Karoubi envelope

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### Context
#### Idempotents
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[[!include idempotents - contents]]
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#### Category theory
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[[!include category theory - contents]]
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# Contents
* table of contents
{: toc}

## Idea ##

The _Karoubi envelope_ or _idempotent completion_ of a [[category]] is the [[universal construction|universal]] enlargement of the category with the property that every [[idempotent]] is a [[split idempotent]].  This is the [[Set]]-enriched version of the more general notion of _[[Cauchy completion]]_ of an [[enriched category]].

A category in which all idempotents split is called *Karoubi complete* or *Cauchy complete* or *idempotent-complete*.  Thus, the Karoubi envelope is a [[completion]] operation into such categories.

## Definition
 {#Definition}

There  is an

* _[Abstract definition](#AbstractDefinition)_

that characterizes idempotent completions. In particular the idempotent completion always exists and is unique up to [[equivalence of categories]]. Explicit constructions include:

* _[Construction in components](#InComponents)_

* _[Construction via the Yoneda embedding](#UnderTheYonedaEmbedding)_

For more constructions and equivalent characterizations see at _[[Cauchy complete category]]_ in the section _[In ordinary category theory](Cauchy+complete+category#InOrdinaryCategoryTheory)_.

### Abstract definition
 {#AbstractDefinition}

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

For $\mathcal{C}$ a [[category]], a [[functor]] $\mathcal{C} \to \tilde \mathcal{C}$ exhibits $\tilde \mathcal{C}$ as an **idempotent completion** of $\mathcal{C}$ if

* $\tilde \mathcal{C}$ is an [[idempotent complete category]];

* $\mathcal{C} \to \tilde \mathcal{C}$ is a [[full and faithful functor]];

* every object in $\tilde \mathcal{C}$ is the [[retract]] of an object in $\mathcal{C}$, under this embedding.

=--

See e.g. ([Lurie, def. 5.1.4.1](#Lurie)).

+-- {: .num_lemma #def2}
###### Lemma
For a fully faithful embedding $i \colon \mathcal{C} \to \mathcal{D}$ to exhibit an idempotent(-splitting) completion of $\mathcal{C}$, it suffices that

* $i(p)$ splits in $\mathcal{D}$ for every idempotent $p$ in $\mathcal{C}$, and

* every object in $\mathcal{D}$ is the retract of an object in $\mathcal{C}$ under $i$.
=--

+-- {: .proof}
###### Proof
We must show that these conditions imply that every idempotent $e \colon D \to D$ in $\mathcal{D}$ splits. Write $D$ as a retract of some $i(C)$, say $r: i(C) \to D$ with right inverse $s$ ($r s = 1_D$). Then $p = s e r \colon i(C) \to i(C)$ is idempotent, and we may split $p$, say as $p = \sigma \pi$ with $\pi \sigma = 1_E$ for some $E$. We claim that the pair

$$\pi s \colon D \to E, \qquad r \sigma \colon E \to D$$

provides a splitting of $e$. Certainly we have

$$(r \sigma)(\pi s) = r s e r s = e,$$

and we also have

$$\sigma(\pi s)(r \sigma) = p s r \sigma = s e r s r \sigma = s e r \sigma = p \sigma = \sigma \pi \sigma = \sigma$$

whence

$$(\pi s)(r \sigma) = \pi \sigma (\pi s)(r \sigma) = \pi \sigma = 1_E,$$

as desired.
=--

### In components
 {#InComponents}

Let $C$ be a [[category]]. We give an elementary construction of the __Karoubi envelope__ $\bar{C}$ which formally splits [[idempotents]] in $C$.

The objects of $\bar{C}$ are pairs $(c, e: c \to c)$ where $e$ is an idempotent on an object $c$ of $C$. Morphisms $(c, e) \to (d, f)$ are morphisms $\phi: c \to d$ in $C$ such that $f \circ \phi = \phi = \phi \circ e$ (or equivalently, such that $\phi = f \circ \phi \circ e$). NB: the identity on $(c, e)$ in $\bar{C}$ is the morphism $e: c \to c$.

There is a functor

$$E: C \to \bar{C}$$

which maps an object $c$ to $(c, 1_c)$. This functor is [[fully faithful functor|full and faithful]]: it fully embeds $C$ in $\bar{C}$. If $e: c \to c$ is an idempotent in $C$, then in $\bar{C}$ there are maps

$$p: (c, 1_c) \to (c, e), \, j: (c, e) \to (c, 1_c),$$

both given by $e: c \to c$. It is clear that $p \circ j$ is the identity $e: (c, e) \to (c, e)$, and that $j \circ p$ is the idempotent $E(e): E(c) \to E(c)$. Thus the pair $(p, j)$ formally splits the idempotent $e: c \to c$. The same argument shows that every idempotent $\phi: (c, e) \to (c, e)$ in $\bar{C}$ splits through $(c,\phi)$. Actually this formal construction does more: it gives a _choice_ of splitting for every idempotent.

Let $D$ be any category in which every idempotent $h: d \to d$ has a chosen splitting $(p_h: d \to d_h, j_h: d_h \to d)$ (using identities to split identities), and let $F: C \to D$ be a functor. Define an extension

$$\bar{F}: \bar{C} \to D$$

by sending $(c, e: c \to c)$ to the underlying object $F(c)_{F(e)}$ of the splitting of $F(e): F(c) \to F(c)$ in $D$. For morphisms $\phi: (c, e) \to (c', e')$, define $\bar{F}(\phi)$ to be the composite

$$F(c)_{F(e)} \overset{F(j_{F(e)})}{\to} F(c) \overset{F(\phi)}{\to} F(c') \overset{F(p_{F(e')})}{\to} F(c')_{F(e')}$$

Then $\bar{F}$ is the unique extension of $F$ which preserves chosen splittings. Thus the Karoubi envelope is universal among functors from $C$ into categories $D$ in which every idempotent has a chosen splitting.

If $D$ is a category in which every idempotent splits, then we can choose a splitting for each idempotent using the [[axiom of choice]] (AC); the extension $\bar{F}$ depends on how we do this but is unique up to unique [[natural isomorphism]].  Alternatively, we can define $\bar{F}$ as an [[anafunctor]]; then no AC is needed, and we still have $\bar{F}$ unique up to unique natural isomorphism.  (It is key here that a splitting of an idempotent is unique up to a coherent isomorphism.)

Essentially the same argument shows that for any $D$ in which idempotents split, the restriction functor $[E, D]: [\bar{C}, D] \to [C, D]$ is an equivalence. The details are spelled out [here](http://ncatlab.org/toddtrimble/published/Karoubi+envelope).

### Under the Yoneda embedding
 {#UnderTheYonedaEmbedding}

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###### Proposition

For $\mathcal{C}$ a [[small category]], write $PSh(\mathcal{C})$ for its [[category of presheaves]] and write $\tilde \mathcal{C} \hookrightarrow PSh(\mathcal{C})$ for the [[full subcategory]] on those presheaves which are [[retracts]] of objects in $\mathcal{C}$, under the [[Yoneda embedding]]. Then the Yoneda embedding

$$
  \mathcal{C} \to \tilde \mathcal{C}
$$

exhibits $\tilde \mathcal{C}$ as [[generalized the|the]] idempotent completion of $\mathcal{C}$.

=--

For instance ([Lurie, proof of prop. 5.1.4.2](#Lurie)).

## Properties

### Finality of the completion
 {#Finality}

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

A functor $\mathcal{C}\to \tilde \mathcal{C}$ exhibiting an idempotent completion, def. \ref{IdempotentCompletionAbstractly}, is a [[final functor]].

=--

For instance ([Lurie, lemma 5.1.4.6](#Lurie)).

### Monadicity over semicategories

The functor that forms idempotent completion is the [[monad]] induced from the [[adjunction]] between categories and [[semicategories]] given by the [[forgetful functor]] $Cat \to SemiCat$ and its [[right adjoint]]. More details on this are at _[Semicategory - Relation to categories](semicategory#RelationToCategories)_.

## Examples

* [[category of Chow motives]]

### The category of smooth manifolds

Let $Man$ be the category of [[smooth manifolds]] and [[smooth maps]], where by a "smooth manifold", we mean a finite-dimensional, second-countable, Hausdorff, $C^\infty$ [[manifold]] without boundary.  Let $i: Open \hookrightarrow Man$ be the [[full subcategory]] whose objects are the [[open subspaces]] of finite-dimensional [[Cartesian spaces]].

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###### Theorem
The subcategory $i: Open \hookrightarrow Man$ exhibits $Man$ as an idempotent-splitting completion of $Open$.
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###### Proof
By lemma \ref{def2}, it suffices to prove that

* Every smooth manifold is a smooth retract of an open set in Euclidean space;

* If $p : U \to U$ is a smooth idempotent on an open set $U \subseteq \mathbb{R}^n$, then the subset $Fix(p) \hookrightarrow U$ is an embedded submanifold.

For the first statement, we use the fact that any manifold $M$ can be realized as a closed submanifold of some $\mathbb{R}^n$, and every closed submanifold has a [[tubular neighborhood theorem|tubular neighborhood]] $U \subseteq \mathbb{R}^n$. In this case $U$ carries a structure of vector bundle over $M$ in such a way that the inclusion $M \hookrightarrow U$ is identified with the zero section, so that the bundle projection $U \to M$ provides a retraction, with right inverse given by the zero section.

For the second statement, assume that the origin $0$ is a fixed point of $p$, and let $T_0(U) \cong \mathbb{R}^n$ be its tangent space (observe the presence of a _canonical_ isomorphism to $\mathbb{R}^n$). Thus we have idempotent linear maps $d p(0), Id-d p(0): T_0(U) \to T_0(U)$ where the latter factors through the inclusion $\ker \; d p(0) \hookrightarrow T_0(U)$ via a projection map $\pi: T_0(U) \to \ker \; d p(0)$. We have a map $f: U \to \mathbb{R}^n$ that takes $x \in U$ to $x - p(x)$; let $g$ denote the composite

$$U \stackrel{f}{\to} \mathbb{R}^n \cong T_0(U) \stackrel{\pi}{\to} \ker\; d p(0).$$

Now we make some easy observations:

1. $Fix(p) \subseteq g^{-1}(0)$.

1. The map $p: U \to U$ restricts to a map $p: g^{-1}(0) \to g^{-1}(0)$, by idempotence of $p$.

1. The derivative $d g(0): T_0(U) \to T_0(\ker \; d p(0)) \cong \ker \; d p(0)$ is $\pi$ again since $Id - d p(0)$ is idempotent. Thus $d g(0)$ has full rank ($m$ say), and so the restriction of $g$ to some neighborhood $V$ has $0$ as a regular value, and $g^{-1}(0) \cap V$ is a manifold of dimension $m$ by the [[implicit function theorem]]. The tangent space $T_0(g^{-1}(0) \cap V)$ is canonically identified with $im(d p(0))$.

1. There are smaller neighborhoods $V'' \subseteq V' \subseteq V$ so that $p$ restricts to maps $p_1, p_2$ as in the following diagram ($i, i', i''$ are inclusion maps, all taking a domain element $x$ to itself):
$$\array{
g^{-1}(0) \cap V'' & \stackrel{i''}{\hookrightarrow} & g^{-1}(0) \\
 _\mathllap{p_2} \downarrow & & \downarrow _\mathrlap{p} \\
g^{-1}(0) \cap V' & \stackrel{i'}{\hookrightarrow} & g^{-1}(0) \\
 _\mathllap{p_1} \downarrow & & \downarrow _\mathrlap{p} \\
g^{-1}(0) \cap V & \stackrel{i}{\hookrightarrow} & g^{-1}(0)
}$$
and such that $p_1, p_2$ are diffeomorphisms by the [[implicit function theorem|inverse function theorem]] (noting here that $d p_i(0): im(d p(0)) \to im(d p(0))$ is the identity map, by idempotence of $p$).

1. Letting $q: g^{-1}(0) \cap V' \to g^{-1}(0) \cap V''$ denote the smooth inverse to $p_2$, we calculate $i' = p \circ i'' \circ q$, and
$$i p_1 = p i' = p p i''q = p i'' q = i',$$
so that $p_1(x) = x$ for every $x \in g^{-1}(0) \cap V'$. Hence $g^{-1}(0) \cap V' \subseteq Fix(p)$.

From all this it follows that $Fix(p) \cap V' = g^{-1}(0) \cap V'$, meaning $Fix(p)$ is locally diffeomorphic to $\mathbb{R}^m$, and so $Fix(p)$ is an embedded submanifold of $\mathbb{R}^n$.
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###### Remark
[Lawvere](#Law) comments on this fact as follows: "For example, if $\mathbf{C}$ is the category of all smooth maps between all open subsets of all Euclidean spaces, then $\widebar{\mathbf{C}}$ $[$the Karoubi envelope$]$
is the category of all smooth manifolds. This powerful theorem justifies bypassing the complicated considerations of charts, coordinate transformations, and atlases commonly offered as a "basic" definition of the concept of manifold. For example the 2-sphere, a manifold but not an open set of any Euclidean space, may be fully specified with its smooth structure by considering any open set $A$ in 3-space $E$ which contains it but not its center (taken to be $0$) and the smooth idempotent endomap of $A$ given by $e(x) = x/{|x|}$. All general constructions (i.e., functors into categories which are Cauchy complete) on manifolds now follow easily (without any need to check whether they are compatible with coverings, etc.) provided they are known on the opens of Euclidean spaces: for example, the tangent bundle on the sphere is obtained by splitting the idempotent $e'$ on the tangent bundle $A \times V$ of $A$ ($V$ being the vector space of translations of $E$) which is obtained by differentiating $e$. The same for cohomology groups, etc."
=--

### Projective modules and vector bundles

The category of projective modules over any ring is the Karoubi envelope of its full subcategory of free modules.

