Knizhnik-Zamolodchikov-Kontsevich construction -- definition

For the Definition \ref{KnizhnikZamolodchikovConnection} of the [[Knizhnik-Zamolodchikov connection]] we need the following notation:

1. **[[configuration spaces of points]]**

   For $N_{\mathrm{f}} \in \mathbb{N}$ write

   \[
     \label{OrderedConfigurationSpaceofnPointsinC}
     \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{R}^2)
     \;\coloneqq\;
     (\mathbb{R}^2)^n \backslash FatDiagonal
   \]

   for the [[ordered configuration space of n points]] in the [[plane]], regarded as a [[smooth manifold]].

   Identifying the plane with the [[complex plane]] $\mathbb{C}$, we have canonical [[holomorphic function|holomorphic]] [[coordinate functions]]

   \[
     \label{CoordinateFunctionsOnConfC}
     (z_1, \cdots, z_{N_{\mathrm{f}}})
     \;\colon\;
     \underset{{}^{\{1,\cdots,n\}}}{Conf}(\mathbb{R}^2)
     \longrightarrow
     \mathbb{C}^{N_{\mathrm{f}}}
     \,.
   \]

1. **[[horizontal chord diagrams]]**

   \[
     \label{HorizontalChordDiagrams}
     \mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}}
     \;\coloneqq\;
     Span\big(\mathcal{D}^{{}^{pb}}_{N_{\mathrm{f}}}\big)/4T
   \]

   for the [[quotient vector space]] of the [[linear span]] of [[horizontal chord diagrams]] on $n$ strands by the [[4T relations]] ([[infinitesimal braid relations]]), regarded as an [[associative algebra]] under [[concatenation]] of strands ([here](horizontal+chord+diagram#AlgebraOfHorizontalChordDiagrams)).

+-- {: .num_defn #KnizhnikZamolodchikovConnection}
###### Definition
**([[Knizhnik-Zamolodchikov form]])**

The _universal [[Knizhnik-Zamolodchikov form]]_ is the [[horizontal chord diagram]]-[[Lie algebra valued differential form|algebra valued differential form]] (eq:HorizontalChordDiagrams) on the [[configuration space of points]]  (eq:OrderedConfigurationSpaceofnPointsinC)

\[
  \label{UniversalKnizhnikZamolodchikovForm}
  \omega_{KZ}
  \;\in\;
  \Omega
  \big(
    \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C})
    \,,
    \mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}}
  \big)
\]

given in the canonical [[coordinates]] (eq:CoordinateFunctionsOnConfC) by:

\[
  \label{DefineUniversalKnizhnikZamolodchikovForm}
  \omega_{KZ}
  \;\coloneqq\;
  \underset{ i \lt j \in \{1, \cdots, n\} }{\sum}
  d_{dR} log\big( z_i - z_j \big)
  \otimes t_{i j}
  \,,
\]

where

is the horizontal chord diagram with exactly one chord, which stretches between the $i$th and the $j$th strand.

Regarded as a connection form for a [[connection on a vector bundle]], this defines the universal _[[Knizhnik-Zamolodchikov connection]]_ $\nabla_{KZ}$, with [[covariant derivative]]

$$
  \nabla \phi
  \;\coloneqq\;
  d \phi + \omega_{KV} \wedge \phi
$$

for any [[smooth function]]

$$
  \phi \;\colon\;
    \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C})
  \longrightarrow
  \mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}} Mod
$$

with values in [[modules]] over the algebra of [[horizontal chord diagrams]] [[quotient space|modulo]] [[4T relations]].

The condition of covariant constancy

$$
  \nabla_{KZ} \phi \;=\; 0
$$

is called the _Knizhnik-Zamolodchikov equation_.

Finally, given a [[metric Lie algebra]] $\mathfrak{g}$ and a [[tuple]] of [[Lie algebra representations]]

$$
  (
    V_1,
      \cdots,
    V_{N_{\mathrm{f}}}
  )
  \;\in\; (\mathfrak{g} Rep_{/\sim})^{N_{\mathrm{f}}}
  \,,
$$

the corresponding [[endomorphism]]-valued [[Lie algebra weight system]]

$$
  w_{V}
  \;\colon\;
  \mathcal{A}^{{}^{pf}}_{N_{\mathrm{f}}}
  \longrightarrow
  End_{\mathfrak{g}}\big( V_1 \otimes \cdots V_{N_{\mathrm{f}}}  \big)
$$

turns the universal [[Knizhnik-Zamolodchikov form]] (eq:UniversalKnizhnikZamolodchikovForm) into a [[endomorphism ring]]-[[Lie algebra valued differential form|valued differential form]]

\[
  \label{LieKnizhnikZamolodchikovForm}
  \omega_{KZ}
  \;\coloneqq\;
  \underset{ i \lt j \in \{1, \cdots, n\} }{\sum}
  d_{dR} log\big( z_i - z_j \big)
  \otimes w_V(t_{i j})
  \;\in\;
  \Omega
  \big(
    \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C})
    \,,
    End\big(V_1 \otimes \cdot V_{N_{\mathrm{f}}} \big)
  \big)

  \,.
\]

=--

The universal formulation (eq:UniversalKnizhnikZamolodchikovForm) is highlighted for instance in [Bat-Natan 95, Section 4.2](Vassiliev+invariant#BarNatan95),
[Lescop 00, p. 7](Kontsevich+integral#Lescop00). Most authors state the version after evaluation in a [[Lie algebra weight system]], e.g. [Kohno 14, Section 5](Kontsevich+integral#Kohno14).

+-- {: .num_prop #KnizhnikZamolodchikovConnectionIsFlat}
###### Proposition
**([[Knizhnik-Zamolodchikov connection]] is [[flat connection|flat]])**

The [[Knizhnik-Zamolodchikov connection]] $\omega_{ZK}$ (Def. \ref{KnizhnikZamolodchikovConnection}) is [[flat connection|flat]]:

$$
  d \omega_{ZK} + \omega_{ZK} \wedge \omega_{ZK}
  \;=\;
  0
  \,.
$$

=--

+-- {: .num_prop #KontsevichIntegralForBraids}
###### Proposition
**([[Kontsevich integral]] for [[braids]])**

The [[Dyson formula]] for the [[holonomy]] of the [[Knizhnik-Zamolodchikov connection]] (Def. \ref{KnizhnikZamolodchikovConnection}) is called the _[[Kontsevich integral]] on [[braids]]_.

=--

(e.g. [Lescop 00, side-remark 1.14](Kontsevich+integral#Lescop00))


Knizhnik–Zamolodchikov connection > history

see [[Knizhnik-Zamolodchikov equation]]


knot

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

# Knots
* table of contents
{: toc}

The theory of knots is very visual. It can provide a link between the concrete and abstract.  Some of the arguments are quite elementary, others very deep, and there are numerous connections with other parts of mathematics.

## Definition

A __knot__ is a [[smooth map|smooth]] (or piecewise-linear) [[embedding]] of the [[circle]] $S^1$ into $\mathbb{R}^3$, or equivalently into the $3$-[[sphere]] $S^3$ (one can also consider knots in other [[3-manifold]]s).

Sometimes, higher dimensional knots are also considered. $n$-dimensional knot (or simply an $n$-knot) is a smooth embedding of $n$-dimensional closed manifold (usually an $n$-sphere) into the $(n+2)$-dimensional sphere $\mathbb{S}^n$.

Typically, knots are considered up to [[ambient isotopy]] (or [[smooth isotopy]]).

Knots that are ambient isotopic are often said to have the *same knot type* or to be *in the same isotopy class*.

## Example

The trefoil knot is the simplest non-trivial knot.  In its simplest representation, it has three crossings.  It is a [[torus knot]], that is it can be embedded on the surface of a solid torus, itself embedded in $S^3$. Here is a picture.

+--{: style="text-align:center"}
[[!include trefoil knot - SVG]]
=--

Classifying knots up to isotopy is usually done using [[knot invariant|knot invariants]]. Some of these are simple to define (these tend to be geometric and also tend to be hard to calculate) others are harder to define and to show they are invariants but are easier to calculate. A few are reasonably easy to define and to calculate... Yippee!

It is often useful to consider the domain circle of a knot as being oriented.  This is then represented by putting a directional arrow on diagrams of the knot. Such oriented knots are usually considered up to [[ambient isotopy]] in which the isotopy is orientation preserving. This leads to the idea of [[invertible knot|invertible knots]].  It is also possible to take the [[mirror reflection]] of  knots and thus to introduce the concept of [[knot chirality]], a [[knot invariant]]; Knots that remain equivalent to their mirror images possess a certain symmetry called [[achiral knot|achiral knots]] or equivalently, [[amphicheiral]]. An alternative definition of this notion is the following:
A knot $K$ is [[amphicheiral]] if there exists an orientation-reversing [[homeomorphism]] of $R^3$ mapping $K$ to itself.

## Relevant nLab Pages

### Knots, Links, and other Variants

The theory of knots can be extended to include various similar things:

* [[links]]
* [[braids]]
* [[strings]]
* [[tangles]]
* [[singular knots]]

### Invariants

A major line in the study of knots is to look for [[knot invariants]] (see also [[link invariants]]).

### Ancillary pages

There are various pages related to knot theory that are linked from the main articles.

* [[Vassiliev skein relations]]
* [[Reidemeister moves]]

### Images

The study of knots is very pictorial.  There are various knot-related SVGs that can be included in to nLab pages.

* [[SVG images]]

## Related concepts

* [[space of knots]]

* [[knot complement]]

* [[knot diagram]]

* [[isotopy]], [[smooth isotopy]]

* [[hyperbolic knot]]

* [[Kirby calculus]]

* [[surface knot]]

* [[MKR dictionary]] in [[arithmetic topology]]

* [[chord diagram]]

* [[Wilson loop]]

* [[knots-quivers correspondence]]

* [[quantum topology]]

[[!include chord diagrams and weight systems -- table]]

## References

### General

Expositions:

* Hoste, Thistlethwaite and Weeks, _The First 1,701,936 Knots_, Scientific American, 20, No. 4, 1998. ([link to pdf](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.215.7894))

* [[Aaron Lauda]], _Knot theory explained (1:24 min lightning idea)_,  USC Dornsife College of Letters, Arts and Sciences ([video](https://www.youtube.com/embed/kxaWqKM5JyQ))

* [[Abhijit Champanerkar]], _The geometry of knot complements_ ([pdf](https://www.math.csi.cuny.edu/~abhijit/talks/knot-geometry-h.pdf), [[ChampanerkarKnotComplements.pdf:file]])

General:

* [[R. H. Crowell]], [[R. H. Fox]], _Introduction to knot theory_, Springer, Graduate Texts 57, 1963.

* G. Burde, H. Zieschang, _Knots_, De Gruyter (1989).

* [[Michael Atiyah]], _The Geometry and Physics of Knots_,  Cambridge University Press 1990 ([doi:10.1017/CBO9780511623868](https://doi.org/10.1017/CBO9780511623868))

* [[Nick Gilbert]], [[Tim Porter]], *Knots and surfaces*, Oxford University Press (1994) [[ISBN:9780198514909](https://global.oup.com/academic/product/knots-and-surfaces-9780198514909?cc=de&lang=en&)]

* [[Dale Rolfsen]], _Knots and links_, AMS Chelsea __346__ (2003) [[ams:chel-346-h](https://bookstore.ams.org/chel-346-h), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/rolfsen.pdf)]

Historically, a motivation for [[Peter Tait]] to start thinking about classification of [[knots]] was the book

* [[Lord Kelvin]], _[[On Vortex Atoms]]_

which presented the speculation in [[physics]] that  [[atoms]]/[[elementary particles]] are fundamentally [[vortices]] in a [[spacetime]]-filling fluid-like substance.

