori motive

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

> under construction

#Contents#
* table of contents
{:toc}

## Idea

[[Madhav Nori]] invented an original approach to [[mixed motives]], in the spirit of Tannakian theory, but based on a specific [[Nori's Tannakian theorem]]. Unfortunately his original work, which has been lectured on some summer schools and conferences is not widely available in written form. [[Maxim Kontsevich]] based his approach to [[period]]s and their appearance in [[deformation quantization]] partly on Nori's insight.

The construction of Nori [[motives]] themselves has been generalised to categories over a base S by Arapura and Ivorra. Arapura's approach
is based on [[constructible sheaves]]. His categories allow pull-back and push-
forward, the latter being a deep result. The same paper also constructs the
weight filtration on Nori motives and establishes the equivalence between
Nori motives and André pure motives. Ivorra's approach is based on [[perverse sheaves]]. Compatibility under the [[six functors formalism]] is open in his
setting [Huber-Stach]{#HS}

It is known that Nori's and Ayoub's Motivic galois groups agree.

The subcategory of 1-motives in Nori motives agrees with Deligne's 1-motives, and hence also with 1-motives in Voevodsky's category.

## References

* Annette Huber, [[Stefan Müller-Stach]], with contributions of Benjamin Friedrich and Jonas von Wangenheim, [Periods and Nori motives](http://home.mathematik.uni-freiburg.de/arithgeom/preprints/buch/mantel.html), (Book in preparation: first version, September 2015, 323 p. ){#HS}

* Annette Huber, [[Stefan Müller-Stach]], _On the relation between Nori motives and Kontsevich periods_, [1105.0865](http://arxiv.org/abs/1105.0865)

* Madhav Nori, TIFR notes on motives, unpublished, [pdf](https://web.archive.org/web/20160922233016/http://www.arithgeo.ethz.ch/alpbach2011/Nori_TIFR) (archived)

* MathOverflow: [kontsevichs-conjectures-on-the-grothendieck-teichmuller-group](http://mathoverflow.net/questions/42148/kontsevichs-conjectures-on-the-grothendieck-teichmuller-group)

* [Alain Bruguieres](http://www.math.univ-montp2.fr/~bruguieres/recherche.html), _On a [[Tannaka duality|tannakian]] result due to Nori_, [dvi](http://www.math.univ-montp2.fr/%7Ebruguieres/ntan.dvi), [ps](http://www.math.univ-montp2.fr/%7Ebruguieres/ntan.ps), [pdf](http://www.math.univ-montp2.fr/%7Ebruguieres/ntan.pdf), [djvu](http://www.math.univ-montp2.fr/%7Ebruguieres/ntan.djvu)

* Jonas von Wangenheim, _Nori-Motive und Tannaka-Theorie_, [arxiv/1111.5146](http://arxiv.org/abs/1111.5146)

* [[Isamu Iwanari]], _Tannakization in derived algebraic geometry_, [arxiv/1112.1761](http://arxiv.org/abs/1112.1761)

* [[Joseph Ayoub]], [[Luca Barbieri-Viale]], _Nori 1-motives_ ([arXiv:1206.5923](http://arxiv.org/abs/1206.5923))

* [[Utsav Choudhury]], [[Martin Gallauer de souza]], _An isomorphism of Motivic galois groups_ ([arXiv:1410.6104](http://arxiv.org/abs/1410.6104))

* [[Joseph Ayoub]], [[Luca Barbieri-Viale]], _Nori 1-motives_, [pdf](https://www.math.uzh.ch/fileadmin/user/ayoub/publikation/N_1_Mot.pdf)

Nori motives over a base S

* [[Florian Ivorra]], _Perverse Nori motives_ ([hal-00978893](https://hal.archives-ouvertes.fr/hal-00978893/document))

* [[Donu Arapura]], _An Abelian category of Motivic sheaves_ ([arXiv:0801.0261](http://arxiv.org/abs/0801.0261))

Characterization of the universality of Nori motives via Barr completion of the [[syntactic category]] of the [[theory]] (in the [[formal logic|formal logical]] sense) of [[singular cohomology]] of [[schemes]] is in

* {#BVCL15} [[Luca Barbieri-Viale]], [[Olivia Caramello]], [[Laurent Lafforgue]], _Syntactic categories for Nori motives_ ([arXiv:1506.06113](http://arxiv.org/abs/1506.06113))

For more on this [[topos theory]] perspective, see

* [[Luca Barbieri-Viale]], _T-Motives_ ([arXiv:1602.05053](http://arxiv.org/abs/1602.05053))

* [[Olivia Caramello]], _Motivic toposes_ ([arXiv:1507.06271](http://arxiv.org/abs/1507.06271))

[[!redirects Nori motives]]
[[!redirects Nori's motive]]
[[!redirects Nori's motives]]


Nori's Tannakian theorem

* [Alain Bruguieres](http://imag.umontpellier.fr/~bruguieres/recherche.html), _On a [[Tannaka duality|tannakian]] result due to Nori_, [dvi](http://imag.umontpellier.fr/~bruguieres/docs/ntan.dvi), [ps](http://imag.umontpellier.fr/~bruguieres/docs/ntan.ps), [pdf](http://imag.umontpellier.fr/~bruguieres/docs/ntan.pdf), [djvu](http://imag.umontpellier.fr/~bruguieres/docs/ntan.djvu)


Noriaki Ikeda

## Related entries

Noriaki Ikeda was the first to introduce both the [[Poisson sigma-model]] as well as the [[Courant sigma-model]].

category: people


Norihiko Minami

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=63526)

* [institute page](http://researcher.nitech.ac.jp/html/316_en.html)

## Selected writings

On detection of [[weak homotopy equivalences]] and [[equivariant weak homotopy equivalences]] in terms of free homotopy sets:

* [[Takao Matumoto]], [[Norihiko Minami]], [[Masahiro Sugawara]], *On the set of free homotopy classes and Brown's construction*, Hiroshima Math. J. 14(2): 359-369 (1984) ([doi:10.32917/hmj/1206133043](https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-14/issue-2/On-the-set-of-free-homotopy-classes-and-Browns-construction/10.32917/hmj/1206133043.full))

category: people


Norihiro Iizuka

* [insitute page](http://www.dma.jim.osaka-u.ac.jp/view?l=en&u=10001728)

## Selected writings

Introducing the [[nuclear matrix model]] for [[baryons]]/[[nucleons]] in the [[Witten-Sakai-Sugimoto 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))

category: people


Norihiro Yamada

* [website](https://www.cs.ox.ac.uk/people/norihiro.yamada/)

* [website](https://www.norileo.com/)

## Publications

* [[Norihiro Yamada]], [[Samson Abramsky]], (2020). *Dynamic game semantics*. Mathematical Structures in Computer Science, Volume 30, Issue 8, September 2020, pp. 892 - 951 ([doi:10.1017/S0960129520000250](https://doi.org/10.1017/S0960129520000250))

## Talks

* [[Norihiro Yamada]], *Game semantics of homotopy type theory*, [[Homotopy Type Theory Electronic Seminar Talks]], 11 February 2021 ([video](https://www.youtube.com/watch?v=uY5LubwJ1Xo), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Yamada-2021-02-11-HoTTEST.pdf))

category: people


norm

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### On an abelian group

For $(G,+)$ an [[abelian group]], then a _norm_ on the group is a [[function]]

$$
  {\vert-\vert} \;\colon\; G \longrightarrow \mathbb{R}
$$

to the [[real numbers]], such that

1. (positivity) $(g \neq 0) \Rightarrow (\vert g\vert \gt 0)$

1. ([[triangle inequality]])  ${\vert g + h\vert}\leq {\vert g\vert} + {\vert h\vert}$

1. (linearity) ${\vert k g\vert} = {\vert k\vert} {\vert g\vert}$ for all $k \in \mathbb{Z}$.

Here ${\vert k\vert} \in \mathbb{N}$ denotes the [[absolute value]].

A group with a norm is a _[[normed group]]_, see there for more.

In [[constructive mathematics]], it is common to replace the [[denial inequality]] with a [[tight apartness relation]] in the positivity condition.

### On a vector space

For $k$ a [[field]] equipped with a [[valuation]] (most usually, a [[local field]] such as $\mathbb{R}$, $\mathbb{C}$, or a [[p-adic field|p-adic]] [[complete field|completion]] of a [[number field]]), a **norm** on a $k$-[[vector space]] $V$ is a [[function]]

$$
  {\vert-\vert} \colon V \to \mathbb{R}
$$

such that for all $\lambda \in k$, $v,w \in V$ we have

1.  ${\vert \lambda v \vert} = {\vert \lambda\vert} {\vert v \vert}$ (where $\vert \lambda \vert$ denotes the [[valuation]])

1.  ${\vert v + w\vert } \leq {\vert v \vert } + {\vert w \vert}$ ("[[triangle inequality]]")

1.  if ${\vert v\vert} = 0$ then $v = 0$.

