Victor Ostrik

__Victor Ostrik__ is a mathematicain at University of Oregon ([web](http://pages.uoregon.edu/vostrik/)). He has been one of the main figures in the development of the theory of [[fusion categories]].

## Selected writings

On [[tensor categories]]:

* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], _Topics in Lie theory and Tensor categories -- 9 Tensor categories_, Lecture notes (spring 2009) ([pdf](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec09.pdf) [web](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/))

* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) $[$[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205)$]$

On [[Deligne's theorem on tensor categories]]:

* {#Ostrik04} [[Victor Ostrik]], _Tensor categories (after P. Deligne)_ ([arXiv:math/0401347](http://arxiv.org/abs/math/0401347))

On [[braided monoidal category|braided]] [[fusion categories]]:

* {#EtingofNikshychOstrik05} [[Pavel Etingof]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On fusion categories*, Annals of Mathematics Second Series, Vol. 162, No. 2 (Sep., 2005), pp. 581-642 ([arXiv:math/0203060](http://arxiv.org/abs/math/0203060), [jstor:20159926](https://www.jstor.org/stable/20159926))

* {#DrinfeldGelakiNikshychOstrik10} [[Vladimir Drinfeld]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On braided fusion categories I*, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 ([doi:10.1007/s00029-010-0017-z](https://doi.org/10.1007/s00029-010-0017-z))

On [[Frobenius exact categories]]:

* {#EtingofOstrik21} [[Pavel Etingof]], [[Victor Ostrik]], *On the Frobenius functor for symmetric tensor categories in positive characteristic*, J. für die reine und angewandte Mathematik **773** (2021) 165–198 $[$[arXiv:1912.12947](https://arxiv.org/abs/1912.12947)$]$

and in view of [[Deligne's theorem on tensor categories]]:

* {#CoulembierEtingofOstrik21} [[Kevin Coulembier]], [[Pavel Etingof]], [[Victor Ostrik]], _On Frobenius exact symmetric tensor categories_ $[$[arXiv:2107.02372](https://arxiv.org/abs/2107.02372)$]$

## Related entries

* [[monoidal category]],

* [[fusion category]], [[weak Hopf algebra]]

category: people

[[!redirects Ostrik]]
[[!redirects V. Ostrik]]

[[!redirects Viktor Ostrik]]


Victor Petrov

* [webpage](http://thd.pnpi.spb.ru/~victorp)

## Selected writings

* [[Dmitri Diakonov]], [[Victor Petrov]], _Exotic baryon resonances in the Skyrme model_ ([arXiv:0812.1212](https://arxiv.org/abs/0812.1212), [doi:10.1142/9789814704410_0004](https://doi.org/10.1142/9789814704410_0004)), Chapter 3 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

> It is astounding that [[Tony Skyrme|Skyrme]] had suggested [[Skyrmion|his model]] as early as in 1961 before it has been generally accepted that [[pions]] are (pseudo) [[Goldstone bosons]] associated with the [[spontaneous symmetry breaking|spontaneous breaking]] of [[chiral symmetry]], and of course long before [[QCD|Quantum Chromodynamics (QCD)]] has been put forward as the microscopic theory of [[strong nuclear force|strong interactions]].

> The revival of the Skyrme idea in 1983 is [due to Witten](Skyrmion#Witten83) who explained the _raison d'ˆetre_ of the [[Skyrmion|Skyrme model]] from the viewpoint of [[QCD]]. In the [[chiral perturbation theory|chiral limit]] when the light [[quark]] masses $m_u$, $m_d$, $m_s$ tend to zero, such that the octet of the pseudoscalar mesons [[pion|π]], [[kaon|K]] , η become nearly massless (pseudo) [[Goldstone bosons]], they are the lightest degrees of freedom of [[QCD]]. The [[effective field theory|effective]] chiral Lagrangian (EχL) for pseudoscalar mesons, understood as an infinite expansion in the derivatives of the pseudoscalar (or chiral) fields, encodes, in principle, full information about QCD. The famous two-term Skyrme Lagrangian can be understood as a low-energy truncation of this infinite series. Witten has added an important four-derivative [[WZW-term|Wess–Zumino term]] to the original Skyrme Lagrangian and pointed out that the overall coefficient in front of the EχL is proportional to the number of quark [[color charge|colours]] $N_c$.

> $[...]$

> Soon after Witten's work it has been realized that it is possible to bring the Skyrme model and the Skyrmion even closer to QCD and to the more customary language of constituent quarks. It has been first noticed $[$[6](#DiakonovEides83), [7a](#DharWadia84), [7b](#DarShankarWadia85), [8](#DiakonovPetrov86)$]$ that a simple chiral invariant Lagrangian for massive (constituent) quarks $Q$ interacting with the octet chiral field $\pi_A$ $(A = 1, ..., 8)$,

> $$\mathcal{L} = \overline{Q} \left( \mathrlap{\partial}/ - M e^{ \tfrac{i \pi^A \lambda^A \gamma_5}{F_\pi}  }  \right) Q$$

> induces, via a quark loop in the external pseudoscalar fields (see Fig. 3.1), the EχL whose lowest-derivative terms coincide with the Skyrme Lagrangian, including
automatically the Wess–Zumino term, with the correct coefficient!

> $[...]$

> The condition that the winding number of the trial field is unity needs to be imposed to get a deeply bound state, that is to guarantee that the baryon number is unity. $[$[9](#DiakonovPetrovPobylitsa88)$]$ The Skyrmion is, thus, nothing but the **mean chiral field binding quarks in a baryon**.

\linebreak

* {#DiakonovEides83} [[Dmitri Diakonov]], Michael I. Eides, _Chiral Lagrangian from a functional integral over quarks_, JETP Letters 38.7 (1983): 433-436 ([pdf](http://jetpletters.ac.ru/ps/1483/article_22635.pdf), [[DiakonovEides83.pdf:file]])

* {#DharWadia84} A. Dhar, Spenta R. Wadia, _Nambu—Jona-Lasinio Model: An Effective Lagrangian for Quantum Chromodynamics at Intermediate Length Scales_, Phys. Rev. Lett. 52, 959 (1984) ([doi:10.1103/PhysRevLett.52.959](https://doi.org/10.1103/PhysRevLett.52.959))

* {#DarShankarWadia85} Avinash Dhar, R. Shankar, Spenta R. Wadia, _Nambu–Jona-Lasinio–type effective Lagrangian: Anomalies and nonlinear Lagrangian of low-energy, large-N QCD_, Phys. Rev. D 31, 3256 (1985) ([doi:10.1103/PhysRevD.31.3256](https://doi.org/10.1103/PhysRevD.31.3256))

* {#DiakonovPetrov86} [[Dmitri Diakonov]], [[Victor Petrov]], _A theory of light quarks in the instanton vacuum_, Nuclear Physics B Volume 272, Issue 2, 21 July 1986, Pages 457-489 (doi:10.1016/0550-3213(86)90011-8)

* {#DiakonovPetrovPobylitsa88} [[Dmitri Diakonov]], [[Victor Petrov]], P.V. Pobylitsa, _A Chiral Theory of Nucleons_, Nucl. Phys. B306 (1988) 809 ([spire:247700](http://inspirehep.net/record/247700), doi:10.1016/0550-3213(88)90443-9)

category: people


Victor Porton

Victor Porton is the owner of a [math site](http://www.mathematics21.org/) with some research (mainly in [[general topology]]) and a [math weblog](https://portonmath.wordpress.com/2009/05/). He wishes to be nominated for the Abel Prize ([link](http://www.mathematics21.org/abel-prize.html)).

category: people


Victor Snaith

__Victor P. Snaith__ was an algebraist, algebraic topologist and algebraic geometer (born 1944, died 2021), prof. emeritus at Sheffield

* [personal webpage](http://victor-snaith.staff.shef.ac.uk),

* [institute page](http://maths.dept.shef.ac.uk/maths/staff_info.php?id=34).

## Selected writings

On [[infinite loop spaces]]:

* [[Ib Madsen]], [[Victor Snaith]], [[Jørgen Tornehave]], *Infinite loop maps in geometric topology*, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 81, Issue 3, (1977)([doi:10.1017/S0305004100053482](https://doi.org/10.1017/S0305004100053482))

On [[algebraic cobordism]] and introducing [[Snaith's theorem]]:

* Algebraic cobordism and $K$-theory. Mem. Amer. Math. Soc. __21__ (1979), no. 221, vii+152 pp.

* Algebraic $K$-theory and localised stable homotopy theory. Mem. Amer. Math. Soc. __43__ (1983), no. 280, xi+102 pp.

* Stable homotopy around the Arf-Kervaire invariant, Progress in Mathematics __273__, Birkhäuser Basel, 2009, ISBN10:3764399031

* _Infinite loop maps and the complex J-homomorphism_, Bull. Amer. Math. Soc. __82__, 3 (1976), 508-510 [MR0410741](http://www.ams.org/mathscinet-getitem?mr=0410741), [euclid](http://projecteuclid.org/euclid.bams/1183537922)

On the [[J-homomorphism]]:

* _The complex J-homomorphism. I_, Proc. Lond. Math. Soc., III. Ser. 34, 269-302 (1977), (Zbl 0344.55016)

* _Geometric dimension of complex vector bundles_, Serie
notas у simposia del Мех. Mat. Soc. 1 (1975) 199-227.

* _Towards algebraic cobordism_, Bull. A, M.. Soc. 83, 3
(1977).

* D. Gepner, V. Snaith, _On the motivic spectra representing algebraic cobordism and algebraic K-theory_, Doc. Math., 14:359–396 (electronic), 2009, [pdf](http://www.math.uni-bielefeld.de/documenta/vol-14/14.pdf)

## Related pages

* [[algebraic cobordism]]

* [[Snaith's theorem]]

[[!redirects V. P. Snaith]]
[[!redirects Snaith]]


Victor Tourtchine

* [webpage](https://www.math.ksu.edu/~turchin/)

## Selected writings

On [[graph complex]]-models for the [[rational homotopy type]] of [[configuration spaces of points]]:

* [[Pascal Lambrechts]], [[Victor Tourtchine]], _Homotopy graph-complex for configuration and knot spaces_, Transactions of the AMS, Volume 361, Number 1, January 2009, Pages 207–222 ([arxiv:math/0611766](https://arxiv.org/abs/math/0611766))

[[!redirects Victor Turchin]]
[[!redirects Виктор Турчин]]


Victor V. Albert

* [personal page](https://sites.google.com/site/victorvalbert)

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

## Selected writings

On ([[quantum error correction|quantum]]) [[error correcting codes]]:

* [errorcorrectionzoo.org](https://errorcorrectionzoo.org)

Specifically on [[Absorption-Emission codes]]:

* Shubham Jain, Eric Hudson, Wesley Campbell, [[Victor V. Albert]], *Æ codes* [[arxiv:2311.12324](https://arxiv.org/abs/2311.12324)]

category: people

[[!redirects Victor Albert]]


Victoria Lebed

[website](http://www.maths.tcd.ie/~lebed/)

##Slides:

[Leeds 5-8 July 2016](http://www1.maths.leeds.ac.uk/~matzk/Modelling%20TPM%20Resources_files/Crossed.pdf)

category:people


Vidas Regelskis

Vidas Regelski is an EPRSC research fellow in [[mathematics]] at University of Surrey.

* [webpage](http://www.surrey.ac.uk/maths/people/vidas_regelskis/)

## related $n$Lab entries

* [[AdS3-CFT2 and CS-WZW correspondence]]

category: people


Vidhyanath K. Rao

* [institute page](https://math.osu.edu/people/rao.3)

## Selected writings

On [[category theory and foundations]]:

* {#Rao06} [[Vidhyanath K. Rao]], _On Doing Category Theory within Set Theoretic Foundations_ in: Giandomenico Sico (ed.), _What is category theory?_, Polimetrica (2006) [[[Rao-CategoryTheory.pdf:file]], [semanticscholar](https://www.semanticscholar.org/paper/What-is-category-theory-Sica/7737401202b220f89ae9c11cf9af65995a4dcf50)],

category: people

[[!redirects Vidhyanath Rao]]


vielbein

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

#Contents#
* table of contents
{:toc}

## Idea

A _vielbein_ or _solder form_ on a [[manifold]] $X$ is a linear identification of the [[tangent bundle]] with a [[vector bundle]] with explicit ([[pseudo-orthogonal structure|pseudo-]])[[orthogonal structure]].

Any such choice encodes a ([[pseudo-Riemannian metric|pseudo-]])[[Riemannian metric]] on $X$.

## Definition

There are different equivalent perspectives on the notion of vielbein that are closely related:

* [In terms of Cartan geometry](#InTermsOfCartanGeometry)

* [In terms of orthogonal structure](#InTermsOfOrthogonalStructure)

### In terms of Cartan geometry
 {#InTermsOfCartanGeometry}

Let $X$ be a [[smooth manifold]] of [[dimension]] $d$. For definiteness we assume here that $X$ is [[orientation|oriented]], but this is not necessary.

