Kan fibration

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
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## Definition

A Kan fibration is one of the notions of [[fibrations of simplicial sets]]:

\begin{definition}
A _Kan fibration_ is a [[morphism]] $\pi \colon Y \to X$ of [[simplicial set|simplicial sets]] with the [[lifting property]] for all [[horn]]-inclusions.
\end{definition}

This means that for

$$
 \array{
   \Lambda^k[n] &\longrightarrow& Y
   \\
   \big\downarrow && \big\downarrow\mathrlap{{}^\pi}
   \\
   \Delta^n &\longrightarrow& X
 }
$$

a [[commuting square]] with $n\ge 1$ and $0\le k\le n$, there always exists a [[lift]]

\[
  \label{HornFiller}
 \array{
   \Lambda^k[n] &\longrightarrow& Y
   \\
   \big\downarrow &\nearrow& \big\downarrow\mathrlap{{}^\pi}
   \\
   \Delta^n &\longrightarrow& X
 }
 \,.
\]

In terms of the canonical [[powering]] of simplicial sets over sets, this is equivalent to the morphisms

$$
  Y^{\Delta[n]}
  \to
  Y^{\Lambda^k[n]} \times_{X^{\Lambda^k[n]}} X^{\Delta^k[n]}
$$

all being [[epimorphisms]]. (Here, for instance, $Y^{\Lambda^k[n]}$ is the set of tuples of $(n-1)$-cells in $Y$ that glue along their boundaries to an image of the $k$th $n$-[[horn]].)

## Illustration

Kan fibrations are combinatorial analogs of [[Serre fibration|Serre]] [[fibrations]]  of [[topological spaces]].  In fact, under the [[Quillen equivalence]] of the standard [[model structure on topological spaces]] and the standard [[model structure on simplicial sets]], Kan fibrations map to Serre fibrations.

Recall the shape of the [[horns]] in low dimension.

* -**$n=1$**- The horns $\Lambda^1_0$ and $\Lambda^1_1$ of the 1-[[simplex]] are just copies of the 0-[[simplex]] $\Delta^0$ regarded as the left and right endpoint of $\Delta^1$. For $n= 1$ the above condition says that for $\pi : Y \to X$ a Kan fibration we have
  $$
    \array{
       Y &\ni &
       y
       \\
      \downarrow^\pi
       \\
       X &\ni& \pi(y) &\stackrel{\forall f}{\to}& x
    }
    \;\;\;\;\;\;
    \Rightarrow
    \;\;\;\;\;\;
    \array{
       Y &\ni&
       y &\stackrel{\exists \hat f}{\to}& \exists \hat x
       \\
       \downarrow^\pi
       \\
       X &\ni& \pi(y) &\stackrel{f = \pi(\hat f)}{\to}& x = \pi(x)
    }
  $$
  corresponding to the lifting diagram
  $$
   \array{
     \Lambda_1^1 &\stackrel{y}{\to}& Y
     \\
     \downarrow &{}^{\hat f}\nearrow& \downarrow^\pi
     \\
     \Delta^1 &\stackrel{f}{\to}& X
   }
   \,.
  $$

* -**$n=2$**- the horn $\Lambda^2_1$ consists of the
two top sides of a triangle. For this the Kan condition
says that for any two composable 1-cells in $Y$ that
have a "composite up to a 2-cell" in $X$, there exists
a corresponding "composite up to a 2-cell" in $Y$ that
projects down to the one in $X$:

  $$
  \array{
    &&&&& y_2
    \\
    &&&& \nearrow && \searrow
    \\
    Y &\ni& & y_1 &&&& y_3
    \\
    \downarrow^\pi
    \\
    X &\ni& &&& \pi(y_2)
    \\
    &&& & \nearrow &\Downarrow^{\forall h}& \searrow
    \\
    &&& \pi(y_1) &&\to&& \pi(y_2)
  }
  \;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;
  \array{
    &&&&& y_2
    \\
    &&&& \nearrow &\Downarrow^{\exists \hat h}& \searrow
    \\
    Y &\ni& & y_1 &&\stackrel{\exists}{\to}&& y_3
    \\
    \downarrow^\pi
    \\
    X &\ni& &&& \pi(y_2)
    \\
    &&& & \nearrow &\Downarrow^{h = \pi(\hat h)}& \searrow
    \\
    &&& \pi(y_1) &&\to&& \pi(y_2)
  }
  $$
  This corresponds to the lifting diagram
  $$
   \array{
     \Lambda_2^1 &\stackrel{y}{\to}& Y
     \\
     \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi
     \\
     \Delta^2 &\stackrel{h}{\to}& X
   }
   \,.
  $$

  * Crucial is this condition for the _outer horns_ $\Lambda^n_0$ and $\Lambda^n_n$, where it says that the above works not only when edges are composable, but also when they meet just at  their sources or their targets. For instance for the horn $\Lambda^2_2$ the picture is
    $$
    \array{
      &&&&& y_2
      \\
      &&&& && \searrow
      \\
      Y &\ni& & y_1 &&\to&& y_3
      \\
      \downarrow^\pi
      \\
      X &\ni& &&& \pi(y_2)
      \\
      &&& & \nearrow &\Downarrow^{\forall h}& \searrow
      \\
      &&& \pi(y_1) &&\to&& \pi(y_2)
    }
    \;\;\;\;\;\;
    \Rightarrow
    \;\;\;\;\;\;
    \array{
      &&&&& y_2
      \\
      &&&& {}^\exists\nearrow &\Downarrow^{\exists \hat h}
      &    \searrow
      \\
      Y &\ni& & y_1 &&\stackrel{\exists}{\to}&& y_3
      \\
      \downarrow^\pi
      \\
      X &\ni& &&& \pi(y_2)
      \\
      &&& & \nearrow &\Downarrow^{h = \pi(\hat h)}& \searrow
      \\
      &&& \pi(y_1) &&\to&& \pi(y_2)
    }
    $$
    This corresponds to the lifting diagram
    $$
     \array{
       \Lambda_2^2 &\stackrel{y}{\to}& Y
       \\
       \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi
       \\
       \Delta^2 &\stackrel{h}{\to}& X
     }
     \,.
    $$

## Variants

### Minimal Kan fibration

A Kan fibration $p : E \to B$ is called
a **[[minimal Kan fibration]]**
if for all cells $x,y : \Delta[n] \to E$
the condition $p(x) = p(y)$ and $\partial_i x = \partial_i y$ implies for all $k$ that $\partial_k x = \partial_k y$.

### Quasi-fibration

A **quasi-fibration** or **weak Kan fibration** or **inner Kan fibration** of simplicial sets is defined as above, but with the lifting property only imposed in _inner horns_: $\Lambda^n_k$ with $0 \lt k \lt (n-1)$, not the _outer horns_ $\Lambda^n_0$ and $\Lambda^n_n$.

This weakened condition then says that _composition_ of cells may be lifted through the quasi-fibration, but not necessarily [[inverse|inversion]] of 1-cells.  See [[fibrations of quasi-categories]] for more details.

### Left and right Kan fibration

Similarly, a **left Kan fibration** is one that has the lifting property for all horns except possibly the last one.
and a **right Kan fibration** is one that has the lifting property for all horns except possibly the first one.  See [[fibrations of quasi-categories]] for more details.

## Examples
 {#Examples}

\begin{example}\label{EmptyBundleIsKanFibration}
**([[empty bundle]] is [[Kan fibration]])**
Every [[empty bundle]] $\varnothing \to X$ is a [[Kan fibration]], since none of the [[commuting squares]] that one would have to [[lifting property|lift in]] actually exist

  $$
    \array{
      \Lambda_k^n &\overset{ \not \exists }{\longrightarrow}& \varnothing
      \\
      \big\downarrow && \big\downarrow
      \\
      \Delta^n &\longrightarrow& B
    }
  $$

(keeping in mind that the [0-simplex has no horns](horn#ZeroSimplexHasNoHorns), hence that all horns are [[inhabited set|inhabited]], so that there is *no* [[morphism]] from any horn to the [[empty simplicial set]], this being a [[strict initial object]]).
\end{example}

## Properties

### Surjective Kan fibrations
 {#SurjectiveKanFibrations}

Notice that a Kan fibration need *not* be a [[surjection]] in any sense:
\begin{example}
 \label{KanFibrationNeedNotBeSurjective}
  For $X \,\in\, sSet$ any [[simplicial set]], the unique morphism from the [[initial object|initial]] simplicial set (which is the [[empty set]] in each degree) is a Kan fibration (the [[empty bundle]]):
$$
  \varnothing \underset{\in KanFib}{\longrightarrow} X
  \,.
$$
This is due to the fact that *[the 0-simplex has no horns](horn#ZeroSimplexHasNoHorns)*, namely that all [[horns]] appearing in the Kan condition (eq:HornFiller) are [[inhabited set|inhabited]], so that none of them has any morphism to $\varnothing$ (which is a [[strict initial object]]), so that the horn filler condition for the [[empty bundle]] is vacuous and hence satisfied.
\end{example}

But:
\begin{lemma}
  \label{KanFibrationWhichIsEpiInDegreeZeroIsEpi}
  As soon as a Kan fibration is a [[surjection]] in degree 0, then it is a [[surjection]] in all degrees (and [hence](SimpSet#EpimorphismsOfSimplicialSetsDetectedDegreewise) an [[epimorphism]] of simplicial sets):
$$
  X \overset{f}{\underset{\in KanFib}{\longrightarrow}} Y
  \;\;\text{and}\;\;
  f_0 \; \text{is surjective}
  \;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;
  \underset{n \in \mathbb{N}}{\forall}
  \;\;
  f_n \; \text{is surjective}
$$
\end{lemma}
\begin{proof}
  Let $\Delta[n] \xrightarrow{\sigma_n} Y$ be any [[n-simplex|$n$-simplex]] (seen, under the [[Yoneda lemma]], as a morphism from the standard [[n-simplex|$n$-simplex]]). We equivalently need to show that this [[lift|lifts]] through $f$, in that we have a [[commuting diagram]] of this form:
$$
  \array{
    && X
    \\
    &
    \mathllap{{}^{\widehat{\sigma_n}}}\nearrow
    &
    \big\downarrow \mathrlap{ {}^{f} }
    \\
    \Delta[n]
    &\underset{\sigma_n}{\longrightarrow}&
    Y
    \mathrlap{\,.}
  }
$$
Now, by the assumption that $f_0$ is surjective, we do have such a lift at least for any [[vertex]] $v$ of $\sigma_n$, hence we have a [[commuting diagram]] of this form:
$$
  \array{
    \Delta[0]
    &\overset{\widehat{v}}{\longrightarrow}&
    X
    \\
    \big\downarrow
    &&
    \big\downarrow\mathrlap{{}^{f}}
    \\
    \Delta[n]
    &\underset{\sigma_n}{\longrightarrow}&
    Y
    \mathrlap{\,.}
  }
$$
But the left morphism is an [[acyclic cofibration]] in the [[classical model structure on simplicial sets]]. These have the [[left lifting property]] against Kan fibrations, meaning that this last square has a [[lift]] $\widehat{\sigma_n}$. This exhibits the desired preimage of $\sigma_n$.
\end{proof}

In fact:
\begin{lemma}
  \label{KanFibrationWhichIsEpiOnComponentsIsEpi}
  As soon as a Kan fibration $f$ is a [[surjection]] on [[connected components]], $\pi_0(f) \colon \pi_0(X_\bullet) \twoheadrightarrow \pi_0(Y_\bullet)$, then it is a [[surjection]] in all degrees (and [hence](SimpSet#EpimorphismsOfSimplicialSetsDetectedDegreewise) an [[epimorphism]] of simplicial sets):
\end{lemma}
\begin{proof}
  By definition (of [[simplicial homotopy groups]]), all [[vertices]] in a connected component are connected by some [[zig-zag]] $Z$ of [[edges]]. But the inclusion of an endpoint vertex $\Delta[0] \hookrightarrow Z$ into (the abstract shape of) such a zig-zag is evidently an [[acyclic cofibration]], hence has the [[left lifting property]] against the given Kan fibration (by the [[classical model structure on simplicial sets]]). Evaluating this lift at the other endpoint shows that the Kan fibration is surjective on vertices as soon as it is surjective on connected components. Therefore the claim follows with Lemma \ref{KanFibrationWhichIsEpiInDegreeZeroIsEpi}.
\end{proof}

