machine learning

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Machine learning_ is a branch of [[computer science]] which devises algorithms to learn from data so as to perform tasks without being explicitly programmed to do so. Notable approaches include [[neural networks]], support vector machines and Bayesian networks.

## Related concepts

* [[neural network]]

* [[kernel method]]

* [[topological data analysis]]

* [[topological machine learning]]

* [[statistical learning theory]]

* [[singular learning theory]]

* [[information geometry]]

## References

### Classical machine learning

Textbook accounts:

* [[Sumio Watanabe]], _Algebraic geometry and statistical learning theory_, CRC Press (2009) [[doi:10.1017/CBO9780511800474]( https://doi.org/10.1017/CBO9780511800474)]

* [[Sumio Watanabe]], _Mathematical theory of Bayesian statistics_, Cambridge University Press (2018) [[ISBN:9780367734817](https://www.routledge.com/Mathematical-Theory-of-Bayesian-Statistics/Watanabe/p/book/9780367734817), [pdf](http://196.189.45.87/bitstream/123456789/36917/1/Sumio%20Watanabe_2018.pdf)]

* [[Shai Shalev-Shwartz]], [[Shai Ben-David]], _Understanding machine learning: from theory to algorithms_, Cambridge University Press (2014) [[doi:10.1017/CBO9781107298019](https://doi.org/10.1017/CBO9781107298019)]

* Ian Goodfellow, Y. Bengio, A. Courville, _Deep learning_, MIT Press (2016) [[web](https://www.deeplearningbook.org/), [pdf](https://github.com/janishar/mit-deep-learning-book-pdf), [ISBN:9780262035613](https://mitpress.mit.edu/9780262035613/)]

See also:

* Wikipedia, *[Machine learning](https://en.wikipedia.org/wiki/Machine_learning)*
* Juergen Schmidhuber (2015), Scholarpedia, 10(11):32832 [Deep learning](http://www.scholarpedia.org/article/Deep_Learning) (with extensive historical bibliography)

Expository review:

* [[Stephen Wolfram]], *[What Is ChatGPT Doing… and Why Does It Work?](https://writings.stephenwolfram.com/2023/02/what-is-chatgpt-doing-and-why-does-it-work/)* (Feb 2023)

With regards to [[kernel methods]]:

* Thomas Hofmann, [[Bernhard Schölkopf]], Alexander J. Smola, *Kernel methods in machine learning*, Annals of Statistics 2008, Vol. 36, No. 3, 1171-1220 ([arXiv:math/0701907](https://arxiv.org/abs/math/0701907))

* [[Julien Mairal]], [[Jean-Philippe Vert]], *Machine Learning with Kernel Methods*, 2017 ([pdf](http://members.cbio.mines-paristech.fr/~jvert/svn/kernelcourse/slides/master2017/master2017.pdf), [[MarailVertKernelMethods.pdf:file]])

Proposal for applications of [[category theory]] to machine learning:

* Dan Shiebler, [[Bruno Gavranović]], Paul Wilson, _Category Theory in Machine Learning_ ([arXiv:2106.07032](https://arxiv.org/abs/2106.07032))
* [[Yiannis Vlassopoulos]], _Modelling language semantics and probability densities of text continuations_, QNLP 2022 [youtube](https://www.youtube.com/watch?v=c3IOe3JPfq8)

The following framework claims to relate to [[homotopy type theory]]

* Ben Goertzel, _Reflective metagraph [[rewriting system|rewriting]] as a foundation for an AGI "Language of Thought"_, [arXiv:2112.08272](https://arxiv.org/abs/2112.08272)

On a definition of artificial *general* intelligence

* S. Legg, M. Hutter, _Universal intelligence: a definition of machine intelligence_, Minds & Machines 17, 391--444 (2007) [doi](https://doi.org/10.1007/s11023-007-9079-x)

* M. Hutter, _Universal artificial intelligence: sequential decisions based on algorithmic
probability_, Springer 2005; book presentation [pdf](http://www.hutter1.net/ai/suaibook.pdf)

* Shane Legg, _Machine super intelligence_, PhD thesis, 2008 [pdf](https://sonar.ch/documents/317954/files/2008INFO002.pdf)

### Quantum machine learning
 {#ReferencesQuantumMachineLearning}

In view of [[quantum computation]] (ie. quantum machine learning):

* Diego Ristè, Marcus P. da Silva, Colm A. Ryan, Andrew W. Cross, Antonio D. Córcoles, John A. Smolin, Jay M. Gambetta, Jerry M. Chow & Blake R. Johnson,

  *Demonstration of quantum advantage in machine learning*, npj Quantum Information volume 3, Article number: 16 (2017) ([doi:10.1038/s41534-017-0017-3](https://doi.org/10.1038/s41534-017-0017-3))

* [[Jacob Biamonte]], Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, [[Seth Lloyd]], *Quantum Machine Learning*, Nature **549**  (2017) 195–202 ([doi:10.1038/nature23474](https://doi.org/10.1038/nature23474))

  EurekaAlert, *Quantum Machine Learning* [14-Sep-2017](https://www.eurekalert.org/pub_releases/2017-09/iiop-qml091417.php)

* Iris Cong, Soonwon Choi, Mikhail D. Lukin, *Quantum convolutional neural networks*, Nature Physics volume 15, pages 1273–1278 (2019)  ([doi:10.1038/s41567-019-0648-8](https://doi.org/10.1038/s41567-019-0648-8))

  > (on [[quantum neural networks]])

* Yunchao Liu, Srinivasan Arunachalam, Kristan Temme, *A rigorous and robust quantum speed-up in supervised machine learning* ([arXiv:2010.02174](https://arxiv.org/abs/2010.02174))

* [[Melanie Swan]], [[Renato P dos Santos]], [[Frank Witte]], Between Science and Economics, Volume 2: *Quantum Computing Physics, Blockchains, and Deep Learning Smart Networks*, World Scientific 2020 ([doi:10.1142/q0243](https://doi.org/10.1142/q0243))

* Stefano Mangini, Francesco Tacchino, Dario Gerace, Daniele Bajoni, Chiara Macchiavello, *Quantum computing models for artificial neural networks*, EPL (Europhysics Letters) 134(1), 10002 (2021) ([arXiv:2102.03879](https://arxiv.org/abs/2102.03879))

review:

* Kamila Zaman, [[Alberto Marchisio]], Muhammad Abdullah Hanif, [[Muhammad Shafique]], *A Survey on Quantum Machine Learning: Current Trends, Challenges, Opportunities, and the Road Ahead* [[arXiv:2310.10315](https://arxiv.org/abs/2310.10315)]

and with emphasis on [classically controlled](quantum+computation#ClassicalControlQuantumData) [[NISQ]]-computes:

* [[John Preskill]], Section 6.5 of: *Quantum Computing in the NISQ era and beyond*, Quantum  2018-08-06, volume 2, page 79 ([arXiv:1801.00862](https://arxiv.org/abs/1801.00862), [doi:10.22331/q-2018-08-06-79](https://doi.org/10.22331/q-2018-08-06-79))

* Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, Keisuke Fujii, *Quantum Circuit Learning*, Phys. Rev. A 98, 032309 (2018) ([arXiv:1803.00745](https://arxiv.org/abs/1803.00745))

* {#BenedettiLloydSackFiorentini19} Marcello Benedetti, Erika Lloyd, Stefan Sack, Mattia Fiorentini, *Parameterized quantum circuits as machine learning models*, Quantum Science and Technology 4, 043001 (2019) ([arXiv:1906.07682](https://arxiv.org/abs/1906.07682))

* Marco Radic, *Quantum-enhanced Machine Learning in the NISQ era*, 2019 ([pdf](https://elib.uni-stuttgart.de/bitstream/11682/10642/1/main-english.pdf), [[RadicQMLWithNISQ.pdf:file]])

* Dario Gerace, *Quantum Machine Learning on NISQ hardware*,, talk at *INT online program on "Scientific Quantum Computing and Simulation on Near-Term Devices"*  2020 ([pdf](https://www.int.washington.edu/talks/WorkShops/int_20_3/People/Gerace_D/Gerace.pdf), [[GeraceQMLonNISQ.pdf:file]])

* {#MariBromleyIzaacSchuldKilloran20} Andrea Mari, Thomas R. Bromley, Josh Izaac, Maria Schuld, Nathan Killoran,  *Transfer learning in hybrid classical-quantum neural networks*, Quantum 4, 340 (2020) ([arXiv:1912.08278](https://arxiv.org/abs/1912.08278))

* Lucas Munds, *[Quantum Machine Learning: A Roadmap for NISQ Era and Beyond](https://lmunds.medium.com/quantum-machine-learning-a-roadmap-for-nisq-era-and-beyond-f50fdf89c1ce)*, Oct 17, 2020

* Tirthajyoti Sarkar, *[TensorFlow Quantum: Marrying machine learning with quantum computing](https://towardsdatascience.com/tensorflow-quantum-marrying-machine-learning-with-quantum-computing-84533757e07f)*, Mar 11, 2020

* QunaSys, Tech Blog, *Quantum machine learning of quantum data with NISQ devices* ([Feb 15 2021](https://qunasys.medium.com/quantum-machine-learning-of-quantum-data-with-nisq-devices-13ef42943d0a))

* [TensorFlow.org](https://www.tensorflow.org/) (Google), *[Quantum Machine Learning](https://www.tensorflow.org/quantum/concepts#quantum_machine_learning)*

On transformers and large language models (LLM)

* Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, Illia Polosukhin, _Attention is all you need_, in  Advances in Neural Information Processing Systems 30 (NIPS 2017) [pdf](https://proceedings.neurips.cc/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf)
* [[Michael R. Douglas]], _Large language models_, [arXiv:2307.05782](https://arxiv.org/abs/2307.05782)

### Applications

There are or will be innumerable applications. Here are some:

To [[mathematical structures]] in [[algebraic geometry]], [[representation theory]], [[number theory]] and [[combinatorics]]:

* [[Yang-Hui He]], *Machine-Learning Mathematical Structures*, International Journal of Data Science in the Mathematical Sciences [[arXiv:2101.06317](https://arxiv.org/abs/2101.06317), [doi:10.1142/S2810939222500010](https://doi.org/10.1142/S2810939222500010)]

reviewed in:

* [[Yang-Hui He]], *Machine-Learning Mathematical Structures*, talk at *[M-Theory and Mathematics 2023](https://ncatlab.org/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (Jan 2023) [[web](/nlab/show/M-Theory+and+Mathematics#He2023)]

To the [[conformal bootstrap]]:

* Gergely Kántor, Vasilis Niarchos, [[Constantinos Papageorgakis]], *Solving conformal field theories with artificial intelligence* ([arXiv:2108.08859](https://arxiv.org/abs/2108.08859))

(...)

[[!redirects machine-learning]]

[[!redirects quantum machine learning]]

[[!redirects artificial intelligence]]


Maciej Dunajski

* [webpage](https://www.damtp.cam.ac.uk/user/md327/)

## Selected writings

On [[twistors]] and [[twistor string theory]]:

* [[Michael Atiyah]], [[Maciej Dunajski]], [[Lionel J. Mason]], _Twistor theory at fifty: from contour integrals to twistor strings_,  Proc. R. Soc. A473: 20170530 ([arXiv:1704.07464](https://arxiv.org/abs/1704.07464), [doi:10.1098/rspa.2017.0530](http://dx.doi.org/10.1098/rspa.2017.0530))

category: people


Maciej Nowak

* [webpage](http://cs.if.uj.edu.pl/nowak/)

## Selected writings

On [[nuclear physics]] via [[chiral perturbation theory]] ([[quantum hadrodynamics]]):

* [[Maciej Nowak]], [[Mannque Rho]], [[Ismail Zahed]], _[[Chiral Nuclear Dynamics]]_,  World Scientific 1996 ([doi:10.1142/1681](https://doi.org/10.1142/1681))

category: people


Maciej Piróg

* [personal page](https://maciejpirog.github.io/)

## Selected writings

On [[monad transformers]] for [[monadic effects]]:

* [[Maciej Piróg]], *Eilenberg--Moore Monoids and Backtracking Monad Transformers*, EPTCS **207** (2016) 23-56 [[ar](https://arxiv.org/abs/1604.01184), [doi:10.4204/EPTCS.207.2](https://doi.org/10.4204/EPTCS.207.2)&rbrack

category: people

[[!redirects Maciej Pirog]]


Maciej Zworski

* [webpage](https://math.berkeley.edu/~zworski/)

## related $n$Lab entries

* [[integrable PDE]]

* [[formally integrable PDE]]

category: people


Mackey functor

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Generally, the term _Mackey functor_ refers to an [[additive functor]] from a ([[subcategory]] of) a [[category of correspondences]] (in a [[disjunctive category]] $\mathcal{C}$) to possibly any other [[additive category]] which however usually is the "base" abelian category. More generally the term now refers to the fairly obvious [[homotopy theory|homotopy theoretic]] or [[higher category theory|higher categorical]] refinements of this concept.

