Geometry, Topology and Physics

This page collects material related to the textbook

* [[Mikio Nakahara]],

  _**Geometry, Topology and Physics**_,

  First Edition: Institute of Physics Publishing, 1990.  ISBN 0-85274-094-8 (hardcover), 0-85274-095-6 (paperback)

  Second Edition: Institute of Physics Publishing, 2003.  ISBN: 9780750306065 (paperback), 9781138413368 (hardcover)

  [doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826),

  pdf

on [[geometry]] and [[topology]] in [[mathematical physics]], such as in [[fiber bundles in physics]].

#Contents#
* table of contents
{:toc}

## Quantum Physics

* [[quantum physics]]

## Mathematical Preliminaries

## Homology Groups

* [[ordinary homology]]

* [[homology groups]]

## Homotopy Groups

### Abelian groups

* [[abelian groups]]

### Simplices and simplicial complexes

* [[simplex]]

* [[simplicial complex]]

* [[triangulation]]

### Homology groups of simplicial complexes

* [[ordinary homology]]

* [[homology groups]]

* [[homotopy groups]]

### General properties of homology groups

* [[Betti number]]

## Manifolds

* [[smooth manifolds]]

## de Rham Cohomology Groups

* [[de Rham cohomology]]

## Riemannian Geometry

* [[Riemannian geometry]]

## Complex Manifolds

* [[complex manifolds]]

## Fibre Bundles

* [[fiber bundles]]

* [[principal bundles]]

## Connections on Fiber Bundles

### Connections on principal bundles

* [[connection on a bundle]]

### Holonomy

* [[holonomy]]

### Curvature

* [[curvature]]

### The covariant derivative

* [[covariant derivative]]

### Gauge theories

* [[gauge theory]]

  * [[electromagnetism]]

  * [[magnetic monopole]]

  * [[Aharonov-Bohm effect]]

  * [[Yang-Mills theory]]

  * [[instantons]]

### Berry's phase

* [[Berry's phase]]

## Characteristic Classes

* [[Chern-Weil homomorphism]]

* [[characteristic classes]]

* [[Chern class]]

* [[Pontrjagin class]]

## Index Theorems

* [[index theory]]

## Anomalies in Gauge Field Theories

* [[quantum anomaly]]

* [[anomaly cancellation]]

## Bosonic String Theory

* [[string theory]]

category: reference


geometryofphysicscontents

* About this page

  * Scope and perspective
  * Layers of exposition
  * The full story in a few formal words

* Coordinate systems

  * Model Layer

    * The continuum real (world-)line
    * Cartesian spaces and smooth functions
    * The fundamental theorems about smooth functions

  * Semantic Layer

    * The algebraic theory of smooth algebras
    * The coverage of differentially good open covers
    * The slice category

  * Syntactic Layer

    * Judgments about types and terms
    * Natural deduction rules for product types
    * Natural deduction rules for dependent sum types
    * Dictionary: type theory / category theory

* Smooth spaces

  * Model Layer

    * Plots of smooth spaces and their gluing
    * Homomorphisms of smooth spaces
    * Products and fiber products of smooth spaces
    * Smooth mapping spaces and smooth moduli spaces
    * The smooth moduli space of smooth functions
    * Outlook

  * Semantic Layer

    * Toposes
    * Subobjects
    * Slice toposes
    * Local, connected and cohesive toposes
    * The topos of smooth spaces

      * Connectedness
      * Locality
      * Cohesion

  * Syntactic Layer

    * Natural deduction rules for dependent product types
    * Internal logic of a toposThe type of propositions
    * Cohesive modality I: Sharp types

* Differential forms

  * Model Layer

    * Differential forms on abstract coordinate systems
    * Differential forms on smooth spaces

  * Semantic Layer

    * Concrete smooth spaces
    * Smooth space of differential forms on a smooth space

  * Syntactic Layer

    * Images

* Differentiation

  * Model Layer

    * Differentiation of smooth functions and differential forms

      * Differentiation on coordinate patches
      * Differentiation on smooth spaces
      * Example: The electromagnetic field strength

    * Variational calculus

      * Discrete points of a smooth space
      * Smooth functionals
      * Functional derivative / variational derivative
      * Euler-Lagrange equations

    * $\mathcal{D}$-geometry

      * Infinitesimal smooth loci
      * de Rham space
      * Jet bundles

  * Semantic Layer

    * Synthetic differential geometry
    * Tangent bundle
    * Differential equations
    * Differential cohesion of the topos of smooth spaces
    * Differential cohesion
    * de Rham space
    * Jet bundle
    * Formally étale / formally unramified / formally smooth

  * Syntactic Layer

    * Differential homotopy type theory

* Smooth homotopy types

  * Model Layer

    * Gauge transformations in electromagnetism
    * Groupoids
    * Smooth groupoids
    * Differential 1-forms are smooth incremental path measures

  * Semantic Layer

    * ∞-Toposes
    * Slice ∞-toposes
    * Local, ∞-connected and cohesive ∞-toposes
    * The ∞-topos of smooth ∞-groupoids
    * Local ∞-Connectedness
    * Locality
    * Cohesion

  * Syntactic Layer

    * Identity types
    * Homotopy type theory
    * Cohesive modality II: flat types

* Groups

  * Model Layer

    * Groups
    * Dold-Kan correspondence

  * Semantic Layer

    * A ∞-types
    * ∞-Groups

  * Syntactic Layer

    * Pointed connected types
    * Identity types of connected types

* Principal bundles

  * Model Layer

    * Connected groupoids
    * Universal principal bundle
    * Principal bundles
    * Weakly principal simplicial bundles

  * Semantic Layer

    * Principal ∞-bundles

  * Syntactic Layer

* Actions, representations and associated bundles

  * Model Layer

    * Actions
    * Associated bundle
    * Representations up to coherent homotopy

  * Semantic Layer

    * ∞-Actions
    * Associated ∞-bundles

  * Syntactic Layer

    * The context of a pointed connected type: representation theory
    * Dependent product over a pointed connected type: invariants
    * Dependent sum over a pointed connected type: quotients

* Manifolds

  * Model Layer

    * Smooth manifolds
    * Tangent bundle

  * Semantic Layer

    * Manifolds modeled on an object

  * Syntactic Layer

* $G$-Structure

  * Model Layer

    * Reduction of structure group
    * Vielbein, orthogonal structure
    * Almost complex structure
    * Almost Hermitean structure
    * Almost symplectic structure
    * Metaplectic structure
    * Metalinear structure
    * Generalized complex geometry
    * Type II geometry
    * Generalized Calabi-Yau structure
    * Exceptional generalized geometry
    * Spin structure, String structure, Fivebrane structure

  * Semantic Layer
  * Syntactic Layer

* Riemannian geometry

  * Model Layer
  * Semantic Layer
  * Syntactic Layer

* Integration

  * Model Layer

    * Integration
    * Integration of differential forms
    * Integration in ordinary differential cohomology
    * Lie integration
    * Transgression
    * Transgression of differential forms
    * Transgression of circle n-bundles with connection
    * Action functionals from transgression

  * Semantic Layer
  * Syntactic Layer

* Flat connections

  * Model Layer
  * Semantic Layer
  * Syntactic Layer

* de Rham Coefficients

  * Model Layer

    * Lie-algebra valued differential 1-forms
    * The de Rham complex

  * Semantic Layer

    * De Rham coefficient objects
    * Recovering smooth differential forms from cohesive de Rham coefficients

  * Syntactic Layer

* Maurer-Cartan forms

  * Model Layer

    * Maurer-Cartan form on a Lie group

  *Semantic Layer

    * Maurer-Cartan form on a cohesive ∞-group
    * Maurer-Cartan forms on smooth ∞-groups
    * Cohesive differentiation
    * Universal curvature characteristic forms

  * Syntactic Layer

* Circle-principal connections

  * Model Layer

    * Circle-principal connections
    * Dirac charge quantization and the electromagnetic field
    * Deligne cohomology and Cheeger-Simons differential characters
    * Circle-principal 2-connection
    * Magnetic charge and the B-field
    * Circle-principal 3-connection
    * The 3d Chern-Simons action functional and the C-field

  * Semantic Layer

    * Smooth differential cohomology
    * Higher holonomy

  * Syntactic Layer

* Characteristic classes

  * Model Layer
  * Semantic Layer
  * Syntactic Layer

* Circle-principal n-connection

  * Model Layer
  * Semantic Layer
  * Syntactic Layer

* Action functionals for Chern-Simons-type gauge theories

  * Model Layer
  * Semantic Layer
  * Syntactic Layer

* Abelian Chern-Simons theory

  * Model Layer
  * Semantic Layer
  * Syntactic Layer

* Principal connections

  * Model Layer

    * Yang-Mills theory

  * Semantic Layer
  * Syntactic Layer

* Associated bundles

  * Model Layer

    * Associated vector bundle
    * Spin geometry

  * Semantic Layer
  * Syntactic Layer

* Covariant derivative
* Einstein-Yang-Mills theory
* Symplectic geometry
* Geometric quantization
* Supergeometric coordinate systems

  * Model Layer
  * Semantic Layer
  * Syntactic Layer

* References

  * General
  * Differential forms and parallel transport
  * Mathematical quantum field theory
  * Further


Georg Biedermann

* [institute page](https://www.uninorte.edu.co/web/departamento-de-matematicas-y-estadistica/profesores)

* [old home page](https://www.math.univ-paris13.fr/~biedermann/)

* [old home webpage](http://www.home.uni-osnabrueck.de/gbiederm/)

## Selected writings

On [[Goodwillie calculus]] via [[model structures on excisive functors]] and [[model structure for n-excisive functors]]:

* {#BiedermannChornyRondings06} [[Georg Biedermann]], [[Boris Chorny]], [[Oliver Röndigs]], _Calculus of functors and model categories_, Advances in Mathematics **214** (2007) 92-115 [[arXiv:math/0601221](http://arxiv.org/abs/math/0601221), [doi:10.1016/j.aim.2006.10.009](https://doi.org/10.1016/j.aim.2006.10.009)]

* [[Georg Biedermann]], [[Oliver Röndigs]], _Calculus of functors and model categories II_, Algebr. Geom. Topol. **14** (2014) 2853-2913 [[arXiv:1305.2834v2](http://arxiv.org/abs/1305.2834), [doi:10.2140/agt.2014.14.2853](https://doi.org/10.2140/agt.2014.14.2853)]

Proving the [[Blakers-Massey theorem]] in any [[(infinity,1)-topos|$(\infty,1)$-topos]] and with the [[(n-connected, n-truncated) factorization system]] allowed to be replaced by more general [[modal homotopy type theory|modalities]]:

* {#AnelBiedermanFinsterJoyal17a} [[Mathieu Anel]], [[Georg Biedermann]], [[Eric Finster]], [[André Joyal]], _A Generalized Blakers-Massey Theorem_, Journal of Topology **13** 4 (2020) 1521-1553 $[$[arXiv:1703.09050](https://arxiv.org/abs/1703.09050), [doi:10.1112/topo.12163](https://doi.org/10.1112/topo.12163)$]$

## Related entries

* [[Joyal locus]]

* [[tangent (infinity,1)-topos|tangent $\infty$-topos]]

category: people


Georg Cantor

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

category: people


Georg Christoph Lichtenberg

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

## Quotes

> Zweifle an allem wenigstens einmal, und wäre es auch der Satz: zwei mal zwei ist vier.

> Doubt everything at least once.

but

> Zweifel darf nichts weiter sein als Wachsamkeit, sonst kann er gefährlich werden.