A [[Lie algebra valued 1-form]]

$$
  (E,\Omega) : T X \to \mathfrak{iso}(d)
$$

with values in the [[Poincaré Lie algebra]] encodes a [[pseudo-Riemannian metric]] on $X$ (if non-degenerate, at least). In this context the component

$$
  E : T X \to \mathbb{R}^d
$$

of the [[connection on a bundle|connection]] 1-form is called the _vielbein_ . It encodes the metric by

$$
  g = \langle E \otimes E\rangle  \in Sym^2_{C^\infty(X)} \Gamma(T^* X)
  \,,
$$

where $\langle -,-\rangle : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ is the canonical [[bilinear form]].

In other words, Given an $(SO(d) \hookrightarrow ISO(d))$-[[Cartan connection]] on $X$, the _vielbein_ is the isomorphism in the definition of Cartan connection.

For $d=4$ this is the _vierbein_ , for $d = 3$ the _dreibein_ , etc.

This terminology is used notably in the context of the _[[first-order formulation of gravity]]_.

### In terms of orthogonal structure
 {#InTermsOfOrthogonalStructure}

We discuss here how a choice of vielbein on a [[manifold]] is equivalently the [[reduction of structure groups|reduction of the structure group]] of the [[tangent bundle]] from the [[general linear group]] $GL(n)$ to its [[maximal compact subgroup]], the [[orthogonal group]].

The following also introduces the description of this in terms of [[smooth infinity-groupoid|smooth]] [[twisted cohomology|twisted]] [[cohomology]]. While of course this is not necessary to understand vielbeins, it does give a very natural conceptual description with the advantage that it seamlessly generalizes to notions of _[[generalized vielbein]]_ fields and generally to [[twisted differential c-structures]].

#### The class of the tangent bundle
 {#ClassOfTheTangentBundle}

For completeness, we first review how the [[tangent bundle]] of a [[smooth manifold]] is naturally incarnated as a [[cocycle]] in $GL(n)$-valued [[Cech cohomology]] and how this in turn is naturally formulated in terms of [[Lie groupoids]]/[[differentiable stack|smooth]] [[moduli stacks]]. The reader familiar with these basics should skip to the [next section](#ReductionOfTheStructureGroup).

Let $X$ be a [[smooth manifold]] of [[dimension]] $n$.

By definition this means that there is an [[atlas]] $\{\phi_i^{-1} : \mathbb{R}^n \simeq U_i \hookrightarrow X\}$ of [[coordinate charts]]. On each overlap $U_i \cap U_j$ of two [[coordinate charts]] the [[partial derivatives]] of the corresponding [[coordinate transformations]]

$$
  \phi_j\circ \phi_i^{-1}
  :
  U_i \cap U_j \subset \mathbb{R}^n \to \mathbb{R}^n
$$

form the [[Jacobian matrix]] of [[smooth functions]]

$$
  ((\lambda_{i j})^{\mu}{}_{\nu})
  \coloneqq
  \left[\frac{d}{d x^\nu} \phi_j \circ \phi_i^{-1} (x^\mu)
  \right]
  :
  U_i \cap U_j
  \to
  GL_n
$$

with values in invertible [[matrices]], hence in the
[[general linear group]] $GL(n)$. By construction (by the [[chain rule]]), these functions satisfy on triple overlaps of coordinate charts the [[matrix product]] equations

$$
  (\lambda_{i j})^\mu{}_\lambda (\lambda_{j k})^\lambda{}_{\nu}
  =
  (\lambda_{i k})^\mu{}_{\nu}
  \,,
$$

hence the equation

$$
  \lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}
$$

in the [[group]] $C^\infty(U_i \cap U_j, GL(n))$ of smooth $GL(n)$-valued functions on the chart overlap.

This is the [[cocycle]] condition for a smooth [[Cech cohomology|Cech cocycle]] in degree 1 with coefficients in $GL(n)$ (precisely: with coefficients in the [[sheaf]] of smooth functions with values in $GL(n)$ ):

$$
  [(\lambda_{i j})] \in H^1_{smooth}(X, GL_n)
  \,.
$$

It is useful to formulate this statement in the language of [[Lie groupoids]]/[[differentiable stacks]].

* $X$ itself is trivially a Lie groupoid $(X \stackrel{\to}{\to} X)$;

* from the atlas $\{U_i \to X\}$ we get the corresponding [[Cech groupoid]]
  $$
    C(\{U_i\}) =
    (\coprod_{i, j} U_i \cap U_j \stackrel{\to}{\to} \coprod_i U_i)
  $$

* any [[Lie group]] $G$ induces its [[delooping]] [[Lie groupoid]]

  $$
    \mathbf{B}G
    =
    \left(
       G \stackrel{\to}{\to} *
    \right)
    \,.
  $$

The above situation is neatly encoded in the existence of a [[diagram]] of Lie groupoids of the form

$$
  \array{
     C(\{U_i\}) &\stackrel{\lambda}{\to}& \mathbf{B} GL(n).
     \\
     {}^{\mathllap{\simeq}}\downarrow
     \\
     X
  }
  \,,
$$

where

* the left morphism is [[stalk]]-wise (around small enough [[neighbourhoods]] of each point) an [[equivalence of groupoids]];

* the horizontal functor has as components the functions $\lambda_{i j}$ and its functoriality is the cocycle condition $\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}$.

We want to think of such a diagram as being directly a morphism of [[smooth infinity-groupoid|smooth groupoids]]

$$
  T X : X \to \mathbf{B} GL(n) \;\; \in \mathbf{H}
  \,.
$$

This is true in the [[(2,1)-category]] $\mathbf{H}$ in which stalkwise equivalences $W \subset Mor(PSh(SmthMfd, Grpd))$ have been [[simplicial localization|formally inverted]] to become [[homotopy equivalences]].

Since all real [[vector bundles]] on $X$ are encoded by such morphisms, as are their [[gauge transformations]], we say that $\mathbf{B} GL(n)$ is the [[moduli stack]] for real vector bundles.

Of course there is a "smaller" Lie groupoid that also classifies real vector bundles. Passing to this "smaller" Lie groupoid is what the choice of vielbein accomplishes, to which we now turn.

#### Reduction of the structure group
 {#ReductionOfTheStructureGroup}

Consider the defining inclusion of the [[orthogonal group]] into the [[general linear group]]

$$
  O(n) \hookrightarrow GL(n)
  \,.
$$

We may understand this inclusion geometrically in terms of the canonical [[metric]] on $\mathbb{R}^n$, but we may also understand it purely Lie theoretically as the
the inclusion of the [[maximal compact subgroup]] of $GL(n)$. This makes manifest that the inclusion is trivial at the level of [[homotopy theory]] (it is a [[homotopy equivalence]]) and hence _only_ encodes geometric information.

The inclusion induces a corresponding morphism of moduli stacks

$$
  \mathbf{c} : \mathbf{B} O(n) \to \mathbf{B} GL(n)
  \,.
$$

A choice of [[orthogonal structure]] on $T X$ a [[G-structure]] for $G = O(n)$, hence is a factorization of the above $GL(n)$-valued cocycle through $\mathbf{c}$, up to a smooth [[homotopy]].

$$
  \array{
     X &&\stackrel{h}{\to}&& \mathbf{B} O(n)
     \\
     & {}_{\mathllap{\lambda}}\searrow &\swArrow_{E^{-1}}& \swarrow_{\mathrlap{\mathbf{c}}}
     \\
     && \mathbf{B}GL(n)
  }
  \,.
$$

This consists of two pieces of data

* the morphism $h$ is a $O(n)$-valued 1-cocycle -- a collection of _orthogonal transition functions_ --  hence on each overlap of coordinate patches a smooth function

  $$
    ((h_{i j}){}^a{}_b) : U_i \cap U_j \to O(n)
  $$

  such that

  $$
    h_{i j} \cdot h_{j k} = h_{i k}
    \,;
  $$

* the homotopy $E$ is on each chart a function

$$
  E_i  = ((E_i)^a{}_\mu) : U_i \to GL(n)
$$

* such that on each double overlap it intertwines the transition functions $\lambda$ of the tangent bundle with the new orthogonal transition functions, meaning that the equation

  $$
    (E_i)^a{}_{\mu} (\lambda_{i j})^{\mu}{}_\nu
    =
    (h_{i j})^a{}_b (E_j]^b{}_\nu
  $$

  holds. This exhibits the [[natural transformation|naturality]] [[diagram]] of $E$:

  $$
    \array{
       * &\stackrel{E_i}{\to}& *
       \\
       {}^{\mathllap{h_{i j}}}\downarrow && \downarrow^{\mathrlap{\lambda_{i j}}}
       \\
       * &\stackrel{E_j}{\to}& *
    }
  $$

Such a lift $(h,E)$ is an [[orthogonal structure]] on $T X$.
The component $E$ is called the corresponding **vielbein**. It exhibits an [[isomorphism]]

$$
  E : T X \stackrel{\simeq}{\to} V
$$

between a [[vector bundle]] $V \to X$ with [[structure group]] explicitly being the [[orthogonal group]] and the [[tangent bundle]], hence it exhibits the [[reduction of structure groups|reduction of the structure group]] of $T X$ from $GL(n)$ to $O(n)$.

#### Moduli space of orthogonal structures: twisted cohomology
 {#ModuliSpaceOfOrhtogonalStructures}

In order to understand the space of choices of vielbein fields on a given tangent bundle, hence the _[[moduli space]]_ or _[[moduli stack]]_ of [[orthogonal structures]]/[[Riemannian metrics]] on $X$, it is useful to first consider the [[homotopy fiber]] of the morphism $\mathbf{c} : \mathbf{B}O(n) \to \mathbf{B}GL(n)$. One finds that this is the [[coset]] $O(n) \backslash GL(n)$. We may think of the [[fiber sequence]]

$$
  \array{
    GL(n)/O(n) &\to& \mathbf{B} O(n)
    \\
    && \downarrow
    \\
    && \mathbf{B} GL(n)
  }
$$

as being a bundle in $\mathbf{H}$ over the [[moduli stack]] $\mathbf{B}GL(n)$ with typical fiber $GL(n)/O(n)$. It is  the smooth [[associated infinity-bundle|associated bundle]] to the smooth [[universal principal bundle|universal GL(n)-bundle]] induced by the canonical [[action]] of $GL(n)$ on $O(n)\backslash GL(n)$.

This means that if the tangent bundle $T X$ is trivializable, then the coset space $O(n)\backslash GL(n)$ is the moduli space for vielbein fields on $T X$, in that the space of these is

$$
  \mathbf{H}(X, O(n)\backslash GL(n)) = C^\infty(X, O(n)\backslash GL(n))
  \,.
$$

However, if $T X$ is not trivial, then this is true only locally: there is then an [[atlas]] $\{U_i \to X\}$ such that restricted to each $U_i$ the moduli space of vielbein fields is $C^\infty(U_i, GL(n)/ O(n))$, but globally these now glue together in a non-trivial way as encoded by the tangent bundle: we may say that

the tangent bundle _twists_ the functions $X \to GL(n)/O(n)$. If -- as we may -- we think of an ordinary such function as a cocycle in degree-0 cohomology, then this means that a vielbein is a cocycle in $T X$-_[[twisted cohomology]]_ relative to the _twisting coefficient bundle_ $\mathbf{c}$.

We can make this more manifest by writing equivalently

$$
  \array{
    O(n)\backslash GL(n) &\to& (O(n)\backslash GL(n)) // GL(n)
    \\
    && \downarrow
    \\
    && \mathbf{B}GL(n)
  }
  \,,
$$

where now on the right we have inserted the [[fibration]] resolution of the morphism $\mathbf{c}$ as provided by the [[factorization lemma]]: this is the morphism out of the [[action groupoid]] of the action of $GL(n)$ on $O(n)\backslash GL(n)$.

The pullback

$$
  \array{
    T X \times_{GL(n)} (O(n)\backslash GL(n))
      &\to&
    O(n)\backslash GL(n) // GL(n)
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{T X}{\to}& \mathbf{B}GL(n)
  }
$$

give the non-linear $T X$-associated bundle whose space of sections is the "twisted $O(n)\backslash GL(n)$-0-cohomology", hence the space of inequivalent vielbein fields.