The category of (locally trivial, finite dimensional) vector bundles over any fixed paracompact space is the Karoubi envelope of its full subcategory of trivial bundles.

Both examples are related by the [[Serre-Swan theorem]]. In fact both these facts together with the observation that the global sections functor is an equivalence from the category trivial bundles over $X$ to the category of free modules over $C(X)$ prove the Serre-Swan theorem itself.

## Related concepts

* [[Karoubian category]], [[pseudo-abelian category]]

* [[Cauchy completion]]

* [[free coproduct completion]]

* [[free cocompletion]]

* [[idempotent complete (infinity,1)-category]]

For Karoubi envelopes of Kleisli categories, see the references on:

* [[split idempotent]]

## References
 {#References}

The term "Karoubi envelope" appears to have been introduced in:

* {#Hayashi1985} [[Susumu Hayashi]], Def. 1.2 in: *Adjunction of semifunctors: categorical structures in nonextensional lambda calculus*, Theoretical Computer Science **41** (1985) 95-104 [doi:10.1016/0304-3975(85)90062-3]

and apparently in reference to the [[universal construction]] of [[pseudo-abelian categories]] (now also: "[[Karoubian categories]]") in:

* {#Karoubi78} [[Max Karoubi]], Theorem 6.10 of: *K-Theory -- An introduction*, Grundlehren der mathematischen Wissenschaften **226**, Springer (1978) [[pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3)]

  > (in the context of [[topological K-theory]]).

A classical account is for instance in

* {#BorceuxDejean} [[Francis Borceux]] and D. Dejean, _Cauchy completion in category theory_  Cahiers Topologie Géom. Différentielle Catégoriques, 27:133–146, (1986) ([numdam](http://www.numdam.org/item?id=CTGDC_1986__27_2_133_0))

For more classical references see at  _[[Cauchy complete category]]_.

Karoubi envelopes for [[(infinity,1)-category|(∞,1)-categories]] are discussed in section 4.4.5 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_
 {#Lurie}

Some discussion of the stable version is in section 4.1.2 of

* [[David Ben-Zvi]], John Francis, David Nadler, _Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry_ ([arXiv](http://arxiv.org/abs/0805.0157), [[geometric infinity-function theory|nLab entry]])

and section 2.3 of

* [[David Ben-Zvi]], David Nadler, _The Character Theory of a Complex Group_ ([arXiv](http://arxiv.org/abs/0904.1247))

In section 3.1.2 of latter are also given
references (to Neeman and Lurie) for an important result of Neeman's about Karoubi closure and compact generators.

The Karoubi envelope for the additive case (see also [[additive envelope]]) is covered at

* [[Dror Bar-Natan]], [[Scott Morrison]], _The Karoubi envelope and Lee's degeneration of Khovanov homology_ ([arXiv](http://arxiv.org/abs/math/0606542))

Discussion for [[triangulated categories]] is in

* Paul Balmer, Marco Schlichting, _Idempotent completion of triangulated categories_ ([pdf](http://www.math.ucla.edu/~balmer/research/Pubfile/IdempCompl.pdf))

The proof that idempotents split in the category of manifolds was adapted from this MO answer:

* Zack (http://mathoverflow.net/users/300/zack), Idempotents split in category of smooth manifolds?, URL (version: 2014-04-06): http://mathoverflow.net/q/162556 ([web](http://mathoverflow.net/a/162556/2926))

Which provides a solution to exercise 3.21 in

* F. W. Lawvere, _Perugia Notes - Theory of Categories over a Base Topos_ , Ms. Università di Perugia 1973.

The accompanying above remark of Lawvere appears on page 267 of

* [[F. William Lawvere]], _Qualitative distinctions between some toposes of generalized graphs_, Contemporary Mathematics 92 (1989), 261-299. ([pdf](http://www.ams.org/books/conm/092/1003203/conm092-1003203.pdf))
 {#Law}

A comparison of the Karoubi envelope to other completions can be found here:

* [[Marta Bunge]], _Tightly Bounded Completions_ , TAC **28** no. 8 (2013) pp.213-240. ([pdf](http://www.tac.mta.ca/tac/volumes/28/8/28-08.pdf))

Formalization in [[homotopy type theory]]:

* {#Shulman14} [[Mike Shulman]], _[Splitting idempotents](http://homotopytypetheory.org/2014/12/08/splitting-idempotents/)_

A generalization of the Karoubi envelope for [[n-categories]] is in

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], _Condensations in higher categories_, ([arXiv:1905.09566](https://arxiv.org/abs/1905.09566))

[[!redirects Karoubi envelopes]]

[[!redirects Karoubi completion]]
[[!redirects Karoubi complete category]]
[[!redirects idempotent-complete category]]
[[!redirects idempotent complete category]]

[[!redirects idempotent completion]]
[[!redirects idempotent completions]]

[[!redirects idempotent-splitting completion]]
[[!redirects idempotent-splitting completions]]

[[!redirects pseudo-abelian completion]]
[[!redirects Karoubian envelope]]
[[!redirects Karoubianization]]
[[!redirects Karoubianization functor]]
[[!redirects Karoubinization functor]]


Karoubi K-theory

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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
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Karoubi defined [[K-theory]] classes given by [[Clifford module bundles]], where a $Cl_n$-module represents a class in $K^n$ and represents the trivial class if it extends to a $Cl_{n+1}$-module.

(Over the point this is the [[Atiyah-Bott-Shapiro isomorphism]].)

## Idea

We have a sequence of [[Clifford algebras]] $Cl_n$ which are generated by $n$ anticommuting square roots of $\pm 1$. The sequence is periodic up to [[Morita equivalence]]; $Cl_8$ is $\mathbb{R}(16)$, the algebra of $16 \times 16$ real matrices, which is Morita equivalent to $\mathbb{R}$, and from then on it repeats every 8 with extra matrix dimensions thrown in -- this is _[[Bott periodicity]]_.
(Notice that here we treat Clifford algebras as $\mathbb{Z}_2$-[[graded algebras]]: while $Cl_6$ is Morita equivalent to $\mathbb{R}(8)$ as an algebra, it is not so as a graded algebra.)

It turns out that $K^n(X)$ can be represented geometrically by [[Clifford module bundles]] over $X$. Start with $K^0$; we know that elements of $K^0(X)$ are 'formal differences' $V - W$ of vector bundles over $X$ ([[virtual vector bundles]]). We can model the formal difference $V - W$ with an honest geometric object by using the $\mathbb{Z}_2$-graded vector bundle $V \oplus W$, where $V$ is even and $W$ is odd. Such a thing should represent the zero class in K-theory just when $V$ and $W$ are isomorphic; this can be rephrased as saying that there exists an odd operator $e$ on $V \oplus W$ (hence, taking $V$ to $W$ and vice versa) such that $e^2 = 1$. But this just says that $V \oplus W$ has an action of the first Clifford algebra $Cl_1$.

More generally, Karoubi proved that for any $n$, $K^{-n}X$ can be represented by $Cl_n$-module bundles on $X$ modulo those such that the $Cl_n$-action extends to a
$Cl_{n+1}$-action. When $n = 0$ this is what we had above, since a $Cl_0$-module is just a vector space. This allows us to deduce Bott periodicity for K-groups from the algebraic periodicity (up to Morita equivalence) of Clifford algebras.

## Categorification

[[Chris Douglas]] and [[André Henriques]] have proposed that this description of K-theory has a good [[vertical categorification|categorification]] that might be relevant for [[tmf]] ([DougHen](#DougHen)).

[Here](http://golem.ph.utexas.edu/category/2007/10/higher_clifford_algebras.html#c012972) is a report by [[Mike Shulman]] on a talk by Chris Douglas on this topic, from which also part of the above text is taken.

## Related concepts

* [[topological K-theory]]

* [[Kasparov K-theory]]

* [[conformal nets]]

* [[mass term]]

* [[K-theory classification of topological phases of matter]]

## References

* {#AtiyahSinger69} [[Michael Atiyah]], [[Isadore Singer]], *Index theory for skew-adjoint Fredholm operators*, Publications Mathématiques de l'IHÉS, Tome **37** (1969) 5-26 $[$[numdam:PMIHES_1969__37__5_0](http://www.numdam.org/item/PMIHES_1969__37__5_0)$]$

* {#Karoubi70} [[Max Karoubi]], *Espaces Classifiants en K-Théorie*, Transactions of the American Mathematical Society **147** 1 (Jan., 1970) 75-115 $[$[doi:10.2307/1995218](https://doi.org/10.2307/1995218)$]$

* [[Max Karoubi]], _Twisted K-theory old and new_, ([pdf](http://www.math.uiuc.edu/K-theory/0840/TwistedK-theory_mwd.pdf))

* [[Peter Donovan]], [[Max Karoubi]], _Graded Brauer groups and $K$-theory with local coefficients_ ([pdf](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1970__38_/PMIHES_1970__38__5_0/PMIHES_1970__38__5_0.pdf))

Relation to [[twisted K-theory]]:

* [[Max Karoubi]], _Clifford modules and twisted K-theory_, Proceedings of the International Conference on Clifford algebras (ICCA7) ([arXiv:0801.2794](http://arxiv.org/abs/0801.2794))

Categorification using [[conformal nets]]:

* {#DougHen} [[Chris Douglas]], [[André Henriques]], _Topological modular forms and conformal nets_, [arXiv:1103.4187](https://arxiv.org/abs/1103.4187)

[[!redirects Karoubi K-theories]]


Karoubian category

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###Context###
#### Additive and abelian categories
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[[!include additive and abelian categories - contents]]
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#Contents#
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## Idea

A _pseudo-abelian category_ [[Karoubi (1978), Def. 6.7](#Karoubi78)] or *Karoubian category* is a [[pre-additive category]] $C$ such that every [[idempotent]] morphism $p \colon A \to A$ in $C$ has a [[kernel]], and hence (one can easily show) also a [[cokernel]].

This is stronger than [[pre-additive category|pre-additivity]] but weaker than [[abelian category|abelianness]], which requires that _every_ morphism has a [[kernel]] and [[cokernel]].

## Definition

Let $C$ be a category and $p : X \to X$ an [[idempotent]] [[endomorphism]] of an object $X$.  One says that $p$ **admits an image** if the functor $Ker(id_X, p)$ is [[representable]], and the representing object is called the image of $p$.  Here $Ker(id_X, p)$ is the functor $C^{op} \to \underline{Set}$ mapping

$$ Y \mapsto Ker(Hom(Y, X) \rightrightarrows Hom(Y, X)); \qquad (\ast) $$

in other words the image of $p$ is the [[equaliser|difference kernel]] of $(id_X, p)$, when it exists.

Now $C$ is called **Karoubian** if every idempotent $p$ admits an image. Since $p: X \to X$ is idempotent iff $id_X - p$ is idempotent, this is the same as saying every idempotent has a kernel.

## Properties

### General

One can show that for any idempotent $p$, $Ker(id_X, p)$ is representable if and only if $Coker(id_X, p)$ is, and that in fact their representing objects are canonically isomorphic.

Recall that one says $p$ **[[split idempotent|splits]]** if there exists an object $Y$, and morphisms $f : X \to Y$, $g : Y \to X$, such that $f \circ g = id_Y$ and $g \circ f = p$.  Observe that when $p$ admits an image $K$, it splits: by definition there are functorial isomorphisms $\Phi_Y$ for all $Y$ between the image of the functor $(\ast)$ and $\Hom(Y, K)$; now take $f : X \to K$ the morphism corresponding to $p$ via $\Phi_X$, $g : K \to X$ the morphism corresponding to $id_K \in Hom(K, K)$ via $\Phi_K$.  Conversely, if $p$ splits via a pair $(f, g)$, then $g: Y \to X$ is a difference kernel of $(id_X, p)$: we have $g = g \circ f \circ g = p \circ g$, and if $h: Z \to X$ satisfies $h = p \circ h = g \circ f \circ h$, then $h$ clearly factors through $g$, and uniquely so since sections $g$ are monomorphisms.

### Karoubi envelope

There is a [[reflective subcategory|universal functor]] from the category of (say, small) [[preadditive categories]] to the category of Karoubian categories, the __[[Karoubinization functor]]__; its value on a preadditive category $C$ is also called the __[[Karoubian envelope]]__ or the __[[pseudo-abelian completion]]__ of $C$.

In more detail, there exists a Karoubian category $kar(C)$ associated to any category $C$, and a fully faithful functor $\varphi : C \to kar(C)$, which is [[universal property | universal]] in the sense that for any Karoubian category $C'$, the functor

$$\underline{Hom}(kar(C), C') \to \underline{Hom}(C, C')$$

taking a functor $F : kar(C) \to C'$ to the composite $F \circ \varphi$ is an [[equivalence]] of categories.  $kar(C)$ is called the **[[Karoubi envelope]]** of $C$ (aka the _[[Cauchy completion]]_, or the _[[idempotent-splitting completion]]_).  It can be realized explicitly by taking as objects pairs $(X, p)$, with $p$ idempotent, and as morphisms $(X, p) \to (Y, q)$ the morphisms $f : X \to Y$ that satisfy $f = q \circ f \circ p$.