In summary:
\begin{proposition}\label{EveryPiZeroEpiIsResolvedByASurjectiveKanFibration}
  Let
  $X \xrightarrow{f} Y$ be any morphism of [[simplicial sets]] which is a [[surjection]] on [[connected components]]: $\pi_0(f) \,\colon\, \pi_0(X) \twoheadrightarrow \pi_0(Y)$ (hence presenting an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] of [[infinity-groupoids|$\infty$-groupoids]], by [this Prop.](effective+epimorphism+in+an+infinity1-category#EffectiveEpisOfInfinityGroupoids)). Then $f$ factors as a [[simplicial weak equivalence]] followed by a Kan fibration which is a degreewise [[surjection]] (and [hence](SimpSet#EpimorphismsOfSimplicialSetsDetectedDegreewise) an [[epimorphism]] of [[simplicial sets]]).
\end{proposition}
\begin{proof}
  The factorization follows by standard constructions such as the [[factorization lemma]] or the existence of the [[classical model structure on simplicial sets]]. With this the claim follows by Lemma \ref{KanFibrationWhichIsEpiOnComponentsIsEpi}.
\end{proof}

\begin{remark}\label{SurjectivityOfAcyclicKanFibrations}
The assumption in Prop. \ref{EveryPiZeroEpiIsResolvedByASurjectiveKanFibration} is met in particular for [[acyclic Kan fibrations]]. In this special case the above factorization is trivial in that acyclic Kan fibrations are already degreewise surjective themselves (see the discussion [there](acyclic+Kan+fibration#AcyclicKanFibrationsAreSurjective)).
\end{remark}

\linebreak

### Acyclic Kan fibrations and weak homotopy equivalences

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

The [[acyclic Kan fibrations]] morphisms $f : X \to Y$ of Kan complexes that are both Kan fibrations as well as [[model structure on simplicial sets|weak equivalences]] in that they induce isomorphisms on all [[simplicial homotopy groups]] (i.e. the __[[acyclic fibrations]]__ of Kan complexes) are precisely the morphisms that have the [[weak factorization system|right lifting property]] with respect to all [[boundary of a simplex|boundary inclusions]] $\partial \Delta^n \hookrightarrow \Delta^n$:
  $$
    \array{
      \partial \Delta[n] &\to& X
      \\
      \downarrow &{}^\exists\nearrow& \downarrow^f
      \\
      \Delta[n] &\to& Y
    }
    \,.
  $$

=--

e.g. ([Goerss-Jardine, chapter I](#GoerssJardine96))

+-- {: .num_cor}
###### Corollary

Kan fibrations and acyclic Kan fibrations are both stable under [[pullback]].

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

Because every class of morphisms defined by a [[weak factorization system|right lifting property]] is stable under pullback.

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+-- {: .num_remark}
###### Remark
From this it follows readily that [[Kan complexes]] form a Brownian [[category of fibrant objects]].
=--

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

A Kan fibration $f\colon X\to Y$ is acyclic precisely if the [[fiber]] $f^{-1}(y)$ over each vertex $y$ is [[contractible]].

=--

Purely combinatorial proofs of this statement include ([Joyal 2008, prop. 8.23](#Joyal08), [Riehl-Verity 13, lemma 5.4.16](#RiehlVerity13))

### Pullback and homotopy pullback

+-- {: .num_lemma #PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence}
###### Lemma

Let $p \colon X \longrightarrow Y$ be a [[Kan fibration]], def. \ref{KanFibration}, and let $f_1,f_2 \colon A \longrightarrow X$ be two morphisms. If there is a [[left homotopy]] $f_1 \rightarrow f_2$ between these maps, then there is a fiberwise [[homotopy equivalence]], between the [[pullback]] fibrations $f_1^\ast X \simeq f_2^\ast X$.

=--

(e.g. [Goerss-Jardine 96, chapter I, lemma 10.6](#GoerssJardine96))

See also at _[[homotopy pullback]]_.

### On nerves of groupoids

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

A [[functor]] $F \colon C \to D$ between [[groupoids]] is [[k-surjective functor|k-surjective]] for all $k$ and hence a surjective [[equivalence of categories]] precisely if under the [[nerve]] $N(F) : N(C) \to N(D)$ it induces an acyclic fibration of Kan complexes;

=--

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

We know that both $N(C)$ and $N(D)$ are Kan complexes. By the above theorem it suffices to show that $N(f)$ being a surjective equivalence is the same as having all lifts

$$
  \array{
    \delta \Delta[n] &\to& N(C)
    \\
    \downarrow &{}^\exists\nearrow& \downarrow^{N(F)}
    \\
    \Delta[n] &\to& N(D)
  }
  \,.
$$

We check successively what this means for increasing $n$:

* $n= 0$. In degree 0 the boundary inclusion is that of the empty set into the [[nLab:point|point]] $\emptyset \hookrightarrow {*}$. The lifting property in this case amounts to saying that every point in $N(D)$ lifts through $N(F)$.
  $$
    \array{
      \emptyset &\to& N(C)
      \\
      \downarrow &{}^\exists\nearrow& \downarrow^{N(F)}
      \\
      {*} &\to& N(D)
    }
    \Leftrightarrow
    \array{
      && N(C)
      \\
      &{}^\exists\nearrow& \downarrow^{N(F)}
      \\
      {*} &\to& N(D)
    }
    \,.
  $$
  This precisely says that $N(F)$ is surjective on 0-cells and hence that $F$ is surjective on objects.

* $n=1$. In degree 1 the boundary inclusion is that of a pair of points as the endpoints of the interval
  $\{\circ, \bullet\} \hookrightarrow \{\circ \to \bullet\}$. The lifting property here evidently is  equivalent to saying that for all objects $a,b \in Obj(C)$ all elements in $Hom(F(a),F(b))$ are hit. Hence that $F$ is a [[nLab:full functor|full functor]].

* $n=2$. In degree 2 the boundary inclusion is that of the triangle as the boundary of a filled triangle. It is sufficient to restrict attention to the case that the map $\partial \Delta[2] \to N(C)$ sends the top left edge of the triangle to an identity. Then the lifting property here evidently is  equivalent to saying that for all objects $a,b \in Obj(C)$ the map $F_{a,b} : Hom(a,b) \to Hom(F(a),F(b))$ is injective. Hence that $F$ is a [[nLab:faithful functor|faithful functor]].
  $$
    \left(
    \array{
      && a
      \\
      & {}^{Id_a}\nearrow && \searrow^{f}
      \\
      a &&\stackrel{g}{\to}&& b
    }
    \right)
    \stackrel{N(F)}{\mapsto}
    \left(
    \array{
      && a
      \\
      & {}^{Id_a}\nearrow &\Downarrow^=& \searrow^{F(f)}
      \\
      a &&\stackrel{F(g)}{\to}&& b
    }
    \right)
  $$

=--

### Universal Kan fibration

See at _[[universal Kan fibration]]_.

## Relation to other concepts

* Kan fibrations and quasi-fibrations are fibrations in two common [[model structure on simplicial sets|model structures on simplicial sets]].

* Recall that the [[horn]] $\Lambda^k[n]$ is the boundary of the $n$-[[simplex]] $\Delta^n$ with one face removed. If in the above definition one replaces horns with the full boundaries of simplices, one obtaines the definition of a [[hypercover]], the acyclic fibrations in the classical [[model structure on simplicial sets]].

* A simplicial set $X$ for which the unique morphism $X \to pt$ to the [[terminal object|terminal simplicial set]] is a Kan fibration is called a [[Kan complex]].

* A simplicial set $X$ for which the unique morphism $X \to pt$ to the [[terminal object|terminal simplicial set]] is a quasi-fibration/weak Kan fibration is called a [[quasi-category]].

* Just as the underlying simplicial set of a [[simplicial group]] is a [[Kan complex]] (see algorithm at [[simplicial group]]), so also given any simplicial morphism $f : G\to H$ of simplicial groups for which in each dimension, $n$, the homomorphism $f_n : G_n \to H_n$ is an [[epimorphism]], then the underlying simplicial map of simplicial sets is a Kan fibration. (Apart from a careful choice of section in each dimension, the proof can be constructed from the algorithm given in [[simplicial group]].)

* A morphism of simplicial sets that has the left [[lifting property]] with respect to all Kan fibrations is called an [[anodyne morphism]].

## Related concepts

* **Kan fibration**, [[anodyne morphism]]

  * [[acyclic Kan fibration]]

* [[Kan complex]]

* [[right/left Kan fibration]], [[right/left anodyne map]]

* [[inner fibration]]

* [[Cartesian fibration]]

* [[simplicial homotopy theory]]

* [[homotopy Kan fibration]]

## References

The original definition is due to [[Daniel M. Kan]], see Definition 3.1 in

* [[Daniel M. Kan]], _A combinatorial definition of homotopy groups_, The Annals of Mathematics 67:2 (1958), 282–312.  [doi](http://dx.doi.org/10.2307/1970006).

This was extended to arbitrary [[simplicial objects]] via the [[Yoneda embedding]] in Definition 3.2 of

* [[Daniel M. Kan]], _On c.s.s. categories_, Boletín de la Sociedad Matemática Mexicana 2 (1957), 82–94.  [PDF](https://dmitripavlov.org/scans/kan-on-css-categories.pdf).

A standard textbook account is

* {#GoerssJardine96} [[Paul Goerss]], [[Rick Jardine]], _[[Simplicial homotopy theory]]_, 1996

That [[topological realization]] takes Kan fibrations to [[Serre fibrations]] is due to:

* {#Quillen68} [[Daniel Quillen]], *The geometric realization of a Kan fibration is a Serre fibration*, Proc. AMS 19 (1968) 1499-1500 [[doi:1968-019-06/S0002-9939-1968-0238322-1](https://www.ams.org/journals/proc/1968-019-06/S0002-9939-1968-0238322-1), [pdf](https://www.ams.org/journals/proc/1968-019-06/S0002-9939-1968-0238322-1/S0002-9939-1968-0238322-1.pdf)]

reviewed in

* {#GoerssJardine99} [[Paul Goerss]], [[J. F. Jardine]], Thm. 10.10 in: *[[Simplicial homotopy theory]]*, Progress in Mathematics, Birkhäuser (1999), Modern Birkhäuser  Classics (2009) [[doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/)]

See also

* {#Joyal08} [[André Joyal]], *[[The Theory of Quasi-Categories and its Applications]]*, lectures at _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, CRM 2008 ([pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [[JoyalTheoryOfQuasiCategories.pdf:file]])

* {#RiehlVerity13} [[Emily Riehl]], [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_, ([arXiv:1310.8279](http://arxiv.org/abs/1310.8279))

A structured/constructive analogue suited to purposes like [[homotopy type theory]] with equivalent homotopy theory is the notion of effective Kan fibration from

* Benno van den Berg, Eric Faber, _Effective Kan fibrations in simplicial sets_, [arXiv:2009.12670](https://arxiv.org/abs/2009.12670)
* Benno van den Berg, _Effective Kan fibrations in simplicial sets_,  Bohemian Logical & Philosophical Café, Feb 2021, [yt](https://www.youtube.com/watch?v=i1TkX_86rLo)

[[!redirects Kan fibrations]]


Kan lift

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _Kan lift_ in a [[2-category]] or in a [[bicategory]] is [[duality|dual]] to the notion of [[Kan extension]], by a process of turning the 1-cells around, much in the same way as a [[homotopy lifting property]] is dual to a [[homotopy extension property]] (in [[model category|model category theory]], for instance).