If the third property is not required, one speaks of a **seminorm**.

If the triangle identity is strengthened to

* ${\vert v + w\vert } \leq max ({\vert v\vert}, {\vert w\vert})$

one speaks of a **[[non-archimedean]]** seminorm, otherwise of an [[archimedean]] one.

A vector space equipped with a norm is a **normed vector space**. A vector of norm 1 is a [[unit vector]].

Each seminorm determines a [[topology]], which is [[Hausdorff space|Hausdorff]] precisely if it is a norm.

A [[topological vector space]] is called **(semi-)normed** if its [[topology]] can be induced by a (semi-)norm.

Two seminorms ${\vert - \vert}_1$ and ${\vert - \vert}_2$ are called **equivalent** if there are $0 \lt C, C' \in \mathbb{R}$ such that for all $v$ we have

$$
  C {\vert v \vert}_1 \leq {\vert v \vert}_2
  \leq C' {\vert v \vert}_1
  \,.
$$

Equivalent seminorms determine the same [[topology]].

The collection of (bounded) multiplicative seminorms on a ([[Banach space|Banach]]) [[ring]] is called its [[analytic spectrum]] (see there for details).

## Examples

### General

* The standard [[absolute value]] is a norm on the [[real numbers]].

* More generally, on any [[Cartesian space]] $\mathbb{R}^n$ the **Euclidean norm** is given by

   $$
     \vert \vec x\vert
       \;\coloneqq\;
     \sqrt{
     \underoverset{i = 1}{n}{\sum}
       (x_i)^2
     }
     \,.
   $$

1. more generally, for $n \in \mathbb{N}$, and $p \in \mathbb{N}$, $p \geq 1$, then the [[Cartesian space]] $\mathbb{R}^n$ carries the _[[p-norm]]_

   $$
     {\vert \vec x \vert}_p  \coloneqq \root p {\sum_i {|x_i|^p}}
   $$

1. The [[p-norm]] generalizes to [[sequence spaces]] and [[Lebesgue spaces]].

* [[operator norm]] in [[operator algebra]]/[[functional analysis]]

### Minkowski Functionals

Let $V$ be a vector space and $B \subseteq V$ an [[absorbing]] [[absolutely convex]] subset.  The **Minkowski functional** of $B$ is the function $\mu_B \colon V \to \mathbb{R}$ defined by:

$$
\mu_B(v) = \inf\{t \gt 0 : v \in t B\}
$$

This is a semi-norm on $V$.

## Properties

* [[Hahn-Banach theorem]]

The (open or closed) [[unit ball]] of a seminormed vector space is a [[convex set]], a [[balanced set]] and an [[absorbing set]]. The first two of these properties make the unit ball (or even any ball of positive radius) an [[absolutely convex set]].

### Uniqueness {#dreamUnique}

In [[dream mathematics]], a given real [[vector space]] (with no topological structure) can have at most one complete norm, up to topological equivalence ([[homeomorphism]] of the [[identity function]]).  It can have multiple inequivalent complete seminorms and incomplete norms, but their Hausdorff quotients and completions must be different.  For example, the various [[Lebesgue norms]] on a [[Cartesian space]] $\mathbb{R}^n$ for finite $n$ are complete and equivalent; on $\mathbb{R}^\infty$, they are inequivalent but incomplete.

As dream mathematics includes [[excluded middle]] and [[dependent choice]], the existence of inequivalent complete norms on a given vector space cannot be proved without a stronger form of the [[axiom of choice]], enough to disprove the [[Baire property]] (which is the only classically false axiom needed in the proof of uniqueness).  In _[[HAF]]_, it is argued that this explains why, in [[applied mathematics]], there tends to be only one norm considered on any particular vector space (after Hausdorff completion).

This theorem applies more generally to [[F-norms]] but not to [[G-norms]] (even on a real vector space).

## Norms valued in another Archimedean integral domain

In [[constructive mathematics]], the notion of "real numbers" bifurcates: the [[Dedekind real numbers]] are different from the [[modulated Cauchy real numbers]], which are different from the [[HoTT book real numbers]], which are different from the [[localic real numbers]], and so on. As a result, there are multiple sets of real numbers in which a metric could be valued in.

In [[predicative mathematics]], the issue becomes even worse: there is no longer one set of Dedekind real numbers, but a whole hierarchy of Dedekind real numbers, one set for every [[universe]] in the [[foundations of mathematics|foundations]]. As a result, one cannot resort to merely using the [[Dedekind real numbers]] for defining the norm as in impredicative mathematics, one has to define norms and normed spaces more generally.

Thus, given an [[Archimedean integral domain]] $R$, for $k$ a [[field]] equipped with a [[valuation]], an **$R$-norm** on a $k$-[[vector space]] $V$ is a [[function]]

$$
  {\vert-\vert} \colon V \to R
$$

such that for all $\lambda \in k$, $v,w \in V$ we have

1.  ${\vert \lambda v \vert} = {\vert \lambda\vert} {\vert v \vert}$ (where $\vert \lambda \vert$ denotes the [[valuation]])

1.  ${\vert v + w\vert } \leq {\vert v \vert } + {\vert w \vert}$ ("[[triangle inequality]]")

1.  if ${\vert v\vert} = 0$ then $v = 0$.

One could define $R$-seminorms, non-archimedean $R$-norms, and $R$-normed vector spaces in the same way as above.

## Related concepts

* [[metric]]

* [[quotient norm]]

* [[p-norm]]

* [[F-norm]], [[G-norm]]

* in [[representation theory]]: [[norm map]]

[[!include analytic geometry ingredients -- table]]

## References

* Wikipedia, _[Normed vector space](https://en.wikipedia.org/wiki/Normed_vector_space)_

* {#BoschGuntzerRemmert84} [[Siegfried Bosch]], [[Ulrich Güntzer]], [[Reinhold Remmert]], _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

* _[[HAF]]_, §27.47.b (for uniqueness in dream mathematics)

[[!redirects norms]]
[[!redirects seminorm]]
[[!redirects seminorms]]
[[!redirects semi-norm]]
[[!redirects semi-norms]]
[[!redirects Minkowski functional]]
[[!redirects Minkowski functionals]]
[[!redirects normed vector space]]
[[!redirects normed vector spaces]]

[[!redirects normed vector space]]
[[!redirects normed vector spaces]]
[[!redirects normable vector space]]
[[!redirects normable vector spaces]]
[[!redirects pseudonormed vector space]]
[[!redirects pseudonormed vector spaces]]
[[!redirects seminormed vector space]]
[[!redirects seminormed vector spaces]]

[[!redirects normed space]]
[[!redirects normed spaces]]
[[!redirects normable space]]
[[!redirects normable spaces]]
[[!redirects pseudonormed space]]
[[!redirects pseudonormed spaces]]
[[!redirects seminormed space]]
[[!redirects seminormed spaces]]

[[!redirects vector measure]]
[[!redirects vector measures]]


norm closure

The term _norml closure_ refers to [[topological closure]] under a [[metric topology]] induced from a given [[norm]].

[[!redirects norm closures]]


norm cofibration

## Related concepts

* [[TC]]

## References

* Ib Madsen, p. 6 and prop. 2.4.3 of _Algebraic K-theory and traces_, [pdf](http://math.mit.edu/~nrozen/juvitop/madsen.pdf)


norm map

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

#Contents#
* table of contents
{:toc}

## Idea

In [[representation theory]] a _norm map_ is a canonical morphism from [[coinvariants]] to [[invariants]] of a given [[action]] which, in suitably well behaved cases is given by [[group averaging]].

(e.g [Lurie, constructions 6.1.6.4, 6.1.6.8, 6.1.6.18](#Lurie))

For [[∞-actions]] of [[finite groups]] $G$ on objects $E$ in [[stable (∞,1)-categories]], then the [[homotopy cofiber]] $X^{t G}$ of the norm map is called the _[[Tate construction]]_, sitting in a [[homotopy fiber sequence]]

$$
  X_G \longrightarrow X^G \longrightarrow X^{t G}
  \,.
$$

(e.g [Lurie, def. 6.1.6.24](#Lurie))

For the [[stable (∞,1)-category of spectra]] this is accordingly called the _[[Tate spectrum]]_.