(Apparently there is no actual source available for this quote: The careful page [here](https://www.quotez.net/german/georg_christoph_lichtenberg.htm) distinguishes between confirmed and merely attributed quotes, and these two are listed as attributed but not as confirmed.)

category: people


Georg H. von Wright

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

## Selected writings

Early discussion of [[modal logic]]:

* [[Georg H. von Wright]], *An Essay in Modal Logic*, North-Holland Publishing (1951)

category: people

[[!redirects Georg von Wright]]
[[!redirects Georg Henrik von Wright]]


Georg Hamel

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

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=7349)

## Selected writings

Original discussion of a [[basis of a vector space|linear basis]] (a [[basis theorem]]) for the [[real numbers]] regarded as a [[rational vector space]] after which the notion of *[[Hamel bases]]* is named:

* {#Hamel1905} [[Georg Hamel]], *Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: $f(x + y) = f(x) + f(y)$*, Mathematische Annalen **60** (1905) 459–462 [[doi:10.1007/BF01457624](https://doi.org/10.1007/BF01457624)]

category: people


Georg Hegel

+-- {: .rightHandSide}
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### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

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

* [hegel.net](http://hegel.net)

> Daran mitzuarbeiten, daß die Philosophie der Form der Wissenschaft näher komme – dem Ziele, ihren Namen der Liebe zum Wissen ablegen zu können und wirkliches Wissen zu sein –, ist es, was ich mir vorgesetzt. ([PdG, Vorr. §5](Phenomenology+of+Spirit#Vorrede5))

#Contents#
* table of contents
{:toc}

## Influences
 {#Influences}

> The importance of [[mathematics]] for Hegel should not be underestimated. [[geometry|Geometry]] was among his earliest intellectual interests (cf. Jaeschke , Hegel-Handbuch, 1) and he devoted intensive study to [[Euclidean geometry|Euclid 's Elements]] in the period leading up to his move from Frankfurt to Jena (cf. Johannes Hoffmeister[ed.], Dokumente zu Hegel's Entwicklung [ Stuttgart-Bad Cannstatt : Fromann 1936 ], 288–300)... In 1805/6 he taught courses in arithmetic and geometry both in Jena and at the Nuremberg Gymnasium while he was working on the [[Science of Logic]]. (_Hegel and the Metaphysics of Absolute Negativity_ by Brady Bowman, p. 171)

## Perception of Hegel's Naturphilosophie
 {#PerceptionOfHegelsPhilosophy}

Some comments on the perception of Hegel's [[metaphysics]]. For much more discussion of this see ([Redding 12](#Redding12)).

Hegel's [[natural philosophy]] as laid out in particular in his _[[Science of Logic]]_ may easily seem rather mysterious and was accordingly criticized. Being partly a creation myth in the spirit of [[Heraclitus]]'s _[[logos]]_ and being all thoroughly arranged in terms of [[unity of opposites]] that taken at face value are typically plain self-contradictions, to a large extent it seems more a work of [[mysticism]] than of [[philosophy]] or [[science]]. Certainly Hegel does not try to provide observational evidence or rational argument for his statements about, say, the _[[becoming]]_ of _[[nothing]]_ into _[[being]]_, that would allow readers to confirm his conclusions, even though it is precisely his claim that his statements are based on some kind of observation ("speculation" in the original latin sense). Rather he states all his natural philosophy as if a collection of self-evident truths which are all confirmed in themselves, as if a set of [[axioms]] for [[natural philosophy]].

In reaction to this, the rise of the school of [[analytic philosophy]] in England defined itself in its opposition to, and its rejection of, Hegel's [[philosophy]], especially as presented by the British Hegelians. On the other hand, later when the foundations of science underwent the now famous revolutions, for instance with the principle of "[[general relativity]]" in the foundations of [[physics]] and the bolstering of [[intuitionistic logic]] via [[topos theory]] in the [[foundations of mathematics]], it was observed here and there that some of Hegel's old thoughts are in curious harmony with these new developments. A systematic attempt to extract hard mathematical and scientific truth from Hegel's natural philosophy is implicit in the life work of [[William Lawvere]]. He effectively suggested that Hegel's notorious [[unity of opposites]] may usefully be formalized as [[adjoint pairs]] of [[idempotent monad|idempotent]] ([[comonad|co]]-)[[monads]]/[[modalities]] in [[category theory]]/[[topos theory]]. See at _[[category of being]]_ and see  at _[Science of Logic -- Formalization dictionary](http://ncatlab.org/nlab/show/Science%20of%20Logic#formalization_dictionary)_ for an idea of how such a dictionary can be used to read Hegel's natural philosophy as being a kind of pseudocode for an actual axiomatization of [[physics]] in [[type theory]]/[[topos theory]].

The opposition against Hegel's philosophy is expressed for instance in

* John Stuart Mill, letter of November 4, 1867 to Alexander Bain:

> Besides these I have been toiling through Stirling's "Secret of Hegel". It is right to learn what Hegel is & one learns it only too well from Stirling's book. I say "too well" because I found by actual experience of Hegel that conversancy with him tends to deprave one's intellect. The attempt to unwind an apparently infinite series of self–contradictions, not disguised but openly faced & coined into [illegible word] science by being stamped with a set of big abstract terms, really if persisted in impairs the acquired delicacy of perception of false reasoning & false thinking which has been gained by years of careful mental discipline with terms of real meaning. For some time after I had finished the book all such words as reflexion, development, evolution, &c., gave me a sort of sickening feeling which I have not yet entirely got rid of.

and in

* [[Bertrand Russell]], Chapter XXXI of _[[A History of Western Philosophy]]_ (1945)

> In philosophy ever since the time of Pythagoras there has been an opposition between the men whose thought was mainly inspired by mathematics and those who were more influenced by the empirical sciences. Plato, Thomas Aquinas, Spinoza, and Kant belong to what may be called the mathematical party; Democritus, Aristotle, and the modern empiricist from Locke onwards, belong to the opposite party. In our day a school of philosophy has arisen which sets to work to eliminate Pythagoreanism from the principles of mathematics, and to combine empiricism with an interest in the deductive parts of human knowledge. The aims of this school are less spectacular than those of most philosophers in the past, but some of its achievements are as solid as those of the men of science.

> The origin of this philosophy is in the achievements of mathematicians who set to work to purge their subject of fallacies and slipshod reasoning. The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibnitz believed in actual infinitesimals, but although this belief suited his metaphysics it has no sound basis in mathematics. Weierstrass, soon after the middle of the nineteenth century, showed how to establish the calculus without infinitesimals, and thus at last made it logically secure. Next came [[Georg Cantor]], who developed the theory of continuity and infinite number. "Continuity" had been, until he defined it, a vague word, convenient for philosophers like [[Hegel]], who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicist. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated.

(Here it is worth remarking that not much later [[infinitesimal objects]] as anticipated by [[Gottfried Leibniz]] were given a neat formal basis in [[mathematics]], by formalisms such as [[synthetic differential geometry]] and [[nonstandard analysis]].)

and in

* Hans Reichenbach, p. 72 of _The rise of scientific philosophy_, 1951.

> Hegel has been called the successor of Kant; that is a serious misunderstanding of Kant and an unjustified elevation of Hegel. Kant's system, though proved untenable by later developments, was the attempt of a great mind to establish rationalism on a scientific basis. Hegel's system is the poor construction of a fanatic who has seen one empirical truth and attempts to make it a logical law within the most unscientific of all logics. Whereas Kant's system marks the peak of the historical line of rationalism, Hegel's  system belongs in the period of decay of speculative philosophy which characterizes the nineteenth century.

The point that Hegel's controversial views on the foundations of physics were later partly vindicated by modern developments is made by [[R. G. Collingwood]] in his book _An Essay on Metaphysics_ (1940). He writes concerning the views expressed by the physicist [[Arthur Eddington]] in the 1928 Gifford lectures:

> The discovery by a very distinguished scientist that there are grains of sense in Hegel's _Naturphilosophie_, and that he feels himself obliged to apologize for having made the discovery, is a sign of the times. How far was the habitual and monotonous execration of Hegel by nineteenth century scientists due to the fact that he violently disliked the science of his own day, and demanded the substitution for it of a physics, which it turns out, was to be in effect the physics that we have now?

This refers to Hegel's complaint about [[Newton's laws of motion]] which give the picture of the planets kept in their orbits by the operation of [[forces]] "like stupid children pushed and pulled into place at drill", while rather they ought to be thought of as moving freely and Eddington's suggestion that according to [[Einstein's laws of motion]] in [[general relativity]] this idea of "free" motion may indeed be argued to be closer to the physical truth.

Another place where Hegel is seen to have have expressed a vision of insights scientifically available only in a distant future is in

* Slavoj Zizek, _[We need thinking](http://bigthink.com/videos/we-need-thinking)_, May 2012

> I claim that what is happening, for example, in quantum physics in the last 100 of years -- these things which are so daring, incredible, that we cannot include into our conscious view of reality -- that Hegel's philosophy, with all it's dialectical paradoxes, can be of some help here.  I claim that reading quantum physics through Hegel and vice versa is very productive.

[[Charles Peirce]] had a more ambivalent view of Hegel:

>My three categories are nothing but Hegel's three grades of thinking. (CP 8.213),

but

>A _Phenomenology_ which does not reckon with pure mathematics, a science hardly come to years of discretion when Hegel wrote, will be the same pitiful club-footed affair that Hegel produced. (CP 5.40).

## Related concepts

* [[idealism]],

  [[objective idealism]], [[subjective idealism]]

  [[absolute idealism]]

* [[unity of opposites]], [[Aufhebung]],

  [[abstract general, concrete general and concrete particular]]

## Texts

* _[[Encyclopedia of the Philosophical Sciences]]_

* _[[Science of Logic]]_

* _[[Philosophy of Nature]]_

* _[[Phenomenology of Spirit]]_

* _[[Lectures on the History of Philosophy]]_

* _[[Lectures on the Philosophy of History]]_

* _[[Lectures on the Philosophy of Religion]]_

## References

* {#Redding} [[Paul Redding]],  _Georg Wilhelm Friedrich Hegel_, The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.) ([web](http://plato.stanford.edu/entries/hegel/))

* Alexander Prähauser, *Hegel in Mathematics* ([pdf](https://github.com/nameiwillforget/hegel-in-mathematics/blob/main/Hegel_in_Mathematics.pdf))

category: people

[[!redirects Georg Wilhelm Friedrich Hegel]]
[[!redirects Georg Wilhelm Hegel]]
[[!redirects Georg Hegel]]
[[!redirects Hegel]]
[[!redirects G. W. F. Hegel]]


Georg Kreisel

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

## Selected writings

On [[mathematical logic]]:

* [[Georg Kreisel]], *Mathematical Logic*,  in T. Saaty et al. (ed.), *Lectures on Modern Mathematics III*, Wiley New York (1965) 95-195

category: people

[[!redirects Kreisel]]


Georg Peschke

* [personal page](https://sites.ualberta.ca/~gepe/)

* [institute page](https://apps.ualberta.ca/directory/person/gepe)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=44750)

## Selected writings

On the notion of [[topological ends]] and [[end compactification]]:

* {#Peschke90}  [[Georg Peschke]], _The Theory of Ends_, Nieuw Archief voor Wiskunde, 8 (1990), 1-12 [[pdf](https://www.ualberta.ca/~gepe/pdf/Peschke_TheoryOfEnds.pdf), [[Peschke-Ends.pdf:file]]]

On [[localization of a space|localization of spaces]] with respect to sel-maps of the [[circle]]:

* [[Carles Casacuberta]], [[Georg Peschke]], *Localizing with respect to self maps of the circle*, Trans. Amer. Math. Soc. **339** (1993) 117-140  [[doi:10.1090/S0002-9947-1993-1123451-X](https://doi.org/10.1090/S0002-9947-1993-1123451-X), [pdf](http://www.ub.edu/topologia/casacuberta/articles/cpes.pdf)]

category: people


Georg Tamme

* [webpage](http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/tamme/)

## Selected writings

Introducing [[differential algebraic K-theory]] with application to [[higher regulator|higher]] [[Beilinson regulators]]:

* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_, Advances in Mathematics Volume 285, 5 November 2015, Pages 1853-1969
([arXiv:1209.6451](http://arxiv.org/abs/1209.6451), [doi:10.1016/j.aim.2015.08.004](https://doi.org/10.1016/j.aim.2015.08.004))

* {#BunkeTamme13} [[Ulrich Bunke]], [[Georg Tamme]], _Multiplicative differential algebraic K-theory and applications_,  Ann. K-Theory 1(3): 227-258 (2016)   ([arXiv:1311.1421](http://arxiv.org/abs/1311.1421), [doi:10.2140/akt.2016.1.227](https://doi.org/10.2140/akt.2016.1.227))

category: people


George Bergman

__George M. Bergman__ is an [[algebraist]] at the University of California at Berkeley.

* [personal page](http://math.berkeley.edu/~gbergman) at Berkeley

* [institute page](http://math.berkeley.edu/index.php?module=mathfacultyman&MATHFACULTY_MAN_op=sView&MATHFACULTY_id=5)

## Selected writings

* partial [bibliography](http://bibserver.berkeley.edu/cgi-bin/bibs7?source=http://math.berkeley.edu/~gbergman/bibserver.bib)

On [[diagram chasing]] and [[elements in an abelian category]]:

* {#Bergman} [[George Bergman]], _A note on abelian categories -- translating element-chasing proofs, and exact embedding in abelian groups_ (1974) [[pdf](http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf), [[Bergman-ElementChasing.pdf:file]]]

On [[cogroups]] and [[corings]] [[internalization|internal to]] [[associative rings]]:

* {#BergmanHausknecht96} [[George M. Bergman]], Adam O. Hausknecht, _Cogroups and co-rings in categories of associative rings_, A.M.S. Math. Surveys and Monographs __45__ (1996) [ISBN 0-8218-0495-2, [ams:surv-45](https://bookstore.ams.org/surv-45), [MR 97k:16001](http://www.ams.org/mathscinet-getitem?mr=97k:16001), [errata and updates](http://math.berkeley.edu/~gbergman/papers/updates/coalg.html)]

Introducing the [[diamond lemma]] in [[ring theory]]:

* [[George M. Bergman]], _The diamond lemma for ring theory_, Advances in Mathematics **29** 2 (1978) 178-218 [doi:10.1016/0001-8708(78)90010-5]

Introducing the [[Salamander lemma]]:

* {#Bergman12} [[George Bergman]], _On diagram-chasing in double complexes_, Theory and Applications of Categories **26** (2012) 60-96 [[arXiv:1108.0958](http://arxiv.org/abs/1108.0958), [tac:26-03](http://www.tac.mta.ca/tac/volumes/26/3/26-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/26/3/26-03.pdf)]

category: people

[[!redirects George Bergman]]
[[!redirects George M Bergman]]
[[!redirects George M. Bergman]]


George Boole

George Boole (2 November 1815 – 8 December 1864) was a mathematician and logician, who was the first professor of mathematics at the then Queen's College Cork in Ireland (now University College Cork, (UCC)). He is best known mathematically for his work on algebraic logic which led to the development of [[Boolean algebra]]s. His work is often cited as being one of the first steps towards the invention of computers.

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

* [St Andrews History archive article](http://www-history.mcs.st-andrews.ac.uk/Biographies/Boole.html)

* [Boole page, UCC](https://georgeboole200.ucc.ie/boole/)

* [Boole archive (UCC)](http://booleweb.ucc.ie/index.php?pageID=264)

## Selected writings

On [[formal logic]] (for which [[George Boole|Boole]] became widely famous, cf. *[[Boolean algebra]]*) but also on [[probability theory]], where  (as observed by [Pitowsky 1989a](Bell's+inequalities#Pitowsky89a), reviewed, e.g., in [Khrennikov 2008](Bell's+inequalities#Khrennikov08)) Boole's book derives the inequalities that later became known as *[[Bell's inequalities]]* in [[quantum probability theory]]:

* [[George Boole]]: *An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities* (1854) republished by: Cambridge University Press (2009)
[[doi:10.1017/CBO9780511693090](https://doi.org/10.1017/CBO9780511693090), [Wikipedia entry](https://en.wikipedia.org/wiki/The_Laws_of_Thought), [Project Gutenberg eBook](https://gutenberg.org/ebooks/15114)]

category:people


George E. Uhlenbeck

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

## Selected writings

Proposing the concept of intrinsic [[spin]] of the [[electron]] (making it a "[[spinor]]"):

* [[Samuel A. Goudsmit]], [[George E. Uhlenbeck]]: *Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons*, Naturwissenschaften **13** 47 (1925) 953 [pdf, [[GoudsmitUhlenbeck-ElektronSpin.pdf:file]]]

category: people

[[!redirects George Uhlenbeck]]


George Ellis

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

## Selected writings:

On [[inhomogeneous cosmology]]:

* {#EllisStoeger87} [[George Ellis]], W. Stoeger, _The 'fitting problem' in cosmology_,  Classical and Quantum Gravity, Volume 4, Number 6, 1987 ([doi:10.1088/0264-9381/4/6/025](https://iopscience.iop.org/article/10.1088/0264-9381/4/6/025))

* {#Ellis18} [[George Ellis]], summary talk at [CosmoBack 2018](https://cosmoback.sciencesconf.org/) ([[EllisCosmoBack18.pdf:file]])

category: people

[[!redirects George Francis Rayner Ellis]]


George H. Mealy

George H. Mealy was an American [[mathematics|mathematician]] and [[computer science|Computer Scientist]].  He defined the type of [[finite state automaton]] known as a [[Mealy machine]].

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

## Related $n$Lab entries

* [[Moore machine]]

* [[Mealy machine]]

* [[Mealy morphism]]

category: people


George Janelidze

**George Janelidze** is a Georgian-South African category theorist, currently a professor at the University of Cape Town. He studied in [Tbilisi](http://en.wikipedia.org/wiki/Tbilisi) under [[Hvedri Inassaridze]] with the dissertation "Satellites and Galois Extensions of Commutative Rings".

His research includes [[nonabelian homological algebra]], the discovery of [[categorical Galois theory]], [[semiabelian categories]], [[internal actions]] and [[internal crossed modules]], [[categorical algebra|categorical]] [[universal algebra]] etc.

* [website](http://www.mth.uct.ac.za/prof-george-janelidze-0)

* [old website](http://www.rmi.ge/~gjanel)

* [MathGenealogy entry](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=80837)

## Selected writings

On ([[monadic descent|monadic]]) [[descent]]:

* {#JanelidzeTholen94} [[George Janelidze]], [[Walter Tholen]], *Facets of descent I*, Applied Categorical Structures **2** 3 (1994) 245-281 [[doi:10.1007/BF00878100](https://doi.org/10.1007/BF00878100)]

* {#JanelidzeTholen97} [[George Janelidze]], [[Walter Tholen]], *Facets of descent II*, Applied Categorical Structures **5** 3 (1997) 229-248 [[doi:10.1023/A:1008697013769](https://doi.org/10.1023/A:1008697013769)]

On [[reflective factorization systems]]:

* {#CJKP} [[Aurelio Carboni]], [[George Janelidze]], [[Max Kelly]], [[Robert Paré]], _On localization and stabilization for factorization systems_, Appl. Categ. Structures 5 (1997), 1--58 ([doi:10.1023/A:1008620404444](https://doi.org/10.1023/A:1008620404444))

On [[Galois theory]]:

* [[Francis Borceux]], [[George Janelidze]]: *[[Galois Theories]]*, Cambridge Studies in Advanced Mathematics **72**, Cambridge University Press (2001) [[doi:10.1017/CBO9780511619939](https://doi.org/10.1017/CBO9780511619939)]

On [[monoid objects in monoidal categories]]:

* [[Francis Borceux]], [[George Janelidze]], [[Gregory Maxwell Kelly]], _Internal object actions_, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 ([dml:249553](https://eudml.org/doc/249553))

category: people

[[!redirects G. Janelidze]]
[[!redirects Janelidze]]
[[!redirects G. Z. Dzhanelidze]]


George Kac

__George I. Kac__ (also spelled Kats) was an algebraist with contributions to noncommutative algebra, group rings, Hopf algebras, extensions of Tannaka-Krein duality etc. So called Kac algebras (especially suitable for [[Tannaka duality]] in certain generality) are named after him.

The standard monograph on [[Kac algebra]]s is

* M. Enock, J. M. Schwartz, _Kac algebras and duality of locally compact groups_,  Springer-Verlag, 1992, , x+257 pp. [gBooks](http://books.google.com/books/about/Kac_algebras_and_duality_of_locally_comp.html?id=U6e6aD1gj3oC), [MR94e:46001](https://mathscinet.ams.org/mathscinet-getitem?mr=1215933)

## Selected publications

A famous 8-dimensional noncommutative Hopf algebras is first exhibited in

* G. I. Kac, V. G. Paljutkin, _Finite ring groups_, Trans. Moscow Math. Soc. 15 (1966) 251-294
* F. A. Berezin, G. I. Kats, _Lie groups with commuting and anticommuting parameters_, Matematicheskii Sbornik, __82__ (124):3(7) (1970) 343-350 [doi](https://doi.org/10.1070/SM1970v011n03ABEH001137) [references](https://www.mathnet.ru/links/025930de46ccc91599a8a5b612873ea6/sm_3454_refs_eng.pdf)
* G. I. Kac, _Extensions of groups to group rings_, Mathematics of the USSR-Sbornik, 1968, __5__:3 (1968) 451-474 [doi](https://doi.org/10.1070/SM1968v005n03ABEH003627)

category: people

[[!redirects G. I. Kac]]
[[!redirects G. I. Kats]]


George Lakoff

George Lakoff is a researcher into language, culture and cognition.

* wikipedia: [George Lakoff](https://en.wikipedia.org/wiki/George_Lakoff)

Cognitive, [[linguistics|linguistic]] and cultural aspects of mathematics (which are also of relevance for learning) are emphasized in

* George Lakoff, Rafael E. Núñez, _Where mathematics comes from_, Basic Books 2000, xviii+493 pp. [MR2001i:00013](http://www.ams.org/mathscinet-getitem?mr=1794854) (cf. also wikipedia: [George Lakoff#Mathematics](https://en.wikipedia.org/wiki/George_Lakoff#Mathematics) and [Where Mathematics Comes From](https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From))

Cognitive role of metaphors is a recurring theme in works of Lakoff.

* George Lakoff, Mark Johnson, _Metaphors We Live By_, University of Chicago Press 1980; 2003. 2004 edition contains an 'Afterword' (see also wikipedia: [Metaphors We Live By](https://en.wikipedia.org/wiki/Metaphors_We_Live_By))

Lakoff is one of the founders of generative semantics (a viewpoint on syntactic analysis (see [wikipedia](https://en.wikipedia.org/wiki/Generative_semantics)) taking into account semantics as a basis).

* George Lakoff, _Toward a generative semantics_, 1963, 1976 [pdf](https://escholarship.org/content/qt64m2z2b1/qt64m2z2b1.pdf)

category: people


George Lusztig

__George Lusztig__ is a mathematician at MIT, born in Romania. He has many deep contributions to Lie theory and representation theory, especially using and developing the methods of [[geometric representation theory]]. He discovered the canonical bases of quantum and Lie groups (in a sense dual to the crystal basis of [[Masaki Kashiwara]]).

* $n$Lab: [[Kazhdan-Lusztig theory]]
* [homepage](http://math.mit.edu/~gyuri/)
* list of [papers](http://www-math.mit.edu/~gyuri/pub.html) and his own [comments](http://www-math.mit.edu/~gyuri/papers/comm.html) on some papers
* [wikipedia](http://en.wikipedia.org/wiki/George_Lusztig)

## Selected works

* George Lusztig, _Intersection cohomology methods in representation theory_, Proc. ICM 1990, [pdf](http://mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0155.0174.ocr.pdf)

[[Kazhdan-Lusztig theory]]:

* [[David Kazhdan]], [[George Lusztig]], _Representations of Coxeter groups and Hecke algebras_, Invent. Math. __53__:2 (1979) 165--184, [MR81j:20066](https://www.ams.org/mathscinet-getitem?mr=560412), [doi](https://doi.org/10.1007/BF01390031)

On [[total positivity]] in the context of [[reductive group]]s

* George Lusztig, _Total positivity in reductive groups. In: Brylinski, JL., Brylinski, R., Guillemin, V., Kac, V. (eds) Lie Theory and Geometry. Progress in Mathematics __123__; Birkhäuser 1994 [doi](https://doi.org/10.1007/978-1-4612-0261-5_20)

On [[quantum group]]s (and relations to quivers and canonical bases)

* George Lusztig, _Quivers, perverse sheaves, and quantized enveloping algebras_ J. Amer. Math. Soc., 4(2):365--421 (1991)
* G. Lusztig, _Canonical bases arising from quantized enveloping algebras_, J. Amer. Math. Soc. 3(2),
447–498 (1990)
* G. Lusztig, _Quivers, perverse sheaves, and quantized enveloping algebras_, J. Amer. Math. Soc. 4(2),
365–421 (1991)
* G. Lusztig, _Introduction to Quantum Groups_, Birkhauser (1992)
* G. Lusztig, _Canonical bases and Hall algebras_, In: Representation Theories and Algebraic Geometry
(Montreal, PQ, 1997), 365--399, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514. Kluwer (1998)

