span trace

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## Idea

In the context of [[integral transforms on sheaves]] one thinks of a [[span]] as a generalized linear map. The _span trace_ is the corresponding generalization of the notion of a trace of a linear map.

This is just the general [[trace]] of an [[endomorphism]] which is definable in any [[compact closed category|compact/autonomous]] [[symmetric monoidal category|symmetric monoidal (2-)category]], of which $Span$ is an example (as described below).

In the context of [[FQFT]] a useful aspect of the span trace is that it is manifestly dual to the [[co-span co-trace]], which, as described there, corresponds under the interpretation of spans as [[cobordism]]s to gluing of the two ends of a cobordism.

## Definition

### For Spans

For

$$
  \array{
    && R
    \\
    & {}^x\swarrow && \searrow^{y}
    \\
    X &&&& X
  }
$$

a [[span]] with identical left and right index object $X$, the simplest way to define its __span trace__ $tr(R)$ is as by regarding it as a map $R\to X\times X$, then pulling back along the [[diagonal morphism]] $X\to X\times X$.

This can be expressed in terms of the [[bicategory]] [[Span]] in several ways.  For instance, we can regard it as the composite of the result

$$
  \array{
    && R
    \\
    & {}^{x \times y}\swarrow && \searrow
    \\
    X \times X &&&& pt
  }
$$

of dualizing one leg of the span with the span

$$
  \array{
    && X
    \\
    & {}^{}\swarrow && \searrow^{Id \times Id}
    \\
    pt &&&& X \times X
  }
$$

i.e. the [[pullback]]

$$
  \array{
    &&&&
    \mathrm{tr}R
    \\
    &&& \swarrow && \searrow
    \\
    && X &&&& R
    \\
    & {}^{}\swarrow && \searrow^{Id \times Id}
    && {}^{x \times y}\swarrow && \searrow
    \\
    pt &&&& X \times X &&&& pt
  }
$$

regarded as a span from the point to the point

$$
  \array{
    && tr(R)
    \\
    & {}^{}\swarrow && \searrow
    \\
    pt &&&& pt
  }
  \,.
$$

### For multispans

More generally, the trace of a [[multispan]] over $n$ identical of its index objects $X$ is the composite with the multispan

$$
  \array{
    & X
    \\
    & {}^{Id}\swarrow \downarrow^{Id} & \cdots
    \\
    X & X & \cdots & X & \cdots
  }
$$

## Examples

### Trace of Set-valued matrices

Let the ambient category be [[Set]], let $X$ be a finite set and $R \to X \times X$ an $|X| \times |X|$-matrix of finite sets, regarded under [[groupoid cardinality]] as a [[groupoidification|groupoidified]] $|X| \times |X|$-matrix with entries in $\mathbb{N}$.

The trace of the span
$$
  \array{
    && R
    \\
    & {}^x\swarrow && \searrow^{y}
    \\
    X &&&& X
  }
$$

is the pullback

$$
  \array{
     tr(R) &\to& R
     \\
     \downarrow && \downarrow^{x,y}
     \\
     X &\stackrel{Id \times Id}{\to}& X \times X
  }
$$

which is the [[coproduct]] set $tr(R) = \sqcup_{x \in X} R_{x,x}$. Under [[groupoid cardinality]] this is indeed the trace
$|tr(R)| = \sum_{x} |R_{x,x}| $
of the matrix $|R|$ represented by $R$.

### Loop objects from homotopy span traces

Let $C$ be a [[category of fibrant objects]] with [[interval object]] $I$. Recall that for every object $B$ of $C$ its free [[loop space object]] is the part of the [[path object]] $B^I = [I,B]$ which consists of closed paths, i.e. the pullback.
$$
  \array{
    \Omega B &\to& [I,B]
    \\
    \downarrow && \downarrow^{d_0 \times d_1}
    \\
    B &\stackrel{Id \times Id}{\to}& B \times B
  }
  \,.
$$
This can be understood as the _homotopy span trace_ of the identity span on $B$

$$
  \Omega B = hotr(Id_B) = hotr\left(
    \array{
      && B
      \\
      & {}^{Id}\swarrow && \searrow^{Id}
      B &&&& B
    }
  \right)
  \,,
$$
where the homotopy span trace is computed like the span trace but with the pullback replaced by a [[homotopy limit|homotopy pullback]]:
$$
  hotr(Id_B) =
  holim
  \left(
    \array{
      B &&&& B
      \\
      & {}_{Id \times Id}\searrow && \swarrow_{Id \times Id}
      \\
      &&
      B \times B
    }
  \right)
  \,.
$$
According to the example described at [[homotopy limit]] and  using that we assume that we are in a [[category of fibrant objects]] we can compute this homotopy limit, up to weak equivalence, as the ordinary limit of the weakly equivalent pullback diagram
$$
  \array{
     F
     &&&&
     B &\stackrel{Id \times Id}{\to}& B \times B
     &\stackrel{Id \times Id}{\leftarrow}& B
     \\
     \downarrow^{\simeq}
     &&&&
     \downarrow^{\simeq} && \downarrow^{Id} && \downarrow^{Id}
     \\
     F'
     &&&&
     [I,B] &\stackrel{d_0 \times d_1}{\to}&
     B \times B &\stackrel{Id \times Id}{\leftarrow}&
     B
  }
$$
where we replace $B$ by its [[path object]] $B^I = [I,B]$ using the factorization of $B \stackrel{Id \times Id}{\to} B \times B$ as $B \stackrel{\simeq}{\to} [I,B] \stackrel{d_0 \times d_1}{\to} B \times B$ guaranteed to exist in a [[category of fibrant objects]], where $[I, B] \stackrel{d_0 \times d_1}{\to} B \times B$ is a _fibration_:

$$
  holim_D F \stackrel{\simeq}{\to} lim_D F'
  \,.
$$

But by the above $lim_D F' = \Omega B$.

### Categorical trace from homotopical span trace

The [[categorical trace]] on a 1-[[endomorphism]] in a 2-category $C$ is the homotopy trace on the span given by that endomorphism.

This should be true quite generally, but here are the details just for the special case that connects to the above example:

Let $C = $ [[Grpd]] with the standard interval object $I = \{a \stackrel{\simeq}{\to} b\}$. This is a [[category of fibrant objects]] with respect to the [[folk model structure]].

Notice that [[natural transformation]]s $\eta : F \to G$ between two functors $F, G : C \to D$ are in bijection with commuting diagrams
$$
  \array{
     X &\stackrel{\eta}{\to}& [I,Y]
     \\
     \downarrow^{Id} && \downarrow^{d_0 \times d_1}
     \\
     X &\stackrel{F \times G}{\to}& Y \times Y
  }
  \,.
$$

Now, the homotopy trace on the span corresponding to an ednofunctor $F : B \to B$ is

$$
  hotr(F)
  =
  hotr\left(
    \array{
      && B
      \\
      & {}^{Id}\swarrow && \searrow^{F}
      \\
      B &&&& B
    }
  \right)
  =
  holim\left(
    \array{
      B &&&& B
      \\
      & {}_{Id \times Id}\searrow && \swarrow{F \times Id}
      \\
      && B \times B
    }
  \right)
  \,.
$$

Since we are in a category of fibrant objects the assumptions of the example discussed at [[homotopy limit]] apply and the above homotopy limit is again computed, up to weak equivalence, by the ordinary limit of

$$
  \cdots
  \stackrel{\simeq}{\to}
  lim
  \left(
    \array{
      [I,B] &&&& B
      \\
      & {}_{d_0 \times d_1}\searrow && \swarrow{Id \times F}
      \\
      && B \times B
    }
  \right)
  \,.
$$

By the above, every cone over the pullback diagram with a functor $h : Q \to B$, on the right defines a natural transformation $h^*(Id_B \Rightarrow F)$. By the universal property of the limit, it _represents_ the collection of these transformations.

## Related concepts

* The dual notion is that of [[co-span co-trace]].

## References

That the canonical trace on $Span$ is compatible with the interpretation of spans as linear maps in the context of [[groupoidification]], and that it corresponds under duality  (in terms of the [[co-span co-trace]]) to the gluing of ends of [[cobordism]]s was mentioned in

* Urs Schreiber, [(co)-traces](http://golem.ph.utexas.edu/category/2008/05/hopkinslurie_on_baezdolan.html#c021537) (2008)

More discussion is in

* [[David Ben-Zvi]], [[David Nadler]], _Nonlinear traces_ ([arXiv:1305.7175](http://arxiv.org/abs/1305.7175))
 {#BenZviNadler13}

[[!redirects span traces]]


Spanier-Whitehead category

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## Idea

Motivated by the [[Freudenthal suspension theorem]], the _suspension category_ of ([Spanier-Whitehead 53](#SpanierWhitehead53)) has as [[objects]] [[pointed topological space|pointed]]  [[CW-complexes]], and as [[hom-sets]] it has [[colimits]]

$$
  Hom(X,Y) \coloneqq \underset{\longrightarrow}{\lim}_q [\Sigma^q X , \Sigma^q Y]
$$

over [[homotopy classes]] of [[continuous functions]] between their arbitrary high [[suspensions]].

More generally one considers the category whose objects are pairs $(X,n)$ of a [[pointed topological space|pointed]]  [[CW-complex]] $X$ and an [[integer]] $n$ and whose [[hom-sets]] are

$$
  Hom((X_1,n_1), (X_2,n_2)) \coloneqq \underset{\longrightarrow}{\lim}_{q \gt max({\vert n_1\vert},{\vert n_2\vert})} [\Sigma^{q+n_1} X_1 , \Sigma^{q+n_2} X_2]
  \,.
$$

This is a [[triangulated category]].

But the Spanier-Whitehead category lacks other desirable properties, for instance it does not have all [[coproducts]] and the canonical functor from the homotopy category of [[pointed topological spaces]] does not preserve the coproducts that already exist. As a consequence, in particular a [[Brown representability theorem]] does not hold in the SW-category.

Later it was realized (see e.g. [Whitehead 62](#Whitehead62)) that this all this is fixed by regarding the SW-category for [[finite CW complexes]] as a [[full subcategory]] on the (shifted) [[suspension spectra]]  inside the larger category of [[spectra]]: the [[stable homotopy category]] (e.g. [Schwede 12, chapter II theorem 7.2](#Schwede12)). As such it is the full subcategory on the [[finite spectra]] (e.g. [Schwede 12, chapter II theorem 7.4](#Schwede12)).

## Related concepts

* [[Spanier-Whitehead duality]]

* [[prestable (∞,1)-category]]

## References

The definition is due to

* {#SpanierWhitehead53} [[Edwin Spanier]], [[George Whitehead]], _A first approximation to homotopy theory_ Proc. Nat. Acad. Sci. U.S.A., 39 (1953), 655-660.

Survey:

* {#Whitehead62} [[George Whitehead]], _Some aspects of stable homotopy theory_, International Confress of Mathematics 1962 ([pdf](http://www.mathunion.org/ICM/ICM1962.1/Main/icm1962.1.0502.0506.ocr.pdf))

* H. R. Margolis, _Spectra and the Steenrod algebra_, volume 29 of North-Holland
Mathematical Library. North-Holland Publishing Co., Amsterdam, 1983

* {#Schwede12} [[Stefan Schwede]], chapter II, section 7 of  _[[Symmetric spectra]]_, 2012  ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 20 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

Discussion in the abstract generality of categories equipped with an abstract suspension-like functor is in

* {#Heller68} [[Alex Heller]], _Stable homotopy categories_, Bull. Amer. Math. Soc. Volume 74, Number 1 (1968), 28-63. ([Euclid](https://projecteuclid.org/euclid.bams/1183529378))

* [[Ivo Dell'Ambrogio]], _The Spanier-Whitehead category is always triangulated_ ([pdf](http://math.univ-lille1.fr/~dellambr/diploma.pdf))

[[!redirects suspension category]]


Spanier-Whitehead duality

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#### Duality
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#Contents#
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## Idea

The [[stable homotopy category]] $Ho(Spec)$ is a [[symmetric monoidal category]] via the [[symmetric smash product of spectra]]. Monoidal [[dual object|duality]] in $Ho(Spec)$ is called _Spanier-Whitehead duality_ or _[[S-duality]]_ .

## Examples

* For $X$ a [[compact space|compact]] [[smooth manifold]]. The dual of its [[suspension spectrum]] $\Sigma^\infty_+ X$ is given by the [[Thom spectrum]] $Th(N X)$ of its stable [[normal bundle]]. This is also called **Atiyah duality** (due to [Atiyah](#Atiyah)).

 The explicit interpretation in terms of monoidal duality is ([DoldPuppe, theorem 3.1](#DoldPuppe)).

  Using this one shows that the [[trace]] on the [[identity]] on $\Sigma^\infty_+ X$ -- its categorical [[dimension]] -- is the [[Euler characteristic]] of $X$.