Therefore the concept of Mackey functors is similar to that of [[sheaves with transfer]] and as such appears (implicitly) in the discussion of [[motives]] (explicitly e.g. in [Kahn-Yamazaki 11, section 2](#KahnYamazaki11), where $\mathcal{C}$ is a category of suitable [[schemes]]).

Specifically, the concept was introduced and named as such in the context of [[representation theory]] ([Dress 71](#Dress71)) and [[equivariant homotopy theory]]/[[equivariant cohomology]] ([May 96](#May96)). Here the underlying [[category of correspondences]] is that in [[finite set|finite]] [[G-sets]], called the _[[Burnside category]]_.

The [[equivariant homotopy groups]] $\pi_n(E)$ of a (genuine) [[G-spectrum]] $E$ organize into a Mackey functor on the [[Burnside category]] with values in [[abelian groups]]. This plays a key role in the [[equivariant Whitehead theorem]].  In fact, genunine [[G-spectra]] themselves are equivalent to Mackey [[(∞,1)-functors]] from the [[Burnside category]] to the [[(∞,1)-category of spectra]] ([GuillouMay 11](#GuillouMay11), [Barwick 14](#Barwick14)).

## Definition

We follow the modern account in ([Barwick 14](#Barwick14)).

Let $\mathcal{C}$ be a [[disjunctive (∞,1)-category]] and write $Corr_1(\mathcal{C})^\otimes$ for the [[(∞,1)-category of correspondences]] in $\mathcal{C}$, regarded as a [[symmetric monoidal (∞,1)-category]] with respect to its [[coproduct]] (which is a [[biproduct]] by disjunctiveness of $\mathcal{C}$).

Write $\mathcal{A} = $[[Spectra]]${}^{\oplus}$ for the [[(∞,1)-category of spectra]] regarded as a [[symmetric monoidal (∞,1)-category]] with respect to [[direct sum]].  More generally $\mathcal{A}$ could be any [[symmetric monoidal (∞,1)-category|symmetric monoidal]] [[stable (∞,1)-category]]

Then a (spectral) _Mackey functor_ on $\mathcal{C}$ is a [[monoidal (∞,1)-functor]] of the form

$$
  S \;\colon\; Corr_1(\mathcal{C})^\otimes \longrightarrow \mathcal{A}^{\oplus}
  \,.
$$

Notice that this means that $S$ is in particular

1. a covariant [[(∞,1)-functor]] $(-)_\ast \colon\mathcal{C} \to \mathcal{A}$;

1. a contravariant [[(∞,1)-functor]], hence  $(-)^\ast \colon\mathcal{C}^{op} \to \mathcal{A}$;

1. satisfying the [[Beck-Chevalley condition]].

(More generally one may specify suitably chosen sub-$(\infty,1)$-categories $\mathcal{C}^\dagger, \mathcal{C}_\dagger \subset \mathcal{C}$ and restrict $Corr_1$ to [[correspondences]] whose left leg is in $\mathcal{C}_\dagger$ and whose right leg is in $\mathcal{C}^\dagger$ ([Barwick 14, section 5](#Barwick14)).)

## Examples

### Dress' Mackey functors

For $\mathcal{A}$ taken to be (the [[derived category]]) of an [[abelian category]] (or better: postcomposed with a [[homological functor]] ) this definition reduces ([Barwick 14](#Barwick14)) to that of Mackey functors as originally defined in ([Dress 71](#Dress71)).

### Equivariant spectra
 {#EquivariantSpectra}

Let $G$ be a [[finite group]]. Let $\mathcal{C}= G Set$ be its category of [[G-sets]]. Then $Corr_1(\mathcal{C})$ is essentially what is called the [[Burnside category]] of $G$ (possibly after abelianizing/stabilizing the hom-spaces suitably, but as ([Barwick 14](#Barwick14)) highlights, this is unnecessary when one is mapping out of this into something abelian/stable, as is the case here).

For $G$ finite, Mackey functors on $\mathcal{C}$ are equivalent to  genuine [[G-spectra]] ([Guillou-May 11, theorem 0.1](#GuillouMay11), [Barwick 14, below example B.6](#Barwick14)) (Notice that this equivalence does not in general hold if $G$ is not a finite group.)

(...)

For $E$ a [[genuine G-spectrum]], the corresponding spectral Mackey functor is given by the [[fixed point spectra]] of $E$

$$
  G/H \mapsto E(G/H) =  [\Sigma^\infty_+ G/H, E]^G \simeq E^H
  \,,
$$

where on the right we have the $G$-equivariant [[mapping spectrum]] from the [[equivariant suspension spectrum]] of the [[transitive action|transitive]] [[G-set]] $G/H$ to $E$.

(e.g. ([Guillou-May 11, remark 2.5](#GuillouMay11)), see also ([Schwede 15, p. 16](#Schwede15)) for restriction and section 4 culminating on p. 37 for transfer and compatibility).

Further, the corresponding abelian-group valued Mackey functor is

$$
  \pi_n(E) \colon G/H \mapsto [G/H_+\wedge S^n, X]_G
  \,,
$$

where now on the right we have just the [[homotopy classes]] of maps, i.e. the morphisms in the [[equivariant stable homotopy category]] (e.g. [Greenlees-May 95, p. 43](#GreenleesMay95))

(...)

## Cohomology with coefficients in a Mackey functor
 {#Cohomology}

We discuss [[cohomology]] of [[topological G-spaces]] with [[coefficients]]
in a Mackey functor, following notation and conventions as in ([May 96, sections IX, X](#May96)). See also ([Greenlees-May 95, p. 9](#GreenleesMay95)).

+-- {: .num_defn #CohomologyOfGSpace}
###### Defininition

For $X$ a [[pointed object|pointed]] [[G-CW complex]], define the [[chain complex]] $C_\bullet(X)$ of Mackey functors to be given by the stable [[equivariant homotopy groups]] of the quotient spaces $X^{\bullet}/X^{\bullet-1}$:

$$
  C_n(X) \coloneqq \pi_n(X^n/X^{n-1})
  \,,
$$

Then for $A$ any Mackey functor, the ordinary cohomology of $X$ with [[coefficients]] in $A$ is the [[cochain cohomology]] of the complex of homs of Mackey functors $C_n(X) \to A$:

$$
  H_G^n(X,A) \coloneqq H^n( Hom(C_\bullet(X), A) )
  \,.
$$

More generally, for $V$ a G-[[representation]], the $(n-V)$-[[RO(G)-grading|RO(G)-graded]] cohomology of $X$ with coefficients in $A$ is

$$
  H_G^{n-V}(X,A) = H_G^n(S^V \wedge X,A)
  \,.
$$

=--

([May 96, section X.4 def. 4.1, def. 4.2 ](#May96))

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

The corresponding [[reduced cohomology]] $\tilde H^n(-,A)$ is represented by maps into the [[Eilenberg-MacLane space|Eilenberg-MacLane G-space]]:

$$
  \tilde H^n(X,A) \simeq [X,K(A,n)]_G
  \,.
$$

=--

([Greenlees-May 95](#GreenleesMay95))

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

For this kind of cohomology, there is equivariant [[Serre spectral sequence]] ([Kronholm 10](#Kronholm10)).

=--

## Related concepts

* [[Tambara functor]]

## References

### Plain Mackey functors

The original article is

* {#Dress71} A. W. M. Dress, _Notes on the theory of representations of finite groups. Part I: The Burnside ring of a finite group and some AGN-applications, Bielefeld_, 1971,

Reviews and surveys:

* {#Blumberg17} [[Andrew Blumberg]], section 3.2 of _The Burnside category_, 2017 ([pdf](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf), [GitHub](https://github.com/adebray/equivariant_homotopy_theory))

* {#tomDieck87} [[Tammo tom Dieck]], Section II.9 of: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

* {#GreenleesMay95} [[John Greenlees]], [[Peter May]], _Equivariant stable homotopy theory_, in I.M. James (ed.), _Handbook of Algebraic Topology_ , pp. 279-325. 1995. ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

* {#May96} [[Peter May]], section IX.4 of _Equivariant homotopy and cohomology theory_ CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. ([pdf](http://www.math.rochester.edu/u/faculty/doug/otherpapers/alaska1.pdf))

* {#Schwede15} [[Stefan Schwede]], around p. 16 of _[[Lectures on Equivariant Stable Homotopy Theory]]_, 2015 ([pdf](http://www.math.uni-bonn.de/people/schwede/equivariant.pdf))

*  Peter Webb, _A Guide to Mackey Functors_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/WebbMF.pdf))

* [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], section 4 of _The Arf-Kervaire problem in algebraic topology: Sketch of the proof_ ([[HHRKervaire.pdf:file]])

  (with an eye towards application to the [[Arf-Kervaire invariant problem]])

* [[Megan Shulman]], chapter 2 of _Equivariant local coefficients and the $RO(G)$-graded cohomology of classifying spaces_ ([arXiv:1405.1770](http://arxiv.org/abs/1405.1770))

See also

* [[Tammo tom Dieck]], _Transformation groups_, Studies in Mathematics, vol. 8, Walter de Gruyter, Berlin, New York, 1987, x + 311 pp.,

* Serge Bouc, chapter 1 of _Green Functors and G-sets_, LNM 1671 (1997; paperback 2008) [doi:10.1007/BFb0095821](http://dx.doi.org/10.1007/BFb0095821)

* [[Tammo tom Dieck]], Equivariant homology and Mackey functors, Mathematische Annalen __206__, no.1, pp. 67--78, 1973 [doi:10.1007/BF01431529](http://dx.doi.org/10.1007/BF01431529)

* {#GreenleesMay95} [[John Greenlees]], [[Peter May]], appendix A of _Generalized Tate cohomology_, Mem. Amer. Math. Soc. 113 (1995) no 543 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/GM-Tate-543.pdf))

* D. Tambara, _The Drinfeld center of the category of Mackey functors_, J. Algebra __319__, 10, pp. 4018-4101 (2008) [doi:10.1016/j.jalgebra.2008.02.011](http://dx.doi.org/10.1016/j.jalgebra.2008.02.011)

* Elango Panchadcharam, _Categories of Mackey Functors_, PhD thesis, Macquarie Univ. 2006

* {#Kronholm10} [[William Kronholm]], _The $RO(G)$-graded Serre spectral sequence_, Homology Homotopy Appl. Volume 12, Number 1 (2010), 75-92. ([pdf](http://www.swarthmore.edu/NatSci/wkronho1/serre.pdf), [Euclid](https://projecteuclid.org/euclid.hha/1296223823))

For Mackey functors [[enriched]] over [[closed category|closed]] [[multicategories]]:

* [[Niles Johnson]], [[Donald Yau]], _Homotopy Theory of Enriched Mackey Functors_ ([arXiv:2212.04276](https://arxiv.org/abs/2212.04276))

Relation of Mackey functors to [[sheaves with transfer]] in the theory of [[motives]]:

* {#KahnYamazaki11} [[Bruno Kahn]], Takao Yamazaki, _Voevodsky's motives and Weil reciprocity_, Duke Mathematical Journal 162, 14 (2013) 2751-2796 ([arXiv:1108.2764](http://arxiv.org/abs/1108.2764))

[[categorification|Categorification]] to Mackey [[2-functors]] is discussed found in

* [[Paul Balmer]], [[Ivo Dell'Ambrogio]], _Mackey 2-functors and Mackey 2-motives_, ([arXiv:1808.04902](https://arxiv.org/abs/1808.04902))

### Spectral Mackey functors

The general concept of spectral Mackey functors

* {#Barwick14} [[Clark Barwick]], _Spectral Mackey functors and equivariant algebraic K-theory (I)_, Adv. Math., 304:646–727,  2017 ([doi:10.1016/j.aim.2016.08.043](https://doi.org/10.1016/j.aim.2016.08.043), [arXiv:1404.0108](http://arxiv.org/abs/1404.0108))

* {#Barwick15} [[Clark Barwick]], [[Saul Glasman]], Jay Shah, _Spectral Mackey functors and equivariant algebraic K-theory (II)_ ([arXiv:1505.03098](http://arxiv.org/abs/1505.03098))

* {#Glasman15} [[Saul Glasman]], _Stratified categories, geometric fixed points and a generalized Arone-Ching theorem_ ([arXiv:1507.01976](https://arxiv.org/abs/1507.01976), [talk notes pdf](http://www-users.math.umn.edu/~sglasman/strattalk.pdf))

#### In equivariant stable homotopy theory

The construction of [[equivariant stable homotopy theory]] in terms of spectral Mackey functors is originally due to

* {#GuillouMay11} [[Bert Guillou]], [[Peter May]],  _Models of $G$-spectra as presheaves of spectra, ([arXiv:1110.3571](http://arxiv.org/abs/1110.3571))

* [[Bert Guillou]], [[Peter May]], _Permutative $G$-categories in equivariant infinite loop space theory_, Algebr. Geom. Topol. 17 (2017) 3259-3339 ([arXiv:1207.3459](http://arxiv.org/abs/1207.3459))

* {#Nardin12} [[Denis Nardin]], section 2.6 and appendix A of _Stability and distributivity over orbital ∞-categories_, 2012 ([hdl.handle.net/1721.1/112895]( http://hdl.handle.net/1721.1/112895), [pdf](https://www.math.univ-paris13.fr/~nardin/thesis.pdf))

The generalization of [[K-theory of permutative categories]] to spectral Mackey functors is discussed in

* {#BohmannOsorno14} [[Anna Marie Bohmann]], [[Angélica Osorno]], _Constructing equivariant spectra via categorical Mackey functors_, Algebraic & Geometric Topology 15.1 (2015): 537-563 ([arXiv:1405.6126](http://arxiv.org/abs/1405.6126))

#### In Goodwillie calculus

Discussion of [[Goodwillie calculus]] via [[spectral Mackey functors]]

* {#Glasman16} [[Saul Glasman]], _Goodwillie calculus and Mackey functors_  ([arXiv:1610.03127](https://arxiv.org/abs/1610.03127))

[[!redirects Mackey functors]]

[[!redirects spectral Mackey functor]]
[[!redirects spectral Mackey functors]]


MacNeille completion

# MacNeille completion
* tic
{: toc}

## Idea

The MacNeille or Dedekind--MacNeille (*not* 'Mac Neille') [[completion]] of a [[lattice]] (or even [[poset]]) $L$ is the [[universal construction|universal]] [[complete lattice]] containing $L$. In [[Pos]], the MacNeille completion of an object is its [[injective hull]].

