Klaus Hepp

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

## Related $n$Lab entries

* [[renormalization]]

* [[Wick's lemma]]

category: people


Klaus Jänich

* [German Wikipedia entry](https://de.m.wikipedia.org/wiki/Klaus_J%C3%A4nich)

## Selected writings

On [[equivariant differential topology]] ([[topological G-space|G-]][[smooth manifold|manifolds]]) and [[manifolds with corners]]:

* [[Klaus Jänich]],  *On the classification of $O(n)$-manifolds*, Math. Ann. 176, 53–76 (1968) [[doi:10.1007/BF02052956](https://doi.org/10.1007/BF02052956)]

On [[differential topology]]:

* [[Theodor Bröcker]], [[Klaus Jänich]], *Introduction to differentiable topology* (1982) [[ISBN:9780521284707](https://www.cambridge.org/ae/universitypress/subjects/mathematics/differential-and-integral-equations-dynamical-systems-and-co/introduction-differential-topology?format=PB&isbn=9780521284707)]

  > (translated from the German 1973 edition)

On [[general topology|general]] [[topology]]:

* [[Klaus Jänich]], *Topology*, Undergraduate Texts in Mathematics, Springer (1984, 1999) [[ISBN:9780387908922](https://link.springer.com/book/9780387908922), [doi:10.1007/978-3-662-10574-0](https://doi.org/10.1007/978-3-662-10574-0), Chapters 1-2: [pdf](http://topologicalmedialab.net/xinwei/classes/readings/Janich/Janich_Topology_ch1-2.pdf)]

category: people


Klaus Keimel

* [personal page](https://www2.mathematik.tu-darmstadt.de/~logik/keimel/)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=21571)

* Karl Hofmann, Jimmie Lawson, *In memoriam: Klaus Keimel (1939–2017)*, Semigroup Forum **96** (2018) 199–202 [[doi:10.1007/s00233-018-9929-2](https://doi.org/10.1007/s00233-018-9929-2)]

## Selected writings

On [[category theory|category theoretic]] [[probability theory]] (cf. *[[convex powerset of distributions monad]]*):

* [[Regina Tix]], [[Klaus Keimel]], [[Gordon D. Plotkin]]: *Semantic domains for combining probability and non-determinism*, Electronic Notes in Theoretical Computer Science, **222** (2009) 3–99 [[doi:10.1016/j.entcs.2009.01.002](https://doi.org/10.1016/j.entcs.2009.01.002)]

category: people


Klaus Künnemann

* [webpage](http://homepages.uni-regensburg.de/~kuk22111/)

## relate $n$Lab entries

* [[Arakelov geometry]]

category: people

[[!redirects Klasu Kuennemann]]


Klaus Lamotke

* [Wikipedia entry](https://de.wikipedia.org/wiki/Klaus_Lamotke)

##Publications

including

* _Semisimpliziale Algebraische Topologie_, Springer, Berlin, Heidelberg 1968

* _Regular Solids and Isolated Singularities_, Vieweg, Braunschweig, Wiesbaden 1986.

## related $n$Lab entries

* [[classification of finite rotation groups]]

* [[ADE singularity]]

* [[platonic solid]]

category: people


Klaus Mainzer

* [Institute page](https://www.professoren.tum.de/en/mainzer-klaus)

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

## Selected writings

On [[formal proof]] and [[software verification]]:

* [[Klaus Mainzer]], *Proof and Computation. Perspectives for Mathematics, Computer Science, and Philosophy* ([doi:10.1142/9789813270947_0001](https://worldscientific.com/doi/10.1142/9789813270947_0001)), Chapter 1 in Klaus Mainzer, Peter Schuster, Helmut Schwichtenberg (eds.) *Proof and Computation -- Digitization in Mathematics, Computer Science, and Philosophy*, World Scientific 2018 ([doi:10.1142/11005](https://doi.org/10.1142/11005))

with highlighting of [[homotopy type theory]]:

* [[Klaus Mainzer]], *From Proof Theory to Proof Assistants -- Challenges of Responsible Software and AI*, talk at *[Arbeitstagung Bern-München, ABM Spring 2019](https://cj-xu.github.io/abm19/)* ([pdf](https://cj-xu.github.io/abm19/mainzer.pdf), [[MainzerResponsibleSoftware2019.pdf:file]])

* [[Klaus Mainzer]], *Proof and Computation. Perspectives for Mathematics, Computer Science, and Philosophy*, Chapter 1 in: Klaus Mainzer, Peter Schuster, Helmut Schwichtenberg (eds.), *Proof and Computation II -- From Proof Theory and Univalent Mathematics to Program Extraction and Verification*, World Scientific 2021 ([doi:10.1142/12263](https://doi.org/10.1142/12263))

category: people


Klaus von Klitzing

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

## Selected writings

Introducing the [[quantum Hall effect]]:

* [[Klaus von Klitzing]], G. Dorda, and M. Pepper, _New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance_, Phys. Rev. Lett. 45, 494 (1980) ([doi:10.1103/PhysRevLett.45.494](https://doi.org/10.1103/PhysRevLett.45.494))

Review:

* {#vonKlitzing86} [[Klaus von Klitzing]], _The quantized Hall effect_, Rev. Mod. Phys. 58, 519 (1986) ([doi:10.1103/RevModPhys.58.519](https://doi.org/10.1103/RevModPhys.58.519))

category: people


Klaus Weihrauch

* [webpage](http://www.fernuni-hagen.de/weihrauch/)

## related $n$Lab entries

* [[computable analysis]], [[Type Two Theory of Effectivity]]

* [[computable physics]]

category: people


Klaus Wirthmüller

* [webpage](http://www.mathematik.uni-kl.de/agag/mitglieder/privatdozenten/wirthm/)

## related $n$Lab entries

* [[Wirthmüller context]]

* [[topological K-theory]]

category: people


Kleene algebra

> This page is about [[de Morgan algebras]] satisfying an extra condition.  For Kleene star algebras in relation to [[regular expressions]], see there.

# Kleene algebra

* table of contents
{: toc}

## Definition

A **Kleene algebra** is a [[de Morgan algebra]] $D$ satisfying $x \wedge \neg x \le y \vee \neg y$ for all $x,y\in D$. Since the order is definable in terms of the lattice operators, this can be stated as the equation

$$ x \wedge \neg x \wedge (y \vee \neg y) = x \wedge \neg x. $$

## Examples

* Any [[Boolean algebra]] is a Kleene algebra, with $\neg$ the logical negation.
* The unit interval $[0,1]$ is a Kleene algebra, with $\neg x = (1-x)$.

## Related concepts

* Kleene algebras are used in one form of [[cubical type theory]]

[[!redirects Kleene algebras]]


Kleene equality

In the setting of [[partial function]]s and operations, saying that an equality between [[term|terms]] such as
$$ s = t $$
is a **Kleene equality** means that it asserts that if either side of the equality is defined, then so is the other, and they are equal.  Sometimes another symbol is used for Kleene equality, such as
$$ s \simeq t .$$

## Elementary examples

For example, in elementary algebra (or more generally, the first-order theory of a [[field]]),
\[ \label{correct} \frac x {x^2} = \frac 1 x \]
is always true as  Kleene equality, even when $x = 0$, because then *both* sides are undefined.  In contrast,
\[ \label{wrong} \frac {x^2} x = x \]
is not always true, even though the two sides are equal whenever both are defined, because the right-hand side is defined when $x = 0$ but the left-hand side is not.

The example (eq:wrong) can be fixed in either of two ways.  First, we can make its statement contingent on some condition under which the equality holds, as follows:
\[ \label{weak} \frac {x^2} x = x, \text{ if } x \ne 0 .\]
This states that (eq:wrong) holds whenever $x \ne 0$, which is correct.  Alternatively, we can add a clause to the term on the right-hand side that forces it to be undefined when $x = 0$, as follows:
\[ \label{strong} \frac {x^2} x = (x, x \ne 0) .\]
The idea here is that $x, x \ne 0$ is a term (like a [[piecewise-defined function|piecewise-defined]] expression with only one piece) that is defined only when $x \ne 0$.  Note that (eq:strong) is a *stronger* statement than (eq:weak); while (eq:weak) states only that (eq:wrong) holds when $x \ne 0$, (eq:strong) states additionally that the left-hand side of (eq:wrong) is defined only when $x \ne 0$.

The version usually seen in elementary algebra textbooks,
\[ \label{ambiguous} \frac {x^2} x = x,\; x \ne 0 ,\]
is ambiguous and could mean either (eq:weak) or (eq:strong).  In any case, there is *no* need to add anything to (eq:correct) if one adopts Kleene equality; in practice, some textbooks do and some don\'t.

## Models

In general, if $\Gamma$ is a [[context]] and $A$ is a [[type]], then the set of terms of type $A$ with free variables given by $\Gamma$ can be thought of as something like the set of [[sections]] of a [[product]] projection $\Gamma \times A \to \Gamma$.  Aside from the fact that we only get *definable* operations this way, we also get a finer notion of equality, as several distinct terms may denote equal operations.  However, that is where the proposition '$s = t$' comes in; it states that the terms $s$ and $t$ denote equal operations.

If we now allow for terms for undefined expressions, then we get sections of the projection $\Gamma \times A_\bot \to \Gamma$ (where $X_\bot$ is the set of [[subsingletons]] of $X$).  Then Kleene equality denotes equality of these operations.

If we instead use the [[partial order]] of inclusion of subsingletons, then we have __[[directed equality]]__ (which is not [[symmetric relation|symmetric]]).

[[!redirects Kleene equality]]
[[!redirects kleene equality]]


Kleene's fixed point theorem

#Contents#
* table of contents
{:toc}

## Idea

_Kleene's fixed point theorem_ theorem constructs least [[fixed points]] of [[endofunctions]] on [[posets]] by [[composition|iterating]] them. [[Adámek's fixed point theorem]] generalizes this to constructing [[initial algebras]].

## Construction

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

Let $f : P \to P$ be a [[monotone function]] on a [[poset]] $P$. If $P$ has a [[least element]] $\bot$ and [[joins]] of increasing sequences, and if $f$ [[preserved colimit|preserves]] joins of increasing sequences, then a least [[fixed point]] of $f$ can be constructed as the join of the increasing sequence:

$$
  \bot
  \;\leq\;
  f(\bot)
  \;\leq\;
  f^2(\bot)
  \;\leq\;
  \cdots
  \,.
$$

=--

## Related entries

* [[fixed point]]

* [[Adámek's fixed point theorem]]

* [[Knaster-Tarski's fixed point theorem]]

* [[Lawvere's fixed point theorem]]

* [[Lefschetz fixed point theorem]]

* [[Brouwer's fixed point theorem]]

* [[Atiyah-Bott fixed point formula]]

## References

Named after [[Stephen Kleene]].

See also:

* Wikipedia, *[Kleene fixed point theorem](https://en.wikipedia.org/wiki/Kleene_fixed-point_theorem)*

[[!redirects Kleene fixed point theorem]]
[[!redirects Kleene's fixed-point theorem]]
[[!redirects Kleene fixed-point theorem]]


Kleene's second algebra

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### Context
#### Constructivism, Realizability, Computability
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# Contents
* table of contents
{: toc}

## Idea

Kleene's second algebra is a [[partial combinatory algebra]] based on [[Baire space (computability)|Baire space]]. It was introduced by [[Kleene]] to formalize [[Brouwer]]'s notion of [[choice sequence]].

## Definition

Let $\mathbb{N}$ be the set of natural numbers equipped with the discrete topology, so that the [[exponential object|function space]] $\mathbb{N}^\mathbb{N}$ is a countable product of copies of $\mathbb{N}$. By means of [[continued fraction]]s, this space is homeomorphic to the Baire space $B$ of [[irrational numbers]] between $0$ and $1$.

To define Kleene's second algebra, we need several ingredients:

1. There is a function $p \colon B \to 1 + \mathbb{N}$ which takes the constant function at $0$ to $\ast \in 1$, and any other function $\alpha \colon \mathbb{N} \to \mathbb{N}$ to the predecessor of the first non-zero $\alpha(k)$.

2. Each irrational number $\beta \in B$ releases a stream of rational approximants $q_1(\beta), q_2(\beta), \ldots$ by successive truncations of the continued fraction of $\beta$. By coding rational numbers by natural numbers, we get a corresponding stream of natural numbers $(\beta_1, \beta_2, \ldots) \in \mathbb{N}^\mathbb{N}$. The map
$$\phi \colon B \to B$$
that sends $\beta$ to $(\beta_1, \beta_2, \ldots)$ is continuous.

3. $B$ is the [[terminal coalgebra]] of the endofunctor $- \times \mathbb{N}$ on $Top$, so there is an isomorphism $\xi \colon B \to B \times \mathbb{N}$ whose inverse is denoted $prefix$.

4. Composition of functions $\mathbb{N} \to \mathbb{N}$ defines a map $comp \colon B \times B \to B$.

Now consider the composite

$$B \times B \times \mathbb{N} \stackrel{1_B \times prefix}{\to} B \times B \stackrel{1_B \times \phi}{\to} B \times B \stackrel{comp}{\to} B \stackrel{p}{\to} 1 + \mathbb{N}$$

and curry this to a map $\psi \colon B \times B \to (1 + \mathbb{N})^\mathbb{N}$. Let $i \colon \mathbb{N} \to 1 + \mathbb{N}$ be the inclusion map.