## Related entries

* [[S-theory]]

* [[Anderson duality]]

* [[Brown-Comenetz duality]]

* [[Spanier-Whitehead category]]

* [[stable Cohomotopy]]

## References

The original references are

* [[Edwin Spanier]], ; [[Henry Whitehead]],  (1953), _A first approximation to homotopy theory_, Proc. Nat. Acad. Sci. U.S.A. 39: 655–660, MR0056290

* [[Edwin Spanier]], ; [[Henry Whitehead]],  (1955), _Duality in homotopy theory_ , Mathematika 2: 56–80, MR0074823

The interpretation of the duality as ordinary monoidal duality in the stable homotopy category is apparently due to

* {#DoldPuppe} [[Albrecht Dold]], [[Dieter Puppe]], _Duality, trace and transfer_ , Proceedings of the Steklov Institute of Mathematics, (1984), issue 4

An English version of that paper is:

* {#DoldPuppeEnglish} [[Albrecht Dold]], [[Dieter Puppe]], _Duality, trace and transfer_ , Proceedings of the International Conference on Geometric Topology (Warsaw, 1978), pp. 81–102, PWN, Warsaw, 1980. [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/doldpup2.pdf)

Atiyah duality is due to

* {#Atiyah} [[Michael Atiyah]], _Thom complexes_ , Proc. London Math. Soc. (3) , no. 11 (1961), 291–310.

Lecture notes include

* {#Adams74} [[Frank Adams]], part III, section 5 of _[[Stable homotopy and generalised homology]]_, 1974

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 22 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

Further discussion of Atiyah duals is in

* [[Ralph Cohen]], _Multiplicative properties of Atiyah duality_ ([arXiv:0403486](http://arxiv.org/abs/math/0403486))

For [[equivariant stable homotopy theory]] Spanier-Whitehead duality is discussed on pages 23 onwards of

* [[John Greenlees]], [[Peter May]], _Equivariant stable homotopy theory_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

The Spanier-Whitehead duality has been studied in the new category of [[cospectra]] with ideas close to the later ones of [[shape theory]] in

* Elon L. Lima, _The Spanier-Whitehead duality in new homotopy categories_, Summa Brasil. Math. 4 1959 91–148 (1959) [MR0116332](http://www.ams.org/mathscinet-getitem?mr=116332)

See also

* Roy Joshua, _Spanier-Whitehead duality in etale homotopy_ Transactions of the American Mathematical Society
Vol. 296, No. 1 (Jul., 1986), pp. 151-166
(article consists of 16 pages)  ([jstor](http://www.jstor.org/pss/2000566))

[[!redirects Spanier-Whitehead dual]]
[[!redirects Atiyah duality]]
[[!redirects S-dual]]
[[!redirects S-duality]]

[[!redirects Alexander-Whitehead-Atiyah duality]]


spatial locale

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### Context
#### Topos Theory
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#### Topology
+-- {: .hide}
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# Spatial locales
* tables of contents
{: toc}

## Idea

A spatial locale is a [[locale]] that comes from a [[topological space]].  This is an [[extra property]] of locales, a property of having enough [[point of a locale|points]].

## Definitions

Let $X$ be a [[topological space]].  Then we may define a [[locale]], denoted $\Omega(X)$, whose [[frame of opens]] is precisely the [[frame]] of [[open subspaces]] of $X$.

A locale is __spatial__ if it is [[isomorphism|isomorphic]] to $\Omega(X)$ for some topological space $X$.

A locale $L$ has __enough points__ if, given any two opens $U$ and $V$ in $L$, $U = V$ if (hence iff) precisely the same [[point of a locale|points]] of $L$ belong to $U$ as belong to $V$.

## Properties

The following conditions are all logically equivalent on a [[locale]] $L$:

1.  $L$ is spatial, as defined above.
2.  $L$ has enough points, as defined above.
3.  Given any two opens $U$ and $V$ in $L$, $U \leq V$ if (hence iff) every point of $L$ that belongs to $U$ also belongs to $V$.
4.  $L$ is isomorphic to $\Omega(pt(L))$, where $pt(L)$ is the [[space of points]] of $L$.
5.  The natural morphism $\eta_L\colon \Omega(pt(L)) \to L$ (the [[counit]] of the [[adjunction]] from [[Top]] and [[Loc]]) is an [[isomorphism]].

(It would be nice to state this as a theorem and put in a proof.)

Basically, what is going on here is that we have an [[idempotent adjunction]] from topological spaces to locales, and the spatial locales comprise the image of this adjunction.  The corresponding condition on topological spaces is being [[sober space|sober]].

Therefore, the [[full subcategory]] of $Loc$ on the spatial locales is [[equivalence of categories|equivalent]] to the full subcategory of $Top$ on sober spaces.

## Terminology

The term 'spatial locale' can be confusing; it suggests a locale [[internalization|in]] [[Top]] or in some category [[Sp]] of spaces, which is not correct.  Instead, the adjective 'spatial' should be taken in the same vein as 'localic' in '[[localic topos]]' or 'topological' in 'topological [[convergence space|convergence]]'.  These two terms also suggest that these other locales are not [[spaces]], which is incorrect.

The really clear term for a spatial locale is 'locale with enough points to separate the opens', but 'locale with enough points' should be unambiguous.  However, it is still a bit long.  Occasionally one sees 'spacial' instead of 'spatial'.

## Criteria for spatiality

1. Assuming the [[axiom of choice]], [[locally compact locales]] are spatial.
In particular, [[compact regular locales]] are locally compact, hence automatically spatial.

2. Any [[coherent locale]] is also spatial.

2. More generally, the meet of a countable family of open sublocales (i.e., a $G_\delta$-sublocale) of a [[compact regular locale]] is spatial.

3. The completion of a [[uniform locale]] with a countable basis of uniformity is spatial.

4. [[Stonean locale|Stonean locales]] are spatial.

## Related concepts

* [[spatial topos]], [[localic topos]]

* [[locale]]

* [[topological space]]

* [[ionad]]

* [[topos]]

[[!redirects topological locale]]
[[!redirects topological locales]]
[[!redirects spatial locales]]
[[!redirects spacial locale]]
[[!redirects spacial locales]]
[[!redirects locale with enough points]]
[[!redirects locales with enough points]]
[[!redirects locale of opens]]
[[!redirects locales of opens]]
[[!redirects locale of open subspaces]]
[[!redirects locales of open subspaces]]
[[!redirects locale of open subsets]]
[[!redirects locales of open subsets]]
[[!redirects locale of open sets]]
[[!redirects locales of open sets]]


spatial tensor product

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{:toc}

## Idea ##
There are several different concepts of [[tensor products]] for [[C-star algebras]], because there are different norms one can put on the algebraic tensor product that turns it into a [[C-star algebra]]. The spatial tensor product uses the smallest norm of all possible norms. There is also a maximal norm and it is a nontrivial theorem that all norms fall in between these two.

If $A$ is a [[C-star-algebra]] it may happen that for every other [[C-star-algebra]] $B$, the maximal and minimal tensor norms on the algebraic tensor product $A\otimes B$ coincide, forcing the $*$-algebra $A\otimes B$ to have a **unique** C-star norm. Such an $A$ is said to be [[nuclear C*-algebra|nuclear]].

## Definition ##
Let $\mathcal{A}_1, ..., \mathcal{A}_k$ be unital $C^*$-algebras faithfully represented on the [[Hilbert spaces]] $H_1, ..., H_k$. Let $H$ be the tensor product of these Hilbert spaces,
$$
H := \otimes_{i=1}^k H_k
$$
The set of operators of finite sums of $A_1 \otimes ... \otimes_k A_k$ form a $*$-subalgebra of $\mathcal{B}(H)$. The norm closure of this set is the **spatial tensor product** of the given $C^*$-algbras, and is sometimes denoted by $A_1\otimes_{min} \dots \otimes_{min} A_k$.

Remark: The spatial tensor product does not depend on the chosen faithful representations, see references.

## Properties ##
+-- {: .un_theorem}
###### Theorem

**states extend to the spatial tensor product**

Let $\rho_1, ..., \rho_k$ be states on the unitary $C^*$-algebras. Then there is a unique state $\rho$ on the spatial tensor product such that
$$
\rho(A_1 \otimes ... \otimes_k A_k) = \rho_1(A_1) \cdots \rho_k(A_k)
$$
=--

The spatial tensor product is injective, in the sense of Grothendieck's approach to tensor norms, but in general it is not projective.

In the category of operator spaces and completely contractive linear maps (completely short in nLab terminology?), one can introduce the projective tensor product and the injective tensor product in analogy with Grothendieck's definitions for the category of Banach spaces and short linear maps. Then if $A$ and $B$ are $C^*$-algebras, the injective tensor product of their underlying operator spaces is canonically isomorphic to the underlying operator space of $A\otimes_{min} B$.

## Related pages

* [[tensor product of Banach spaces]]
* [[inductive tensor product]]

## References ##

Appendix T in the book

* Niels Erik Wegge-Olsen: _K-theory and $C^*$-algebras: a friendly approach._ ([ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0780.46038&format=complete))

[[!redirects Creating spatial tensors]]

[[!redirects tensor product of C*-algebras]]
[[!redirects tensor product of C-star-algebras]]
[[!redirects tensor products of C*-algebras]]
[[!redirects tensor products of C-star-algebras]]


spatial topos > history

see at _[[category of sheaves on a topological space]]_


spatial type theory

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## Idea

A [[modal type theory|modal]] [[dependent type theory]] with the [[sharp modality]] $\sharp$ and the [[flat modality]] $\flat$. Spatial type theory which is also a [[homotopy type theory]] is called **spatial homotopy type theory**.

## Presentation

In this presentation of [[spatial type theory]], we assume a [[dependent type theory]] with crisp term judgments $a::A$ in addition to the usual (cohesive) type and term judgments $A \; \mathrm{type}$ and $a:A$, as well as context judgments $\Xi \vert \Gamma \; \mathrm{ctx}$ where $\Xi$ is a list of crisp term judgments, and $\Gamma$ is a list of cohesive term judgments. A crisp type is a type in the context $\Xi \vert ()$, where $()$ is the empty list of cohesive term judgments.

### Identity types

In addition, we also assume the dependent type theory has [[typal equality]]:

* Formation rule for identity types:
$$\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi \vert \Gamma \vdash a:A \quad \Xi \vert \Gamma \vdash b:A}{\Xi \vert \Gamma \vdash a =_A b \; \mathrm{type}}$$

* Introduction rule for identity types:
$$\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash \mathrm{refl}_A(a) : a =_A a}$$

* Elimination rule for identity types:
$$\frac{\Xi \vert \Gamma, x:A, y:A, p:a =_A b \vdash C \; \mathrm{type} \quad \Xi \vert \Gamma, z:A \vdash t:C[z/a, z/b, \mathrm{refl}_A(z)/p] \quad \Xi \vert \Gamma \vdash a:A \quad \Xi \vert \Gamma \vdash b:A \quad \Xi \vert \Gamma \vdash q:a =_A b}{\Xi \vert \Gamma \vdash \mathrm{ind}_{=_A}^{x,y,p.C}(z.t, a, b, q):C[a, b, q/x, y, p]}$$

* Computation rules for identity types:
$$\frac{\Xi \vert \Gamma, x:A, y:A, p:a =_A b \vdash C \; \mathrm{type} \quad \Xi \vert \Gamma, z:A \vdash t:C[z/a, z/b, \mathrm{refl}_A(z)/p] \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash \beta_{=_A}^{x,y,p.C}(a):\mathrm{ind}_{=_A}^{x,y,p.C}(z.t, a, a, \mathrm{refl}(a)) =_{C[a, a, \mathrm{refl}_A(a)/x, y, p]} t}$$

### Sharp modality

The [[sharp modality]] is given by the following rules:

* Formation rule for sharp types:

$$\frac{\Xi, \Gamma \vert () \vdash A \; \mathrm{type}}{\Xi \vert \Gamma \vdash \sharp A \; \mathrm{type}}\sharp-\mathrm{form}$$

* Introduction rule for sharp types:

$$\frac{\Xi, \Gamma \vert () \vdash a:A}{\Xi \vert \Gamma \vdash a^\sharp:\sharp A}\sharp-\mathrm{intro}$$

* Elimination rule for sharp types:

$$\frac{\Xi \vert () \vdash a:\sharp A}{\Xi \vert \Gamma \vdash a_\sharp:A}\sharp-\mathrm{elim}$$

* Computation rule for sharp types:

$$\frac{\Xi \vert () \vdash a:A}{\Xi \vert \Gamma \vdash \beta_{\sharp A}(a):(a^\sharp)_\sharp =_A a}\sharp-\mathrm{comp}$$

* Uniqueness rule for sharp types:

$$\frac{\Xi \vert () \vdash a:\sharp A}{\Xi \vert \Gamma \vdash \eta_{\sharp A}(a):(a_\sharp)^\sharp =_{\sharp A} a}\sharp-\mathrm{uniq}$$

### Flat modality

The [[flat modality]] is given by the following rules:

* Formation rule for flat types:

$$\frac{\Xi, \Gamma \vert () \vdash A \; \mathrm{type}}{\Xi \vert \Gamma \vdash \flat A \; \mathrm{type}}\flat-\mathrm{form}$$

* Introduction rule for flat types:

$$\frac{\Xi, \Gamma \vert () \vdash a:A}{\Xi \vert \Gamma \vdash a^\flat:\flat A}\flat-\mathrm{intro}$$

* Elimination rule for flat types:

$$\frac{\Xi \vert \Gamma, x:\flat A \vdash C \; \mathrm{type} \quad \Xi \vert \Gamma \vdash M:\flat A \quad \Xi, u::A \vert \Gamma \vdash N : C[u^\flat/x]}{\Xi \vert \Gamma \vdash (\mathrm{let}\; u^\flat \coloneqq M \;\mathrm{in}\; n):C[M/x]}\flat-\mathrm{elim}$$

* Computation rule for flat types:

$$\frac{\Xi \vert \Gamma, x:\flat A \vdash C \; \mathrm{type} \quad \Xi \vert () \vdash M:\flat A \quad \Xi, u::A \vert \Gamma \vdash N : C[u^\flat/x]}{\Xi \vert \Gamma \vdash \beta_{\flat}(M, N):(\mathrm{let}\; u^\flat \coloneqq M \;\mathrm{in}\; N) =_{C[M^\flat/x]} N[M/u]}\flat-\mathrm{comp}$$

## Examples

* [[homotopy type theory]] is trivially a spatial type theory where every type is [[discrete]]

A [[cohesive homotopy type theory]] is a spatial homotopy type theory which additionally has an [[axiom of cohesion]] and a [[shape modality]]. Examples of this include

* [[real-cohesive homotopy type theory]]
* [[A1-cohesive homotopy type theory]]

## See also

* [[dependent type theory]]

* [[modal type theory]]

* [[cohesive homotopy type theory]]

## References

* [[David Corfield]], *Modal Homotopy Type Theory*, ... [reference to be completed]

A formal presentation of spatial type theory is found in

* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))


Spec(S)

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

#Contents#
* table of contents
{:toc}

## Idea

$Spec(\mathbb{S})$ is the spectrum, in the sense of [[spectrum of a commutative ring]] in [[E-∞ geometry]], of the [[sphere spectrum]] $\mathbb{S}$, in the sense of [[stable homotopy theory]], regarded canonically as the [[initial object in an (∞,1)-category|initial]] [[E-∞ ring]].