### Relation to physics

Relation of knot theory to [[physics]]/[[quantum field theory]]:

* [[Louis Kauffman]], _Knots and physics_, Series on _Knots and Everything_, Volume 1,  World Scientific, 1991 ([doi:10.1142/1116](https://doi.org/10.1142/1116))

* [[Louis Kauffman]] (ed.) _The Interface of Knots and Physics_, Proceedings of Symposia in Applied Mathematics
Volume 51 (1996) ([pdf](http://www.csee.umbc.edu/~lomonaco/kelvin/kelvin23.pdf), [doi:10.1090/psapm/051](https://doi.org/10.1090/psapm/051))

In [[string theory]] ([[NS5-branes]]/[[M5-branes]]):

* [[Edward Witten]], _Fivebranes and Knots_, Quantum Topology, Volume 3, Issue 1, 2012, pp. 1-137 ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216), [doi:10.4171/QT/26](https://www.ems-ph.org/journals/show_abstract.php?issn=1663-487X&vol=3&iss=1&rank=1))

### Higher dimensional knots

* D. Roseman, _Reidemeister-type moves for surfaces in four dimensional space_, Banach Center Publication, __42__ (1998), 347-380 [pdf](http://matwbn.icm.edu.pl/ksiazki/bcp/bcp42/bcp42124.pdf) [doi](https://doi.org/10.4064/-42-1-347-380)

* J. S. Carter, M. Saito, _Knotted surfaces and their diagrams_, Mathematical Surveys and Monographs __55__, Amer. Math. Soc., Providence, RI, 1998

* V. A. Rohlin, _The embedding of non-orientable three-manifolds into five-dimensional Euclidean space_, Dokl. Akad. Nauk SSSR __160__ (1965) 549–551 (in Russian; English transl.: Soviet Math. Dokl. 6 (1965), 153–156)

* I.G. Korepanov, G.I. Sharygin, D.V. Talalaev, _Cohomology of the tetrahedral complex and quasi-invariants of 2-knots_, [arxiv/1510.03015](https://arxiv.org/abs/1510.03015)

* J. E. Fischer, Jr. _2-Categories and 2-knots_, Duke Math. J. 75 (1994), 493–596.

category: knot theory

[[!redirects knot]]
[[!redirects knots]]
[[!redirects Knot]]
[[!redirects Knots]]

[[!redirects Knot Theory]]
[[!redirects knot theory]]


knot complement

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

#Contents#
* table of contents
{:toc}

## Idea

By a _knot complement_ one means the [[complement]] of a [[knot]]. More precisely:  The [[complement]] in the [[3-sphere]] $S^{3}$ or in [[Euclidean space]]/[[Cartesian space]] $\mathbb{R}^{3}$ of a [[tubular neighbourhood]] of a [[knot]].

## Properties

### Homology

The [[ordinary homology]] of the [[complement]] of a [[knot]] $K \colon S^1 \hookrightarrow S^3$ in the [[3-sphere]] is concentrated in degrees 0 and 1:

$$
  H_n\big( S^3\setminus K, \mathbb{Z}\big)
  \;=\;
  \left\{
  \array{
    \mathbb{Z} &\vert& n \in \{0,1\}
    \\
    0 &\vert& n \geq 2
  }
  \right.
$$

(see for instance [Greene 13, Prop. 3.1](#Greene13), [Haney 16, Cor. 1.6](#Haney16))

## Related concepts

* [[hyperbolic knot]]

## References

Exposition:

* [[Abhijit Champanerkar]], _The geometry of knot complements_ ([pdf](https://www.math.csi.cuny.edu/~abhijit/talks/knot-geometry-h.pdf), [[ChampanerkarKnotComplements.pdf:file]])

Lecture notes:

* {#Greene13} Joshua Greene, _Combinatorial Methods in Knot Theory_, 2013 ([pdf](https://www2.bc.edu/joshua-e-greene/MT885S13/Lecture%203.pdf))

* {#Haney16} Zach Haney, _Knot Complements_, 2016 ([pdf](https://web.northeastern.edu/beasley/MATH7375/Lecture4.pdf))

See also

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

On the [[ordinary homology]] of [[configuration spaces of points]] in knot complements in $\mathbb{R}^3$:

* [[Carl-Friedrich Bödigheimer]], [[Fred Cohen]], L. Taylor, Examples 5.2 of: _On the homology of configuration spaces_, Topology Vol. 28 No. 1, p. 111-123 1989 ([pdf](https://core.ac.uk/download/pdf/82124359.pdf))

[[!redirects knot complements]]


knot group

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

#Contents#
* table of contents
{:toc}

##The Knot Group of a Knot##

The **knot group**, $G(K)$, of a [[knot]] $K$ is the [[fundamental group]] of the [[knot complement|complement of the knot]].  Taking this apart, the knot, $K$, is an embedding of $S^1$ into $\mathbb{R}^3$ or $S^3$. We tend to abuse terminology and to think (and to write) of $K$ as  the image of the embedding rather than the embedding itself.  We can therefore write $\mathbb{R}^3 \setminus K$ for the space outside the knot, i.e. its complement.  This is clearly arcwise connected so we do not need to worry about a choice of base point any point will do.  We make the

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

$G(K) \coloneqq \pi_1(\mathbb{R}^3 \setminus K)$.

=--

To see that this is an invariant of the knot type, note that if you perform an ambient [[isotopy]] on the knot then it deforms this complement into the complement of the isotopic knot.

This is an unambiguous definition, but does not tell us how to calculate anything about this group for a given knot. Luckily there are two neat algorithms for giving [[group presentation|presentations]] of $G(K)$ for arbitrary knots working from a diagram of that knot, one due to Wirtinger, the other to Dehn.

##The Dehn Presentation##

Start by orienting the knot diagram.  The diagram divides the plane into various faces.  Label these faces with distinct letters. For example, $x_1,\ldots, x_n$. These face labels will be the generators in the presentation.  The relations are given by the crossings in the following diagram.

A typical crossing will have a configuration something like this:

+-- {: style="text-align:center"}
<svg width="214" height="214" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="75641">
 <g>
  <title>Layer 1</title>
  <line fill="none" stroke="#7f0000" stroke-width="4" x1="102" y1="214" x2="102" y2="14" id="svg_75641_1" marker-end="url(#se_marker_end_svg_75641_1)"/>
  <rect fill="#ffffff" stroke="#7f0000" stroke-width="0" x="81" y="106" width="43" height="17" id="svg_75641_3"/>
  <line fill="none" stroke="#7f0000" stroke-width="4" x1="2" y1="114" x2="202" y2="114" id="svg_75641_2" marker-end="url(#se_marker_end_svg_75641_2)"/>
  <foreignObject x="27" y="39" id="svg_75641_4" font-size="20" width="50" height="50">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <msub>
       <mi>x</mi>
       <mi>i</mi>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">x_i</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject x="27" y="164" font-size="20" width="50" height="50" id="svg_75641_5">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <msub>
       <mi>x</mi>
       <mi>j</mi>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">x_j</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject x="127" y="39" font-size="20" width="50" height="50" id="svg_75641_13">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <msub>
       <mi>x</mi>
       <mi>l</mi>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">x_l</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject x="127" y="164" font-size="20" width="50" height="50" id="svg_75641_21">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <msub>
       <mi>x</mi>
       <mi>k</mi>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">x_k</annotation>
    </semantics>
   </math>
  </foreignObject>
 </g>
 <defs>
  <marker id="se_marker_end_svg_75641_1" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50">
   <path id="svg_75641_6" d="m100,50l-100,40l30,-40l-30,-40l100,40z" fill="#7f0000" stroke="#7f0000" stroke-width="10"/>
  </marker>
  <marker id="se_marker_end_svg_75641_2" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50">
   <path id="svg_75641_7" d="m100,50l-100,40l30,-40l-30,-40l100,40z" fill="#7f0000" stroke="#7f0000" stroke-width="10"/>
  </marker>
 </defs>
</svg>
=--

Look at the underpass arc that leaves the crossing, and write down the face labels going anticlockwise around the crossing, and alternating in exponent between $+1$ and $-1$.  Here that gives $x_i x_j^{-1}x_k x_l^{-1}$.  After doing this for every crossing, set any _one_ face label to be $1$. (This last relation effectively deletes that generator from the presentation.)

That method gives the following presentation, where $x_m$ is the choice of generator to be set to $1$.

$$\langle x_1, \ldots, x_n\mid \text{relations from crossings}, x_m=1\rangle.$$

This is a presentation of $G(K)$.

The element in the fundamental group corresponding to a particular face is given as follows.  We set the basepoint to a point above the plane and draw a path which goes from the basepoint and through the face.  It then goes under the plane (thus, under the knot) to the face whose label was set to $1$, and through this face back to the basepoint.

##Example of a Dehn presentation##

 We will calculate the Dehn presentation for the cinquefoil knot.  This is the next knot in the family of $(2,k)$-torus knots after the [[trefoil knot]].  It has five lobes when drawn in the following nice symmetric form. We orient the diagram clockwise and label the faces of the diagram with the letters from $a$ up to $g$.

+-- {: style="text-align:center"}
<svg width="150" height="150" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="75642">
 <g>
  <title>Layer 1</title>
  <g id="svg_75642_1">
   <g id="svg_75642_2">
    <g stroke="rgb(0.0%,0.0%,0.0%)" id="svg_75642_3">
     <g fill="rgb(0.0%,0.0%,0.0%)" id="svg_75642_4">
      <g stroke-width="0.4pt" id="svg_75642_5">
       <g id="svg_75642_6"/>
       <g id="svg_75642_7">
        <g id="svg_75642_8">
         <g id="svg_75642_9">
          <g stroke="rgb(100.0%,0.0%,0.0%)" id="svg_75642_10">
           <g fill="rgb(100.0%,0.0%,0.0%)" id="svg_75642_11">
            <g stroke-width="2.0pt" id="svg_75642_12">
             <g id="svg_75642_13">
              <g id="svg_75642_14">
               <g transform="matrix(1, 0, 0, -1, 0, -115.306)" id="svg_75642_15">
                <g fill="rgb(100.0%,0.0%,0.0%)" id="svg_75642_16">
                 <text transform="matrix(1, 0, 0, -1, 84, -242)" text-anchor="middle" font-size="10" id="svg_75642_17" y="-4.0977" x="-9.08893"/>
                </g>
               </g>
              </g>
             </g>
            </g>
           </g>
          </g>
         </g>
        </g>
        <g id="svg_75642_18">
         <g id="svg_75642_19">
          <g stroke="rgb(100.0%,0.0%,0.0%)" id="svg_75642_20">
           <g fill="rgb(100.0%,0.0%,0.0%)" id="svg_75642_21">
            <g stroke-width="2.0pt" id="svg_75642_22">
             <g id="svg_75642_23">
              <g id="svg_75642_24">
               <g transform="matrix(0.30902, -0.95107, -0.95107, -0.30902, 40.5906, -144.797)" id="svg_75642_25">
                <g fill="rgb(100.0%,0.0%,0.0%)" id="svg_75642_26">
                 <text transform="matrix(1, 0, 0, -1, -204.196, -154.668)" text-anchor="middle" font-size="10" id="svg_75642_27" y="-9.91021" x="1.08848"/>
                </g>
               </g>
              </g>
             </g>
            </g>
           </g>
          </g>
         </g>
        </g>
        <g id="svg_75642_28">
         <g id="svg_75642_29">
          <g stroke="rgb(100.0%,0.0%,0.0%)" id="svg_75642_30">
           <g fill="rgb(100.0%,0.0%,0.0%)" id="svg_75642_31">
            <g stroke-width="2.0pt" id="svg_75642_32">
             <g id="svg_75642_33">
              <g id="svg_75642_34">
               <g transform="matrix(-0.80902, -0.58778, -0.58778, 0.80902, 25.0861, -192.514)" id="svg_75642_35">
                <g fill="rgb(100.0%,0.0%,0.0%)" id="svg_75642_36">
                 <text transform="matrix(1, 0, 0, -1, -210.201, 146.41)" text-anchor="middle" font-size="10" id="svg_75642_37" y="-2.02719" x="9.76166"/>
                </g>
               </g>
              </g>
             </g>
            </g>
           </g>
          </g>
         </g>
        </g>
        <g id="svg_75642_38">
         <g id="svg_75642_39">
          <g stroke="rgb(100.0%,0.0%,0.0%)" id="svg_75642_40">
           <g fill="rgb(100.0%,0.0%,0.0%)" id="svg_75642_41">
            <g stroke-width="2.0pt" id="svg_75642_42">
             <g id="svg_75642_43">
              <g id="svg_75642_44">
               <g transform="matrix(-0.80902, 0.58778, 0.58778, 0.80902, -25.0861, -192.514)" id="svg_75642_45">
                <g fill="rgb(100.0%,0.0%,0.0%)" id="svg_75642_46">
                 <text transform="matrix(1, 0, 0, -1, 74.2852, 245.157)" text-anchor="middle" font-size="10" id="svg_75642_47" y="8.65743" x="4.9446"/>
                </g>
               </g>
              </g>
             </g>
            </g>
           </g>
          </g>
         </g>
        </g>
        <g id="svg_75642_48">
         <g id="svg_75642_49">
          <g stroke="rgb(100.0%,0.0%,0.0%)" id="svg_75642_50">
           <g fill="rgb(100.0%,0.0%,0.0%)" id="svg_75642_51">
            <g stroke-width="2.0pt" id="svg_75642_52">
             <g id="svg_75642_53">
              <g id="svg_75642_54">
               <g transform="matrix(0.30902, 0.95107, 0.95107, -0.30902, -40.5906, -144.797)" id="svg_75642_55">
                <g fill="rgb(100.0%,0.0%,0.0%)" id="svg_75642_56">
                 <text transform="matrix(1, 0, 0, -1, 256.11, 5.1069)" text-anchor="middle" font-size="10" id="svg_75642_57" y="7.37776" x="-6.70566"/>
                </g>
               </g>
              </g>
             </g>
            </g>
           </g>
          </g>
         </g>
        </g>
        <g id="svg_75642_58">
         <g stroke="rgb(100.0%,0.0%,0.0%)" id="svg_75642_59">
          <g fill="rgb(100.0%,0.0%,0.0%)" id="svg_75642_60">
           <g stroke-width="2.0pt" id="svg_75642_61">
            <path d="m74.911072,79.916748m2.207169,44.886215c33.106552,33.106613 61.056396,12.800827 38.38343,-31.697487c-5.668228,-11.124565 -13.551338,-35.385445 -15.016228,-44.634148m17.797394,43.216988c41.717148,-21.255791 31.040665,-54.112576 -18.285698,-46.299858c-12.331566,1.953182 -37.840569,1.953182 -47.089268,0.488281m46.600918,-3.571201c-7.324356,-46.243438 -41.871078,-46.243438 -49.683781,3.08292c-1.953197,12.331573 -9.8363,36.592453 -14.087509,44.935829m11.00457,-45.424076c-46.243429,-7.324471 -56.919909,25.532314 -12.4216,48.205265c11.124573,5.668251 31.762173,20.661942 38.383503,27.283272m-39.800533,-24.502029c-21.255939,41.717072 6.693913,62.022858 42.007633,26.709129c8.82843,-8.82843 29.466042,-23.822121 37.809471,-28.07325" fill="none" id="svg_75642_62"/>
           </g>
          </g>
         </g>
        </g>
       </g>
       <g id="svg_75642_63"/>
      </g>
     </g>
    </g>
   </g>
  </g>
  <foreignObject x="52" y="20.32608" id="svg_75642_64" font-size="16" width="48" height="20">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <mi>a</mi>
     </mrow>
     <annotation encoding="application/x-tex">a</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject x="99" y="48" font-size="16" width="48" height="20" id="svg_75642_65">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <mi>b</mi>
     </mrow>
     <annotation encoding="application/x-tex">b</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject x="83" y="108" font-size="16" width="48" height="20" id="svg_75642_71">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <mi>c</mi>
     </mrow>
     <annotation encoding="application/x-tex">c</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject x="21" y="108" font-size="16" width="48" height="20" id="svg_75642_77">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <mi>d</mi>
     </mrow>
     <annotation encoding="application/x-tex">d</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject x="5" y="48" font-size="16" width="48" height="20" id="svg_75642_83">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <mi>e</mi>
     </mrow>
     <annotation encoding="application/x-tex">e</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject x="52" y="69.32608" font-size="16" width="48" height="20" id="svg_75642_89">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <mi>f</mi>
     </mrow>
     <annotation encoding="application/x-tex">f</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject x="5" y="13.32608" font-size="16" width="48" height="20" id="svg_75642_95">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <mi>g</mi>
     </mrow>
     <annotation encoding="application/x-tex">g</annotation>
    </semantics>
   </math>
  </foreignObject>
 </g>

