Verdier duality

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#### Duality
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#### Homological algebra
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## Idea

What is called _Verdier duality_  is the refinement of [[Poincaré duality]] from [[ordinary cohomology]] to [[abelian sheaf cohomology]].
The Grothendieck [[six operations]] formalism is a formalization of aspects of Verdier duality.

In general abstract formulation one says that a locally compact [[site]] $X$ (e.g. a [[locally compact topological space]]) _satisfies Verdier duality_ if there exists a [[derived functor|derived]] [[right adjoint]] $\mathbb{L}p^!$ ([[exceptional inverse image]]) to the operation $\mathbb{R}p_!$ of [[direct image with compact support]] for the terminal map $X \to \ast$. More generally, in the relative situation, for $f \colon X \to Y$ a map, Verdier duality of $f$ means that $\mathbb{R}f_!$ has a derived right adjoint. If here $X$ is a [[scheme]] and $f_! \simeq f_\ast$ coincides with the [[direct image]] ("[[Grothendieck context]]") then this is called _[[Grothendieck duality]]_. Therefore one often speaks of _Verdier-Grothendieck duality_.

If these adjoints exist, then one has the _[[dualizing object in a closed category]]_ ([[dualizing module]]) $\omega \coloneqq \mathbb{L}p^! k$ (the image under the extra right adjoint of the [[unit object]]) and for every [[abelian sheaf]] $V$ on $X$ can define the [[dual object in a closed category]]

$$
  \mathbb{D}V \coloneqq [V,\omega]
  \,.
$$

The analog of [[Poincaré duality]] is then the statement that the [[abelian sheaf cohomology]] with [[coefficients]] in $V$ is dual to the abelian sheaf cohomology with [[compact support]] with coefficients in $\mathbb{D}V$.

## Implementations

A fairly general class of implementations of Verdier-Grothendieck duality is discussed in ([Joshua, theorem 1.3](#Joshua)).

With assumption ([Joshua, 4.3](#Joshua)) this derives that dualization intertwines the two pushforwards as $\mathbb{D} \circ f_\ast \simeq f_! \circ \mathbb{D}$ ([Joshua, corollary 5.4](#Joshua)).

## Related concepts

* [[Verdier-Grothendieck context]]

* [[Grothendieck duality]]

* [[Poincaré duality]]

## References

The duality is named after [[Jean-Louis Verdier]].

Accounts of the traditional notion include

* [[Amnon Neeman]], _The Grothendieck duality theorem via Bousfield's techniques and Brown representability_, Jour. Amer. Math. Soc. 9 (1996), 205–236. ([JSTOR](http://www.jstor.org/stable/2152846))
 {#Neeman96}

* Wikipedia, _[Verdier duality](http://en.wikipedia.org/wiki/Verdier_duality)_

* Liviu Niculaescu, _The derived category of sheaves and the Poincaré-Verdier duality_ ([pdf](http://www3.nd.edu/~lnicolae/Verdier-ams.pdf))

* [[Jacob Lurie]], lecture notes on _Verdier duality_ ([2011 pdf](http://www.math.harvard.edu/~lurie/287xnotes/Lecture19.pdf), [2013 pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.348.5739&rep=rep1&type=pdf), [2014 pdf](http://people.math.harvard.edu/~lurie/282ynotes/LectureXXI-Verdier.pdf))

A general abstract formalization is in

* [[Roy Joshua]], _Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories_ ([[JoshuaDuality.pdf:file]])
 {#Joshua}

Discussion in the context of [[higher algebra]]/[[higher geometry]] is in

* [[Roy Joshua]], _Generalised Verdier duality for presheaves of spectra—I_, Journal of Pure and Applied Algebra Volume 70, Issue 3, 29 March 1991, Pages 273–289

* [[Jonathan Block]] and [[Andrey Lazarev]], _Homotopy Theory and Generalized Duality for Spectral Sheaves_, IMRN International Mathematics Research Notices
1996, No. 20 ([pdf](http://www.math.upenn.edu/~blockj/papers/BlockLazarevSpectralSheaves.pdf))

* [[Vladimir Drinfeld]], [[Dennis Gaitsgory]], section 5.3 _On some finiteness questions for algebraic stacks_ ([arXiv:1108.5351](http://arxiv.org/abs/1108.5351))

* [[Jacob Lurie]], section 4.2 of _[[Representability theorems]]_

Formulation in terms of an equivalence between sheaves and cosheaves is in

* [[Peter Schneider]], _Verdier duality on the building_, Journal reine angew. Math. 494, 1998

* [[Justin Curry]], _Sheaves, Cosheaves and Applications_ ([arXiv:1303.3255](http://arxiv.org/abs/1303.3255))

* [[Jacob Lurie]], section 5.5.5 of _[[Higher Algebra]]_

Discussion in the context of prestacks and diagrams of schemes

* [[Dennis Gaitsgory]], _The Atiyah-Bott formula for the cohomology of the moduli space of bundles on a  curve_ ([arXiv:1505.02331](http://arxiv.org/abs/1505.02331))

* Alexey Kalugin, _On categorical approach to Verdier Duality_ ([arXiv:1505.06922](http://arxiv.org/abs/1505.06922))

[[!redirects Verdier dualities]]


Verdier product

## Definition

Verdier product is an analogue of the [[Gabriel product]] in triangulated context.

## Literature

* [[Wendy Lowen]], [[Michel Van den Bergh]], _On compact generation of deformed schemes_, Adv. Math. __244__ (2013) 441--464 [doi](https://doi.org/10.1016/j.aim.2013.04.024)
* Gustavo Jasso, [[Bernhard Keller]], [[Fernando Muro]], _The triangulated Auslander--Iyama correspondence_, [arXiv:2208.14413](https://arxiv.org/abs/2208.14413)


Verdier site

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#### Topos Theory
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## Idea

A _Verdier site_ (named after [[Jean-Louis Verdier]]) is a [[site]] with an extra condition on the [[coverage]] that ensures that one can refine every [[hypercover]] in the context of [[simplicial presheaf|simplicial presheaves]] on that site in a particularly nice.

For a discussion of the use of Verdier sites see [[descent]].

## Definition

A **Verdier site** is a [[site]] such that for every morphism $U \to X$ which is part of a covering [[sieve]] on $X$, also

$$
  U \stackrel{(Id_U,Id_U)}{\to} U \times_X U
$$

is in a covering [[sieve]] of $U \times_X U$.

## References

section 9 of

* [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], _Hypercovers and simplicial presheaves_ ([arXiv](http://arxiv.org/abs/math/0205027))

[[!redirects Verdier sites]]


Verdier's abelianization functor

Given a [[triangulated category]] $T$ there is an [[abelian category]] $A$ and an [[additive functor]] $T\to A$ satisfying certain universal property; this functor is due to [[Verdier]] and is called the Verdier's abelianization functor. A readable and comprehensive introduction is in the monograph

* A. Neeman, _Triangulated categories_, Annals of Mathematics Studies __148__,  Princeton University Press 2001, viii+449 pp.

[[!redirects Verdier abelianization functor]]


Verdier-Grothendieck context

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#Contents#
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## Idea

The specialization of a context of [[six operations]] $(f^\ast \dashv f_\ast)$, $(f_! \dashv f^!)$ to the case that the "[[projection formula]]" $Y \otimes f_! X \simeq f_!(f^\ast Y \otimes X)$ holds naturally in $X,Y$. ([May 05, def. 2.12](#May05))

## Properties

In a Verdier-Grothendieck context, [[dual object in a closed category|duality]] intertwines $f_!$ with $f_\ast$ and $f^!$ with $f^\ast$.

([May 05, section 3](#May05), [Joshua, corollary 5.4](#Joshua))

## Related concepts

* [[Verdier duality]], [[Grothendieck duality]]

* [[Wirthmüller context]], [[Grothendieck context]]

## References

A general abstract discussion of the axioms and their consequences is in

* {#May05} [[Halvard Fausk]], [[Po Hu]], [[Peter May]],  *Isomorphisms between left and right adjoints*, Theory and Applications of Categories, **11** 4 (2003) 107-131 [[tac:11-04](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), [pdf](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf)]

A fairly general class of implementations is in

* [[Roy Joshua]], _Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories_ ([[JoshuaDuality.pdf:file]])
 {#Joshua}

[[!redirects Verdier-Grothendieck contexts]]

[[!redirects Grothendieck-Verdier context]]
[[!redirects Grothendieck-Verdier contexts]]


This page collect material related to the article

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

  English translation by M. W. Haskell:

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

on foundations of [[differential geometry]] in the notion of [[group actions]].

For more see at _[[Erlangen program]]_.

category: people


Verity on descent for strict omega-groupoid valued presheaves

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This entry is about a theorem by [[Dominic Verity]] that characterizes the [[descent]] condition for [[(∞,1)-sheaves]]/[[∞-stack]]s that take values not in arbitrary [[∞-groupoid]]s, but in [[strict ∞-groupoid]]s.

#Contents#
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## Verity's theorem

The details are here:

* [[Dominic Verity]], _[[VerityDescent.pdf:file]]_ (pdf)

Here is an abstract that served as an abstract for a talk on this at the _Australian Category Seminar_ at Macquarie University on Wednesday 27th of May 2009.

Abstract

>In the literature one can find a number of different [[limit]] notions which one might refer to as a "[[descent]] construction". Generally speaking, these may all be regarded as a kind of [[weak limit|lax, pseudo]] or [[homotopy limit]] of a [[simplicial object|co-simplicial diagram]] of objects in some theory of "spatially-enriched" categories. While each of these notions certainly deserves to bear the [[descent]] name, it is not necessarily immediately clear how they may be related in any more specific mathematical sense.

>Recently I was asked by [[Urs Schreiber]] if I knew how a couple of these descent notions might be related formally, and so spent a little time contemplating this problem. My hope is that this talk might achieve two things, firstly I hope to provide a little of the intuition which leads us to define and study such descent constructions. Then I would like to discuss a specific answer to Urs' question, which gives a precise relationship between [[Ross Street]]'s [descent construction](http://arxiv.org/abs/math.CT/0303175) for [[strict omega-category|strict ∞-categories]] (or more precisely [[strict ∞-groupoid]]s in this case) and the [[descent for simplicial presheaves|simplicial descent construction]] used to characterise the fibrant objects in [[model structure on simplicial presheaves|model categories of simplicial sheaves]].

## Idea

Generally, [[models for ∞-stack (∞,1)-toposes]] are provided by a [[model structure on simplicial presheaves|model structure on presheaves with values in simplicial sets]].

As for all [[combinatorial simplicial model category|combinatorial simplicial model categories]] the $(\infty,1)$-topos [[presentable (∞,1)-category|presented]] by this model structure is the full [[SSet]]-[[enriched category|enriched subcategory]] on fibrant and cofibrant objects.

By a theorem by Dugger-Isaksen-Hollander on the projective [[local model structure on simplicial presheaves]] the _fibrant_ simplicial presheaves are those that

* take values in [[∞-groupoid]]s (i.e. the [[simplicial set]]s assigned by them are [[Kan complex]]es)

* and satisfy [[descent]].

While general [[∞-groupoid]]s are useful due to their generality and conceptual simplicity, for many concrete computations it is useful to get a more concrete [[algebraic definition of higher category|algebraic model]] and consider just [[strict ∞-groupoid]]s. Under the [[oriental]]-[[nerve]]

$$
  Str \omega Grpd \stackrel{N}{\hookrightarrow} \infty Grpd
$$

the strict $\omega$-groupoids form a [[subcategory]] of all [[∞-groupoid]]s. This is to be regarded as a refinement of the [[Dold-Kan correspondence]] which embeds strict $\omega$-groupoids _with abelian group structure_ equivalently modeled as [[chain complex]]es into all $\infty$-groupoids

$$
  Ch_+
  \stackrel{\simeq}{\to} StrAb \omega Grpd
  \hookrightarrow
  Str \omega Grpd \stackrel{N}{\hookrightarrow} \infty Grpd
  \,.
$$

It is a familiar process to restrict general [[∞-stack]]s to those that factor through the entire inclusion: this is the topic of [[homological algebra]] and restricts the general notion of [[cohomology]] to that of [[abelian sheaf cohomology]].

What we are interested in here is a notion in between the fully strictly abelian context and the fully general context: that of strict $\omega$-groupoid valued $\infty$-stacks inside all $\infty$-stacks. This may be thought of as [[nonabelian algebraic topology|nonabelian homological algebra]] that uses not [[chain complex]]es of sheaves but [[crossed complex]]es.