**Kleene's second algebra** is the applicative structure or partial binary operation on $B$ defined by the pullback

$$\array{
D & \to & \mathbb{N}^\mathbb{N} \\
\downarrow & & \downarrow^\mathrlap{i^\mathbb{N}} \\
B \times B & \underset{\psi}{\to} & (1 + \mathbb{N})^\mathbb{N}
}$$

## Properties

### Relation to function realizability

Kleene's second algebra is an abstraction of [[function realizability]] introduced for the purpose of extracting computational content from proofs in [[exact analysis|intuitionistic analysis]]. (e.g. [Streicher 07, p. 17](#Streicher07))

## Related concepts

* [[realizability]]

* [[Kleene's first algebra]]

[[!include computable mathematics -- table]]

## References

* [[Stephen Kleene]], _Introduction to Meta-Mathematics_ (1952)

* {#Streicher07} [[Thomas Streicher]], example 3.4 of _Realizability_ (2017/18) ([pdf](https://www2.mathematik.tu-darmstadt.de/~streicher/REAL/REAL.pdf))

* {#Bauer05} [[Andrej Bauer]], section 2 of _Realizability as connection between constructive and computable mathematics_, in T. Grubba, P. Hertling, H. Tsuiki, and [[Klaus Weihrauch]], (eds.) _CCA 2005 - Second International Conference on Computability and Complexity in Analysis_, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, , vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. ([pdf](http://math.andrej.com/data/c2c.pdf))

A presentation of neighborhood functions using algebras and coalgebras:

* {#GhaniHP09} [[Neil Ghani]], Peter Hancock, Dirk Pattinson, _Continuous Functions on Final Coalgebras_, 2009. ([pdf](https://personal.cis.strath.ac.uk/neil.ghani/papers/ghani-mfps09.pdf))

[[!redirects Kleene's second algebra]]
[[!redirects Kleene's second partial combinatory algebra]]


Kleene-Vesley topos

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#### Topos Theory
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[[!include topos theory - contents]]
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#Contents#
* table of contents
{:toc}
## Idea

The effective part of the [[function realizability topos]] $RT(\mathcal{K}_2)$.

## References

This construction in realizability goes back to

* [[Stephen Kleene]], R. E. Vesley, _The foundations of intuitionistic mathematics_,North-Holland (1965)

The Kleene-Vesley topos as such is discussed for instance in

* {#vanOosten08} [[Jaap van Oosten]], page 279 of _Realizability: an introduction to its categorical side_, Studies in Logic and the Foundations of Mathematics, vol. 152, Elsevier, 2008 ([preface pdf](http://www.staff.science.uu.nl/~ooste110/boekbegin.pdf))


Klein 2-geometry

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[[!include synthetic differential geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Klein 2-geometry is an attempt to [[vertical categorification|categorify]] Klein's [[Erlangen program]] by replacing groups with [[2-group|2-groups]], which are like groups but categories instead of sets.

The idea of Klein's original Erlangen program is to study geometries that have some group of symmetries, $G$.  If this group acts transitively on some kind of geometrical figure - e.g. points, lines, etc. - the space of figures of this kind is $G/H$, where $H \subseteq G$ is the stabilizer of such a figure: that is, the subgroup of symmetries that preserve it.

To [[vertical categorification|categorify]] this we'd like to replace $G$ with a [[2-group]], and replace $H$ with a "sub-2-group".  We then need to define the suitable analogue of the [[quotient]] $G/H$, and see in what sense $G$ acts transitively on this.

A suitable notion of sub-2-group is a [[faithful functor|faithful]], but not necessarily full, functor into $G$. Quotients may be thought of as homogeneous [[orbifold|orbifolds]].

## Related concepts

[[!include local and global geometry - table]]

## Discussion

{#blog} Blog entries: [I](http://math.ucr.edu/home/baez/corfield/2006/05/klein-2-geometry.html), [II](http://math.ucr.edu/home/baez/corfield/2006/06/klein-2-geometry-ii.html), [III](http://math.ucr.edu/home/baez/corfield/2006/07/klein-2-geometry-iii.html), [IV](http://math.ucr.edu/home/baez/corfield/2006/08/klein-2-geometry-iv.html), [V](http://golem.ph.utexas.edu/category/2006/09/klein_2geometry_v.html), [VI](http://golem.ph.utexas.edu/category/2006/10/klein_2geometry_vi.html), [VII](http://golem.ph.utexas.edu/category/2006/11/klein_2geometry_vii.html), [VIII](http://golem.ph.utexas.edu/category/2006/12/klein_2geometry_viii.html), [VIIIS](http://golem.ph.utexas.edu/category/2007/05/klein_2geometry_viiis.html), [IX](http://golem.ph.utexas.edu/category/2007/10/klein_2geometry_ix.html), [X](http://golem.ph.utexas.edu/category/2008/04/klein_2geometry_x.html), [XI](http://golem.ph.utexas.edu/category/2008/06/klein_2geometry_xi.html), [XII](http://golem.ph.utexas.edu/category/2011/08/klein_2geometry_xii.html), and [stabilizer](http://golem.ph.utexas.edu/category/2009/03/the_stabilizer_of_a_subcategor.html).


Klein bottle

## Idea

A [[closed manifold|closed]] [[surface]] that does not admit an [[orientation]].

## Properties

\begin{proposition}
The [[topological complexity]] of the [[Klein bottle]] is $4$ (with convention $TC(*)=0$).
\end{proposition}

([Cohen & Vandembrouq 16, Theorem 1](#CohenVandembroucq16))

## Related concepts

* [[Möbius strip]]

## References

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

Discussion of Klein bottles appearing in [[topological data analysis]]:

* [[Robert Ghrist]], _Barcodes: The Persistent Topology of Data_, Bull. Amer. Math. Soc. 45 (2008), 61-75 ([doi:10.1090/S0273-0979-07-01191-3](https://doi.org/10.1090/S0273-0979-07-01191-3), [pdf](https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf))

On [[topological complexity]] of the Klein bottle:

* {#CohenVandembroucq16} [[Daniel C. Cohen]], [[Lucile Vandembroucq]], _Topological Complexity of the Klein bottle_ (2018), [arXiv:1612.03133v3](https://arxiv.org/abs/1612.03133v3)

[[!redirects Klein bottles]]

[[!redirects Klein's bottle]]
[[!redirects Klein's bottles]]


Klein four-group

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#Contents#
* table of contents
{:toc}

## Idea

The [[direct product group]] of the [[group of order 2]] with itself is known as the _Klein four group_:

$$
  \mathbb{Z}/2 \times \mathbb{Z}/2
  \,.
$$

## Properties

### General
 {#General}

Besides the [[cyclic group of order 4]] $\mathbb{Z}/4$, the Klein group $\mathbb{Z}/2 \times \mathbb{Z}/2$ is the only other [[group]] of [[order]] 4, up to [[isomorphism]]. (This follows, for instance, by the [[fundamental theorem of finitely generated abelian groups]], as in [this example](fundamental+theorem+of+finitely+generated+abelian+groups#TwoFiniteGroupsOfOrderp2)).

In particular the Klein group is _not_ itself a [[cyclic group]], and it is in fact the smallest non-[[trivial group|trivial]] [[group]] which is not a [[cyclic group]].

### ADE-Classification

In the [[ADE-classification]] of [[finite subgroups of SO(3)]], the Klein four-group is the smallest in the D-series, labeled by [[D4]].

[[!include ADE -- table]]

## References

See also

* Wikipedia, _[Klein four-group](https://en.wikipedia.org/wiki/Klein_four-group)_

[[!redirects Klein 4-group]]


Klein geometry

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#Contents#
* table of contents
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## Idea

The notion of  _Klein geometry_ is essentially that of _[[homogeneous space]]_ ([[coset space]]) $G/H$ in the context of [[differential geometry]]. This is named "Klein geometry" due to its central role in [[Felix Klein]]'s [[Erlangen program]], see below at _[History](#History)_.

Klein geometries form the _local models_ for [[Cartan geometries]].

For the generalization of Klein geometry to [[higher category theory]] see [[higher Klein geometry]].

## Definition

A **Klein geometry** is a pair $(G, H)$ where $G$ is a
[[Lie group]] and $H$ is a closed Lie subgroup of $G$ such that the (left) [[coset space]]

$$
   X \coloneqq G/H
$$

is [[connected space|connected]]. $G$ [[action|acts]] [[transitive action|transitively]] on the homogeneous space $X$. We may think of $H\hookrightarrow G$ as the [[stabilizer subgroup]] of a point in $X$.

See there at _[Examples -- Stabilizers of shapes / Klein geometry](stabilizer+group#KleinGeometry)_.

## History
 {#History}

In ([Klein 1872](#Klein1872)) (the "[[Erlangen program]]") is first of all, in section 1, considered the general idea of (what in modern language one would call) the [[action]] of a [[Lie group]] "of transformations" on a [[smooth manifold]]. The group of all such transformations

> by which the geometric properties of configurations in space remain entirely unchanged

is called the _Hauptgruppe_,  _principal group_.

Then in ([Klein 1872, end of section 5](#Klein1872)) it says:

> Suppose in space some group or other, the principal group for instance, be given. Let us then select a single configuration, say a point, or a straight line, or even an ellipsoid, etc., and apply to it all the transformations of the principal group. We thus obtain an infinite manifoldness with a number of dimensions in general equal to the number of arbitrary parameters contained in the group, but reducing in special cases, namely, when the configuration originally selected has the property of being transformed into itself by an infinite number of the transformations of the group. Every manifoldness generated in this way may be called, with reference to the generating group, a body.

This means in modern language, that if $G$ is the given group acting on a given space $X$, and if $S \hookrightarrow X$ is a given subspace (a configuration), then the "body" generated by this is the [[coset]] $G/Stab_G(S)$ of $G$ by the [[stabilizer subgroup]] $Stab_G(X)$ of that configuration. See also there at _[Stabilizer of shapes -- Klein geometry](stabilizer%20group#KleinGeometry)_.

The text goes on to argue that spaces of this form $G/Stab_G(S)$ are of fundamental importance:

> If now we desire to base our investigations upon the group, selecting at the same time certain definite configurations as space-elements, and if we wish to represent uniformly things which are of like characteristics, we must evidently choose our space-elements in such a way that their manifoldness either is itself a body or can be decomposed into bodies.

## Examples

\begin{example}
For $G = E(n)$, the [[Euclidean group]] in $n$-dimensions; $H = O(n)$, the [[orthogonal group]]; then, $X$ is $n$-dimensional [[Cartesian space]].
\end{example}

\begin{example}
Analogously, for $G = Iso(d,1)$ the [[Poincare group]] of $(d+1)$-dimensional [[Minkowski space]], and $H = O(d,1)$ the [[Lorentz group]], then $X = \mathbb{R}^{d+1}$ is [[Minkowski space]] itself.

  Passing to the corresponding [[Cartan geometry]] -- by what physicists call _gauging_ -- yields the [[first order formulation of gravity]].
\end{example}

\begin{example}\label{ExampleCrystallography}
**([[crystallography]])**
\linebreak
The study of [[crystallographic groups]] in [[crystallography]] is much in the spirit of Klein geometry/the [[Erlangen program]] (see for instance [Weyl 1938](Erlangen+program#Weyl38); [Grünbaum & Shephard 2010](Erlangen+program#GruenbaumShephard10); [Engel 1986](crystal#Engel86)).

Concretely, the [[quotient space]]/[[quotient orbifold]] of the space of [[wave vectors]]/[[momenta]] in a [[crystal]] lattice by the (dual) [[crystallographic group]] is the [[Brillouin torus]](-[[orbifold]]), in terms of which much of [[condensed matter theory]] is formulated (see for instance the [[electron energy bands]], the [[valence bundle]], and the [[K-theory classification of topological phases]]).
\end{example}

## Related concepts

[[!include local and global geometry - table]]

* [[Clifford-Klein space form]]

* [[spherical space form]]

* [[(d,d) Klein space]]

## References

### General

The notion of _Klein geometry_ goes back to

* {#Klein1872} [[Felix Klein]], _[[Vergleichende Betrachtungen über neuere geometrische Forschungen]]_ (1872) Mathematische Annalen volume 43, pages 63–100 1893  ([doi:10.1007/BF01446615](https://doi.org/10.1007/BF01446615))

  English translation by M. W. Haskell:

  _A comparative review of recent researches in geometry_, Bull. New York Math. Soc. 2, (1892-1893), 215-249. ([euclid:1183407629](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-2/issue-10/A-comparative-review-of-recent-researches-in-geometry/bams/1183407629.full), LaTeX version retyped by Nitin C. Rughoonauth: [arXiv:0807.3161](https://arxiv.org/abs/0807.3161))

in the context of what came to be known as the [[Erlangen program]].