## Related concepts

* [[Tate spectrum]]

* [[cyclotomic spectrum]]

* _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_

## References

The general abstract construction is due to

* {#Lurie} [[Jacob Lurie]], section 6.1.6 of _[[Higher Algebra]]_

Review with an eye towards discussion of [[topological cyclic homology]] is in section I.1 of

* {#NikolausScholze17} [[Thomas Nikolaus]], [[Peter Scholze]], _On topological cyclic homology_, ([arXiv:1707.01799](https://arxiv.org/abs/1707.01799))

[[!redirects norm maps]]


norm residue isomorphism theorem

A self-contained proof is given by

* [[Christian Haesemeyer]] and [[Charles Weibel]], _The Norm Residue Theorem in Motivic Cohomology_ ([pdf](http://sites.math.rutgers.edu/~weibel/BK.pdf))

## References

* Wikipedia, _[Norm residue isomorphism theorem](http://en.wikipedia.org/wiki/Bloch-Kato_conjecture)_

[[!redirects norm residue isomoprhism theorem]]
[[!redirects norm residue isomoprhism theorem]]
[[!redirects Bloch-Kato conjecture]]


norm topology

The [[topology]] induced on a collection of [[linear operators]] by the [[operator norm]].

[[!redirects norm topologies]]


normal bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn}
###### Definition

For $i : X \hookrightarrow Y$ an [[immersion]] (notably an [[embedding]]) of [[smooth manifolds]], the **normal bundle** of $X$ in $Y$ relative to $i$ is the  [[vector bundle]]

$$
  N_i X  \to X
$$

defined as the [[fiber|fiberwise]] [[quotient]] bundle

$$
  N_i X =\frac{i^* T Y}{T X}
$$

If $Y$ is equipped with [[Riemannian metric]] structure then this is equivalently the space of [[tangent vectors]] to $Y$ which are [[normal vectors]] to the [[tangent vectors]] to $X$, whence the name.

=--

The pullback $i^* T Y$ can be of course interpreted as the restriction $T Y|_X$. The normal bundle is [[fiber]]wise the [[quotient]] of the fiber of the [[tangent bundle]] of $Y$ by the fiber of the [[tangent bundle]] of $X$: for $x \in X$

$$
  N_i (X)_x =  T_{i(x)}Y/T_x(X)
  \,.
$$

The dual notion is that of [[conormal bundle]]. The notion also makes sense for some other contexts, e.g. for smooth algebraic varieties.

+-- {: .num_remark}
###### Remark

There is always an [[isomorphism]]

$$
  T X \oplus N_i X  \simeq T Y|_X
  \,,
$$

but it is not canonically given, hence in particular cannot it general be chosen [[natural transformation|naturally]]. Except if certain additional structure is given:

If $Y$ is equipped with a [[Riemannian metric]] $g$, then we may identify the normal bundle with the bundle of [[vector]]s that are _orthogonal_  to ("normal to") the vectors in $T X$:

$$
  N_i(X) \simeq (T X)^\perp :=
  \{
    (x,v) \in T Y|_{X} | \forall w \in T_x X :  g(v,w) = 0
  \}
  \,.
$$

=--

Let $M^n$ be a smooth compact $n$-dimensional manifold without boundary, then the question of triviality of the normal bundle for an embedding $M^n\hookrightarrow \mathbf{R}^{n+r}$ for sufficiently large $r$ does not depend on the embedding. For this one uses the fact that any two such embeddings are regularly homotopic (this means the existence of a smooth homotopy $H(x,t)$ which is immersion for every $t \in [0,1]$ and which induces on the level of differentials a homotopy for the tangent bundles) and that any two regular homotopies are themselves homotopic through regular homotopies leaving end points fixed. Then one just uses the homotopy invariance of vector bundles. Thus, if $M^n$ admits an embedding into $\mathbf{R}^{n+r}$ with a trivial normal bundle then one says that $M^n$ has a __stably trivial normal bundle__. In that case, if $M^n_+$ is the union of $M$ with a disjoint base point, then there is a homeomorphism $T (M^n\times \mathbf{R}^{r})\cong \Sigma^r M^n_+$ where $\Sigma^r$ denotes the $r$-fold (reduced) suspension of based spaces $(S^r\times M^n_+)/(S^r\wedge M_+)$.

## Related concepts

* Normal bundle plays a central role for instance in the theory of [[fiber integration]] by means of [[Pontrjagin-Thom collapse map]]s.

* [[tubular neighbourhood]], [[conormal bundle]]

## Literature

* [[Victor Snaith]], _Stable homotopy around the Arf-Kervaire invariant_, Birkhauser 2009

[[!redirects normal bundles]]


normal closure

Given a [[group]] $G$ and a subset $S\subset G$, the **normal closure** of $S$ in $G$ is the smallest [[subgroup]] containing the set of all conjugates of elements in $S$, i.e. the subgroup [[Moore closure|generated by]] the set of all elements of the form $g^{-1} s g$, where $g\in G$ and $s\in S$. The normal closure is clearly a [[normal subgroup]] of $G$.

The normal closure, also called __conjugate closure__ (see [Wikipedia](http://en.wikipedia.org/wiki/Conjugate_closure)), should be distinguished from the [[normalizer]] of $S$ in $G$. In [[combinatorial group theory]] a group $G$ is [[group presentation|presented]] via specifying a set $X$ of generators and a set $R$ of relations. This means that $G=F/N$ where $F$ is the [[free group]] generated by $X$ and $N$ is the normal closure of $R$ in $F$.


normal complex of groups

#Contents#
* table of contents
{:toc}

## Idea

[[chain complex|Chain complexes]] are most often encountered in an Abelian context, but the definition makes sense even when the [[group]]s involved need not be [[abelian group|abelian]]. What does not work well is the formation of the [[homology]] of such a chain complex of groups, except if it is 'normal' in the following sense:

## Definition

A _chain complex_ of groups is a sequence (of any length, finite or infinite) of groups and homomorphisms, for instance,

$$\ldots \to C_n \stackrel{\partial_n}{\longrightarrow} C_{n-1} \stackrel{\partial_{n-1}}{\longrightarrow}C_{n-2}\longrightarrow \ldots ,$$

in which each composite $\partial_{n-1} \circ \partial_n $ is the trivial homomorphism.

The chain complex is __normal__ if each image $\partial_n C_n$ is a [[normal subgroup]] of the next group $C_{n-1}$.

## Example

If $G$ is a [[simplicial group]] then its [[Moore complex]] is a normal complex of groups. See there.

## Remark

There is an obvious generalisation to normal complex of groupoids, provided one keeps to working with groupoids all having the same set of objects and morphisms which are the identity on objects.

Likewise, one can talk about normal complexes in a [[semi-abelian category]].

[[!redirects normal complexes of groups]]


normal eigenvalue

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

## Idea

(...)

## Definition

(...)

## Related concepts

* [[eigenvalue]]

* [[operator spectrum]], [[discrete spectrum]]

## References

See also:

* Wikipedia, *[Normal eigenvalue](https://en.wikipedia.org/wiki/Normal_eigenvalue)*

[[!redirects normal eigenvalues]]


normal field extension

A [[field extension]] $K\subset L$ is said to be **normal** if $L$ is the [[splitting field]] of a family of [[polynomials]] in $K[x]$.


normal form

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In a [[rewriting system]], a term is said to be of **normal form** if it does not admit any further rewrites.

In [[type theory]], a [[canonical form]] is a normal form, but the converse need not hold.

A term is weakly normalizing if there exists at least one particular sequence of rewrites that eventually yields a normal form. A rewrite has weak normalization if every term has this property.

A term is strongly normalizing if every sequence of rewrites eventually terminates with a normal form. If every term has this property, the rewrite system has strong normalization.

In the [[untyped lambda calculus]] the term $(\lambda x . x x x) (\lambda x . x x x)$ is neither strongly nor weakly normalizing.

One method for obtaining normal forms, in the context of [[dependent type theory]], is [[normalisation by evaluation]].

## Related concepts

* [[canonical form]]

[[!redirects normal forms]]


normal framing

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

#Contents#
* table of contents
{:toc}

## Definition

A _normal framing_ is a [[trivial vector bundle|trivialization]] of a [[normal bundle]].

Specifically, if $X$ is a [[smooth manifold]] and $\Sigma \overset{\iota}{\hookrightarrow} X$ is a [[submanifold]], then a normal framing for $\Sigma$ is a [[trivial vector bundle|trivialization]] of the [[normal bundle]] $N_\iota(X)$.

A [[submanifold]] equipped with such normal framing is a _normally framed submanifold_ ([Pontrjagin 55, Sec. 6](#Pontrjagin55) e.g. [Kosinski 93, IX (2.1)](#Kosinski93)). Beware that this is often called just a _framed submanifold_, despite the potential class with "[[framed manifold]]".

