variational bicomplex

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

# Contents
* table of contents
{: toc}

## Idea

For $X$ a ([[spacetime]]) [[manifold]] and $E \to X$ a [[bundle]] (in [[physics]] called the _[[field bundle]]_) with [[jet bundle]] $Jet(E) \to X$, the **variational bicomplex** is essentially the [[de Rham complex]] $(\Omega^\bullet(Jet( E)),\mathbf{d})$ of $Jet(E)$ with [[differential forms]] $\Omega^n(Jet(E)) = \bigoplus_{h+v=n} \Omega^{h,v}(E)$ bigraded by [[horizontal differential form|horizontal]] degree $h$ (with respect to $X$) and vertical degree $v$ (along the [[fiber]]s of $j_\infty E$)). Accordingly the [[differential]] decomposes as

$$
  \mathbf{d} = d + \delta
  \,,
$$

where $\mathbf{d}$ is the [[de Rham differential]] on $Jet(E)$, $d$ is called the **horizontal differential** and $\delta$ is called the **vertical differential**.

With $E \to X$ thought of as a [[field bundle]] over [[spacetime]]/[[worldvolume]], then $d$ is a measure for how quantities change over spacetime, while $\delta$ is the [[variational calculus|variational]] differential that measures how quantities change as the field configurations are varied.

Accordingly, much of [[classical mechanics]] and [[classical field theory]] on $X$ is formalized in terms of the variational bicomplex. For instance

* a field configuration is a [[section]] of $E$;

* a [[Lagrangian]] is an element $L \in \Omega^{n,0}(E)$;

* a [[local action functional]] is a map

  $$
    S : \Gamma(E) \to \mathbb{R}
  $$

  of the form

  $$
    S(\phi) = \int_X L(j^\infty \phi)
    \,,
  $$

* the [[Euler-Lagrange equation]] is

  $$
    E(L) := \delta L \mod im d = 0
  $$

* the [[covariant phase space]] is the locus

  $$
    \{
     \phi \in \Gamma(E) | E(L)(j^\infty \phi) = 0
   \}
  $$

* a [[conserved current]] is an element $\eta\in \Omega^{n-1,0}(E)$ that is  horizontally closed on the covariant phase space

  $$
    d \eta = 0 \mod E(L)
  $$

* a [[symmetry]] is an evolutionary vector field $v$ such that

  $$
    v(L) = 0 \mod im d
  $$

* [[Noether's theorem]] asserts that every [[symmetry]] induces a [[conserved current]].

## Definition

Let $X$ be a [[smooth manifold]] and $p : E \to X$ some smooth [[bundle]] over $X$. Write $Jet(E) \to X$ for the corresponding [[jet bundle]].

### The bicomplex

The spaces of [[sections]] $\Gamma(E)$ and $\Gamma(Jet(E))$ canonically inherit a generalized smooth structure that makes them [[diffeological spaces]]: we have a [[pullback]] [[diagram]] of diffeological spaces

$$
  \array{
     \Gamma(E) & \longrightarrow & *
     \\
     \downarrow && \downarrow^{id}
     \\
     [X,E] &\stackrel{p_*}{\longrightarrow}& [X,X]
  }
  \,.
$$

This induces the [[evaluation map]]

$$
  X \times \Gamma(E) \to E
  \,.
$$

and composed with the jet prolongation

$$
  j^\infty : \Gamma(E)  \to \Gamma(Jet(E))
$$

it yields a smooth map ([[homomorphism]] of [[diffeological spaces]])

\[
  \label{EProlongation}
  e_\infty : X \times \Gamma(E) \stackrel{(id,j^\infty)}{\to}
   X \times \Gamma(Jet(E))
   \stackrel{ev}{\to}
  Jet(E)
  \,.
\]

Write

$$
  \Omega^{\bullet, \bullet}(X \times \Gamma(E))
$$

for the [[cochain complex]] of smooth differential forms on the [[product]] $X \times \Gamma(E)$, bigraded with respect to the differentials on the two factors

$$
  \mathbf{d} \coloneqq d + \delta
  \,,
$$

where the $\mathbf{d}$, $d$ and $\delta$, are the [[de Rham differentials]] of $X\times\Gamma(E)$, $X$ and $\Gamma(E)$, respectively.

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

The **variational bicomplex** of $E \to X$ is the sub--bi-complex of $\Omega^{\bullet, \bullet}(X \times \Gamma(E))$ that is the [[image]] of the pullback of forms along the map $e_\infty$ (eq:EProlongation):

$$
  e_\infty^* : \Omega^{\bullet}(Jet(E)) \to
   \Omega^\bullet(X \times \Gamma(E))
  \,.
$$

We write

$$
  \Omega^{\bullet, \bullet}_{loc}
  \coloneqq
  im (e_\infty^*)
$$

and speak of the bicomplex of **local forms** on sections on $E$.

=--

The bicomplex structure on $\Omega^{\bullet, \bullet}_{loc}$ is attributed in ([Olver 86](#Olver86)) to ([Takens 79](#Takens79)). The above formulation as a sub-bicomplex of the evident bicomplex of forms on $X \times \Gamma(E)$ is due to [[Zuckerman 87, p. 5](#Zuckerman87)].

<img src="https://ncatlab.org/nlab/files/VariationalBicomplex.jpg" width="550">

### More on the horizontal differential complex
 {#MoreOntheHorizontalComplex}

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

In terms of a [[coordinate chart]] $(x^i, u^\alpha,u^\alpha_i,u^\alpha_{i j},\cdots)$ of $E$ covering a coordinate chart $(X^i)$ of $X$, the action of the horizontal differential on functions $f \in C^\infty(Jet(E))$ is given by the formula for the [[total derivative]] operation, but with concrete differentials substituted by the respective jet coordinates:

$$
  d_h f
    \;\coloneqq\;
  \sum_i
  \left(
    \frac{\partial f}{\partial x^i}
    +
    \frac{\partial f}{\partial u^\alpha}u^\alpha_i
    +
    \sum_j \frac{\partial f}{\partial u^\alpha_j} u^\alpha_{j i}
    +
    \cdots
  \right)
  d x^i
  \,.
$$

([Anderson 89, p. 10](#Anderson89)).

=--

More abstractly, the horizontal differential is characterized as follows:

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

The horizontal differential takes horizontal forms to horizontal forms, and for all sections $\phi \in \Gamma(E)$ it respects [[pullback of differential forms]] along the jet prolongation $j_\infty \phi \in \Gamma(Jet(E))$

$$
  (j^\infty \phi)^\ast \circ d_h = d \circ (j^\infty \phi)^\ast
$$

(where on the right we have the ordinary de Rham differential on the base space).

=--

Yet more abstractly, the horizontal complex may be understood in terms of [[differential operators]] and the [[jet comonad]] as follows.

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

A horizontal differential $n$-form $\alpha$ on $Jet(E) \to X$ is equivalently a homomorphism of [[bundles]] over $X$

$$
  \alpha \colon Jet(E) \longrightarrow \wedge^n T^\ast X
$$

from the [[jet bundle]] $Jet(E)$ to the [[exterior bundle]] $\wedge^n T^\ast X$. This in turn is, by the discussion there, equivalently a [[differential operator]] $\alpha \colon E \to \wedge^n T^\ast X$.

Now of course also the [[de Rham differential]] $d_X$ on $X$ is a differential operator $\wedge^n T^\ast X \to \wedge^n T^\ast X$. In view of this, the horizontal differential of the variational bicomplex is just the [[composition]] operation of differential operators, with horizontal forms regarded as differential operators as above.

By the fact that differential operators are the [[co-Kleisli morphisms]] of the [[Jet comonad]], this means that the horizontal differential is

$$
  d_H \alpha
  \colon
  Jet(F)
  \longrightarrow
  Jet(Jet(F))
  \stackrel{Jet(\alpha)}{\longrightarrow}
  Jet(\wedge^n T^\ast X)
  \stackrel{\tilde d_X}{\longrightarrow}
  \wedge^n T^\ast X
  \,.
$$

=--

(e.g. [Krasil'shchik-Verbovertsky 98, around def. 3.27](#KrasilshchikVerbovertsky98), [Krasil'shchik-Vinogradov 99, ch 4, around def. 1.8](#KrasilshchikVinogradov99))

### Evolutionary vector fields

Vector fields on $J^\infty E$ also split into a direct sum of **vertical** and **horizontal** ones, respectively being annihilated by contraction with any horizontal $1$-forms or with any vertical $1$-forms, $\mathfrak{X}(J^\infty E) = \mathfrak{X}_H(J^\infty E) \oplus \mathfrak{X}_V(J^\infty E)$. A special kind of vertical vector field $v \in \mathfrak{X}_V(J^\infty E)$ is called an **[[evolutionary vector field]]** provided it satisfies $\mathcal{L}_v d = d \mathcal{L}_v$ and $\mathcal{L}_v = \iota_v \delta + \delta \iota_v$, we denote the subspace of evolutionary vector fields as $\mathfrak{X}_{ev}(J^\infty E) \subset \mathfrak{X}_V(J^\infty E)$.

## Properties

### Horizontal, vertical, and total cohomology

Let $E \to X$ be a smooth [[fiber bundle]] over a base [[smooth manifold]] $X$ of [[dimension]] $n.$ Write $J^\infty E \to X$ for the [[jet bundle]] of $E\to X$.

Write

$$
  \mathcal{F}^s(J^\infty E) \coloneqq I (\Omega^{n,s}(J^\infty E))
$$

for the projection of $(n,s)$-forms to the image of the "interior Euler operator" ([Anderson 89, p. 21 (50/318)](#Anderson89)).

+-- {: .num_prop #ELComplexHasSameCohomologyAsDeRhamComplex}
###### Proposition
**(Takens acyclicity theorem)**

The [[cochain cohomology]] of the [[Euler-Lagrange complex]]

$$
 0 \to \mathbb{R}
 \to
  \Omega^{0,0}(J^\infty E)
  \stackrel{d_H}{\to}
  \Omega^{1,0}(J^\infty E)
  \stackrel{d_H}{\to}
  \cdots
  \stackrel{d_H}{\to}
  \Omega^{n,0}(J^\infty E)
  \stackrel{E}{\to}
  \mathcal{F}^1(J^\infty E)
  \stackrel{\delta_V}{\to}
  \mathcal{F}^2(J^\infty E)
  \stackrel{\delta_V}{\to}
  \cdots
$$

is [[isomorphism|isomorphic]] to the [[de Rham cohomology]] of the total space $E$ of the given fiber bundle.

=--

For smooth functions of locally bounded jet order this is due to ([Takens 79](#Takens79)). A proof is also in ([Anderson 89, theorem 5.9](#Anderson89)).

For smooth functions of globally bounded order and going up to the Euler-Lagrange operator $E$, this is also shown in ([Deligne 99, vol 1, p.188](#Deligne99)).

### The fundamental variational formula

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

A **source form** is an element $\alpha$ in $\Omega^{n,1}_{loc}$ such that

$$
  \alpha_\phi(\delta \phi)
$$

depends only on the 0-jet of $\delta \phi$.

=--

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

Let $L \in \Omega^{n,0}_{loc}$.

Then there is a unique source form $E(L) $ such that

$$
  \delta L = E(L) - d \Theta
  \,.
$$

Moreover

* $E(L)$ is independent of changes of $L$ by $d$-exact terms:

  $$
    E(L) = E(L + d K)
    \,.
  $$

* $\Theta$ is unique up to $d$-exact terms.

=--

This is [[Zuckerman 87, theorem 3](#Zuckerman87)].

Here $E$ is the _[[Euler-Lagrange equations|Euler-Lagrange operator]]_ .

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

Write

$$
  \Omega = \delta \Theta
  \,.
$$

=--

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

By prop. \ref{VariationOfLagrangian} have

$$
  d \Omega = -\delta E(L)
  \,.
$$

=--

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

$\delta E$ vanishes when restricted to vertical tangent vectors based in [[covariant phase space]] (but not necessarily tangential to it).

$$
  \delta E(L) |_{T_{E(L) = 0} \Gamma(E)} = 0
  \,.
$$

=--

This is [[Zuckerman 87, lemma 8](#Zuckerman87)].