## Definition

A **MacNeille completion** of $L$ is any [[complete lattice]] $C$ containing $L$ as a [[subobject|sublattice]], with $L$ both join-dense and meet-dense in $C$ (that is, each element of $C$ is both a [[join]] of elements of $L$ and a [[meet]] of elements of $L$), and such that if $C'$ is another such lattice then there is a unique isomorphism from $C$ to $C'$ fixing $L$.  In particular, this allows us to speak of [[the]] MacNeille completion of $L$.

A MacNeille completion may be constructed as follows.  Let $P(L)$ be the [[power set|set of all subsets]] of $L$, ordered by inclusion. For any $X \in P(L)$ let

* $X^{\Delta} = \{\alpha \in L \mid \forall x \in X . \alpha \geq x \}$

* $X^{\nabla} = \{\alpha \in L \mid \forall x \in X . \alpha \leq x \}$

These form a [[Galois connection]] on $P(L)$; the condition $\phi(X) = (X^{\Delta})^{\nabla}$ defines a [[Moore closure|closure]] operation $\phi$ on $P(L)$. The lattice $C$ of all $\phi$-closed subsets of $L$ is complete. For any $x \in L$ the set $i(x) = (\{x\}^{\Delta})^{\nabla}$ is the [[principal ideal]] generated by $x$. Then $i$ is an isomorphic embedding of $L$ into the complete lattice $C$ that preserves all least upper bounds (joins) and greatest lower bounds (meets) existing in $L$.

## Examples

When applied to the ordered set of [[rational number]]s, the construction described above gives the completion of the set of rational numbers by Dedekind sections, *including* the empty sections.  Thus we get, not the [[real line]] $\mathbb{R} = {]{-\infty},\infty[}$, but the [[extended real line]] $\overline{\mathbb{R}} = [-\infty,\infty]$.  In [[constructive mathematics]], we may get additional finite elements, since the Dedekind sections are not required to be located; if we then drop $\pm\infty$, we get the (bounded) [[MacNeille real numbers]].

## Related pages

* The MacNeille completion can be [[categorification|categorified]] to the category of self-dual (saturated) objects in the [[Isbell envelope]]; see [this MO question and answers](https://mathoverflow.net/q/59291/).

## References

* [MacNeille completions and canonical extensions](http://www.math.nmsu.edu/mgehrke/gehharven.pdf)
* [Encyclopedia of Mathematics](http://eom.springer.de/c/c024070.htm)
* Section 21H of [[The Joy of Cats]] ([web](http://katmat.math.uni-bremen.de/acc/acc.pdf))

[[!redirects MacNeille completion]]
[[!redirects MacNeille completions]]
[[!redirects Mac Neille completion]]
[[!redirects Mac Neille completions]]
[[!redirects Dedekind-MacNeille completion]]
[[!redirects Dedekind-MacNeille completions]]
[[!redirects Dedekind–MacNeille completion]]
[[!redirects Dedekind–MacNeille completions]]
[[!redirects Dedekind--MacNeille completion]]
[[!redirects Dedekind--MacNeille completions]]


MacNeille real number

# MacNeille real numbers
* table of contents
{: toc}

## Idea

The MacNeille real numbers are a version of [[real numbers]] defined by forming the [[MacNeille completion]] of the [[rational numbers]], and then perhaps excluding the top and bottom element ($\pm\infty$).  In [[classical mathematics]], this produces the ordinary [[real numbers]] (and if we include $\pm\infty$ it produces the [[extended real numbers]]).  However, in [[constructive mathematics]], it yields something different (and less well behaved); it is not a constructive theorem that every bounded MacNeille number is located (hence a real number in the usual sense of constructive mathematics).

## Definition

Let $\mathbb{Q}$ be the [[poset]] of [[rational numbers]] with its usual [[Archimedean order]].

There are two obvious ways to define the MacNeille real numbers from $\mathbb{Q}$, one based on closed [[Dedekind cut|cuts]] (which matches the general theory of [[MacNeille completions]]) and one based on open cuts (which is compatible with other types of real numbers).  Fortunately, they are equivalent.

Outside of these definitions themselves, we will adopt the open definition, since it relates better to other kinds of real numbers.

### With closed sets

A pair $(L,U)$ of [[subsets]] of $\mathbb{Q}$ is called a __closed Dedekind--MacNeille cut__ in $\mathbb{Q}$ if:

* $U$ is the set of [[upper bounds]] of $L$: that is, $s \in U$ iff for all $r \in L$, $s \geq r$; and
* $L$ is the set of [[lower bounds]] of $U$: that is, $r \in L$ iff for all $s \in U$, $r \leq s$.

Given two such cuts, define $(L_1, U_1) \leq (L_2, U_2)$ to mean:

* $L_1 \subseteq L_2$; and
* $U_1 \supseteq U_2$.

(Actually, either of these assertions alone implies the other.)

Then the set of such cuts becomes a [[poset]], in fact a [[complete lattice]], called the set of **extended MacNeille real numbers**.  The **MacNeille real numbers** are the sub-poset consisting of the pairs $(L,U)$ such that $L$ and $U$ are both [[inhabited set|inhabited]].

### With open sets

A pair $(L,U)$ of [[subsets]] of $\mathbb{Q}$ is called an __open Dedekind--MacNeille cut__ in $\mathbb{Q}$ if:

* $L$ is an extended [[lower real number]] and $U$ is an extended [[upper real number]].  That is, $L$ is an open down-set and $U$ is an open up-set.
* $U$ is the interior of the complement of $L$: that is, $s \in U$ iff there is an $r\lt s$ with $r\notin L$.
* $L$ is the interior of the complement of $U$: that is, $r \in L$ iff there is an $s\gt r$ with $s\notin U$.

Given two such cuts, define $(L_1, U_1) \leq (L_2, U_2)$ to mean:

* $L_1 \subseteq L_2$; and
* $U_1 \supseteq U_2$.

(Actually, either of these assertions alone implies the other.)

Then the set of such cuts becomes a [[poset]], in fact a [[complete lattice]], called the set of **extended MacNeille real numbers**.  The **MacNeille real numbers** are the sub-poset consisting of the pairs $(L,U)$ such that $L$ and $U$ are both [[inhabited set|inhabited]].

### Comparing definitions

The open and closed definitions above are equivalent, even in constructive mathematics.

In one direction, if $(L,U)$ is a closed extended MacNeille real number, define $L^o$ and $U^o$ to be the interiors of $L$ and $U$ respectively.  Then $L^o$ is an extended lower real and $U^o$ is an extended upper real.  If $s\in U^o$, then by definition there is a $t\lt s$ with $t\in U$.  Then if $r$ is such that $t\lt r \lt s$, we have $r\notin L$ (and hence $r\notin L^o$), since if $r\in L$ we would have $r\le t$.  Conversely, if we have $r\lt s$ with $r\notin L^o$, then there does not exist a $t\gt r$ with $t\in L$, which is to say (since the rationals satisfy trichotomy) that for every $t\in L$ we have $t\le r$, i.e. $r\in U$, and hence $s\in U^o$.  The dual proof shows the corresponding axiom for $L^o$, so $(L^o,U^o)$ is an open extended MacNeille real number, which is bounded if $(L,U)$ is.

In the other direction, if $(L,U)$ is an open extended MacNeille real number, define $L^c$ to be the set of lower bounds of $U$, and $U^c$ to be the set of upper bounds of $L$.  (The notation is admittedly a little misleading, since $L^c$ is a function of $U$ rather than $L$.)  Then if $r\in L^c$ and $s\in U^c$ and $r\gt s$, then for a $t$ such that $s\lt t\lt r$ we would have $t\notin U$ and $t\notin L$, hence by the MacNeille condition $r\in L$ and $s\in U$, a contradiction.  Thus, if $r\in L^c$ and $s\in U^c$ then $r\le s$, so $L^c$ consists of lower bounds for $U^c$ and vice versa.  Conversely, if $r$ is a lower bound of $U^c$, then it is also a lower bound of $U$, and so $r\in L^c$.  Thus $(L^c,U^c)$ is a closed extended MacNeille real number, which is bounded if $(L,U)$ is.

To show that these operations are inverses, suppose first $(L,U)$ is a closed extended MacNeille real.  Then $(L^o)^c$ is the set of lower bounds of the interior of $U$.  Any lower bound of $U$ is of course a lower bound for its interior, so $L\subseteq (L^o)^c$.  On the other hand, if $r$ is a lower bound for $U^o$, to show $r\in L$ it suffices to show $r\le s$ for all $s\in U$.  But if $r\gt s$, then $r\in U^o$, a contradiction since $U^o$ is open.  Thus $L= (L^o)^c$, and dually $U = (U^o)^c$.

On the other side, suppose $(L,U)$ is an open extended MacNeille real.  Then $(L^c)^o$ is the interior of the set of lower bounds of $U$.  Since $L$ consists of lower bounds for $U$, and $L$ is open, we have $L \subseteq (L^c)^o$.  On the other hand, if $r\in (L^c)^o$ then there is an $s\in L^c$ with $r\lt s$.  Then for any $t$ with $r\lt t\lt s$ we have $t\notin U$ (since $s$ is a lower bound for $U$), hence by the MacNeille condition $r\in L$.  Thus $L= (L^o)^c$, and dually $U = (U^o)^c$.

On the rest of this page, we will adopt the open definition, since it relates better to other kinds of real numbers.

## Properties

### Relation to other kinds of real numbers

Any (extended) [[located real number]] is an (extended) MacNeille real number.  The locatedness condition says that if $r\lt s$ then $r\in L$ or $s\in U$, which certainly implies that $(r\notin L)\Rightarrow (s\in U)$ and $(s\notin U)\Rightarrow (r\in L)$.  This gives an order-embedding.

Similarly, if $(L,U)$ is an (extended) MacNeille real number, then by definition $L$ is an (extended) [[lower real number]] and $U$ is an (extended) [[upper real number]].  Since $L$ and $U$ in a MacNeille real are recoverable from each other, this gives two order-embeddings.

If we define an **interval** to be a pair $(L,U)$ where $L$ is an extended lower real, $U$ is an extended upper real, and $L\cap U = \emptyset$, then we can embed both lower reals and upper reals into the poset of intervals by taking the interior of the complement as the other half.  That is, we map $L$ to $(L,int(\neg L))$ and $U$ to $(int(\neg U),U)$ --- two kinds of "singleton" intervals.  These are both order-embeddings, and in fact the lower reals are a [[coreflective subcategory|coreflective]] sub-poset and the upper reals are a [[reflective subcategory|reflective]] sub-poset.  Thus we have an [[adjunction]] between extended lower reals and extended upper reals, and the extended MacNeille reals are the jointly fixed sub-poset.  They are also the set-theoretic [[intersection]] of the lower and upper reals inside the intervals.

### Completeness

Since the extended MacNeille reals are a reflective sub-poset of the extended lower reals and a coreflective sub-poset of the extended upper reals, they are closed in the former under [[infima]] and in the latter under [[suprema]].  In particular, they are a complete lattice.  Note, though, that neither infima nor suprema are constructed as set-theoretic unions (unlike infima of upper reals and suprema of lower reals), which makes them less useful for purposes of constructive analysis.  (On the other hand, the extended located real numbers are not, constructively, a complete lattice at all.)

### Strict ordering

Recall that the MacNeille reals are a poset in which $(L_1, U_1) \leq (L_2, U_2)$ iff $L_1 \subseteq L_2$ iff $U_1 \supseteq U_2$.  This relation $\leq$ may be called the _weak ordering_ on MacNeille reals.

There is also a _strict (or strong) ordering_ $\lt$ defined by

*  $(L_1, U_1) \lt (L_2, U_2)$ iff $U_1$ meets $L_2$ (meaning that the [[intersection]] $U_1 \cap L_2$ is [[inhabited subset|inhabited]]).