## Properties

### Pontryagin's theorem

For $X$ a [[closed manifold|closed]] [[smooth manifold]] of [[dimension]] $D$, the [[Pontryagin theorem]] (e.g. [Kosinski 93, IX.5](#Kosinski93)) identifies the [[set]]

$$
  Cob_{Fr}^{d}(X)
$$

of [[cobordism classes]] of  [[closed manifold|closed]] and [[normally framed submanifolds]] $\Sigma \overset{\iota}{\hookrightarrow} X$ of [[dimension]] $d$ inside $X$ with the [[cohomotopy]] $\pi^{D-d}(X)$ of $X$ in degree $D- d$

$$
  Cob_{Fr}^{d}(X)
    \underoverset{\simeq}{PT}{\longrightarrow}
  \pi^{D-d}(X)
  \,.
$$

(e.g. [Kosinski 93, IX Theorem (5.5)](#Kosinski93))

In particular, by this [[bijection]] the canonical [[group]] [[structure]] on [[cobordism groups]] in sufficiently high [[codimension]] (essentially given by [[disjoint union]] of [[submanifolds]]) this way induces a group structure on the cohomotopy sets in sufficiently high degree.

## Related concepts

* [[framed manifold]]

* [[normal twisted framing]]

## References

* {#Pontrjagin38} [[Lev Pontrjagin]], _[[Classification of continuous maps of a complex into a sphere]]_, Dokl. Akad. Nauk SSSR 19 (1938), 361-363

* {#Pontrjagin55} [[Lev Pontrjagin]], Section 6 of: _[[Smooth manifolds and their applications in homotopy theory]]_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955, AMS Translation Series 2, Vol. 11, 1959 ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont001.pdf))

* {#Kosinski93} [[Antoni Kosinski]], chapter IX of _Differential manifolds_, Academic Press 1993 ([pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf))

[[!redirects normal framings]]

[[!redirects normally framed submanifold]]
[[!redirects normally framed submanifolds]]

[[!redirects framed submanifold]]
[[!redirects framed submanifolds]]


normal modal logic

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

# Uniform and normal logics
* table of contents
{: toc}

A [[modal logic]] is **uniform** if it is closed under the rule of uniform substitution of $\mathcal{L}_\omega(n)$-formulae for propositional variables and is **normal** if it also contains the axiom schemata:

(K)  $\Diamond_i(\psi \vee \chi) \to \Diamond_i(\psi)\vee \Diamond_i(\chi)$

(N) $\neg \Diamond_i(\bot)$

and monotonicity (for each $i$):

if $\psi \to \chi \in \Lambda$ then $\Diamond_i \psi \to \Diamond_i \chi \in \Lambda$.

The smallest normal modal logic with $m$ 'agents' is [[the logic K(m)|K(m)]]. (The diamonds correspond to the $M_i$ of that entry.)

[[!redirects normal modal logic]]
[[!redirects normal modal logics]]

[[!redirects uniform modal logic]]
[[!redirects uniform modal logics]]


normal monomorphism

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

# Normal (mono/epi)morphisms
* table of contents
{: toc}

## Idea

The concept of _normal monomorphism_ generalizes the concept of _[[normal subgroup]] inclusions_.

One often considers [[regular monomorphisms]] and [[regular epimorphisms]].  In the theory of [[abelian categories]], one often equivalently uses _normal_ monomorphisms and epimorphisms.  In general, the concept makes sense in a category with [[zero morphisms]], that is in a [[category enriched]] over [[pointed sets]].

## Definition

A [[monomorphism]] $f\colon A \to B$ is __normal__ if it is the [[kernel]] of some morphism $g\colon B \to C$, that is if it is the [[equalizer]] of $g$ and the [[zero morphisms]] $0_{B,C}\colon B \to C$.

An [[epimorphism]] $f\colon B \to A$ is __normal__ if it is the [[cokernel]] of some morphism $g\colon C \to B$, that is if it is the [[coequalizer]] of $g$ and the [[zero morphism]] $0_{C,B}\colon C \to B$.

Note that a normal monomorphism in $C$ is the same as a normal epimorphism in the [[opposite category]] $C^{op}$, and vice versa.

A [[category]] is __normal__ if every monomorphism is normal; it is __conormal__ if every epimorphism is normal, and it is __binormal__ if it is both normal and conormal.  Note that $C$ is normal if and only if $C^{op}$ is conormal, and vice versa.

## Examples

The [[inclusion function]] of a [[subgroup]] into a [[group]] is normal if and only if the subgroup is [[normal subgroup|normal]]; this is the origin of the terminology for normal monomorphisms.

Every normal monomorphism or epimorphism is clearly regular; the converse holds in any [[Ab-enriched category]] (the equalizer of $f$ and $g$ is the kernel of $f-g$).

Every [[abelian category]] must be binormal; this is one of the axioms in a common definition of abelian category.

## Related concepts

* [[normal]]

## References

Discussion of [[normalizer]] constructions for monomorphisms is in

* {#Gray14} James Richard Andrew Gray, _Normalizers, centralizers and action representability in semiabelian categories_, Applied Categorical Structures 22(5-6), 981–1007, 2014.

[[!redirects normal monomorphisms]]
[[!redirects normal epimorphism]]
[[!redirects normal epimorphisms]]
[[!redirects normal category]]
[[!redirects conormal category]]
[[!redirects binormal category]]
[[!redirects normal categories]]
[[!redirects conormal categories]]
[[!redirects binormal categories]]


normal operator

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

#Contents#
* table of contents
{:toc}

## Definition

An element $a \in A$ in a [[C-star algebra]] is called _normal_ if it commutes with its star-adjoint:

$$
  a \cdot a^*
  \;=\; a^* \cdot a
  \,.
$$

## Properties

The [[spectral theorem]] asserts, roughly, that (bounded) normal operators can be diagonalized.

## Related concepts

* [[positive operator]]

[[!redirects normal operators]]


normal space

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

# Normal spaces
* table of contents
{: toc}

## Idea

A normal space is a space (typically a [[topological space]]) which satisfies one of the stronger [[separation axioms]].

[[!include main separation axioms -- table]]

## Definitions

A [[topological space]] $X$ is __normal__ if it satisfies:

*  $T_4$: for every two [[closed subspace|closed]] [[disjoint subsets]] $A,B \subset X$ there are (optionally [[open subspace|open]]) [[neighborhoods]] $U \supset A$, $V \supset B$ such that their [[intersection]] $U \cap V$ is [[empty set|empty]].

By [[Urysohn's lemma]] this is equivalent to the condition

* for every two [[closed subspace|closed]] [[disjoint subsets]] $A,B \subset X$ there exists an [[Urysohn function]] that separates them.

Often one adds the requirement

*  $T_1$: every [[point]] in $X$ is closed.

(Unlike with [[regular spaces]], $T_0$ is not sufficient here.)

One may also see terminology where a __normal space__ is any space that satisfies $T_4$, while a __$T_4$-space__ must satisfy both $T_4$ and $T_1$.  This has the benefit that a $T_4$-space is always also a $T_3$-space while still having a term available for the weaker notion.  On the other hand, the reverse might make more sense, since you would expect any space that satisfies $T_4$ to be a $T_4$-space; this convention is also seen.

If instead of $T_1$, one requires

*  $R_0$: if $x$ is in the [[topological closure|closure]] of $\{y\}$, then $y$ is in the [[topological closure|closure]] of $\{x\}$,

then the result may be called an __$R_3$-space__.

Any space that satisfies both $T_4$ and $T_1$ must be [[Hausdorff space|Hausdorff]], and every Hausdorff space satisfies $T_1$, so one may call such a space a __normal Hausdorff space__; this terminology should be clear to any reader.

Any space that satisfies both $T_4$ and $R_0$ must be [[regular space|regular]] (in the weaker sense of that term), and every regular space satisfies $R_0$, so one may call such a space a __normal regular space__; however, those who interpret 'normal' to include $T_1$ usually also interpret 'regular' to include $T_1$, so this term can be ambiguous.

Every normal Hausdorff space is an [[Urysohn space]], a fortiori regular and a fortiori Hausdorff.

It can be useful to rephrase $T_4$ in terms of only open sets instead of also closed ones:

*  $T_4$: if $G,H \subset X$ are [[open subspace|open]] and $G \cup H = X$, then there exist open sets $U,V$ such that $U \cup G$ and $V \cup H$ are still $X$ but $U \cap V$ is empty.

This definition is suitable for generalisation to [[locales]] and also for use in [[constructive mathematics]] (where it is not equivalent to the usual one).

To spell out the localic case, a __normal locale__ is a [[frame]] $L$ such that

*  $T_4$: if $G,H \in L$ are opens and $G \vee H = \top$, then there exist opens $U,V$ such that $U \vee G$ and $V \vee H$ are still $\top$ but $U \wedge V = \bot$.

