music theory

[[!redirects music]]
[[!redirects music]]
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### Context
#### Philosophy
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#Contents#
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## Idea

The application of mathematics in the theory of music in the West is generally thought to start with Pythagoras and progress though the works of Euler, Mersenne, Helmholtz, and others. Recently, **mathematical music theory** has gained impetus through, among others, the Neo-Riemannian theory of harmony, David Lewin's research on transformational theory, and Guerino Mazzola's application of topos theory.

As well, the application of science and math to the theory of human perception of [[sound]] is known as _psychoacoustics_ and has had a major impact in today's musically digital world. All of these viewpoints have greatly enhanced the variety of mathematical concepts employed in the study of music that today include tools from [[set]], [[group]], and [[category theory]].

## Related entries

* [[dihedral group]]

* [[harmonic analysis]]

* [[Music of the Spheres]]

* [[canonical formula of myth]]

* [[acoustics]]

* [[pitch-class]]

## Links

* John Baez: _This Week's Finds (Week 234)_, June 12, 2006. ([link](http://math.ucr.edu/home/baez/week234.html))

* n-Café: _The Mathematics of Music at Chicago_, May 31, 2009. ([link](https://golem.ph.utexas.edu/category/2009/05/the_mathematics_of_music_at_ch.html))

* For applications of mathematics to music, see [Thomas M. Fiore\'s page](http://www-personal.umd.umich.edu/~tmfiore/1/music.html).

* For discussions on the interaction of category theory and music, see the [MaMux seminar](http://recherche.ircam.fr/equipes/repmus/mamux/) at IRCAM, Paris.

## References

A comprehensive anthology of Riemannian and Neo-Riemannian music theory is:

* Edward Gollin, Alexander Rehding (eds.), _The Oxford Handbook of Neo-Riemannian Music Theories_, Oxford UP 2014.

An excellent textbook on the mathematics of music that ranges from Fourier analysis over digital music to compositional symmetries is

* David J. Benson, _Music - A Mathematical Offering_, Cambridge UP 2007. ([pdf](https://homepages.abdn.ac.uk/d.j.benson/pages/html/music.pdf))

Comparable in range though presumably more digestible for the non mathematical reader is

* Gareth Loy, _Musimathics - the mathematical foundations of music. Vols. 1,2_, MIT Press 2007.

Some canonical music theory texts that rely on mathematics:

* Allen Forte, _The Structure of Atonal Music_, Yale University Press 1977.

> "The Structure of Atonal Music ranks as one of the most important contributions to music theory in the twentieth century. His use of the computer as well as traditional means of analysis has led to fuller knowledge of musical structure and thereby enhanced both the understanding and the enjoyment of music."

* David Lewin, _Generalized Musical Intervals and Transformations_, Oxford University Press 2010.

> "David Lewin's Generalized Musical Intervals and Transformations is recognized as the seminal work paving the way for current studies in mathematical and systematic approaches to music analysis."

* Joseph N. Straus, _Introduction to Post-Tonal Theory (third edition)_, Pearson 2004.

> "The third edition stays abreast of recent theoretical developments by including discussions of transformational networks and graphs, contour theory, atonal voice leading, triadic post-tonality (including neotonality), inversional symmetry, and interval cycles."

* Arnold Schönberg, _Theory of Harmony: 100TH ANNIVERSARY EDITION_, University of California Press 2010.

A nice first introduction to music theory and what groups can do for it is

* [[Alissa Crans|Alissa S. Crans]], [[Thomas M. Fiore]], Ramon Satyendra, _Musical Actions of Dihedral Groups_, arXiv:0711.1873 (2008). ([abstract](http://arxiv.org/abs/0711.1873))

More mathematically inclined are the following

* Emilio Lluis-Puebla, Ocatavio A. Augustín-Aquino (eds.), _Memoirs of the
Fourth International Seminar on
Mathematical Music Theory_, Sociedad Matemática Mexicana 2010. ([pdf](http://sociedadmatematicamexicana.org.mx/SEPA/ECMS/resumen/PME1_1.pdf))

* Guerino Mazzola, _Geometrie der Töne_, Birkhäuser Basel 1990.

* {#Mazzola02} Guerino Mazzola, _The Topos of Music_, Birkhäuser Basel 2002.

* Guerino Mazzola, _Mathematical Music Theory: Status Quo 2010_, pp.11-42 in [Lluis-Puebla, Augustín-Aquino (2010)](#LPAA10).

* Dimitri Tymoczko, _A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice_, Oxford University Press 2011.

* Dmitri Tymoczko, _Generalizing Musical Intervals_, Journal of Music Theory **53** no.2 (2009) pp.227-254. ([draft](http://dmitri.mycpanel.princeton.edu/files/publications/lewin.pdf))

An introduction to group theory with applications to neo-Riemannian theory is

* Ocatavio A. Augustín-Aquino, Janine du Plessis, Emilio Lluis-Puebla, Mariana Montiel, _Una introducción a la Teoría de Grupos
con aplicaciones en la Teoría Matemática de la Música_, Sociedad Matemática Mexicana 2009. ([pdf](http://pesmm.org.mx/Serie%20Textos_archivos/T10.pdf))

Besides the monumental [Mazzola (2002)](#Mazzola02), the following employ category theory:

* Thomas Noll, _The topos of triads_, pp.1-26 in Fripertinger, Reich (eds.),  _Colloquium on Mathematical Music Theory_,
Grazer Math. Ber. **347** (2005). ([citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.134.5669))

* Alexandre Popoff, Moreno Andreatta, Andree Ehresmann, _Groupoids and wreath products of musical transformations: A categorical approach from poly-Klumpenhouwer networks_, pp. 33-45 in Mathematics and Computation in Music. MCM 2019. Lecture Notes in Computer Science, vol 11502. Springer. ([pdf](https://hal.archives-ouvertes.fr/hal-02392889/document))

* Alexandre Popoff, Jason Yust, _Meter networks: a categorical framework for metrical analysis_, Journal of Mathematic and Music. ([pdf](https://doi.org/10.1080/17459737.2020.1836687))

Methods of physics are employed to study tonality in

* Peter beim Graben, Reinhard Blutner, _Toward a Gauge Theory of Musical Forces_, pp. 99-111 in LNCS **10106** (2017).  ([draft](http://www.blutner.de/GrabenBlutnerQI16.pdf))

Hegel's philosophy of music can be found in his lectures on aesthetics though these reflect to a large extent the view of his pupil Heinrich Gustav Hotho who compiled the posthumeous publication:

* [[Georg Hegel|G. W. F. Hegel]], _Vorlesungen über die Ästhetik II_, Verlag Das Europäische Buch Westberlin $^4$1985. (pp. 258-326)

A development of the basic concepts of music theory along his ideas on dialectics can be found in

* [[Moritz Hauptmann]], _Die Natur der Harmonik und Metrik_, Breitkopf & Härtel Leipzig 1853. ([wikisource](https://de.wikisource.org/wiki/Moritz_Hauptmann))

The sonata form from a [[Hegel|Hegelian]] perspective:

* Ernst Nobbe, _Die thematische Entwicklung der Sonatenform im Sinne der Hegel'schen Philosophie betrachtet_, Triltsch Würzburg (1941)

The following introductory texts on psychoacoustics help provide a foundation to better understand how sound becomes emotion becomes music:

* Arthur H. Benade, _Fundamentals of Musical Acoustics_  Dover New York 1976.

* Hugo Fastl, Eberhard Zwicker, _Psychoaccoustics: facts and models_, Springer, Heidelberg 1990.

* Juan G. Roederer, _The Physics and Psychophysics of Music_, Springer Heidelberg 2008$^4$.

A theory of emotions which applies beyond music in narrative arts is in

* David Huron, _Sweet Anticipation - Music and the Psychology of Expectation_, MIT Press 2006.

An introduction to [[Haskell]] programming systematically employing examples from music is

* Paul Hudak, _The Haskell School of Music - From Signals to Symphonies_, ms. Yale University 2012. ([pdf](http://www.cs.yale.edu/homes/hudak/Papers/HSoM.pdf))

Other "mathemusical" topics include:

* Milton Babbit, _The Function of Set Structure in the Twelve-Tone System_, Ph.D. dissertation, Princeton University 1946.

* E. Chew, _Towards a Mathematical Model of Tonality_, Ph.D. dissertation, MIT Cambridge 2000.

* David Clampitt, Thomas Noll, _Modes, the Height-Width Duality, and Handschin's Tone Character_, Music Theory Online **17** no.1 (2011).

* [[Thomas M. Fiore]], Ramon Satyendra, _Generalized Contextual Groups_, Music Theory Online **11** no.3 (2005).

* J. Hook, _Uniform triadic transformations_, Journal of Music Theory **46**
no.1-2 (2002) pp.57–126.

* T. A. Johnson, _Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals_, Scarecrow Press 2008.

* Fred Lerdahl, _Tonal Pitch Space_, Oxford University Press 2004.

* Miguel F. M. Lima, J. A. Tenreiro Machado, António C. Costa, _A Multidimensional Scaling Analysis of Musical Sounds Based on Pseudo Phase Plane_, Abstract and Applied Analysis Vol. 2012. (Article ID 436108)

* Thomas Noll, _Sturmian sequences and morphisms: a music theoretical application_, Journée annuelle (2008) pp.79–102.

* R. N. Shepard,  _Geometrical approximations to the structure of musical pitch_, Psychological Review, **89** no.4 (1982) 305.

*  J. Tennenbaum,  _The Foundations of Scientific Musical Tuning_, in Staff, Sigerson (eds.), _A Manual On the rudiments of Tuning and Registration_, Schiller Institute Washington 1992.

* J. S. Walker, D. W. Gary, _Mathematics and Music: Composition, Perception, and Performance_, Taylor&Francis Boca Raton 2013.

[[!redirects music]]
[[!redirects mathematical music theory]]


mutant knot

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## Idea

In [[knot theory]], the known problem of [[unwinding knots|unwinding]] [[knots]] gives rise to the factorization of knots generated by composing [[prime knots]] and their __mutants__. The computational process of computing [[knot invariants]], namely the polynomials describing these [[invariants]], has been postulated as a basis for "attacks" in [[quantum cryptography]]. Classes of knots called _mutants_ would hinder these attacks and remain undetected by a [[quantum computer]], at their present maturity.  The _mutants_ of a given [[knot]] are indistinguishable since they all have the same Jones' type [[invariants]]. _Mutants_ also appear in the field-theoretic framework manifest in the expectation values of [[Wilson loop operators]].

## Definition

Beginning with the [[tangle]] representation of an [[oriented knot]] $K$, one can obtain a __mutant__ $K'$ by replacing one of the [[tangle]]s with a rotated version of the former tangle. It is important to note that the process of mutation can be executed on various subsets of the same [[knot diagram]] containing at least one [[crossing]].

## References

* Marzuoli and Palumbo, _Post Quantum Cryptography from Mutant Prime Knots_, 2011, Int. J. Geom. Methods Mod. Phys. ([arXiv: 1010.2055/math-ph](http://arxiv.org/abs/1010.2055))

[[!redirects mutant knots]]


mutations of exceptional collections

see also [Kauer-Roggenkamp](https://github.com/The-Singularity-Research/cirq_hybrid_codes/blob/master/higher-dimensional-orders-graph-orders-and-derived-equivalences.pdf) for general form of mutations of graph orders, as well as [Gnedin](https://arxiv.org/abs/1908.08895) for Calabi-Yau properties. This relates to the construction of a AQFT related to Artin L-functions and dessins d'enfants. This is related to [Seiberg duality](https://ncatlab.org/nlab/show/quiver+gauge+theory) in type IIB string theory derived from quiver gauge theories. See also [https://arxiv.org/abs/0803.1328](https://arxiv.org/abs/0803.1328) for combinatorial constructions.


mutually unbiased bases

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#Contents#
* table of contents
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## Idea

In [[quantum information theory]] a pair of [[orthonormal basis|orthonormal]] [[linear bases]] of a given (usually [[finite number|finite]]-[[dimension of a vector space|dimensional]]) [[Hilbert space]] [[space of quantum states|of quantum states]] is called *mutually unbiased* if the [[absolute value]] of the [[inner products]] of any one element in one basis with any in the other basis all have the same value.

## Examples

In the 2-dimensional Hilert space of [[qbits]] $\simeq \mathbb{C}^2$ the pair consisting of

* the "measurement basis" ([[Pauli-Z]]-[[eigenstates]])

  $$
    \Big\{
      \left\vert 0 \right\rangle
      ,\,
      \left\vert 1 \right\rangle
    \Big\}
  $$

* the [[Hadamard gate|Hadamard]] basis ([[Pauli-X]]-[[eigenstates]])

  $$
    \Big\{
      \tfrac{1}{\sqrt{2}}
      \big(
        \left\vert 0 \right\rangle
        \pm
        \left\vert 1 \right\rangle
      \big)
    \Big\}
  $$

are (each [[orthonormal basis|orthonormal]] and) mutually unbiased: The [[absolute value]]-[[square]] of the mutual [[inner product]] is

$$
  \tfrac{1}{\sqrt{2}}
  \left\vert
  \left\langle
     b
  \vert
     0
  \right\rangle
  \,\pm\,
  \left\langle
     b
  \vert
     1
  \right\rangle
  \right\vert
  \;\;\;
    =
  \;\;\;
  \tfrac{1}{\sqrt{2}}
$$

for all $b \,\in\, \{0,1\}$ and $\pm \in \{+,-\}$.

