semistandard Young tableau

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

#Contents#
* table of contents
{:toc}

## Idea

In [[combinatorics]] a *(semi-)standard Young tableau* is a labelling of the boxes of a [[Young diagram]] with [[positive number|positive]] [[natural numbers]] (a [[Young tableau]]) satisfying extra conditions, at the *minimum* that labels do not decrease to the right and do increase downwards.

The number of (semi-)standard Young tableau of given shape (underlying [[Young diagram]]) govern various objects in the [[representation theory]] [[representation theory of the symmetric group|of the symmetric-]] [[representation theory of the general linear group|and general linear group]].

## Definition

\begin{definition}\label{YoungDiagram}
**([[Young diagram]])** \linebreak
For $n \in \mathbb{N}$ a [[positive number|positive]] [[natural number]],  a *[[Young diagram]] with $n$ boxes* is a [[partition]] of $n$, hence a sequence of weakly decreasing [[positive number|positive]] [[natural numbers]]

$$
  (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_{rows(\lambda)} )
  \,,
  \;\;
  \lambda_i \in \mathbb{N}_+
$$

whose [[sum]] is $n$:

\[
  \label{NumberOfBoxes}
  n \;=\;  \underset{i}{\sum} \lambda_i
  \,,
\]

\end{definition}

\begin{definition}\label{SemistandardYoungTableau}
**([[semistandard Young tableau]])** \linebreak

Given a [[Young diagram]]/[[partition]] $\lambda$ (Def. \ref{YoungDiagram}),
a *semistandard Young tableau* $T$ *of shape* $\lambda$

* is an [[indexed set]] of [[positive number|positive]] [[natural numbers]] of the following form (a *[[Young tableau]]*):

  \[
    \big(
      T_{i, j}
      \in
      \mathbb{N}_+
    \big)_{
      {1 \leq i \leq rows(\lambda)} \atop { 1 \leq j \leq \lambda_i }
    }
    \,;
  \]

* such that these labels are:

  1. weakly increasing to the right/along rows;

     \[
       \label{RowsAreWeaklyIncreasing}
       \underset{
         i
       }{\forall}
       \;\;\;\;\;\;
         j_1 \lt j_2 \;\Rightarrow\; T_{i, j_1} \leq T_{i, j_2}
     \]

  1. strictly increasing downwards/along columns:

     \[
       \label{ColumnsAreStrictlyIncreasing}
       \underset{ j }{\forall}
       \;\;\;\;\;\;
       i_1 \lt i_2 \;\Rightarrow\; T_{i_1, j} \lt T_{i_2,  j}
     \]

\end{definition}

\begin{remark}\label{RemarkOnSSYTAxioms}
Beware that in a semistandard Young tableau (Def. \ref{SemistandardYoungTableau}):

* the label $T_{1,  1}$ in the top left box is *not required* to be 1;

* in *some* applications (e.g. (eq:RelationToSchurPolynomialsOfFinitelyManyVariables) below), though not in general, the labels are in addition required to be bounded $T_{i, j} \leq k$.

\end{remark}

\begin{definition}\label{StandardYoungTableau}
**([[standard Young tableau]])**
\linebreak
A [[semistandard Young tableau]] (Def. \ref{SemistandardYoungTableau})
of $n$ boxes (eq:NumberOfBoxes)
is called a *[[standard Young tableau]]* if every element of $\big\{1, \cdots, n\}$ appears exactly once as a label.
\end{definition}

\begin{remark}\label{RemarkOnSYTAxioms}
In particular, in a standard Young tableau (Def.\ref{StandardYoungTableau}) we have that:

1. not just the columns but also the rows are strictly increasing (cf. (eq:RowsAreWeaklyIncreasing)),

1. the first label of a standard Young tableau must be $T_{1,1} = 1$ (cf. Rem. \ref{RemarkOnSSYTAxioms}).

\end{remark}

## Examples

The following is a semistandard Young tableau that does happen to start with 1 but is not a standard Young tableau:

$$
\array{
  1 & 1 & 1 & 3
  \\
  2 & 3
  \\
  4
}
$$

## Properties

Given a [[partition]] $\lambda \in Part(n)$, we write

\[
  \label{SetOfssYTOfFixedShape}
  ssYT_\lambda
  \;in\;
  Set
\]

for the [[set]] of semistandard Young tableaux (Def. \ref{SemistandardYoungTableau}) whose underlying [[Young diagram]] (i.e. forgetting its labels) is given by $\lambda$.

Given moreover a [[natural number]] $N \in \mathbb{N}$ we write

\[
  \label{SetOfssYTOfFixedShapeWithBoundedLabels}
  ssYT_\lambda(N)
  \;\subset\;
  ssYT_\lambda
\]

for the subset on those semistandard Young tableau $T$ whose labels are bounded by $N$ in that $\underset{i,j}{\forall}\; T_{i, j } \;\leq\; N$.

### Relation to Schur polynomials
 {#RelationToSchurPolynomials}

Given a [[semistandard Young tableau]] $T$, we write $X^T$ for the [[monomial]] which contains one factor of the [[variable]] $x_k$ for each occurrence of $k$ in the Young tableau:

\[
  \label{MonomialAssociatedWithSemistandardYoungTableau}
  x^T
  \;\coloneqq\;
  x_1^{\#1s}
  x_2^{\#2s}
  \cdots
  \,.
\]

\begin{prop}
  For $n \in \mathbb{N}$ and $\lambda$ a [[partition]] of $n$, the corresponding [[Schur polynomial]] $s_\lambda$ is equal to the [[sum]] over the [[monomials]] (eq:MonomialAssociatedWithSemistandardYoungTableau) associated with all [[semistandard Young tableau]] of shape $\lambda$ (eq:SetOfssYTOfFixedShape):

$$
  s_\lambda(x_1, x_2, \cdots)
  \;\;=\;\;
  \underset{
    {T \in ssYT_\lambda},
  }{\sum}
  x^T
  \,.
$$
\end{prop}

([Sagan 01, Def. 4.4.1](#Sagan01), review in [Sagan Enc., p. 1](#SaganEnc))

This means that Schur polynomials in a [[finite set]] $\{x_1, \cdots, x_N\}$ of [[variables]] count semistandard Young tableau (of fixed shape and) with bounded labels (eq:SetOfssYTOfFixedShapeWithBoundedLabels):

\[
  \label{RelationToSchurPolynomialsOfFinitelyManyVariables}
  s_\lambda(x_1, \cdots, x_N)
  \;\;=\;\;
  \underset{
    {T \in ssYT_\lambda(N)},
  }{\sum}
  x^T
  \,.
\]

### Hook-length formula

See at *[[hook length formula]]*.

### Hook-content formula
 {#HookContentFormula}

The *[[hook-content formula]]* expresses the number of semistandard Young-Tableau $T$ with

1. shape $\lambda = (\lambda_1 \geq \cdots \geq \lambda_i)$ (a [[partition]]),

1. labels in $\{1, \cdots, N\}$,

1. sum of labels $\sum_{i,j} T_{i,j}$

through the following identification of the *generating function* for numbers of semistandard Young tableaux (ssYT) as a [[polynomial]] in a [[variable]] $q$:

$$
  \underset{
    T \in ssYT_\lambda(N)
  }{\sum}
  q^{ \sum_{i,j} T_{i,j} }
  \;\;=\;\;
  q^{ \sum_i i \cdot \lambda_i }
  \underset{
    { i \in \{1, \cdots, rows(\lambda)\} }
    \atop
    {j \in \{1, \cdots, \lambda_j\}}
  }{\prod}
  \frac{
    1 - q^{N + content(i,j)}
  }{
    1 - q^{\ell hook(i,j)}
  }
  \,,
$$

where

$$
  content(i,j) \;\coloneqq\; j - 1
  \,,
  {\phantom{AAAA}}
  \ell hook(i,j)
  \;=\;
  \text{length of hook at (i,j)th box}
  \,.
$$

See at *[[hook-content formula]]* for more.

### Relation to dimensions of irreps

[[!include hook length and content formulas -- table]]

### Number of sYT with bounded number of rows
  {#NumberOfSYTWithBoundedNumberOfRows}

We discuss formulas for the number

$$
  \left\vert
    sYT_n(N)
  \right\vert
  \;=\;
  \underset{
    { \lambda \in Part(n) }
    \atop
    { rows(\lambda) \leq N }
  }{
    \sum
  }
  \left\vert
     sYT_n
  \right\vert
$$

of standard Young tableaux with $n$ boxes and $\leq N$ rows.

#### Asymptotic formulas for large $n$

{#AsymptoticFormulaForNumberOfSYTWIthBoundedHeight} Asymptotically for large $n$ this is ([Regev 81, (F.4.5.1)](#Regev81)):

\[
 \label{AsymoptoticsOfSYTOfBoundedHeightRegevForm}
  \begin{aligned}
  \left\vert
    sYT_n(N)
  \right\vert
  &
  \;
  \overset{
    n \to \infty
  }{\sim}
  \;
  \left(
    \frac{N}{n}
  \right)^{ \tfrac{1}{4} N(N-1) }
  \cdot
  \frac
    {N^n}
    {N!}
  \cdot
  \big(
    \underset{
       \sqrt{\pi}/2
    }{
    \underbrace{
      \Gamma(3/2)
    }
    }
  \big)^{- N}
  \cdot
  \underoverset
    {j=1}
    {N}
    {\prod}
    \Gamma
    \left(
      1 + \tfrac{1}{2}j
    \right)
  \end{aligned}
\]

Later this appears again as a conjecture in [Kotěšovec 13](#Kotesovec13):

\[
 \label{AsymoptoticsOfSYTOfBoundedHeightKotesovecForm}
  \begin{aligned}
  \left\vert
    sYT_n(N)
  \right\vert
  &
  \;\sim\;
  \underoverset
    {j = 1}
    {N}
    {\prod}
  \underset{
    \Gamma(1 + j/2) \tfrac{2}{j}
  }{
  \underbrace{
    \Gamma(j/2)
  }
  }
  \cdot
  \frac{N^n}{\pi^{N/2}}
  \cdot
  \left(
    \frac{N}{n}
  \right)^{ \tfrac{1}{4}(N(N-1)) }
  \\
  & \;=\;
  \underoverset
    {j = 1}
    {N}
    {\prod}
  \Gamma
  \big(
    1 + j/2
  \big)
  \cdot
  \frac{2^N}{N!}
  \cdot
  \frac{N^n}{\pi^{N/2}}
  \cdot
  \left(
    \frac{N}{n}
  \right)^{ \tfrac{1}{4}(N(N-1)) }
  \,.
  \end{aligned}
\]

(The first line is the expression conjectured in [Kotěšovec 13](#Kotesovec13), under the brace we use the [translation formula](Gamma+function#eq:TranslationFormula), and the second line shows agreement with [Regev 81 (F.4.5.1)](#Regev81) as in (eq:AsymoptoticsOfSYTOfBoundedHeightRegevForm).)

