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Alexander Campbell | https://ncatlab.org/nlab/source/Alexander+Campbell | Alexander Campbell is an Australian category theorist.
* [webpage](https://acmbl.github.io/)
## Selected writings
On [[gerbe (in nonabelian cohomology)|gerbes]] in [[nonabelian cohomology]] via [[tricategories]]:
* [[Alexander Campbell]], _A higher categorical approach to Giraud's non-abelian cohomology_, PhD thesis, Macquarie University (2016) [[hdl:1959.14/1261186](http://hdl.handle.net/1959.14/1261186)]
On [[double categories]]:
* {#CampbellGreg} [[Alexander Campbell]], *The gregarious model structure for double categories* (2020) [[talk slides pdf](https://acmbl.github.io/greg_slides.pdf)]
On the [[straightening theorem]]:
* [[Alexander Campbell]], *A modular proof of the straightening theorem*, talk at Macquarie University (2020) [[pdf](https://acmbl.github.io/straight_slides.pdf), [[Campbell-ModularProofofStraightening.pdf:file]]]
category: people |
Alexander Efimov | https://ncatlab.org/nlab/source/Alexander+Efimov | [[!redirects Alexander I. Efimov]]
__Alexander I. Efimov__ is a mathematician at Steklov Mathematical Institute in Moscow and has involvement into the laboratory for algebraic geometry ([Efimov page](http://www.hse.ru/en/org/persons/26335030)) which is lead by [[Fedor Bogomolov]].
* _Cohomological Hall algebra of a symmetric quiver_, Compositio Mathematica __148__:4 (2012) 1133-1146 [doi](https://doi.org/10.1112/S0010437X12000152) [arXiv:1103.2736](https://arxiv.org/abs/1103.2736)
* other [arXiv papers](http://arxiv.org/find/math/1/au:+Efimov_A/0/1/0/all/0/1)
[[!redirects Alexander I. Efimov]] |
Alexander Engel | https://ncatlab.org/nlab/source/Alexander+Engel |
* [webpage](http://www.uni-regensburg.de/mathematik/mathematik-engel/)
## Selected writings
On ([[equivariant K-theory|equivariant]]) [[KK-theory]] as the [[homotopy category of an (infinity,1)-category|homotopy category]] of a [[stable (infinity,1)-category|stable $(\infty,1)$-category]]:
* {#BunkeEngelLand21} [[Ulrich Bunke]], [[Alexander Engel]], [[Markus Land]], *A stable $\infty$-category for equivariant KK-theory* $[$[arXiv:2102.13372](https://arxiv.org/abs/2102.13372)$]$
## Related entries
* [[bornological coarse space]]
category: people |
Alexander Givental | https://ncatlab.org/nlab/source/Alexander+Givental |
* [webpage](http://math.berkeley.edu/~giventh/)
## Selected writings
On [[quantum cohomology rings]]:
* [[Alexander Givental]], *A tutorial on Quantum Cohomology* [[pdf](https://math.berkeley.edu/~giventh/papers/lqc.pdf), [[Givental-QuantumCohomologyTutorial.pdf:file]]]
Introducing a [[quantum K-theory]]-analog of [[quantum cohomology rings]]:
* [[Alexander B. Givental]], *On the WDVV-equation in quantum K-theory*, Michigan Math. J. **48** 1 (2000) 295-304 [[arXiv:math/0003158](https://arxiv.org/abs/math/0003158), [doi:10.1307/mmj/1030132720](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-48/issue-1/On-the-WDVV-equation-in-quantum-K-theory/10.1307/mmj/1030132720.full)]
On [[quantum K-theory rings]] for some form of [[equivariant K-theory]]:
* [[Alexander B. Givental]], *Permutation-equivariant quantum K-theory* [[webpage](https://math.berkeley.edu/~giventh/perm/perm.html)]
*I. Definitions. Elementary K-theory of $\overline{\mathcal{M}}_{0,n}/S_n$* [[arXiv:1508.02690](https://arxiv.org/abs/1508.02690)]
*II. Fixed point localization* [[arXiv:1508.04374](https://arxiv.org/abs/1508.04374)]
*III. Lefschetz' formula on $\overline{\mathcal{M}}_{0,n}/S_n$ and adelic characterization* [[arXiv:1508.06697](https://arxiv.org/abs/1508.06697)]
*IV. $D_q$-modules* [[arXiv:1508.06697](https://arxiv.org/abs/1508.06697)]
*V. Toric $q$-hypergeometric functions* [[arXiv:1509.03903](https://arxiv.org/abs/1509.03903)]
*VI. Mirrors* [[arXiv:1509.07852](https://arxiv.org/abs/1509.07852)]
*VII. General theory* [[arXiv:1510.03076](https://arxiv.org/abs/1510.03076)]
*VIII. Explicit reconstruction* [[arXiv:1510.06116](https://arxiv.org/abs/1510.06116)]
*IX. Quantum Hirzebruch-Riemann-Roch in all genera* [[arXiv:1709.03180](https://arxiv.org/abs/1709.03180)]
*X. Quantum Hirzebruch-Riemann-Roch in genus 0*, SIGMA **16** 031 (2020) [[arXiv:1710.02376](https://arxiv.org/abs/1710.02376), [doi:10.3842/SIGMA.2020.031](https://doi.org/10.3842/SIGMA.2020.031)]
*XI. Quantum Adams-Riemann-Roch* [[arXiv:1711.04201](https://arxiv.org/abs/1711.04201)]
category: people
[[!redirects Alexander B. Givental]]
|
Alexander Gnedin | https://ncatlab.org/nlab/source/Alexander+Gnedin |
* [institute page](https://www.qmul.ac.uk/maths/profiles/gnedina.html)
* [Mathematics Genealogy page](https://www.qmul.ac.uk/maths/profiles/gnedina.html)
## Selected writings
On [[character of a linear representation|characters]] on the [[symmetric group]] depending only on [[Cayley distance]]:
* {#GnedinGorinKerov11} [[Alexander Gnedin]], [[Vadim Gorin]], [[Sergei Kerov]], *Block characters of the symmetric groups*, Journal of Algebraic Combinatorics, 38, no. 1 (2013), 79-101 ([arXiv:1108.5044](https://arxiv.org/abs/1108.5044), [doi:10.1007/s10801-012-0394-9](https://doi.org/10.1007/s10801-012-0394-9))
category: people
|
Alexander Goncharov | https://ncatlab.org/nlab/source/Alexander+Goncharov |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Alexander_Goncharov)
category: people |
Alexander Gorsky | https://ncatlab.org/nlab/source/Alexander+Gorsky |
* [InSpire page](https://inspirehep.net/authors/1007902)
## Selected writings
On [[Seiberg-Witten theory]] in relation to [[integrable systems]]:
* [[Alexander Gorsky]], [[Igor Krichever]], [[Andrei Marshakov]], [[Andrei Mironov]], [[Andrey Morozov]], *Integrability and Seiberg-Witten Exact Solution*, Phys. Lett. B **355** (1995) 466-474 [[arXiv:hep-th/9505035](https://arxiv.org/abs/hep-th/9505035), <a href="https://doi.org/10.1016/0370-2693(95)00723-X">doi:10.1016/0370-2693(95)00723-X</a>]
category: people
|
Alexander Green | https://ncatlab.org/nlab/source/Alexander+Green |
* [haskellers page](https://www.haskellers.com/user/alexandersgreen)
* [github page](https://github.com/alexandersgreen)
## Selected writings
Introducing the [[quantum IO monad]]:
* [[Alexander Green]], *The Quantum IO Monad*, Nottingham (2007) [[pdf](http://drinkupthyzider.co.uk/asg/pdfs/bctcs07.pdf), [[Green-QIOMonad.pdf:file]] ]
* {#AltenkirchGreen10} [[Thorsten Altenkirch]], [[Alexander Green]], *The quantum IO monad*, Ch. 5 of: Simon Gay, Ian Mackie (eds.): *Semantic Techniques in Quantum Computation* (2010) 173-205 [[pdf](http://www.cs.nott.ac.uk/~txa/publ/qio-chapter.pdf), [doi:10.1017/CBO9781139193313.006](https://doi.org/10.1017/CBO9781139193313.006)]
Implementation in [[Haskell]]:
* [[Alexander Green]], *[hackage.haskell.org/package/QIO](https://hackage.haskell.org/package/QIO)*
On [[software verification|formally verified]] [[quantum programming languages]] like [[QML]]:
* [[Alexander Green]], *Towards a formally verified functional quantum programming language* (2010) [[eprints:11457](http://eprints.nottingham.ac.uk/11457)]
Introducing the [[functional programming language|functional]] [[quantum programming language]] [[Quipper]]:
* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: A Scalable Quantum Programming Language_, ACM SIGPLAN Notices 48(6):333-342, 2013 ([arXiv:1304.3390](https://arxiv.org/abs/1304.3390))
* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _An Introduction to Quantum Programming in Quipper_, Lecture Notes in Computer Science 7948:110-124, Springer, 2013 ([arXiv:1304.5485](https://arxiv.org/abs/1304.5485))
category: people |
Alexander Grothendieck | https://ncatlab.org/nlab/source/Alexander+Grothendieck |
The European mathematician __Alexander Grothendieck__ (in French sometimes Alexandre Grothendieck), created a very influential body of work foundational for ([[algebraic geometry|algebraic]]) [[geometry]] but also for modern [[mathematics]] more generally. He is widely regarded as a singularly important figure of 20th century mathematics and his ideas continue to be highly influential in the 21st century.
