Messier 87

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## Idea

A [[galaxy]].

## References

See also

* Wikipedia, _[Messier 87](https://en.wikipedia.org/wiki/Messier_87)_

Resolved image of the direct vicinity of the [[event horizon]] of the [[black hole]] in the center of the[[galaxy]] [[Messier 87]]:

* {#EHT19} [[Event Horizon Telescope Collaboration]], _First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole_,  The Astrophysical Journal Letters, Volume 875, Number 1, 2019 ([doi:10.3847/2041-8213/ab0ec7](https://iopscience.iop.org/article/10.3847/2041-8213/ab0ec7))


meta topological space > history


metabelian group

# Metabelian groups
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## Idea

A _metabelian_ or _meta-abelian_ group is a [[group]] that is one step _beyond_ (Greek 'μετά') being _[[abelian group|abelian]]_.  The steps here are those in the [[normal series]] of a [[solvable group]].

## Definition

Let $G$ be a [[group]].  Then the following conditions are all equivalent:

*  $G$ satisfies the [[equational law]] that every element of the form $a b a' b' c d c' d' b a b' a' d c d' c'$ is the [[identity element]] (where primes indicate [[inverse element|inverses]]).

*  $G$ satisfies the equational law that $[[a,b],[c,d]]$ is the identity, where brackets indicate [[commutators]].

*  The [[commutator subgroup]] of $G$ is [[abelian group|abelian]].

*  $G$ has an abelian [[subgroup]] which is [[normal subgroup|normal]] and whose [[quotient group|quotient]] is also abelian.

*  $G$ has an abelian quotient group whose [[kernel]] is also abelian.

*  $G$ is [[solvable group|solvable]] with solvability at most $2$.

In this case, we say that $G$ is __meta-abelian__, or __metabelian__ for short.

## Properties

The solvable groups with solvability _exactly_ $2$ are precisely the groups that are metabelian but not [[abelian group|abelian]].  (Then the groups with solvability $1$ are those that are abelian but not trivial, and the [[trivial group]] is [[the]] only group of solvability $0$.)

Being metabelian is [[hereditary property|hereditary]] and [[cohereditary property|cohereditary]].  That is, [[subgroups]] and [[quotient groups]] of metabelian groups are also metabelian.

## Examples

Every group of [[cardinality|order]] less than $24$ is metabelian; the smallest non-metabelian group is the [[symmetric group]] $S_4$.

## References

*  English Wikipedia.  [web](https://en.wikipedia.org/wiki/Metabelian_group).

[[!redirects metabelian group]]
[[!redirects metabelian groups]]
[[!redirects metaabelian group]]
[[!redirects metaabelian groups]]
[[!redirects meta-abelian group]]
[[!redirects meta-abelian groups]]
[[!redirects meta-Abelian group]]
[[!redirects meta-Abelian groups]]


metacategory

In _[[Categories for the Working Mathematician]]_, [[Saunders Mac Lane]] uses the term 'metacategory' to mean any [[model theory|model]] of the [[first-order logic|first-order]] [[theory]] of [[categories]], and reserves the word '[[category]]' for a metacategory whose [[objects]] and [[morphisms]] form [[sets]].  He then assumes the existence of one [[Grothendieck universe]] $U$ and calls sets and categories in $U$ [[small category|small]] and categories not in $U$ [[large category|large]].  Thus, for him, there is:

* the large category [[Cat]] of all small categories, and
* the metacategory [[CAT]] of all (possibly large) categories.

However, this usage has largely fallen out of favor among modern authors.  Most category theorists nowadays are happy to either use 'category' in the more general sense for which Mac Lane used 'metacategory', and/or to assume the existence of enough universes that any (meta)category has sets of objects and morphisms.  For instance, if we assume the existence of two universes $U \in U'$, we may call:

* sets and categories in $U$ 'small',
* those not in $U$ 'large',
* those in $U'$ '[[moderate category|moderate]]',
* and those (still with a set of objects and morphisms but) not in $U'$ 'very large',

then we have:

* the large but moderate category [[Cat]] of all small categories, and
* the very large category [[CAT]] of all moderate categories.

The authors of _[[The Joy of Cats]]_ use the term 'quasicategory' similarly to how Mac Lane uses 'metacategory', but this usage is even more strongly to be avoided, since it conflicts with the more common modern usage (following Joyal) of '[[quasicategory]]' for a particular model for [[(∞,1)-categories]].

[[!redirects metacategory]]
[[!redirects metacategories]]


metacompact space

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## Definition

A [[topological space]] is metacompact if every [[countable set|countable]] [[open cover]] has a point finite open refinement which covers $X$.

## Examples

* Every [[paracompact topological space]] is metacompact.

* Every [[sigma-compact topological space|sigma-compact]] [[Hausdorff space|Hausdorff]] space is metacompact.

## Related concepts and theorems

[[!include topology - countability axioms]]

## References

* [Metacompact](https://topology.jdabbs.com/properties/P000031), $\pi$-Base.

[[!redirects metacompact spaces]]

[[!redirects metacompactness]]


Metadata properties


metal

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#### Solid state physics
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## Idea

(...)

## Related concepts

* [[Landau's Fermi liquid theory]]

* [[semi-metal]]

* [[strange metal]]

## References

See also:

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

[[!redirects metals]]


metalanguage

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#### Foundations
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#### Type theory
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# Contents
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## Idea

In [[formal logic]], a _metalanguage_ is a [[language]] (formal or informal) in which the symbols and rules for manipulating another (formal) language -- the _object language_ -- are themselves formulated.  That is, the metalanguage is the language used when talking *about* the object language.

For instance the symbol $\phi$ may denote a [[proposition]] in a [[deductive system]], but the statement "$\phi$ can be proven" is not itself a proposition in the deductive system, but a statement in the metalanguage, often denoted by a [[sequent]] of the form

$$
  \vdash \phi \, true
$$

and then called a _[[judgement]]_ instead (in [[type theory]] one might also omit the "$true$", see at _[[propositions-as-types]]_ for details on this). Or, more generally, if the truth of $\phi$ depends on the truth of some other proposition $\psi$ then

$$
  \psi \, true \vdash \phi \, true
$$

is the [[hypothetical judgement]] in the metalanguage that there is a [[proof]] of $\phi$ in the [[context]] in which $\psi$ is assumed.

In contrast, the [[implication]] expression $(\psi \to \phi)$ may denote another [[proposition]] in the object language, a [[conditional statement]]. A [[deduction theorem]] connects the two, in that it says that if the [[judgement]]

$$
  \psi \, true \vdash \phi \, true

$$

holds in the metalanguage, then the judgement

$$
  \vdash (\psi \to \phi) \, true
$$

may be [[deduction|deduced]]; the converse is the rule of [[modus ponens]].  (Actually, both the deduction theorem and modus ponens are slightly more general, being relativized to an arbitrary [[context]], but we needn\'t get into that here.)

## Examples

* A system of [[natural deduction]] with its [[type formation]]/[[term introduction]]/[[term elimination]] and [[computation rules]] is the metalanguage for [[type theory]].

## References

Detailed discussion of the difference between judgements in the metalanguage and propositions in the object language is in the foundational lectures

* [[Per Martin-Löf]], _On the meaning of logical constants and the justifications of the logical laws_, leture series in Siena (1983) ([web](http://docenti.lett.unisi.it/files/4/1/1/6/martinlof4.pdf))
 {#Martin-Löf83}

[[!redirects object language]]
[[!redirects object languages]]
[[!redirects object theory]]
[[!redirects object theories]]

[[!redirects metalanguage]]
[[!redirects metalanguages]]
[[!redirects meta-language]]
[[!redirects meta-languages]]


metalinear group

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## Idea

For $n \in \mathbb{N}$ The _metalinear group_ is a [[Lie group]] that is a $\mathbb{Z}_2$-[[group extension]] of the [[general linear group]] $GL(n, \mathbb{R})$.

## Definition

Inside the [[symplectic group]] $Sp(2n, \mathbb{R})$ sits the [[general linear group]]

$$
  Gl(n,\mathbb{R}) \hookrightarrow Sp(2n, \mathbb{R})
$$

as the subgroup that preserves the standard [[Lagrangian submanifold]] $\mathbb{R}^n \hookrightarrow \mathbb{R}^{2n}$. Restriction of the metaplectic [[group extension]] along this inclusion defines the [[metalinear group]] $Ml(n)$

$$
  \array{
    Ml(n, \mathbb{R}) &\hookrightarrow& Mp(2n,\mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    Gl(n,\mathbb{R}) &\hookrightarrow& Sp(2n, \mathbb{R})
  }
  \,.
$$

Hence a [[metaplectic structure]] on a [[symplectic manifold]] induces a [[metalinear structure]] on its [[Lagrangian submanifolds]].

## Related concepts

* [[metaplectic group]], [[spin group]]

* [[metalinear structure]]


metalinear structure

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## Definition

A _metalinear structure_ on a [[smooth manifold]] of [[dimension]] $n$ is a [[lift of the structure group]] of the [[tangent bundle]] along the [[group extension]] $Ml(n) \to GL(n)$ of the [[general linear group]] by the [[metalinear group]].

## Properties

### Obstruction and existence

A metalinear structure on a manifold $Q$ of [[dimension]] $n$ exists precisely if the [[Chern class]] of the [[canonical bundle]] $\wedge^n T^*Q$ is divisible by 2. So a metalinear structure is equivalent to the existence of a [[square root]] [[line bundle]] $\sqrt{\wedge^n T^* Q}$ ( _[[Theta characteristic]]_ ).

This means that for $E \to Q$ any hermitean [[line bundle]], [[sections]] of the tensor product $E \otimes \sqrt{\wedge^n T^* Q}$ have a canonical [[inner product]] (if $Q$ is compact and orientable). This is the use of metalinear structure in [[metaplectic correction]].

### Relation to metaplectic structure

+-- {: .num_theorem}
###### Theorem

Let $(X,\omega)$ be a [[symplectic manifold]] and $L \subset T X$ a subbundle of [[Lagrangian subspaces]] of the [[tangent bundle]]. Then $T X$ admits a [[metaplectic structure]] precisely if $L$ admits a metalinear structure.

=--

([Bates-Weinstein, theorem 7.16](#BatesWeinstein))

## Related concepts

* [[metaplectic structure]], [[metaplectic correction]]

* [[spin structure]]

[[!include square roots of line bundles - table]]

## References

Lecture notes include

* {#BatesWeinstein} Sean Bates, [[Alan Weinstein]], _[[Lectures on the geometry of quantization]]_, ([pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf))

Discussion with an eye towards [[Theta characteristics]] is in

* [[Andrei Tyurin]], _Quantization, classical and quantum field theory and Theta-functions_ ([arXiv:math/0210466v1](http://arxiv.org/abs/math/0210466v1))

[[!redirects metalinear structures]]


Metamath

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#### Constructivism, Realizability, Computability
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## Idea

[Metamath](http://us.metamath.org/) is a [[proof assistant]] for creating databases of formally verified [[proofs]].  The proof language is extremely parsimonious using textual [[substitution]] as its only [[rule of inference]] (augmented by distinct [[variable]] constraints so that unfortunate variable captures can be prohibited).