### Presymplectic covariant phase space

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

The form $\Omega$ is a [[conserved current]].

=--

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

By remark \ref{HorizontalDOfOmega} and
prop. \ref{SecondOrderVariationVanishesOnShell}.

=--

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

For $\Sigma \subset X$ a [[compact space|compact]] [[closed manifold|closed]] submanifold of [[dimension]] $n-1$, one says that

$$
  \omega := \int_\Sigma \Omega \in \Omega^{0,2}_{loc}
$$

is the [[presymplectic structure]] on [[covariant phase space]] relative to $\Sigma$.

=--

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

The 2-form $\omega$ is indeed closed

$$
  \delta \omega = 0
$$

and in fact exact:

$$
  \theta := \int_\Sigma \Theta
$$

is its _presymplectic potential_ .

$$
  \delta \theta  = \omega
  \,.
$$

=--

### Symmetries

Let $L \in \Omega^{n,0}_{loc}$.

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

An evolutionary vertical vector field $v \in \mathfrak{X}_{ev}(J^\infty E)$ is a **[[symmetry]]** if

$$
  v(L) = 0 \mod im d
  \,.
$$

=--

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

The presymplectic form $\omega$ from def. \ref{ThePresymplecticForm} is annihilated by the Lie derivative of the vector field on $\Gamma(E)$ induced by a symmetry.

=--

This appears as [[Zuckerman 87, theorem 13](#Zuckerman87)].

## Elementary formalization in differential cohesion
 {#FormalizationInDifferentialCohesion}

We discuss aspects of an [[elementary (infinity,1)-topos|elementary]]
formalization in [[differential cohesion]] of the concept of the variational bicomplex .

> under construction

Let $\mathbf{H}$ be a context of [[cohesion]] and [[differential cohesion]], with

* [[flat modality]] denoted $\flat$;

* [[infinitesimal shape modality]] denoted $\Im$.

Choose

1. an [[object]] $\Sigma \in \mathbf{H}$, the _base space_ (or _[[spacetime]]_ or _[[worldvolume]]_);

1. an object $E \in \mathbf{H}_{/\Sigma}$, the _[[field bundle]]_,

1. an object $\mathbf{A} \in Stab(\mathbf{H}_{/\Sigma}) \stackrel{\Omega}{\to} \mathbf{H}_{/\Sigma}$, the _[[differential cohomology|differential]] [[coefficients]]_.

Write

* $\mathbf{H}_{/\Sigma}
    \stackrel{\overset{\sum_\Sigma}{\longrightarrow}}{\stackrel{\overset{\Sigma^\ast}{\longleftarrow}}{\underset{\prod_\Sigma}{\longrightarrow}}}
    \mathbf{H}$
  for the [[base change]] [[adjoint triple]] over $\Sigma$, the [[étale geometric morphism]] of the [[slice (infinity,1)-topos]] $\mathbf{H}_{/\Sigma}$;

* $\Gamma_X \coloneqq \flat \circ \prod_\Sigma \colon \mathbf{H}_{/\Sigma} \to \mathbf{H}$ for the external [[space of sections]] functor;

* $i \colon \Sigma \longrightarrow \Im(\Sigma)$ for the $\Sigma$-component of the [[unit of a monad|unit]] of $\Im$;

* $Jet_\Sigma \coloneqq i^\ast i_\ast$ for the induced [[jet comonad]];

* $\mathbf{H}_{/\Sigma} \stackrel{\overset{}{\longleftarrow}}{\underset{\iota}{\longrightarrow}} PDE(\mathbf{H})_{\Sigma}$ for the [[Eilenberg-Moore category]] of $Jet_\Sigma$-[[algebras over a monad|coalgebras]] (the objects are [[differential equations]] with [[variables]] in $\Sigma$, the morphisms are [[differential operators]] between these, preserving solution spaces), manifested as a [[topos of coalgebras]] over $\mathbf{H}$;

  the (non-full) [[direct image]] of this [[geometric morphism]] is the [[co-Kleisli category]] of the [[jet comonad]] and so for $\phi \colon free(E) \to free(F)$ a morphism in $PDE(\mathbf{H})_\Sigma$, we write $\tilde f \colon Jet(E) \to F$ for the corresponding co-Kleisli morphism in $\mathbf{H}_{/\Sigma}$;

We record the following simple fact,
which holds generally since the [[jet comonad]] $Jet_\Sigma$ is a [[right adjoint]]
(to the [[infinitesimal disk bundle]] functor), hence preserves [[terminal objects]], and $\Sigma \in \mathbf{H}_{/\Sigma}$ is the [[terminal object]]:

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

The essentially unique morphism

$$
  Jet(\Sigma) \stackrel{\simeq}{\longrightarrow} \Sigma
$$

in $\mathbf{H}_{/\Sigma}$ in an [[equivalence]].

=--

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

The _jet prolongation_ map

$$
  j \colon \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(Jet(E))
$$

is the the Jet functor itself, regarded,
in view of prop. \ref{JetSigmaPreservesSigma},
as taking [[sections]] to sections via

$$
  (\Sigma \stackrel{\sigma}{\to} E)
  \;\;\mapsto
  \;\;
  \left(
  \Sigma \stackrel{\simeq}{\to} Jet(\Sigma)
  \stackrel{Jet(\sigma)}{\longrightarrow}
  Jet\left(E\right)
  \right)
  \,.
$$

=--

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

For $E \in \mathbf{H}_{\Sigma}$ a [[bundle]] over $\Sigma$,
then a _horizontal $\mathbf{A}$-form_ on the [[jet bundle]]
$Jet(E)$ is a morphism in $PDE(\mathbf{H})_{\Sigma}$ of the form

$$
  \alpha \colon \iota E \to \iota  \mathbf{A}
  \,.
$$

For $d \colon \iota Et_\Sigma\Sigma^\ast \mathbf{A}\to \iota Et_\Sigma\Sigma^\ast \mathbf{A}'$ a morphism in $\mathbf{PDE}(\mathbf{H})_{\Sigma}$, then
the induced _horizontal differential_ is the operation of horizontal forms sending $\alpha$ to the composite

$$
  d \alpha \colon \iota E
     \stackrel{\alpha}{\longrightarrow}
  \iota \mathbf{A}
   \stackrel{d}{\longrightarrow} \iota \mathbf{A}'
  \,.
$$

=--

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

Since all objects in def. \ref{HorizontalFormInDifferentialCohesion}
are in the [[co-Kleisli category]] of the [[jet comonad]], the morphism $\alpha$ there is equivalently a morphism in $\mathbf{H}_{/\Sigma}$ of the form

$$
  \tilde \alpha \colon Jet(E) \longrightarrow \mathbf{A}
  \,.
$$

For the special case that $E = \Sigma$ in def. \ref{HorizontalFormInDifferentialCohesion}, then $Jet_{\Sigma}(\Sigma)\simeq\Sigma$ and so a horizontal $\mathbf{A}$-form on $\Sigma$ we call just a an $\mathbf{A}$-form.

=--

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

The horizontal differential of def. \ref{HorizontalFormInDifferentialCohesion} commutes with pullback of horizontal differential forms $\alpha$ along the jet prolongation, def. \ref{JetProlongationInDifferentialCohesion}, of any field section $\sigma \in \Gamma_X(E)$.

In detail: for

* $d \colon \iota Et_\Sigma\Sigma^\ast \mathbf{A} \longrightarrow \iota Et_\Sigma\Sigma^\ast \mathbf{A}'$ a morphism,

* $\alpha \colon \iota E \to \iota Et_\Sigma\Sigma^\ast \mathbf{A}$ a horizontal $\mathbf{A}$-form on $Jet(E)$, def. \ref{HorizontalFormInDifferentialCohesion};

* $\sigma \in \Gamma_\Sigma(E)$ a [[field (physics)|field]] [[section]],

then there is a [[natural equivalence]]

$$
  j(\sigma)^\ast (d \alpha)
   \simeq
  d (j(\sigma)^\ast \alpha)
  \,.
$$

=--

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

Since all objects are in the [[direct image]] $free\colon \mathbf{H} \to PDE(\mathbf{H})_\Sigma$, this is an equivalence of morphisms in the [[co-Kleisli category]] of the [[jet comonad]], hence is equivalently an equivalence of co-Kleisli composites of morphisms in $\mathbf{H}$.

As such, the left hand side of the equality is given in $\mathbf{H}$ by the composite morphism

$$
  \Sigma
  \stackrel{\simeq}{\to}
  Jet(\Sigma)
   \stackrel{Jet(\sigma)}{\longrightarrow}
  Jet(E)
  \stackrel{}{\to}
  Jet(Jet(E))
  \stackrel{Jet(\tilde \alpha)}{\longrightarrow}
  Jet(\mathbf{A})
  \stackrel{\tilde d}{\longrightarrow}
  \mathbf{A}'
  \,,
$$

thought of as bracketed to the right. By [[natural transformation|naturality]] of the Jet-counit this is equivalently

$$
  Jet(\Sigma) \stackrel{\simeq}{\to} Jet(Jet(\Sigma))
  \stackrel{Jet(Jet(\sigma))}{\longrightarrow}
  Jet(Jet(E))
  \stackrel{Jet(\tilde \alpha)}{\longrightarrow}
  Jet(\mathbf{A})
  \stackrel{\tilde d}{\longrightarrow}
  \mathbf{A}'
  \,,
$$

By functorality of $Jet(-)$ this is equivalent to

$$
  Jet(\Sigma)
  \stackrel{\simeq}{\to}
  Jet(Jet(\Sigma)) \stackrel{Jet ( \tilde \alpha \circ Jet(\sigma) )}{\longrightarrow}
  Jet(\mathbf{A})
  \stackrel{\tilde d}{\to}
  \mathbf{A}'
$$

which is the right hand side of the equivalence to be proven.

=--

## Related concepts

* [[local Lagrangian]], [[Euler-Lagrange form]], [[source form]], [[Lepage form]]

* [[variational calculus]]

* [[secondary differential calculus]]

* [[variational sequence]]

* [[BV-BRST complex]]

* [[BV-BRST variational bicomplex]]

* [[presymplectic current]], [[covariant phase space]]

## References

The variational bicomplex was introduced independently in

* [[Alexandre Vinogradov]], _A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints_ , Sov. Math. Dokl. 19 (1978) 144–148.

* W. M. Tulczyjew, _The Euler-Lagrange resolution_ , in Lecture Notes in Mathematics 836 22–48 (Springer-Verlag, New York 1980).

* T. Tsujishita, _On variation bicomplexes associated to differential equations_, Osaka J. Math. 19 (1982), 311–363.