## Examples
 {#Examples}

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

Let $(X,d)$ be a [[metric space]] regarded as a [[topological space]]
via its [[metric topology]]. Then this is a normal Hausdorff space.

=--

+-- {: .proof}
###### Proof

We need to show that given two [[disjoint subsets|disjoint]] [[closed subsets]] $C_1, C_2 \subset X$, there exist [[disjoint subset|disjoint]] [[open neighbourhoods]] $U_{C_1} \supset C_1$
and $U_{C_2} \supset C_2$.

Consider the function

$$
  d(S,-) \colon X \to \mathbb{R}
$$

which computes [[distances]] from a subset $S \subset X$, by forming the [[infimum]] of the distances to all its points:

$$
  d(S,x) \coloneqq inf\left\{ d(s,x) \vert s \in S \right\}
  \,.
$$

If $S$ is closed and $x \notin S$, then $d(S, x) \gt 0$. Then the
[[unions]] of [[open balls]]

$$
  U_{C_1}
    \coloneqq
  \underset{x_1 \in C_1}{\bigcup} B^\circ_{x_1}( \frac1{2}d(C_2,x_1) )
$$

and

$$
  U_{C_2}
    \coloneqq
  \underset{x_2 \in C_2}{\bigcup} B^\circ_{x_2}( \frac1{2}d(C_1,x_2) )
  \,.
$$

have the required properties. For if there exist $x_1 \in C_1, x_2 \in C_2$ and $y \in B^\circ_{x_1}( \frac1{2}d(C_2,x_1) ) \cap B^\circ_{x_2}( \frac1{2}d(C_1,x_2) )$, then

$$d(x_1, x_2) \leq d(x_1, y) + d(y, x_2) \lt \frac1{2} (d(C_2, x_1) + d(C_1, x_2)) \leq \max\{d(C_2, x_1), d(C_1, x_2)\}$$

and if $d(C_1, x_2) \leq d(C_2, x_1)$ say, then $d(x_2, x_1) = d(x_1, x_2) \lt d(C_2, x_1)$, contradicting the definition of $d(C_2, x_1)$.

=--

+-- {: .num_prop}
###### Proposition

Every [[regular space|regular]] [[second countable space]] is normal.

=--

See [[Urysohn metrization theorem]] for details.

+-- {: .num_prop}
###### Proposition
**(Dieudonné's theorem)**

Every [[paracompact Hausdorff space]], in particular every [[compact Hausdorff space]], is normal.

=--

See _[[paracompact Hausdorff spaces are normal]]_ for details.

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

The [[real numbers]] equipped with their [[K-topology]] $\mathbb{R}_K$ are a [[Hausdorff topological space]] which is not a [[regular Hausdorff space]] (hence in particular not a normal Hausdorff space).

=--

+-- {: .proof}
###### Proof

By construction the K-topology is [[finer topology|finer]] than the usual [[Euclidean space|euclidean]] [[metric topology]]. Since the latter is Hausdorff, so is $\mathbb{R}_K$. It remains to see that $\mathbb{R}_K$ contains a point and a [[disjoint subset|disjoint]] closed subset such that they do not have disjoint [[open neighbourhoods]].

But this is the case essentially by construction: Observe that

$$
  \mathbb{R} \backslash K
  \;=\;
  (-\infty,-1/2)
    \cup
  \left(
    (-1,1) \backslash K
  \right)
    \cup
  (1/2, \infty)
$$

is an open subset in $\mathbb{R}_K$, whence

$$
  K = \mathbb{R} \backslash ( \mathbb{R} \backslash K  )
$$

is a [[closed subset]] of $\mathbb{R}_K$.

But every [[open neighbourhood]] of $\{0\}$ contains at least $(-\epsilon, \epsilon) \backslash K$ for some positive real number $\epsilon$. There exists then $n \in \mathbb{N}_{\geq 0}$ with $1/n \lt \epsilon$ and $1/n \in K$. An open neighbourhood of $K$ needs to contain an open interval around $1/n$, and hence will have non-trivial intersection with $(-\epsilon, \epsilon)$. Therefore $\{0\}$ and $K$ may not be separated by disjoint open neighbourhoods, and so $\mathbb{R}_K$ is not normal.

=--

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

If $\omega_1$ is the first un-[[countable ordinal]] with the [[order topology]], and $\widebar{\omega_1}$ its [[one-point compactification]], then $X = \omega_1 \times \widebar{\omega_1}$ with the [[product topology]] is not normal.

Indeed, let $\infty \in \widebar{\omega_1}$ be the unique point in the [[complement]] of $\omega_1 \hookrightarrow \widebar{\omega_1}$; then it may be shown that every open set $U$ in $X$ that includes the closed set $A = \{(x, x): x \neq \infty\}$ in $X$ must somewhere intersect the closed set $\omega_1 \times \{\infty\}$ which is disjoint from $A$. For if that were false, then we could define an increasing sequence $x_n \in \omega_1$ by recursion, letting $x_0 = 0$ and letting $x_{n+1} \in \omega_1$ be the least element that is greater than $x_n$ and such that $(x_n, x_{n+1}) \notin U$. Then, letting $b \in \omega_1$ be the [[supremum]] of this increasing sequence, the sequence $(x_n, x_{n+1})$ converges to $(b, b)$, and yet the neighborhood $U$ of $(b, b)$ contains none of the points of this sequence, which is a contradiction.
=--

This example also shows that general subspaces of normal spaces need not be normal, since $\omega_1 \times \widebar{\omega_1}$ is an open subspace of the compact Hausdorff space $\widebar{\omega_1} \times \widebar{\omega_1}$, which is itself normal.

+-- {: .num_example}
###### Counter-Example
An uncountable product of infinite discrete spaces $X$ is not normal. More generally, a product of $T_1$ spaces $X_i$ uncountably many of which are not [[limit point compact space|limit point compact]] is not normal.

Indeed, by a simple application of Remark \ref{retracts} below and the fact that closed subspaces of normal Hausdorff spaces are normal Hausdorff, it suffices to see that the archetypal example $\mathbb{N}^{\omega_1}$ is not normal. For a readable and not overly long account of this result, see [Dan Ma's blog](https://dantopology.wordpress.com/2009/10/13/the-uncountable-product-of-the-countable-discrete-space-is-not-normal/).
=--

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

A [[Dowker space]] is an example of a normal space which is not [[countably paracompact topological space|countably paracompact]].

=--

## Properties

### Basic properties

+-- {: .num_prop #T4InTermsOfTopologicalClosures}
###### Proposition
**(normality in terms of topological closures)**

A [[topological space]] $(X,\tau)$ is normal Hausdorff, precisely if all points are closed and for all [[closed subsets]] $C \subset X$ with [[open neighbourhood]] $U \supset C$ there exists a smaller open neighbourhood $V \supset C$ whose [[topological closure]] $Cl(V)$ is still contained in $U$:

$$
  C \subset V \subset Cl(V) \subset U
  \,.
$$

=--

+-- {: .proof}
###### Proof

In one direction, assume that $(X,\tau)$ is normal, and consider

$$
  C \subset U
  \,.
$$

It follows that the [[complement]] of the open subset $U$ is closed and disjoint from $C$:

$$
  C \cap X \setminus U = \emptyset
  \,.
$$

Therefore by assumption of normality of $(X,\tau)$, there exist open neighbourhoods  with

$$

  V \supset C
  \,, \phantom{AA}
  W \supset X \setminus U
  \phantom{AA}
    \text{with}
  \phantom{AA}
  V \cap W = \emptyset
  \,.
$$

But this means that

$$
  V \subset X \setminus W
$$

and since the [[complement]] $X \setminus W$ of the open set $W$ is closed, it still contains the closure of $V$, so that we have

$$
  C \subset V \subset Cl(V) \subset X \setminus W \subset U
$$

as required.

In the other direction, assume that for every open neighbourhood $U \supset C$ of a closed subset $C$ there exists a smaller open neighbourhood $V$ with

$$
  C \subset V \subset Cl(V) \subset U
  \,.
$$

Consider disjoint closed subsets

$$
  C_1, C_2 \subset X
  \,,
  \phantom{AAA}
  C_1 \cap C_2 = \emptyset
  \,.
$$

We need to produce disjoint open neighbourhoods for them.

From their disjointness it follows that

$$
  X \setminus C_2 \supset C_1
$$

is an open neighbourhood. Hence by assumption there is an open neighbourhood $V$ with

$$
  C_1 \subset V \subset Cl(V) \subset X \setminus C_2
  \,.
$$

Thus

$$
  V \supset C_1
    \,, \phantom{AAAA}
  X \setminus Cl(V) \supset C_2
$$

are two disjoint open neighbourhoods, as required.