One disadvantage of this philosophy is that [[definitions]], [[syntax]] and [[axioms]] are all [[axioms]]. In particular, the user is responsible for ensuring that no ambiguities or contradictions are inadvertently introduced.

Metamath proof verifiers can be very small and simple, so many have been implemented in a wide variety of computer languages. Perhaps the most interesting was created by Stephen O'Rear using a language that makes Turing machines optimised for few states. This was used to [reduce the bound of the smallest Busy Beaver Number that ZFC cannot prove to exist](https://www.scottaaronson.com/blog/?p=2725). A side effect was a small turing machine that [halts iff the Riemann Hypothesis is False](https://www.scottaaronson.com/blog/?p=2741) (and gives the smallest counterexample)

## Related entries

[[!include proof assistants and formalization projects -- list]]

## References

Metamath was originally designed by Norman Megill, but many people have contributed over the years.

* [Metamath home page](http://us.metamath.org/)

There are several Metamath databases on the Metamath website.  The most developed covers classical ZFC set theory, but there is a very nice NF set theory database and a couple of other less well developed ones.

See also

* Wikipedia, _[Metamath](https://en.wikipedia.org/wiki/Metamath)_
* Mario Carneiro's paper _[Metamath Zero: The Cartesian Theorem Prover](https://arxiv.org/abs/1910.10703)_ discusses a metamath variant that _bootstraps_ ie proves it's own correctness.


metaphysics

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## Idea

From ([Hegel, Science of Logic, §85](Science+of+Logic#85)):

> Die objektive Logik tritt damit vielmehr an die Stelle der vormaligen Metaphysik, als welche das wissenschaftliche Gebäude über die Welt war, das nur durch Gedanken aufgeführt seyn sollte.—Wenn wir auf die letzte Gestalt der Ausbildung dieser Wissenschaft Rücksicht nehmen, so ist erstens unmittelbar die Ontologie, an deren Stelle die objektive Logik tritt,—der Theil jener Metaphysik, der die Natur des Ens überhaupt erforschen sollte;—das Ens begreift sowohl Seyn als Wesen in sich, für welchen Unterschied unsere Sprache glücklicherweise den verschiedenen Ausdruck gerettet hat.—Alsdann aber begreift die objektive Logik auch die übrige Metaphysik insofern in sich, als diese mit den reinen Denkformen die besondern, zunächst aus der Vorstellung genommenen Substrate, die Seele, die Welt, Gott, zu fassen suchte, und die Bestimmungen des Denkens das Wesentliche der Betrachtungsweise ausmachten.

> The objective logic, then, takes the place rather of the former metaphysics which was intended to be the scientific construction of the world in terms of thoughts alone. If we have regard to the final shape of this science, then it is first and immediately ontology whose place is taken by objective logic — that part of this metaphysics which was supposed to investigate the nature of ens in general; ens comprises both being and essence, a distinction for which the German language has fortunately preserved different terms. But further, objective logic also comprises the rest of metaphysics in so far as this attempted to comprehend with the forms of pure thought particular substrata taken primarily from figurate conception, namely the soul, the world and God; and the determinations of thought constituted what was essential in the mode of consideration.

## Related concepts

* [[ontology]]

* [[logos (in philosophy)]], [[Nous]]

* [[philosophy of physics]]

* [[category of being]]

* [[道德經]]

* [[hermeticism]]

## References

* [[Aristotle]], _[[Metaphysics (Aristotle)|Metaphysics]]_

* [[Georg Hegel]], _[[Science of Logic]]_

* [[Michael Dummett]], _The logical basis of metaphysics_, Harvard University Press ([publisher](http://www.hup.harvard.edu/catalog.php?isbn=9780674537866), [introduction pdf](http://philosophy.fas.nyu.edu/docs/IO/27735/Dummett.pdf))

Discussion from a point of view of [[type theory]] is in

* {#MartinLoef90} [[Per Martin-Löf]], _A path from logic to metaphysics_, talk at _Nuovi problemi della logica e della filosofia della scienza_, Jan 1990 ([pdf](https://github.com/michaelt/martin-lof/raw/master/pdfs/A-path-from-logic-to-metaphysics-1991.pdf))

* [[Tao Te Ching]]

* [[Hegel's "Logic" as Modal Type Theory]]

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


Metaphysics (Aristotle)

* [[Aristotle]]

  _Metaphysics_

  ~ 4th century BC

on [[metaphysics]].

## Some content

* 1011b24: "Of any one subject, one thing must be either asserted or denied"

  ([[principle of excluded middle]], [[classical logic]])

## References

* Wikipedia, _[Metaphysics (Aristotle)](https://en.wikipedia.org/wiki/Metaphysics_%28Aristotle%29)_

category: reference


metaplectic correction (in geometric quantization)

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#Contents#
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## Idea

In the context of _[[geometric quantization]]_ a _metaplectic correction_ is a choice of [[metaplectic structure]] on the given [[symplectic manifold]]. It allows to make the [[space of states (in geometric quantization)|space of states]] into a [[Hilbert space]].

It is called a _correction_ mostly for historical reasons, since it was not included in all constructions from the beginning.

## Properties

### Induced inner product / Hilbert space structure
 {#InducedInnerProduct}

A [[metaplectic structure]] on a [[symplectic manifold]] $(X, \omega)$ induces a [[metalinear structure]] on each [[Lagrangian submanifold]] $Q \hookrightarrow X$ of a given [[foliation]] by Lagrangian submanifolds ([[polarization]]). This allows to form a [[square root]] line bundle $\sqrt{\Lambda^n T^* Q}$ of the [[canonical bundle]] of $Q$ (a "[[Theta characteristic]]", see [below](#RelationToSpinCStructure)) and hence induces an [[inner product]] on [[sections]] of the [[tensor product]] $E|_Q \otimes \sqrt{\Lambda^n T^* Q}$ with the restriction of any [[line bundle]] $E$ on $X$ (a prequantum line bundle, notably).

### Relation to $Spin$-structure and $Spin^c$-structure
 {#RelationToSpinCStructure}

Let $(X,\omega)$ be a [[compact topological space|compact]] [[symplectic manifold]] equipped with a [[Kähler polarization]] $\mathcal{P}$ hence a [[Kähler manifold]] structure $J$. A metaplectic structure is now a choice of [[square root]] $\sqrt{\Omega^{n,0}}$ of the [[canonical line bundle]] $\Omega^{n,0}$ (a [[Theta characteristic]] for the [[complex analytic space]] $X$). This is equivalently a [[spin structure]] on $X$ (see the discussion at _[spin structure -- over Kähler manifolds](spin%20structure#OverAKahlerManifold)_).

Now given a [[prequantum line bundle]] $L_\omega$, in this case the [Dolbault quantization](geometric+quantization#IndexOfDolbeaultDiracOperator) of $L_\omega$ coincides with the [[spin^c quantization]] of the [[spin^c structure]] induced by $J$ and $L_\omega \otimes \sqrt{\Omega^{n,0}}$.

This appears as ([Paradan 09, prop. 2.2](#Paradan09)).

### Relation to geometric quantization

See at

* _[geometric quantization -- Quantum space of states](#geometric+quantization#GeometricQuantizationProper)_.

## Related concepts

* [[metaplectic group]]

[[!include square roots of line bundles - table]]

## References

For general discussion see the references listed at _[[geometric quantization]]_, for instance the introduction in section 7.2 of

* Sean Bates, [[Alan Weinstein]], _Lectures on geometric quantization_ ([pdf](http://math.berkeley.edu/~alanw/GofQ.pdf))

or

* [[Martin Schottenloher]], _Metaplectic reduction_ ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/GQ/12%20Metaplectic%20Correction.1.1.pdf))

* Alexander Cardona, _Geometric and metaplectic quantization_ ([pdf](http://pentagono.uniandes.edu.co/~acardona/GMQ.pdf))

Relation to [[spin^c quantization]] is discussed in

* [[Paul-Emile Paradan]], _Spin-quantization commutes with reduction_, J. Symplectic Geom. Volume 10, Number 3 (2012), 389-422. ([arXiv:0911.1067](http://arxiv.org/abs/0911.1067), [Euclid](http://projecteuclid.org/euclid.jsg/1350392491))
 {#Paradan09}

Discussion with an eye towards [[Theta characteristics]] is in

* [[Andrei Tyurin]], _Quantization, classical and quantum field theory and Theta-functions_ ([arXiv:math/0210466v1](http://arxiv.org/abs/math/0210466v1))

Further references include

* L. Charles, _Semi-classical properties of geometric quantization with metaplectic correction_ ([arXiv:math.SG/0602168](http://arxiv.org/abs/math.SG/0602168))

[[!redirects metaplectic correction]]
[[!redirects metaplectic corrections]]


metaplectic group

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#### Group Theory
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## Definition

### Double cover $Mp$ of $Sp$
 {#DoubleCover}

For $(V,\omega)$ a [[symplectic vector space]], the _metaplectic group_ $Mp(V,\omega)$ is the [[Lie group]] which is [[generalized the|the]] [[universal cover|universal]] [[double cover]] of the [[symplectic group]] $Sp(V,\omega)$.

This has various more explicit presentations. One is by [[quadratic Hamiltonians]]: The metaplectic group is that subgroup of the [[quantomorphism group]] of the [[symplectic manifold]] $(V,\omega)$ whose elements are given by [[paths]] of [[Hamiltonians]] that are [[homogeneously quadratic Hamiltonians]] (due to [Leray 81, section 1.1](#Leray81), see also [Robbin-Salamon 93, sections 9-10](#RobbinSalamon93)). (The more general subgroup given by possibly inhomogeneous [[quadratic Hamiltonians]] this way is the [[extended affine symplectic group]]. The subgroup given by linear Hamiltonians is the [[Heisenberg group]] $Heis(V,\omega)$.)

### Circle extension $Mp^c$ of $Sp$
  {#CircleExtension}

There is also a nontrivial  [[circle group]]-[[group extension|extension]] of the [[symplectic group]], called $Mp^c$. This is the circle extension [[associated bundle|associated]] to the plain metaplectic group $Mp$ [above](#DoubleCover), via the canonical [[action]] of $\mathbb{Z}_2$ on $U(1)$ (by [[complex conjugation]]): ([Forger-Hess 79 (2.4)](#ForgerHess79))

$$
  \begin{aligned}
    Mp^c(V,\omega)
     & \coloneqq
    Mp(V,\omega) \times_{\mathbb{Z}_2} U(1)
    \coloneqq
    ( Mp(V,\omega) \times U(1) )/\mathbb{Z}_2
  \end{aligned}
$$

where the last line denotes the [[quotient group]] by the [[diagonal action]] of $\mathbb{Z}_2$.

(This is in direct [[analogy]] to the group [[Spin^c]] and its relation to [[Spin]].)