See also

* {#Takens79} [[Floris Takens]], _A global version of the inverse problem of the calculus of variations_ J. Diff. Geom. 14 (1979) 543-562

* {#Deligne99} [[Pierre Deligne]] et al. _[[Quantum Fields and Strings]]_

An introduction is in

* [[Ian Anderson]], _Introduction to the variational bicomplex_, in _Mathematical aspects of classical field theory_, Contemp. Math. 132 (1992) 51–73, [gBooks](http://books.google.com/books?id=NuiJ0c72_gQC&lpg=PA51&ots=-zAVQLViUn&dq=Zuckerman%20variational%20problems)

A careful discussion that compares the two versions (one over smooth functions globally of finite jet order, one over smooth functions locally of finite jet order) is in

* G. Giachetta, L. Mangiarotti, [[Gennadi Sardanashvily]], _Cohomology of the variational bicomplex on the infinite order jet space_, Journal of
Mathematical Physics 42, 4272-4282 (2001) ([arXiv:math/0006074](http://arxiv.org/abs/math/0006074))

Textbook accounts include

* {#Olver86} [[Peter Olver]], section 5.4 of _Applications of Lie groups to differential equations_, Springer Graduate Texts in Mathematics 107 (1986)

* {#Anderson89} [[Ian Anderson]], _The variational bicomplex_, Utah State University 1989 ([[AndersonVariationalBicomplex.pdf:file]])

* {#KrasilshchikVerbovetsky98} [[Joseph Krasil'shchik]], [[Alexander Verbovetsky]], _Homological Methods in Equations of Mathematical Physics_ ([arXiv:math/9808130](http://arxiv.org/abs/math/9808130))

* {#KrasilshchikVinogradov99} [[Joseph Krasil'shchik]],  [[Alexandre Vinogradov]] et al. (eds.) _Symmetries and  Conservation Laws for Differential Equations of Mathematical Physics_, AMS 1999

Other surveys include

* Juha Pohjanpelto, _Symmetries, Conservation Laws, and Variational Principles for Differential Equations_ (2014) ([pdf slides](http://math.usask.ca/~cheviakov/bluman2014/talks/Pohjanpelto.pdf))

An early discussion with application to [[covariant phase spaces]] and their [[presymplectic structure]] is in

* {#Zuckerman87} [[Gregg Zuckerman]], *Action principles and global geometry*, in: [[Shing-Tung Yau]] (ed.) *Mathematical Aspects of String Theory*, World Scientific (1987) 259-284 [[[ZuckermanVariation.pdf:file]], [doi:10.1142/0383](https://doi.org/10.1142/0383)]

An invariant version (under group action) is in

* [[Irina Kogan]], [[Peter Olver]], _The invariant variational bicomplex_, [pdf](http://www4.ncsu.edu/~iakogan/papersPDF/sivKoganOlver.pdf)

A more detailed version of this is in

* [[Irina Kogan]], [[Peter Olver]], _Invariant Euler-Lagrange Equations and the Invariant Variational Bicomplex_, [pdf](http://www4.ncsu.edu/~iakogan/papersPDF/ivbKoganOlver-cor.pdf)

See also

* [[Victor Kac]], _An explicit construction of the complex of variational calculus and Lie conformal algebra cohomology_, talk at Algebraic Lie Theory, Newton Institute 2009, [video](http://sms.cam.ac.uk/media/538761)

An application to [[multisymplectic geometry]] is discussed in

* Thomas Bridges, Peter Hydon, Jeffrey Lawson, _Multisymplectic structures and the variational bicomplex_  ([pdf](http://personal.maths.surrey.ac.uk/st/T.Bridges/PAPERS/MPCPS-Paper-09024.pdf))

Discussion in the context of [[supergeometry]] is in

* [[Gennadi Sardanashvily]], _Grassmann-graded Lagrangian theory of even and odd variables_ ([arXiv:1206.2508](https://arxiv.org/abs/1206.2508))

Discussion in the convenient context of [[smooth sets]]:

* {#GiotopoulosSati23} [[Grigorios Giotopoulos]], [[Hisham Sati]], §5.1 & §7.1 in: *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

[[!redirects horizontal derivative]]
[[!redirects horizontal derivatives]]

[[!redirects vertical derivative]]
[[!redirects vertical derivatives]]

[[!redirects variational differential form]]
[[!redirects variational differential forms]]

[[!redirects total spacetime derivative]]
[[!redirects total spacetime derivatives]]

[[!redirects horizontal variational complex]]
[[!redirects horizontal variational complexes]]


variational calculus

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

# Contents
* table of contents
{: toc}

## Idea

**Variational calculus** -- sometimes called **secondary calculus** -- is a version of [[differential calculus]] that deals with local extremization of [[nonlinear functionals]]: extremization of differentiable functions on non-finite dimensional spaces such as [[mapping spaces]], [[spaces of sections]] and hence [[spaces of histories]] of [[field (physics)|fields]] in [[field theory]].

Specifically, it studies the _[[critical points]]_ , i.e. the points where the first variational derivative of a functional vanishes, for functionals on spaces of [[sections]] of [[jet bundles]]. The kinds of equations specifying these critical points are [[Euler-Lagrange equations]].

This applies to, and is largely motivated from, the study of [[action functionals]] in [[physics]]. In [[classical physics]] the critical points of a specified action functional on the space of field configurations encode the physically observable configurations.

There are strong [[cohomology|cohomological]] tools for studying variational calculus, such as the [[variational bicomplex]] and [[BV-BRST formalism]].

## In terms of smooth spaces
 {#InTermsOfSmoothSpaces}

We discuss some basics of variational calculus of functional in terms of [[smooth spaces]] and in particular in terms of [[diffeological spaces]].

### Smooth functionals
 {#SmoothFunctionals}

Let $X$ be a [[smooth manifold]].
Let $\Sigma$ be a [[smooth manifold|smooth]] [[manifold with boundary]] $\partial \Sigma \hookrightarrow \Sigma$.

Write

$$
  [\Sigma, X] \in Smooth0Type
$$

for the [[smooth space]] (a [[diffeological space]]) which is the [[mapping space]] from $\Sigma$ to $X$, hence such that for each $U \in $ [[CartSp]] we have

$$
  [\Sigma, X](U) = C^\infty(U \times \Sigma, X)
  \,.
$$

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

Write

$$
  [\Sigma, X]_{\partial \Sigma}
  \coloneqq
  [\Sigma, X] \times_{[\partial \Sigma,X]} \flat [\partial \Sigma,X]
$$

for the [[pullback]] in smooth spaces

$$
  \array{
    [\Sigma,X]_{\partial \Sigma}
    &\to&
    \flat [\partial \Sigma, X]
    \\
    \downarrow && \downarrow
    \\
    [\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X]
  }
  \,,
$$

where

* the bottom morphism is the restriction $[\partial \Sigma \hookrightarrow \Sigma, X]$ of configurations to the boundary;

* the right vertical morphism is the [[counit of an adjunction|counit]] of the $(Disc \dashv \Gamma)$-[[adjunction]] on smooth spaces.

=--

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

The [[smooth space]] $[\Sigma, X]_{\partial \Sigma}$ is a [[diffeological space]] whose underlying set is $C^\infty(\Sigma,X)$ and whose $U$-plots for $U \in $ [[CartSp]] are smooth functions
$\phi \colon U \times \Sigma \to X$ such that $\phi(-,s) \colon U \to X$ is the constant function for all $s \in \partial \Sigma \hookrightarrow \Sigma$.

=--

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

A **functional** on the mapping space $[\Sigma, X]$ is a [[homomorphism]] of smooth spaces

$$
  S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R}
  \,.
$$

=--

### Functional derivative
 {#FunctionalDerivative}

Write

$$
  \mathbf{d} \colon \mathbb{R} \to \Omega^1
$$

for the [[de Rham differential]] incarnated as a [[homomorphism]] of [[smooth spaces]] from the [[real line]] to the smooth [[moduli space]] of [[differential 1-forms]].

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

The **[[functional derivative]]**

$$
  \mathbf{d}S
  \in
  \Omega^1([\Sigma,X]_{\partial \Sigma})
$$

of a functional $S$, def. \ref{SmoothFunctional}, is simply its [[de Rham differential]] as a homomorphism of smooth spaces, hence the composite

$$
  \mathbf{d} S \colon [
  \Sigma, X]_{\partial \Sigma}
     \stackrel{S}{\to}
  \mathbb{R}
    \stackrel{\mathbf{d}}{\to}
  \Omega^1
  \,.
$$

=--

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

This means that for each smooth parameter space $U \in $ [[CartSp]] and for each smooth $U$-parameterized collection

$$
  \phi \colon U \times \Sigma \to X
$$

of smooth functions $\Sigma \to X$, constant in the parameter $U$ in some neighbourhood of the boundary of $\Sigma$,

$$
  S_\phi \colon [\Sigma,X]_{\partial \Sigma}(U) \to C^\infty(U,\mathbb{R})
$$

is the function that sends each such $U$-collection of configurations to the $U$-collection of their values under $S$. Then

$$
  (\mathbf{d}S)_\phi \in \Omega^1(U)
$$

is the smooth [[differential 1-form]]

$$
  (\mathbf{d}S)_\phi = \mathbf{d}(S(\phi))
  \,.
$$

=--

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

Let $\Sigma = [0,1] \hookrightarrow \mathbb{R}$ be the standard [[interval]]. Let

$$
  L(-,-) \mathbf{d}t \in \Omega^1([0,1], C^\infty(\mathbb{R}^2))
$$

and consider the functional

$$
  S
  \colon
  ([0,1] \stackrel{\gamma}{\to} X)
  \mapsto
  \int_{0}^1 L(\gamma(t), \dot \gamma(t)) d t
  \,.
$$

Then for $U = \mathbb{R}$ and any smooth $U$-parameterized collection $\{\gamma_{u} \colon \Sigma \to X\}_{u \in I}$ the functional derivative takes the value

$$
  \begin{aligned}
   \mathbf{d}S_{\gamma_{(-)}}
   & =
   \left(
     \frac{d}{d u} \int_0^1 L(\gamma_u(t), \dot \gamma_u(t)) dt
   \right)
   \mathbf{d}u
   \\
    & =
     \int_{0}^1
     \left(
       \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{d \gamma_u(t)}{d u}
      +
       \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{\partial \dot \gamma_u(t)}{\partial u}
     \right)
     \mathbf{d} u
     \\
     & =
     \int_{0}^1
     \left(
       \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{d \gamma_u(t)}{d u}
      +
       \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{\partial }{\partial t}\frac{\partial \gamma_u(t)}{\partial u}
     \right)
     \mathbf{d} u
     \\
     & =
     \int_{0}^1
     \left(
       \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
      -
       \frac{\partial}{\partial t}\frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
     \right)
     \frac{\partial \gamma_u(s)}{\partial u}
     \mathbf{d}u
  \end{aligned}
  \,.
$$

Here we used [[integration by parts]] and used that the boundary term vanishes

$$
  \begin{aligned}
    \int_{0}^1 \frac{\partial}{\partial t}
    \left(
      \frac{\partial}{\partial \dot\gamma} L(\gamma_u(s), \dot \gamma_u(s))
      \frac{\partial \gamma_u(s)}{\partial u}
    \right)
    d s
    & =
    \left(
      \frac{\partial}{\partial \dot\gamma} L(\gamma_u(1), \dot \gamma_u(1))
      \frac{\partial \gamma_u(1)}{\partial u}
    \right)
    -
    \left(
      \frac{\partial}{\partial \dot\gamma} L(\gamma_u(0), \dot \gamma_u(0))
      \frac{\partial \gamma_u(0)}{\partial u}
    \right)
    \\
    & = 0
  \end{aligned}
$$

since by prop. \ref{PlotsOfMappingSpaceWithNonVaryingBoundary} $\gamma_{(-)}(1)$ and $\gamma_{(-)}(0)$ are constant.

=--

## In terms of the variational bicomplex

In the special case that the functional to be varied comes from a [[Lagrangian density]], then its variational derivative is the image under [[transgression of variational differential forms|transgression]] of the [[vertical derivative]] in the [[variational bicomplex]] of differential forms on the given [[jet bundle]].

(...)

## Related concepts

* [[Euler-Lagrange equations]], [[shell]]

* [[source form]], [[evolutionary vector field]], [[evolutionary derivative]]

* [[de Donder-Weyl formalism]]

* [[phase space]]

* [[variational bicomplex]], [[secondary differential calculus]]

* [[Lagrange multiplier]]

* [[variational sequence]]

## References

Exposition of variational calculus in terms of [[jet bundles]] and [[Lepage forms]] and aimed at examples from [[physics]] is in

* {#MusilovaHronek16} [[Jana Musilová]],  [[Stanislav Hronek]], _The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories_, Communications in Mathematics, Volume 24, Issue 2 (Dec 2016) ([doi.org/10.1515/cm-2016-0012]( https://doi.org/10.1515/cm-2016-0012))

Fundamental texts on variational calculus include

* [[Ian Anderson]], _The variational bicomplex_, ([[AndersonVariationalBicomplex.pdf:file]])

* Hubert Goldschmidt, [[Shlomo Sternberg]], _The Hamilton-Cartan formalism in the calculus of variations_, Annales de l'institut Fourier 23 no. 1 (1973), p. 203-267 [numdam](http://www.numdam.org/item?id=AIF_1973__23_1_203_0)

* [[Peter Olver]], _Applications of Lie groups to differential equations_, Springer; _Equivalence, invariants, and symmetry_, Cambridge Univ. Press 1995.