That is, $x \lt y$ iff for some rational number $q$, $x \lt q \lt y$.

This restricts to the usual strict order $\lt$ on Dedekind reals.  We also have for any rational number $q$ that $x \lt q$ by this new strict order iff $x \lt q$ in the previously known sense (that $q$ belongs to the upper set of $x$), and similarly for $x \gt q$.

For Dedekind reals, the strict ordering is often taken to be the more basic concept, as the weak ordering can be derived from it ($\leq$ is the [[negation]] of $\gt$).  However, this is not so for the MacNeille reals, and the two orderings (weak and strict) exist separately.

They are not entirely unrelated, however; we do have $x \lt z$ whenever $x \lt y \leq z$ or $x \leq y \lt z$.  Also, while neither of $\leq$ and $\gt$ is the negation of the other, they are mutually exclusive, and it is impossible for both to be false.  As such, the pair of relations $({\leq},{\gt})$ makes the MacNeille real numbers into a tight interpretation of the classical theory of [[total orders]] in the [[antithesis interpretation]].

## References

* Section D4.7 of [[Sketches of an Elephant]]

* In [[Anne Troelstra|Troelstra]] and [[Dirk van Dalen|van Dalen]], *[[Constructivism in Mathematics]]*, section 5.5, the extended MacNeille real numbers are called the "extended real numbers" (and the MacNeille real numbers themselves are called the "bounded extended real numbers").

[[!redirects MacNeille real number]]
[[!redirects MacNeille real numbers]]
[[!redirects MacNeille real]]
[[!redirects MacNeille reals]]
[[!redirects MacNeille number]]
[[!redirects MacNeille numbers]]


Macquarie University

* [webpage](http://mq.edu.au/)


Madeleine Birchfield

## Selected writings

On the [[locale of real numbers]]:

* [[Madeleine Birchfield]], [[Simon Henry]], *The field structure on the locale of real numbers* (2022) [[MO:q/434706](https://mathoverflow.net/q/434706/381)]

category: people


Madhav Nori

* [Maths Genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=38122)

## Related pages

* [[Nori motive]]

category: people


Magari algebra

[[!redirects empty 158]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
#### Foundations
+--{: .hide}
[[!include higher algebra - contents]]
[[!include foundations - contents]]
=--
=--
=--

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

[[!redirects diagonizable algebra]]
[[!redirects Magari Algebra]]
[[!redirects Magari algebras]]

## Idea

A  _diagonizable_ or **Magari algebra** is a [[Boolean algebra]] together with a modal operator internalizing the diagonalization properties involved in [[incompleteness theorem|Gödel's second incompleteness theorem]].

## Definition
A _Magari algebra_ is a pair $(M,\diamond)$ where $M$ is a [[Boolean algebra]] and $\diamond:M\to M$ is an operator satisfying:

1. $\diamond\bot=\bot$ and $\diamond (x\vee y) =\diamond x\vee \diamond y$ .

2. $\diamond x=\diamond (x\wedge \neg\diamond x)$.

### Remark

Suppose that $T$ is a 'Gödelian' theory containing enough arithmetics in order to admit a formula $\mathsf{Cons}$ internalizing consistency. $\mathsf{Cons}$ induces an operator $\diamond_T$ on the [[Lindenbaum-Tarski algebra]] of provable equivalence classes of formulas of $T$ by $[\varphi]\to [\mathsf{Cons}(T+\varphi)]$.

Then the condition M2 for $\diamond_T$ can be viewed as expressing the non-provability of the consistency of the theory $T'=T+\varphi$ since $\mathsf{Cons}(T')$ says that $T'$ is consistent whereas $\mathsf{Cons}(T'+\neg\mathsf{Cons}(T'))$ says that $T'+\neg\mathsf{Cons}(T')$ is consistent i.e. not $T'\vdash\mathsf{Cons}(T')$ (For more details see [Beklemishev-Gabelaia 2012](#BG12)).

## Properties

From M1 it follows that $\diamond$ is monotone: $x\leq y$ implies $\diamond x\leq \diamond y$.

Moreover, $\diamond$ satisfies the following **transitivity property**:
$$\diamond\diamond x\leq\diamond x\quad ,\forall x\in M.$$

To see this, consider $\diamond\diamond x\leq\diamond\diamond x\vee\diamond x=\diamond (x\vee \diamond x)$ where the last equation follows from the commutativity of $\vee$ together with M1.

Set $y=x\vee\diamond x$ whence sofar we have got : $\diamond\diamond x\leq\diamond y$. Now we want to show that $\diamond y\leq\diamond x$:

From M2 plus the definition of $y$: $\diamond y=\diamond(y\wedge\neg\diamond y)=\diamond((x\vee\diamond x)\wedge \neg\diamond y)=\diamond((x\wedge\neg\diamond y)\vee(\diamond x\wedge\neg\diamond y))$. But $\diamond x\wedge\neg\diamond y=\bot$ since the expanded left side is a $\wedge$ containing a factor $\diamond x\wedge\neg\diamond x$. Whence $\diamond((x\wedge\neg\diamond y)\vee(\diamond x\wedge\neg\diamond y))=\diamond((x\wedge\neg\diamond y)\vee\bot)=\diamond(x\wedge\neg\diamond y)$ but $\diamond(x\wedge\neg\diamond y)\leq\diamond x$ since $x\wedge\neg\diamond y\leq x$ and $\diamond$ is monotone. Whence all in all we get $\diamond y\leq\diamond x$ and we are done.

## Examples

* Let $X$ be a [[topological space]] and $d:2^X\to 2^X$ the **Cantor-Bendixson derivative** that maps a subset $A\subseteq X$ to the set $d A$ of all its [[accumulation point|accumulation points]]. Then $(2^X,d)$ is a Magari algebra iff $X$ is a [[scattered space]]. This result is due to L. Esakia and, independently, H. Simmons (cf. [Beklemishev-Gabelaia 2012](#BG12)).

## Related entries

* [[provability logic]]

* [[scattered space]]

* [[scattered topos]]

* [[diagonal argument]]

* [[Gödel's incompleteness theorem]]

* [[arithmetic pretopos]]

## References

* {#BG12}L. Beklemishev, D. Gabelaia, _Topological interpretations of provability logic_ , arXiv:1210.7317 (2012).  ([abstract](http://arxiv.org/abs/1210.7317))

* A. Macintyre, H. Simmons, _Gödel's diagonalization technique and related properties of theories_ , Coll. Math. **28** no.2 (1973) pp.165-180. ([pdf](http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.desklight-8bd6a8c7-d547-43f0-b220-eda51556c1ad/c/cm28_2_01.pdf))


Magdalena Kedziorek

* [former webpage](http://maths.dept.shef.ac.uk/maths/staff_info_447.html)

* [webpage](http://people.epfl.ch/magdalena.kedziorek)

## Selected writings

On left-[[transferred model structures]] and [[model structures on functors]]:

* [[Marzieh Bayeh]], [[Kathryn Hess]], [[Varvara Karpova]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *Left-induced model structures and diagram categories*, in: *Women in Topology: Collaborations in Homotopy Theory*, Contemporary Mathematics **641** American Mathematical Society (2015) [[arXiv:1401.3651](http://arxiv.org/abs/1401.3651), [ISBN:978-1-4704-2495-4](https://bookstore.ams.org/conm-641)]

* {#HKRS15} [[Kathryn Hess]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *A necessary and sufficient condition for induced model structures*, J. Topology **10** 2  (2017) 324-369 [[arXiv:1509.08154](http://arxiv.org/abs/1509.08154), [doi:10.1112/topo.12011](https://doi.org/10.1112/topo.12011)]

  > (this contains an error which is corrected in [Garner, Kedziorek & Riehl 2018](#GKR18))

* {#GKR18} [[Richard Garner]], [[Magdalena Kedziorek]], [[Emily Riehl]], *Lifting accessible model structures*, J. Topology **13** 1 (2020) 59-76 [[arXiv:1802.09889](https://arxiv.org/abs/1802.09889), [doi:10.1112/topo.12123](https://doi.org/10.1112/topo.12123)]

On [[rational equivariant stable homotopy theory]]:

* [[Magdalena Kedziorek]], _Algebraic models for rational G-spectra_, thesis 2014 ([web](http://etheses.whiterose.ac.uk/7699/))

* [[David Barnes]], [[Magdalena Kedziorek]], _An introduction to algebraic models for rational G-spectra_ ([arXiv:2004.01566](https://arxiv.org/abs/2004.01566))

and specifically for [[equivariance group]] (a [[subgroup]] of) [[SO(3)]]:

* [[Magdalena Kedziorek]], _An algebraic model for rational $SO(3)$–spectra_, Algebraic & Geometric Topology 17 (2017) 3095–3136 ([arXiv:1611.08415](https://arxiv.org/abs/1611.08415), [doi:10.2140/agt.2017.17.3095](https://doi.org/10.2140/agt.2017.17.3095))

On [[commutative ring]]-structure ([[E-infinity ring|$E_\infty$-ring]]-structure) in [[rational equivariant stable homotopy theory]] in general and in [[rational equivariant K-theory]] in particular:

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Naive-commutative structure on rational equivariant K-theory for abelian groups* ([arXiv:2002.01556](https://arxiv.org/abs/2002.01556))

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Genuine-commutative ring structure on rational equivariant K-theory for finite abelian groups* ([arXiv:2104.01079](https://arxiv.org/abs/2104.01079))

On [[simply connected homotopy type|simply connected]] [[homotopy types]] as the [[comodule over a comonad|comodules]] in [[spectra]] over the [[comonad]] $\Sigma^\infty \Omega^\infty$ (the [[exponential modality]], see [there](!-modality#RealizationInLinearHomotopyTypeTheory)) induced by the [[stabilization]] [[adjunction]]:

* {#HessKedziorek17} [[Kathryn Hess]], [[Magdalena Kedziorek]], Thm. 3.11 in: *The homotopy theory of coalgebras over simplicial comonads*, Homology, Homotopy and Applications **21** 1 (2019) [[arXiv:1707.07104](https://arxiv.org/abs/1707.07104), [doi:10.4310/HHA.2019.v21.n1.a11](https://dx.doi.org/10.4310/HHA.2019.v21.n1.a11)]

## Related entries

* [[rational equivariant stable homotopy theory]]

category: people

[[!redirects Magdalena Kędziorek]]


magic pyramid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The classification of pure [[supergravity]] theories in dimensions 3,4,6,10 and 4,5,7,11 is parameterized by the real [[normed division algebras]], while the classification of [[super Yang-Mills theories]] is parameterized by pairs of real [[normed division algebras]], see at _[[supersymmetry and division algebra]]_ and at _[[spin representation]]_.

Combining this yields a classification of the extended super [[Einstein-Yang-Mills theories]] by triples of real normed division algebras. called a _magic pyramid_ of supergravity theories ([ABDHN 13](#ABDHN13)).

<img src="https://ncatlab.org/nlab/files/MagicPyramid.jpg" width = "600">

(from [ABDHN 13, p. 3](#ABDHN13))

In this diagram vertical axis labels the spacetime division algebra (hence vertical label $\mathbb{R}$ corresponds to $d = 2+1$, vertical label $\mathbb{C}$ to $d = 3+1$, $\mathbb{H}$ to $d = 5+1$ and $\mathbb{O}$ to $d = 9+1$) , while the horizontal axes label the algebras associated with the number of left and right supersymmetries. The entries display the corresponding [[U-duality]] groups. Each horizontal level is a "[[magic square]]".

## Related concepts

* [[magic supergravity]]

* [[KLT relations]]

## References

* {#ABDHN13} A. Anastasiou, [[Leron Borsten]], [[Mike Duff]], L. J. Hughes, S. Nagy, _A magic pyramid of supergravities_, Journal of High Energy Physics, April 2014, 2014:178, ([arXiv:1312.6523](https://arxiv.org/abs/1312.6523))

based on

* [[Leron Borsten]], [[Mike Duff]], L. J. Hughes, S. Nagy, _A magic square from Yang-Mills squared_, Phys. Rev. Lett. 112, 131601 (2014) [arXiv:1301.4176](http://arxiv.org/abs/1301.4176)

* {#ABDH13} A. Anastasiou, [[Leron Borsten]], [[Mike Duff]], L. J. Hughes, S. Nagy, _Super Yang-Mills, division algebras and triality_ ([arXiv:1309.0546](http://arxiv.org/abs/1309.0546))

[[!redirects magic pyramids]]


magic supergravity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[magic square|Magic Squares]] (MS's), arrays of [[Lie algebras]] enjoying remarkable symmetry properties under reflection with respect to their main diagonal, were discovered long time ago by [[Freudenthal magic square|Freudenthal]], Rozenfeld and Tits, and their structure and fascinating properties have been studied extensively in mathematics and mathematical physics, especially in relation to exceptional Lie algebras.