Again, this has various more explicit presentations.

The _[[Segal-Shale-Weil representation]]_ is the following. By the [[Stone-von Neumann theorem]] there is an essentially unique [[irreducible representation|irreducible]] [[unitary representation]] $W$ of the [[Heisenberg group]] $Heis(V,\omega)$. This being essentially unique implies that for each element $g\in Sp(V,\omega)$ of the [[symplectic group]], there is a unique [[unitary operator]] $U_g$ such that for all $v\in V$

$$
  W(g(v)) = U_g W(v) U^{-1}_g
  \,.
$$

The group $Mp^c$ is the [[subgroup]] of the [[unitary group]] of all such $U_g$ for $g\in Sp(V,\omega)$. The map $U_g \mapsto g$ exhibits this as a [[group extension]] by the [[circle group]]

$$
  U(1)\longrightarrow Mp^c(V,\omega) \longrightarrow Sp(V,\omega)
  \,.
$$

e.g. ([Robinson-Rawnsley 89, p. 19](#RobinsonRawnsley89), [Dereziński-Gérard 13, def. 10.24](#DerezinskiGérard13))

Alternatively, there is again a characterization by [[quadratic Hamiltonians]] ([Robinson-Rawnsley 89, theorem (2.4)](#RobinsonRawnsley89)

### Circle extension $MU^c$ of $U$
 {#CirclExtensionMUc}

A [[symplectic vector space]] $(V,\omega)$ has a compatible [[complex structure]] $J$. Write

$$
  U(V,J) \hookrightarrow Sp(V,\omega)
$$

for the corresponding [[unitary group]].

+-- {: .num_prop #MUc}
###### Definition

The restriction ([[pullback]]) of $Mp^c$ [above](#CircleExtension) to this subgroup is denoted $MU^c$ in ([Robinson-Rawnsley 89, p. 22](#RobinsonRawnsley89))

$$
  \array{
      U(1) &=& U(1)
      \\
      \downarrow && \downarrow
      \\
      MU^c(V,J) &\hookrightarrow& Mp^c(V,\omega)
      \\
      \downarrow && \downarrow
      \\
     U(V,J) &\hookrightarrow& Sp(V,\omega)
  }
$$

=--

(beware the notational clash with the [[Thom spectrum]] [[MU]], which is unrelated).

## Properties

### Relation to the metalinear group

Inside the [[symplectic group]] $Sp(2n, \mathbb{R})$ in dimension $2n$ sits the [[general linear group]] in dimension $n$

$$
  Gl(n,\mathbb{R}) \hookrightarrow Sp(2n,\mathbb{R})
$$

as the subgroup that preserves the standard [[Lagrangian submanifold]] $\mathbb{R}^n \hookrightarrow \mathbb{R}^{2n}$. Restriction of the metaplectic [[group extension]] along this inclusion defines the [[metalinear group]] $Ml(n)$

$$
  \array{
    Ml(n, \mathbb{R}) &\hookrightarrow& Mp(2n, \mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    Gl(n, \mathbb{R}) &\hookrightarrow& Sp(2n, \mathbb{R})
  }
  \,.
$$

Hence a [[metaplectic structure]] on a [[symplectic manifold]] induces a [[metalinear structure]] on its [[Lagrangian submanifolds]].

### (Non-)Triviality of extensions
 {#NonTrivialityOfExtensions}

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

The extension

$$
  U(1)\to Mp^c(V,\omega) \to Sp(V,\omega)
$$

is nontrivial (does not give a [[split exact sequence]]).

=--

([Robinson-Rawnsley 89, theorem (2.8)](#RobinsonRawnsley89))

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

The extension

$$
  U(1)\to MU^c(V,J) \to U(V,J)
$$

is trivial (does give a [[split exact sequence]]).

=--

([Robinson-Rawnsley 89, theorem (2.9)](#RobinsonRawnsley89))

+-- {: .num_cor #SpStructureLiftsToMpcStructure}
###### Corollary

Every [[symplectic manifold]] admits a [[metaplectic structure]].

=--

([Robinson-Rawnsley 89, theorem (6.2)](#RobinsonRawnsley89))

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

Since the [[unitary group]] $U(V,J)$ is the [[maximal compact subgroup]] of the [[symplectic group]] (see [here](maximal+compact+subgroup#ExamplesForLieGroups)) every $Sp(V,\omega)$-[[principal bundle]] has a [[reduction of the structure group|reduction]] to a $U(V,J)$-principal bundle. By prop. \ref{MUcExtensionSplits} this reduction in turn lifts to a $MU^c(V,J)$-structure. By def. \ref{MUc} this induces an $Mp^c$-structure under inclusion along $MU^c \hookrightarrow Mp^c$.

=--

## Related concepts

* [[metaplectic structure]]

* [[metaplectic correction (in geometric quantization)]]

* [[metaplectic representation]]

## References

Original references include

* [[Andre Weil]], _Sur certains groupes d'opérateurs unitaires_, Acta Math. 111: 143–211. (1964).

* {#KashiwaraVergne78} M. Kashiwara; [[Michèle Vergne]], _On the Segal-Shale-Weil Representations and Harmonic Polynomials_, Inventiones mathematicae (1978) ([EuDML](https://eudml.org/doc/142517), [pdf](http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN356556735_0044&divID=LOG_0008&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN356556735_0044%7C&targetFileName=PPN356556735_0044_LOG_0008.pdf&))

* {#ForgerHess79} [[Michael Forger]], Harald Hess, _Universal metaplectic structures and geometric quantization_, Comm. Math. Phys. Volume 64, Number 3 (1979), 269-278. ([euclid:cmp/1103904723](https://projecteuclid.org/euclid.cmp/1103904723))

* {#Leray81} [[Jean Leray]], _Lagrangian analysis and quantum mechanics_, MIT press 1981 [pdf](http://www.maths.ed.ac.uk/~aar/papers/leraybook.pdf)

Further discussion includes

* {#RobinsonRawnsley89} P. L. Robinson, [[John Rawnsley]], _The metaplectic representation, $Mp^c$-structures and geometric quantization_, 1989 ([doi:10.1090/memo/0410](http://dx.doi.org/10.1090/memo/0410))

* {#RobbinSalamon93} [[Joel Robbin]],  [[Dietmar Salamon]], _Feynman path integrals on phase space and the metaplectic representation_  Math. Z. __221__ (1996), no. 2, 307–335, ([MR98f:58051](http://www.ams.org/mathscinet-getitem?mr=98f:58051), [doi](http://dx.doi.org/10.1007/BF02622118), [[RobbinSalamonMetaplectic.pdf:file]]), also  in [[Dietmar Salamon]] (ed.), _Symplectic Geometry_, LMS Lecture Note series 192 (1993)

* [[John Rawnsley]], _On the universal covering group of the real symplectic group_, Journal of Geometry and Physics 62 (2012) 2044–2058 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/rawnsley.pdf))

* {#CahenGutt13} [[Michel Cahen]], [[Simone Gutt]], _$Spin^c$, $Mp^c$ and Symplectic Dirac Operators_, Geometric Methods in Physics Trends in Mathematics 2013, pp 13-28 ([pdf](http://homepages.ulb.ac.be/~sgutt/spinbiel.pdf))

* {#DerezinskiGérard13} Jan Dereziński, Christian Gérard, _Mathematics of Quantization and Quantum Fields_, Cambridge University Press, 2013

[[!redirects metaplectic groups]]

[[!redirects Mp]]
[[!redirects Mp^c]]


metaplectic quantization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### Geometric quantization
+--{: .hide}
[[!include geometric quantization - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}
## Idea

A variant of [[geometric quantization]]. Given a [[symplectic manifold]] equipped with [[Mp^c structure]], metaplectic quantization forms instead of the geometric [[prequantum line bundle]] the [[symplectic spinor bundle]] and lets a symplectic Dirac operation induced by a choice of symplectic connection act on its sections ([[wavefunctions]]).

## References

The original article is

* K. Habermann,  _An $Mp^c$-spinorial approach to a geometric-type quantization for symplectic manifolds_, in Fritz Gesztesy,  et al. (eds.), _Stochastic processes, physics and geometry: New interplays, vol. II._ CMS Conf. Proc. 29, 259-268 (2000).

Review includes

* Alexander Cardona, _Geometric and Metaplectic quantization_ ([pdf](http://pentagono.uniandes.edu.co/~acardona/GMQ.pdf))


metaplectic representation

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

#Contents#
* table of contents
{:toc}

## Idea

A [[representation]] of the [[metaplectic group]].

The [Segal-Shale-Weil metaplectic representation](#SegalShaleWeilRepresentation) is also called the _[[symplectic spinor]]_ representation.

## Segal-Shale-Weil representations
 {#SegalShaleWeilRepresentation}

The _Segal-Shale-Weil representation_ is the following. By the [[Stone-von Neumann theorem]] there is an essentially unique [[irreducible representation|irreducible]] [[unitary representation]] $W$ of the [[Heisenberg group]] $Heis(V,\omega)$ of a given [[symplectic vector space]].  This being essentially unique implies that for each element $g\in Sp(V,\omega)$ of the [[symplectic group]], there is a unique (up to multiplication by a constant of modulus one) [[unitary operator]] $U_g$ such that for all $v\in V$

$$
  W(g(v)) = U_g W(v) U^{-1}_g
  \,.
$$

The group $Mp^c$ is the [[subgroup]] of the [[unitary group]] of all such $U_g$ for $g\in Sp(V,\omega)$. The map $U_g \mapsto g$ exhibits this as a [[group extension]] by the [[circle group]]

$$
  U(1)\longrightarrow Mp^c(V,\omega) \longrightarrow Sp(V,\omega)
  \,.
$$

## Related concepts

* [[spin representation]]

* [[related Langlands program]]

## References

* {#KashiwaraVergne78} M. Kashiwara; [[Michèle Vergne]], _On the Segal-Shale-Weil Representations and Harmonic Polynomials_, Inventiones mathematicae (1978) ([EuDML](https://eudml.org/doc/142517), [pdf](http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN356556735_0044&divID=LOG_0008&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN356556735_0044%7C&targetFileName=PPN356556735_0044_LOG_0008.pdf&))

* Gérard Lion, [[Michèle Vergne]], _The Weil representation, Maslov index and theta series_, Progress in Math. __6__, Birkhäuser 1980. vi+337 pp.