* [[Demeter Krupka]], _Introduction to global variational geometry_, 2015

* Olga Krupková, _The geometry of ordinary variational equations_, Springer 1997, 251 p.

* Robert Hermann, _Some differential-geometric aspects of the Lagrange variational problem_, Illinois J. Math. __6__, 1962, 634–673 [MR145457](http://www.ams.org/mathscinet-getitem?mr=145457) [euclid](http://projecteuclid.org/euclid.ijm/1255632711);
_Differential geometry and the calculus of variations_, Acad. Press 1968

* J. Jost, X. Li-Jost, _Calculus of variations_, CUP 1998

* {#Zuckerman} [[Gregg Zuckerman|G. J. Zuckerman]], _Action Principles and Global Geometry_ , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259€284. ([[ZuckermanVariation.pdf:file]])

Zuckerman's ideas are used in

* [[Marco Benini]], [[Alexander Schenkel]], _Poisson algebras for non-linear field theories in the Cahiers topos_, [arxiv/1602.00708](http://arxiv.org/abs/1602.00708)

Examples: [[Jürgen Jost]], _Variational problems from physics and geometry_, [pdf](http://www.mis.mpg.de/fileadmin/jjost/variational_problems_from_physics_and_geometry.pdf)

* J. J. Duistermaat, _On the Morse index in variational calculus_, Adv. Math. __21__ (1976), 2, 173--195 [pdf](http://www.maths.ed.ac.uk/~aar/papers/duistermaat.pdf).

Some new results are in

* [[E. Getzler]], _A Darboux theorem for Hamiltonian operators in the formal calculus of variations_, Duke Math. J. __111__, n. 3 (2002), 535-560, [MR2003e:32026](http://www.ams.org/mathscinet-getitem?mr=1885831), [doi](http://dx.doi.org/10.1215/S0012-7094-02-11136-3)
* Alberto De Sole, Victor G. Kac, _The variational Poisson cohomology_, [arxiv/1106.0082](http://arxiv.org/abs/1106.0082)

Geometric extremization problems (e.g. minimal surfaces), see also [[geometric measure theory]]:

* [[Jürgen Jost]], _The geometric calculus of variations: a short survey and a list of open problems_, Exposition. Math. __6__ (1988), no. 2, 111–143, [MR89h:58036](http://www.ams.org/mathscinet-getitem?mr=938179)
* H. Federer, _Geometric measure theory_, Springer 1969(especially appendices to Russian transl.)
* Frederick J., Jr. Almgren, Almgren's big regularity paper (book form of a 1970s preprint)

Discussion in the context of [[BV formalism]]:

* Arthemy V. Kiselev, _The geometry of variations in Batalin-Vilkovisky formalism_,  [http://arxiv.org/abs/1312.1262](http://arxiv.org/abs/1312.1262)

Other references

* J. C. P. Bus, _The Lagrange multiplier rule on manifolds and optimal control of nonlinear systems_, SIAM J. Control and Optimization __22__, 5, 1984, 740-757 [pdf](http://oai.cwi.nl/oai/asset/2552/2552A.pdf)

In the [[covariant phase space]]-perspective:

* L. Vitagliano, _Secondary calculus and the covariant phase space_, [pdf](http://diffiety.ac.ru/preprint/2008/02-08.pdf)
 {#Vitagliano}

On [[smooth sets]] as a [[convenient category of spaces|convenient category]] for [[variational calculus]] of [[Lagrangian quantum field theory|Lagrangian]] [[classical field theory]]:

* [[Grigorios Giotopoulos]], [[Hisham Sati]], *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

### By functorial analysis and $\mathcal{D}$-geometry

See also references at [[diffiety]].

A formalism for variational calculus based on [[functorial analysis]] (with a precise relation with functional analytic methods and jet formalism) and a long list of examples of variational problems arising in [[classical mechanics]] and [[quantum field theory]] are collected in

* [[Frédéric Paugam]], _Towards the mathematics of quantum field theory_ ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics.pdf))

The formulation of variational calculus in terms of [[diffeological spaces]] is mentioned for instance in section 1.65 of

* [[Patrick Iglesias-Zemmour]], _Diffeology_ ([pdf](http://math.huji.ac.il/~piz/documents/Diffeology.pdf#page=64))
{#PIZ}

* [[Frédéric Paugam]], _Histories and observables in covariant field theory_ ([arXiv:1010.3210](http://arxiv.org/abs/1010.3210)), sec. 2.4
{#Paugam}

following section 2.3.20 of

* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _[[Chiral Algebras]]_

For variational calculus in [[nonstandard analysis]] see survey

* Vítor Neves, _Nonstandard calculus of variations, a survey_, [pdf](http://www2.mat.ua.pt/vneves/nsa/CalcVar-vitor6.pdf)

category: analysis, physics
[[!redirects calculus of variations]]
[[!redirects variational derivative]]
[[!redirects variational derivatives]]


variational calculus - contents

**[[variational calculus]]**

## Differential geometric version

* [[variational bicomplex]]

* [[local Lagrangian]], [[local action functional]]

  * [[Euler-Lagrange form]], [[source form]]

  * [[Lepage form]]

* [[principle of extremal action]], [[Euler-Lagrange equations]], [[de Donder-Weyl formalism]]

* [[covariant phase space]]

* [[Noether's theorem]]

  * [[conserved current]], [[charge]]

  * [[symmetry]]

## Derived differential geometric version

* [[BV-BRST complex]]

* [[D-module]]

[Edit this sidebar](/nlab/edit/variational+calculus+-+contents)


variational sequence

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[jet bundle]] $J^r E$ of jets of finite order $r \in \mathbb{Z}$, its _variational sequence_ ([Krupka 90](#Krupka90)) is a [[resolution]] of its [[constant sheaf]] of [[locally constant functions]] with values in the [[real numbers]] by a [[chain complex]] of quotient of sheaves of [[differential forms]] by combinations of [[contact forms]] and their [[de Rham differentials]]. (e.g. [PRWM 15, section 2.2](#PRWM15)).

The variational sequence is a sub-sequence of the [[Euler complex]] as obtained from the theory of the [[variational bicomplex]] ([Krupka 90](#Krupka90), see e.g. [PRWM 15,theorem 1](#PRWM15)).

## Related concepts

* [[Poincaré lemma]]

* [[Lepage form]]

## References

The concept was introduced in

* {#Krupka90} [[Demeter Krupka]], _Variational Sequences on Finite Order Jet Spaces_, Proc. Diff. Geom. Appl.; J. Janyška, D. Krupka eds., World Sci. (Singapore, 1990) 236–254.

Review includes

* [[Demeter Krupka]], _Introduction to Global Variational Geometry_

* {#PRWM15} M. Palese, O. Rossi, E. Winterroth, J. Musilova, _Variational sequences, representation sequences and applications in physics_ ([arXiv:1508.01752](http://arxiv.org/abs/1508.01752))

[[!redirects variational sequences]]


variety

## Idea

A *variety* may denote

* [[algebraic variety]]
* [[analytic variety]]
* [[variety of algebras]] in [[universal algebra]]

In the first two cases, at least locally, varieties are given by satisfaction of a family of equations, given by polynomials (algebraic variety), analytic functions (analytic variety). The  variety of algebras consists to some extent similarly of algebras satisfying some list of equations for algebraic operations true for any elements in the place of free variables.

## Related concepts

* [[complex variety]]

* [[locus]]

[[!redirects varieties]]


variety of algebras

* tic
{: toc}

## Idea

The notion of *variety of algebras* is a classical notion from [[universal algebra]] that subsumes nearly all of the usual kinds of [[algebra|algebraic]] objects, such as [[groups]], [[rings]], [[vector spaces]] (over a fixed [[field]]), and so on.  Those algebraic objects that do not form a variety of algebras, such as [[fields]], can still usually be easily described as a subcategory of some variety of algebras (in this case, [[commutative rings]]).

A variety of algebras is not to be confused with an [[algebraic variety]], although in fact there is a connection: both concepts are (roughly speaking) given by a set of equations, but an algebraic variety is the set of elements (of a fixed generic structure) that satisfy the equations, while a variety of algebras is the class of structures in which *every* element satisfies the equations.

##  Definitions

A __signature__ (in this context) is a [[set]], whose elements are called __operations__, to each of which is assigned a [[cardinal number]] ($0, 1, 2, \ldots$) called its [[arity]].  A signature is __finitary__ if its set of operations is [[finite set|finite]] (in the strictest sense for the purposes of [[constructive mathematics]]) and every arity is finite (so a [[natural number]]); similar definitions apply to other [[arity classes]].  Given a signature $\sigma$ and a set $V$, whose elements are called __variables__[^1], a __word__ (in $\sigma$ using $V$) is a (well-founded directed rooted) [[tree]] in which each node is labelled by either a variable or an operation, such that every node labelled by a variable has no branches away from the root and every node labelled by an operation $o$ has as many branches away from the root as the arity of $o$.  An __axiom__ (in $\sigma$ using $V$) is a pair of such words; we write the axiom consisting of the words $v$ and $w$ as $v = w$ or $\vdash v = w$.  A __theory__ (in our sense) consists of a signature $\sigma$ and a set of axioms in $\sigma$.  An algebraic theory is __finitary__ if its signature is finitary and its set of equational axioms is finite ([[Kuratowski-finite]] in constructive mathematics); analogous remarks apply to other arity classes.

[^1]: For a finitary signature, we may without loss of generality use the set $V \coloneqq \{x_0, x_1, x_2, \ldots\}$, effectively the set of [[natural numbers]], as the only set of variables; similar remarks apply to other arity classes.

A __variety of algebras__ is given precisely by a theory in this sense, but it is more usual to think of the variety as being the class (or [[category]]) of models, so we continue with the definitions.

Given a theory $T$, a __model__ or __algebra__ of $T$ consists of a [[set]] $A$ together with, for each operation of $T$ with arity $\kappa$, a [[function]] to $A$ from its $\kappa$th [[cartesian power]] $A^\kappa$ such that, for each axiom $\vdash v = w$ and each assignment of elements of $A$ to the variables in that axiom, the equation holds that is given by applying the operations to the elements of $A$ as indicated by the trees defining $v$ and $w$ (the meaning of which we hope is obvious).  We say that $A$ __carries__ the operations and __satisfies__ the axioms.

Given a algebras $A$ and $B$ of the same (quasi)-algebraic theory, a __homomorphism__ from $A$ to $B$ consists of a [[function]] $f\colon A \to B$ such that, for each operation, the diagram
$$ \array {
   A^\kappa   & \stackrel{f^\kappa}\rightarrow & B^n \\
   \downarrow &                                & \downarrow \\
   A          & \stackrel{f}\rightarrow        & B
} $$
commutes.  We say that $f$ __preserves__ the operations.

It is fairly straightforward to define a model (and homomorphisms) in any [[cartesian monoidal category]] $C$, replacing [[Set]] in the two paragraphs above.  But note that the operations and axioms still form sets, not objects of $C$; the theory $T$ is independent of the modelling category.  In any case, given any theory $T$ and any cartesian monoidal category $C$, the models of $T$ in $C$ (as [[objects]]) and the homomorphisms between them (as [[morphisms]]) form a [[category]], which is the __variety of $T$-algebras in $C$__.

## Generalisations

A __sorted variety__ or __many-sorted variety__ is a generalisation.  A __sorted signature__ or __many-sorted signature__ consists of a set, whose elements are called __sorts__, together with operations and arities as before, except that now an arity is a [[list]] of __input__ sorts instead of merely a natural number (the length of the list) together with one __output__ sort.  Each variable in an axiom comes equipped with a sort, which we may also classify as an output sort; then each branch in (the tree reprsenting) a word must link output sort to input sort.  The root of a word gives the entire word a sort; the two words in an equational axiom must have the same sort.  Then a model of a sorted variety consists of one set for each sort, and the rest should be obvious.  A sorted variety is __finitary__ if it satisfies the finitarity conditions of a variety and its set of sorts is also finite.  An ordinary variety as above may be called __unsorted__ or __single-sorted__ to avoid invoking the [[red herring principle]]; it is the same thing as a sorted variety with exactly one sort.