* [Wikipedia article](http://en.wikipedia.org/wiki/Alexander_Grothendieck)
Initially working on [[topological vector spaces]] and [[analysis]], Grothendieck then made revolutionary advances in [[algebraic geometry]] by developing [[sheaf and topos theory]] and [[abelian sheaf cohomology]] and formulating algebraic geometry in these terms ([[locally ringed spaces]], [[schemes]]). Later [[topos theory]] further developed independently and today serves as the foundation also for other kinds of [[geometry]]. Notably its [[homotopy theory|homotopy theoretic]] refinement to [[higher topos theory]] serves as the foundation for modern [[derived algebraic geometry]].
#Contents#
* table of contents
{:toc}
## Selected writings
Grothendieck's geometric work is documented in texts known as [[EGA]] (with [[Dieudonné]]), an early account [[FGA]], and the many volume account [[SGA]] of the seminars at l'IHÉS, Bures-sur-Yvette, where he was based at the time. (See the [wikipedia article](http://en.wikipedia.org/wiki/Alexander_Grothendieck) for some indication of the story from there until the early 1980s.)
On [[fiber bundles]] and [[nonabelian cohomology|nonabelian]] [[Čech cohomology]]:
* [[Alexander Grothendieck]], *A General Theory of Fibre Spaces With Structure Sheaf*, University of Kansas, Report No. 4 (1955, 1958) [[pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Kansasnotes.pdf), [[Grothendieck-FibreSpaces.pdf:file]]]
In the late 1970s and early 1980s Grothendieck wrote several documents that have been of outstanding importance in the origins of the theory that underlies the [[nPOV]]. These include
* _La Longue Marche à travers la Théorie de Galois_ (1600 manuscript pages written between January and June 1981, plus addenda etc. which double its length!) (see [[Long March]] for some discussion of the ideas.)
* _[[Esquisse d'un programme]]_, (January 1984), in which Grothendieck sketches out a vaste programme of research, incorporating many of the ideas from [[Long March]]. A copy is available [here](http://www.math.jussieu.fr/~leila/grothendieckcircle/EsquisseFr.pdf). It is discussed in brief at [[Esquisse d'un programme|Grothendieck's Esquisse]].
* _À la poursuite des Champs_ (also entitled ''[[Pursuing Stacks]]'').
It starts with a short (12 page) letter to Quillen, dated 19 Feb. 1983, but then discusses a wide ranging vision of homotopy theory and its applicability to problems in algebraic and arithmetic geometry.
* _[[Les Dérivateurs]]_ (another 2000 page manuscript taking up some of the themes in Pursuing Stacks, section 69) Dating from the end of 1990 and the start of 1991.
In the same time he also wrote voluminous intellectual memoirs _Récoltes et Semailles_.
En Guise de Programme [p I](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Guise1.jpg) [p II](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Guise2.jpg), a text written by Grothendieck as a course description while teaching in Montpellier "Introduction à la recherche".
A chronological bibliography of Grothendieck's published mathematical writings ([pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/GrothBiblio.pdf)).
## Texts about Grothendieck
For an account of his work, including some of the work published in the 1980s, see the [English Wikipedia entry](http://en.wikipedia.org/wiki/Alexander_Grothendieck).
The video of a talk by W. Scharlau on his life can be seen [here](http://www.dailymotion.com/video/x8juek_colloque-grothendieck-winfried-scha_tech).
* Reminiscences of Grothendieck, 2007 conversations of [[Sasha Beilinson]], [[Luc Illusie]], [[Vladimir Drinfel'd]], [[Spencer Bloch]], ([pdf](http://www.math.uchicago.edu/~mitya/langlands/reminiscences1.pdf))
* [[Pierre Cartier]], _Alexander Grothendieck: A Country Known Only by Name_ ([pdf](http://www.ams.org/notices/201504/rnoti-p373.pdf))
* [[Luca Barbieri-Viale]], _Alexander Grothendieck: Enthusiasm and creativity_ ([pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Mathbiographies/barbierieng.pdf))
* [[Pierre Deligne]], _Quelques idées maîtresses de l'œuvre de A Grothendieck_ ([pdf](http://www.math.jussieu.fr/~leila/grothendieckcircle/Mathbiographies/Deligne.pdf))
* [[Luc Illusie]], _Alexandre Grothendieck, le magicien des foncteurs_ ([pdf](http://www.cnrs.fr/insmi/IMG/pdf/Alexandre-Grothendieck.pdf))
* _Une entrevue avec [[Jean Giraud]], à propos d'Alexandre Grothendieck_ ([pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/giraud.pdf)).
A recent article in French on Grothendieck is to be found [here](http://images.math.cnrs.fr/Alexandre-Grothendieck.html).
There were two articles on Grothendieck's life and work in the Notices AMS in 2004:
* Allyn Jackson, _Comme Appelé du Néant_, As If Summoned from the Void:
The Life of Alexandre Grothendieck, Part 1, [Notices AMS](http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf)
* Allyn Jackson, _Comme Appelé du Néant_, As If Summoned from the Void:
The Life of Alexandre Grothendieck, Part 2, [Notices AMS](http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf)
Grothendieck obituary in the Notices AMS (Michael Artin, Allyn Jackson, David Mumford, and John Tate, Coordinating Editors):
* "_The obituary begins here with a brief sketch of Grothendieck's life,
followed by a description of some of his most outstanding work in mathematics._" Alexandre Grothendieck 1928–2014, ([Part 1] (http://www.ams.org/journals/notices/201603/rnoti-p242.pdf))
* "_set of reminiscences by some of the many mathematicians who knew Grothendieck and were influenced by him._" Alexandre Grothendieck 1928–2014, ([Part 2](http://www.ams.org/publications/journals/notices/201604/rnoti-p401.pdf)).