=--

[[!include main separation axioms -- as lifting properties]]

### Tietze extension and lifting property
 {#TietzeEtensionAndLiftingProperty}

The [[Tietze extension theorem]] applies to normal spaces.

In fact the Tietze extension theorem can serve as a basis of a [[category theory|category theoretic]] characterization of normal spaces: a (Hausdorff) space $X$ is normal if and only if every function $f \colon A \to \mathbb{R}$ from a [[closed subspace]] $A \subset X$ admits an [[extension]] $\tilde{f}: X \to \mathbb{R}$, or what is the same, every [[regular monomorphism]] into $X$ in $Haus$ has the [[left lifting property]] with respect to the map $\mathbb{R} \to 1$. (See _[[separation axioms in terms of lifting properties]]_ ([Gavrilovich 14](#Gavrilovich14)) for further categorical characterizations of various topological properties in terms of lifting problems.)

### The category of normal spaces
 {#CategoryOfNormalSpaces}

Although normal (Hausdorff) spaces are "[[nice topological spaces]]" (being for example [[Tychonoff spaces]], by [[Urysohn's lemma]]), the [[category]] of normal topological spaces with [[continuous maps]] between them seems not to be very well-behaved. (Cf. the rule of thumb expressed in [[dichotomy between nice objects and nice categories]].) It admits [[equalizers]] of pairs of maps $f, g: X \rightrightarrows Y$ (computed as in $Top$ or $Haus$; one uses the easy fact that closed subspaces of normal spaces are normal). However it curiously does not have [[products]] -- or at least it is not closed under products in $Top$, as shown by Counter-Example \ref{Morse}. It follows that this category is not a [[reflective subcategory]] of $Top$, as $Haus$ is.

+-- {: .num_remark #retracts}
###### Remark
A small saving grace is that the category of normal spaces is [[Cauchy complete category|Cauchy complete]], in fact is closed under retracts in $Top$. This is so whether or not the Hausdorff condition is included. (If $r: Y \to X$ retracts $i: X \to Y$, then $r$ is a quotient map and $i$ is a subspace inclusion. If $A, B$ are closed and disjoint in $X$, then $r^{-1}(A), r^{-1}(B)$ are closed and disjoint in $Y$. Separate them by disjoint open sets $U \supseteq r^{-1}(A), V \supseteq r^{-1}(B)$ in $Y$; then $i^{-1}(U), i^{-1}(V)$ are disjoint open sets separating $i^{-1} r^{-1}(A) = A, i^{-1} r^{-1}(B) = B$ in $X$.)
=--

More at the page _[[colimits of normal spaces]]_.

## References

The class of normal spaces was introduced by Tietze (1923) and Aleksandrov--Uryson (1924).

* Ryszard Engelking, _General topology_, (Monographie Matematyczne, tom 60) Warszawa 1977; expanded Russian edition Mir 1986.

Discussion of [[separation axioms in terms of lifting properties]] is due to:

* {#Gavrilovich14} [[Misha Gavrilovich]], *Point set topology as diagram chasing computations*, ([arXiv:1408.6710](https://arxiv.org/abs/1408.6710)).

[[!redirects normal space]]
[[!redirects normal spaces]]
[[!redirects Normal space]]
[[!redirects Normal spaces]]

[[!redirects normal topological space]]
[[!redirects normal topological spaces]]

[[!redirects T4]]

[[!redirects T4 space]]
[[!redirects T4 spaces]]
[[!redirects T4-space]]
[[!redirects T4-spaces]]
[[!redirects T-4 space]]
[[!redirects T-4 spaces]]
[[!redirects T-4-space]]
[[!redirects T-4-spaces]]

[[!redirects normal Hausdorff space]]
[[!redirects normal Hausdorff spaces]]
[[!redirects normal Hausdorff topological space]]
[[!redirects normal Hausdorff topological spaces]]

[[!redirects R3 space]]
[[!redirects R3 spaces]]
[[!redirects R3-space]]
[[!redirects R3-spaces]]
[[!redirects R-3 space]]
[[!redirects R-3 spaces]]
[[!redirects R-3-space]]
[[!redirects R-3-spaces]]

[[!redirects normal regular space]]
[[!redirects normal regular spaces]]

[[!redirects normal locale]]
[[!redirects normal locales]]


normal subgroup

+-- {: .rightHandSide}
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###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### Normal subgroups

A [[subgroup]] $N$ of a [[group]] $G$ is **normal** if the [[conjugation action|conjugation]] $n\mapsto g^{-1}n g$ by any element $g\in G$ leaves $N$ [[invariant]], i.e. $g^{-1}N g \coloneqq \{g^{-1}n g\,|\,n\in N\} = N$.

A subgroup $N$ is normal iff the [[partition]] of the group into left *[[cosets]]* of the subgroup $N$, that is the sets $g N = \{ g n\,|\,n\in N\}$, is *stable* in the sense that the left coset $g_1 g_2 N$ of the product $g_1 g_2$ of any two elements $g_1,g_2\in G$ depends only on the coset $g_1 N$, $g_2 N$. Thus there is well defined product on the set of cosets making the set of left cosets $N\backslash G$ a group. By $g N = g N g^{-1}g = N g$ the set of left cosets and the set of right cosets of a normal subgroup coincide; thus the induced group structure on the right coset set $G/N$ is the same and called the [[quotient group]] (see [[quotient object]]).

Thus normal subgroups may also be defined as [[kernels]] $f^{-1}(e)$ of group homomorphisms $G \to H$ ($e \in H$ is the identity), and this is largely the point of normal subgroups: they are equivalent to [[congruence relations]] in the category of groups.

### Normal subobject in a semiabelian category

A normal subgroup is a [[normal subobject]] of a group in the [[Grp|category of groups]]: the more general notion of 'normal subobject' makes sense in [[semiabelian category|semiabelian categories]] and some other setups. If we consider a group as a special case of an $\Omega$-[[Omega-group|group]], then a normal subgroup corresponds to an [[ideal of an Omega-group|ideal]].

### Normal morphisms of $\infty$-groups
 {#NormalMorphismOfInfinityGroups}

The notion of normal subgroups generalizes from groups to [[∞-groups]].

We may take as the characteristic propery of normal subgroup inclusions $K \hookrightarrow G$ that the [[quotient]] $G/K$ inherits a group structure. This quotient may be identified with the [[homotopy fiber]] of the induced morphism of [[delooping]] [[groupoids]] $\mathbf{B}K \to \mathbf{B}G$ (see example \ref{CrossedModulesAreHomotopyNormalGroupMaps} below). The following definition takes this as the defining property of "normality" of morphisms.

+-- {: .num_defn}
###### Definition

Let $\mathbf{H}$ be an [[(∞,1)-topos]] of [[homotopy dimension]] 0 (for instance a [[cohesive (∞,1)-topos]]) and let $K,G$ be [[∞-group]] objects in $\mathbf{H}$.

A [[morphism]] $f : K \to G$ of [[∞-groups]] in $\mathbf{H}$ is **normal** if its [[delooping]] is the [[homotopy fiber]] of a morphism to a [[0-connected]] object, hence if it fits into a [[fiber sequence]] of the form

$$
  \mathbf{B}K \stackrel{\mathbf{B}f}{\to} \mathbf{B}G \to \mathbf{B}(G\sslash K)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

Here the object on the right is any [[0-connected]] [[∞-groupoid]]. By the assumption of [[homotopy dimension]] 0 and by the discussion at [[looping and delooping]] this is necessarily the [[delooping]] of some [[∞-group]], to be denoted $G\sslash K$. By the discussion at [[fiber sequence]] it follows that $G\sslash K \simeq \Omega \mathbf{B}(G \sslash K)$ is the [[homotopy fiber]] of $\mathbf{B}f$, hence that we have a long fiber sequence

$$
 G\sslash K \to \mathbf{B}K \stackrel{\mathbf{B}f}{\to}\mathbf{B}G \to \mathbf{B}(G\sslash K)
  \,.
$$

Therefore equivalently this says that $f : K \to G$ is normal precisely if $\mathbf{B}f : \mathbf{B}K \to \mathbf{B}G$ is a [[principal ∞-bundle]]. The above fiber sequence says that this principal $\infty$-bundle has typical [[fiber]] $G\sslash K$ and is classified by the [[cocycle]] $\mathbf{B}G \to \mathbf{B}(G\sslash K)$.

=--

For the case $\mathbf{H} = $ [[∞Grpd]] -- hence for [[discrete ∞-groups]] -- and with [[∞Grpd]] [[presentable (∞,1)-category|presented]] by the standard [[model structure on topological spaces]], this notion is discussed in ([Prezma](#Prezma)). The further special where $f$ is a morphism of [[discrete group|discrete 1-groups]], such that $G\sslash K$ is a [[2-group]] (example \ref{CrossedModulesAreHomotopyNormalGroupMaps} below) is discussed in ([Farjoun-Segev](#FarjounSegev)).