Another generalisation is a __quasivariety__.  While an axiom in a variety gives an equation, an axiom in a quasivariety is a [[Horn clause]], stating that one equation (the __consequent__) holds whenever each equation in some set of equations (the __premises__) holds.  A Horn clause is __finitary__ if this set of premises is (Kuratowski)-finite; a quasivariety is __finitary__ if it satisfies the fintarity conditions of a variety and additionally each Horn clause is finitary.  An ordinary variety is a quasivariety in which each Horn clause has an [[empty set]] of premises.

##  Examples

For each definition of an appropriate algebraic object, there is a corresponding variety of algebras, a few of which are given below:

*  The variety of _[[groups]]_ is given by a theory with three operations $m,e,i$ (traditionally called multiplication, identity, and inverse), with respective arities $2,0,1$ (so multiplication is a binary operation, identity is a nullary operation, and inverse is a unary operation).  There are five axioms, given by the following pairs of trees (with their traditional names):

   *  associative law:
      \[ \label{ass} \array {
           &          &   &          & m \\
           &          &   & \nearrow &   & \nwarrow \\
           &          & m &          &   &          & z \\
           & \nearrow &   & \nwarrow \\
         x &          &   & y
      } \;\;\;=\;\;\; \array {
           &          & m \\
           & \nearrow &   & \nwarrow \\
         x &          &   &          & m \\
           &          &   & \nearrow &   & \nwarrow \\
           &          & y &          &   &          & z
      } \]
   *  right unit law:
      \[ \label{runit} \array {
           &          & m \\
           & \nearrow &   & \nwarrow \\
         x &          &   &          & e
      } \;\;\;=\;\;\; \array {
         x
      } \]
   *  left unit law:
      \[ \label{lunit} \array {
           &          & m \\
           & \nearrow &   & \nwarrow \\
         e &          &   &          & x
      } \;\;\;=\;\;\; \array {
         x
      } \]
   *  right inverse law:
      \[ \label{rinv} \array {
           &          & m \\
           & \nearrow &   & \nwarrow \\
         x &          &   &          & i \\
           &          &   &          & \uparrow \\
           &          &   &          & x
      } \;\;\;=\;\;\; \array {
         e
      } \]
   *  left inverse law:
      \[ \label{linv} \array {
                  &          & m \\
                  & \nearrow &   & \nwarrow \\
         i        &          &   &          & x \\
         \uparrow \\
         x
      } \;\;\;=\;\;\; \array {
         e
      } \]

   Then a model of this variety is a set $A$ together with functions $m\colon A^2 \to A$, $e\colon 1 \to A$ (which may be identified with an element of $A$), and $i\colon A \to A$ such that, for all $x,y,z$ in $A$, $m(m(x,y),z) = m(x,m(y,z))$, $m(e,x) = x$, $m(x,e) = x$, $m(x,i(x)) = e$, and $m(i(x),x) = e$.  In other words, an algebra in this variety is simply a [[group]].  Similarly, a homomorphism of such algebras is simply a [[group homomorphism]].  Also, an algebra in this variety in the category $C$ is a [[group object]] in $C$.

Note that it is not necessary to draw explicit trees; these can be recovered from the succinct equations like $m(m(x,y),z) = m(x,m(y,z))$; we will do this below.

*  The variety of _[[monoids]]_ is given by a subtheory of the theory of groups.  It has only the operations $m,e$ and the axioms (eq:ass),(eq:runit),(eq:lunit).  An algebra in this theory is a monoid, and a homomorphism of such algebras is a monoid homomorphism (including the condition that it preserve the identity).

*  There is no variety of _[[cancellative monoid]]s_, but there is a quasivariety of them.  In addition to the axioms for a monoid, we have an axiom stating that the equation $x = y$ follows from the equation $m(x,z) = m(y,z)$, which we write succinctly as $m(x,z) = m(y,z) \vdash x = y$.

*  The variety of _[[abelian groups]]_ is given by a supertheory of the theory of groups.  In addition to the operations and axioms of the variety of groups, it has an additional axiom, the commutative law: $m(x,y) = m(y,x)$.

*  The variety of (associative unital) _[[rings]]_ has five operations $a,z,n,m,e$, with respective arities $2,0,2,0,1$, and ten axioms.  Five of these axioms are those of groups, applied to $a,z,n$ in place of $m,e,i$; three of these axioms are those of monoids (applied to $m,e$ as before).  The remaining two axioms are:
   *  left distributive law:  $m(x,a(y,z)) = a(m(x,y),m(x,z))$,
   *  right distributive law:  $m(a(x,y),z) = a(m(x,z),m(y,z))$.

*  There is no variety or quasivariety of [[fields]]; the elements with multiplicative inverses are not identified by an equation.  One can, however, define fields as certain rings and then get the correct notion of field homomorphism as a ring homomorphism between rings that happen to be fields.

*  The variety of (real) _[[vector spaces]]_ has infinitely many operations and axioms (so is not finitary).  First, there are operations $a,z,n$ with the arities and axioms of an abelian group (as with rings).  In addition, there is one operation $m_r$ of arity $1$ for each [[real number]] $r$, satisfying these axioms:
   *  left distributive law:  for each real number $r$, $m_r(a(x,y)) = a(m_r(x),m_r(y))$,
   *  right distributive law:  for each real number $r$ and real number $s$, $m_{r+s}(x) = a(m_r(x),m_r(y))$ (where $r + s$ is the sum of $r$ and $s$ as real numbers),
   *  associative law:  for each real number $r$ and real number $s$, $m_{r s}(x) = m_r(m_s(x))$ (where $r s$ is the product of $r$ and $s$ as real numbers),
   *  unit law:  $m_1(x) = x$ (where $1$ the real number [[1]]).

*  There is a similar variety of vector spaces over any given [[ground field]], but there is no variety of vector spaces over arbitrary fields, since we would have to specify the field as well as the vector space.  The problem is not really that there is no variety of fields; there is no variety of (left) [[modules]] over arbitrary [[commutative rings]] either.  However, there is a _sorted_ variety of modules over arbitrary rings, with two types, one for the ring and one for the module.  Even though this variety includes the non-finitary variety of real vector spaces, it is itself a finitary sorted variety.

##  As algebraic theories

A variety of algebras is one way to look at an [[algebraic theory]].

As a [[logic]]al theory, everything that we wish to say can be said in (typed or untyped, to match the variety) [[cartesian logic]] (although one might want to use something stronger, such as full classical or intuitionistic predicate logic with equality).  We use one function symbol (of matching arity) for each operation, and each axiom in the variety becomes an axiom in the logic, a universally quantified equation.  Then the [[models]] of this logical theory are the same as the algebras in the variety.

As a [[Lawvere theory]], an untyped finitary variety of algebras defines a category $L$ whose objects are of the form $A^n$ for $n$ a natural number.  The morphisms of the category are those which are necessary to make $A^n$ the $n$-fold [[product]] of $A$, along with one morphism for each operation in the variety.  Each axiom of the variety corresponds to a commutative diagram, and $L$ is freely generated by these as a [[sketch]].  Then the models of the variety in a cartesian monoidal category $C$ are the same as the [[product-preserving functor|product-preserving]] functors from $L$ to $C$.  Conversely, any [[locally small category|locally small]] (and hence [[small category|small]]) cartesian monoidal category whose objects are all of the form $A^n$ may be thought of as a Lawvere theory and defines an untyped variety of algebras, with operation for each morphism to $A$ and one axiom for each commutative diagram.  (This can all be generalised to sorted varieties as well.)

A variety of algebras can be seen as a cartesian version of a (typed or untyped) [[operad]]; whereas an operad has models in any [[symmetric monoidal category]], a variety of algebras has models only in a cartesian monoidal category.

A variety of algebras is traditionally identified with its category $M$ of models in $Set$ (or even with simply the [[class]] of objects of this category), but it then becomes unclear what an algebra in the variety would be in some other category $C$.  However, it is worth noting that $M$ is always an [[algebraic category]]; see below for the construction of free objects.

## Free algebras

Given a variety $V$ and a [[cartesian monoidal category]] $C$ (such as $Set$), let $C^V$ be the category of algebras in $V$ in $C$, with homomorphisms as morphisms.  There is an obvious [[forgetful functor]] $U\colon C^V \to C$; $C$ is a [[cocomplete category|cocomplete]] $W$-[[W-pretopos|pretopos]] (or if $C$ is just a $W$-pretopos if $V$ is finitary), then $U$ has a [[left adjoint]], the [[free functor]] $F\colon C \to C^V$.  In fact, this adjunction is [[monadic adjunction|monadic]]; we can identify the models of $V$ in $C$ with the [[module of a monad|modules/algebras]] of the [[monad]] $U \circ F$.

Given an object $A$ of $C$, we construct the free algebra $F(A)$ as follows.  First, let $W(A)$ be the [[inductive type]] in $C$ given by the signature of $V$, that is the [[initial algebra]] of the [[endofunctor]] $A \mapsto \sum_o A^{\#o}$, where $o$ ranges over the operations of $V$ and $\#o$ is the arity of $o$.  Then each axiom specifies an internal binary [[relation]] on $W(A)$; the [[colimit]] of these is $F(A)$.  Note that, if $C$ is $Set$ and $A$ is a set, then $W(A)$ consists precisely of the words (defined as trees above) with variables from $A$.

The __word problem__ for a variety $V$ consists of deciding, whenever $A$ is a [[finite set]], whether two elements of $W(A)$ are equal in $F(A)$.  As a problem in algorithmic decidability, the word problem can be very difficult; for example, it is:

*  trivial for [[Boolean algebras]],
*  impossible for [[Heyting algebras]],
*  nontrivial but possible for [[lattices]],

even though each of these is a supertheory of the next.

## Literature

* [[Jiří Adámek]], [[Lawvere|F.W. Lawvere]], J. Rosický, _On the duality between varieties and algebraic theories_, Algebra universalis 2003, __49__:1, pp 35–49 [doi](https://doi.org/10.1007/s000120300002)

[[!redirects variety of algebras]]
[[!redirects varieties of algebras]]
[[!redirects quasivariety of algebras]]
[[!redirects quasivarieties of algebras]]

[[!redirects word problem]]
[[!redirects word problems]]


Varmo Vene

* [personal page](https://kodu.ut.ee/~varmo/)

## Selected writings

On [[comonads in computer science]]:

* {#UustaluVene08}  [[Tarmo Uustalu]], [[Varmo Vene]], *Comonadic Notions of Computation*, Electronic Notes in Theoretical Computer Science **203** 5 (2008) 263-284 [[doi:10.1016/j.entcs.2008.05.029](https://doi.org/10.1016/j.entcs.2008.05.029)]

category: people


Varun Gupta

## Selected writings

On [[codimension]]=2 [[defect brane]] [[brane intersection|intersections]] of [[M5-brane|M5]]-[[probe branes]] in an [AdS7-CFT6](AdS-CFT+correspondence#AdS7CFT6) [[background field|background]] (i.e. in the [[near horizon limit]] of [[black brane|black]] [[M5-branes]], cf. the [[super 3-brane in 6d]]):

* [[Varun Gupta]], *Holographic M5 branes in $AdS_7 \times S^4$*, J. High Energ. Phys. **2021** 32 (2021) [[arXiv:2109.08551](https://arxiv.org/abs/2109.08551), doi:10.1007/JHEP12(2021)032]

* [[Varun Gupta]], *More Holographic M5 branes in $AdS_7 \times S^4$* [[arXiv:2301.02528](https://arxiv.org/abs/2301.02528)]

category: people


Varvara Karpova

* [webpage](http://hessbellwald-lab.epfl.ch/page-77160-en.html)

## Selected writings

Reviewing [[topological K-theory]]:

* {#Karpova09} [[Varvara Karpova]], _Complex topological K-theory_, 2009  ([pdf](http://infoscience.epfl.ch/record/162450/files/karpova.semestre.hess2.pdf), [[KarpovaTopologicalKTheory.pdf:file]])

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)]

## Related $n$Lab entries

* [[topological K-theory]]

category: people

[[!redirects Karpova Varvara]]


Vasilis Niarchos

* [institute page](https://www.physics.uoc.gr/en/faculty/v.niarchos)

* [InSpire page](https://inspirehep.net/authors/995738)

## Selected writings

On the *[[blackfold]]*-approach to describing [[black branes]]:

* [[Roberto Emparan]], [[Troels Harmark]], [[Vasilis Niarchos]], [[Niels A. Obers]], *Essentials of Blackfold Dynamics*, JHEP 1003 063 (2010) [[arXiv:0910.1601](https://arxiv.org/abs/0910.1601), doi:10.1007/JHEP03(2010)063]

* [[Roberto Emparan]], [[Troels Harmark]], [[Vasilis Niarchos]], [[Niels A. Obers]], *Blackfolds in Supergravity and String Theory*, JHEP 1108 (2011) 154 [[arXiv:1106.4428](https://arxiv.org/abs/1106.4428)doi:10.1007/JHEP08(2011)154]

On [[black brane]] [[M2-M5 brane bound states]]:

* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *M2-M5 blackfold funnels*, Journal of High Energy Physics **2012** 175 (2012) [[arXiv:1205.1535](https://arxiv.org/abs/1205.1535), doi:10.1007/JHEP06(2012)175]

* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *Entropy of the self-dual string soliton*, Journal of High Energy Physics **2012** 134 (2012) [[arXiv:1206.2935](https://arxiv.org/abs/1206.2935), doi:10.1007/JHEP07(2012)134]

* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *The black M2-M5 ring intersection spins* [[arXiv:1302.0854](https://arxiv.org/abs/1302.0854)]

* [[Vasilis Niarchos]], *Localizing the black M2-M5 intersection* (2012) [[pdf](https://hep.physics.uoc.gr/Slides/Fall_2012/VasilisNiarchos.pdf), [[Niarchos-BlackM2M5Branes.pdf:file]]]

* Jay Armas, [[Vasilis Niarchos]], [[Niels A. Obers]], Section 2 of: _Thermal transitions of metastable M-branes_, J. High Energ. Phys. (2019) 2019: 128 [[arXiv:1904.13283](https://arxiv.org/abs/1904.13283)]

category: people


Vasily Dolgushev

* [webpage](https://www.math.temple.edu/~vald/)

## Selected writings

On [[orbifold cohomology]] of [[symplectic orbifolds]]:

* [[Vasily Dolgushev]], [[Pavel Etingof]], _Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology_, Int. Math. Res. Not. 2005, no. 27, 1657-1688 ([arXiv:math/0410562](https://arxiv.org/abs/math/0410562))

On [[graph complexes]] and the [[Lie algebra]] of the [[Grothendieck-Teichmüller group]]:

* {#DolgushevRogers12} [[Vasily Dolgushev]], [[Christopher Rogers]], _Notes on Algebraic Operads, Graph Complexes, and Willwacher's Construction_, In: Mathematical aspects of quantization 583 (2012): 25-145. ([arXiv:1202.2937](https://arxiv.org/abs/1202.2937))

* [[Vasily Dolgushev]], _A manifestation of the Grothendieck-Teichmuellergroup in geometry_ ([slides pdf](https://math.temple.edu/~vald/AMSslides.pdf))

## Related entries

* [[deformation quantization]], [[Grothendieck-Teichmüller group]]

category: people


Vasily Pestun

* [personal webpage](https://pestun.ihes.fr/)

* arXiv [hep-th author page](https://arxiv.org/search/hep-th?searchtype=author&query=Pestun%2C+V), [cs author page](https://arxiv.org/search/cs?searchtype=author&query=Pestun%2C+V)

## Selected writings

On [[equivariant de Rham cohomology]] and [[equivariant localization]]:

* [[Vasily Pestun]], _Review of localization in geometry_ ([arXiv:1608.02954](https://arxiv.org/abs/1608.02954)), chapter in: [[Vasily Pestun]], [[Maxim Zabzine]] (eds.) _Localization techniques in quantum field theories_,  J. Phys. A: Math. Theor. 50 440301 ([doi:10.1088/1751-8121/aa63c1](https://iopscience.iop.org/article/10.1088/1751-8121/aa63c1), [arXiv:1608.02952](https://arxiv.org/abs/1608.02952), [full pdf](http://pestun.ihes.fr/pages/LocalizationReview/LocQFT.pdf))

On [[tensor networks]] for [[machine learning]] [[natural language]]:

* [[Vasily Pestun]], [[Yiannis Vlassopoulos]], *Tensor network language model*, [[arXiv:1710.10248](https://arxiv.org/abs/1710.10248)]

category: people


Vassili Gorbounov

* [webpage](http://www.abdn.ac.uk/ncs/profiles/v.gorbunov)

category: people


Vassiliev invariant

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

# Vassiliev knot invariants
* table of contents
{: toc}

## Idea ##

The space of [[knots]] in the [[Euclidean space]] $\mathbb{R}^3$ (or in the [[3-sphere]] $S^3$) is an [[open subset|open]] [[submanifold]] of the [[smooth loop space]].  [[knot invariant|Knot invariants]] are [[locally constant functions]] on this manifold.  The [[complement]] of the space of knots is called the [[discriminant]] and consists of all [[singular knots]].

If we consider those singular knots with only a finite number of double points, we can build a [[cubical complex]] from this data.  The vertices in the complex are labelled by the isotopy classes of knots, and more generally the $n$-cubes by the isotopy classes of singular knots with $n$ double points (and a few other technical pieces of information).  The boundary operator resolves a double crossing either upwards or downwards according to the orientation at the crossing.

A **Vassiliev invariant** is simply a cubical morphism from this complex to an abelian group that vanishes above a certain degree.

## Definition ##

One does not need the language of cubical complexes to _define_ Vassiliev invariants.  Rather, there is a general method whereby a [[knot invariant]] can be extended to all [[singular knots]] with only finitely many double points (and no other singularities) using the [[Vassiliev skein relations]].

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

A **Vassiliev invariant** of degree (or order) $\le n$ is a knot invariant whose extension to singular knots (with double points) vanishes on all singular knots with more than $n$ double points.

=--

As is standard, it is of degree $n$ if it is of degree $\le n$ but not $\le n - 1$.  Vassiliev invariants are also called **finite type invariants**.

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

The degree of Vassiliev invariants defines a filtration on the space of knots (and more particularly, on the [[algebra of knots]]).  Two knots are $n$-equivalent if all the Vassiliev invariants of degree $\le n$ agree on them.  In particular, a knot that is $n$-equivalent to the unknot is said to be $n$-trivial.

=--

## Properties

### Relation to Chord diagrams and weight systems

A function which is constant on nonsingular knots may be extended to a Vassiliev invariant of degree 0 by applying the [[Vassiliev skein relations]], and conversely, any Vassiliev invariant of degree 0 must be constant on nonsingular knots.  Likewise, any Vassiliev invariant of degree 1 must be constant on nonsingular knots.

Any [[singular knot]] $f : S^1 \to \mathbb{R}^3$ with $n$ distinct double points $x_1,\dots,x_n \in \mathbb{R}^3$ gives rise to a [[chord diagram]] of order $n$, consisting of the circle $S^1$ with a chord connecting each pair of points $f^{-1}(x_1), \dots, f^{-1}(x_n)$.

The importance of this construction for singular knots comes from the fact that any finite type invariant determines a function on chord diagrams:

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

Let $v$ be a Vassiliev invariant of degree $\le n$.  Then the value of $v$ on a singular knot with $n$ distinct double points depends only on the [[chord diagram]] of the singular knot, and not on the knot itself.

=--

Conversely, one can ask which functions on chord diagrams come from finite type invariants.  The answer is that Vassiliev invariants (of degree $\le n$) can essentially be identified with _[[weight systems]]_ (of order $n$), which are functions on chord diagrams (of order $n$) satisfying two properties called the "[[1T relation]]" (or "framing independence") and the "[[4T relation]]": see Theorem 1 of [Bar-Natan 95](#BarNatan95) (or Theorem 6.2.13 of [Lando & Zvonkin](#LandoZvonkin)):

+-- {: .num_prop #WeightSystemsAreAssociatedGradedOfVassilievInvariants}
###### Proposition
**([[weight systems are associated graded of Vassiliev invariants]])**

For [[ground field]] $k = \mathbb{R}, \mathbb{C}$ the [[real numbers]] or [[complex numbers]], there is for each [[natural number]] $n \in \mathbb{N}$ a canonical [[linear isomorphism]]

$$
  \mathcal{V}_n/\mathcal{V}_{n-1}
  \underoverset{\simeq}{\phantom{AAAA}}{\longrightarrow}
  \big(
    \mathcal{A}_n^u
  \big)^\ast
$$

from

1. the [[quotient vector space]] of order-$n$ [[Vassiliev invariants]] of [[knots]] by those of order $n-1$

1. to the space of [[unframed weight systems]] of order $n$.

In other words, in [[characteristic zero]], the [[graded vector space]] of [[unframed weight systems]] is the [[associated graded vector space]] of the [[filtered vector space]] of [[Vassiliev invariants]].

=--

([Bar-Natan 95, Theorem 1](#BarNatan95), following [Kontsevich 93](#Kontsevich93))

### Relation to homology of loop spaces of configuration spaces of points
  {#RelationToHomologOfLoopSpacesOfConfigurationSpaces}

We discuss the relation between Vassiliev invariants and the [[Euler characteristic]] of the [[ordinary homology]] of [[loop spaces]] of [[configuration spaces of points]]:

For $n, q \in \mathbb{N}$ and $q \geq 1$, write

1. $\underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big)$ for the [[configuration space of points|configuration space of n ordered points]] in [[Euclidean space]] $\mathbb{R}^{q+2}$;

1. $\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big)$ for the corresponding [[based loop space]] (for any choice of base point);

1. $H_\bullet\Big(\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big)$ for the [[ordinary homology]] of this loop space, with [[coefficients]] in the [[complex numbers]];

1. $\chi H_\bullet\Big(\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big)$ for the [[Euler characteristic]]-series of the homology

Write also

1. $V^n_k$ for the [[complex vector space]] of [[Vassiliev invariants]] of order $k$ for [[pure braids]] with $n$ strands;;

1. $A^n_k$ for the [[complex vector space]] spanned by the horizontal [[chord diagrams]] with $n$ vertical strands modulo the "horizontal 4T relation"

such that there is an [[linear isomorphism]]

$$
  V^n_k/V^n_{k-1}
  \simeq
  (A^n_k)^\ast
$$

between the [[quotient vector space]] of [[Vassiliev invariants]] and the [[dual vector space]] of [[chord diagrams]].

Then:

The [[Euler characteristic]]-series (...) of the homology of the loop spaces of configuration spaces

$$
  \chi H_\bullet\Big(\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big)
  \;=\;
  \Big[
  \big(
    1 - t^q
  \big)
  \cdot
  \big(
    1 - 2 t^q
  \big)
  \cdots
  \big(
    1 - (n-1) t^q
  \big)
  \Big]^{-1}
$$

and is related to the complex [[dimensions]] of spaces of Vassiliev invariants according to

\[
  \label{HomotopyOfOmegaConfR3IsBraidVassiliev}
  \chi
  H_\bullet
  \Big(
    \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{3} \big), \mathbb{C}
  \Big)
  \;=\;
  \underset{k \in \mathbb{N}}{\sum}
  dim_{\mathbb{C}}\big( A^n_k \big)
  t^k
\]

([Cohen-Gitler 01, Prop. 9.1](#CohenGitler01), based on [Cohen 76](#Cohen76) and [Kohno 94](#Kohno94))

\linebreak

Alternatively, we have the combination of the following two facts, via [[weight systems]]:

+-- {: .num_prop #WeightSystemsAreCohomologyOfLoopSpaceOfConfigurationSpace}
###### Proposition
**([[weight systems are cohomology of loop space of configuration space]])**

For [[ground field]] $k = \mathbb{R}$ the [[real numbers]],
there is a canonical [[injection]] of the [[real vector space]] $\mathcal{W}$ of [[framed weight systems]] (eq:SpaceOfWeightSystemsAsLinearDual) into the [[real cohomology]] of the [[based loop spaces]] of the ordered [[configuration spaces of points]] in 3-[[dimension|dimensional]] [[Euclidean space]]:

$$
  \mathcal{W}
  \;\overset{\;\;\;\;}{\hookrightarrow}\;
  H^\bullet
  \Big(
    \Omega
    \underset{{}^{\{1,\cdots,n\}}}{Conf}
    \big(
      \mathbb{R}^3
    \big)
  \Big)
$$

=--

This is stated as [Kohno 02, Theorem 4.2](#Kohno02).