A review is for instance in

* Vladimir Kisil, _Erlangen Programme at Large: An Overview_ ([arXiv:1106.1686](http://arxiv.org/abs/1106.1686))

### Alternative terminology
 {#ReferencesAlternativeTerminology}

Beware that some authors say "Kleinian geometry" for [[pseudo-Riemannian geometry]] of signature $(--++)$:

* [[John Barrett]], [[Gary W. Gibbons]], M. J. Perry, [[Christopher N. Pope]], P. J. Ruback, *Kleinian Geometry and the $N=2$ Superstring*, Int. J. Mod. Phys. **A9** (1994) 1457-1494  [[doi:10.1142/S0217751X94000650](https://doi.org/10.1142/S0217751X94000650), [arXiv:hep-th/9302073](https://arxiv.org/abs/hep-th/9302073)]

[[!redirects Klein geometries]]

[[!redirects Kleinian geometry]]
[[!redirects Kleinian geometries]]


Klein quadric

#Contents#
* table of contents
{:toc}

## Idea

A [[projective variety]] that is moreover a [[quadric]] given by zeroes of the quadratic form
$$
x_0x_5 + x_1 x_4 + x_2 x_3
$$
where $[x_0:x_1:x_2:x_3:x_4:x_5]$ are [[projective space|homogeneous coordinates]] on $\mathbb{P}^5$. It was studied in ([Klein 1868](#Klein)), together with what is now called the [[Klein correspondence]], by [[Felix Klein]] in his 1868 dissertation under [[Julius Plücker]].

## Properties

The [[conformal compactification]], of _complexified_ [[Minkowski spacetime]] $\mathbb{R}^{3,1}$, is given by the Klein quadric. (eg, [Fioresi-Lledo-Varadarajan 07, section 2](#FioresiLledoVaradarajan07)).

## Related concepts

* [[twistor correspondence]]

## References

* {#Klein} [[Felix Klein]], _Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische Form_, dr. Phil. thesis (1868), published as Math. Ann **23** (1884), 539-578. doi:[10.1007/BF01446603](dx.doi.org/10.1007/BF01446603) ([English translation pdf](http://neo-classical-physics.info/uploads/3/0/6/5/3065888/klein_-_canonical_forms.pdf))

* _The Klein quadric and triality_ ([pdf](http://www.maths.qmul.ac.uk/~pjc/pps/pps8.pdf))

* {#FioresiLledoVaradarajan07} R. Fioresi, M. A. Lledo, [[Veeravalli Varadarajan]], _The Minkowski and conformal superspaces_, J.Math.Phys.48:113505,2007 ([arXiv:math/0609813](https://arxiv.org/abs/math/0609813))


Klein-Gordon equation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Klein-Gordon equation_ is the [[linear differential equation|linear]] [[partial differential equation]] which is the [[equation of motion]] of a [[free field|free]] [[scalar field|scalar]] [[field (physics)|field]] of possibly non-vanishing [[mass]] $m$ on some (possibly [[curved spacetime|curved]]) [[spacetime]] ([[Lorentzian manifold]]): it is the relativistic [[wave equation]] with inhomogeneity the mass $m^2$.

The structure of the Klein-Gordon equation appears also in the [[equations of motion]] of richer [[field (physics)|fields]] than just [[scalar fields]], where now the underlying [[field bundle]] may more generally be some [[vector bundle]]. Therefore the [[fundamental solutions]] of the Klein-Gordon equation, called the _[[propagators]]_ (see [below](#FundamentalSolutions)) pervades all of relativistic [[perturbative quantum field theory]].

## Definition

Given a [[spacetime]] $(X,g)$ (a [[pseudo-Riemannian manifold]]) and a [[real number]] $m \in \mathbb{R}_{\geq 0}$, then the _Klein-Gordon equation_ is the [[differential equation]] on [[smooth functions]] $\phi \colon X \to \mathbb{R}$ given by

$$
  \left(
    \Box_g
      -
    \left(
      \tfrac{m c}{\hbar}
    \right)^2
  \right) \phi
   \;=\; 0
  \,,
$$

where $\Box_g$ denotes the _[[wave operator]]_ on $(X,g)$ (the analog of the [[Laplace operator]] in [[Lorentzian geometry]]) and where $\tfrac{m c }{\hbar}$ is for the purpose of pure [[PDE]] theory just a [[real number]], while regarded as equipped with [[physical units]] it is the inverse _[[Compton wavelength]]_ for [[mass]] $m$.

This is the [[equation of motion]] of the [[free field|free]] [[scalar field]] on $X$, of [[mass]] $m$ and subject to a [[background field]] of [[gravity]] as encoded in the metric $g$.

## Examples

If $(X,g) = \mathbb{R}^{p,1}$ is [[Minkowski spacetime]] equipped with its canonical [[coordinate functions]] $x^0 = c t$ and $\{x^i\}_{i = 1}^p$, then the Klein-Gordon equation reads as follows (using [[Einstein summation convention]])

$$
  \left(
    \eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu}
      -
    \left(
      \tfrac{m c}{\hbar}
    \right)^2
  \right) \phi
   \;=\;
  0
$$

hence

$$
  \left(
    -\tfrac{1}{c^2} \frac{\partial^2}{\partial t^2}
    +
    \underoverset{i = 1}{p}{\sum}\frac{\partial}{\partial x^i} \frac{\partial}{\partial x^i}
      -
    \left(
      \tfrac{m c}{\hbar}
    \right)^2
  \right) \phi
   \;=\;
  0
$$

## Properties

### Formal self-adjointness

+-- {: .num_example #FormallySelfAdjointKleinGordonOperator}
###### Example
**([[Klein-Gordon operator]] is [[formal adjoint differential operator|formally self-adjoint]])**

Let $\Sigma = \mathbb{R}^{p,1}$ be [[Minkowski spacetime]] with [[Minkowski metric]] $\eta$ and let $E \coloneqq \Sigma \times \mathbb{R}$ be the [[trivial line bundle]]. The canonical [[volume form]] $dvol_\Sigma$ induces an [[isomorphism]] $\tilde E^\ast \simeq E$.

Consider then the [[Klein-Gordon operator]]

$$
  (\Box - m^2)
    \;\colon\;
  \Gamma_\Sigma(\Sigma \times \mathbb{R})
     \longrightarrow
  \Gamma_\Sigma(\Sigma \times \mathbb{R}) \otimes \langle dvol_\Sigma\rangle
  \,.
$$

This is its own [[formal adjoint differential operator|formal adjoint]] witnessed by the bilinear differential operator given by

$$
  K(\Phi_1, \Phi_2)
  \;\coloneqq\;
  \left(
    \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2
    -
    \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu}
  \right)
  \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma
  \,.
$$

=--

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

$$
  \begin{aligned}
    d K(\Phi_1, \Phi_2)
    & =
    d
    \left(
      \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2
      -
      \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu}
    \right)
    \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma
    \\
    &=
    \left(
    \left(
      \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2
      +
      \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\mu} \frac{\partial \Phi_2}{\partial x^\nu}
    \right)
    -
    \left(
      \eta^{\mu \nu}
      \frac{\partial \Phi_1}{\partial x^\nu} \frac{\partial \Phi_2}{\partial x^\mu}
      +
      \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu}
    \right)
    \right)
    dvol_\Sigma
    \\
    & =
    \left(
      \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2
      -
      \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu}
    \right)
    dvol_\Sigma
    \\
    & =
    \Box(\Phi_1) \Phi_2 - \Phi_1 \Box (\Phi_2)
  \end{aligned}
$$

=--

### Bicharacteristic flow and propagation of singularities

The [[bicharacteristic strips]] of the Klein-Gordon operator are [[cotangent vectors]] along [[lightlike]] [[geodesics]] ([this example](bicharacteristic+flow#BicharachteristicFlowOfKleinGordonOperator)).

### Fundamental solutions
 {#FundamentalSolutions}

On a [[globally hyperbolic spacetime]] $M$ the Klein-Gordon equation has unique advanced and retarded [[Green functions]], $\Delta_R \in \mathcal{D}'(M\times M)$ and $\Delta_A \in \mathcal{D}'(M\times M)$ respectively.

The [[advanced and retarded Green functions]] are uniquely distinguished by their [[support of a distribution|support]] properties. Namely, $(x,y) \in \operatorname{supp} \Delta_R$ only if $x$ is in the [[causal future]] of $y$, while $(x,y) \in \operatorname{supp} \Delta_A$ only if $x$ is in the [[causal past]] of $y$.

Their difference $\Delta_S = \Delta_R - \Delta_A$ is a bisolution known as the [[causal propagator]], which is the [[Peierls bracket]] which gives the [[Poisson bracket]] on the [[covariant phase space]] of the [[free field|free]] [[scalar field]].  This in turn defines the [[Wick algebra]] of the free scalar field, which yields the [[quantization]] of the free scalar field to a [[quantum field theory]].

Other important Green functions or bisolutions include any (anti-)[[Feynman propagator]] ($\Delta_{\bar{F}}$) $\Delta_F$ and [[Hadamard propagator]]. Unfortunately, it is not possible to identify them by a simple support condition. On Minkowski space, they are identified by the support of their Fourier transform. On curved spacetimes, there are two possibilities. One specifies the asymptotic expansion $\Delta(x,y)$ in a geodesically convex neighborhood of the diagonal $x=y$ to be of a special _Hadamard form_. The other specifies constraints on the [[wavefront set]] $WF(\Delta)$. The possibilities were proven to be equivalent in [Radzikowski 96](#Radzikowski96), which made essential use of the relevant notions of microlocal analysis and of _distinguished parametrices_ introduced in [DuistermaatHörmander 72](#DuistermaatHoermander72).

According to [Radzikowski 96](#Radzikowski96), the constraints on the wavefront sets of important Green functions and bisolutions can be diagrammatically illustrated as follows below. Note that the primed [[wavefront set]] of a distribution on $M\times M$ is defined as $WF'(\Delta) = \{ (x,y;p,-q) \in T^*(M\times M) \mid (x,y;p,q) \in WF(\Delta) \}$. The diagrams illustrate tuples $(x,y; p,q) \in T^*(M\times M)$, where the vertex of a cone corresponds to $y$ and the cone illustrates all points $x$ linked to $y$ by null geodesics; the arrows illustrate the allowed directions of $p$, with $p$ and $q$ linked by parallel transport and both tangent to null geodesic linking $x$ and $y$.

[[!include propagators - table]]

These propagators govern the construction of the [[Wick algebra]] of [[quantum observables]] of the [[free field|free]] [[scalar field]] on the given [[globally hyperbolic spacetime]], as well as the further [[deformation quantization]] to [[interaction|interacting]] [[perturbative quantum field theory]] [[quantum field theory on curved spacetimes|on curved spacetimes]] via [[causal perturbation theory]]. See at _[[locally covariant perturbative quantum field theory]]_ for more on this.

###  Relation to Schrödinger equation

Sometimes the Klein-Gordon equation is thought of as a [[general relativity|relativistic]] refinement of the [[Schrödinger equation]] (as one passes from the non-relativistic to the [[relativistic particle]]). But this requires some care. A priori the Klein-Gordon equation takes as arguments a [[field (physics)|field]] on [[spacetime]], whereas the Schrödinger equation takes as argument a [[wave function]] on [[phase space]].

## Related concepts

* [[wave equation]], [[wave]]

* [[plane wave]]

  [[wave vector]], [[wavelength]], [[frequency]]

* [[Fourier analysis]]

* [[heat equation]]

* [[wave equation]]

* [[Schrödinger equation]]

* [[Dirac equation]]

* [[sine-Gordon equation]]

## References

The Klein-Gordon equation is named after [[Oskar Klein]] and [[Walter Gordon]].

An overview over the KG [[propagators]] on [[Minkowski spacetime]] is given in

* {#Kocic16} [[Mikica Kocic]], _Invariant Commutation and Propagation Functions Invariant Commutation and Propagation Functions_, 2016 ([[KGPropagatorsOnMinkowskiTable.pdf:file]])

The [[Hadamard propagator]] for the Klein-Gordon equation on general [[globally hyperbolic spacetimes]] was found in

* {#Radzikowski96} [[Marek Radzikowski]], _Micro-local approach to the Hadamard condition in quantum field theory on curved space-time_, Commun. Math. Phys. 179 (1996), 529–553 ([Euclid](http://projecteuclid.org/euclid.cmp/1104287114))

The original reference on the relevant notions of microlocal analysis and distinguished parametrices of the Klein-Gordon equation is

* {#DuistermaatHoermander72} [[Johann Duistermaat]], [[Lars Hörmander]], _Fourier integral operators. II_, Acta Mathematica 128 (1972), 183-269 ([doi](http://dx.doi.org/10.1007/bf02392165))

Textbook accounts include

* {#BaerGinouxPfaeffle07} [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], _Wave Equations on Lorentzian Manifolds and Quantization_, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover ([arXiv:0806.1036](https://arxiv.org/abs/0806.1036))

* {#Ginoux08} [[Nicolas Ginoux]], _Linear wave equations_,  Ch. 3 in [[Christian Bär]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Spacetimes: Concepts and Methods_, Lecture Notes in Physics, Vol. 786, Springer, 2009

See also

* Wikipedia, _[Klein-Gordon equation](https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation)_

[[!redirects Klein-Gordon equations]]

[[!redirects Klein-Gordon operator]]
[[!redirects Klein-Gordon operators]]


Kleinian group

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

#Contents#
* table of contents
{:toc}

## Definition

A _Kleinian group_ is a [[discrete group|discrete]] [[subgroup]] of the [[projective general linear group]] [[PSL(2,C)]] over the [[complex numbers]] $PGL(2,\mathbb{C}) \simeq PSL(2,\mathbb{C})$.