+-- {: .num_remark}
###### Remark

Such a normal morphism equivalently exhibits an [[∞-group extension]] $G$ of $G \sslash K$ by $K$. See there for more details.

=--

+-- {: .num_remark}
###### Remark

Every ordinary normal subgroup inclusion $K \hookrightarrow G$ is also a normal morphism of ∞-groups, but there are more morphisms of 1-groups that are normal as morphisms of $\infty$-groups. See example \ref{CrossedModulesAreHomotopyNormalGroupMaps} below.

=--

## Properties

* The lattice of normal subgroups of a group $G$ is a [[modular lattice]], because the category of groups is a [[Mal'cev category]] and, as mentioned earlier, normal subgroups are tantamount to congruence relations. (N.B.: it need not be true that the lattice of subgroups is modular: take for example the lattice of subgroups of the [[dihedral group]] of order $8$, which contains a forbidden pentagon.)

### Recognition of homotopy-normal maps
 {#Recognition}

A recognition principle for normality of morphisms of [[∞-groups]] is ([Prezma, theorem 6.2](#Prezma)).

## Examples

### Normal sub-1-groups

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

Every [[subgroup]] of an [[abelian group]] is normal, trivially.

=--

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

For $G$ a group equipped with an [[action]] on another group $N$ by group [[automorphisms]] $\rho : G \to Aut(N)$, the canonical inclusion

$$
  N \hookrightarrow G \ltimes N
$$

exhibits $N$ as a normal subgroup of the [[semidirect product group]] $G \ltimes N$.

=--

+-- {: .num_prop #inverse}
###### Proposition
If $N$ is a normal subgroup of $H$ and $\phi: G \to H$ is a group homomorphism, then the [[inverse image]] $\phi^{-1}(N)$ is normal in $G$ and $\phi$ induces a group homomorphism $G/f^{-1}(N) \to H/N$.
=--

The proof is entirely straightforward and will be omitted.

### Normal sub-2-groups

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

Let $f : K \to G$ be a morphism of [[discrete groups]] (not necessarily a [[monomorphism]]) regarded as a morphisms of [[0-truncated]] [[discrete ∞-groups]]. Then the [[homotopy fiber]] of its [[delooping]] is the [[action groupoid]]

$$
  G\sslash K =
  \left(
    G \times K
    \stackrel{\overset{(-)\cdot f(-)}{\to}}{\underset{p_1}
   {\to}}
    G
  \right)
  \,.
$$

(This follows for instance by computing the [[homotopy pullback]] via the [[factorization lemma]].)

Since $G\sslash K$ is a [[homotopy type|1-type]], this being an [[∞-group]] means that it is a [[2-group]], hence (see the discussion there) that $f : K \to G$ makes a [[crossed module]] of groups.

So [normal morphisms](#NormalMorphismOfInfinityGroups) of [[0-truncated]] [[discrete ∞-groups]] are equivalently morphisms underlying [[crossed modules]] of [[discrete groups]].

=--

This observation apparently goes back to [[Whitehead]].

## References

Normal morphisms of [[discrete ∞-groups]] are discussed in

* [[Matan Prezma]], _Homotopy Normal Maps_ (2010) ([arXiv:1011.4708](http://arxiv.org/abs/1011.4708))
 {#Prezma}

The special case of this for morphisms of 1-groups is discussed in

* E. D. Farjoun and Y. Segev, _Crossed modules as homotopy normal maps_, Topology and its applications 157 pp. 359–368 (2010).
 {#FarjounSegev}

[[!redirects normal subgroups]]

[[!redirects normal morphism of ∞-groups]]
[[!redirects normal morphism of infinity-groups]]

[[!redirects normal morphisms of ∞-groups]]
[[!redirects normal morphisms of infinity-groups]]

[[!redirects normal sub Lie algebra]]
[[!redirects normal sub Lie algebras]]


normal subobject

The notion of a normal subobject is the proper generalization of a [[normal subgroup]] to other [[algebraic categories]]. The notion was found relatively late. In the [[Grp|category of groups]], there are two equivalent descriptions of a normal subgroup: as the [[kernel]] of a homomorphism of groups and as the equivalence class of the unit of some (necessarily unique) [[congruence]].

Given a category $C$ admitting [[finitely complete category|finite limits]], one says that a morphism $f:X\to Y$ in $C$ is __normal__ to the [[internal equivalence relation]] $r: R\hookrightarrow Y\times Y$ if it factors through the monomorphism $r$ (i.e. $\exists \bar{f}$ such that $r\circ\bar{f}=f$ and ...)

A **normal subobject** is a monomorphism which is normal to some equivalence relation.

(to be finished later, need to switch off)

A [[protomodular category]] is defined in such a way that it possesses an intrinsic notion of normal subobject.

See also [[normal monomorphism]]


normal twisted framing

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### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

\begin{defn}\label{TwistedNormalFraming}
Consider a [[differentiable manifold|differentiable]]/[[smooth manifold]] $M^D$ of [[dimension]] $D$ and equipped with a [[real vector bundle]]

\[
  \label{TwistingVectorBundle}
  \mathcal{V} \to M^d
  \;\;\;\;
  \in
  VectorBundles_{\mathbb{R}}(M)
\]

of [[rank]] $n \leq D$, with [[classifying map]] to be denoted $\tau \;\colon\; M \longrightarrow B \mathrm{O}(n)$.

Then a _$\tau$-twisted normal framing_ on a [[submanifold]] $\Sigma^d \hookrightarrow M$ is an [[isomorphism]] of [[real vector bundles]]

$$
  N\Sigma \overset{\simeq}{\longrightarrow} \mathcal{V}_{\vert \Sigma}
$$

between the [[normal bundle]] of the submanifold and the [[pullback bundle]] of $\mathcal{V}$ along its inclusion.
\end{defn}

(e.g. [Cruickshank 99, Def. 6.0.7](#Cruickshank 99) [Cruickshank 03, Def. 5.1](#Cruickshank03))

For this to exist it is necessary that

$$
  n \;=\; D - d
$$

is the [[codimension]] of $\Sigma^d \hookrightarrow M^D$.

\begin{example}
In the special case that $\mathcal{V}$ (eq:TwistingVectorBundle) is the [[trivial bundle|trivial]] real vector bundle $n$, the notion of twisted normal framing (Def. \ref{TwistedNormalFraming}) reduces to that of _[[normal framing]]_.

\end{example}

## Properties

### Twisted Pntrjagin theorem

The [[twisted Pontrjagin theorem]] identifies [[cobordism classes]] of twisted-framed submanifolds in $M^D$ (Def. \ref{TwistedNormalFraming}) with the [[twisted Cohomotopy]] of $M^d$.

## References

* {#Cruickshank99} [[James Cruickshank]], _Twisted  Cobordism and its Relationship to Equivariant Homotopy Theory_, 1999 ([pdf](http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/PQDD_0030/NQ46823.pdf), [[Cruickshank99.pdf:file]])

* {#Cruickshank03} [[James Cruickshank]], _Twisted homotopy theory and the geometric equivariant 1-stem_, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (doi:10.1016/S0166-8641(02)00183-9)

[[!redirects normal twisted framings]]

[[!redirects normal twisted-framing]]
[[!redirects normal twisted-framings]]

[[!redirects normally twisted-framed submanifold]]
[[!redirects normally twisted-framed submanifolds]]


normal variety

#Contents#
* table of contents
{:toc}

## Idea

An [[algebraic variety]] or an algebraic [[scheme]] $X$ is __normal__ if at every point $p$ the [[stalk]] $\mathcal{O}_{X,x}$ is an [[integrally closed domain]]. For every integral scheme $X$ there is a scheme $\mathcal{X}$ and a morphism $\mathcal{X} \to X$ which is universal among all dominant morphisms from normal schemes into $X$; $\mathcal{X}$ (together with the universal morphism) is said to be the __[[normalization]] of__ $X$.

category: algebraic geometry

[[!redirects normal scheme]]


normal vector

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### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given an [[inner product space]] $(V, \langle -,-\rangle)$ then two [[vectors]] $v,w \in W$ are _normal_ to each other if they are _[[orthogonality|orthogonal]]_ to each other, in that their [[inner product]] vanishes:  $\langle v,w\rangle = 0$.

Given an [[embedding of differentiable manifolds]] $i_X \colon X \hookrightarrow Y$ where $Y$carries [[Riemannian geometry]] structure, then the [[normal bundle]] of $i_X$ is the [[vector bundle]] of [[tangent vectors]] in $Y$ that are normal to [[tangent vectors]] of $X$.