This is the refinement to [[E-∞ arithmetic geometry]] of what [[Spec(Z)]] is in [[arithmetic geometry]].

It is the absolute base space, in that it is the [[terminal object in an (∞,1)-category|terminal object]] among [[E-∞ schemes]]. Notably for $E$ any other [[E-∞ ring]], then the essentially unique $E_\infty$-ring homomorphism $\mathbb{S} \longrightarrow E$ (the unit, exhibiting $E$ as an [[E-∞ algebra]] over $\mathbb{S}$) corresponds to an essentially unique morphism

$$
  Spec(E) \longrightarrow Spec(\mathbb{S})
  \,.
$$

The [[1-image]] of such a morphism is known as (the [[formal dual]] of) the $E$-[[nilpotent completion]] of $\mathbb{S}$.

For instance for $E = $ [[HA|H]]$\mathbb{F}_p$, then the 1-image of $Spec(H\mathbb{F}_p) \to Spec(\mathbb{S})$ is $Spec(\mathbb{S}_{(p)})$, the [[formal dual]] of the [[p-completion]] of $\mathbb{S}$, hence the [[infinitesimal neighbourhood]] of $(p)$ in $Spec(S)$. The tool that computes $\mathbb{S}_{(p)}$ (hence the $p$-primary [[stable homotopy groups of spheres]]) by regarding it this way is the original $\mathbb{F}_p$-[[Adams spectral sequence]].

Or for instance for $E = $ [[MU]], then the 1-image of $Spec(MU) \to Spec(\mathbb{S})$ is already all of $Spec(S)$, hence $Spec(MU)$ is [[covering]]. The tool that computes $\mathbb{S}$, hence the full [[stable homotopy groups of spheres]], using this covering is the [[MU]]-[[Adams spectral sequence]], hence the [[Adams-Novikov spectral sequence]].

See at _[Adams spectral sequence -- As derived descent](Adams%20spectral%20sequence#DefinitionInHigherAlgebra)_ for more on this.

## Related concepts

* [spectrum of the category of spectra](prime+spectrum+of+a+symmetric+monoidal+stable+%28?,1%29-category#OfCategoryOfSpectra)

* [[Spec(Z)]], [[field with one element]]

* [[moduli stack of formal groups]]

* [[chromatic homotopy theory]]

* [[The Music of the Spheres]]


Spec(Z)

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

#Contents#
* table of conntents
{:toc}

## Idea

$Spec(\mathbb{Z})$ denotes the [[spectrum of a commutative ring|spectrum]] of the [[commutative ring]] $\mathbb{Z}$ of [[integers]]. Its [[closed points]] are the [[maximal ideals]] $(p)$, for each [[prime number]] $p$ in $\mathbb{Z}$, which are [[closed point|closed]], and the non-maximal [[prime ideal]] $(0)$, whose closure is the whole of $Spec(\mathbb{Z})$. For details see at _[[Zariski topology]]_ [this example](Zariski+topology#SpecZ).

Since $\mathbb{Z}$ is the [[initial object]] in the [[category]] [[CRing]] of [[commutative rings]], $Spec(\mathbb{Z})$ is the [[terminal object]] in the [[category]] of [[affine schemes]].

The [[gros topos|gros]] [[etale topos]] over $Spec(\mathbb{Z})$ is the context for [[arithmetic geometry]]. By the discussion at _[[Borger's absolute geometry]]_ it sits via an [[essential geometric morphism]] over the [[F1]]-[[topos]]:

$$
  Et(Spec(\mathbb{Z}))\longrightarrow Et(Spec(\mathbb{F}_1))
$$

## Properties

### As a 3-dimensional space
 {#As3dSpaceContainingKnots}

There are some phenomena that may be interpreted as $Spec(\mathbb{Z})$ behaving like a [[3-manifold]] in some ways.

#### As a 3-sphere containing knots

Several properties of $Spec(\mathbb{Z})$ make it behave as if of [[dimension]] 3. For instance $Spec(\mathbb{Z}) \cup \{\infty\}$ has [[étale cohomology|étale]] [[cohomological dimension]] equal to 3, up to 2-torsion ([Mazur 73](#Mazur73)). Moreover the [[étale fundamental group]] $\hat \pi_1(Spec(\mathbb{Z}) \cup \{\infty\})$ is trivial, and hence Mazur suggested that $Spec(\mathbb{Z}) \cup \{\infty\}$ is in fact analogous to the [[3-sphere]].

Similarly, the spectra $Spec(\mathbb{F}_q)$ of [[finite fields]]  look like compact 1-dimensional spaces -- [[circles]] -- in that their [[étale cohomology]] with $\mathbb{Z}_l$-[[coefficients]] for $l$ coprime to $q$ is $\mathbb{Z}_l$ in degrees 0 and 1 and vanishes in all higher degrees.

From this it is [[folklore]] (going back to [Mazur](#Mazur) and Manin, review includes [Deninger 05, section 8](#Deninger05), [Kohno-Morishita 06](#KohnoMorishita06)) that the spectra of [[prime fields]] with their canonical embedding into $Spec(\mathbb{Z})$

$$
  Spec(\mathbb{F}_p) \hookrightarrow Spec(\mathbb{Z})
$$

(formally dual to the canonical mod-$p$ [[projection]] $\mathbb{Z}\to \mathbb{F}_p$) are analogous to [[knots]] inside this 3-dimensional space (a good exposition is in [LeBruyn](#LeBruyn10)).

Observations like this give rise to the field of [[arithmetic topology]].

#### As a hyperbolic 3-manifold containing prime geodesics

However, in view of the [analogy between the Selberg zeta function and the Artin L-function](Selberg%20zeta%20function#AnalogyWithArtinLFunction) it might be more appropriate to think of $Spec(\mathbb{Z})$ as analogous to a  [[hyperbolic manifold]] of dimension 3 (see also [Fujiwara 07, slide 7](#Fujiwara07)) and then to think of finite field spectra as analogous to the [[prime geodesics]] in the manifold. This does not change the fact that every single $Spec(\mathbb{F}_p) \hookrightarrow Spec(\mathbb{Z})$ is like an embedded circle, hence like a knot, but it affects the perspective on which role these play. For instance there does not seem to be a differential geometric analog situation where one considers [[infinite products]] over all knots in a 3-space, but there are such situations where one considers infinite products over all [[prime geodesics]] in a space, namely the [[Selberg zeta function]] analogous to the [[Artin L-function]] with its product over [[prime ideals]].

### Function field analogy

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

[[!redirects Et(Spec(Z))]]

## Related concepts

* [[Spec(S)]]

* [[field with one element]]

## References

* {#Mazur73} [[Barry Mazur]], _Notes on étale cohomology of number fields_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 6 no. 4 (1973), p. 521-552 ([NUMDAM](http://www.numdam.org/numdam-bin/fitem?id=ASENS_1973_4_6_4_521_0))

* {#Mazur} [[Barry Mazur]], _Remarks on the Alexander polynomial_, unpublished note

* {#Deninger05} [[Christopher Deninger]], _Arithmetic Geometry and Analysis on Foliated Spaces_ ([arXiv:math/0505354](http://arxiv.org/abs/math/0505354))

* {#KohnoMorishita06} [[Toshitake Kohno]], [[Masanori Morishita]] (eds.), _Primes and Knots_, Contemporary Mathematics, AMS 2006 ([web](http://www.ams.org/bookstore-getitem/item=CONM-416))

* {#LeBruyn10} [[Lieven LeBruyn]], talk 2010 ([pdf slides](http://win.ua.ac.be/~lebruyn/LeBruyn2010d.pdf) (35 mb), [MO comment](http://mathoverflow.net/a/50995/381) (with more details))

* {#Fujiwara07} K. Fujiwara, _$p$-adic gauge theory in number theory_, 2007 ([pdf slides](http://www.ms.u-tokyo.ac.jp/~t-saito/conf/rv/Leopoldt.pdf))

* [[Bertrand Toën]], [[Michel Vaquié]], *Au-dessous de $Spec \mathbb{Z}$*, Journal of K-Theory **3** 3 (2009) 437-500 [[doi:10.1017/is008004027jkt048](https://doi.org/10.1017/is008004027jkt048)]

[[!redirects Spec Z]]


Specht module

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

#Contents#
* table of contents
{:toc}

## Idea

*Specht modules* are [[linear representations]] of the [[symmetric group]] $Sym(n)$ (for any $ \in \mathbb{N}$), hence [[modules]] over the [[group ring]] $k(Sym(n))$, which are indexed by the [[partitions]] of $n$.

For every partition (in other words [[Young diagram]]) $\lambda \vdash n$, the Specht module associated to $\lambda$ is usually denoted $S^{\lambda}$.

In [[characteristic 0]], they are [[irreducible representations|irreducible]] and exhaust the [[isomorphism classes]] of [[irreps]] (e.g. [Sagan 01, Thm. 2.4.6](#Sagan01)).

In this case, any finite dimensional representation $M$ of $Sym(n)$ is of the form
$$
M \cong \underset{\lambda \vdash n}{\bigoplus} M^{\lambda}
$$
where $M^{\lambda}$ is the isotypic component of $M$ associated to $\lambda$ which is of the form $M^{\lambda} \cong (S^{\lambda})^{\oplus^{m_{\lambda}}}$, with $m_{\lambda}$ called the multiplicity of $S^{\lambda}$ in $M$.

Over a field of [[positive characteristic]] $p$, where $p \mid n!$, the Specht modules are not irreducible, but every [[irreducible module]] does appear as the [[cosocle]] of a Specht module.

## Related concepts

* [[representation theory of the symmetric group]]

## References

Textbook accounts:

* {#Sagan01} [[Bruce Sagan]], Section 2.3 in: _The symmetric group_, Springer 2001 ([doi:10.1007/978-1-4757-6804-6](https://link.springer.com/book/10.1007/978-1-4757-6804-6), [pdf](http://math.sfsu.edu/federico/Clase/RepTh/sagan.pdf))

* {#LodayVallette12} [[Jean-Louis Loday]], [[Bruno Vallette]], Annex A in: *Algebraic Operads*, Grundlehren der mathematischen Wissenschaften **346**, Springer 2012 ([ISBN 978-3-642-30362-3](https://www.springer.com/gp/book/9783642303616), [pdf](https://pure.mpg.de/rest/items/item_3121746_1/component/file_3121747/content))

See also

* Wikipedia, *[Specht module](https://en.wikipedia.org/wiki/Specht_module)*

[[!redirects Specht modules]]


special characters

# Special characters
* table of contents
{: toc}

## Introduction

This page is to help you copy and paste special characters if you want them.  For the moment, it is discouraged to put these in [page names](http://ncatlab.org/nlab/show/HowTo#naming), although you may want to use them in [[redirected pages]].  It is also usually better to use iTeX when appropriate (mathematical characters outside of wiki links).

## Text symbols

Č (more to come!)

## Mathematical symbols

These are alphabetical by the LaTeX name.  If you don\'t know the LaTeX name, try [Detexify](http://detexify.kirelabs.org/) (requires Javascript).  Not included are commands like `\arccos` that produce ASCII, commands like `\textbullet` that are meant for text mode (use mathmode\'s `\bullet` instead), and commands like `\Bbbk` that produce font and size variants.

Although this is not intended as a list of iTeX commands (since the [itex symbols list](http://xbeta.org/wiki/show/itex) is pretty comprehensive and well-organised), inevitably people will use it that way too.  So please place an asterisk by names that are *not* recognised by iTeX (which you will discover for yourself when you add it).