## Related concepts

* [[Fuchsian group]]

* [[hyperbolic 3-manifolds]]

## References

* William James Harvey, _Kleinian groups (a survey)_, Séminaire Bourbaki : vol. 1976/77, exposés 489-506,Séminaire Bourbaki no. 19 (1978), Talk no. 491, p. 30-45 ([numdam:SB_1976-1977__19__30_0](http://www.numdam.org/item/SB_1976-1977__19__30_0))

* Gaven J. Martin, _The Geometry and Arithmetic of Kleinian Groups_ ([arXiv:1311.2636](https://arxiv.org/abs/1311.2636))

See also

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

[[!redirects Kleinian groups]]


Kleisli 2-category

#Contents#
* table of contents
{:toc}

## Idea

The notion of _Kleisli 2-category_ over a [[2-monad]] is the generalization of the notion of [[Kleisli category]] over a [[monad]] from [[category theory]] to [[2-category]] theory (or from 2- to [[3-category]] theory, depending on perspective).

## References

Section 2.1.2 of

* Ignacio Lopez Franco, _Autonomous Pseudomonoids_ ([pdf](http://www.dspace.cam.ac.uk/bitstream/1810/219201/1/phdthesis.pdf))

* [[Susan Niefield]], _Exponentiability in homotopy slices of Top and pseudo-slices of Cat_ ([TAC](http://www.tac.mta.ca/tac/volumes/19/1/19-01abs.html))

[[!redirects Kleisli bicategory]]

[[!redirects Kleisli 2-categories]]
[[!redirects Kleisli bicategories]]


Kleisli category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Kleisli category
* table of contents
{: toc}

## Idea

Given a [[monad]] $T$ on some [[category]] $\mathcal{C}$, then its _Kleisli category_ is the [[full subcategory]] of the [[Eilenberg-Moore category]] of $T$, hence the category of [[algebras over a monad|T-algebras]], on those that are *[free](algebra+over+a+monad#FreeAlgebras)* [[algebra for a monad|T-algebras]] (free $T$-[[modules]]).

Explicitly one may describe (Prop. \ref{KleisliEquivalence} below) the _Kleisli category_ of $T$ (Def. \ref{KleisliCategory}) to have as [[objects]] the objects of $\mathcal{C}$, and a morphism $X \to Y$ in the Kleisli category is a morphism in $\mathcal{C}$ of the form $X \to T(Y)$ in $\mathcal{C}$. The monad structure induces a natural [[composition]] of such "$T$-shifted" morphisms.

The Kleisli category is also characterized by the following [[universal property]]:

Since every [[adjunction]] gives rise to a [[monad]] on the [[domain]] of its [[left adjoint]], we might ask if every monad may be construed as arising from an adjunction. This is in fact true, and the [[initial object|initial]] such adjunction in the [[adjoint functor#RelationToMonads|category of adjunctions]] for the given monad has the Kleisli category as the codomain of its left adjoint.

## Definition

Let $\mathbf{T} = (T,\mu,\eta)$ be a [[monad]] in [[Cat]], where $T \colon \mathcal{C} \longrightarrow \mathcal{C}$ is an [[endofunctor]] with

* multiplication $\mu \colon T T \to T$,

* [[unit of a monad|unit]] $\eta \colon Id_C \to T$.

### In terms of free algebras

\begin{definition}\label{FreeTAlgebras}

A __free $\mathbf{T}$-[[algebra|algebra over a monad]]__ (or free $\mathbf{T}$-module) is a $\mathbf{T}$-algebra (module) of the form $(T(M),\mu_M)$, where the [[action]] is the component of multiplication transformation $\mu_M : T(T(M))\to T(M)$.

\end{definition}

\begin{definition}\label{KleisliCategory}

The __Kleisli category__ $C_{\mathbf{T}}$ of the monad $\mathbf{T}$ is the [[full subcategory]] of the [[Eilenberg-Moore category]] $C^{\mathbf{T}}$ on the free $\mathbf{T}$-algebras (Def. \ref{FreeTAlgebras}).

\end{definition}

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

If $U:C^{\mathbf{T}}\to C$ is the [[forgetful functor]] and $F: C\to C^{\mathbf{T}}$ is the [[free functor|free algebra functor]] $F: M\mapsto (T M,\mu_M)$, then the Kleisli category is simply the [[full subcategory]] of $C^{\mathbf{T}}$ containing those objects in the image of $F$.

=--

### In terms of Kleisli morphisms
 {#InTermsOfKleisliMorphisms}

As another way of looking at this, we can keep the same objects as in $C$ but redefine the morphisms. This was the original Kleisli construction:

\begin{definition}\label{KleisliCategory}

The **Kleisli category** $C_{\mathbf{T}}$ of a [[monad]] $T$ on a [[category]] $\mathcal{C}$ has:

1. as [[objects]] the objects of $\mathcal{C}$,

1. as [[morphisms]] $M \to N$ the [[morphisms]] of the form

   \[
     \label{KleisliMorphism}
     M \longrightarrow{\;\;} T(N)
   \]

   in $\mathcal{C}$, called **Kleisli morphisms**;

and

* [[composition]] of $M \xrightarrow{f} T N$ with $N \xrightarrow{ g } T P$ is given by the **Kleisli composition** rule

  \[
    \label{KleisliComposition}
    g \circ_{Kleisli} f
    \;\coloneqq\;
    \mu_P \circ T(g) \circ f
    \;\colon\;
    M
      \overset{f}{\longrightarrow}
    T (N)
      \overset{T (g)}{\longrightarrow}
    T \big(T (P)\big)
      \overset{\mu_P}{\longrightarrow}
    T (P)
    \,;
  \]

* the [[identity morphisms]] on $M$ is the Kleisli morphism which is the [[unit of a monad|$T$-unit]] $M \xrightarrow{ \eta_M } T M$.

\end{definition}

\begin{proposition}\label{KleisliEquivalence}
**([[Kleisli equivalence]])**
\linebreak
The construction which sends a Kleisli morphism $X \xrightarrow{f} T Y$ (eq:KleisliMorphism) to

$$
  T(X)
    \overset{T(f)}{\longrightarrow}
  T^2(Y)
    \overset{\mu_Y}{\longrightarrow}
  T(Y)
$$

constitutes a [[fully faithful functor]]

$$
  \mathcal{C}_{T} \xhookrightarrow{\phantom{--}} \mathcal{C}^{T}
$$

from the $T$-Kleisli category (Def. \ref{KleisliCategory}) to the category of [[algebra over a monad|$T$-algebras]], hence constitutes an [[equivalence of categories]] onto its  [[essential image]] (the free $T$-algebras).

\begin{tikzcd}[sep=0pt]
    \mathcal{C}_{T}
    \ar[
      rr,
      hook
    ]
    &&
    \mathcal{C}^{T}
    \\
    X
    &\mapsto&
    \big(
      T(X)
      ,\,
      \mu_X
    \big)
    \\[+5pt]
    \mathcal{C}_{T}
    (X,\, Y)
    \ar[
      rr,
      <->,
      "{ \sim }"
    ]
    &&
    \mathcal{C}^{T}
    \Big(
      \big(T(X),\mu_X\big)
      ,\,
      \big(T(Y),\mu_{Y}\big)
    \Big)
    \\
    \Big(
      X
      \xrightarrow{
        \hspace{22pt}
        f
        \hspace{22pt}
      }
      \mathcal{E}(Y)
    \Big)
    &\mapsto&
    \Big(
      \mathcal{E}(X)
      \xrightarrow{ \mathcal{E}(f)}
      \mathcal{E}\big(\mathcal{E}(Y)\big)
      \xrightarrow{ \mu_{Y} }
      \mathcal{E}(Y)
    \Big)
    \\
    \Big(
    X
    \xrightarrow{
      \eta^{\mathcal{E}}_{X}
    }
    \mathcal{E}(X)
    \xrightarrow{ \phi }
    \mathcal{E}(Y)
    \Big)
    \ar[rr, phantom, "{\mapsto}"{rotate=180}]
    &&
    \Big(
    \mathcal{E}(X)
    \xrightarrow{
      \hspace{39pt}
      \phi
      \hspace{39pt}
    }
    \mathcal{E}(Y)
    \Big)
\end{tikzcd}

\end{proposition}

(eg. [Borceux (1994), Prop. 4.1.6](#Borceux94))

\begin{proof}
\label{ProofOfKleisliEquivalence}
To see that the functor is [[full functor|full]], hence that $f \mapsto \mu_Y \circ T(f)$ is [[surjective]], oberve that any homomorphism $g \colon T(X) \to T(Y)$ of [[algebra over a monad|algebras]] is the [[image]] of $X \stackrel{\eta_X}{\to} T(X) \stackrel{g}{\to} T(Y)$, as shown by the following [[commuting diagram]]:

\begin{tikzcd}
  T(X)
  \ar[r, "{ T(\eta_X) }"]
  \ar[dr, equals]
  &
  T T(X)
  \ar[r, "{ T(g) }"]
  \ar[d, "{ \mu_X }"]
  &
  T T(Y)
  \ar[d, "{ \mu_Y }"]
  \\
  &
  T(X)
  \ar[r, "{ g }"]
  &
  T(Y)
  \mathrlap{\,.}
\end{tikzcd}

Here the triangle on the left is the [[unit law]] of the monad, while the commutativity of the square is the fact that $G$ is a [[homomorphism]] of [[algebra over a monad|algebras]].

To see that the functor is [[faithful functor|faithful]], hence that $f \mapsto \mu_Y \circ T(f)$ is [[injectivity|injective]], notice that

$$
  \big( \mu_Y \circ T(f) \big) \circ \eta_X
  \;=\;
  f
  \,,
$$

by [[naturality square|naturality]] of the [[unit of a monad|unit]] $\eta_X$ combined with its [[unit law]]:

\begin{tikzcd}
  X
    \ar[r, "{ f }"]
    \ar[d, "{ \eta_X }"]
  &
  T(Y)
    \ar[d, "{ \eta_Y }"]
  \ar[dr, equals]
  \\
  T(X)
  \ar[r, "{ T(f) }"{swap}]
  &
  T T(Y)
  \ar[r, "{ \mu_Y }"{swap}]
  &
  T(Y)
  \mathrlap{\,,}
\end{tikzcd}

whence

$$
  \mu_Y \circ T(f)
  \,=\,
  \mu_Y \circ T(g)
  \;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;
  \mu_Y \circ T(f) \circ \eta_X
  \,=\,
  \mu_Y \circ T(g) \circ \eta_X
  \;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;
  f \,=\, g
  \,.
$$
\end{proof}

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

This Kleisli composition plays an important role in [[computer science]]; for this, see the article at [[monad (in computer science)]].

=--

### Two-sided Kleisli category
 {#TwoSidedKleisliCategory}

\begin{proposition}
\label{ExistenceOfTwoSidedKleisliCategory}
If in addition to the given monad $\mathcal{E}$ there is a [[comonad]] $\mathcal{C}$ on the same category $\mathbf{C}$, equipped with a [[distributivity law]] (see [there](distributive+law#ComonadDistributingOverMonad))

$$
  distr^{\mathcal{C}, \mathcal{E}}
  \;\;\colon\;\;
  \mathcal{C}
  \big(
    \mathcal{E}(D)
  \big)
  \longrightarrow
  \mathcal{E}
  \big(
    \mathcal{C}(D)
  \big)
$$

then there is a two-sided ("double") [[Kleisli category]] whose [[objects]] are those of $\mathbf{C}$, and whose morphisms $D_1 \to D_2$ are morphisms in $\mathbf{C}$ of the form
$$
  prog_{12}
  \;\colon\;
  \mathcal{C}(D_1)
    \longrightarrow
  \mathcal{E}(D_2)
$$

with two-sided Kelisli composition

$$
  prog_{12} \text{>=>} prog_{23}
  \;\;
  \colon
  \;\;
  \mathcal{C}(D_1)
  \longrightarrow
  \mathcal{E}(D_3)
$$

given by the (co-)[[Kleisli extension|bind-operation]] on the factors connected by the distributivity transformation:

\begin{imagefromfile}
    "file_name": "CoMonKleisliComposition-230930b.pdf",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

\end{proposition}
([Brookes & Van Stone 1993 Thm. 2](#BrookesVanStone93))

\begin{proposition}
\label{MonadTransformationsOnTwoSidedKleisli}
In the situation of Prop. \ref{ExistenceOfTwoSidedKleisliCategory}, given in addition:

1. $\mathcal{E}'$ another monad on $\mathbf{C}$

   * also equipped with distributivity $distr^{\mathcal{C}, \mathcal{E}'} \,\colon\, \mathcal{C}\circ \mathcal{E}' \to \mathcal{E}' \circ \mathcal{C}$ over the given comonad $\mathcal{C}$,

1. a [[monad transformer|monad transformation]] $trans^{\mathcal{E} \to \mathcal{E}'} \,\colon\, \mathcal{E} \to \mathcal{E}'$