This may further be re-expressed ([Kotěšovec 13, p. 2](#Kotesovec13)) in terms of the [[Barnes G-function]] $G(-)$ (see [there](Barnes+G-function#MultiplicationFormulaForGammaFunctionInTermsOfGFunction)):

\[
  \label{AsymoptoticsOfSYTOfBoundedHeightInTermsOfBarnesGFunction}
  \begin{aligned}
  \left\vert
    sYT_n(N)
  \right\vert
  &
  \;\underset{n \to \infty}{\sim}\;
  \underset{
  \mathclap{
    \frac
      {
        G(N/2 + 1)
        \cdot
        G(N/2 + 1/2)
      }
      {
        G(1/2)
      }
  }
  }{
    \underbrace{
    \underoverset
      {j = 1}
      {N}
      {\prod}
    \Gamma(j/2)
    }
  }
  \cdot
  \frac{N^n}{\pi^{N/2}}
  \cdot
  \left(
    \frac{N}{n}
  \right)^{ \tfrac{1}{4}(N(N-1)) }
  \end{aligned}
\]

#### Asymptotic formulas for large $n$ and large $N$

Using the large-$N$ asymptotic expansion ([here](Barnes+G-function#AsymptoticExpansion)) of the [[Barnes G-function]]

$$
  G(N/2 + 1)
  \underset{N \to \infty}{\sim}
  \exp
  \left(
    \tfrac{1}{8}
    N^2 ln (N)
    -
    \left(
      \tfrac{3}{16}
      +
      \tfrac{ln(2)}{8}
    \right)
    N^2
    +
    \tfrac{ln(2\pi)}{4} N
    -
    \tfrac{1}{12} ln(N)
    +
    \big(
      \zeta^'(-1)
      -
      \tfrac{ln(2)}{12}
    \big)
  \right)
$$

inside the large-$n$ asymptotic expansion (eq:AsymoptoticsOfSYTOfBoundedHeightInTermsOfBarnesGFunction), we obtain the doubly asymptotic expansion of the number of standard Yound tableaux with $n$ boxes and $\leq N$ rows:

\[
  \label{DoublyAsymoptoticsOfSYTOfBoundedHeight}
  \begin{aligned}
  &
  ln
  \big(
  \left\vert
    sYT_n(N)
  \right\vert
  \big)
  \\
  &
  \;
  \underset{
    { n \to \infty }
    \atop
    { N \to \infty }
  }{\sim}\;
   n ln(N)
   -
   \tfrac{1}{4}N^2 ln(n)
   +
   \tfrac{1}{4}N ln(n)
   \\
   &
    \phantom{
      \underset{
        { n \to \infty }
        \atop
        { N \to \infty }
      }{\sim}\;
    }
   +
   \tfrac{1}{2}
    N^2 ln(N)
   -
    \left(
      \tfrac{3}{8}
      +
      \tfrac{ln(2)}{4}
    \right)
    N^2
   -
   \tfrac{1}{4} N ln(N)
   +
    \tfrac{ln(2)}{2}
    N
    -
    \tfrac{1}{6} ln(N)
    \\
    &
    \;
    \phantom{
      \underset{
        { n \to \infty }
        \atop
        { N \to \infty }
      }{\sim}\;
    }
    +
    \big(
      2 \zeta^'(-1)
      -
      \tfrac{ln(2)}{6}
      -
      ln G(1/2)
    \big)
  \end{aligned}
\]

#### Exact formulas for small $N$

Exact formulas for small $N$:

$$
  \left\vert
    sYT_{2n}(2)
  \right\vert
  \;=\;
  \frac{
    (2n)!
  }{
    (n!)^2
  }
  \,,
  \;\;\;\;\;\;
  \left\vert
    sYT_{2n+1}(2)
  \right\vert
  \;=\;
  \frac{
    (2n + 1)!
  }{
    n! (n+1)!
  }
$$

$$
  \left\vert
    sYT_{n}(3)
  \right\vert
  \;=\;
  \underoverset
    {i = 0}
    { \rfloor n/2 \lfloor }
    {\sum}
    \left(
      n \atop {2i}
    \right)
    Catalan(i)
  \;=\;
   Motzkin(n)
  \,,
$$

where

* $Catalan(k) \;=\; \frac{(2k)!}{k! (k+1)!}$ are the [[Catalan numbers]];

* $Motzkin(-)$ are the [[Motzkin numbers]].

This is attributed in [Gouyou-Beauchamps 89](#Gouyou-Beauchamps89) to [Regev 81](#Regev81).

For $N = 4,5$:

$$
  \left\vert
    sYT_{2n-1}(4)
  \right\vert
  \;=\;
  Catalan(n) Catalan(n)
  \,,
  \;\;\;\;\;\;
  \left\vert
    sYT_{2n}(4)
  \right\vert
  \;=\;
  Catalan(n) Catalan(n+1)
$$

due to [Gouyou-Beauchamps 89](#Gouyou-Beauchamps89).

## References

### General

Textbook account:

* {#Sagan01} [[Bruce Sagan]], Def. 2.9.5 of: _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))

Survey in the context of [[Schur functions]]:

* {#SaganEnc} [[Bruce E. Sagan]], _Schur functions_, in: [[Michiel Hazewinkel]] (ed.), *[[eom|Encyclopaedia of Mathematics]]*, Springer 2002 ([pdf](http://www.mth.msu.edu/~sagan/Papers/Old/schur.pdf), [[SaganSchurFunctions.pdf:file]])

Specifically for standard Young tableaux:

* [[Ron M. Adin]], [[Yuval Roichman]], *Enumeration of Standard Young Tableaux*, Chapter 14 in: Miklós Bóna, *Handbook of Enumerative Combinatorics*, CRC Press 2015  ([arXiv:1408.4497](https://arxiv.org/abs/1408.4497), [ISBN:9781482220858](https://www.routledge.com/Handbook-of-Enumerative-Combinatorics/Bona/p/book/9781482220858))

See also the references at *[[Young tableau]]*.

### Counting for bounded number of rows

* {#Regev81} [[Amitai Regev]], *Asymptotic values for degrees associated with strips of young diagrams*, Advances in Mathematics Volume 41, Issue 2, August 1981, Pages 115-136 (doi:10.1016/0001-8708(81)90012-8)

* {#Gouyou-Beauchamps89} [[Dominique Gouyou-Beauchamps]], *Standard Young Tableaux of Height 4 and  5*, Europ. J. Combinatorics (1989) 10, 69-82 (doi:10.1016/S0195-6698(89)80034-4, [pdf](https://core.ac.uk/download/pdf/82423471.pdf))

* Marilena Barnabei, Flavio Bonetti, and Matteo Silimbani: *Combinatorial properties of the numbers oftableaux of bounded height* ([arXiv:0803.2112](https://arxiv.org/abs/0803.2112), [pdf](http://amsacta.unibo.it/2443/1/bounded_height.pdf))

* {#Kotesovec13} Václav Kotěšovec, *Asymptotic of Young tableaux of bounded height*, 2013 ([pdf](http://www.kotesovec.cz/math_articles/kotesovec_young_tableaux_conjecture.pdf), [[KotesovecAsymptoticsOfBoundedYoungTableaux.pdf:file]])

[[!redirects semistandard Young tableaux]]

[[!redirects semi-standard Young tableau]]
[[!redirects semi-standard Young tableaux]]

[[!redirects semi standard Young tableau]]
[[!redirects semi standard Young tableaux]]

[[!redirects standard Young tableau]]
[[!redirects standard Young tableaux]]


semitransitive coring

[[!redirects semitransitive coring]]
Let $k$ be a (commutative) [[field]] and $R$ a $k$-algebra.

An $R$-[[coring]] $C$ is semi-transitive if the category  $M^C_f$ of $C$-comodules which are finitely generated as modules over $R$

* $M^C_f$ is artinian and noetherian (has all objects of finite composition length), all its objects are projective as $R$-modules, and all its hom-spaces are finite dimensional over $k$
* the category of all $C$-comodules is Ind-generated by $M^C_f$ (Every $C$-comodule is a filtered colimit of $C$-comodules finitely generated as $R$-modules)

It appears that semi-transitive corings are those corings which can be reconstructed from fiber functors on categories which are artinian, noetherian and with finite dimensional hom-spaces.

* A. Bruguières, _Théorie tannakienne non commutative_, Comm. Algebra __22__, 5817–5860, 1994

* K. Szlachanyi, _Fiber functors, monoidal sites and Tannaka duality for bialgebroids_, [arxiv/0907.1578](http://arxiv.org/abs/0907.1578)


Semjon Gerschgorin

Semjon Aronowitsch Gerschgorin (Russian: Семён Аронович Гершгорин)
was a Russian mathematician.