</svg>
=--

We start with the crossing in the top left. The underpass is going from left to right and has $a$ to its left as it exits from under the overpass. Reading off the relation for this we get $a g^{-1} e f^{-1}$. We go to the next (clockwise) crossing and similarly get $b g^{-1} a f^{-1}$; continuing we get $c g^{-1} b f^{-1}$, then $d g^{-1}c f^{-1}$, and for the final crossing $e g^{-1}d f^{-1}$.

We will choose to kill off $g$ setting it equal to $1$ (and eliminating it from the presentation). This gives

$$\langle a,b,c,d,e,f \mid b a f^{-1}=c b f^{-1}= d c f^{-1}=e d f^{-1}=a e f^{-1}=1\rangle.$$

This is 'the' **Dehn presentation** of $G(K)$ for this knot

We can eliminate $f$ as it can be expressed in terms of the other generators.  This gives us

$$\langle a,b,c,d,e\mid b a = c b =d c= e d = a e\rangle.$$

This corresponds to something neat.  The 'geometric' significance of the faces is they represent a path that starts above the 'page' goes down through that face then comes up through the face labelled $g$ (as we set that equal to $1$). The presentation in its current form shows that going down through any of the lobes, coming back up at the outside, then going down through the next one clockwise is the same as going through the middle! It can be useful to keep $f$ in the presentation as we will see.

This group presentation is [[Tietze transformation|Tietze equivalent]] to $\langle f,y\mid f^5 = y^2\rangle$. (Use the substitution $y=e d c b a$.)  Note that the nature of the cinquefoil as a (2,5)-[[torus knot]] can be seen in this presentation.

##The Wirtinger presentation##
To  get the Wirtinger presentation of $G(K)$, you label the arcs of the knot diagram rather than the faces. The arc labels will give the generators in this presentation and the crossings will give the relations.

Suppose the crossing looks like

+-- {: style="text-align:center"}
<svg width="214" height="214" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="63639">
 <g>
  <title>Layer 1</title>
  <line marker-end="url(#se_marker_end_svg_75641_1)" id="svg_75641_1" y2="12" x2="102" y1="212" x1="102" stroke-width="4" stroke="#7f0000" fill="none"/>
  <rect id="svg_75641_3" height="17" width="43" y="104" x="81" stroke-width="0" stroke="#7f0000" fill="#ffffff"/>
  <line marker-end="url(#se_marker_end_svg_75641_2)" id="svg_75641_2" y2="112" x2="202" y1="112" x1="2" stroke-width="4" stroke="#7f0000" fill="none"/>
  <foreignObject height="50" width="50" font-size="20" id="svg_75641_4" y="80" x="20">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <msub>
       <mi>x</mi>
       <mi>i</mi>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">x_i</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject id="svg_75641_5" height="50" width="50" font-size="20" y="40" x="95">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <msub>
       <mi>x</mi>
       <mi>j</mi>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">x_j</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject id="svg_75641_21" height="50" width="50" font-size="20" y="140" x="95">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <msub>
       <mi>x</mi>
       <mi>k</mi>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">x_k</annotation>
    </semantics>
   </math>
  </foreignObject>
 </g>
 <defs>
  <marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_end_svg_75641_1">
   <path stroke-width="10" stroke="#7f0000" fill="#7f0000" d="m100,50l-100,40l30,-40l-30,-40l100,40z" id="svg_63639_1"/>
  </marker>
  <marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_end_svg_75641_2">
   <path stroke-width="10" stroke="#7f0000" fill="#7f0000" d="m100,50l-100,40l30,-40l-30,-40l100,40z" id="svg_63639_2"/>
  </marker>
 </defs>
</svg>
=--

Pick one of the quadrants and think of a rectanglar path going anti-clockwise about the crossing.  (We will assume we have started this path in the NW quadrant, to the left of the exit.)
Traverse this rectangular path and write down the $\text{arc label}$ if the arc is entering the crossing and $(\text{arc label})^{-1}$ if it is leaving it.  In the above situation we get $x_i  x_k x_i^{-1} x_j^{-1}$.  This is set equal to $1$. Now repeat for all the other crossings.

The **Wirtinger presentation** is then

$$\langle \text{arc labels} \mid \text{crossing relations}\rangle$$

It is worth noting that any one of the crossing relations is a consequence of the others.

The element in the fundamental group corresponding to a strand is the following.  We fix a basepoint above the plane.  The path comes down from this basepoint, goes under the strand, and then back up again.  The direction in which it goes under the strand is determined by the orientation of the strand: to someone sitting on the strand facing in the direction of the orientation, the path will go down on the left and come up on their right.

**Hint:** If doing an example, do not throw away a crossing relation just because it is redundant.  It is a good idea to keep it and it will act as a check on the final form of the presentation when it has been 'processed' through some Tietze transformations. It is easy, at least to start with, to make a slip in the calculation and often the presentation for nice simple knots can be reduced to a form with two generators and one relation.  The above trick keeps in a duplicate relation for comparison.

##Related entries

The classical treatment of the [[Alexander polynomial]] as found, for instance in the book by [[Crowell]] and [[Ralph Fox|Fox]], uses the Wirtinger presentation and [[Fox derivatives]] to derive an Alexander matrix which is then processed to give the polynomial.

Alexander's original paper uses a method which is more closely related to the Dehn presentation.

category: knot theory


knot invariant

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

#Contents#
* table of contents
{:toc}

## Idea

A __knot invariant__ is map from [[isotopy]] [[equivalence class]]es of [[knots]] to any kind of structure you could imagine.  These are helpful because it is often much easier to check that the structures one maps to (numbers, groups, etc.) are different than it is to check that knots are different.  To define a knot invariant, it suffices to define its value on [[knot diagrams]] and check that this value is preserved under the [[Reidemeister moves]] (possibly with the exception of the first Reidemeister move, in the case of an invariant of [[framed knots]]).

## Examples

* [[crossing number]]
* [[bridge number]]
* [[unknotting number]]
* [[colorability]]
* [[knot group]]= the [[fundamental group]] of the complement of the knot.
* [[quandles]] and [[racks]]
* [[knot genus]]
* [[Jones polynomial]]
* [[HOMFLY polynomial]]
* [[Alexander polynomial]]
* [[Reshetikhin-Turaev invariant]]s for any object in a [[ribbon category]]
* [[Khovanov homology]]
* [[Kauffman bracket]] (an invariant of [[framed links]] closely related to the [[Jones polynomial]])

Many of these extend to [[link invariant|link invariants]] or have variants that depend on the knot being oriented.

## References

### General

(...)

### In string theory

Knot invariants arising in [[string theory]]/[[M-theory]]:

#### Via 5-brane BPS states

Discussion of [[knot invariants]] in terms of [[BPS states]] of [[M5-branes]]:

* [[Edward Witten]], _Fivebranes and Knots_, Quantum Topology, Volume 3, Issue 1, 2012, pp. 1-137 ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216), [doi:10.4171/QT/26](https://www.ems-ph.org/journals/show_abstract.php?issn=1663-487X&vol=3&iss=1&rank=1))

* [[Davide Gaiotto]], [[Edward Witten]], _Knot Invariants from Four-Dimensional Gauge Theory_, Advances in Theoretical and Mathematical Physics, Volume 16 (2012) Number 3 ([arxiv:1106.4789](https://arxiv.org/abs/1106.4789))

* [[Sergei Gukov]], [[Marko Stošić]], _Homological algebra of knots and BPS states_ ([arXiv:1112.0030](http://arxiv.org/abs/1112.0030))

* Ross Elliot, [[Sergei Gukov]], _Exceptional knot homology_ ([arXiv:1505.01635](http://arxiv.org/abs/1505.01635))

* Satoshi Nawata, Alexei Oblomkov, _Lectures on knot homology_ ([arXiv:1510.01795](http://arxiv.org/abs/1510.01795))

#### Via D$p$/D$(p+2)$ bound states / monopoles

Relation of [[Dp-D(p+2)-brane bound states]]/[[Yang-Mills monopoles]] to [[knot invariants]] via [[chord diagrams]]:

* S. Ramgoolam, B. Spence, S. Thomas, Section 3.2 of: _Resolving brane collapse with $1/N$ corrections in non-Abelian DBI_, Nucl. Phys. B703 (2004) 236-276 ([arxiv:hep-th/0405256](https://arxiv.org/abs/hep-th/0405256))

* S. McNamara, Constantinos Papageorgakis, S. Ramgoolam, B. Spence, Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))

* Constantinos Papageorgakis, p. 161-162 of: _On matrix D-brane dynamics and fuzzy spheres_, 2006 ([[Papageorgakis06.pdf:file]])

category: knot theory

[[!redirects knot invariants]]
[[!redirects Knot Invariant]]
[[!redirects Knot Invariants]]
[[!redirects Knot invariant]]
[[!redirects Knot invariants]]


knot theory - contents

**[[knot theory]]**

**[[knot]]**, **[[link]]**

* [[isotopy]]

* [[knot complement]]

* [[knot diagrams]], [[chord diagram]]

* [[Reidemeister move]]

**Examples/classes:**

* [[trefoil knot]]

* [[torus knot]]

* [[singular knot]]

* [[hyperbolic knot]]

* [[Borromean link]]

* [[Whitehead link]]

* [[Hopf link]]

**Types**

* [[prime knot]]

* [[mutant knot]]

**[[knot invariants]]**

* [[crossing number]]

* [[bridge number]]

* [[unknotting number]]

* [[colorability]]

* [[knot group]]

* [[knot genus]]

* polynomial knot invariants

  ([[quantum observables|observables]] of [[non-perturbative quantum field theory|non-perturbative]] [[Chern-Simons theory]])

  * [[Jones polynomial]]

  * [[HOMFLY polynomial]]

  * [[Alexander polynomial]]

  * [[Reshetikhin-Turaev invariants]]

* [[Vassiliev knot invariants]]

  ([[quantum observables|observables]] of [[perturbative quantum field theory|pertrubative]] [[Chern-Simons theory]])

* [[Khovanov homology]]

* [[Kauffman bracket]]

[[link invariants]]

* [[Milnor mu-bar invariants]]

* [[linking number]]

**Related concepts:**

* [[Vassiliev skein relation]]

* [[Seifert surface]]

* [[virtual knot theory]]

* [[Dehn surgery]], [[Kirby calculus]]

* [[volume conjecture]]

* [[arithmetic topology]]

category: knot theory


knots-quivers correspondence

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

#Contents#
* table of contents
{:toc}

## Idea

The *knots-quivers correspondence* [[KRSS19](#KRSS19)] is a certain identification between [[knot invariants|invariants]] of [[knots]] and data encoded by [[quiver representation|representations]] of corresponding [[quivers]].