## Students
* [[Pierre Gabriel]]
* [[Michel Demazure]]
* [[Jean Giraud]]
* [[Jean-Louis Verdier]]
* [[Monique Hakim]]
* [[Michel Raynaud]]
* [[Jean-Pierre Jouanolou]]
* [[Luc Illusie]]
* [[William Messing]]
* [[Pierre Berthelot]]
* [[Pierre Deligne]]
* [[Michèle Raynaud]]
* [[Neantro Saavedra-Rivano]]
* [[Hamet Seydi]]
* [[Hoàng Xuân Sính]]
* [[Yves Ladegaillerie]]
* [[Marcus Wanderley]]
* [[Carlos Contou-Carrère]]
See _[Mathematics Genealogy for Grothendieck](http://www.genealogy.ams.org/id.php?id=31245)_
## Correspondence
[The Grothendieck-Serre correspondence](http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Letters/GS.pdf)
[The Grothendieck-Mumford correspondence](http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Letters/AGMumford.pdf)
## External links
[Grothendieck circle](http://www.grothendieckcircle.org/)
[Grothendieck's Angle] (https://www.dropbox.com/s/9u7n6xkwytocogf/grothendieckangle.pdf?dl=0) by G. Aiello
[A. Grothendieck](http://matematicas.unex.es/~navarro/res/) by J. A. Navarro
[A. Grothendieck](https://agrothendieck.github.io/) by M. Carmona
A. Grothendieck, una guía a la obra matemática y filosófica ([pdf](http://matematicas.unex.es/~navarro/res/zalamea.pdf)) por F. Zalamea
## Quotes
On [[K-theory]]:
> The way I first visualized a K-group was as a group of "classes of objects" of an abelian (or more generally, additive) category, such as coherent sheaves on an algebraic variety, or vector bundles, etc. I would presumably have called this group $C(X)$ ($X$ being a variety or any other kind of "space"), $C$ the initial letter of 'class', but my past in functional analysis may have prevented this, as $C(X)$ designates also the space of continuous functions on $X$ (when $X$ is a topological space). Thus, I reverted to $K$ instead of $C$, since my mother tongue is German, Class = Klasse (in German), and the sounds corresponding to $C$ and $K$ are the same.
from Grothendieck's letter to Bruce Magurn, on 9th February 1985, quoted after:
* {#Bak87} A. Bak, Editorial, K-theory 1 (1987), 1 ([doi:10.1007/BF00533984](https://access.portico.org/stable?au=pgg197frtxd))
## Related $n$Lab entries
* [[The Rising Sea]]
* [[Grothendieck group]]
category: people
[[!redirects Grothendieck]]
[[!redirects A. Grothendieck]]
[[!redirects Alexander Grothendieck]]
[[!redirects Alexandre Grothendieck]] |
Alexander Haupt | https://ncatlab.org/nlab/source/Alexander+Haupt |
* [webpage](https://www.math.uni-hamburg.de/home/haupt/)
## Selected writings
On [[M-theory on Calabi-Yau 5-folds]] and [[supersymmetric quantum mechanics]]:
* [[Alexander Haupt]], [[Andre Lukas]], [[Kellogg Stelle]], _M-theory on Calabi-Yau Five-Folds_, JHEP 0905:069, 2009 ([arXiv:0810.2685](https://arxiv.org/abs/0810.2685))
category: people |
Alexander Jahn | https://ncatlab.org/nlab/source/Alexander+Jahn |
* [Institute page](https://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-eisert/people/jahn/index.html)
## Selected writings
Introducing [[Majorana dimer codes]] ([[quantum error correcting codes]] induced by [[tensor networks]] exhibiting [[holographic entanglement entropy]]):
* [[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))
based on:
* [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], *Holography and criticality in matchgate tensor networks*, Science Advances 5, eaaw0092 (2019) ([arXiv:1711.03109](https://arxiv.org/abs/1711.03109), [doi:10.1126/sciadv.aaw0092](https://advances.sciencemag.org/content/5/8/eaaw0092))
Review:
* [[Alexander Jahn]], [[Jens Eisert]], _Holographic tensor network models and quantum error correction: A topical review_ ([arXiv:2102.02619](https://arxiv.org/abs/2102.02619))
Discussion of [[asymptotic boundaries]] of [[hyperbolic space|hyperbolic]] [[tensor networks]] as conformal [[quasicrystals]] (see also at [[AdS/CFT in solid state physics]]):
* [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], _Central charges of aperiodic holographic tensor network models_, Phys. Rev. A 102, 042407 ([arXiv:1911.03485](https://arxiv.org/abs/1911.03485))
* [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], _Tensor network models of AdS/qCFT_ ([arXiv:2004.04173](https://arxiv.org/abs/2004.04173))
More on holographic [[quantum error correcting codes]]:
* Matthew Steinberg, Sebastian Feld, [[Alexander Jahn]], *Holographic Codes from Hyperinvariant Tensor Networks* [[arXiv:2304.02732](https://arxiv.org/abs/2304.02732)]
category: people
|
Alexander K. Zvonkin | https://ncatlab.org/nlab/source/Alexander+K.+Zvonkin | [website](https://www.labri.fr/perso/zvonkin/)
## Related entries
* [[embedded graph]]
* [[combinatorial map]]
* [[child's drawing]]
* [[chord diagram]]
|
Alexander Kahle | https://ncatlab.org/nlab/source/Alexander+Kahle |
* [website](http://www.uni-math.gwdg.de/kahle/)
category: people |
Alexander Karabegov | https://ncatlab.org/nlab/source/Alexander+Karabegov |
* [InSpire page](https://inspirehep.net/authors/2332229)
## Selected writings
On [[Fedosov deformation quantization]] for [[almost Kähler structures]]:
* {#KarabegovSchlichenmaier01} [[Alexander Karabegov]], [[Martin Schlichenmaier]], _Almost-Kähler deformation quantization_, Letters in Mathematical Physics, **57** 2 (2001) 135–148 [[doi:10.1023/A:1017993513935](https://doi.org/10.1023/A:1017993513935), [arXiv:0102169](https://arxiv.org/abs/math/0102169)]
category: people |
Alexander Kirillov | https://ncatlab.org/nlab/source/Alexander+Kirillov |
* [webpage](http://www.math.sunysb.edu/~kirillov/)
category: people |
Alexander Kupers | https://ncatlab.org/nlab/source/Alexander+Kupers | [[!redirects Sander Kupers]]
* [website](http://math.harvard.edu/~kupers/)
## Selected writings
On the [[h-principle]] and [[microflexible sheaf|microflexible sheaves]]:
* [[Alexander Kupers]], Section 2 of: *Three applications of delooping to H-principles*, Geom Dedicata **202** (2019) 103–151 ([arXiv:1701.06788](https://arxiv.org/abs/1701.06788), [doi:10.1007/s10711-018-0405-7](https://doi.org/10.1007/s10711-018-0405-7))
## Related pages
* [[string topology]]
category: people
|
Alexander Kurz | https://ncatlab.org/nlab/source/Alexander+Kurz | Alexander Kurz is a Professor in the department of Computer Science at Chapman University, Orange, Southern California.. He has written extensively on the applications of [[coalgebra]] techniques in [[modal logic]].
* [Faculty Webpage](https://www.chapman.edu/engineering/about/faculty/program-faculty/alex-kurz.aspx).
Publication list (from previous post at Leicester University, UK.):
* [Publication List](http://www.cs.le.ac.uk/people/akurz/works.html)
###References
* C. Cirstea, A. Kurz, D. Pattinson, L. Schröder, Y. Venema: Modal Logics are Coalgebraic. [BCS Visions in Computer Science 2008](www.cs.le.ac.uk/people/akurz/Papers/BCS08/ModalCoalg.pdf).
* A. Kurz : _Coalgebras and Modal Logic._ Course Notes for ESSLLI 2001, Version of October 2001. Appeared on the CD-Rom ESSLLI'01, Department of Philosophy, University of Helsinki, Finland, available from [site](http://www.cs.le.ac.uk/people/akurz/works.html).
category:people
[[!redirects A. Kurz]] |
Alexander Kuznetsov | https://ncatlab.org/nlab/source/Alexander+Kuznetsov | **Alexander Kuznetsov** is a former student of [[Alexei Bondal]].