#### Moduli stack of orthogonal structures

The above says that the space of vielbein fields is the [[cohomology]] of $T X$ in the [[over-(infinity,1)-topos|slice (2,1)-topos]] $\mathbf{H}_{/\mathbf{B}GL(n)}$ with coefficients in $\mathbf{c} : \mathbf{B}O \to \mathbf{B}GL(n)$

$$
  O Struc_{TX}(X) \simeq \mathbf{H}_{/\mathbf{B}GL(n)}(T X, \mathbf{c})
  \,.
$$

But also this space of choices of vielbein fields has a smooth structure, it is itself a smooth [[moduli stack]]. This is obtained by forming the [[internal hom]] in the slice over $\mathbf{B}GL(n)$ of the [[locally cartesian closed (infinity,1)-category|locally cartesian closed (2,1)-category]] $\mathbf{H}$.

$$
  O \mathbf{Struc}_{T X}(X)
  =
  [X, \mathbf{B}O]_{\mathbf{B}GL(n)}
$$

#### Differential refinement: Spin connection

We may further lift this discussion to [[differential cohomology]] to get genuine _differential_ $T X$-twisted $\mathbf{c}$-structures.

Write $\mathbf{B}G_{conn}$ for [[groupoid of Lie-algebra valued forms]]. As an object of $\mathbf{H} = $ [[smooth infinity-groupoid|SmoothGrpd]] this the [[moduli stack]] of $G$-[[connection on a bundle|connections]].

The morphism $\mathbf{c}$ has an evident differential refinement

$$
  \mathbf{c}_{conn} : \mathbf{B}O(n)_{conn} \to \mathbf{B}GL(n)_{conn}
  \,.
$$

The [[homotopy fiber]] of this differential refinement is still the same as before

$$
  \array{
    GL(n)/ O(n) &\to& \mathbf{B} O(n)_{conn}
    \\
    && \downarrow
    \\
    && \mathbf{B} GL(n)_{conn}
  }
  \,,
$$

so that the moduli space of "differential vielbein fields" is the same as that of plain vielbein fields.

Consider an [[affine connection]]

$$
  \nabla_{T X} : X \to \mathbf{B}GL(n)
$$

hence a $GL(n)$-[[principal connection]] which locally on out atas is given by the [[Christoffel symbols]]

$$
  \Gamma_i = ((\Gamma_i)_\mu{}{}^{\alpha}{}_\beta)
  \in
  \Omega^1(U_i, \mathfrak{gl}(n))
  \,.
$$

A lift $(\nabla_V, E)$ in

$$
  \array{
    X &&\stackrel{\nabla_{V}}{\to}&& \mathbf{B}O_{conn}
    \\
    & {}_{\mathllap{\nabla_{T X}}}\searrow
     &\swArrow_{E^{-1}}&
    \swarrow_{\mathbf{c}_{conn}}
    \\
    && \mathbf{B}GL(n)_{conn}
  }
$$

is in components a "[[spin connection]]"

$$
  \omega_\mu = E d E^{-1} + E \Gamma_\mu E^{-1}
$$

$$
  \omega_\mu{}^a{}_b
   =
   E^a{}_\nu \partial_\mu E^\nu{}_b
   +
   E^a{}_\nu \Gamma_\mu{}^\nu{}_\lambda E^\lambda{}_b
   \,.
$$

This is the standard formula for the relation between the [[Christoffel symbols]] and the [[spin connection]] in terms of the vielbein.

#### Generalized and exceptional vielbein fields

The above discussion seamlessly generalizes to many other related cases. For instance

1. For the coefficient bundle

   $$
     \array{
        O(n)\backslash O(n,n) /O(n) &\to& \mathbf{B} (O(n) \times O(n))
      \\
      && \downarrow
      \\
      && \mathbf{B}O(n,n)
     }
   $$

   one gets the [[generalized vielbein]] of [[type II geometry]];

1. for the coefficient bundle

   $$
     \array{
        H_n\backslash E_n &\to& \mathbf{B} H_n
      \\
      && \downarrow
      \\
      && \mathbf{B} E_n
     }
   $$

   coming from the inclusion of the [[maximal compact subgroup]] of an [[exceptional Lie group]] one gets generalized vielbein fields for [[exceptional generalized geometry]];

1. for the coefficient bundle

   $$
     \array{
       && \mathbf{B} E_8
      \\
      && \downarrow^{\mathbf{a}}
      \\
      && \mathbf{B}^3 U(1)
     }
   $$

   coming from the second smooth universal [[Chern class]] of [[E8]] one gets part of the geometry of the [[supergravity C-field]]

and so on. More examples are discussed for instance at
_[[twisted smooth cohomology in string theory]]_.

### In terms of basic forms on the frame bundle
 {#InTermsOfBasicFormsOnFrameBundle}

A [[G-structure]] on $X$ for $G = O(n)$ the [[orthogonal group]] is equivalently an $O(n)$-[[principal bundle|principal]] subbundle of the [[frame bundle]] $\pi \colon Fr(X)\to X$.

This frame bundle carries a universal "basic" $\mathbb{R}^n$-valued differential form

$$
  \tau_{b} \in \Omega^1(Fr(X), \mathbb{R}^n)
$$

defined on a [[tangent vector]] $v\in \Gamma_{f \in Fr(X)}$ by

$$
  \tau_b(v) \coloneqq f^{-1}(d \pi(v))
  \,,
$$

where $d\pi \colon T Fr(X)\to T X$ is the [[differential]] of the bundle projection $\pi$ and $f$ is the given frame regarded as a linear [[isomorphism]] $f\colon \mathbb{R}^n \stackrel{\simeq}{\longrightarrow} T_x X$.

Then given an orthogonal structure in the form of an $O(n)$-subbundle $i \colon Fr_O(X) \hookrightarrow Fr(X)$ and given finally a local section $\sigma$ of $Fr_O(X)$, then the vielbein field with respect to that local trivialization is the [[pullback of differential forms|pullback]] form

$$
 \tau = \sigma^\ast i^\ast \tau_b
  \,.
$$

(exposition of this in the wider context of [[integrability of G-structures]] includes [Lott 90, p. 4](#Lott90)).

## Related concepts

* [[reduction of structure groups]], [[orthogonal structure]]

* [integrability of G-structures -- Examples -- Orthogonal structure](integrability+of+G-structures#ExamplesOrthogonalStructure)

* [[super-vielbein]]

* [[generalized vielbein]], [[exceptional generalized geometry]]

See also at [[field (physics)]] the section on _[Ordinary gravity](field%20%28physics%29#OrdinaryGravity)_.

## References

Discussion in the general context of [[G-structures]] includes

* {#Lott90} [[John Lott]], _The Geometry of Supergravity Torsion Constraints_, Comm. Math. Phys. 133 (1990), 563–615, (exposition in [arXiv:0108125](http://arxiv.org/abs/math/0108125))

[[!redirects vielbeine]]
[[!redirects vielbeins]]

[[!redirects vierbein]]
[[!redirects vierbein field]]

[[!redirects soldering form]]
[[!redirects soldering forms]]

[[!redirects vielbein field]]
[[!redirects vielbein fields]]

[[!redirects solder form]]
[[!redirects solder forms]]

[[!redirects soldering form]]
[[!redirects soldering forms]]


Vienna Circle

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

The **Vienna Circle** (in German, *Der Wiener Kreis*) were a group of philosophers, mathematicians, logicians and scientists who were active in Vienna in the 1920s and 1930s where they promoted a positivist view of the world and the use of formal languages to resolve or dissolve philosophical disputes, hence the name of their position _Logical Positivism_. Prominent members included [[Rudolf Carnap]], Moritz Schlick, Hans Hahn, Otto Neurath, Olga Hahn-Neurath, Friedrich Waismann, Philipp Frank, and Herbert Feigl.

In 1929, Hahn, Neurath and Carnap published the Circle's Manifesto, [The Scientific Conception of the World: The Vienna Circle](https://www.manchesterism.com/the-scientific-conception-of-the-world-the-vienna-circle/):

>The scientific world conception is characterised not so much by theses of its own, but rather by its basic attitude, its points of view and direction of research. The goal ahead is _unified science_. The endeavour is to link and harmonise the achievements of individual investigators in their various fields of science. From this aim follows the emphasis on _collective efforts_, and also the emphasis on what can be grasped intersubjectively; from this springs the search for a neutral system of formulae, for a symbolism freed from the slag of historical languages; and also the search for a total system of concepts. Neatness and clarity are striven for, and dark distances and unfathomable depths rejected. In science there are no 'depths'; there is surface everywhere: all experience forms a complex network, which cannot always be surveyed and, can often be grasped only in parts. Everything is accessible to man; and man is the measure of all things. ...The scientific world-conception knows _no unsolvable riddle_. Clarification of the traditional philosophical problems leads us partly to unmask them as pseudo-problems, and partly to transform them into empirical problems and thereby subject them to the judgment of experimental science. The task of philosophical work lies in this clarification of problems and assertions, not in the propounding of special 'philosophical' pronouncements. The method of this clarification is that of _logical analysis_; of it, Russell says (*Our Knowledge of the External World*, p. 4) that it "has gradually crept into philosophy through the critical scrutiny of mathematics… It represents, I believe, the same kind of advance as was introduced into physics by Galileo: the substitution of piecemeal, detailed and verifiable results for large untested generalities recommended only by a certain appeal to imagination."

## References

* SEP, _[Vienna Circle](https://plato.stanford.edu/entries/vienna-circle/)_

* Wikipedia, _[Vienna Circle](http://en.wikipedia.org/wiki/Vienna_Circle)_


Vietoris complex

#Contents#
* table of contents
{:toc}

##Idea##

In the 1920s homology and cohomology were known for simplicial complexes and there were attempts to extend the definitions to first of all compact metric spaces and then more general spaces.  [[Leopold Vietoris]] (1927) came up with a construction and then shortly after Alexandrov and Čech gave a different one involving [[Čech nerve|the nerve that now bears Čech's name]]. The input of the  two methods is the same, we have a space $X$ and an open cover $\mathcal{U}$ of $X$.

It was noted that all the calculations of Vietoris homology gave the same answer as  Čech homology.  In 1952, C. H. Dowker showed why.  His [[Dowker's theorem|result]] is a beautiful mix of  abstraction and concrete explicit calculation (and is not that well known unfortunately).

First some abstraction:

*  We take the set $X$ and the open cover $\mathcal{U}$, together with the relation $x\in U$, i.e. an element $x$ will be considered to be _related to_ the set $U$ in the cover $\mathcal{U}$, if it is an element of it!

*  We replace this by an abstract setting of a set $X$, another set $Y$ and a relation $R\subseteq X\times Y$, from $X$ to $Y$.

+--{: .query}
Thinks: does what follows have a good generalisation to [[spans]]?
=--

##Definition:##
The **Vietoris complex** of a relation $R$, as above,  is the [[simplicial complex]] specified by

* Vertex set: the set of vertices is the set $X$;

* Simplices : a $p$-simplex of $K$ is a set $\{x_0, \ldots, x_p\}  \subseteq X$ such that there is some $y \in Y$ with $x_i R y$ for $i  = 0, 1, \ldots, p$

The Vietoris complex of $R$ will be denoted $V(R)$.

The special, and original, case where $R$ comes from an open cover $\mathcal{U}$ of $X$ will be denoted $V(X,\mathcal{U})$ or simply $V(\mathcal{U})$ and will be called the _Vietoris complex of the open covering $\mathcal{U}$_.

##Remarks##
In fact each relation will give **two** simplicial complexes, one as above, the other obtained by reversing the roles of $X$ and $Y$ in the above. In other words, in the usual way define the _[[opposite relation]]_ by ' $R^{op}\subseteq Y\times X$ is the relation in which $yR^{op}x$ if and only if $xRy$'.

We define $C(R) = V(R^{op})$ and call it the **Čech nerve** or **Čech complex** of the relation $R$.

If we look at this in the case of the relation coming from a space $X$ and an open covers $\mathcal{U}$ you get exactly the nerve of $\mathcal{U}$ as discussed in [[Čech methods]].

There are natural questions that arise here. One is how to turn these [[simplicial complexes]] into [[simplicial sets]], to which _one_
 reply is take a total order (or at least pick a partial order that any simplex is ordered in that partial order; another reply would be to take each $n$-simplex $(n+1)!$-times, one for each possible order.

Another obvious question is to compare $V(R)$ and $C(R)$: are they closely related?  That is discussed more in [[Dowker's Theorem]].  It looks very much a situation for a combinatorial duality result.

A final question (and one I can answer!) is: are there useful applications of this construction other than in (co)homology.  The answer is most decidedly 'yes'.

##Volodin spaces

One of the ways of constructing the [[algebraic K-theory]] of a ring $R$ is to construct a simplicial set / complex from the stable [[general linear group]] and the family of cosets of subgroups of upper triangular matrices. (This method is due to _[[I.A. Volodin]].)

The method can be applied to other groups with given families of subgroups as is described at [[Volodin spaces]].  It uses the cosets of the subgroups in the family to get a covering of the set of elements and then applies the Vietoris construction (but without mention of that source).