* [[Joel Robbin]], [[Dietmar Salamon]], _Feynman path integrals and the metaplectic representation_, Math. Z. __221__ (1996), no. 2, 307–-335, ([MR98f:58051](http://www.ams.org/mathscinet-getitem?mr=98f:58051), [doi](http://dx.doi.org/10.1007/BF02622118), [[RobbinSalamonMetaplectic.pdf:file]])

* Maurice de Gosson, _Maslov classes, metaplectic representation and Lagrangian quantization_, Math. Research 95, Akademie-Verlag, Berlin, 1997. 186 pp.
*  José M. Gracia-Bondía, _On the metaplectic representation in quantum field theory_,  Classical and quantum systems (Goslar, 1991), 611–614, World Sci. 1993
* R. Ranga Rao, _The Maslov index on the simply connected covering group and the metaplectic representation_, J. Funct. Anal. __107__ (1992), no. 1, 211–233, [MR93g:22013](http://www.ams.org/mathscinet-getitem?mr=1165870), doi
* G. Burdet, M. Perrin, _Weyl quantization and metaplectic representation_,
Lett. Math. Phys. 2 (1977/78), no. 2, 93–99, [MR473105](http://www.ams.org/mathscinet-getitem?mr=473105), [doi](http://dx.doi.org/10.1007/BF00398573)
* Y. Flicker, [[D. Kazhdan]], G. Savin, _Explicit realization of a metaplectic representation_, J. Analyse Math. __55__ (1990), 17–39, [MR92c:22036](http://www.ams.org/mathscinet-getitem?mr=1094709), [doi](http://dx.doi.org/10.1007/BF02789195)

* [[Gerald B. Folland]], Chapter 4 in: *A course in abstract harmonic analysis*, Studies in Advanced Mathematics. CRC Press (1995) [[pdf](https://sv.20file.org/up1/1415_0.pdf), [gBooks](http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC)]

[[!redirects metaplectic representations]]

[[!redirects Segal-Shale-Weil representation]]


metaplectic structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $(X, \omega)$ a [[symplectic manifold]] a _metaplectic structure_ on $X$ is a [[G-structure]] for $G$ the [[metaplectic group]], hence a [[lift of structure groups]] of the [[tangent bundle]] from the [[symplectic group]] to the [[metaplectic group]] through the [[double cover]] map $Mp(2n, \mathbb{R}) \to Sp(2n, \mathbb{R})$:

$$
  \array{
    && \mathbf{B}Mp(2n, \mathbb{R})
    \\
    & {}^{\mathllap{metaplectic \atop structure}}\nearrow & \downarrow
    \\
    X &\stackrel{T X}{\to}& \mathbf{B} Sp(2n, \mathbb{R})
  }
  \,.
$$

Analogously for the [[Mp^c]]-group one considers _$Mp^c$-structures_.

## Properties

### Relation to metalinear structure

+-- {: .num_theorem}
###### Theorem

Let $(X,\omega)$ be a [[symplectic manifold]] and $L \subset T X$ a subbundle of [[Lagrangian subspaces]] of the [[tangent bundle]]. Then $T X$ admits a metaplectic structure precisely if $L$ admits a [[metalinear structure]].

=--

([Bates-Weinstein, theorem 7.16](#BatesWeinstein))

### Existence of $Mp^c$-structures

+-- {: .num_theorem}
###### Theorem

Every [[symplectic group|Sp]]-[[principal bundle]] has a [[lift of structure group|lift]] to an [[Mp^c]]-[[principal bundle]].

=--

([Robinson-Rawnsley 89, theorem (6.2)](#RobinsonRawnsley89))

For more details, see at [metaplectic group -- (Non-)Triviality of Extensions](metaplectic+group#NonTrivialityOfExtensions).

## Related concepts

* [[metaplectic group]]

* [[spin structure]]

* [[metaplectic correction (in geometric quantization)]]

## References

* {#ForgerHess79} [[Michael Forger]], Harald Hess, _Universal metaplectic structures and geometric quantization_, Comm. Math. Phys. Volume 64, Number 3 (1979), 269-278. ([EUCLID](https://projecteuclid.org/euclid.cmp/1103904723))

* R. J. Plymen, _The Weyl bundle_, Journal of Functional Analysis 49, 186-197 (1982) ([journal](http://www.sciencedirect.com/science/article/pii/0022123682900799))

* {#RobinsonRawnsley89} P. L. Robinson, [[John Rawnsley]], _The metaplectic representation, $Mp^c$-structures and geometric quantization_, 1989

* Michel Cahen, [[Simone Gutt]], [[John Rawnsley]], _Symplectic Dirac Operators and $Mp^c$-structures_ ([arXiv:1106.0588](http://arxiv.org/abs/1106.0588))

* {#BatesWeinstein} Sean Bates, [[Alan Weinstein]], _[[Lectures on the geometry of quantization]]_, ([pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf))

[[!redirects metaplectic structures]]

[[!redirects Mp^c structure]]
[[!redirects Mp^c structures]]
[[!redirects Mp^c-structure]]
[[!redirects Mp^c-structures]]


meter

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

#Concepts#
* table of concepts
{:toc}

## Idea

The _meter_  or _metre_ (in European spelling) is a [[physical unit]] of [[length]]. It is defined by taking the fixed numerical value of the [[speed of light]] in [[vacuum]] $c$ to be $299792458$ when expressed in the [[physical unit|unit]] $m \cdot s^{-1}$ (meters per second), where the [[second]] is previously defined.

## Related concepts

* [[micrometer]]

* [[nanometer]]

* [[femtometer]]

* [[Planck length]]

## References

See also

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

[[!redirects meters]]

[[!redirects metre]]
[[!redirects metres]]

[[!redirects centimeter]]
[[!redirects centimeters]]
[[!redirects centimetre]]
[[!redirects centimetres]]

[[!redirects cm]]


Methods of homological algebra

This page has material related to the book

* [[Sergei Gelfand]], [[Yuri Manin]]:

  _Methods of homological algebra_,

  transl. from the 1988 Russian (Nauka Publ.) original,

  Springer 1996. xviii+372 pp. 2nd corrected ed. 2002

  [doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5)

on [[homological algebra]] and  [[homotopical algebra]] (including [[triangulated categories]], [[t-structures]] and also [[rational homotopy theory]] via [[model structure on dgc-algebras]]).

\linebreak

Related references:

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))

Review references

* {#Hess06} [[Kathryn Hess]], p. 6 of: _Rational homotopy theory: a brief introduction_, contribution to _[Summer School on Interactions between Homotopy Theory and Algebra](https://jdc.math.uwo.ca/summerschool/)_, University of Chicago, July 26-August 6, 2004, Chicago ([arXiv:math.AT/0604626](http://arxiv.org/abs/math.AT/0604626)), chapter in Luchezar Lavramov, [[Dan Christensen]], [[William Dwyer]], [[Michael Mandell]], [[Brooke Shipley]] (eds.), _Interactions between Homotopy Theory and Algebra_, Contemporary Mathematics 436, AMS 2007 ([doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436))

* {#Moerman15} [[Joshua Moerman]], Section II of: _Rational Homotopy Theory_, 2015 ([pdf](https://www.ru.nl/publish/pages/813282/rational_homotopy_theory.pdf), [[MoermanRationalHomotopyTheory.pdf:file]])

category: reference

[[!redirects Methods of Homological Algebra]]


Methods of Modern Mathematical Physics

This page compiles material related to:

\linebreak

* [[Michael Reed]] and [[Barry Simon]]:

  **Methods of Modern Mathematical Physics**

  Academic Press (1978)

on [[mathematical physics]].

  * Volume I: *Functional Analysis*

    [ISBN:9780125850506](https://www.elsevier.com/books/i-functional-analysis/reed/978-0-08-057048-8), [pdf](http://www.astrosen.unam.mx/~aceves/Metodos/ebooks/reed_simon1.pdf)

    on [[functional analysis]]

  * Volume II: *Fourier Analysis, Self-Adjointness*

    [ISBN:9780125850025](https://www.elsevier.com/books/ii-fourier-analysis-self-adjointness/reed/978-0-08-092537-0), [pdf](http://www.astrosen.unam.mx/~aceves/Metodos/ebooks/reed_simon2.pdf)

    on [[Fourier analysis]]

  * Volume III: *Scattering Theory*

    [ISBN:9780080925387](https://www.elsevier.com/books/iii-scattering-theory/reed/978-0-08-092538-7)

    on [[scattering theory]]

  * Volume IV: *Analysis of Operators*

    [ISBN:9780080570457](https://www.elsevier.com/books/iv-analysis-of-operators/reed/978-0-08-057045-7)

    on [[spectral theory]]

    * Section XIII.16: *Schrödinger operators with periodic potentials*

      on [[Bloch-Floquet theory]] and [[electron band theory]]

category: reference


metric abstract elementary class

A version of the concept of [[abstract elementary class]] involving distance estimates.

* Åsa Hirvonen, _Homogenous model theory of metric structures_, thesis, intro part [pdf](http://helda.helsinki.fi/bitstream/handle/10138/21263/homogene.pdf?sequence=1)
* Åsa Hirvonen, Tapani Hyttinen, _Metric abstract elementary classes with perturbations_, Fund. Math. (2012) 217:123-170 [preprint pdf](http://wiki.helsinki.fi/download/attachments/33885362/Preprint517.pdf?version=1&modificationDate=1306925208551)
* PH Zambrano Ramírez, _Around superstability in metric abstract elementary classes_, [pdf](http://www.bdigital.unal.edu.co/6452/1/830044.2011.pdf)
* Monica VanDieren, _Categoricity in abstract elementary classes with no maximal models_, [arxiv/math.LO/0510579](http://arxiv.org/abs/math/0510579), Annals of Pure and Applied Logic
* Will Boney, _A presentation theorem for continuous logic and metric abstract elementary classes_, [pdf](http://homepages.math.uic.edu/~wboney/BoneyTdense.pdf)

category: model theory


metric cone over complex projective 3-space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[metric cone]] over [[complex projective 3-space]] carries the [[mathematical structure|structure]] of  a [[G2-manifold]] whose [[Riemannian metric]] is [[invariant]] under the canonical [[Sp(2)]] [[action]] by left-[[matrix multiplication]] on homomogeneous coordinates in $\mathbb{H}^2 \simeq_{\mathbb{R}} \mathbb{C}^4 \to \mathbb{C}P^3$ ([Bryant-Salamon 89](#BryantSalamon89), see also [Acharya-Bryant-Salamon 20](#AcharyaBryantSalamon20)).

## Related concepts

* [[twistor fibration]]

## References

* {#BryantSalamon89} [[Robert Bryant]], [[Simon Salamon]], _On the construction of some complete metrics with exceptional holonomy_, Duke Math. J. Volume 58, Number 3 (1989), 829-850 ([euclid:euclid.dmj/1077307681](https://projecteuclid.org/euclid.dmj/1077307681))

* {#AcharyaBryantSalamon20} [[Bobby Acharya]], [[Robert Bryant]], [[Simon Salamon]], _A circle quotient of a $G_2$ cone_, Differential Geometry and its Applications Volume 73, December 2020, 101681  ([arXiv:1910.09518](https://arxiv.org/abs/1910.09518), [doi:10.1016/j.difgeo.2020.101681](https://doi.org/10.1016/j.difgeo.2020.101681))

As a [[KK-compactification|compactification]]-space for [[M-theory on G2-manifolds]]:

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]], _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

[[!redirects metric cone on complex projective 3-space]]


metric jet

# Metric jets
* table of contents
{: toc}

## Idea

The notions of **metric tangency** and **metric jet** are generalizations of notions from [[differential calculus]] such as [[tangent vectors]] and [[jet spaces]] to the setting of arbitrary [[metric spaces]].