Remember to use iTeX itself when appropriate.  You mostly will only need to copy and paste these symbols to produce redirects (and most of the time you shouldn\'t have to come here since the symbol will be on the page already).

`*`: $*$

`-`: $-$

`|`: $|$

`'`: $'$

\* `<`: $\lt$

\* `>`: $\gt$

`\|`: $\|$

### A

* `\aleph`: $\aleph$
* `\alpha`: $\alpha$
* `\amalg`: $\amalg$
* `\angle`: $\angle$
* `\approx`: $\approx$
* \* `\approxcolon`: <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><mo lspace='mediummathspace' rspace='mediummathspace'>≈∶</mo></math> (no single Unicode character)
* \* `\approxcoloncolon`: <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><mo lspace='mediummathspace' rspace='mediummathspace'>≈∷</mo></math> (no single Unicode character)
* `\approxeq`: $\approxeq$
* \* `\ast`: $∗$
* `\asymp`: $\asymp$

### B

* `\backepsilon`: $\backepsilon$
* `\backprime`: $\backprime$ (outside of the superscript)
* `\backsim`: $\backsim$
* `\backsimeq`: $\backsimeq$
* `\backslash`: $\backslash$
* \* `\baro`: $⌽$
* \* `\barwedge`: $⌅$
* \* `\bbslash`: $⑊$
* `\because`: $\because$
* `\beta`: $\beta$
* `\beth`: $\beth$
* `\between`: $\between$
* \* `\bigbox`: $⬜$
* `\bigcap`: $\bigcap$
* `\bigcirc`: $\bigcirc$
* `\bigcup`: $\bigcup$
* \* `\biginterleave`: $⫼$
* `\bigodot`: $\bigodot$
* `\bigoplus`: $\bigoplus$
* `\bigotimes`: $\bigotimes$
* \* `\bigparallel`: $║$
* \* `\bigsqcap`: $⨅$
* `\bigsqcup`: $\bigsqcup$
* `\bigstar`: $\bigstar$
* `\bigtriangledown`: $\bigtriangledown$
* `\bigtriangleup`: $\bigtriangleup$
* `\biguplus`: $\biguplus$
* `\bigvee`: $\bigvee$
* `\bigwedge`: $\bigwedge$
* \* `\binampersand`: $\&$
* \* `\bindnasrepma`: $\parr$
* `\blacklozenge`: $\blacklozenge$
* `\blacksquare`: $\blacksquare$
* `\blacktriangle`: $\blacktriangle$
* `\blacktriangledown`: $\blacktriangledown$
* \* `\blacktriangleleft`: $◂$
* \* `\blacktriangleright`: $▸$
* `\bot`: $\bot$
* `\bottom`: $\bottom$
* `\bowtie`: $\bowtie$
* `\Box`: $\Box$
* \* `\boxast`: $⧆$
* \* `\boxbox`: $⧈$
* \* `\boxbslash`: $⧅$
* \* `\boxcircle`: $⧇$
* \* `\boxdot`: $⊡$
* \* `\boxempty`: $\Box$
* `\boxminus`: $\boxminus$
* `\boxplus`: $\boxplus$
* \* `\boxslash`: $⧄$
* `\boxtimes`: $\boxtimes$
* `\bullet`: $\bullet$
* `\bumpeq`: $\bumpeq$
* `\Bumpeq`: $\Bumpeq$

### C

* `\cap`: $\cap$
* \* `\Cap`: $⋒$
* `\cdot`: $\cdot$
* \* `\cdotp`: $\cdot$
* `\cdots`: $\cdots$
* \* `\centerdot`: $·$
* \* `\checkmark`: $✓$
* `\chi`: $\chi$
* `\circ`: $\circ$
* `\circeq`: $\circeq$
* `\circlearrowleft`: $\circlearrowleft$
* `\circlearrowright`: $\circlearrowright$
* \* `\circledast`: $⊛$
* \* `\circledcirc`: $⊚$
* `\circleddash`: $\circleddash$
* \* `\circledR`: $®$
* \* `\circledS`: $Ⓢ$
* `\clubsuit`: $\clubsuit$
* \* `\complement`: $∁$
* `\coloneq`: $\coloneq$
* `\coloneqq`: $\coloneqq$
* `\cong`: $\cong$
* `\coprod`: $\coprod$
* \* `\copyright`: $©$
* `\cup`: $\cup$
* \* `\Cup`: $⋓$
* `\curlyeqprec`: $\curlyeqprec$
* `\curlyeqsucc`: $\curlyeqsucc$
* \* `\curlyvee`: $⋎$
* \* `\curlywedge`: $⋏$
* `\curvearrowleft`: $\curvearrowleft$
* `\curvearrowright`: $\curvearrowright$

### D

* `\dag`: $\dagger$
* `\dagger`: $\dagger$
* \* `\daleth`: $ℸ$
* `\dasharrow`: $\dashrightarrow$
* `\dashleftarrow`: $\dashleftarrow$
* `\dashrightarrow`: $\dashrightarrow$
* `\dashv`: $\dashv$
* `\ddag`: $\ddagger$
* `\ddagger`: $\ddagger$
* `\ddots`: $\ddots$
* \* `\degree`: $°$
* `\delta`: $\delta$
* `\Delta`: $\Delta$
* \* `\diagdown`: $\setminus$
* \* `\diagup`: $∕$
* `\diamond`: $\diamond$
* `\Diamond`: $\Diamond$
* `\diamondsuit`: $\diamondsuit$
* `\digamma`: $\digamma$
* `\div`: $\div$
* \* `\divideontimes`: $⋇$
* `\doteq`: $\doteq$
* \* `\Doteq`: $\doteqdot$
* `\doteqdot`: $\doteqdot$
* \* `\dotplus`: $∔$
* `\dots`: $\dots$
* \* `\doublecap`: $⋒$
* \* `\doublecup`: $⋓$
* \* `\doublebarwedge`: $⌆$
* `\downarrow`: $\downarrow$
* `\Downarrow`: $\Downarrow$
* `\downdownarrows`: $\downdownarrows$
* `\downharpoonleft`: $\downharpoonleft$
* `\downharpoonright`: $\downharpoonright$

### E

* `\ell`: $\ell$
* `\emptyset`: $\emptyset$
* `\epsilon`: $\epsilon$
* `\eqcirc`: $\eqcirc$
* \* `\eqsim`: $≂$
* `\eqslantgtr`: $\eqslantgtr$
* `\eqslantless`: $\eqslantless$
* `\equiv`: $\equiv$
* `\eta`: $\eta$
* `\eth`: $\eth$
* `\exists`: $\exists$

### F

* `\fallingdotseq`: $\fallingdotseq$
* \* `\fatsemi`: $⨟$
* \* `\fint`: $⨍$
* \* `\Finv`: $Ⅎ$
* `\flat`: $\flat$
* `\forall`: $\forall$
* `\frown`: $\frown$

### G

* \* `\Game`: $⅁$
* `\gamma`: $\gamma$
* `\Gamma`: $\Gamma$
* `\ge`: $\ge$
* `\geq`: $\geq$
* `\geqq`: $\geqq$
* `\geqslant`: $\geqslant$
* \* `\gets`: $\leftarrow$
* `\gg`: $\gg$
* `\ggg`: $\ggg$
* \* `\gggtr`: $\ggg$
* \* `\gimel`: $ℷ$
* `\gnapprox`: $\gnapprox$
* `\gneq`: $\gneq$
* `\gneqq`: $\gneqq$
* `\gnsim`: $\gnsim$
* `\gt`: $\gt$
* `\gtrapprox`: $\gtrapprox$
* `\gtrdot`: $\gtrdot$
* `\gtreqless`: $\gtreqless$
* `\gtreqqless`: $\gtreqqless$
* `\gtrless`: $\gtrless$
* `\gtrsim`: $\gtrsim$

### H

* `\hbar`: $\hbar$
* \* `\hdots`: $\cdots$
* `\heartsuit`: $\heartsuit$
* `\hookleftarrow`: $\hookleftarrow$
* `\hookrightarrow`: $\hookrightarrow$
* \* `\hslash`: $\hbar$

### I

* \* `\iff`: $⟺$ (present but incorrect in iTeX)
* `\iiiint`: $\iiiint$
* `\iiint`: $\iiint$
* `\iint`: $\iint$
* `\Im`: $\Im$
* `\imath`: $\imath$
* `\implies`: $\implies$
* `\in`: $\in$
* `\infty`: $\infty$
* \* `\inplus`: $⨭$
* `\int`: $\int$
* \* `\intercal`: $⊺$
* \* `\interleave`: $⫴$
* `\iota`: $\iota$

### J

* `\jmath`: $\jmath$
* \* `\Join`: $\bowtie$

### K

* `\kappa`: $\kappa$

### L

* `\lambda`: $\lambda$
* `\Lambda`: $\Lambda$
* \* `\land`: $\wedge$
* `\langle`: $\langle$
* `\lceil`: $\lceil$
* `\ldots`: $\ldots$
* `\le`: $\le$
* \* `\leadsto`: $↝$
* `\leftarrow`: $\leftarrow$
* `\Leftarrow`: $\Leftarrow$
* `\leftarrowtail`: $\leftarrowtail$
* \* `\leftarrowtriangle`: $⇽$
* \* `\leftharpoondown`: $↼$
* \* `\leftharpoonup`: $↽$
* `\leftleftarrows`: $\leftleftarrows$
* `\leftrightarrow`: $\leftrightarrow$
* `\Leftrightarrow`: $\Leftrightarrow$
* `\leftrightarrows`: $\leftrightarrows$
* \* `\leftrightarrowtriangle`: $⇿$
* `\leftrightharpoons`: $\leftrightharpoons$
* `\leftrightsquigarrow`: $\leftrightsquigarrow$
* \* `\leftslice`: $⪦$
* \* `\leftthreetimes`: $⋋$
* `\leq`: $\leq$
* `\leqq`: $\leqq$
* `\leqslant`: $\leqslant$
* `\lessapprox`: $\lessapprox$
* `\lessdot`: $\lessdot$
* `\lesseqgtr`: $\lesseqgtr$
* `\lesseqqgtr`: $\lesseqqgtr$
* `\lessgtr`: $\lessgtr$
* `\lesssim`: $\lesssim$
* `\lfloor`: $\lfloor$
* \* `\lgroup`: $〔$
* `\lhd`: $\lhd$
* \* `\LHD`: $◄$
* `\lightning`: $☇$
* `\ll`: $\ll$
* \* `\llbracket`: $⟦$
* `\Lleftarrow`: $\Lleftarrow$
* `\lll`: $\lll$
* \* `\llless`: $\lll$
* \* `\llparenthesis`: $⦇$
* \* `\lmoustache`: $⎰$
* `\lnapprox`: $\lnapprox$
* `\lneq`: $\lneq$
* `\lneqq`: $\lneqq$
* \* `\lnot`: $\neg$
* `\lnsim`: $\lnsim$
* `\longleftarrow`: $\longleftarrow$
* \* `\Longleftarrow`: $⟸$
* \* `\longleftrightarrow`: $⟷$
* \* `\Longleftrightarrow`: $⟺$ (present but incorrect in iTeX)
* \* `\longmapsfrom`: $↤$
* \* `\Longmapsfrom`: $⤆$
* \* `\longmapsto`: $↦$
* \* `\Longmapsto`: $⤇$
* `\longrightarrow`: $\longrightarrow$
* \* `\Longrightarrow`: $⟹$
* `\looparrowleft`: $\looparrowleft$
* `\looparrowright`: $\looparrowright$
* \* `\lor`: $\vee$
* `\lozenge`: $\lozenge$
* `\Lsh`: $\Lsh$
* `\lt`: $\lt$
* `\ltimes`: $\ltimes$
* \* `\lvert`: $\vert$
* \* `\lVert`: $\Vert$

### M

* \* `\mapsfrom`: $↤$
* \* `\Mapsfrom`: $⤆$
* `\mapsto`: $\mapsto$
* \* `\Mapsto`: $⤇$
* \* `\mathellipsis`: $\ldots$
* \* `\mathparagraph`: $¶$
* \* `\mathsection`: $§$
* \* `\mathsterling`: $£$
* `\measuredangle`: $\measuredangle$
* \* `\merge`: $⨇$
* `\mho`: $\mho$
* `\mid`: $\mid$
* `\models`: $\models$
* `\mp`: $\mp$
* `\mu`: $\mu$
* `\multimap`: $\multimap$