## Examples

The requirement that, say, a [[dg-category]] or a [[triangulated category]] be Karoubian is a natural requirement in a number of contexts.

The Karoubian envelope is also used in the construction of the [[category of pure motives]], and in [[K-theory]].

## Related concepts

* [[Karoubi envelope]]

* [[semi-abelian category]]

* [[quasi-abelian category]]

* [[abelian category]]

## References

* [[Alexandre Grothendieck]], [[Jean-Louis Verdier]].  Exercise 7.5 in _Topos_, Exposé IV of [[SGA 4]], volume 1.

* {#Karoubi78} [[Max Karoubi]], Def. 6.7 of: *K-Theory -- An introduction*, Grundlehren der mathematischen Wissenschaften **226**, Springer (1978) [[pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3)]

  > (in the context of [[topological K-theory]], see also at *[[Karoubi envelope]]*)

The terminology "Karoubian category" is used for instance in:

* [[Bruno Kahn]], Appendix A.1 of: *Zeta and L-Functions of Varieties and Motives*, Cambridge University Press (2020) [[doi:10.1017/9781108691536](https://doi.org/10.1017/9781108691536)]

* [[Stacks Project]], *Karoubian categories *[[tag:09SF](https://stacks.math.columbia.edu/tag/09SF)]

[[!redirects pseudo-abelian category]]
[[!redirects pseudoabelian category]]

[[!redirects pseudo-abelian categories]]
[[!redirects pseudoabelian categories]]


Kashiwara-Vergne conjecture

Quoting the seminar [page](https://math.temple.edu/~vald/KV.html),

> The Kashiwara-Vergne conjecture is a subtle property of the Campbell-[[Hausdorff series]] which implies that the [[Duflo isomorphism]] theorem can be extended to the germs of bi-invariant distributions on a Lie group. This problem is related to the combinatorics of multiple zeta values and to the Deligne-Drinfeld conjecture. It is also related to some open questions in [[deformation quantization]]. An important corollary of the Kashiwara-Vergne conjecture was proved in papers of M. Andler, A. Dvorsky, S. Sahi, and C. Torossian. For an arbitrary Lie algebra the conjecture was established by A. Alekseev and E. Meinrenken. A. Alekseev and C. Torossian built a nice algebraic framework for the Kashiwara-Vergne problem. Using this framework they showed that solutions of the Kashiwara-Vergne problem are closely related to Drinfeld's associators. In fact, starting from any [[Drinfeld associator]] one can produce a solution of the Kashiwara-Vergne problem.

* [[M. Kashiwara]], M. Vergne, _The Campbell-Hausdorff formula and invariant hyperfunctions, Inventiones math. __47__, 249--272 (1978)  [pdf](http://www.math.polytechnique.fr/~vergne/publications2/Remplacer/78thecampbellhaussdorff.pdf)
* Anton Alekseev, Charles Torossian, _On the Kashiwara-Vergne conjecture_, Invent. Math. 164(3), 615–634 (2006) [doi](https://doi.org/10.1007/s00222-005-0486-4); _The Kashiwara-Vergne conjecture and Drinfeld's associators_, Ann. of Math. __175__:2 (2012) 415–463 [pdf](http://www.math.jussieu.fr/~torossian/AT1.pdf);  _Kontsevich deformation quantization and flat connections_ [arxiv/0906.0187](https://arxiv.org/abs/0906.0187) [doi](https://doi.org/10.1007/s00220-010-1106-8)
* A. Alekseev, B. Enriquez, C. Torossian, _Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations_, Publ. Math. IHES __112__ (2010), 143–189 [arxiv/0903.4067](https://arxiv.org/abs/0903.4067)
* M. Andler, A. Dvorsky, S. Sahi, _Kontsevich quantization and invariant distributions on Lie groups_, [arxiv/9910104](https://arxiv.org/abs/math/9910104)
* M. Andler, S. Sahi, C. Torossian, _Convolution of invariant distributions: proof of the Kashiwara-Vergne conjecture_, [math.QA/0104100](https://arxiv.org/abs/math/0104100)
* D. Bar-Natan, _On associators and the Grothendieck-Teichmuller Group I_, [q-alg/9606021](https://arxiv.org/abs/q-alg/9606021)
* V.G. [[Drinfeld]], _On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal(\bar{\mathbf{Q}}/\mathbf{Q})$_, (Russian) Algebra i Analiz __2__:4, 149–181 (1990); translation in Leningrad Math. J. 2:4, 829–860 (1991)
* [[Thomas Willwacher]], _M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra_ [arxiv/1009.1654](https://arxiv.org/abs/1009.1654)
* Maria Podkopaeva, _On the Jacobson element and generators of the Lie algebra_ [arxiv/0812.0772](https://arxiv.org/abs/0812.0772)
* [[Anton Alekseev]], Florian Naef, Xiaomeng Xu, [[Chenchang Zhu]], _Chern–Simons, Wess–Zumino and other cocycles from Kashiwara–Vergne and associators_, Letters in Mathematical Physics __108__:3 (2018) 757–778 [doi](https://doi.org/10.1007/s11005-017-0985-4)


Kasper Peeters

* [webpage](http://maths.dur.ac.uk/users/kasper.peeters/)

## Selected writings

On [[higher curvature corrections]] to [[D=11 supergravity]]:

* [[Kasper Peeters]], Pierre Vanhove, Anders Westerberg, _Supersymmetric $R^4$ actions and quantum corrections to superspace torsion constraints_ ([arXiv:hep-th/0010182](https://arxiv.org/abs/hep-th/0010182))

On [[higher curvature corrections]] in [[D=11 supergravity]]/[[M-theory]] via analysis of [[superparticle]] [[scattering amplitudes]]:

* [[Kasper Peeters]], [[Jan Plefka]], [[Steffen Stern]], _Higher-derivative gauge field terms in the M-theory action_, JHEP 0508 (2005) 095 ([arXiv:hep-th/0507178](https://arxiv.org/abs/hep-th/0507178))

On [[Euclidean field theory]]:

* [[Kasper Peeters]], [[Marija Zamaklar]], _Euclidean Field Theory_, Lecture notes 2009-2011 ([web](http://maths.dur.ac.uk/users/kasper.peeters/eft.html), [pdf](http://maths.dur.ac.uk/users/kasper.peeters/pdf/eft.pdf))

## Related $n$Lab entries

* [[Euclidean field theory]]

* [[lattice gauge theory]]

* [[string theory]]

category: people


Katarzyna Rejzner

* [webpage](http://rejzner.com/)

## Selected

On [[BV-BRST formalism]] in [[perturbative AQFT]]:

* Thesis: _Batalin-Vilkovisky formalism in locally covariant field theory_ ([arXiv:1111.5130](http://arxiv.org/abs/1111.5130))

On [[locally covariant perturbative AQFT]]:

* textbook: _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

Introduction to [[AQFT]]:

* {#FewsterRejzner19} [[Christopher Fewster]], [[Kasia Rejzner]], _Algebraic Quantum Field Theory - an introduction_ ([arXiv:1904.04051](https://arxiv.org/abs/1904.04051))

On the [[perturbative algebraic quantum field theory|locally covariant pAQFT approach]] to effective [[quantum gravity]] and applications to experiment:

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Kasia Rejzner]], *Locally covariant approach to effective quantum gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07800](https://arxiv.org/abs/2212.07800)]

## Related $n$Lab entries

* [[AQFT]]

* [[locally covariant perturbative quantum field theory]]

* [[deformation quantization]]

* [[BRST-BV formalism]]

category: people

[[!redirects Kasia Rejzner]]


Kate Gruher

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=135548)

## Selected writings

On the [[free loop space of a classifying space]]:

* {#Gruher07} [[Kate Gruher]], *A proof of $L B G \simeq Ad(E G)$*, Appendix A in: *String Topology of Classifying Spaces*, 2007 ([[Gruher_FreeLoopSpaceOfClassifyingSpace.pdf:file]])

category: people


Kate Ponto

* [website](http://www.ms.uky.edu/~kate/)

## Selected writings

On the [[Dennis trace]] on [[algebraic K-theory]] and relating to [[bicategorical trace]]:

* {#CLMPZ20} [[Jonathan Campbell]], [[John Lind]], [[Cary Malkiewich]], [[Kate Ponto]], [[Inna Zakharevich]], _K-theory of endomorphisms, the TR-trace, and zeta functions_ [[arXiv:2005.04334](https://arxiv.org/abs/2005.04334)]

category: people


Kathryn Hess

* [website](http://sma.epfl.ch/~hessbell/)

## Selected writings

* *Homotopic Hopf-Galois extensions: foundations and examples*, [arxiv/0902.3393](http://arxiv.org/abs/0902.3393)

On [[descent]]:

* *A general framework for homotopic descent and codescent* ([arXiv/1001.1556](http://arxiv.org/abs/1001.1556))

On [[rational homotopy theory]]:

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

On the [[Adams-Hilton model]] for the [[Pontrjagin ring]]-structure on the [[singular chain complex]] of a [[based loop space]]:

* [[Kathryn Hess]], [[Paul-Eugène Parent]], [[Jonathan Scott]], [[Andrew Tonks]], *A canonical enriched Adams-Hilton model for simplicial sets*, Advances in Mathematics **207** 2 (2006) 847-875 [[doi:10.1016/j.aim.2006.01.013](https://doi.org/10.1016/j.aim.2006.01.013), [arXiv:math/0408216](https://arxiv.org/abs/math/0408216)]

On [[free loop spaces]] in [[algebraic topology]] and [[physics]] ([[quantum field theory]]):

* [[Kathryn Hess]], *Free loop spaces in topology and physics*, talk at *Meeting of Edinburgh Math. Soc. Glasgow* (14 Nov 2008) [[pdf](http://sma.epfl.ch/~hessbell/Pub_Edinburgh.pdf), [[Hess-FreeLoopSpaces.pdf:file]]]

On [[model categories]] of [[coalgebras over a comonad]] via left-[[transferred model category|left transfer]] (left-induction):

* [[Kathryn Hess]], [[Brooke Shipley]], *The homotopy theory of coalgebras over a comonad*, Proceedings of the London Mathematical Society **108** 2 (2014) [[arXiv:1205.3979](https://arxiv.org/abs/1205.3979), [doi:10.1112/plms/pdt038](https://doi.org/10.1112/plms/pdt038)]

Further on left-[[transferred model structures]] and [[model structures on functors]]:

* [[Marzieh Bayeh]], [[Kathryn Hess]], [[Varvara Karpova]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *Left-induced model structures and diagram categories*, in: *Women in Topology: Collaborations in Homotopy Theory*, Contemporary Mathematics **641** American Mathematical Society (2015) [[arXiv:1401.3651](http://arxiv.org/abs/1401.3651), [ISBN:978-1-4704-2495-4](https://bookstore.ams.org/conm-641)]

* {#HKRS15} [[Kathryn Hess]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *A necessary and sufficient condition for induced model structures*, J. Topology **10** 2  (2017) 324-369 [[arXiv:1509.08154](http://arxiv.org/abs/1509.08154), [doi:10.1112/topo.12011](https://doi.org/10.1112/topo.12011)]

  > (this contains an error which is corrected in [Garner, Kedziorek & Riehl 2018](transferred+model+structure#GKR18))

On [[simply connected homotopy type|simply connected]] [[homotopy types]] as the [[comodule over a comonad|comodules]] in [[spectra]] over the [[comonad]] $\Sigma^\infty \Omega^\infty$ (the [[exponential modality]], see [there](!-modality#RealizationInLinearHomotopyTypeTheory)) induced by the [[stabilization]] [[adjunction]]:

* {#HessKedziorek17} [[Kathryn Hess]], [[Magdalena Kedziorek]], Thm. 3.11 in: *The homotopy theory of coalgebras over simplicial comonads*, Homology, Homotopy and Applications **21** 1 (2019) [[arXiv:1707.07104](https://arxiv.org/abs/1707.07104), [doi:10.4310/HHA.2019.v21.n1.a11](https://dx.doi.org/10.4310/HHA.2019.v21.n1.a11)]

## Related entries

* [[rational homotopy theory]]

* [[generalized universal bundle]]

* [[homotopy coinvariants functor]]

category: people


Kathryn Van Stone

* [personal page](http://www.cs.cmu.edu/~kvs/)

## Selected writings

On ([[comonad|co-]])[[monads (in computer science)]], their [[distributive laws]] and their combined (double) [[Kleisli categories]]:

* {#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]]]

category: people


Katrin Becker

* [webpage](http://people.physics.tamu.edu/kbecker/)

## Selected writings

On [[T-duality]]:

* [[Katrin Becker]], [[Aaron Bergman]], _Geometric Aspects of D-branes and T-duality_ ([arXiv:0908.2249](http://arxiv.org/abs/0908.2249))

Textbook account of [[perturbative string theory]] and [[M-theory]]/[[F-theory]]:

* [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], _String Theory and M-Theory: A Modern Introduction_, Cambridge University Press 2006 ([doi:10.1017/CBO9780511816086](https://doi.org/10.1017/CBO9780511816086))

On the [[BFSS matrix model]]:

* [[Katrin Becker]], [[Melanie Becker]], _A Two-Loop Test of M(atrix) Theory_, Nucl.Phys. B506 (1997) 48-60 ([arXiv:hep-th/9705091](https://arxiv.org/abs/hep-th/9705091))