## Related concepts

* [[quantum measurement]]

* [[ZX-calculus]]

## References

Original articles:

* [[William K. Woooters]], [[Brian D. Fields]], *Optimal state-determination by mutually unbiased measurements*, Annals of Physics **191** 2 (1989) 363-381 [doi:10.1016/0003-4916(89)90322-9]

Review:

* [[Ingemar Bengtsson]], *Three ways to look at mutually unbiased bases*, AIP Conference Proceedings **889** (2007) 40-51 [[arXiv:quant-ph/0610216](https://arxiv.org/abs/quant-ph/0610216), [doi:10.1063/1.2713445](https://doi.org/10.1063/1.2713445)]

See also:

* Wikipedia, *[Mutually unbiased bases](https://en.wikipedia.org/wiki/Mutually_unbiased_bases)*


MV algebras

# MV algebras
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{: toc}

## Idea

An MV-algebra  is an algebraic structure which models Luzasiewicz multivalued logic, and  the fragment of that calculus which deals with the basic logical connectives "and", "or", and "not", but in a multivalued context.

## Definitions

An MV-algebra consists of

* a non-empty set, $A$;

* a binary operation, $\oplus$, on $A$;

* a unary operation, $\neg$, on $A$; and a constant, $0$, such that

1. $\langle A, \oplus, 0\rangle$ is a commutative [[monoid]];

1. $\neg\neg x =  x$ for all $x \in A$;

1. $x\oplus \neg 0 = \neg 0$ for all $x\in A$; and

1. $\neg(\neg x\oplus y)\oplus y = \neg(\neg y\oplus x)\oplus x$ for all $x, y$ in $A$.

The last axiom is more difficult to interpret but is clarified by some examples.

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

Let $A = [0,1]$ be the unit interval.  Define

$$x\oplus y = min(1,x+y)$$

$$\neg x = 1-x.$$

This gives a commutative monoid easily enough and the double negation and absorption axioms are easy to check. Finally the last axiom divides into two cases: $x \lt y$ and $y \lt x$ and only one of these needs checking!

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

Any [[Boolean algebra]] defines a MV-algebra with $\oplus = \vee$, and $\neg$ being the complement operation. The expressions in the last axiom evaluate to $x\vee y$.

=--

## Properties

Define $x \odot y \coloneqq \neg (\neg x \oplus \neg y)$, and define $x \Rightarrow y \coloneqq \neg x \oplus y$.

+-- {: .un_prop}
######Proposition 1
Each MV algebra carries a lattice structure, where the join and meet operations are defined by

$$x \vee y = (x \odot \neg y) \oplus y$$

$$x \wedge y = x \odot (\neg x \oplus y)$$

and where $0$ and $1 = \neg 0$ are the bottom and top elements.
=--

Let $\leq$ be the partial ordering for the lattice structure, where $x \leq y$ if $x = x \wedge y$.

+-- {: .un_prop}
######Proposition 2
Each MV algebra is a [[residuated lattice]], i.e., a closed monoidal poset, where the monoidal product is $\odot$ and
$$x \odot y \leq z \qquad iff \qquad x \leq y \Rightarrow z.$$
In particular, for all elements $x$ and $y$ we have $1 = y \Rightarrow y$, $1 \Rightarrow y = y$, and $x \leq (x \Rightarrow y) \Rightarrow y$.
=--

In fact, an MV algebra is a $\ast$-[[star-autonomous category|autonomous]] poset, where $\neg x = x \Rightarrow 0$.

+-- {: .un_prop}
######Proposition 3
The equational variety of MV algebras is a [[Mal'cev variety]], where the Mal'cev operation is defined by

$$t(x, y, z) = ((x \Rightarrow y) \Rightarrow z) \wedge ((z \Rightarrow y) \Rightarrow x).$$

In other words, $t(x, y, y) = x = t(y, y, x)$.
=--

+-- {: .un_thm}
######Completeness theorem
The variety of MV algebras is generated by $[0, 1]$ with its standard MV algebra structure. Consequently, an identity holds for all MV algebras if and only if it holds in $[0, 1]$.
=--

##References

*  D. Mundici, _MV-algebras, a short tutorial_, available [here](http://www.matematica.uns.edu.ar/IXCongresoMonteiro/spanish.htm#cursos).

A recent preprint is

* [[Mark Lawson]], [[Philip Scott]], _AF inverse monoids and the structure of countable MV-algebras_, [ArXiv 1408.1231v2](http://arxiv.org/abs/1408.1231)


MV-algebra

\tableofcontents

## Idea

MV-algebras are the algebraic semantics for [[propositional logic|propositional]] [[Łukasiewicz logic]].

## Definition

An **MV-algebra** is a [[commutative monoid]] $(S, 0, \oplus)$ with an [[involution]] $x \mapsto \neg x:S \to S$ such that for all $x \in S$ and $y \in S$, $x \oplus \neg 0 = \neg 0$ and $\neg (\neg x \oplus y) \oplus y = \neg (\neg y \oplus x) \oplus x$.

An MV-algebra [[homomorphism]] between MV-algebras $A$ and $B$ is a [[function]] $f:A \to B$ such that $f(0) = 0$, for all $x \in A$, $\neg f(x) = f(\neg x)$, and for all $x \in A$ and $y \in A$, $f(x) \oplus f(y) = f(x \oplus y)$.

## Examples

* The [[unit interval]] $[0, 1]$ is an MV-algebra where $x \oplus y$ is defined as $\max(x + y, 1)$ and $\neg x$ is defined as $1 - x$.

* Every [[Boolean algebra]] is a MV-algebra whose monoid operation is idempotent.

* Any [[singleton]] $1$ is a trivial MV-algebra where $x \oplus y$ is defined as the unique [[binary function]] on $1$ and $\neg x$ defined as the unique unary function on $1$, the [[identity function]] on $1$. It is also the [[terminal object|terminal]] MV-algebra in the category of MV-algebras and MV-algebra homomorphisms.

## Related concepts

* [[Łukasiewicz logic]]

## References

* Wikipedia, [MV-algebra](https://en.wikipedia.org/wiki/MV-algebra)


Myers effect

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#Contents#
* table of contents
{:toc}

## Idea

What is called the _Myers effect_ ([Myers 99](#Myers99)) in [[string theory]] is the phenomenon that given $N$  [[D0-branes]] in a constant [[background field|background]] [[RR field]] $F_4$ (the [[field strength]] associated with [[D2-brane]] [[D-brane charge|charge]]) with, crucially, nonabelian effects included via a [[nonabelian DBI action]], then these D0-branes expand into a [[fuzzy 2-sphere]] which represents a spherical [[D0-D2 brane bound state]] of a [[D2-brane]] and $N$ [[D0-branes]] ([Myers 99, section 6, see p. 21 (22 of 33)](#Myers99), [Myers 03, section 4](#Myers03)).

Analogous polarization effects are thought to exists for branes of other dimensions, for instance polarizing [[D2-branes]] into [[D4-branes]] etc, and notably for [[M2-branes]] to polarize into [[M5-branes]] ([Bena 00](#Bena00), review in [BLMP 13, Section 6](#BaggerLambertMukhiPapageorgakis13)). In each case the higher dimensional brane ends up [[wrapped brane|wrapping]] a [[cycle]] which is topologically trivial (hence in principle shrinkable) but [[flux]] through this cycle exerts a force that stabilizes the brane against collapsing on this cycle.

A closely related stabilization mechanism is that of branes turned into "[[giant gravitons]]", where the stabilizing role of [[flux]] is instead taken by [[angular momentum]] (typically along the [[equator]] of an [[n-sphere]] in a [[Freund-Rubin compactification]]).

## Related concepts

* [[giant graviton]]

* [[fuzzy funnel]]

* [[dyon]]

* [[BFSS matrix model]], [[BMN matrix model]]

* [[wrapped brane]], [[brane intersection]]

\linebreak

[[!include brane bound states -- table]]

## References

### Polarization of D-branes

Precursor discussion appears in

* [[Roberto Emparan]], _Born-Infeld Strings Tunneling to D-branes_, Phys. Lett. B423 (1998) 71-78 ([arXiv:hep-th/9711106](https://arxiv.org/abs/hep-th/9711106))

The effect now known as the "Myers effect" in [[D-brane]] theory was first described in:

* {#Myers99} [[Robert Myers]], _Dielectric-Branes_, JHEP 9912 (1999) 022 ([arXiv:hep-th/9910053](https://arxiv.org/abs/hep-th/9910053))

for the case of [[D0-branes]] [[D-brane polarization|polarizing]] to [[D0-D2 brane bound states]].

Review:

* {#Myers03} [[Robert Myers]], Section 4 of: _Nonabelian Phenomena on D-branes_, Class. Quant. Grav. 20 (2003) S347-S372 ([arXiv:hep-th/0303072](https://arxiv.org/abs/hep-th/0303072))

See also:

* Z. Guralnik, [[Sanyaje Ramgoolam]], _On the Polarization of Unstable D0-Branes into Non-Commutative Odd Spheres_, JHEP 0102:032, 2001 ([arXiv:hep-th/0101001](https://arxiv.org/abs/hep-th/0101001))

* Pedro J. Silva, _Quantum Myers effect and its supergravity dual for D0/D4 systems_ ([arXiv:hep-th/0109112](https://arxiv.org/abs/hep-th/0109112))

* Yoshifumi Hyakutake, _Gravitational Dielectric Effect and Myers Effect_, Phys. Rev. D71:046007, 2005 ([arXiv:hep-th/0401026](https://arxiv.org/abs/hep-th/0401026))

Discussion of [[D-brane polarization]] on curved spacetime in context of the [[AdS/QCD correspondence]]:

* [[Joseph Polchinski]], [[Matthew Strassler]], _The String Dual of a Confining Four-Dimensional Gauge Theory_ ([arXiv:hep-th/0003136](https://arxiv.org/abs/hep-th/0003136))

* [[Ofer Aharony]], [[Arvind Rajaraman]], _String Theory Duals for Mass-deformed $SO(N)$ and $USp(2N)$ $\mathcal{N}=4$ SYM Theories_, Phys. Rev. D62:106002, 2000 ([arXiv:hep-th/0004151](https://arxiv.org/abs/hep-th/0004151))

* [[Iosif Bena]], Stanislav Kuperstein, _Brane polarization is no cure for tachyons_, JHEP09 (2015) 112 ([arXiv:1504.00656](https://arxiv.org/abs/1504.00656))

Discussion of [[D2-branes]] [[D-brane polarization|polarizing]] to [[D2-D4 brane bound states]]:

* [[Iosif Bena]], Aleksey Nudelman, _Warping and vacua of $(S)YM_{3+1}$_, Phys. Rev. D62 (2000) 086008 ([arXiv:hep-th/0005163](https://arxiv.org/abs/hep-th/0005163))

* [[Iosif Bena]], Aleksey Nudelman, _Exotic polarizations of D2 branes and oblique vacua of $(S)YM_{2+1}$_, Phys. Rev. D62 (2000) 126007 ([arXiv:hep-th/0006102](https://arxiv.org/abs/hep-th/0006102))

Polarization into [[torus]] shape:

* Yoshifumi Hyakutake, _Torus-like Dielectric D2-brane_, JHEP 0105:013, 2001 ([arXiv:hep-th/0103146](https://arxiv.org/abs/hep-th/0103146))

* Tatsuma Nishioka, Tadashi Takayanagi, _Fuzzy Ring from M2-brane Giant Torus_, JHEP 0810:082, 2008 ([arXiv:0808.2691](https://arxiv.org/abs/0808.2691))

Discussion of [[D4-branes]] polarizing into [[NS5-branes]]:

* [[Iosif Bena]], Calin Ciocarlie, _Exact $\mathcal{N}=2$ Supergravity Solutions With Polarized Branes_, Phys. Rev. D70 (2004) 086005 ([arXiv:hep-th/0212252](https://arxiv.org/abs/hep-th/0212252))

* [[Iosif Bena]], [[Radu Roiban]], _$\mathcal{N}=1^\ast$ in 5 dimensions: Dijkgraaf-Vafa meets Polchinski-Strassler_, JHEP 0311 (2003) 001 ([arXiv:hep-th/0308013](https://arxiv.org/abs/hep-th/0308013))

Polarization of [[fractional D-branes]]:

* Timothy J. Hollowood, S. Prem Kumar, _World-sheet Instantons via the Myers Effect and $\mathcal{N} = 1^\ast$ Quiver Superpotentials_, JHEP 0210:077, 2002 ([arXiv:hep-th/0206051](https://arxiv.org/abs/hep-th/0206051))

### D-brane polarization into supertubes
 {#ReferencesSupertubes}

* [[David Mateos]], [[Paul Townsend]], _Supertubes_, Phys. Rev. Lett. 87 (2001) 011602 ([arXiv:hep-th/0103030](https://arxiv.org/abs/hep-th/0103030))

* [[Martin Kruczenski]], [[Robert Myers]], [[Amanda Peet]], David J. Winters, _Aspects of supertubes_, JHEP 0205:017, 2002 ([arXiv:hep-th/0204103](https://arxiv.org/abs/hep-th/0204103))

### Polarization of M2-branes into M5-branes

The Myers effect in [[M-theory]] for [[M2-branes]] polarizing into [[M5-branes]] of ([[fuzzy 3-sphere|fuzzy]]) [[3-sphere]]-shape ([[M2-M5 brane bound states]]):

* {#Bena00} [[Iosif Bena]], _The M-theory dual of a 3 dimensional theory with reduced supersymmetry_, Phys. Rev. D62:126006, 2000 ([arXiv:hep-th/0004142](https://arxiv.org/abs/hep-th/0004142))

* [[Iosif Bena]], [[Nicholas Warner]], _A harmonic family of dielectric flow solutions with maximal supersymmetry_, JHEP 0412:021, 2004 ([arXiv:hep-th/0406145](https://arxiv.org/abs/hep-th/0406145))

* Masato Arai, [[Claus Montonen]], Shin Sasaki, _Vortices, Q-balls and Domain Walls on Dielectric M2-branes_, JHEP 0903:119, 2009 ([arXiv:0812.4437](https://arxiv.org/abs/0812.4437))

* [[Iosif Bena]], [[Mariana Graña]], Stanislav Kuperstein, Stefano Massai, _Tachyonic Anti-M2 Branes_, JHEP 1406:173, 2014 ([arXiv:1402.2294](https://arxiv.org/abs/1402.2294))

With emphasis on the role of the [[Page charge]]/[[Hopf WZ term]]:

* [[Krzysztof Pilch]], Alexander Tyukov, [[Nicholas Warner]], _Flowing to Higher Dimensions: A New Strongly-Coupled Phase on M2 Branes_, JHEP11 (2015) 170 ([arXiv:1506.01045](https://arxiv.org/abs/1506.01045))

The corresponding D2-NS5 bound state under [[duality between M-theory and type IIA string theory]]:

* [[Iosif Bena]], Aleksey Nudelman, _Warping and vacua of $(S)YM_{3+1}$_, Phys. Rev. D62 (2000) 086008 ([arXiv:hep-th/0005163](https://arxiv.org/abs/hep-th/0005163))

Via the [[mass-deformed ABJM model]]:

* [[Jaume Gomis]], Diego Rodriguez-Gomez, [[Mark Van Raamsdonk]], [[Herman Verlinde]], _A Massive Study of M2-brane Proposals_, JHEP 0809:113, 2008 ([arXiv:0807.1074](https://arxiv.org/abs/0807.1074))

and identifying the polarization into $S^3$ as a [[fuzzy sphere]]-version of the [[complex Hopf fibration]]:

* [[Horatiu Nastase]], [[Constantinos Papageorgakis]], [[Sanjaye Ramgoolam]], _The fuzzy $S^2$ structure of M2-M5 systems in ABJM membrane theories_, JHEP 0905:123, 2009 ([arXiv:0903.3966](https://arxiv.org/abs/0903.3966))

Review in:

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 6 of: _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

### Polarization of M5-branes into MK6s

More general [[M-brane polarizations]] and polarization of [[M5-branes]] into [[MK6]]s:

* [[Yolanda Lozano]], _Non-commutative Branes from M-theory_, Phys. Rev. D64 (2001) 106011 ([arXiv:hep-th/0012137](https://arxiv.org/abs/hep-th/0012137))

* [[Iosif Bena]], Diana Vaman, _The polarization of M5 branes and little string theories with reduced supersymmetry_, JHEP 0111 (2001) 032 ([arXiv:hep-th/0101064](https://arxiv.org/abs/hep-th/0101064))

Via the [[ABJM model]]:

* Wung-Hong Huang, _KK6 from M2 in ABJM_, JHEP 1105:054, 2011 ([arXiv:1102.3357](https://arxiv.org/abs/1102.3357))

* Wung-Hong Huang, _M2-KK6 System in ABJM Theory: Fuzzy $S^3$ and Wrapped KK6_ ([arXiv:1107.2030](https://arxiv.org/abs/1107.2030))

### Relation to giant gravitons

On the relation of polarized branes to [[giant gravitons]]:

* [[Iosif Bena]], Douglas Smith,  _Towards the solution to the giant graviton puzzle_, 	Phys.Rev. D71 (2005) 025005 ([arXiv:hep-th/0401173](https://arxiv.org/abs/hep-th/0401173))

### In the BMN matrix model

On solutions of the [[BMN matrix model]] in relation to the [[Myers effect]] and [[D0-D2 brane bound states]]:

* Hai Lin, _The Supergravity Dual of the BMN Matrix Model_, JHEP 0412:001, 2004 ([arXiv:hep-th/0407250](https://arxiv.org/abs/hep-th/0407250))

[[!redirects Myers' effect]]

[[!redirects polarized brane]]
[[!redirects polarized branes]]

[[!redirects D-brane polarization]]
[[!redirects D-brane polarizations]]

[[!redirects polarized D-brane]]
[[!redirects polarized D-branes]]

[[!redirects M-brane polarization]]
[[!redirects M-brane polarizations]]

[[!redirects M2-brane polarization]]
[[!redirects M2-brane polarizations]]

[[!redirects polarized M-brane]]
[[!redirects polarized M-branes]]

[[!redirects polarized M2-brane]]
[[!redirects polarized M2-branes]]

[[!redirects supertube]]
[[!redirects supertubes]]


Myers-Steenrod theorem

#Contents#
* automatic table of contents goes here
{:toc}

## Description

In their 1939 paper, Myers and Steenrod
proved two theorems on [[Riemannian manifolds]].

The first theorem shows that distance-preserving maps between Riemannian manifolds are differentiable isometries.

In 1956 Palais simplified the proof and extended the result to show that a Riemannian manifold can be reconstructed from its [[metric space]].
That is to say, the functor that sends Riemannian manifolds and isometries to metric spaces and isometries is a [[fully faithful functor]].

The second theorem proves that the group of isometries of a Riemannian manifold is a [[Lie group]].

## Idea

Recall that an [[isometry]] between two [[Riemannian manifolds]] $(M,g), (N,h)$ is a [[diffeomorphism]] $\phi: M \to N$ such that $\phi^* h = g$.  In other words, $\phi$ respects the Riemannian structure as well as the differentiable structure.

It follows that if $d,d'$ are the metrics on $M,N$ induced by the Riemannian metrics $g,h$, then $d'(\phi(x),\phi(y)) = d(x,y)$ for $x,y \in M$---that is, $\phi$ is distance-preserving.
Interestingly, a version of  the converse is true, according to the Myers--Steenrod theorem:

+-- {: .un_theorem}
###### Theorem (Myers--Steenrod)
If $\phi: M \to N$ is distance-preserving and surjective, then it is an [[isometry]] (in particular, it is smooth).
=--

For notational simplicity, assume $N=M$.

$\phi$ is evidently a homeomorphism.  Now pick $p \in M$ and a neighborhood $D_r(p)$ such that for any $q \in D_r(p)$, there is a unique geodesic in that neighborhood connecting $p,q$.  Call it $\gamma$.   I claim that $\phi \circ \gamma$ is a geodesic in $N$.

The [[geodesic]] $\gamma$ can be assumed to be parametrized by unit length. We have for all $t$,
\[ d(p,q) = d(\gamma(t),p) + d(\gamma(t), q).\]
Thus
\[ \label{streq} d(\phi(p),\phi(q)) = d(\phi(\gamma(t)),\phi(p)) + d(\phi(\gamma(t)), \phi(q)) .\]
In other words, we have strict equality in the triangle inequality.

##Geodesic preservation

In a Riemannian manifold $M$, if $q,r$ lie in a small neighborhood of $p$ and
\[ d(q,r) + d(r,p)  = d(q,p) ,\]
then $r$ lies on a geodesic betwen $p,q$.
Indeed, draw a geodesic $\alpha$ from $q$ to $r$ and a geodesic $\beta$ from $r$ to $p$ that travel at unit speed and minimze the distances; we can do this locally even without the assumption of completeness.  The catenation $\alpha \beta$ minimizes the length from $q$ to $p$, so it is an unbroken geodesic.  Thus $r$ lies on the geodesic.

So, returning to the discussion above, we see that by (eq:streq) $\phi$ *sends geodesics to geodesics*.

## Exponential coordinates

Now fix open $U \subset T_p(M),V \subset T_q(N)$ containing the origin such that $\exp_p$ takes $U$ diffeomorphically to a neighborhood $D_r(p)$, and $\exp_q$ takes $V$ diffeomorphically to $D_r(q)$.  There is an expression for $\phi$ as a map $\tilde{\phi}: U \to V$ in these exponential coordinates.
It is obtained as follows: if $v \in U$, take a geodesic $\gamma_v$ with $\gamma_v'(0) = v$ and consider $(\phi \circ \gamma_v)'(0)$.  In fact this induces more generally a map
\[ \tilde{\phi}: T_p(M) \to T_q(M).\]
Since $\gamma_v$ moves at constant speed $|v|$ and $\phi \circ \gamma_v$ moves at the same speed, it follows that
$\tilde{\phi}$ is norm-preserving.  Also $\tilde{\phi}(0) = 0$.  If we can show that $\tilde{\phi}$ is linear, then we'll get smoothness in this exponential coordinate system---hence smoothness in general.

## $\tilde{\phi}$ is an isometry

It can be checked that
\[ \boxed{\lim_{A,B \to 0 \in T_p(M)} \frac{d(\exp_p(A),\exp_p(B)) }{|A-B|} } = 1.\]
We will postpone this for now.

This is what we will use to show that $\tilde{\phi}: T_p(M) \to T_q(M)$ is an isometry, where each tangent space is given the norm from the Riemannian metric.  Pick $A,B \in T_p(M)$ and consider the    geodesics $\gamma_{A},\gamma_{B}$ at $p$ and $\gamma_{A'}, \gamma_{B'}$ where $A'=\tilde{\phi}(A), B' = \tilde{\phi}(B)$.
Then $d(\gamma_{A}(t), \gamma_{B}(t)) = d( \gamma_{A'}(t), \gamma_{B'}(t))$.
So
\[ 1=  \lim_{t \to 0} \frac{d(\exp_p(t A),\exp_p(t B)) }{t|A-B|}
=  \lim_{t \to 0} \frac{d(\exp_q(t A'),\exp_q(t B')) }{t|A-B|}  = \frac{|A'-B'|}{|A-B|}.\]

To show that $\tilde{\phi}$ is linear, we appeal to a general fact:

 Let $X,Y$ be normed linear spaces and $T: X \to Y$ a map such that $|Tx - Tx'|  = |x-x'|$ for $x,x' \in X$. Then $T$ is linear if $T(0) = 0$.

##Conclusion of the proof

Now we have seen that $\phi$ is smooth, and that for $v \in T_p(M)$, $|v|_p = |\phi_*(v)|_{\phi(p)}$, i.e. $\phi$ preserves lengths on tangent vectors.  By the polarization identity, $\phi$ preserves the inner product, and is thus an isometry.

##A lemma

We never proved a fact  about the exponential map---the equality
\[ \lim_{A,B \to 0 \in T_p(M)} \frac{d(\exp_p(A),\exp_p(B)) }{|A-B|} = 1.\]
We will briefly sketch the idea here.  $|A-B|$ is the length of the linear  path from $A $ to $B$ in $T_p(M)$, so it will be sufficient to show:

+-- {: .un_lemma}
###### Lemma
If $c:(0,1) \to T_p(M)$ is a path in $T_p(M)$, then as $c((0,1)) \to 0$ with the derivative $|c'|$ staying bounded,
\[ \frac{ l(\exp_p(c))}{l(c)} = 1.\]
=--

We are  abusing notation quite a bit here, but it should  not cause confusion.

The reason is that
\[ l(c) = \int |c'| \]
while
\[ l(\exp_p(c)) = \int | (\exp_p)_{*c(t)} c'(t) |_{\exp_p(c(t))} \]
where in the second equation we are referring to the norms induced by the metric on the various tangent spaces of $M$.
The difference between these two is $o(l(c))$, because $\exp_p$ has derivative the identity at $0 \in T_p(M)$, and the metric on $T_{\exp_p(c(t))}(M)$ is very close (say by a factor of $1 \pm \epsilon$) to that of $T_p(M)$ if we work in local coordinates and agree to identify the tangent spaces.