## Definition

Let $M$ and $M'$ be [[metric spaces]], $f,g:M\to M'$ two maps, and $a\in M$.

+-- {: .num_defn #Tangency}
###### Definition
We say that $f$ and $g$ are **tangent at $a$** if $f(a)=g(a)$ and the function $C_a:M\to \mathbb{R}_+$ defined by
$$
C_a(a) = 0 \qquad
C_a(x) = \frac{d(f(x),g(x))}{d(x,a)} \forall x\neq a
$$
is continuous at $x=a$.
=--

Now let $a'\in M'$ be another point.

+-- {: .num_defn #Jets}
###### Definition
The set of **jets from $(M,a)$ to $(M',a')$** is the [[quotient set]] of the set of maps $f:M\to M'$ which are [[locally Lipschitz map|locally Lipschitz]] at $a$ and satisfy $f(a)=a'$ by the [[equivalence relation]] of tangency at $a$.
=--

## References

* [[Elisabeth Burroni]] and [[Jacques Penon]], "A metric tangential calculus", [TAC](http://www.tac.mta.ca/tac/volumes/23/10/23-10abs.html).

[[!redirects metric jet]]
[[!redirects metric jets]]
[[!redirects metric tangency]]


metric Lie algebra

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

#Contents#
* table of contents
{:toc}

## Defintion
 {#Definition}

A _metric Lie algebra_ or _quadratic Lie algebra_ over some [[ground field]] $\mathbb{F}$ is

1. _Lie algebra_ $\mathfrak{g}$

1. a [[bilinear form]] $\mathfrak{g} \otimes \mathfrak{g} \overset{g}{\longrightarrow} \mathbb{F}$

such that $g$

1. is [[symmetric bilinear form|symmetric]] and [[invariant]] under the [[adjoint action]], hence is an [[invariant polynomial]] on $\mathfrak{g}$;

1. is non-degenerate as a [[bilinear form]], in that its [[tensor product of vector spaces|tensor product]]-[[adjunct]] $\mathfrak{g} \to \mathfrak{g}^\ast$ is a [[linear isomorphism]].

(But $g$ may be [[indefinite bilinear form|indefinite]].)

The following table shows the data in a metric Lie representation equivalently

1. in [[category theory]]-notation;

1. in [[Penrose notation]] ([[string diagrams]]);

1. in index notation:

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

## Examples

* Every [[semisimple Lie algebra]] is a metric Lie algebra via its [[Killing form]].

## Applications

### Lie algebra weight systems on chord diagrams

"Most" [[weight systems]] on [[chord diagrams]] come from [[metric Lie representations]] over [[metric Lie algebras]]: these are the _[[Lie algebra weight systems]]_.

## Related concepts

* [[metric Lie representation]]

* [[Lie algebra weight system]]

* [[tensor network state]]

## References

### General

On the [[Faulkner construction]]:

* {#Faulkner73} [[John Faulkner]], _On the geometry of inner ideals_, Journal of Algebra, Volume 26, Issue 1, July 1973, Pages 1-9 (doi:10.1016/0021-8693(73)90032-X)

See also

* Wikipedia, _[Quadratic Lie algebra](https://en.wikipedia.org/wiki/Quadratic_Lie_algebra)_

### Relation to M2-brane 3-Lie algebras

The full generalized axioms on the M2-brane 3-algebra and first insights into their relation to [[Lie algebra representations]] of [[metric Lie algebras]] is due to

* {#CherkisSaemann08} [[Sergey Cherkis]], [[Christian Saemann]], _Multiple M2-branes and Generalized 3-Lie algebras_, Phys. Rev. D78:066019, 2008 ([arXiv:0807.0808](https://arxiv.org/abs/0807.0808))

The full identification of M2-brane 3-algebras with dualizable [[Lie algebra representations]] over [[metric Lie algebras]] is due to

* {#dMFFMER08} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], [[Patricia Ritter]], _On the Lie-algebraic origin of metric 3-algebras_, Commun. Math. Phys. 290:871-902, 2009 ([arXiv:0809.1086](http://arxiv.org/abs/0809.1086))

reviewed in

* [[José Figueroa-O'Farrill]], slide 145 onwards in: _Triple systems and Lie superalgebras_ in _M2-branes, ADE and Lie superalgebras_, talk at IPMU 2009 ([pdf](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/Hongo.pdf), [[FigueroaOFarrillM2Branes02.pdf:file]])

further explored in

* [[José Figueroa-O'Farrill]], _Simplicity in the Faulkner construction_,  Journal of Physics A: Mathematical and Theoretical, Volume 42, Number 44 ([arXiv:0905.4900](https://arxiv.org/abs/0905.4900))

and putting to use the [[Faulkner construction]] that was previously introduced in ([Faulkner 73](#Faulkner73))

See also:

* Sam Palmer, [[Christian Saemann]], section 2 of _M-brane Models from Non-Abelian Gerbes_, JHEP 1207:010, 2012 ([arXiv:1203.5757](http://arxiv.org/abs/1203.5757))

* {#SaemannRitter13} [[Patricia Ritter]], [[Christian Saemann]], section 2.5 of _Lie 2-algebra models_, JHEP 04 (2014) 066 ([arXiv:1308.4892](http://arxiv.org/abs/1308.4892))

* {#Saemann16} [[Christian Saemann]], appendix A of _Lectures on Higher Structures in M-Theory_ ([arXiv:1609.09815](https://arxiv.org/abs/1609.09815))

[[!redirects metric Lie algebras]]

[[!redirects quadratic Lie algebra]]
[[!redirects quadratic Lie algebras]]


metric Lie representation

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[Lie algebra]] $(\mathfrak{g}, [-,-])$ and a [[Lie algebra representation]] $\mathfrak{g} \otimes V \overset{\rho}{\to} V$, and given non-degenerate [[inner products]], hence "metrics", both on $\mathfrak{g}$ and on $V$, one may ask that all [[mathematical structure|structure]] is compatible with these metrics. For the Lie algebra this means to have a _[[metric Lie algebra]]_ and for the representation this means to have an [[orthogonal Lie algebra|orthogonal]] representation. Hence together this is an  _orthogonal representation of a metric Lie algebra_, or _metric Lie representation_, for short.

## Definition
 {#Definition}

The following table shows the data in a metric Lie representation equivalently

1. in [[category theory]]-notation;

1. in [[Penrose notation]] ([[string diagrams]]);

1. in index notation:

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

## Properties

### Structures induced from metric Lie representations

The following [[mathematical structures]] are induced from the data of metric Lie representations:

* [[Lie algebra weight systems]] on ([[horizontal chord diagram|horizontal]]) [[chord diagrams]];

* [[M2-brane 3-algebras]] (the [[Faulkner construction]]);

* ...

See there for more.

## Related concepts

* [[metric Lie algebra]]

* [[Lie algebra weight system]]

* [[M2-brane 3-algebra]]

## References

On the [[Faulkner construction]] (which became known as the [[M2-brane 3-algebra]] constructed from a [[metric Lie representation]]):

* {#Faulkner73} [[John Faulkner]], _On the geometry of inner ideals_, Journal of Algebra, Volume 26, Issue 1, July 1973, Pages 1-9 (doi:10.1016/0021-8693(73)90032-X)

[[!redirects metric Lie representations]]

[[!redirects orthogonal representation of a metric Lie algebra]]
[[!redirects orthogonal representations of a metric Lie algebra]]
[[!redirects orthogonal representations of metric Lie algebras]]


metric locale

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## Definition

A metric locale is a [[premetric locale]] such that for all $U:\mathcal{O}(x)$,

$$U =_{\mathcal{O}(x)} \bigvee_{V:\sum_{V':\mathcal{O}(x)} V' \lhd U} V$$

where

$$\sum_{V':\mathcal{O}(x)} V' \lhd U$$

is the [[subtype]] of all $V':\mathcal{O}(x)$ such that $V' \lhd U$.

## See also

* [[locale]]

* [[premetric locale]]

* [[metric space]]

## References

* [[Simon Henry]], Localic Metric spaces and the localic Gelfand duality ([arXiv:1411.0898v1](https://arxiv.org/abs/1411.0898v1))

* Graham Manuell, *Uniform locales and their constructive aspects*, ([arXiv:2106.00678](https://arxiv.org/abs/2106.00678))

[[!redirects metric locale]]
[[!redirects metric locales]]


metric space

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## Idea

A **metric space** is a [[set]] which comes equipped with a [[function]] which measures _distance_ between points, called a _metric_. The metric may be used to generate a [[topology]] on the set, the _[[metric topology]]_, and a [[topological space]] whose topology comes from some metric is said to be _[[metrizable]]_.

## Definitions

Traditionally, a __metric space__ is defined to be a [[set]] $X$ equipped with a __distance__ [[function]]
$$ d\colon X \times X \to [0, \infty) $$
(valued in [[nonnegative number|non-negative]] [[real numbers]]) satisfying the following axioms:

* [[triangle inequality|Triangle inequality]]: $d(x, y) + d(y, z) \geq d(x, z)$;

* Point inequality: $0 \geq d(x, x)$ (so $0 = d(x,x)$);

* Separation: $x = y$ if $d(x, y) = 0$ (so $x = y$ iff $d(x,y) = 0$);

* Symmetry: $d(x, y) = d(y, x)$.

Given a metric space $(X, d)$ and a point $x \in X$, the __[[open ball]]__ centered at $x$ of radius $r$ is
$$
  B_r(x) \coloneqq \{y \in X : d(x, y) \lt r\}
$$
and it may be shown that the open balls form a [[basis for a topology|basis]] for a [[topological structure|topology]] on $X$, the _[[metric topology]]_. In fact, metric spaces are examples of [[uniform spaces]], and much of the general theory of metric spaces, including for example the notion of [[complete space|completion]] of a metric space, can be extrapolated to uniform spaces and even [[Cauchy spaces]].

### As a Richman premetric space

In [[constructive mathematics]], a **metric space** is a [[Richman premetric space]] $(X, \sim)$ such that for all non-negative rational numbers $q \in \mathbb{Q}_{\geq 0}$, and $r \in \mathbb{Q}_{\geq 0}$, if $q \lt r$, than for every $x \in X$ and $y \in X$, either $x \sim_r y$ or $\neg(x \sim_q y)$. Classically, every Richman premetric space is a metric space, but constructively, Richman premetric spaces are only "metric spaces" that are valued in the lower Dedekind real numbers.

### Metrizable spaces

A **[[metrizable topological space]]** is a [[topological space]] $X$ which admits a metric such that the  [[metric topology]] agrees with the topology on $X$. In general, many different metrics (even ones giving different [[uniform space|uniform structures]]) may give rise to the same topology; nevertheless, metrizability is manifestly a topological notion.

Metrizable spaces enjoy a number of separation properties: they are [[Hausdorff space|Hausdorff]], [[regular space|regular]], and even [[normal space|normal]]. They are also [[paracompact space|paracompact]].