Following the seminal papers ([Gunyadin-Sierra-Townsend 83](#GunyadinSierraTownsend83), [Gunyadin-Sierra-Townsend 83](#GunyadinSierraTownsend84)) [[magic squares]]  have been related to
the generalized electric-magnetic [[U-duality]] symmetries of particular classes of [[Einstein-Maxwell theory|Maxwell-Einstein]] [[supergravity]]
theories, called _magic supergravities_. In particular, non-compact, real forms of Lie algebras, corresponding to non-compact symmetries of (super)gravity theories, have
become relevant as symmetries of the corresponding rank-3 simple [[Jordan algebras]], defined over real [[normed division algebra]] $(A = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O})$ or split $(A = \mathbb{R}, \mathbb{C}_S, \mathbb{H}_S, \mathbb{O}_S)$ composition algebras. Later on, some other MS's have been constructed in literature through the exploitation of Tits' formula.

(from [Cacciatori-Cerchiai-Marrani 12](#CacciatoriCerchiaiMarrani12))

## Related concepts

* [[magic pyramid]]

## References

* {#GunyadinSierraTownsend83} [[Murat Gunaydin]], G. Sierra and [[Paul Townsend]], _Exceptional Supergravity Theories and the Magic Square_, Phys. Phys. Lett. B 133, 72 (1983) ([spire](http://inspirehep.net/record/192025/))

* {#GunyadinSierraTownsend84} [[Murat Gunaydin]], G. Sierra, [[Paul Townsend]], _The Geometry of $\mathcal{N}=2$ Maxwell-Einstein Supergravity and Jordan Algebras_, Nucl.Phys. B242 (1984) 244-268

* [[Murat Gunaydin]], G. Sierra and [[Paul Townsend]], Nucl. Phys. B 253, 573 (1985).

* [[Murat Gunaydin]], S. McReynolds, M. Zagermann, _Unified $\mathcal{N}=2$ Maxwell-Einstein and Yang-Mills-Einstein Supergravity Theories in Four Dimensions_, JHEP 0509:026,2005 ([arXiv:hep-th/0507227](https://arxiv.org/abs/hep-th/0507227))

* Massimo Bianchi, [[Sergio Ferrara]], _Enriques and Octonionic Magic Supergravity Models_,  JHEP 0802:054,2008 ([arXiv:0712.2976](https://arxiv.org/abs/0712.2976))

* {#CacciatoriCerchiaiMarrani12} Sergio L. Cacciatori, [[Bianca Cerchiai]], [[Alessio Marrani]], _Squaring the Magic_ ([arXiv:1208.6153](http://arxiv.org/abs/1208.6153))

* Sergio L. Cacciatori, [[Bianca Cerchiai]], [[Alessio Marrani]], _Magic coset decompositions_, Adv. Theor. Math. Phys. Volume 17, Number 5 (2013), 1077-1128. ([euclid:atmp/1408626511^](https://projecteuclid.org/euclid.atmp/1408626511))

[[!redirects magic supergravities]]

[[!redirects magic supergravity theory]]
[[!redirects magic supergravity theories]]


magic triangle

##Idea

A triangular array of [[Lie groups]] which extends the [[Freudenthal magic square]], due to Predrag Cvitanovic, Pierre Deligne and Benedict Gross.

##References

* Predrag Cvitanovic, _Tracks, Lie's, and Exceptional Magic_, ([pdf](http://www.nbi.dk/GroupTheory/refs/cvitanovicLH03.pdf)) (Fig. 1, p. 154)

* [[Pierre Deligne]], [[Benedict Gross]], _On the exceptional series, and its descendants_, ([pdf](https://www.math.ias.edu/files/deligne/ExcepSeries.pdf)) (Table 1, p. 7)


magma

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

# Binary operations
* table of contents
{: toc}

## Definitions

A __binary operation__ on a [[set]] $S$ is a [[function]] $(-)\cdot (-) \colon S \times S \to S$ from the [[Cartesian product]] $S \times S$ to $S$.  A __magma__ (or binary algebraic structure, or, alternatively, a mono-binary algebra) $(S,\cdot)$ is a set equipped with a binary operation on it.

A magma is called

* _[[unital magma|unital]]_ if it has a [[identity element|neutral element]]; that is, an element $1 \in S$ such that $1 \cdot x = x = x \cdot 1$. Some authors mean by 'magma' what we call a unital magma (cf. [[Borceux-Bourn]] Def. 1.2.1). One can consider one-sided unital elements separately: $1_L \cdot x = x$ and/or $x = x \cdot 1_R$. Note that units may be far from unique.

* _[[commutative magma|commutative]]_ if the binary operation takes the same value when its two arguments are interchanged: $x \cdot y = y \cdot x$.

* _[[associative magma|associative]]_ if the binary operation satisfies the [[associativity]] condition $(x \cdot y) \cdot z = x \cdot (y \cdot z)$.

* _[[invertible magma|invertible]]_ if it has an [[inverse element]].

* an _[[absorption magma]]_ if it has an element $0 \in S$ such that the binary operation satisfies the absorption condition: $0 \cdot x = x \cdot 0 = 0$.

* a *[[quasigroup]]* if one-sided multiplication by any element is a [[bijection]].

A magma has a [[square root]] [[function]] $\sqrt{-}:S \to S$ if for all $x \in S$, $\sqrt{x \cdot x} = x$ and $\sqrt{x} \cdot \sqrt{x} = x$.

The term 'magma' is from [[Bourbaki]] and is intended to suggest the fluidity of the concept; special cases include [[unital magmas]], [[semigroups]]/[[monoids]], [[quasigroups]], [[groups]], and so on.  The term 'groupoid' is also used, but here that word means [[groupoid|something else]] (see also related discussion at [[historical notes on quasigroups]]).

More generally, in any [[multicategory]] $M$, a **magma object** or **magma in $M$** is an [[object]] $X$ of $M$ equipped with a [[multimorphism]] $m: X, X \to X$ in $M$.  Here the multimorphism from $X$ and $X$ to $X$ is a __binary operation in $M$__.  In particular, for $M$ a [[monoidal category]], a magma structure on $X$ is a [[morphism]] $m\colon X \otimes X \to X$, and in a [[closed category]], a magma structure on $X$ is a morphism $m\colon X \to [X, X]$.

## Properties

### Square function {#square function}

Every magma $(M,\cdot)$ has a morphism $(-)^2:M \to M$ called the __square__ and defined as $x^2 := x \cdot x$ for all $x \in M$.

### Magmas as actions

An [[action]] of a set $A$ on another set $B$ is a function $act:A \times B \to B$. So this means that a magma is just an action of a set on itself.

### Opposite magmas

There exists a function on the binary operation set $B:(M\times M\to M)\to (M\times M\to M)$ called the __[[braiding]]__ that takes every binary operation on the set to its opposite binary operation, where for every magma operation $m:M\times M\to M$ and for all elements $a,b$ in $M$, $B(m)(a,b) = m(b,a)$. The set $M$ with the opposite binary operation is the [[opposite magma]] of $M$. $B$ is an [[involution]]; the opposite of an opposite magma is the original magma itself; this follows from the fact that [[Set]] is a [[symmetric monoidal category]]. Fixed points of $B$ are called __[[commutative magma|commutative]]__ binary operations.

## Free magmas

The free magma on one generator $(M, \cdot, 1)$ is a model of nesting parentheses around a magma, and is important in the study of [[higher category theory]] as [[composition]] of [[morphisms]] in [[(n,r)-categories]] and [[(infinity,n)-categories]] are not [[associative]], but rather satisfy [[coherence laws]] such as the [[pentagon identity]] which relate the various ways to nest parentheses around the [[tensor product]] magma in an [[(infinity,n)-category]] with respect to [[homotopy equivalence]]. Many other higher categorical objects have [[coherence theorems]] which also deal with nested parentheses around a binary operation.

## Extensional magmas

In [[foundations of mathematics]] such as [[intensional type theory|intensional]] Martin-Loef [[dependent type theory]], where [[functions]] do not [[function extensionality|preserve equality]], one could distinguish between ordinary magmas as defined above, and __extensional magmas__, whose binary operation preserves equality.

For a magma $(A, \cdot)$ and for all elements $a, b, c:A$, a magma is __left extensional__ if $a = c$ implies $a \cdot b = c \cdot b$, and a magma is __right extensional__ if $a = c$ implies $b \cdot a = b \cdot c$. A magma is __extensional__ if it is both left and right extensional.

## Related concepts

* [[magmoidal groupoid]]

* [[magmoidal category]]

* [[n-ary operation]]

[[!include oidification - table]]

## Literature

On the history of the notion:

* {#Hollings09} Christopher Hollings, *The Early Development of the Algebraic Theory of Semigroups*, Archive for History of Exact Sciences **63** (2009) 497–536 [[doi:10.1007/s00407-009-0044-3](https://doi.org/10.1007/s00407-009-0044-3)]

The wikipedia entry [magma](http://en.wikipedia.org/wiki/Magma_%28algebra%29) is quite useful, having a list and a table of various subclasses of magmas, hence of binary algebraic structures.

* [[eom]]: [magma](http://www.encyclopediaofmath.org/index.php?title=m/m110040)
* R.H. Bruck, _A survey of binary systems_, Springer-Verlag 1958

Formalization of magmas as [[mathematical structures]] in [[proof assistants]]:

in a context of plain [[Agda]]:

* [[Martín Escardó]], *[The Types of Magmas and Monoids](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#magmasandmonoids)*, §3.4 in: *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580),  [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html)]

in a context of [[cubical type theory|cubical]] [[Agda]]:

* [[1lab]]: *[Algebra.Magma](https://1lab.dev/Algebra.Magma.html)*

category: algebra

[[!redirects magmas]]

[[!redirects binary operation]]
[[!redirects binary operations]]

[[!redirects binary algebraic structure]]

[[!redirects magma object]]
[[!redirects magma objects]]

[[!redirects square function]]
[[!redirects square functions]]


magmoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Just as a [[groupoid]] is the [[oidification]] of a [[group]] and a [[ringoid]] is the oidification of a [[ring]], a [[magmoid]] should be the oidification of a [[magma]].

## Definition

### With a set of morphisms

Let $Q$ be a [[quiver]] with a collection of objects $Ob(Q)$, a set of morphisms $Mor(Q)$, a function $s: Mor(Q) \to Ob(Q)$ called the __source__ or __domain__ and a function $t: Mor(Q) \to Ob(Q)$ called the __target__ or __codomain__. A __magmoid__ is a quiver $Q$ with a [[partial map|partial binary operation]]

$$\left(-\right)\circ\left(-\right): Mor(Q) \times Mor(Q) \to Mor(Q) + 1$$

with coproduct injections $inj:Mor(Q) \to Mor(Q) + 1$ and $pt: 1 \to Mor(Q) + 1$, such that for $f, g \in Mor(Q)$, there is a function $h$ in the inverse image of $inj(h) = f \circ g$ if $s(f) = t(g)$, and $f \circ g = pt(\bullet)$ otherwise.

### With a family of sets of morphisms

Let $Q$ be a [[quiver]] with a collection of objects $Ob(Q)$ and a [[Set]]-valued functor $Mor: Ob(Q) \times Ob(Q) \to Set$ for all objects $a, b \in Ob(Q)$. A __magmoid__ is a quiver $Q$ with a binary operation

$$\left(-\right)\circ\left(-\right): Mor(b,c) \times Mor(a,b) \to Mor(a,c)$$

for all $a,b,c \in Ob(Q)$.

### Weak and strict magmoids

A __weak magmoid__ is a magmoid $Q$ whose collection of objects $Ob(Q)$ form a [[groupoid]], while a __strict magmoid__ is a magmoid $Q$ whose collection of objects $Ob(Q)$ form a [[set]].

### Enriched magmoids

Let $V$ be a [[monoidal category]] (or a [[monoidal (infinity,1)-category]]) and let $Q$ be a $V$-enriched [[quiver]], with a collection of objects $Ob(Q)$ and a $V$-valued functor $Mor: Ob(Q) \times Ob(Q) \to V$ for all objects $a, b \in Ob(Q)$. A __$V$-enriched magmoid__ is a $V$-enriched quiver $Q$ with a binary operation

$$\left(-\right)\circ\left(-\right): Mor(b,c) \times Mor(a,b) \to Mor(a,c)$$

for all $a,b,c \in Ob(Q)$.

## Examples

A [[transitive relation]] is a magmoid [[enriched category|enriched]] in [[truth values]], or a magmoid $M$ where there is at most one morphism from every object $a$ to another object $b$ in $M$.