+-- {: .num_prop #WeightSystemsAreAssociatedGradedOfVassilievInvariants}
###### Proposition
**([[weight systems are associated graded of Vassiliev invariants]])**

For [[ground field]] $k = \mathbb{R}, \mathbb{C}$ the [[real numbers]] or [[complex numbers]], there is for each [[natural number]] $n \in \mathbb{N}$ a canonical [[linear isomorphism]]

$$
  \mathcal{V}_n/\mathcal{V}_{n-1}
  \underoverset{\simeq}{\phantom{AAAA}}{\longrightarrow}
  \big(
    \mathcal{A}_n^u
  \big)^\ast
$$

from

1. the [[quotient vector space]] of order-$n$ [[Vassiliev invariants]] of [[knots]] by those of order $n-1$

1. to the space of [[unframed weight systems]] of order $n$.

In other words, in [[characteristic zero]], the [[graded vector space]] of [[unframed weight systems]] is the [[associated graded vector space]] of the [[filtered vector space]] of [[Vassiliev invariants]].

=--

([Bar-Natan 95, Theorem 1](#BarNatan95), following [Kontsevich 93](#Kontsevich93))

## Examples

1. The $n$th coefficient of the [[Conway polynomial]] is a Vassiliev invariant of order $\le n$.

(...)

## Related concepts ##

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

* [[Kontsevich integral]]

* [[universal Vassiliev invariant]]

* [[knot theory]]

* [[Jones polynomial]]

* [[singularity]]

## References

### General

The original articles are

* [[Viktor Vassiliev]], _Complements of discriminants of smooth maps: topology and applications_, Amer. Math. Soc. 1992 ([ams:mmono-98](https://bookstore.ams.org/mmono-98))

* {#Kontsevich93} [[Maxim Kontsevich]], _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf), [[KontsevichVassilievInvariants93.pdf:file]])

* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

Lecture notes:

* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_, in: *Geometry and Physics*, Lecture Notes in Pure and Applied Mathematics, volume 184, Marcel Dekker Inc. 1996 ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009), [ISBN:0-8247-9791-4](https://users-math.au.dk/swann/Aarhus-proceedings-1995/index.html))

Textbook accounts:

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

* [[David Jackson]], [[Iain Moffat]], _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))

* {#Lescop20} [[Christine Lescop]], _Invariants of links and 3-manifolds from graph configurations_ ([arXiv:2001.09929](https://arxiv.org/abs/2001.09929))

Further review:

* [[Simon Willerton]], _On the Vassiliev invariants for knots and for pure braids_, 1997 ([hdl:1842/11581](http://hdl.handle.net/1842/11581), [ethos.663801](https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.663801), [[WillertonVasdilievInvariants1997.pdf:file]])

* [[Dror Bar-Natan]], _Finite Type Invariants_, in:  J.-P. Francoise, G.L. Naber and Tsou S.T. (eds.) Encyclopedia of Mathematical Physics, Oxford: Elsevier, 2006, volume 2 page 340
([arXiv:math/0408182](https://arxiv.org/abs/math/0408182))

* {#LandoZvonkin} Sergei K. Lando and Alexander K. Zvonkin, Chapter 6 of: _Graphs on Surfaces and Their Applications_, Springer, 2004.

More literature is listed at

* [[Dror Bar-Natan]], [[Sergei Duzhin]], _[Bibliography of Vassiliev Invariants](http://www.pdmi.ras.ru/~duzhin/VasBib/Long)_

See also

* Mathworld, _[Vassiliev invariant](http://mathworld.wolfram.com/VassilievInvariant.html)_

Concrete computations:

* [[Jan Kneissler]], _On spaces of connected graphs I: Properties of Ladders_, Proc. Internat. Conf. "Knots in Hellas '98", Series on Knots and Everything, vol. 24 (2000), 252-273 ([arXiv:math/0301018](https://arxiv.org/abs/math/0301018))

* [[Jan Kneissler]], _On spaces of connected graphs II: Relations in the algebra Lambda_, Jour. of Knot Theory and its Ramif. vol. 10, no. 5 (2001), 667-674 ([arXiv:math/0301019](https://arxiv.org/abs/math/0301019))

* [[Jan Kneissler]], _On spaces of connected graphs III: The Ladder Filtration_, Jour. of Knot Theory and its Ramif. vol. 10, no. 5 (2001), 675-686 ([arXiv:math/0301020](https://arxiv.org/abs/math/0301020))

* [[Pierre Vogel]], _Algebraic structures on modules of diagrams_, Journal of Pure and Applied Algebra Volume 215, Issue 6, June 2011, Pages 1292-1339 ([pdf](https://webusers.imj-prg.fr/~pierre.vogel/diagrams.pdf))

### Relation to Jones polynomial:

Relation to the [[Jones polynomial]]:

* Joan S. Birman; Xiao-Song Lin, _Knot polynomials and Vassiliev's invariants_,  Inventiones mathematicae (1993) Volume: 111, Issue: 2, page 225-270 ([https://dml:144077](https://eudml.org/doc/144077))

Relation to other polynomial [[knot invariants]]:

* Myeong-Ju Jeong, Chan-Young Park, _Polynomial invariants and Vassiliev invariants_, Geom. Topol. Monogr. 4 (2002) 89-101 ([arxiv:math/0211045](https://arxiv.org/abs/math/0211045))

### Relation to homology of loop spaces of configuration spaces

Relation to the [[Euler characteristic]] of the [[ordinary homology]] of [[loop spaces]] of [[configuration spaces of points]]

* {#CohenGitler01} [[Fred Cohen]], [[Samuel Gitler]], _Loop spaces of configuration spaces, braid-like groups, and knots_, In: Jaume Aguadé, Carles Broto, [[Carles Casacuberta]]  (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel 2001 ([doi:10.1007/978-3-0348-8312-2_7](https://doi.org/10.1007/978-3-0348-8312-2_7))

based on

* {#Kohno94} [[Toshitake Kohno]], _Vassiliev invariants and de Rham complex on the space of knots_, In: Yoshiaki Maeda, Hideki Omori and [[Alan Weinstein]] (eds.), _Symplectic Geometry and Quantization_, Contemporary Mathematics 179 (1994): 123-123 ([doi:10.1090/conm/179](http://dx.doi.org/10.1090/conm/179))

* {#Cohen76} [[Fred Cohen]], _The homology of $\mathcal{C}_{n+1}$-Spaces, $n \geq 0$. In: The Homology of Iterated Loop Spaces. Lecture Notes in Mathematics, vol 533. Springer, Berlin, Heidelberg  1976 ([doi:10.1007/BFb0080467](https://doi.org/10.1007/BFb0080467))

See also

* {#Kohno02} [[Toshitake Kohno]], _Loop spaces of configuration spaces and finite type invariants_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arXiv:math/0211056](https://arxiv.org/abs/math/0211056))

* [[Toshitake Kohno]], Section 3.1 in: *Conformal field theory and topology*, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs __210__. Iwanami Series in Modern Mathematics. Amer. Math. Soc. 2002 $[$[AMS:mmono-210](https://bookstore.ams.org/mmono-210)$]$

### As Chern-Simons amplitudes

Discussion of higher order Vassiliev invariants as [[Chern-Simons theory]]-[[correlators]], hence as [[configuration space of points|configuration space]]-[[integrals]] of [[wedge products]] of [[Chern-Simons propagators]] assigned to [[edges]] of [[Feynman diagrams]] in the [[graph complex]]:

* [Kontsevich 93, Section 5](#Kontsevich93)

* {#AF96} [[Daniel Altschuler]], [[Laurent Freidel]], _Vassiliev knot invariants and Chern-Simons perturbation theory to all orders_, Commun. Math. Phys. **187** (1997) 261-287 [[arXiv:q-alg/9603010](https://arxiv.org/abs/q-alg/9603010), [doi:10.1007/s002200050136](https://doi.org/10.1007/s002200050136)]

* [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Configuration spaces and Vassiliev classes in any dimension_, Algebr. Geom. Topol. 2 (2002) 949-1000 ([arXiv:math/9910139](https://arxiv.org/abs/math/9910139))

* [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Algebraic structures on graph cohomology_, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 ([arXiv:math/0307218](https://arxiv.org/abs/math/0307218))

Reviewed in:

* {#Volic13} [[Ismar Volić]], Section 4 of: _Configuration space integrals and the topology of knot and link spaces_, [Morfismos, Vol 17, no 2, 2013](https://fdocuments.co/amp/document/morfismos-vol-17-no-2-2013.html) ([arxiv:1310.7224](https://arxiv.org/abs/1310.7224))

See also

* J. de-la-Cruz-Moreno, H. García-Compeán, E. López, _Vassiliev Invariants for Flows Via Chern-Simons Perturbation Theory_ ([arXiv:2004.13893](https://arxiv.org/abs/2004.13893))

### As observables on fuzzy spheres

Relation of [[Dp-D(p+2)-brane bound states]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]) to [[Vassiliev braid invariants]] via [[chord diagrams]] computing [[radii]] of [[fuzzy spheres]]:

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

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

* [[Simon McNamara]], Section 4 of: _Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels_, 2006 ([spire:1351861](http://inspirehep.net/record/1351861), [pdf](https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf), [[McNamara06.pdf:file]])

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

### Vassiliev invariants of braids

Vassiliev invariants of [[braid group|braids]] via [[horizontal chord diagrams]]:

* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))

* Louis Funar, _Vassiliev invariants I: Braid groups and rational homotopy theory_ ([arXiv:q-alg/9510008](https://arxiv.org/abs/q-alg/9510008))

category: geometry, topology

[[!redirects Vassiliev knot invariant]]
[[!redirects Vassiliev knot invariants]]

[[!redirects Vassiliev invariants]]

[[!redirects Vassiliev finite type invariants]]
[[!redirects Vassiliev finite type invariant]]
[[!redirects Vassiliev finite-type invariants]]
[[!redirects Vassiliev finite-type invariant]]
[[!redirects finite type invariants]]
[[!redirects finite type invariant]]
[[!redirects finite-type invariants]]
[[!redirects finite-type invariant]]

[[!redirects Vassiliev link invariant]]
[[!redirects Vassiliev link invariants]]

[[!redirects Vassiliev invariants of braids]]

[[!redirects Vassiliev braid invariant]]
[[!redirects Vassiliev braid invariants]]


Vassiliev skein relation

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## Idea

The **Vassiliev skein relation** is a way to extend [[knot invariants]] to [[singular knots]] (at least, to singular knots where the only singularities are double points).  If $v$ is a knot invariant that takes values in an abelian group, then it is extended to singular knots using the relation
$$
v(L_d) = v(L_+) - v(L_-)
$$
where $L_d$ is a singular knot with a double point and $L_+$, respectively $L_-$, are formed from $L_d$ by replacing the double point by a positively oriented, respectively negatively oriented, crossing.

\begin{tikzcd}[sep={between origins, 30pt}]
  &
  {} && {}
  &[+30pt] &
  {} \ar[-Latex, from=ddrr, line width=1.5pt]  && {}
  &[+30pt] &
  {} && {}
  \\
  L_d =
  &
  & \raisebox{-2.5pt}{\scalebox{1.8}{$\bullet$}}
  &
  &
  L_+ =
  &
  &
  &
  &
  L_- =
  \\
  &
  {} \ar[-Latex, uurr, line width=1.5pt]
  &&
  {} \ar[-Latex, uull, line width=1.5pt]
  &&
  {} \ar[-Latex, uurr, line width=1.5pt, crossing over]
  &&
  {}
  &&
  {} \ar[-Latex, uurr, line width=1.5pt]
  &&
  {} \ar[-Latex, uull, line width=1.5pt, crossing over]
\end{tikzcd}

## References

General discussion:

* Wikipedia, _[Skein relation](https://en.wikipedia.org/wiki/Skein_relation)_

Discussion in the context of [[quantization of 3d Chern-Simons theory]]:

* {#GelcaUribe10a} [[Razvan Gelca]], [[Alejandro Uribe]], _From classical theta functions to topological quantum field theory_  ([arXiv:1006.3252](http://arxiv.org/abs/1006.3252), [slides pdf](http://www.math.ttu.edu/~rgelca/berk.pdf))

* {#GelcaUribe10b} [[Razvan Gelca]], [[Alejandro Uribe]], _Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$_ ([arXiv:1007.2010](http://arxiv.org/abs/1007.2010))

category: knot theory

[[!redirects Vassiliev skein relations]]

[[!redirects skein relation]]
[[!redirects skein relations]]


Vassily Gorbounov

**Vassily Gorbounov** is a professor of mathematics at Aberdeen. Much of his mathematical work is inspired by [[quantum field theory]], especially [[topological field theory|topological]] and [[conformal field theory]] as well as with [[superstring theory]].