In his work

* [[Ross Street]], _Categorical and combinatorial aspects of descent theory_ ([arXiv](http://arxiv.org/abs/math.CT/0303175))

Ross Street had proposed a formulation of the [[descent]] condition for such $Str \omega Grpd$-valued $\infty$-stacks (see the corresponding section at [[descent]] for the details).

The question remained open how that definition of descent on $Str \omega Grpd$-valued presheaves relates to the general one of $\infty Grpd$-valued presheaves under the above inclusion.

It is this question that Dominic Verity's theorem answers.

In words, Verity's theorem says that Ross Street's descent conditon on a $Str \omega Grpd$-valued presheaf $A$ is the correct one if the [[hypercover]] $\pi : Y \to X$  along which one checks descent is sufficiently well behaved -- in that the [[simplicial object|cosimplicial]] $\infty$-groupoid $A(Y)$ is [[Reedy category|Reedy fibrant]].

[[!redirects Verity on descent for strict omega-groupoid valued presheave]]


Verity-Gray tensor product

#Idea#

The _Verity-Gray tensor product_ or _lax Gray tensor product of [[stratified simplicial set]]s_ is a [[tensor product]] on the [[category]] $Strat$ of [[stratified simplicial set]]s which when restricted to [[complicial set]]s, i.e. [[oriental|omega-nerve]]s of [[strict omega-category|strict omega-categories]] reproduces the [[Crans-Gray tensor product]] on strict $\omega$-categories.

#Definition#

Let $(X, t X)$ and $(Y, t Y)$ be [[stratified simplicial set]]s. Then their **Verity-Gray tensor product** $(X, t X) \otimes (Y, t Y)$ is given by

$$
  (X, t X) \otimes (Y, t Y)
  :=
  (X \times Y, q(t X, t Y))
  \,,
$$

where $X \times Y$ is the [[cartesian product]] of [[simplicial set]]s (hence the standard monoidal structure on [[SSet]]), while $q(t X, t Y)$, the set of thin cells, is $tX\times tY$ for the Gray product, and for the lax-Gray product  is enlarged as described in the paper.

#References#

definition 128 of

* Dominic Verity, _Complicial sets_ ([arXiv](http://arxiv.org/abs/math/0410412))

definition 59, page 32 of

* Dominic Verity, _Weak complicial sets I_ ([arXiv](http://arxiv.org/abs/math/0604414))

slide 60 of

* Dominic Verity, _Weak complicial sets and internal quasi-categories_ ([arXiv](http://www.mat.uc.pt/~categ/ct2007/slides/verity.pdf))


Verlinde formula

#Contents#
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{:toc}

## Idea

Generally, a _Verlinde formula_ gives the [[dimension]] of a [[space of states]] of [[Chern-Simons theory]] for a given [[gauge group]] $G$. Depending on which one of various different algebraic means to expresses these spaces is used, the Verlinde formula equivalently computes the dimension of spaces of non-abelian [[theta functions]], the dimension of objects in a [[modular tensor category]] and so forth.

There are also Verlinde formulas in algebraic geometry (proved by Faltings) and a related one in the theory of [[vertex operator algebra]]s (proved only in very special cases).

## Related concepts

* [[fusion ring]].

## References

Original article:

* [[Erik Verlinde]], *Fusion rules and modular transformations in 2D conformal field theory*, Nuclear Physics B **300** (1988) 360-376 $[$doi:10.1016/0550-3213(88)90603-7$]$

Proofs of the Verlinde formula:

* {#TUY89} A. Tsuchiya, [[Kenji Ueno]], and Y. Yamada, *Conformal field theory on the universal family of stable curves with gauge symmetry* In: *Conformal field theory and solvable lattice models* Adv. Stud. Pure Math. **16** (1989), 297–372.

* {#Faltings94} [[Gerd Faltings]], *A proof for the Verlinde formula*, J. Alg. Geom. **3** (1994) 347–374

* {#Huang} [[Yi-Zhi Huang]], *Vertex operator algebras, the Verlinde conjecture and modular tensor categories*, Proc. Nat. Acad. Sci. **102** (2005) 5352-5356 $[$[arXiv:math/0412261](http://arxiv.org/abs/math/0412261), [doi:10.1073/pnas.0409901102](https://doi.org/10.1073/pnas.0409901102)$]$

Review in:

* [[Martin Schottenloher]], *Mathematical Aspects of the Verlinde Formula*, in: *A Mathematical Introduction to Conformal Field Theory*, Lecture Notes in Physics **759**, Springer (2008) $[$[doi:10.1007/978-3-540-68628-6](https://link.springer.com/book/10.1007/978-3-540-68628-6), [web](https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/), [pdf](https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/11_978-3-540-68625-5_Ch11_23-08-08.pdf)$]$

where it says (p. 213):

> Several mathematicians have worked on the problem of proving the Verlinde formula, starting with $[$[TUY89](#TUY89)$]$ and coming to a certain end with $[$[Fal94](#Faltings94)$]$. These proofs are all quite difficult to understand.

* [[Ralph Blumenhagen]], [[Erik Plauschinn]], §4.3 of: *Introduction to Conformal Field Theory -- With Applications to String Theory*, Lecture Notes in Physics **779**, Springer (2009) [[doi:10.1007/978-3-642-00450-6](https://doi.org/10.1007/978-3-642-00450-6)]

Introduction:

* [[Shigeru Mukai]], _An introduction to invariants and moduli_, Cambridge Univ. Press 2003

See also:

* Dowker, _On Verlinde's formula for the dimensions of vector bundles on moduli spaces_, [iopscience](http://iopscience.iop.org/0305-4470/25/9/033)

* Juergen Fuchs, Christoph Schweigert, _A representation theoretic approach to the WZW Verlinde formula_, 1997

A generalization to [[logarithmic 2d CFTs]] has been suggested in:

* [[Thomas Creutzig]], [[Terry Gannon]], _Logarithmic conformal field theory, log-modular tensor categories and modular forms_, J. Phys. A __50__, 404004 (2017) ([doi:10.1088/1751-8121/aa8538](https://doi.org/10.1088/1751-8121/aa8538), [arXiv:1605.04630](https://arxiv.org/abs/1605.04630))

[[!redirects Verlinde formulas]]


Verlinde ring

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## Idea

For $G$ a suitable [[Lie group]], the _Verlinde ring_ is the collection of [[isomorphism classes]] of [[positive energy representations]] of the corresponding [[loop group]], equipped with the "fusion" [[tensor product]].

The Verlinde ring is also understood as being the ring of [[equivariant cohomology|equivariant]] [[twisted K-theory]] classes on $G$ ([FHT](#FHT)) and, essentially equivalently, of [[Chan-Paton gauge fields]] over [[D-branes]] in the [[WZW model]].

## Related concepts

* [[fusion ring]]

* [[twisted ad-equivariant K-theory]]

* [[twisted ad-equivariant Tate K-theory]]

## References

### General

Due to

* [[Erik Verlinde]], _Fusion rules and modular transformations in 2D conformal field theory_, Nuclear Physics B __300__ (3): 360–376, (1988) (doi:10.1016/0550-3213(88)90603-7)

> We study conformal field theories with a finite number of primary fields with respect to some chiral algebra. It is shown that the fusion rules are completely determined by the behavior of the characters under the modular group. We illustrate with some examples that conversely the modular properties of the characters can be derived from the fusion rules. We propose how these results can be used to find restrictions on the values of the central charge and conformal dimensions.

See also

* [[Domenico Fiorenza]], [[Alessandro Valentino]], _$(3,2,1)$-TQFTs and Verlinde algebras_  ([MO question](http://mathoverflow.net/q/74593/381), [MO answer](http://mathoverflow.net/a/263829/381))

* Wikipedia, _[Verlinde algebra](https://en.wikipedia.org/wiki/Verlinde_algebra)_

### Relation to twisted ad-equivariant K-theory

On [[twisted ad-equivariant K-theory]] of [[compact Lie groups]] and the identification with the [[Verlinde ring]] of [[positive energy representations]] of their [[loop group]]:

* [[Daniel S. Freed]], _The Verlinde algebra is twisted equivariant K-theory_, Turk. J. Math 25 (2001) 159-167 ([arXiv:math/0101038](https://arxiv.org/abs/math/0101038))

{#FHT}
* [[Daniel S. Freed]], [[Michael Hopkins]], [[Constantin Teleman]],

  1. _[[Loop Groups and Twisted K-Theory]] I_,

     J. Topology, 4 (2011), 737-789

     ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906), [doi:10.1112/jtopol/jtr019](https://doi.org/10.1112/jtopol/jtr019))

  1. _[[Loop Groups and Twisted K-Theory]] II_,

     J. Amer. Math. Soc. 26 (2013), 595-644

     ([arXiv:0511232](http://arxiv.org/abs/math/0511232), [doi:10.1090/S0894-0347-2013-00761-4](https://doi.org/10.1090/S0894-0347-2013-00761-4))

  1. _[[Loop Groups and Twisted K-Theory]] III_,

     Annals of Mathematics, Volume 174 (2011) 947-1007

     ([arXiv:math/0312155](http://arxiv.org/abs/math/0312155), [doi:10.4007/annals.2011.174.2.5](http://dx.doi.org/10.4007/annals.2011.174.2.5))

* [[Daniel S. Freed]], [[Constantin Teleman]],

  _Dirac families for loop groups as matrix factorizations_,

  Comptes Rendus Mathematique, Volume 353, Issue 5, May 2015, Pages 415-419

  ([arxiv/1409.6051](http://arxiv.org/abs/1409.6051), [doi:10.1016/j.crma.2015.02.011](https://doi.org/10.1016/j.crma.2015.02.011))

Review:

* [[Valentin Zakharevich]], _K-Theoretic Computation of the Verlinde Ring_, thesis 2018 ([hdl:2152/67663](http://hdl.handle.net/2152/67663), [pdf](http://www.math.jhu.edu/~vzakharevich/research/Dissertation.pdf))

[[!redirects Verlinde rings]]

[[!redirects Verlinde algebra]]
[[!redirects Verlinde algebras]]


Verma module

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## Idea

Given a [[semisimple Lie algebra]] $\mathfrak{g}$ over a field $F$ with [[Borel subalgebra]] $\mathfrak{b}\subset\mathfrak{g}$ a _Verma module_ is the [[induced module]]
$$
U(\mathfrak{g})\otimes_{U(\mathfrak{b})} F_\lambda
$$
where $F_\lambda$ is one dimensional [[representation]] of $U(\mathfrak{b})$ corresponding to the [[character]] $\lambda$ of $\mathfrak{b}$.

This construction is used also for many other Lie algebras with triangular decomposition, for quantized enveloping algebras and many other generalizations.

## Related concepts

* [[BGG resolution]]

[[!redirects Verma modules]]


Veronika Hubeny

* [webpage](https://hubeny.physics.ucdavis.edu/)

## Selected writings

Introducing the [[fluid/gravity correspondence]]:

* [[Sayantani Bhattacharyya]], [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _Nonlinear Fluid Dynamics from Gravity_, JHEP 0802:045, 2008 ([arXiv:0712.2456](https://arxiv.org/abs/0712.2456))

More on the [[fluid/gravity correspondence]]:

* [[Veronika Hubeny]], _The Fluid/Gravity Correspondence: a new perspective on the Membrane Paradigm_, Class. Quant. Grav. 28:114007, 2011 ([arXiv:1011.4948](https://arxiv.org/abs/1011.4948))

* [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _The fluid/gravity correspondence_ ([arXiv:1107.5780](https://arxiv.org/abs/1107.5780))

category: people


Veronique Godin

* [website](http://math.ucalgary.ca/profiles/veronique-godin)

## Related entries

* [[string topology]], [[HCFT]]

* [[ribbon graph]]

category: people


Verschiebung morphism

(under construction)

##Idea

The *Verschiebung morphism* (German for 'shift') is an endomorphism, related to the [[Frobenius morphism]].

##Witt vectors

For a commutative ring $A$, let $\mathbb{W}(A)$ denote the set $1+T A[[T]]$
of power series in $A$ with constant term one. Given a positive integer $n$, the Verschiebung homomorphism is the additive homomorphism $V_n : \mathbb{W}(A) \to \mathbb{W}(A)$ defined by the formula $V_n(f(T)) = f(T^n)$.

[[!redirects Verschiebung morphisms]]


vertex

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

## Idea

A __vertex__ is a _[[point]]_ in a [[graph]] or [[simplicial set]] or similar. In the context of [[simplicial sets]] it is a 0-[[dimension|dimensional]] [[simplex]].