##Higher generation by subgroups

It is notable that the same situation  but with corresponding Čech complex is used by Abels and Holz, (1993) for the calculation of syzygies and is applied to various settings with linear groups.  They give induction methods for homological finiteness criteria for the groups.  This is discussed in more detail in [[higher generation by subgroups]].

##'Hidden Stasheff polytopes ... '

Kapranov and Saito have studied the syzygies of the usual presentation of the Steinberg group. These are expected to give a combinatorial construction of the classifying space for the algebraic K-theory of rings. They use Volodin's construction but their combinatorial construction would seem to be related to that of Abels and Holz. They conjecture various results in this. but there is still some mystery about what results have been proved, beyond those published in their paper.

##Gavin Wraith's puzzle##

Gavin Wraith posted a puzzle on the $n$Café whose solution uses the fact that the simplicial complexes $V(R)$ and $C(R)$ are homotopy equivalent:

* [A Puzzle From Gavin Wraith](http://golem.ph.utexas.edu/category/2009/07/a_puzzle_from_gavin_wraith.html)

Perhaps this fact is [[Dowker's Theorem]]?

## Related concepts

* [[persistent homotopy]]

## References

Discussion in [[persistent homotopy theory]]:

* [[J. F. Jardine]], *Data and homotopy types* $[$[arXiv:1908.06323](https://arxiv.org/abs/1908.06323)$]$

See also:

* Wikipedia, *[Vietoris-Rips complex](https://en.wikipedia.org/wiki/Vietoris–Rips_complex)*

* [[I. Volodin]], _Algebraic K-theory as extraordinary homology theory on the category of associative  rings with unity_, Izv. Akad. Nauk. SSSR, 35, (Translation: Math. USSR Izvestija Vol. 5 (1971)
No. 4, 859-887).

* H. Abels and S. Holz, _Higher generation by subgroups_, J. Alg, 160, (1993), 311– 341

*  M. Kapranov and M. Saito, 1999, Hidden Stasheff polytopes in algebraic K-theory and in the  space of Morse functions , in Higher homotopy structure in topology and mathematical physics
(Poughkeepsie, N.Y. 1996) , volume 227 of Contemporary Mathematics , 191–225, AMS.

[[!redirects Vietoris complexes]]

[[!redirects Vietoris-Rips complex]]
[[!redirects Vietoris-Rips complexes]]


Vigleik Angeltveit

* [webpage](http://maths-people.anu.edu.au/~angeltveit/)

## Selected writings

On [[enriched model category|enriched]] [[Reedy model categories]]:

* {#Angeltveit} [[Vigleik Angeltveit]], *Enriched Reedy categories*, Proceedings of the American Mathematical Society **136** 7 (2008)  [[arXiv:math/0612137](http://arxiv.org/abs/math/0612137), [AMS:S0002-9939-08-09185-5](https://www.ams.org/journals/proc/2008-136-07/S0002-9939-08-09185-5/home.html), [jstor:20535419](https://www.jstor.org/stable/20535419)]

category: people


Vijay Balasubramanian

* [webpage](https://www.physics.upenn.edu/people/standing-faculty/vijay-balasubramanian)

## related $n$Lab entries

* [[moduli stabilization]]

category: people


Vijay Patodi

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

## Related $n$Lab entries

* [[index theory]]

* [[e-invariant]]

category: people


Viktor Vassiliev

__Viktor A. Vassiliev__, a Russian geometer, a pupil of [[Vladimir Arnold]]. He is also active in the questions of education and the well-being of the academic community in contemporary Russia.

* [webpage](http://www.hse.ru/en/org/persons/1297545) at Higher School of Economics, Moscow

## Selected writings

Introducing [[Vassiliev knot invariants]]:

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

Further:

* _Ramified integrals, singularities and lacunas_, Kluwer 1994, 289+xvii pp. Russian version: Ветвящиеся интегралы, Moscow 2002, 432 pp.
* _Applied Picard-Lefschetz theory_,  Mathematical Surveys and Monographs __97__. Amer. Math. Soc.  2002. xii+324 pp. [MR2003k:32043](http://www.ams.org/mathscinet-getitem?mr=1930577)
* _Stratified Picard-Lefschetz theory_,
Selecta Math. (N.S.) 1 (1995), no. 3, 597–621, [MR96i:32037](http://www.ams.org/mathscinet-getitem?mr=1366625), [doi](http://dx.doi.org/10.1007/BF01589499)
* _Complements of discriminants of smooth maps: topology and applications_, Amer. Math. Soc. 1992.
* _Complexes of connected graphs_, [arxiv/1409.5999](http://arxiv.org/abs/1409.5999)

On [[cocycle spaces]] in ([[equivariant cohomotopy|equivariant]]) [[Cohomotopy]]:

* [[Victor Vassiliev]], _Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials_ ([arXiv:1809.05632](https://arxiv.org/abs/1809.05632))

* [[Victor Vassiliev]], *Cohomology of spaces of Hopf equivariant maps of spheres* ([arXiv:2102.07157](https://arxiv.org/abs/2102.07157))

## Related entries

* [[Vassiliev knot invariant]]

* [[Picard-Lefschetz theory]]

* [[singularity theory]]

* [[Kontsevich integral]]

category: people

[[!redirects V. A. Vassiliev]]
[[!redirects Viktor A. Vassiliev]]

[[!redirects Victor Vassiliev]]


Viktoriya Ozornova

* [website](https://www.ruhr-uni-bochum.de/ffm/Lehrstuehle/Lehrstuhl-XIII/ozornova.html)

## Selected writings

A [[model category]] structure on [[stratified simplicial sets]] modelling [[(infinity,n)-categories|$(\infty,n)$-categories]] in the guise of [[n-complicial sets|$n$-complicial sets]]:

* [[Viktoriya Ozornova]], [[Martina Rovelli]], *Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces*, Algebr. Geom. Topol. **20** (2020) 1543-1600 [[arxiv:1809.10621](https://arxiv.org/abs/1809.10621), [doi:10.2140/agt.2020.20.1543](https://doi.org/10.2140/agt.2020.20.1543)]

On classes of [[categories]] which are [[fibrant object|fibrant]] in the [[Thomason model structure]]:

* [[Lennart Meier]], [[Viktoriya Ozornova]], _Fibrancy of partial model categories_,  Homology, Homotopy and Applications, Volume 17 (2015) Number 2
([arXiv:1408.2743](https://arxiv.org/abs/1408.2743), [doi:10.4310/HHA.2015.v17.n2.a5](http://dx.doi.org/10.4310/HHA.2015.v17.n2.a5))

On the [[adjunction unit]] of [[total décalage]]:

* [[Viktoriya Ozornova]], [[Martina Rovelli]], *The unit of the total décalage adjunction*, Journal of Homotopy and Related Structures **15** (2020) 333–349 [[doi.org/10.1007/s40062-020-00257-1](https://doi.org/10.1007/s40062-020-00257-1)]

A [[model category]] structure on [[stratified simplicial sets]] modelling [[(infinity,n)-categories|$(\infty,n)$-categories]] in the guise of [[n-complicial sets|$n$-complicial sets]]:

* [[Viktoriya Ozornova]], [[Martina Rovelli]], *Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces*, Algebr. Geom. Topol. **20** (2020) 1543-1600 [[arxiv:1809.10621](https://arxiv.org/abs/1809.10621), [doi:10.2140/agt.2020.20.1543](https://doi.org/10.2140/agt.2020.20.1543)]

A [[Quillen adjunction]] relating [[n-complicial sets|$n$-complicial sets]] to [[n-fold complete Segal spaces|$n$-fold complete Segal spaces]]:

* [[Viktoriya Ozornova]], [[Martina Rovelli]], *A Quillen adjunction between globular and complicial approaches to $(\infty,n)$-categories*, Advances in Mathematics **421** (2023) 108980 [[doi:10.1016/j.aim.2023.108980](https://doi.org/10.1016/j.aim.2023.108980)]

category:people


Ville Lahtinen

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

## Selected writings

On [[topological quantum computation]]:

* {#LahtinenPachos17} [[Ville Lahtinen]], [[Jiannis K. Pachos]], _A Short Introduction to Topological Quantum Computation_,  SciPost Phys. **3** 021 (2017) [[arXiv:1705.04103](https://arxiv.org/abs/1705.04103), [doi:10.21468/SciPostPhys.3.3.021](https://scipost.org/SciPostPhys.3.3.021)]

category: people


Vin de Silva

* [institute page](https://www.pomona.edu/directory/people/vin-de-silva)

## Selected writings

Introducing [[zigzag persistence]], connecting [[persistent homology]] to [[quiver representation]]-theory, re-proving [[Gabriel's theorem]] for the case of A-type [[quivers]] and introducing the [[diamond principle]]:

* {#CarlssonDeSilva10} [[Gunnar Carlsson]], [[Vin de Silva]], *Zigzag Persistence*, Found Comput Math **10** (2010) 367–405 $[$[arXiv:0812.0197](https://arxiv.org/abs/0812.0197), [doi:10.1007/s10208-010-9066-0](https://doi.org/10.1007/s10208-010-9066-0)$]$

and applied to [[level sets]]:

* [[Gunnar Carlsson]], [[Vin de Silva]], [[Dmitriy Morozov]], *Zigzag persistent homology and real-valued functions*, in: *SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry (2009) 247–256 $[$[doi:10.1145/1542362.1542408](https://doi.org/10.1145/1542362.1542408)$]$

* [[Gunnar Carlsson]], [[Vin de Silva]], [[Sara Kališnik]], [[Dmitriy Morozov]], *Parametrized Homology via Zigzag Persistence*, Algebr. Geom. Topol. **19** (2019) 657-700 $[$[arXiv:1604.03596](https://arxiv.org/abs/1604.03596), [doi:10.2140/agt.2019.19.657](https://doi.org/10.2140/agt.2019.19.657)$]$

On [[persistence modules]] and [[stability of persistence diagrams]]:

* [[Frédéric Chazal]], [[Vin de Silva]], [[Marc Glisse]], [[Steve Oudot]],  *The structure and stability of persistence modules*, SpringerBriefs in Mathematics, Springer (2016) $[$[arXiv:1207.3674](https://arxiv.org/abs/1207.3674), [doi:10.1007/978-3-319-42545-0](https://doi.org/10.1007/978-3-319-42545-0)$]$

category: people


Vincent Bouchard

* [webpage](http://www.perimeterinstitute.ca/people/vincent-bouchard)

## related $n$Lab entries

* [[topological recursion]]

* [[topological string]]

category: people


Vincent Braunack-Mayer

[[!redirects Vincent Schlegel]]

> (formerly V. Schlegel)

* [webpage](http://www.math.uzh.ch/index.php?assistenten&key1=9218)

## Selected writings

On dg-algebraic models for [[rational parameterized stable homotopy theory]]:

* [[Vincent Braunack-Mayer]]

  _**[[schreiber:thesis Braunack-Mayer|Rational parametrised stable homotopy theory]]**_

  Zurich 2018

  [pdf](https://ncatlab.org/schreiber/files/VBM_RPSHT.pdf)

* [[Vincent Braunack-Mayer]],

  **Strict algebraic models for rational parametrised spectra I**

  Algebraic & Geometric Topology 21 (2021) 917–1019

  [arXiv:1910.14608](https://arxiv.org/abs/1910.14608)

  [doi:10.2140/agt.2021.21.917](https://doi.org/10.2140/agt.2021.21.917)

On [[combinatorial model category|combinatorial]] [[model structures for parameterized spectra]]:

* [[Vincent Braunack-Mayer]],

  **Combinatorial parametrised spectra**

  Algebr. Geom. Topol. **21** (2021) 801-891

  [[arXiv:1907.08496](https://arxiv.org/abs/1907.08496), [doi:10.2140/agt.2021.21.801](https://doi.org/10.2140/agt.2021.21.801)]

On [[rational parameterized stable homotopy theory]] applied to the mathematical analysis of the [[duality between M-theory and type IIA string theory]]:

* [[nLab:Vincent Braunack-Mayer]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]]

  _**[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of Super M-Branes rational parameterized stable homotopy theory]]**_

  Communications in Mathematical Physics (2019)

  [doi:10.1007/s00220-019-03441-4](https://doi.org/10.1007/s00220-019-03441-4)

  [arXiv:1806.01115](https://arxiv.org/abs/1806.01115)

* [[nlab:Vincent Braunack-Mayer]]

  _**[[schreiber:Parametrised homotopy theory and gauge enhancement]]**_

  talk at

  _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_

  Durham Symposium 2018

  Fortschritte der Physik (2019)

  [doi:10.1002/prop.201910003](https://onlinelibrary.wiley.com/doi/abs/10.1002/prop.201910003)

  [arXiv:1903.02862](https://arxiv.org/abs/1903.02862)

## Related $n$Lab entries

* [[tangent (∞,1)-topos]], [[parametrized homotopy theory]]

* [[rational parameterized stable homotopy theory]]