On Kazhdan-Luzstig correspondence and monoidal structures

* George Lusztig, [[David Kazhdan]], _Tensor structures arising from affine Lie algebras_, J. Amer. Math. Soc. __6__ (1993) 905-947 [doi](https://doi.org/10.1090/S0894-0347-1993-99999-X), Part II: J. Amer. Math. Soc. (1993), 949-1011 [doi](https://doi.org/10.1090/S0894-0347-1993-1186962-0); Part III: J. Amer. Math. Soc. (1994) 335-381 [doi](https://doi.org/10.1090/S0894-0347-1994-1239506-X); Part IV: J. Amer. Math. Soc. (1994) 383-453 [doi](https://doi.org/10.1090/S0894-0347-1994-1239507-1)
* G. Lusztig, _Comments on my papers_, [arXiv:1707.09368](https://arxiv.org/abs/1707.09368), 55 pages, contains historical notes on some of Lusztig's papers and the list of 247 publications till mid 2017

[[!redirects G. Lusztig]]
category: people


George Mackey

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

## Links

* _George Mackey (1916–2006)_, Notices of the American Mathematical Society; vol. 54, no. 7 (August 2007) ([pdf](http://www.ams.org/notices/200707/tx070700824p.pdf)).
* [[David Mumford]], [George Mackey](http://www.amphilsoc.org/sites/default/files/proceedings/1520411.pdf).
* [Harvard commemorative website](http://www.math.harvard.edu/history/mackey/index.html)

## Selected writings

On [[Pontrjagin duality]]:

* {#Mackey48} [[George Mackey]], *The Laplace Transform For Locally Compact Abelian Groups*, Proceedings of the National Academy of Sciences of the United States of America Vol. 34, No. 4 (Apr. 15, 1948), pp. 156-162 ([jstor:87968](https://www.jstor.org/stable/87968))

On the [[mathematical physics]] and [[quantum logic]] of [[quantum mechanics]]:

* [[George Mackey]], *The Mathematical Foundations of Quantum Mechanics: a Lecture-note Volume*, Mathematical physics monograph series, Benjamin (1963), Dover (2004) [[google books](https://books.google.de/books?id=qlpb2mWYmfYC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)]

On [[harmonic analysis]]:

* _Harmonic analysis as the exploitation of symmetry - a historical survey_, Bull. Amer. Math. Soc. (N.S.) Volume 3, Number 1, Part 1 (1980), 543-698 ([pdf](http://www.ams.org/journals/bull/1980-03-01/S0273-0979-1980-14783-7/S0273-0979-1980-14783-7.pdf)).

This is reprinted with other essays in

* _The Scope and History of Commutative and Noncommutative Harmonic Analysis_, AMS, 1992.

##Related concepts

* [[induced representation]], [[Wigner classification]]

* [[Mackey functor]]

category: people

[[!redirects George W. Mackey]]


George Mostow

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

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=28467)

## Selected writings

On the [[slice theorem]]:

* [[George Mostow]], _Equivariant embeddings in Euclidean space_,  Ann. of Math. 65 (1957), 432-446 ([jhir:1774.2/46183](http://jhir.library.jhu.edu/handle/1774.2/46183), [pdf scan](https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/46183/Mostow_Mar57.pdf?sequence=1&isAllowed=y))

On the [[Mostow rigidity theorem]]:

* {#Mostow68} [[George Mostow]], _Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms_,  Publications Mathématiques de l'IHÉS, Volume 34  (1968), p. 53-104 ([numdam:PMIHES_1968__34__53_0](http://www.numdam.org/item/PMIHES_1968__34__53_0))

category: people


George Papadopoulos

George Papadopoulos is professor for mathematics at King's College London.

* [webpage](http://www.kcl.ac.uk/nms/depts/mathematics/people/atoz/papadopoulosg.aspx)

## Seletced writings

On [[Killing spinors]] in [[heterotic supergravity]]:

* [[Ulf Gran]], [[George Papadopoulos]], Diederik Roest, _Supersymmetric heterotic string backgrounds_, Phys.Lett.B656:119-126, 2007 ([arXiv:0706.4407](https://arxiv.org/abs/0706.4407))

On [[Killing spinors]] [[D=6 supergravity]] related to [[D=6 N=(1,0) SCFT]]:

* Mehmet Akyol, [[George Papadopoulos]], _Spinorial geometry and Killing spinor equations of 6-D supergravity_, Class. Quant. Grav. 28:105001, 2011 ([arXiv:1010.2632](https://arxiv.org/abs/1010.2632))

* Mehmet Akyol, [[George Papadopoulos]], _$(1,0)$ superconformal theories in six dimensions and Killing spinor equations_, JHEP 07 (2012) 070 ([arXiv:1204.2167](https://arxiv.org/abs/1204.2167))

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

* {#ILPT95} [[José M. Izquierdo]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], *Dyonic Membranes*, Nucl. Phys. B **460** (1996) 560-578 [[arXiv:hep-th/9508177](https://arxiv.org/abs/hep-th/9508177), doi:10.1016/0550-3213(95)00606-0]

* [[Michael Green]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], _Dyonic $p$-branes from self-dual $(p+1)$-branes_, Phys.Lett.B384:86-92, 1996 ([arXiv:hep-th/9605146](https://arxiv.org/abs/hep-th/9605146))

* [[George Papadopoulos]], [[Dimitrios Tsimpis]], _The holonomy of the supercovariant connection and Killing spinors_, JHEP 0307:018, 2003 ([arXiv:hep-th/0306117](https://arxiv.org/abs/hep-th/0306117))

On [[black brane|black]]$\,$[[D8-brane]]-solutions in [[massive type IIA supergravity]]/[[massive type IIA string theory]]:

* {#BergshoessdeRooGreenPapadopoulosTownsend96} [[Eric Bergshoeff]], [[Mees de Roo]], [[Michael Green]], [[George Papadopoulos]], [[Paul Townsend]], _Duality of Type II 7-branes and 8-branes_, Nucl. Phys. B470:113-135, 1996 ([arXiv:hep-th/9601150](https://arxiv.org/abs/hep-th/9601150))

## Related $n$Lab entries

* [[exceptional geometry]]

category: people


George Peschke

* [webpage](https://www.math.ualberta.ca/pi/authors/author_peschke.html)

## selected writings

* {#Peschke94} [[George Peschke]], _Degree of certain equivariant maps into a representation sphere_, Topology and its Applications Volume 59, Issue 2, 30 September 1994, Pages 137-156 (doi:10.1016/0166-8641(94)90091-4)

  (on [[equivariant cohomotopy]])

## related $n$Lab entries

* [[equivariant cohomotopy]]

category: people


George Polya

__George Pólya__ (December 13, 1888 – September 7, 1985) was a Hungarian mathematician (natively: Pólya György) who worked in many fields of mathematics, including [[combinatorics]], analysis, probability and [[arithmetic]]. He was the author of a number of books on the heuristics of mathematical research, that is, methods to formulate and solve problems. Against the trend of considering mathematical reasoning to be wholly deductive, Pólya wrote on analogical and [[inductive reasoning]] in mathematics. He was also writing about mathematics education.

* wikipedia [George Pólya](http://en.wikipedia.org/wiki/George_P%C3%B3lya), [Pólya enumeration theorem](http://en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem)
* [[Pólya enumeration theorem]]

category: people


George R. Livesay

* [Mathematics Genealogy page](https://genealogy.math.ndsu.nodak.edu/id.php?id=4034)

## Selected writings

On [[fixed point]]-free [[involutions]], hence [[free action|free]] [[group actions on n-spheres|$\mathbb{Z}/2$-actions on]], the [[3-sphere]]:

* {#Livesay60} [[George R. Livesay]]  *Fixed Point Free Involutions on the 3-Sphere*, Annals of Mathematics Second Series, Vol. 72, No. 3 (Nov., 1960), pp. 603-611 ([jstor:1970232](https://www.jstor.org/stable/1970232))

category: people

[[!redirects G. R. Livesay]]


George Raptis

* [personal page](https://graptismath.net/)

## Selected writings

On a [[Thomason model structure]] on (just) [[posets]]:

* [[George Raptis]], *Homotopy theory of posets*, Homology Homotopy Appl. **12** 2 (2010) 211-230 [[euclid:hha/1296223882](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-2/Homotopy-theory-of-posets/hha/1296223882.full)]

On [[combinatorial model categories]]:

* [[George Raptis]], *Notes on combinatorial model categories* 2014/2020 ([pdf](https://graptismath.net/files/notes-comb-model-cat.pdf) [[RaptisCombinatorialModelCategories.pdf:file]])

On [[epimorphisms in an (∞,1)-category]]:

* {#Raptis17} [[George Raptis]], *Some characterizations of acyclic maps*, Journal of Homotopy and Related Structures volume 14, pages 773–785 (2019) 2017 ([arxiv:1711.08898](https://arxiv.org/abs/1711.08898), [doi:10.1007/s40062-019-00231-6](https://doi.org/10.1007/s40062-019-00231-6))

On the [[adjoint (∞,1)-functor theorem]]:

* {#NRS18} Hoang Kim Nguyen, [[George Raptis]], Christoph Schrade, _Adjoint functor theorems for ∞-categories_, Journal of the London Mathematical Society **101** 2 (2019) 659-681
([arXiv:1803.01664](https://arxiv.org/abs/1803.01664))

category: people


George Spencer Brown

* [Wikipedia entry](http://en.wikipedia.org/wiki/G._Spencer-Brown)

category: people


George Strecker

George E. Strecker

* [website](http://www.math.ksu.edu/~strecker/)

## Selected writings

On [[category theory]] with emphasis on [[concrete categories]]:

* [[Jiri Adamek|Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]], *[[Abstract and Concrete Categories -- The Joy of Cats]]*, John Wiley and Sons, New York (1990), reprinted in: Reprints in Theory and Applications of Categories **17** (2006) 1-507 ([tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html), [webpage](http://katmat.math.uni-bremen.de/acc/), [pdf](http://katmat.math.uni-bremen.de/acc/acc.pdf))

On [[category theory|category theoretic]] aspects of [[general topology]]:

* [[Horst Herrlich]], [[George Strecker]], *Categorical topology -- Its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971* ([pdf](https://link.springer.com/content/pdf/10.1007%2F978-94-017-0468-7_15.pdf)), pages 255-341 in: C. E. Aull, R Lowen (eds.), *Handbook of the History  of General Topology. Vol. 1* , Kluwer 1997 ([doi:10.1007/978-94-017-0468-7](https://link.springer.com/book/10.1007/978-94-017-0468-7))

category: people

[[!redirects George E. Strecker]]


George Thompson

* [personal page](http://users.ictp.it/~thompson/)

* [InSpire page](https://inspirehep.net/authors/986179)

* [GoogleScholar page](https://scholar.google.com/citations?user=0BW2JXIAAAAJ&hl=en)

## Selected writings

On [[massive Yang-Mills theory]]:

* {#DelbourgoTwiskThompson88} Robert Delbourgo, S. Twisk, [[George Thompson]], _Massive Yang-Mills theory: Renormalizability versus unitarity_, Int. J. Mod. Phys. A **3** 2 (1988) 435-449 [[doi:10.1142/S0217751X88000163](https://doi.org/10.1142/S0217751X88000163)]

On [[topological field theory]]:

* Danny Birmingham, [[Matthias Blau]], Mark Rakowski, [[George Thompson]], *Topological field theory*, Physics Reports **209** 4–5 (1991) 129-340 \[doi:10.1016/0370-1573(91)90117-5\]

On the [[supersymmetry|supersymmetric]] [[WZW model]] using [[group algebra]]:

* [[Matthias Blau]], [[George Thompson]], *Equivariant Kähler Geometry and Localization in the $G/G$ Model*, Nucl. Phys. B **439** (1995) 367-394 [doi:10.1016/0550-3213(95)00058-Z, [arXiv:hep-th/9407042](https://arxiv.org/abs/hep-th/9407042)]

category: people


George Whitehead

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

> Not to be confused with [[J. H. C. Whitehead]], who also worked in [[algebraic topology]].