Metrizable spaces are closed under topological [[coproducts]] and of course [[subspaces]] (and therefore [[equalizers]]); they are closed under [[countable family|countable]] [[products]] but not general products (for instance, a product of uncountably many copies of the [[real line]] $\mathbb{R}$ is not a normal space).

Fundamental early work in point-set topology established a number of metrization theorems, i.e., theorems which give sufficient conditions for a space to be metrizable. One of the more useful theorems is due to [[Urysohn]]:

\begin{theorem}\label{UrysohnMetrizationTheorem}
**([[Urysohn metrization theorem]])**
\linebreak
A [[separation axiom|regular, Hausdorff]] [[second-countable space]] is metrizable.
\end{theorem}

\begin{example}
  A [[compact Hausdorff space]] that is second-countable is metrizable.
\end{example}

\begin{remark}
**([[counter-example]])**
\linebreak
A [[compact Hausdorff space]] that is merely [[separable space|separable]] need not be metrizable.

One [[counter-example]] is the [[Stone-Cech compactification]] $\beta(\mathbb{N})$, in which the [[discrete topological space|discrete space]] $\mathbb{N}$ of [[natural numbers]] is [[dense subspace|dense]], whle no non-principal [[ultrafilter]] $U$ is the [[limit of a sequence]] $X = \{x_n\} \subseteq \mathbb{N}$ of principal ultrafilters.

Namely: Supposing it were, choose complementary subsets $A, B \subseteq \mathbb{N}$ such that $A \cap X$ and $B \cap X$ are infinite. By the decomposition $\beta(\mathbb{N}) = \beta(A + B) = \beta(A) + \beta(B)$, we have $U$ belonging to exactly one of $\beta(A), \beta(B)$, say $\beta(B)$. But since the subsequence $A \cap X$ converges to $U$, the basic open neighborhood $\beta(B) = \{V \in \beta(\mathbb{N}): B \in V\}$ of $U$ intersects $A \cap X$ non-trivially, i.e., $B$ is contained in some principal ultrafilter $prin(a)$ with $a \in A \cap X$. Then $a$ lies in disjoint sets $B$ and $A \cap X$, contradiction.
\end{remark}

### Variations

If we allow $d$ to take values in $[0,\infty]$ (the nonnegative [[lower reals]]) instead of just in $[0,\infty)$, then we get __extended metric spaces__.  If we drop separation, then we get __pseudometric spaces__.  If we drop the symmetry condition, then we get __quasimetric spaces__.  Thus the most general notion is that of an extended quasipseudometric space, which are also called __Lawvere metric spaces__ for the reasons below.

{#LawvereMetricSpacesDefinition}On the other hand, if we strengthen the [[triangle inequality]] to
$$ max(d(x,y), d(y,z)) \geq d(x,z) ,$$
then we get __ultrametric spaces__, a more restricted concept.  (This include for example $p$-[[adic completion]]s of [[number fields]].)  Extended quasipseudoultrametric spaces can also be called __Lawvere ultrametric spaces__.

### Metrics valued in another Archimedean integral domain

In some cases, we might want our [[codomain]] of our metric to take values in values other than the real numbers $\mathbb{R}$. This is already important in the definition of the real numbers as [[modulated Cauchy real numbers]], where the metric used is valued in the [[non-negative number|non-negative]] [[rational numbers]] $\mathbb{Q}_{\geq 0}$.

This becomes even more essential in [[constructive mathematics]]: while in [[classical mathematics]], the [[modulated Cauchy real numbers]] are equivalent to the [[Dedekind real numbers]], in [[constructive mathematics]], those definitions are inequivalent to the [[Dedekind real numbers]], and so there are multiple sets of real numbers in which a metric could be valued in. Something similar happens in [[predicative mathematics|predicative]] [[constructive mathematics]] with the [[axiom of fullness]]: the Dedekind real numbers could be defined as [[multivalued Cauchy real numbers]], which is unique up to field isomorphism.

In [[predicative mathematics|predicative]] [[constructive mathematics]] without the [[axiom of fullness]], the issue becomes even worse: there is no longer one set of Dedekind real numbers. In some [[foundations of mathematics|foundations]], there is a whole hierarchy of Dedekind real numbers, one set for every [[universe]] in the foundations. In other foundations, there isn't even a single set of Dedekind real numbers. As a result, one cannot resort to merely using the [[Dedekind real numbers]] for defining the metric as in impredicative mathematics, one has to define metrics and metric spaces more generally.

Thus, given an [[Archimedean integral domain]] $R$, an **$R$-metric** on a set $X$ is a function $d:X \times X \to R_{\geq 0}$, where $R_{\geq 0}$ is the set of non-negative elements in $R$, such that

* [[triangle inequality|Triangle inequality]]: $d(x, y) + d(y, z) \geq d(x, z)$;

* Point inequality: $0 \geq d(x, x)$ (so $0 = d(x,x)$);

* Separation: $x = y$ if $d(x, y) = 0$ (so $x = y$ iff $d(x,y) = 0$);

* Symmetry: $d(x, y) = d(y, x)$.

This is equivalently an $(R_{\geq 0}, \geq, +, 0)$-[[enriched set]].

Given any [[Archimedean integral domain]] $R$ and an Archimedean integral subdomain $S \subseteq R$, then $S$ is an $(R_{\geq 0}, \geq, +, 0)$-enriched set.

### In homotopy type theory

In [[homotopy type theory]], metric and pseudometric spaces could be defined as any [[type]], not just 0-truncated types. Let $\mathbb{R}$ be some suitable notion of real number, and
$$\mathbb{R}_{\geq 0} \coloneqq \sum_{a:\mathbb{R}} a \geq 0$$
be the non-negative real numbers.

\begin{definition}
A **pseudometric space** is a type $X$ with a function $d:X \times X \to \mathbb{R}_{\geq 0}$, which comes with  witnesses for symmetry, the triangle identity, and the identity-to-zero-distance condition:
$$\mathrm{sym}(x, y):d(x, y) = d(y, x)$$
$$\mathrm{triangle}(x, y, z):d(x, z) \leq d(x, y) + d(y, z)$$
$$\mathrm{idtozerodis}(x, y):(x = y) \to (d(x, y) = 0)$$
\end{definition}

The last axiom is equivalent to the point inequality axiom
$$\mathrm{ptineq}(x):d(x, x) = 0$$
by the induction principle of [[identity types]]. Every pseudometric can be shown to have an [[equivalence relation]]
$$x \sim y \coloneqq d(x, y) = 0$$
through the axioms of a pseudometric space.

In set theory, the separation axiom is usually defined as the converse of the identity-to-zero-distance axiom. However, in type theory, since not all identity types are propositions, merely postulating a function
$$\mathrm{sepax}(x, y):(d(x, y) = 0) \to (x = y)$$
does not result in a metric space, since there might be many such functions into the identity type $x = y$. Instead, one must strengthen the condition to a [[Rezk completion|Rezk completeness]] condition for pseudometric spaces, similar to the type-theoretic antisymmetry condition in [[posets]] from [[preorder|preordered types]]:

\begin{definition}
A pseudometric space is a **metric space** if the axiom
$$\mathrm{idtozerodis}(x, y):(x = y) \to (d(x, y) = 0)$$
is an [[equivalence in homotopy type theory|equivalence of types]].
\end{definition}

Since any type of non-negative real numbers is a set, the Rezk completeness condition automatically implies that every metric space is a set.

### Category of metric spaces

When working with metric spaces one encounters a broad variety of different maps including non-[[continuous map|continuous]] ones like almost isometries. For a convenient categorical set-up one often restricts to __[[short map|short maps]]__, i.e. maps that do not increase any distance or, equivalently, have are of [[Lipschitz map|Lipschitz norm]] not greater than 1. The category of Lawvere metric spaces and short maps forms the [[category of metric spaces]] $Met$. The restriction to ordinary metric spaces is denoted by $Met_{ord}$.

## Lawvere metric spaces
  {#LawvereMetricSpace}

In [Lawvere, 73](#Lawvere73), it was pointed out that [Lawvere metric spaces](#LawvereMetricSpacesDefinition) are precisely [[enriched category|categories enriched]] in the [[monoidal category|monoidal]] [[partial order|poset]] $([0, \infty], \geq)$, where the [[tensor product]] is taken to be addition.  The [[composition]] operation in this enriched category identifies with the [[triangle identity]] in the metric space (see at _[triangle inequality -- Interpretation in enriched category theory](triangle+inequality#InterpretationInEnrichedCategoryTheory)_).

Alternatively, taking the monoidal product to be [[supremum]] instead, enriched categories amount to Lawvere ultrametric spaces.

Thus generalized, many constructions and results on metric spaces turn out to be special cases of yet more general constructions and results of [[enriched category theory]].  This includes for example the notion of [[Cauchy complete category|Cauchy completion]], which in general enriched category theory is related to [[Karoubi envelope|Karoubi envelopes]] and [[Morita equivalence]].

Imposing the symmetry axiom then gives us enriched $\dagger$-[[dagger category|categories]].  Note that when enriching over a [[cartesian monoidal category|cartesian monoidal]] poset, there is no difference between a $\dagger$-category and a [[groupoid]], so ultrametric spaces can also be regarded as [[enriched groupoid|enriched groupoids]], which is perhaps a more familiar concept.

(The requisite axioms for an enriched groupoid do not make sense when the enriching category is not cartesian, but one might argue that since in a poset "they would commute automatically anyway", it makes sense to call any poset-enriched $\dagger$-category also an "enriched groupoid".  However, perhaps it makes more sense just to speak about enriched $\dagger$-categories.)

In the presence of the symmetry axiom, the "separation" axiom "$x=y$ if $d(x,y)=0$" is equivalent to [[skeletal category|skeletality]] of an enriched category.  That is, a pseudo-metric space is a metric space precisely when it is skeletal.  But in the non-symmetric case, this separation axiom is stronger than skeletality; the latter would say only "$x=y$ if $d(x,y)=d(y,x)=0$".  That is, a quasi-pseudo-metric space can be skeletal without being a quasi-metric space, at least the way the latter term is usually used.

Note that like any kind of enriched category, Lawvere metric spaces are [[monads]] in a [[bicategory]] of "matrices", whose objects are sets and whose morphisms from $X$ to $Y$ are functions $d:X\times Y \to [0,\infty]$.  This sort of perspective can be generalized to many other kinds of topological structures; an exposition can be found in the book [Monoidal topology](#HST).

The category of metric spaces and categories of random maps as generalised metric spaces were studied in the [thesis](#Meng) of Lawvere's student Xiao-qing Meng.

## Motivation for the axioms

The [[triangle inequality|triangle axiom]] is the fundamental idea behind a metric space; it goes back (at least) to [[Euclid]] and captures the idea that we are discussing the *shortest* distance between two points.  Given the triangle inequality, we have the polygon inequality
$$ d(x_0,x_1) + \cdots + d(x_{n-1},x_n) \geq d(x_0,x_n) $$
for all $n \gt 0$; the point inequality extends this to $n = 0$.