* [[twisted differential cohomology]]

category: people


Vincent Danos

* [GoogleScholar page](https://scholar.google.com/citations?user=Q0aaJVUAAAAJ&hl=en)

## Selected writings

On [[measurement-based quantum computation]]:

* [[Vincent Danos]], [[Elham Kashefi]], [[Prakash Panangaden]], *The Measurement Calculus*, Journal of the ACM, **54** 2 (2007) [[arXiv:0704.1263](https://arxiv.org/abs/0704.1263), [doi:10.1145/1219092.1219096](https://doi.org/10.1145/1219092.1219096)]

category: people


Vincent Lafforgue

* [webpage](http://www.math.jussieu.fr/~vlafforg/)

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

category: people


Vincent Laurence Rouleau

* [webpage](http://137.122.43.51/cgi-bin/mat/people/create_list?lastname=Laurence%20Rouleau&firstnames=Vincent&language=Francais)

## related entries

* [[transcendental syntax]]

## writings

* _Towards an understanding of Girard's transcendental syntax: Syntax by testing_, PhdD thesis 2013 ([pdf](https://www.ruor.uottawa.ca/fr/bitstream/handle/10393/23680/Laurence_Rouleau_Vincent_2013_thesis.pdf?sequence=3))

category: people


Vincent Mourik

* [institute page](https://www.fqt.unsw.edu.au/staff/vincent-mourik-0)

* [GoogleScholar page](https://scholar.google.com/citations?user=VbOBK0YAAAAJ&hl=en)

## Selected writings

On (non-)detection of [[Majorana zero modes]] in [[experiment]]:

* {#FrolovMourik22} [[Sergey M. Frolov]], [[Vincent Mourik]], *We cannot believe we overlooked these Majorana discoveries* [[arXiv:2203.17060](https://arxiv.org/abs/2203.17060), [doi:10.5281/zenodo.6364928](https://doi.org/10.5281/zenodo.6364928), conclusion on: [p. 7](https://arxiv.org/ftp/arxiv/papers/2203/2203.17060.pdf#page=7)]

* {#Frolov22} [[Sergey M. Frolov]], [[Vincent Mourik]], *Majorana Fireside Podcast, Episode 1: The Microsoft TGP paper live review* [[video](https://www.youtube.com/watch?v=RnYghkDaHH0), conclusion at: [1:01:31](https://www.youtube.com/watch?v=RnYghkDaHH0&t=3691s)]

category: people


Vincent Pasquier

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

## Selected writings

On [[Drinfeld doubles]] of [[quasi-Hopf algebras]], motivated from [[rational conformal field theory|rational]] [[orbifold]] [[conformal field theories]]:

* [[Robbert Dijkgraaf]], [[Vincent Pasquier]], [[Philippe Roche]], *QuasiHopf algebras, group cohomology and orbifold models*, Nucl. Phys. B Proc. Suppl **18** (1990) 60-72 [doi:10.1016/0920-5632(91)90123-V]

* [[Robbert Dijkgraaf]], [[Vincent Pasquier]], [[Philippe Roche]], _Quasi-quantum groups related to orbifold models_, *International Colloquium on Modern Quantum Field Theory* (Bombay, 1990), 375-383, World Sci. (1991) [[cds:206306](https://cds.cern.ch/record/206306?ln=ka), [[DPR-QuasiQuantumGroups.pdf:file]]]

category: people


Vincent Vargas

* [personal page](https://www.math.ens.fr/~vargas/)

## Selected writings

On the rigorous construction of [[Liouville theory]]:

* [[François David]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Liouville Quantum Gravity on the Riemann sphere_, Communications in Mathematical Physics volume 342, pages869–907 (2016) ([arxiv:1410.7318](https://arxiv.org/abs/1410.7318))

* [[Colin Guillarmou]], [[Rémi Rhodes]], [[Vincent Vargas]], *Polyakov's formulation of 2d bosonic string theory*, Publ. Math. IHES **130** (2019) 111–185 [[arXiv:1607.08467](https://arxiv.org/abs/1607.08467), [doi:10.1007/s10240-019-00109-6](https://doi.org/10.1007/s10240-019-00109-6)]

* [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Integrability of Liouville theory: proof of the DOZZ Formula_, Annals of Mathematics, Pages 81-166 from Volume 191 (2020), Issue 1, ([arxiv:1707.08785](https://arxiv.org/abs/1707.08785), [doi:10.4007/annals.2020.191.1.2](https://doi.org/10.4007/annals.2020.191.1.2))

* [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *The DOZZ formula from the path integral*, Journal of High Energy Physics volume 2018, Article number: 94 (2018) ([arXiv:1803.05418](https://arxiv.org/abs/1803.05418) doi:10.1007/JHEP05(2018)094)

review:

* [[Vincent Vargas]], *Lecture notes on Liouville theory and the DOZZ formula* ([arXiv:1712.00829](https://arxiv.org/abs/1712.00829))

and via the [[conformal bootstrap]]:

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Conformal bootstrap in Liouville Theory_ ([arxiv:2005.11530](https://arxiv.org/abs/2005.11530))

and as a [[functorial field theory]] following [Segal 1988](conformal+field+theory#Segal88):

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *Segal's axioms and bootstrap for Liouville Theory* [[arXiv:2112.14859](https://arxiv.org/abs/2112.14859)]

category: people


vine

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##Idea

[Lavers (1997)](#Lavers97) introduced the [[monoid]] $V_n$ of *$n$-vines*,  whose elements are thought of as paths between two sets of $n$ distinct points in $\mathbb{R}^3$ which are allowed to merge into a single path, but not separate again. Some examples, from Lavers' original paper, are depicted below.

\begin{imagefromfile}
        "file_name": "vinesJPEG.jpg",
        "width": 300
    \end{imagefromfile}

Notice that $V_n$ is not a group because a general vine cannot be untangled to give the trivial vine which is $n$ paths straight down. However, via pre- and post-composition, the vine monoid is acted on by the [[braid group|braid group on $n$ strands]]. When two $n$-vines are composed any resulting strands which are only attached to one endpoint get retracted down to that endpoint, as in the following image (again from [Lavers 97](#Lavers97)):

\begin{imagefromfile}
        "file_name": "vinescompose.jpg",
        "width": 300
    \end{imagefromfile}

More generally, one can consider vines from $n$ points to $m$ points. As a result vines can be assembled into a category, as described below.

## Definition

What is called the *category of vines* is the [[free construction|free]] [[braided monoidal category|braided]] [[strict monoidal category]] containing a [[braided monoid]] (i.e. an [[E2-algebra|$E_2$-algebra]]). It is also the [[PRO|PROB]] for braided monoids.

Concretely:

\begin{defn}
Let

* $P_I \coloneqq \big\{ (i,0,1) \,\big\vert\, i=1, 2, \ldots, m \big\}$

* $P_T \coloneqq \big\{ (i,0,0) \,\big\vert\, i=1, 2, \ldots,n \big\}$

be [[linear order|linearly ordered]] collections of points in [[Cartesian space|$\mathbb{R}^3$]].

Then an *$(m,n)$-vine* is a set of arcs $\{v_1,\ldots,v_m\}$ (i.e. piecewise linear maps from $[0,1]$ to $\mathbb{R}^3$) with the following properties:

1. $v_i(0)=(i,0,1)$,

1. $v_i(1)\in P_T$,

1. for all $h\in [0,1]$, $v_i(h)$ has $z$-coordinate $1-h$,

1. if $v_i(t)=v_j(t)$ for any $t\in (0,1]$ then $v_i(s)=v_j(s)$ for all $t\leq s\leq 1$.

\end{defn}

\begin{defn}\label{CategoryV}
Let $\mathbb{V}$ be the category with set of [[objects]] the [[natural numbers]] with set of [[morphisms]] $\mathbb{V}(m,n)$ being $(m,n)$-vines modulo a suitable notion of ambient isotopy that allows arcs to be deformed up to homotopy but not pass through one another.
\end{defn}

##Properties

The [[monoidal category|monoidal structure]] of $\mathbb{V}$ is given by addition of [[natural numbers]] and juxtaposition of vines. Because $\mathbb{V}$ contains the [[braid category]] as a [[subcategory]], it cannot be [[symmetric monoidal]], but it is [[braided monoidal category|braided monoidal]]. The braiding is the same as that of the braid category, the "$m$-over-$n$" braid depicted below.

\begin{imagefromfile}
        "file_name": "braiding.jpg",
        "width": 300
    \end{imagefromfile}

\begin{exercise}
Show that the above is a braided monoidal structure on $\mathbb{V}$.
\end{exercise}

## References

* {#Lavers97} T. G. Lavers, *The theory of vines*, Comm. Algebra **25** 4 (1997) 1257–1284 [[doi:10.1080/00927879708825919](https://doi.org/10.1080/00927879708825919)]

* {#Weber2015}[[Mark Weber]], §6.3 of: _Internal algebra classifiers as codescent objects of crossed internal categories_. Theory and Applications of Categories, **30** 50 (2015) 1713–1792 [[tac:30/50](http://www.tac.mta.ca/tac/volumes/30/50/30-50abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/30/50/30-50.pdf)]

[[!redirects vines]]

[[!redirects vine monoid]]
[[!redirects vine monoids]]

[[!redirects vine category]]
[[!redirects vine categories]]


Vipul Naik

* [webpage](https://vipulnaik.com/)

## Selected writings

On [[separation axioms]]:

* {#Naik} [[Vipul Naik]], _Topology: The journey into separation axioms_  [[[Naik-SeparationAxioms.pdf:file]]]

category: people


Virasoro algebra

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## Idea

The __Virasoro algebra__ or _Virasoro [[Lie algebra]]_ is the nontrivial [[central extension]] of the [[Witt Lie algebra]] (the Lie algebra of the [[diffeomorphism group|group of diffeomorphisms]] of the [[circle]]). It is of central importance in some questions of [[complex analysis]], in [[conformal field theory]] and the study of [[affine Lie algebras]].

## Definition

The generators $L_n$ of the Virasoro algebra are indexed by an integer $n \in \mathbb{Z}$, and they satisfy the commutation relation

$$
[L_m, L_n] = (m - n) L_{m+n} + \frac{c}{12}(m^3 - m) \delta_{m+n,0}.
$$

Here, $c$ denotes the element of the algebra known as the **[[central charge]]**; it commutes with each generator,

$$
[L_n, c] = 0 \forall n.
$$

The factor of 1/12 is conventional and chosen for normalisation purposes.

## Related concepts

* [[W-algebra]]

[[!redirects Virasoro Lie algebra]]


Virasoro-Shapiro amplitude

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#Contents#
* table of contents
{:toc}

## Idea

The _Virasoro-Shapiro amplitude_ is the [[string scattering amplitude]] [[n-point function|4-point function]] for four [[tachyon]] field insertions.

## Related entries

* [[Veneziano amplitude]]

* [[vacuum amplitude]]

## References

E.g. (16.47) in

* [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], _String Scattering Amplitudes and Low Energy Effective Field Theory_, chapter 16 in _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013

[[!redirects Virasoro-Shapiro amplitudes]]

[[!redirects Virasoro amplitude]]
[[!redirects Virasoro amplitudes]]


virtual cohomological dimension

#Contents#
* table of contents
{:toc}

## Idea

In [[group cohomology]] theory, if $G$ is  a  [[group]]  which  has  [[torsion-free module|torsion-free]]  subgroups  of  [[finite index subgroup|finite index]], then all such subgroups have the same [[cohomological dimension]];  this common dimension  is  called  the  **virtual cohomological dimension** of $G$ and denoted $vcd(G)$.

For example, $SL_n(\mathbb{Z})$ has infinite cohomological dimension, and yet

$$
vcd(SL_n(\mathbb{Z})) = \binom{n}{2}.
$$

## Related concepts

* [[Farrell-Tate cohomology]]

[[!include notions of dimension -- contents]]


virtual double category

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# Contents
* tic
{: toc}

## Idea

A **virtual double category** or **$fc$-multicategory** is a common generalization of a [[monoidal category]], a [[bicategory]], a [[double category]], and a [[multicategory]].  It contains:

* objects
* vertical arrows, which form a category
* horizontal arrows, which do not have identities or composites, and
* 2-cells which have
  * a horizontal source and target, which are vertical arrows,
  * a vertical target, which is a horizontal arrow, and
  * a vertical source, which is a composable string of horizontal arrows.

2-cells are usually drawn like this:

[[virtual-double-category-cell.png:pic]]

Note that this includes the case when $n=0$, i.e. a cell of "nullary" source.  In this case, we must have $X_0 = X_n$.