## Selected writings

Introducing the [[J-homomorphism]]:

* {#Whitehead42} [[George Whitehead]], _On the homotopy groups of spheres and rotation groups_, Annals of Mathematics. Second Series 43 (4): 634–640 (1942) ([jstor:1968956](http://www.jstor.org/stable/1968956))

Computing the [[second stable homotopy group of spheres]]:

* [[George Whitehead]], _The $(n+2)$nd Homotopy Group of the $n$-Sphere_, Annals of Mathematics Second Series, Vol. 52, No. 2 (Sep., 1950), pp. 245-247  ([jstor:1969466](https://www.jstor.org/stable/1969466))

On [[homotopy theory]]:

* [[George Whitehead]], _Elements of homotopy theory_, Springer 1978 ([doi:10.1007/978-1-4612-6318-0](https://link.springer.com/book/10.1007/978-1-4612-6318-0))

## Related $n$Lab entries

* [[Spanier-Whitehead category]]

* [[Spanier-Whitehead duality]]

* [[stable homotopy category]]

* [[generalized (Eilenberg-Steenrod) cohomology]]

* [[Whitehead principle]]

category: people

[[!redirects George W. Whitehead]]


George Wilson

On [[integrable system|integrable]] hierarchies in terms of infinitedimensional [[Sato-Segal-Wilson Grassmannian]]

* G. Segal, [[George Wilson]], _Loop groups and equations of KdV type_, Publ. Math. IHÉS __61__, 5--65 (1985) [numdam](http://www.numdam.org/item/PMIHES_1985__61__5_0) [MR87b:58039](https://mathscinet.ams.org/mathscinet-getitem?mr=87b:58039) [Zbl0592.35112](https://zbmath.org/?q=an:0592.35112)

On [[Calogero model|Calogero-Moser integrable system]]s

* George Wilson, _Collisions of Calogero-Moser particles and an adelic Grassmannian_, With an appendix by I. G. Macdonald. Invent. Math. __133__:1 (1998) 1--41 [MR99f:58107](https://mathscinet.ams.org/mathscinet-getitem?mr=1626461), [doi](https://doi.org/10.1007/s002220050237)
* [[Yuri Berest]], G. Wilson, _Ideal classes of the Weyl algebra and noncommutative projective geometry_ (with an Appendix by [[M. Van den Bergh]]), Internat. Math. Res. Notices __26__ (2002) 1347--1396
* Yu. Berest, G. Wilson, _Mad subalgebras of rings of differential operators on curves_, Advances in Math. __212__ no. 1 (2007) 163--190

category: people


George Zweig

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

## Selected writings

* {#Zweig64} [[George Zweig]], _An SU(3) model for strong interaction symmetry and its breaking_,  version 1 is CERN preprint 8182/TH.401, Jan. 17, 1964, version 2 in _Developments in the Quark Theory of Hadrons_ Volume 1. Edited by D. Lichtenberg and S. Rosen. Nonantum, Mass., Hadronic Press, 1980. pp. 22-101 ([spire:4674](http://inspirehep.net/record/4674))

  (introducing the [[quark]] [[model (in theoretical physics)]])

* [[George Zweig]], _Origins of the quark model_, 1980 ([pdf](https://authors.library.caltech.edu/18969/1/Origins_of_the_Quark_Model_Final_Zweig%5B1%5D.pdf), [[ZweigQuarkModel1980.pdf:file]])

## Related $n$Lab entry

* [[quark]]

* [[QCD]]

category: people


Georges de Rham

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

## Selected writings

On [[differential forms]] and introducing their [[current (distribution theory)|currents]]:

* [[Georges de Rham]], *Variétés différentiables, formes, courants, formes harmoniques*, Act. Sci. et Ind. **1222**, Hermann, Paris (1955) [[GBooks](https://books.google.de/books/about/Vari%C3%A9t%C3%A9s_diff%C3%A9rentiables.html?id=2lLvAAAAMAAJ&redir_esc=y)]

On [[differential geometry]]: [[differentiable manifolds|differentiable]]/[[smooth manifolds]], [[differential forms]] ([[de Rham complex]]), [[harmonic forms]], [[current (distribution theory)|currents]] and their [[ordinary homology|homology]]:

* [[Georges de Rham]], *Differentiable Manifolds -- Forms, Currents, Harmonic Forms*, Grundlehren **266**, Springer (1984) [[doi:10.1007/978-3-642-61752-2](https://doi.org/10.1007/978-3-642-61752-2)]

See also:

* [[Georges de Rham]], *Quelques souvenirs des années 1925-1950*, Cahiers du séminaire d'histoire des mathématiques, Serie 1, Volume 1 (1980) 19-36 [[numdam:CSHM_1980__1__19_0](http://www.numdam.org/item/?id=CSHM_1980__1__19_0)]

## Related entries

* [[de Rham cohomology]]

* [[de Rham theorem]]

category: people


Georges Lemaître

* [Wikipedia entry](http://en.wikipedia.org/wiki/Georges_Lema%C3%AEtre)

## Selected writings

Introducing what came to be known as *[[Hubble's law]]*:

* [[Georges Lemaître]], *Un Univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques*, Annales de la Société Scientifique de Bruxelles, A **47** (1927) 49-59 [[adsabs:1927ASSB...47...49L](https://adsabs.harvard.edu/full/1927ASSB...47...49L)]

## Related entries

* [[FLRW model]]

category: people


Georges Maltsiniotis

Georges Maltsiniotis is a Greek mathematician, who works at the Institut de Mathématiques de Jussieu, Université Paris 7 Denis Diderot.

He has worked for many years now editing,  extending and completing the work of [[Grothendieck]] on *[[Pursuing Stacks]]*, and areas such as *[[Les Dérivateurs]]* and [[homotopy theory of Grothendieck|Grothendieck's homotopy theory]]. Most materials available at his

* [homepage](https://webusers.imj-prg.fr/~georges.maltsiniotis/)

## Selected writings

* _La K-théorie d'un dérivateur triangulé_ (suivi d'un appendice par B. Keller) dans "Categories in Algebra, Geometry and Mathematical Physics", Contemp. Math. __431__, pp. 341-368, (2007); Version preprint: [ps](http://people.math.jussieu.fr/~maltsin/ps/streetfest.ps) (2005)

* _Catégories triangulées supérieures_, Pré-preprint [ps](http://people.math.jussieu.fr/~maltsin/ps/triansup.ps) (2005)

On [[Grothendieck ∞-groupoids]]:

* _Grothendieck $\infty$-groupoids, and still another definition of $\infty$-categories_, [[arXiv:1009.2331](http://arxiv.org/abs/1009.2331)]

[[!redirects Maltsiniotis]]
category: people


Georges Reeb

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

## Selected writings

Introducing the [[Reeb sphere theorem]] as well as the concept now known as *[[Reeb graphs]]*:

* [[Georges Reeb]], *Sur les points singuliers d'une forme de Pfaff completement integrable ou d'une fonction numerique*, Comptes Rendus Acad. Sciences Paris **222** (1946) 847-849 $[$[crid:1571417125676878592](https://cir.nii.ac.jp/crid/1571417125676878592)$]$

## Related nLab entries

* [[Reeb vector field]]

category: people


Georges Skandalis

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

category: people

[[!redirects Skandalis]]
[[!redirects G. Skandalis]]


Georges Vincent

## Selected writings

On [[free action|free]] [[orthogonal group|orthgonal]] [[group actions on n-spheres]]:

* {#Vincent47} [[Georges Vincent]], *Les groupes linéaires finis sans points fixes*, Commentarii Mathematici Helvetici 20.1 (1947): 117-171 ([pdf](https://link.springer.com/content/pdf/10.1007/BF02568125.pdf))

category: people


Georgia Benkart

__Georgia Benkart__ is a mathematician, professor emeritus at the University of Wisconsin at Madison. Her interests are in algebra, Lie theory and representation theory.

Related entries: [[down-up algebra]]

category: people
[[!redirects G. Benkart]]


Georgia Triantafillou

* [webpage](https://www.math.temple.edu/cgi-bin/get_person?uid=georgiat)

## Selected writings

On [[equivariant rational homotopy theory]] (see [[model structure on equivariant dgc-algebras]]):

* [[Georgia Triantafillou]], _Equivariant rational homotopy theory_, chapter III of [[Peter May]], _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comeza˜na, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. ([pdf](http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf), [ISBN: 978-0-8218-0319-6](https://bookstore.ams.org/cbms-91))

* [[Georgia Triantafillou]], _Equivariant minimal models_, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) ([jstor:1999119](http://www.jstor.org/stable/1999119))

category: people


Georgios Zikos

* [GoogleScholar page](https://scholar.google.com/citations?user=6MVjQA8AAAAJ&hl=en)

## Selected writings

On [[topological quantum compilation]] and approximating [[topological quantum computation|topological]] [[braid group]] [[quantum gates]] by "weaves":

* [[Layla Hormozi]], [[Georgios Zikos]], [[Nick E. Bonesteel]], [[Steven H. Simon]], *Topological quantum compiling*, Phys. Rev. B 75, 165310 ([doi:10.1103/PhysRevB.75.165310](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.75.165310), [arXiv:quant-ph/0610111](https://arxiv.org/abs/quant-ph/0610111))

category: people


Georgiy E. Shilov

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

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=22253)

## Selected writings

Early comprehensive account of [[generalized functions]] ([[distributions]]):

* [[Israel M. Gelfand]], [[Georgiy E. Shilov]], *Generalized Functions* Vol. 1-6 (1958), Acad. Press  (1966-1968), AMS (2016) [[ISBN:978-1-4704-2885-3](https://bookstore.ams.org/view?ProductCode=CHELGELFSET)]

category: people

[[!redirects Georgiy Shilov]]


Gepner model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Gepner model_ ([Gepner 87](#Gepner87)) is a [[rational CFT|rational]] [[2d SCFT]]  which is a [[tensor product]] of $N = 2$ super-[[minimal model CFT]].

This means that Gepner models are "[[non-geometric string vacua]]" in that they do not arise as [[sigma-models]] with [[target space]] a [[smooth manifold]]. Indeed the Gepner models appear as the limiting cases of sigma-models with [[target space]] a 6d [[Calabi-Yau manifold]] at singular points in the [[moduli space]] of the CY target: the _[[flop transition]]_.

As such the Gepner models are directly analogous to the purely algebraically defined non-classical fibers in the [[Connes-Lott-Chamseddine-Barrett model]] (it is a "[[2-spectral triple]]"-analog of the [[spectral triples]] in the Connes-Lott model, see [there](Connes-Lott-Chamseddine-Barrett+model#Schreiber16)) and, accordingly, plays a central role in [[string phenomenology]] (for review see e.g. [Reppel 07](#Reppel07)).

The Gepner models are a basic building block for [[rational conformal field theory]].

## Properties

### Boundary states
 {#BoundaryStates}

> All the known [[rational CFT|rational]] [[boundary states]] for [[Gepner models]] can be regarded as [[permutation branes]].

([Enger-Recknagel-Roggenkamp 05](#EngerRecknagelRoggenkamp05))

### Phenomenology
 {#Phenomenology}

Discussion of [[string phenomenology]] of [[intersecting D-brane models]] [[KK-compactification|KK-compactified]] with [[non-geometric background|non-geometric]] [[fibers]] such that the would-be string [[sigma-models]] with these [[target spaces]] are in fact [[Gepner models]] (in the sense of _[Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_) is in ([Dijkstra-Huiszoon-Schellekens 04a](#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04a)):

>  A plot of [[standard model of particle physics|standard model]]-like [[coupling constants]] in a computer scan of [[Gepner model]]-[[KK-compactification]] of [[intersecting D-brane models]] according to [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b).

>  The blue dot indicates the couplings in $SU(5)$-[[GUT]] theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.

## Related concepts

* [[WZW model]]

* [[Kazama-Suzuki model]]

* [[heterotic string]], [[type II superstring]]

* [[KK-compactification]]

## References

The original article is

* {#Gepner87} [[Doron Gepner]], _Space-time supersymmetry in compactified string theory and superconformal models_, Nucl. Phys. B296 (1987) 757.

Lecture notes include

* [[Doron Gepner]], _Lectures On $N=2$ String Theory_, 1989 ([[Gepner89.pdf:file]], [spire:277718/](http://inspirehep.net/record/277718/))

Further discussion in

* [[Maximilian Kreuzer]], _Heterotic $(0,2)$ Gepner Models and Related Geometries_, in _Fundamental Interactions_, pp. 335-362 World Scientific (2009) ([arXiv:0904.4467](https://arxiv.org/abs/0904.4467), [doi:10.1142/9789814277839_0019](https://doi.org/10.1142/9789814277839_0019))

See also the references at _[[flop transition]]_ for more.