## Terminological remarks

Vertices are also sometimes called **points** (somewhat old-fashioned, e.g. [Harary, Chapter 2](#HararyGraphTheory)) or **nodes** (more frequently in [[computer science]]).  In specialized contexts other terms are used.  For instance, the two classes of vertices of a [[Petri net]] are called "places" or "species" and "transitions" or "reactions", respectively;
in an [[automaton]] the vertices are called "states";
and for [[quasicategories]] (particular simplicial sets) the vertices are also called [[objects]].

Historically, the use of "vertex" for a point of an abstract graph probably derives from the natural connections between [[graph theory]] and the theory of [[polyhedra]], which have "vertices".  One such connection is [[Ernst Steinitz|Steinitz's]] [[Steinitz's theorem|1916 theorem]] on the one-dimensional skeleta of [[polyhedra]] in three-dimensional [[Euclidean space]].

## Related concepts

* [[graph]], [[directed graph]], [[quiver]], [[digraph]]

* [[simplicial set]]

* [[edge]]

* **vertex**, [[simplex]], [[dendrex]]

## References

* Frank Harary: Graph Theory. Addison-Wesley. 1969
{#HararyGraphTheory}

[[!redirects vertex]]
[[!redirects vertices]]
[[!redirects node]]
[[!redirects nodes]]


vertex coloring

A (proper) **vertex coloring** of a [[graph]] is a labelling of its vertices with the property that no two vertices sharing an edge have the same label, while an **$n$-coloring** is a vertex coloring using at most $n$ distinct labels. The **chromatic number** $\chi(G)$ of a graph $G$ is the least $n$ such that $G$ has an $n$-coloring.

Vertex coloring may also be expressed in terms of graph homomorphisms: a $n$-coloring of $G$ is the same thing as a graph homomorphism $G \to K_n$ into the complete graph on $n$ vertices.
This formulation makes clear the duality between colorings and _cliques_: an $n$-[[clique]] in $G$ is the same thing as a homomorphism $K_n \to G$.

## Related concepts

* [[knot coloring]]

## References

* Chris Godsil and Gordon Royle (2001), _Algebraic Graph Theory_, Springer.

* Pavol Hell and Jaroslav Nešetřil (2001), _Graphs and Homomorphisms_, Oxford University Press.

Two posts by [[Tom Leinster]] at the n-Category Café on vertex-coloring and a categorical formulation of _Hedetniemi's conjecture_:

* [[Tom Leinster]], ["Colouring a Graph"](https://golem.ph.utexas.edu/category/2013/04/colouring_a_graph.html), n-Category Café post of April 12, 2013.

* [[Tom Leinster]], ["Graph Colouring and Cartesian Closed Categories"](https://golem.ph.utexas.edu/category/2014/12/graph_colouring_and_cartesian.html), n-Category Café post of December 5, 2014.

Other references:

* Wikipedia: [Graph coloring](https://en.wikipedia.org/wiki/Graph_coloring)


vertex operator algebra

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#### Algebra
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#### AQFT
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[[!include AQFT and operator algebra contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

Vertex operator algebras (or "vertex algebras", for short) are [[algebras]] with a product-operation parameterized by points in the [[complex plane]].

Vertex operator algebras equipped with an [[action]] of the [[Virasoro algebra]] encode the local ([[genus of a surface|genus-0]] behaviour) of [[2-dimensional conformal field theories]]. Here one may think of the [[complex plane]] as the [[Riemann sphere]] and of the $z$-parameterized product operation in the vertex algebras as being the [[n-point function|3-point function]] of the [[2d CFT]] with field insertions at the points 0, $z$ and $\infty$. In fact one vertex operator algebra encodes (only) one chiral/holomorphic half of such a genus-0 2d CFT; a full genus-0 2d CFT is given by the combination of two vertex operator algebras called a _[[full field algebra]]_.

The traditional definition of _vertex operator algebra_ (VOA) is long and tends to be somewhat unenlightening. But due to ([Huang 91](#Huang)) it is now known that vertex operator algebras have equivalently an [[FQFT]]-type characterization which manifestly captures this relation to [[n-point functions]] in the [[2d CFT]]:

There is a [[monoidal category]] or [[operad]] whose [[morphism|morphisms]] are conformal spheres with $n$-punctures marked as incoming and one puncture marked as outgoing (each puncture equipped with a conformally parametrized annular neighborhood). Composition of such spheres is by gluing along punctures. This can be regarded as a category $2Cob_{conf}^0$ of 2-dimensional genus-0 conformal [[cobordism|cobordisms]].

As shown by theorems by [[Yi-Zhi Huang]] and [[Liang Kong]], a vertex operator algebra is precisely a _[[holomorphic representation]]_ of this category, or [[algebra over an operad]] for this [[operad]] i.e. an genus-0 [[conformal field theory|conformal]] [[FQFT]], hence a [[monoidal functor]]

$$
  V : 2Cob_{conf}^0 \to Vect
$$

such that its component $V_1$ is a holomorphic function on the moduli space of conformal punctured spheres.

## Standard definition

(under construction) A __vertex algebra__ consists of the following data:

* nonnegatively graded complex vector space $V = \oplus_{n =0}^\infty V_n$
* vacuum vector $|0\rangle\in V_0$
* a shift operator $T: V\to V$
* operation $Y = Y(-,z) : V\to End V [ [z,z^{-1}] ]$

This data satisfy three axioms:

(vacuum)

(translation axiom)

(locality)

In practice, the most important case is when the shift $T$ is related to a conformal vector "the energy momentum tensor" $\omega$ whose Fourier components are related to the generators $L_i$ of the [[Virasoro Lie algebra]] $Vir$. In that case we talk about a __conformal vertex algebra__.

There is a notion of a module over a vertex algebra. A conformal vertex algebra $A$ is said to be __rational__ if every $A$-module is completely reducible. Then it follows that there are only finitely many nonisomorphic simple $A$-modules.

## Properties
 {#Properties}

### Category of vertex operator algebras
 {#CategoryOfVOAs}

Vertex operator algebras naturally form a [[category]] (see section 2.4 of ([FrenkenHuangLepowsky](#FrenkenHuangLepowsky)). This is naturally a [[monoidal category]] with respect to [[tensor product]] of VOAs (section 2.5).

This is [[equivalence of categories|equivalent]] to

* the category of algebras over the holomorphic punctured sphere operad ([Huang](#HuangCFT));

* the category of vertex operator coalgebras ([Hubbard](#Hubbard)).

### Modular category of modules over a VOA
 {#ModularTensorCategories}

The [[category]] of [[modules]]/[[representations]] over a given vertex operator algebra is a [[modular tensor category]], ([Huang](#Huang))

### Goddard-Thorn theorem

* [[Goddard-Thorn theorem]]

### Relation to $A_\infty$-algebras and RG fixed points
 {#RelationToAInfinityAlgebras}

A [[functor]] from the category of  [[BRST complex|BRST]]-VOAs to that of [[A-infinity algebra]]s is described in

* [[Anton Zeitlin]], _Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory_ ([arXiv](http://arxiv.org/abs/0910.3652))

and argued to algebraically encode the [[effective QFT|effective]] [[string theory]] background encoded by the [[CFT]] given by the VOA.

### Relation to conformal nets

Subject to some conditions, from a vertex operator algebra one may induce a [[conformal net]] and conversely ([Carpi-Kawahigahshi-Longo-Weiner 15](#CarpiKawahigahshiLongoWeiner15), [Carpi 16](#Carpi16)) also ([Gui 18](#Gui18)).

## Examples

A class of examples are _[[current algebras]]_ .

A database of examples is given by ([Gannon-Höhn](#GannonHoehn)).

The [[Moonshine]] module over the [[Griess algebra]] admits the structure of a vertex operator algebra, which has

* rank 24;

* is a self-[[dual object]] in the category of VOAs;

* has trivial degree-1 subspaces.

A conjecture due to

* [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, _Vertex operator algebras and the monster_, Pure and Applied Mathematics __134__, Academic Press, New York 1998.

says that the Moonshine VOA is, up to isomorphism the unique VOA with these properties. See at _[[monster vertex algebra]]_.

## Variants

* [[Poisson vertex algebra]]

* [[rational vertex operator algebra]]

* [[super vertex operator algebra]]

* [[sheaf of vertex operator algebras]], [[vertex operator algebroid]]

## Related concepts

* [[operator product expansion]], [[conformal bootstrap]]

* [[affine Lie algebra]]

* [[conformal field theory]],

* [[factorization algebra]],

* [[automorphism of a vertex operator algebra]]

## References

### General

* [[Victor Kac]], *Vertex algebras for beginners*, Amer. Math. Soc., Univ. Lecture Series **10** (1998)  [[ams:ulect-10-r](https://bookstore.ams.org/ulect-10-r), [ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0924.17023&format=complete)]

* [[Edward Frenkel]], [[David Ben-Zvi]]: _Vertex algebras and algebraic curves_, Math. Surveys and Monographs __88__, AMS (2001) xii+348 pp. [[review](http://www.ams.org/journals/bull/2002-39-04/S0273-0979-02-00955-2/S0273-0979-02-00955-2.pdf), [ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1106.17035&format=complete)]

* {#FrenkenHuangLepowsky} [[Igor Frenkel]], [[Yi-Zhi Huang]], [[James Lepowsky]], _On Axiomatic approaches to Vertex Operator Algebras and Modules_, Memoirs of the AMS Vol 104, No 494 (1993)

Brief review:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], pp. 4 of: *Algebraic structures in two-dimensional conformal field theory*, Encyclopedia of Mathematical Physics [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

See also:

* [what-is-the-motivation-for-a-vertex-algebra](http://mathoverflow.net/questions/53988/what-is-the-motivation-for-a-vertex-algebra) - a MathOverflow discussion on the motivation

* [The online database of Vertex Operator Algebras and Modular Categories](http://www.math.ksu.edu/~gerald/voas)

* [[Keith Hubbard]], _The duality between vertex operator algebras and coalgebras, modules and comodules_, [[pdf](http://www.faculty.sfasu.edu/hubbardke/duality.pdf)]

Relation to [[sporadic groups]]:

* [[John Duncan]], _Vertex operator algebras and sporadic groups_, [pdf](http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.1306v1.pdf); _Moonshine for Rudvalis's sporadic group I_, [pdf](http://arxiv.org/PS_cache/math/pdf/0609/0609449v2.pdf)

Vertex operator algebras over [[non-archimedean fields]]:

* [[Victor Kac]], _Non-Archimedean vertex algebras_ [[arXiv:2304.09651](https://arxiv.org/abs/2304.09651)]

Classification of strongly rational holomorphic vertex operator algebras of [[central charge]] $\leq 24$ (of relevance in [[heterotic string theory]] and [[monstrous moonshine]]):

* [[Adrian Norbert Schellekens]], *Meromorphic $c = 24$ Conformal Field Theories*, Commun. Math. Phys. **153** (1993) 159-186 [[doi:10.1007/BF02099044](https://doi.org/10.1007/BF02099044)]

* [[Chongying Dong]], [[Geoffrey Mason]], *Holomorphic vertex operator algebras of small central charges*, Pacific Journal of Mathematics **213** 2 (2004) 253--266 [[arXiv:math/0203005](https://arxiv.org/abs/math/0203005), [doi:10.2140/pjm.2004.213.253](http://dx.doi.org/10.2140/pjm.2004.213.253)]

* Philip Boyle Smith, Ying-Hsuan Lin, Yuji Tachikawa, Yunqin Zheng, *Classification of chiral fermionic CFTs of central charge $\leq 16$* [[arXiv:2303.16917](https://arxiv.org/abs/2303.16917)]

* Brandon C. Rayhaun, *Bosonic rational conformal field theories in small genera, chiral fermionization, and symmetry/subalgebra duality* [[arXiv:2303.16921](https://arxiv.org/abs/2303.16921)]

* [[Gerald Höhn]], [[Sven Möller]], *Classification of self-dual vertex operator superalgebras of central charge at most 24* [[arXiv:2303.17190](https://arxiv.org/abs/2303.17190)]

### As algebras over the holomorphic sphere operad
 {#AsOperadAlgebras}

The original article with the interpretation of vertex operator algebras as holomorphic algebras over the holomorphic punctured sphere operad is

* {#Huang91} [[Yi-Zhi Huang]], _Geometric interpretation of vertex operator algebras_, Proc. Natl. Acad. Sci. USA __88__ (1991) pp. 9964-9968 ([doi:10.1073/pnas.88.22.9964](https://doi.org/10.1073/pnas.88.22.9964))

A standard textbook summarizing these results is

* {#HuangCFT} [[Yi-Zhi Huang]], _Two-dimensional conformal geometry and vertex operator algebras_, Progr. in Math. Birkhauser (1997) [gbooks](http://books.google.hr/books?isbn=0817638296)

As mentioned in the [acknowledgements](http://books.google.com/books?id=SUUdknTpYjEC&pg=PR11&lpg=PR11&dq=%22vertex+operator+algebras%22+%22trimble%22&source=bl&ots=v3GHx_ra2M&sig=TW5MzAtDi4n_gJjvUHb6eELoAXQ&hl=en&ei=9jdnSoXqKJiCtge6xqDuDw&sa=X&oi=book_result&ct=result&resnum=4) there, [[Todd Trimble]] and [[Jim Stasheff]] had a hand in making the operadic picture manifest itself here.