### N

* `\nabla`: $\nabla$
* \* `\napprox`: $≉$
* `\natural`: $\natural$
* `\ncong`: $\ncong$
* `\ne`: $\ne$
* `\nearrow`: $\nearrow$
* `\neg`: $\neg$
* `\neq`: $\neq$
* `\nequiv`: $\nequiv$
* `\nexists`: $\nexists$
* `\ngeq`: $\ngeq$
* `\ngtr`: $\ngtr$
* `\ni`: $\ni$
* \* `\nin`: $\notin$
* \* `\niplus`: $⨮$
* `\nleftarrow`: $\nleftarrow$
* `\nLeftarrow`: $\nLeftarrow$
* `\nleftrightarrow`: $\nleftrightarrow$
* `\nLeftrightarrow`: $\nLeftrightarrow$
* `\nleq`: $\nleq$
* `\nless`: $\nless$
* `\nmid`: $\nmid$
* `\notin`: $\notin$
* `\nparallel`: $\nparallel$
* `\nprec`: $\nprec$
* `\nrightarrow`: $\nrightarrow$
* `\nRightarrow`: $\nRightarrow$
* `\nsim`: $\nsim$
* \* `\nsimeq`: $≄$
* `\nsubseteq`: $\nsubseteq$
* `\nsucc`: $\nsucc$
* `\nsupseteq`: $\nsupseteq$
* \* `\ntriangleleft`: $⋪$
* \* `\ntrianglelefteq`: $⋬$
* \* `\ntriangleright`: $⋫$
* \* `\ntrianglerighteq`: $⋭$
* `\nu`: $\nu$
* `\nvdash`: $\nvdash$
* `\nvDash`: $\nvDash$
* \* `\nVdash`: $⊮$
* `\nVDash`: $\nVDash$
* `\nwarrow`: $\nwarrow$

### O

* \* `\oast`: $⊛$
* \* `\obar`: $⦶$
* \* `\oblong`: $⫾$
* \* `\ocircle`: $⊚$
* `\odot`: $\odot$
* \* `\ogreaterthan`: $⧁$
* \* `\ohm`: $\Omega$
* \* `\oiiint`: $∰$
* \* `\oiint`: $∯$
* `\oint`: $\oint$
* \* `\ointclockwise`: $∲$
* \* `\ointcounterclockwise`: $∳$
* \* `\olessthan`: $⧀$
* `\omega`: $\omega$
* `\Omega`: $\Omega$
* \* `\ominus`: $⊖$
* `\oplus`: $\oplus$
* \* `\oslash`: $⊘$
* `\otimes`: $\otimes$
* \* `\owns`: $\ni$

### P

* \* `\P`: $¶$
* `\parallel`: $\parallel$
* `\partial`: $\partial$
* \* `\permil`: $‰$
* `\perp`: $\perp$
* `\phi`: $\phi$
* `\Phi`: $\Phi$
* `\pi`: $\pi$
* `\Pi`: $\Pi$
* `\pitchfork`: $\pitchfork$
* `\pm`: $\pm$
* \* `\pounds`: $£$
* `\prec`: $\prec$
* `\precapprox`: $\precapprox$
* `\preccurlyeq`: $\preccurlyeq$
* `\preceq`: $\preceq$
* `\precnapprox`: $\precnapprox$
* \* `\precneqq`: $⪵$
* `\precnsim`: $\precnsim$
* `\precsim`: $\precsim$
* `\prime`: $\prime$ (outside of the superscript)
* \* `\Prime`: $″$ (outside of the superscript)
* `\prod`: $\prod$
* `\propto`: $\propto$
* `\psi`: $\psi$
* `\Psi`: $\Psi$

### Q

* `\qprime`: $⁗$ (outside of the superscript)

### R

* `\rangle`: $\rangle$
* `\rceil`: $\rceil$
* `\Re`: $\Re$
* \* `\restriction`: $\upharpoonright$
* `\rfloor`: $\rfloor$
* \* `\rgroup`: $〕$
* `\rhd`: $\rhd$
* \* `\RHD`: $►$
* `\rho`: $\rho$
* `\rightarrow`: $\rightarrow$
* `\Rightarrow`: $\Rightarrow$
* `\rightarrowtail`: $\rightarrowtail$
* \* `\rightarrowtriangle`: $⇾$
* \* `\rightharpoondown`: $⇁$
* \* `\rightharpoonup`: $⇀$
* `\rightleftarrows`: $\rightleftarrows$
* `\rightleftharpoons`: $\rightleftharpoons$
* `\rightrightarrows`: $\rightrightarrows$
* \* `\rightslice`: $⪧$
* `\rightsquigarrow`: $\rightsquigarrow$
* \* `\rightthreetimes`: $⋌$
* `\risingdotseq`: $\risingdotseq$
* \* `\rmoustache`: $⎱$
* \* `\rrbracket`: $⟧$
* `\Rrightarrow`: $\Rrightarrow$
* \* `\rrparenthesis`: $⦈$
* `\Rsh`: $\Rsh$
* `\rtimes`: $\rtimes$
* \* `\rvert`: $\vert$
* \* `\rVert`: $\Vert$

### S

* \* `\S`: $§$
* `\searrow`: $\searrow$
* `\setminus`: $\setminus$
* `\sharp`: $\sharp$
* `\sigma`: $\sigma$
* `\Sigma`: $\Sigma$
* `\sim`: $\sim$
* `\simeq`: $\simeq$
* `\smallsetminus`: $\smallsetminus$
* `\smile`: $\smile$
* `\spadesuit`: $\spadesuit$
* `\sphericalangle`: $\sphericalangle$
* `\sqcap`: $\sqcap$
* `\sqcup`: $\sqcup$
* `\sqsubset`: $\sqsubset$
* `\sqsubseteq`: $\sqsubseteq$
* `\sqsupset`: $\sqsupset$
* `\sqsupseteq`: $\sqsupseteq$
* `\square`: $\square$
* `\sslash`: $\sslash$
* `\star`: $\star$
* `\subset`: $\subset$
* `\Subset`: $\Subset$
* `\subseteq`: $\subseteq$
* `\subseteqq`: $\subseteqq$
* `\subsetneq`: $\subsetneq$
* `\subsetneqq`: $\subsetneqq$
* `\succ`: $\succ$
* `\succapprox`: $\succapprox$
* `\succcurlyeq`: $\succcurlyeq$
* `\succeq`: $\succeq$
* `\succnapprox`: $\succnapprox$
* \* `\succneqq`: $⪶$
* `\succnsim`: $\succnsim$
* `\succsim`: $\succsim$
* `\sum`: $\sum$
* `\supset`: $\supset$
* `\Supset`: $\Supset$
* `\supseteq`: $\supseteq$
* `\supseteqq`: $\supseteqq$
* `\supsetneq`: $\supsetneq$
* `\supsetneqq`: $\supsetneqq$
* `\surd`: $√$
* `\swarrow`: $\swarrow$

### T

* \* `\talloblong`: $⫿$
* `\tau`: $\tau$
* `\therefore`: $\therefore$
* `\theta`: $\theta$
* `\Theta`: $\Theta$
* `\times`: $\times$
* `\to`: $\to$
* `\top`: $\top$
* \* `\tprime`: $‴$ (outside of the superscript)
* `\triangle`: $\triangle$
* `\triangledown`: $\triangledown$
* `\triangleleft`: $\triangleleft$
* \* `\trianglelefteq`: $⊴$
* \* `\triangleq`: $≜$
* `\triangleright`: $\triangleright$
* \* `\trianglerighteq`: $⊵$
* `\twoheadleftarrow`: $\twoheadleftarrow$
* `\twoheadrightarrow`: $\twoheadrightarrow$

### U

* `\unlhd`: $\unlhd$
* `\unrhd`: $\unrhd$
* `\uparrow`: $\uparrow$
* `\Uparrow`: $\Uparrow$
* `\updownarrow`: $\updownarrow$
* `\Updownarrow`: $\Updownarrow$
* `\upharpoonleft`: $\upharpoonleft$
* `\upharpoonright`: $\upharpoonright$
* `\uplus`: $\uplus$
* `\upsilon`: $\upsilon$
* `\Upsilon`: $\Upsilon$
* `\upuparrows`: $\upuparrows$

### V

* `\varepsilon`: $\varepsilon$
* `\varkappa`: $\varkappa$
* `\varnothing`: $\varnothing$
* `\varphi`: $\varphi$
* `\varpi`: $\varpi$
* `\varpropto`: $\varpropto$
* `\varrho`: $\varrho$
* `\varsigma`: $\varsigma$
* `\vartheta`: $\vartheta$
* `\vdash`: $\vdash$
* `\vDash`: $\vDash$
* `\Vdash`: $\Vdash$
* `\VDash`: $\VDash$
* `\vdots`: $\vdots$
* `\vee`: $\vee$
* \* `\veebar`: $⊻$
* `\vert`: $\vert$
* `\Vert`: $\Vert$
* `\Vvdash`: $\Vvdash$

### W

* `\wedge`: $\wedge$
* `\wp`: $\wp$
* `\wr`: $\wr$

### X

* `\xi`: $\xi$
* `\Xi`: $\Xi$

### Y

### Z

* `\zeta`: $\zeta$

category: meta

[[!redirects unicode]]
[[!redirects Unicode]]
[[!redirects special character]]
[[!redirects special characters]]


special function

#Contents#
* table of contents
{:toc}

## Idea

__Special functions__ is more a name of a subject than a denotation of a well-defined class; they denote a _historically ever expanding class of [[functions]] of a [[complex numbers|complex]] [[variable]]_ coming from one of the several standard sources, most important being the solutions of second order [[differential equations]] with rational coefficients, as certain type of an infinite [[power series]], as members of a Fourier basis with respect to some orthogonality relation on an interval ("orthogonal polynomials"), by satisfaction of some "triangular" recursion relations, a type of integral representations (e.g. [[Selberg integral]]s) or arise in constructions of representation theory as certain subspaces of functions with prescribed covariance properties. Usually to accept some function as a special function one desires its appearance in several of the mentioned sources simultaneously. More rarely one looks at finite difference versions of differential equations, at functions of many variables or with parameters and similar generalizations.

Most of the examples of the classical special functions are special cases of the __hypergeometric series__ ${}_p F_q(a_1,\ldots,a_p; b_1,\ldots,b_q;x)$. While [[hypergeometric function]] is the sum of a hypergeometric series (analytically continued beyond the region of its convergence) sometime one says hypergeometric function for the case ${}_2 F_1$ as opposed to the general case of hypergeometric series ${}_p F_q$. There is a version in which some poles of a hypergeometric function collide, so called [[confluent hypergeometric function]]. There is also a very general [[elliptic hypergeometric function]].

Basic [[elementary function]]s (e.g. exponential, logarithmic, sine, cosine) are also special cases of hypergeometric series for certain values of parameters.

Very important class are so called __orthogonal polynomials__; they are neatly classified in so called "Askey-Wilson scheme", while they can be expressed as special values of hypergeometric series, they include special cases like Jacobi, Hermite, [[Legendre polynomial|Legendre]], Laguerre, Chebyshev polynomials etc.

Some other important special functions include [[gamma function]], [[Euler beta function]], [[theta functions]] ( [[Jacobi theta functions]], [[elliptic theta functions]]), Weierstrass elliptic $\mathcal{P}$-function, elliptic integrals, hyperelliptic integrals, [[mock theta functions]], [[Riemann zeta function]], ultraspherical polynomials, Whittaker functions, [[Airy funtion]], [[dilogarithm]], [[Bessel functions]]...

...and $q$-version of many of the mentioned functions (e.g. [[Ramanujan]]'s $q$/beta integral);

...many interesting functions with parameters or with many variables (e.g. [[Hall polynomial]]s, [[Macdonald polynomial]]s, Kostka polynomials...).

Unlike [[hypergeometric functions]] which are solutions of linear second order ODEs, the [[Painlevé transcendents]] are solutions of certain nonlinear ODEs.

## References

* G. E. Andrews, R. Askey, R. Roy, _Special functions_, Enc. of Math. and its Appl. 71 (1999)
* Н. Я. Виленкин, _Специальные функции и теория представления групп_

* Richard Askey, James Wilson, _A set of hypergeometric orthogonal polynomials_, SIAM J. Math. Anal. __13__ (1982), no. 4, 651–655, [doi](http://dx.doi.org/10.1137/0513043); _A recurrence relation generalizing those of Apéry_, J. Austral. Math. Soc. Ser. A 36 (1984), no. 2, 267–278;

* _Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials_, Mem. Amer. Math. Soc. __54__ (1985), no. 319, iv+55 pp.

Lecture notes motivated by [[partial differential equations]] appearing in [[mathematical physics]]:

* [[Cliff P. Burgess]], §10 in: *Primer on Partial Differential Equations for Physicists*, lecture notes (1990) [[pdf](https://physics.mcmaster.ca/~cburgess/Notes/mathphys.pdf), [[Burgess-PDEs.pdf:file]]]

[[!redirects special functions]]


special geometry

393
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[field theory]] with [[scalar field|scalar fields]], a frequent question is whether it can be reinterpreted as a [[sigma-model]], where the scalars are the coordinates of some [[target space]]. For [d=4 N=2 supergravity](https://ncatlab.org/nlab/show/D%3D4+supergravity#on___supergravity_2) and [[D=5 supergravity|d=5 N=2 supergravity]], such target spaces correspond to smooth manifolds with structures collectively known as *special geometry*.

## History

The term *special geometry* first appears in ([Strominger (1990)](#Strominger90)) in the context of d=4 N=2 theories. The structure described therein eventually became known as *special Kähler*, allowing to relate structures other than Kähler to the "special" structure, which essentially refers to the existence of some *special* [[affine space|affine coordinates]]. These variants lead to different [[supersymmetry]] multiplets.

## Definition

The starting point of special geometries is an [[Hermitian manifold]] (see ([Lopes Cardoso & Mohaupt (2020)](#CMhes20))). One then introduces additional compatible structure to get the different variants of special geometries.