Informally, a Kan lift is a best approximate solution to the problem of finding a lift $\widetilde{f}: A \to B$ of an arrow ([[morphism]]) $f: A \to C$ through an arrow $p: B \to C$, as in the [[diagram]]

$$\array{
 & & B \\
 & & \downarrow p \\
A & \overset{f}{\to} & C
}$$

Of course, lifts typically don't literally exist in the sense of an equation $p \circ \widetilde{f} = f$ or an [[isomorphism]] $p \circ \widetilde{f} \cong f$. But in good situations, one may have the next best thing: a [[2-morphism|2-cell]] $p \circ \widetilde{f} \Rightarrow f$ which is universal among 2-cells of this form. This gives the notion of _right_ Kan lift. The notion of _left_ Kan lift is similar, but with 2-cells in the opposite direction.

### Terminology ###

Usually, when working within a 2-categorical context, Kan lifts are simply refered to as _lifts/liftings_, just as what happens with [[Kan extension|Kan extensions]].

## Definition ##

As in the discussion at [[Kan extension]], there is a local notion of Kan lift and a global notion. Expanding on the informal description above, we give the local notion first, followed by the (conceptually more perspicuous) global notion in terms of [[adjoint functor]]s.

### Local description ###

Given 1-cells $p: B \to C$, $f: A \to C$ in a [[bicategory]], a **right Kan lift** of $f$ through $p$, denoted $Rift_p f$, is a 1-cell $\widetilde{f}: A \to B$ equipped with a 2-cell

$$\varepsilon: p \circ \widetilde{f} \Rightarrow f$$

satisfying the [[universal property]]: given any pair $(g: A \to B, \eta: p \circ g \Rightarrow f)$, there exists a unique 2-cell

$$\zeta: g \Rightarrow \widetilde{f}$$

such that $\varepsilon \bullet (p \circ \zeta) = \eta$. (Here $\circ$ refers to composition across a 0-cell, and $\bullet$ to composition across a 1-cell.) As with any universal description, the pair $(Rift_p f, \varepsilon)$ is unique up to unique 2-cell isomorphism.

A **left Kan lift** of $f$ through $p$, denoted $Lift_p f$, is a 1-cell $\widetilde{f}: A \to B$ equipped with a 2-cell

$$\eta: f \Rightarrow p \circ \widetilde{f}$$

such that given any pair $(g: A \to B, \theta: f \Rightarrow p \circ g)$, there exists a unique 2-cell

$$\zeta: \widetilde{f} \Rightarrow g$$

such that $(p \circ \zeta) \bullet \eta = \theta$.

### Global description ###

Given $p: B \to C$ and a 0-cell $A$ in a bicategory, if the right Kan lift $Rift_p f$ exists for any $f: A \to C$, then we speak of a **global** Kan lift. When this is the case, we may define a functor between hom-categories

$$Rift_p: [A, C] \to [A, B]$$

which at the object level is of course $f \mapsto Rift_p f$. At the morphism level, given a 2-cell

$$\eta: f \Rightarrow g: A \to C$$

there is an induced 2-cell $Rift_p f \Rightarrow Rift_p g$, the one which corresponds (by the universal property of $Rift_p g$) to the composite

$$p \circ Rift_p f \overset{\varepsilon}{\Rightarrow} f \overset{\eta}{\Rightarrow} g.$$

A standard universality argument shows that $Rift_p$ thus defined is functorial.

**Proposition:** $Rift_p$ is right adjoint to the functor
$[A, p]: [A, B] \to [A, C]$ obtained by postcomposing with $p$, i.e., the functor $f \mapsto p \circ f$.

This is the easier, more conceptual way to remember right Kan lifts: in a global sense, they are right adjoint to postcomposition. Similarly, global left Kan lifts are left adjoint to postcomposition.

* In a [post](http://golem.ph.utexas.edu/category/2009/06/kan_lifts.html) at the nCafé, [[David Corfield]] put the question: why are Kan extensions more frequently mentioned than Kan lifts? One proposed [answer](http://golem.ph.utexas.edu/category/2009/06/kan_lifts.html#c024705) is that in many cases a left or right Kan lift will exist for a trivial reason: namely that $p$ itself has a left or right adjoint. Other examples will be given below.

## Properties ##

Dual to the situation in Kan extensions, one is interested in whether a Kan lift is _respected_ by a 1-cell with codomain the domain of the lift. This is defined as follows:

- given $(\widetilde{f}, \varepsilon) = \mathop{Rift}_p f$, $g$ is said to _respect_ this right lift if $(\widetilde{f}g, \varepsilon \bullet g) = \mathop{Rift}_p(f g)$

and analogously for left Kan lifts. A Kan lift $\widehat{f}$ is _absolute_ if it is respected by any 1-cell into $\mathop{dom}(\widehat{f})$. Absolute Kan lifts subsume adjunctions and relative adjunctions, and are prominently present in the axioms of a [[Yoneda structure]]; for more see the examples below.

## Examples ##

* Let $Mod$ denote the [[framed bicategory]] of small categories $C$, bimodules $r: C \to D$ (i.e., functors $r: D^{op} \times C \to Set$), and transformations between such bimodules. Then global right Kan lifts exist (as do global right Kan extensions). These may be computed as [[weighted limit]]s or via [[end]] formulas, e.g.,
$$(Rift_r s)(b, c) = \int_{d: D} hom(r(d, c), s(d, b))$$

Examples of this construction abound in mathematics, especially when generalized to the [[enriched category theory]] context. For example, in the bicategory $Rel$, which corresponds to enrichment in $\mathbf{2}$, the right Kan lift is essentially a universally quantified predicate of the form

$$\forall_{d: D} r(d, c) \Rightarrow s(d, b)$$

("for all $d$ satisfying condition $r$, we impose condition $s$").

* More generally, a [[biclosed bicategory]] is precisely a bicategory where global right Kan extensions and right Kan lifts exist for every 1-cell $p$. Monoidal bicategories provide instances of this.

* adjunctions in a 2-category can be defined in terms of Kan lifts: a 1-cell $u\colon A \to B$ has a left adjoint iff $\mathop{Lift}_u 1_B$ exists and is absolute; in this case putting $(f,\iota) = \mathop{Lift}_u 1_B$ we have $f \dashv u$ with unit $\iota \colon 1_B \Rightarrow u f$. The universal property of the left Kan lift plus absoluteness are enough to construct the counit and to verify the triangular equations. There's of course a dual definition in terms of absolute right Kan lifts.

* relative adjoints in $\mathbf{Cat}$ can also be expressed as absolute kan lifts; see [[relative adjoint]] for a precise statement.

* representably fully faithful 1-cells, meaning those for which $B(X,f)$ is fully faithful in $\mathbf{Cat}$ for every object $X\colon B$, are those for which $(1_A, 1_f) = \mathop{Lift}_f f$, and this lifting is absolute.

* In $Cat$, if $A$ is small and $B$ is locally small, and if $F: A \to B$ is a functor, then we have a Yoneda embedding $y: A \to P A = Set^{A^{op}}$ and a functor $B(F-, -): B \to P A$, and there is a canonical map
$$y_A \to B(F-, -) \circ f$$
(essentially, $hom(a, b) \to hom(F a, F b)$ taking $f: a \to b$ to $F f: F a \to F b$). This arrow exhibits $F$ as a left Kan lift of $y$ through $B(F-, -)$, which is moreover absolute. This example is important in the theory of Yoneda structures, due to Street and Walters; see Mark Weber's updated [development](http://www.pps.jussieu.fr/~weber/Two-toposes4.pdf) in the context of 2-topos theory.

## References

Kan lifts were mentioned in passing in Definition I.12.6 of:

* [[John Gray]], *[[Adjointness for 2-Categories|Formal category theory: adjointness for $2$-categories]]*, Lecture Notes in Mathematics, **391**, Springer (1974) [[doi:10.1007/BFb0061280](https://doi.org/10.1007/BFb0061280)]

where it is mentioned the concept does not have a name since they had never been considered before in [[Cat]].

Kan lifts were called "liftings" in:

* [[Ross Street]], [[Bob Walters]], *Yoneda structures on 2-categories*,  JPAA **50** (1978) 350-379 [doi:10.1016/0021-8693(78)90160-6]

The terminology "Kan lifting" is used in:

* Charles Grellois, *Algebraic theories, monads, and arities* (2011) [[arXiv:1110.3294](https://arxiv.org/abs/1110.3294), [hal:00670841](https://inria.hal.science/hal-00670841)]

The terminology "Kan lift" seems to first appear in:

* Weng Kin Ho, *Characterising E-projectives via Co-monads*, Electronic Notes in Theoretical Computer Science **301** (2014) 61-77 [[doi:10.1016/j.entcs.2014.01.006](https://doi.org/10.1016/j.entcs.2014.01.006)]

[[!redirects Kan lifts]]
[[!redirects Kan lifting]]
[[!redirects Kan liftings]]


Kan object

# Kan objects
* table of contents
{: toc}

## Idea

A Kan object $X$ in a category $C$ is a [[simplicial object]] in $C$ satisfying a generalized [[Kan complex|Kan condition]].

This generalization of the notion of [[Kan complex]] takes into account that the category in which $X$ is a simplicial object (namely $C$) is now not equal to, but only enriched over, the category in which the [[horn|horns]] on which the horn-filler condition (the Kan condition) is imposed are simplicial objects (namely $Set$).

Note that a Kan object in a category other than $\Set$ is not per se (a model for) an [[internal infinity-groupoid|internal ∞-groupoid]] (discussed there).

## Motivation

Recall that in the unenriched case (i.e. $C=Set$) a [[simplicial set]] $X$ is a [[Kan complex]] if the following equivalent conditions are satisfied.

1. The map $X\to *$ is a [[Kan fibration]]. This means for all $n\in\mathbb{N}$ and for all $0\le k\le n$ and for all $h_{k , n}$ the is a morphism $\exists_{h_{k, n}}$ making $\array{\Lambda^k[n]&\xrightarrow{h_{k , n}}&X\\\downarrow^{\iota_{k,n}}&\nearrow_{\exists_{h_{k,n}}}&&\\\Delta[n]&&}$ commute where the $\iota_{k,n}$ are the [[horn|horn inclusions]].

1. $hom_\sSet(\Delta[n],X)\to \hom_\sSet(\Lambda^k[n],X)$ is an epimorphism in $\Set$ for all $n$ and all $0\le k\le n$ where $\sSet$ is regarded as a $\Set$-[[enriched category]].

## Definition

+-- {: .num_defn}
###### Definition
Let $C$ be a category, and let $X:\Delta^{op}\to C$ be a [[simplicial object]] in $C$.

The **object of k-horns of n-simplices of**  $X$ is defined to be the [[weighted limit]] $\X^{\Lambda_n^k}\coloneqq \lim_{\Lambda_n^k}X$

$X$ is called a **Kan object** (in $C$) if $X[n]\to \X^{\Lambda_n^k}$ is an [[epimorphism]] for all $n$ and all $0\le k\le n$.  We obtain a family of related notions by requiring these maps to be different kinds of epimorphisms (regular, split, etc.).

=--

Note that this condition---called the **internal horn-filler condition**---coincides with the usual horn-filler condition (i.e. the Kan condition) if $C=\Set$, since for $V$-enriched functors $F:K\to C$ and $W:K\to V$ we have in the case $V=C$ that the [[weighted limit]] $\lim_W F=[K,V](W,F)$ coincides with the [[hom object]]; so in particular $X^{\Lambda_n^k}=\sSet(\Lambda_n^k,X)$.

## Examples

[[Kan simplicial manifolds]] form one possible generalization of [[Lie groupoids]].

If a [[variety of algebras]] $V$ contains a [[Malcev operation]],
then every [[simplicial object]] in $V$ is Kan.