* [[Katrin Becker]], [[Melanie Becker]], [[Joseph Polchinski]], [[Arkady Tseytlin]], _Higher Order Graviton Scattering in M(atrix) Theory_, Phys.Rev.D56:3174-3178,1997 ([arXiv:hep-th/9706072](https://arxiv.org/abs/hep-th/9706072))

On [[M-theory on 8-manifolds]] and [[C-field tadpole cancellation]]:

* [[Katrin Becker]], [[Melanie Becker]], _M-Theory on Eight-Manifolds_, Nucl.Phys. B477 (1996) 155-167 ([arXiv:hep-th/9605053](https://arxiv.org/abs/hep-th/9605053))

On [[M5-branes]], [[M2-branes]] and [[non-perturbative effects]] ([[M5-brane instantons]], [[M2-brane instantons]]) in [[string theory]]/[[M-theory]]:

* [[Katrin Becker]], [[Melanie Becker]], [[Andrew Strominger]], section 2.1 of _Five-branes, membranes and nonperturbative string theory_, Nucl. Phys. B 456, 130 (1995) ([hep-th/9507158](http://arxiv.org/abs/hep-th/9507158))

Discussion of [[Cayley forms]] and [[wrapped branes]] on [[associative submanifolds]]:

* [[Katrin Becker]], [[Melanie Becker]], D.R.Morrison, [[Hirosi Ooguri]], Y.Oz, Z.Yin, _Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds_, Nucl.Phys.B480:225-238,1996 ([arXiv:hep-th/9608116](http://arxiv.org/abs/hep-th/9608116))

On [[moduli stabilization]] in [[non-geometric string vacua]]:

* [[Katrin Becker]], [[Melanie Becker]], [[Cumrun Vafa]], [[Johannes Walcher]], _Moduli Stabilization in Non-Geometric Backgrounds_,  	Nucl.Phys.B770:1-46,2007 ([arXiv:hep-th/0611001](https://arxiv.org/abs/hep-th/0611001))

On [[supersymmetry breaking]] in [[flux compactifications]] of [[M-theory]]:

* [[Katrin Becker]], [[Melanie Becker]], _Supersymmetry Breaking, M-Theory and Fluxes_, JHEP 0107:038,2001 ([arXiv:hep-th/0107044](https://arxiv.org/abs/hep-th/0107044))

On [[D=5 supergravity]] formulated in $\mathcal{N} = 1/2$ [[superspace]]:

* [[Katrin Becker]], [[Melanie Becker]], Daniel Butter, William D. Linch III, Stephen Randall, _Five-dimensional Supergravity in N = 1/2 Superspace_ ([arXiv:1909.09208](https://arxiv.org/abs/1909.09208))

## Related $n$Lab entries

* [[non-perturbative effect]]

* [[BFSS matrix model]]

category: people


Katrin Wehrheim

* [website](http://www-math.mit.edu/~katrin/)

## Selected writings

On the [[symplectic category]]:

* [[Katrin Wehrheim]], [[Chris T. Woodward]], _Functoriality for Lagrangian correspondences in Floer theory_, Quantum Topology **1** 2 (2010) 129-170 [[doi:10.4171/qt/4](https://doi.org/10.4171/qt/4), [arXiv:0708.2851](https://arxiv.org/abs/0708.2851)]

## Related entries

* [[Lagrangian correspondences and category-valued TFT]]

* [[polyfold]]

category: people


Katrin Wendland

* [website](http://www.math.uni-augsburg.de/geo/mitarbeiter/wendland/)

## Selected writings

On [[2d CFT]]:

* [[Katrin Wendland]], _Snapshots of Conformal Field Theory_, in:
Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies. Springer 2015
([arXiv:1404.3108](http://de.arxiv.org/abs/1404.3108), [doi:10.1007/978-3-319-09949-1_4](https://doi.org/10.1007/978-3-319-09949-1_4))

category: people


Katsumi Nomizu

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

## Selected writings

On [[differential geometry]]:

* [[Shoshichi Kobayashi]], [[Katsumi Nomizu]], _Foundations of differential geometry_, Volume 1 (1963), Volume 2 (1969) Interscience Publishers, reprinted by Wiley Classics Library (1996) [[ISBN:978-0-470-55558-3](https://www.wiley.com/en-us/Foundations+of+Differential+Geometry%2C+2+Volume+Set-p-9780470555583), [Wikipedia entry](https://en.wikipedia.org/wiki/Foundations_of_Differential_Geometry)]

## Related entries

* [[differential geometry]]

* [[G-structure]]

category: people


Kawahara Yasuo

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=166219)

## Selected writings

Introducing the notion of *[[meroi]]* for [[relational set theory]]:

* {#Kawahara95} [[Kawahara Yasuo]], _Relational Set Theory_, in Proceedings of the 6th International Conference on Category Theory and Computer Science, 1995 ([doi:10.1007/3-540-60164-3_19](http://dx.doi.org/10.1007/3-540-60164-3_19), [pdf](https://www.researchgate.net/profile/Yasuo-Kawahara/publication/226561376_Relational_set_theory/links/569f960908ae21a564270542/Relational-set-theory.pdf))

category: people


Kay Rülling

* [webpage](http://www.mi.fu-berlin.de/users/ruelling/)

* [[Chow groups with modulus]]

category: people


Kaya Arro

[[!redirects Kaya Ferendo]]
I'm a visiting assistant professor at UC Riverside. Before that, I was a graduate student at Brown University under [[Tom Goodwillie]]. Before that, I was an undergraduate at Amherst College, where I studied under [[Michael Ching]]. [Here](https://kayaarro.site) is my webpage.

My research interest is [[higher category theory]], and more specifically functor calculi, and more specifically functor calculi in the mold of [[Michael Weiss]]' [[orthogonal calculus]].

I can be reached at kaya (dot) arro (at) ucr (dot) edu.

## Selected  writings
 {#SelectedWritings}

On [[FI-representations]] and their [[representation stability]] in the generality of [[infinity-representations|$\infty$-representations]] in [[stable infinity-categories|stable $\infty$-categories]] and their analysis via a functor calculus analogous to [[orthogonal calculus]]:

* [[Kaya Arro]], *FI-Calculus and Representation Stability* (2023) [[arXiv:2306.13597](https://arxiv.org/abs/2306.13597)]

category: people


Kazama-Suzuki model

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## Idea

Kazama-Suzuki models are [[2d SCFTs]] which generalize the class of [[Gepner models]]. Instead of trivial tensoring of [[minimal models]], Kazana-Suzuki models are built by a superconformal [[coset space]] method to yield a [[central charge]] as needed for internal consistency of Superstring [[KK-compactification|compactification]].

The method involves a generalized form of N=1 SCFT super [[current algebra]] to yield [[OPEs]] of [[number of supersymmetries|N=2]] [[2d SCFT]]. This yields a class of [[2d SCFTs]] based on [[Hermitian manifold]] [[symmetric spaces]].

## Related entries

* [[Gepner model]]

* [[WZW model]]

## References

The original article is

* [[Yoichi Kazama]], [[Hisao Suzuki]], _$N = 2$ superconformal field theories and superstring compactification_, Nucl. Phys. B 321 (1989), pp. 232-268 ([http://inspirehep.net/record/263185)](http://inspirehep.net/record/263185), doi:10.1016/0550-3213(89)90250-2)

See also

* S. E. Parkhomenko, _Kazama-Suzuki Models of N=2 Superconformal Field Theory and Manin triples_ ([arxiv:1410.2977](https://arxiv.org/abs/1410.2977))

* [[Davide Gaiotto]], _Kazama-Suzuki models and BPS domain wall junctions in $N=1$ $SU(n)$ Super Yang-Mills_ ([arxiv:1306.5661](https://arxiv.org/abs/1306.5661))

[[!redirects Kazama-Suzuki models]]


Kazhdan's property (T)

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## Idea

(...)

## Examples

* [[U(ℋ)]] ([Bekka 2003](#Bekka03))

* ...

## Related concepts

* [[locally compact topological group]]

* [[discrete group]]

## References

* [[David Kazhdan]], _Connection of the dual space of a group with the structure of its close subgroups_, Funct Anal Its Appl (1967) 1: 63 ([doi:10.1007/BF01075866](https://doi.org/10.1007/BF01075866))

* Annabel Deutsch, _Kazhdan's property (T) and related properties locally compact and discrete groups_, 1992 ([era:1842/13631](https://era.ed.ac.uk/handle/1842/13631))

* [[Bachir Bekka]], P. de la Harpe and A. Valette, _Kazhdan's Propert (T)_, 2007 ([pdf](https://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf), [[BekkaHarpeValetteOnKashdanPropertyT.pdf:file]])

See also

* Wikipedia, _Kazhdan's property (T)_

The example of

* {#Bekka03} [[Bachir Bekka]], *Kazhdan's Property (T) for the unitary group of a separable Hilbert space*, Geom. funct. anal. 13, 509–520 (2003) ([doi:10.1007/s00039-003-0420-0](https://doi.org/10.1007/s00039-003-0420-0))

[[!redirects Kazhdan's property T]]


Kazhdan-Lusztig theory

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## Idea

Kazhdan-Lusztig theory is about special recursive combinatorics which appears in several setups in mathematics, most notably in representation theory where it concerns the Jordan-Hölder coefficients of certain modules. As a phenomenon it has been discovered by [[David Kazhdan]] and [[George Lusztig]], and some partial aspects independently by Deodhar. A central result is the Kazhdan-Lusztig conjecture, proved by Borho-Brylinski and by [[Masaki Kashiwara]] using [[D-module]]s and [[perverse sheaves]].

## References

* [[David Kazhdan]], [[George Lusztig]], _Representations of Coxeter groups and Hecke algebras_, Invent. Math. __53__ (1979), no. 2, 165–184, [MR81j:20066](http://www.ams.org/mathscinet-getitem?mr=560412), [doi](http://dx.doi.org/10.1007/BF01390031)

* D. Kazhdan, G. Lusztig, _Schubert varieties and Poincaré duality, in: Geometry of the Laplace operator, 185–203, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc. 1980.

* _Kazhdan-Lusztig theory_, chapter 8 in James E. Humphreys, _Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$_, Graduate Studies in Mathematics __94__, Amer. Math. Soc. 2008. xvi+289 pp.

* [[Wolfgang Soergel]], _Kazhdan-Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln_, Represent. Theory 1 (1997) 37-68, [pdf](http://www.ams.org/ert/1997-001-04/S1088-4165-97-00006-X/S1088-4165-97-00006-X.pdf); engl. version Kazhdan-Lusztig polynomials and a combinatoric for tilting modules. Represent. Theory 1 (1997) 83-114, [pdf](http://www.ams.org/ert/1997-001-06/S1088-4165-97-00021-6/S1088-4165-97-00021-6.pdf).

* R. Hotta, K. Takeuchi, T. Tanisaki, _D-modules, perverse sheaves, and representation theory_, Progress in Mathematics __236__, Birkhäuser, Boston 2008.

* Jean-Luc Brylinski, [[Masaki Kashiwara]], _Démonstration de la conjecture de Kazhdan-Lusztig sur les modules de Verma_, C. R. Acad. Sci. Paris Sér. A-B __291__ (1980), no. 6, A373–A376, [MR81k:17004](http://www.ams.org/mathscinet-getitem?mr=596075)

* Walter Borho, Jean-Luc Brylinski, _Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules._
Invent. Math. __69__ (1982), no. 3, 437–476, [MR84b:17007](http://www.ams.org/mathscinet-getitem?mr=679767), [doi](http://ams.mpim-bonn.mpg.de/leavingmsn?url=http://dx.doi.org/10.1007/BF01389364); _II. Relative enveloping algebras._, Bull. Soc. Math. France __117__ (1989), no. 2, 167–210, [MR90j:17023](http://www.ams.org/mathscinet-getitem?mr=1015807), [numdam](http://www.numdam.org/item?id=BSMF_1989__117_2_167_0)

* N. Chriss, [[V. Ginzburg]], _Representation theory and complex geometry_, Birkhäuser 1997. x+495 pp.

* Vinay V. Deodhar, _On a construction of representations and a problem of Enright_,  Invent. Math. __57__ (1980), no. 2, 101–118, [MR81f:17004](http://www.ams.org/mathscinet-getitem?mr=567193), [doi](http://dx.doi.org/10.1007/BF01390091)

* Vinay V. Deodhar, _On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells_, Invent. Math. __79__ (1985), no. 3, 499–511, [MR86f:20045](http://www.ams.org/mathscinet-getitem?mr=782232), [doi](http://dx.doi.org/10.1007/BF01388520); _II. The parabolic analogue of Kazhdan-Lusztig polynomials_, [MR89a:20054](http://www.ams.org/mathscinet-getitem?mr=916182), [doi](http://dx.doi.org/10.1016/0021-8693(87)90232-8)

* [[Anthony Joseph]], _The Enright functor on the Bernstein-Gelʹfand-Gelʹfand category $\mathcal{O}$_, Invent. Math. __67__ (1982), no. 3, 423–445, [MR84j:17005](http://www.ams.org/mathscinet-getitem?mr=664114), [doi](http://dx.doi.org/10.1007/BF01398930)

* _Kazhdan-Lusztig theory and related topics_, Proc. of the AMS Special Session at Loyola Univ., Chicago 1989. Edited by Vinay Deodhar. Contemporary Mathematics __139__, Amer. Math. Soc. 1992.