* [website](http://www.mi.ras.ru/~akuznet/)
* [[derived categories of coherent sheaves]]
* [[derived noncommutative algebraic geometry]]
category: people |
Alexander Körschgen | https://ncatlab.org/nlab/source/Alexander+K%C3%B6rschgen |
* [webpage](http://www.math.uni-bonn.de/people/alex)
## Selected writings
On [[orbispaces]] in [[global equivariant homotopy theory]]:
* {#Koerschgen16} [[Alexander Körschgen]], _A Comparison of two Models of Orbispaces_, Homology, Homotopy and Applications, vol. 20(1), 2018, pp. 329--358 ([arXiv:1612.04267](https://arxiv.org/abs/1612.04267))
category: people |
Alexander L. Fetter | https://ncatlab.org/nlab/source/Alexander+L.+Fetter |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alexander_Fetter)
* [GoogleScholar page](https://scholar.google.com/citations?user=RKnDQh4AAAAJ)
## Selected writings
A textbook on [[quantum theory]]/[[quantum field theory]] of many-particle systems with application to [[condensed matter theory]]:
* [[Alexander L. Fetter]], [[John Dirk Walecka]], *Quantum theory of many-particle systems*, Mcgraw-Hill (1991); Dover (2003) [[archive.org](https://archive.org/details/quantum-theory-of-many-particle-systems-by-alexander-l.-fetter-john-dirk-walecka-physics-z-lib.org/mode/thumb)]
[[!redirects Alexander Fetter]]
category: people
|
Alexander Lenz | https://ncatlab.org/nlab/source/Alexander+Lenz |
* [webpage](https://www.ippp.dur.ac.uk/profile/lenz)
* [webpage](https://www.dur.ac.uk/research/directory/staff/?mode=staff&id=10956)
## Selected writings
Experimental constraints on the existence of a [[4th generation of fermions]]:
* {#Lenz13} [[Alexander Lenz]], _Constraints on a fourth generation of fermions from Higgs Boson searches_, Adv. High Energy Phys. 2013 (2013) 910275 ([doi:10.1155/2013/910275](https://doi.org/10.1155/2013/910275))
* [[Alexander Lenz]], _How to really kill a physics model_, talk notes 2012 ([pdf](https://www.ippp.dur.ac.uk/~lenz/Talk_Bern_Lenz.pdf), [[Lenz4thGenerationConstraints.pdf:file]])
category: people |
Alexander Merkurjev | https://ncatlab.org/nlab/source/Alexander+Merkurjev |
* [personal page](https://www.math.ucla.edu/~merkurev/)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alexander_Merkurjev)
## Selected writings
On [[quadratic forms]]:
* [[Richard Elman]], [[Nikita Karpenko]], [[Alexander Merkurjev]], *Algebraic and Geometric Theory of Quadratic Forms*, Colloquium Publication **56**, AMS (2008) [[ams:coll-56](https://bookstore.ams.org/coll-56), [pdf](https://www.math.ucla.edu/~rse/old_book/Kniga.pdf)]
category: people
|
Alexander Odesskii | https://ncatlab.org/nlab/source/Alexander+Odesskii | [[!redirects A. Odesskii]]
[[!redirects Aleksandr Odesskii]]
__Alexander/Aleksandr V. Odesskii__ is a mathematical physicist.
* A. V. Odesskii, B.L. Feigin, _Sklyanin’s elliptic algebras_, Funkc. Anal. i Pril. 23 (3) (1989) 45--54
* A. V. Odesskii, _Elliptic algebras_, Uspekhi Mat. Nauk (2002) 57:6 (348) 87--122 (Russian) [doi](https://doi.org/10.4213/rm573); Engl. transl. Russian Math. Surveys 2002, 57:6, 1127--1162 [doi](https://doi.org/10.1070/RM2002v057n06ABEH000573)
* A. Odesskii, [[V. Rubtsov]], V. Sokolov, _Double Poisson brackets on free associative algebras_, in: Noncommutative Birational Geometry, Representations and Combinatorics, Contemp. Math. __592__, Amer. Math. Soc. (2013) 225--239 [doi](https://arxiv.org/abs/1208.2935)
* A. Odesskii, [[V. Rubtsov]], V. Sokolov, _Parameter-dependent associative Yang–Baxter equations and Poisson brackets_, Int. J. Geom. Meth. Mod. Phys. __11__:09, 1460036 (2014) Proc. XXII IFWGP, Univ. of Évora, Portugal, 2013 [doi](https://doi.org/10.1142/S0219887814600366)
* A. V. Odesskii, [[V. Rubtsov|V. N. Rubtsov]], V. V. Sokolov, _Bi-hamiltonian ordinary differential equations with matrix variables_, Theor. and Math. Phys. 171(1): 442--447 (2012)
* [[M. Kontsevich]], A. Odesskii, _$p$-Determinants and monodromy of differential operators_, Sel. Math. New Ser. __28__, 52 (2022) [doi](https://doi.org/10.1007/s00029-022-00770-6) [arXiv:2009.12159](https://arxiv.org/abs/2009.12159)
* [[M. Kontsevich]], A. Odesskii, _Multiplication kernels_, Lett. Math. Phys. __111__, 152 (2021) [arxiv:2105.04238](https://arxiv.org/abs/2105.04238) [doi](https://doi.org/10.1007/s11005-021-01491-1)
category: people
|
Alexander Ostrowski | https://ncatlab.org/nlab/source/Alexander+Ostrowski |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Alexander_Ostrowski)
category: people
|
Alexander Polishchuk | https://ncatlab.org/nlab/source/Alexander+Polishchuk |
* [Webpage](http://pages.uoregon.edu/apolish/)
## Selected writings
On [[t-structures]] on [[derived categories of coherent sheaves]]:
* [[Dan Abramovich]], [[Alexander Polishchuk]], *Sheaves of t-structures and valuative criteria for stable complexes*, J. reine angew. Math. **590** (2006) 89-130 [[arXiv:math/0309435](https://arxiv.org/abs/math/0309435), [doi:10.1515/CRELLE.2006.005](https://doi.org/10.1515/CRELLE.2006.005)]
* [[Alexander Polishchuk]], *Constant families of t-structures on derived categories of coherent sheaves*, Moscow Math. J. __7__ (2007) 109-134 [[arXiv:math/0606013](https://arxiv.org/abs/math/0606013)]
Discussion of traditional [[algebraic geometry]] for [[super-schemes]]:
* [[Ugo Bruzzo]], Daniel Hernandez Ruiperez, [[Alexander Polishchuk]], _Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes_ ([arXiv:2008.00700](https://arxiv.org/abs/2008.00700))
category: people
|
Alexander Polyakov | https://ncatlab.org/nlab/source/Alexander+Polyakov |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Alexander_Markovich_Polyakov)
* [old personal page](https://phy.princeton.edu/people/alexander-polyakov) at Princeton (archived [screenshot](https://ncatlab.org/nlab/files/APolyakovOldHomepage.jpg))
> "My main interests this year [1993?] were directed towards [[string theory]] of [[quark]] [[confinement]]. The problem is to find the string Lagrangian for the Faraday's "[[flux tube|lines of force]]", which would reproduce perturbative corrections from the [[Yang-Mills theory]] to the Coulomb law at small distances and would give permanent [[confinement]] of quarks at large distances."