   * which is compatible with the distributive laws in that that

\begin{tikzcd}
    \mathcal{C}\big(
      \mathcal{E}(-)
    \big)
    \ar[
      dd,
      "{
        \mathrm{distr}
          ^{ \mathcal{C}, \mathcal{E} }
          _{ (-) }
      }"{description}
    ]
    \ar[
      rr,
      "{
        \mathcal{C}\big(
          \mathrm{trans}
            ^{ \mathcal{E} \to \mathcal{E}' }
            _{ (-) }
        \big)
      }"
    ]
    &&
    \mathcal{C}\big(
      \mathcal{E}'(-)
    \big)
    \ar[
      dd,
      "{
        \mathrm{distr}
          ^{ \mathcal{C}, \mathcal{E}' }
          _{ (-) }
      }"{description}
    ]
    \\
    \\
    \mathcal{E}\big(
      \mathcal{C}(-)
    \big)
    \ar[
      rr,
      "{
        \mathrm{trans}
          ^{ \mathcal{E} \to \mathcal{E}' }
          _{ \mathcal{C}(-) }
      }"{swap}
    ]
    &&
    \mathcal{E}'\big(
      \mathcal{C}(-)
    \big)
\end{tikzcd}

then the usual compatibility of the one-sided-Kleisli category under monad transformations (see [here](monad+transformer#eq:RespectForBindOperation)) passes to the two-sided Kleisli category, in that

\[
  \label{TransformationActingOnTwoSidedKleisliComposition}
  \big(
    trans^{ \mathcal{E} \to \mathcal{E}' }_{D_2}
    \circ
    prog_{12}
  \big)
  \;\;
  \text{>=>}
  \;\;
  \big(
    trans^{ \mathcal{E} \to \mathcal{E}' }_{D_3}
    \circ
    prog_{23}
  \big)
  \;\;\;
  =
  \;\;\;
  trans^{ \mathcal{E} \to \mathcal{E}' }_{D_3}
  \circ
  \big(
    prog_{12}
    \;\text{>=>}\;
    prog_{23}
  \big)
  \,.
\]
\end{proposition}
\begin{proof}
Consider the following diagram:

\begin{imagefromfile}
    "file_name": "MonadTransfOnTwoSidedKleisli-290801.jpg",
    "width": 750,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

Here all squares commute by assumption on the monad transformation and hence the entire [[commuting diagram|diagram commutes]]. Now the total top and right composite is the right hand side of (eq:TransformationActingOnTwoSidedKleisliComposition), while the total left and bottom composite is the left hand side of (eq:TransformationActingOnTwoSidedKleisliComposition), thus proving their equality.
\end{proof}

## Properties

### Universal properties
 {#UniversalProperties}

In more general 2-categories the [[universal properties]] of [[Kleisli objects]] are dual to the universal properties of [[Eilenberg-Moore category#Definition|Eilenberg-Moore objects]].

In particular, $C_{\mathbf{T}}$ is [[initial object|initial]] in the [[adjoint functor#RelationToMonads|category of adjunctions]] for $\mathbf{T}$ (whereas $C^{\mathbf{T}}$ is [[terminal objects|terminal]]). For a proof, see _[[Category Theory in Context]]_ Proposition 5.2.12.

## Examples

### General

\begin{example}\label{CallByValueProgramming}
In [[type theory|typed]] [[functional programming]], the Kleisli category is used to model [[call-by-value]] functions with [[side effects]] and [[computation]].
Dually, the [[co-Kleisli category]] of a [[comonad]] may be used to model [[call-by-name]] programming , see [there](Kleisli+category+of+a+comonad#CallByNameProgramming).

Generally, see at _[[monad (in computer science)]]_ for more on this.
\end{example}

### Specific

\begin{example}
**([[matrix multiplication]] as (co-)Kleisli composition)**
\linebreak
See [here](Kleisli+category+of+a+comonad#MatrixmultiplicationAsCoKleisliComposition).
\end{example}

## Related concepts

* [[Kleisli triple]]

* [[co-Kleisli category]]

* [[Kleisli object]]

* [[extension system]]

* [[monad in computer science]]

* [[Kleisli 2-category]]

* [[thunk-force category]]

## References

The original articles:

* {#Kleisli65} [[Heinrich Kleisli]], *Every standard construction is induced by a pair of adjoint functors*, Proc. Amer. Math. Soc. **16**, AMS (1965) 544-546 [[ISSN0002-9939](http://www.ams.org/journals/proc/1965-016-03/S0002-9939-1965-0177024-4/), [jstor:2034693](https://www.jstor.org/stable/2034693)]

* {#Maranda66} [[Jean-Marie Maranda]], *On Fundamental Constructions and Adjoint Functors*, Canadian Mathematical Bulletin **9** 5  (1966) 581-591 [[doi:10.4153/CMB-1966-072-9](https://doi.org/10.4153/CMB-1966-072-9)]

Early accounts (together with the [[Eilenberg-Moore category]]):

* [[Fred Linton]], §7 in: *An outline of functorial semantics*, in *[[Seminar on Triples and Categorical Homology Theory]]*, Lecture Notes in Mathematics **80**, Springer (1969) 7-52 [[doi:10.1007/BFb0083080](https://doi.org/10.1007/BFb0083080)]

* {#MacLane71} [[Saunders MacLane]], §VI.5 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5**  Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Michael Barr]], [[Charles Wells]], §3.2 of: *[[Toposes, Triples, and Theories]]*, Grundlehren der math. Wissenschaften **278**, Springer (1983) [[TAC:12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)]

* Jenö Szigeti, _On limits and colimits in the Kleisli category_, Cahiers de Topologie et Géométrie Différentielle Catégoriques **24** 4 (1983) 381-391 [[numdam:CTGDC_1983__24_4_381_0](http://www.numdam.org/item?id=CTGDC_1983__24_4_381_0)]

The [[equivalence of categories]] between the [[Kleisli category]] over a given [[monad]] with the [[co-Kleisli category]] of an [[adjoint functor|adjoint]] [[comonad]] (if it exists):

* [[Mark Kleiner]],  *Adjoint monads and an isomorphism of the Kleisli categories*, Journal of Algebra **133** 1 (1990) 79-82 [doi:10.1016/0021-8693(90)90069-Z]

The terminology "[[Kleisli triple]]" for a [[monad]] presented as an "[[extension system]]" and relation to [[computation]] with effects (see at *[[monads in computer science]]*):

* {#Moggi91} [[Eugenio Moggi]], *Notions of computation and monads*, Information and Computation, **93** 1 (1991) [doi:10.1016/0890-5401(91)90052-4, [pdf](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf)]

Textbook account making explicit the [[Kleisli equivalence]]:

* {#Borceux94} [[Francis Borceux]], pp. 191 in: *[[Handbook of Categorical Algebra]]*, Vol 2 *Categories and Structures*, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]

Lecture notes:

* [[Thomas Streicher]], pp. 54 in: *Introduction to Category Theory and Categorical Logic* (2003) [[pdf](https://www2.mathematik.tu-darmstadt.de/~streicher/CTCL.pdf), [[Streicher-CategoryTheory.pdf:file]]]

Discussion of cases where the inclusion of the Kleisli category into the [[Eilenberg-Moore category]] is a [[reflective subcategory]]:

* [[Marcelo Fiore]], [[Matias Menni]], _Reflective Kleisli subcategories of the category of Eilenberg-Moore algebras for factorization monads_, Theory and Applications of Categories, Vol. 15, CT2004, No. 2, pp 40-65. ([TAC](http://www.tac.mta.ca/tac/volumes/15/2/15-02abs.html))

{#TwoSidedReferences} Discussion of combined "double" or "two-sided" Kleisli categories, combining the Kleisli category of a [[monad]] with the [[co-Kleisli category]] of a [[comonad]] that [[distributive law|distributes]] over it:

* {#BrookesVanStone93} [[Stephen Brookes]],  [[Kathryn Van Stone]], *Monads and Comonads in Intensional Semantics* (1993) [[dtic:ADA266522](https://apps.dtic.mil/sti/citations/ADA266522), [pdf](https://www.cs.cmu.edu/~brookes/papers/MonadsComonads.pdf), [[BrookesVanStone-CoMonads.pdf:file]]]

and generalization to [[2-monads]]:

* {#Garner08} [[Richard Garner]], pp. 11 of: *Polycategories via pseudo-distributive laws*, Advances in Mathematics **218** 3 (2008) 781-827 [[arXiv:0606735](https://arxiv.org/abs/math/0606735), [doi:10.1016/j.aim.2008.02.001](https://doi.org/10.1016/j.aim.2008.02.001)]

Discussion in [[internal category]] theory:

* Tomasz Brzeziński, Adrian Vazquez-Marquez, _Internal Kleisli categories_, Journal of Pure and Applied Algebra **215** 9 (2011) 213-147 [[arXiv:0911.4048](http://arxiv.org/abs/0911.4048)]

Discussion in a context of [[categorical systems theory]]:

* {#MyersBook} [[David Jaz Myers]], §2.3 of: *Categorical systems theory*, book project [[github](https://github.com/DavidJaz/DynamicalSystemsBook/tree/master/book), [pdf](http://davidjaz.com/Papers/DynamicalBook.pdf)]

Discussion of Kleisli categories in [[type theory]] is in

* [[Alex Simpson]], _Recursive types in Kleisli Categories_ ([pdf](https://www.research.ed.ac.uk/en/publications/recursive-types-in-kleisli-categories))

[[!redirects Kleisli category]]
[[!redirects Kleisli categories]]

[[!redirects Kleisli morphism]]
[[!redirects Kleisli function]]
[[!redirects Kleisli map]]

[[!redirects Kleisli morphisms]]
[[!redirects Kleisli functions]]
[[!redirects Kleisli maps]]

[[!redirects Kleisli composition]]
[[!redirects Kleisli compositions]]

[[!redirects Kleisli composite]]
[[!redirects Kleisli composites]]

[[!redirects Kleisli construction]]
[[!redirects Kleisli constructions]]

[[!redirects Kleisli equivalence]]


Kleisli category of a comonad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[formal dual|Formally dually]] to how a [[monad]] has a [[Kleisli category]] so also a [[comonad]] $P \colon \mathcal{C}\to\mathcal{C}$ has a Kleisli category: its [[objects]] are the objects of $\mathcal{C}$, a [[morphism]] $f \colon c_1 \to c_2$ in the Kleisli category is a morphism

$$
  \tilde f \colon P(c_1) \longrightarrow c_2
$$

in $\mathcal{C}$, and the composition of two such in the Kleisli category is represented by the morphism in $\mathcal{C}$ given by

$$
  \widetilde{f_2 \circ f_1}
  \colon
  P(c_1) \longrightarrow P(P(c_1)) \stackrel{P(\tilde f_1)}{\longrightarrow}
  P(c_2) \stackrel{\tilde f_2}{\longrightarrow}
  c_3
 \,.
$$

Sometimes the term *co-Kleisli category* is encountered, but this is redundant, since there is only one notion of Kleisli category for a comonad.

## Examples
 {#Examples}

### General

\begin{example}
For an [[idempotent comonad]] the Kleisli category is the [[coreflective subcategory]] of its [[modal types]].
\end{example}

\begin{example}\label{CallByNameProgramming}
In [[type theory|typed]] [[functional programming]],
the Kleisli category of a comonad can be used to model [[call-by-name]] programming languages.
Dually, the [[Kleisli category]] of a monad may be used to model [[call-by-value]]-programming, see [there](Kleisli+category#CallByValueProgramming).

Generally, see at *[[monad (in computer science)]]* for more on this.
\end{example}

### Specific

\begin{example}\label{MatrixmultiplicationAsCoKleisliComposition}
**([[matrix multiplication]] as Kleisli composition)**
\linebreak
  For

  * $k$ a [[field]]

  * $B$ be a [[finite set]]

  consider the [[base change]] [[adjoint triple]] of $B$-[[indexed sets]] of $k$-[[vector spaces]] ($k$-[[vector bundles]] over $B$):

\begin{tikzcd}
  k\mathrm{Vec}_B
  \ar[
    r,
    shift left=18pt,
    "{ (p_B)_! }"
  ]
  \ar[
    from=r,
    "{ (p_B)^\ast }"{description}
  ]
  \ar[
    r,
    shift right=18pt,
    "{ (p_B)_\ast }"{swap}
  ]
  \ar[
    r,
    phantom,
    shift left=10pt,
    "{ \scalebox{.65}{$\bot$} }"
  ]
  \ar[
    r,
    phantom,
    shift right=10pt,
    "{ \scalebox{.65}{$\bot$} }"
  ]
  &
  k\mathrm{Vec}
  \mathrlap{\,.}
\end{tikzcd}

By the assumption that $B$ is [[finite set|finite]] and since [[Vect]] has [[biproducts]], the [[colimit]]/[[coproduct]]

$$
  (p_B)_! \mathscr{V}_\bullet
  \;=\;
  \coprod_{b : B} \mathscr{V}_b
$$

and the [[limit]]/[[product]]

$$
  (p_B)_\ast \mathscr{V}_\bullet
  \;=\;
  \prod_{b : B} \mathscr{V}_b
$$

agree (we have an [[ambidextrous adjunction]]) and are both equivalent to the [[direct sum]]:

$$
  (p_B)_! \mathscr{V}_\bullet
    \;\simeq\;
  (p_B)_\ast \mathscr{V}_\bullet
  \;=\;
    \underset{b \colon B}{\bigoplus}
  \mathcal{V}_b
  \,.
$$

Accordingly, the underlying [[endofunctors]] of the induced [[monad]] and [[comonad]] on $k Vec_B$

$$
  \Box_B
  \;\coloneqq\;
  (p_B)^\ast (p_B)_\ast
  \;\;\vdash\;\;
  \lozenge_B
  \;\coloneqq\;
  (p_B)^\ast (p_B)_!
  \;\;\colon\;\;
  k Vec_B
  \longrightarrow
  k Vec_B
$$

agree, to make a [[self-adjoint functor]]

$$
  (p_B)^\ast
  \bigotimes_B
  \;\;\vdash\;\;
  (p_B)^\ast
  \bigotimes_B
  \;\;\;\colon\;\;\;
  k Vec
  \longrightarrow
  k Vec
$$

which carries the [[structure]] both of a [[monad]] and of a [[comonad]].