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

## Selected writings

Introducing the [[Gershgorin circle theorem]] in [[matrix analysis]]:

* {#Gershgorin31} [[S. Gerschgorin]], *Über die Abgrenzung der Eigenwerte einer Matrix*, Bulletin de l'Académie des Sciences de l'URSS. Classe des sciences mathématiques et na, 1931, Issue 6, Pages 749–754 ([mathnet:eng/izv5235](http://mi.mathnet.ru/eng/izv5235))

category: people

[[!redirects Semyon Aronovich Gershgorin]]
[[!redirects Semyon A. Gershgorin]]
[[!redirects S. A. Gershgorin]]
[[!redirects S. Gershgorin]]

[[!redirects Semyon Aronovich Gerschgorin]]
[[!redirects Semyon A. Gerschgorin]]
[[!redirects S. A. Gerschgorin]]
[[!redirects S. Gerschgorin]]

[[!redirects Semyon Aronovich Geršgorin]]
[[!redirects Semyon A. Geršgorin]]
[[!redirects S. A. Geršgorin]]
[[!redirects S. Geršgorin]]

[[!redirects Семён Аронович Гершгорин]]
[[!redirects Семён А. Гершгорин]]
[[!redirects Семён Гершгорин]]
[[!redirects С. Гершгорин]]


Semën S. Kutateladze

* [Wikipedia entry](https://en.wikipedia.org/wiki/Sem%C3%ABn_Samsonovich_Kutateladze)

* [arXiv entry](https://arxiv.org/a/kutateladze_s_1.html)

## Selected writings

On [[nonstandard analysis]]

* E. I. Gordon, A. G. Kusraev, [[Semën S. Kutateladze]], *Infinitesimal Analysis*, Mathematics and its Applications **544**, Springer (2002) [[doi:10.1007/978-94-017-0063-4](https://doi.org/10.1007/978-94-017-0063-4)]
* Piotr Blaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, [[Semën S. Kutateladze]], David Sherry, _Toward a history of mathematics focused on procedures_, [arxiv/1609.04531](https://arxiv.org/abs/1609.04531)

and on [[monads in nonstandard analysis]]:

* [[Semën S. Kutateladze]], *Leibniz's Definition of Monad*, NeuroQuantology **4** 3 (2006) 249-241 [[arXiv:math/0608298](https://arxiv.org/abs/math/0608298), [pdf](https://www.researchgate.net/profile/Semen-Kutateladze/publication/2130297_Leibniz%27s_Definition_of_Monad/links/0912f5058405405dbb000000/Leibnizs-Definition-of-Monad.pdf)]

and the [[monad (disambiguation)|monad terminology]] in particular:

* {#Kutateladze11} [[Semën S. Kutateladze]], *Leibnizian, Robinsonian, and Boolean valued monads*, Journal of Applied and Industrial Mathematics **5** 3 (2011) 365-373 [[arxiv/1106.2755](http://arxiv.org/abs/1106.2755), [doi:10.1134/S1990478911030082](https://doi.org/10.1134/S1990478911030082)]

See also:

* Piotr Blaszczyk, Vladimir Kanovei, Karin U. Katz, [[Mikhail Katz]], [[Semen S. Kutateladze]], David Sherry, *Toward a history of mathematics focused on procedures*, Foundations of Science **22** (2017) 763–783 [[arxiv/1609.04531](http://arxiv.org/abs/1609.04531), [doi:10.1007/s10699-016-9498-3](https://doi.org/10.1007/s10699-016-9498-3)]

category: people

[[!redirects Semen Kutateladze]]
[[!redirects Semën Samsonovich Kutateladze]]
[[!redirects Semen Samsonovich Kutateladze]]
[[!redirects Semen S. Kutateladze]]
[[!redirects S. S. Kutateladze]]


Sen limit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The limit in which [[F-theory]] reduces to the [[perturbative string theory]] of [[type IIB string theory|type IIB]] is called the _Sen limit_.

Essentially this is the limit where one approaches the [[nodal curve]] [[singularities]] of the F-theory [[elliptic fibration]]. A detailed analysis is in ([Clingher-Donagi-Wijnholt 12](#ClingherDonagiWijnholt12)).

## Related concepts

[[!include F-branes -- table]]

## References

* {#Sen96} [[Ashoke Sen]], _F-theory and Orientifolds_, Nucl.Phys.B475:562-578,1996 ([arXiv:hep-th/9605150](http://arxiv.org/abs/hep-th/9605150))

* {#Sen97} [[Ashoke Sen]], _Orientifold Limit of F-theory Vacua_, Nucl. Phys. Proc. Suppl. 68 (1998) 92 (Nucl. Phys. Proc. Suppl. 67 (1998) 81) ([arXiv:hep-th/9702165](http://arxiv.org/abs/hep-th/9702165))

* {#ClingherDonagiWijnholt12} [[Adrian Clingher]], [[Ron Donagi]], [[Martin Wijnholt]], _The Sen Limit_ ([arXiv:1212.4505](http://arxiv.org/abs/1212.4505))


Sen's conjecture

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

#Contents#
* table of contents
{:toc}

## Idea

### In bosonic string theory - The tachyon vacuum

The [[tachyon]] mode in the [[open string|open]] [[bosonic string]] spectrum is an indication that its [[effective quantum field theory|effective]] [[perturbative quantum field theory]] describes [[perturbation theory|perturbation]] about an _[[unstable vacuum]]_ which will [[vacuum decay|decay]] via [[tachyon condensation]].

It was argued in [Sen 99, Section 4](#Sen99) that, since the [[open string|open]] [[bosonic string]] technically propagates with its endpoints on the space-filling [[D25-brane]], that instability must be the instability of the D25, which wants to decay to a "true bosonic string vacuum", often called the _tachyon vacuum_.

This conjecture became famous as **Sen's conjecture**. It was subsequently checked to be true by numerical means in [[open bosonic string field theory]]. Moreover, using string field theory it became possible to study that non-perturbative state in which the [[D25-brane]] is indeed gone. There were several arguments that indeed in the vicinity of this state the perturbative open string has disappeared and turned into the closed bosonic string.

A breakthrough happened in 2005, when [[Martin Schnabl]] found an analytic expression for this "tachyon vacuum" ([Schnabl 05](#Schnabl05)).

### In superstring theory -- The K-theory conjecture
 {#InSuperstringTheory}

Sen's conjecture actually originates in [[superstring theory]] ([Sen 98](#Sen98)). While here the tachyon mode for [[open string|open]] [[superstrings]] on [[supersymmetry|supersymmetric]] [[D-brane]]-configurations is famously absent, signifying the [[vacuum stability|stability]] of [[perturbative superstring vacua]],
there are still [[tachyon]] modes in [[open string|open]] [[superstrings]] stretching between [[D-branes]] and _[[anti-D-branes]]_.

\begin{imagefromfile}
    "file_name": "AntiBraneAnnihilationAsKGroupEquivRelation.jpg",
    "float": "right",
    "width": 300,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 20
    },
    "caption": "from [Sch 18](https://ncatlab.org/schreiber/show/Equivariant+Stable+Cohomotopy+and+Branes)"
\end{imagefromfile}

Here the conjecture states that the respective [[tachyon condensation]] witnesses the annihilation of the brane/anti-brane pair ([Sen 98](#Sen98)).

This conjectured process is one motivations for the further [[conjecture]] that [[D-brane charge]] is [[Dirac charge quantization|charge quantized]] in [[topological K-theory]] ([Witten 98, Section 3](D-brane#Witten98)), since this decay plausibly implements on the D-branes's [[Chan-Paton gauge field|Chan-Paton]] [[vector bundles]] the defining [[equivalence relation]] ([here](topological+K-theory#eq:DefiningEquivalenceRelation)) of [[topological K-theory]].

## References

### General

Review:

* [[Ashoke Sen]], _Tachyon Dynamics in Open String Theory_, Int. J. Mod. Phys. A20:5513-5656, 2005 ([arXiv:hep-th/0410103](https://arxiv.org/abs/hep-th/0410103))

For more references in the context of [[string field theory]], see there at:

* _[Tachyon dynamics, decaying D-branes and Sen's conjecture](string+field+theory#BosonicStringVacuumAndSenConjecture)

### For superstrings
 {#ReferencesForSuperstrings}

Sen's conjecture on the decay of [[D-brane]]/[[anti-D-brane]] pairs in [[superstring theory]] via [[open string|open]] [[superstring]] [[tachyon condensation]]:

* {#Sen98} [[Ashoke Sen]], _Tachyon Condensation on the Brane Antibrane System_, JHEP 9808 012 (1998) [[arXiv:hep-th/9805170](https://arxiv.org/abs/hep-th/9805170), [doi:10.1088/1126-6708/1998/08/012](https://doi.org/10.1088/1126-6708/1998/08/012)]

* {#Sen99} [[Ashoke Sen]], _Universality of the Tachyon Potential_, JHEP 9912:027, 1999 ([arXiv:hep-th/9911116](http://arxiv.org/abs/hep-th/9911116))

The resulting conjecture that [[D-brane charge]] is [[Dirac charge quantization|charge quantized]] in [[topological K-theory]]:

* [Witten 98, Section 3](D-brane#Witten98)

Discussion via [[superstring field theory]]:

* {#Erler13} [[Theodore Erler]], _Analytic Solution for Tachyon Condensation in Berkovits' Open Superstring Field Theory_, JHEP 1311 (2013) 007 ([arXiv:1308.4400](https://arxiv.org/abs/1308.4400))

which concludes (on [p. 32](https://arxiv.org/pdf/1308.4400.pdf#page=32)) with:

> It would also be interesting to see if these developments can shed light on the long-speculated relation between string field theory and the K-theoretic description of D-brane charge $[$[75, 76, 77](D-brane#DBraneChargeQuantizationInTopologicalKTheory)$]$. We leave these questions for future work.

Similarly in

* {#Erler19} [[Theodore Erler]], _Four Lectures on Analytic Solutions in Open String Field Theory_ ([arXiv:1912.00521](https://arxiv.org/abs/1912.00521), [spire:1768105](https://inspirehep.net/literature/1768105))

which lists (on [p. 112](https://arxiv.org/pdf/1912.00521.pdf#page=113)) among open problems of [[string field theory]]:

> Are there topological invariants of the open string star algebra representing D-brane charges?

### For bosonic strings

The conjecture for [[bosonic strings]] originates with

* [Sen 99, Section 4](#Sen99)

An analytic check was found in

* {#Schnabl05} [[Martin Schnabl]], _Analytic solution for tachyon condensation in open string field theory_, Adv. Theor. Math. Phys.
Volume 10, Number 4 (2006), 433-501. ([arXiv:hep-th/0511286](http://arxiv.org/abs/hep-th/0511286))

* {#EllwoodSchnabl06} Ian Ellwood, [[Martin Schnabl]], _Proof of vanishing cohomology at the tachyon vacuum_, JHEP 0702:096,2007 ([arXiv:hep-th/0606142](http://arxiv.org/abs/hep-th/0606142))


sentence

In a [[language]] $L$, for example a [[first-order language]], a *sentence* is a [[formula]] in which all [[variables]] are [[bound variable|bound]].

When a sentence is interpreted in terms of a [[structure in model theory|structure]] of $L$, it becomes [[true]] or [[false]].