> (cf. *[[Polyakov gauge-string duality]]*)
* Valery Pokrovsky, *Hidden Sasha Polyakov’s life in Statistical and Condensed Matter Physics*, in: *Polyakov’s String: Twenty Five Years After*, Proceedings of an International Workshop at Chernogolovka (2005) 2-15 [[arXiv:hep-th/0510214](https://arxiv.org/abs/hep-th/0510214)]
## Selected writings
Discussing what came to be called the *[[Polyakov action]]* for the [[bosonic string]] and the resulting [[Liouville theory]], in the non-critical case:
* [[Alexander Polyakov]], *Quantum geometry of bosonic strings*, Phys. Lett. B **103** (1981) 207-210 [<a href="https://doi.org/10.1016/0370-2693(81)90743-7">doi:10.1016/0370-2693(81)90743-7</a>, [pdf](http://qft.itp.ac.ru/polyakov-2.pdf)]
and for the [[superstring]]:
* [[Alexander Polyakov]], *Quantum geometry of fermionic strings*, Physics Letters B **103** 3 (1981) 211-213 [<a href="https://doi.org/10.1016/0370-2693(81)90744-9">doi:10.1016/0370-2693(81)90744-9</a>]
Introducing [[conformal field theory]]:
* [[Alexander Belavin]], [[Alexander Polyakov]], [[Alexander Zamolodchikov]], _Infinite conformal symmetry in two–dimensional quantum field theory_, Nuclear Physics B Volume 241, Issue 2, 23 July 1984, Pages 333-380 (<a href="https://doi.org/10.1016/0550-3213(84)90052-X">doi:10.1016/0550-3213(84)90052-X</a>)
Proposing the identification of [[flux tubes]] in [[confinement|confined]] [[QCD]] with the [[strings]] of [[string theory]], hence of the [[holographic principle]] in what came to be known as the [[AdS-QCD correspondence]] (see *[[Polyakov gauge-string duality]]*):
* [[Alexander Polyakov]], *String representations and hidden symmetries for gauge fields*, Physics Letters B **82** 2 (1979) 247-250 [<a href="https://doi.org/10.1016/0370-2693(79)90747-0">doi:10.1016/0370-2693(79)90747-0</a>]
* [[Alexander Polyakov]], *Gauge fields as rings of glue*, Nuclear Physics B **164** (1980) 171-188 [<a href="https://doi.org/10.1016/0550-3213(80)90507-6">doi:10.1016/0550-3213(80)90507-6</a>]
* [[Alexander Polyakov]], *Gauge Fields and Strings*, Routledge, Taylor and Francis (1987, 2021) [[doi:10.1201/9780203755082](https://doi.org/10.1201/9780203755082), [oapen:20.500.12657/50871](https://library.oapen.org/handle/20.500.12657/50871)]
* {#Polyakov98} [[Alexander Polyakov]], *String Theory and Quark Confinement*, Nucl. Phys. Proc. Suppl. **68** (1998) 1-8 [[arXiv:hep-th/9711002](https://arxiv.org/abs/hep-th/9711002), <a href="https://doi.org/10.1016/S0920-5632(98)00135-2">doi:10.1016/S0920-5632(98)00135-2</a>]
* [[Alexander Polyakov]], *The wall of the cave*, Int. J. Mod. Phys. A **14** (1999) 645-658 [[arXiv:hep-th/9809057](https://arxiv.org/abs/hep-th/9809057), [doi:10.1142/S0217751X99000324](https://doi.org/10.1142/S0217751X99000324)]
eventually leading to the rules of the [[AdS-CFT correspondence]]:
* {#GubserKlebanovPolyakov98} [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], *Gauge theory correlators from non-critical string theory*, Physics Letters B **428** 105-114 (1998) [[hep-th/9802109](http://arxiv.org/abs/hep-th/9802109), <a href="https://doi.org/10.1016/S0370-2693(98)00377-3">doi:10.1016/S0370-2693(98)00377-3</a>]
> Relations between gauge fields and strings present an old, fascinating and unanswered question. The full answer to this question is of great importance for theoretical physics. It will provide us with a theory of quark confinement by explaining the dynamics of color-electric fluxes.
* [[Alexander Polyakov]], [[Vyacheslav Rychkov]] *Gauge fields-strings duality and the loop equation*, Nucl. Phys. B **581** (2000) 116-134 $[$[arXiv:hep-th/0002106](https://arxiv.org/abs/hep-th/0002106), <a href="https://doi.org/10.1016/S0550-3213(00)00183-8">doi:10.1016/S0550-3213(00)00183-8</a>$]$
and specifically between [[single trace operators]] and [[superstring]]-excitations:
* {#Polyakov02} [[Alexander Polyakov]], _Gauge Fields and Space-Time_, Int. J. Mod. Phys. A17S1 (2002) 119-136 ([arXiv:hep-th/0110196](https://arxiv.org/abs/hep-th/0110196)):
> The picture which slowly arises from the above considerations is that of the space-time gradually disappearing in the regions of large curvature. The natural description in this case is provided by a gauge theory in which the basic objects are the texts formed from the gauge-invariant words. The theory provides us with the expectation values assigned to the various texts, words and sentences.
> These expectation values can be calculated either from the gauge theory or from the strongly coupled 2d sigma model. The coupling in this model is proportional to the target space curvature. This target space can be interpreted as a usual continuous space-time only when the curvature is small. As we increase the coupling, this interpretation becomes more and more fuzzy and finally completely meaningless.
* [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], *A semi-classical limit of the gauge/string correspondence*, Nucl. Phys. B **636** (2002) 99-114 [[arXiv:hep-th/0204051](https://arxiv.org/abs/hep-th/0204051), <a href="https://doi.org/10.1016/S0550-3213(02)00373-5">doi:10.1016/S0550-3213(02)00373-5</a>]
Historical reminiscences:
* [[Alexander Polyakov]], *Confinement and Liberation*, in [[Gerardus ’t Hooft]] (ed.) *50 Years of Yang-Mills Theory* (2005) 311-329 [[arXiv:hep-th/0407209](https://arxiv.org/abs/hep-th/0407209), [doi:10.1142/9789812567147_0013](https://doi.org/10.1142/9789812567147_0013), [doi:10.1142/5601](https://doi.org/10.1142/5601)]
* {#Polyakov07} [[Alexander M. Polyakov]], *Beyond Space-Time*, in *The Quantum Structure of Space and Time*, Proceedings of the 23rd Solvay Conference on Physics, World Scientific (2007) [[arXiv:hep-th/0602011](https://arxiv.org/abs/hep-th/0602011), [pdf](http://www.solvayinstitutes.be/pdf/Proceedings_Physics/23rd_Solvay_Physics.pdf)]
* {#Polyakov08} [[Alexander M. Polyakov]], *From Quarks to Strings* [[arXiv:0812.0183](https://arxiv.org/abs/0812.0183)]
published as *Quarks, strings and beyond*, section 44 in: [[Paolo Di Vecchia]] et al. (ed.), *The Birth of String Theory*, Cambridge University Press (2012) 544-551 [[doi:10.1017/CBO9780511977725.048](https://doi.org/10.1017/CBO9780511977725.048)]
> "By the end of ’77 it was clear to me that I needed a new strategy [for understanding confinement] and I became convinced that the way to go was the [[gauge/string duality]]. [...]"
> "I kept thinking about [[gauge/string duality|gauge/strings dualities]]. Soon after the Liouville mode was discovered it became clear to many people including myself that its natural interpretation is that random surfaces in 4d are described by the strings flying in 5d with the Liouville field playing the role of the fifth dimension. The precise meaning of this statement is that the wave function of the general string state depends on the four center of mass coordinates and also on the fifth, the Liouville one. In the case of minimal models this extra dimension is related to the matrix eigenvalues and the resulting space is flat."
> "Since this 5d space must contain the flat 4d subspace in which the gauge theory resides, the natural ansatz for the metric is just the Friedman universe with a certain warp factor. This factor must be determined from the conditions of conformal symmetry on the world sheet. Its dependence on the Liouville mode must be related to the renormalization group flow. As a result we arrive at a fascinating picture -- our 4d world is a projection of a more fundamental 5d string theory. [...]"