Note that the "empty" case of a string of horizontal arrows this has a single object $X$ in which case the 2-cell looks like:

\begin{tikzcd}% https://q.uiver.app/?q=WzAsNCxbMSwwLCJYIl0sWzAsMiwiWV8wIl0sWzIsMiwiWV8xIl0sWzEsMSwiXFxEb3duYXJyb3cgXFxhbHBoYSJdLFswLDEsImYiLDJdLFswLDIsImciXSxbMSwyLCJxIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiYmFycmVkIn19fV1d
	& X \\
	& {\Downarrow \alpha} \\
	{Y_0} && {Y_1}
	\arrow["f"', from=1-2, to=3-1]
	\arrow["g", from=1-2, to=3-3]
	\arrow["q"', "\shortmid"{marking}, from=3-1, to=3-3]
\end{tikzcd}

Finally, the 2-cells can be composed in a more or less evident way, akin to composition in a multicategory:

[[virtual-double-category-composite.png:pic]]

Virtual double categories are related to double categories precisely as ordinary multicategories are related to monoidal categories (see [[generalized multicategory]] and [[tensor product]]).

## Definition

A virtual double category can be defined in two equivalent ways:

* It is a $T$-[[generalized multicategory|multicategory]], in the sense of Leinster, relative to the monad $T$ on [[quiver|directed graphs]] whose algebras are categories.  For this reason, Leinster originally called them **fc-multicategories**, where "fc" is a name for this monad $T$ which stands for "free-category."

* It is a generalized multicategory, in the sense of Hermida, Cruttwell-Shulman, and others, relative to the monad $T$ on graphs-internal-to-Cat whose algebras are double categories.  This is the origin of the name "virtual double category," in line with the general terminology "virtual $T$-algebra" of Cruttwell-Shulman for such generalized multicategories.

We can also give an explicit definition, which was more or less already given in the "Idea" section: all that is missing are identities and associativity for 2-cell composition.

## Examples

* Any double category is an example, and thus also any bicategory
viewing the arrows as horizontal.
* For any [[monoidal category]] $V$, there is a virtual double category
of $V$-matrices whose objects are sets, vertical arrows are functions
and a horizontal arrow $p : X \to Y$ is a family of objects $p_{y,x}
\in V$ for each $x\in X, y \in Y$, and a 2-cell from $X_0
\overset{p_1}{\to} X_1 \to\dots \to X_n$ to $Y_0 \overset{q}{\to} Y_1$
along $f : X_0 \to Y_0, g : X_n \to Y_n$ is a family of arrows
$\alpha_{x_0,...} : p_1(x_1,x_0)\otimes p_2(x_2,x_1)\otimes\dots \to
q(g(x_n),f(x_0))$ in $V$ (using the unit of the monoidal category if the source string is empty).
If $V$ has certain colimits that are preserved by $\otimes$ then
composites exist and this virtual double category is pseudo.
* Given a monad on a virtual double category, the [[generalized multicategory| horizontal kleisli
double category]] produces a virtual double
category that is only pseudo under strong conditions on the monad. In
particular, "free monoid" monad on the double category of sets and
spans does not produce a pseudo double category.

## Higher categories of virtual double categories

There are notions of **functor**, **transformation**, and **profunctor** between virtual double categories.  The neatest way to define all of these notions at once is to use the general framework of [[generalized multicategories]]: from the monad $fc$ on the [[virtual equipment]] $Span = Span(Set)$ we can construct a new virtual equipment $vDblProf = KMod(Span,fc)$ whose objects are virtual double categories, whose arrows are functors between them, whose proarrows are profunctors between them, and whose cells are transformations.  But we can also give explicit definitions of all of these notions.

### Functors and transformations

A **functor** of virtual double categories is fairly obvious; it takes each kind of morphism/cell to the same kind, preserving sources, targets, composition, and identities.

The relevant **transformations** are a "virtual" version of [[vertical transformations]] between ordinary double categories.  Specifically, a transformation $\alpha\colon F\to G$ has a vertical arrow component $\alpha_X\colon F X\to G X$ for each object $X$ of the domain, and a cell component
$$\array{F X & \overset{F p}{\to} & F Y\\
  ^{\alpha_X}\downarrow & \Downarrow ^{\alpha_p}& \downarrow^{\alpha_Y}\\
  G X& \underset{G p}{\to} & G Y}$$
for each horizontal arrow $p\colon X\to Y$ in the domain.  These must be natural with respect to vertical composition of arrows and of 2-cells, where we must of course allow composites with arbitrary arities in the latter case.

Virtual double categories, functors, and transformations form a [[strict 2-category]], and thus we can apply all notions of 2-category theory to it.  In particular, we have a notion of a [[monad]] *on* a virtual double category, which is the starting point for one theory of [[generalized multicategories]].

### Profunctors

The **profunctors** between virtual double categories are a similar "virtualization" of the notion of [[double profunctor]] between double categories.  Explicitly, a profunctor $H\colon C ⇸ D$ consists of:

* An ordinary [[profunctor]] $H_0\colon C_0 ⇸ D_0$ between the categories of objects and vertical arrows.
* For each string of horizontal arrows $X_0 \overset{p_1}{\to} X_1 \to\dots \to X_n$ in $D$, each horizontal arrow $Y_0 \overset{q}{\to} Y_1$ in $C$, and each pair of elements $f \in H_0 (X_0,Y_0)$ and $g\in H_0(X_n,Y_1)$, a set of "hetero-cells" of shape

  [[virtual-double-category-cell.png:pic]]

* The hetero-cells are acted on by the 2-cells of $D$ on the top, and by the 2-cells of $C$ on the bottom, in an evident way, respecting the given action of vertical arrows of $D$ and $C$ on the elements of $H_0$.

Every [[double profunctor]] induces such a profunctor in an evident way, but even if $C$ and $D$ are (non-virtual) double categories, not every "virtual double profunctor" from $C$ to $D$ need be a double functor; only those for which the "hetero-cells" also factor uniquely through the opcartesian cells in $D$ which make it "representable."

As mentioned above in the context of the abstract definition, virtual double categories, functors, transformations, and profunctors form another virtual double category, which is in fact a [[virtual equipment]].

### Monads on virtual double categories {#Monads}

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

A _monad on a virtual double category_  is a [[monad]] in the [[2-category]] [[vDbl]].

=--

So a monad on a $X \in vDbl$ consists of a functor

$$
  T : \mathbb{X} \to \mathbb{X}
$$

and transformations $\eta : Id  \to T$ and $\mu : T T \to T$ satisfying [[associativity]] and [[unitality]].

#### Monoids and modules

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

For $T$ a monad on $\mathbb{X} \in $ [[vDbl]], a **$T$-monoid** is

* an object $X_0 \in \mathbb{X}$;

* a horizontal morphism $X_0 \stackrel{X}{⇸} T X_0$

* an [[action]] [[2-morphism]]

  $$
    \array{
       X_0 &\stackrel{X}{⇸} & T X_0
        & \stackrel{T X}{⇸} T^2 &  X_0
       \\
       {}^{\mathllap{=}}\downarrow && \Downarrow^{\bar x}
       &&  \downarrow^{\mathrlap{\mu}}
       \\
       X_0 && \underset{X}{⇸}  && T X_0
    }
  $$

  and a [[unit]] 2-morphism

  $$
    \array{
       && X_0
       \\
       & {}^{\mathllap{=}}\nearrow &\Downarrow^{\bar x}& \searrow^{\mathrlap{\eta}}
       \\
       X_0 &&\underset{X}{⇸}&& T X_0
    }
  $$

satisfying the evident compatibility conditions.

=--

This is ([CruttwellShulman, def. 4.2](#CruttwellShulman)).

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

A $T$-monoid $X_0 \stackrel{X}{⇸} T X_0$ is called **normalized** if its unit 2-morphism

$$
  \array{
    X_0 &\stackrel{U_{X_0}}{\to}& X_0
    \\
    {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{\eta}}
    \\
    X_0 &\underset{X}{⇸}& T X_0
  }
$$

is a [cartesian 2-morphism](#...).

=--

#### Generalized multicategories

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

A **[[generalized multicategory]]** is a normalized $T$-monoid for some monad $T$ on a [[virtual equipment]] $\mathbb{X} \in $ [[vDbl]].

=--

This is ([CruttwellShulman, page 7](#CruttwellShulman)).

## Enriching categories

Virtual double categories can be viewed as "the natural place in which to enrich categories."  Specifically, for any set $A$, there is a virtual double category $A_{ch}$ which has $A$ as its objects, only identity vertical arrows, exactly one horizontal arrow from every object to every other object, and exactly one 2-cell in every possible niche.  For any other virtual double category $W$, a functor $A_{ch}\to W$ of virtual double categories is the same as a $W$-enriched category with object set $A$.

## Related pages

* [[double category]]
* [[virtual equipment]]
* [[hypervirtual double category]]

## References

Virtual double categories were first considered by [[Albert Burroni]] under the name **multicatégorie** on page 61 of:

* [[Albert Burroni]], _$T$-catégories (catégories dans un triple)_, Cahiers de topologie et géométrie différentielle catégoriques **12**.3 (1971) 215-321 [[dml:91097](https://eudml.org/doc/91097), [pdf](http://www.numdam.org/article/CTGDC_1971__12_3_215_0.pdf)]

Later, they were considered in:

* [[Tom Leinster]], _Higher Operads, Higher Categories_, [link](http://www.maths.gla.ac.uk/~tl/book.html), [arXiv:0305049](http://arxiv.org/abs/math/0305049)

* [[Tom Leinster]], **fc**-multicategories, [arxiv](https://arxiv.org/abs/math/9903004) (1999)

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

On a [[string diagram]]-calculus for ([[virtual double category|virtual]]) [[double categories]] with ([[virtual equipment|virtual]]) [[2-category equipped with proarrows|pro-arrow equipments]]:

* {#Myers16} [[David Jaz Myers]], _String Diagrams For Double Categories and (Virtual) Equipments_ [[arXiv:1612.02762](https://arxiv.org/abs/1612.02762)]

* [[David Jaz Myers]], *String Diagrams for (Virtual) Proarrow Equipments* (2017) [slides: [pdf](http://www.mat.uc.pt/~ct2017/slides/myers_d.pdf), [[Myers-StringDiagrams2017.pdf:file]]]

[[!redirects virtual double categories]]
[[!redirects fc-multicategory]]
[[!redirects fc-multicategories]]


virtual equipment

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## Idea

A _proarrow equipment on a virtual double category_ or _virtual equipment_ for short is a [[virtual double category]] equipped with the same kind of structure that makes an ordinary [[double category]] into a [[proarrow equipment]].

Virtual equipments are the structures that support structures of [[generalized multicategories]].

## Definition

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

A **virtual equipment** is a [[virtual double category]] in which all units and all restrictions exist.

=--

This is ([CruttwellShulman, def. 7.6](#CruttwellShulman)).

## References

* {#CruttwellShulman} [[Geoff Cruttwell]], [[Mike Shulman]], Section 7 of: _A unified framework for generalized multicategories_, [[arXiv:0907.2460](http://arxiv.org/abs/0907.2460)]

On a [[string diagram]]-calculus for ([[virtual double category|virtual]]) [[double categories]] with ([[virtual equipment|virtual]]) [[2-category equipped with proarrows|pro-arrow equipments]]:

* {#Myers16} [[David Jaz Myers]], _String Diagrams For Double Categories and (Virtual) Equipments_ [[arXiv:1612.02762](https://arxiv.org/abs/1612.02762)]

* [[David Jaz Myers]], *String Diagrams for (Virtual) Proarrow Equipments* (2017) [slides: [pdf](http://www.mat.uc.pt/~ct2017/slides/myers_d.pdf), [[Myers-StringDiagrams2017.pdf:file]]]

[[!redirects virtual equipments]]
[[!redirects virtual proarrow equipment]]


virtual fundamental class

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{:toc}

## Idea

The notion of _virtual fundamental class_ is a generalization of that of _[[fundamental class]]_ from [[manifolds]] to more general spaces in [[higher geometry]], notably to [[orbifolds]] and their equivalent incarnations as [[stacks]], as well as to [[derived manifolds]].

This plays a central role when these stacks serve as [[moduli stacks]] for certain structures on some space (certain maps into that space regarded as a [[target space]], notably) and one is interested in the relevant "[[path integral]]" over all these structures (to produce [[invariants]] of target space). This is manifestly so for instance in the application to [[Gromov-Witten invariants]]. In these cases the pairing of [[cocycles]] against the virtual fundamental class plays the role of [[integration]] over the given moduli stack.

## Definition

### For derived Schemes
Given a [[locally noetherian scheme|locally Noetherian]] [[derived scheme]], $(X, \mathcal{O}_X)$ with underlying [[scheme]] $t_0X$, $\pi_i\mathcal{O}_X$ [[coherent module|coherent]], and $\pi_i\mathcal{O}_X = 0$ for $i\gg0$, the _virtual fundamental class_ is defined by first constructing a class in $G_0(t_0X)$ (the [[K-theory]] of [[coherent sheaves]]) and then using this to produce an element of the [[Chow homology]] of $t_0X$. In our hearts we all know that the fundamental class should be something tautological, just like how the fundamental class of a triangulated manifold in simplicial homology is _the manifold itself_. We can use this idea once we have the following theorem:

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

The natural map of [[derived schemes]] $j: t_0X \rightarrow X$ induces an [[isomorphism]] $j_*: G_0(t_0X) \rightarrow G_0(X)$.