## Related concepts

* [[Riemannian manifold]]

* [[Lie group]]

* [[isometry]]

## References

* [[S. B. Myers]], [[N. E. Steenrod]], _The Group of Isometries of a Riemannian Manifold_, The Annals of Mathematics 40:2 (1939), 400–416.  [doi](http://dx.doi.org/10.2307/1968928).

A simplified proof that isometries are differentiable is given by

* [[Richard S. Palais]], _On the differentiability of isometries_, Proceedings of the American Mathematical Society 8:4 (1956), 805–807.  [doi](http://dx.doi.org/10.1090/s0002-9939-1957-0088000-x).

[[!redirects Myers–Steenrod theorem]]
[[!redirects Myers--Steenrod theorem]]


Mykola Dedushenko

* [InSpire page](https://inspirehep.net/authors/1501846)

## Selected writings

On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[4-manifolds]] to [[2d CFTs]]:

* [[Mykola Dedushenko]], [[Sergei Gukov]], [[Pavel Putrov]], *Vertex algebras and 4-manifold invariants*, Chapter 11 in: *Geometry and Physics: Volume I*  (2018) 249–318 [[arXiv:1705.01645](https://arxiv.org/abs/1705.01645), [doi:10.1093/oso/9780198802013.003.0011](https://doi.org/10.1093/oso/9780198802013.003.0011)]

category: people


Myles Tierney

Myles Tierney was a Rutgers faculty member for thirty-four years, coming to Rutgers as an Associate Professor in 1968 following positions at Rice University (1965-6) and at the ETH-Forschungsinstitut für Mathematik, Zürich (1966-68). He received his B.A. from Brown University in 1959 and his Ph.D. from Columbia in 1965.

Myles began his career as an [[algebraic topology|algebraic topologist]], moved toward [[category theory]] and was responsible (together with [[Bill Lawvere|F.W. Lawvere]]) for the introduction of a new field within category theory: [[elementary topoi]].

He had six Ph.D. students: Ira Wolf (1971), Carol Ann Keller (1983), Norman Adams (1984), Terence Lindgren (1984), [[Todd Trimble]] (1994), and Luca Mauri (1998). Myles retired as of January 1, 2003 and was honored by the Department at a retirement party on December 10, 2002.

> (Taken from [this website](http://www.math.rutgers.edu/newsletter/newsletter2003.html))

At Dalhousie he had a further Ph.D. student: Radu Diaconescu (1973).

Myles Tierney died on October 5, 2017 having turned 80 in September. He made important contributions to category theory in collaboration with [[Bill Lawvere]] and with [[André Joyal]].

## Selected writings

On [[monads]] in [[universal algebra]] and ([[cohomology|co-]])[[homology]]-theory:

* [[H. Applegate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]: _[[Seminar on Triples and Categorical Homology Theory]]_, ETH 1966/67, edited by [[Beno Eckmann]] and [[Myles Tierney]], **[[LNM 80]]**, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

On ([[idempotent monad|idemponent]]) [[monads]]:

* [[Harry Applegate]], [[Myles Tierney]], *Categories with models*, p. 156-244 In: [[Beno Eckmann]] (ed.) *[[Seminar on Triples and Categorical Homology Theory]]* Lecture Notes in Mathematics, **80**, Springer  1969  [[doi:10.1007/BFb0083086](https://doi.org/10.1007/BFb0083086), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0083086)]

On [[comonads]]:

* {#ApplegateTierney70} [[Harry Applegate]], [[Myles Tierney]], _Iterated cotriples_, Lecture Notes in Math. **137** (1970) 56-99 ([doi:10.1007/BFb0060440](https://doi.org/10.1007/BFb0060440))

On [[forcing]] via [[classifying toposes]] and the [[classifying topos of a localic groupoid]]:

* {#JoyalTierney84} [[André Joyal]], [[Myles Tierney]], *An extension of the Galois theory of Grothendieck*,  Mem. Amer. Math. Soc. **309** (1984) [[ISBN:978-1-4704-0722-3](https://bookstore.ams.org/memo-51-309)]

  (historical note: according to MR756176 (86d:18002) by [[Peter Johnstone]], the main results of this monograph were obtained by the authors around 1978-1979, typed version circulated from 1982, and the results influenced the field much before the actual publication)

On [[stacks]] and [[classifying spaces]]:

* [[André Joyal]] and [[Myles Tierney]], _Strong stacks and classifying spaces_,  Category theory (Como, 1990),  213--236, Lecture Notes in Math. __1488__, Springer 1991.

On the [[Dwyer-Kan loop groupoid]]-construction ([[path groupoid|path]]-[[simplicial groupoids]]):

* [[André Joyal]], [[Myles Tierney]], *On the theory of path groupoids*, Journal of Pure and Applied Algebra **149** 1 (2000) 69-100 [doi:10.1016/S0022-4049(98)00164-9]

On the [[Quillen equivalence]] between the [[model categories]] for [[quasi-categories]] and [[complete Segal spaces]]:

* {#JoyalTierney07} [[Andre Joyal]], [[Myles Tierney]], _Quasi-categories vs. Segal spaces_, in *Categories in Algebra, Geometry and Mathematical Physics*, Contemporary Mathematics **431** (2007) [[arXiv:math/0607820](http://arxiv.org/abs/math/0607820), [doi:10.1090/conm/431](https://doi.org/10.1090/conm/431)]

On [[simplicial homotopy theory]], the [[classical model structure on simplicial sets]] and the [[classical model structure on topological spaces]]:

* {#JoyalTierney08} [[André Joyal]], [[Myles Tierney]] _Notes on simplicial homotopy theory_, Lecture at _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, CRM 2008  ([[JoyalTierneyNotesOnSimplicialHomotopyTheory.pdf:file]])

* {#JoyalTierney09} [[André Joyal]], [[Myles Tierney]], _An introduction to simplicial homotopy theory_, 2009  ([web](http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01), [[JoyalTierneySimplicialHomotopyTheory.pdf:file]])

category: people

[[!redirects Miles Tierney]]
[[!redirects M. Tierney]]
[[!redirects Tierney]]
[[!redirects Miles Teirney]]


mysterious duality

+-- {: .rightHandSide}
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#### Duality in string theory
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[[!include duality in string theory -- contents]]
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=--

# Contents
* table of contents
{:toc}

## Idea

What is called the **mysterious duality** in [Vafa 00](#Vafa00), [Iqbal-Neitzke-Vafa 01](#INV01) is a kind of "[[duality in string theory]]" in the form of a correspondence between [[torus|toroidal]] [[KK-compactifications]] of [[M-theory]] and [[del Pezzo surfaces]]:
Here M-theory on the $k$-[[torus]] $T^k$ corresponds to the [[complex projective space]] $\mathbb{C}P^2$ [[blow up|blown up]] at $k$ [[generic points]]. In particular, for $k = 1$ this corresponds to  [[type IIA string theory]] (via [[duality between M-theory and type IIA string theory]]). [[type II string theory|Type IIB]]  corresponds to $\mathbb{P}^1 \times \mathbb{P}^1$.

Moreover, the [[moduli]] of [[KK-compactifications]] of [[M-theory]] on rectangular tori are mapped to Kähler moduli of del Pezzo surfaces. The [[U-duality]] group of M-theory corresponds to a group of classical symmetries of the del Pezzo represented by global [[diffeomorphisms]]. The $\frac{1}{2}$-[[BPS state|BPS]] [[brane]] charges of M-theory correspond to [[spheres]] in the del Pezzo, and their tension to the exponentiated volume of the corresponding spheres.

The [[S-duality]] of type IIB in 10 dimensions corresponds to the exchange of the two [[complex projective curves]] $\mathbb{C}P^1$s in $\mathbb{C}P^1 \times \mathbb{C}P^1$.

## References

The original observation:

* {#Vafa00} [[Cumrun Vafa]], from slide 42 in: _Mirror symmetry_, Talk at [String Theory at the Millennium](http://theory.caltech.edu/conf2000/), Caltech, January 2000 ([slides html](https://web.archive.org/web/20100612073150/http://www.theory.caltech.edu/conf2000/Vafa/fs1.html))

* {#INV01} [[Amer Iqbal]], [[Andrew Neitzke]], [[Cumrun Vafa]], _A mysterious duality_, ([arXiv:hep-th/0111068](https://arxiv.org/abs/hep-th/0111068))

Relation to [[Borcherds algebras]]:

* Pierre Henry-Labordere, [[Bernard Julia]], Louis Paulot, _Borcherds symmetries in M-theory_, ([arXiv:hep-th/0203070](https://arxiv.org/abs/hep-th/0203070))

{#SatiVoronov} Relation to [[Hypothesis H]] via [[automorphisms]] of [[iterated loop space|iterated]] [[cyclic loop spaces]] of the [[4-sphere]]:

* [[Hisham Sati]], [[Alexander Voronov]], *Mysterious Triality and Rational Homotopy Theory*, Comm. Math. Phys. **400** (2023) 1915-1960 [[arXiv:2111.14810](https://arxiv.org/abs/2111.14810), [doi:10.1007/s00220-023-04643-7](https://doi.org/10.1007/s00220-023-04643-7)]

* [[Hisham Sati]], [[Alexander Voronov]], *Mysterious Triality and M-Theory* [[arXiv:2212.13968](https://arxiv.org/abs/2212.13968)]

review:

* {#Voronov21} [[Alexander Voronov]] (joint with [[Hisham Sati]]), *Mysterious Duality*, talk at Texas Tech 2021 ([abstract pdf](https://www.math.ttu.edu/~lhoang/PureMath/abstract-2021-03-15-Voronov.pdf) [[Voronov_MysteriousAbstract.pdf:file]], [slides pdf](https://www.math.ttu.edu/~lhoang/PureMath/Slides-2021-03-15-Voronov.pdf), [[Voronov_MysteriousSlides.pdf:file]])

* [[Alexander Voronov]]: *The $E_k$ symmetry of dimensional reductions of M-theory*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (2023) [[web](/nlab/show/M-Theory+and+Mathematics#Voronov2023)]

See also at:

* [[U-duality]]

[[!redirects Mysterious Duality]]


mysticism

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### Context
#### Philosophy
+-- {: .hide}
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=--

#Contents#
* table of contents
{:toc}

## Relation to speculative philosophy

According to [[Georg Hegel]], _[[Encyclopedia of the Philosophical Sciences]]_, medieval mysticism is the same as the [[speculative philosophy]] of the 19th century:

> [Enc§82d](Science+of+Logic#Enc82d)  Hinsichtlich der Bedeutung des Spekulativen ist hier noch zu erwähnen, daß man darunter dasselbe zu verstehen hat, was früher, zumal in Beziehung auf das religiöse Bewußtsein und dessen Inhalt, als das Mystische bezeichnet zu werden pflegte.  Wenn heutzutage vom Mystischen die Rede ist, so gilt dies in der Regel als gleichbedeutend mit dem Geheimnisvollen und Unbegreiflichen, und dies Geheimnisvolle und Unbegreifliche wird dann, je nach Verschiedenheit der sonstigen Bildung und Sinnesweise, von den einen als das Eigentliche und Wahrhafte, von den anderen aber als das dem Aberglauben und der Täuschung Angehörige betrachtet.

> Hierüber ist zunächst zu bemerken, daß das Mystische allerdings ein Geheimnisvolles ist, jedoch nur für den Verstand, und zwar einfach um deswillen, weil die abstrakte Identität das Prinzip des Verstandes, das Mystische aber (als gleichbedeutend mit dem Spekulativen) die konkrete Einheit derjenigen Bestimmungen ist, welche dem Verstand nur in ihrer Trennung und Entgegensetzung für wahr gelten.
Wenn dann diejenigen, welche das Mystische als das Wahrhafte anerkennen, es gleichfalls dabei bewenden lassen, daß dasselbe ein schlechthin Geheimnisvolles sei, so wird damit ihrerseits nur ausgesprochen, daß das Denken für sie gleichfalls nur die Bedeutung des abstrakten Identischsetzens hat und daß man um deswillen, um zur Wahrheit zu gelangen, auf das Denken verzichten oder, wie auch gesagt zu werden pflegt, daß man die Vernunft gefangennehmen müsse.

> [Enc§82e](Science+of+Logic#Enc82e) Nun aber ist, wie wir gesehen haben, das abstrakt verständige Denken so wenig ein Festes und Letztes, daß dasselbe sich vielmehr als das beständige Aufheben seiner selbst und als das Umschlagen in sein Entgegengesetztes erweist, wohingegen das Vernünftige als solches gerade darin besteht, die Entgegengesetzten als ideelle Momente in sich zu enthalten. Alles Vernünftige ist somit zugleich als mystisch zu bezeichnen, womit jedoch nur so viel gesagt ist, daß dasselbe über den Verstand hinausgeht, und keineswegs, daß dasselbe überhaupt als dem Denken unzugänglich und unbegreiflich zu betrachten sei.