Seok Kim

* [webpage](http://insti.physics.sunysb.edu/conf/simonswork7/participants/Kim.Seok.html)

## Selected writings

On [[E-strings]] and emergence of [[SU(2)]]  [[flavor (particle physics)|flavor]]-[[chiral symmetry|symmetry]] on [[M5-branes]] in [[heterotic M-theory]] (in the [[D=6 N=(1,0) SCFT]] on [[small instantons]] in [[heterotic string theory]]):

* {#GHKKLV15} [[Abhijit Gadde]], [[Babak Haghighat]], [[Joonho Kim]], [[Seok Kim]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _6d String Chains_, JHEP **1802** (2018) 143 [[arXiv:1504.04614](https://arxiv.org/abs/1504.04614)]

On [[black holes in string theory]]:

* {#CKJN18} Sunjin Choi, [[Joonho Kim]], [[Seok Kim]], June Nahmgoong, _Large AdS black holes from QFT_ ([arxiv:1810.12067](https://arxiv.org/abs/1810.12067))

category: people


separable algebra

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

# Contents
* table of contents
{: toc}

## The idea ##

The concept of *separable algebras* is a strengthening of the concept of *[[semisimple algebras]]*, and a generalization of the concept of *[[separable field extensions]]*.

## Definition ##

There are a several equivalent characterizations of separable algebras.  For all of these we fix a [[field]] $k$.  In what follows, all $k$-algebras will be assumed [[associative algebra|associative]] and [[unital algebra|unital]].

First, a $k$-algebra $A$ is defined to be **separable** if for every [[field extension]] $K$ of $k$, the [[tensor product]] algebra $A \otimes_k K$ is [[semisimple algebra|semisimple]].

Second, a $k$-algebra $A$ is separable if and only if it is [[flat module|flat]] when considered as a right [[module]] of $A^e = A \otimes_k A^{op}$ in the obvious (but perhaps not quite standard) way.

Third, a $k$-algebra $A$ is separable if and only if it is [[projective module|projective]] when considered as a left [[module]] of $A^e$ in the usual way.

Fourth, a $k$-algebra $A$ is separable if and only if the $A^e$-module morphism
$$ \array{
m : & A^e &\to & A  \\
     & a \otimes b & \mapsto & a b
}
$$
has a right inverse, that is a $A^e$-module morphism
$$  f: A \to  A^e  $$
with $m f = 1_A$.

It is easy to see that the third and fourth definitions are equivalent.  We have an epimorphism of $A^e$-modules
$$  A^e \stackrel{m}{\longrightarrow} A \to 0 $$
If $f$ as above exists, this splits, so $A$ is a summand of a free $A^e$-module, namely $A^e$ itself, so $A$ is projective as an $A^e$-module.  Conversely, if $A$ is projective, any epimorphism to $A$ splits.

We can also state the fourth characterization in a more grungy way in terms of the element $p = f(1)$.  Namely, a $k$-algebra $A$ is separable if and only if there exists an element
$$ p = \sum_{i=1}^n a_i \otimes b_i  \in A^e $$
such that
$$ \sum_{i=1}^n a_i b_i = 1 $$
and
$$ a p = p a $$
for all $a \in A$.  Such an element $p$ is called a **separability idempotent**, since it satisfies $p^2 = p$.
While grungy, finding a separability idempotent is a practical way to prove that an algebra is separable.

## Properties

### Classification

There is a classification theorem for separable algebras: separable algebras are the same as finite products of [[matrix algebras]] over finite-dimensional [[division algebras]] whose [[centers]] are [[finite-dimensional vector space|finite-dimensional]] [[separable field extension|separable field extensions]] of the [[field]] $k$.

A [[perfect field]] is one for which every extension is [[separable field extension|separable]].  Examples include fields of characteristic zero, or finite fields, or algebraically closed fields, or extensions of perfect fields.  If $k$ is a perfect field, separable algebras are the same as finite products of matrix algebras over finite-dimensional division algebras over field $k$.  In other words, if $k$ is a perfect field, there is no difference between a separable algebra over $k$ and a finite-dimensional [[semisimple algebra]] over $k$.

### Relation to Frobenius algebras
 {#RelationToFrobeniusAlgebras}

A result of [Eilenberg & Nakayama (1955)](#EilenbergNakayama55) states that any [[separable algebra]] over a field $k$ can be given the structure of a [[Frobenius algebra|symmetric Frobenius algebra]].  Since the [[underlying]] [[vector space]] of a Frobenius algebra is [[isomorphism|isomorphic]] to its [[dual vector space|dual]], any Frobenius algebra is necessarily [[finite dimensional vector space|finite dimensional]], and so the same is true for separable algebras.  For more details, see [Eilenberg & Nakayama (1955)](#EilenbergNakayama55), [Endo & Watanabe (1967), in particular Thm. 4.2.](#EndoWatanabe67).

A separable algebra is said to be **strongly separable** if there exists a separability idempotent $p$ that is **symmetric**, meaning
$$ p = \sum_{i=1}^n a_i \otimes b_i = \sum_{i=1}^n b_i \otimes a_i $$
An algebra is strongly separable if and only if it can be made into a [[Frobenius algebra|special Frobenius algebra]].  When this can be done, it can be done in a unique way.

There is an equivalent characterization of strongly separable algebras which makes this fact clearer.  Any element $a$ of an associative unital algebra gives a left multiplication map
$$ \array{
L_a : &A &\to& A \\
      &b &\mapsto& a b
}
$$
When $A$ is finite-dimensional, there is a bilinear pairing $g: A \times A \to k$ defined by
$$ g(a,b) = tr(L_a L_b) $$
An algebra $A$ is strongly separable if and only if $g$
is **nondegenerate**, i.e., if there is a vector space isomorphism $A \to A^*$ given by
$$ a \mapsto g(a, -)  $$
In this case, there is just one way to make $A$ into a special Frobenius algebra, namely by defining the counit to be
$$ \epsilon(a) = tr(L_a) $$

Here are some examples of strongly separable algebras:

* the algebra of $n \times n$ matrices with entries in a field $k$ is always separable, but it is strongly separable if and only if $n$ is not divisible by the characteristic of $k$.

* the group algebra $k[G]$ of a finite group is strongly separable if and only if the order of $G$ is not divisible by the characteristic of $k$.  (It is also semisimple precisely in this case, so it is separable precisely in this case.)

For more details, see Aguiar's note below.

### Over commutative rings

More generally, if $k$ is any [[unital ring|unital]] [[commutative ring]], we can define a separable $k$-algebra to be an algebra $A$ such that $A$ is projective as a module over $A^e = A \otimes_k A^{op}$.

As in the case of algebras over a field, an algebra $A$ over a commutative ring $k$ is separable if and only if the $A^e$-module epimorphism
$$
  \array{
    m \colon & A^e &\to & A
    \\
     & a \otimes b & \mapsto & a b
  }
$$
splits, and this in turn is equivalent to the existence of a separability idempotent.

If a separable algebra $A$ is also projective as a module over $k$, it must be finitely generated as a $k$-module.  For more details see DeMeyer-Ingraham.

If the center of a separable algebra $A$ over a commutative ring $k$ is $k$, $A$ is called an [[Azumaya algebra]].

### Separable extensions of noncommutative rings

The ring extension $S$ over $R$ is said to be a separable extension if all short exact sequences of $S$-$S$-bimodules that are split as $S$-$R$-bimodules also split as $S$-$S$-bimodules. This is equivalent to the statement that the relative Hochschild cohomology $HH^n(S,R;M) = 0$ for all $n\gt 0$ and all coefficient bimodules $M$.

### In algebraic geometry

Commutative separable algebras play an important role in [[algebraic geometry]], where the concept of *[[etale morphism|étale cover]]* is a way of implementing the idea of [[covering space]], but any étale cover of the [[spectrum of a ring|spectrum]] of a field corresponds to a commutative separable algebra over that field.  Lieven Le Bruyn has written "in categorical terms, studying the monoidal category of commutative separable $k$-algebras is the same as studying the étale site of $k$".   This idea is partially captured by the so-called *fundamental theorem of Grothendieck Galois theory*: the  category of commutative separable algebras over a field $k$ is contravariantly equivalent to the category of continuous actions on finite sets of the profinite fundamental group of $k$.   As a group, the *profinite fundamental group* of $k$ is the Galois group $Gal(k_s|k)$ where $k_s$ is a *separable closure* of $k$, that is, a maximal [[separable field extension]] of $k$.

Separable algebras play a major role in the [[Galois theory]] of extensions of algebras.  There are further generalizations, leading to [[separable functor|separable functors]]...

## Related entries

* [[separable functor]]

* [[separable field extension]]

* [[separable coring]]

## References
 {#References}

* {#EilenbergNakayama55} [[Samuel Eilenberg]], [[Tadasi Nakayama]], *On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings*, _Nagoya Math. J._ **9** (1955) 1-16  [[euclid](https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-9/issue-none/On-the-dimension-of-modules-and-algebras-II-Frobenius-algebras/nmj/1118799677.pdf)]

* K. Hirata, K. Sugano, _On semisimple and separable extensions of noncommutative rings_, J. Math. Soc. Japan 18 (1966), 360-373.

* {#EndoWatanabe67} Shizuo Endo, Yutaka Watanabe, *On separable algebras over a commutative ring*, _Osaka J. Math._ **4**(2) (1967) 233-242 [[euclid](https://projecteuclid.org/download/pdf_1/euclid.ojm/1200691953)]

* F. DeMeyer and E. Ingraham, _Separable algebras over commutative rings_, Lecture Notes in Mathematics **181**, Springer, Berlin (1971)

* [[Marcelo Aguiar]], *A note on strongly separable algebras*, Boletín de la Academia Nacional de Ciencias (Córdoba, Argentina), special issue in honor of Orlando Villamayor, **65** (2000) 51-60  [[pdf](http://www.math.cornell.edu/~maguiar/strongly.pdf)]

See also:

* Wikipedia, *[Separable algebra](http://en.wikipedia.org/wiki/Separable_algebra)*

An explicit proof of the Grothendieck Galois theory statement that the category of commutative separable algebras over a field $K$ is anti-equivalent to the category of continuous actions on finite sets of the profinite fundamental group of $K$:

* Federico G. Lastaria, _On separable algebras in Grothendieck Galois theory_, Le Mathematiche 51:3, 1996, [link](http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/464/0)

[[!redirects separable algebras]]
[[!redirects separable ring extension]]
[[!redirects separable ring extensions]]


separable C*-algebra

#Contents#
* table of contents
{:toc}

## Definition

A _separable $C^\ast$-algebra_ is  a [[C*-algebra]] whose underlying [[topological space]] is a [[separable topological space]]

## References

* Chun-Yen Chou, _Notes on separability of $C^\ast$-algebras_, Taiwanese Journal of mathematics, Vol 16, No 2 (2012) ([web](http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/1786))

* George A. Elliott, Ilijas Farah, Vern Paulsen, Christian Rosendal, Andrew S. Toms, Asger Törnquist, _The isomorphism relation for separable C*-algebras_ ([arXiv:1301.7108](http://arxiv.org/abs/1301.7108))

[[!redirects separable C*-algebras]]
[[!redirects separable C-star algebra]]
[[!redirects separable C-star algebras]]


separable closure

#Contents#
* table of contents
{:toc}

## Definition

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

Let $\bar K$ be a fixed [[algebraic closure]] of $K$. If $F \subset K[X] - \{0\}$ is any collection of non-zero [[polynomial]]s, the **splitting field** of $F$ over $K$ is the subfield of $\bar K$ generated by $K$ and the zeros of the polynomials in $F$.