Review of application in [[string phenomenology]] includes

* {#Reppel07} Christian Reppel, _Phenomenological Aspects of Gepner Models_, 2007 ([pdf](https://www.ru.nl/publish/pages/760962/christiaan_reppel.pdf))

[[D-branes]] in [[string theory]] [[vacua]] defined by Gepner model SCFTs are discussed, via [[boundary conformal field theory]] in

* {#FuchsSchweigertWalcher00} [[Jürgen Fuchs]], [[Christoph Schweigert]], J. Walcher, _Projections in string theory and boundary states for Gepner models_, Nucl.Phys. B588 (2000) 110-148 ([arXiv:hep-th/0003298](https://arxiv.org/abs/hep-th/0003298))

  (with emphasis on [[GSO projections]])

* {#FuchsKasteLercheLutkenSchweigert00} [[Jürgen Fuchs]], P. Kaste, [[Wolfgang Lerche]], C. Lutken, [[Christoph Schweigert]], J. Walcher, _Boundary Fixed Points, Enhanced Gauge Symmetry and Singular Bundles on K3_, Nucl.Phys.B598:57-72, 2001 ([arXiv:hep-th/0007145](https://arxiv.org/abs/hep-th/0007145))

See also

* [[Andreas Recknagel]], [[Volker Schomerus]], _D-branes in Gepner models_, Nucl.Phys. B531 (1998) 185-225 ([arXiv:hep-th/9712186](http://arxiv.org/abs/hep-th/9712186))

* {#BraunSchaeferNameki05} [[Volker Braun]], [[Sakura Schafer-Nameki]], _D-Brane Charges in Gepner Models_, J. Math. Phys. 47 (2006) 092304 ([arXiv:hep-th/0511100](https://arxiv.org/abs/hep-th/0511100))

Discussion of [[permutation D-branes]] for Gepner models, via [[boundary conformal field theory]], includes

* {#EngerRecknagelRoggenkamp05} Håkon Enger, [[Andreas Recknagel]], [[Daniel Roggenkamp]], _Permutation branes and linear matrix factorisations_, JHEP0601:087, 2006 ([arXiv:hep-th/0508053](https://arxiv.org/abs/hep-th/0508053))

Gepner model [[orientifolds]]:

* Brandon Bates, [[Charles Doran]], Koenraad Schalm, _Crosscaps in Gepner Models and the Moduli space of T2 Orientifolds_, Advances in Theoretical and Mathematical Physics, Volume 11, Number 5, 839-912, 2007 ([arXiv:hep-th/0612228](https://arxiv.org/abs/hep-th/0612228))

Specifically [[string phenomenology]] and the [[landscape of string theory vacua]] of Gepner model [[orientifold]] compactifications:

* {#DijkstraHuiszoonSchellekens04a} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models_, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th/0403196](https://arxiv.org/abs/hep-th/0403196))

* {#DijkstraHuiszoonSchellekens04b} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Supersymmetric Standard Model Spectra from RCFT orientifolds_, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th/0411129](https://arxiv.org/abs/hep-th/0411129))

* [[Fernando Marchesano]], [[Bert Schellekens]], [[Timo Weigand]], Sec. 4 of: *D-brane and F-theory Model Building* [[arXiv:2212.07443](https://arxiv.org/abs/2212.07443)]

[[!redirects Gepner models]]

[[!redirects Gepner point]]
[[!redirects Gepner points]]


Gerald B. Folland

* [personal page](https://sites.math.washington.edu//~folland/Homepage/)

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

## Selected writings

* [[Gerald B. Folland]], *A course in abstract harmonic analysis*, Studies in Advanced Mathematics. CRC Press (1995) [[pdf](https://sv.20file.org/up1/1415_0.pdf), [gBooks](http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC)]

category: people

[[!redirects Gerald Folland]]


Gerald Dunne

* [webpage](https://www.phys.uconn.edu/~dunne/)

## Selected writings

On the [[Schwinger effect]]:

* [[Gerald Dunne]], _The Heisenberg-Euler Effective Action: 75 years on_, Int. J. Mod.Phys. A27 (2012) 1260004 ([arXiv:1202.1557](https://arxiv.org/abs/1202.1557))

* {#Dunne04} [[Gerald Dunne]], _Heisenberg-Euler Effective Lagrangians: Basics and Extensions_, in:  [[Misha Shifman]], [[Arkady Vainshtein]], [[John Wheater]] (eds.), _[[From Fields to Strings -- Circumnavigating Theoretical Physics]]_, pp. 445-522, World Scientific 2005 ([arXiv:hep-th/0406216](https://arxiv.org/abs/hep-th/0406216), [doi:10.1142/9789812775344_0014](https://doi.org/10.1142/9789812775344_0014))

Discussion of the [[Schwinger effect]] via [[worldline formalism]]:

* [[Gerald Dunne]], [[Christian Schubert]], _Worldline Instantons and Pair Production in Inhomogeneous Fields_, Phys. Rev. D72 (2005) 105004 ([arXiv:hep-th/0507174](https://arxiv.org/abs/hep-th/0507174))

Discussion of [[experiments]] that could/should see the [[Schwinger effect]]:

* {#Dunne09} [[Gerald Dunne]], _New Strong-Field QED Effects at ELI: Nonperturbative Vacuum Pair Production_, Eur. Phys. J.D55:327-340, 2009 ([arXiv:0812.3163](https://arxiv.org/abs/0812.3163))

On the appearance of [[Bernoulli numbers]] in [[perturbative quantum field theory]] and [[string theory]]:

* [[Gerald Dunne]], [[Christian Schubert]], _Bernoulli Number Identities from Quantum Field Theory and Topological String Theory_, Communications in Number Theory and Physics, Volume 7 (2013) Number 2, 225 - 249 ([arXiv:math/0406610](https://arxiv.org/abs/math/0406610))

On [[resurgence]]:

* [[Gerald Dunne]], _Introduction to Resurgence, Trans-series and Non-perturbative Physics_, 2018 ([pdf](https://www.icts.res.in/sites/default/files/NUMSTRINGS2018-2018-01-27-Gerald-Dunne.pdf))

* {#DunneUnsal14} [[Gerald Dunne]], Mithat Unsal, _Uniform WKB, Multi-instantons, and Resurgent Trans-Series_, Phys. Rev. D 89, 105009 (2014) ([arXiv:1401.5202](https://arxiv.org/abs/1401.5202))

On [[Borel summation]]:

* {#CostinDunne17} Ovidiu Costin, [[Gerald Dunne]], _Convergence from Divergence_ ([arXiv:1705.09687](https://arxiv.org/abs/1705.09687))

On [[effective action functionals]]:

* [[Gerald Dunne]], _Heisenberg-Euler Effective Lagrangians : Basics and Extensions_, From Fields to Strings: Circumnavigating Theoretical Physics: pp. 445-522. ([arXiv:hep-th/0406216](https://arxiv.org/abs/hep-th/0406216), [doi:10.1142/9789812775344_0014](https://doi.org/10.1142/9789812775344_0014))

category: people


Gerald E. Brown

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

* [inspire page](https://inspirehep.net/authors/1015185)

## Selected writings

On [[Skyrmions]]:

* [[Ismail Zahed]], [[Gerald E. Brown]] *The Skyrme model*, Physics Reports **142** 1–2 (1986) 1-102 [doi:10.1016/0370-1573(86)90142-0]

* {#BrownRho88} [[Gerald E. Brown]], [[Mannque Rho]], *The Chiral Bag*, Comments Nucl. Part. Phys. **18** 1 (1988) 1-29 [[spire:18025](http://inspirehep.net/record/18025)]

category: people

[[!redirects Gerald Brown]]


Gerald Höhn

* [website](http://www.math.ksu.edu/~gerald/)

## Selected writings

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

* [[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)]

category: people


Geraldine Brady

* [webpage](http://people.cs.uchicago.edu/~brady/)

## Some writings

* with [[Todd Trimble]], _[[A string diagram calculus for predicate logic]]_

category: people


Gerard J. Milburn

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

* [Institute page](https://smp.uq.edu.au/profile/194/gerard-milburn)

## Selected writings

Introducing [[measurement-based quantum computation]]:

* [[Emanuel Knill]], [[Raymond Laflamme]], [[Gerard J. Milburn]], *A scheme for efficient quantum computation with linear optics*, Nature **409** (2001) 46–52 [[doi:10.1038/35051009](https://doi.org/10.1038/35051009)]

On the *[[DQC1]]* [[quantum complexity theory|quantum complexity]] [[complexity class|class]] ("one clean qbit"):

* [[David Poulin]], [[Raymond Laflamme]], [[Gerard J. Milburn]], [[Juan Pablo Paz]], *Testing integrability with a single bit of quantum information*, Phys. Rev. A **68** 22302 (2003) [[arXiv:quant-ph/0303042](https://arxiv.org/abs/quant-ph/0303042), [doi:10.1103/PhysRevA.68.022302](https://doi.org/10.1103/PhysRevA.68.022302)]

category: people

[[!redirects Gerard Milburn]]


Gerard Murphy

* Wikipedia entry

## Selected writings

On [[C*-algebra]]] and [[operator algebra]]:

* [[Gerard Murphy]], _$C^\ast$-algebras and Operator Theory_, Academic Press 1990 ([doi:10.1016/C2009-0-22289-6](https://doi.org/10.1016/C2009-0-22289-6))

category: people


Gerard van der Geer

* [webpage](http://gerard.vdgeer.net)

category: people


Gerard ’t Hooft

* [webpage](http://www.staff.science.uu.nl/~hooft101/)

## Selected writings

Proving the non-[[renormalizable interaction|renormalizability]] of Einstein-[[gravity]] ([[general relativity]]):

* {#tHooftVelrman74} [[Gerard 't Hooft]], [[Martinus Veltman]], _One loop divergencies in the theory of gravitation_, Ann. Inst. Poincar é 20 (1974) 69 ([numdam:AIHPA_1974__20_1_69_0](AIHPA_1974__20_1_69_0))

Introducing [['t Hooft double line notation]] and the [[large N limit]] of [[Yang-Mills theory]]/[[QCD]] at fixed [['t Hooft coupling]] in terms of [[planar graph|planar]] [[Feynman diagrams]]:

* [[Gerard 't Hooft]], _A Planar Diagram Theory for Strong Interactions_, Nucl. Phys. B72 (1974) 461 ([spire:80491](http://inspirehep.net/record/80491), doi:10.1016/0550-3213(74)90154-0)

* [[Gerard 't Hooft]], _Large N_, workshop lecture ([hep-th/0204069](http://arxiv.org/abs/hep-th/0204069))

On [[D=2 QCD]] ([['t Hooft model]]):

* {#tHooft74} [[Gerard ’t Hooft]], _A Two-Dimensional Model For Mesons_, Nucl. Phys. B75, 461 (1974) (doi:10.1016/0550-3213(74)90088-1)

Original argument that [[magnetic monopoles]] may appear in [[grand unified theories]]:

* [[Gerard ’t Hooft]], *Magnetic Monopoles in Unified Gauge Theories*, Nucl. Phys. B **79** (1974) 276-284 [doi:10.1016/0550-3213(74)90486-6]

On [[naturalness]] and [[chiral symmetry breaking]] (and introducing the concept that became known as *[['t Hooft anomalies]]*):

* {#tHooft1980} [[Gerard 't Hooft]], *Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking* In: G. 'Hooft et al., *Recent Developments in Gauge Theories*, NATO Advanced Study Institutes Series **59**, Springer (1980) [[doi:10.1007/978-1-4684-7571-5_9](https://doi.org/10.1007/978-1-4684-7571-5_9)]

On [[3d gravity]] as a [[Chern-Simons theory]] and its [[holography|holographic]] relation to a [[2d CFT]] [[boundary field theory]] (well before [[AdS/CFT]] was conceived from [[string theory]]):

* [[Stanley Deser]], [[Roman Jackiw]], [[Gerard 't Hooft]], _Three-dimensional Einstein gravity: Dynamics of flat space_, Ann. Phys. 152 (1984) 220 (doi:10.1016/0003-4916(84)90085-X)

On [[instantons]] in [[chiral perturbation theory]]:

* [[Gerard 't Hooft]], _How instantons solve the $U(1)$ problem_, Physics Reports **142** 6  (1986) 357-387 [doi:10.1016/0370-1573(86)90117-1]

On [[non-perturbative quantum field theory]]:

* [[Gerard ’t Hooft]], [[Arthur Jaffe]], [[Gerhard Mack]], [[P. K. Mitter]], [[Raymond Stora]] (eds.) *Nonperturbative Quantum Field Theory*,  NATO Science Series B **185** (1988) [[doi:10.1007/978-1-4613-0729-7](https://doi.org/10.1007/978-1-4613-0729-7)]

Introducing the [[holographic principle]]:

* [[Gerard 't Hooft]], _Dimensional Reduction in Quantum Gravity_, talk at *[Conference on Highlights of Particle and Condensed Matter Physics (SALAMFEST)](https://inspirehep.net/conferences/968293)* (1993) [[gr-qc/9310026](http://arxiv.org/abs/gr-qc/9310026), [spire:36137](https://inspirehep.net/literature/36137)]

## Related $n$Lab entries

* [[Yang-Mills theory]]

* [[piecewise flat spacetime]]

* [[’t Hooft operator]]

category: people

[[!redirects Gerard 't Hooft]]

[[!redirects Gerardus 't Hooft]]
[[!redirects Gerardus ’t Hooft]]

[[!redirects 't Hooft]]
[[!redirects 't Hooft]]


Gerardo L. Rossini

* [institute page](http://www.fisica.unlp.edu.ar/Members/rossini)

* [InSpire page](https://inspirehep.net/authors/991174)

## Selected writings

On the [[CS-WZW correspondence]] (with motivation by [[Laughlin wavefunctions]] for [[anyons]]):

* [[Daniel C. Cabra]], [[Gerardo L. Rossini]], *Explicit connection between conformal field theory and 2+1 Chern-Simons theory*, Mod. Phys. Lett. A **12** (1997) 1687-1697 [[arXiv:hep-th/9506054](https://arxiv.org/abs/hep-th/9506054), [doi:10.1142/S0217732397001722](https://doi.org/10.1142/S0217732397001722)]

category: people

[[!redirects Gerardo Rossini]]


Gerardo Ortiz

* [personal page](https://ortizg.pages.iu.edu/)

* [institute page](https://physics.indiana.edu/about/directory/all-faculty-scientists/ortiz-gerardo.html)

## Selected writings

On the [[quantum adiabatic theorem]] in the generality of non-abelian [[Berry phases]] such as needed for [[anyon]]-[[braiding]]:

* [[Gustavo Rigolin]], [[Gerardo Ortiz]], *The Adiabatic Theorem for Quantum Systems with Spectral Degeneracy*, Phys. Rev. A **85** 062111 (2012) [[arXiv:1111.5333](https://arxiv.org/abs/1111.5333), [doi:10.1103/PhysRevA.85.062111](https://doi.org/10.1103/PhysRevA.85.062111)]

category: people


Gerardus Mercator

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

> (not to be confused with [[Nicholas Mercator]])

category: people


gerbe

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### General

In full generality, we have the following definition of _gerbe_ .

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

Given an [[(∞,1)-topos]] $\mathcal{X}$, a **gerbe** in $\mathcal{X}$ is an [[object]] $\mathcal{G} \in \mathcal{X}$ that is

1. [[1-truncated]]

1. [[1-connective]] (= [[connected]]).

=--

The first condition says that a gerbe is an object in the [[(2,1)-topos]] $\tau_{\leq 1 } \mathcal{X} \hookrightarrow \mathcal{X}$ inside $\mathcal{X}$. This means that for $C$ any [[(∞,1)-site]] of definition for $\mathcal{X}$, a gerbe is a [[(2,1)-sheaf]] on $C$, $\mathcal{G} \in Sh_{(2,1)}(C)$: a [[stack]] on $C$.

The second condition says that  a gerbe is a stack that _locally_ looks like the [[delooping]] of a [[sheaf]] of [[group]]s. More precisely, it says that

* the morphism $\mathcal{G} \to *$ to the [[terminal object in an (∞,1)-category|terminal object]] of $\mathcal{X}$ is an [[effective epimorphism in an (∞,1)-category|effective epimorphism)]];

* the 0th [[categorical homotopy groups in an (∞,1)-topos|categorical homotopy]] group $\pi_0 \mathcal{G}$ is isomorphic to the terminal object $*$ as objects in the [[sheaf topos]] $\tau_{\leq 0} \mathcal{X} = Sh_{(1,1)}(C)$. Here $\pi_0 \mathcal{G}$ is the [[sheafification]] of the presheaf of connected components of the groupoids that $\mathcal{G} : C^{op} \to Grpd \hookrightarrow \infty Grpd$ assigns to each object in the site.

Traditionally this is phrased before sheafification as saying that a gerbe is a stack that is _locally_  non-empty and _locally_ connected . This is the traditional definition, due to [Giraud](#Giraud).

Also traditionally gerbes are considered in the [[little topos|little (2,1)-toposes]] $\tau_{\leq 1} \mathcal{X}$ of a [[topological manifold]] or [[smooth manifold]] $X$ or a [[topological stack]] or [[differentiable stack]] $X$. One then speaks of a _gerbe over $X$_ .

More precisely, we may associate to any $X \in C :=$ [[Top]] or $X \in C :=$ [[Diff]] the corresponding [[big site]] $C/X$ and form the [[(2,1)-topos]] $\tau_{leq} \mathcal{X} := Sh_{(2,1)}(C/X)$. In terms of this a gerbe is given by a collection of groupoids assigned to patches of $X$, satisfying certain conditions.

Equivalent to this is the [[over-(∞,1)-topos|over-(2,1)-topos]] $\tau_{\leq 1} \mathcal{H}/j(X)$, where $\tau_{\leq 1}\mathcal{H} := Sh_{(2,1)}(C)$ is the [[big topos|big]] [[(2,1)-topos]] over $C$ (and $j$ denotes its [[(∞,1)-Yoneda embedding|(2,1)-Yoneda embedding]]).

Since this $\mathcal{H}$ is a [[cohesive (∞,1)-topos]] we may think of its objects a general [[topological ∞-groupoid|continuous ∞-groupoid]]s or [[∞-Lie groupoid|smooth ∞-groupoid]]s. In large parts of the literature coming after [Giraud](#Giraud) gerbes, or related structures equivalent to them, are described this way in terms of [[topological groupoid]]s and [[Lie groupoid]]s. This perspective is associated with the notion of a _[[bundle gerbe]]_ .

### $G$-Gerbes

We discuss gerbes that have a "strucure group" $G$ akin to a [[principal bundle]]. Indeed, while not the same concept, these _$G$-gerbes_ are [[equivalence in an (∞,1)-category|equivalent]] to $AUT(G)$-[[principal 2-bundle]]s, for $AUT(G)$ the [[automorphism 2-group]] of $G$.

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

The definition \ref{GeneralDef} of _gerbe_ is almost verbatim that of _[[Eilenberg-MacLane object]]_ in degree 1. The only difference is that the latter is required to have not only the homotopy sheaf $\pi_0 = *$, but even have a "global section" in the form of a morphism $* \to P$.

First consider this locally. A gerbe (as any [[1-connected]] object) necessarily has _local_ sections:

for

$$
  (x^* \dashv x_*) :  \infty Grpd
   \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}}
  \mathcal{X}
$$

any [[point of a topos|topos point]], the [[stalk]] functor $x^*$, being an [[inverse image]] is [[left exact functor|left exact]] and hence preserves [[categorical homotopy groups in an (∞,1)-topos|homotopy sheaves]] and [[terminal object]]s. It follows that the 0th homotopy sheaf is trivial

$$
  \pi_1 x^* P \simeq x^* \pi_1(P) \simeq x^* * \simeq *
$$

as are all the degree-$p$ homotopy sheaves for $p \gt 1$. Therefore $x^* P$ is a [[groupoid]] with a single object: the [[delooping]] groupoid of a [[group]] $G_x$:

$$
  x^* P \simeq B G_x
  \,.
$$

More generally, by the discussion at [[looping and delooping]] we have in an [[equivalence of (∞,1)-categories]]

$$
  (\Omega \dashv \mathbf{B})
  :
  \infty Gpr(\mathcal{X})
   \stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}
  \mathcal{X}_{pt, \geq 1}
$$

between the [[∞-group]] objects in the ambient [[(∞,1)-topos]] $\mathcal{X}$ and the [[pointed object|pointed]] [[connected]] objects.

It follows that for a gerbe $P$ that admits a global section $* \to P$ the above relation holds not only [[stalk]]-wise, but globally: it is the [[delooping]] of its own first [[categorical homotopy groups in an (∞,1)-topos|sheaf of homotopy groups]]

$$
  P \simeq \mathbf{B} \pi_1(P)
  \,.
$$

=--

The following definition characterizes gerbes that are _locally_ of the form of remark \ref{RelationToEMObjects}.

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

Let $G \in Grp(\mathcal{X})$ be a group object. A gerbe $P \in \mathcal{X}$ is a **$G$-gerbe** if there exists an [[effective epimorphism]] $U \to *$ and an [[equivalence in an (∞,1)-category|equivalence]]

$$
  P|_U \simeq \mathbf{B}(G|_U)
  \,,
$$

where $P|_U := P \times U$ and $G|_U := G \times U$.

=--

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

In a typical application one considers gerbes over some [[topological space]] $X$. In that case

* $\mathcal{X} = Sh_{(\infty,1)}(Op(X))$ is the [[(∞,1)-category of (∞,1)-sheaves]] on the [[category of open subsets]] of $X$;

* the [[terminal object]] of $\mathcal{X}$ _is_ the space $X$, regarded as an object in its own $(\infty,1)$-topos, hence we can write $X := * \in \mathcal{X}$;

* a [[group object]] $G \in \mathcal{X}$ is [[sheaf]] of [[group]]s on $X$;

* an [[effective epimorphism]] $U \to *$, hence $U \to *$ is obtained from any [[open cover]] $\{U_i \to X\}$ by setting $U := \coprod_i U_i$;

* with such a choice of effective epimorphism, $G|_U = \coprod_i G|_{U_i}$ is simply the restriction of the sheaf of groups $G$ to each [[open subset]] that is a member of the cover;

* $\mathbf{B}G_{U} \in \mathcal{X}/U$ is the [[stack]] of $G_{U}$-[[principal bundle]]s on $U$.

=--

## Properties

### Equivalence of $G$-gerbes to $AUT(G)$-2-bundles

Let $\mathcal{X}$ be any ambient [[(∞,1)-topos]].

Let $G \in Grp(\mathcal{X}) \subset \infty Grpd(\mathcal{X})$ be a [[group object]] (a [[0-truncated]] [[∞-group]]).

Write

$$
  G Gerbe \subset \mathcal{X}
$$

for the [[core]] of the full [[sub-(∞,1)-category]] on $G$-gerbes in $\mathcal{X}$.

Write

$$
  AUT(G) :=
  Aut_{\mathcal{X}_{*}}(\mathbf{B}G)
  \in
  2 Grp(\mathcal{X}) \subset \infty Grp(\mathcal{X})
$$

for the [[2-group]] object called the [[automorphism 2-group]] of $G$.

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

$G$-gerbes in $\mathcal{X}$ are classified by first $AUT(G)$-[[nonabelian cohomology]]

$$
  \pi_0 G Gerbe \simeq \pi_0 \mathcal{X}(*, \mathbf{B} AUT(G))
   =:
   H_{\mathcal{X}}^1(X,AUT(G))
  \,.
$$

=--

In the general perspective of [[(∞,1)-topos theory]] this appears as  ([JardineLuo, theorem 23](#JardineLuo)).

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

Since [[nonabelian cohomology]] with coefficients in $AUT(G)$ also classified $AUT(G)$-[[principal 2-bundles]] it follows that also

$$
  \pi_0 G Gerbe \simeq AUT(G) 2Bund(*)
  \,.
$$

Notice that under this equivalence a $G$-gerbe is not identified with the total space object of the corresponding $AUT(G)$-[[principal 2-bundle]]. The latter differs by an $Aut(H)$-factor. Where a $G$-gerbe is locally equivalent to

$$
  \mathbf{B}(G|_U)
  =
  G|_U \stackrel{\to}{\to} *|_U
$$

an $AUT(G)$-principal 2-bundle is locally equivalent to

$$
  AUT(G|_U)
  =
  Aut(G|_U) \times G
   \stackrel{\overset{Ad(p_2) \cdot p_1}{\to}}{\underset{p_1}{\to}}
  Aut(G|_U)
  \,.
$$

Instead, under the above equivalence a gerbe is identified with the [[associated ∞-bundle]] with fibers $\mathbf{B}G$
that is associated via the canonical [[action]] of $AUT(G) = Aut(\mathbf{B}G)$ on $\mathbf{B}G$.