The correspondence has been conjectured from [[geometric engineering]] via [[intersecting D-brane models]] of both [[knot invariants]] and [[quiver gauge theory]], motivated by comparing the framework of Ooguri-Vafa large $N$ duality [[OV19](#OV19)] with the motivic generating series for symmetric quivers [[R20](#R20)][[E20](#E20)][[KS20](#KS20)].

Beware that the correspondence is far from [[bijective]]: Most [[quivers]] correspond to no knot and most [[knots]] correspond to many different quivers at a time.

## Related concepts

* [[McKay correspondence]]

* [[AGT correspondence]]

* [[3d-3d correspondence]]

## References

The original articles:

* {#KRSS19} [[Piotr Kucharski]], [[Markus Reineke]], [[Marko Stosic]], [[Piotr Sułkowski]], *Knots-quivers correspondence*, Adv. Theor. Math. Phys. **23** 7 (2019) 1849-1902 [[arXiv:1707.04017](https://arxiv.org/abs/1707.04017), [doi:10.4310/ATMP.2019.v23.n7.a4](https://doi.org/10.4310/ATMP.2019.v23.n7.a4)]

* [[Piotr Kucharski]], [[Markus Reineke]], [[Marko Stosic]], [[Piotr Sułkowski]], *BPS states, knots and quivers*, Phys. Rev. D **96** 121902 (2017) [[arXiv:1707.02991](https://arxiv.org/abs/1707.02991), [doi:10.1103/PhysRevD.96.121902](https://doi.org/10.1103/PhysRevD.96.121902)]

Review:

* [[Piotr Sułkowski]], Section 5 of: *Quantum fields, strings, and physical mathematics*, PoS **390** (2021) [[arXiv:2104.03350](https://arxiv.org/abs/2104.03350), [doi:10.22323/1.390.0042](https://doi.org/10.22323/1.390.0042)]

based on

* {#OV19} [[Hirosi Ooguri]], [[Cumrun Vafa]], *Knot invariants and topological strings*, Nucl. Phys. B **577** (2000), 419-438 [[arXiv:hep-th/9912123](https://arxiv.org/abs/hep-th/9912123), doi:10.1016/S0550-3213(00)001188]

* {#R20} [[Markus Reineke]], *Degenerate Cohomological Hall algebra and quantized Donaldson-Thomas invariants for m-loop quivers*, [[arXiv:1102.3978](https://arxiv.org/abs/1102.3978)]

* {#E20} [[Alexander I. Efimov]], *Cohomological Hall algebra of a symmetric quiver*, Compositio Mathematica, **148** 4 (2012) 1133-1146 [[arXiv:1103.2736](https://arxiv.org/abs/1103.2736), [doi:10.1112/S0010437X12000152](https://doi.org/10.1112/S0010437X12000152)]

* {#KS20} [[Maxim Kontsevich]], [[Yan Soibelman]], *Cohomological Hall algebra exponential Hodge structures and motivic Donaldson-Thomas invariants*, Commun. Num. Theor. Phys. **5** (2011) 231-352 [[arXiv:1006.2706](https://arxiv.org/abs/1006.2706)]

Further discussion:

* [[Vivek Kr. Singh]], S. Chauhan, A. Dwivedi, P. Ramadevi, B.P. Mandal, S. Dwivedi, *Knot-Quiver correspondence for double twist knots* [[arXiv:2303.07036](https://arxiv.org/abs/2303.07036)]

* [[Marko Stošić]], *Generalized knots-quivers correspondence* [[arXiv:2402.03066](https://arxiv.org/abs/2402.03066)]

[[!redirects knot-quiver correspondence]]


Knut B. Haus

* [personal page](https://www.ntnu.edu/employees/knut.b.haus)

## Selected writings

A geometric cocycle model for [[Hodge-filtered differential cohomology|Hodge-filtered differential]] [[MU]]-[[cobordism cohomology theory]]:

* [[Knut Bjarte Haus]], *Geometric Hodge filtered complex cobordism*, PhD thesis (2022) [[ntnuopen:3017489](https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/3017489)]

* [[Knut B. Haus]], [[Gereon Quick]], *Geometric Hodge filtered complex cobordism* [[arXiv:2210.13259](https://arxiv.org/abs/2210.13259)]

On [[Umkehr maps]] in [[Hodge-filtered differential cohomology|Hodge-filtered differential]] [[complex cobordism cohomology]]:

* [[Knut Bjarte Haus]], [[Gereon Quick]], *Geometric pushforward in Hodge filtered complex cobordism and secondary invariants* [[arXiv:2303.15899](https://arxiv.org/abs/2303.15899)]

category: people

[[!redirects Knut Haus]]
[[!redirects Knut Bjarte Haus]]


Ko Sanders

* [webpage](http://home.uni-leipzig.de/tet/?page_id=834)

## related $n$Lab entries

* [[Green hyperbolic differential equation]]

category: people


KO-dimension

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

#Contents#
* table of contents
{:toc}

## Idea

What is called _KO-dimension_ ([Connes 95](#Connes95)) is one notion of [[dimension]] which may be associated with a [[space]] that is given by a [[spectral triple]] with "real structure".

The definition is motivated by the fact that the [[dimension]] $d$ of an ordinary [[closed manifold]] $X$ is seen in the grading shift involved in the [[Poincaré duality]] that the manifold induces on its [[ordinary homology]]/[[ordinary cohomology]]. Now since in [[spectral geometry]] ("[[noncommutative geometry]]") a [[space]] is represented by a [[spectral triple]] and hence by a kind of [[Dirac operator]] which naturally defines a class not in [[ordinary homology]] but in [[K-homology]], so the idea of KO-dimension is that it is the shift in the grading on [[K-theory]] which is involved in a [[Poincaré duality]] for [[spectral triples]].

Now since [[complex K-theory]] is 2-[[periodic cohomology theory|periodic]] this sees such a dimension only modulo 2, and hence only sees whether the dimension is even or odd. But [[KO-theory]] is 8-periodic and hence sees dimension at least modulo 8.

The exact definition of KO-dimension given in ([Connes 95, def. 3](#Connes95)) moreover requires that the [[Poincaré duality]] is exhibited by a class in [[KR-homology]].

## Examples

For classical [[Riemannian manifolds]] KO-dimension coincides with the traditional concept of [[dimension]] of manifolds, modulo 8.

The [[Podlés spheres]] have KO-dimension 2, but classical dimension 0.

The spectral [[Kaluza-Klein compactification]] considered in the [[Connes-Lott-Chamseddine model]] ([Connes 06](#Connes06)) is taken to be along [[fibers]] with KO-dimension 6 and classical dimension 0 (just as [[perturbative string theory vacuum|perturbative superstring vacua]])

## Related concepts

* [[dimension]]

  * [[cohomological dimension]]

  * [[homotopy dimension]]

  * [[covering dimension]]

  * [[Heyting dimension]]

## References

### General

The original source is def. 3 in

* {#Connes95} [[Alain Connes]], _Noncommutative geometry and reality_, J. Math. Phys. 36 (11), 1995 ([pdf](https://alainconnes.org/wp-content/uploads/reality.pdf))

### In Connes-Lott models and superstring vacua

With an eye towards [[phenomenology|phenomenological]] considerations of the [[spectral action]] (the [[Connes-Lott-Chamseddine model]]) this is recalled as def. 7.2 in

* {#Connes06} [[Alain Connes]],  _Noncommutative Geometry and the standard model with neutrino mixing_, JHEP0611:081,2006 ([arXiv:hep-th/0608226](http://arxiv.org/abs/hep-th/0608226))

From p. 8 there:

> {#Connes06OnRelationToStringVacua} When one looks at the table (7.2) of Appendix 7 giving the [[KO-dimension]] of the finite space $[$ i.e. the [[noncommutative geometry|noncommutative]] [[KK-compactification]]-[[fiber]] $F$ $]$ one then finds that its [[KO-dimension]] is now equal to 6 [[modulo]] 8 (!). As a result we see that the [[KO-dimension]] of the [[Cartesian product|product space]] $M \times F$ $[$ i.e. of 4d [[spacetime]] $M$ with the [[noncommutative geometry|noncommutative]] [[KK-compactification]]-[[fiber]] $F$$]$ is in fact equal to $10  \sim 2$ [[modulo]] 8. Of course the above 10 is very reminiscent of string theory, in which the finite space $F$ might bea good candidate for an "[[effective field theory|effective]]" [[KK-compactification|compactification]] at least for low energies. But 10 is also 2 [[modulo]] 8 which might be related to the observations of [Lauscher-Reuter 06](#LauscherReuter06) about [[gravity]].

For more on this see at _[[2-spectral triple]]_.

[[!redirects KO-dimensions]]


KO-theory

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

#Contents#
* table of contents
{:toc}

## Idea

The [[generalized cohomology theory]] [[Brown representability theorem|represented]] by the [[KO]]-[[spectrum]], hence the "orthogonal" version of [[complex K-theory]].

This is supposed to be the [[generalized cohomology theory]] which measures [[D-brane charge]] in [[type I string theory]]/on [[orientifold planes]].

## Propertis

### Homotopy groups
 {#HomotopyGroups}

The [[stable homotopy groups]] of [[KO]]

$$
  \pi_n(KO) \,\simeq\, \widetilde {KO}^0( S^n )
$$

are:

| $n = $ | $8k + 0$ | $8k + 1$ | $8k + 2$ | $8k + 3$ | $8k + 4$ | $8k + 5$ | $8k + 6$ | $8k + 7$ |
|--------|-----|-----|-----|-----|-----|-----|-----|-----|
| $\pi_n(KO) = $ | $\mathbb{Z}$ | $\mathbb{Z}/2$ | $\mathbb{Z}/2$ | $0$ | $\mathbb{Z}$ | $0$ | $0$ | $0$ |

With [[Bott periodicity]] 8.

## Related concepts

* [[KO-dimension]]

* [[quaternionic oriented cohomology theory]]

* [[KR-theory]], [[KU-theory]], [[KQ-theory]]

[[!include string theory and cohomology theory -- table]]

## References

### General

On the [[differential K-theory]] for KO:

* {#GradySati18} [[Daniel Grady]], [[Hisham Sati]], _Differential KO-theory: Constructions, computations, and applications_, Advances in Mathematics Volume 384, 25 June 2021, 107671 ([arXiv:1809.07059](https://arxiv.org/abs/1809.07059), [doi:10.1016/j.aim.2021.107671](https://doi.org/10.1016/j.aim.2021.107671))

* [[Kiyonori Gomi]], Mayuko Yamashita, *Differential KO-theory via gradations and mass terms* ([arXiv:2111.01377](https://arxiv.org/abs/2111.01377))

On the full [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[KO-theory|orthogonal]] [[topological K-theory|K-theory]]:

* {#GradySati19} [[Daniel Grady]], [[Hisham Sati]], _Twisted differential KO-theory_ [[arXiv:1905.09085](https://arxiv.org/abs/1905.09085)]

### For D-brane charge

The original observation that [[D-brane charge]] for [[orientifolds]] should be in [[KR-theory]], hence in [[KO-theory]] right on the [[O-planes]], is due to

* [Witten 98, section 5](#Witten98)

and was then re-amplified in

* {#Gukov99} [[Sergei Gukov]], _K-Theory, Reality, and Orientifolds_, Commun.Math.Phys. 210 (2000) 621-639 ([arXiv:hep-th/9901042](https://arxiv.org/abs/hep-th/9901042))

* {#BergmanGimonSugimoto01} [[Oren Bergman]], E. Gimon, [[Shigeki Sugimoto]], _Orientifolds, RR Torsion, and K-theory_, JHEP 0105:047, 2001 ([arXiv:hep-th/0103183](https://arxiv.org/abs/hep-th/0103183))

With further developments in

* [[Varghese Mathai]], [[Michael Murray]], [[Daniel Stevenson]], _Type I D-branes in an H-flux and twisted KO-theory_, JHEP 0311 (2003) 053 ([arXiv:hep-th/0310164](https://arxiv.org/abs/hep-th/0310164))

Discussion of orbi-orienti-folds using [[equivariant K-theory|equivariant]] [[KO-theory]] is in

* N. Quiroz, [[Bogdan Stefanski]], _Dirichlet Branes on Orientifolds_, Phys.Rev. D66 (2002) 026002 ([arXiv:hep-th/0110041](https://arxiv.org/abs/hep-th/0110041))

* [[Volker Braun]], [[Bogdan Stefanski]], _Orientifolds and K-theory_ ([arXiv:hep-th/0206158](https://arxiv.org/abs/hep-th/0206158))

* H. Garcia-Compean, W. Herrera-Suarez, B. A. Itza-Ortiz, O. Loaiza-Brito, _D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory_, JHEP 0812:007, 2008 ([arXiv:0809.4238](https://arxiv.org/abs/0809.4238))

An elaborate proposal for the correct flavour of real equivariant K-theory needed for [[orientifolds]] is sketched in

* {#Precis} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

[[!redirects KO theory]]

[[!redirects orthogonal K-theory]]

[[!redirects KO-cohomology]]


Kobayashi-Hitchin correspondence

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

#Contents#
* table of contents
{:toc}

## Idea

The _Kobayashi-Hitchin correspondence_ states that over suitable [[complex manifolds]] the [[moduli space]] of [[stable vector bundle|semi-stable vector bundles]] and that of [[Hermite-Einstein connections]] are essentially the same.