We call $f \in K[X]- \{0\}$ **separable** if it has no multiple zero in $\bar K$.

We call $\alpha \in \bar K$ **separable over $K$** if the [[irreducible polynomial]] $f^\alpha_K$ of $\alpha$ over $K$ is separable.

A subfield $K \subset L \subset \bar K$ is called **separable over $K$** if each $\alpha \in L$ is separable over $K$.

=--

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

Let $K$ be a [[field]] and $\bar K$ an [[algebraic closure]] of $K$. The **separable closure**  $K_S$ of $K$ is defined by

$$
  K_S \simeq \{x \in \bar K | x \; is \; separable \; over \; K\}
  \,.
$$

=--

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

We have that $K_S$ is a subfield of $\bar K$ and that $K_S \simeq \bar K$ precisely if $K$ is a [[perfect field]], in particular if the [[characteristic]] of $K$ is 0.

From xyz it follows that the inclusion $K \subset K_S$ is Galois.

=--

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

The Galois group $Gal(K_S/K)$ is called the **[[absolute Galois group]]** of $K$.

=--

## Related concepts

* [[algebraic closure]]

* [[geometric point]]

* [[field extension]]

* [[absolute Galois theory]]

## References

* Wikipedia, _[Separable closure](http://en.wikipedia.org/wiki/Algebraic_closure#Separable_closure)_

[[!redirects spearable closed field]]
[[!redirects spearable closed fields]]


separable coring

This is a certain analogue of the notion of [[separable algebra]]. See also [[separable functor]].

* [[T. Brzeziński]], L. Kadison, R. Wisbauer, _On coseparable and biseparable corings_ In: Hopf algebras in  non-commutative geometry and physics,  S. Caenepeel, F. Van Oystaeyen (eds.), Lecture Notes in Pure and  Applied Mathematics __239__, Marcel Dekker, New York, 2004, pp. 71-88, [arXiv:math.RA/0208122](http://arxiv.org/abs/0208122).

* J. Gómez-Torrecillas, _Separable functors in [[corings]]_,
Int. J. of Math. and Math. Sci. __30__ (2002), 4, 203-225, [doi](http://dx.doi.org/10.1155/S016117120201270X), [pdf](http://www.emis.de/journals/HOA/IJMMS/Volume30_4/225.pdf)
[[!redirects separable corings]]


separable field extension

#Contents#
* table of contents
{:toc}

## Definition

A [[polynomial]] $P$ over a [[field]] $K$ is __separable__ if all its irreducible factors have distinct roots over the [[algebraic closure]] $\bar{K}$ of $K$.

An [[field extension|extension]] $K\subset L$ of fields is __separable__ if every element $x\in L$ is a root of a separable polynomial over $K$.

## Properties

Every finite separable field extension is an [[étale morphism]] of rings.

If $K\subset L\subset M$ are fields and $K\subset M$ is separable, then $L\subset M$ is also separable.

[[!redirects separable extension]]
[[!redirects separable extensions]]
[[!redirects separable field extensions]]

[[!redirects separable extension of fields]]
[[!redirects separable extensions of fields]]


separable functor

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

#Contents#
* table of contents
{:toc}

## Motivation

The original motivation for the notion of *separable functors* [[Năstăsescu, van den
Bergh & van Oystaeyen (1989)](#NăstăsescuVanDenBerghVanOystaeyen89)]
was the functorial form of [[Maschke's theorem]] over [[rings]] and its various cousins: namely the classical Maschke's theorem about finite [[group rings]] over [[fields]], generalizes to the statements that when the [[order of a group|order of the group]] is [[invertible element|invertible]] in the [[ground ring]] then the [[split exact sequence|splitting]] of an [[exact sequence]] of [[group representation|$k G$-modules]] can be obtained [[functor|functorially]] from its splitting as an exact sequence of [[underlying]]
[[module|$k$-modules]].

This "Maschke property" of the  [[forgetful functor]] ${}_{k G} Mod\to {}_{k}Mod$ is what is generalized by the notion of separable functors.

A further example appears in the study of [[ring extensions]] $f \colon R\to S$: Such is separable iff the corresponding [[restriction of scalars]]-functor $\operatorname{Res}_R^S \colon {}_{S}Mod \to {}_{R}Mod$ is a separable functor.

## Definition

Let $F \colon C\to D$ be a [[functor]] with [[hom-set]] components
$$
\array{
F_{x,y}
\colon
&
C(x,y)&\longrightarrow& D(F x,F y)
\\
&
(x\stackrel{f}\to y)&\mapsto& (F x\stackrel{F f}\to F y)
\mathrlap{\,.}
}
$$
Then $F$ is called *separable* [[Năstăsescu, van den
Bergh & van Oystaeyen (1989), p. 398](#NăstăsescuVanDenBerghVanOystaeyen89)] if each $F_{x,y}$ has a [[section]] which is "natural" in $x$ and $y$, in a suitable sense.

## Properties

\begin{theorem}
**(Rafael's theorem)**
Let $F\dashv G$ be a pair of [[adjoint functors]]. Then

* $F$ is separable iff the [[adjunction unit]] $\eta \con id \to G F$ has a [[section]] (= a [[natural transformation]] $\nu$ which is its right inverse, $\eta\circ\nu = 1$),

* $G$ is separable iff the [[adjunction counit]] $\epsilon \colon F G \to id$ has a [[retraction]] (i.e. a natural transformation $\zeta$ that is its left inverse, $\zeta\circ\epsilon =1$).

\end{theorem}

The latter condition is reminiscent of one of the many equivalent definitions of a [[separable algebra]] $A$ over a field, namely one for which multiplication, viewed as an $(A,A)$-[[bimodule]] map $A \otimes A^{\mathrm{op}} \to A$, has a  right inverse.

## Literature

Separable functors were defined in

* {#NăstăsescuVanDen BerghVanOystaeyen89} [[Constantin Năstăsescu]], [[Michel van den Bergh]], [[F. van Oystaeyen]], _Separable functors applied to graded rings_, J. Algebra __123__ (1989) 397-413 [doi:10.1016/0021-8693(89)90053-7]

Now there is a monograph available:

* S. Caenepeel, G. Militaru, S. Zhu, _Frobenius and separable functors for generalized module categories and nonlinear equations_, Lec. Notes in Math. __1787__, Springer (2002) xiv+354 pp.

Other references

* M. D. Rafael, _Separable functors revisited_, Comm. in Algebra __18__ (1990), 1445-1459.

* S. Caenepeel, B. Ion, G. Militaru, _The structure of Frobenius algebras and separable algebras_,  $K$-Theory __19__ (2000),  no. 4, 365--402.

* J. Gómez-Torrecillas, _Separable functors in [[corings]]_,
Int. J. of Math. and Math. Sci. __30__ (2002), 4, Pages 203-225, [doi](http://dx.doi.org/10.1155/S016117120201270X), [pdf](http://www.emis.de/journals/HOA/IJMMS/Volume30_4/225.pdf)

* A. Ardizzoni, _Separable functors and [[formal smoothness]]_,  J. K-Theory  1  (2008),  no. 3, 535--582, [math.QA/0407095](http://arxiv.org/abs/math.QA/0407095), [doi](http://dx.doi.org/10.1017/is007011015jkt012), [MR2009k:16069](http://www.ams.org/mathscinet-getitem?mr=2009k:16069)

* A. Ardizzoni, [[G. Böhm]], C. Menini, _A Schneider type theorem for Hopf algebroids_, J. Algebra __318__ (2007), no. 1, 225–269 [MR2008j:16103](http://www.ams.org/mathscinet-getitem?mr=2363132) [doi](http://dx.doi.org/10.1016/j.jalgebra.2007.05.017) [math.QA/0612633](http://arxiv.org/abs/math/0612633) (arXiv version is unified, corrected); _Corrigendum_, J. Algebra __321__:6 (2009) 1786-1796 [MR2010b:16060](http://www.ams.org/mathscinet-getitem?mr=2498268) [doi](http://dx.doi.org/10.1016/j.jalgebra.2008.11.040)

The following article studies formal smoothness and generalizes the separable functors to the context of the so-called [[S-category|S-categories]] which are introduced therein:

* [[T. Brzeziński]], _Notes on formal smoothness_, *in*: Modules and Comodules (series _Trends in Mathematics_). T Brzeziński, JL Gomez Pardo, I Shestakov, PF Smith (eds), Birkhäuser, Basel, 2008, pp. 113-124 ([doi](http://dx.doi.org/10.1007/978-3-7643-8742-6), [arXiv:0710.5527](http://arxiv.org/abs/0710.5527))

[[!redirects separable functors]]


separable Hilbert space

#Contents#
* table of contents
{:toc}

## Definition

A [[Hilbert space]] $H$ over a field $F$ of real or complex numbers and with inner product $(|)$ is __separable__ if it has a countable topological base, i. e. a family of vectors $e_i$, $i\in I$ where $I$ is at most countable, and such that every vector $v\in H$ can be uniquely represented as a series $v = \sum_{i\in I} a_i e_i$ where $a_i\in F$ and the sum converges in the norm $\|x\| = \sqrt{(x|x)}$.