[[!include oidification - table]]

## Related concepts

* [[magmoidal category]]
* [[H-magmoid]]

## References

The terminology seems to have been introduced in:

* A. Arnold, M. Dauchet, *Théorie des magmoïdes (I)*, RAIRO -- Informatique Théorique **12** 3 (1978) 235-257 [[numdam:ITA_1978__12_3_235_0](http://www.numdam.org/item/?id=ITA_1978__12_3_235_0), [eudml:92079](https://eudml.org/doc/92079)]

* A. Arnold, M. Dauchet, *Théorie des magmoïdes (II)*, RAIRO -- Informatique Théorique **13** 2 (1979) 135-154 [[numdam:ITA_1979__13_2_135_0](http://www.numdam.org/item/ITA_1979__13_2_135_0)]

Further discussion:

* Miklos Bartha, p. 101 in: *An algebraic model of synchronous systems*, Information and Computation
**97** 1 (1992) 97-131

* Dan Jonsson, Def. 2.2 in: *On Group-Like Magmoids* [[arXiv:1902.06109](https://arxiv.org/abs/1902.06109)]

* {#Torzewska22} [[Fiona Torzewska]], §2.1 of: *Topological quantum field theories and homotopy cobordisms* [[arXiv:2208.14504](https://arxiv.org/abs/2208.14504)]

[[!redirects magmoids]]

[[!redirects enriched magmoid]]
[[!redirects enriched magmoids]]


magmoidal category

[[!redirects magma category]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Just as a [[monoidal category]] is the [[categorification]] of a [[monoid]] and a [[rig category]] is the categorification of a [[rig]], a *magmoidal category* should be the categorification of a [[magma]].

## Definition

A __magmoidal category__ or __magmal category__ is a [[category]] $C$ with a [[functor]]

$$\otimes: C \times C \to C$$

from the [[product category]] of $C$ with itself.

If there is a specified object $I$, and natural isomorphisms $\lambda_A:\colon A\otimes I\stackrel{\simeq}{\to} A$ and $\rho_A:\colon I\otimes A\stackrel{\simeq}{\to} A$ satisfying $\rho_I = \lambda_I\colon I\otimes I\stackrel{\simeq}{\to} I$, then it is a **magmoidal category with unit**.

Note that this coherence condition appeared in [[Mac Lane]]'s original definition of a [[monoidal category]], but was proven to follow from the other coherence conditions for a monoidal category by [[Max Kelly]]. (The coherence condition usually placed on unitors in [[monoidal categories]] cannot even be stated in the context of a magmoidal category, since there is no associator to relate $A\otimes (I\otimes B)$ and $(A\otimes I)\otimes B$.)

## Examples

For example, [[monoidal categories]] such as [[braided monoidal categories]], [[symmetric monoidal categories]], [[cartesian monoidal categories]], [[cocartesian monoidal categories]], and [[rig categories]] are all magmoidal categories.

Every [[magma]] is a [[discrete category|discrete]] magmoidal category.

An [[Ackermann groupoid]] is a particular sort of [[poset]] that is magmoidal (where 'groupoid' is here a synonym for '[[magma]]').

## Related concepts

* [[magma]]

* [[magmoidal groupoid]]

* [[categorification]]

* [[magmoid]]

## References

* [[David Roberts]], *Substructural fixed-point theorems and the diagonal argument: theme and variations*, Compositionality **5** issue 8 (2023) doi:[10.32408/compositionality-5-8](https://doi.org/10.32408/compositionality-5-8),
([arXiv:2110.00239](https://arxiv.org/abs/2110.00239)).

The concept seems to first be named in:

* [[Alexei Davydov]], _Nuclei of categories with tensor products_, Theory and Applications of Categories, Vol. 18, 2007, No. 16, pp 440-472. [tac:18-16](http://www.tac.mta.ca/tac/volumes/18/16/18-16abs.html)

The terminology __magmal category__ first appears in:

* [[Ross Street]], _Wood fusion for magmal comonads_, [arXiv:2311.07088](https://arxiv.org/abs/2311.07088) (2023).

[[!redirects magmoidal categories]]
[[!redirects magmal category]]
[[!redirects magmal categories]]


magmoidal groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[category theory]], the notion of a *magmoidal groupoid* is the [[groupoidal categorification]] of the notion of *[[magma]]*, hence a magmoidal groupoid is simply a [[groupoid]] equipped with a "[[binary operation]]" on it.
Similarly, under the [[relation between category theory and type theory]],
in [[homotopy type theory]] a *magmoidal groupoid* is a [[1-truncated type]] equipped with a binary operation.

## Definitions

### In type theory

In [[intensional type theory]] with [[function extensionality]], a [[groupoid]] is a [[n-truncated|1-truncated]] [[type]] and a __binary operation__ on a [[groupoid]] $G$ is a [[function]] $(-)\otimes (-) \colon G \times G \to G$ from the [[product type|product groupoid]] $G \times G$ to $G$. A __magmoidal groupoid__ $(G,\otimes)$ is a [[groupoid]] equipped with a binary operation on it.

### In category theory

In [[category theory]], a [[groupoid]] is a [[dagger category]] where every [[morphism]] is [[unitary morphism|unitary]], and a __binary operation__ on a [[groupoid]] $G$ is a [[dagger functor]] $(-)\otimes (-) \colon G \times G \to G$ from the [[product groupoid]] $G \times G$ to $G$. A __magmoidal groupoid__ $(G,\otimes)$ is a [[groupoid]] equipped with a binary operation on it.

## See also

* [[magma]]
* [[groupoid]]
* [[H-spatial groupoid]]
* [[magmoidal category]]

[[!redirects magmoidal groupoids]]


magnetic charge

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Maxwell's equations for the [[electromagnetic field]] $F \in \Omega^2(X)$ on a $d$-dimensional [[spacetime]] $X$ in the presence of [[electric current]] $j_{el} \in \Omega^{d-1}(X)$ read

$$
  d F = 0
$$

$$
  d \star F = j_{el}
  \,.
$$

Formally this suggests an immediate generalization where a [[conserved current]] called the **magnetic current** $j_{mag} \in \Omega^{3}(X)$ is introduced and the first of the equations above is replaced by

$$
  d F = j_{mag}
  \,.
$$

The corresponding [[charge]] is the [[magnetic charge]].

One thing to notice about this is that

* net magnetic charge has not been observed in nature.

The other thing is a more conceptual problem: the [[electromagnetic field|Dirac quantization condition]] says that the $2$-form $F$ is not entirely arbitrary, but constrained to be the characteristic curvature $2$-form of a degree-$2$ cocycle in [[ordinary differential cohomology]], for instance the curvature $2$-form of a $U(1)$-[[principal bundle]] [[connection on a bundle|with connection]]. But this necessarily implies that $d F = 0$.

Indeed, to circumvent dealing with this problem Dirac, in his original argument, has considered removing from $X$ the support of any magnetic charge density.

It was [[Dan Freed]] in ([Freed](#Freed)) who discussed that the global description of the [[electromagnetic field]] does make sense even in the presence of electric current if one generalizes the model of a degree-$2$ [[ordinary differential cohomology|differential cocycle]] to that of a [[twisted cohomology|twisted]] cocycle.

The magnetic current $3$-form $j_{mag}$ is then realized as the curvature characteristic $3$-form of a degree-$3$ cocycle in [[ordinary differential cohomology]] $\hat j_{mag}$, the [[electromagnetic field]] $F$ is the curvature characteristic $2$-form of a degree-$2$ [[twisted cohomology|twisted]] differential cocycle $\hat F$ and the equation

$$
  d F = j_{mag}
$$

expresses the twisting of $\hat F$ by $\hat j_{mag}$ at the level of curvature forms.

This means that Dirac almost found [[bundle gerbe]]s already in 1931, had he not discussed only the neighbourhood of magnetic monopoles.

## Magnetic charge anomaly
 {#MagneticChargeAnomaly}

Short of an experimental detection of magnetic monopoles the above considerations are of little practical relevance for ordinary [[electromagnetism]]. In their (straightforward) generalization to higher abelian [[gauge theory]] they do, however, serve to provide a more complete conceptual picture that provides the conceptual bases for effects such as the [[Green–Schwarz mechanism]].

Namely from the very fact that in the presence of magnetic charge the gauge field $\hat F$ is a twisted cocycle, it follows that the standard [[path integral|action functional]] for electrically charged quantum particles -- the one whose consideration led Dirac to his Dirac quantization condition -- in general has a [[quantum anomaly]].

To see this, consider the following.

In the simplest case we consider a single electrically charged particle with worldline $\gamma : [0,1] \to X$ propagating inside a single patch $U_i  \to X$ of some [[cover]] over which the [[electromagnetic field]] is represented by a [[Deligne cohomology|Deligne cocycle]] with $1$-form $A_i \in \Omega^1(U_i)$. Then then action functional for the coupling of the particle to the background [[electromagnetic field]] is

$$
  (\hat F, \gamma) \mapsto
  \exp( -2 \pi i \int_\gamma A_i) \in U(1)
  \,.
$$

This may be rewritten by introducing the corresponding distributional $(d-1)$-form current $(j_{el})_i \in \Omega^{d-1}(U_i)$ as

$$
  (\hat F, \gamma) \mapsto
  \exp( -2 \pi i \int_{U_i} (j_{el})_i \wedge A_i) \in U(1)
  \,.
$$

This integrand may be recognized as the local connection $d$-form of the [[cup product]] of the Deligne $(d-1)$-cocycle $\hat {j_{el}}$ and the Deligne $2$-cocycle $\hat F$. This way

$$
  (\hat F, \gamma) \mapsto
  \exp( -2 \pi i \int_X \hat{j_{el}} \cdot \hat F) \in U(1)
  \,,
$$

where now the integral including the prefactor of $- 2 \pi i$ denotes push-forward in cohomology along $X \to {*}$, and we canonically identify degree-$0$ [[Deligne cohomology]] of the point with $U(1)$.

So far this is the formulation in [[differential cohomology]] of the ordinary action functional for the coupling of electric charges $\hat {j_{el}}$ to the [[electromagnetic field]] $\hat F$.

But in this formulation one can discuss what happens to this when a non-vanishing magnetic current $\hat {j_{mag}}$ is present. In that case $\hat F$ is no longer a plain $2$-cocycle, but a twisted $2$-cocycle. Accordingly, the above push-forward to the point produces not a $U(1)$-valued _function_ of $(\hat F, \hat {j_{el}})$, but a "twisted function", i.e. a section of a [[vector bundle|line bundle]].

Since $\hat F$ is twisted by $\hat {j_{mag}}$, this line bundle is the $2$-cocycle

$$
  \exp( -2 \pi i \int_X \hat{j_{el}} \cdot \hat {j_{mag}})  \,.
$$

(...discussion of how this is a $2$-cocycle on configuration space goes here...)

Comparing with the discussion at [[quantum anomaly]], it follows that the non-triviality of this $2$-cocycle is the _higher gauge theory anomaly_ of the ordinary action functional describing the coupling of electric charges to the [[electromagnetic field]] in the presense of magnetic charges.

This phenomenon has traditionally been known somewhat implicitly in the context of the [[Green–Schwarz mechanism]].

## Related entries

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

* [[electric charge]], **magnetic charge**

* [[color charge]]

* [[flux]]

* [[dyon]]

* [[magnetic field]]

* [[magnetic moment]]

## References

Magnetic charge for general [[compact Lie groups]] as [[gauge groups]] was first discussed in

* [[François Englert]], P. Windey, _Quantization Condition For 't Hooft Monopoles In Compact Simple Lie Groups_, Phys. Rev. D14 (1976) 2728-2731.

In

* {#GNO77} P. Goddard, J. Nuyts, and [[David Olive]], _Gauge Theories And Magnetic Charge_, Nucl. Phys. B125 (1977) 1-28.

it was first noticed that when [[electric charge]] takes values in the [[weight lattice]] of the [[gauge group]], then [[magnetic charge]] takes values in the lattice of what is now called the [[Langlands dual group]]. (This led to Montonen-Olive's [[S-duality]] [[conjecture]]).

The interpretation of magnetic charge in terms of degree-3 cocycles in [[ordinary differential cohomology]] is in the introduction-section of

* [[Dan Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_ ([arXiv](http://arxiv.org/abs/hep-th/0011220))

[[!redirects magnetic current]]
[[!redirects magnetic charges]]


magnetic dipole moment

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[anomalous magnetic moment]]

* [[electric dipole moment]]

* [[magnetic field]]

* [[magnetic charge]]

## References

### General

See also

* Wikipedia, _[Magnetic moment](https://en.wikipedia.org/wiki/Magnetic_moment)_

[[!redirects magnetic dipole moments]]

[[!redirects magnetic moment]]
[[!redirects magnetic moments]]


magnetic eigenbrane

#Contents#
* table of contents
{:toc}

## Idea

A [[brane]] that is taken to a tensor-multiple of itself under the action of a  [['t Hooft operators]].

[[!include geometric Langlands QFT -- table]]

## References

* {#KapustinWitten06} [[Anton Kapustin]], [[Edward Witten]], _Electric-Magnetic Duality And The Geometric Langlands Program_ , Communications in number theory and physics, Volume 1, Number 1, 1–236 (2007) ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))

[[!redirects magnetic eigenbranes]]


magnetic field

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

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] the [[electromagnetic field]] may be decomposed, over a choice of [[coordinate chart]] of [[spacetime]], into two components: the [[electric field]] and the **magnetic field**.

See there [there](electromagnetic+field#FieldStrengthAsClosed2Form) for more.