* V. Deodhar, _A brief survey of Kazhdan-Lusztig theory and related topics_,  Algebraic groups and their generalizations: classical methods (University Park, PA, 1991),  105–124, Proc. Sympos. Pure Math. __56__, Part 1, Amer. Math. Soc. 1994.

* Vinay V. Deodhar, [[Ofer Gabber]], [[Victor Kac]], _Structure of some categories of representations of infinite-dimensional Lie algebras_, Adv. in Math. __45__ (1982), no. 1, 92–116. [MR83i:17012](http://www.ams.org/mathscinet-getitem?mr=663417), [doi](http://dx.doi.org/10.1016/S0001-8708(82)80014-5)

* O. Gabber, A Joseph, _Towards the Kazhdan-Lusztig conjecture_, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 261–302, [MR83e:17009](http://www.ams.org/mathscinet-getitem?mr=644519), [numdam](http://www.numdam.org/item?id=ASENS_1981_4_14_3_261_0)

The following article proves a conjecture from above article of Deodhar, Gabber and Kac:

* Luis Casian, _Proof of the Kazhdan-Lusztig conjecture for Kac-Moody algebras (the characters_ $\mathrm{ch}\,L_{\omega\rho-\rho})$, Adv. Math. __119__ (1996), no. 2, 207–281,  [MR97k:17033](http://www.ams.org/mathscinet-getitem?mr=1390798), [doi](http://dx.doi.org/10.1006/aima.1996.0031],

* A. V. Zelevinskiĭ, _The $p$-adic analogue of the Kazhdan-Lusztig conjecture_,   Funktsional. Anal. i Prilozhen. __15__ (1981), no. 2, 9–21, 96.

* Luis G. Casian, David H. Collingwood, _The Kazhdan-Lusztig conjecture for generalized Verma modules_, Math. Z. __195__ (1987), no. 4, 581–600, [MR88i:17008](http://www.ams.org/mathscinet-getitem?mr=900346), [doi](http://dx.doi.org/10.1007/BF01166705)

[[!redirects Kazhdan-Lusztig polynomial]]


Kazimierz Kuratowski

[[!redirects Kasimierz Kuratowski]]
[[!redirects Kuratowski]]
**Kazimierz Kuratowski** (b. February 2, 1896, d. June 18, 1980) was a Polish mathematician who worked primarily in [[general topology]] and [[mathematical logic]].

He is probably best known for a celebrated theorem in [[graph]] theory: that a graph $G$ is [[planar graph|planar]] iff neither the complete graph $K_5$ on $5$ vertices nor the complete bipartite graph $K_{3, 3}$ on a pair of $3$-element sets appear as edge [[subdivisions]] of subgraphs of $G$ (closely related to Wagner's theorem that this is the case iff neither $K_5$ nor $K_{3, 3}$ appear as [[graph minors]] of $G$).

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

## Selected writings

Early discussion of [[topology]]/[[topological spaces]]:

* [[Kazimierz Kuratowski]], *Sur l'opération Ā de l'Analysis Situs*, Fundamenta Mathematicae **3** (1922) 182–199 [[doi:10.4064/fm-3-1-182-199](https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/fundamenta-mathematicae/all/3/0/92454/sur-l-operation-a-de-l-analysis-situs)]

See also:

* C. Kuratowski, _Sur la notion d'ensemble fini_ , Fund. Math. **I** (1920) pp.130-131. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm1/fm1117.pdf))

* Casimir Kuratowski (1930), "Sur le problème des courbes gauches en Topologie", _Fundamenta Mathematicae_, 15 (1):271-283. ([eudml](https://eudml.org/doc/212352))

## Related entries

* [[closed subspace]]

* [[finite set]]

[[!redirects Kuratowski]]
[[!redirects C. Kuratowski]]
[[!redirects Casimir Kuratowski]]

category:people


Kazuhiko Aomoto

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

* [InSpire page](https://inspirehep.net/authors/2433264)

## Selected writings

On the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) (via [[Gauss-Manin connections]]) of solutions to the [[Knizhnik-Zamolodchikov equation]]:

* [[Kazuhiko Aomoto]], *Gauss-Manin connection of integral of difference products*,  J. Math. Soc. Japan **39** 2 (1987) 191-208  $[$[doi:10.2969/jmsj/03920191](https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-39/issue-2/Gauss-Manin-connection-of-integral-of-difference-products/10.2969/jmsj/03920191.full)$]$

## Related entries

* [[hypergeometric function]]

category: people


Kazuhisa Shimakawa

## Selected writings

On [[equivariant algebraic K-theory]]:

* [[Kazuhisa Shimakawa]], _Note on the equivariant $K$-theory spectrum_, Publ. RIMS, Kyoto Univ. **29** (1993), 449-453 ([pdf](http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=29&iss=3&rank=5), [doi:10.2977/prims/1195167052](https://doi.org/10.2977/prims/1195167052))

On [[universal principal bundle|universal]] [[equivariant principal bundles]]:

* [[Mitutaka Murayama]], [[Kazuhisa Shimakawa]], _Universal equivariant bundles_, Proc. Amer. Math. Soc. 123 (1995), 1289-1295 ([doi:10.1090/S0002-9939-1995-1231040-9](https://doi.org/10.1090/S0002-9939-1995-1231040-9))

On the [[adjunction between topological spaces and diffeological spaces]]:

* {#SYH10} [[Kazuhisa Shimakawa]], K. Yoshida, [[Tadayuki Haraguchi]], _Homology and cohomology via enriched bifunctors_, Kyushu Journal of Mathematics, 2018 Volume 72 Issue 2 Pages 239-252 ([arXiv:1010.3336](https://arxiv.org/abs/1010.3336), [doi:10.2206/kyushujm.72.239]( https://doi.org/10.2206/kyushujm.72.239))

On a [[model structure on diffeological spaces]]:

* [[Tadayuki Haraguchi]], [[Kazuhisa Shimakawa]], _A model structure on the category of diffeological spaces_ ([arXiv:1311.5668](https://arxiv.org/abs/1311.5668))

* [[Tadayuki Haraguchi]], [[Kazuhisa Shimakawa]], _On homotopy types of diffeological cell complexes_ ([arXiv:1912.05359](https://arxiv.org/abs/1912.05359))

* {#HaraguchiShimakawa20} [[Tadayuki Haraguchi]], [[Kazuhisa Shimakawa]], _A model structure on the category of diffeological spaces, I_ ([arXiv:2011.12842](https://arxiv.org/abs/2011.12842))


Kazuo Hosomichi

* [InSpire page](https://inspirehep.net/authors/1005531)

## Selected writings

On [[black brane]] [[M2-M5 brane bound states]] and a corresponding [[Hanany-Witten effect]]:

* {#Hosomichi00} [[Kazuo Hosomichi]], *On Branes Ending on Branes in Supergravity*, JHEP 0006 (2000) 004 [[arXiv:hep-th/0002069](https://arxiv.org/abs/hep-th/0002069), [doi:10.1088/1126-6708/2000/06/004](https://doi.org/10.1088/1126-6708/2000/06/004)]

category: people


Kazushige Terui


Kazuya Kato

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

## related $n$Lab entries

* [[Bloch-Kato conjecture]]

* [[logarithmic geometry]]

category: people


Kazuya Kawasetsu

* [personal page](https://www.sci.kumamoto-u.ac.jp/math/staff/kawasetsu/eng_index.html)

## Selected writings

On [[fractional-level WZW models]] and [[logarithmic CFT]]:

* [[Kazuya Kawasetsu]], [[David Ridout]], *Relaxed highest-weight modules I: rank 1 cases*, Commun. Math. Phys. **368** (2019) 627–663  $[$[arXiv:1803.01989](https://arxiv.org/abs/1803.01989), [doi:10.1007/s00220-019-03305-x](https://doi.org/10.1007/s00220-019-03305-x)$]$

* [[Kazuya Kawasetsu]], [[David Ridout]], *Relaxed highest-weight modules II: classifications for affine vertex algebras*, Communications in Contemporary Mathematics, **24** 05 (2022) 2150037 $[$[arXiv:1906.02935](https://arxiv.org/abs/1906.02935), [doi:10.1142/S0219199721500371](https://doi.org/10.1142/S0219199721500371)$]$

category: people


Kazuya Yonekura

* [InSpire page](https://inspirehep.net/authors/1056904)

## Selected writings

On [[D=6 N=(2,0) SCFT]] and [[D=6 N=(1,0) SCFT]] ([[E-string]]-theories) and their [[anomaly cancellation]] via the [[Green-Schwarz mechanism]]:

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], [[Kazuya Yonekura]], _Anomaly polynomial of general 6d SCFTs_, Progress of Theoretical and Experimental Physics, Volume 2014, Issue 10, October 2014, 103B07 [[arXiv:1408.5572](https://arxiv.org/abs/1408.5572)]

On [[anomaly inflow]]:

* [[Edward Witten]], [[Kazuya Yonekura]], *Anomaly Inflow and the $\eta$-Invariant* [[arXiv:1909.08775](https://arxiv.org/abs/1909.08775), [spire:1755070](http://inspirehep.net/record/1755070)]

On the mod-2 [[elliptic genus]], both as a [[partition function]] of a [[superstring]] as well as via the [[string-orientation of tmf]]:

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], [[Kazuya Yonekura]], *Remarks on mod-2 elliptic genus* [[arXiv:2302.07548](https://arxiv.org/abs/2302.07548)]

category: people


KBC void

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#### Physics
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* table if contents
{:toc}

## Idea

In astrophysics/[[cosmology]], the _KBC void_ ([KBC13](#KBC13)) refers to a large neighbourhood of the [[Milky Way]] whose average matter density is considerably smaller than on average in the [[observable universe]].

In the context of [[inhomogeneous cosmology]], this means that the [[Hubble constant]] effectively observed inside the void appears larger than that given by the [[cosmological constant]] of a large-scale homogeneous [[FRW model]]. It has been suggested that this might explain away the need for the [[dark energy]] hypothesis, or at least help explain recent tensions in different measurements of the [[Hubble constant]], but the issue remains debated ([Wu-Huterer 17](#WuHuterer17)).

## Related concepts

* [[structure formation]]

## References

Due to

* {#KBC13} Ryan C. Keenan, Amy J. Barger, Lennox L. Cowie, _Evidence for a ~300 Mpc Scale Under-density in the Local Galaxy Distribution_, 2013, ApJ, 775, 62 ([arXiv:1304.2884](https://arxiv.org/abs/1304.2884))

Review in:

* {#Peebles22} [[Phillip James E. Peebles]], §4 of: *Anomalies in Physical Cosmology* [[arXiv:2208.05018](https://arxiv.org/abs/2208.05018)]

See also:

* Wikipedia, _[KBC void](https://en.m.wikipedia.org/wiki/KBC_Void)

Discussion in view of the observed [[Hubble constant]]:

* {#WuHuterer17} Hao-Yi Wu, Dragan Huterer, _Sample variance in the local measurements of the Hubble constant_, Monthly Notices of the Royal Astronomical Society, Volume 471, Issue 4, November 2017, Pages 4946–4955 ([arXiv:1706.09723](https://arxiv.org/abs/1706.09723))

Discussion in view of [[MOND]]:

* Moritz Haslbauer, Indranil Banik, Pavel Kroupa, _The KBC void and Hubble tension contradict ΛCDM on a Gpc scale − Milgromian dynamics as a possible solution_, Monthly Notices of the Royal Astronomical Society, Volume 499, Issue 2, December 2020, Pages 2845–2883 ([arXiv:2009.11292](https://arxiv.org/abs/2009.11292))


Kefeng Liu

* [webpage](http://www.math.ucla.edu/~liu/)

## Selected writings

On [[elliptic genera]] and the [[rigidity theorem for elliptic genera]]:

* [[Kefeng Liu]], _On modular invariance and rigidity theorems_, J. Differential Geom. Volume 41, Number 2 (1995), 343-396 ([euclid:jdg/1214456221](https://projecteuclid.org/euclid.jdg/1214456221))

* [[Kefeng Liu]], _On elliptic genera and theta-functions_, Topology Volume 35, Issue 3, July 1996, Pages 617-640 (doi:10.1016/0040-9383(95)00042-9)

* {#KL96} [[Kefeng Liu]], _Modular forms and topology_, Proc. of the AMS Conference on the Monster and Related Topics, Contemporary Math. (1996) ([pdf](https://www.math.ucla.edu/~liu/Research/vam.pdf), [[LiuModularFormsTopology.pdf:file]], [doi:10.1090/conm/193](http://dx.doi.org/10.1090/conm/193))

via [[super vertex operator algebras]]:

* {#DLM02} Chongying Dong, [[Kefeng Liu]], Xiaonan Ma, _Elliptic genus and vertex operator algebras_, Algebr. Geom. Topol. 1 (2001) 743-762 ([arXiv:math/0201135](http://arxiv.org/abs/math/0201135), [doi:10.2140/agt.2001.1.743](http://dx.doi.org/10.2140/agt.2001.1.743))

On the [[volume conjecture]]:

* {#ChenLiuZhu15} [[Qingtao Chen]], [[Kefeng Liu]], Shengmao Zhu, _Volume conjecture for $SU(n)$-invariants_ ([arXiv:1511.00658](https://arxiv.org/abs/1511.00658))

On the [[Green-Schwarz mechanism]] in [[heterotic string theory]] and [[Hořava-Witten theory]]:

* [[Fei Han]], [[Kefeng Liu]], [[Weiping Zhang]], _Anomaly Cancellation and Modularity. II: $E_8 \times E_8$ case_, Sci. China Math. 60, 985–994 (2017) ([arXiv:1209.4540](https://arxiv.org/abs/1209.4540), [doi:10.1007/s11425-016-9034-1](https://doi.org/10.1007/s11425-016-9034-1))

On [[gravitino]] [[anomaly cancellation]] in [[D=11 N=1 supergravity]]:

* {#HanHuangLiuZhang20} [[Fei Han]], [[Ruizhi Huang]], [[Kefeng Liu]], [[Weiping Zhang]], *Cubic forms, anomaly cancellation and modularity*, Advances in Mathematics **394** (2022) 108023 [[arXiv:2005.02344](https://arxiv.org/abs/2005.02344), [doi:10.1016/j.aim.2021.108023](https://doi.org/10.1016/j.aim.2021.108023)]

category: people


Kehe Zhu

* [personal page](https://www.albany.edu/~kzhu/english.html)

## Selected writings

On [[operator algebra]]:

* [[Kehe Zhu]], *An Introduction to Operator Algebras*, CRC Press 1993 ([ISBN:9780849378751](https://www.routledge.com/An-Introduction-to-Operator-Algebras/Zhu/p/book/9780849378751))

category: people


Keisler-Shelah isomorphism theorem

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#### Model theory
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#Contents#
* table of contents
{:toc}

## Idea
The _Keisler-Shelah isomorphism theorem_ characterizes the partially syntactic concept of elementary equivalence in purely semantic form with the help of [[ultrapower|ultrapowers]].