## Related concepts

* [[separable space]]

* [[U(ℋ)|$U(\mathcal{H})$]]

[[!redirects separable Hilbert space]]


separable metacompact spaces are Lindelöf

## Statement

Recall that a [[topological space]] $X$ is [[metacompact space|metacompact]] if every open cover has a point-finite open refinement which covers $X$. Further a space is called [[Lindelöf topological space|Lindelöf]] if every [[open cover]] has a [[countable cover|countable]] sub-cover.

\begin{theorem}
Assuming the law of [[excluded middle]]:

Every [[separable space|separable]] metacompact [[topological space]] $X$ is Lindelöf .
\end{theorem}

\begin{proof}
The proof goes by contradiction: Assume there is an open cover $\{U_i\}_{i\in I}$ with no countable subcover. Let $\{U_j\}_{j\in J}$ be a point finite refinement, which again must not be countable. Let $D\subset $ be a countable dense set. Then $\{U_j\}_{j\in J}$ is uncountable and so some point $x \in D$ is in $U_j$ for infinetly (even uncountably) many $j \in J$.
\end{proof}

## Related statements and properties

[[!include topology - global countability axioms]]


separable metric space

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

#Contents#
* table of contents
{:toc}

## Idea

In [[topology]], a _separable metric space_ is a [[topological space]] that is both [[separable topological space|separable]] and [[metrizable topological space|metrizable]].

## Properties

### Dimension
 {#PropertiesDimension}

\begin{prop}
For separable metric spaces, the following notions of [[dimension]] all (exist and) coincide and are thus uniformly referred to as the _[[dimension of a separable metric space]]_:

1. small and large [[inductive dimension]];

1. [[covering dimension]].

\end{prop}

(e.g. [Engelking 78, Theorem 1.7.7](#Engelking78))

## References

* {#Engelking78} [[Ryszard Engelking]], _Dimension Theory_, Mathematical Library **19**, North-Holland Publishing/Polish Scientific Publishers 1978 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/engelking.pdf))

* {#Engelking95} [[Ryszard Engelking]], _Theory of Dimensions -- Finite and Infinite_, Sigma Series in Pure Mathematics **10**, Helderman 1995 ([pdf](http://www.gbv.de/dms/goettingen/187241074.pdf))

[[!redirects separable metric spaces]]


separable space

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

# Separable spaces
* table of contents
{: toc}

## Definitions

A [[topological space]] is __separable__ if it has a [[countable set|countable]] [[dense subspace|dense]] subset.

To be explicit, $X$ is __separable__ if there exists an [[infinite sequence]] $a\colon \mathbb{N} \to X$ such that, given any point $b$ in $X$ and any [[neighbourhood]] $U$ of $b$, we have $a_i \in U$ for some $i$.

## Properties

A [[second-countable space]] is separable (and trivially it is
[[first-countable space|first-countable]]). However, first-countability plus separability do not imply second-countability; a counterexample is $\mathbb{R}$ with the half-open topology (see [Munkres](#Munk), page 192), denoted $\mathbb{R}_l$.

An arbitrary [[product]] of separable spaces need not be separable, but a product of as many as a continuum number of separable spaces _is_ separable (with the product computed in [[Top]]); a proof of the more general Hewitt-Marczewski-Pondiczery theorem can be found [here](#Brandsma). In particular, the space $\mathbb{R}^\mathbb{R}$ of _all_ [[functions]] $\mathbb{R} \to \mathbb{R}$ under pointwise convergence is separable, but is not even first-countable (and thus not second-countable either). A first-countable space need not be separable; a simple example of that is a [[discrete space]] of uncountable cardinality.

+-- {: .num_remark}
###### Remark
Although an arbitrary product of separable spaces need not be separable, an arbitrary product of separable spaces does satisfy the [[countable chain condition]] (see there for more discussion). This is somewhat remarkable, since it is not necessarily true that even a finite product of spaces satisfying the countable chain condition also satisfies the countable chain condition (whether it does is independent of [[ZFC]]).
=--

Subspaces of separable spaces need not be separable. Example: the product $\mathbb{R}_l \times \mathbb{R}_l$, also called the [[Sorgenfrey plane]], is separable, but the subspace defined by the equation $y = -x$ is uncountable and discrete and therefore not separable. However, open subspaces of separable spaces are separable.

Many results in [[analysis]] are easiest for separable spaces.  This is particularly true if one wishes to avoid using strong forms of the [[axiom of choice]], or to use arguments [[predicative mathematics|predicative]] over the natural numbers. For example, the [[Hahn-Banach theorem]] for _separable_ [[Banach spaces]] can be established using only mild forms of choice, e.g., [[dependent choice]]. More precisely, it can be established in the weak subsystem $WKL_0$ of [[second-order arithmetic]]; see [Brown-Simpson](#BS).

## Separable metric spaces

A classical fact is

+-- {: .num_theorem #classical}
###### Theorem
Using [[countable choice]], then:

For a metric space $X$ the following are equivalent

1. $X$ is separable;
1. $X$ is [[second-countable space|second-countable]];
1. $X$ is [[Lindelöf space|Lindelöf]], i.e. any open cover admits a countable subcover.
=--

+-- {: .proof}
###### Proof
The second property is implied by the first: given any dense subset $\{x_n \mid n\in \mathbb{N}\}$, form the countable system of sets $\{B_{1/m}(x_n) \mid n,m\in\mathbb{N}\}$. To see that this is indeed a [[base]] for the topology of $X$, take any open set $U\subset X$, a point $x\in U$ and a radius $1/k$ such that $B_{1/k}(x)\subset U$. By separability there is some $n$ such that $x_n\in B_{1/(2k)}(x)$ and therefore $x\in B_{1/(2k)}(x_i)\subset U$.

To show (2) implies (3), let $\{U_n\}$ be a countable base of the topology.
Given any open cover $\{V_\lambda\}$ of $X$, we can form the index set $I\subset \mathbb{N}$ of those $n$ that are contained in some $V_\lambda$.
By assumption $\bigcup_{i\in I} U_{i} = \bigcup_\lambda V_\lambda = X$.
The axiom of [[countable choice]] provides now a section of $\bigsqcup_{i\in I} \{\lambda \mid U_i \subset V_\lambda\}\to I$.

Finally, we prove that (3) implies (1).
Consider the open covers $\{B_{1/1}(x) \mid x\in X\}$, $\{B_{1/2}(x) \mid x\in X\}$, ...
From each extract a countable subcover corresponding to collection of centers $A_1, A_2, \ldots$. We claim that that the union $A_1\cup A_2\cup\ldots$ forms a dense set.
Indeed, given any $y\in X$ and $n$ the point $x$ has to be contained in some $B_{1/n}(x)$ for some $x\in A_n$.
=--

\begin{remark}
In the proof any variant of the [[axiom of choice]] is only used for the implication $(2)\Rightarrow(3)$. On the other hand, assuming [[countable choice]], this implication [[second-countable spaces are Lindelöf|holds in every topological space]].
The implication $(2)\Rightarrow(1)$ holds in any topological space as well (see Example \ref{2ndCountablImplSeparable})
\end{remark}

Similar in spirit to (1)$\Leftrightarrow$(2) but less well-known is the following.

+-- {: .num_theorem}
###### Theorem
A [[metric space]] $X$ is separable iff every open set is a countable union of balls.
=--

+-- {: .proof}
###### Proof
One direction is not hard: if $x_i$ is a countable dense subset of $X$ and $r_j$ is an enumeration of the rationals, then according to the proof of Theorem \ref{classical}, the balls $B_{r_j}(x_i)$ form a countable base ($X$ is a [[second-countable space]]). Hence every open set is a union of a family of such balls that is at most countable.

The other direction is trickier. (This is based on a [MathOverflow discussion](http://mathoverflow.net/questions/181226/if-any-open-set-is-a-countable-union-of-balls-does-it-imply-separability), which for the moment we record with little adaptation.) Suppose $X$ is not separable; construct by recursion a sequence $x_\beta$ of length $\omega_1$ such that no $x_\beta$ lies in the closure of the set of its predecessors $x_\alpha$ (each such set is countable and therefore not dense, so such $x_\beta$ outside its closure can be found at each stage).

Therefore, for each $x_\beta$ we may choose a rational radius $r_\beta$ such that the ball $B_{r_\beta}(x_\beta)$ contains no predecessor $x_\alpha$. There are uncountably many $\beta$, so some rational radius $r$ was used uncountably many times. The collection of $x_\beta$ for which $r_\beta = r$ forms another $\omega_1$-sequence which we now use instead of the first; since all the radii are the same, this sequence has the property that each $B_r(x_\beta)$ contains no other $x_\alpha$, no matter whether $\alpha$ appears before or after $\beta$ in the sequence.

Provided that there uncountably many such $x_\alpha$ that are non-isolated in $X$, we can construct an open set $U$ that is not a countable union of balls. For around each such $x_\alpha$ we can find a point $p_\alpha \neq x_\alpha$ within distance $r/4$ of $x_\alpha$; put $U = \bigcup_\alpha B_{d(x_\alpha, p_\alpha)}(x_\alpha)$. Notice that $p_\alpha \notin U$. If $B_s(x)$ is any ball contained in $U$, then $d(x, x_\gamma) \lt r/4$ for some $\gamma$. Supposing we have distinct $x_\alpha, x_\beta \in B_s(x)$, then $r \lt d(x_\alpha, x_\beta) \leq d(x_\alpha, x) + d(x, x_\beta) \lt s + s$, so $r/2 \lt s$. But then from $d(x_\gamma, p_\gamma) \lt r/4$ and $d(x, x_\gamma) \lt r/4$, we have $d(x, p_\gamma) \lt r/2 \lt s$ so that $p_\gamma \in B_s(x) \subset U$, a contradiction. We conclude that any ball contained in $U$ contains at most one $x_\alpha$, and so $U$ cannot be covered by countably many balls.