> It should also be  mentioned here that the meaning of the  speculative is  to be understood as being the same as what used in earlier times to be called "mystical", especially with regard to  the religious consciousness and its content. When we speak of the "mystical" nowadays, it is  taken as a rule to be synonymous with what is mysterious and incomprehensible; and, depending on the ways their culture and mentality vary in other respects, some people treat the mysterious and incomprehensible as what is authentic and genuine, whilst others regard it as belonging to the domain of superstition and deception. About this we must remark first that "the mystical" is certainly something mysterious, but only for the understanding, and then only because  abstract identity is the principle of the understanding. But when it is regarded as synonymous with  the speculative, the mystical is the concrete unity of just those determinations that count as true for the understanding only in their separation and opposition. So if those who recognise the mystical as what is genuine say that it is something utterly mysterious, and just leave it at that, they are only declaring that for them, too, thinking has only the Significance of an abstract positing of identity, and that in order to attain the truth we must renounce thinking, or, as they frequently put it, that we must "take reason captive." As we have seen, however, the abstract thinking of the understanding is so far from being something firm and ultimate that it proves itself, on the contrary, to be a constant sublating of itself and an overturning into its opposite, whereas the rational as such is rational precisely because it contains both of the opposites as ideal moments within itself. Thus, everything rational can equally be called "mystical"; but this only amounts to saying that it transcends the understanding. It does not at all imply that what is so spoken of must be considered inaccessible to thinking and incomprehensible.

For Hegel's relation to [[Meister Eckhart]], see [there](Meister+Eckhart#RelationToHegel).

## Bertrand Russell's relation to mysticism

The philosopher [[Bertrand Russell]] concerned himself with mysticism and was, according to himself, influenced by a "mathematical mysticism" in his younger years.

In his essay "Mysticism and Logic" ([Russell 1918](#Russell18)) Russell understands mysticism as "the attempt to conceive the world as a whole by means of thought". He gives four characteristics of it:

1. "belief in insight as against discursive knowledge"
1. a belief in unity of everything
1. denial of the reality of time
1. the claim that evil, and sometimes also good, is a mere appearance

Russell finds at least some of these characteristics in the philosophies of [[Heraclitus]], [[Plato]], [[Hegel]], and [[Spinoza]]. Interestingly, Heraclitus is judged far better than Plato, the former being ascribed "a nature [in which] we see the true union of the mystic and the man of science". In the remainder of the essay Russell discusses the relation of the four characteristics of mysticism to his own views.

In the autobiographical book _Portraits from Memory_ the passage

> Mathematics ... deals with the world of ideas and has in consequence an exactness and perfection which is absent from the everyday world. This kind of mathematical mysticism, which Plato derived from Pythagoras, appealed to me.

may serve as an example how Russell was influenced in young age by the exposure to mathematics; more poetically it is expressed in the essay _The Study of Mathematics_ published along with _Mysticism and Logic_ ([Russell 1918](#Russell18))

> Mathematics, rightly viewed, possesses not only truth, but supreme beauty---a beauty cold and austere, like that of sculpture, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. ... Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great works springs.

> ... Of the austere virtues the love of truth is the chief, and in mathematics, more than elsewhere, the love of truth may find encouragement for waning faith.

According to Ray [Monk's biography](#Monk97) on Russell, the search for a foundation of this "Pythagorean" mysticism governed the first half of his life: Being deeply disappointed by the mathematical curriculum at Cambridge, which did not address his philosophical needs, Russell turned to [[idealism]], which was at that time propounded in Britain by [[J.M. Ellis McTaggart]] following the German tradition of [[Kant]] and [[Hegel]]. Under this influence Russell devised in [[An Essay on the Foundations of Geometry]] a variation of Kant's transcendental aesthetics claiming that the curvature of the space of our perception, i.e. of any ''form of externality'', is a priori constant.

Exposure to the work of the German mathematicians [[Richard Dedekind|Dedekind]], [[Karl Weierstraß|Weierstraß]], and [[Georg Cantor|Cantor]] along with a personal meeting with [[Peano]] caused a turn in Russell's views. He now hoped to find a rigorous [[foundations|foundation of mathematics]] based solely on [[logic|logical]] [[axioms]], thereby justifying a kind of [[platonism|Platonic]] view on mathematics. But over the years he was frustrated in this attempt by many problems he encountered, especially [[Russell's paradox]]. He gradually gave up all his Platonic beliefs; arriving in the end (after [[Wittgenstein|Wittgenstein's]] [[Tractatus Logico-Philosophicus|Tractatus]]) at [[formalism]].

He remarks on the power of rigorous mathematical definition:

> "Continuity" had been, until he [Cantor] 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. (Chapter XXXI of "[[A History of Western Philosophy]]" (1945))

## Brouwer

[[L.E.J. Brouwer]] had a long standing interest in the theory and practice of mysticism, setting out his manifesto while still a graduate student in the essay [Life, Art, and Mysticism](#LAM05) (1905). The essay quotes the Bhagavad Gita, [[Meister Eckhart]], and [[Jakob Böhme]].

However, he did not consider his [[intuitionism]] to depend upon his mystical views as such. For Brouwer, mysticism is the domain of "consciousness' deepest home", wherein there is no subject-object distinction. The first act of intuitionism, the intuition of time, propels us away from this pure state of "wisdom". It is in fact the precondition for any experience of an outer world or experience of the self as something distinct. Mathematics thus comes into being in this exodus from the home of consciousness, so on Brouwer's view, "mathematics is in a sense the very opposite of mysticism" ([Van Atten, 2004](#vanAtten04)), and mystical insight is only possible quite apart from the machinations of the intellect.

Brouwer even went so far as to remark on the perniciousness of mathematics in its keeping us from the home of consciousness. Thus in the notebooks in which he conceived his 1907 thesis we find: "One could see as the goal of one's life: abolition and delivery from all mathematics".

## Related entries

* [[hermeticism]], [[gnosticism]]

* [[Meister Eckhart]]

## References

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

* [[Georg Hegel]], _[[Lectures on the Philosophy of Religion]]_ 1832

* {#LAM05} [[L.E.J. Brouwer]], [_Life, Art, and Mysticism_](https://projecteuclid.org/euclid.ndjfl/1039886518), 1905

* {#Russell18} [[Bertrand Russell]], _Mysticism and Logic and Other Essays_, 1918 ([on Wikisource](https://en.wikisource.org/wiki/Mysticism_and_Logic_and_Other_Essays)). Containing _Mysticism and Logic_ and _The Study of Mathematics_. The former was published already in 1914 in the _Hibbert Journal_.

* {#CPM} [[L. E. J. Brouwer]], [_Consciousness, Philosophy, and Mathematics_](https://projecteuclid.org/euclid.jsl/1183730638), 1948

* {#Monk97} Ray Monk, _Bertrand Russell -- Mathematics: Dreams and Nightmares_, 1997

* van Atten & Tragesser, [Mysticism and mathematics: Brouwer, Gödel, and the common core thesis](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.99.1853&rep=rep1&type=pdf), 2003

* {#vanAtten04} van Atten, _On Brouwer_, 2004

* {#Melzer14} Arthur Melzer, _Philosophy between the lines -- The lost history of esoteric writing_, University of Chicago Press, 2014

Decent discussion of the history of mysticism (or not) in [[quantum physics]], such as induced by the concept of "[[wavefunction collapse]]" (see also [[interpretation of quantum mechanics]]) is in

* Juan Miguel Marin, _'Mysticism' in quantum mechanics: the forgotten controversy_, European Journal of Physics, Volume 30, Number 4

[[!redirects mystic]]


Mário Ziman

* [institute page](https://www.sav.sk/?lang=en&doc=user-org-user&user_no=3679)

* [GoogleScholar page](https://scholar.google.sk/citations?user=96S9ctwAAAAJ&hl=sk)

## Selected writings

On [[quantum physics]] with emphasis on [[quantum probability theory]]:

* [[Teiko Heinosaari]], [[Mário Ziman]], *The Mathematical Language of Quantum Theory -- From Uncertainty to Entanglement*, Cambridge University Press  (2011) [[doi:10.1017/CBO9781139031103]( https://doi.org/10.1017/CBO9781139031103)]

category: people

[[!redirects Mario Ziman]]


Máximo Bañados

* [webpage](http://srv2.fis.puc.cl/~mbanados)

## Selected writings

Introducing [[BTZ black holes]]:

* [[Máximo Bañados]], [[Claudio Teitelboim]], [[Jorge Zanelli]], _The Black Hole in Three Dimensional Space Time_, Phys. Rev. Lett. 69 (1992) 1849-1851 ([arXiv:hep-th/9204099](https://arxiv.org/abs/hep-th/9204099))

* [[Máximo Bañados]], [[Marc Henneaux]], [[Claudio Teitelboim]], [[Jorge Zanelli]],
_Geometry of the 2+1 Black Hole_, Phys. Rev. D48: 1506-1525, 1993 ([arXiv:gr-qc/9302012](https://arxiv.org/abs/gr-qc/9302012))

category: people

[[!redirects Maximo Banados]]


Måns Henningson

* [webpage](http://fy.chalmers.se/~mans/)

category: people


Möbius inversion

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

#Contents#
* table of contents
{:toc}

## Idea

The classical _Möbius inversion formula_ is a principle originating in [[number theory]], which says that if $f$ and $g$ are two (say, [[complex]]-valued) functions on the positive [[natural numbers]], then
$$f(n) = \sum_{d\mid n} g(d)$$
if and only if
$$g(n) = \sum_{d\mid n} f(d)\mu(n/d)$$
where the sums run over the divisors of $n$, and where $\mu$ is the _Möbius function_:
$$
\mu(n) = \begin{cases}(-1)^r & n\,\text{ is the product of }\,r\,\text{ distinct primes}\\0 & \text{otherwise}\end{cases}
$$
[[Gian-Carlo Rota]] [(1964)](#Rota64) described a vast generalization of the Möbius inversion formula, which was at the same time a generalization of the classical [[inclusion-exclusion]] principle in [[combinatorics]]. Rota's formulation of Möbius inversion applies to functions defined over [[partially ordered sets]], while further category-theoretic generalizations have also been proposed.

## Möbius inversion for posets

### Incidence algebras and the zeta function
 {#IncidenceAlgebrasZetaFn}

The **incidence algebra** of a poset $P$ (relative to a [[commutative ring]] $R$) is an [[associative unital algebra]] defined as follows.
Its elements are functions $f : P \times P\to R$ such that $x \nleq y$ implies $f(x,y) = 0$.
Pointwise addition $f+g$ and scalar multiplication $r\cdot f$ are defined straightforwardly ($(f+g)(x,y) = f(x,y) + g(x,y)$, $(r\cdot f)(x,y) = r\cdot f(x,y)$),
while the _convolution product_ $f*g$ is defined as follows:
$$(f*g)(x,y) = \sum_{x \le z \le y} f(x,z) \cdot g(z,y)$$
Note that the unit of the convolution product is the [[Kronecker delta]]:
$$\delta(x,y) = \begin{cases}1 & x = y \\ 0 & \text{otherwise}\end{cases}$$

We write $I(P)$ for the incidence algebra of $P$.
A special and important element of $I(P)$ is given by
$$\zeta(x,y) = \begin{cases}1 & x \le y \\ 0 & \text{otherwise}\end{cases}$$
referred to as the **zeta function** of $P$.

### The Möbius function and the Möbius inversion formula

A **Möbius function** of $P$ is defined as an inverse to the zeta function with respect to the convolution product. Such an inverse always exists if the poset is [[locally finite poset|locally finite]], that is, if every [[interval]] $[x,y] = \{z \mid x \le z\le y\}$ is finite.

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

[(Rota 1964)](#Rota64):
If $P$ is a locally finite poset, then there exists a function $\mu$ such that $\zeta * \mu = \mu * \zeta = \delta$.

=--

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

$\mu(x,y)$ can be computed by induction on the number of elements in the interval $[x,y]$. In the base case we set $\mu(x,x) = 1$. Otherwise in the case that $x \ne y$, we assume by induction that $\mu(x,z)$ has already been defined for all $z$ in the half-open interval $[x,y)$, and set
$$
\mu(x,y) = -\sum_{x \le z \lt y} \mu(x,z).
$$
From this definition, it is immediate that if
$$x_0 \lt x_1 \lt \cdots \lt x_n$$
is a chain of length $n$ in $P$, where each $x_{i+1}$ [[covering relation|covers]] $x_i$, then for all $i \leq j$,
$$
\mu(x_i,x_j) = \begin{cases}1 & i = j \\ -1 & j - i = 1\\ 0 & j - i \ge 2\end{cases}
$$
It follows that both of the sums
$\sum_{x \le z \le y} \mu(x,z)$ and $\sum_{x \le z \le y} \mu(z,y)$
add to zero unless $x = y$, hence that $\zeta*\mu = \mu*\zeta = \delta$.