=--

A proof of this theorem and a definition of [[G-theory]] for derived schemes in ([Barwick](#Barwick)).

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

The $K$-theory _virtual fundamental class_ of a derived scheme $(X, \mathcal{O}_X)$ is the unique element $[X]$ in $G_0(t_0X)$ such that $j_*[X] = [\mathcal{O}_X]$ where $j: t_0X \rightarrow X$ is the natural map.

=--

If you trace through the definition of $G_0$ and the identification of the [[heart]] of $Coh(X)$ with $Coh(t_0X)$, you can show that $[X] = \sum (-1)^i [\pi_i\mathcal{O}_X]$.

To get from here to a virtual fundamental class in cohomology we need some more assumptions on $X$. For example, let's assume $X$ is finitely-presented over a field $k$ and that $\mathbb{L}_{X/k}$ is concentrated in degrees $0$ and $-1$, locally up to quasi-isomorphism, and a perfect complex. Then $j^*\mathbb{L}_{X/k}$ is a perfect complex over $t_0X$ and the dual is called the virtual tangent sheaf $\mathbb{T}^{vir}$.

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

With the above assumptions and notation, the _virtual fundamental class_ of $X$ is the evaluation of the inverse of the [[Todd class]] $Td(\mathbb{T}^{vir})$ on $\tau([X])$, where $[X]$ is as in def. \ref{KTheoryVirtualFundamentalClass} and $\tau$ is the [[Grothendieck-Riemann-Roch theorem|Grothendieck-Riemann-Roch transformation]].

=--

([Toën 14, p. 23](#Toen14))

## Examples

### Moduli space of stable maps
 {#ModuliSpaceOfStableMaps}

Although the [[moduli space]] of [[stable maps]] is sometimes referred to as a compactifiaction of the space of maps, in analogy with the [[Deligne-Mumford compactification]] of the [[moduli space]] of curves, in
fact it typically has boundary components of higher dimension than the space it was supposed to compactify!

Take for example $\bar M_{1,0}(P^2,3)$.  It ought to be a compactification of the space of degree-3 maps from genus-1 curves to $P^2$, and indeed one of its components has a Zariski open subset birational to the $P^9$ of all plane cubics.  But there is also a 'boundary component' of
higher dimension, namely the boundary component consisting of maps whose domain is a genus-1 curve glued to a nodal rational curve: the nodal curve maps to a rational cubic in $P^2$, while the $g=1$ component contracts to a point on that nodal cubic.

This boundary component has
dimension 10: namely, there are 8 parameters to specify the image nodal cubic, 1 paramenter to determine the point to which the $g=1$ component contracts, and finally there is 1 paramenter for the
[[j-invariant]] for the $g=1$ component.  The topological fundamental class lives in dimension 10 so it is rather useless to integrate against if all your cohomology classes are codimension 9 --- which is the
expected dimension.

The virtual fundamental class always lives in the
expected dimension.

(The expected dimension is often the one you would expect(!) from
naive counts like the above.  More formally it can be computed as dim
$H^0(C,N_f)$, where $f:C\to P^2$ is a moduli point (with [[normal bundle]]
$N_f$) such that $H^1(C,N_f)=0$ (this is to say that the first order [[infinitesimal space|infinitesimal]] deformations are unobstructed).)

The situation is analogous (possibly in fact a special case of) the standard situation in [[intersection theory]] when a section of a [[vector bundle]] is not regular: its zero locus is then of too high dimension
and is of little use to intersect against.  The correct class to work with is then the top [[Chern class]] of the vector bundle (cf.
[Fulton] ch.14), which could be called the virtual class of the zero locus.

In the example above, I don't know right now if the virtual class in fact appears as a top Chern class of a vector bundle --- I think it should, because the excess is just a variation of the standard example $\bart M_{1,1}(X,0)$, and in that example it is true that the virtual class appears as a top [[Chern class]]: there is a so-called [[obstruction bundle]] which in this case is the dual of the [[Hodge bundle]]
from the factor $\bar M_{1,1}$ tensored with the [[tangent bundle]] from $X$.

(The Hodge bundle is the direct image bundle of the canonical bundle of the universal curve, hence of rank $g$, hence just a line bundle in this case.)

The virtual fundamental class is the top Chern class of
the obstruction bundle (cap the topological fundamental class).

In this case, $dim \bar M_{1,1}(X,0) = 1 + dim X$, and the obstruction bundle has rank $dim X$, hence the virtual class has dimension 1.

Perhaps it should be mentioned also that the moduli space of maps can have components of too high dimension even before it is 'compactified', and even without involving contracting curves.  A
famous example is $M_{0,0}(Q,d)$ (no bar needed for this argument) where
$Q$ is a quintic three-fold.  Let's say $d=2$, so we are talking about conics on the quintic three-fold.  Since $Q$ has trivial [[canonical class]] it follows that the expected dimension is always 0 (i.e. in every
degree there ought to be a finite number of rational curves on Q).

But now, $M_{0,0}(Q,2)$ is a space of maps, not a space of curves, and for every one of the famous 2875 lines on $Q$ there is a 2-dimensional family of double covers of the line, which clearly count as stable
degree-2 maps, so $M_{0,0}(Q,2)$ contains 2875 components of dimension 2, in contrast to the virtual dimension 0.

## Related entries

* [[fundamental class]]

* [[Poincaré duality algebra]]

* [[intersection theory]]

* Virtual fundamental classes play a central role in the theory of [[Gromov-Witten invariants]].

## References

The idea of virtual fundamental classes and corresponding picture of derived moduli spaces comes from

* [[Maxim Kontsevich]], _Enumeration of rational curves via torus actions_, in: The moduli space of curves (Texel Island, 1994), 335–368, Progr. Math. __129__, Birkhäuser 1995. MR1363062 (97d:14077), [hep-th/9405035](http://arxiv.org/abs/hep-th/9405035)

A quick overview of virtual fundamental classes for [[algebraic stacks]] is in section 4.4.3 of

* [[Bertrand Toën]], _Higher and derived stacks: a global overview_ ([arXiv:math/0604504](http://arxiv.org/abs/math/0604504))

A more detailed overview with many pointers to the literature is in section 8 "Introduction to the virtual fundamental class" of

* Nicola Pagani, _Introduction to Gromov-Witten theory and quantum cohomology_ ([pdf](http://pcwww.liv.ac.uk/~pagani/DOC/corsogw.pdf))

A review of virtual fundamental classes of [[orbifolds]] in [[differential geometry]] is around page 7 of

* Andrés Angel, _When is a differentiable manifold the boundary of an orbifold?_ ([pdf](http://ems.math.uni-bonn.de/people/aangel79/files/boundary.pdf))

Detailed disucssion for the [[branched manifold]]-variant of [[orbifolds]] is in section 3.4 of

* [[Dusa McDuff]], _Groupoids, branched manifolds and multisections_ ([arXiv:math/0509664](http://arxiv.org/abs/math/0509664))

Virtual fundamental classs of [[derived manifolds]] are discussed in section 13.2 of

* [[Dominic Joyce]], _D-manifolds and d-orbifolds: a theory of derived differential geometry_ ([pdf](http://people.maths.ox.ac.uk/joyce/dmbook.pdf))

with applications to [[moduli spaces]] of coherent sheaves on [[Calabi-Yau space|Calabi-Yau 4-folds]] in

* [[Dominic Joyce]], [[Dennis Borisov]], _Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds_ (2014) ([wev](https://docs.google.com/file/d/0B4KuPQcvke9sOS1mTDBhNWxocnc/edit))

Specifically for applications to [[Gromov-Witten theory]] see for instance

* Ch. Okonek, A. Teleman, _Computing virtual fundamental classes: gauge theoretical Gromov-Witten invariants for toric varieties_ ([arXiv:math/0205137](http://arxiv.org/abs/math/0205137))

The material in _[Moduli space of stable maps](#ModuliSpaceOfStableMaps)_ above originates in a blog discussion [here](http://golem.ph.utexas.edu/category/2009/09/a_seminar_on_gromovwitten_theo.html#c027000)

The definition and description above for derived schemes derives from p.23 of

* [[Bertrand Toën]] Derived Algebraic Geometry, 2013. ([http://arxiv.org/pdf/1401.1044v1.pdf](http://arxiv.org/pdf/1401.1044v1.pdf))
 {#Toen14}

The devissage result needed is proven as Proposition 8.2 here:

* [[Clark Barwick]] _On Exact $\infty$-categories and the Theorem of the Heart ([pdf](http://dl.dropboxusercontent.com/u/1741495/papers/k2-1.pdf))
  {#Barwick}

[[!redirects virtual fundamental classes]]


virtual knot theory

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

# Contents
* table of contents
{: toc}

## Idea

One fundamental tool in a knot theorist's toolbox is the [[knot diagram]].
Classically, a [[knot]] is an embedding of the circle $S^1$ into Euclidean space $\mathbb{R}^3$, and knot diagrams arise by projecting back down to the [[plane]] $\mathbb{R}^2$ to obtain a 4-valent [[plane graph]] (referred to as the _shadow_ of the knot), while keeping track of whether each vertex corresponds to an under-crossing or an over-crossing.
Although the knot is traditionally seen as embedded in $\mathbb{R}^3$, it could equally well be realized inside of a "thickened" [[sphere]] $S^2 \times [0,1]$, with its shadow embedded in $S^2$.
From that perspective, _virtual knot theory_ generalizes classical knot theory by considering knots as embeddings of circles into thickened [[orientable]] surfaces $X \times [0,1]$ of arbitrary [[genus of a surface|genus]].
Abstractly, the shadow of such a knot is a 4-valent graph embedded in $X$ (i.e., a [[topological map]]).
However, if one tries to project the knot onto the plane, the corresponding diagram might contain crossings that do _not_ represent places where the knot passes over/under itself inside the thickened surface, but rather are artifacts of the knot's non-planar shadow.
So, in a _virtual knot diagram_ such crossings are explicitly indicated as "virtual", using a distinct notation from that for under/overcrossings.

## Definitions

There are various equivalent definitions of virtual knots/links:

* As equivalence classes of virtual knot/link diagrams, modulo the "virtual" [[Reidemeister moves]] [(Kauffman 1999)](#Kauffman1999).
* As equivalence classes of "abstract" knot/link diagrams embedded in orientable surfaces, modulo the Reidemeister moves [(Kamada and Kamada 2000)](#KamadaKamada).
* As embeddings of circles into thickened orientable surfaces, considered up to [[isotopy]] [(Kuperberg 2003)](#Kuperberg2003).
* As equivalence classes of [[Gauss codes]], modulo relations corresponding to the Reidemeister moves [(Kauffman 1999)](#Kauffman1999).

## Categorifications

The original work on virtual knot theory was not expressed in categorical language, but a first attempt at categorifying the _virtual braid group_ $VB_n$ was made by [Kauffman and Lambropoulou](#KL2011). [[Victoria Lebed]] studied the question extensively in her thesis [(Lebed 2012)](#LebedThesis), and developed a categorical approach based on the notion of a _[[braided object]]_ in a [[symmetric monoidal category]], that is an object $V$ equipped with an invertible morphism $\sigma : V\otimes V \to V\otimes V$ satisfying the [[Yang-Baxter equation]]
$$(\sigma \otimes V) \circ (V \otimes \sigma) \circ (\sigma \otimes V) = (V \otimes \sigma) \circ (\sigma \otimes V) \circ (V \otimes \sigma)$$
In particular, Lebed shows that the virtual braid group $VB_n$ is isomorphic to the group of endomorphisms $End_{\mathcal{C}_{2br}}(V^{\otimes n})$, where $\mathcal{C}_{2br}$ is the free symmetric monoidal category generated by a single braided object $V$.

The intuition here is that the (symmetric) braiding of the ambient symmetric monoidal category represents virtual crossings "for free", while the braiding $\sigma$ on the object $V$ represents "real" over- and under-crossings. (Compare some [remarks by John Baez](#Baez08), which are similar in spirit.)

## Terminology

Warning: a virtual knot/link has a genus in the sense of the genus of the underlying thickened surface into which it embeds (or equivalently, the genus of its shadow as a topological map), but this is unrelated to the classical notion of _knot genus_, in the sense of the minimal genus of a [[Seifert surface]] whose [[boundary]] is the knot.