For the special case over [[Kähler manifolds]] this is the _[[Donaldson-Uhlenbeck-Yau theorem]]_. For the special case over [[Riemann surfaces]] it is the _[[Narasimhan-Seshadri theorem]]_. See also Deligne's characterization of [[intermediate Jacobians]] (in particular there at [Examples -- Picard variety](intermediate+Jacobian#ExamplePicard)).

## Related concepts

* [[Koszul-Malgrange theorem]]

## References

* Wikipedia, _[Kobayashi-Hitchin correspondence](http://en.wikipedia.org/wiki/Kobayashi–Hitchin_correspondence)_

* {#Kobayashi87} [[Shoshichi Kobayashi]], _Differential geometry of complex vector bundles_, Princeton University Press (1987) ([pdf](http://mathsoc.jp/publication/PublMSJ/PDF/Vol15.pdf))

* {#LubkeTeleman95} [[Martin Lübke]], and [[Andrei Teleman]], _The Kobayashi-Hitchin correspondence_, World Scientific, 1995.


Kobe Wullaert

* [personal page](https://kfwullaert.github.io/)

## Selected writings

On [[monoidal category|monoidal]] [[univalent categories]]:

* [[Kobe Wullaert]], [[Ralph Matthes]], [[Benedikt Ahrens]], *Univalent Monoidal Categories* [[arXiv:2212.03146](https://arxiv.org/abs/2212.03146)]

category: people


Kochen-Specker theorem

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

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum mechanics]], the _Kochen-Specker theorem_ -- developed in 1967 by [[Simon Kochen]] and [[Ernst Specker]] -- is a [[no-go theorem]] that places limits on the types of [[hidden variable theories]] that may be used to explain the (apparent) [[probability theory|probabilistic]] nature of [[quantum mechanics]] in a causal way.  It roughly asserts that it is impossible to assign values to all physical [[observables]] while simultaneously preserving the functional relations between them.  It is a complement to [[Bell's theorem]], developed by [[John Bell]] in 1964, and is related to [[Gleason's theorem]], proven by ([Gleason (1957)](#Gleason57)) (who incidentally is the person who communicated the original Kochen-Specker paper to the _Journal of Mathematics and Mechanics_ ).  Later in ([Butterfield-Hamilton-Isham 98](#ButterfieldHamiltonIsham)) it was observed that the Kochen-Specker theorem is equivalent to the statement that the [[spectral presheaf]] has no [[global elements]], which led to the proposal that the [[phase space]] in [[quantum mechanics]] is naturally to be understood as a ([[ringed topos|ringed]]) [[topos]], the "[[Bohr topos]]".

## Kochen-Specker theorem

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

Let $B(\mathcal{H})$ be the algebra of [[bounded operator]]s on some [[Hilbert space]] $\mathcal{H}$. (In physics $\mathcal{H}$ is the space of states of a [[quantum mechanics|quantum mechanical]] system, and the elements $\hat A \in B(\mathcal{H})$ represent quantum observables.)

A **valuation** on $B(\mathcal{H})$ is a [[function]]

$$
  \lambda : B(\mathcal{H}) \to \mathbb{R}
$$

to the [[real number]]s, satisfying two conditions:

1. _value rule_ -- the value $\lambda(\hat{A})$ belongs to the spectrum of $\hat{A}$;

2. _functional composition principle_ (FUNC) -- for any pair of [[self-adjoint operators]] $\hat{A}$, $\hat{B}$ such that $\hat{B}=h(\hat{A})$ for some real-valued function $h$ we have $\lambda(\hat{B})=h(\lambda(\hat{A}))$.

=--

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

This has the following equivalent reformulation, which is crucial for the [[sheaf and topos theory|sheaf-theoretic]] interpretation discussed below.

Observed that if $\hat{A}_{1}$ and $\hat{A}_{2}$ commute, then it follows from the [[spectral theorem]] that there exists an operator $\hat{C}$ and [[continuous functions]] $h_{1}$ and $h_{2}$ such that $\hat{A}_{1}=h_{1}(\hat{C})$ and $\hat{A}_{2}=h_{2}(\hat{C})$.  Then the [[axiom]] FUNC in def. \ref{Valuation} implies that a valuation satisfies

$$
  \lambda(\hat{A}_{1} + \hat{A}_{2}) = \lambda(\hat{A}_{1}) + \lambda(\hat{A}_{2})
$$

and

$$
  \lambda(\hat{A}_{1}\hat{A}_{2})=\lambda(\hat{A}_{1})\lambda(\hat{A}_{2})
  \,,
$$

hence that on  [[poset of commutative subalgebras|commutative subalgebras]] it is a [[ring]] [[homomorphism]].

Therefore a valuation as in def. \ref{Valuation} is equivalently a function on the algebra which is an algebra homomorphism on each [[poset of commutative subalgebras|commutative subalgebra]].

(Observe the difference to [[quasi-states]] ([[quantum states]]), which are positive linear functions on commutative subalgebras, not necessarily respecting the ring structure.)

=--

Now we have:

+-- {: .num_theorem}
###### Theorem
**(Kochen-Specker)**

No valuations on $B(\mathcal{H})$ exist if dim($\mathcal{H}$)>2.

=--

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

If a valuation _did_ exist and was restricted to a [[poset of commutative subalgebras|commutative sub-algebra of operators]], it would be an element of the [[Gelfand spectrum]] of the commutative sub-algebra.  Since such elements _do_ exist, valuations do exist on any commutative sub-algebra of operators even if not on the whole _non_-commutative algebra, $\mathcal{B}(\mathcal{H})$, of all bounded operators.  Isham calls these valuations on commutative subalgebras _local_. In the [[Bohr topos]] of the algebra of observables (see there for more), the local valuations are just the [[internal language|internal]] valuations.

=--

## Sheaf-theoretic interpretation

[[Chris Isham]] and [[Jeremy Butterfield]] gave a [[topos theory|topos theoretic]] reformulation of the Kochen-Specker theorem as follows.

+-- {: .num_defn}
###### Definition
**(category of contexts)**

Let $\mathcal{V}(\mathcal{H})$ be a [[category]] (the [[poset of commutative subalgebras]] of the algebra $B(\mathcal{H})$ of [[bounded operator]]s) whose

* [[object]]s are _commutative_ [[von Neumann algebra|von Neumann subalgebras]] $V \subset B(\mathcal{H})$;

* [[morphism]]s $V_1 \to V_2$ are inclusions $V_1 \subset V_2$.

=--

Isham calls this the **category of (classical) contexts** of $B(\mathcal{H})$.
Each commutative algebra is viewed as a context within which to view a quantum system in an essentially classical way in the sense that the physical quantities in any such algebra can be given consistent values (as they can in a classical context). These classical contexts were maybe first amplified by [[Niels Bohr]] as being the contexts through which all of [[quantum mechanics]] is to be perceived. (Therefore the word "[[Bohr topos]]" for the concept that is meant to formalize this perspective of Bohr.)

+-- {: .num_defn}
###### Definition
**(spectral presheaf)**

Let $\Sigma : \mathcal{V}(\mathcal{B})^{op} \to Set$ be the [[presheaf]] on the category of contexts such that

* to $V \subset B(\mathcal{H})$ it assigned the set underlying the
  spectrum of $V$: the set of multiplicative [[linear functional]]s
  $\kappa : V \to \mathbb{R}$;

* to an inclusion $i : V_1 \hookrightarrow V_2$ it assigns the corresponding
  function $i^* : \Sigma(V_2) \to \Sigma(V_1)$ that sends a functional
  $V_2 \stackrel{\kappa}{\to} \mathbb{R}$ to its restriction
  $V_1 \hookrightarrow V_2 \stackrel{\kappa}{\to} \mathbb{R}$.

This is called the _[[spectral presheaf]]_.

=--

Recall that the [[terminal object]], $* = 1_{Set^{\mathcal{V}(\mathcal{H})^{op}}}$ in the [[category of presheaves]] on $\mathcal{V}(\mathcal{H})$ is the [[presheaf]] that assigns the singleton set $*$ (the [[terminal object]] in [[Set]]) to each commutative algebra.

A [[global element]] of the spectral presheaf $\Sigma$ is a morphism $e : * \to \Sigma$ in the presheaf topos. Being a [[natural transformation]] of functors, such a global element $\lambda : 1_{Set^{\mathcal{V}(\mathcal{H})^{op}}} \to \underline{\Sigma}$ of the spectral presheaf, would associate an element of the [[Gelfand spectrum]] of an algebra $V$ to that algebra such that all local valuations are global, i.e. for $V \subseteq W$ valuations on $V$ are local valuations on $W$ but global on $V$.

Because a multiplicative linear functional $\kappa : V \to\mathbb{R}$ satisfies the axioms of a valuation, def. \ref{Valuation}, when restricted to the self-adjoint elements of $V$.

By the Kochen-Specker theorem these cannot exist, hence a global element of $\Sigma$ cannot exist. Hence we have:

+-- {: .num_prop}
###### Proposition
(Hamilton, Isham, Butterfield)

The Kochen-Specker theorem is equivalent to the statement that the [[spectral presheaf]] $\Sigma$ of the algebra of bounded operators has no [[global elements]] if the dimension of the Hilbert space is greater than 2.

=--

([Butterfield-Hamilton-Isham 98](#ButterfieldHamiltonIsham))

For more see at _[[Bohr topos]]_.

## Related theorems

Other theorems about the foundations of [[quantum mechanics]] include:

* [[order-theoretic structure in quantum mechanics]]

  * [[Gleason's theorem]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Bell's theorem]]

* [[Wigner theorem]]

* [[Bub-Clifton theorem]]

## Related concepts

* [[beable]]

* [[quantum contextuality]]

* [[Bohr topos]]

* [[interpretation of quantum mechanics]]

  * [[hidden variable theory]]

* [[Bell's inequalities]]

Other theorems about the foundations and [[interpretation of quantum mechanics]] include:

* [[order-theoretic structure in quantum mechanics]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Fell's theorem]]

* [[Gleason's theorem]]

* [[Wigner theorem]]

* [[Bell's theorem]]

* [[Bub-Clifton theorem]]

* [[Kadison-Singer problem]]

## References

The original article:

* {#KochenSpecker67} [[Simon Kochen]], [[Ernst Specker]], _The problem of hidden variables in quantum mechanics_, Journal of Mathematics and Mechanics **17** (1967) 59-87 [[pdf](http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1968/17/17004/pdf), [jstor:24902153](https://www.jstor.org/stable/24902153), [doi:10.1007/978-94-010-1795-4_17](https://doi.org/10.1007/978-94-010-1795-4_17)]

Alternative proofs:

* Del Rajan, Matt Visser, _Kochen-Specker theorem revisited_ ([arXiv:1708.01380](https://arxiv.org/abs/1708.01380))

Review:

* Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, Jan-Åke Larsson, *Kochen-Specker Contextuality* [[arXiv:2102.13036](https://arxiv.org/abs/2102.13036)]

The [[sheaf and topos theory|sheaf-theoretic]] interpretation of the theorem was proposed in

* {#IshamButterfield98} [[Chris Isham]], [[Jeremy Butterfield]], _A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations_ ([arXiv:quant-ph/9803055](http://arxiv.org/abs/quant-ph/9803055))

The formulation in terms of presheaves on the category of commutative sub-algebra of $B(\mathcal{H})$ was proposed in part III of

* {#ButterfieldHamiltonIsham} [[Jeremy Butterfield]], John Hamilton, [[Chris Isham]], _A topos perspective on the Kochen-Specker theorem_,

  _I. quantum states as generalized valuations_, Internat. J. Theoret. Phys. 37(11):2669--2733, 1998, [MR2000c:81027](http://www.ams.org/mathscinet-getitem?mr=1669557), [doi](http://dx.doi.org/10.1023/A:1026680806775);

  _II. conceptual aspects and classical analogues_ Int. J. of Theor. Phys. 38(3):827--859, 1999, [MR2000f:81012](http://www.ams.org/mathscinet-getitem?mr=1697983), [doi](http://dx.doi.org/10.1023/A:1026652817988);

  _III. Von Neumann algebras as the base category_, Int. J. of Theor. Phys. 39(6):1413--1436, 2000, [arXiv:quant-ph/9911020](http://arxiv.org/abs/quant-ph/9911020), [MR2001k:81016](http://www.ams.org/mathscinet-getitem?mr=1788498),[doi](http://dx.doi.org/10.1023/A:1003667607842);

  _IV. Interval valuations_, Internat. J. Theoret. Phys. __41__ (2002), no. 4, 613–639, [MR2003g:81009](http://www.ams.org/mathscinet-getitem?mr=1902067), [doi](http://dx.doi.org/10.1023/A:1015276209768)

The original paper outlining [[Bell's theorem]]:

* [[John S. Bell]], *On the Einstein Podolsky Rosen Paradox*, Physics, [pdf](http://www.drchinese.com/David/Bell_Compact.pdf).