Besides extended metric spaces (where distances may be infinite), one might consider spaces where distances may be negative.  But in fact this gives us nothing new, at least if we have symmetry.  First,
$$ d(x,x) + d(x,x) \geq d(x,x) $$
forces $d(x,x) \geq 0$, so $d(x,x) = 0$; then
$$ 2 d(x,y) = d(x,y) + d(y,x) \geq d(x,x) $$
forces $d(x,y) \geq 0$.  A generalisation to negative distances is possible for quasimetric spaces, however; the simplest example has $2$ elements, with $d(x,y) = -d(y,x)$ (but necessarily $d(x,x) = 0$ still).

We can define a [[preorder]] $\leq$ on the points of a Lawvere metric space:
$$ x \leq y \;\Leftrightarrow\; d(x,y) = 0 .$$
Then the symmetry axiom implies that this relation is [[symmetric relation|symmetric]] and hence an [[equivalence relation]].  The [[quotient set]] under this equivalence relation satisfies separation; in this way, every pseudometric space is [[equivalence of categories|equivalent]] (as an enriched category) to a metric space.  Even for quasimetric spaces, we can still define an equivalence relation:
$$ x \equiv y \;\Leftrightarrow\; d(x,y) = 0 \;\wedge\; d(y,x) = 0 .$$
In [[constructive mathematics]], it works better to use $\lneq$:
$$ x \lneq y \;\Leftrightarrow\; d(x,y) \gt 0 ;$$
then the symmetry axiom implies that this is an [[apartness relation]], which (for quasimetric spaces) we can also define directly:
$$ x \# y \;\Leftrightarrow\; d(x,y) \gt 0 \;\vee\; d(y,x) \gt 0 .$$

## Examples

* Every [[set]] carries the **[[discrete space|discrete metric]]** given by
  $$
    d(x,y) \coloneqq \left\{\array{
      0 & \text{if } x = y
      \\
      1 & \text{otherwise}
    }\right.
  $$

  For certain purposes, it makes more sense to make most the non-zero distance $\infty$ instead of $1$; then one has an extended metric space.

* Every [[normed vector space]] is a metric space by $d(x,y) \coloneqq {\|x - y\|}$; a pseudonormed vector space is a pseudometric space.

* Every [[connected space|connected]] [[Riemannian manifold]] becomes a pseudometric space (or a metric space if, as is often assumed, the manifold is [[Hausdorff space|Hausdorff]]) by taking the distance between two points to be [[infimum]] of the Riemannian lengths of all continuously differentiable [[path|paths]] connecting them:
  $$
    d(x,y) \coloneqq inf_{x \stackrel{\gamma}{\to} y} len(\gamma)
  .$$

  If the manifold might not be connected, then it still becomes an extended metric space.

## Properties

* [[Lebesgue number lemma]]

* [[sequentially compact metric spaces are totally bounded]]

* [[sequentially compact metric spaces are equivalently compact metric spaces]]

* [[countably compact metric spaces are equivalently compact metric spaces]]

* [[metric spaces are fully normal]]

* [[metric spaces are paracompact]]

## Related concepts

* [[induced metric]]

* [[metric jet]] -- a notion of "tangency" for maps of metric spaces

* [[complete metric space]] and [[localic completion]]

* [[separable metric space]]

* [[distance (graph theory)]]

* [[Booij premetric space]]

* [[Richman premetric space]]

* [[enriched poset]]

[[!include generalized uniform structures - table]]

## References

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

*  {#Lawvere73}  [[Bill Lawvere]] (1973).  _Metric spaces, generalized logic and closed categories_.  Reprinted in [[TAC]], 1986.  [Web](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html).

* {#Meng} Xiao-qing Meng, _Categories of convex sets and of metric spaces with applications to stochastic programming and related areas_, PhD thesis ([[Meng.djvu|djvu:file]])

* {#HST} Dirk Hofmann, Gavin J. Seal, and Walter Tholen (editors), *Monoidal topology: A Categorical Approach to Order, Metric, and Topology*,  Cambridge University Press, 2014

* {#Porton} [[Victor Porton]], _[Algebraic General Topology: Book 3: Algebra](https://mathematics21.org/the-algebra-of-general-topology/)_

* [[Fred Richman]], *Real numbers and other completions*, Mathematical Logic Quarterly **54** 1 (2008) 98-108 [[doi:10.1002/malq.200710024](https://onlinelibrary.wiley.com/doi/10.1002/malq.200710024)]

* Auke B. Booij, Analysis in univalent type theory ([pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf))

[[!redirects metric space]]
[[!redirects metric spaces]]
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[[!redirects quasimetrics]]
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[[!redirects quasipseudometric]]
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[[!redirects ultrametric]]
[[!redirects ultrametrics]]
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metric spaces are fully normal

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## Statement

Every [[metric space]], regarded as a [[topological space]] via its [[metric topology]], is a [[fully normal topological space]].

## Applications

Together with _Stone's theorem_ that [[fully normal spaces are equivalently paracompact]] this implies that [[metric spaces are paracompact]].

## References

* ProofWiki, _[Metric Space is Fully Normal](https://proofwiki.org/wiki/Metric_Space_is_Fully_Normal)_


metric spaces are paracompact

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## Statement

Every [[separable space|separable]] [[metrisable topological space]] is [[paracompact topological space|paracompact]].

With the [[axiom of choice]] we have more generally that:

Every [[metrisable topological space]] is [[paracompact topological space|paracompact]].

The orignal **proof** due to ([Stone 48](#Stone48)) used that  [[metric spaces are fully normal]] and showed that [[fully normal spaces are equivalently paracompact]] ("Stone's theorem").

A direct and short proof was later given in ([Rudin 68](#Rudin68)).

## Related statements

* [[paracompact Hausdorff spaces are normal]]

* [[locally compact and sigma-compact spaces are paracompact]]

* [[second-countable regular spaces are paracompact]]

* [[regular Lindelöf spaces are paracompact]]

* [[Hausdorff spaces are sober]]

* [[compact subspaces of Hausdorff spaces are closed]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

* [[CW-complexes are paracompact Hausdorff spaces]]

* [[Hausdorff spaces are sober]]

* [[maps from compact spaces to Hausdorff spaces are closed and proper]]

* [[continuous images of compact spaces are compact]]

* [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]]

## References

Historically, first it was shown that [[fully normal spaces are equivalently paracompact]] in

* {#Stone48} A. H. Stone, _Paracompactness and product spaces_, Bull. Amer. Math. Soc. Volume 54, Number 10 (1948), 977-982. ([Euclid](http://projecteuclid.org/euclid.bams/1183512390))

Since it is easy to see that [[metric spaces are fully normal]] this implies that metric spaces and hence metriable topological spaces are paracompact. Accordingly this statement came to be known as _Stone's theorem_.

A direct and short proof that metric spaces are paracompact was given in

* {#Rudin68} [[Mary Ellen Rudin]], _A new proof that metric spaces are paracompact_, AMS 1968 ([pdf](http://www.ams.org/journals/proc/1969-020-02/S0002-9939-1969-0236876-3/S0002-9939-1969-0236876-3.pdf))


metric topology

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## Definition

Given a [[metric space]] $(X,d)$, the _metric topology_ on $X$ is the structure of a [[topological space]] $\mathcal{T}$ on $X$ which is generated from the [[topological base]] of $\mathcal{T}$ given by the [[open balls]]

$$
  B(x,r) \coloneqq \{y \in X \;|\; d(x,y) \lt r \}
$$

for all $x \in X$ and $r \in (0,\infty) \subset \mathbb{R}$.

A topological space whose topology is the metric topology for some [[metric space]] structure on its underlying set is called a _[[metrizable topological space]]_.

## Related concepts

* [[continuous function]]

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metrisable topological space

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## Definition

A [[topological space]] $(X,\tau)$ is called _metrisable_ if there exists the stucture of a [[metric space]] $(X,d)$ on the underlying set, such that $\tau$ is the corresponding [[metric topology]].

## Properties

Metrizable spaces enjoy a number of separation properties: they are [[Hausdorff space|Hausdorff]], [[regular space|regular]], and even [[normal space|normal]]. They are also [[paracompact space|paracompact]].

Metrizable spaces are closed under topological [[coproducts]] and of course [[subspaces]] (and therefore [[equalizers]]); they are closed under [[countable family|countable]] [[products]] but not general products (for instance, a product of uncountably many copies of the [[real line]] $\mathbb{R}$ is not a normal space).

## Metrisability theorem

Fundamental early work in point-set topology established a number of [[metrization theorem|metrization theorems]], i.e., theorems which give sufficient conditions for a space to be metrisable. One of the more useful theorems is [[Urysohn]] [[Urysohn metrization theorem |metrization theorem]]:
A [[separation axiom|regular, Hausdorff]], and [[second-countable space]] is metrisable.
So, for instance, a compact Hausdorff space that is second-countable is metrisable. Other metrization theorems are:

* Nagata-Smirnov metrization theorem

* Bing metrization theorem

* Moore metrization theorem

## Related concepts

* [[metric space]]

* [[separable metric space]]

* [[topological space]]

* [[Polish space]]

## References

See also

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

[[!redirects metrisable topological spaces]]

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metrization theorem

A _metrization theorem_ is a result that gives sufficient conditions, and sometimes necessary and sufficient conditions, for a [[topological space]] to be [[metrisable topological space|metrisable]], i.e. its topology is induced by a metric.

## Theorems

An optimal metrization theorem, giving a necessary and sufficient, condition is [[Nagata-Smirnov metrization theorem]]:

[[!include Nagata-Smirnov metrization theorem - main theorem]]

A variation of this, directly implied by the fact that [[metrisable spaces have countably locally discrete bases]], is the Bing metrization theorem:

\begin{theorem}
A [[topological space]] $ X $ is [[metrisable topological space|metrisable]] if and only if it is [[regular space|regular]], [[Hausdorff space|Hausdorff]] and has a [[countably locally discrete set of subsets|countably locally discrete]] [[topological base|base]].
\end{theorem}

A historical predecessor and direct implication of these theorems is the [[Urysohn metrization theorem]]:

[[!include Urysohn metrization theorem - main theorem]]

Another variation is the Moore metrization theorem.

## Related questions

A refined and more complicated question is whether or not the topology of a given space is induced by a [[complete space|complete metric]]. In this case the space is called [[topologically complete space|topologically complete]]. A sufficient and neceissary criterion can be given in terms of [[cotopology#Properties|cotopology]].

If one weakens the concept of metric space sufficiently to the notion of [[approach space]], then every topological space is metrisable in this weaker sense.

## Related concepts

* [[metric space]]

* [[metrisable topological space]]

* [[topologically complete space]]

## References

* [[James Munkres]], _Topology_ (2nd edition), Prentice-Hall, 2000.

* [[Ryszard Engelking]], _General Topology_, Heldermann Verlag Berlin, 1989.