> "At this point I was certain that I have found the right language for the [[gauge/string duality]]. I attended various conferences, telling people that it is possible to describe gauge theories by solving Einstein-like equations (coming from the conformal symmetry on the world sheet) in five dimensions. The impact of my talks was close to zero. That was not unusual and didn’t bother me much. What really caused me to delay the publication ([Polyakov 1998](#Polyakov98)) for a couple of years was my inability to derive the asymptotic freedom from my equations. At this point I should have noticed the paper of [Klebanov 1997](https://arxiv.org/abs/hep-th/9702076) in which he related [[D3 branes]] described by the supersymmetric Yang Mills theory to the same object described by supergravity. Unfortunately I wrongly thought that the paper is related to matrix theory and I was skeptical about this subject. As a result I have missed this paper which would provide me with a nice special case of my program. This special case was presented little later in full generality by Juan Maldacena ([Maldacena 1997](AdS-CFT+correspondence#Maldacena97a)) and his work opened the flood gates."
## Related entries
* [[Polyakov gauge-string duality]]
* [[QCD]], [[confinement]]
* [[Polyakov action]], [[harmonic map]]
* [[conformal bootstrap]]
* [[string theory]]
* [[AdS-CFT correspondence]]
* [[AdS-QCD correspondence]]
* [[Polyakov gauge-string duality -- references]]
category: people
[[!redirects Alexander M. Polyakov]]
|
Alexander polynomial | https://ncatlab.org/nlab/source/Alexander+polynomial |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The *Alexander polynomial* ([Alexander 1928](#Alexander28)) is a [[polynomial]] [[knot invariant|invariant]] related to [[braid representation]]-theory (cf. the *[[Burau representation]]*).
There are several ways to look at thid invariant, some of these use the [[knot group]] previously defined by [[Max Dehn]], but there are also various combinatorial methods derived from Alexander's original one. One of the best known methods is via [[Fox derivatives]] and is described in the classical text by [[Richard Crowell]] and [[Ralph Fox]].
(...)
Consider some 3-manifold given as a [[surface]] [[fiber bundle]] over the circle (notice the _[[virtually fibered conjecture]]_).
For a [[fiber]] surface $T$, the translation of the fibre around the base-space circle determines an element in the mapping-class group of $T$, a homeomorphism $h\colon T \to T$ well defined up to isotopy; this element is called the _holonomy_ of the fiber surface; the _Alexander polynomial_ is the [[characteristic polynomial]] of the map the holonomy induces on $H_1(T)$.
([Stallings 87](#Stallings87))
## Properties
### Analogue in number theory
See [Sikora 01, analogy 2.2 (10)](#Sikora01)) for the comparison in [[arithmetic topology]], where Alexander-Fox theory is the analog of [[Iwasawa theory]] ([Morishita, section 7](#Morishita)).
In [Remark 3.3 of Sugiyama 04](#Sugiyama04), the Alexander polynomial is described as the L-function of the knot complement, taken there with the trivial represenation. As such it resembles the local zeta function of a curve.
## Related concepts
* [[Burau representation]]
## References
The original article:
* {#Alexander28} [[James W. Alexander]], *Topological invariants of knots and links*, Trans. Amer. Math. Soc. **30** (1928) 275-306 $[$[doi:10.1090/S0002-9947-1928-1501429-1](https://doi.org/10.1090/S0002-9947-1928-1501429-1)$]$
Textbook accounts:
* [[R. H. Crowell]] and [[R. H. Fox]], *Introduction to Knot Theory* Springer, Graduate Texts **57** (1963)
* [[Nick Gilbert]], [[Tim Porter]], *Knots and surfaces*, Oxford University Press (1994) [[ISBN:9780198514909](https://global.oup.com/academic/product/knots-and-surfaces-9780198514909?cc=de&lang=en&)]
See also:
* {#Stallings87} [[John Stallings]], _Constructions of fibered knots and links_, Proceedings of Symposia in Pure Mathematics **32** (1987) $[$[pdf](http://www.maths.ed.ac.uk/~aar/papers/stallfib2.pdf)$]$
* Wikipedia, *[Alexander polynomial](https://en.m.wikipedia.org/wiki/Alexander_polynomial)*
An analogue in number theory is the *Iwasawa polynomial*. Cf. for number theoretic analogies
* [[Barry Mazur]], _Remarks on the Alexander polynomial_, [pdf](http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf)
* {#Morishita} Masanori Morishita, _Analogies between Knots and Primes, 3-Manifolds and Number Rings_, ([arxiv](http://arxiv.org/abs/0904.3399))
* Masanori Morishita, Knots and primes: an introduction to arithmetic topology, Springer 2012, chapter 12
* Ken-ichi Sugiyama, _The properties of an L-function from a geometric point of view_, 2007 [pdf](http://geoquant2007.mi.ras.ru/sugiyama.pdf)
* {#Sugiyama04} Ken-ichi Sugiyama, _A topological $\mathrm{L}$ -function for a threefold_, 2004 [pdf](http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1376-12.pdf);
* {#Sugiyama06} Ken-ichi Sugiyama _An analog of the Iwasawa conjecture for a compact hyperbolic threefold_, [math.GT/0606010](http://arxiv.org/abs/math/0606010)
* {#Sikora01} Adam S. Sikora, _Analogies between group actions on 3-manifolds and number fields_, [arXiv:0107210](http://arxiv.org/abs/math/0107210)
Other works
* Takefumi Nosaka, _Twisted cohomology pairings of knots I; diagrammatic computation_, [arxiv/1602.01129](http://arxiv.org/abs/1602.01129); _Twisted cohomology pairings of knots II; to classical invariants_, [arxivs/1602.01131](http://arxiv.org/abs/1602.01131)
* V. Mishnyakov, A. Sleptsov, N. Tselousov, _A new symmetry of the colored Alexander polynomial_ ([arXiv:2001.10596](https://arxiv.org/abs/2001.10596))
[[!redirects Alexander polynomials]]
category:knot theory
|
Alexander Popov | https://ncatlab.org/nlab/source/Alexander+Popov |
## Selected writings
A derivation of 4d [[Skyrmions]] by [[KK-compactification]] of [[D=5 Yang-Mills theory]] on a [[closed interval]], motivated by [[M5-branes]] instead of by [[D4/D8-brane intersections]] as in the [[Sakai-Sugimoto model]]:
* [[Tatiana Ivanova]], [[Olaf Lechtenfeld]], [[Alexander Popov]], _Skyrme model from 6d $\mathcal{N}= (2,0)$ theory_, Physics Letters B Volume 783, 10 August 2018, Pages 222-226 ([arXiv:1805.07241](https://arxiv.org/abs/1805.07241))
following
* [[Tatiana Ivanova]], [[Olaf Lechtenfeld]], [[Alexander Popov]], _Non-Abelian sigma models from Yang-Mills theory compactified on a circle_, Physics Letters B Volume 781, 10 June 2018, Pages 322-326 ([arXiv:1803.07322](https://arxiv.org/abs/1803.07322))
category: people |
Alexander Postnikov | https://ncatlab.org/nlab/source/Alexander+Postnikov | *[web](http://math.mit.edu/~apost)
category: people |
Alexander Reznikov | https://ncatlab.org/nlab/source/Alexander+Reznikov |
* [Wikipedia entry](https://de.wikipedia.org/wiki/Alexander_Resnikow)
## Selected writings
On [[secondary characteristic classes]] being [[torsion elements]]:
* [[Alexander Reznikov]], _Rationality of secondary classes_, J. Differential Geom., Volume 43, Number 3 (1996), 674-692. ([arXiv:dg-ga/9407007](https://arxiv.org/abs/dg-ga/9407007), [euclid:jdg/1214458328](https://projecteuclid.org/euclid.jdg/1214458328))
* [[Alexander Reznikov]], _All regulators of flat bundles are torsion_, Annals of Mathematics Second Series, Vol. 141, No. 2 (Mar., 1995), pp. 373-386 (14 pages) ([arXiv:dg-ga/9407006](https://arxiv.org/abs/dg-ga/9407006), [jstor:2118525](https://www.jstor.org/stable/2118525))
category: people |
Alexander Rosenberg | https://ncatlab.org/nlab/source/Alexander+Rosenberg |
There are at least two Alexander Rosenbergs (and one [[Jonathan Rosenberg]]) in mathematics:
* algebraist Alex (Alexander) Rosenberg (1926-2007) known for [[Hochschild-Kostant-Rosenberg theorem]] and for many other articles in [[algebra]] (cf. [notices](http://www.spelman.edu/~colm/alexnotices.html))
* Alexander L. Rosenberg (or Sasha Rosenberg) who was a professor until 2012 at [Kansas State University](http://www.math.ksu.edu).