We claim that

1. the categories of

   1. [free $(p_B)^\ast \oplus_B$-algebras](algebra+over+a+monad#FreeAlgebras)

   1. co-free $(p_B)^\ast \oplus_B$-coalgebras

   are both [[equivalence of categories|equivalent]] to the category [[Vect|$k Vec$]] of $k$-vector spaces,

1. where:

   1. the [[Kleisli category]] of $\bigoplus_B$ exhibits such linear maps as $B$-[[tuples]] of *columns* of $B \times B$-block [[matrices]];

   1. the [[coKleisli category]] of $\bigoplus_B$ exhibits such linear maps as $B$-[[tuples]] of *rows* of $B \times B$-block [[matrices]]

  with (co)[[Kleisli composition]] expressing the operation of block-[[matrix multiplication]].

In particular, on the (co)free algebras of the form $\bigotimes_B (p_B)^\ast k$ this statement holds in the sense of actual $B \times B$-matrices with entries in $k$ (as opposed to block matrices).

Since every [[linear map]] admits *some* $B \times B$-block matrix representation, the second statement implies the first by the Kleisli theorem (see [here](Kleisli+category#InTermsOfKleisliMorphisms)).

Moreover, the second statement follows by direct inspection of the [[Kleisli morphisms]] (and [[formal duality|dually]] for the [[coKleisli morphisms]]):

For this, identify a [[coKleisli morphism]] (where our notation now borrows from [[dependent linear type theory]] in order to express morphisms between $B$-[[indexed sets]] of vector spaces)

$$
  b \colon B
  \;\;
    \vdash
  \;\;
  \Big(
    \underset{b' \colon B}{\bigoplus}
    \mathscr{V}_{b'}
  \Big)
  \xrightarrow{\; F_b \;}
  \mathscr{W}_b
$$

with the $B$-tuple of "block $B \times 1$" matrices $(F_b)_{b \colon B}$ that it evidently encodes.

Observe that the comultiplication on the $\bigotimes_B$-[[comonad]] is given by the [[diagonal map]] into the [[biproduct]], coming from the "randomness unit" (terminology from [modal quantum logic](necessity+and+possibility#ModalQuantumLogic), following [CQTS (2022)](quantum+circuits+via+dependent+linear+types#CQTS22)):

\begin{imagefromfile}
    "file_name": "QCwthLHT-QuantumModalUnits-221021b.jpg",
    "width": "750",
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Using this, one readily sees that the Kleisli construction indeed sends such $B$-tuples $(F_b)_{b \colon B}$ of block-matrix rows to the linear map $F$ which is given by the corresponding full matrix, and that the [[Kleisli composite]] with any other

$$
  b \colon B
  \;\;
    \vdash
  \;\;
  \Big(
    \underset{b' \colon B}{\bigoplus}
    \mathscr{W}_{b'}
  \Big)
  \xrightarrow{\; G_b \;}
  \mathscr{Z}_b
$$

is indeed the row-decomposition of the corresponding [[matrix product]]:

\begin{tikzcd}
  b : B
  \;\;\vdash
  &[-10pt]
  \Big(
    \underset{b' \!\colon\!\! B}{\bigoplus}
    \,
    \mathcal{V}_{b'}
  \Big)
  \ar[
    rr,
    "{
      v
      \;\mapsto\;
      \underset{ b'' \!\colon\!\! B }{\sum}
      \,
      v_{b''}
    }"
  ]
  \ar[
    rrrr,
    bend right=14,
    "{
      v
      \;\mapsto\;
      \underset{ b'' \!\colon\!\! B }{\sum}
      \,
      F_{b''}(v)
      \;=\;
      F(v)
    }"{swap}
  ]
  \ar[
    rrrrrr,
    bend right=25,
    "{
      v
      \;\mapsto\;
      \big(G \circ F(v)\big)_b
    }"{swap}
  ]
  &&
  \Big(
    \underset{b'' \!\colon\!\! B}{\bigoplus}
    \;
    \underset{b' \!\colon\!\! B}{\bigoplus}
    \,
    \mathcal{V}_{b'}
  \Big)
  \ar[
    rr,
    "{
       \underset{b'' \!\colon\!\! B}{\bigoplus}
       \,
       F_{b''}
     }"
  ]
  &&
  \Big(
    \underset{b'' \colon B}{\bigoplus}
    \,
    \mathcal{W}_{b''}
  \Big)
  \ar[
    rr,
    "{ G_b }"
  ]
  &&
  \mathcal{Z}_{b}
  \\
\end{tikzcd}

The [[formal duality|formally dual]] argument (now using the "definiteness counit") shows the analogous statement for Kleisli morphisms and column-decomposition of block matrices:

\begin{tikzcd}
  b \!\colon\! B
  \;\;\vdash
  &
  \mathcal{V}_b
  \ar[
    rr,
    "{
      F_b
    }"
  ]
  \ar[
    rrrrrr,
    bend right=25,
    "{
      v_b
      \;\mapsto\;
      \big(
        G \cdot F_b(v_b)
      \big)
    }"{swap}
  ]
  &&
  \big(
  \underset{
    b'' \!\colon\!\! B
  }{
    \bigoplus
  }
  \mathcal{W}_{b''}
  \big)
  \ar[
    rr,
    "{
      \underset{b'' \!\colon\!\! B}{\bigoplus}
      G_{b''}
    }"
  ]
  \ar[
    rrrr,
    bend right=14,
    "{
      v
      \;\mapsto\;
      \underset{b'' \!\colon\!\! B}{\sum}
      G_{b''}(v)
      \;=\;
      G(v)
    }"{swap}
  ]
  &&
  \big(
  \underset{
    b'' \!\colon\! B
  }{
    \bigoplus
  }
  \;
  \underset{
    b' \!\colon\! B
  }{
    \bigoplus
  }
  \mathcal{Z}_{b'}
  \big)
  \ar[
    rr,
    "{
      \underset{b'' \!\colon\!\! B}{\oplus}
      v_{b''}
      \;\mapsto\;
      \underset{b'' \!\colon\!\! B}{\sum}
      v_{b''}
    }"
  ]
  &&
  \big(
  \underset{
    b' \!\colon\!\! B
  }{
    \bigoplus
  }
  \mathcal{Z}_{b'}
  \big)
\end{tikzcd}

\end{example}

\begin{example}
For $P= Jet$ a [[jet comonad]], then morphisms in its coKleisli category are [[differential operators]] (see [there](jet+comonad#Marvan86)).
\end{example}

## Related concepts

* [[Kleisli category]]
* [[Eilenberg-Moore category]]
* [[monad (in computer science)]]

## References

The [[equivalence of categories]] between the [[Kleisli category]] over a given [[comonad]] with the [[Kleisli category]] of an [[adjoint functor|adjoint]] [[monad]] (if it exists):

* [[Mark Kleiner]],  *Adjoint monads and an isomorphism of the Kleisli categories*, Journal of Algebra
Volume **133** 1 (1990) 79-82 [doi:10.1016/0021-8693(90)90069-Z]

Some introductory material on [[comonads]], [[coalgebras]] and [[co-Kleisli morphisms]]:

* [[Paolo Perrone]], _Notes on Category Theory with examples from basic mathematics_, Chapter 5. ([arXiv](http://arxiv.org/abs/1912.10642))

[[!redirects co-Kleisli category]]
[[!redirects co-Kleisli categories]]
[[!redirects coKleisli category]]
[[!redirects coKleisli categories]]
[[!redirects Kleisli category of a comonad]]
[[!redirects Kleisli category for a comonad]]

[[!redirects co-Kleisli morphism]]
[[!redirects co-Kleisli morphisms]]
[[!redirects coKleisli morphism]]
[[!redirects coKleisli morphisms]]

[[!redirects co-Kleisli map]]
[[!redirects co-Kleisli maps]]
[[!redirects coKleisli map]]
[[!redirects coKleisli maps]]


Kleisli object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given a [[monad]] $t \colon a \to a$ in a [[2-category]] $K$, the **Kleisli object** $a_t$ of $t$ is, if it exists, the [[universal property|universal]] right $t$-[[module over a monad|module]] or $t$-opalgebra.  Equivalently, $a_t$ [[representable functor|represents]] the [[functor]] $RMod(-,t)$ that takes an object $x$ of $K$ to the category of right $t$-modules $a \to x$.

This means that there is a "free" [[1-morphism]] $f_t \colon a \to a_t$ and a [[2-morphism]] $\alpha \colon f_t t \Rightarrow f_t$ that induce an [[isomorphism]] $K(a_t,x) \cong RMod(x,t)$: given a right $t$-module $r \colon a \to x, \rho \colon r t \to r$, there is a unique morphism $h \colon a_t \to x$ whose [[composition|composite]] with $f_t$ (respectively $\alpha$) is equal to $r$ (resp. $\rho$).
\begin{center}
\begin{tikzcd}
a
\arrow[rrr, "r", bend left=60, "" {name=A, xshift=-6.5ex, below} ]
\arrow[rr, "f_t"', bend right=80, "" {name=B, above} ]
\arrow[r, "t"']
&a
\arrow[Rightarrow, "\alpha", shorten >= 0.8ex, shorten <=0.4ex,  to=B]
\arrow[Rightarrow, "\rho", shorten >= 1ex, shorten <=0.4ex,  to=A]
\arrow[rr, "r", bend left ]
\arrow[r, "f_t"']
&a_t
\arrow[r, dashed, "h"']
& x
\end{tikzcd}
\end{center}

## Examples

\begin{example}
The motivating example is that of ordinary [[Kleisli categories]] $C_T$ for monads in the [[2-category]] [[Cat]], hence for ordinary monadic [[endofunctors]] $T$ on [[categories]] $C$.
\end{example}

\begin{example}
In a ([[locally ordered]]) [[bicategory of relations]], the Kleisli object of a monad $t$ is part of a [[splitting]] of $t$ as an [[idempotent]].
\end{example}

\begin{example}
For a monad $T \colon A ⇸ A$ in the bicategory [[Prof]] of [[profunctors]], its Kleisli object consists of a category $A_T$ equipped with a [[bijective-on-objects functor]] $A\to A_T$.  The category $A_T$ has the same objects as $A$, with [[hom-sets]] $A_T(a,b) = T(a,b)$.  Identities and composition are given by the [[monad unit]] and multiplication of $T$.

  Every functor $B \to A$ yields a monad $A(f,f)$ in $Prof$, whose Kleisli object is part of the (bijective on objects, fully-faithful) factorization $B \to A_{A(f,f)} \to A$ of $f$.

  Because of this, we can identify a monad on $A$ in $Prof$ with a bijective-on-objects functor $A \to B$.
\end{example}

## Properties

\begin{remark}
By 2-category theoretic [[formal duality]] (see at *[[opposite 2-category]]*):

1. Kleisli objects in a 2-category $K$ are equivalently [[Eilenberg-Moore objects]] in  $K^{op}$.

1. Kleisli objects for monads in $K^{co}$ are equivalently Kleisli objects for [[comonads]] in $K$.

\end{remark}

\begin{remark}
Kleisli objects for a monad $t$ in $K$ are equivalently a particular sort of [[weighted limit|weighted]] [[2-colimits]], namely the [[lax colimits]] of the [[lax functor]] $\ast \to K$ corresponding to $t$.
\end{remark}

## Related concepts

* [[Eilenberg-Moore object]]

## References

An original article is

* {#Street72} [[Ross Street]], *The formal theory of monads*, Journal of Pure and Applied Algebra **2** 2 (1972) 149-168 [doi:10.1016/0022-4049(72)90019-9]

Generalization from [[monads]] to more general [[categories enriched in a bicategory]] (where "Kleisli objects" are then called "collages") is in

* {#GarnerShulman13} [[Richard Garner]], [[Michael Shulman]], *Enriched categories as a free cocompletion*,
Advances in Mathematics **289** (2016) 1-94
[[arXiv:1301.3191](http://arxiv.org/abs/1301.3191), [doi:10.1016/j.aim.2015.11.012](https://doi.org/10.1016/j.aim.2015.11.012)]

[[!redirects Kleisli objects]]


Kleisli triple

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#### Categorical algebra
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#Contents#
* table of contents
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## Idea

*Kleisli triple* is the term chosen in [Moggi (1991), Def. 1.2](#Moggi91) for what was previously called an *algebraic theory in extension form* in [Manes (1976), Ex. 3.12 (p. 32)](#Manes76) and what later authors refer to as *[[extension systems]]*.

All of these terms refer to the equivalent [[axiom|axiomatization]] of [[monads]] (old terminology: "triples") via the ingredients of their [[Kleisli categories]] under the [Kleisli equivalence](Kleisli+category#KleisliEquivalence).

This is the natural and common incarnation of [[monads in computer science]]; see there for more.