=--

We are now ready to state the Möbius inversion formula(s) for posets.

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

[(Rota 1964)](#Rota64):
Let $f$ and $g$ be functions defined over a locally finite poset $P$. Then
$$f(y) = \sum_{x \le y} g(x)
\qquad\text{if and only if}\qquad
g(y) = \sum_{x \le y} f(x)\mu(x,y).$$
Dually, we also have that
$$f(x) = \sum_{y \ge x} g(y)
\qquad\text{if and only if}\qquad
g(x) = \sum_{y \ge x} \mu(x,y)f(y).$$

=--

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

An easy way of proving the Möbius inversion formulas [(Kung et al.)](#KungRotaYan) is to view $\zeta$ and $\mu$ as matrices acting on the column vectors $f$ and $g$ by multiplication. Then the first proposition simply says that $f = g * \zeta$ iff $f * \mu = g$, while the dual formulation says that $f = \zeta * g$ iff $\mu * f = g$.

=--

## Properties of the Möbius function

### Duality

The [[opposite poset]] $P^{op}$ has the same Möbius function as $P$ except with the arguments exchanged:
$$\mu_{P^{op}}(x,y) = \mu_P(y,x)$$

### The product formula

The Möbius function of the [[cartesian product]] $P \times Q$ of two posets $P$ and $Q$ is the product of their Möbius functions:
$$\mu_{P\times Q}((x,y), (x',y')) = \mu_P(x,x') \cdot \mu_Q(y,y')$$
for all $x,x' \in P$ and $y,y' \in Q$.

### The Galois coconnection theorem

Suppose $P$ and $Q$ are related by an [[adjunction]] (i.e., covariant [[Galois connection]]) $f \dashv g : Q \to P$. Then for all $x \in P$, $y \in Q$, the following equation holds:
$$
\sum_{\substack{u \in P\\ f(u) = y}} \mu_P(x,u) = \sum_{\substack{v \in Q \\ g(v) = x}} \mu_Q(v,y)
$$
(This result essentially goes back to [Rota (1964)](#Rota64) but in a slightly different formulation; for the above formulation, see [Greene (1981)](#Greene81), [Aguiar and Ferrer Santos (2000)](#AguiarFS), [Kung et al. (2009)](#KungRotaYan).)

## Examples of Möbius inversion

### Inclusion-exclusion

Let $\mathcal{P}X$ be the lattice of subsets of a finite set $X$. $\mathcal{P}X$ is isomorphic to the cartesian product of $|X|$ many copies of $\mathbf{2} = \{ 0 \lt 1 \}$, and so by the product rule for the Möbius function, we have
$$\mu(I,J) = (-1)^{|J|-|I|}$$
for any pair of subsets $I,J \subseteq X$ such that $I \subseteq J$. The Möbius inversion formula then says that for any functions $f$ and $g$ defined on $\mathcal{P}X$, we have
$$f(J) = \sum_{I \subseteq J} g(I)
\qquad\text{if and only if}\qquad
g(J) = \sum_{I \subseteq J} (-1)^{|J|-|I|} f(I)$$
or dually that
$$f(I) = \sum_{J \supseteq I} g(J)
\qquad\text{if and only if}\qquad
g(I) = \sum_{J \supseteq I} (-1)^{|J|-|I|} f(J)$$
This is called the _inclusion-exclusion_ principle.

As a textbook example of the inclusion-exclusion principle in action, suppose we want to compute $g(I)$ = the number of permutations of $X$ fixing exactly the elements in $I$. Well, it is easy to compute the number of permutations of $X$ fixing _at least_ the elements in $I$,
$$f(I) = \sum_{J \supseteq I} g(J) = (|X|-|I|)!$$
but then we can compute $g$ in terms of $f$ via Möbius inversion. As a special case, when $|X| = n$ and $I = \emptyset$, we recover the classical formula for the number of _derangements_ of $n$ elements:
$$
!n = \sum_{k=0}^n (-1)^k \binom{n}{k} (n-k)! = \sum_{k=0}^n (-1)^k \frac{n!}{k!}
$$
where a derangement is defined as a permutation fixing no elements.

**Exercise:** Prove that $n! = \sum_{k=0}^n (-1)^k \binom{n}{k} (n-k)^n$.

### Number-theoretic Möbius inversion

The classical Möbius inversion formula is recovered by considering the set of positive natural numbers ordered by divisibility.

## Möbius inversion for categories

## Related concepts

* [[Euler characteristic]]
* [[incidence algebra]]
* [[inclusion-exclusion]]
* [[zeta polynomial]]

## References

The classical reference on Möbius inversion for posets is:

* [[Gian-Carlo Rota]], _On the foundations of combinatorial theory I: theory of Möbius functions_ , Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368.
 {#Rota64}

and a more modern reference is:

* Joseph P. S. Kung, [[Gian-Carlo Rota]], Catherine H. Yan. Combinatorics: The Rota Way. Cambridge, 2009.
 {#KungRotaYan}

Möbius inversion for categories is discussed in:

* [[Tom Leinster]], _Notions of Möbius inversion_, Bulletin of the Belgian Mathematical Society 19 (2012) 911-935, [arXiv:1201.0413v3](http://arxiv.org/abs/1201.0413)

* $n$Cafe: [Möbius inversion for categories](http://golem.ph.utexas.edu/category/2011/05/mbius_inversion_for_categories.html)

Other references include:

* Curtis Greene, _The Möbius Function of a Partially Ordered Set_, NATO Advanced Study Institute, Series C (1981), 555-581.
 {#Greene81}

* [[Joachim Kock]], _Incidence Hopf algebras_, [pdf](http://mat.uab.es/~kock/seminars/incidence-algebras.pdf)

* Imma Gálvez-Carrillo, [[Joachim Kock]], Andrew Tonks, _Decomposition spaces, incidence algebras and Möbius inversion_, [arxiv/1404.3202](http://arxiv.org/abs/1404.3202)

* Metropolis, N.; Rota, Gian-Carlo, _Witt vectors and the algebra of necklaces_, Advances in Mathematics __50__ (2): 95–125 (1983) [doi](http://dx.doi.org/10.1016%2F0001-8708%2883%2990035-X)

* [[Marcelo Aguiar]] and Walter Ferrer Santos, _Galois connections for incidence Hopf algebras of partially ordered sets_, Adv. Math. 151 (2000), 71-100.
  {#AguiarFS}

* wikipedia [Möbius function](http://en.wikipedia.org/wiki/M%C3%B6bius_function), [necklace polynomial](http://en.wikipedia.org/wiki/Moreau%27s_necklace-counting_function)

category: combinatorics

[[!redirects Mobius inversion]]
[[!redirects Möbius function]]
[[!redirects Moebius function]]
[[!redirects Moebius inversion]]
[[!redirects necklace polynomial]]


Möbius space

#Contents#
* table of contents
{:toc}

## Idea

The local model space for [[conformal geometry]] regarded as [[parabolic geometry|parabolic]] [[Cartan geometry]].

## Related concepts

[[!include local and global geometry - table]]

## References

* R. Sulanke, _Differential geometry of the Möbius space I_ ([pdf](http://www-irm.mathematik.hu-berlin.de/~sulanke/diffgeo/moebius/mdg.pdf))

* [[Felipe Leitner]], part 1, section 6 of _Applications of Cartan and Tractor Calculus to Conformal and CR-Geometry_, 2007 ([pdf](http://elib.uni-stuttgart.de/opus/volltexte/2009/3922/pdf/HabilKopf.pdf))

* [[Pierre Anglès]], section 2.2.1.2 of _Conformal Groups in Geometry and Spin Structures_, Progress in Mathematical Physics 2008

[[!redirects Möbius spaces]]

[[!redirects Möbius sphere]]
[[!redirects Möbius spheres]]


Möbius strip

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

#Contents#
* table of contents
{:toc}

## Idea

  graphics grabbed from
  Lawson 03

The _Möbius strip_ is the [[topological space]] obtained from the "open strip", hence the [[square]], by gluing two of its opposite sites but after applying half of a full rotation to one of them.

## Realizations

### As a quotient of the open strip

As a [[topological space]], the _Möbius strip_ is the [[quotient topological space]] obtained from the [[square]] $[0,1]^2$ by the [[equivalence relation]] which identifies two of the opposite sides, but  with opposite [[orientation]]

  $$
    ((x_1,y_1) \sim (x_2,y_2))
     \;\;\Leftrightarrow \;\;
     \left(
       (x_1,y_1) = (x_2, y_2)
        \;\text{or}\;
       \left(
         x_1 = (1-x_2) \in \{0,1\}
         \;\text{and}\;
         y_1 = (1-y_2)
       \right)
    \right)
  $$

### As the tautological line bundle over $\mathbb{R}P^1$
 {#AsTautologicalLineBundle}

Regarded a [[vector bundle]] over the circle, the Möbius strip is the [[tautological line bundle]] over the 1-dimensional [[real projective space]] $\mathbb{P}P^1$.

\begin{imagefromfile}
  "file_name": "TautologicalLineBundleOverRP1.jpg",
  "width": 700
\end{imagefromfile}

For more discussion of the [[topological vector bundle]] structure see [this example](topological+vector+bundle#MoebiusStrip) and [this prop.](tautological+line+bundle#WellDefinedTautologicalTopologicalLineBundle)

## Properties

Regarded as a [[manifold]], the Möbius strip is among the simplest examples of a [[manifold]] that is not [[orientable]].

Regarded as a [[real vector bundle]] over the [[circle]], the Möbius strip is among the simplest examples of a non-trivial [[vector bundle]].

## Related concepts

* [[cylinder]]

* [[Klein bottle]]

## References

Named after _[[August Möbius]]_.

* {#Lawson03} Terry Lawson, _Topology: A Geometric Approach_, Oxford University Press (2003) ([pdf](http://users.metu.edu.tr/serge/courses/422-2014/supplementary/TGeometric.pdf))

[[!redirects Möbius strips]]

[[!redirects Moebius strip]]
[[!redirects Moebius strips]]

[[!redirects Möbius band]]
[[!redirects Möbius bands]]


Møller operator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[scattering theory]] the _Møller operator_ intertwines the [[observables]] of the [[free field theory|free]] [[theory (physics)|theory]] with those of the [[interaction|interacting]] theory.

## Details

### In quantum mechanics

The [[quantum states]] $\vert \psi(t)\rangle_I$ in the [[interaction picture]] of [[quantum mechanics]] are by definition ([this equation](Dirac+interaction+picture#eq:StateInTheInteractionPicture)) related to the asymptotic free states $\vert\psi\rangle$ by

$$
  \vert \psi(t)\rangle_I
  \;=\;
  \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right)
  \exp\left({\tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right)
  \vert \psi \rangle
$$

and conversely

$$
  \vert \psi \rangle
  \;=\;
  \underbrace{
    \exp\left({\tfrac{- t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right)
    \exp\left({\tfrac{+ t}{i \hbar} H_{free}}\right)
  }
  \vert \psi(t)\rangle_I
$$

the suitable [[limit of a sequence|limit]] for $t \to \mp \infty$ of the operator under the brace is called the _Møller operator_

$$
  \Omega{\pm}
  \;\coloneqq\;
  \underset{t \to \mp \infty}{\lim}
    \exp\left({\tfrac{- t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right)
    \exp\left({\tfrac{+ t}{i \hbar} H_{free}}\right)
$$

(e.g. [BEM 01](#BEM01))

### In quantum field theory

In [[perturbative quantum field theory]] the maps that intertwine the [[Wick algebra]] of [[quantum observables]] of the [[free field theory]] with the [[interacting field algebra]] are, on [[regular polynomial observables]]. the [[derivatives]] of the [[Bogoliubov formula]] of the given [[S-matrix]] $\mathcal{S}$ for the given [[interaction]] $S_{int}$ with respect to [[source fields]]:

$$
  \mathcal{R}(-)
    \;\coloneqq\;
  \mathcal{S}(S_{int})^{-1}
    \star_H
  (\mathcal{S}(S_{int}) \star_F (-))
$$

(Here $\star_H$ denotes the [[star product]] induced by the [[Wightman propagator]], hence the [[Wick algebra]]-product, while $\star_F$ denotes the [[star product]] induced by the [[Feynman propagator]], hence the [[time-ordered product]]. The [[inverse]] $(-)^{-1}$ is taken with respect to $\star_H$.)

This $\mathcal{R}(-)$ is referred to as the _quantum Møller operator_ in ([Hawkins-Rejzner 16, below def. 5.1](#HawkinsRejzner16)). (But notice that in many previous articles in [[perturbative AQFT]], by the same authors and by others, the very same operator is referred to just as the "intertwining operator", or similar.)