* [institute page](https://www.hse.ru/en/org/persons/216359816)

## Selected writings

* [[Imma Gálvez]], Vassily Gorbounov, [[Andrew Tonks]], _Homotopy Gerstenhaber structures and vertex algebras_, [math/0611231.QA](https://arxiv.org/abs/math/0611231)

On the [[quaternionic cobordism ring]]:

* {#GorbunovRay92} [[Vassily Gorbounov]], [[Nigel Ray]], _Orientations of $Spin$ Bundles and Symplectic Cobordism_, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 ([[GorbunovRaySpinBundles.pdf:file]], [doi: 10.2977/prims/1195168855](https://www.ems-ph.org/journals/show_abstract.php?issn=0034-5318&vol=28&iss=1&rank=4))

* [[Vassily Gorbounov]], [[Nigel Ray]], [[Paul Turner]], _On the Hopf Ring for a Symplectic Oriented Spectrum_, American Journal of Mathematics
Vol. 117, No. 4 (Aug., 1995), pp. 1063-1088 ([doi:10.2307/2374960](https://doi.org/10.2307/2374960))

On [[chiral differential operators]]:

* [[Vassily Gorbounov]], [[Fyodor Malikov]], [[Vadim Schechtman]], _Gerbes of chiral differential operators_  Math. Res. Lett. __7__ (2000),  no. 1, 55--66, [MR2002c:17040](https://mathscinet.ams.org/mathscinet-getitem?mr=2002c:17040), [math.AG/9906117](https://arxiv.org/abs/math/9906117);

* _Gerbes of chiral differential operators. II. Vertex algebroids_, Invent. Math. __155__ (2004), no. 3, 605--680, [MR2005e:17047](http://www.ams.org/mathscinet-getitem?mr=2005e:17047), [math.AG/0003170](http://arxiv.org/abs/math/0003170), [doi](https://doi.org/10.1007/s00222-003-0333-4);

* _Gerbes of chiral differential operators. III_, in: The orbit method in geometry and physics (Marseille, 2000),  73--100, Progr. Math. __213__, Birkhäuser 2003, [MR2005a:17028](https://mathscinet.ams.org/mathscinet-getitem?mr=2005a:17028), [math.AG/0005201](https://arxiv.org/abs/math/0005201),

* _On chiral differential operators over homogeneous spaces_, Int. J. Math. Math. Sci. __26__ (2001), no. 2, 83--106, [MR2002g:14020](https://mathscinet.ams.org/mathscinet-getitem?mr=2002g:14020), [math.AG/0008154](https://arxiv.org/abs/math/0008154), [doi](http://doi.org/10.1155/S0161171201020051)

* Vassily Gorbounov, [[Owen Gwilliam]], Brian Williams, _Chiral differential operators via Batalin-Vilkovisky quantization_, [pdf](http://people.mpim-bonn.mpg.de/gwilliam/cdo.pdf)

category: people

[[!redirects Vassili Gorbounov]]


Vassily Gorbunov > history


Vaughan Jones

* [website](http://math.berkeley.edu/~vfr/)

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

## related $n$Lab entries

* [[knot invariant]]

* [[Jones polynomial]]

* [[planar algebra]]

category: people


Vaughan Pratt

Vaughan Pratt is Professor Emeritus of Computer Science at Stanford University.  Prior to his retirement in 2000 his research focused on theory and technical applications of computers.  After retirement he started Tiqit Computers to promote what are now called netbooks.  His current interests are in [[universal algebra]] and [[category theory]], signal processing, principles of written and spoken communication, thin film optics, and philosophy of mind.  His most recent four Ph.D. students are in acoustic signal processing, including Parham Aarabi, named to MIT's TR35 for 2005, and Keyvan Mohajer, CEO of Melodis Corp.  He is responsible for 38 of Sophus Lie's 990 descendants.

He has written extensively on [[Chu spaces]], taking simple examples to show the potential of the idea, and encouraging the use of generalisations of it.

* [website](http://boole.stanford.edu/pratt.html)

* [Chu Space Web Site (HTML)](http://chu.stanford.edu/)

## Selected writings

On [[quantum programming]] via [[quantum logic]] understood as [[linear type theory]] [[categorical semantics|interpreted in]] [[symmetric monoidal categories]]:

* {#Pratt92} [[Vaughan Pratt]], *Linear logic for generalized quantum mechanics*, in Proceedings of *[Workshop on Physics and Computation (PhysComp'92)](https://www.computer.org/csdl/proceedings/phycmp/1992/12OmNx19jVS)* (1992) 166-180 [[doi:10.1109/PHYCMP.1992.615518](https://doi.ieeecomputersociety.org/10.1109/PHYCMP.1992.615518), [pdf](http://boole.stanford.edu/pub/ql.pdf), [[Pratt-LinearLogicForQuantum.pdf:file]]]

On [[linear logic]] and the [[Chu construction]]:

* {#Pratt93} [[Vaughan Pratt]], *The second calculus of binary relations*, Mathematical Foundations of Computer Science 1993. MFCS 1993, Lecture Notes in Computer Science **711**, Springer (1993) [[doi:10.1007/3-540-57182-5_9](https://doi.org/10.1007/3-540-57182-5_9)]

  > "Linear logic is seen in its best light as the
realization of the [[Curry-Howard isomorphism]] for [[linear algebra]]"

On [[Chu spaces]]:

* [[Vaughan Pratt]], *Chu Spaces* (1999) [[pdf](http://boole.stanford.edu/pub/coimbra.pdf), [[Pratt-ChuSpaces.pdf:file]]]

* [[Vaughan Pratt]], *Linear process algebra*, in: *Distributed Computing and Internet Technology ICDCIT 2011*,  Lecture Notes in Computer Science **6536**, Springer (2011) [[doi:10.1007/978-3-642-19056-8_6](https://doi.org/10.1007/978-3-642-19056-8_6), [pdf](http://boole.stanford.edu/pub/bhub.pdf)]

On the relation between [[toposes]] and [[abelian categories]] (cf. *[[AT-categories]]*):

* [[Vaughan Pratt]] in conversation with [[Peter Freyd]], *Topos-abelian categories that are neither*, mailing list comments (1999) [[catlist:1999/atcat](http://www.mta.ca/~cat-dist/catlist/1999/atcat), [[Freyd_AT-categories.txt:file]]]

category: people


vDbl

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## Definition

By $vDbl$ is denoted the [[2-category]] of [[virtual double categories]].

## References

* [[Geoff Cruttwell]] and [[Mike Shulman]], _A unified framework for generalized multicategories_, [arXiv:0907.2460](http://arxiv.org/abs/0907.2460)
{#CruttwellShulman}

category: category


Vect

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## Definition

Given a [[field]] $k$, the _category of $k$-vector spaces_ **$Vect_k$**  is the [[category]] whose

1. [[object|objects]] are [[vector space|vector spaces]],

1. [[morphisms]] are [[linear maps]].

If the field $k$ is understood, one often just writes $Vect$.

Via [[direct sum]] and [[tensor product of vector spaces]] $\otimes_k$, this becomes a [[symmetric monoidal category]] in two compatible ways, making it a [[distributive monoidal category]], in particular a [[rig category]].

The study of $Vect$ is called _[[linear algebra]]_.

## Properties

For a field $k$, the category $Vect_k$ is [[ cocomplete category | cocomplete ]].

### Splitting lemma

Assuming the [[axiom of choice]] (and essentially by the [[basis theorem]]):
\begin{proposition}\label{ShortExactSequencesSplit}
In $Vect$ every [[short exact sequence]] [[split short exact sequence|splits]].
\end{proposition}
(See [there](split+exact+sequence#SESOfVectorSpacesSplits).)

\begin{remark}
On [[FinDimVect]] this is a [[categorification]] of the [[rank-nullity theorem]].
\end{remark}

## Related categories

### Finite-dimensional vector spaces

The [[full subcategory]] of Vect consisting of [[finite-dimensional vector spaces]] may be denoted **[[FinDimVect]]**.

This is a [[compact closed category]] (see [here](finite-dimensional+vector+space#CompactClosure)).

$FinDimVect$ is where most of ordinary [[linear algebra]] lives, although much of it makes sense in all of $Vect$.  See also at _[[quantum information theory in terms of dagger-compact categories]]_.

On the other hand, anything involving transposes or [[inner products]] really takes place in $Fin$ [[Hilb]].

### Modules

More generally, for $R$ any [[ring]] (not necessarily a [[field]]) then the analog of $Vect$ is the category $R$[[Mod]] of $R$-[[modules]] and module homomorphisms between them.

### Vector bundles

For $X$ a suitable [[space]] of sorts, there is the category [[Vect(X)]] of [[vector bundles]] over $X$. Specifically for $X$ a [[topological space]], there is the category of [[topological vector bundles]] over $X$. For $X = \ast$ the [[point space]], then this is [[equivalence of categories|equivalently]] the category of plain vector spaces:

$$
  Vect(\ast) \simeq Vect
  \,.
$$

category: category

[[!redirects category of vector spaces]]
[[!redirects categories of vector spaces]]

[[!redirects category of vector bundles]]
[[!redirects categories of vector bundles]]

[[!redirects VectorSpaces]]


Vect(X)

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## Definition

For $X$ a suitable [[space]] of sorts, the _category of vector bundles over $X$_ is the [[category]] denoted $Vect(X)$  whose

1. [[objects]] are [[vector bundles]] over $X$,

1. [[morphisms]] are vector bundle [[homomorphisms]] over $X$.

Specifically for $X$ a [[topological space]], there is the category of [[topological vector bundles]] over $X$.

Via [[direct sum of vector bundles]] and [[tensor product of vector bundles]] this becomes a [[symmetric monoidal category]] in two compatible ways, making it a [[distributive monoidal category]], in particular a [[rig category]].

For $X$ a [[compact Hausdorff space]] then the [[Grothendieck group]] of $Vect(X)$ is the [[topological K-theory]] group $K(X)$.

## Relation to other categories

### Vector spaces

For $X = \ast$ the [[point space]], then this is [[equivalence of categories|equivalently]] the category [[Vect]] of plain vector spaces:

$$
  Vect(\ast) \simeq Vect
  \,.
$$

### Vector bundles over general bases

See *[[VectBund]]*

### Higher vector bundles

An analog in [[homotopy theory]]/[[higher category theory]] is the [[(infinity,1)-category]] of [[(infinity,1)-module bundles]].

[[!redirects VectBund(X)]]

[[!redirects VectBund(B)]]