## Properties

[[!include flux quantization in superconductors -- section]]

## Related oncepts

* [[field (physics)]]

* [[magnetic charge]]

* [[magnetic dipole moment]]

## References

See also:

* Wikipedia, *[Magnetic field](https://en.wikipedia.org/wiki/Magnetic_field)*

In [[holographic QCD]]:

* Umut Gursoy, *Holographic QCD and magnetic fields* ([arXiv:2104.02839](https://arxiv.org/abs/2104.02839))

[[!redirects magnetic fields]]


magnetohydrodynamics

#Contents#
* table of contents
{:toc}

## Idea

__Magnetohydrodynamics__ is a physical description of a charged fluid (typically a plasma) in the presence of an [[electromagnetic field]].

## Related concepts

* [[electromagnetism]]

* [[hydrodynamics]]

## References

* [[Lev Landau]], [[Evgeny Lifshitz]], Section VIII of: *Electrodynamics of Continuous Media*, volume 8 of: *[[Course of Theoretical Physics]]*, Pergamon Press (1960, 1984) [[ISBN:9780750626347](https://shop.elsevier.com/books/electrodynamics-of-continuous-media/koch/978-0-08-057060-0), [archive](https://archive.org/details/ost-physics-landaulifshitz-electrodynamicsofcontinuousmedia/mode/2up),  [pdf](https://www.aldebaran.cz/studium/books/1984_Landau-Electrodynamics.pdf)]

See also:

* wikipedia: [magnetohydrodynamics](http://en.wikipedia.org/wiki/Magnetohydrodynamics)

* Søren Bertil F. Dorch, [Magnetohydrodynamics](http://www.scholarpedia.org/article/Magnetohydrodynamics) at scholarpedia

category: physics


magnitude of an enriched category

* table of contents
{: toc}

## Idea

The **magnitude** of an [[enriched category]] is a canonical measure of its size.

## Examples

- The magnitude of an ordinary category is the [[Euler characteristic]] of its [[nerve]].
- The magnitude of a [[set]] (as a [[discrete category]]) is its [[cardinality]].

## References

* [[Tom Leinster]], _The Euler characteristic of a category_ ([arXiv:0610260](http://arxiv.org/abs/math/0610260))
  {#Leinster}

* [[Tom Leinster]], _The Euler characteristic of a category as a sum of a divergent series_ ([arXiv:0707.0835](http://arxiv.org/abs/0707.0835))

- n-category café post on [The Magnitude of an Enriched Category](https://golem.ph.utexas.edu/category/2011/06/the_magnitude_of_an_enriched_c.html).

[[!redirects magnitude]]


Magnus Bakke Botnan

* [personal page](http://www.few.vu.nl/~botnan/)

## Selected writings

Proof of the [[stability of persistence diagrams|algebraic stability theorem]] for [[zigzag persistence modules]]:

* {#BotnanLesnick18} [[Magnus Bakke Botnan]], [[Michael Lesnick]], *Algebraic Stability of Zigzag Persistence Modules*, Algebr. Geom. Topol. **18** (2018) 3133-3204 $[$[arXiv:1604.00655](https://arxiv.org/abs/1604.00655), [doi:10.2140/agt.2018.18.3133](https://doi.org/10.2140/agt.2018.18.3133)$]$

category: people


Magnus Carlson

* [webpage](https://mathematics.huji.ac.il/people/magnus-carlson)

## Selected writings

On [[BF-theory]] in [[arithmetic geometry]]:

* [[Magnus Carlson]], [[Minhyong Kim]], _A note on abelian arithmetic BF-theory_, ([arXiv:1911.02236](https://arxiv.org/abs/1911.02236))

category: people


Magnus Forrester-Barker

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=93617)

## Selected writings

Review of [[internalization]] for the cases of [[group objects]], [[action objects]] and [[internal categories]]:

* [[Magnus Forrester-Barker]], *Group Objects and Internal Categories* [[arXiv:math/0212065](https://arxiv.org/abs/math/0212065)]

On [[2-representations]] of [[crossed modules]] on [[2-vector spaces]]:

* {#Forrester-Barker} [[Magnus Forrester-Barker]], *Representations of Crossed Modules and $Cat^1$-Groups*, PhD thesis U. Wales Bangor (2004) [[pdf](http://www.maths.bangor.ac.uk/research/ftp/theses/forrester-barker.pdf)]

category: people


Mahlo cardinal

\tableofcontents

## Idea

A [[Mahlo cardinal]] is a [[large cardinal]] in [[set theory]] needed to make the set theory equivalent to a [[type theory]], like [[Agda]], with [[type universes]] with [[inductive-recursive types]] inside of the universe.

## Definition

The [[ordinal]] $\kappa$ is Mahlo iff for any function $F : \kappa \to \kappa$, there is an [[inaccessible cardinal]] $\lambda$ such that $\lambda \lt \kappa$ and $\lambda$ is closed under $F$.

We can phrase this more structurally as follows: $\kappa$ is Mahlo iff for any functor $F : Core(Set_{\lt\kappa}) \to Core(Set_{\lt\kappa})$ on the category of sets strictly smaller than $\kappa$ and bijections, there is a [[universe]] $U$ with $|U| \lt \kappa$ which is closed under $F$.

## See also

* [[large cardinal]]

## External links

* Wikipedia, [Mahlo cardinal](https://en.wikipedia.org/wiki/Mahlo_cardinal)


Mahmoud Zeinalian

* [webpage](https://www.zeinalian.com/)

## Selected publications

On higher [[Hochschild homology]] and [[factorization algebras]]:

* [[Grégory Ginot]], [[Thomas Tradler]], [[Mahmoud Zeinalian]], _Derived higher Hochschild homology, topological chiral homology and factorization algebras_ ([arxiv/1011.6483](http://arxiv.org/abs/1011.6483))

Identifying the cobar construction on [[singular homology|singular chains]] on a [[topological space]] with the dg-algebra of chains on its [[loop space]]:

* [[Manuel Rivera]], [[Mahmoud Zeinalian]], *Cubical rigidification, the cobar construction, and the based loop space*, Algebr. Geom. Topol. **18** (2018) 3789-3820 [[arXiv:1612.04801](https://arxiv.org/abs/1612.04801), [doi:10.2140/agt.2018.18.3789](https://doi.org/10.2140/agt.2018.18.3789)]

On [[infinity-local systems|$\infty$-local systems]] in the sense of [[(infinity,1)-vector bundles|$(\infty,1)$-vector bundles]] with [[flat infinity-connections|flat $\infty$-connections]]:

* [[Manuel Rivera]], [[Mahmoud Zeinalian]], §5 of: *The colimit of an $\infty$-local system as a twisted tensor product*, Higher Structures **4** 1 (2020) 33-56  [[arXiv:1805.01264](https://arxiv.org/abs/1805.01264), [higher-structures:Vol4Iss1](https://higher-structures.math.cas.cz/api/files/issues/Vol4Iss1/RiveraZeinalian)]

On [[rational homotopy theory]] for possibly non-[[nilpotent spaces]] via [[deck transformation|deck]]-[[Borel-equivariant rational homotopy theory]]:

* [[Manuel Rivera]], [[Felix Wierstra]], [[Mahmoud Zeinalian]], *Rational homotopy equivalences and singular chains*, Algebr. Geom. Topol. **21** (2021) 1535-1552 [[arXiv:1906.03655](https://arxiv.org/abs/1906.03655). [doi:10.2140/agt.2021.21.1535](https://doi.org/10.2140/agt.2021.21.1535)]

On a kind of [[BV-quantization]] of the [[Loday-Quillen-Tsygan theorem]] and relating to the [[large N limit|large $N$-limit]] of [[Chern-Simons theory]]:

* [[Grégory Ginot]], [[Owen Gwilliam]], [[Alastair Hamilton]], [[Mahmoud Zeinalian]], _Large $N$ phenomena and quantization of the Loday-Quillen-Tsygan theorem_, Adv. Math. __409A__ (2022) 108631 [[arXiv:2108.12109](https://arxiv.org/abs/2108.12109), [doi:10.1016/j.aim.2022.108631](https://doi.org/10.1016/j.aim.2022.108631)]

Relating the [[Dwyer-Kan loop groupoid]]-construction to the [[homotopy coherent nerve]]-construction:

* {#MinichielloRiveraZeinalian23} [[Emilio Minichiello]], [[Manuel Rivera]], [[Mahmoud Zeinalian]], Def. 3.1 in: *Categorical models for path spaces*, Advances in Mathematics **415** (2023) 108898 [[arXiv:2201.03046](https://arxiv.org/abs/2201.03046), [doi:10.1016/j.aim.2023.108898](https://doi.org/10.1016/j.aim.2023.108898)]

Weak homotopy equivalences among path connected pointed topological spaces can be detected by singular chain functor $C_*$, provided $C_*(X)$ is taken with the structure of a dg-coalgebra,

* [[Manuel Rivera]], [[Felix Wierstra]], Mahmoud Zeinalian, _The functor of singular chains detects weak homotopy equivalences_, Proc. Amer. Math. Soc. 147 (2019) 4987--4998 [MR4011530](https://mathscinet.ams.org/mathscinet-getitem?mr=4011530) [doi](https://doi.org/10.1090/proc/14555) [arXiv/1807.06410](https://arxiv.org/abs/1807.06410)

[[!redirects M. Zeinalian]]

category: people


Maia Lesosky

* [Google Scholar](https://scholar.google.com/citations?user=80YvrH4AAAAJ&hl=en)

## Selected writings

[[operator algebra|Operator algebraic]] formulation of [[quantum error correction]]:

* [[David W. Kribs]], [[Raymond Laflamme]], [[David Poulin]], [[Maia Lesosky]], *Operator quantum error correction*, Quant. Inf. & Comp., 6 (2006), 383-399 ([arXiv:quant-ph/0504189](https://arxiv.org/abs/quant-ph/0504189))

category: people


main separation axioms -- as lifting properties

### In terms of lifting properties
 {#SeparationAxiomInTermsOfLiftingProperties}

The [[separation conditions]] $T_0$ to $T_4$ may equivalently be understood as [[lifting properties]] against certain maps of [[finite topological spaces]], among others.

This is discussed at *[[separation axioms in terms of lifting properties]]*, to which we refer for further details. Here we just briefly indicate the corresponding lifting diagrams.

In the following diagrams, the relevant [[finite topological spaces]] are indicated explicitly by illustration of their [[underlying]] point set and their [[open subsets]]:

* points (elements) are denoted by $\bullet$ with subscripts indicating where the points map to;

* boxes are put around [[open subsets]],

* an arrow $\bullet_u \to \bullet_c$  means that $\bullet_c$ is in the [[topological closure]] of $\bullet_u$.

In the lifting diagrams for $T_2-T_4$ below, an arrow out of the given [[topological space]] $X$ is a [[map]] that  determines (classifies) a decomposition of $X$  into a [[union]] of [[subsets]] with properties indicated by the picture of the finite space.

Notice that the diagrams for $T_2$-$T_4$ below do not in themselves imply [[T1|$T_1$]].

\begin{proposition}
**(Lifting property encoding $T_0$)**
\linebreak
The following [[lifting property]] in [[Top]] equivalently encodes the [[separation axiom]] [[T0|$T_0$]]:

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
    \{\boxed{\bullet_0\leftrightarrow \bullet_1}\}
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[dd]
  &&
  X
  \ar[dd,"{ (T_0) }"]
  \\
  \\
  \bullet_{0=1}
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \bullet
\end{tikzcd}

\end{proposition}

\begin{proposition}
**(Lifting property encoding $T_1$)**
\linebreak
The following [[lifting property]] in [[Top]] equivalently encodes the [[separation axiom]] [[T1|$T_1$]]:

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
\{\boxed{ \overset{\boxed{\bullet_{0}}}{}\searrow\underset{\bullet_1}{} }\}
%    \{\boxed{ \boxed{ {\bullet_0} }\rightarrow{\bullet_1} }\}
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[dd]
  &&
  X
  \ar[dd,"{ (T_1) }"]
  \\
  \\
  \bullet_{0=1}
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \bullet
\end{tikzcd}
\end{proposition}

\begin{proposition}
**(Lifting property encoding $T_2$)**
\linebreak
The following [[lifting property]] in [[Top]] equivalently encodes the [[separation axiom]] [[T2|$T_2$]]:

\begin{tikzcd}
[
  column sep={between origins, 80pt},
  row sep={between origins, 40pt}
]
  \boxed{\{\boxed{\bullet_x}, \boxed{\bullet_y}\}}
  \ar[
    rr,
    "{ }"
  ]
  \ar[dd, hook, "{ \forall  }"{left}, ,"{ (T_2) }"{right} ]
  &&
  \boxed{
    \overset{
      \boxed{
        \boxed{\bullet_x}
        \;\;
        \,
        \;\;
        \boxed{\bullet_y}}
      }{
        \underset{
           \bullet_X
        }
        {
          \searrow \;\, \swarrow
        }
      }
  }
  \ar[dd]
  \\
  \\
  X
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
   \{\boxed{\bullet_{x=X=y}}\}
\end{tikzcd}
\end{proposition}

\begin{proposition}
**(Lifting property encoding $T_3$)**
\linebreak
The following [[lifting property]] in [[Top]] equivalently encodes the [[separation axiom]] [[T3|$T_3$]]:

\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \{\boxed{\bullet_x} \}
  \ar[dd,"{ \forall }"{left}, "{ (T_3) }"{right}]
  \ar[rr]
  &&
\{
\boxed{
 \boxed{
\overset{\boxed{\bullet_x}}{}
\searrow
\underset{\bullet_X}{} \swarrow
\overset{\boxed{\bullet_U}}{}
}\!\!\!\!\!\!\!
{\,\,\,\,\,\,\searrow\underset{\bullet_F}{} }
}
\}
  \ar[dd]
  \\
  \\
  X
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
      \{\boxed{ \overset{\boxed{\bullet_{x=X=U}}}{}\searrow\underset{\bullet_F}{} }\}
\end{tikzcd}
\end{proposition}

\begin{proposition}
**(Lifting property encoding $T_4$)**
\linebreak
The following [[lifting property]] in [[Top]] equivalently encodes the [[separation axiom]] [[T4|$T_4$]]:

\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[dd, "{ (T_4) }"{right}]
  \ar[rr]
  &&
\boxed{
\boxed{\underset{\bullet_x}{}\swarrow
\,\,\,\,\,\,\,}
\!\!\!\!\!\!\!\!
\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\,\,\,\,
\,\,\,\,\,}
\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!
\!\!\!\!
\boxed{
\boxed{
\overset{\boxed{\bullet_u}}{}
\searrow
\underset{\bullet_X}{} \swarrow
\boxed{
\overset{\boxed{\bullet_v}}{}
\!\!\!\!\!\!\!
{\,\,\,\,\,\,\searrow\underset{\bullet_y}{} }
}\!\!\!\!\!\!\!%%%%%%%%
\!\!\!\!\!}
%\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,}
  \ar[dd]
  \\
  \\
  X
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \{
\boxed{
\underset{\bullet_x}{}{\swarrow}
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
%\,\,
%\,\,\,\,\,\,\,
}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
%\!\!%\!\!\!\!\!\!\!
\boxed{\overset{\boxed{\bullet_{U=X=V}}}{}
{\searrow}
\underset{\bullet_y}{}
}
\}
\end{tikzcd}
\end{proposition}


main separation axioms -- table

the **main [[separation axioms]]**

| number | name | statement | reformulation |
|--------|------|-----------|---------------|
| $T_0$  | [[Kolmogorov space|Kolmogorov]] | given two distinct points, at least one of them has an [[open neighbourhood]] not containing the other point |  every [[irreducible closed subset]] is the [[topological closure|closure]] of at most one point |
| $T_1$  |      | given two distinct points, both have an [[open neighbourhood]] not containing the other point | all points are [[closed point|closed]] |
| $T_2$  | [[Hausdorff topological space|Hausdorff]] | given two distinct points, they have [[disjoint subset|disjoint]] [[open neighbourhoods]] | the [[diagonal]] is a [[closed map]]  |
| $T_{\gt 2}$ |  |  $T_1$ and... | all points are [[closed point|closed]] and... |
| $T_3$  | [[regular Hausdorff topological space|regular Hausdorff]] |  ...given a point and a [[closed subset]] not containing it, they have [[disjoint subset|disjoint]] [[open neighbourhoods]]  | ...every [[neighbourhood]] of a point contains the [[topological closure|closure]] of an [[open neighbourhood]] |
| $T_4$  | [[normal Hausdorff topological space|normal Hausdorff]] | ...given two [[disjoint subset|disjoint]] [[closed subsets]], they have [[disjoint subset|disjoint]] [[open neighbourhoods]] | ...every [[neighbourhood]] of a [[closed set]] also contains the [[topological closure|closure]] of an [[open neighbourhood]] <br/> ... every pair of [[disjoint subset|disjoint]] [[closed subsets]] is separated by an [[Urysohn function]] |


main theorem of classical Galois theory

+-- {: .un_theorem }
###### Main theorem of classical Galois theory

Let $K \subset L$ be a [Galois extension](#GaloisExtension) of fields with [Galois group](#GaloisGroup) $G$. Then the intermediate fields of $K \subset L$ correspond bijectively to the closed subgroups of $G$.

More precisely, the maps

$$
  \{E | E\;is\;a\;subfield\;of\;L\;containing\;K\}
  \stackrel{\overset{\phi}{\to}}{\underset{\psi}{\leftarrow}}
   \{H|H\;is\;a\;closed\;subgroup\;of\;G\}
$$

defined by

$$
  \phi(E) = Aut_E(L)
$$

and

$$
  \psi(H) = L^H
$$

are bijective and inverse to each other. This correspondence reverses the inclusion relation: $K$ corresponds to $G$ and $L$ to $\{id_L\}$.

If $E$ corresponds to $H$, then we have

1. $K \subset E$ is finite precisely if $H$ is open (in the profinite topology on $G$)

   $[E:K] \simeq index[G:K]$ if $H$ is open;

1. $E \subset L$ is Galois with $Gal(L/E) \simeq H$ (as [[topological group]]s);

1. for every $\sigma \in G$ we have that $\sigma[E]$ corresponds to $\sigma H \sigma^{-1}$;

1. $L \subset E$ is Galois precisely if $H$ is a [[normal subgroup]] of $G$;

   $Gal(E/K) \simeq G/H$ (as [[topological group]]s) if $K \subset E$ is Galois.

=--

This appears for instance as [Lenstra, theorem 2.3](#Lenstra).

This suggests that more fundamental than the subgroups of a Galois group $G$ are its quotients by subgroups, which can be identified with transitive $G$-sets.  This naturally raises the question of what corresponds to non-transitive $G$-sets.

category: Galois theory


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

#Contents#
* table of contents
{:toc}

## Idea

A central result in the construction of [[perturbative quantum field theories]] via the method of [[causal perturbation theory]] is called the _main theorem of perturbative renormalization theory_. This says that given a suitable [[free field theory|free field]] [[vacuum]] to be [[perturbation theory|perturbed]] ([this def.](S-matrix#VacuumFree)), then any two _[[renormalization schemes]]_ for [[perturbative quantum field theory]] around this free field theory, hence any two solutions $\mathcal{S}$, $\mathcal{S}'$ to the [[induction|inductive]] construction of the [[perturbative S-matrix]] scheme as a [[function]]

$$
  \mathcal{S}
   \;\colon\;
  LocObs(E)[ [\hbar , g ] ]\langle g , j\rangle
    \longrightarrow
  PolyObs(E)_{mc}((\hbar))[ [ g, j] ]
$$

from [[local observables]] $g S_{int} + j A$, regarded as [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functionals]], to [[scattering amplitude]] [[Wick algebra]] elements $S\mathcal{S}(\mathbf{L}_{int})$, are related by a unique _perturbative_ transformation

$$
  \mathcal{Z}
    \;\colon\;
  LocObs(E)[ [\hbar, g, j] ]\langle g, j\rangle
    \longrightarrow
  LocObs(E)[ [\hbar, g, j] ]\langle g, j\rangle
$$

of the space of [[local observable|local]] [[interaction]] [[action functionals]] via [[precomposition]]

$$
  \mathcal{S}'
    \;=\;
  \mathcal{S} \circ \mathcal{Z}
  \,.
$$

The collection of these operations $\mathcal{Z}$ forms a [[group]], called the _[[Stückelberg-Petermann renormalization group]]_. Hence the space of [[renormalization schemes]] is a [[torsor]] over this group.

The precise nature of this group depends on which set of _[[renormalization conditions]]_ one imposes. The larger this set, the smaller the corresponding [[renormalization group]] ([Dütsch 18, remark 3.102](#Duetsch18)).

Beware the terminology: Contrary to common practice, the construction of a single $\mathcal{S}$ is more properly called a choice of _normalization_ rather a "re"-normalization (e. g. [Scharf 95, section 4.3](causal+perturbation+theoryscatt#Scharf95)), but the "main theorem" above says that the elements in the [[Stückelberg-Petermann renormalization group]] are precisely that: _re_-normalizations, passing from one choice of normalization to another.

## Details

See at _[[renormalization]]_ _[this theorem](renormalization#PerturbativeRenormalizationMainTheorem)_.

## References

The theorem is originally due to

* G. Popineau, [[Raymond Stora]], _A Pedagogical Remark on the Main Theorem of Perturbative Renormalization Theory, Nucl. Phys. B 912 (2016), 70–78, preprint:
LAPP–TH, Lyon (1982)

In various variants it has been (re-)proved in the following articles:

* G. Pinter, section 6 of _The Action Principle in Epstein Glaser Renormalization and Renormalization of the S-Matrix in $\phi^4$-Theory_, Annalen Phys. 10 (2001) 333-363 ([arXiv:hep-th/9911063](https://arxiv.org/abs/hep-th/9911063))

* [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], section 4.1 of _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics ([arXiv:0901.2038](https://arxiv.org/abs/0901.2038))

* [[Michael Dütsch]], [[Klaus Fredenhagen]], section 4.2 of _Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity_,  Rev. Math. Phys. 16, 1291 (2004) ([arXiv:hep-th/0403213](https://arxiv.org/abs/hep-th/0403213))

* [[Stefan Hollands]], theorem 2 in _Renormalized Quantum Yang-Mills Fields in Curved Spacetime_, Rev. Math. Phys. 20:1033-1172, 2008 ([arXiv:0705.3340](https://arxiv.org/abs/0705.3340))

Review is in

* {#Rejzner16} [[Katarzyna Rejzner]], section 6.3 of _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

* [[Michael Dütsch]], section 3.5.2 of _[[From classical field theory
to perturbative quantum field theory]]_, 2018

[[!redirects main theorem of perturbative renormalization theory]]


Maissam Barkeshli

* [institute page](https://umdphysics.umd.edu/people/faculty/current/item/718-maissam.html)

## Selected writings

On [[symmetry protected topological phases]] and their [[topological orders]]:

* {#BBCW19} [[Maissam Barkeshli]], [[Parsa Bonderson]], [[Meng Cheng]], [[Zhenghan Wang]], *Symmetry Fractionalization, Defects, and Gauging of Topological Phases*, Phys. Rev. B **100** 115147 (2019) $[$[arXiv:1410.4540](https://arxiv.org/abs/1410.4540), [doi:10.1103/PhysRevB.100.115147](https://doi.org/10.1103/PhysRevB.100.115147)$]$

category: people


Majorana dimer

#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]] and [[quantum many-body physics]], and specifically in the context of [[topological phases of matter]],
by a *[[Majorana dimer]]* one means a [[dimer]] $a \to b$ (an [[edge]] in some [[graph]])  that is labeled by an [[entangled state]] $\psi_{a b}$ of two [[qbits]] ("spins") characterized by the equation

$$
  \big(
    \gamma_a + i p_{a b} \gamma_b
  \big)
  \left\vert \psi \right\rangle
  \;=\;
  0
  \,,
$$

where

* $\gamma_i = f^\dagger_i + f_i$ are [[Clifford algebra]] [[operators]] defined in terms of the [[canonical anti-commutation relations]] $\{f_j, f^\dagger_j\} = \delta_{i j}$ that define [[qbit]] [[spaces of quantum states]] $\simeq \mathbb{C}^2$ at each of any given lattice sites $i,j, \cdots$,

* $p_{a b} \in \{\pm 1\}$ is a sign (a direction/orientation of the dimer)

(e.g. [JGPE 19, III.A](#JGPE19))

## References

The notion is due to:

* Brayden Ware, Jun Ho Son, [[Meng Cheng]], Ryan V. Mishmash, [[Jason Alicea]], and Bela Bauer, *Ising anyons in frustration-free Majorana-dimer models*, Phys. Rev. B 94, 115127 2016 ([arXiv:10.1103/PhysRevB.94.115127](https://doi.org/10.1103/PhysRevB.94.115127))

  (in relation to [[Ising anyons]])

* Nicolas Tarantino and Lukasz Fidkowski, *Discrete spin structures and commuting projector models for two-dimensional fermionic symmetry-protected topological phases*, Phys. Rev. B 94, 115115 2016 ([doi:10.1103/PhysRevB.94.115115](https://doi.org/10.1103/PhysRevB.94.115115))

Majorana dimer states are used (not under that name though) for discussion of the [[SYK model]] and its [[AdS2-CFT1 duality|AdS2/CFT1 dual]] in:

* Ioanna Kourkoulou, [[Juan Maldacena]], around (2.2) of: *Pure states in the SYK model and nearly-$AdS_2$ gravity* ([arXiv:1707.02325](https://arxiv.org/abs/1707.02325))

Application of Majorana dimers to [[quantum error correcting codes]] and [[holographic entanglement entropy]] in the [[Majorana dimer code]]:

* {#JGPE19} [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], *Majorana dimers and holographic quantum error-correcting code*, Phys. Rev. Research 1, 033079 (2019) ([arXiv:1905.03268](https://arxiv.org/abs/1905.03268))

[[!redirects Majorana dimers]]