## Related concepts

* [[quantum master equation]]

## References

Discussion in [[quantum mechanics]]:

* {#BEM01} A. Baute, I. Egusquiza, J. Muga, _Møller operators and Lippmann-Schwinger equations for step-like potentials_ ([arXiv:quant-ph/0104043](https://arxiv.org/abs/quant-ph/0104043))

Discussion in [[relativistic field theory|relativistic]] [[perturbative quantum field theory]] in the rigorous formulation of [[causal perturbation theory]]/[[perturbative AQFT]]:

* {#HawkinsRejzner16} [[Eli Hawkins]], [[Kasia Rejzner]], _The Star Product in Interacting Quantum Field Theory_ ([arXiv:1612.09157](https://arxiv.org/abs/1612.09157))

[[!redirects Møller operators]]

[[!redirects quantum Møller operator]]
[[!redirects quantum Møller operators]]

[[!redirects Moller operator]]
[[!redirects Moller operators]]

[[!redirects quantum Moller operator]]
[[!redirects quantum Moller operators]]


Münchhausen trilemma

[[!redirects Munchhaüsen trilemma]]

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

\tableofcontents

## Idea

The **Münchhausen trilemma** or the **Agrippan trilemma** is a thought experiment in [[epistemology]] which states that it is theoretically impossible to prove any truth, because one has to resort to one of three unsatisfactory arguments

* Circular reasoning, in which the proof of a statement relies on said statement itself

* Infinite regress, in which one needs an [[infinite sequence]] of statements, each of which proves the previous statement

* Foundationalism, in which one selects some statements as unjustified axioms or rules.

## Relevance to the foundations of mathematics

The [[foundations of mathematics]] by definition uses foundationalism to resolve the Münchhausen trilemma in [[mathematics]]. However, different communities of mathematicians disagree with which unjustified axioms and rules to use for their foundations. For example, classical mathematicians use [[classical logic]], while constructive mathematicians use [[intuitionistic logic]] and reject [[excluded middle]].

Subfields of mathematics also use foundationalism by way of [[synthetic mathematics]], in which unjustified axioms and rules are used as a basic foundation for the entire subfield. Examples include [[synthetic topology]], [[synthetic geometry]], [[synthetic differential geometry]], and [[synthetic (infinity,1)-category theory]].

## See also

* [[epistemology]]
* [[philosophy of mathematics]]

## External links

* Wikipedia, *[Münchhausen trilemma](https://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma)*

[[!redirects Munchhausen trilemma]]
[[!redirects Münchhausen trilemma]]
[[!redirects Agrippan trilemma]]


n-angulated category

__$n$-angulated categories__ are a generalization to integers $n\geq 3$ of a [[triangulated category]], which is obtained for $n = 3$. They are introduced in

* C. Geiss, [[B. Keller]], S. Oppermann, _$n$-angulated categories_, [arxiv/1006.4592](http://arxiv.org/abs/1006.4592), to appear in J. Reine Angew. Math.

Its abstract reads:

> We define $n$-angulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller's parametrization of pre-triangulations extends to pre-$n$-angulations. We obtain a large class of examples of $n$-angulated categories by considering $(n-2)$-cluster tilting subcategories of triangulated categories which are stable under the $(n-2)$nd power of the suspension functor. As an application, we show how $n$-angulated Calabi-Yau categories yield triangulated Calabi-Yau categories of higher Calabi-Yau dimension. Finally, we sketch a link to algebraic geometry and string theory.

Other works are

* P.A. Bergh, M. Thaule, _The axioms for $n$-angulated categories_, [arXiv:1112.2533](http://arxiv.org/abs/1112.2533); _The Grothendieck group of an $n$-angulated category_, [arxiv/1205.5697](http://arxiv.org/abs/1205.5697)

* Gustavo Jasso, _$n$-abelian and $n$-exact categories_, [arxiv/1405.7805](http://arxiv.org/abs/1405.7805)

* Zengqiang Lin, _$n$-angulated quotient categories induced by mutation pairs_, [arxiv/1409.2716](http://arxiv.org/abs/1409.2716)


n-ary group

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## Idea

A generalization of [[associative quasigroup]] to [[n-ary operation]]s.

## Definition

Given a natural number $n$, an $n$-ary group is a set $S$ that both an [[n-ary semigroup]] and an [[n-ary quasigroup]].

\begin{remark}
A binary group is *not* a [[group]]. Binary groups are [[associative quasigroups]], as they can be empty.
\end{remark}

## Properties

\begin{definition}
An *n-ary identity element* is an element $e$ such that any string of $n$ elements consisting of all $e$'s, apart from one place, is mapped to the element at that place.
\end{definition}

\begin{theorem}
Every n-ary group with an n-ary identity element is a group, with the n-ary group operation being simply repeated application of a group's binary operation.
\end{theorem}

\begin{remark}
The above theorem is why most authors typically do not require the existence of $n$-ary identity elements in the definition of $n$-ary groups, and why binary groups are just associative quasigroups.
\end{remark}

## See also

* [[associative quasigroup]]

* [[n-ary magma]]

* [[n-ary semigroup]]

* [[n-ary quasigroup]]

* [[group]]

## References

* W. Dörnte, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Mathematische Zeitschrift, vol. 29 (1928), pp. 1-19.

* E. L. Post, Polyadic groups, Transactions of the American Mathematical Society 48 (1940), 208–350.

* W. A. Dudek, "On some old and new problems in n-ary groups", Quasigroups and Related Systems, 8 (2001), 15–36.

* Wiesław A. Dudek, Remarks to Głazek's results on n-ary groups, Discussiones Mathematicae. General Algebra and Applications 27 (2007), 199–233.

* Wiesław A. Dudek and Kazimierz Głazek, Around the Hosszú-Gluskin theorem for n-ary groups, Discrete Mathematics 308 (2008), 486–4876.

* Wikipedia, [n-ary group](https://en.wikipedia.org/wiki/N-ary_group)


n-ary quasigroup

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## Idea

A generalization of [[quasigroup]] to [[n-ary operation]]s.

## Definition

Given a natural number $n$, an $n$-ary quasigroup is a set $S$ with an $n$-ary operation $f:S^n \to S$ and $n$ $n$-ary operations $f_i^{-1}:S^n \to S$ for natural numbers $i \lt n$, such that, for $0 \lt m \lt n - 1$,

$$f(f_0^{-1}(a_1, \ldots a_{n - 1}, b), \ldots a_{n - 1}) = b$$

$$f(a_0, \ldots a_{m - 1}, f_m^{-1}(a_0, \ldots a_{m - 1}, a_{m + 1}, \ldots a_{n - 1}, b), a_{m + 1}, \ldots a_{n - 1}) = b$$

$$f(a_0, \ldots a_{n - 2}, f_{n - 1}^{-1}(a_0, \ldots a_{n - 2}, b)) = b$$

## See also

* [[quasigroup]]

* [[n-ary magma]]

* [[n-ary semigroup]]

* [[n-ary group]]

## References

* Wikipedia, [n-ary group](https://en.wikipedia.org/wiki/N-ary_group)


n-ary semigroup

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## Idea

A generalization of [[semigroup]] to [[n-ary operation]]s.

## Definition

Given a natural number $n$, an $n$-ary semigroup is a set $S$ with an $n$-ary operation $f:S^n \to S$, such that every one of the $n$ possible bracketings of a list consisting of $2n - 1$ distinct elements with any $n$ consecutive elements bracketed are equal to each other.

## See also

* [[semigroup]]

* [[n-ary magma]]

* [[n-ary quasigroup]]

* [[n-ary group]]

## References

* Wikipedia, [n-ary group](https://en.wikipedia.org/wiki/N-ary_group)


n-category

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## Idea
 {#Idea}

In broad generality, under *$n$-categories* one understands [[higher categories]] with non-trivial [[higher morphisms]] up to dimension $n \in \mathbb{N} \sqcup \{\infty\}$. For $n = 1$ these are the ordinary [[categories]] (or *[[1-categories]]*, for emphasis) of [[category theory]], for $n = 2$ they are the [[2-categories]] (traditionally: "[[bicategories]]") and/or [[double categories]] of [[2-category theory]], for $n = 3$ one also speaks of [[tricategories]], etc. The case $n = \infty$ would broadly refer to the most general notion of [[infinity-categories|$\infty$-categories]] where there is no constraint on the dimensionality of [[higher morphisms]].

Especially as $n$ increases, there is a plethora of different definitions of $n$-categories, some differing in generality others different-looking but secretly equivalent.
A (woefully incomplete) list is given [below](#ListOfDefinitions), with pointers to dedicated entries. Part of the subject of [[higher category theory]] is to understand, organize, systematize and, last not least, apply these definitions. (It is the "$n$" in "$n$-category" that gives the *[[HomePage|nLab]]* its name.)

Often it is important to consider the more fine-grained notion of *[[(n,r)-categories|$(k,n)$-categories]]*, where now $n$ refers only to the maximal dimension of non-[[invertible morphisms|invertible]] [[higher morphisms]] (non-[[equivalences]]) while the [[invertible morphisms|invertible]] [[higher morphisms]] are allowed to range in dimension up to $k \in \mathbb{N} \sqcup \{\infty\}$.

Particularly important here is the limiting case ($k = \infty$) of [[(infinity,n)-categories|$(\infty,n)$-categories]]:
Since [[(infinity,0)-categories|$(\infty,0)$-categories]] aka *[[infinity-groupoids|$\infty$-groupoids]]* are the bare [[homotopy types]] of "spaces" as considered in [[homotopy theory]], [[(infinity,n)-categories|$(\infty,n)$-categories]] are the natural notion of $n$-categories [[internalization|internal]] to [[homotopy theory]] (also: internal to [[homotopy type theory]], up some technical issues) where the notoriously intricate [[coherence laws]] of [[higher category theory]] typically have a particularly natural (if maybe non-explicit) formulation. Therefore many authors these days use "$\infty$-category" as a pseudonym for *[[(infinity,1)-category|$(\infty,1)$-category]]* and some use "$n$-category" as a tacit pseudonym for [[(infinity,n)-categories|$(\infty,n)$-categories]].

In this [[homotopy theory|homotopy theoretic]]-perspective on $n$-categories they are naturally understood as (models for)
*[[directed spaces]]* (with an $n$-fold notion of "direction") and in principle the topics of *[[directed homotopy theory]]* (and *[[directed homotopy type theory]]*) ought to be thought of as essentially synonymous to [[(infinity,n)-categories|$(\infty,n)$-category]] theory, though a genuine perspective of [[directed homotopy theory|directed homotopy]]/[[directed homotopy type theory]] is still in its infancy.

Conversely, for concrete computations it is at times convenient to have *less* flexible and more rigid definitions or at least presentations/models of $n$-categories. In as far as these are still suitably equivalent to the general notion one speaks of *[[semi-strict n-categories]]* and much of [[higher category theory]] revolves around identifying and comparing such *rigidifications*. On the other hand, perfectly *[[strict n-categories|strict $n$-categories]]* are typically too rigid to play more than a niche role in [[higher category theory]].

Finally, it is sometimes useful to subsume degenerate cases in the general pattern of $n$-categories, such as [[0-categories]] ([[sets]]), [[(0,1)-categories]] ([[posets]]) and even *[[(-1)-categories]]* ([[truth values]]) and *[[(-2)-categories]]* (the [[terminal category|singleton category]]).

## Categories of $n$-categories

Just as the collection of all ([[small category|small]]) sets is the prototypical example of a category, so the collection of all small $n$-categories is the prototypical example of an $(n+1)$-category.

Actually, if you define things cleverly, then you can get an $(n+1)$-category of *all* $n$-categories.  If one assumes the [[Grothendieck universe|Axiom of Universes]], then there is a sequence of Grothendieck universes
$$U_0 \subset U_1 \subset U_2 \subset \cdots$$
and we can say a set is **$U_i$-small** if it is an element of $U_i$.  This allows us to make the following definitions:

* $\Set$ is the category of all $U_0$-small sets;
* $\Cat$ is the 2-category of all $U_1$-small categories;
* $2\Cat$ is the 3-category of all $U_2$-small 2-categories;
* etc.

This is a convenient way to settle size questions once and for all for finite $n$, but it doesn\'t really work for $\infty$-categories.

For more, see the discussion at [sci.logic](http://groups.google.com/group/sci.logic/browse_thread/thread/6891c85c2d9b2caf).

## Definitions {#defn}

Here is a list of (some of) the proposed definitions of (weak) $n$-category, with references, and also a list of (some of) the comparisons that have been done.

### List of definitions
 {#ListOfDefinitions}

Many of these definitions are actually "truncations" of definitions of (weak) [[∞-categories]] (aka [[∞-categories]]).  Some others are truncations of a definition of [[(∞,n)-categories]].  A nice overview of (many) of these can be found in Tom Leinster's paper "A survey of definitions of $n$-category."