## Related entries

* [[Kan complex]]

* [[Kan simplicial manifold]]

* [[Malcev operation]]

[[!redirects Kan object]]
[[!redirects Kan objects]]

category: ∞-groupoid

category: simplicial object


Kan simplicial manifold

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## Idea

A _Kan-fibrant simplicial manifold_ is a [[simplicial manifold]] (for instance [[simplicial object|simplicial]] [[topological manifold]] or simplicial [[smooth manifold]]) which satisfies a suitable analog of the [[Kan complex]]-condition on a [[simplicial set]].
(Typically, to get an interesting theory, Kan-fibrancy on simplicial manifolds is imposed in a suitable local sense, meaning that [[horns]] have continuous/smooth fillers in a  [[open neighbourhoods]] of all points, but possibly not globally.)

Motivated by the standard way (see at _[[homotopy hypothesis]]_) in which bare [[Kan complexes]] (hence Kan-fibrant [[simplicial sets]]) present [[geometrically discrete ∞-groupoids]] and given that the [[nerve]] of a [[Lie groupoid]] is an example of a locally Kan-fibrant simplicial manifold (a [[1-truncated]] one), such Kan-fibrant simplicial manifolds are often referred to _[[Lie infinity-groupoids]]_ (or _[[Lie n-groupoids]]_ for finite [[n-truncated object in an (infinity,1)-category|truncation]]) ([Zhu 06](#Zhu06)).

With such a suitably local definition, there should be the structure of a [[homotopical category]] on Kan-fibrant simplicial manifolds which embeds homotopically full and faithful into a local [[model structure on simplicial presheaves]] over a suitable [[site]] of manifolds, hence such that this inclusion presents and [[full sub-(∞,1)-category]] of the [[(∞,1)-sheaf (∞,1)-topos]] over manifolds ("[[smooth ∞-groupoids]]").

Some care is needed in correctly interpreting the "Lie" condition in view of the [[homotopy theory]]. For instance _every_ [[∞-stack]] on the category of smooth manifolds ("[[smooth ∞-groupoid]]") is presented by a [[simplicial manifold]], just not in general by a suitably Kan-fibrant simplicial manifold ([NSS 12, section 2.2](#NSS12)).

A homotopy-correct characterization of the sub-$\infty$-category presented by the Kan-fibrant simplicial objects as that of _[[geometric ∞-stacks]]_ modeled on manifolds is in ([Pridham 09](#Pridham09)) (see around p. 17 for the differential geometric version).

Kan-fibrant simplicial manifolds have received particular attention as the result of [[Lie integration]] of [[L-∞ algebroids]]. See at _[[Lie integration]]_ for more on that.

## Examples

1. Any ordinary [[manifold]], interpreted as a constant [[simplicial object]].

2. The [[nerve]] of a [[Lie groupoid]]. In particular, the [[delooping]] of any [[Lie group]], which represents [[principal bundles]] with this [[Lie group]] as a structure group.

3. The [[Dold–Kan functor]] $\Gamma$ applied to any nonnegatively graded [[chain complex]] of abelian Lie groups.

4. In particular, applying $\Gamma$ to the [[chain complex]] $\mathrm{U}(1)[n]$, we get the Kan simplicial manifold representing bundle $(n-1)$-gerbes.

5. The nonabelian analogue of $\Gamma$ applied to any [[crossed module]] whose two constituent groups are [[Lie groups]] and the involved homomorphisms and actions are smooth.

6. The nonabelian analogue of $\Gamma$ applied to any [[hypercrossed complex]] whose constituent groupoids are [[Lie groupoids]] and the involved homomorphisms and actions are smooth.

7.  As a special case of the previous example, any [[simplicial Lie group]] is a Kan simplicial manifold.

## Related concepts

* [[simplicial manifold]]

* [[higher differential geometry]]

* [[internal ∞-groupoid]]

* [[smooth ∞-groupoid]]

## References

Early appearances of the concept include

* [[André Henriques]], _Integrating $L_\infty$-algebras_, Compositio Mathematica, 144, (2008), no. 4, 1017–1045 ([arXiv:math/0603563](http://arxiv.org/abs/math/0603563))

* {#Zhu06} [[Chenchang Zhu]], _$n$-groupoids and stacky Lie groupoids_, Int Math Res Notices (2009) 2009 (21): 4087-4141. ([arXiv:math/0609420](http://arxiv.org/abs/math/0609420))

* {#Zhu08} [[Chenchang Zhu]], _Kan replacement of simplicial manifolds_, Lett Math Phys (2009) 90:383–405, [arXiv:0812.4150](http://arxiv.org/abs/0812.4150)

Characterization of the homotopy theory of Kan-fibrant simplicial manifolds as [[geometric ∞-stacks]] modeled on smooth manifolds is in (see aroung p. 17 for the differential geometric version)

* {#Pridham09} [[Jonathan Pridham]], _Presenting higher stacks as simplicial schemes_, Advances in Mathematics, Volume 238, Pages 184-245 ([arXiv:0905.4044](http://arxiv.org/abs/0905.4044))

Discussion of [[principal ∞-bundles]] in [[Smooth∞Grpd]] $= Sh_\infty(SmoothMfd)$ which are represented by locally Kan-fibrant simplicial manifolds is in

* {#NSS12} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], section 4.2 of _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- Presentations]]_, Journal of Homotopy and Related Structures, March 2014 ([arXiv:1207.0249](http://arxiv.org/abs/1207.0249))

* [[Jesse Wolfson]], _Descent for $n$-Bundles_ ([arXiv:1308.1113](http://arxiv.org/abs/1308.1113))

[[!redirects Kan simplicial manifolds]]
[[!redirects Kan-fibrant simplicial manifold]]
[[!redirects Kan-fibrant simplicial manifolds]]


Kan-Thurston theorem

[[!redirects Kan-Thurston Theorem]]

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## Idea

The _Kan-Thurston theorem_ says that every [[path-connected topological space]] is [[ordinary homology|homology]]-[[equivalence|equivalent]] to the [[classifying space]] $K(G,1)$ of a [[discrete group]] $G$.

## Statement

For every path connected  space $X$ with base point there exists a ([[Serre fibration|Serre-]])-[[fibration]]

$$
  t X \colon T X \rightarrow X
$$

which is natural with respect to $X$ and has the following properties.

(i) The map $t X$ induces an [[isomorphism]] on ([[singular homology|singular]]-)[[homology]] and [[singular cohomology|cohomology]] with [[local coefficient bundle|local coefficients]]

$$H_*(T X;A)\cong H_*(X;A),\;\;  H^*(T X;A)\cong H^*(X;A)$$
for every local coefficient system $A$ on $X$, and

(ii) $\pi_i\;(T X)$ is trivial for $i\neq 1$ and $\pi_1\;t X$ is onto.

Furthermore the [[homotopy type]] of $X$ is completely determined by the pair of groups $(G_X, P_X)$ where
$G_X  = \pi_1T X,\;\; P_X = ker\;\pi_1t X$ and $P_X$ is a
[[perfect group|perfect]] [[normal subgroup|normal]] [[subgroup]] of $G_X$.

More precisely, $X$ can be recovered, up to homotopy, by applying the [[plus construction]] to $K(G_X,1)$ with respect to $P_X$.

More generally, the Kan-Thurston theorem
gives an embedding of the homotopy category of topological spaces into the category of presheaves on the category whose objects are groups and whose morphisms are conjugacy classes of [[group homomorphisms]].  Under this embedding a topological space $X$ is sent to the presheaf $G \mapsto [B G, X]$.

## References

* [[Daniel Kan]] and [[William Thurston]], _Every connected space has the homology of a $K(\pi,1)$_, Topology  Vol.  15.  pp.  253--258, 1976.


Kane-Mele invariant

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## Idea

In [[solid state physics]], the *Kane-Mele invariant* ([Kane & Mele 05b](#KaneMele05b)) is an element in [[cyclic group of order two|$\mathbb{Z}/2$]] naturally assigned to [[time-reversal symmetry|time-reversal symmetric]] spinful [[insulators]], whose non-triviality partially detects whether the insulator is in a non-trivial [[topological phase of matter|topological phase]], hence whether it is a *[[topological insulator]]*.

Concretely, for 3d materials the Kane-Mele invariant is obtained from the [[Chern-Simons invariant]] of the [[Berry connection]] on the [[Brillouin torus]] (eg. [KLW 16, Sec. 4.3](#KLW16)).

Under the expected [[K-theory classification of topological phases of matter]], [[time-reversal symmetry|time-reversal symmetric]] and spinful [[topological insulators]] are classified by the class of their [[valence bundle]] in [[KO-theory]] of degree 4 (hence in [[KQ-theory]]) equipped with the time-reversal [[involution]] $\tau$. The Kane-Mele invariant is a  non-trivial group homomorphism from that K-theory group to $\mathbb{Z}/2$ ([Bunk & Szabo 20, Prop. 2.25, Pro. 5.22(4)](#BunkSzabo20)):

$$
  K Q
  \big(
    \mathbb{T}^d, \tau
  \big)
  \xrightarrow{\text{KM invariant}}
  \mathbb{Z}/2
  \,.
$$

## Related concepts

* [[Haldane model]]

## References

The idea of the [[Kane-Mele invariant]] originates in:

* {#KaneMele05b} [[Charles Kane]], [[Eugene Mele]],  *$Z_2$ Topological Order and the Quantum Spin Hall Effect*, Phys. Rev. Lett. **95** (2005) 146802 ([doi:10.1103/PhysRevLett.95.146802](https://doi.org/10.1103/PhysRevLett.95.146802))

This was motivated by the observation that [[graphene]] appears as a [[topological insulator]] with a non-trivial MK-invariant when its [[spin-orbit coupling]] is resolved, due to a [[quantum spin Hall effect]]:

* {#KaneMele05a} [[Charles Kane]], [[Eugene Mele]], _Quantum Spin Hall Effect in Graphene_, Phys. Rev. Lett. **95** 226801 (2005) ([arXiv:cond-mat/0411737](https://arxiv.org/abs/cond-mat/0411737), [doi:10.1103/PhysRevLett.95.226801](https://doi.org/10.1103/PhysRevLett.95.226801))

On the relation of the Kane-Mele invariant to the [[K-theory classification of topological phases of matter]]:

* {#KLW16} [[Ralph M. Kaufmann]], Dan Li, Birgit Wehefritz-Kaufmann, *Notes on topological insulators*, Rev. Math. Phys. **28** 10 (2016) 1630003 ([arXiv:1501.02874](https://arxiv.org/abs/1501.02874), [doi:10.1142/S0129055X1630003X](https://doi.org/10.1142/S0129055X1630003X))

* {#BunkSzabo20} [[Severin Bunk]], [[Richard Szabo]], *Topological Insulators and the Kane-Mele Invariant: Obstruction and Localisation Theory*,  Reviews in Mathematical Physics **32** 06 (2020) 2050017  ([arXiv:1712.02991](https://arxiv.org/abs/1712.02991), [doi:10.1142/S0129055X20500178](https://doi.org/10.1142/S0129055X20500178))

[[!redirects Kane-Mele invariants]]

[[!redirects Kane-Mele model]]
[[!redirects Kane-Mele models]]


Kantaro Ohmori

* [webpage](https://www.ias.edu/scholars/kantaro-ohmori)

## Selected writings

On [[D=6 N=(2,0) SCFT]] and [[D=6 N=(1,0) SCFT]] ([[E-string]]-theories)

* [[Kantaro Ohmori]], _Six-Dimensional Superconformal Field Theories and Their Torus Compactifications_, Springer Theses 2018 ([springer:book/9789811330919](https://www.springer.com/gp/book/9789811330919))

their [[KK-compactification]] to [[D=5 super Yang-Mills theory]]:

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], _$S^1/T^2$ Compactifications of 6d $\mathcal{N} = (1,0)$ Theories and Brane Webs_, J. High Energ. Phys. (2016) 2016: 24 ([arXiv:1509.03195](https://arxiv.org/abs/1509.03195))

and their [[anomaly cancellation]] via the [[Green-Schwarz mechanism]]:

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], _Anomaly polynomial of E-string theories_, J. High Energ. Phys. 2014, 2 (2014) ([arXiv:1404.3887](https://arxiv.org/abs/1404.3887))

* [[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 the [[WZW term]] in [[chiral perturbation theory]]:

* Yasunori Lee, [[Kantaro Ohmori]], [[Yuji Tachikawa]], _Revisiting Wess-Zumino-Witten terms_ ([arXiv:2009.00033](https://arxiv.org/abs/2009.00033))

Determination of the [[2-group]] structure constants for all the six-dimensional [[little string theories]] (LSTs) geometrically engineered in [[F-theory]] without frozen singularities:

* [[Michele Del Zotto]] and [[Kantaro Ohmori]], _2-Group Symmetries of 6D Little String Theories and T-Duality_, Annales Henri Poincaré volume 22, pp. 2451–2474 (2021) ([doi:10.1007/s00023-021-01018-3)](https://doi.org/10.1007/s00023-021-01018-3), [arXiv:2009.03489](https://arxiv.org/abs/2009.03489))

On [[anomaly cancellation]] via the [[Green-Schwarz mechanism]] from the point of view of [[higher gauge theory]]:

* [[Yasunori Lee]], [[Kantaro Ohmori]], [[Yuji Tachikawa]], *Matching higher symmetries across Intriligator-Seiberg duality* ([arXiv:2108.05369](https://arxiv.org/abs/2108.05369))

category: people


Kantor-Koecher-Tits construction

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## Idea

The _Kantor-Koecher-Tits construction_ produces a [[Lie algebra]] from a [[Jordan algebra]].