See also:

* [[Yi-Zhi Huang]], _Meromorphic open-string vertex algebras_, J. Math. Phys. __54__, 051702 (2013) [[doi:10.1063/1.4806686](http://dx.doi.org/10.1063/1.4806686)]

Other operadic approaches are known, e.g. an earlier one in:

* Bojko Bakalov, Alessandro D'Andrea, [[Victor Kac]], _Theory of finite pseudoalgebras_, Adv. Math. __162__ (2001), no. 1, 1--140, [MR2003c:17020](http://www.ams.org/mathscinet-getitem?mr=2003c:17020)

and even earlier treatments via coloured [[operads]] and [[generalized multicategories]], for references (Snydal, Soibelman, Beilinson-Drinfeld) see [[relaxed multicategory]].

More recently Huang's student [[Liang Kong]] has been developing these results further, generalizing them to open-closed CFTs and to non-chiral, i.e. completely general CFTs. See

* {#Kong08} [[Liang Kong]], _Open-closed field algebras_ Commun. Math. Physics. 280, 207-261 (2008), [math.QA/0610293](http://arxiv.org/abs/math/0610293).

### As factorization algebras

Discussion of vertex operator algebras as [[factorization algebras of observables]] is in section 6.3 and section 6.5 of

* [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis ([pdf](http://math.berkeley.edu/~gwilliam/thesis.pdf))
 {#Gwilliam}

* Emily Cliff, _Universal factorization spaces and algebras_, [arxiv:1608.08122](https://arxiv.org/abs/1608.08122)

>  We introduce categories of weak factorization algebras and factorization spaces, and prove that they are equivalent to the categories of ordinary factorization algebras and spaces, respectively. This allows us to define the pullback of a factorization algebra or space by an \'etale morphism of schemes, and hence to define the notion of a universal factorization space or algebra. This provides a generalization to higher dimensions and to non-linear settings of the notion of a vertex algebra.

### As coalgebras over a comonad
 {#ReferencesAsCoalgebrasOverAComonad}

On vertex operator algebras as [[coalgebras over a comonad]]:

* [[Jethro van Ekeren]], *A quantum field comonad* [[arXiv:2305.18223](https://arxiv.org/abs/2305.18223)]

### Relation to modular tensor categories

The [[representation categories]] of (rational) vertex operator algebras ([[modular tensor categories]]) are discussed in

* {#Huang} [[Yi-Zhi Huang]], *Vertex operator algebras, the Verlinde conjecture and modular tensor categories*, Proc. Nat. Acad. Sci. **102** (2005) 5352-5356 $[$[arXiv:math/0412261](http://arxiv.org/abs/math/0412261), [doi:10.1073/pnas.0409901102](https://doi.org/10.1073/pnas.0409901102)$]$

* {#GannonHoehn} [[Terry Gannon]], [[Gerald Höhn]], Hiroshi Yamauchi, _[The online database of Vertex Operator Algebras and Modular Categories](http://www.math.ksu.edu/~gerald/voas/)_

### As chiral algebras

An algebrogeometric version is due Beilinson and Drinfel'd and called the [[chiral algebra]].

* E. Frenkel, N. Reshetikhin, _Towards deformed chiral algebras_, [q-alg/9706023](http://arXiv.org/abs/q-alg/9706023)

* Ruthi Hortsch, [[Igor Kriz]], Ales Pultr, _A universal approach to vertex algebras_, [arxiv/1006.0027](http://arxiv.org/abs/1006.0027)

Much algebraic insight to algebaric structures in CFT is in unfinished notes

* [[A. Beilinson]], [[B. Feigin]], B. Mazur, _Notes on conformal field theory_, <http://www.math.sunysb.edu/~kirillov/manuscripts.html>

### Relation to 2d conformal field theory

Relation to [[2d CFT]]:

* [[Yi-Zhi Huang]], _Two-dimensional conformal geometry and vertex operator algebras_  Birkhäuser (1997) ([doi:10.1007/978-1-4612-4276-5](https://link.springer.com/book/10.1007/978-1-4612-4276-5))

For [[orbifolds]]:

* [[Yi-Zhi Huang]], _Representation theory of vertex operator algebras and orbifold conformal field theory_ ([arXiv:2004.01172](https://arxiv.org/abs/2004.01172))

Relation specifically to [[conformal nets]]:

* {#CarpiKawahigahshiLongoWeiner15} [[Sebastiano Carpi]], [[Yasuyuki Kawahigashi]], [[Roberto Longo]], Mihály Weiner, _From vertex operator algebras to conformal nets and back_, Memoirs of the American Mathematical Society **254** 1213 (2018) [[doi:10.1090/memo/1213](https://doi.org/10.1090/memo/1213), [arXiv:1503.01260](http://arxiv.org/abs/1503.01260)]

* {#Carpi16} [[Sebastiano Carpi]], _Operator algebras and vertex operator algebras_, Contribution to the Proceedings of the 14th Marcel Grossmann Meeting - MG14 (Rome, 2015) [[arXiv:1603.06742](http://arxiv.org/abs/1603.06742), [doi:10.1142/9789813226609_0508](https://doi.org/10.1142/9789813226609_0508)]

and for [[super vertex operator algebras]]:

* [[Sebastiano Carpi]], [[Tiziano Gaudio]], [[Robin Hillier]], *From vertex operator superalgebras to graded-local conformal nets and back* [[arXiv:2304.14263](https://arxiv.org/abs/2304.14263)]

See also

* James E. Tener, _Representation theory in chiral conformal field theory: from fields to observables_ ([arXiv:1810.08168](https://arxiv.org/abs/1810.08168))

Relation of the corresponding [[ribbon categories]]:

* {#Gui18} Bin Gui, _Categorical extensions of conformal nets_ ([arXiv:1812.04470](https://arxiv.org/abs/1812.04470))

### Deformations

There is an interesting theory of deformation quantization of VOAs from

* [[Pavel Etingof]], [[David Kazhdan]], _Quantization of Lie bialgebras. V. Quantum vertex operator algebras_,
Selecta Math. (N.S.) 6 (2000), no. 1, 105--130, [MR2002i:17022](http://www.ams.org/mathscinet-getitem?mr=2002i:17022)

Deformation quantization of chiral algebras are studied by

* [[Dmitry Tamarkin]], _Deformations of chiral algebras_, Proceedings of the ICM, Beijing 2002, vol. 2, 105--118

A class of "free" vertex algebras are also quantized using [[Batalin-Vilkovisky formalism]], with results important for understanding [[mirror symmetry]], in

* [[Si Li]], _Vertex algebras and quantum master equation_ ([arxiv/1612.01292](https://arxiv.org/abs/1612.01292))

[[!redirects vertex algebra]]
[[!redirects vertex algebras]]
[[!redirects Vertex operator algebra]]

[[!redirects vertex operator algebras]]

[[!redirects VOA]]
[[!redirects VOAs]]


vertex operator algebroid

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* table of contents
{:toc}

## Idea

A [[sheaf of vertex operator algebras]] which in degree-0 coincides with the [[structure sheaf]] of the base space is a kind of "[[horizontal categorification]]" of [[vertex operator algebra]], hence also called a _vertex operator algebroid_ or _vertex algebroid_ for short.

## References

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

* [[Paul Bressler]], _Vertex algebroids I_ ([arXiv:math/0202185](http://arxiv.org/abs/math/0202185))

[[!redirects vertex operator algebroids]]

[[!redirects vertex algebroid]]
[[!redirects vertex algebroids]]


vertical categorification

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# Contents
* table of contents
{:toc}

## Idea

Roughly speaking, _vertical categorification_ is a procedure in which structures are generalized from the context of [[set theory]] to [[category theory]] or from category theory to [[higher category theory]].

What precisely that means may depend on circumstances and authors, to some extent. The following lists some common procedures that are known as categorification. They are in general different but may in cases lead to the same categorified notions, as discussed in the examples.

See also [[categorification in representation theory]].

## Variants

### As a section of decategorification

What has a more specific definition is the process of [[decategorification]]: this concretely quotients out [[k-morphism]]s in a [[n-category]] to produce a $k$-category for some $k \lt n$, in particular a (possibly large) [[set]] (the set of isomorphism classes or equivalence classes of objects) when $k = 0$.

One may understand _vertical categorification_ as any operation that is a [[section]] of this [[decategorification]] operation, and many examples in the literature are of this kind.

#### Examples

* The maybe archetypical example of categorification as something that is taken by decategorification to the identity operation is the categorification of the [[set]] $\mathbb{N}$ of [[natural numbers]] to the [[category]] [[FinSet]] of [[finite sets]]:

  The set $\mathbb{N}$ is a [[0-category]], while $FinSet$ is a [[1-category]]. The [[decategorification|isomorphism classes]] of $FinSet$ however are in canonical bijection with the elements of $\mathbb{N}$:

  $$ \mathbb{N} \simeq FinSet_\sim .$$

  So $\mathbb{N}$ is the [[decategorification]] of $FinSet$ and accordingly $FinSet$ is a (vertical) categorification of $\mathbb{N}$.

  From the point of view of [[category theory]], there is some justification to the converse of this statement: the study of the natural numbers is nothing but the study of the isomorphism classes in $FinSet$. In a way, the very notion of **counting** is about this. This point is made nicely in [BaDo98](http://arxiv.org/abs/math.QA/9802029).

  But there are other categorifications of the natural numbers, for instance the category [[FinDimVect]] of [[finite dimensional vector spaces]] (over any given [[field]]). This is not equivalent to [[FinSet]], but still, its set of [[isomorphism classes]] is also canonically in [[bijection]] with the [[natural numbers]]:

  $$ \mathbb{N} \;\simeq\; FinDimVect_{/\sim} \,. $$

### As internalization in $n Cat$

Common structures such as [[group]]s, [[ring]]s, etc. may be defined entirely diagrammatically as a collection of objects, morphisms and [[2-morphisms]] in a [[category]] like [[Set]] (the 2-morphisms then are the identities required to hold between the given morphisms).

For instance a [[group]] is an object $G$ in [[Set]] equipped with a morphism $\mu : G \times G \to G$ and so on, and equipped with a 2-morphism from $\mu \circ Id_G \times \mu $ to $\mu \circ \mu \times Id_G$.

In such diagrammatic incarnation, these definitions may be [[internalization|internalized]] into other categories. For instance a group internal to [[Diff]] is a [[Lie group]].

But similarly one can also internalize in categories of higher categories. Let [[Cat]] be the category of (small) categories, regarded as an ordinary category. Then a group internal to [[Cat]] is a [[strict 2-group]]. This is thought of as a notion of a categorified group.

And under the decategorification functor $Cat \to Set$ every categorified group $\mathbf{G}$ in this sense maps to an ordinary group $G$ and one could speak of $\mathbf{G}$ being "a categorification" of $G$. But notice again how highly non-unique such categorification is. In particular, every ordinary group may trivially be regarded as a strict 2-group. As such every ordinary group is at the same time one of its own categorifications, in this sense.

Not every [[2-group]] is a [[strict 2-group]] that is a group object internal to the 1-category [[Cat]]. In general, a [[2-group]] is a _weak_ group object in [[Cat]]: where in [[Set]] all 2-morphisms in the diagrammatic description of the concept of group necessarily had to be identities, in [[Cat]] they could be taken to be non-trivial. But if they are, one will usually want 3-morphisms (which now are necessarily identities) to relate various combinations of these 2-morphisms.