The original paper outlining [[Gleason's theorem]] is

* {#Gleason57} [[Andrew M. Gleason]], _Measures on the closed subspaces of a Hilbert space_, Journal of Mathematics and Mechanics,  Indiana Univ. Math. J. 6 No. 4 (1957), 885–893 ([web](http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050))

A technical discussion on the interplay of [[Gleason's theorem|Gleason's]] and Kochen-Specker theorems and various issues regarding non-contextuality, locality and [[Bell's inequality]]:

* [[Valter Moretti]], §5.1.2 of: *Fundamental Mathematical Structures of Quantum Theory: Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation*
Springer (2019) [[doi:10.1007/978-3-030-18346-2](https://doi.org/10.1007/978-3-030-18346-2)]

The hierarchy of strengths of no-go theorems in quantum context is studied from sheaf theoretic perspective in

* [[Samson Abramsky]], Adam Brandenburger, _The sheaf-theoretic structure of non-locality and contextuality_, [arxiv/1102.0264](https://arxiv.org/abs/1102.0264)

Strengthening of the Kochen-Specker theorem and relation to the "[[free will theorem]]":

* [[Simon Kochen]], *On the Free Will Theorem* [[arXiv:2207.06295](https://arxiv.org/abs/2207.06295)]

  > The new theorem [...] does require any use of free will on the experimenter’s part. The theorem also strengthens the Kochen-Specker Theorem

Relation of Kochen-Specker to [[Bell's inequalities]]:

* [[Leandro Aolita]], Rodrigo Gallego, Antonio Acín, Andrea Chiuri, Giuseppe Vallone, Paolo Mataloni, Adán Cabello, *Fully nonlocal quantum correlations*, Phys. Rev. A **85** 032107 (2012) [[arXiv:1105.3598](https://arxiv.org/abs/1105.3598), [doi:10.1103/PhysRevA.85.032107](https://doi.org/10.1103/PhysRevA.85.032107)]

[[!redirects Kochen-Specker theorems]]

[[!redirects Bell-Kochen-Specker theorem]]
[[!redirects Bell-Kochen-Specker theorems]]

[[!redirects free will theorem]]


Kock field

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

\tableofcontents

## Idea

A definition of [[field]] in [[constructive mathematics]] which uses [[denial inequality]]. It is primarily used in [[synthetic differential geometry]].

## Definition

A **field in the sense of Kock**, or **Kock field** for short, is a [[commutative ring]] $R$ such that [[Anders Kock]]'s Postulate K is satisfied:

1. $R$ is nontrivial ($0 \neq 1$)

2. For all [[natural numbers]] $n \in \mathrm{N}$ and [[functions]] $x \colon \mathrm{Fin}(n) \to R$, if $x(i) \neq 0$ for all elements $i \in \mathrm{Fin}(n)$, then there exists an element $j \in \mathrm{Fin}(n)$ such that $x(j)$ is invertible.

> (Here $Fin(n) \simeq \{1, \cdots, n\}$ denotes any [[finite set]] with $n$ elements.)

## Properties

Every Kock field is a [[local ring]] where the [[denial inequality]] $\neq$ is the [[apartness relation]] defined by $a \neq b$ if and only if $b - a$ is [[invertible]]. Thus, it is a [[weak local ring]] with an [[equivalence relation]] $a \approx b$ defined by the [[double negation]] of [[equality]].

However, unlike a [[Heyting field]], a Kock field cannot in general be proven to have [[stable equality]], since its [[apartness relation]] is not tight.

Every Kock field with [[decidable equality]] is a [[discrete field]].

## See also

* [[local ring]]
* [[ordered local ring]]
* [[discrete field]]
* [[Heyting field]]
* [[weak Heyting field]]

## References

* [[David Jaz Myers]], Axiom 3 in Sec. 4.1 of: *Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory* [[arXiv:2205.15887](https://arxiv.org/abs/2205.15887)]

[[!redirects Kock field]]
[[!redirects Kock fields]]

[[!redirects Postulate K]]

[[!redirects field in the sense of Kock]]
[[!redirects fields in the sense of Kock]]


Kock-Lawvere axiom

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

#Kock--Lawvere axiom#
* table of contents
{:toc}

## Idea

The Kock--Lawvere axiom is the crucial axiom for the theory
of [[synthetic differential geometry]].

Imposed on a [[topos]] equipped with an [[internalization|internal]] [[algebra]]
object $R$ over an internal [[ring]] object $k$, the Kock--Lawvere axiom says essentially that morphisms
$D \to R$ from the
[[infinitesimal space|infinitesimal interval]] $D \subset R$ into $R$ are necessarily _linear_
maps, in that they always and uniquely extend to linear maps $R \to R$.

This linearity condition is what in [[synthetic differential geometry]]
allows to identify the [[tangent bundle]]
$T X \to X$ of a space $X$ with its fiberwise linearity
by simply the [[internal hom]] object $X^D \to X$.

Put the other way round, the Kock--Lawvere axiom axiomatizes the
familiar statement that "to first order every [[smooth map]] is linear".

## Details

### KL axiom for the infinitesimal interval

The plain Kock--Lawevere axiom on a [[ring]] object $R$ in a [[topos]] $T$
is that for $D = \{x \in R| x^2 = 0\}$ the [[infinitesimal space|infinitesimal interval]]
the canonical map

$$
  R \times R \to R^D
$$

given by

$$
  (x,d) \mapsto  (\epsilon \mapsto x + \epsilon d)
$$

is an [[isomorphism]] of objects in $T$.

### KL axiom for spectra of internal Weil algebras
 {#ForWeilAlgebras}

We  can consider the [[internalization|internal]] $R$-algebra object
$R \oplus \epsilon R \coloneqq (R \times R, \cdot, +)$ in $T$, whose underlying object is
$R \times R$, with addition $(x,q)+(x',q') \coloneqq (x+x',q+q')$ and multiplication $(x, q ) \cdot (x', q') = (x x',x q ' + q x')$.

For $A$ an algebra object in $T$, write $Spec_R(A) \coloneqq Hom_{R Alg(T)}(A,R) \subset R^A$
for the object of $R$-algebra homomorphisms from $A$ to $R$.

Then one checks that

$$
  D = Spec(R \oplus \epsilon R)
  \,.
$$

The element $q \in D \subset R$, $q^2 = 0$ corresponds to the algebra homomorphism
$(a,d) \mapsto a + q d$.

Using this, we can rephrase the standard Kock--Lawvere axiom by saying that the
canonical morphism

$$
  R \oplus \epsilon R \to R^{Spec_R(R \oplus \epsilon R)}
$$

is an [[isomorphism]].

Notice that $(R \oplus \epsilon R)$ is a
[[infinitesimally thickened point|Weil algebra]]/[[Artin algebra]]:
an $R$-algebra
that is finite dimensional and whose underlying [[ring]] is a local ring, i.e.
of the form $W = R \oplus m$, where $m$ is a maximal nilpotent ideal finite dimensional over $R$.

Then the general version of the Kock--Lawvere axiom for all Weil algebras says that

For all Weil algebra objects $W$ in $T$ the canonical morphism

$$
  W \to R^{Spec_R(W)}
$$

is an [[isomorphism]] of objects in $T$. Note that because the canonical morphism is also a homomorphism of $R$-algebras, and inverses of algebra homomorphisms are algebra homomorphisms, this gives an isomorphism of internal $R$-algebras.

## References

The Kock-Lawvere axiom was introduced in

* {#Kock77} [[Anders Kock]], _A simple axiomatics for differentiation_,  Mathematica Scandinavica Vol. 40, No. 2 (October 24, 1977), pp. 183-193 ([JSTOR](http://www.jstor.org/stable/24491223))

Textbook accounts are in

* [[Anders Kock]], section I.12 of _Synthetic differential geometry_, Cambridge University Press, London Math. Society Lecture Notes Series No. 333 (1981, 2006) ([pdf](http://home.imf.au.dk/kock/sdg99.pdf))

* [[Anders Kock]], section 1.3 of _Synthetic geometry of manifolds_, Cambridge Tracts in Mathematics, 180 (2010) ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

[[!redirects Kock-Lawvere axioms]]

[[!redirects Kock--Lawvere axiom]]
[[!redirects Kock–Lawvere axiom]]
[[!redirects Kock Lawvere axiom]]


Kodaira embedding theorem

## References

* Wikipedia, _[Kodaira embedding theoerem](http://en.wikipedia.org/wiki/Kodaira_embedding_theorem)_


Kodaira vanishing theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The _Kodaira vanishing theorem_ for [[complex geometry]] says that if $X$ is a [[Kähler manifold]] and $L$ a [[holomorphic line bundle]] on $X$ which is [[positive line bundle|positive]], then the [[abelian sheaf cohomology]] of $X$ with [[coefficients]] in the sheaf of [[sections]] of the [[tensor product]]

$$
  \Omega^{n,0}_X(L) \simeq L \otimes \Omega^{n,0}_X
$$

with the [[canonical line bundle]] $\Omega^{n,0}_X$ is concentrated in degree 0:

$$
  (L \; positive)
  \;\;\Rightarrow\;\;
  H^{\bullet \geq 1}(X, \Omega^{n,0}_X(L))
  = 0
  \,.
$$

## Related concepts

* [[Kodaira-Spencer theory]]

* [geometric quantization -- Dolbeault quantization](geometric%20quantization#EulerCharacteristicOfSheafCohomology)

## References

The statement is due to [[Kunihiko Kodaira]].

Lecture notes include

* [[Hélène Esnault]], [[Eckart Viehweg]], _Lectures on vanishing theorems_ (1992) DMV Seminar 20 ([pdf](http://www.uni-due.de/%7Emat903/books/esvibuch.pdf))

[[!redirects Kodeira vanishing theorems]]


Kodaira-Spencer theory

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

#Contents#
* table of contents
{:toc}

## Idea

[[deformation theory]] of [[complex manifolds]].