## Related concepts

* [[extension system]]

## References

* {#Moggi89} [[Eugenio Moggi]], Def. 1.2 in: *Computational lambda-calculus and monads*, in: *Proceedings of the Fourth Annual Symposium on Logic in Computer Science* (1989) 14-23 [[doi:10.1109/LICS.1989.39155](https://doi.org/10.1109/LICS.1989.39155)]

* {#Moggi91} [[Eugenio Moggi]], Def. 1.2 in: *Notions of computation and monads*, Information and Computation, **93** 1 (1991) [doi:10.1016/0890-5401(91)90052-4, [pdf](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf)]

following

* {#Manes76} [[Ernest G. Manes]], Sec 3, Ex. 12 (p. 32) of: *Algebraic Theories*, Springer (1976) [[doi:10.1007/978-1-4612-9860-1](https://doi.org/10.1007/978-1-4612-9860-1)]

Lecture notes:

* [[Thomas Streicher]], p. 55 in: *Introduction to Category Theory and Categorical Logic* (2003) [[pdf](https://www2.mathematik.tu-darmstadt.de/~streicher/CTCL.pdf), [[Streicher-CategoryTheory.pdf:file]]]

[[!redirects Kleisli triples]]


KLT relations

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### Context
#### Quantum Field Theory
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#### Duality in string theory
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[[!include duality in string theory -- contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]] the _KLT relations_ ([Kawai-Lewellen-Tye 86](#KawaiLewellenTye86)) express [[tree level]] [[scattering amplitudes]] in ([[supergravity|super]]-)[[gravity]] equivalently as squares of scattering amplitudes in ([[super Yang-Mills theory|super]]-)[[Yang-Mills theory]].

These relations are manifest in [[perturbative string theory]]-[[UV-completions]] of [[Einstein-Yang-Mills theory]], where they were originally found and from which the QFT relations have been deduced: in string theory the [[graviton]] is an excitation of the [[closed string]] and the [[gluon]] of the [[open string]]. But [[cylinder]]-shaped [[worldvolume]] may be read in two different ways: either as a [[closed string]] [[propagator]] or as an [[open string]] [[vacuum diagram]]. This [[open/closed string duality]] of [[string scattering amplitudes]] yields the KLT relations.

Moreover, a color-kinematics duality suggest that these relations extend from [[tree level]] to all [[loop order]].

Its [[classical field theory]] counterpart is named [[classical double copy]].

## Related entries

* [[magic pyramid]]

* [[classical double copy]]

* [[string theory results applied elsewhere]]

* [[duality in physics]]

* [[effective QFT]] incarnations of [[open/closed string duality]],

  relating ([[supergravity|super]]-)[[gravity]] to ([[super Yang-Mills theory|super]]-)[[Yang-Mills theory]]:

  * [[KLT relations]]

  * [[BCJ relations]]

  * [[classical double copy]]

## References

### KLT Relations

The original article is

* {#KawaiLewellenTye86} [[Hikaru Kawai]], D. C. Lewellen, S. H. H. Tye, _A Relation Between Tree Amplitudes of Closed and Open Strings_, Nucl. Phys. B269 (1986) 1-23 (doi:10.1016/0550-3213(86)90362-7)

See also

* [[Zvi Bern]], _Perturbative Quantum Gravity and its Relation to Gauge Theory_, LivingRev.Rel.5:5,2002 ([arXiv:gr-qc/0206071](http://arxiv.org/abs/gr-qc/0206071))

* N. E. J. Bjerrum-Bohr, K. Risager, _String theory and the KLT-relations between gravity and gauge theory including external matter_ ([arXiv:hep-th/0407085](http://arxiv.org/abs/hep-th/0407085))

* [[Zvi Bern]], _The S-matrix reloaded: Twistors, Unitarity, Gauge theories and Gravity_, 2005 ([pdf](http://online.kitp.ucsb.edu/online/strings05/bern/))

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Lance Dixon]], [[Henrik Johansson]], [[Radu Roiban]], _Amplitudes and Ultraviolet Behavior of $N=8$ Supergravity_, Fortschr. Phys. **59** (2011) 7-8 [[arXiv:1103.1848](http://arxiv.org/abs/1103.1848), [doi:10.1002/prop.201100037](https://doi.org/10.1002/prop.201100037)]

  > (in [[D=4 N=8 supergravity]])

* Bo Feng, Song He, Rijun Huang, Yin Jia, _Note on New KLT relations_ ([arXiv:1008.1626](http://arxiv.org/abs/1008.1626))

* Dhritiman Nandan, [[Jan Plefka]], Oliver Schlotterer, Congkao Wen, _Einstein-Yang-Mills from pure Yang-Mills amplitudes_ ([arXiv:1607.05701](http://arxiv.org/abs/1607.05701))

* Luiz Antonio Barreiro, Ricardo Medina, _The origin of the KLT relations and nonlinear relations for Yang-Mills amplitudes_ ([arxiv:1910.13519](https://arxiv.org/abs/1910.13519))

* Pierre Vanhove, Federico Zerbini, _Building blocks of closed and open string amplitudes_ ([arXiv:2007.08981](https://arxiv.org/abs/2007.08981))

As an [[isomorphism]] of [[Lie algebras]]:

* Hadleigh Frost, *The Algebraic Structure of the KLT Relations for Gauge and Gravity Tree Amplitudes*, SIGMA **17** 101 (2021) ([arXiv:2111.07257](https://arxiv.org/abs/2111.07257))

### Gravity amplitudes as double copy Yang-Mills amplitudes

The extension of the KLT relation beyond [[tree level]] to "quantum gravity is Yang-Mills squared" ("double copy" approach) originates with

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Henrik Johansson]], _Perturbative Quantum Gravity as a Double Copy of Gauge Theory_, Phys. Rev .Lett. 105:061602,2010 ([arXiv:1004.0476](https://arxiv.org/abs/1004.0476))

and is further discussed in

* David C. Dunbar, Paul S. Norridge, _Calculation of Graviton Scattering Amplitudes using String-Based Methods_, Nucl.Phys. B433 (1995) 181-208 ([arXiv:hep-th/9408014](https://arxiv.org/abs/hep-th/9408014))

* [[Zvi Bern]], D.C. Dunbar, T. Shimada, _String-Based Methods in Perturbative Gravity_, Phys.Lett.B312:277-284, 1993 ([arXiv:hep-th/9307001](https://arxiv.org/abs/hep-th/9307001))

* [[Alexandros Anastasiou]], [[Leron Borsten]], [[Mike Duff]], [[Alessio Marrani]], [[Silvia Nagy]], M. Zoccali, _Are all supergravity theories Yang-Mills squared?_ ([arXiv:1707.03234](https://arxiv.org/abs/1707.03234))

* [[Alexandros Anastasiou]], [[Leron Borsten]], [[Mike J. Duff]], M. J. Hughes, A. Marrani, [[Silvia Nagy]], M. Zoccali, _Twin Supergravities from Yang-Mills Squared_, Phys. Rev. D 96, 026013 (2017) ([arXiv:1610.07192](https://arxiv.org/abs/1610.07192))

* {#ABDNZ18} A. Anastasiou, L. Borsten, M. J. Duff, S. Nagy, M. Zoccali, _BRST squared_ ([arXiv:1807.02486](https://arxiv.org/abs/1807.02486))

* L. Borsten, _Gravity as the square of gauge theory: a review_, ([doi:10.1007/s40766-020-00003-6](https://doi.org/10.1007/s40766-020-00003-6))

Discussion in terms of [[superstring scattering amplitudes]] is in

* [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude I: Pure spinors and superfield kinematics_ ([arXiv:1812.10969](https://arxiv.org/abs/1812.10969))

* [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude II: Worldsheet functions and their duality to kinematics_ ([arXiv:1812.10970](https://arxiv.org/abs/1812.10970))

* [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure_ ([arXiv:1812.10971](https://arxiv.org/abs/1812.10971))

Application to computation of (classical) [[gravitational wave]]-signatures from [[relativistic binary]]-mergers for used at [[LIGO]]:

* {#BCRSSZ19} [[Zvi Bern]], Clifford Cheung, [[Radu Roiban]], Chia-Hsien Shen, Mikhail P. Solon, Mao Zeng, _Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order_, Phys. Rev. Lett. 122, 201603 (2019) ([arXiv:1901.04424](https://arxiv.org/abs/1901.04424))

Review:

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Marco Chiodaroli]], [[Henrik Johansson]], [[Radu Roiban]], *Supergravity amplitudes, the double copy and ultraviolet behavior*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2304.07392](https://arxiv.org/abs/2304.07392)]

[[!redirects KLT relation]]

[[!redirects Yang-Mills squared]]

[[!redirects double copy]]


KMS state

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

## Idea

A _KMS state_ (named after [Kubo 57](#Kubo57), [Martin-Schwinger 59](#MartinSchwinger59)) is a [[vacuum state]] for a [[relativistic quantum field theory]] describing [[thermal equilibrium]] at some [[positive number|positive]] [[temperature]] $T$.

The final formulation of the [[KMS condition]], now widely accepted, is due to [Haag-Hugenholtz-Winnink 67](#HaagHugenholtzWinnink67).

Under [[Wick rotation]] (if applicable) a relativistic field theory on [[Minkowski spacetime]] $\mathbb{R}^{d,1}$ in a KMS state at [[temperature]] $T$ is identified with a [[Euclidean field theory]] on the [[product space]] $\mathbb{R}^d \times S^1_{\beta}$ with a "compact/periodic Euclidean time" [[circle]] $S^1_\beta$ of [[length]] $\beta = 1/T$, this is essentially the _[[KMS condition]]_ (see [Fulling-Ruijsenaars 87](#FullingRuijsenaars87)). This relation is the basis of _[[thermal quantum field theory]]_, see there for more.

## Related concepts

[[!include states and observables -- content]]

## References

The formulation of the KMS condition is due to

* {#Kubo57} R. Kubo _Statistical-Mechanical Theory of Irreversible Processes  I. General Theory and Simple Applications to Magnetic and Conduction Problems_, Journal of the Physical Society of Japan 12, 570-586 1957

* {#MartinSchwinger59} Paul C. Martin, [[Julian Schwinger]], _Theory of Many-Particle Systems.  I_, Physical Review 115, 1342-1373 (1959)

and found its final, now generally accepted, form in

* {#HaagHugenholtzWinnink67} [[Rudolf Haag]], N. M. Hugenholtz, M. Winnink, _On the equilibrium states in quantum statistical mechanics_, Comm. Math. Phys. Volume 5, Number 3 (1967), 215-236 ([euclid:1103840050](https://projecteuclid.org/euclid.cmp/1103840050))

A good quick survey, putting the KMS states into context with [[Wick rotation]] and [[thermal quantum field theory]], is

* {#FullingRuijsenaars87} S.A. Fulling, S.N.M. Ruijsenaars, _Temperature, periodicity and horizons_, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 ([pdf](https://www1.maths.leeds.ac.uk/~siru/papers/p26.pdf), doi:10.1016/0370-1573(87)90136-0)

See also

* Wikipedia _[KMS state](http://en.wikipedia.org/wiki/KMS_state)_, [Tomita-Takesaki theory](http://en.wikipedia.org/wiki/Tomita%E2%80%93Takesaki_theory)

* Martin Bordemann, Hartmann Römer, [[Stefan Waldmann]], _KMS states and star product quantization_, Reports on Mathematical Physics __44__, n. 1-2 (1999) 45-52, doi

* W. Pusz, [[S. L. Woronowicz]], _Passive states and KMS states for general quantum systems_, Commun. Math. Phys. __58__ (1978), no. 3, 273–290, [MR471796](http://www.ams.org/mathscinet-getitem?mr=471796), [euclid](http://projecteuclid.org/euclid.cmp/1103901491)

* [[S. L. Woronowicz]], _On the existence of KMS states_, Letters on Math. Phys. __10__, no 1, 29--31 (1985) [MR86m:46066](http://www.ams.org/mathscinet-getitem?mr=796996), [euclid](http://projecteuclid.org/euclid.cmp/1103901491), [doi](http://dx.doi.org/10.1007/BF00704583)

* [[Alain Connes]], [[Matilde Marcolli]], _Noncommutative geometry, quantum fields and motives_, Amer. Math. Soc. Coll. Publ. __55__, 2008. xxii+785 pp. [gBooks](http://books.google.com/books?id=a6PdFik-a9QC), [draft pdf](http://www.alainconnes.org/docs/bookwebfinal.pdf)

* Matthew Eric Bassett, math blog: [KMS states and symmetries](http://mebassett.blogspot.com/2010/08/im-still-looking-at-quantum-statistical.html), [Class Field Theory and the Bost-Connes System](http://mebassett.blogspot.com/2011/01/class-field-theory-bost-connes-system.html)

* Neverendingbooks blog [KMS, Gibbs and zeta function](http://www.neverendingbooks.org/index.php/kms-gibbs-zeta-function.html)
[[!redirects Kubo-Martin-Schwinger state]]

[[!redirects KMS states]]

[[!redirects thermal vacuum]]
[[!redirects thermal vacua]]


Knizhnik-Zamolodchikov equation

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> under construction

#Contents#
* table of contents
{:toc}

## Idea

### General

Given a suitable [[Lie algebra]] $\mathfrak{g}$ a __Knizhnik-Zamolodchikov equation__ is the [[equation]] expressing [[flat connection|flatness]] of certain class of [[vector bundles]] [[connection on a bundle|with connection]] on the [[configuration space of points]] of $N$ distinct points in the [[plane]] $\mathbb{C}$. It appeared in the study of [[Wess-Zumino-Novikov-Witten model]] (WZNW model) of [[2d CFT]] in ([Knizhnik-Zamolodchikov 84](#KnizhnikZamolodchikov84)).