Applied to the [[exceptional Jordan algebra]] this construction yields the Lie algebra of the [[exceptional Lie group]] [[E7]].

## References

The original articles are

* {#Kantor64} [[Isaiah Kantor]], _Classification of irreducible transitive differential groups_, Dokl. Akad. Nauk SSSR, 158:6 (1964), 1271–1274 ([mathnet:dan30260](http://mi.mathnet.ru/eng/dan30260))

(...)

See also:

* Wikipedia, _[Kantor-Koecher-Tits construction](https://en.wikipedia.org/wiki/Kantor%E2%80%93Koecher%E2%80%93Tits_construction)_

Further discussion:

* [[Jakob Palmkvist]], _A generalization of the Kantor-Koecher-Tits construction_, J. Gen. Lie Theory Appl. 2, (2008) 226 ([arXiv:1308.3761](https://arxiv.org/abs/1308.3761))


Kantorovich monad

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## Idea

The Kantorovich monad is a [[probability monad]] on the category of [[complete metric spaces]] and 1-Lipschitz (or [[Lipschitz map|Lipschitz]]) maps.
Its functor part assigns to each complete metric space $X$ the space of [[Radon measure|Radon]] [[probability measures]] with finite first moment, equipped with the [[Kantorovich-Wasserstein distance]].

Its [[algebra over a monad|algebras]] are [[closed set|closed]] [[convex set|convex]] subsets of [[Banach spaces]].

The construction was first given by van Breugel for the compact case (see [vB '05](#vanbreugel)).

There is an [[order theory|ordered]] version of the Kantorovich monad, making use of the [[stochastic order]]. Its [[algebra over a monad|algebras]] are [[closed set|closed]] [[convex set|convex]] subsets of [[ordered Banach spaces]].

## Wasserstein spaces

Let $X$ be a [[metric space]]. A [[probability measure]] on $X$ has **finite first moment** if
$$
\int_{X\times X} d(x,y) \, dp(x)\,dp(y) \;\lt\; +\infty,
$$
where $d(-,-)$ denotes the [[distance]].

Let $p$ and $q$ be  [[Radon measure|Radon]] [[probability measures]] on $X$ of finite first moment. The **[[Kantorovich-Wasserstein distance]]** is given by
$$
d(p,q) \;\coloneqq\; \inf_{r\in\Gamma(p,q)} \int_{X\times X} d(x,y) \,dr(x,y) ,
$$
where the infimum (which is attained if $X$ is complete) is taken over the set $\Gamma(p,q)$ of [[joint measure|couplings]] of $p$ and $q$, i.e. the measures on $X\times X$ which admit $p$ and $q$ as their [[marginal measure|marginals]].

Equivalently, the distance can be written as
$$
d(p,q) \;=\; \sup_{f\in[X,\mathbb{R}]} \int f\, dp - \int f\, dq,
$$
where the supremum (which is usually not attained) is taken over the set of 1-[[Lipschitz map|Lipschitz]] functions $X\to \mathbb{R}$.
(This equivalence is an instance of the celebrated [[Kantorovich duality]].)

The **1-Wasserstein space** over $X$ is the space $P X$ of Radon probability measures on $X$ with finite first moment, equipped with the Wasserstein metric.

### Properties

* If $X$ is [[complete metric space|complete]], so is $P X$.

* If $X$ is [[separable space|separable]], so if $P X$.

* If $X$ is [[compact]], so is $P X$.

More information on Wasserstein spaces can be found in [Villani '08](#villani).

## Functor, unit and multiplication

As it is customary for [[probability monads]], the functor assigns to a [[morphism]] $f:X\to Y$ (in this case, a [[Lipschitz map|Lipschitz]] or 1-Lipschitz map) the map $P f: P X \to P Y$ induced by the [[pushforward of measures]].
The pushforward of a Radon probability measure of finite first moment along a Lipschitz map is again Radon and of finite first moment. Moreover, the resulting map $P f$ is again Lipschitz, with the same Lipschitz constant as $f$. Therefore we have an endofunctor $P$ on the category of [[complete metric spaces]] and [[Lipschitz maps]] (or 1-Lipschitz).

Again, as it happens in general with [[probability monads]], the unit $\delta:X\to P X$ is given by forming the [[Dirac measures]] and the multiplication $E:P P X\to P X$ is given by [[integration]]. More formally, given $\xi\in P P X$, the measure $E \xi\in PX$ is defined as the one which assigns to a [[Borel set]] $A\subseteq X$ the number
$$
E \xi(A) \;\coloneqq\; \int p(A) \,d\xi(p) .
$$

The maps $\delta$ and $E$ satisfy the usual [[axioms]] of a [[monad]] (see [F-P '19](#kantorovich19)), which is known as the **Kantorovich monad**.

## Algebras

As it is the case for most [[probability monad]], the algebras are [[convex spaces]] of a particular kind. For the Kantorovich monad, the objects of the underlying category are complete metric spaces, and so the [[closed set|closed]] [[convex set|convex]] subsets of [[Banach spaces]] are an ideal candidate: they are [[metric spaces]], they are [[complete metric space|complete]], and they have a well-defined [[convex space|convex combination operation]].
It turns out that this is exactly the case.

More in detail, every closed convex subset $A$ of a Banach space comes equipped with a map $e:P A\to A$ given by vector-valued [[integration]],
$$
p\;\mapsto\; \int a \,dp(a) .
$$
The integral exists since $p$ has finite first moment.
Conversely, it can be proven that every $P$-algebra is of this form.

The [[Eilenberg-Moore category|morphisms of algebras]] are maps between algebras $f:A\to B$ which commute with the integration map. It can be shown that equivalently these are exactly the **affine** maps, i.e.~those maps which satisfy
$$
f\big(\lambda\,x + (1-\lambda)\,y\big) \;=\; \lambda\,f(x) + (1-\lambda)\,f(y).
$$

For more details, see [F-P '19](#kantorovich19).

## Duality

(...)

## The ordered case

(Work in progress. For now see [F-P '18](#orderedkantorovich).)

## For Lawvere metric spaces

(...)

## See also

* [[monads of probability, measures, and valuations]]

* [[Giry monad]], [[Radon monad]], [[extended probabilistic powerdomain]]

* [[Kantorovich-Wasserstein distance]], [[optimal transport]]

* [[metric space]], [[complete metric space]]

* [[Banach space]], [[ordered Banach space]], [[convex space]]

## References

* {#vanbreugel} Franck van Breugel, _The metric monad for probabilistic nondeterminism_, unpublished, 2005. ([pdf](http://www.cse.yorku.ca/~franck/research/drafts/monad.pdf))

* {#kantorovich19} [[Tobias Fritz]] and [[Paolo Perrone]], _A probability monad as the colimit of spaces of finite samples_, Theory and Applications of Categories 34, 2019. ([pdf](http://www.tac.mta.ca/tac/volumes/34/7/34-07.pdf))

* {#orderedkantorovich} [[Tobias Fritz]] and [[Paolo Perrone]], _Stochastic order on metric spaces and the ordered Kantorovich monad_, Advances in Mathematics 366, 2020. ([arXiv:1808.09898](https://arxiv.org/abs/1808.09898))

* {#villani} [[Cedric Villani]], _Optimal transport: old and new_, Springer, 2008.

* {#behavioral} Paolo Baldan, Filippo Bonchi, Henning Kerstan and Barbara König, _Coalgebraic behavioral metrics_, Logical Methods in Computer Science 14(3), 2018. ([doi: 10.23638/LMCS-14(3:20)2018](https:doi.org/10.23638/LMCS-14%283:20%292018))


kaon

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#### Fields and quanta
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## Idea

A _kaon_ is a type of [[meson]]:

The [[Lorentz group|Lorentz]] [[trivial representation|scalar]] and [[isospin]] [[vector representation]] [[bound state|bound]] from [[up-quark|up]]/[[down-quarks]] in the [[first generation of fundamental particles]] and a [[strange quark]] in the [[second generation of fundamental particles]].  Hence a [[heavy meson]] with respect to 2-flavor [[chiral perturbation theory]].

Together with the [[K* mesons]], which are the corresponding [[vector mesons]], these are also called _K-mesons_ (alongside [[D-mesons]] and [[B-mesons]], etc.)

The class of K-mesons goes alongside that of [[D-mesons]] ([[light mesons]] with a [[charm quark]]) and [[B-mesons]] (mesons with a [[bottom quark]]).

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

\linebreak

[[!include flavors of fundamental particles -- table]]

## Related concepts

* [[K* meson]]

## References

### General

See also

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

Via [[chiral perturbation theory]]:

* V. Cirigliano, G. Ecker, H. Neufeld, A. Pich, J. Portoles, _Kaon Decays in the Standard Model_, Rev. Mod. Phys. 84 (2012) 399 ([arXiv:1107.6001](https://arxiv.org/abs/1107.6001))

On [[kaon]]-[[K*-meson]]-[[photon]] [[interaction]] via the [[WZW term]]:

* S. Ozaki, H. Nagahiro, [[Atsushi Hosaka]], Equations (3) and (9) in: _Magnetic interaction induced by the anomaly in kaon-photoproductions_, Physics Letters B Volume 665, Issue 4, 24 July 2008, Pages 178-181 ([arXiv:0710.5581](https://arxiv.org/abs/0710.5581), [doi:10.1016/j.physletb.2008.06.020](https://doi.org/10.1016/j.physletb.2008.06.020))

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

### Flavour anomalies

On [[flavour physics]] and potential [[flavour anomalies]] in [[kaon]]-[[scattering|decays]]:

* {#Buras16} [[Andrzej Buras]], _The Revival of Kaon Flavour Physics_,  EPJ Web Conf. Volume 129, 2016  ([arXiv:1609.05711](https://arxiv.org/abs/1609.05711), [doi:10.1051/epjconf/201612900050](https://doi.org/10.1051/epjconf/201612900050))

* {#Buras18} [[Andrzej Buras]], _The Return of Kaon Flavour Physics_, Acta Phys. Pol. B 49, 1043 (2018) ([doi:10.5506/APhysPolB.49.1043](https://www.actaphys.uj.edu.pl/index_n.php?I=R&V=49&N=6#1043))

* Xiao-Gang He, Xiao-Dong Ma, Jusak Tandean, German Valencia, _Breaking the Grossman-Nir Bound in Kaon Decays_ ([arXiv:2002.05467](https://arxiv.org/abs/2002.05467))

* Teppei Kitahara, Takemichi Okui, Gilad Perez, Yotam Soreq, Kohsaku Tobioka, _New physics implications of recent search for $K_L \to \pi_0 \nu \bar \nu$ at KOTO_ ([arXiv:1909.11111](https://arxiv.org/abs/1909.11111))

[[!redirects kaons]]

[[!redirects K-meson]]
[[!redirects K-mesons]]

[[!redirects K meson]]
[[!redirects K mesons]]


kappa-symmetry

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## Idea

What is called _$\kappa$-symmetry_ in [[string theory]]/[[M-theory]] is a certain fermionic [[symmetry]] of [[Green-Schwarz action functionals]] for [[super p-branes]] whose effect is to [[gauge symmetry|gauge away]] half of the [[spinor|spinorial]] [[sigma-model]] [[field (physics)|fields]].