### Affine special real geometry

\begin{definition}
An affine special real manifold is a [[Hessian manifold]] $(M,g,\nabla)$ such that the Hessian potential is cubic polynomial in $\nabla$-affine (i.e. special) coordinates.
\end{definition}

This structure appears in the context of five-dimensional theories with vector multiplets ([CM20](#CMhes20)).

### Special Kähler geometry

There are at least two definitions of special Kähler geometry on a [[smooth manifold]] $M$. One is local in nature, phrased in terms of the existence of [[coordinates]] satisfying certain conditions, first appearing in ([de Wit, Lauwers & van Proeyen (1985)](#DWLvP85)). The other definition is global, which we recall below. These two definitions are shown to be equivalent in [Strominger (1990)](#Strominger90).

\begin{definition}

Let $M$ be an $n$-dimensional [[Kähler manifold]] such that its Kähler form $J$ represents in [[de Rham cohomology]] the [[first Chern class]] of a complex [[line bundle]] $L$. Let $H$ be a [[holomorphic vector bundle|holomorphic]] $\text{Sp}(2n+2,\mathbb{R})$ (the [[symplectic group]]) vector bundle over $M$ and $(-,-)$ a compatible [[Hermitian form|Hermitian]] [[inner product|metric]] on $H$. Then $M$ is a **special Kähler manifold**, meaning it admits special geometry, if there exists a [[holomorphic function|holomorphic]] section $\Omega$ of $H\otimes L$ such that

$$
J
  \,=\,
  -
  \frac{\mathrm{i}}{2\pi}
 \partial\bar{\partial}
 \text{ln}
 (\Omega,\bar{\Omega})
  \,.
$$

\end{definition}

A more general notion is __special complex geometry__ defined in [ACD02](#AlekseevskyCortésDevchand02):  a complex manifold $(M,J)$ with a flat torsionfree connection $\nabla$ such that $\nabla J$ is symmetric.

## Examples

* The [[moduli spaces]] of [[3d Calabi-Yau object
|Calabi-Yau threefolds]] and of $c=9$ $(2,2)$ [[SCFTs]] both admit special geometries.

* The base of any algebraically completely [[integrable system]] has canonically defined special geometry and conversely, any special Kähler manifolds is a base of some integrable system ([Freed 1999](#Freed99)).

## Related concepts

* [[supergravity]]

* [[Hessian manifold]]

## References

* {#dWLvP85} [[Bernard de Wit]], P.G. Lauwers, [[Antoine Van Proeyen]]. *Lagrangians of N = 2 supergravity-matter systems*, Nuclear Physics B **255** (1985) 569--608 [doi](https://doi.org/10.1142/9789814542340_0040)

* {#Strominger90} [[Andrew Strominger]], _Special geometry_,  Comm. Math. Phys. __133__:1 (1990), 163--180 [[euclid:cmp/1104201320](http://projecteuclid.org/euclid.cmp/1104201320)]

* {#Freed99} [[Daniel Freed]], _Special Kähler manifolds_, Comm. Math. Phys. __203__ (1999) 31--52 [[doi:10.1007/s002200050604](https://doi.org/10.1007/s002200050604)]

* {#CMhes20} Gabriel Lopes Cardoso, [[Thomas Mohaupt]]. *Special geometry, Hessian structures and applications*. Physics Reports __855__ (2020) 1--141. ([doi](https://doi.org/10.1016/j.physrep.2020.02.002))

* B. Craps, F. Roose, W. Troost, [[Antoine Van Proeyen]], *What is special Kähler geometry?* , Nuclear Physics B __503__:3 (1997) 565--613 (doi arXiv:[hep-th/9703082](https://arxiv.org/abs/hep-th/9703082))

> We show equivalences of some definitions and give examples which show that earlier definitions are not equivalent, and are not sufficient to restrict the Kähler metric to one that occurs in N = 2 supersymmetry.

* [[Pietro Fré]], _Lectures on special Kähler geometry and electric-magnetic duality rotations_, Nucl. Phys. B Proc. Suppl. __45__:2-3, pp. 59--114 (1996) ([hep-th/9512043](https://arxiv.org/abs/hep-th/9512043) doi)

Freed's approach to special Kähler geometry is generalized to special complex geometry in

* {#AlekseevskyCortésDevchand02} [[Dmitry Vladimirovich Alekseevsky|D. V. Alekseevsky]], V. Cortés, C. Devchand, _Special complex manifolds_, Journal of Geometry and Physics __42__:1-2 (2002) 85--105 doi

[[!redirects special geometries]]


special holonomy

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

#Contents#
* table of contents
{:toc}

## Idea

For $X$ a [[space]] equipped with a $G$-[[connection on a bundle]] $\nabla$ (for some [[Lie group]] $G$) and for $x \in X$ any point, the [[parallel transport]] of $\nabla$ assigns to each [[curve]] $\gamma : S^1 \to X$ in $X$ starting and ending at $x$ an element $ hol_\nabla(\gamma) \in  G$: the [[holonomy]] of $\nabla$ along that curve.

The **holonomy group** of $\nabla$ at $x$ is the subgroup of $G$ on these elements.

If $\nabla$ is the [[Levi-Civita connection]] on a [[Riemannian manifold]] and the holonomy group is a proper subgroup $H$ of the [[special orthogonal group]], one says that $(X,g)$ is a manifold of **special holonomy** .

## Properties

### Classification

[[Berger's theorem]] says that if a [[manifold]] $X$ is

* [[simply connected topological space|simply connected]]

* neither locally a product nor a [[symmetric space]]

then the possible special holonomy groups are the following

[[!include special holonomy table]]

### Relation to $G$-reductions

A manifold having special holonomy means that there is a corresponding
[[reduction of structure groups]].

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

Let $(X,g)$ be a [[connected topological space|connected]] [[Riemannian manifold]]
of [[dimension]] $n$ with [[holonomy group]]  $Hol(g) \subset O(n)$.

For $G \subset O(n)$ some other [[subgroup]], $(X,g)$ admits a torsion-free
[[G-structure]] precisely if $Hol(g)$ is [[adjoint action|conjugate]] to
a [[subgroup]] of $G$.

Moreover, the space of such $G$-structures is the [[coset]] $G/L$,
where $L$ is the group of elements suchthat conjugating $Hol(g)$ with them
lands in $G$.

=--

This appears as ([Joyce prop. 3.1.8](#Joyce))

### Via $\mathbb{O}$-Riemannian manifolds

[[!include normed division algebra Riemannian geometry -- table]]

## Related concepts

* [[stable form]], [[definite form]]

* [[connection on a bundle]]

  * [[connection on a 2-bundle]], [[connection on an infinity-bundle]]

* [[parallel transport]], [[higher parallel transport]]

* [[holonomy]]

  * [[holonomy group]]

  * **special holonomy**, [[reduction of structure groups]], [[G-structure]],  [[exceptional geometry]], [[Walker coordinates]]

## References

### General

The classification in [[Berger's theorem]] is due to

* {#Berger} [[Marcel Berger]], _Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes_, Bull. Soc. Math. France 83 (1955)

For more see

* [[Nigel Hitchin]], _Special holonomy and beyond_, Clay Mathematics Proceedings ([pdf](https://people.maths.ox.ac.uk/hitchin/hitchinlist/clay.pdf))

* {#Joyce} [[Dominic Joyce]], _Compact manifolds with special holonomy_ , Oxford Mathematical Monographs (2000)

* Luis J. Boya, _Special holonomy manifolds in physics_ Monografías de la Real Academia de Ciencias de Zaragoza. 29: 37–47, (2006). ([pdf](http://www.unizar.es/acz/05Publicaciones/Monografias/MonografiasPublicadas/Monografia29/037.pdf))

Discussion of the relation to [[Killing spinors]] includes

* [[Andrei Moroianu]], [[Uwe Semmelmann]], _Parallel spinors and holonomy groups_, Journal of Mathematical Physics 41 (2000), 2395-2402 ([arXiv:math/9903062](https://arxiv.org/abs/math/9903062))

Discussion in terms of [[Riemannian geometry]] modeled on [[normed division algebras]] is in

* {#Leung02} [[Naichung Conan Leung]], _Riemannian geometry over different normed division algebras_, J. Diff. Geom. __61__:2 (2002) 289--333([euclid](https://projecteuclid.org/euclid.jdg/1090351387))

See also

* {#Freudenthal65} [[Hans Freudenthal]], _Lie groups in the foundations of geometry_, Advances in Mathematics, __1__ (1965) 145--190 ([dspace](http://dspace.library.uu.nl/handle/1874/17442))

On [[special holonomy orbifolds]]:

* [[Jeff Cheeger]], [[Gang Tian]], _Anti-self-duality of curvature and degeneration of metrics with special holonomy_, Commun. Math. Phys. __255__, 391--417 (2005) ([doi:10.1007/s00220-004-1279-0](https://doi.org/10.1007/s00220-004-1279-0))

### In supergravity and string theory

Discussion of special holonomy manifolds in [[supergravity]] and [[superstring theory]] as fiber-spaces for [[KK-compactifications]] preserving some [[number of supersymmetries]]:

* {#Gubser01} [[Steven Gubser]], _Special holonomy in string theory and M-theory_, In [[Steven Gubser]], Joseph Lykken (eds.) _Strings, Branes and Extra Dimensions - TASI 2001_, World Scientific 2004 ([arXiv:hep-th/0201114](https://arxiv.org/abs/hep-th/0201114), [doi:10.1142/5495](https://doi.org/10.1142/5495))

[[!redirects manifold with special holonomy]]
[[!redirects manifolds with special holonomy]]
[[!redirects special holonomy manifold]]

[[!redirects special holonomy orbifold]]
[[!redirects special holonomy orbifolds]]


special holonomy table

**classification of [[special holonomy]] [[manifolds]] by [[Berger's theorem]]:**

|  | $\,$[[G-structure]]$\,$ | $\,$[[special holonomy]]$\,$ | $\,$[[dimension]]$\,$ | $\,$preserved [[differential form]]$\,$ |
|--|--|--|--|--|
| $\,\mathbb{C}\,$ | $\,$[[Kähler manifold]]$\,$ | $\,$[[unitary group|U(n)]]$\,$ | $\,2n\,$ | $\,$[[Kähler forms]] $\omega_2\,$ |
|  | $\,$[[Calabi-Yau manifold]]$\,$ | $\,$[[special unitary group|SU(n)]]$\,$ | $\,2n\,$ | |
| $\,\mathbb{H}\,$ | $\,$[[quaternionic Kähler manifold]]$\,$ | $\,$[[Sp(n).Sp(1)]]$\,$ | $\,4n\,$ | $\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$   |
| | $\,$[[hyper-Kähler manifold]]$\,$ | $\,$[[symplectic group|Sp(n)]]$\,$ | $\,4n\,$ | $\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,$ ($a^2 + b^2 + c^2 = 1$)  |
| $\,\mathbb{O}\,$ | $\,$[[Spin(7) manifold]]$\,$ | $\,$[[Spin(7)]]$\,$ | $\,$8$\,$ | $\,$[[Cayley form]]$\,$ |
|  | $\,$[[G2 manifold]]$\,$ | $\,$[[G2]]$\,$ | $\,7\,$ | $\,$[[associative 3-form]]$\,$ |


special Lagrangian submanifold

(...)

## Related concepts

* [[Lagrangian submanifold]]

* [[submanifold]]

## References

* Wikipedia, _[Special Lagrangian submanifolds](https://en.wikipedia.org/wiki/Symplectic_manifold#Special_Lagrangian_submanifolds)_

[[!redirects special Lagrangian submanifolds]]

[[!redirects special lagrangian submanifold]]
[[!redirects special lagrangian submanifolds]]


special lambda-ring

+-- {: .query}

[[John Baez]]: I believe 'special $\lambda$-ring' is an old-fashioned term for what almost everyone now calls a [[lambda-ring]], the 'nonspecial' ones having been found to be too general.  This, at least, is what Hazewinkel says in his article cited on our page about [[lambda-ring]]s.   So I believe this page here should be folded in with [[lambda-ring]].

At the very least, we should give both definitions of $\lambda$-ring --- special and unspecial --- over at [[lambda-ring]].  When I last checked, that page did not include a definition.

=--

For any [[commutative ring]] $A$ we can consider the set $\Phi(A)$ of [[power series]] in an indeterminate $t$ with coefficients in $A$ whose constant term is $1$:

  $f(t) = 1 + a_1t + a_2t^2 + \ldots$

These form an [[abelian group]] under multiplication with the constant power-series $1$ as unit. What may be less familiar is that there is a commutative associative binary operation $\circ$ on this set that distributes over multiplication making $\Phi(A)$ into a commutative ring, for which the function $\epsilon_A: \Phi(A) \rightarrow A$ taking $f(t)$ to $f'(0)$ is a homomorphism. So what is usually called multiplication of power-series becomes _addition_ in this ring, with $1$ as zero; very confusing. Of course, $\Phi$ is a functor from rings to rings and $\epsilon$ is a natural transformation. In fact it is the counit of a [[comonad]].