We are therefore left to deal with the case where there are at most countably many non-isolated $x_\beta$. Discard these, so without loss of generality we may suppose all the $x_\beta$ are isolated points of $X$ and that for some fixed $r$ the ball $B_r(x_\beta)$ contains no other $x_\alpha$. Let $Z$ be this set of $x_\beta$. For each $\beta$, let $t_\beta$ be the supremum over all $t$ such that $B_t(x_\beta) \cap Z$ is countable. It follows that $B_{t_\beta}(x_\beta) \cap Z$ is itself countable, as is $\{\alpha: \alpha \lt \beta\}$. At each stage $\beta$, there is a countable set $C_\beta \subset Z$ disjoint from $B_{t_\beta}(x_\beta) \cup \{x_\alpha: \alpha \lt \beta\}$ such that for each $s \gt t_\beta$, there exists $y \in C_\beta$ with $d(x_\beta, y) \lt s$. By transfinite induction, we can construct a cofinal subset $I$ of $\omega_1$ such that $x_\beta \notin C_\alpha$ whenever $\alpha, \beta \in I$ and $\alpha \lt \beta$. The set $Y = \{x_\alpha: \alpha \in I\}$ is open in $X$, and we claim that it is not a countable union of balls. For suppose otherwise. Let $F$ be such a countable family of balls; then there is some minimal $\alpha$ for which $B_t(x_\alpha) \in F$ is uncountable, so that $t \gt t_\alpha$. By construction of $C_\alpha$, there exists $z \in C_\alpha \cap B_t(x_\alpha)$. But since $Y = \bigcup F$, we have that $z = x_\beta$ for some $\beta \in I$, and this contradicts our condition on $I$.
=--

## Examples

\begin{example}\label{2ndCountablImplSeparable}
Every [[second-countable topological space]] is separable.
To see this take a countable cover and select a point in each member of the cover (using [[countable choice]]).
\end{example}

## Related countability properties

[[!include topology - countability axioms]]

## Related concepts

* [[decidable object]]

* [[separable metric space]]

* [[separable Hilbert space]]

* [[Hausdorff topological space]]

## References

* {#Munk} [[James Munkres]], _Topology, a first course_, Prentice-Hall (1975).

* {#BS} D. K. Brown and S. G. Simpson, _Which set existence axioms are needed to prove the separable Hahn-Banach theorem?_, Annals of Pure and Applied Logic 31 (1986), pp. 123-144.

* {#Brandsma} Henno Brandsma, Thread on Hewitt-Marczewski theorem, Ask a Topologist (2010) ([link](http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2010;task=show_msg;msg=0487.0001))

[[!redirects separable space]]
[[!redirects separable spaces]]

[[!redirects separable topological space]]
[[!redirects separable topological spaces]]


Separable topological vector space


separated (2,1)-presheaf

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

A  **separated** [[(2,1)-presheaf]]/[[prestack]] over a [[(2,1)-site]] $C$ is a [[(2,1)-presheaf]] $X : C^{op} \to Grpd$ such that [[covering]] families $\{U_i \to U\}$ in $C$ the [[descent]] morphism

$$
  X(U) \to PSh_{(2,1)}(S(\{U_i\}), X)
$$

is a [[full and faithful functor]] and hence exhibits a [[full subcategory]].

(Here $S(\{U_i\})$ denotes the [[sieve]] associated to the cover).

If this morphism is even an [[equivalence of categories]], then $X$ is even a [[(2,1)-sheaf]]/[[stack]].

=--

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

The term _prestack_ is used in two different ways in the literature: some authors use it synonymously with just [[(2,1)-presheaf]], others with _separated $(2,1)$-presheaf_ .

=--

## Examples

Let [[Mfd]] be the [[site]] of [[topological manifold]]s. Let $G$ be a [[topological group]] and $\bar W G$ the [[(2,1)-presheaf]] on $Mfd$ represented by the [[nerve]] of the [[delooping]]-[[groupoid]] (see [[simplicial group]] for te notation). Let $G Bund$ be the [[(2,1)-sheaf]] of all $G$-[[principal bundle]]s. This is the [[(2,1)-sheafification]] of $\bar W G$. The canonical morphism

$$
  \bar W G \to G Bund
$$

includes over each $U \in Mfd$ the single object of $(\bar W G)(U)$ as the trivial $G$-[[principal bundle]]. Its [[automorphism]]s are given by [[continuous function]]s $C(U,G)$. This is the same on both sides, hence $(\bar W G)(U) \to G Bund(U)$ is a [[full and faithful functor]] and $\bar W G$ is a separated $(2,1)$-presheaf.

## Related concepts

* [[separated presheaf]]

* **separated $(2,1)$-presheaf**

* [[separated (∞,1)-presheaf]]

[[!redirects prestack]]

[[!redirects separated prestack]]

[[!redirects separated (2,1)-presheaves]]

[[!redirects prestacks]]


separated (infinity,1)-presheaf

[[!redirects separated (∞,1)-presheaf]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

A  **separated** [[(∞,1)-presheaf]] over an [[(∞,1)-site]] $C$ is a [[(∞,1)-presheaf]] $X : C^{op} \to $ [[∞Grpd]] such that [[covering]] families $\{U_i \to U\}$ in $C$ the [[descent]] comparison morphism

$$
  X(U) \to PSh_{(\infty,1)}(S(\{U_i\}), X)
$$

is a [[full and faithful (∞,1)-functor]] and hence exhibits a full
[[sub-(∞,1)-category]].

(Here $S(\{U_i\})$ denotes the [[sieve]] associated to the cover).

More generally, $X$ is **$k$-separated** for $k \in \mathbb{N}$ if the  descent morphism is a $(k-2)$-[[n-truncated|truncated morphism]].

=--

Notice that this means that a _0-separated $(\infty,1)$-presheaf_ is one whose descent morphisms are [[equivalence in an (∞,1)-category|equivalences]], hence those which are [[(∞,1)-sheaves]].

## Related concepts

* [[separated presheaf]]

* [[separated (2,1)-presheaf]]

* **separated (∞,1)-presheaf**

[[!redirects separated (∞,1)-presheaves]]


separated geometric morphism

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

#Contents#
* table of contents
{:toc}

## Definition

A [[geometric morphism]] $f : \mathcal{X} \to \mathcal{Y}$ of [[toposes]] is **separated** if the [[diagonal]] $\mathcal{X} \to \mathcal{X} \times_{\mathcal{Y}} \mathcal{X}$ is a [[proper geometric morphism]].

In particular if $\mathcal{Y}$ is the [[terminal object]] in [[Topos]], hence the canonical [[base topos]] [[Set]], we say that a topos $\mathcal{X}$ is a **Hausdorff topos** if $\mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is a proper geometric morphism.

More generally, since there is a hierarchy of notions of _[[proper geometric morphism]]_, there is accordingly a hierarchy of separatedness conditions.

## Examples

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

For $G$ a [[discrete group]] and $\mathbf{B}G = (G \stackrel{\to}{\to} *)$ its [[delooping]] [[groupoid]], the [[presheaf topos]] $G Set \simeq [\mathbf{B}G, Set]$ is Hausdorff precisely if $G$ is a [[finite group]].

=--

In ([Johnstone](#Johnstone)) this is example C3.2.24

## Related concepts

* [[Hausdorff topological space]]
* [[quasi-separated scheme]]
* [[separated scheme]], [[separated morphism of schemes]]

## References

Chapter II of

* [[Ieke Moerdijk]], Jacob Vermeulen,  _Relative compactness conditions for toposes_ ([pdf](http://igitur-archive.library.uu.nl/math/2001-0702-142944/1039.pdf)) and _Proper maps of toposes_ , American Mathematical Society (2000)
  {#MoerdijkVermeulen}

Around def. C3.2.12 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
 {#Johnstone}

[[!redirects Hausdorff topos]]
[[!redirects strongly Hausdorff topos]]

[[!redirects Hausdorff toposes]]
[[!redirects strongly Hausdorff toposes]]

[[!redirects Hausdorff topoi]]
[[!redirects strongly Hausdorff topoi]]

[[!redirects separated geometric morphisms]]


separated morphism

#Contents#
* table of contents
{:toc}

## Idea

The general notion is that of

* _[[separated geometric morphism]]_ between [[toposes]].

Special cases include the notion

* [[quasi-separated morphism]] of schemes

which is a bit weaker than

* _[[separated morphism of schemes]]_.

## Related concepts

* [[open morphism]]

* [[closed morphism]]

* [[étale morphism]]

category: algebraic geometry

[[!redirects separated morphisms]]


separated morphism of schemes

#Contents#
* table of contents
{:toc}

## Definition

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

Let $f : X \to Y$ be a morphism of [[schemes]]. Write $\Delta : X \to X \times_Y X$ for the [[diagonal]] morphism.

* The morphism $f$ is called __separated__ if $\Delta(X)$ is a [[closed subspace]] of $X \times_Y X$.

* A [[scheme]] $X$ is called __separated__ if the [[terminal object|terminal]] morphism $X \to \operatorname{Spec} \mathbb{Z}$ is separated.

=--

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

Let $X$ be a [[scheme]] (resp. a [[locally noetherian scheme]]),
 $f: X\to Y$ a morphism of schemes (resp. a morphism locally of finite type). The following conditions are equivalent.

1. $f$ is separated.

1. The [[diagonal]] morphism $X\to X\times_Y X$ is [[quasicompact morphism|quasicompact]], and for every affine scheme $Y' = Spec A$ in which $A$ is a [[valuation ring]] (resp. a discrete valuation ring), any two morphisms from $Y'\to X$ which coincide at the generic point of $Y'$ are equal.

1. The diagonal morphism $X\to X\times_Y X$ is quasicompact, and for every affine scheme of the form  $Y' = Spec A$ in which $A$ is a valuation ring (resp. a discrete valuation ring), any two sections of $X' = X(Y')$ which coincide at the generic point of $Y'$ are equal.

=--

This is the _[[valuative criterion of separatedness]]_. See Hartshorne or [[EGA II]] for more details.

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

The definition of a separated scheme is formally similar to the definition of a [[Hausdorff space]] which says that the diagonal $\Delta(X) \subseteq X \times X$ is closed; the same pattern is followed in the definition of a [[Hausdorff locale]], [[Hausdorff topos]], etc. More generally, the definition of a separated morphism of schemes is formally similar to e.g. a [[separated geometric morphism]]. This leads to these properties having similar formal properties. Nevertheless, because finite products and pullbacks in these categories do not necessarily agree, these notions of separation also vary. For example, the underlying topological space of a separated scheme is typically not Hausdorff.

=--

## Properties

(...)

## Related concepts

* [[separated geometric morphism]]

* [[Hausdorff topos]], [[Hausdorff topological space]]

category: algebraic geometry

[[!redirects separated morphisms of schemes]]
[[!redirects separated scheme]]
[[!redirects separated schemes]]

[[!redirects locally separated]]


separated presheaf

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

#Contents#
* table of contents
{:toc}

## Separated presheaf

### Idea

The condition that a [[presheaf]] be a [[sheaf]] may be seen as a condition of unique existence.  A presheaf is separated if it satisfies the uniqueness part.