## Statement of theorem
In practice, the theorem is usually stated as follows: let $A$ and $B$ be [[first-order logic|first-order]] $\mathcal{L}$-[[structures]]. Then $A$ and $B$ are [[elementary equivalence|elementarily equivalent]], written $A \equiv B$, if and only if there is an ultrafilter $\mathcal{U}$ on some index set $I$ such that there is an isomorphism of [[ultrapowers]] $A^{\mathcal{U}} \simeq B^{\mathcal{U}}$.

Keisler proved, assuming [[GCH]], that when $A \models T$ and $B \models T$ have cardinality $\leq 2^{|T|}$ then they have isomorphic $|T|$-indexed ultrapowers. Shelah removed the assumption of GCH at the cost of exhibiting the isomorphism for only $2^{|T|}$-indexed ultrapowers instead.

## Examples
* The [[Ax-Kochen-Ershov theorem]] states that for any non-principal ultrafilter $\mathcal{U}$ on the primes, the valued fields (viewed as structures in [[ACVF]], where $\mathbb{Q}_p$ is the [[p-adic]] field and $\mathbb{F}_p((t))$ is the field of [[formal Laurent series]] over the [[finite field]] $\mathbb{F}_p$)

$$

\displaystyle \prod_{p} \mathbb{Q}_p/\mathcal{U} \equiv \displaystyle \prod_{p} \mathbb{F}_p((t)) / \mathcal{U}

$$

are elementarily equivalent. Assuming the continuum hypothesis (this is an example of where this technical distinction is vital), they are also isomorphic.

## Remarks
* One could view this theorem as a generalization/variant of the [[nonstandard analysis|transfer principle]] from nonstandard analysis: given any two structures with the same theory, there exists a single "nonstandard model" linking the two via their diagonal embeddings into their ultrapowers.

## Related concepts
[[ultraroot]]

[[ultrapower]]

[[ultraproduct]]

[[Los ultraproduct theorem]]

## References
* See e.g. section 10 of [this survey article](https://ms.mcmaster.ca/~hanj18/jessewebpage_files/keisler-ultraproducts.pdf) by H. Jerome Keisler.

[[!redirects Keisler-Shelah theorem]]


Keith Conrad

* [webpage](http://www.math.uconn.edu/~kconrad/)

## Selected writings

On [[tensor products]] ([[tensor product of modules|of modules]]):

* [[Keith Conrad]], *Tensor products* [part I:[pdf](http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf), [[Conrad-TensorProductsI:file]]; part II: [pdf](https://kconrad.math.uconn.edu/blurbs/linmultialg/tensorprod2.pdf), [[Conrad-TensorProductsII:file]]]

On [[bilinear forms]]:

* [[Keith Conrad]], *Bilinear forms* [[pdf](https://kconrad.math.uconn.edu/blurbs/linmultialg/bilinearform.pdf), [[Conrad-BilinearForms.pdf:file]]]

On [[splitting fields]]:

* [[Keith Conrad]], *Isomorphisms of Splitting Fields*, Galois theory notes [[[Conrad-IsoSplittingFields.pdf:file]]]

On [[cyclotomic field extensions]]:

* [[Keith Conrad]], *Cyclotomic extensions* [[pdf](https://kconrad.math.uconn.edu/blurbs/galoistheory/cyclotomic.pdf), [[Conrad-CyclotomicExtensions.pdf:file]]]

On [[SL(2,Z)]]:

* [[Keith Conrad]], *$SL_2(\mathbb{Z})$* [pdf, [[Conrad-SL2Z.pdf:file]]]

## Related entries

* [[cyclotomic field]]

category: people


Keith Dienes

* [website](http://math.arizona.edu/~www-main-2003/people/profile.php?n=dienes)

## Selected writings

On the [[Higgs field]] in [[heterotic string theory]]:

* Steven Abel, [[Keith Dienes]],
*Calculating the Higgs Mass in String Theory* ([arXiv:2106.04622](https://arxiv.org/abs/2106.04622))

## Related $n$Labentries

* [[supersymmetry]]

* [[string theory]]

category: people


Keith Hannabus

* [personal webpage](http://users.ox.ac.uk/~kch/personal.html)

* [institute webpage](https://www.maths.ox.ac.uk/people/keith.hannabuss)

## related $n$Lab entries

* [[topological T-duality]]

category: people


Keith Hardie

Keith Hardie is a South African mathematician, who has published work on homotopy theory and category theory.

* [Genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=42132)

## Selected writings

On the [[Toda bracket]] understood via [[homotopy coherent diagram|homotopy-coherent]] [[pasting diagrams]] in a [[pointed (infinity,1)-category|pointed]] [[homotopy 2-category]]:

* {#HardieKampsKieboom99} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Rudger  Kieboom]], _Higher homotopy groupoids and Toda brackets_, Homology Homotopy Appl. Volume 1, Number 1 (1999), 117-134 ([euclid:hha/1139840198](https://projecteuclid.org/euclid.hha/1139840198))

* [[Keith Hardie]], [[Howard Marcum]], [[Nobuyuki Oda]], _Bracket operations in the homotopy theory of a 2-category_, Rend. Ist. Mat. Univ. Trieste 33, 19–70 (2001) ([rendiconti:33/02](https://rendiconti.dmi.units.it/volumi/33/02.pdf))

* {#HardieKampsMarcum02} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Howard Marcum]], _The Toda bracket in the homotopy category of a track bicategory_, Journal of Pure and Applied Algebra Volume 175, Issues 1–3, 8 November 2002, Pages 109-133 (doi:10.1016/S0022-4049(02)00131-7)

On the [[fundamental 2-groupoid]] of [[Hausdorff topological spaces]] as a [[strict 2-groupoid]] ([[Grpd-enriched category]]):

* [[Keith A. Hardie]], [[Klaus H. Kamps]], [[Rudger Kieboom]], *A homotopy 2-groupoid of a Hausdorff space*. Papers in honour of Bernhard Banaschewski (Cape Town, 1996). Appl. Categ. Structures **8** (2000) 209-234 [[doi:10.1023/A:1008758412196](https://doi.org/10.1023/A:1008758412196)]

and as a [[weak 2-groupoid]]:

* [[Keith A. Hardie]], [[Klaus H. Kamps]], [[Rudger Kieboom]], *A Homotopy Bigroupoid of a Topological Space*, Applied Categorical Structures **9** (2001) 311-327 [[doi:10.1023/A:1011270417127](https://doi.org/10.1023/A:1011270417127)]

category:people

[[!redirects K.A. Hardie]]

[[!redirects K. A. Hardie]]
[[!redirects Keith A. Hardie]]


Keith Hubbard

* [webpage](http://www.faculty.sfasu.edu/hubbardke/)

category: people


Keith Olive

* [webpage](https://www.physics.umn.edu/people/olive.html)

## Selected writings

On the [[Starobisnky model of cosmic inflation]] and its embedding into [[supergravity]]:

* {#ENOV19} [[John Ellis]], [[Dimitri Nanopoulos]], [[Keith Olive]], Sarunas Verner, _A Unified No-Scale Model of Modulus Fixing, Inflation, Supersymmetry Breaking and Dark Energy_, Phys. Rev. D 100, 025009 (2019) ([arXiv:1903.05267](https://arxiv.org/abs/1903.05267), [doi:10.1103/PhysRevD.100.025009](https://doi.org/10.1103/PhysRevD.100.025009))

category: people


Kellogg Stelle

Kellogg Stelle is a professor of theoretical [[physics]] at Imperial College London.

* [webpage](http://www.imperial.ac.uk/people/k.stelle/)

* [ResearchGate page](https://www.researchgate.net/profile/Kellogg-Stelle)

## Selected writings

On [[black brane|black]] [[M2-brane]]-solutions to [[D=11 supergravity]]:

* {#DuffStelle91} [[Mike Duff]], [[Kellogg Stelle]], _Multi-membrane solutions of $D = 11$ supergravity_, Phys. Lett. B 253, 113 (1991) ([spire:299386](http://inspirehep.net/record/299386), doi:10.1016/0370-2693(91)91371-2)

On the [[Hopf-Wess-Zumino term]] of the [[M5-brane]]:

* {#KalkkinenStelle02} [[Jussi Kalkkinen]], [[Kellogg Stelle]], Section 3.2 of: _Large Gauge Transformations in M-theory_, J. Geom. Phys. 48 (2003) 100-132 ([arXiv:hep-th/0212081](https://arxiv.org/abs/hep-th/0212081))

On [[higher curvature corrections]] to [[D=11 supergravity]] and [[M-theory on G2-manifolds]]:

* H. Lu, [[Christopher Pope]], [[Kellogg Stelle]], [[Paul Townsend]], _Supersymmetric Deformations of $G_2$ Manifolds from Higher-Order Corrections to String and M-Theory_, JHEP 0410:019, 2004 ([arXiv:hep-th/0312002](https://arxiv.org/abs/hep-th/0312002))

* H. Lu, [[Christopher Pope]], [[Kellogg Stelle]], [[Paul Townsend]], _String and M-theory Deformations of Manifolds with Special Holonomy_, JHEP 0507:075, 2005 ([arXiv:hep-th/0410176](https://arxiv.org/abs/hep-th/0410176))

On [[M-theory on Calabi-Yau 5-folds]] and [[supersymmetric quantum mechanics]]:

* [[Alexander Haupt]], [[Andre Lukas]], [[Kellogg Stelle]], _M-theory on Calabi-Yau Five-Folds_, JHEP 0905:069, 2009 ([arXiv:0810.2685](https://arxiv.org/abs/0810.2685))

On the work of [[Mike Duff]]:

* [[Leron Borsten]], [[Alessio Marrani]], [[Christopher N. Pope]], [[Kellogg Stelle]], *Introduction to the special issue dedicated to Michael J. Duff FRS on the occasion of his 70th birthday*, Proceedings of the Royal Society A,
**478** (2022) 2259 [[doi:10.1098/rspa.2022.0166](https://doi.org/10.1098/rspa.2022.0166)]

## Related entries

* [[BPS charge]]

category: people

[[!redirects Kellogg S. Stelle]]


Kelly-Mac Lane graph

# Contents
* table of contents
{: toc}

## Idea

A diagrammatic calculus for [[closed categories]], akin to [[proof nets]] in linear logic and more generally [[string diagrams]] for [[monoidal categories]] and related categorical structures of "[[linear type theory|linear]]" type.

## Related concepts

* [[proof net]]

* [[Trimble rewiring]]

## References

The concept was actually introduced in

* [[Samuel Eilenberg]], [[Max Kelly]], _A generalization of the functorial calculus_, J. Algebra 3 (1966), 366--375.