How is $\circ$ defined? We impose the condition

$(1 + a t)\circ f(t) = f(a t)$

for $a\in A$ and $f(t)\in \Phi(A)$. It is now clear from distributivity that if $g(t) = \Pi_j(1+a_j t)$ then $g(t)\circ f(t) = \Pi_j f(a_j t)$. But what happens if $g(t)$ is not a product of linear factors? Newton's theorem on [[symmetric polynomial]]s comes to the rescue. For we note that the coefficient of $t^n$ in $\Pi_j f(a_j t)$ is a symmetric function in the $a_j$ and so can be expressed as a polynomial over the integers in the first $n$ coefficients of $g(t)$ and of $f(t)$. In this way we have indicated that a universal formula for multiplication in $\Phi(A)$ exists, though we may not have written it down explicitly.

We use the same trick in defining the comultiplication

$\mu_A : \Phi(A) \rightarrow \Phi(\Phi(A))$

but now our old-fashioned notation and use of indeterminates starts to cause trouble. A power-series in
$\Phi(A)$ is really just a sequence $(a_1, a_2, \ldots)$.
We demand that

$\mu_A (a,0,0, \ldots) = ((a,0,0, \ldots),(0,0, \ldots), \ldots )$

Again, to define $\mu_A$ on an arbitrary power-series, factorize it formally into linear factors, apply the rule and distributivity, and apply Newton's theorem.

A **special $\lambda$-ring** is a [[coalgebra]] for the comonad $\Phi$ above. If $\xi : A \rightarrow \Phi(A)$ is the costructure map for such a coalgebra, we define unary operations $\lambda^n$ on $A$ by the formula

$\xi(a) = 1 + \lambda^1(a)t + \lambda^2(a)t^2 + \ldots$

The counit condition forces $\lambda^1(a) = a$. It is also traditional to denote $\xi(a)$ by $\lambda_t (a)$. Note that $\lambda_t(a_1 + a_2) = \lambda_t(a_1)\lambda_t(a_2)$ and that $\lambda_t(a_1 a_2) = \lambda_t(a_1)\circ\lambda_t(a_2)$.

The functor $\Phi$ is representable by a commutative Hopf algebra $\Lambda$, and so has a left adjoint. The underlying ring of $\Lambda$ is $\mathbb{Z}[\lambda^1,\lambda^2, \ldots ]$, the free special $\lambda$-ring on one generator ($\lambda^1$).

In the terminology of [[bimodel|bimodels]] $\Phi = \Lambda\Rightarrow?$ and its left adjoint is $\Lambda\otimes?$.
So the theory of special $\lambda$-rings is a monadic extension of the theory of rings.

[[!redirects SpecialLambdaRing]]
[[!redirects special lambda-rings]]
[[!redirects special ∞-ring]]
[[!redirects special ∞-rings]]


special linear group

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## Idea

Given a [[field]] $k$ and a [[natural number]] $n \in \mathbb{N}$, the **special linear group** $SL(n,k)$ (or $SL_n(k)$) is the [[subgroup]] of the [[general linear group]] $SL(n,k) \subset GL(n,k)$ consisting of those linear transformations that preserve the [[volume form]] on the vector space $k^n$.  It can be canonically identified with the group of $n\times n$ [[matrix|matrices]] with entries in $k$ having [[determinant]] $1$.

This group can be considered as a sub[[variety]] of the [[affine scheme|affine space]] $M_{n\times n}(k)$ of square matrices of size $n$ carved out by the equations saying that the [[determinant]] of a matrix is 1.  This [[variety]] is an [[algebraic group]] over $k$, and if $k$ is the field of [[real number|real]] or [[complex numbers]] then it is a [[Lie group]] over $k$.

## Properties

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

The [[special linear group]] $SL_n(\mathbb{F})$ is a [[perfect group]] for any [[field]] $\mathbb{F}$ and any $n \geq 1$, except for the cases of the [[prime fields]] $\mathbb{F}_2$ and $\mathbb{F}_3$.

=--

See for example [here](http://people.brandeis.edu/~igusa/Math101b/SL.pdf), or [Lang 02, theorems XIII 8.3 and 9.2](#Lang02).

## Examples

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

The first case admitted by Prop. \ref{SLOverFieldIsPerfect} is the [[binary icosahedral group]] ([this Prop.](icosahedral+group#IsomorphismToSL25)):

$$
  SL(2,\mathbb{F}_5)
  \;\simeq\;
   2I
$$

=--

* [[SL(2,Z)]]

* not quite an example: [[SL(2,O)]] (over the [[octonions]])

## Related concepts

* [[general linear group]],

* [[projective special linear group]]

* [[special unitary group]]

* [[congruence subgroup]]

* [[special linear Lie algebra]]

## References

* {#Lang02} [[Serge Lang]], _Algebra_, $3^{rd}$ edition, Springer 2002 ([pdf](https://math24.files.wordpress.com/2013/02/algebra-serge-lang.pdf))

[[!redirects special linear group]]
[[!redirects special linear groups]]

[[!redirects SL(n)]]


special linear Lie algebra

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## Idea,

For $k$ a [[field]] and $n$ a [[natural number]], the _special linear Lie algebra_ $\mathfrak{sl}(n,k)$ is the [[Lie algebra]] of [[trace]]-free $n\times n$-[[matrices]] with entries in $k$, with [[Lie bracket]] being the [[commutator]] of [[matrix multiplication]].

## Examples

* [[sl(2)]]

## Related concepts

* [[special linear group]]

* [[special unitary Lie algebra]]

* [[special orthogonal Lie algebra]]

* [[Lie algebra weight system]]

[[!redirects special linear Lie algebras]]

[[!redirects sl(n)]]


special orthogonal group

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## Definition

The _special orthogonal group_ or _rotation group_, denoted $SO(n)$, is the [[group]] of [[rotations]] in a  [[Cartesian space]] of [[dimension]] $n$.

This  is one of the [[classical Lie groups]]. It is the [[connected component]] of the neutral element in the [[orthogonal group]] $O(n)$.

For instance for $n=2$ we have $SO(2)$ the [[circle group]].

It is the first step in the [[Whitehead tower]] of $O(n)$

$$
  \cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n)
  \,,
$$

the next step of which is the [[spin group]].

In [[physics]] the rotation group is related to [[angular momentum]].

## Properties

### Homology and cohomology
 {#HomologyAndCohomology}

On the [[ordinary cohomology]] of the [[classifying spaces]] $B O(n)$ and $B SO(n)$

* with [[cyclic group of order 2|$\mathbb{Z}_2$]] [[coefficients]]:

  generated by the [[Stiefel-Whitney classes]]

  (e.g. [Milnor & Stasheff 74, Theorem 7.1. & Theorem 12.4.](#MilnorStasheff74))

* with [[integer]] [[coefficients]] ([[integral cohomology]]):

  generated by the [[integral Stiefel-Whitney classes]] and the [[Pontrjagin classes]]

  (e.g. [Brown 82](#Brown82), [Feshbach 83](#Feshbach83), [Pittie 91](#Pittie91), [Rudolph-Schmidt 17, Theorem 4.2.23](#RudolphSchmidt17))

### As part of the ADE pattern

[[!include ADE -- table]]

### Relation to orientation of manifolds

For $X$ an $n$-dimensional [[manifold]] a lift of [[generalized the|the]] classifying map $X \to \mathcal{B}O(n)$ of the $O(n)$-[[principal bundle]] to which the [[tangent bundle]] $T X$ is [[associated bundle|associated]] is the same as a choice of [[orientation]] of $X$.

## Examples

* For the almost degenerate case $n = 2$ there are exceptional [[isomorphisms]] of [[Lie groups]]

  $$
    SO(2) \simeq U(1) \simeq Spin(2)
  $$

  with the [[circle group]] and [[spin group]] in dimension 2.

* the case of [[SO(8)]] is special, since in the [[ADE classification]] of [[simple Lie groups]] it corresponds to [[D4]], which makes its [[representation theory]] enjoy _[[triality]]_.

\linebreak

[[!include low dimensional rotation groups -- table]]

\linebreak

## Related concepts

$\cdots \to$ [[Fivebrane group]] $\to$ [[string group]] $\to$ [[spin group]] $\to$ **special orthogonal group** $\to$ [[orthogonal group]].

[[!include table of orthogonal groups and related]]

* [[Euler angles]]

* [[finite rotation group]]

## References
 {#References}

For general references see also at _[[orthogonal group]]_.

[[ordinary cohomology]] of the [[classifying spaces]]:

* {#MilnorStasheff74} [[John Milnor]], [[Jim Stasheff]], _Characteristic classes_, Princeton Univ. Press (1974) [[ISBN:9780691081229](https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76), [doi:10.1515/9781400881826](https://doi.org/10.1515/9781400881826), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf)]

* {#Brown82} [[Edgar H. Brown]], _The Cohomology of $B SO_n$ and $BO_n$ with Integer Coefficients_, Proceedings of the American Mathematical Society Vol. 85, No. 2 (Jun., 1982), pp. 283-288 ([jstor:2044298](https://www.jstor.org/stable/2044298))

* {#Feshbach83} Mark Feshbach, _The Integral Cohomology Rings of the Classifying Spaces of $\mathrm{O}(n)$ and $\mathrm{SO}(n)$,_ Indiana Univ. Math. J.  32 (1983), 511-516 ([doi:10.1512/iumj.1983.32.32036](https://doi.org/10.1512/iumj.1983.32.32036))

* {#Pittie91} [[Harsh Pittie]], _The integral homology and cohomology rings of SO(n) and Spin(n)_, Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (doi:10.1016/0022-4049(91)90108-E)

* [[Howard Georgi]], §21 & 22 in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

  > with an eye towards application to [[spinors]] in (the [[standard model of particle physics|standard model]] of) [[particle physics]]

* {#RudolphSchmidt17} [[Gerd Rudolph]], [[Matthias Schmidt]], around Theorem 4.2.23 of: _Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields_, Theoretical and Mathematical Physics series, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

See also

* {#Stasheff13} [[Jim Stasheff]], _The topology and algebra of $SO(n-1) \to SO(n) \to S^{n-1}$_, Herman's seminar July 2013 ([[StasheffSOn.pdf:file]])

[[!redirects special orthogonal groups]]

[[!redirects rotation group]]
[[!redirects rotation groups]]

[[!redirects SO(n)]]


special relativity

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## Idea

In [[physics]] the [[theory (physics)|theory]] of _special relativity_ describes dynamics in [[physics|physical]] [[spacetime]] in absence of a field of [[gravity]].

Specifically, it models

* [[spacetime]] as a flat time-oriented [[pseudo-Riemannian manifold]]:
  [[Minkowski space]];

* [[trajectories]] of point [[particles]] as timelike [[curves]] in Minkowski space.

* the [[electromagnetic field]] as a closed 2-[[differential form|form]] on Minkowski space.

* the [[Lorentz force]] exerted by the electromagnetic field on a charged particle as the contraction of that 2-form with the tangent vector of the trajectory of the particle.

If instead of restricting to Minkowski space, spacetime is allowed to be an arbitrary [[pseudo-Riemannian manifold]] one speaks of the resulting framework as the theory of _[[general relativity]]_.

Mathematically, special relativity is essentially the theory of [[Minkowski space]] as a [[pseudo-Riemannian manifold]], hence equipped with the constant [[metric]] [[tensor]] which at each point is the [[bilinear form]] with [[signature of a quadratic form|signature]] $(-,+, \cdots, +)$ or else $(+,-,\cdots, -)$ (depending on convention). The null-vectors in this metric characterize the [[speed of light]]. The linear [[isometries]] of this metric are the [[Lorentz transformations]] and the fact that with the Minkowski metric these also preserve, hence, null rays is the mathematical model for the famous [[phenomenology|phenomenological]] observation that "the speed of light is the same in all reference systems".

## Related concepts

* [[speed of light]]

  [[Michelson-Morley experiment]]

* [[theory (physics)]]

* [[electromagnetism]]

* [[general relativity]], [[gravity]]

* [[relativistic particle]]

* [[relativistic field theory]]

## References

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

[[!redirects special relativity theory]]

[[!redirects theory of special relativity]]


special unitary group

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## Idea

The _special unitary group_ is the [[subgroup]] of the [[unitary group]] on the elements with [[determinant]] equal to 1.

For $n$ a [[natural number]],  the **special unitary group** $SU(n)$ is the [[group]] of [[isometry|isometries]] of the $n$-dimensional [[complex number|complex]] [[Hilbert space]] $\mathbb{C}^n$ which preserve the [[volume form]] on this space. It is the [[subgroup]] of the [[unitary group]] $U(n)$ consisting of the $n \times n$ [[unitary matrix|unitary]] [[matrix|matrices]] with [[determinant]] $1$.

More generally, for $V$ any complex [[vector space]] equipped with a nondegenerate [[Hermitian form]] $Q$, $SU(V,Q)$ is the group of isometries of $V$ which preserve the volume form derived from $Q$.  One may write $SU(V)$ if $Q$ is obvious, so that $SU(n)$ is the same as $SU(\mathbb{C}^n)$.  By $SU(p,q)$, we mean $SU(\mathbb{C}^{p+q},Q)$, where $Q$ has $p$ positive [[eigenvalue]]s and $q$ negative ones.

## Properties

### As part of the ADE pattern

[[!include ADE -- table]]

### Representation theory

See at *[[representation theory of the special unitary group]]*.