## As super-diffeomorphisms in the super-embedding approach

In a completely super-covariant formulation of the [[Green-Schwarz action functionals]] -- called the _[[super-embedding formalism]]_ -- this $\kappa$-symmetry is simply the odd-graded part of the [[supermanifold|super]]-[[worldvolume]] [[super-diffeomorphism]] symmetry of the [[sigma-model]] ([Sorokin-Tkach-Volkov 89](#SorokinTkachVolkov89), review includes [Sorokin 99, section 4.3](#Sorokin99), [Howe-Sezgin 04, section 4.3](#HoweSezgin04)): If

1. $X$ denotes a [[superspacetime]] locally modeled on [[super-Minkowski spacetime]] $\mathbb{R}^{d-1,1\vert \mathbf{N}}$

1. $\Sigma$ denotes a [[supermanifold|super]]-[[worldvolume]] of a [[super p-brane]] locally modeled on [[super-Minkowski spacetime]] $\mathbb{R}^{p,1\vert \mathbf{N}/2}$

1. so that a [[sigma-model]] [[field (physics)|field]] configuration for a [[super p-brane]] of shape $\Sigma$ to popagate in $X$ is a morphism of [[supermanifolds]] of the form

   $$
     \array{
        \Sigma && \text{super-worldvolume}
        \\
        \downarrow^{\mathrlap{\phi}} &&\text{sigma-model super-field}&
        \\
        X && \text{super-spacetime}
     }
   $$

then:

1. the postcomposition [[action]] of [[spacetime]] [[super-isometries]] $X \stackrel{\simeq}{\longrightarrow} X$ is in even degree the action of [[spacetime]] [[isometries]] and in odd degree the action of **[[spacetime]] [[supersymmetry]]** on the sigma-model fields;

1. the precomposition action of [[worldvolume]] [[super-diffeomorphism]] $\Sigma \stackrel{\simeq}{\to} \Sigma$ is in even degree the action of bosonic worldvolume [[diffeomorphism]] and in odd degree the action of **$\kappa$-symmetry**:

$$
  \array{
      \Sigma  &\underoverset{\simeq}{\kappa\text{-symmetry}}{\longrightarrow}& \Sigma
      \\
      \downarrow^{\mathrlap{\phi}} && \downarrow^{\mathrlap{\phi'}}
      \\
      X &\underoverset{\simeq}{\text{spacetime supersymmetry}}{\longrightarrow}&
    X
  }
  \,.
$$

Notice here the assumption that the number of odd directions on the [[worldvolume]] is half that of the [[target spacetime]]. This is the default assumption for fundamental [[super p-branes]], and it directly reflects the statement that the corrresponding [[black brane]] solutions are $1/2$ [[supergravity]] [[BPS states]].

For example, consider the embedding

$$
  \mathbb{R}^{2,1} \hookrightarrow \mathbb{R}^{10,1}
$$

of 2+1d [[Minkowski spacetime]], thought of as the [[worldvolume]] of a [[membrane]], into 11d Minkowski spacetime, linearly along the coordinate axis. Any such embedding breaks the [[isometry group]] of $\mathbb{R}^{10,1}$ from the 11d [[Poincaré group]] $Iso(10,1)$ to the [[product group]]

$$
  Iso(2,1) \times SO(8)
    \hookrightarrow
  Iso(10,1)
$$

(meaning that this [[subgroup]] is the [[stabilizer subgroup]] of the embedding).

Now consider instead [[super Minkowski spacetime]] $\mathbb{R}^{10,1\vert \mathbf{32}}$ (with $\mathbf{32}$ the irreducible [[Majorana spinor]] representation in 11), hence the local model [[superspace]] for [[super spacetimes]] in [[11-dimensional supergravity]]. We are to ask what subspace of the [[spin representation]] $\mathbf{32}$ preserves the embedding in that the [spinor bilinear pairing](Majorana+spinor#TheSpinorPairingToVectors) $\overline{\psi}_1 \Gamma \psi_2$ on that subspace lands in $\mathbb{R}^{2,1} \hookrightarrow Iso(2,1) \hookrightarrow Iso(10,1)$ ([Sorokin 99, section 5.1](#Sorokin99)). This is found to be the case for a half-dimensional subspace, and hence we may lift the above to a super-embedding of the form

$$
  \label{M2WordvolumeInSpacetime}
  \mathbb{R}^{2,1\vert 8 \otimes \mathbf{2}} \hookrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}
$$

(where now $\mathbf{2}$ is the [[irrep|irreducible]] [[Majorana spinor]] representation in 3d, and $8 \otimes \mathbf{2}$ denotes the [[direct sum]] of 8 copies of it)
such that the induced [[stabilizer group|stabilizer]] [[supergroup]] inside the [[super Poincaré group ]] now is

$$
  Iso(\mathbb{R}^{2,1\vert 8 \otimes \mathbf{2}}) \times Spin(8)
    \hookrightarrow
  Iso(\mathbb{R}^{10,1\vert \mathbf{32}})
  \,.
$$

It is in this sense that the membrane "breaks exactly half the supersymmetry", namely from $\mathbf{32}$ to $8 \otimes \mathbf{2}$.

If one now thinks of this not as inclusions of global spacetimes, but of their super tangent spaces at the points where the membrane sits in spacetime, then this reflects the local structure of $\kappa$-symmetry: the $\kappa$-symmetries are locally generated by the 16 odd dimensions in $Iso(\mathbb{R}^{2,1\vert 8 \otimes \mathbf{2}} )$, being super-translations along the membrane worldvolume.

This explains why $\kappa$-symmetry in [[Green-Schwarz sigma models]] is taken to quotient out precisely half the spinor components, hence why, in the fully super-covariant formulation, one takes the worldvolume of a super $p$-brane in a superspacetime locally modeled on $\mathbb{R}^{d-1,1\vert \mathbf{N}}$ to be $\mathbb{R}^{p,1\vert \mathbf{N}/2}$.
But notice that this is not a mathematical necessity. One may consider the worldvolume instead to have fewer odd directions. This then describes [[sigma models]] for "non-BPS super $p$-branes" (or rather "non-half-BPS" ).

## History

Historically $\kappa$-symmetry was originally observed and considered for [[Green-Schwarz sigma models]] whose [[worldvolume]] is regarded as an ordinary (bosonic) [[smooth manifold]]. Then $\kappa$-symmetry is a "hidden" symmetry, with no evident geometric interpretation. As such it was first observed for the [[super-particle]] ([Azcarraga-Lukierski 82](#AzcarragLukierski82), [Siegel 83](#Siegel83)) and then for the [[super 1-brane in 3d]] ([Siegel 84](#Siegel84)).

Based on this the [[Green-Schwarz action functional]] for the [[superstring]] in 10d ([[heterotic string]], [[type II superstring]]) was found in ([Green-Schwarz 84](#GreenSchwarz84)) by first observing that the plain [[Nambu-Goto action]] for a string on a [[supermanifold]] has twice as many fermionic [[shell|on-shell]] [[degrees of freedom]] as the [[NSR-string]] and then by adding a term to the action (the [[WZW-term]]) to correct this defect, by ensuring that the sum of the NG-action with the WZW term enjoys $\kappa$-symmetry.
By the same recipe later $\kappa$-symmetric Green-Schwarz-type [[sigma-model]] actions for all the other [[super p-branes]] were found, for instance for the [[M2-brane]] in ([Bergshoeff-Sezgin-Townsend 87](#BergshoeffSezginTownsend87)).

## Related entries

* [[geometry of physics -- supersymmetry]]

## References

Originally, $\kappa$-symmetry was observed for the [[super-particle]] in

* {#AzcarragaLukierski82} [[J. de Azcárraga]], J. Lukierski, _Supersymmetric particles_, Phys. Lett. B113 (1982) 170; Phys. Rev. D28 (1983) 1337.

* {#Siegel83} Warren Siegel, _Hidden Local Supersymmetry In The Supersymmetric Particle Action_ Phys. Lett. B 128, 397 (1983)

and for the [[super 1-brane in 3d]] in

* {#Siegel84} Warren Siegel, _Light Cone Analysis Of Covariant Superstring_ , Nucl. Phys. B 236, 311 (1984).

Then it was used to define/construct the manifestly spacetime supersymmetric [[Green-Schwarz action functional]] for the [[superstring]] in 10d in

* {#GreenSchwarz84} [[Michael Green]], [[John Schwarz]], _Covariant description of superstrings_, Phys. Lett. B136 (1984), 367–370 ([web](http://adsabs.harvard.edu/abs/1984PhLB..136..367G), [spire:193596](http://inspirehep.net/record/193596), doi:10.1016/0370-2693(84)92021-5)

and then for the [[M2-brane]] in 11d in

* {#BergshoeffSezginTownsend87} [[Eric Bergshoeff]], [[Ergin Sezgin]], and [[Paul Townsend]], _Supermembranes and eleven-dimensional supergravity_, Phys. Lett. B189 (1987) 75–78. ([pdf](http://streaming.ictp.trieste.it/preprints/P/87/010.pdf))

and so forth (see the references at _[[super p-brane]]_).

The [[supergeometry|super-geometric]] interpretation of $\kappa$-symmetry as the odd-graded part of the action of [[super-diffeomorphism]] on the [[super p-brane]] [[worldvolume]], regarded itself as a [[supermanifold]] was first suggested in

* {#SorokinTkachVolkov89} [[Dmitri Sorokin]], V. Tkach and [[Dmitrij Volkov]], _Superparticles, twistors and Siegel symmetry_, Mod. Phys. Lett. A4 (1989) 901-908 ([spire:271923](http://inspirehep.net/record/271923), [doi:10.1142/S0217732389001064](https://doi.org/10.1142/S0217732389001064))

Review of this perspective includes

* {#Sorokin99} [[Dmitri Sorokin]], _Superbranes and Superembeddings_, Phys.Rept.329:1-101,2000 ([arXiv:hep-th/9906142](http://arxiv.org/abs/hep-th/9906142))

* {#HoweSezgin04} [[Paul Howe]], [[Ergin Sezgin]], section 4.3 of _The supermembrane revisited_, Class.Quant.Grav. 22 (2005) 2167-2200 ([arXiv:hep-th/0412245](http://arxiv.org/abs/hep-th/0412245))

[[!redirects kappa symmetry]]


Kapranov's noncommutative geometry

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## Idea

[[Mikhail Kapranov]] introduced a framework of [[noncommutative algebraic geometry]], which could be interpreted as the study of [[formal geometry|infinitesimal thickenings]] of commutative [[schemes]] within bigger [[noncommutative geometry|noncommutative spaces]]. The idea is to look at commutative schemes together with a sheaf of nilpotent noncommutative O-algebras; the formalism utilizes the filtrations relating to the commutator expansions from [[Maslov-Feynman calculus]].

## Related concepts

* [[noncommutative algebraic geometry]]

* [[derived noncommutative geometry]]

## References

* M. Kapranov, _Noncommutative geometry based on commutator expansions_,  J. reine und angew. Math. __505__ (1998), 73-118, [math.AG/9802041](https://arxiv.org/abs/math/9802041).