* Classical explicit definitions of "fully weak" $n$-category exist for $n\le 4$.  Weak 0-categories are [[sets]], weak 1-categories are simply [[categories]] (due to [[Sammy Eilenberg|Eilenberg]] and [[Saunders Mac Lane|Mac Lane]]), weak 2-categories are [[bicategories]] (due to [[Jean Benabou|Benabou]]), weak 3-categories are [[tricategories]] (due to [[Robert Gordon|Gordon]]--[[John Power|Power]]--[[Ross Street|Street]]), and weak 4-categories are [[tetracategories]] (due to [[Todd Trimble]]).  Going on in this way is generally admitted to be infeasible beyond $n=4$.

* [[Ross Street|Street's]] definition: an $n$-category is a [[simplicial set]] satisfying certain horn-filling conditions.  See [[weak complicial set]] and [[simplicial model for weak ∞-categories]].  This is a truncation of a definition of $\omega$-category.  It can be specialized to yield a notion of $(\infty,n)$-category.  The resulting notion of $(\infty,1)$-category is a [[quasicategory]], and the resulting notion of $\infty$-groupoid is a [[Kan complex]].

* [[John Baez|Baez]]--[[James Dolan|Dolan]] definition: an $n$-category is an [[opetopic set]] having enough $n$-universal fillers.  Alternate definitions of opetopes (aka multitopes) have been given by [[Claudio Hermida|Hermida]]--[[Michael Makkai|Makkai]]--[[John Power|Power]] and [[Tom Leinster|Leinster]]; a comparison is due to [[Eugenia Cheng]], see [these](http://arxiv.org/abs/math.CT/0304277) [three](http://arxiv.org/abs/math.CT/0304279) [papers](http://arxiv.org/abs/math.CT/0304284).  Makkai's version can do $\omega$.

* [[Jacques Penon|Penon]]'s definition: (someone describe this please!)  Penon's original definition turned out to be too strict (see [Batanin](http://web.science.mq.edu.au/~mbatanin/Penon1.ps) and [Cheng--Makkai](http://arxiv.org/abs/0907.3961)) because it used [[reflexive globular set|reflexive globular sets]], but a modification of it using [[globular sets]] is still a contender.

* [[Michael Batanin|Batanin]]--[[Tom Leinster|Leinster]] definition: an $n$-category is an $n$-[[globular set]] with an action of a suitable [[globular operad]].  This is a truncation of a definition of $\omega$-category; see [[Batanin ∞-category]].

* [[Todd Trimble|Trimble]]-style definition: An $n$-category is a category weakly enriched over $(n-1)$-categories, where the weakness is parametrized by an [[operad]].  This definition is inductive and thus cannot do $\omega$ in an obvious way, but it has been accomplished using terminal coalgebras; see [[Trimble n-category]].  Alternately, by starting with enrichment in [[spaces]] or [[simplicial sets]], one can obtain directly a notion of [[(∞,n)-category]].  The resulting notion of [[(∞,1)-category]] is an $A_\infty$-[[A∞-category|category]].

* [[Tamsamani]]--[[Carlos Simpson|Simpson]] definition: An $n$-category is a simplicial object in $(n-1)$-categories satisfying object-discreteness and the [[Segal condition]].  This definition is inductive (it is a different way of formalizing "iterated weak enrichment") and thus cannot do $\omega$ in an obvious way.  It does have a natural extension to $(\infty,n)$-categories, and the resulting notion of [[(∞,1)-category]] reduces to a [[Segal category]].  The iterated version of this is that of [[Segal n-category]]. This notion of "weak enrichment" in a [[cartesian model category]] is studied carefully in Simpson's book [[Homotopy Theory of Higher Categories]].

* [[Ieke Moerdijk|Moerdijk]] and [[Ittay Weiss|Weiss]]'s definition uses yet another way of formalizing "iterated weak enrichment," using [[dendroidal sets]] and [[quasi-operad]]s.

* [[Andre Joyal|Joyal]]'s definition: An $n$-category is an $n$-[[cellular set]] satisfying horn-filling conditions.  This definition can do $\omega$ by using $\omega$-cellular sets instead of $n$-cellular sets, and it can do $(\infty,n)$ by requiring different horn-filling conditions on $n$-cellular sets.  The notion of [[(∞,1)-category]] one obtains in this way is a [[quasicategory]], and the resulting notion of $\infty$-groupoid is a [[Kan complex]].  For $n\gt 1$, however, the obvious "horn-filling conditions" are not quite right; [[Dimitri Ara]] has shown how to correct them (albeit not very explicitly), obtaining a definition he calls an [[n-quasicategory]], which form a model category Quillen equivalent to Rezk's definition (below).

* [[Clark Barwick|Barwick]]'s definition (popularized by [[Jacob Lurie|Lurie]] in solving the Baez--Dolan [[cobordism hypothesis]]): an $(\infty,n)$-category is an $n$-fold [[simplicial object|simplicial]] [[topological space]] satisfying completeness and the [[Segal condition]].  See [[n-fold complete Segal space]].  An $n$-category is again defined as an $(\infty,n)$-category in which all $k$-cells are essentially unique for $k\gt n$.  It is not clear whether this definition can do $\omega$.  An $(\infty,1)$-category with this definition is also the same as a [[complete Segal space]].

* [[Charles Rezk|Rezk]]'s definition: An $(\infty,n)$-category is a *simplicial* $n$-cellular set satisfying [[fibrancy]], completeness, and the [[Segal condition]].  An $n$-category can then be defined as an $(\infty,n)$-category in which all $k$-cells are essentially unique for $k\gt n$.  This definition can potentially do $\omega$, although it seems not to have been written down yet.  An $(\infty,1)$-category with this definition is the same as a [[complete Segal space]]. See [[Theta space]].

* [[Georges Maltsiniotis|G. Maltsiniotis]] has apparently extracted a definition of $\infty$-groupoid from [[Pursuing Stacks]] and generalized it to a definition of $\infty$-category; see [this](http://people.math.jussieu.fr/~maltsin/ps/infgrart.pdf) and [this](http://people.math.jussieu.fr/~maltsin/ps/infctart.pdf).

* [[Simona Paoli]]: [[weakly globular n-fold categories]]

### Comparisons

* All definitions produce the correct well-known notion of [[1-category]], up to minor inessential details.

* Since all the common definitions of [[(∞,1)-category]] are known to be equivalent (give references!), the definitions of Street, Trimble, Tamsamani--Simpson, Joyal, Barwick, and Rezk can be said to agree for $(\infty,1)$-categories.

* Julie Bergner has [shown](http://arxiv.org/abs/0710.2254) that all the notions of $\infty$-groupoid obtained from the common notions of $(\infty,1)$-category are equivalent, so the definitions of Street, Trimble, Tamsamani--Simpson, Joyal, Barwick, and Rezk can also be said to agree for $(\infty,0)$-categories.

* It is known that the notions of $(n,0)$-category obtained from categories, bicategories, and tricategories model all [[homotopy n-types]] for $n\le 3$.  Thus, in these cases, the classical definitions can be said to agree with those listed in the previous example.

* In Tom Leinster's paper, proofs are sketched showing that the notion of [[2-category]] obtained in each case looks somewhat like the notion of [[bicategory]].

* Nick Gurski has shown in "Nerves of bicategories as stratified simplicial sets" that Street's definition is correct for $n=2$ (that is, it agrees with bicategories).

* Eugenia Cheng has [shown](http://arxiv.org/abs/math/0304285) that the opetopic definition is correct for $n=2$ (that is, it agrees with bicategories).

* Eugenia Cheng has more recently also [shown](http://arxiv.org/abs/0809.2070) that from any sequence of operads used for iterated enrichment in a Trimble-style definition, one can construct a Batanin--Leinster-style globular operad whose algebras are the $n$-categories obtained in the Trimblean inductive manner.  Not all globular operads can be obtained in this way, however, since those that arise have strict interchange.

## Related concepts

* [[0-category]], [[(0,1)-category]], [[Set]]

* [[category]]

  * [[category object in an (∞,1)-category]]

* [[2-category]], [[bicategory]]

* [[3-category]], [[tricategory]]

* [[4-category]], [[tetracategory]]

* **$n$-category**

  * [[n-category object in an (∞,1)-category]]

* [[(n,1)-category]]

* [[(∞,1)-category]]

* [[(∞,2)-category]]

* [[(∞,n)-category]]

* [[(n,r)-category]]

## References

See also the references at *[[higher category theory]]* and especially at *[[(infinity,n)-category]]*.

Exposition of the original [[algebraic definition of higher categories|algebraic notions]] of $n$-categories:

* [[John Baez]], *[The Tale of n-Categories](http://math.ucr.edu/home/baez/week73.html#tale)*

* [[John Baez]], *An Introduction to $n$-Categories*,  in *7th Conference on Category Theory and Computer Science*, Lecture Notes in Computer Science **1290**. Springer (1997)

* [[Eugenia Cheng]], [[Aaron Lauda]], *Higher-Dimensional Categories: An Illustrated Guidebook* [[pdf](http://eugeniacheng.com/wp-content/uploads/2017/02/cheng-lauda-guidebook.pdf)]

* [[Tom Leinster]], _A Survey of Definitions of $n$-Category_ [[arXiv:math.CT/0107188](http://arxiv.org/abs/math.CT/0107188)]

* [[Tom Leinster]], _Higher Operads, Higher Categories_ [[arXiv:math/0305049](http://arxiv.org/abs/math/0305049)]

[[!redirects $n$-category]]
[[!redirects n-categories]]
[[!redirects weak n-category]]
[[!redirects weak n-categories]]


n-Category Café

\tableofcontents

## Idea

The **n-Category Café** is a blog about [[category theory]]. It was created in 2006 by [[John Baez]], [[David Corfield]], and [[Urs Schreiber]].

## References

* [website](https://golem.ph.utexas.edu/category/)

[[!redirects nCafé]]
[[!redirects nCafe]]

[[!redirects n-Category Café]]
[[!redirects n-Category Cafe]]

[[!redirects n-category café]]
[[!redirects n-category cafe]]


n-category object in an (infinity,1)-category

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## Idea

The notion of [[n-category]] formulated [[internalization|internal]] to an ambient [[(∞,1)-category]] is supposed to externally be a kind of [[(∞,n)-category]].

For $n =1$ this is a [[category object in an (∞,1)-category]].

A model along these lines for general with the [[simplex category]] generalized to the [[Theta category|Theta_n category]] and the ambient $(\infty,1)$-category being [[∞Grpd]] are _[[Theta_n spaces]]_.

Another model is that of [[n-fold complete Segal space]] objects.

## Definition

For the moment see at _[category object in an (infinity,1)-category -- n-Categories](category+object+in+an+%28infinity%2C1%29-category#InternalnCategories)_

## Related concepts

* [[n-fold category]]

* [[(∞,n)-sheaf]]

[[!redirects n-category object in an (∞,1)-category]]
[[!redirects internal (infinity,n)-category]]

[[!redirects internal (∞,n)-category]]
[[!redirects internal (∞,n)-categories]]


N-complex

# $N$-complexes
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## Idea

While (co)[[chain complexes]] involve $d^2 = 0$, _(co)chain $N$-complexes_ involve a differential $d$ satisfying $d^N = 0$, where $N$ is a [[positive integer]] bigger than $2$.

## References

The $N=3$ case has been studied in

* W. Mayer, _A new homology theory, I, II_, Annals of Math. __43__ (1942), 370–380, 594–605.

The modern attention to the subject started in

* [[Mikhail Kapranov|M. M. Kapranov]], _On the q-analog of homological algebra_, [q-alg/9611005](http://arxiv.org/abs/q-alg/9611005)

* Michel Dubois-Violette, _Lectures on differentials, generalized differentials and on some examples related to theoretical physics_, [math.QA/0005256](http://arxiv.org/abs/math/0005256)

* Michel Dubois-Violette, _Tensor product of N-complexes and
generalization of graded differential algebras_, Bulg. J. Phys. __36__ (2009) 227–236 [pdf](http://www.bjp-bg.com/papers/bjp2009_3_227-236.pdf)

* Michel Dubois-Violette, Marc Henneaux, _Tensor fields of mixed Young symmetry type and N-complexes_, [math.QA/0110088](http://arxiv.org/abs/math/0110088) [doi](http://dx.doi.org/10.1007/s002200200610)

In the following article the $N$-homological versions of [[Tor]] and [[Ext]] functors are expressed in terms of classical Tor and Ext (of course, with shifted indices). As a consequence of this type of result, the interest in the $N$-homological algebra somewhat diminished.

* Christian Kassel, Marc Wambst, _Algèbre homologique des N-complexes et homologie de Hochschild aux racines de l'unité_, Publ. RIMS, Kyoto Univ. __34__ (1998), p. 91-114 [pdf](http://www.euro-math-soc.eu/jemsaccess/journals/show_pdf.php?issn=0034-5318&vol=34&iss=2&rank=1) [q-alg/9705001](http://arxiv.org/abs/q-alg/9705001)

category: algebra

[[!redirects N-complex]]
[[!redirects N-complexes]]


n-congruence

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## Related concepts

* [[congruence]]

* [[2-congruence]]

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