One speaks of this process of categorification using weak internalization of diagrammatic descriptions as categorification of structures _up to coherent higher equivalences_ or _up to coherent higher homotopies_ . One way to make this systematic is discussed below.

#### Examples

In the sense of "weakly internalizing in a higher categorical context" we have the following examples of categorification:

* the categorification of [[sheaf]] is [[stack]], 2-stack, 3-stack, ... [[∞-stack]].

* accordingly, the categorification of ([[sheaf]]-)[[topos]] is ([[stack]]-)2-topos, ... [[∞-stack]] [[(∞,1)-topos]]

* using these [[∞-stack]] [[(∞,1)-topos]] notions one can then speak of [[∞-space]]s [[Lie ∞-groupoid]]s and the like.

* Since [[simplicial object]]s in any category usually [[model category|model]] [[higher category theory|higher categorial]] structures, many simplicial objects may be regarded as secretly being objects in [[higher category theory]]. For instance [[simplicial group]]s, being [[Kan complex]]es, may be thought of as [[∞-groupoid]]s equipped with a strict group structure, i.e. as _strict $\infty$-groups_. By the [[Dold-Kan correspondence]] it follows then that many objects in [[homological algebra]] such as [[chain complex]]es are equivalent incarnations of these simplicial (abelian) groups. This way large parts of [[homological algebra]] may be regarded as being secretly about "categorified linear algebra" in some sense. For instance the [[dg-category]] of [[chain complex]]es in many contexts plays the role of a [[stable (∞,1)-category]] of "$(\infty,1)$-vector spaces". Notably [[L-∞-algebra]]s may be thought of as [[Lie algebra]]s weakly internalized in $(\infty,1)$-vector spaces understood in this sense.

* When understanding higher linear algebra in this sense, there are important $\infty$-categorifications of notions like integral transforms, [[Fourier transform]]s etc. Examples of such categorified linear algebra are [[Fourier-Mukai transform]]s or more generally [[integral transform]]s. A general theory of this is described at [[geometric ∞-function theory]]. Also [[geometric Langlands]] duality fits into this context.

* [categorified Gram-Schmidt process](Gram-Schmidt+process#CategorifiedGramSchmidtProcess)

* [[(∞,1)-category theory|(∞,1)-]]categorification of [[differential calculus]] is [[Goodwillie calculus]]

  [[(∞,1)-category theory|(∞,1)-]]categorification of [[derivative]] is [[Goodwillie derivative]]

  [[(∞,1)-category theory|(∞,1)-]]categorification of [[chain rule]] is [[Goodwillie chain rule]]

  etc.

### As homotopy coherent resolution

The above process of categorification by coherently weak internalization into higher categorical contexts can be made systematic at least in some cases.

If the structures being defined are [[algebra over an operad|algebras over]] an [[operad]] $T$ one may think of regarding $T$ as an [[(∞,1)-operad]]s and then consider the structures of its algebras as algebras over an $(\infty,1)$-operad. These are infinity-categorified versions of the original structures.

#### Examples

For instance this way

* the notion of ordinary associative [[algebra]] is sent to that of $A_\infty$-[[A-∞-algebra|algebra]]

* the notion of ordinary [[Lie algebra]] to $L_\infty$-[[L-∞-algebra|algebra]].

## Contrast to horizontal categorification

Some people also speak of [[horizontal categorification]] as categorification. This is to be distinguished from vertical categorification.

Some people just say 'oidification' for horizontal categorification, in which case it is consistent to speak of vertical categorificaton as just _categorification_ .

## Homotopification versus laxification

(Vertical) categorification can often be usefully decomposed into two operations.

* In **groupoidal categorification**, which may also be called **homotopification** or **groupoidification** (although the latter term also has a [[groupoidification|different meaning]], we allow objects to come with automorphisms, those automorphisms to come with automorphisms, and so on.  In the limit, this involves replacing sets by [[∞-groupoids]] or [[homotopy types]] (hence the name "homotopification").

* In **directed categorification**, which may also be called **directification** or **laxification**, we allow morphisms that were previously required to be invertible to instead be noninvertible (i.e. "directed").

If you like [[negative thinking]], then instead of saying that categorification 'replaces [[sets]] by [[categories]]' (to quote Wikipedia), you can say that we replace [[truth values]] by sets, especially the truth values of [[equality|equations]].  That is, we acknowledge that there may be many different ways in which something may be true, and in particular many different ways in which two things may be the same.  And then it is meaningful to ask whether two ways in which these things are the same are the same way (and if so, whether two ways that *they* are the same are the same way, etc).

However, when we apply "replace truth values by sets" to the truth values of the equality relation of a set, we end up with a groupoid, since the equality of a set is symmetric.  Thus, while two [[elements]] of a set simply may (or may not) be equal, two [[objects]] of a [[groupoid]] may be [[isomorphic]] in many different ways. And while two [[parallel morphisms|parallel isomorphisms]] in a groupoid may be equal, two parallel [[equivalences]] in a $2$-groupoid may be isomorphic in many different ways.  Thus, this gives us groupoidal categorification, or homotopification.

To get from groupoids to categories, we need to also allow things which were previously invertible to be noninvertible, i.e. perform "directification."  We could also do this first starting from a set, obtaining a poset (in which the symmetric relation "is equal to" has been replaced by the non-symmetric one "is less than or equal to").  Then when we homotopify a poset, we get a category: while one element $x$ of a [[poset]] may precede an element $y$, there may be many different [[morphisms]] from one object $x$ of a category to an object $y$.

This can also be understood naturally in the language of [[(n,r)-categories]].  Recall that an $(n,r)$-category can be defined as an [[∞-category]] in which all cells above dimension $r$ are invertible, and all cells above dimension $n$ are trivial.  Thus, groupoidal categorification can be understood as increasing $n$ but keeping $r$ constant, while directification can be understood as increasing $r$ but keeping $n$ constant.

## Related entries

* [[(∞,1)-categorification]]

* [[higher structures]]

* [[categorification in representation theory]]

* [[categorified Dold-Kan correspondence]]

## References

The terminology goes back to

* {#CraneYetter96} [[Louis Crane]], [[David Yetter]],
_Examples of categorification_ ([arXiv:q-alg/9607028](http://arxiv.org/abs/q-alg/9607028))

and was further amplified in

* {#BaezDolan98} [[John Baez]], [[James Dolan|James Dolan]], _Categorification_, in [[Ezra Getzler]], [[Mikhail Kapranov]] (eds.) _Higher Category Theory_, Contemp. Math. 230, American Mathematical Society, Providence, Rhode Island, 1998, pp. 1-36 ([arXiv:math.QA/9802029](http://arxiv.org/abs/math.QA/9802029))

Exposition:

* [[John Baez]], _[What is categorification?](http://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html)_

Lecture notes:

* [[Alistair Savage]], *Introduction to categorification* [[arXiv:1401.6037](https://arxiv.org/abs/1401.6037), slides:[pdf](https://alistairsavage.ca/talks/2014-savage-intro-to-categorification.pdf)]

* Dylan Madden, *Introductory Categorification* (2015) [[pdf](https://warwick.ac.uk/fac/sci/maths/people/staff/madden/1102171m.pdf)]

  > (elementary introduction to [[category theory]] leading up to the example of the categorified [[Weyl algebra]])

* [[Mikhail Khovanov]] (notes by [[You Qi]]), *Introduction to categorification*, lecture notes, Columbia University (2010, 2020) [[web](https://www.math.columbia.edu/~khovanov/cat2020/), [web](https://you-qi2121.github.io/mypage/categorificationnotes.html), full:[pdf](https://www.dropbox.com/scl/fi/wdesax1c8il6tgwbi20t4/KhovanovYouQi-Categorification.pdf?rlkey=l5cm3khnzu604ijdnl06o89od&dl=0)]

  > (focus on [[link homology]])

* [[Yuri Ximenes Martins]], part II of _Categorical and Geometrical Methods in Physics_, Master’s thesis, UFMG, 2018. ([pdf](https://repositorio.ufmg.br/bitstream/1843/32053/1/dissertacao_yuri_FINAL.pdf), [slides](https://math-phys.group/wp-content/uploads/2020/08/slides_yuri.pdf))

* [[Yuri Ximenes Martins]], Chapter 4 of _Introdução à Teoria da Homotopia Abstrata_, NEA, 2018. ([arXiv:2008.05302](https://arxiv.org/abs/2008.05302), [hal-02908896v1](https://hal.archives-ouvertes.fr/hal-02908896v1))

A general notion of categorification for structures defined by [[cartesian monads]], which specializes to produce weak [[n-categories]] in the sense of Leinster, can be found here:

* [[D. Bourn]], J. Penon, _Catégorification de structures définies par monade cartésienne_, Cahiers de Topologie et Géométrie Différentielle Catégoriques __46__, no. 1 (2005), p. 2-52, [numdam](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_2005__46_1_2_0).

[[!redirects vertical categorification]]
[[!redirects vertical categorifications]]

[[!redirects categorification]]
[[!redirects categorifications]]
[[!redirects categorify]]
[[!redirects categorified]]

[[!redirects ∞-categorification]]
[[!redirects ∞-categorifications]]

[[!redirects groupoidal categorification]]
[[!redirects groupoidal categorifications]]


vertical composition

+-- {: .rightHandSide}
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#### Higher category theory
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=--
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=--

#Contents#
* table of contents
{:toc}

## Definition

In a [[2-category]] (and more generally in [[higher category theory]]) [[2-morphism]]s have a composition along the 1-morphisms that they go between.

$$
  \array{
     & \nearrow &\Downarrow& \searrow
     \\
     x &&\to&& y
     \\
      & \searrow &\Downarrow& \nearrow
  }
  \,.
$$

This is in contrast to the other composition operation, along objects, which is called [[horizontal composition]].

## Examples

* In the 2-category [[Cat]], 2-morphisms are [[natural transformation]]s and vertical composition is the composition of these, the composition in the corresponding [[functor category]].

* For $A$ an [[abelian group]] and $\mathbf{B}^2 A$ its double [[delooping]] [[2-groupoid]], vertical composition (as well as horizontal composition) is given by the group product operation in $A$.

## References

For the notion of vertical composition of [[natural transformations]] (ie. composition in [[functor categories]]):

* [[Saunders MacLane]], §II.4 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

[[!redirects vertical compositions]]


vertical differential form

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## Definition

### In differential geometry

Let $\pi : P \to X$ be a [[bundle]] in the category [[Diff]] of [[smooth manifold]]s.

The [[dg-algebra]] $\Omega^\bullet_{vert}(P)$ of _vertical differential forms_ on $P$ is the quotient of the [[de Rham complex]] dg-algebra $\Omega^\bullet(P)$ of all forms on $P$, by the dg-ideal of [[horizontal differential forms]], hence of all those forms that vanish when any one vector in their arguments is a [[vertical vector field]] in that it is in the [[kernel]] of the [[differential]] $d \pi : T P \to T X$.

For a trivial bundle $P = X \times F$ the underlying complex of $\Omega^\bullet_{vert}(P)$ is $\wedge^\bullet_{C^\infty(X \times F)} \Gamma(T^* F)$.

## Related concepts

* [[horizontal differential form]]

* [[Lie integration]]

## References

* Philippe Bonnet, Alexandru Dimca, _Relative differential forms and complex polynomials_, Bulletin des Sciences Mathématiques **124** Issue 7 (2000) pp 557-571, doi:[10.1016/s0007-4497(00)01055-1](https://doi.org/10.1016/s0007-4497%2800%2901055-1), ([author pdf](https://web.archive.org/web/20150326174153/https://jones.math.unibas.ch/~bonnet/Publications/Bullscience.pdf))

[[!redirects vertical differential forms]]


vertical tangent Lie algebroid

For $\pi : Y \to X$ a morphism of smooth [[manifold]]s, the **vertical tangent Lie algebroid** with respect to $\pi$ is the sub-[[Lie algebroid]] $T_{vert} Y \hookrightarrow T X $ of the [[tangent Lie algebroid]] $T Y$ of $Y$ whose [[Chevalley-Eilenberg algebra]] is the [[dg-algebra]] of horizontal differential forms on $Y$ with respect to $\pi$.

For more details on the construction see the examples section at [[exterior differential system]]s.

The vertical tangent Lie algebroid is the infinitesimal version of the [[schreiber:Ehresmann ∞-connection|vertical path ∞-groupoid]]. It plays a central role in the context of [[Ehresmann connection]]s and [[schreiber:Cartan-Ehresmann ∞-connection|Cartan-Ehresmann ∞-connections]].


vertical transformation

# Contents
* tic
{: toc}

## Idea

A **vertical transformation** is an analogue of a [[natural transformation]] which goes between [[double functors]] of [[double categories]], and whose components are vertical arrows and squares.  There is a dual notion of **horizontal transformation.