The Knizhnik-Zamolodchikov equation involves what is called the __Knizhnik-Zamolodchikov connection__ and it is related to [[monodromy]] representations of Artin's [[braid group]].

In the standard variant, the basic data involves a given complex [[simple Lie algebra]] $\mathfrak{g}$ with a fixed bilinear  [[invariant polynomial]] $(-,-)$ (the _[[Killing form]]_) and $N$ (not necessarily [[finite-dimensional vector space|finite-dimensional]]) [[representations]] $V_1,\ldots, V_n$ of $\mathfrak{g}$. Let $V \,=\, V_1 \otimes \ldots \otimes V_N$.

(...)

### From geometric quantization of Chern-Simons theory
 {#FromGeometicQuantization}

The existence of the Knizhnik-Zamolodchikov connection can naturally be understood from the [[holographic principle|holographic]] [[quantization]] of the [[WZW model]] on the Lie group $G$ by [[geometric quantization]] of $G$-[[Chern-Simons theory]]:

As discussed there, for a 2-dimensional [[manifold]] $\Sigma$, a choice of [[polarization]] of the [[phase space]] of 3d [[Chern-Simons theory]] on $\Sigma$ is naturally induced by a choice $J$ of [[conformal structure]] on $\Sigma$. Once such a choice is made, the resulting [[space of quantum states]] $\mathcal{H}_\Sigma^{(J)}$ of the Chern-Simons theory over $\Sigma$ is naturally identified with the space of [[conformal blocks]] of the [[WZW model]] [[2d CFT]] on the [[Riemann surface]] $(\Sigma, J)$.

But since, from the point of view of the 3d Chern-Simons theory, the [[polarization]] $J$ is an arbitrary choice, the [[space of quantum states]] $\mathcal{H}_\Sigma^{(J)}$ should not depend on this choice, up to specified [[equivalence]]. Formally this means that as $J$ varies (over the [[moduli space of conformal structures]] on $\Sigma$) the $\mathcal{H}_{\Sigma}^{(J)}$ should form a [[vector bundle]] on this [[moduli space of conformal structures]] which is equipped with a [[flat connection]] whose [[parallel transport]] hence provides equivalences between between the [[fibers]] $\mathcal{H}_{\Sigma}^{(J)}$ of this vector bundle.

This flat connection is the Knizhnik-Zamolodchikov connection. This was maybe first realized and explained in ([Witten 89](#Witten89)).

## Definition

[[!include Knizhnik-Zamolodchikov-Kontsevich construction -- definition]]

## Related entries

* [[Kohno-Drinfeld theorem]]

* [[Hitchin connection]]

* [[Pochhammer loop]]

## References
 {#References}

### General

The original articles:

* {#KnizhnikZamolodchikov84} [[Vadim Knizhnik]], [[Alexander Zamolodchikov]], _Current algebra and Wess–Zumino model in two-dimensions_, Nucl. Phys. __B247__, 83–103 (1984) doi, [MR87h:81129](http://www.ams.org/mathscinet-getitem?mr=87h:81129)

* [[Alexander Belavin]], [[Alexander Polyakov]], [[Alexander Zamolodchikov]], _Infinite conformal symmetry in two-dimensional quantum field theory_ (1984) Nucl. Phys. B 241 (2): 333–80.

* [[Daniel Friedan]], [[Stephen Shenker]], *The analytic geometry of two-dimensional conformal field theory*, Nuclear Physics B281 (1987) ([pdf](http://www.physics.rutgers.edu/~friedan/papers/Nucl_Phys_B281_509_1987.pdf))

Textbook account, specifically for the [[su(2)|$\mathfrak{su}(2)$]]/[[sl(2)|$\mathfrak{sl}(2,\mathbb{Z})$]]-[[WZW model]]:

* [[Philippe Di Francesco]], Pierre Mathieu, David Sénéchal: §15.3.2, §15.4 in: *Conformal field theory*, Springer (1997) [[doi:10.1007/978-1-4612-2256-9](https://doi.org/10.1007/978-1-4612-2256-9)]

Relation to [[braid representations]]:

* {#TodorovHadjiivanov01} [[Ivan Todorov]], [[Ludmil Hadjiivanov]], *Monodromy Representations of the Braid Group*, Phys. Atom. Nucl. **64** (2001) 2059-2068; Yad.Fiz. **64** (2001) 2149-2158 [[arXiv:hep-th/0012099](https://arxiv.org/abs/hep-th/0012099), [doi:10.1134/1.1432899](https://doi.org/10.1134/1.1432899)]

* [[Ivan Marin]], *Sur les représentations de Krammer génériques*, Annales de l'Institut Fourier, **57** 6 (2007) 1883-1925 [[numdam:AIF_2007__57_6_1883_0](http://www.numdam.org/item/AIF_2007__57_6_1883_0)]

and understood in terms of [[anyon statistics]]:

* {#GuHaghighatLiu21} Xia Gu, [[Babak Haghighat]], Yihua Liu, *Ising- and Fibonacci-Anyons from KZ-equations*, J. High Energ. Phys. **2022** 15 (2022) [[arXiv:2112.07195](https://arxiv.org/abs/2112.07195), doi:10.1007/JHEP09(2022)015]

The generalization to higher [[genus of a surface|genus]] [[surfaces]]:

* D. Bernard, _On the Wess-Zumino-Witten models on the torus_, Nucl. Phys. B 303 77-93 (1988)

* D. Bernard, _On the Wess-Zumino-Witten models on Riemann surfaces, Nucl. Phys. B 309 145-174 (1988)

The interpretation of this connection in terms of the [[geometric quantization]] of [[Chern-Simons theory]]/[[Wess-Zumino-Witten model]]:

* {#Witten89} [[Edward Witten]], p. 20 of: *Quantum Field Theory and the Jones Polynomial*, Commun. Math. Phys. **121** 3 (1989) 351–399. $[$[euclid.cmp/1104178138](http://projecteuclid.org/euclid.cmp/1104178138), MR0990772$]$

Review:

* [[Ivan Cherednik]], *Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras*, Mathematical Society of Japan Memoirs **1998** (1998) 1-96 $[$[doi:10.2969/msjmemoirs/00101C010](https://doi.org/10.2969/msjmemoirs/00101C010)$]$

  > (in relation to [[Hecke algebras]])

* [[Toshitake Kohno]], §1.5 and §2.1 in: *Conformal field theory and topology*, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs __210__. Iwanami Series in Modern Mathematics. Amer. Math. Soc. 2002 $[$[AMS:mmono-210](https://bookstore.ams.org/mmono-210)$]$

  > (relating to [[conformal blocks]] and [[braid representations]])

* [[Ralph Blumenhagen]], [[Erik Plauschinn]], §3.5 of: *Introduction to Conformal Field Theory -- With Applications to String Theory*, Lecture Notes in Physics **779**, Springer (2009) [[doi:10.1007/978-3-642-00450-6](https://doi.org/10.1007/978-3-642-00450-6)]

  > (in view of [[2d CFT]] and [[string theory]])

A quick review of the Knizhnik-Zamolodchikov equation in the context of an introduction to [[WZW model]] [[CFT]] is in

* [[Krzysztof Gawędzki]], section 5.6 of: _Conformal field theory: a case study_ ([arXiv:hep-th/9904145](http://arxiv.org/abs/hep-th/9904145))

A review of the definition of the Knizhnik-Zamolodchikov connection on the moduli space of [[genus of a surface|genus]]=0 surfaces with $n$ marked points:

* Shu Oi, Kimio Ueno, section 2 of: _Connection Problem of Knizhnik-Zamolodchikov Equation on Moduli Space $\mathcal{M}_{0,5}$_ ([arXiv:1109.0715](http://arxiv.org/abs/1109.0715))

In relation to [[hypergeometric functions]] and [[quantum groups]] (for more see the references [below](#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces)):

* [[Alexander Varchenko]], _Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups_, Adv. Ser. in Math. Phys. __21__, World Sci. Publ. 1995. x+371 pp. ([doi:10.1142/2467](https://doi.org/10.1142/2467))

* V. Tarasov, [[Alexander Varchenko]], _Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups_, Astérisque __246__ (1997), vi+135 pp. ([arXiv:q-alg/9703044](https://arxiv.org/abs/q-alg/9703044), [numdam:AST_1997__246__R1_0](http://www.numdam.org/item/AST_1997__246__R1_0))

On KZ-equations controlling [[codimension]]$=2$ [[defect QFT|defects]] in [[D=4 super Yang-Mills theory]]:

* [[Nikita Nekrasov]], *BPS/CFT correspondence V: BPZ and KZ equations from $q q$-characters* [[arXiv:1711.11582](https://arxiv.org/abs/1711.11582)]

* [[Nikita Nekrasov]], Alexander Tsymbaliuk, *Surface defects in gauge theory and KZ equation*, Letters in Mathematical Physics **112** 28 (2022) [[arXiv:2103.12611](https://arxiv.org/abs/2103.12611), [doi:10.1007/s11005-022-01511-8](https://doi.org/10.1007/s11005-022-01511-8)]

* Saebyeok Jeong, Norton Lee, [[Nikita Nekrasov]], *Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations*,  J. High Energ. Phys. **2021** 120 (2021) [[arXiv:2103.17186](https://arxiv.org/abs/2103.17186), doi:10.1007/JHEP10(2021)120]

See also

* Wikipedia, *[Knizhnik-Zamolodchikov equations](http://en.wikipedia.org/wiki/Knizhnik–Zamolodchikov_equations)*

* I. B. Frenkel, N. Yu. Reshetikihin, _Quantum affine algebras and holonomic diference equations_, Comm. Math. Phys. __146__ (1992), 1-60, [MR94c:17024](http://www.ams.org/mathscinet-getitem?mr=94c:17024)

* [[Valerio Toledano-Laredo]], _Flat connections and quantum groups_, Acta Appl. Math. 73 (2002), 155-173, [math.QA/0205185](http://arxiv.org/abs/math.QA/0205185)

* [[Toshitake Kohno]], _Conformal field theory and topology_, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs __210__. Iwanami Series in Modern Mathematics. Amer. Math. Soc. 2002 $[$[AMS:mmono-210](https://bookstore.ams.org/mmono-210)$]$

* [[P. Etingof]], N. Geer, _Monodromy of trigonometric KZ equations_, [math.QA/0611003](http://arxiv.org/abs/math.QA/0611003)

* [[Valerio Toledano-Laredo]], _A Kohno-Drinfeld theorem for quantum Weyl groups_, [math.QA/0009181](http://arxiv.org/abs/math.QA/0009181)

* A. Tsuchiya, Y. Kanie, _Vertex operators in conformal field theory on $\mathbf{P}^1$ and monodromy representations of braid group_, Adv. Stud. Pure Math. __16__, pp. 297–372 (1988); Erratum in vol. 19, 675–682

* C. Kassel, _Quantum groups_, Grad. Texts in Math. __155__, Springer 1995

* V. Chari, , A. Pressley, _A guide to quantum groups_, Camb. Univ. Press 1994В.

* А. Голубева, В. П. Лексин, _Алгебраическая характеризация монодромии обобщенных уравнений Книжника–Замолодчикова типа $B_n$_, Монодромия в задачах алгебраической геометрии и дифференциальных уравнений, Сборник статей, Тр. МИАН, 238, Наука, М., 2002, 124–143, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=tm&paperid=349&what=fullt&option_lang=rus); V. A. Golubeva, V. P. Leksin, "Algebraic Characterization of the Monodromy of Generalized Knizhnik–Zamolodchikov Equations of Bn Type", Proc. Steklov Inst. Math., 238 (2002), 115–133

* V. A. Golubeva, V. P. Leksin, _Rigidity theorems for multiparametric deformations of algebraic structures, associated with the Knizhnik-Zamolodchikov equations_, Journal of Dynamical and Control Systems, 13:2 (2007), 161–171, [MR2317452](http://www.ams.org/mathscinet-getitem?mr=2317452)

* V. A. Golubeva, _Integrability conditions for two–parameter Knizhnik–Zamolodchikov equations of type $B_n$ in the tensor and spinor cases_, Doklady Mathematics, 79:2 (2009), 147–149

* V. G. Drinfelʹd, _Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations_, Problems of modern quantum field theory (Alushta, 1989), 1–13, Res. Rep. Phys., Springer 1989.

* R. Rimányi, V. Tarasov, A. Varchenko, P. Zinn-Justin, _Extended Joseph polynomials, quantized conformal blocks, and a $q$-Selberg type integral_, [arxiv/1110.2187](http://arxiv.org/abs/1110.2187)

* E. Mukhin, V. Tarasov, A. Varchenko, _KZ characteristic variety as the zero set of classical Calogero-Moser Hamiltonians_, [arxiv/1201.3990](http://arxiv.org/abs/1201.3990)

On the [[flat vector bundles]] underlying the KZ-equation:

* [[Ivan Marin]], *Knizhnik-Zamolodchikov bundles are topologically trivial* ([arXiv:0809.3590](https://arxiv.org/abs/0809.3590))

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