## Related entries

* [[orthonormal basis]], [[orthogonal basis]]

  [[Gram-Schmidt process]]

* [[normal bundle]], [[normally framed submanifold]], [[Pontryagin's theorem]]

* [[orthogonality]], [[orthogonal structure]]

[[!redirects normal vectors]]


normalization

In [[mathematics]], the concept of an [[object]] being "normal" -- and of its  "normalization" if not -- appears in many different contexts, with different meanings:

* in [[linear algebra]]:

  _[[orthonormal basis]]_

  (see also _[[normal vector]]_, _[[normal bundle]]_)

* in [[category theory]]:

  _[[normal subobject]]_, _[[normal monomorphism]]_

* in [[group theory]]:

  _[[normal subgroup]]_, _[[normalizer]]_

* in [[topology]]:

  _[[normal topological space]]_

* in [[algebraic geometry]]:

  _[[normal variety]]_

* in [[formal logic]]:

  _[[normal form]]_

* in [[operator algebra]]:

  _[[normal operator]]_

category: disambiguation

[[!redirects normalization]]

[[!redirects normal]]


normalized chain complex > history

The _normalized chain complex_ of a [[simplicial group]] is another term for its [[Moore complex]]. See there for details.


normalizer

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#### Group Theory
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[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given a [[subset]] $S$ of (the [[underlying]] [[set]] of) a [[group]] $G$, its __normalizer__ $N(S) = N_G(S)$ is the [[subgroup]] of $G$ consisting of all elements $g\in G$ such that $g S = S g$, i.e. for each $s\in S$ there is $s'\in S$ such that $g s = s'g$.

Notice the similarity but also the difference to the definition of the _[[centralizer subgroup]]_, for which $s' = s$ in the above.

## Properties

### Normalization and Weyl group
 {#NormalizationAndWeylGroup}

If the subset $S$ is in fact a [[subgroup]] of $G$, then it is a [[normal subgroup]] of the normalizer $N_G(S)$; and $N_G(S)$ is the largest subgroup of $G$ such that $S$ is a normal subgroup of it, whence the terminology _normalizer_.

Indeed, if $S$ is already a normal subgroup of $G$, then its normalizer coincides with the whole of $G$, and only then (e.g. [here](https://proofwiki.org/wiki/Normal_Subgroup_iff_Normalizer_is_Group)).

Hence when $S$ is a group then the [[quotient]]

$$
  W_G S \coloneqq N_G(S)/S
$$

is a [[quotient group]]. This is also called the _[[Weyl group]]_ of $S$ in $G$. (This use of terminology is common in [[equivariant stable homotopy theory]] -- see e.g. [May 96, p. 13](#May96) -- but not otherwise.)

## Examples

### Holomorph

Each group $G$ embeds into the [[symmetric group]] $Sym(G)$ on the [[underlying set]] of $G$ by the left [[regular representation]] $g\mapsto l_g$ where $l_g(h) = g h$. The [[image]] is isomorphic to $G$ (that is, the left regular representation of a [[discrete group]] is [[faithful representation|faithful]]).

The normalizer of the image of $G$ in $Sym(G)$ is called the [[holomorph]]. This solves the elementary problem of embedding a group into a bigger group $K$ in which every [[automorphism]] of $G$ is obtained by restricting (to $G$) an [[inner automorphism]] of $K$ that fixes $G$ as a subset of $K$.

## Generalization

In ([Gray 14](#Gray14)) the concept of the normalizer of a subgroup of a group is generalized to the normalizer of a [[monomorphism]] in any [[pointed category]] in terms of a universal decomposition $U \stackrel{u}{\to} N \stackrel{f}{\to}T$ of a [[monomorphism]] $U \to T$ with $u$ a [[normal monomorphism]].

In ([Bourn-Gray 13](#BournGray13)) the condition that $w$ be a monomorphism is dropped.

## Related concepts

* [[stabilizer subgroup]]

* [[centralizer subgroup]]

## References

* {#Bredon72} [[Glen Bredon]], Section 0.1 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

* {#May96} [[Peter May]], _Equivariant homotopy and cohomology theory_ CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. ([pdf](http://www.math.rochester.edu/u/faculty/doug/otherpapers/alaska1.pdf))

In [[semiabelian categories]]:

* {#Gray14} James Richard Andrew Gray, _Normalizers, centralizers and action representability in semiabelian categories_, Applied Categorical Structures 22(5-6), 981–1007, 2014.

* {#BournGray13} [[Dominique Bourn]], James Richard Andrew Gray, _Normalizers and split extensions_ ([arXiv:1307.4845](http://arxiv.org/abs/1307.4845))

[[!redirects normalizers]]

[[!redirects normalizer subgroup]]
[[!redirects normalizer subgroups]]


Norman Margolus

* [personal page](https://people.csail.mit.edu/nhm/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Norman_Margolus)

## Selected writings

Influential early discussion of ([[controlled quantum gate|controlled]]) [[quantum logic gates]]:

* {#BBCDMSSSW95} [[Adriano Barenco]], [[Charles H. Bennett]], [[Richard Cleve]], [[David P. DiVincenzo]], [[Norman Margolus]], [[Peter W. Shor]], [[Tycho Sleator]], [[John A. Smolin]], [[Harald Weinfurter]], *Elementary gates for quantum computation*, Phys. Rev. A**52** (1995) 3457 [[arXiv:quant-ph/9503016](https://arxiv.org/abs/quant-ph/9503016), [doi:10.1103/PhysRevA.52.3457](https://doi.org/10.1103/PhysRevA.52.3457)]

category: people


Norman Steenrod

Normal Earl Steenrod was an algebraic topologist working at Princeton University.

He got his PhD from Princeton in 1936, advised by [[Solomon Lefschetz]].

* [Wikipedia entry](http://en.wikipedia.org/wiki/Norman_Steenrod)

His PhD students include [[George Whitehead]], [[Edwin Spanier]], [[William Massey]], [[Jose Adem]], [[Franklin Peterson]], [[Emery Thomas]], [[Peter Freyd]], [[Samuel Gitler]], [[Wu-Chung Hsiang]].

## Selected writings

On [[twisted ordinary cohomology]]:

* [[Norman Steenrod]], _Homology With Local Coefficients_, Annals of Mathematics, Second Series, Vol. 44, No. 4 (Oct., 1943), pp. 610-627 ([jstor:1969099](https://www.jstor.org/stable/1969099))

Introducing the *[[Eilenberg-Steenrod axioms]]* for [[ordinary cohomology]]:

* [[Samuel Eilenberg]], [[Norman E. Steenrod]], _Axiomatic Approach to Homology Theory_, Proceedings of the National Academy of Sciences **31** 4 (1945) 117–120  [[doi:10.1073/pnas.31.4.117](http://dx.doi.org/10.1073/pnas.31.4.117)]

On [[algebraic topology]]:

* [[Samuel Eilenberg]], [[Norman Steenrod]], _Foundations of Algebraic Topology_, Princeton University Press 1952 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/eilestee.pdf), [ISBN:9780691653297](https://press.princeton.edu/books/hardcover/9780691653297/foundations-of-algebraic-topology))

On [[extensions]] (such as in the [[Tietze extension theorem]]), [[ordinary cohomology]], [[Eilenberg-MacLane spaces]], [[cohomology operations]]:

* [[Norman Steenrod]], _Cohomology operations, and obstructions to extending continuous functions_  Advances in Math. 8, 371–416. (1972)  (doi:10.1016/0001-8708(72)90004-7, [pdf](http://www.maths.ed.ac.uk/%7Eaar/papers/steen6.pdf))

On [[compactly generated topological spaces]] as a [[convenient category of topological spaces]]:

* [[Norman Steenrod]], _A convenient category of topological spaces_, Michigan Math. J. 14 (1967) 133--152 ([euclid:mmj/1028999711](http://projecteuclid.org/euclid.mmj/1028999711))

## Related $n$Lab entries

* [[cohomology operations]]

* [[Steenrod algebra]]

* [[Steenrod squares]]

category: people

[[!redirects Steenrod]]
[[!redirects N. E. Steenrod]]
[[!redirects Norman E. Steenrod]]
[[!redirects Norman Earl Steenrod]]


Norman Swartz

* [personal page](http://www.sfu.ca/~swartz/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Norman_Swartz)

## Selected writings

On [[modal logic]] and its [[possible worlds semantics]]:

* {#BradleySwartz79} [[Raymond D. Bradley]], [[Norman Swartz]], *Possible Worlds -- An Introduction to Logic and its Philosophy*, Hackett Publishing (1979) [[webpage](http://www.sfu.ca/~swartz/pw/), [pdf](https://www.sfu.ca/~swartz/pw/text/pw_all.pdf)]

category: people