## Definition

If $C$ and $D$ are strict double categories regarded as [[internal categories]] in $Cat$ and $F,G\colon C\to D$ are double functors regarded as internal functors in $Cat$, then a transformation between them is simply an internal natural transformation in $Cat$.  Whether this is a vertical or horizontal transformation depends on how we identify double categories with internal categories in $Cat$ (there being two ways).

More explicitly, a vertical transformation $\alpha\colon F\to G$ consists of

* For every object $c\in C$, a vertical arrow
  $$ \array{ F c \\ \downarrow ^{\alpha_c} \\ G c}$$
  in $D$, which are natural with respect to vertical composition of vertical arrows in $C$.

* For every horizontal arrow $p\colon c_1 \to c_2$ in $C$, a square
  $$\array{F c_1 & \overset{F p}{\to} & F c_2\\
    ^{\alpha_{c_1}}\downarrow & \Downarrow^{\alpha_p}& \downarrow^{\alpha_{c_2}}\\
    G c_1& \underset{G p}{\to} & G c_2}$$
  in $D$, which are natural with respect to vertical composition of squares in $C$.

* For each $c\in C$, if $1_c\colon c\to c$ is its horizontal identity, then the square $\alpha_{1_c}$ is equal to $1_{\alpha_c}$, the identity square on the arrow $\alpha_c$.

* For $p\colon c_1\to c_2$ and $q\colon c_2\to c_3$, the horizontal composite of $\alpha_p$ and $\alpha_q$ is equal to $\alpha_{q p}$.

The notion of horizontal transformation is dual.

## The double category of double functors

Another characterization of transformations between double categories comes from observing that the 1-category $DblCat$ is [[cartesian closed category|cartesian closed]], and so any two double categories have an exponential $D^C$.  The objects of $D^C$ are double functors, its vertical arrows are vertical transformations, and its horizontal arrows are horizontal transformations.  Its squares are a sort of "square modification" relating a pair of vertical and a pair of horizontal transformations.

## Generalizations

It is easy to modify the explicit definition to handle the cases when $C$ and $D$ are weak in one direction or the other, and/or when $F$ and $G$ are pseudo functors in one direction or the other, by composing with appropriate coherence constraints.  In this way, we obtain many [[2-categories]] of double categories.

It is also easy to define *vertical* transformations between double functors which are *horizontally* lax or colax, and dually.  In fact, given double categories $C,D,C',D'$, lax functors $F\colon C\to D$ and $F'\colon C'\to D'$, and colax functors $G\colon C\to C'$ and $G'\colon D\to D'$, we can define a vertical transformation having the shape
$$\array{C & \overset{F}{\to} & D \\
  ^G\downarrow & \Downarrow& \downarrow^{G'}\\
  C'& \underset{F'}{\to} & D'}$$
despite the fact that the composites $G' \circ F$ and $F'\circ G$ do not exist as double functors of any sort.  Such transformations are the squares of a large double category $Dbl$ whose objects are double categories, whose horizontal arrows are lax functors, and whose vertical arrows are colax functors.  This, in turn, is a special case of a construction which works for algebras over any [[2-monad]].

Finally, we can also define vertical transformations between functors of (horizontally) [[virtual double categories]].

## References

* [[Robert Paré]], *Yoneda theory for double categories*, TAC, 2011 ([pdf](http://emis.library.cornell.edu/journals/TAC/volumes/25/17/25-17.pdf))

[[!redirects horizontal transformation]]
[[!redirects vertical transformations]]
[[!redirects horizontal transformations]]
[[!redirects vertical natural transformation]]
[[!redirects horizontal natural transformation]]
[[!redirects vertical natural transformations]]
[[!redirects horizontal natural transformations]]


vertical vector field

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#Contents#
* table of contents
{:toc}

## Definition

### Vertical vector fields

Let $\pi : P \to X$ be a [[bundle]] in the [[category]] of [[SmoothManifolds]]. A [[vector field]] $v \in \Gamma(T P)$ is _vertical_ with respect to this bundle if it is in the [[kernel]] of the [[derivative]] $d \pi \colon T P \to T X$.

### Vertical tangent bundle
 {#VerticalTangentBundle}

The collection of vertical vectors forms the _vertical tangent bundle_ inside the full [[tangent bundle]], typically denoted $T_\pi P$.

For $\pi \colon P \to B$ a [[smooth function]] between [[smooth manifolds]], its _vertical tangent bundle_ is the [[fiber]]-wise [[kernel]] of the [[differential]] $d \pi$, as shown in the following diagram (e.g [Tu 17 (27.4)](#Tu17),  [Berglund 20, p. 16](#Berglund20)):

\begin{tikzcd}[column sep=normal, row sep=small]
  T_{\pi} P
  \ar[
   r,
   "
    \mathrm{ker}(d \pi)
   "
  ]
  &
  T P
  \ar[
    r,
    dashed,
    "d \pi"
  ]
  \ar[ddr]
  &
  \pi^\ast
  \big(
    T B
  \big)
  \ar[
    r
  ]
  \ar[
    ddr,
    phantom,
    "
      \mbox{\tiny\textrm(pb)}
    "{
      description
    }
  ]
  \ar[dd]
  &
  T B
  \ar[
    dd
  ]
  \\
  \\
  &
  &
  P
  \ar[
    r,
    "\pi"{
      below
    }
  ]
  &
  B
\end{tikzcd}

### Horizontal differential forms

A [[differential form]] on $P$ is a [[horizontal differential form]] with respect to $P \to X$ it it _vanishes_ on vertical vector fields.

## Properties

\begin{proposition}\label{SplittingOfTotalSpaceTangentBundle}
  If $P \overset{\pi}{\longrightarrow} X$ is a [[surjective submersion]] (for instance a [[smooth manifold|smooth]] [[fiber bundle]]) then the full [[tangent bundle]] of its total space $P$ is [[isomorphism|isomorphic]] to the [[direct sum of vector bundles|direct sum]] of the vertical tangent bundle ([above](#VerticalTangentBundle)) with the [[pullback bundle|pullback]] of the [[tangent bundle]] of the base space:

$$
  T P
  \;\simeq\;
  \big(
    \pi^\ast T B
  \big)
  \oplus_P
  \big(
    T_\pi P
  \big)
$$

\end{proposition}

\begin{proof}
The assumption that $\pi$ is a [[surjective submersion]] implies that $d \pi \colon T P \longrightarrow \pi^\ast T B$ is a [[surjection]] and hence that

$$
  0
    \to
  T_\pi P
    \longrightarrow
  T P
   \overset{d \pi}{\longrightarrow}
  \pi^\ast T B
    \to
  0
$$

is a [[short exact sequence]] of smooth [[real vector bundles]].

Now all [[short exact sequences]] of [[real vector bundles]] over [[paracompact topological spaces]] (such as [[smooth manifolds]]) [[split exact sequence|split]] (by a choice of fiberwise metric, see at _[[short exact sequence of vector bundles]]_), which is the statement to be shown.
\end{proof}

## Examples

\begin{proposition}\label{VerticalTangentBundleOfRealVectorBundle}
**([[vertical tangent bundle]] of a [[real vector bundle]])**
  Let $\pi \colon \mathcal{V} \longrightarrow M$ be a [[real vector bundle]]. Then its [[vertical tangent bundle]] is [[isomorphism|isomorphic]] to the its [[pullback bundle|pullback]] along itself:

$$
  T_\pi \mathcal{V}
  \;\simeq_{{}_{\mathcal{V}}}\;
  \pi^\ast \mathcal{V}
  \,.
$$
\end{proposition}

(e.g. [tomDieck 00 (6.9)](#tomDieck00), [tomDieck 08 (15.6.7)](#tomDieck2008), [Gollinger 16, inside the proof of Prop. 1.1.9](#Gollinger16))

\begin{prop}\label{StableTangentBundleOfUnitSphereBundle}
**([[stable tangent bundle]] of [[unit sphere bundle]])** \linebreak
  The once-[[stable tangent bundle|stabilized]] [[tangent bundle]] of a [[unit sphere bundle]] $S(\mathcal{V})$ in a [[real vector bundle]] $\mathcal{V} \overset{p}{\longrightarrow} M$ (Example \ref{UnitSphereBundles}) over a [[smooth manifold]] $M$ is [[isomorphism|isomorphic]] to the [[pullback bundle|pullback]]  of the [[direct sum of vector bundles|direct sum]] of the [[stable tangent bundle]] of the base manifold with that vector bundle:

$$
  T S(\mathcal{V}) \times \mathbb{R}
  \;
  \simeq_{{}_M}
  \;
  S(p)^\ast
  \big(
    T M
    \oplus_M
    \mathcal{V}
  \big)
  \,.
$$
\end{prop}

This is stated without proof as [Crowley-Escher 03, Fact. 3.1](#CrowleyEscher03), apparently reading between the lines in [Milnor 56, p. 403](#Milnor1956). The key sub-statement that

$$
  T_{S(p)} S(\mathcal{V}) \times \mathbb{R}
  \;\simeq_{{}_M}\;
  \mathcal{V}
$$

is made explicit in [Gollinger 16, Prop. 1.1.9](#Gollinger16)

\begin{proof}\label{ProofOfStableTangentBundleOfUnitSphereBundle}
  Consider first the actual [[tangent bundle]] but to the [[open ball]]/[[disk]]-[[fiber bundle]] $D(\mathcal{V})$ that fills the given sphere-fiber bundle: By the standard splitting ([this Prop.](vertical+vector+field#SplittingOfTotalSpaceTangentBundle)) this is the [[direct sum of vector bundles|direct sum]]

$$
  T
  \big(
   D(\mathcal{V})
  \big)
  \;\simeq\;
  \big(
     D(p)^\ast T M
  \big)
  \oplus_M
  T_p D(\mathcal{V})
  \,,
$$

where $T_p D(\mathcal{V})$ is the [[vertical tangent bundle]] of the disk bundle. But, by definition of disk bundles, that is the restriction of the vertical tangent bundle of the vector bundle $\mathcal{V}$ itself, and that is just the pullback of that vector bundle along itself (by [this Example](vertical+vector+field#SplittingOfTotalSpaceTangentBundle)):

$$
  \begin{aligned}
  \cdots
  &
  \simeq\;
  \big(
    D(p)^\ast T M
  \big)
  \oplus_M
  \big(
    D(p)^\ast \mathcal{V}
  \big)
  \\
  & \simeq\;
  D(p)^\ast
  \big(
    T M \oplus_M \mathcal{V}
  \big)
  \,.
  \end{aligned}
$$

To conclude, it just remains to observe that the [[normal bundle]] of the [[n-sphere]]-boundary inside the $(n+1)$-ball is manifestly [[trivial bundle|trivial]], so that the restriction of the tangent bundle of $D(\mathcal{V})$ to $S(\mathcal{V})$ is the [[stable tangent bundle]] of $S(\mathcal{V})$.
\end{proof}

## Related concepts

* [[evolutionary vector field]]

## References

Textbook accounts:

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]],  Section VII.1 in Volume 1 _De Rham Cohomology of Manifolds and Vector Bundles_, in: _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)  ([ISBN:978-0-12-302701-6](https://www.elsevier.com/books/connections-curvature-and-cohomology-v1/greub/978-0-12-302701-6))

*  {#Tu17} [[Loring Tu]], Section 27.5 in: _Differential Geometry -- Connections, Curvature, and Characteristic Classes_, Springer 2017 ([ISBN:978-3-319-55082-4](https://www.springer.com/gp/book/9783319550824), [pdf](http://www.math.nagoya-u.ac.jp/~richard/teaching/f2018/Tu_geometry.pdf))

See also:

* [[Manifold Atlas]], _Tangent bundles of bundles_

* {#tomDieck00} [[Tammo tom Dieck]], around Satz 6.9 in: _Topologie_, De Gruyter (2000)([doi:10.1515/9783110802542](https://doi.org/10.1515/9783110802542))

* {#tomDieck2008} [[Tammo tom Dieck]],  around (15.6.7) _Algebraic topology_, European Mathematical Society, Zürich (2008) ([doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf))

* {#Gollinger16} William Gollinger, Section 1.1.4 in: _Madsen-Tillmann-Weiss Spectra and a Signature Problem for Manifolds_, Münster 2016 ([pdf](https://repositorium.uni-muenster.de/document/miami/7369e8b5-6ae4-4e42-b6f3-4602ec24427a/diss_gollinger.pdf), [[GollingerMTWSpectra.pdf:file]])

* {#Berglund20} Alexander Berglund, _Characteristic classes for families of bundles_ ([arXiv:2012.12170](https://arxiv.org/abs/2012.12170))

* [MO discussion](https://math.stackexchange.com/a/251668/58526)

[[!redirects vertical vector fields]]

[[!redirects vertical vector]]
[[!redirects vertical vectors]]

[[!redirects vertical tangent bundle]]
[[!redirects vertical tangent bundles]]

[[!redirects vertical vector bundle]]
[[!redirects vertical vector bundles]]

[[!redirects vertical cotangent bundle]]
[[!redirects vertical cotangent bundles]]


very commutative object

> under constrution

Given a [[symmetric monoidal (∞,1)-category]] $\mathcal{C}$ then one may consider [[E-∞ space]]-objects inside it.