Sasha Rosenberg had defended in 1973 dissertation at Moscow State University studying [[Tannaka duality|Tannakian reconstruction theorems]], mainly using methods of [[functional analysis]]. His main goals were in [[representation theory]]. Until leaving Soviet Union around 1987, Rosenberg worked in applied mathematics; on the side however he was developing some methods in representation theory which included functional analysis and categorical and ring/module theoretic methods, and the noncommutative localization in particular. A main construction, presented at a conference at Baikal (1981), was a new spectrum of a ring, so-called left spectrum, later generalized to a [[spectrum of an abelian category]], used to prove (in the quasicompact case) the Gabriel-Rosenberg reconstruction theorem for commutative schemes; more [[spectral theory|spectral]] constructions followed. This work outgrown into a wide ongoing work on foundations of [[noncommutative algebraic geometry]], including a natural definition of [[noncommutative scheme]]. More recently he proposed a new framework for nonabelian derived functors, including a new version of algebraic K-theory. Other related entries in $n$Lab include [[Q-category]], [[neighborhood of a topologizing subcategory]], [[sheaf on a noncommutative space]], [[spectral cookbook]].
Rosenberg's coauthors in pure mathematics works are [[Valery Lunts]] and [[Maxim Kontsevich]]. One should be warned that most of the newly released articles of Rosenberg are not at the arXiv but at the Max Planck Bonn [preprint server](http://www.mpim-bonn.mpg.de/Research/MPIM+Preprint+Series) and many of works are republished in a volume
* [Selected papers on noncommutative geometry](http://newprairiepress.org/ebooks/1), New Prairie Press 2014, which are open access
Rosenberg's available papers and online resources include:
* [web page](http://sasharosenberg.com/), [mathematical work](http://sasharosenberg.com/?page_id=7)
* 3 volume work on noncommutative geometry: _Geometry of Noncommutative 'Spaces' and Schemes_, [pdf](http://www.mpim-bonn.mpg.de/preblob/5148); _Homological algebra of noncommutative 'spaces' I._, [pdf](http://www.mpim-bonn.mpg.de/preblob/5159) (note: longer version than another MPI preprint with the same title mentioned before); _Noncommutative 'Spaces' and 'Stacks'_, [pdf](http://www.mpim-bonn.mpg.de/preblob/5149)
* video of msri lecture _Noncommutative schemes and spaces_ (Feb 2000) can be found at [msri](http://www.msri.org/publications/ln/msri/2000/interact/rosenberg/1/index.html)
* A. L. Rosenberg, _Topics in noncommutative algebraic geometry, homological algebra and K-theory_, preprint MPIM Bonn 2008-57 [pdf](http://www.mpim-bonn.mpg.de/preblob/3589)
* А. Л. Розенберг, _Теоремы двойственности для групп и алгебр Ли_, УМН, 26:6(162) (1971), 253–254, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=5287&what=fullt&option_lang=rus), _Инвариантные алгебры на компактных группах_, Матем. сб., 81(123):2 (1970), 176–184, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=sm&paperid=3368&what=fullt&option_lang=rus)
* A. L. Rosenberg, _Almost quotient categories, sheaves and localizations_, 181 p. Seminar on supermanifolds __25__, University of Stockholm, D. Leites editor, 1988 (in Russian; partial remake in English exists)
* A. L. Rosenberg, _Non-commutative affine semischemes and schemes_, Seminar on supermanifolds __26__, Dept. Math., U. Stockholm (1988)
* A. L. Rosenberg, _Noncommutative algebraic geometry and representations of quantized algebras_, MIA __330__, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9
* A. L. Rosenberg, _Reconstruction of groups_, Selecta Math. N.S. __9__:1 (2003) [doi](http://dx.doi.org/10.1007/s00029-003-0322-x) ($n$lab remark: this paper is on a generalization of Tannaka--Krein and not of the Gabriel--Rosenberg kind of reconstruction)
* A. L. Rosenberg, _The left spectrum, the Levitzki radical, and noncommutative schemes_, Proc. Nat. Acad. Sci. U.S.A. __87__ (1990), no. 21, 8583--8586.
* A. L. Rosenberg, _Noncommutative local algebra_, Geom. Funct. Anal. __4__ (1994), no. 5, 545--585.
* A. L. Rosenberg, _The existence of fiber functors_, The Gelfand Mathematical Seminars, 1996--1999, 145--154, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000 [doi](https://dx.doi.org/10.1007/978-1-4612-1340-6_7)
* A. L. Rosenberg, _The spectrum of abelian categories and reconstructions of schemes_, in Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. __197__, Marcel Dekker, New York, 257--274, 1998; MR99d:18011; and Max Planck Bonn preprint _Reconstruction of Schemes_, [MPIM1996-108](http://www.mpim-bonn.mpg.de/preblob/3948) (1996).
* A. L. Rosenberg, _Spectra of noncommutative spaces_, MPIM2003-110 [ps](http://www.mpim-bonn.mpg.de/preblob/1946) [dvi](http://www.mpim-bonn.mpg.de/preblob/1945) (2003)
* A. L. Rosenberg, _Noncommutative schemes_, Compos. Math. __112__ (1998) 93--125 [doi](http://dx.doi.org/10.1023/A:1000479824211)
* V. A. Lunts, A. L. Rosenberg, _Differential operators on noncommutative rings_, Selecta Math. (N.S.) __3__ (1997), no. 3, 335--359 ([doi](http://dx.doi.org/10.1007/s000290050014)); sequel: _Localization for quantum groups_, Selecta Math. (N.S.) __5__ (1999), no. 1, pp. 123--159 ([doi](http://dx.doi.org/10.1007/s000290050044)).
* V. A. Lunts, A. L. Rosenberg, _Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings_, MPI 1996-53 [pdf](http://www.mpim-bonn.mpg.de/preblob/3894), II. D-Calculus in the braided case. The localization of quantized enveloping algebras, MPI 1996-76 [pdf](http://www.mpim-bonn.mpg.de/preblob/3916)
* V. A. Lunts, A. L. Rosenberg, _Kashiwara theorem for hyperbolic algebras_,
MPIM1999-82, [dvi](http://www.mpim-bonn.mpg.de/preblob/947), [ps](http://www.mpim-bonn.mpg.de/preblob/948)
* M. Kontsevich, A. Rosenberg, _Noncommutative spaces_, preprint MPI-2004-35 ([dvi](http://www.mpim-bonn.mpg.de/preblob/2303),[ps](http://www.mpim-bonn.mpg.de/preblob/2331)), _Noncommutative spaces and flat descent_, MPI-2004-36 [dvi](http://www.mpim-bonn.mpg.de/preblob/2304),[ps](http://www.mpim-bonn.mpg.de/preblob/2332), _Noncommutative stacks_, MPI-2004-37 [dvi](http://www.mpim-bonn.mpg.de/preblob/2305),[ps](http://www.mpim-bonn.mpg.de/preblob/2333)
* M. Kontsevich, A. Rosenberg, _Noncommutative smooth spaces_, The Gelfand Mathematical Seminars, 1996--1999, 85--108, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000; ([arXiv:math/9812158](http://arxiv.org/abs/math/9812158))
* A. Rosenberg, [Homological algebra of noncommutative 'spaces' I](http://www.mpim-bonn.mpg.de/preblob/3623), 199 pages, preprint Max Planck, Bonn: MPIM2008-91.