They have come to be known as "Kelly--Mac Lane graphs" principally on account of their application to coherence problems for closed categories in the treatment

* {#KM71} [[Max Kelly]], [[Saunders MacLane]], _Coherence in closed categories_, JPAA 1 (1971), 97--140 ([web](http://www.sciencedirect.com/science/article/pii/0022404971900132))

Reviews include

* [[Dominic Hughes]], _Simple free star-autonomous categories and full coherence_, Journal of Pure and Applied Algebra, Volume 216, Issue 11, 2012, Pages 2386-2410, [DOI](https://doi.org/10.1016/j.jpaa.2012.03.020), [arxiv](https://arxiv.org/abs/math/0506521)

* [[John Baez]], _Classical versus quantum lambda-calculus. From lambda-terms to string diagrams_, [Nov. 2006](http://math.ucr.edu/home/baez/qg-fall2006/index.html#computation) ([pdf](http://math.ucr.edu/home/baez/qg-fall2006/f06week06b.pdf))

* [[Eugenia Cheng]], section 1 of _A relationship between trees and Kelly–Mac Lane graphs_, Math. Proc. Camb. Phil. Soc. (2006), 141, 33 ([pdf](http://www.sci-prew.inf.ua/v141/1/S0305004105009060.pdf)), [arxiv](https://arxiv.org/abs/math/0304287)

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[[!redirects EKM diagrams]]


Ken Ono

* [website](http://www.mathcs.emory.edu/~ono/)

category: people


Ken Ribet

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

## related $n$Lab entries

* [[Ribet's theorem]]

cateory: people


Ken Richardson

* [webpage](http://faculty.tcu.edu/richardson/)

## related $n$Lab entries

* [[eta invariant]]

category: people


Ken Shiozaki

* [personal page](http://www2.yukawa.kyoto-u.ac.jp/~ken.shiozaki/)

* [GoogleScholar page](https://scholar.google.com/citations?user=mbTEhB4AAAAJ&hl=en)

## Selected writings

On the [[K-theory classification of topological phases of matter]] for [[topological crystalline insulators]]:

* [[Ken Shiozaki]], [[Masatoshi Sato]], [[Kiyonori Gomi]], *Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups*, Phys. Rev. B **95** (2017) 235425 $[$[arXiv:1701.08725](https://arxiv.org/abs/1701.08725), [doi:10.1103/PhysRevB.95.235425](https://doi.org/10.1103/PhysRevB.95.235425)$]$

and using the [[Atiyah-Hirzebruch spectral sequence]]:

* [[Ken Shiozaki]], [[Masatoshi Sato]], [[Kiyonori Gomi]], *Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups*, Phys. Rev. B **106** (2022) 165103 [[arXiv:1802.06694](https://arxiv.org/abs/1802.06694), [doi:10.1103/PhysRevB.106.165103](https://doi.org/10.1103/PhysRevB.106.165103)]

category: people


Kendall tau distance

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

#Contents#
* tabla of contents
{:toc}

## Idea

An adaptation of the [[Kendall tau rank correlation]], the *Kendall tau distance* between two [[permutations]] is the [[minimum]] [[natural number|number]] of [[transpositions]] (swaps) of *adjacent* elements needed to turn one into the other. This quantity can also be defined as the total number of [[discordant pairs]] between the two [[permutations]].

(This is in contrast to the [[Cayley distance]] which counts the minimum number of [[transpositions]] of any pairs of elements, not necessarily adjacent.)

This quantity is also referred to as the *bubble sort distance*, since bubble sorting algorithms repeatedly iterate over a list and swap adjacent elements if they are out of order.

## Definition

For $n \in \mathbb{N}$ consider the [[symmetric group]] $Sym(n)$ with its [[mathematical structure|structure]] as a [[finitely generated group]] exhibited by the [[finite set]] of [[generators]] given by the *adjacent* [[transposition permutations]]

$$
  Gen
  \;\coloneqq\;
  \big\{
    \sigma_{i,i+1}
    \,\vert\,
    1 \leq i \lt n
  \big\}
  \;\subset\;
  Sym(n)
  \,.
$$

Then the *Kendall tau distance* ([Kendall 1938](#Kendall1938), see also [Diaconis 88, p. 112](#Diaconis88)) is the [[distance]] on $Sym(n)$ which is the [[graph distance]] of the corresponding [[Cayley graph]], hence the [[function]]

$$
  d_K(-,-)
  \;\colon\;
   Sym(n) \times Sym(n)
 \longrightarrow
 \mathbb{N}
    \hookrightarrow
  \mathbb{R}
$$

which sends any [[pair]] of [[permutations]] $\sigma_1, \sigma_2 \in Sym(n)$ to the [[minimum]] [[number]] $d(\sigma_1, \sigma_2) \in \mathbb{N}$ of adjacent [[transpositions]] $\sigma_{i_1, i_1 + 1}, \sigma_{i_2, i_2 + 1}, \cdots, \sigma_{i_d, i_d+1}$ such that

$$
  \sigma_2
  \;=\;
  \sigma_{i_d, i_d+1}
  \circ
  \cdots
  \circ
  \sigma_{i_2, i_2 + 1}
  \circ
  \sigma_{i_1, i_1 + 1}
  \circ
  \sigma_1
  \,.
$$

For $\beta \in \mathbb{R}_{\geq 0}$, the [[exponential]] of this distance weighted by $- \beta$ is known as the *Mallows kernel*:

\[
  \label{MallowsKernel}
  (\sigma_1,\sigma_2)
  \;\mapsto\;
  \exp
  \big(
    - \beta d_K(\sigma_1, \sigma_2)
  \big)
\]

## Properties

\begin{theorem}
The Mallows kernel (eq:MallowsKernel) is [[positive definite bilinear form|positive definite]] for all $\beta \geq 0$.
\end{theorem}

([Jiao-Vert 18, Theorem 1](#JiaoVert18))

## Related concepts

* [[Cayley distance]]

* [[Mallows kernel]],

  [[Cayley distance kernel]]

  [[Markov kernel]]

* [[kernel method]]

## References

The distance function is due to:

* {#Kendall1938} [[Maurice Kendall]], *A new measure of rank correlation*, Biometrika, Volume 30, Issue 1-2, June 1938, Pages 81–93 ([doi:10.1093/biomet/30.1-2.81](https://doi.org/10.1093/biomet/30.1-2.81))

The corresponding exponential kernel is named after

* [[C. L. Mallows]], *Non-Null Ranking Models. I*,  Biometrika Vol. 44, No. 1/2 (Jun., 1957), pp. 114-130 ([jstor:2333244](https://www.jstor.org/stable/2333244))

Textbook account:

* {#Diaconis88} [[Persi Diaconis]], Chapter 6B of: *Group Representations  in Probability  and  Statistics*, IMS Lecture Notes Monogr. Ser., 11: 198pp. (1988) ([jstor:i397389](https://www.jstor.org/stable/i397389), [ISBN: 0940600145](https://projecteuclid.org/ebooks/institute-of-mathematical-statistics-lecture-notes-monograph-series/Group-representations-in-probability-and-statistics/toc/10.1214/lnms/1215467407), [pdf](https://jdc.math.uwo.ca/M9140a-2012-summer/Diaconis.pdf))

See also:

* Wikipedia, *[Kendall tau distance](https://en.wikipedia.org/wiki/Kendall_tau_distance)*

* Cicirello_2020, *Kendall tau sequence distance: Extending Kendall tau from ranks to sequences*, EAI Endorsed Transactions on Industrial Networks and Intelligent Systems, vol. 7, no.23, ([DOI=10.4108/eai.13-7-2018.163925](http://dx.doi.org/10.4108/eai.13-7-2018.163925))

Review and [[proof]] that the exponentiated negative Kendall tau distance (the [[Mallows kernel]]) is a [[positive definite bilinear form]]:

* {#JiaoVert18} [[Yunlong Jiao]], [[Jean-Philippe Vert]], *The Kendall and Mallows Kernels for Permutations*, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 40, no. 7, pp. 1755-1769, 1 July 2018 ([doi:10.1109/TPAMI.2017.2719680](https://doi.org/10.1109/TPAMI.2017.2719680), [hal:01279273](https://hal.archives-ouvertes.fr/hal-01279273/document))

Discussion of weighted variants:

* [[Yunlong Jiao]], [[Jean-Philippe Vert]], *The Weighted Kendall and High-order Kernels for Permutations*, Proceedings of the 35th International Conference on Machine Learning, PMLR 80:2314-2322, 2018 ([arXiv:1802.08526](https://arxiv.org/abs/1802.08526), [mlr:v80/jiao18a](http://proceedings.mlr.press/v80/jiao18a.html), [hal:hal-01717385](https://hal.inria.fr/hal-01717385))

[[!redirects Kendall tau distances]]

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Kenji Fukaya

* [website](http://www.math.kyoto-u.ac.jp/~fukaya/fukaya.html)

## related $n$Lab entries

* [[Fukaya category]]

* [[string topology]]

category: people


Kenji Fukumizu

* [personal page](https://www.ism.ac.jp/~fukumizu/)

## Selected writings

On [[Bochner's theorem]] generalized to [[non-abelian groups]] in the context of [[kernel methods]]:

* [[Kenji Fukumizu]], [[Bharath Sriperumbudur]], [[Arthur Gretton]], [[Bernhard Schölkopf]], *Characteristic Kernels on Groups and Semigroups*, Advances in Neural Information Processing Systems 21 : 22nd Annual Conference on Neural Information Processing Systems 2008 ([NIPS 2008](https://neurips.cc/Conferences/2008/)), 473-480 ([mpg:5466](http://www.is.mpg.de/publications/5466), [pdf](http://www.gatsby.ucl.ac.uk/~gretton/papers/FukSriGreSch09.pdf))

Exposition:

* [[Kenji Fukumizu]], *Theory of Positive Definite Kernel and Reproducing Kernel Hilbert Space* 2008 ([pdf](https://www.ism.ac.jp/~fukumizu/H20_kernel/Kernel_7_theory.pdf), [[FukumizuPositiveDefiniteKernels.pdf:file]])

category: people


Kenji Mohri

* [webpage](https://trios.tsukuba.ac.jp/en/researcher/0000000607)

## Selected writings

On the [[Hilbert schemes of points]] of [[ADE-singularities]]:

* [[Kenji Mohri]], _Kähler Moduli Space of a D-Brane at Orbifold Singularities_, Commun. Math. Phys. 202 (1999) 669-699 ([arXiv:hep-th/9806052](https://arxiv.org/abs/hep-th/9806052))

On [[E-strings]]:

* [[Kenji Mohri]], _Exceptional String: Instanton Expansions and Seiberg-Witten Curve_, Rev. Math. Phys. 14 (2002) 913-975 ([arXiv:hep-th/0110121](https://arxiv.org/abs/hep-th/0110121))

category: people


Kenji Ueno

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

## Selected writings

On [[analytic manifolds]]:

* [[Kenji Ueno]] (ed.), _Classification of Algebraic and Analytic Manifolds: Katata Symposium Proceedings 1982_, Birkhäuser 1983 (ISBN:9780817631376)

On [[complex cobordism]] and [[conformal field theory]]:

* Toshiyuki Katsura, Yuji Shimizu, [[Kenji Ueno]], _Complex cobordism ring and conformal field theory over $\mathbb{Z}$_, Mathematische Annalen
March 1991, Volume 291, Issue 1, pp 551-571 ([journal](http://link.springer.com/article/10.1007%2FBF01445226))

Proof that the values of a [[modular functor]] at [[genus of a surface|genus]]=0 (ie. the [[conformal blocks]] on the punctured [[Riemann sphere]]) determine the full modular functor:

* [[Jørgen Ellegaard Andersen]], [[Kenji Ueno]], *Modular functors are determined by their genus zero data*, Quantum Topology **3** 3/4 (2012) 255-291 $[$[doi:10.4171/QT/29](https://doi.org/10.4171/qt/29)$]$

category: people


Kenkichi Iwasawa

* [Wikipedia entry](http://en.wikipedia.org/wiki/Kenkichi_Iwasawa)
*  John Coates, _Kenkichi Iwasawa (1917--1998)_, Notices Amer. Math. Soc. __46__:11, Nov. 1999, [pdf](http://www.ams.org/notices/199910/mem-iwasawa.pdf)

## Related $n$Lab entries

* [[function field analogy]]

category: people


Kenneth Brown

* [webpage](http://www.math.cornell.edu/~kbrown/)

## Selected writings

On abstract [[homotopy theory]] ([[localization]] to [[homotopy categories]]):

* {#Brown65} [[Edgar Brown]], _Abstract homotopy theory_, Trans. AMS 119 no. 1 (1965) $[$[doi:10.1090/S0002-9947-1965-0182970-6](https://doi.org/10.1090/S0002-9947-1965-0182970-6)$]$

Introducing [[categories of fibrant objects]] for discussion of [[homotopy categories]] of [[simplicial presheaves]] (in hindsight: [[(infinity,1)-toposes|$(\infty,1)$-toposes]]):

* K. Brown, _[[BrownAHT|Abstract homotopy theory and generalized sheaf cohomology]]_, Transactions of the American Mathematical Society **186** (1973) 419-458 $[$[jstor:1996573](http://www.jstor.org/stable/1996573)$]$

On [[algebraic K-theory]] and the [[Brown-Gersten property]]:

* [[Kenneth S. Brown]], [[Stephen Gersten]], *Algebraic K-theory as generalized sheaf cohomology*, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 266-292. Lecture Notes in Math. **341** Springer (1973) [[doi:10.1007/BFb0067062]( https://doi.org/10.1007/BFb0067062)]

On [[group cohomology]]:

* [[Kenneth Brown]],  _Cohomology of Groups_, Graduate Texts in Mathematics, **87**, Springer 1982 ([doi:10.1007/978-1-4684-9327-6](https://link.springer.com/book/10.1007/978-1-4684-9327-6))

On [[buildings]]:

*  K. Brown, _Buildings: Theory and applications_ (with Peter Abramenko). Graduate Texts in Mathematics, 248, Springer, 2008.

category: people

[[!redirects Ken Brown]]
[[!redirects Kenneth S. Brown]]


Kenneth Intriligator

[[!redirects Ken Intriligator]]

* [webpage](http://keni.ucsd.edu/)

## Selected writings

Introducing [[mirror symmetry]] of [[D=3 N=4 super Yang-Mills theory]]:

* {#IntriligatorSeiberg96} [[Ken Intriligator]], [[Nathan Seiberg]], _Mirror Symmetry in Three Dimensional Gauge Theories_, Phys. Lett.B 387 : 513-519,1996 ([arXiv:hep-th/9607207](http://arxiv.org/abs/hep-th/9607207))