## Related concepts

* [[Kodaira vanishing theorem]]

* [[Teichmüller theory]]

## References

* [[Kunihiko Kodaira]],  (1986), _Complex manifolds and deformation of complex structures_, Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences 283, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96188-0, MR815922

* Robert F. Coleman, José Felipe Voloch, _Companion forms and Kodaira-Spencer theory_, Inventiones mathematicae __110__:1 (1992) 263--281 [doi](https://doi.org/10.1007/BF01231333)
[pdf](https://web.ma.utexas.edu/users/voloch/Preprints/comp2.pdf)

[[!redirects Kodaira principle]]


Koenraad Schalm

* [institute page](https://www.universiteitleiden.nl/en/staffmembers/koenraad-schalm#tab-1)

## Selected writings

On [[AdS-CFT in condensed matter physics]]:

* [[Jan Zaanen]], [[Yan Liu]], Ya-Wen Sun, [[Koenraad Schalm]], _Holographic Duality in Condensed Matter Physics_, Cambridge University Press 2015 ([doi:10.1017/CBO9781139942492]( https://doi.org/10.1017/CBO9781139942492))

category: people


Kohei Kishida

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

## Selected writings

On [[quantum programming]] via [[dependent linear type theory]]:

* {#FKS20} [[Peng Fu]], [[Kohei Kishida]], [[Peter Selinger]], _Linear Dependent Type Theory for Quantum Programming Languages_, LICS '20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer ScienceJuly 2020 Pages 440–453 ([arXiv:2004.13472](https://arxiv.org/abs/2004.13472), [doi:10.1145/3373718.3394765](https://dl.acm.org/doi/10.1145/3373718.3394765), [pdf](https://depend.cs.uni-saarland.de/lics-icalp/papers/B5.F), [video](https://www.youtube.com/watch?v=GUT8j4V6Nzg))

On [[quantum programming]] via [[dependent linear type theory]] with [[Quipper]]:

* [[Peng Fu]], [[Kohei Kishida]], [[Neil Ross]], [[Peter Selinger]],  _A Tutorial Introduction to Quantum Circuit Programming in Dependently Typed Proto-Quipper_, in I. Lanese, M. Rawski  (eds.) _Reversible Computation_ RC 2020. Lecture Notes in Computer Science, vol 12227 ([arXiv:2005.08396](https://arxiv.org/abs/2005.08396), [doi:10.1007/978-3-030-52482-1_9](https://doi.org/10.1007/978-3-030-52482-1_9))

On [[dependent linear type theory]] and [[categorical semantics]] for versions of "[[Quipper]]":

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *A biset-enriched categorical model for Proto-Quipper with dynamic lifting* [[arXiv:2204.13039](https://arxiv.org/abs/2204.13039)]

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *Proto-Quipper with dynamic lifting* [[arXiv:2204.13041](https://arxiv.org/abs/2204.13041)]

category: people


Kohhei Yamaguchi

* [institute page](http://kjk.office.uec.ac.jp/Profiles/4/0000339/prof_e.html)

## Selected writings

On [[homotopy of rational maps]] between [[real projective spaces]]:

* {#AdamaszekKozlowskiYamaguchi08} [[Michal Adamaszek]], [[Andrzej Kozlowski]], [[Kohhei Yamaguchi]], *Spaces of algebraic and continuous maps between real algebraic varieties*, Quart. J. Math. 62 (2011), 771–790 ([arXiv:0809.4893](https://arxiv.org/abs/0809.4893), [doi:10.1093/qmath/haq029](https://doi.org/10.1093/qmath/haq029))

and from [[real projective space]] to [[complex projective space]]:

* [[Andrzej Kozlowski]], [[Kohhei Yamaguchi]], *Spaces of algebraic maps from real projective spaces into complex projective spaces* ([arXiv:0812.3954](https://arxiv.org/abs/0812.3954)), in: [[Yves Félix]] et al. (eds.) *Homotopy Theory of Function Spaces and Related Topics*, Contemporary Mathematics Volume: 519; 2010 ([ISBN:978-0-8218-4929-3](https://bookstore.ams.org/conm-519))

category: people


Koichi Yamakawi

* [InSpire page](https://inspirehep.net/authors/982854)

## Selected writings

Early discussion of [[discretized light-cone quantization]]:

* {#MaskawaYamakawi76} [[Toshihide Maskawa]], [[Koichi Yamakawi]], *The Problem of $P^+ = 0$ Mode in the Null-Plane Field Theory and Dirac's Method of Quantization*, Progress of Theoretical Physics **56** 1 (1976) 270–283 [[doi:10.1143/PTP.56.270](https://doi.org/10.1143/PTP.56.270)]

category: people


Koji Azuma

* [personal page](http://www.brl.ntt.co.jp/people/azuma/)

* [GoogleScholar page](https://scholar.google.com/citations?user=sPYOgYEAAAAJ&hl=en)

## Selected writings

On [[quantum networks]]:

* [[Koji Azuma]], Stefan Bäuml, Tim Coopmans, [[David Elkouss]], Boxi Li, *Tools for quantum network design* [[arXiv:2012.06764](https://arxiv.org/abs/2012.06764)]

On [[quantum repeaters]] for [[quantum networks]]:

* [[Koji Azuma]], [[Sophia E. Economou]], [[David Elkouss]], Paul Hilaire, Liang Jiang, Hoi-Kwong Lo, Ilan Tzitrin, *Quantum repeaters: From quantum networks to the quantum internet* [[arXiv:2212.10820](https://arxiv.org/abs/2212.10820)]

category: people


Koji Hashimoto

* [webpage](https://www.phys.sci.osaka-u.ac.jp/en/research_groups/group/02-1_hashimoto/hashimoto/index.html)

## Selected writings

Interpretation in [[holographic QCD]] of the [[Schwinger effect]] as exhibited by the [[DBI-action]] on [[flavor branes]]:

* {#HashimotoOka13} [[Koji Hashimoto]], [[Takashi Oka]], _Vacuum Instability in Electric Fields via AdS/CFT: Euler-Heisenberg Lagrangian and Planckian Thermalization_, JHEP 10 (2013) 116 ([arXiv:1307.7423](https://arxiv.org/abs/1307.7423))

* {#HashimotoOkaSonoda14a} [[Koji Hashimoto]], [[Takashi Oka]], Akihiko Sonoda, _Magnetic instability in AdS/CFT : Schwinger effect and Euler-Heisenberg Lagrangian of Supersymmetric QCD_, J. High Energ. Phys. 2014, 85 (2014) ([arXiv:1403.6336](https://arxiv.org/abs/1403.6336))

* {#HashimotoOkaSonoda14b} [[Koji Hashimoto]], [[Takashi Oka]], Akihiko Sonoda, _Electromagnetic instability in holographic QCD_,  J. High Energ. Phys. 2015, 1 (2015) ([arXiv:1412.4254](https://arxiv.org/abs/1412.4254))

* [[Koji Hashimoto]], Shunichiro Kinoshita, Keiju Murata, [[Takashi Oka]], _Electric Field Quench in AdS/CFT_, J. High Energ. Phys. 2014 ([arXiv:1407.0798](https://arxiv.org/abs/1407.0798))

Relation of [[Yang-Mills monopoles]] as [[Dp/D(p+2)-brane intersections]] and [[Yang-Mills instantons]] as [[Dp/D(p+4)-brane intersections]] to the [[K-theory classification of topological phases of matter]] via [[AdS/CFT duality in solid state physics]]:

* [[Koji Hashimoto]], [[Taro Kimura]], _Band spectrum is D-brane_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 1 ([arXiv:1509.04676](https://arxiv.org/abs/1509.04676))

On [[Montonen-Olive duality]] in [[D=4 super Yang-Mills theory]] via [[ABJM-model]] as [[D3-brane]] model after [[double dimensional reduction]] followed by [[T-duality]]:

* [[Koji Hashimoto]], Ta-Sheng Tai, [[Seiji Terashima]], _Toward a Proof of Montonen-Olive Duality via Multiple M2-branes_, JHEP 0904:025 (2009) [[arxiv:0809.2137](https://arxiv.org/abs/0809.2137)]

On [[holographic QCD]]:

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

On the analogy between [[AdS/CFT duality]] and [[deep learning]]:

* {#HashimotoEtAl18} [[Koji Hashimoto]], Sotaro Sugishita, Akinori Tanaka, Akio Tomiya, _Deep Learning and Holographic QCD_, Phys. Rev. D 98, 106014 (2018) ([arXiv:1809.10536](https://arxiv.org/abs/1809.10536))

* Tetsuya Akutagawa, Koji Hashimoto, Takayuki Sumimoto, _Deep Learning and AdS/QCD_ ([arXiv:2005.02636](https://arxiv.org/abs/2005.02636))

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

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

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

holographic QCD_ ([arXiv:2103.03563](https://arxiv.org/abs/2103.03563))

On the binding energy of [[nucleons]] via [[AdS/QCD]]:

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

category: people


Kolmogorov complexity

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

#Contents#
* table of contents
{:toc}

## Idea

The _Kolmogorov complexity_ of a piece of data is a measure for the [[computability]] resources needed to specify it. This is a central concept in [[complexity theory]].

## Related concepts

* [[Chaitin's incompleteness theorem]]

## References

Named after [[Andrey Kolmogorov]].

See also:

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

[[!redirects Kolmogorov-Chaitin complexity]]


Kolmogorov extension theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

One of the main tenets of [[probability theory]] is that in general [[joint and marginal probability|the probability of the product is not the product of the probabilities]], because of [[correlation]] and other [[statistical interactions]].
In particular, the probability of an [[infinite tensor product|infinite product]] is not the infinite product of the probabilities.

However, an infinite product is a [[cofiltered limit]] of finite products, and under some conditions, *the probability of an infinite product is the limit of the probability of the finite products*.
One can interpret this fact as the absence of "infinitary correlations".

Statements of this kind are known collectively as the **Kolmogorov extension theorem**.
They are particularly important in the theory of [[stochastic processes]], since a stochastic process can be seen as a [[joint distribution]] over an [[infinite tensor product|infinite product]].

## In measure-theoretic probability

We look at the classical statement, and then give two (related) ways to express it category-theoretically.

### Classical statement

The classical statement is about [[probability measures]], and roughly says that under some conditions, a probability measure on an [[infinitary tensor product|infinite product]] is uniquely determined by its finite [[marginal distribution|marginals]].

\begin{theorem}\label{kolmogorov}
Let $\{(X_j,\mathcal{A}_j)\}_{j\in J}$ be a collection of [[standard Borel spaces]], indexed by a set $J$ of arbitrary cardinality.

* Denote by $(X_J,\mathcal{A}_J)$ the [[product]] of the $X_j$, equipped with the product [[sigma-algebra]].
* Similarly, for every finite set $F\subseteq J$, denote $(X_F,\mathcal{A}_F)$ the product of the $(X_j,\mathcal{A}_j)$ for $j\in F$.

There is a [[bijection]] between

* probability measures on $(X_J,\mathcal{A}_J)$

and

* families of probability measures $p_F$ on $(X_F,\mathcal{A}_F)$ for each finite subset $F\subseteq J$ such that for every inclusion $F\subseteq G$, the measure $p_F$ is the [[marginal distribution|marginal]] of $p_G$.

In particular, the $p_F$ can be obtained as marginal distributions of the measure $p_J$.

\end{theorem}

Note that

* The same result holds more in general if each $(X_j,\mathcal{A}_j)$ is a [[Hausdorff space]] equipped with its [[Borel sigma-algebra]], and the measures $p_F$ are [[inner regular measure|inner regular]].

* The uncountable product of standard Borel spaces may fail to be standard Borel.

* The [[Borel sigma-algebra]] of an uncountable product of topological spaces is not the product of the Borel sigma-algebras of the factors.

### In the category of Markov kernels

We can restate Kolmogorov's extension theorem as a *limit*, as follows.

Let $\{(X_j,\mathcal{A}_j)\}_{j\in J}$ be a collection of [[standard Borel spaces]], indexed by a set $J$ of arbitrary cardinality.

* Denote by $(X_J,\mathcal{A}_J)$ the categorical [[product]] of the $X_j$ in the [[Meas|category Meas of measurable spaces and measurable maps]].
* Similarly, for every finite set $F\subseteq J$, denote $(X_F,\mathcal{A}_F)$ the product of the $(X_j,\mathcal{A}_j)$ for $j\in F$.

Note that, since infinite products are [[cofiltered limits]] of finite products, $(X_J,\mathcal{A}_J)$ is the limit in [[Meas]] of the diagram formed by the $(X_F,\mathcal{A}_F)$ and their projections.

Now:

\begin{corollary}\label{kolmogorov_kernels}
The object $(X_J,\mathcal{A}_J)$ is the limit of the $(X_F,\mathcal{A}_F)$ also in the [[Stoch|category Stoch of measurable spaces and Markov kernels]].
\end{corollary}

Note that this does not say that $(X_J,\mathcal{A}_J)$ is the product of the $(X_j,\mathcal{A}_j)$ ([[joint and marginal distributions|that would be false even in the finite case]]).
It is just the "[[cofiltered limit|finite to infinite]]" part which is retained in the category of kernels.

\begin{proof}
For finite sets $F\subseteq G\subseteq J$, denote by $\pi_{J,F}:(X_J,\mathcal{A}_J)\to(X_F,\mathcal{A}_F)$ and $\pi_{G,F}:(X_G,\mathcal{A}_H)\to(X_F,\mathcal{A}_F)$ the product projections.
With a slight abuse, denote the [[Markov kernel#kernels_from_deterministic_functions|Markov kernel induced by the functions]] $\pi_{J,F}$ and $\pi_{G,F}$ again by $\pi_{J,F}$ and $\pi_{G,F}$.