If the monoidal structure happens to be Cartesian then one may ask for "more" commutativity than that. [[Jacob Lurie]] called that "very commutative" ([MO comment](http://mathoverflow.net/a/118528/381))

(should come down to whether one interprets commutative monoids as a 1-[[algebraic theory]] or an [[(∞,1)-algebraic theory]], i.e. [[(2,1)-algebraic theory of E-infinity algebras]])


very large (infinity,1)-sheaf (infinity,1)-topos

[[!redirects huge (infinity,1)-sheaf (infinity,1)-topos]]

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{:toc}

## Definition

Fix a [[Grothendieck universe]] $\mathcal{U}$ and a smaller universe $\mathcal{V} \in \mathcal{U}$.  Let $\mathcal{V}$ be the reference-universe, so that sets in $\mathcal{V}$ are called *[[small set]]s*, sets in $\mathcal{U}$ are called *large*, and sets not necessarily in $\mathcal{U}$ are called *very large*.

Write [[∞Grpd]] for the (large) [[(∞,1)-category]] of small [[∞-groupoid]]s and $\infty GRPD$ for the very-large $(\infty,1)$-category of large $\infty$-groupoids.

Then the general procedures of [[universe enlargement]] can be applied to any large $(\infty,1)$-category to produce a very-large one.  Specifically, we have the *locally presentable enlargement*: for $C$ a large [[(∞,1)-category]] with small [[(∞,1)-colimit]]s, write

$$
  \Uparrow C
  \subset
  Func(C^{op}, \infty GRPD)
$$

for the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-functors]] that preserves small [[(∞,1)-limit]]s.  As described at [[universe enlargement]], if $C$ is locally presentable, then $\Uparrow C$ can be identified with the *naive enlargement*, which consists of the large models of the "theory" of which $C$ consists of the small models.

In particular, when $C$ is an [[(∞,1)-sheaf (∞,1)-topos]] $Sh(S)$ of small (∞,1)-sheaves, then $\Uparrow C$ can be identified with a (∞,1)-category of *large* (∞,1)-sheaves on the same site.  (That is, with a suitable accessible left-exact-reflective subcategory of $PSH(S)$ rather than $Psh(S)$---it is not yet known how to specify such a reflective subcategory purely in terms of data on $S$.)  Thus, we refer to $\Uparrow C$ as the **very large $(\infty,1)$-sheaf $(\infty,1)$-topos** on $C$.

Note that since every topos can be identified with the category of sheaves on itself for the [[canonical topology]], it is also reasonable to denote $\Uparrow \mathbf{H}$ by $SH(\mathbf{H})$.  $\Uparrow C$ can also also be identified with the category of [[ind-object in an (∞,1)-category|ind-objects]] of $\mathbf{H}$, for a suitable [[regular cardinal]] $\kappa$ (namely, the cardinal of $\mathcal{V}$).

## Properties

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

For every $(\infty,1)$-topos $\mathbf{H}$ there is an [[(∞,1)-functor]]

$$
  \Uparrow ((\infty,1)Topos)
  \to
  \Uparrow \mathbf{H}
$$

that preserves large [[(∞,1)-colimit]]s and finite [[(∞,1)-limit]]s.  It is defined by sending $F : (\infty,1)Topos^{op} \to \infty GRPD$ to the composite

$$
  \mathbf{H}^{op}
  \simeq
  ((\infty,1)Topos/\mathbf{H})_{et}^{op}
  \to
  (\infty,1)Topos^{op}
  \xrightarrow{F}
  \infty GRPD
$$

=--

This is [[Higher Topos Theory|HTT, lemma 6.3.5.21]].

## References

This is discussed in section 6.3 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .

The definition of $\Uparrow\mathbf{H}$ is in Notation 6.3.5.16 and Remark 6.3.5.17.  The relation to ind-objects appears as remark 6.3.6.18.

[[!redirects huge (infinity,1)-sheaf (infinity,1)-topos]]
[[!redirects huge (∞,1)-sheaf (∞,1)-topos]]
[[!redirects very large (∞,1)-sheaf (∞,1)-topos]]


Vesna Stojanoska

* [webpage](http://math.mit.edu/~vstojanoska/)

category: people


Viatcheslav Mukhanov

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

## selected writings

* {#Mukhanov16} [[Viatcheslav Mukhanov]], _Quantum Universe_, Phys.Usp. 59 (2016) no.10, 1021-1027 ([spire:1507528](http://inspirehep.net/record/1507528), [doi:10.3367/UFNe.2016.07.037857](https://iopscience.iop.org/article/10.3367/UFNe.2016.07.037857/meta), [video recording](https://youtu.be/xUIrj_97i0A))

> In March 2013, following an accurate processing of available measurement data, the Planck Scientific Collaboration published the highest-resolution photograph ever of the early Universe when it was only a few hundred thousand years old. The photograph showed galactic seeds in sufficient detail to test some nontrivial theoretical predictions made more than thirty years ago. Most amazing was that all predictions were confirmed to be remarkably accurate. With no exaggeration, we may consider it established experimentally that quantum physics, which is normally assumed to be relevant on the atomic and subatomic scale, also works on the scale of the entire Universe, determining its structure with all its galaxies, stars, and planets.

## related $n$Lab entries

* [[standard model of cosmology]]

* [[Starobinsky model of cosmic inflation]]

category: people


Vicente Cortés

* [webpage](http://www.math.uni-hamburg.de/home/cortes/)

## selected writings

* {#Cotes05} [[Vicente Cortés]], _Quaternionic K ̈ahler manifolds_, 2005 ([pdf](https://www2.math.hu-berlin.de/gradkoll/Cortes_vorlesung1_handout.pdf))

  (on [[quaternion-Kähler manifolds]])

category: people


Victor Bouniakowsky

[[!redirects Bouniakowsky]]

Victor Bouniakowsky (Russian: Ви́ктор Я́ковлевич Буняко́вский) was a Russian mathematician.

He defended his PhD thesis in 1825 at Sorbonne supervised by [[Augustin Cauchy]].

He is known for publishing a proof of the [[Cauchy inequality]] in 1859,
also known as the Cauchy–Bouniakowsky–Schwarz inequality.

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

[[!redirects Bouniakowsky]]

category: people


Victor Buchstaber

* [webpage](http://www.mi.ras.ru/~buchstab/)

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

## Selected writings

Introducing the [[Chern-Dold character]] (specifically for [[complex cobordism cohomology]]):

* {#Buchstaber70} [[Victor Buchstaber]], _The Chern–Dold character in cobordisms. I_,

  Russian original: Mat. Sb. (N.S.), 1970 Volume 83(125), Number 4(12), Pages 575–595 ([mathnet:3530](http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=3530&option_lang=eng))

  English translation: Mathematics of the USSR-Sbornik, Volume 12, Number 4, AMS 1970 ([doi:10.1070/SM1970v012n04ABEH000939](https://iopscience.iop.org/article/10.1070/SM1970v012n04ABEH000939))

On [[formal groups]] and [[Adams operations]]:

* [[Victor M. Buchstaber]], [[Sergei P. Novikov]], _Formal groups, power systems and Adams operators_, Math. USSR-Sb.13 (1971) 80-116 [[doi:10.1070/SM1971v013n01ABEH001030](https://doi.org/10.1070/SM1971v013n01ABEH001030)]

* [[Victor Buchstaber]], A. P. Veselov, _Chern-Dold character in complex cobordisms and abelian varieties_ ([arXiv:2007.05782](https://arxiv.org/abs/2007.05782))

On [[cobordism cohomology theory]]:

* [[Victor Buchstaber]], _Cobordisms in problems of algebraic topology_,  J Math Sci 7, 629–653 (1977) ([doi:10.1007/BF01084983](https://doi.org/10.1007/BF01084983))

On [[Hopf algebras]] and [[homology of H-spaces]] (such as [[homology of loop spaces]]):

* [[Victor Buchstaber]], [[Jelena Grbić]], *Hopf algebras and homology of loop suspension spaces* ([pdf](http://www.personal.soton.ac.uk/jg1u11/Papers/Buchstaber.Grbic.pdf), [[BuchstaberGrbicHopfAlgebrasAndHomology.pdf:file]]) in: V. Buchstaber et al. (eds.), *Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov’s Seminar 2012–2014*, American Mathematical Society Translations - Series 2
Advances in the Mathematical Sciences, 2014 ([ISBN:978-1-4704-1871-7](https://bookstore.ams.org/trans2-234))

## Related entries

* [[cobordism theory]]

* [[bordism ring]]

* [[Chern-Dold character]]

* [[multivalued group]]

[[!redirects Victor M. Buchstaber]]

[[!redirects Victor Buhštaber]]
[[!redirects Victor M. Buhštaber]]

category: people


Victor Carmona

* [personal page](https://personal.us.es/vcarmona1/)

## Selected writings

On the [[time slice axiom]] in [[homotopical AQFT]]:

* [[Marco Benini]], [[Victor Carmona]], [[Alexander Schenkel]], *Strictification theorems for the homotopy time-slice axiom* [[arxiv:2208.04344](https://arxiv.org/abs/2208.04344)]

category: people


Victor Galitski

* [Institute page](https://umdphysics.umd.edu/people/faculty/current/item/164-galitski.html)

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

* [GoogleScholar page](https://scholar.google.com/citations?user=BJDC0QkAAAAJ&hl=en)

## Selected writings

On the [[adiabatic quantum computation|adiabaticity]] of [[topological quantum computation]] by [[braid group|braiding]] of [defect anyons](braid+group+statistics#AsBraidingOfDefects):

* {#ChengGalitskiDasSarma11} [[Meng Cheng]], [[Victor Galitski]], [[Sankar Das Sarma]], *Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors*, Phys. Rev. B **84** (2011) 104529 $[$[arXiv:1106.2549](https://arxiv.org/abs/1106.2549), [doi:10.1103/PhysRevB.84.104529](https://doi.org/10.1103/PhysRevB.84.104529)$]$

category: people


Victor Ginzburg

**Victor Ginzburg** (in some 1980s articles spelled Ginsburg) is a professor of mathematics at the University of Chicago. His thesis in Moscow was under [[Alexandre Kirillov]]. His main interests are [[representation theory]], especially [[geometric representation theory]], including more recently [[noncommutative algebraic geometry]].

Warning: there is another mathematician (global analysis, [[symplectic geometry]]), *Viktor* Ginzburg (note the English spelling).

* [U. Chicago: faculty research interests](http://www.math.uchicago.edu/research); wikipedia: [Victor Ginzburg](http://en.wikipedia.org/wiki/Victor_Ginzburg); an [article](http://chronicle.uchicago.edu/060608/gta-ginzburg.shtml) in Chicago Chronicle

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

* V. Ginzburg, _Geometric methods in representation theory of Hecke algebras and quantum groups_ (this survey is closely related to the book above), [math.AG/9802004](http://front.math.ucdavis.edu/math.AG/9802004)

* _Lectures on noncommutative geometry_, [math.AG/0506603](http://arxiv.org/abs/math/0506603)

* (with [[Alexander Beilinson|A. Beilinson]]), _[[wall crossing|Wall-crossing]] functors and $D$-modules_, Representation Theory __3__ (electronic), 1--31 (1999)

* A. A. Beĭlinson, V. A. Ginsburg, V. V. Schechtman, _Koszul duality_, J. Geom. Phys. 5 (1988), no. 3, 317–350.

* A. Beilinson, V. Ginzburg, W. Soergel, _Koszul duality patterns in representation theory_, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527.