## Examples

### $SU(2)$
 {#SU2}

We discuss aspects of [[SU(2)]], hence

$$
  SU(2) \coloneqq SU(2,\mathbb{C}) = SU(\mathbb{C}^2)
  \,.
$$

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

As a [[matrix group]] $SU(2)$ is equivalent to the subgroup of the [[general linear group]] $GL(2, \mathbb{C})$ on those of the form

$$
  \left(
    \array{
      u & v
      \\
      - \overline{v} & \overline{u}
    }
  \right)
  \;\;\;
   with
  \;\;
   {\vert u\vert}^2 + {\vert v\vert}^2 = 1
  \,,
$$

where $u,v \in \mathbb{C}$ are [[complex numbers]] and $\overline{(-)}$ denotes [[complex conjugation]].

=--

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

The underlying [[manifold]] of $SU(2)$ is [[diffeomorphism|diffeomorphic]] to the 3-[[sphere]] $S^3$.

=--

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

There is an [[isomorphism]] of [[Lie groups]]

$$
  SU(2) \simeq Spin(3)
$$

with the [[spin group]] in [[dimension]] 3.

=--

See at _[spin group -- Exceptional isomorphisms](spin%20group#ExceptionalIsomorphisms)_.

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

The [[Lie algebra]] $\mathfrak{su}(2)$ as a [[matrix Lie algebra]] is the sub Lie algebra on those matrices of the form

$$
  \left(
    \array{
      i z & x + i y
      \\
      - x + i y & - i z
    }
  \right)
  \;\;\;
  with
  \;\;
  x,y,z \in \mathbb{R}
  \,.
$$

=--

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

The standard [[basis]] elements of $\mathfrak{su}(2)$ given by the above presentation are

$$
  \sigma_1
   \coloneqq
  \left(
   \array{
     0 & 1
     \\
     -1 & 0
   }
  \right)
$$

$$
  \sigma_2
   \coloneqq
  \left(
   \array{
     0 & i
     \\
     i & 0
   }
  \right)
$$

$$
  \sigma_3
   \coloneqq
  \left(
   \array{
     i & 0
     \\
     0 & -i
   }
  \right)
  \,.
$$

These are called the _[[Pauli matrices]]_.

=--

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

The [[Pauli matrices]] satisfy the [[commutator]] relations

$$
  [\sigma_1, \sigma_2] = 2\sigma_3
$$

$$
  [\sigma_2, \sigma_3] = 2\sigma_1
$$

$$
  [\sigma_3, \sigma_1] = 2\sigma_2
  \,.
$$

=--

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

The [[maximal torus]] of $SU(2)$ is the [[circle group]] $U(1)$. In the above [[matrix group]] presentation this is naturally identified with the subgroup of matrices of the form

$$
  \left(
    \array{
      t & 0
      \\
      0 & t^{-1}
    }
  \right)
  \;\;
  with
  t \in U(1) \hookrightarrow \mathbb{C}
  \,.
$$

=--

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

The [[coadjoint orbits]] of the [[coadjoint action]] of $SU(2)$ on $\mathfrak{su}(2)$ are equivalent to the subset of the above matrices with $x^2 + y^2 + z^2 = r^2$ for some $r \geq 0$.

These are _regular_ coadjoint orbits for $r \gt 0$.

=--

### $SU(3)$

* [[SU(3)]]

### $SU(4)$
 {#SU4}

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

There is an [[isomorphism]] of [[Lie groups]]

[[SU(4)]] $\simeq$ [[Spin(6)]]

with the [[spin group]] in [[dimension]] 6.

=--

See at _[spin group -- Exceptional isomorphisms](spin%20group#ExceptionalIsomorphisms)_.

## Related concepts

* For [[SU(2)]]:

  * [[quaternion]]

  * [[geometric quantization of the 2-sphere]]

* [[Euler angles]]

* [[Pauli group]]

## References

* [[Howard Georgi]], §13  in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

  > with an eye towards application to (the [[standard model of particle physics|standard model]] of) [[particle physics]]

[[!redirects special unitary group]]
[[!redirects special unitary groups]]

[[!redirects Pauli matrix]]
[[!redirects Pauli matrices]]

[[!redirects SU(N)]]
[[!redirects SU(n)]]


special unitary Lie algebra

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## Idea

The [[Lie algebra]] $\mathfrak{su}(n)$ of the [[special unitary group]] $SU(n)$. Canonically identified with [[matrices]] with [[complex number]] entries that are skew-hermitean and have vanishing [[trace]].

## Examples

* [[su(2)]]

## Related concepts

* [[unitary Lie algebra]]

* [[special orthogonal Lie algebra]]

* [[special linear Lie algebra]]

## References

Textbook accounts:

* Walter Pfeifer, _The Lie algebras $\mathfrak{su}(N)$_, Birkhäuser Verlag 2003 ([doi:10.1007/978-3-0348-8097-8](https://doi.org/10.1007/978-3-0348-8097-8))

[[!redirects special unitary Lie algebras]]

[[!redirects su(n)]]


special values of L-functions

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## Idea

At certain [[integers]] $s = n$ the values $L(s)$ of  [[L-functions]] $L$ are typically of special interest, these are known as the "special values of L-functions".

For instance some of special values of the [[Riemann zeta function]] found (for the non-trivial region of non-positive integers) by [[Leonhard Euler]] in 1734 and 1749 are

| n | -5 | -3 | -1 |  0 | 2 | 4 | 6 | 8 |
|---|----|----|----|----|---|---|---|---|
| $\zeta(n)$ | $-\frac{1}{252}$ |  $\frac{1}{120}$ | $-\frac{1}{12}$ | $-\frac{1}{2}$ | $\frac{\pi^2}{6}$ | $\frac{\pi^4}{90}$ | $\frac{\pi^6}{945}$ | $\frac{\pi^8}{9450}$

where for instance the value $\zeta(-1) = -\frac{1}{12}$ turns out to be the [Euler characteristic of the moduli stack of complex elliptic curves](moduli+stack+of+elliptic+curves#EulerCharacteristic) and as such controls much of [[string theory]] (notably the critical dimension).

### Periods and Deligne's conjecture

More generally, it is known that for $K$ a [[number field]] and $\zeta_K$ its [[Dedekind zeta function]], then all values $\zeta_K(n)$ for integer $n$ happen to be [[periods]] ([MO comment](http://mathoverflow.net/a/54244/381)).

Based on this ([Deligne 79](#Deligne79)) identified _critical values_ of $L$-functions (at certain integers) and [[conjecture|conjectured]] that these all are algebraic multiples of [[determinants]] of [[matrices]] whose entries are [[periods]]. Review of this relation to periods in in ([Konsevich Zagier, section 3](#KonsevichZagier)).

Notice that under the [[function field analogy]] these arithmetic $L$-functions are supposed to be analogous to [[eta functions]] and [[zeta functions of elliptic differential operators]], which, when regarding these operators as [[Dirac operators]]/[[Hamiltonians]] encode [[partition functions]] and [[path integrals]] of [[quantum mechanical systems]], and that [[periods]] naturally appear as [[expectation values]] in [[quantum field theory]] (highlighted in [Kontsevich 99](deformation%20quantization#Kontsevich99)).

### Regulators and Beilinson's conjectures

Another kind of special value is given by the classical [[class number formula]] which says that the [[residue]] of the [[Dedekind zeta function]] $\zeta_K$ of a [[number field]] $K$ at $n = 1$ is proportional to the [[regulator of a number field|regulator of the number field]].

Also the [[Birch and Swinnerton-Dyer conjecture]] expresses the first non-vanishing [[derivative]] of the $L$-series of an [[elliptic curve]] over $\mathbb{Q}$ at $s= 1$ in terms of the [[regulator of a number field|regulator]].

Motivated by this ([Beilinson 85](#Beilinson85)) generalized both Deligne's conjecture and the statements about regulators by defining [[higher regulators]] ("Beilinson regulators") and conjecturing a general statement that expresses special values of $L$-function in terms of these, now known as _[[Beilinson's conjectures]]_.

### Periods and Scholl's conjecture

However ([Scholl 91](#Scholl91)) observed that also these [[Beilinson regulators]] have an expression in terms of [[periods]] and ([Konsevich Zagier, section 3.2 and 3.5](#KonsevichZagier)) advertises the

**Deligne-Beilinson-Scholl conjecture**: _For $r$ the order of vanishing of a motivic $L$-function at some integer $n$, then the $r$th [[derivative]] of the function at that point is a [[period]]._

### Relative Langlands and boundary conditions in topological quantum field theory

Ongoing by work by [[David Ben-Zvi]], [[Yiannis Sakellaridis]], and [[Akshay Venkatesh]] relate special values of L-functions and periods to boundary conditions in a  [[topological quantum field theory]] approach to the [[Langlands program]] (see [BenZvi](#BenZvi)).

## Related concepts

* [[Beilinson conjectures]]

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## References

Original article on the main conjectures are

* {#Deligne79} [[Pierre Deligne]], _Valeurs de fonctions $L$ et périodes d'intégrales_, in  _Automorphic forms, Representations, and $L$-functions_, Proc. Symp. Pure Math. 33, AMS (1979) pages 313-346

* {#Beilinson85} [[Alexander Beilinson]], _Higher regulators and values of L-functions_, Journal of Soviet Mathematics 30 (1985), 2036-2070 ([web (Russian original)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng))

* {#Scholl91} [[Anthony Scholl]], _Remarks on special values of L-functions_, in J. Coates, M. Taylor (eds.) _$L$-Functins and Arithmetic_, London Math. Soc.  Lecture Notes 153, Cambridge University Press 1991, pages 373-392 ([pdf](https://www.dpmms.cam.ac.uk/~ajs1005/preprints/remarks.pdf))

Review and further developments:

* [[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)

* {#KonsevichZagier} [[Maxim Kontsevich]], [[Don Zagier]], section 3 of _Periods_ ([pdf](http://www.ihes.fr/~maxim/TEXTS/Periods.pdf))

* {#Flach} [[Matthias Flach]], _The equivariant Tamagawa Number Conjecture: A survey_ ([pdf](http://www.math.caltech.edu/papers/baltimore-final.pdf))

* [[Bruno Kahn]], _Fonctions zêta et $L$ de variétés et de motifs_, [arXiv:1512.09250](http://arxiv.org/abs/1512.09250v1).

* {#BenZvi} [[David Ben-Zvi]], _Relative Langlands_, MSRI lectures (notes taken by Jackson van Dyke) [pdf](https://www.msri.org/workshops/918/schedules/28233/documents/50487/assets/88599)

[[!redirects special value of an L-function]]


specialization order

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# The specialisation order
* table of contents
{: toc}

## Idea

The __specialisation order__ is a way of turning any [[topological space]] $X$ into a [[preorder]]ed set (with the same underlying set).

## Definition

+-- {: .un_defn}
###### Definition
Given a topological space $X$ with topology $\mathcal{O}(X)$, the specialization order $\leq$ is defined by either of the following two equivalent conditions:

1. $x \leq y$ if and only if $x$ belongs to the [[topological closure]] of $\{y\}$; we say that $x$ is a __specialisation__ of $y$.

1. $x \leq y$ if and only if $\forall_{U \colon \mathcal{O}(X)} (x \in U) \Rightarrow (y \in U).$

(Note: some authors use the opposite ordering convention.)
=--

## Properties

The specialization order on a topological space is always a [[preorder]], because the specialization order is defined as $\forall_{U \colon \mathcal{O}(X)} (x \in U) \Rightarrow (y \in U)$. Given any two [[sets]] $A$ and $B$ and any [[binary relation]] $R(x, y)$ between $A$ and $B$,  the relation $\forall_{w \colon B} R(x, w) \implies R(y, w)$ is a [[preorder]] on $A$.

$X$ is $T_0$ if and only if its specialisation order is a [[partial order]].  $X$ is $T_1$ iff its specialisation order is [[equality]].  $X$ is $R_0$ (like $T_1$ but without $T_0$) iff its specialisation order is an [[equivalence relation]].  (See [[separation axioms]].)

Given a [[continuous map]] $f: X \to Y$ between topological spaces, it is order-preserving relative to the specialisation order.  Thus, we have a [[faithful functor]] $Spec$ from the category of $\Top$ of topological spaces to the category $\ProSet$ of preordered sets.

In the other direction, to each proset $X$ we may associate a topological space whose elements are those of $X$, and whose open sets are precisely the upward-closed sets with respect to the preorder. This topology is called the [[specialization topology]]. This defines a functor

$$i \colon ProSet \to Top$$

which is a full embedding; the [[essential image]] of this functor is the category of [[Alexandroff spaces]] (spaces in which an arbitrary intersection of open sets is open). Hence the category of prosets is equivalent to the category of Alexandroff spaces.

In fact, we have an adjunction $i \dashv Spec$, making $ProSet$ a [[coreflective subcategory]] of $Top$. In particular, the counit evaluated at a space $X$,

$$i(Spec(X)) \to X,$$

is the identity function at the level of sets, and is continuous because any open $U$ of $X$ is upward-closed with respect to $\leq$, according to the second equivalent condition of the definition of the specialization order.

This [[adjunction]] restricts to an [[adjoint equivalence]] between the categories $\Fin\Pros$ and $\Fin\Top$ of finite prosets and finite topological spaces. The unit and counit are both identity functions at the level of sets, so we in fact have an equivalence between these categories as [[concrete categories]].

## Related entries

* [[finite topological space]]

* [[separation axioms in terms of lifting properties]]

* [[specialization topology]]

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