Given a measurable space $(Y,\mathcal{B})$, we have to show that there is a bijection between [[Markov kernels]] $k_J:Y\to X_J$ and families of Markov kernels $k_F:Y\to X_F$ such that for each inclusion $F\subseteq G$, the outer triangle in the following diagram of [[Stoch]] commutes.
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large,%
]
& & & (X_G,\mathcal{A}_G) \ar{dd}{\pi_{G,F}} \\
(Y,\mathcal{B}) \ar{urrr}{k_G} \ar[dashed]{rr}[near end]{k_J} \ar{drrr}[swap]{k_F} && (X_J,\mathcal{A}_J) \ar{ur}[swap]{\pi_{J,G}} \ar{dr}{\pi_{J,F}} \\
& & & (X_F,\mathcal{A}_F)
\end{tikzcd}

Now for each $y\in Y$, the Markov kernels $k_J(-|y)$ and $k_F(-|y)$ are just [[probability measures]], so that Theorem \ref{kolmogorov} applies. We only need to show that $k_J$ is measurable in $y$ if and only if the $k_F$ are. Now if $k_J$ is measurable, so is $k_F$, since the composition of measurable functions is measurable. The converse is true as well, since measurability of Markov kernels can be equivalently tested on a [[pi-system]].
\end{proof}

### In terms of the Giry monad

The result in terms of [[Markov kernels]] can be restated in terms of the [[Giry monad]] as follows.

Once again, recall that $(X_J,\mathcal{A}_J)$ is the [[cofiltered limit]] in [[Meas]] of the diagram formed by the $(X_F,\mathcal{A}_F)$ and their projections.

Now, if the $X_j$ are [[standard Borel]]:

\begin{corollary}\label{kolmogorov_giry}
The Giry monad preserves this limit.
\end{corollary}

\begin{proof}
We have to show that given a measurable space $Y$, there is a bijection between [[measurable functions]] $f_J:Y\to X_J$ and families of measurable functions $f_F:Y\to X_F$ such that for each inclusion $F\subseteq G$, the outer triangle in the following diagram of [[Meas]] commutes.
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large,%
]
& & & P(X_G) \ar{dd}{P\pi_{G,F}} \\
Y \ar{urrr}{f_G} \ar[dashed]{rr}[near end]{f_J} \ar{drrr}[swap]{f_F} && P(X_J) \ar{ur}[swap]{P\pi_{J,G}} \ar{dr}{P\pi_{J,F}} \\
& & & P(X_F)
\end{tikzcd}
Now since [[Stoch]] is the [[Kleisli category]] of the [[Giry monad]], notice that there is a bijection
$$
Stoch(Y,X) \cong Meas(Y,PX)
$$
natural in $X$, so that the universal property that we want to prove is just a restatement of the proof of Corollary \ref{kolmogorov_kernels}.
\end{proof}

## In Markov categories

In [[Markov categories]] the Kolmogorov extension theorem appears in the form of an [[axiom]] that the category can satisfy, an abstraction of the universal property in the [[category of Markov kernels]] given above.

\begin{definition}
Let $C$ be a Markov category, and let $(X_j)_{j\in J}$ be a family of objects. A **Kolmogorov product** of them is an [[infinite tensor product]]
$$
X_J=\bigotimes_{j\in J} X_j
$$
such that all finite projections (marginalizations) $X_J\to X_F$, with $F\subseteq J$ finite, are [[Markov category#deterministic_morphisms|deterministic morphisms]].
\end{definition}

The category [[BorelStoch]], for example, has all countable Kolmogorov products. (The uncountable product of [[standard Borel spaces]], in general, is not standard Borel.)

See [[Markov category#kolmogorov_products|Markov category - Kolmogorov products]] for more on this, as well as [Fritz-Rischel'20](#fritzrischel).

## See also

* [[probability theory]], [[categorical probability]]
* [[filtered category]], [[filtered limit]]
* [[infinitary tensor product]]
* [[probability monad]], [[Giry monad]]
* [[Markov category]], [[infinitary tensor product]]
* [[joint and marginal distributions]]
* [[stochastic dependence and independence]]

## References

* {#fritzrischel} [[Tobias Fritz]] and Eigil Fjeldgren Rischel, _Infinite products and zero-one laws in categorical probability_, Compositionality 2(3) 2020. ([arXiv:1912.02769](http://arxiv.org/abs/1912.02769))

* {#vb_martingales} Ruben Van Belle, _A categorical treatment of the Radon-Nikodym theorem and martingales_, 2023. ([arXiv](https://arxiv.org/abs/2305.03421))

category: probability

[[!redirects Kolmogorov's extension theorem]]


Kolmogorov product

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!i
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _Kolmogorov product_ is a way of forming infinite [[tensor products]] in a [[semicartesian monoidal category]] which is not necessarily [[cartesian monoidal category|cartesian]]. In a [[cartesian monoidal category]] it coincides with the usual infinite [[cartesian product]] (if it exists).

If the category in question is a [[poset]], Kolmogorov products correspond to a [[finitary complete monoid]] structure (up to reversing the arrows).

## Definition

In what follows, let $(C,\otimes 1)$ be a [[symmetric monoidal category|symmetric]] [[semicartesian monoidal category]].

Let $X_1$ and $X_2$ be objects of $C$. The unique map $!:X_1\to 1$ induces a map
\begin{tikzcd}
X_1\otimes X_2 \ar{r}{!\otimes X_2} & X_2 ,
\end{tikzcd}
sometimes called the **projection** (or _[[marginal]]_ in the probability literature). This projection is natural in $X_2$, and there is a commutative diagram
\begin{tikzcd}
& X_1\otimes X_2 \ar{dl}[swap]{X_1\otimes !} \ar{dr}{!\otimes X_2} \\
X_1 \ar{dr}[swap]{!} && X_2 \ar{dl}{!}\\
& 1
\end{tikzcd}

More generally, let $\{X_1, \dots, X_n\}$ be a finite set of objects of $C$. All the possible projections of their product onto the factors form a finite [[Boolean lattice]], or more precisely, a functor $B \to C$ from a finite Boolean lattice $B$ (considered as a [[thin category]]) to $C$.
\begin{tikzcd}[column sep=small, nodes={scale=1.25}]
&& X_1\otimes \dots \otimes X_n \ar{d} \ar{dll} \ar{drr} \\
X_1\otimes \dots \otimes X_{n-1} \ar{d} \ar{dr} && \dots \ar{dl} \ar{dr} && X_2\otimes \dots \otimes X_n \ar{dl} \ar{d} \\
\dots \ar{d} & \dots \ar{dl} \ar{dr} && \dots \ar{dl} \ar{dr} & \dots \ar{d} \\
X_1 \ar{drr} && \dots \ar{d} && X_n \ar{dll} \\
&& 1
\end{tikzcd}
We call this lattice the **lattice of projections**.

Even more generally, let now $\{X_i\}_{i\in I}$ be a set of objects of $C$, possibly infinite. We can form the lattice of all the finite products of the $X_i$ and their projections, with the arrows in the form
$$
\bigotimes_{i\in F} X_i \to \bigotimes_{j\in S} X_j ,
$$
where $F$ is a finite subset of $I$, and $S\subseteq F$.
This is again a lattice (it may be infinite, but it is closed under finite [[joins]] and [[meets]]), which we call the **lattice of finite projections**. In particular, this is a [[filtered category|cofiltered diagram]].
We call the **Kolmogorov product** of the set $\{X_i\}_{i\in I}$ the [[cofiltered limit]] of this diagram, if it exists, and if it is preserved by the functor $-\otimes Y$ for every object $Y$ of $C$. We denote it by
$$
\bigotimes_{i\in I} X_i .
$$

## Properties

* The Kolmogorov product of a finite set of objects is just their $n$-ary [[tensor product]].

* In a [[cartesian monoidal category]], the Kolmogorov product coincides with the possibly infinite [[cartesian product]].

## Examples

(...)

## See also

* [[filtered category]], [[filtered limit]]
* [[finitary complete monoid]]
* [[monoidal category]], [[cartesian monoidal category]], [[semicartesian monoidal category]]
* [[Kolmogorov extension theorem]]

## References

(...)

* [[Tobias Fritz]] and Eigil Fjeldgren Rischel, _The zero-one laws of Kolmogorov and Hewitt--Savage in categorical probability_, 2019. ([arXiv:1912.02769](http://arxiv.org/abs/1912.02769))

[[!redirects Kolmogorov products]]


Kolmogorov topological space

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

#Contents#
* table of contents
{:toc}

## Definition

A [[topological space]] $(X,\tau)$ is called a _Kolmogorov space_ if it satisfies the $T_0$-[[separation axiom]], hence if for $x_1 \neq x_2 \in X$ any two distinct points, then at least one of them has an [[open neighbourhood]] $U_{x_i} \in \tau$ which does not contain the other point. This is equivalent to its [[contrapositive]]: for all points $x_1, x_2 \in X$, if every [[open neighbourhood]] $U_{x_i} \in \tau$ which contains one of the points also contains the other point, then the two points are equal $x_1 = x_2$.

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

## Properties
 {#Properties}

### Specialization order

Every topological space $(X, O(X))$ is a [[preorder]] with respect to the [[specialization order]] $\forall_{U:O(X)} (x \in U) \implies (y \in U)$.

\begin{proposition}
A topological space is $T_0$ precisely when it is a [[partial order]] with respect to the [[specialization order]] of the topological space.
\end{proposition}

The proof follows from the [[contrapositive]] of the definition of a $T_0$-space.

### Alternative Characterizations

Regard [[Sierpinski space]] $\Sigma$ as a [[frame]] object in the category of topological spaces (meaning: the representable functor $Top(-, \Sigma): Top^{op} \to Set$ lifts through the monadic forgetful functor $Frame \to Set$), hence as a [[dualizing object]] that induces a contravariant adjunction between frames and topological spaces. Thus for a space $X$, the frame $Top(X, \Sigma)$ is the frame of [[open sets]]; for a frame $A$, there is an accompanying topological space of points $Frame(A, \Sigma)$, a subspace of the product space $\Sigma^{{|A|}}$. The unit of the adjunction is called the *double dual embedding*.

\begin{proposition}
A [[topological space]] $X$ is $T_0$ precisely when the double dual embedding $X \to Frame(Top(X, \Sigma), \Sigma)$ is a [[monomorphism]].
\end{proposition}

The proof is trivial: the monomorphism condition translates to saying that for any points $x, y \in X$, if the truth values of $x \in U$ and $y \in U$ agree for every open set $U$, then $x = y$.

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

### Reflection

+-- {: .num_prop #KolmogorovQuotient}
###### Proposition
**(Kolmogorov quotient)**

Let $(X,\tau)$ be a [[topological space]]. Consider the [[relation]] on the underlying set by which $x_1 \sim x_1$ precisely if neither $x_i$ has an [[open neighbourhood]] not containing the other. This is an [[equivalence relation]]. The [[quotient topological space]] $X \to X/\sim$ by this
equivalence relation is a $T_0$-space.

This construction is the reflector exhibiting Kolmogorov spaces as a [[reflective subcategory]] of the [[category]] [[Top]] of all topological spaces.

=--

## Related concepts

* [[Hausdorff topological space]]

* [[nice topological space]]

| [[preorder]] | [[partial order]] | [[equivalence relation]] | [[equality]] |
| - | - | - | - |
| [[topological space]] | [[ Kolmogorov topological space]] | [[symmetric topological space]] | [[accessible topological space]] |

## References

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

[[!redirects T0]]
[[!redirects T_0]]

[[!redirects Kolmogorov space]]
[[!redirects Kolmogorov spaces]]
[[!redirects T0 space]]
[[!redirects T0 spaces]]
[[!redirects T0-space]]
[[!redirects T0-spaces]]
[[!redirects T_0 space]]
[[!redirects T_0 spaces]]
[[!redirects T_0-space]]
[[!redirects T_0-spaces]]

[[!redirects Kolmogorov topological space]]
[[!redirects Kolmogorov topological spaces]]
[[!redirects T0 topological space]]
[[!redirects T0 topological spaces]]
[[!redirects T_0 topological space]]
[[!redirects T_0 topological spaces]]
[[!redirects T0-topological space]]
[[!redirects T0-topological spaces]]
[[!redirects T_0-topological space]]
[[!redirects T_0-topological spaces]]

[[!redirects Kolmogorov quotient]]
[[!redirects Kolmogorov quotients]]
[[!redirects T0 quotient]]
[[!redirects T0 quotients]]
[[!redirects T0-quotient]]
[[!redirects T0-quotients]]
[[!redirects T_0 quotient]]
[[!redirects T_0 quotients]]
[[!redirects T_0-quotient]]
[[!redirects T_0-quotients]]


Konrad Osterwalder

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

## related $n$Lab entries

* [[Osterwalder-Schrader theorem]]

category: people


Konrad Schmüdgen

* [webpage](http://www.math.uni-leipzig.de/~schmuedgen/)

## Selected writings

On [[operator algebra]]:

* {#Schmuedgen90} [[Konrad Schmüdgen]], _Unbounded  operator  algebras  and  representation  theory_, Operator  theory, advances and applications, vol. 37. Birkhäuser, Basel (1990) ([doi:10.1007/978-3-0348-7469-4](https://link.springer.com/book/10.1007/978-3-0348-7469-4))

category: people

[[!redirects Konrad Schmuedgen]]
[[!redirects Konrad Schmudgen]]