* V. Ginsburg, _Characteristic varieties and vanishing cycles_, Inv. Math. __84__, 327--402 (1986) [MR87j:32030](http://www.ams.org/mathscinet-getitem?mr=833194), [doi](http://dx.doi.org/10.1007/BF01388811)
* V.G. _Lectures on D-modules_, 1998 Chicago notes, writeup by V. Baranovsky, [pdf](https://www.math.harvard.edu/~gaitsgde/grad_2009/Ginzburg.pdf)
* P. Etingof, V. Ginzburg, _Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism_, Invent. math. __147__, 243--348 (2002) [doi](https://doi.org/10.1007/s002220100171) arXiv:[math.AG/0011114](https://arxiv.org/abs/math/0011114)
* [[Pavel Etingof]], Wee Liang Gan, Victor Ginzburg, Alexei Oblomkov, _Harish-Chandra homomorphisms and [[symplectic reflection algebra]]s for wreath-products_, arXiv:[math.RT/0511489](https://arxiv.org/abs/math/0511489)
* Victor Ginzburg, [[Dmitry Kaledin]], _Poisson deformations of symplectic quotient singularities_, Adv. Math. __186__(1) 1--57 (2003) [doi](https://doi.org/10.1016/j.aim.2003.07.006)
* [[William Crawley-Boevey]], [[Pavel Etingof]], [[Victor Ginzburg]], _Noncommutative geometry and quiver algebras_,  Adv. Math. __209__:1 (2007) 274--336 [doi](https://doi.org/10.1016/j.aim.2006.05.004)
* V. Ginzburg, [[T. Schedler]], _Differential operators and BV structures in noncommutative geometry_, Sel. Math. New Ser. 16, 673--730 (2010) [doi](https://doi.org/10.1007/s00029-010-0029-8)
* V. Ginzburg, [[T. Schedler]], _A new construction of cyclic homology_, Proc. Lon. Math. Soc 112 (2016), no. 3, 549--587; [doi](https://doi.org/10.1112/plms/pdw001; arXiv:1201.6635)

He introduced the notion of $d$-Calabi-Yau algebra (and also a construction of a class of examples, [[Ginzburg dg-algebra]]s) in

* _Calabi-Yau algebras_, [arXiv:math.AG/0612139](https://arxiv.org/abs/math/0612139)

category: people

[[!redirects Victor Ginzburg]]
[[!redirects Victor Ginsburg]]
[[!redirects V. Ginzburg]]

[[!redirects Viktor Ginzburg]]


Victor Gugenheim

* [obituatries](https://www.maths.ed.ac.uk/~v1ranick/surgery/uicc/vkamg.txt)

* [biographical information](http://www.maths.ed.ac.uk/~aar/surgery/uicc/vkamg.txt)

## Selected writings

On [[twisting functions]] and ([[semisimplicial set|semi]]-)[[simplicial fiber bundles]]:

* [[Michael G. Barratt]], [[Victor K.A.M. Gugenheim]], [[John C. Moore]], _On semisimplicial fibre-bundles_, Amer. J. Math. __81__ (1959) 639-657 [[doi:10.2307/2372920](https://doi.org/10.2307/2372920), [jstor:2372920](https://www.jstor.org/stable/2372920), MR0111028]

On [[rational homotopy theory]] and the [[fundamental theorem of dg-algebraic rational homotopy theory]]:

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))

category: people

[[!redirects Victor K.A.M. Gugenheim]]


Victor Guillemin

* [webpage](http://math.mit.edu/~vwg/)

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

## Selected writings

On [[differential geometry]] with focus of [[geometric optics]] and [[current (distribution theory)|currents]]:

* [[Victor Guillemin]], [[Shlomo Sternberg]], *Geometric asymptotics*, Mathematical Surveys and Monographs **14**, AMS (1977) [[ams:surv-14](https://bookstore.ams.org/surv-14)]

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

* {#GuilleminSternberg99} [[Victor Guillemin]], [[Shlomo Sternberg]], _Supersymmetry and equivariant de Rham theory_, Springer, (1999) ([doi:10.1007/978-3-662-03992-2](https://link.springer.com/book/10.1007/978-3-662-03992-2))

On [[differential geometry]] with emphasis on [[differential forms]] and their [[de Rham cohomology]]:

* {#GuilleminHaine19} [[Victor Guillemin]], [[Peter Haine]], _Differential Forms_, World Scientific (2019) [[doi:10.1142/11058](https://doi.org/10.1142/11058), draft:[pdf](https://math.mit.edu/classes/18.952/2018SP/files/18.952_book.pdf)]

## Related entries

* [[symplectic geometry]]

* [[geometric quantization]]

* [[deformation quantization]]

* [[Guillemin-Sternberg geometric quantization conjecture]]

* [[integrability of G-structures]]

* [[Weil algebra]]

* [[equivariant de Rham cohomology]]

category: people


Victor Gurarie

* [personal page](https://spot.colorado.edu/~gurarie/research.htm)

* [institute page](https://www.colorado.edu/physics/victor-gurarie)

## Selected writings

The triplet algebra of the $c = -2$ Virasoro model as a [[logarithmic CFT]]:

* [[Victor Gurarie]], *Logarithmic Operators in Conformal Field Theory*,  	Nucl.Phys. B **410** (1993) 535-549 $[$[arXiv:hep-th/9303160](https://arxiv.org/abs/hep-th/9303160), doi:10.1016/0550-3213(93)90528-W$]$

On [[2d CFT|2d]] [[logarithmic CFT]] for generalized [[Laughlin wavefunctions]]:

* {#GurarieFlohrNayak97} [[Victor Gurarie]], [[Michael Flohr]], [[Chetan Nayak]], *The Haldane-Rezayi Quantum Hall State and Conformal Field Theory*, Nucl. Phys. B **498** (1997) 513-538 $[$[arXiv:cond-mat/9701212](https://arxiv.org/abs/cond-mat/9701212), [doi:10.1016/S0550-3213%2897%2900351-9](https://doi.org/10.1016/S0550-3213%2897%2900351-9)$]$

category: people


Victor Kac

* [wikipedia page](http://en.wikipedia.org/wiki/Victor_Kac)

## Selected writings

On [[super Lie algebras]], [[Jordan superalgebras]]:

* {#Kac77a} [[Victor Kac]], _Lie superalgebras_, Advances in Math. 26 (1977), no. 1, 8--96 (doi:10.1016/0001-8708(77)90017-2)

* {#Kac77b} [[Victor Kac]], _A sketch of Lie superalgebra theory_, Comm. Math. Phys. Volume 53, Number 1 (1977), 31-64. ([euclid:cmp/1103900590](https://projecteuclid.org/euclid.cmp/1103900590))

* {#Kac77} [[Victor Kac]], _Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras_, Comm. Algebra **5** (1977), 1375–1400, ([doi: 10.1080/00927877708822224]( https://doi.org/10.1080/00927877708822224))

See also:

* _The idea of locality_ [[q-alg/9709008](http://arxiv.org/abs/q-alg/9709008)]

On [[affine Lie algebras]]:

* [[Victor G. Kac]], *Infinite Dimensional Lie Algebras*,  Progress in Mathematics **44** Springer 1983 [[doi:10.1007/978-1-4757-1382-4](https://link.springer.com/book/10.1007/978-1-4757-1382-4)), Cambridge University Press (1990) [[doi:10.1017/CBO9780511626234](https://doi.org/10.1017/CBO9780511626234)]

their relation to [[modular forms]]:

* [[Victor G. Kac]], [[Dale H. Peterson]], *Affine Lie algebras and Hecke modular forms*,  Bull. Amer. Math. Soc. (N.S.) **3** 3 (1980) 1057-1061 [[bams:1183547694](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-3/issue-3/Affine-Lie-algebras-and-Hecke-modular-forms/bams/1183547694.full)]

* [[Victor G. Kac]], [[Dale H. Peterson]], *Infinite-dimensional Lie algebras, theta functions and modular forms*, Advances in Mathematics **53** 2 (1984) 125-264 [doi:10.1016/0001-8708(84)90032-X]

* [[Victor G. Kac]], [[Minoru Wakimoto]], *Modular and conformal invariance constraints in representation theory of affine algebras*, Advances in Mathematics **70** 2 (1988) 156-236 [doi:10.1016/0001-8708(88)90055-2, [spire:275458](https://inspirehep.net/literature/275458)]

On non-integrable but "admissible" [[irreps]] of [[affine Lie algebras]] (cf. *[fractional level WZW models](Wess-Zumino-Witten+model#FractionalLevelWZWModelReferences)*):

* [[Victor G. Kac]], [[Minoru Wakimoto]], *Modular invariant representations of infinite-dimensional Lie algebras and superalgebras*,  PNAS **85** 14 (1988)  4956-4960 [[doi:10.1073/pnas.85.14.4956](https://doi.org/10.1073/pnas.85.14.4956)]

* [[Victor G. Kac]], [[Minoru Wakimoto]], *Classification of modular invariant representations of affine algebras*, p. 138-177 in V. G. Kač (ed.): *Infinite dimensional Lie algebras and groups*, Advanced series in Mathematical physics **7**, World Scientific (1989) [[pdf](https://math.mit.edu/~kac/not-easily-available/admissible.pdf),  [cds:268092](https://cds.cern.ch/record/268092)]

On [[vertex operator algebras]]:

* [[Victor Kac]], *Vertex algebras for beginners*, Amer. Math. Soc., Univ. Lecture Series **10** (1998)  [[ams:ulect-10-r](https://bookstore.ams.org/ulect-10-r), [ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0924.17023&format=complete)]

On [[vertex operator algebras]] over [[non-archimedean fields]]:

* [[Victor Kac]], _Non-Archimedean vertex algebras_ [[arXiv:2304.09651](https://arxiv.org/abs/2304.09651)]

category: people

[[!redirects V. G. Kac]]
[[!redirects Victor G. Kac]]

[[!redirects Victor Kac]]
[[!redirects V. G. Kac]]
[[!redirects Victor G. Kac]]


Victor Maslov

A Soviet and Russian mathematical physicist. Discovered
[[Maslov index]], [[Fourier integral operator]]s (in terms of "canonical operator"), [[Maslov-Feynman calculus]], systematic approach to [[WKB method]] to all orders, corresponding "Maslov" quantization conditions and the complex WKB method. He also works on idempotent analysis.

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

* _[Fiftieth anniversary of research and teaching by Victor Pavlovich Maslov]_, Theoretical and Mathematical Physics, 155(2): 674--677 (2008), transl. from Teoreticheskaya i Matematicheskaya Fizika __155__:2, pp. 197--201 (2008) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/maslovbio.pdf))

## Literature

### Related entries

* [[Lagrangian submanifold]]
* [[Maslov index]]
* [[Maslov dequantization]]
* [[semiclassical approximation]]

### Books

* V. P. Maslov, _Theory of perturbations and asymptotic methods_ (in Russian), Mos. Gos. Univ., Moscow (1965)

### Papers

* [[V. P. Maslov]], O. Yu. Shvedov, _Large-N expansion as a semiclassical approximation to the third-quantized theory_, Physical Review D60(10) 105012 [doi](https://doi.org/10.1103/PhysRevD.60.105012)
* G. L. Litvinov, V.P. Maslov, _The correspondence principle for Idempotent Calculus and some computer applications_, In book Idempotency J. Gunawardena (Editor), Cambridge University Press, Cambridge, 1998, p.420-443

category: people

[[!redirects Victor Pavlovich Maslov]]
[[!redirects V. P. Maslov]]

[[!redirects Maslov]]


Victor Nistor

* [institute page](https://iecl.univ-lorraine.fr/membre-iecl/nistor-victor-2/)

* [Google Scholar page](https://scholar.google.com/citations?user=F9gokPUAAAAJ&hl=en)

## Selected writings

On the [[cyclic homology]] of the [[groupoid convolution algebra]] of [[étale groupoid|étale]] [[Lie groupoids]], hence of [[orbifolds]]:

* [[Jean-Luc Brylinski]], [[Victor Nistor]], *Cyclic cohomology of étale groupoids*,  K-theory 8.4 (1994): 341-365 ([pdf](https://www.researchgate.net/profile/Victor-Nistor/publication/226520448_Cyclic_cohomology_of_etale_groupoids/links/5509d9d30cf20ed529e227b0/Cyclic-cohomology-of-etale-groupoids.pdf), [[BrylinskiNistor_CyclicCohomology.pdf:file]])

category: people