On [[ADE-singularities]] in [[string theory]]:

* [[Kenneth Intriligator]], _New String Theories in Six Dimensions via Branes at Orbifold Singularities_, Adv. Theor. Math. Phys.1:271-282, 1998 ([arXiv:hep-th/9708117](https://arxiv.org/abs/hep-th/9708117))

On [[D=3 N=4 super Yang-Mills theories]] with [[compact hyperkähler manifold]] [[Coulomb branches]] obtained by [[KK-compactification]] of [[little string theories]]:

* {#Intriligator99} [[Kenneth Intriligator]], _Compactified Little String Theories and Compact Moduli Spaces of Vacua_, Phys. Rev. D61:106005, 2000 ([arXiv:hep-th/9909219](https://arxiv.org/abs/hep-th/9909219))

On the [[Hopf-Wess-Zumino term]] and the [[Whitehead integral formula]] for [[M5-branes]]:

* {#Intriligator00} [[Kenneth Intriligator]], _Anomaly Matching and a Hopf-Wess-Zumino Term in 6d, N=(2,0) Field Theories_, Nucl.Phys. B581 (2000) 257-273 ([arXiv:hep-th/0001205](https://arxiv.org/abs/hep-th/0001205))

On the [[Green-Schwarz mechanism]] in [[D=6 N=(1,0) SCFT]]:

* [[Kenneth Intriligator]], _$6d$, $\mathcal{N}=(1,0)$ Coulomb Branch Anomaly Matching_, J. High Energ. Phys. 2014, 162 (2014) ([arXiv:1408.6745](https://arxiv.org/abs/1408.6745), doi:10.1007/JHEP10(2014)162)

and from a [[higher gauge theory]]-perspective:

* [[Clay Cordova]], [[Thomas Dumitrescu]], [[Kenneth Intriligator]], from p. 18 in: _2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories_ ([arXiv:2009.00138](https://arxiv.org/abs/2009.00138))

On [[generalized global symmetries]]:

* [[Clay Cordova]], [[Thomas T. Dumitrescu]], [[Kenneth Intriligator]], [[Shu-Heng Shao]], *Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond* [[arXiv:2205.09545](https://arxiv.org/abs/2205.09545)]

## Related  entries

* [[string theory]]

* [[higher gauge theory]]

* [[Green-Schwarz mechanism]]

* [[3d mirror symmetry]]

* [[N=4 D=3 super Yang-Mills theory]]

* [[SCFT]]

* [[7-sphere]]

* [[ALE space]]

* [[conifold transition]]

* [[M-theory on G2-manifolds]]

* [[symplectic duality]]

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

category: people


Kenneth Kunen

__Kenneth Kunen__ is a mathematician, Prof. Emeritus at University of Wisconsin-Madison. His interests are in foundations and logic, model theory, set theory and set-theoretical aspects of general topology, non-associative algebraic systems and artificial intelligence.

* [web](http://www.math.wisc.edu/~kunen)
[[!redirects Ken Kunen]]
[[!redirects K. Kunen]]


Kenneth Ross

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

## Selected writings

On [[harmonic analysis]]:

* [[Edwin Hewett]], [[Kenneth Ross]], *Abstract Harmonic Analysis*

  Volume I: *Structure of Topological Groups, Integration Theory, Group Representations*,  Grundlehren der mathematischen Wissenschaften **115**, Springer 1979 ([doi:10.1007/978-1-4419-8638-2](https://link.springer.com/book/10.1007/978-1-4419-8638-2))

  Volume II: *Structure and Analysis for Compact Groups, Analysis on Locally Compact Abelian Groups*,  Grundlehren der mathematischen Wissenschaften **152** ([doi:10.1007/978-3-662-26755-4](https://link.springer.com/book/10.1007/978-3-662-26755-4))

category: people


Kenneth Slonneger

* [personal page](https://homepage.divms.uiowa.edu/~slonnegr/)

* [institute page](https://cs.uiowa.edu/people/ken-slonneger)

## Selected writings

On [[syntax]] and [[semantics]] of [[programming languages]]:

* [[Kenneth Slonneger]], [[Barry Kurtz]], *Formal Syntax and Semantics of Programming Languages*, Addison-Wesley (1995) [[webpage](https://homepage.divms.uiowa.edu/~slonnegr/plf/Book/), [pdf](https://doc.lagout.org/science/Artificial%20Intelligence/Natural%20Language%20Processing/Formal%20Syntax%20and%20Semantics%20of%20Programming%20Languages%20-%20Kenneth%20Slonneger.pdf)]

## Related entries

* [[denotational semantics]]

category: people

[[!redirects Ken Slonneger]]


Kenneth Wilson

* [Wikipedia entry](http://en.wikipedia.org/wiki/Kenneth_G._Wilson)

* [[Michael Peskin]] et al., *Ken Wilson Memorial Volume -- Renormalization, Lattice Gauge Theory, the Operator Product Expansion and Quantum Fields*, World Scientific (2015) [[doi:10.1142/9279](https://doi.org/10.1142/9279)]

  * [[Alexander Polyakov]], *Kenneth Wilson in Moscow* [[arXiv:1502.03502](https://arxiv.org/abs/1502.03502)]

## Selected writings

On [[renormalization]] and the [[renormalization group]]:

* {#Wilson71} [[Kenneth Wilson]], _Renormalization group and critical phenomena 1. Renormalization  group  and  the  Kadanoff  scaling  picture_ , , Physical review B 4(9) (1971) ([doi:10.1103/PhysRevB.4.3174](https://doi.org/10.1103/PhysRevB.4.3174))

* {#Wilson71b} [[Kenneth Wilson]], _Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior_, Phys. Rev. B4 , 3184 (1971). [doi:10.1103/PhysRevB.4.3184](https://doi.org/10.1103/PhysRevB.4.3184)

On [[confinement]] of [[quarks]] and early suggestion of [Gauge/String duality](AdS-QCD+correspondence#PolyakovGaugeStringDualityReferences) ("[[Wilson loops]]"):

* [[Kenneth G. Wilson]], *Confinement of quarks*, Phys. Rev. D **10** (1974) 2445 $[$[doi:10.1103/PhysRevD.10.2445](https://doi.org/10.1103/PhysRevD.10.2445)$]$

On research with [[Kenneth Wilson]]:

* [[Alexander Polyakov]], *Kenneth Wilson in Moscow*, in: *Ken Wilson Memorial Volume -- Renormalization, Lattice Gauge Theory, the Operator Product Expansion and Quantum Fields*, World Scientific (2015) [[doi:10.1142/9279](https://doi.org/10.1142/9279), [arXiv:1502.03502](https://arxiv.org/abs/1502.03502)]

## Related entries

* [[effective field theory]]

* [[Wilsonian RG flow]]

* [[confinement]]

* [[lattice gauge theory]]

category: people

[[!redirects Kenneth G. Wilson]]


Kenny Courser

* [institute page](https://mathdept.ucr.edu/people/kenny-courser)

## Selected writings

On [[structured cospans]]:

* [[John C. Baez]], [[Kenny Courser]], _Structured cospans_, Theory and Applications of Categories **35** 48 (2020) 1771-1822 $[$[arXiv:1911.04630](https://arxiv.org/abs/1911.04630), [tac:35-48](http://www.tac.mta.ca/tac/volumes/35/48/35-48abs.html)$]$

* [[John C. Baez]], [[Kenny Courser]], *Structured cospans* talk notes (2019) ([pdf](http://math.ucr.edu/home/baez/SYCO4/courser_syco4.pdf))

category: people


Kent R. Fuller

* [institute page](https://math.uiowa.edu/people/kent-fuller)

## Selected writings

On [[rings]] and [[modules]] ([[algebra]]):

* [[Frank W. Anderson]], [[Kent R. Fuller]], _Rings and Categories of Modules_, Graduate Texts in Mathematics, **13** Springer (1992) [[doi:10.1007/978-1-4612-4418-9](https://doi.org/10.1007/978-1-4612-4418-9)]

category: people

[[!redirects Kent Fuller]]


Kenta Cho

* [personal page](https://group-mmm.org/~kenta/)

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

## Selected writings

On [[categorical semantics]] of [[quantum programming languages]] on [[von Neumann algebras]]

for the [[quantum lambda-calculus]]:

* [[Kenta Cho]], [[Abraham Westerbaan]], *Von Neumann Algebras form a Model for the Quantum Lambda Calculus*, QPL 2016 [[arXiv:1603.02133](https://arxiv.org/abs/1603.02133), [pdf](https://www.cs.ru.nl/K.Cho/papers/model-qlc.pdf), slides:[pdf](http://qpl2016.cis.strath.ac.uk/pdfs/2cho-final.pdf), [[Cho-vNQuantumCalculus.pdf:file]]]

and for [[QPL]]:

* [[Kenta Cho]], *Semantics for a Quantum Programming Language by Operator Algebras*, EPTCS **172** (2014) 165-190 [[arXiv:1412.8545](https://arxiv.org/abs/1412.8545), [doi:10.4204/EPTCS.172.12](https://doi.org/10.4204/EPTCS.172.12)]

category: people


Kentaro Hori

* [website](http://www.perimeterinstitute.ca/index.phpindex.php?option=com_content&task=view&id=30&Itemid=7&view_directory=1&pi=1766)

## Selected writings

On [[D=3 N=2 super Yang-Mills theory]]:

* [[Jan de Boer]], [[Kentaro Hori]], [[Yaron Oz]], _Dynamics of N=2 Supersymmetric Gauge Theories in Three Dimensions_, Nucl. Phys. B500:163-191, 1997 ([arXiv:hep-th/9703100](https://arxiv.org/abs/hep-th/9703100))

On [[mirror symmetry]] between [[Higgs branches]]/[[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]] (with emphasis of [[Hilbert schemes of points]]):

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], _Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes_, Nucl. Phys. B493:101-147, 1997 ([arXiv:hep-th/9611063](https://arxiv.org/abs/hep-th/9611063))

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Zheng Yin, _Mirror Symmetry in Three-Dimensional Gauge Theories, $SL(2,\mathbb{Z})$ and D-Brane Moduli Spaces_, Nucl. Phys. B493:148-176, 1997 ([arXiv:hep-th/9612131](https://arxiv.org/abs/hep-th/9612131))

On [[geometric engineering of QFT]] on [[D4-D6 brane intersections]] subject to the [[s-rule]]:

* [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Section 3 of: _Strong Coupling Dynamics of Four-Dimensional N=1 Gauge Theories from M Theory Fivebrane_, Adv. Theor. Math. Phys.1:1-52, 1998 ([arXiv:hep-th/9706082](https://arxiv.org/abs/hep-th/9706082))

On [[string theory]] [[perturbative string theory vacuum|vacua]] [[KK-compactification|compactified]] on [[orbifolds]] and [[orientifolds]]:

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 3 of _Triples, Fluxes, and Strings_, Adv.Theor.Math.Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170))

category: people


Kenzo

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#### Algebraic topology
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#### Computation
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#Contents#
* table of contents
{:toc}

## Idea

*Kenzo* is a [[computer program|software]] for computations in [[constructive mathematics|constructive]] [[algebraic topology]] ([[computational topology]]).

## Related entries

* [[list of mathematics software]]

## References

* [Kenzo Homepage](https://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/)

* Jonathan Heras, Vico Pascual, Ana Romero, [[Julio Rubio]], *Integrating multiple sources to answer questions in Algebraic Topology*, Lectures Notes in Artificial Intelligence **6167** (2010) [[arXiv:1005.0749](https://arxiv.org/abs/1005.0749)]

Based on :

* [[Julio Rubio]], [[Francis Sergeraert]], *Constructive Algebraic Topology*, Bulletin des Sciences Mathématiques
**126** 5 (2002) 389-412 [[arXiv:math/0111243](https://arxiv.org/abs/math/0111243), doi:10.1016/S0007-4497(02)01119-3]

Exposition:

* [[Julio Rubio]], *Formalization of Mathematics: why Algebraic Topology?*, MAP Spring School 2012 [[pdf](http://www-sop.inria.fr/manifestations/MapSpringSchool/contents/JulioRubio.pdf)]

A rewrite of Kenzo in [[Haskell]]:

* [[Mitchell Riley]]: [github.com/mvr/at](https://github.com/mvr/at)

category: reference


Kenzo Ishikawa

## Selected writings

On finite [[wave packet]] effects ([[Stokes phenomena]]) in [[scattering matrix]]-elements:

* [[Kenzo Ishikawa]], Yutaka Tobita, *Matter-enhanced transition probabilities in quantum field theory*, Annals Phys. 344 (2014) 118-178 ([arXiv:1206.2593](https://arxiv.org/abs/1206.2593))

* [[Kenzo Ishikawa]], Kenji Nishiwaki, Kin-ya Oda, *New effect in wave-packet scatterings of quantum fields: Saddle points, Lefschetz thimbles, and Stokes phenomenon* ([arXiv:2102.12032](https://arxiv.org/abs/2102.12032))

category: people


Kepler's conjecture

#Contents#
* table of contents
{:toc}

## Idea

The [[conjecture]] that the densest [[sphere packing]] in 3-dimesnions is given by spheres centered on the cubical lattice.

Proven with [[computer proof assistants]] by the [[Flyspeck project]].

## References

See also

* Wikipedia. _[Kepler conjecture](https://en.wikipedia.org/wiki/Kepler_conjecture)_

[[!redirects Kepler conjecture]]