* lecture at msri 2000: _Noncommutative neighborhoods and noncommutative Fourier transform_, [link](http://msri.org/publications/ln/msri/2000/interact/kapranov/1/index.html)

* Guillermo Cortiñas, _The structure of smooth algebras in Kapranov's framework for noncommutative geometry_, [math.RA/0002177](https://arxiv.org/abs/math/0002177).

* G. Cortiñas, _De Rham and infinitesimal cohomology in Kapranov's model for noncommutative algebraic geometry_, Compositio Mathematica __136__, 171-208, 2003, [math.AG/0102133](http://arxiv.org/abs/math/0102133).

* Shilin Yu, _Dolbeault dga of a formal neighborhood_, [arxiv/1206.5155](http://arxiv.org/abs/1206.5155); _The Dolbeault dga of the formal neighborhood of a diagonal_, [arxiv/1211.1567](http://arxiv.org/abs/1211.1567)

* Alexander Polishchuk, Junwu Tu. _DG-resolutions of NC-smooth thickenings and NC-Fourier-Mukai transforms_, [arXiv/1308.4244](https://arxiv.org/abs/1308.4244)


Kapranov-Voevodsky 2-vector space

A **Kapranov--Voevodsky 2-vector space** is a kind of [[2-vector space]], in this case a [[category]] [[equivalence of categories|equivalent]] to [[Vect]]${}^n$ for some finite $n$.  For details, see:

* [[Mikhail Kapranov]] and [[Vladimir Voevodsky]], 2-categories and Zamolodchikov tetrahedra equations, in _Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods_, _Proc. Sympos. Pure Math._ **56**, Part 2, AMS, Providence, RI, 1994, pp. 177--259. ([pdf](https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/1994_Kapranov_Voevodsky.pdf))

* Josep Elgueta, A strict totally coordinatized version of Kapranov and Voevodsky's 2-category $2Vect$.  ([arXiv](http://arxiv.org/abs/math/0406475))

There is also a more abstract characterization of Kapranov--Voevodsky 2-vector spaces, described here:

* Martin Neuchl, [_Representation Theory of Hopf Categories_](http://math.ucr.edu/home/baez/neuchl.ps),
Ph.D. dissertation, University of Munich, 1997.

* [[David Yetter]], [Categorical linear algebra---a setting for questions from physics and low-dimensional topology](http://math.ucr.edu/home/baez/yetter.pdf), Kansas State University preprint.

Namely, they are semisimple $k$-linear [[abelian categories]] with finitely many simple objects.  We may also drop the finiteness condition here to define a class of 'infinite-dimensional' Kapranov--Voevodsky 2-vector spaces.  For further discussion and more references, see:

* [[John Baez]], Aristide Baratin, Laurent Freidel and Derek Wise, Infinite-dimensional representations of 2-groups.  ([arXiv](http://arxiv.org/abs/0812.4969))

For representations of 2-groups on Kapranov--Voevodsky 2-vector spaces see:

* Josep Elgueta, Representation theory of 2-groups on finite dimensional 2-vector spaces.   ([arXiv](https://arxiv.org/abs/math/0408120))

[[!redirects Kapranov-Voevodsky 2-vector spaces]]
[[!redirects Kapranov--Voevodsky 2-vector space]]
[[!redirects Kapranov–Voevodsky 2-vector space]]
[[!redirects Kapranov--Voevodsky 2-vector spaces]]
[[!redirects Kapranov–Voevodsky 2-vector spaces]]


Kapustin-Witten TQFT

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## Idea

The _Kapustin-Witten TQFT_ is the [[4d TQFT]] obtained by  [[topologically twisted super Yang-Mills theory|topological twisting]] from  [[N=4 D=4 super Yang-Mills theory]] on a 4-dimensional manifold of the form $M = \Sigma\times C$, where $\Sigma$ is a [[Riemann surface]], possibly [[manifold with boundary|with boundary]] and $C$ is a Riemann surface of [[genus of a surface|genus]] $g\geq 2$. Its [[S-duality]] is supposed to encode, as a special case, [[geometric Langlands duality]] for the Riemann surface $C$.

Upon [[Kaluza-Klein mechanism|compactification]] on $C$ (i.e. $C$ is taken to be much smaller than $\Sigma$) down to 2d the resulting [[effective field theory]] on $\Sigma$ reproduces, at certain parameters, the [[A-model]] and the [[B-model]] [[topological string theory]], whose [[target space|target]] as a [[sigma model]] is the Hitchin moduli space $\mathcal{M}_{H}(G)$, where $G$ is the [[gauge group]] of the theory.

The Hitchin moduli space is a [[hyperkähler manifold]]. In [[complex structure]] $I$ it can be seen as the moduli space of [[Higgs bundles]] on $C$, and in complex structures $J$ and $K$ it can be seen as the moduli space of $G$-[[local systems]] on $C$. This is a form of [[nonabelian Hodge theory]].

[[S-duality]] manifests as [[mirror symmetry]] between $\mathcal{M}_{H}(G)$ and $\mathcal{M}_{H}({}^{L}G)$, where ${}^{L}G$ is the [[Langlands dual group]] of $G$. It exchanges [[B-branes]] on $\mathcal{M}_{H}({}^{L}G)$ in complex structure $J$ with [[A-branes]] on $\mathcal{M}_{H}(G)$ with complex structure $K$. It also exchanges [[Wilson loop]] operators and [['t Hooft operators]] (which correspond to [[Hecke correspondences]]).

The connection to the geometric Langlands correspondence may now be made more explicit as follows. We start with a zerobrane (a B-brane) $\mathcal{B}$ on $\mathcal{M}_{H}({}^{L}G)$ in complex structure $J$, which is an eigenbrane for the Wilson loop operators, corresponding to a local system on $C$. S-duality gives us an A-brane $\mathcal{A}$ on $\mathcal{M}_{H}(G)$ in complex structure $K$, which is an eigenbrane for the 't Hooft operators. Now we perform a hyperkahler rotation and consider $\mathcal{A}$ as an A-brane on $\mathcal{M}_{H}(G)$, in complex structure $I$. Let $\mathcal{A}_{cc}$ be the canonical isotropic brane on $\mathcal{M}_{H}(G)$. The [[sheafification]] of $\mathrm{Hom}(\mathcal{A}_{cc},\mathcal{A})$ gives us a [[D-module]] for the sheaf of differential operators given by the sheafification of $\mathrm{Hom}(\mathcal{A}_{cc},\mathcal{A}_{cc})$ (see also the related topic of [[quantization via the A-model]]). The resulting D-module is expected to be the [[Hecke eigensheaf]] predicted by the geometric Langlands correspondence.

[[!include gauge theory from AdS-CFT -- table]]

## Related concepts

* [[geometric Langlands duality]]

* [[A-model]], [[B-model]]

* [[quantization via the A-model]]

## References

The TQFT was introduced in

* [[Anton Kapustin]], [[Edward Witten]], _Electric-Magnetic Duality And The Geometric Langlands Program_ , Communications in number theory and physics, Volume 1, Number 1, 1–236 (2007) ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))
 {#KapustinWitten}

Reviews include

* [[Anton Kapustin]], _Langlands Duality and Topological Field Theory_ (2007) ([pdf](http://www.icmat.es/seminarios/langlands/school/handouts/kapustin.pdf))

* [[Edward Frenkel]], _Gauge Theory and Langlands Duality_ (2009) ([arXiv:0906.2747](https://arxiv.org/abs/0906.2747))

* [[Edward Witten]], _Mirror Symmetry, Hitchin's Equations, and Langlands Duality_ (2009) ([arXiv:0802.0999](https://arxiv.org/abs/0802.0999))

The link between mirror symmetry and geometric Langlands was explored in the earlier paper

* [[Tamas Hausel]], [[Michael Thaddeus]], _Mirror Symmetry, Langlands Duality, and the Hitchin System_ ([arXiv:0205236](https://arxiv.org/abs/math/0205236))

The 0-1-2 [[extended QFT]] version of $GL$-twisted [[N=4 D=4 super Yang-Mills theory]] is considered in

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

A discussion formalized in [[BV quantization]] of [[factorization algebras]] is in

* [[Kevin Costello]], _Notes on supersymmetric and holomorphic field theories in dimension 2 and 4_ ([pdf](http://www.math.northwestern.edu/~costello/sullivan.pdf))


Karapet Mkrtchyan

* [institute page](https://ifs.am/dr-karapet-mkrtchyan)

* [GoogleScholar page](https://scholar.google.co.kr/citations?user=GUg-rREAAAAJ&hl=en)

## Selected writings

A [[Lagrangian density]] for the [[boson|bosonic sector]] of [[type II supergravity]] making the nature of the [[pregeometric RR-fields]] and their [[self-dual higher gauge theory|self-duality]] manifest:

* [[Karapet Mkrtchyan]], [[Fridrich Valach]], *Democratic actions for type II supergravities*, Phys.Rev.D **107** 6 (2023) 066027 [[arXiv:2207.00626](https://arxiv.org/abs/2207.00626), [doi:10.1103/PhysRevD.107.066027](https://doi.org/10.1103/PhysRevD.107.066027)]

category: people


Karel Lambert

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

## related $n$Lab pages

* [[logic]]

category: people


Karen Uhlenbeck

* [webpage](http://www.ma.utexas.edu/users/uhlen/)

## Selected writings

On [[instantons]] and [[4-manifolds]]:

* [[Daniel Freed]], [[Karen Uhlenbeck]], _Instantons and Four-Manifolds_, Mathematical Sciences Research Institute Publications, Springer 1991 ([doi:10.1007/978-1-4613-9703-8](https://link.springer.com/book/10.1007/978-1-4613-9703-8))

On [[Yang-Mills theory]] ([[gauge theory]]) and [[Yang-Mills equations]]:

* {#Uhlenbeck12} [[Karen Uhlenbeck]], notes by [[Laura Fredrickson]], _Equations of Gauge Theory_, lecture at Temple University, 2012 ([pdf](https://web.stanford.edu/~ljfred4/Attachments/TempleLectures.pdf), [[UhlenbeckGaugeTheory.pdf:file]])

category: people


Karen Yeats

* [personal page](https://www.math.uwaterloo.ca/~kayeats/index.html)

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

## Selected writings

On [[round chord diagrams]] organizing [[Dyson-Schwinger equations]]:

* Nicolas Marie, [[Karen Yeats]], *A chord diagram expansion coming from some Dyson-Schwinger equations*, Communications in Number Theory and Physics, 7(2):251291, 2013 ([arXiv:1210.5457](https://arxiv.org/abs/1210.5457))

* Markus Hihn, [[Karen Yeats]], *Generalized chord diagram expansions of Dyson-Schwinger equations*, Ann. Inst. Henri Poincar Comb. Phys. Interact. 6 no 4:573-605 ([arXiv:1602.02550](https://arxiv.org/abs/1602.02550))

category: people


Kari Vilonen

__Kari Vilonen__ is a mathematician at Northwestern University, specialized in [[geometric representation theory]].

* [homepage](http://www.math.northwestern.edu/~vilonen)

category: people


Karl Heinrich Hofmann

* [webpage](https://www.mathematik.tu-darmstadt.de/fb/personal/details/karl_heinrich_hofmann.en.jsp)

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

## Selected writings

Introducing [[Baire lattices]]:

 * [[Karl Heinrich Hofmann]], _A note on Baire spaces and continuous lattices_ 1980. Bulletin of the Australian Mathematical Society, 21(2), pp. 265-279. ([doi:10.1017/S0004972700006080]( https://doi.org/10.1017/S0004972700006080))

Monograph on [[domain theory]] and [[Baire lattices]]:

 * G. Gierz, [[Karl H. Hofmann]], K. Keimel, J. D. Lawson, [[Michael W. Mislove]], [[Dana S. Scott]], *Continuous Lattices and Domains* in *Encyclopedia of Mathematics and its Applications* 93, Cambridge University Press (2003)  [[doi:10.1017/CBO9780511542725](https://doi.org/10.1017/CBO9780511542725)]

On [[compact subgroups]] of [[Lie groups]] and [[locally compact groups]]:

* [[Karl Heinrich Hofmann]], [[Christian Terp]], *Compact Subgroups of Lie Groups and Locally Compact Groups*, Proceedings of the American Mathematical Society Vol. 120, No. 2 (Feb., 1994), pp. 623-634 ([jstor:2159906](https://www.jstor.org/stable/2159906), [doi:10.2307/2159906](https://doi.org/10.2307/2159906))

category: people

[[!redirects Karl Hofmann]]
[[!redirects Karl H. Hofmann]]


Karl Kraus

* Wikipedia entry

## Selected writings

Introducing the notion of [[quantum operations]] and what came to be called their *[[Kraus decomposition]]*:

* {#Kraus71} [[Karl Kraus]], *General state changes in quantum theory*, Ann. Physics **64** 2 (1971) 311-335 [doi:10.1016/0003-4916(71)90108-4]

On the theoretical foundations of [[quantum physics]] ([[quantum states]], [[effect algebras]], and [[quantum operations]]):

* {#Kraus} [[Karl Kraus]], *States, Effects, and Operations -- Fundamental Notions of Quantum Theory*, Lecture Notes in Physics **190** Springer (1983) [[doi:10.1007/3-540-12732-1](https://doi.org/10.1007/3-540-12732-1)]

category: people


Karl Landsteiner

* [personal page](https://members.ift.uam-csic.es/karllandsteiner/index.html)

* [GoogleScholar page](https://scholar.google.com/citations?user=BLukMW0AAAAJ&hl=en)

## Selected writings

Introducing the holographic descriptions of [[topological semimetals]] via the [[AdS-CMT correspondence]]:

* [[Karl Landsteiner]], [[Yan Liu]], *The holographic Weyl semi-metal*, Physics Letters B **753** (2016) 453-457 [[arXiv:1505.04772](https://arxiv.org/abs/1505.04772), [doi:10.1016/j.physletb.2015.12.052](https://doi.org/10.1016/j.physletb.2015.12.052)]

* [[Karl Landsteiner]], [[Yan Liu]], Ya-Wen Sun, *Quantum phase transition between a topological and a trivial semimetal from holography*, Phys. Rev. Lett. **116** 081602 (2016) [[arXiv:1511.05505](https://arxiv.org/abs/1511.05505), [doi:10.1103/PhysRevLett.116.081602](https://doi.org/10.1103/PhysRevLett.116.081602)]

category: people


Karl Popper

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

category: people


Karl Rosenkranz

**Karl Rosenkranz** (1805-1879) was a German philosopher and a leading exponent of the _Hegel school_. A student, editor and biographer of [[G. W. F. Hegel]], Rosenkranz is probably the top candidate for the title of greatest Hegel scholar of all time.

Particularly noteworthy are his writings on logic as they present a careful (and eminently readable!) reworking of Hegel's system and come as close to a 'third edition' of the latter's _Wissenschaft der Logik_ - [[Science of Logic]] as can be.

As is suitable for a member of the philosophical faculty of the university of Königsberg where he held the chair of [[J. F. Herbart]], Rosenkranz stressed the continuity between the Hegelian system and [[Kant]]'s enterprise of transcendental philosophy as initiated in the _Kritik der reinen Vernunft_.

## Related entries

* [[G. W. F. Hegel]]

* [[Dieter Wandschneider|D. Wandschneider]]

* [[Stephan Wannowski]]

## Links

* [Wikipedia entry (German)](http://de.wikipedia.org/wiki/Karl_Rosenkranz)

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

## References

Scans of the editions of his writings in fracture print are available online (follow the link on the English wikipedia page above!).

The most famous of Rosenkranz' writings is:

* K. Rosenkranz, _Ästhetik des Hässlichen_, Reclam Stuttgart 2007[1853].

For his contribution to the Hegelian system consult e.g.:

* K. Rosenkranz, _Wissenschaft der logischen Idee_ - 2 vols., Königsberg 1858-59.

* K. Rosenkranz, _Hegels Naturphilosophie und die Bearbeitung derselben durch den italienischen Philosophen Augusto Véra_, Olms Hildesheim 1979[1868].

His views on Kant are expressed in

* {#Rosenkranz40} [[Karl Rosenkranz]], _Geschichte der Kant'schen Schule_, Akademie-Verlag Berlin 1987[1840].

For a recent assessment of his contribution see

* V. Hösle, _Hegels System - Der Idealismus der Subjektivität und das Problem der Intersubjektivität_, Meiner Hamburg 1998.

category:people


Karl Stein

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

## Selected writings

Introducing the concept of [[Stein manifolds]]:

* {#Stein51} [[Karl Stein]],  _Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem_, Math. Ann. (in German) 123: 201–222, (1951) ([doi:10.1007/BF02054949](https://doi.org/10.1007/BF02054949), MR 0043219)

and in relation to open [[Riemann surfaces]]:

* {#BehnkeStein47} [[Heinrich Behnke]], [[Karl Stein]], *Entwicklung analytischer Funktionen auf Riemannschen Flächen*, Mathematische Annalen volume 120, pages 430–461 (1947) ([doi:10.1007/BF01447838](https://doi.org/10.1007/BF01447838))

## Related $n$Lab entries

* [[Stein manifold]]

category: people


Karl Weierstraß

[[!redirects Karl Weierstrass]]

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

## related $n$Lab entries

* [[Stone-Weierstrass theorem]]

* [[Weierstrass preparation theorem]]

* [[Weierstrass elliptic function]]

* [[Weierstrass sigma-function]]

* [[epsilontic analysis]]

category: people


Karl-Georg Schlesinger

* [webpage](http://physik.univie.ac.at/index.php?id=1598)

category: people


Karl-Hermann Neeb

* [webpage](http://www3.mathematik.tu-darmstadt.de/hp/algebra-ehemalige/neeb-karl-hermann/startseite.html)

## Selected writings

On [[Lie theory]]

on [[Lie semigroups]]:

* [[Joachim Hilgert]], [[Karl-Hermann Neeb]], *Lie Semigroups and their Applications*. Lecture Notes in Mathematics **1552**, Springer 1993 ([doi:10.1007/BFb0084640](https://link.springer.com/book/10.1007/BFb0084640))

on [[Lie groups]];

* {#HilgertNeeb12} [[Joachim Hilgert]], [[Karl-Hermann Neeb]], _Structure and Geometry of Lie Groups_, Springer Monographs in Mathematics, Springer-Verlag New York, 2012 ([doi:10.1007/978-0-387-84794-8](https://link.springer.com/book/10.1007/978-0-387-84794-8))

on [[infinite-dimensional Lie groups]]:

* [[Karl-Hermann Neeb]], _Monastir summer school: Infinite-dimensional Lie groups_, 2005 ([pdf](http://www.math.uni-hamburg.de/home/wockel/data/monastir.pdf), [[Neeb_MonastirLieGroups.pdf:file]])

* [[Karl-Hermann Neeb]], _Towards a Lie theory of locally convex groups_, Japanese Journal of Math. 1 (2006), 291-468 ([arXiv:1501.06269](https://arxiv.org/abs/1501.06269))

## Related $n$Lab entries

* [[infinite-dimensional Lie group]]

* [[universal central extension]]

* [[Spin(n).Spin(m)]]

category: people


Karlheinz Seeger

* [Wikipedia entry](https://de.wikipedia.org/wiki/Karlheinz_Seeger) (in Deutsch)

## Selected writings

On [[solid state physics]] (such as [[electron band theory]]) in view of [[semiconductor]] theory:

* [[Karlheinz Seeger]], *Semiconductor Physics*, Advanced texts in physics, Springer (2004) $[$[doi:10.1007/978-3-662-09855-4](https://doi.org/10.1007/978-3-662-09855-4)$]$

category: people


Karol Borsuk

Karol Borsuk was a Polish topologist, who was famous for numerous theorems of [[general topology|classical topology]].

He was the initial developer of [[shape theory]], and then was leader of a very influential group of researchers in that area.

* [St. Andrews History page](http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Borsuk.html)

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

* [Wikipedia entry (Polish)](http://pl.wikipedia.org/wiki/Karol_Borsuk)

## Selected writings

Introducing the notion of [[absolute neighbourhood retracts]]:

* {#Borsuk32} [[Karol Borsuk]], _Über eine Klasse von lokal zusammenhängenden Räumen_, Fund. Math. **19** (1932) 220-242 ([dml:212574](https://eudml.org/doc/212574), [pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm19/fm19119.pdf))

On the [[nerve theorem]]:

* {#Borsuk48} [[Karol Borsuk]], _On the imbedding of systems of compacta in simplicial complexes_ , Fund. Math. 35, (1948) 217–234 ([dml:213158](https://eudml.org/doc/213158))

More on [[absolute neighbourhood retracts]]:

* [[Karol Borsuk]], _Concerning the classification of topological spaces from the stand point of the theory of retracts_, Fundamenta Mathematicae **46** (3) (1959) 321-330 ([dml:213516](https://eudml.org/doc/213516))

Introducing unstable [[Cohomotopy]]:

* {#Borsuk36} [[Karol Borsuk]], _Sur les groupes des classes de transformations continues_, C. R. Acad. Sci. Paris 202.1400-1403 (1936)

On [[shape theory]]:

* [[Karol Borsuk]], _Concerning homotopy properties of compacta_, Fund Math. 62 (1968) 223-254

* [[Karol Borsuk]], _Theory of Shape_, Monografie Matematyczne Tom 59, Warszawa 1975 ([mathscinet:0418088](https://mathscinet.ams.org/mathscinet-getitem?mr=0418088))

* [[Karol Borsuk]], [[Jerzy Dydak]], _What is the theory of shape?_, Bulletin of the Australian Mathematical Society, 1980, 22(2), 161-198, ([doi:10.1017/S000497270000647X](https://doi.org/10.1017/S000497270000647X))

## Related $n$Lab entries

* [[shape theory]]

* [[nerve theorem]]

* [[cohomotopy]]

* [[retract]]

category: people

[[!redirects K. Borsuk]]


Karol Szumiło

* [webpage](http://www1.maths.leeds.ac.uk/~pmtks/)

## Selected writings

On the [[constructive model structure on simplicial sets]]:

* {#GambinoSattlerSzumilo19} [[Nicola Gambino]], [[Christian Sattler]], [[Karol Szumiło]], _The constructive Kan-Quillen model structure: two new proofs_ ([arXiv:1907.05394](https://arxiv.org/abs/1907.05394))

category: people

[[!redirects Karol Szumilo]]


Karol Życzkowski

* [personal page](https://chaos.if.uj.edu.pl/~karol/)

## Selected writings

On [[quantum mechanics]] with focus on the [[geometry]] of ([[mixed quantum state|mixed]]) [[quantum state spaces]] and their [[quantum channels]], culminating in a chapter on [[quantum entanglement]]:

* {#ŻyczkowskiBengtsson04} [[Karol Życzkowski]], [[Ingemar Bengtsson]], *On Duality between Quantum Maps and Quantum States*, Open Systems & Information Dynamics **11** 01 (2004) 3-42 [[doi:10.1023/B:OPSY.0000024753.05661.c2](https://doi.org/10.1023/B:OPSY.0000024753.05661.c2)]

* [[Ingemar Bengtsson]], [[Karol Życzkowski]], *Geometry of Quantum States --- An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [[doi:10.1017/CBO9780511535048](https://doi.org/10.1017/CBO9780511535048), [ResearcgGate](https://www.researchgate.net/publication/266435541_Geometry_of_Quantum_States_An_Introduction_to_Quantum_Entanglement)]

category: people

[[!redirects Karol Zyczkowski]]