=--

### Banded gerbes

For $G \in Grp(\mathcal{G})$, the [[automorphism 2-group]] $AUT(G)$ has a canonical morphism to its [[0-truncated|0-truncation]], the ordinary [[outer automorphism]] group object of $G$:

$$
   \to AUT(G) \to \pi_0(Aut(G)) =:  Out(G)
  \,.
$$

Therefore every $AUT(G)$-[[cocycle]] has an underlying $Out(G)$-cocycle (an $Out(G)$-[[principal bundle]]):

$$
  \mathcal{X}(* , \mathbf{B}AUT(G))
   \to
  \mathcal{X}(* , \mathbf{B}Out(G))
  \,.
$$

By prop. \ref{ClassificationOfGGerbes} this an assignment of $Out(G)$-cohomology classes to $G$-gerbes:

$$
  Band : \pi_0 ( G Gerbe ) \to H_{\mathcal{X}}^1(X,Out(G))
  \,.
$$

For $P \in G Gerbe$ one says that $Band(P)$ is its **band**.

Sometimes in applications one considers not just the restriction from all gerbes to $G$-gerbes for some $G$, but further to $K$-banded $G$-gerbes for some $K \in H_{\mathcal{X}}^1(X,Out(G))$.

The [[groupoid]] $G Gerbe_K(X)$ of $K$-banded gerbes is the $K$-[[twisted cohomology|twisted]] $\mathbf{B}^2 Z(G)$-cohomology of $X$ (where $Z(G)$ is the [[center]] of $G$): it is the [[homotopy pullback]]

$$
  \array{
    G Gerbe_K(X) &\to& {*}
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{K}}
    \\
    \mathcal{X}(X,\mathbf{B}AUT(G))
    &\to&
    \mathcal{X}(X, \mathbf{B}Out(G))
  }
 \,.
$$

## Sub-entries

More details on gerbes is at the following sub-entries:

* [[gerbe (as a stack)]]

* [[gerbe (in nonabelian cohomology)]]

* [[gerbe (in differential geometry)]]

* [[bundle gerbe]]

* [[gerbe (general idea)]]

## Examples

* [[determinantal gerbe]]

## Related concepts

* [[principal bundle]] / [[torsor]] / [[associated bundle]]

* [[principal 2-bundle]] / **gerbe** / [[bundle gerbe]]

* [[principal 3-bundle]] / [[2-gerbe]] / [[bundle 2-gerbe]]

* [[principal ∞-bundle]] / [[associated ∞-bundle]]  / [[∞-gerbe]]

## References

The definition of _gerbe_ goes back to (see also [[nonabelian cohomology]])

* [[Jean Giraud]], _Cohomologie non abélienne_ , Springer  (1971) ([doi:10.1007/978-3-662-62103-5](https://link.springer.com/book/10.1007/978-3-662-62103-5))

Review:

* {#Breen} [[Lawrence Breen]], _Notes on 1- and 2-gerbes_ in [[John Baez]], [[Peter May]] (eds.) _[[Towards Higher Categories]]_, The IMA Volumes in Mathematics and its Applications, vol 152. Springer 2009 ([arXiv:math/0611317](http://arxiv.org/abs/math/0611317), [doi:10.1007/978-1-4419-1524-5_5](https://doi.org/10.1007/978-1-4419-1524-5_5)).

* [[Ieke Moerdijk]], _Introduction to the language of stacks and gerbes_ ([arXiv:math/0212266](http://arxiv.org/abs/math/0212266))

A discussion from the point of view of [[(∞,1)-topos theory]] is in

* {#JardineLuo} [[Rick Jardine]], Z. Luo, _Higher order principal bundles_ , K-theory (2004) ([web](http://www.math.uiuc.edu/K-theory/0681/))

The definition for $n$-gerbes as $n$-truncated and $n$-connected objects (see [[∞-gerbe]]) is in

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

[[!redirects gerbes]]
[[!redirects gerb]]

[[!redirects nonabelian gerbe]]
[[!redirects nonabelian gerbes]]

[[!redirects non-abelian gerbe]]
[[!redirects non-abelian gerbes]]


gerbe (as a stack)

This is a sub-entry for [[gerbe]].

For related entries see

* [[bundle gerbe]]
* [[gerbe (general idea)]]

##Idea##

(The definitions of [[sheaf]] and [[stack]] are elsewhere in the nLab but we will have to review them briefly. There is also  [[motivation for sheaves, cohomology and higher stacks]], which provides other insights. For simplicity of exposition, we will initially look at sheaves, stacks and gerbes over a base space $B$ rather than in the more general setting of a topos.)

Naively a [[sheaf]] is a [[family of sets]] indexed 'continuously' by the points of a space $B$.  It corresponds either to a [[presheaf]] satisfying a 'gluing condition' (which is the version of [[descent]] in one of its simplest cases), or to an [[etale space|étalé space]] over $B$.  The relationship between them is that the presheaf is the presheaf of local sections of the étalé space, and the 'gluing condition' is that local sections that agree on the intersections of  open sets can be glued uniquely to give a section over the union of the open sets.

An important class of sheaves are the [[torsor|torsors]].  Let $G$ be a sheaf of groups on $B$.   The category, $Tors(B;G)$ of $G$-torsors on $B$ is a groupoid.  The notion of $G$-torsor 'localises' well, so that if $U$ is an open set of $B$, then we can restrict $G$ to a sheaf on $U$, and look at the torsors over $U$ using the restricted $G$. (We abuse notation and just write $Tors(U;G)$ for the corresponding groupoid. If $V$ is another open set contained in $U$, there is a restriction functor from $Tors(U;G)$ to $Tors(V;G)$, so it looks as if we have a presheaf of groupoids on $B$, but things are not quite right here.

The assignment of $Tors(G)_U$ to $U$ only gives a [[pseudofunctor]] not a functor from the category $Open(B)^{op}$ to the category of groupoids. It thus corresponds to a [[Grothendieck fibration]] or **fibred category** over $Open(B)$.  It does have quite nice 'gluing properties' however, it is a [[stack]] of groupoids.  (Roughly 'morphisms glue, objects glue up to isomorphism'. For enlightenment note that there are almost presheaves of objects and of morphisms in this 'pseudo' presheaf.) This stack will be called $\mathcal{T}ors(G)$.

There is a **stackcompletion** functor from fibred categories to stacks.  If we take the sheaf of groups $G$, think of it as a presheaf of groupoids $BG$, in the usual way, then it is a pseudo-functor from $Open(B)^{op}$ to the category of groupoids. If we stack complete it, we get ... $\mathcal{T}ors(G)$.

We thus can think of a stack of groupoids as a 'lax' generalisation of a sheaf of groupoids.  What about **gerbes**?

(It should be mentioned that often in topological settings, the sheaf of groups is actually a constant sheaf and that in that case a $G$-torsor is just a principal $G$-bundle.)

Groups 'are' groupoids with a single object, but groupoids are not 'groups with many objects' (although that is a nice phrase to use when introducing them). A groupoid need not have any objects ... if it is empty! Of course, we think of the vertex groups of a groupoid, but, if the groupoid is not connected, there may be many different non-isomorphic ones. So a group *is* a very special type of groupoid.

Similarly, the term 'gerbe' refers to a special sort of stack of groupoids.

+-- {: .standout}
A gerbe is to a general stack what, up to equivalence, a group is to a general groupoid. It is non-empty and connected.
=--

+--{: .query}

[[David Roberts]]: This is reflected in the fact that the fibres of bundle gerbes (either abelian or $G$-bundles gerbes) are transitive groupoids. I believe Tim and I had a bit of discussion of this sort of thing last year (or before - mists of time...). Or maybe this comment should go in [[gerbe (general idea)]]?
=--

To make this precise we use some additional notions:

We will have a pseudofunctor $F : Open(B)^{op}\to Grpd$, and this will be a stack.

##Definitions##

*   A stack of groupoids, $F$, on $B$ is *locally non-empty* if there is an open covering $\mathcal{U}$ of $B$ for which each groupoid $F(U)$ is non-empty, for $U \in \mathcal{U}$.

*   A stack of groupoids, $F$, on $B$ is said to be *locally connected*  if for each open $U$ in $B$ there is an open covering $\mathcal{U}$ of $B$ such that all elements in $F(U)$ become connected in all $F(U_i)$.

and finally:

*  A *gerbe* $F$ on $B$ is a locally non-empty, locally connected stack of groupoids on $B$.

It is important to note that it does not state in the definition of a gerbe that the open cover that we have over which it is non-empty is or is not one over which it is connected.

_Local connectedness_ can be well stated by saying that for the various $U$, if $x$ and $y$ are local objects defined over $U$, the set $F(U_i)(x,y)$ is not empty. (Translation: a 'local object', or 'locally defined object', of $F$ is a 'local section'  of $Ob(F)$, say, over $U$, in other words, an element in $Ob(F(U))$.

## Important example##

+-- {: .un_proposition}

###### Proposition
$\mathcal{T}ors(G)$ is a gerbe on $B$.
=--

+-- {: .proof}
######Proof
To see _locally non-empty_: If $U$ is any open set in $B$, then as $\mathcal{T}ors(G)(U) = Tors(U;G)$, the category of $G_U$-torsors over $U$, it has at least the [[trivial torsor|trivial G-torsor]] (over $U$) amongst its objects, so $\mathcal{T}ors(G)$ is locally non-empty.

Next look at $\mathcal{T}ors(G)(U)$ again.  Any two $G_U$-torsors are locally isomorphic to each other, since they are both locally isomorphic to the trivial $G_U$-torsor, so, if $F$ and $F^\prime$ are two $G_U$-torsors, there is an open cover such that over that cover $F$ and $F^\prime$ are isomorphic, hence $\mathcal{T}ors(G)$ is locally connected.  We thus have that $\mathcal{T}ors(G)$ is a gerbe.
=--

##$\mathcal{A}$-Gerbes##

The notion of a $G$-gerbe arises in the article [[gerbe (general idea)|gerbe ]], but one needn't use just a group $G$. Fix a sheaf of abelian (possibly not necessary) groups $\mathcal{A}$ on $B$. Then an $\mathcal{A}$-gerbe is a gerbe $F$ on $B$ such that for any open $U$ on $B$ we have a functorial isomorphism $\mathcal{A}(U)\stackrel{\sim}{\to} \text{Aut}(s)$ for all $s\in F(U)$.

Note that since $F$ is a stack, $\text{Aut}(s)$ is a sheaf, so by isomorphism we mean an isomorphism as sheaves, and by functorial we mean given another object $t\in F(U)$, the isomorphism commutes

$$
\begin{matrix} \mathcal{A}(U) & \to & \text{Aut}(s) \\
Id \downarrow &  & \downarrow \\
\mathcal{A}(U) & \to & \text{Aut}(t) \end{matrix}
$$

In particular, we get that for any two objects $C, D\in F(U)$ we have that the sheaf $Isom(C,D)$ is an $\mathcal{A}$-torsor. This gives that if there is some object over $U$, namely that $F(U)\neq \emptyset$, then the set of isomorphism classes of obects in $F(U)$ is in natural bijection with $H^1(U, \mathcal{A}_U)$.

For example, consider the stack of rank 1 vector bundles on a scheme $X$, $\text{Vect}_1$. One can check that $\text{Vect}_1$ is a $\mathbb{G}_m$-gerbe by noting that the automorphism group of any vector bundle over $U$ will be precisely $\mathcal{O}_X(U)^\times$, and everything is functorial. By the interpretation in cohomology, we see that the global vector bundles (up to isomorphism) are in correspondence with $H^1(X, \mathcal{O}_X^\times)$ which is just $\text{Pic}(X)$.

One can form the *classifying stack*, $B\mathcal{A}$ from the important example above by taking $B\mathcal{A}(U)=\mathcal{T}ors(\mathcal{A}(U))$. A basic theorem about $\mathcal{A}$-gerbes is that an $\mathcal{A}$-gerbe, $F$, is isomorphic to $B\mathcal{A}$ if and only if $F(B)\neq \emptyset$. This says that $F$ is isomorphic to the classifying stack if and only if it has a global object.

For information on how gerbes play a role in differential geometry see [[gerbe (in differential geometry)]].

##References

There is a lengthier description of gerbes (at this level of generality) in the Menagerie notes that are available from [[Tim Porter|Tim Porter's]] home page.

Other material available online includes the following:

* I. Moerdijk, _Introduction to the language of stacks and gerbes_ ([arXiv](http://arxiv.org/abs/math/0212266))

* Larry Breen, _Notes on 1- and 2-gerbes_ ([arXiv](http://arxiv.org/abs/math/0611317))

Further references are given in the other entries on gerbes.