### Definition

Let $S$ be a [[site]].

Recall that a [[sheaf]] on $S$ is a presheaf $A \in PSh_S$ such that for all [[local isomorphism]]s $Y \to X$ the induced morphism  $PSh_S(X,A) \to PSh_S(Y,A)$ (under the [[hom-functor]] $PSh_S(-,A)$) is an [[isomorphism]].  (For an arbitrary class of morphisms $V$, the corresponding condition is called being a [[local object]].)
It is sufficient to check this on the [[dense monomorphism]]s instead of all local isomorphisms.  This is equivalent to checking [[cover]]ing [[sieve]]s.

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

A [[presheaf]] $A \in PSh(S)$ is called **separated** (or a **monopresheaf**) if for all [[local isomorphism]]s $Y \to X$ the induced morphism $Hom(X,A) \to Hom(Y,A)$ is a [[monomorphism]].

More generally, for a class $V$ of arrows in a category $C$, an object $A\in C$ is $V$-**separated** if for all morphisms $Y\to X$ in $V$, the induced morphism $Hom(X,A)\to Hom(Y,A)$ is a monomorphism.

=--

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

As for sheaves, it is sufficient to check the separation condition on the [[dense monomorphism]]s, hence on the [[sieve]]s.

For $\{p_i : U_i \to U\}$ a [[covering]] family of an object $U \in S$, the condition is that if $a,b \in A(U)$ are such that for all $i$ we have $A(p_i)(a) = A(p_i)(b)$ then already $a = b$.

=--

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

The definition generalizes to any system of [[local isomorphisms]] on any [[topos]], such as that obtained from any [[Lawvere-Tierney topology]], or equivalently any [[subtopos]].

=--

### Example

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

Let $(S,J)$ be a [[site]] for which every $J$-[[covering family]] is [[inhabited]]. Then for any [[set]] $X$, the [[constant presheaf]] $S\ni a \mapsto X$ is separated.

=--

See also at _[[locally connected site]]_.

### Properties

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

The full [[subcategory]]

$$
  i : Sep(S) \hookrightarrow PSh(S)
$$

of separated presheaves in a presheaf category is

* a [[reflective subcategory]];

* a [[quasitopos]].

=--

Being a reflective subcategory means that there is a [[left adjoint]] functor to the inclusion

$$
  (L_{sep} \dashv i) : Sep(S) \stackrel{\overset{L_{sep}}{\leftarrow}}{\hookrightarrow}
  PSh_S
  \,.
$$

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

For $A \in PSh_S$ the **separafication** $L_{sep}A$ of $A$ is the presheaf that assigns equivalence classes
$$L_{sep}A : U \mapsto A(U)/\sim_U,$$
where $\sim_U$ is the [[equivalence relation]] that relates two elements $a \sim b$ iff there exists a [[covering]] $\{p_i : U_i \to U\}$ such that $A(p_i)(a) = A(p_i)(b)$ for all $i$.

This construction extends in the evident way to a functor
$$L_{sep} : PSh(S) \to Sep(S).$$

=--

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

This functor $L_{sep}$ is indeed a [[left adjoint]] to the inclusion $i : Sep(S) \hookrightarrow PSh(S)$.

=--

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

Let $A \in PSh(S)$ and $B \in Sep(S) \hookrightarrow PSh(S)$. We need to show that morphisms $f : A \to B$ in $PSh_C$ are in natural bijection with morphisms $L_{sep} A \to B$ in $Sep(S)$.

For $f$ such a morphism and $f_U : A(U) \to B(U)$ its component over any object $U \in S$, consider any covering $\{U_i \to U\}$, let $S(U_i) \to U$  be the corresponding [[sieve]] and consider the [[commuting diagram]]
$$
  \array{
    \{(a_i \in A(U_i)) | \cdots \}
    &\to&
   \{(b_i \in F(U_i)) | \cdots \}
    \\
    \uparrow && \uparrow
    \\
    A(U) &\stackrel{f_U}{\to}& B(U)
  }
$$
obtained from the naturality of $PSh_S(S(U_i) \to U, A \stackrel{f}{\to} B)$.

If for $a,a' \in A(U)$ two elements that are not equal their restrictions to the cover become equal in that  $\forall i :  a|_{U_i} = a'|_{U_i}$, then also $f(a|_{U_i}) = f(a'|_{U_i})$ and since the right vertical morphism is monic there is a _unique_ $b \in B(U)$ mapping to the latter. The commutativity of the diagram then demands that $f(a) = f(a') = b$.

Since this argument applies to all covers of $U$, we have that $f_U$ factors uniquely through the projection map $A(U) \to A(U)/\sim_U =: L_{sep}(U)$ onto the quotient. Since this is true for every object $U$ we have that $f$ factors uniquely through $A \to L_{sep}A$.

=--

## Biseparated presheaf

### Idea

Often one is interested in separated presheaves with respect to one [[coverage]] that are sheaves with respect to another coverage. These are called _biseparated presheaves_ .

This typically arises if a [[reflective subcategory]]
$$C \stackrel{\stackrel{}{\leftarrow}}{\hookrightarrow} Sh(S)$$
of a [[sheaf topos]] is given. This is the [[localization]] at a set $W$ of morphisms in $Sh(S)$, with $C$ the full subcategory of all [[local object]]s $c$: objects such that $Sh(S)(w,c)$ is an isomorphism for all $w \in W$. A $W$-separated object is then called a _biseparated presheaf_ on $S$ and their collection $BiSep(S)$ factors the reflective inclusion as
$$
  C
    \stackrel{\leftarrow}{\hookrightarrow}
  BiSep(S)
    \stackrel{\leftarrow}{\hookrightarrow}
  Sh(S).
$$

### Definition

\begin{definition}\label{Biseparated}

A **bisite** is a [[small category]] $S$ equipped with two [[coverage]]s: $J$ and $K$ such that $J \subset K$.

A presheaf $A \in PSh_S$ is called $(J,K)$-**biseparated** if it is

* a [[sheaf]] with respect to $J$;

* a separated presheaf with respect to $K$.

Write
$$
  BiSep_{(J,K)}(S) \hookrightarrow Sh_J(S) \hookrightarrow PSh(S)
$$
for the full [[subcategory]] on biseparated presheaves.

\end{definition}

### Examples
 {#Examples}

\begin{example}\label{DiffeologicalSpaces}
[[diffeological space|Diffeological spaces]] are biseparated presheaves (Def. \ref{Biseparated}) for the bisite structure on [[cartesian spaces]], where $J$ is the usual topology of open covers and $K$ is the topology given by the families

$$
  \coprod_{u\in U}
  \mathbb{R}^0
    \longrightarrow
  U
$$

for every [[cartesian space]] $U$.

The [[sheaf]] condition with respect to $J$ yields a [[smooth set]], whereas the [[separated presheaf]] condition with respect to $K$ makes it into a [[diffeological space]].
\end{example}

### Properties

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

Biseparated presheaves form a [[reflective subcategory]] of all sheaves
$$
  BiSep_{(J,K)}(S)
  \stackrel{\stackrel{L^K_{sep}}{\leftarrow}}{\hookrightarrow}
  Sh_J(S)
  \,.
$$

=--

See [[quasitopos]] for the proof.

## Related concepts

* **separated presheaf**

* [[epipresheaf]]

* [[separated (2,1)-presheaf]]

* [[separated (∞,1)-presheaf]]

## References

The general theory of biseparated presheaves and Grothendieck quasitoposes is in section C.2.2 of

* [[Peter Johnstone]], [[Sketches of an Elephant]] .

A concrete description of separafication appears on page 43 of

* [[Angelo Vistoli]], _Notes on Grothendieck topologies, fibered categories and descent theory_ ([pdf](http://homepage.sns.it/vistoli/descent.pdf#Page=43))

category: sheaf theory

[[!redirects monopresheaf]]
[[!redirects separated presheaves]]
[[!redirects separated object]]
[[!redirects separated objects]]
[[!redirects biseparated presheaf]]
[[!redirects biseparated presheaves]]
[[!redirects biseparated object]]
[[!redirects biseparated objects]]


separating vector

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

#Contents#
* automatic table of contents goes here
{:toc}

## Definition

Let $\mathcal{M}$ be a [[von Neumann algebra]] acting on a [[Hilbert space]] $\mathcal{H}$.

A vector $x \in \mathcal{H}$ is a **separating vector** if $M(x) = 0$ implies $M = 0$ for all $M \in \mathcal{M}$.

## Properties

The notions of separating vector is dual to that of [[cyclic vector]] with respect to the [[commutant]] $\mathcal{M}'$, that is a vector is cyclic for $\mathcal{M}$ iff it is separating for $\mathcal{M}'$.

## Applications

In the context of [[AQFT]] separating vectors appear as _vacuum [[state]]s_ . See [[Reeh-Schlieder theorem]].

[[!redirects separating vectors]]


separation algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
####
+--{: .hide}
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Motivation

Several generalizations of [[effect algebras]] exist, based on the needs of various logics; these may not have a clear relationship to the quantum foundations of effect algebras, but deal with informationally similar situations where the "resources" of the deductions rules are subject to certain restrictions. _Separation algebras_  correspond to [[separation logic]] in the way effect algebras correspond to [[quantum logic]], and are often used as models for memory allocation and program modularity, where the outcome of a program must be guaranteed with respect to specified memory states. Separation algebra therefore has a role in [[certified programming]].

## Definition

+-- {: .num_defn #SA}
###### Definition
**((Generalized) Separation Algebra)**

A _generalized separation algebra_ (GSA) is a partial [[monoid]] satisfying the following equations:

(1) $ab = ac \rightarrow b=c$,

(2) $ab=cb \rightarrow a=c$,

and the non-equational formula:

(3) $\forall a, b (\exists d : da=b \iff \exists c : ac = b)$

A _separation algebra_ is a commutative GSA, which removes the axiom (3). The real interest of (3) lies in other order-theoretic properties.
=--

This reveals effect algebras to be those separation algebras with the positivity property.

### A Family of Partial Algebras
  {#familytree}

(...)

## Examples

Aside from low-dimensional cases found in a model generator such as Mace, many natural examples which are not effect algebras arise from ordered bisemigroups.

(...)

## Related Concepts

* [[separation logic]]

* [[Hoare logic]]

* [[effect algebras]]

## References

(...)

[[!redirects separation algebra]]
[[!redirects separation-algebra]]