* Wikipedia, _Metrization theorem_. ([link](http://en.wikipedia.org/wiki/Metrization_theorem))

[[!redirects metrization]]
[[!redirects metrization theorems]]
[[!redirects Bing metrization theorem]]


MeV

A [[physical unit]] of [[energy]]: mega-[[electronvolt]].

## Related entries

* [[GeV]]

* [[TeV]]

## References

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


Meyer Bockstein

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

## Related $n$Lab entries

* [[Bockstein homomorphism]]

* [[Bockstein spectral sequence]]

category: people


MHV amplitude

[[!redirects MHV amplitudes]]

#Contents#
* table of contents
{:toc}

## Idea

Certain [[scattering amplitudes]] in [[N=4 D=4 super Yang-Mills theory]] and neighbouring [[gauge theories]] -- called _maximally helicity violating amplitudes_ ("MHV amplitudes") simplify drastically as compared to the generic situation and in fact are controled by a certain [[twistor string theory]] whose [[target space]] is a [[twistor space]]. See ([Monteiro](#Monteiro)) for a review.

## References

* Gustavo Machado Monteiro, _MHV tree amplitudes in Super-Yang-Mills and in Superstring Theory_ (2010) ([pdf](http://www.ift.unesp.br/posgrad/gustavo.pdf))
  {#Monteiro}
*  [[Edward Witten]], _Perturbative gauge theory as a string theory in twistor space_, Commun. Math. Phys. **252** (2004) 189-258 [[arXiv:hep-th/0312171](http://arxiv.org/abs/hep-th/0312171)]

A [[light-cone quantization|light-cone QCD]]--[[Lagrangian density]] adapted to MHV -amplitudes:

* Hiren Kakkad, Piotr Kotko, Anna Stasto, *Quantum correction to a new Wilson line-based action for Gluodynamics* [[arXiv:2311.04101](https://arxiv.org/abs/2311.04101)]


Mia J. Hughes

* [spire page](https://inspirehep.net/authors/1258972)

## Selected writings

On [[Freudenthal magic square|magic squares]] of [[magic supergravities]]  and [[U-duality]] via the [[classical double copy]]:

* {#BorstenDuffHughesNagy13} [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic square from Yang-Mills squared*, Phys. Rev. Lett. **112** (2014) 131601  [[arXiv:1301.4176](http://arxiv.org/abs/1301.4176)]

* {#ABDHN} [[Alexandros Anastasiou]], [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic pyramid of supergravities*, JHEP (2014) 178 [[arXiv:1312.6523](http://arxiv.org/abs/1312.6523)]

category: people

[[!redirects M. J. Hughes]]
[[!redirects L. J. Hughes]]


Micha Berkooz

* [webpage](http://www.weizmann.ac.il/particle/content/micha-berkooz)

## Selected writings

On [[chiral fermions]] in [[intersecting D-brane models]]:

* {#BerkoozDouglasLeigh96} [[Micha Berkooz]], [[Michael Douglas]], [[Robert Leigh]], _Branes Intersecting at Angles_, Nucl. Phys. B480:265-278, 1996 ([arXiv:hep-th/9606139](https://arxiv.org/abs/hep-th/9606139))

On [[multi trace operators]] in [[AdS/CFT]]:

* [[Ofer Aharony]], [[Micha Berkooz]], [[Eva Silverstein]], _Multiple-Trace Operators and Non-Local String Theories_, JHEP 0108:006, 2001 ([arXiv:hep-th/0105309](https://arxiv.org/abs/hep-th/0105309))

* [[Ofer Aharony]], [[Micha Berkooz]], [[Eva Silverstein]], _Non-local string theories on $AdS_3$ times $S^3$ and stable non-supersymmetric backgrounds_, Phys. Rev. D65:106007, 2002 ([arXiv:hep-th/0112178](https://arxiv.org/abs/hep-th/0112178))

On [[Lorentzian orbifolds]] as [[target spacetimes]] in [[string theory]]:

* [[Micha Berkooz]], Dori Reichmann, _A Short Review of Time Dependent Solutions and Space-like Singularities in String Theory_, Nucl. Phys. Proc. Suppl. 171:69-87, 2007 ([arXiv:0705.2146](https://arxiv.org/abs/0705.2146))

On [[thermal AdS3]]/[[BTZ black holes]]:

* [[Micha Berkooz]], Zohar Komargodski, Dori Reichmann, _Thermal $AdS_3$, BTZ and competing winding modes condensation_, JHEP 0712:020, 2007 ([arXiv:0706.0610](https://arxiv.org/abs/0706.0610))

On [[weight systems]] on [[chord diagrams]] as [[single trace observables]] in [[SYK-model]]-like [[AdS-CFT duality|dual]] descriptions of [[black hole thermodynamics]] (see at [[weight systems on chord diagrams in physics]]):

* {#BerkoozNarayanSimon18} [[Micha Berkooz]], [[Prithvi Narayan]], [[Joan Simón]], Section 2.1 of: _Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction_, JHEP 08 (2018) 192 ([arxiv:1806.04380](https://arxiv.org/abs/1806.04380))

Specifically on [[chord diagrams]] encoding [[SYK model]] [[correlators]] as representing [[hyperbolic space|hyperbolic]]/[[AdS-CFT|holographic content]]:

* [[Micha Berkooz]], Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 5 of: _Towards a full solution of the large $N$ double-scaled SYK model_, JHEP 03 (2019) 079 ([arxiv:1811.02584](https://arxiv.org/abs/1811.02584))

## Related $n$Lab entries

* [[intersecting D-brane model]]

* [[BFSS matrix model]]

* [[6d (2,0)-supersymmetric QFT]]

category: people


Michael A. Fligner

## Selected writings

On [[geometric group theory]] on the [[symmetric group]] ([[Cayley distance]], [[Kendall distance]], [[Mallows kernel]], etc.):

* [[Michael A. Fligner]], [[Joseph S. Verducci]], Section 4 of: *Distance Based Ranking Models*, Journal of the Royal Statistical Society. Series B (Methodological) Vol. 48, No. 3 (1986), pp. 359-369 ([jstor:2345433](https://www.jstor.org/stable/2345433))

* [[Michael A. Fligner]], [[Joseph S. Verducci]] (eds.), *Probability Models and Statistical Analyses for Ranking Data*, Lecture Notes in Statistics **80**, Springer 1993 ([doi:10.1007/978-1-4612-2738-0](https://link.springer.com/book/10.1007/978-1-4612-2738-0))

category: people

[[!redirects Michael Fligner]]
[[!redirects M. A. Fligner]]


Michael A. Nielsen

* [personal page](https://michaelnielsen.org/)

* [blog](https://michaelnielsen.org/blog/)

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

## Selected writings

On [[quantum computation]] and [[quantum information]]:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

On the [[Solovay-Kitaev theorem]]:

* [[Christopher M. Dawson]], [[Michael A. Nielsen]], *The Solovay-Kitaev algorithm*, Quantum Information & Computation **6** 1 (2006) 81–95 [[arXiv:quant-ph/0505030](https://arxiv.org/abs/quant-ph/0505030), [doi:10.5555/2011679.2011685](https://dl.acm.org/doi/10.5555/2011679.2011685)]

On [[quantum circuits]] for [[quantum computation]] as ([[geodesic]], if optimal) [computational paths](computation#ReferencesConceptualizationViaPathLifting):

* {#NielsenDowlingGuDoherty06} [[Michael A. Nielsen]], [[Mark R. Dowling]], [[Mile Gu]], [[Andrew C. Doherty]], *Quantum Computation as Geometry*, Science, **311** 5764 (2006) 1133-1135 [[doi:10.1126/science.1121541](https://doi.org/10.1126/science.1121541), [arXiv:quant-ph/0603161](https://arxiv.org/abs/quant-ph/0603161)]

* [[Mark R. Dowling]], [[Michael A. Nielsen]], *The geometry of quantum computation*, Quantum Information & Computation **8** 10 (2008) 861–899 [[doi:10.5555/2016985.2016986](https://dl.acm.org/doi/abs/10.5555/2016985.2016986), [arXiv:quant-ph/0701004](https://arxiv.org/abs/quant-ph/0701004)]

category: people

[[!redirects Michael Nielsen]]


Michael Abbott

* [institute page](https://www.cs.le.ac.uk/people/mabbott/)

## Selected writings

On [[inductive types]] and [[W-types]]:

* [[Michael Abbott]], [[Thorsten Altenkirch]], [[Neil Ghani]]: *Representing Nested Inductive Types using W-types*, in: *Automata, Languages and Programming*, ICALP 2004, Lecture Notes in Computer Science, **3142**, Springer (2004) [[doi:10.1007/978-3-540-27836-8_8](https://doi.org/10.1007/978-3-540-27836-8_8), [pdf](http://www.cs.nott.ac.uk/~psztxa/publ/icalp04.pdf)]

  exposition: _Inductive Types for Free -- Representing nested inductive types using W-types_, talk at ICALP (2004) [[pdf](http://www.cs.nott.ac.uk/~txa/talks/icalp-slides.pdf)]

category: people


Michael Albanese

A geometer that got lost on the internet.


Michael Arbib

Michael A. Arbib  is a Professor of Computer Science, as well as a Professor of Biological Sciences, Biomedical Engineering, Electrical Engineering, Neuroscience and Psychology at the University of Southern California (USC). He has used category theory extensively in his work on computer science and brain function.

* [Home Page](http://www.usc.edu/programs/neuroscience/faculty/profile.php?fid=16)

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

category:people


Michael Artin

__Michael Artin__ (born 1934) is an algebraist and algebraic geometer at MIT. Early in his career he worked within Grothendieck's school, where his works on Grothendieck topologies, etale cohomology and algebraic stacks are especially well known, including [[Artin's representability theorem]] in theory of stacks and the notion of Artin's stack. Some of the SGA volumes are prepared with his contribution. In 1990-s Artin worked intensively at the interface between noncommutative algebra and algebraic geometry.

His father was [[Emil Artin]].

* [wikipedia](http://en.wikipedia.org/wiki/Michael_Artin), [MIT page](http://math.mit.edu/people/profile.php?pid=9)

## Selected writings

On [[algebraic stacks]] and introducing what came to be known as *[[Artin stacks]]*:

* [[Michael Artin]], *Versal deformations and algebraic stacks*, Invent Math **27** (1974) 165–189 [[doi:10.1007/BF01390174](https://doi.org/10.1007/BF01390174), [eudml:142310](https://eudml.org/doc/142310),  [pdf](http://math.uchicago.edu/~drinfeld/Artin_on_stacks.pdf)]

Introducing [[Artin-Mazur formal groups]]:

* {#ArtinMazur77} [[Michael Artin]], [[Barry Mazur]], _Formal Groups Arising from Algebraic Varieties_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131  ([numdam:ASENS_1977_4_10_1_87_0](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663))

## Related entries

* [[comparison theorem (étale cohomology)]]

[[!redirects M.Artin]]
[[!redirects M. Artin]]

category: people