## Related concepts

* [[Gauss diagram]]
* [[embedded graph]]

## References

The original paper on virtual knot theory and some early followup work:

* {#Kauffman1999} [[Louis Kauffman]], Virtual Knot Theory, _European Journal of Combinatorics_ (1999) 20, 663-691. [pdf](http://homepages.math.uic.edu/~kauffman/VKT.pdf)

* {#KamadaKamada} Naoko Kamada and Seiichi Kamada, Abstract Link Diagrams and Virtual Knots, _Journal of Knot Theory and its Ramifications_, Vol. 9, No. 1 (2000), 93-106. [doi](http://dx.doi.org/10.1142/S0218216500000049)

* {#Kuperberg2003} [[Greg Kuperberg]], What is a virtual link?, _Algebraic & Geometric Topology_ Volume 3 (2003), 587-591. [pdf](http://arxiv.org/pdf/math/0208039v2.pdf)

Miscellaneous papers:

* {#Kauffman2012} [[Louis Kauffman]], Introduction to Virtual Knot Theory. July 2012. [arXiv](http://arxiv.org/abs/1101.0665)

* {#Viro} Oleg Viro, Virtual Links, Orientations of Chord Diagrams and Khovanov Homology, Proceedings of 12th Gökova Geometry-Topology Conference, pp. 184–209, 2005. [pdf](http://www.pdmi.ras.ru/~olegviro/ggt05-viro.pdf)

* {#ManturovIlyutko} Vassily Olegovich Manturov and Denis Petrovich Ilyutko, _Virtual Knots: The State of the Art_.  Series on Knots and Everything (vol. 51), World Scientific, 2013.

* {#Kauffman2015} [[Louis Kauffman]], Rotational Virtual Knots and Quantum Link Invariants, 14 Oct 2015. [arxiv:1509.00578](http://arxiv.org/abs/1509.00578)

On the question of categorifying virtual knot theory, see:

* {#Baez08} [[John Baez]], [comment at the n-Café](https://golem.ph.utexas.edu/category/2008/10/categorification_in_new_scient.html#c019490) following a post titled "Categorification in New Scientist", October 2008.

* {#KL2011} [[Louis Kauffman]] and Sofia Lambropoulou, A Categorical Structure for the Virtual Braid Group, _Comm. Algebra_, 39(12):4679–4704, 2011. ([pdf](http://www.math.ntua.gr/~sofia/publications/A22%20VirtualPureMia-my%20pdf.pdf))

* {#LebedThesis} [[Victoria Lebed]], _Objets tressés: une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles_, Thèse, Université Paris Diderot, 2012. ([pdf](http://www.maths.tcd.ie/~lebed/thesis_LEBED.pdf)) Note that the title is in French ("Braided objects: a unifying study of algebraic structures and a categorification of virtual braids") but the main text of the thesis is in English.

* {#Lebed13} [[Victoria Lebed]], Categorical Aspects of Virtuality and Self-Distributivity, _Journal of Knot Theory and its Ramifications_, 22 (2013), no. 9, 1350045, 32 pp. ([doi](http://dx.doi.org/10.1142/S0218216513500454))  According to the author, [arXiv:1206.3916](http://arxiv.org/abs/1206.3916) is "an extended version of the above JKTR publication, containing in particular a chapter on free virtual shelves and quandles".


virtual machine

In [[computer science]], a virtual machine is a simulation of a fixed architecture executing some sort of signaling or software. It can abstract an entire or part of hardware,  firmware, a file system with some ports, devices and/or CPU and installed operating system, but it can be simply an isolated environment executing some specified class of software, or simply any installation in hardware or in another virtual machine of a specified execution model. For example, Java Virtual Machine executes formally specified "bytecode" to ensure identical execution on all hardware.

See also [[certified programming]], [[WebAssembly]]

## Virtual machines for cloud

### Virtual machine simulating machine with operating system

...

### Containers

...

### Blockchain runtimes for multiple blockchains

Microsoft Azure develops a runtime framework intended to work on various blockchains. This is the CoCo project (see whitepaper 2017 at github [pdf](https://github.com/Azure/coco-framework/blob/master/docs/Coco%20Framework%20whitepaper.pdf), [announcement](https://azure.microsoft.com/en-us/blog/announcing-microsoft-s-coco-framework-for-enterprise-blockchain-networks) and a presentation on [Medium](https://medium.com/@jrodthoughts/with-coco-framework-microsoft-wants-to-become-the-redhat-of-the-blockchain-5b35f8ac8707)).

## Languages for simulation of devices

...

## Bytecode virtual machines

### Java Virtual Machine

Java VM has a specification which executes Java bytecode. Several languages compile to JVM including Java, Kotlin and Scala.

* [java (language)](http://en.wikipedia.org/wiki/Java_%28programming_language%29), [java (platform)](http://en.wikipedia.org/wiki/Java_%28software_platform%29)
* [kotlin](https://kotlinlang.org)
* [scala](http://en.wikipedia.org/wiki/Scala_%28programming_language%29)
* [scalaz](https://github.com/scalaz/scalaz) - library for functional programming in scala

### WebAssembly VM

WebAssembly is optimized for small compiling time and near native execution time on major architectures (like 86 series). It appeared first as new VM standard
on web browsers, backed by major internet companies; it is also used or planned on a number of [[blockchain]] projects.

* WebAssembly [wikipedia](https://en.wikipedia.org/wiki/WebAssembly), [spec](https://webassembly.github.io/spec), [webassembly.org](https://webassembly.org), [msdn](https://developer.mozilla.org/en-US/docs/WebAssembly) docs, [rust-to-wasm](https://developer.mozilla.org/en-US/docs/WebAssembly/Rust_to_wasm), [github](tps://github.com/WebAssembly)
* [WebAssembly-Links](https://wiki.parity.io/WebAssembly-Links) in Parity Tech Documentation
* AssemblyScript (maps a subset of javascript code to wasm) [github](https://github.com/AssemblyScript/assemblyscript), [news](https://www.infoworld.com/article/3224006/assemblyscript-compiles-typescript-to-webassembly.html)

[[Rust]] has small runtime, which is desirable in common applications of WebAssembly. Thus Rust commonly compiles either to native code or to wasm.

* Rust & WebAssembly with Nick Fitzgerald [yt](https://www.youtube.com/watch?v=ZiiTRxWk8gA)

Ethereum flavoured version of wasm VM specification is at github/[ewasm](https://github.com/ewasm), see also github/[ewasm/design](https://github.com/ewasm/design). eWasm has a [testnet](http://ewasm.ethereum.org/explorer). According to article _[ewasm explained](https://www.mycryptopedia.com/ewasm)_,

> The ewasm specification consists of a subset of WebAssembly components suitable for Ethereum’s needs, namely determinism and relevant features. It also includes a number of system smart contracts that provide access to Ethereum platform features.

### IELE

Part of [[zoranskoda:Cardano]] project, IELE executes and verifies [[smart contract]]s as well as providing a human-readable language for blockchain developers.

* IELE, _Semantics based compilation_, at [iohkdev.io](https://testnet.iohkdev.io/iele/about/semantics-based-compilation)

### Ethereum VM

Primarily used for smart contract execution on Ethereum blockchain (and some others blockchains, like [[zoranskoda:Hyperledger]] Fabric). Ethereum community has plans to supercede it with Ethereum version of wasm VM (ewasm).

### TON VM

Used for executing smart contracts on [[Telegram Open Network]]. See

* [[Nikolai Durov]], _Telegram Open Network Virtual Machine_, Sep. 2018, 148 pp. [pdf](https://www.docdroid.net/R3vEKBY/telegram-open-network-virtual-machine-september-5-2018.pdf)

### CKB VM (Nervos)

This is a [RISC-V](https://en.wikipedia.org/wiki/RISC-V) VM for a Nervos blockchain design.  "Uses rv64imc architecture: it is based on core RV64I ISA with M standard extension for integer multiplication and division, and C standard extension for RCV(RISC-V Compressed Instructions)." No floating point.

* github [mb](https://github.com/nervosnetwork/rfcs/blob/master/rfcs/0002-ckb/0002-ckb.md), [code](https://github.com/nervosnetwork/ckb)

[[!redirects Virtual Machine]]


virtual particle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]], it is suggestive to think of the [[edges]] in the [[Feynman diagrams]] as [[worldlines]] of "virtual particles" and of the [[vertices]] as the points where they collide and transmute. (Care must be exercised not to confuse this with concepts of real [[particles]].)

This intuition is made precise by the _[[worldline formalism]]_ of [[perturbative quantum field theory]] ([Strassler 92](worldline+formalism#Strassler92)). This is the perspective on [[perturbative QFT]] which directly relates [[perturbative QFT]] to [[perturbative string theory]] ([Schmidt-Schubert 94](worldline+formalism#SchmidtSchubert94)). In fact the [[worldline formalism]] for [[perturbative QFT]] was originally found by taking thre point-particle limit of [[string scattering amplitudes]] ([Bern-Kosower 91](worldline+formalism#BernKosower91), [Bern-Kosower 92](worldline+formalism#BernKosower92)).

## Related concepts

* [[relativistic particle]], [[spinning particle]], [[superparticle]]

* [[fundamental particle]], [[antiparticle]]

* [[radiative correction]]

## References

* [[Arnold Neumaier]], _[The Physics of Virtual Particles](https://www.physicsforums.com/insights/physics-virtual-particles/)_

* [[Arnold Neumaier]], _[Misconceptions about Virtual Particles](https://www.physicsforums.com/insights/misconceptions-virtual-particles/)_

See also

* Wikipedia, _[Virtual particle](https://en.wikipedia.org/wiki/Virtual_particle)_

[[!redirects virtual particles]]


virtual representation

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

#Contents#
* table of contents
{:toc}

## Idea

A _virtual representation_ of a [[group]] is a formal difference with respect to [[direct sum]] of two ordinary [[representations]], hence the [[isomorphism class]] of a virtual representation is an element of the [[Grothendieck group]] of $(G Rep, \oplus)$. Equivalence classes of virtual representations form the elements of the [[representation ring]] of the group, see there for more.

If we regard an ordinary [[representation]] as an [[equivariant vector bundle]] over the point, then a virtual representation is a corresponding equivariant [[virtual vector bundle]]. Accordingly the [[representation ring]] of a [[finite group]] is its [[equivariant K-theory]] of the point.

## Related concepts

* [[representation ring]]

* [[Burnside ring]]

[[!redirects virtual representations]]


virtual vector bundle

#Contents#
* table of contents
{:toc}

## Idea

A formal difference (with respect to [[direct sum]]/[[Whitney sum]]) of [[vector bundles]]. Appears as representative for classes in [[topological K-theory]]. A virtual vector bundle $[E] - [F]$ has a _virtual rank_ $rk([E]-[F])$, which is the integer $rk(E) - rk(F)$.

## Properties

+-- {: .num_prop}
###### Proposition
Let X be a space that is locally compact and regular. Then any virtual vector bundle of virtual rank 0 has some tensor power that is 0 in K-theory.
=--

The following argument comes from ([Karoubi 1978](#Karoubi78), Theorem II.5.9), but generalised to take into account non-compact spaces which are introduced in the following pages.

+--{: .proof}
###### Proof
For closed subsets $Y_1$ and $Y_2$, then the ring structure on K-theory means
$$
K_{Y_1}(X)\cdot K_{Y_2}(X) \subset K_{Y_1\cup Y_2}(X)
$$
where $K_{Y_i}(X) := ker(K(X) \to K(Y_i))$. We can likewise consider the same for a sequence $Y_1,\ldots ,Y_n$.

If $e = [E] - [\underline{rk(E)}]$ is in $ker(rk)$ (i.e. reduced K-theory $\tilde K(X)$) with $supp(e)$ in some compact set $C$ (recall that vector bundle K-theory elements are differences of vector bundles that are isomorphic outside a compact set. Without loss of generality such an element can be represented as $[E] - [\underline{V}]$ where $\underline{V}$ is a trivial bundle with fibre $V$), cover $C$ by finitely many closed neighbourhoods $Y_1,\ldots ,Y_n$ over which $E$ trivialises. Thus $e$ is in $K_{Y_i}(X)$ for each $i=1,\ldots n$.

Thus $e$ is in the intersection of all the $K_{Y_i}(X)$ as well as $K_Z(X)$ for $Z$ the complement of $supp(e)$. Thus $e^{n+1}$ is in $K_{\cup Y_i \cup Z}(X) = K_X(X) = 0$.

Hence for every element in $ker(rk)$, some power of it is zero, hence $ker(rk)$ is a [[ideal|nil ideal]].
=--

## Related concepts

* [[virtual representation]]

## References

* {#Karoubi78} [[Max Karoubi]], _K-Theory: An Introduction_, Grundlehren der mathematischen Wissenschaften 226, (1978) Springer-Verlag

[[!redirects virtual vector bundles]]

[[!redirects virtual vector space]]
[[!redirects virtual vector spaces]]