* A. Rosenberg, _Underlying spaces of noncommutative schemes_, MPIM2003-111, [dvi](http://www.mpim-bonn.mpg.de/preblob/1947), [ps](http://www.mpim-bonn.mpg.de/preblob/1948)
* A. Rosenberg, _Noncommutative spaces and schemes_,
MPIM1999-84, [dvi](http://www.mpim-bonn.mpg.de/preblob/949), [ps](http://www.mpim-bonn.mpg.de/preblob/950)
[[!redirects Sasha Rosenberg]]
[[!redirects Sasha Rozenberg]]
[[!redirects A. L. Rosenberg]]
[[!redirects Rosenberg]]
category: people, algebraic geometry, noncommutative geometry |
Alexander Schenkel | https://ncatlab.org/nlab/source/Alexander+Schenkel |
* [webpage](http://www.aschenkel.eu)
## Selected writings
On construction and axiomatization of [[homotopical AQFT]] via [[homotopy theory]] and [[homotopical algebra]]:
* {#BDS} [[Marco Benini]], [[Claudio Dappiaggi]], [[Alexander Schenkel]], _Quantized Abelian principal connections on Lorentzian manifolds_, Communications in Mathematical Physics 2013 ([arXiv:1303.2515](http://arxiv.org/abs/1303.2515))
* {#BeniniSchenkelSzabo15} [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Homotopy colimits and global observables in Abelian gauge theory_ ([arXiv:1503.08839](http://arxiv.org/abs/1503.08839))
* {#BeniniSchenkel16} [[Marco Benini]], [[Alexander Schenkel]], _Quantum field theories on categories fibered in groupoids_, Communications in Mathematical Physics November 2017, Volume 356, Issue 1, pp 19–64 ([arXiv:1610.06071](https://arxiv.org/abs/1610.06071))
* [[Simen Bruinsma]], [[Christopher J. Fewster]], [[Alexander Schenkel]], *Relative Cauchy evolution for linear homotopy AQFTs* ([arXiv:2108.10592](https://arxiv.org/abs/2108.10592))
On [[self-dual higher gauge theory]] on [[Lorentzian spacetimes]] via [[ordinary differential cohomology]]:
* [[Christian Becker]], [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Abelian duality on globally hyperbolic spacetimes_, Commun. Math. Phys. **349** (2017) 361-392 [[arXiv:1511.00316](https://arxiv.org/abs/1511.00316), [doi:10.1007/s00220-016-2669-9](https://doi.org/10.1007/s00220-016-2669-9)]
On the [[stack]] of [[Yang-Mills theory|Yang-Mills]] [[gauge fields]]:
* {#BeniniSchenkelSchreiber17} [[Marco Benini]], [[Alexander Schenkel]], [[Urs Schreiber]], _The stack of Yang-Mills fields on Lorentzian manifolds_, Commun. Math. Phys. 359, 765–820 (2018) ([arXiv:1704.01378](https://arxiv.org/abs/1704.01378), [doi:10.1007/s00220-018-3120-1](https://doi.org/10.1007/s00220-018-3120-1))
Review and exposition:
* {#Schenkel17} [[Alexander Schenkel]], _Towards Homotopical Algebraic Quantum Field Theory_, talk at _[Foundational and Structural Aspects of Gauge Theories](https://indico.mitp.uni-mainz.de/event/76/overview)_, Mainz Institute for Theoretical Physics, 29 May – 2 June 2017. ([pdf](http://aschenkel.eu/Mainz17.pdf))
* {#Schenkel14} [[Alexander Schenkel]], _On the problem of gauge theories in locally covariant QFT_, talk at _[Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html)_ Trento, 2014 ([[SchenkelTrento2014.pdf:file]]) (with further emphasis on this point in the companion talk [Schreiber 14](field+bundle#Schreiber14))
* {#Schenkel17b} [[Alexander Schenkel]], _From Fredenhagen's universal algebra to homotopy theory and operads_, talk at _Quantum Physics meets Mathematics_, Hamburg, December 2017 ([pdf slides](http://aschenkel.eu/Hamburg17.pdf))
* {#BeniniSchenkel19} [[Marco Benini]], [[Alexander Schenkel]], _Higher Structures in Algebraic Quantum Field Theory_, Proceedings of LMS/EPSRC Symposium _[[Higher Structures in M-Theory 2018]]_, Fortschritte der Physik 2019 ([arXiv:1903.02878](https://arxiv.org/abs/1903.02878), [doi:10.1002/prop.201910015]( https://doi.org/10.1002/prop.201910015))
An [[operad]] for [[local nets of observables]] in [[AQFT]]
* {#BeniniSchenkelWoike17} [[Marco Benini]], [[Alexander Schenkel]], [[Lukas Woike]], _Operads for algebraic quantum field theory_ ([arXiv:1709.08657](https://arxiv.org/abs/1709.08657))
and its [[model structure on algebras over an operad]] (with respect to the [[model structure on chain complexes]]) is discussed in
* {#BeniniSchenkelWoike18} [[Marco Benini]], [[Alexander Schenkel]], [[Lukas Woike]], _Homotopy theory of algebraic quantum field theories_, Lett. Math. Phys. 109, 1487-1532 (2019) ([arXiv:1805.08795](https://arxiv.org/abs/1805.08795), [doi:10.1007/s11005-018-01151-x](https://doi.org/10.1007/s11005-018-01151-x))
On relating [[homotopy AQFT|homotopy]] [[algebraic quantum field theory]] via [[local nets of observables]] to [[factorization algebras]]:
* [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], _Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifolds_, Communications in Mathematical Physics volume 377, pages 971–997 (2020) ([arXiv:1903.03396v2](https://arxiv.org/abs/1903.03396), [doi:10.1007/s00220-019-03561-x](https://doi.org/10.1007/s00220-019-03561-x))
* [[Marco Benini]], Giorgio Musante, [[Alexander Schenkel]], *Quantization of Lorentzian free BV theories: factorization algebra vs algebraic quantum field theory* [[arXiv:2212.02546](https://arxiv.org/abs/2212.02546)]
Discussion of [[orbifold|orbifolding]] via [[categorification]], in [[homotopical algebraic quantum field theory]]:
* [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], [[Lukas Woike]], _Categorification of algebraic quantum field theories_, Lett. Math. Phys. 2021 ([arXiv:2003.13713](https://arxiv.org/abs/2003.13713))
On rigorous [[semi-topological 4d Chern-Simons theory]] via [[homotopical AQFT]]:
* [[Marco Benini]], [[Alexander Schenkel]], Benoit Vicedo, _Homotopical analysis of 4d Chern-Simons theory and integrable field theories_ ([arXiv:2008.01829](https://arxiv.org/abs/2008.01829))
On 1d [[AQFT]] and [[smooth stacks]]:
* [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], *Smooth 1-dimensional algebraic quantum field theories* ([arXiv:2010.13808](https://arxiv.org/abs/2010.13808))
On [[non-perturbative quantum field theory|non-perturbative]] aspects of the [[BV-formalism]]:
* [[Marco Benini]], [[Pavel Safronov]], [[Alexander Schenkel]], *Classical BV formalism for group actions* ([arXiv:2104.14886](https://arxiv.org/abs/2104.14886))
On [[2d CFT]] in terms of [[AQFT on curved spacetimes]]:
* [[Marco Benini]], [[Luca Giorgetti]], [[Alexander Schenkel]], *A skeletal model for 2d conformal AQFTs* ([arXiv:2111.01837](https://arxiv.org/abs/2111.01837))
On the [[time slice axiom]] in [[homotopical AQFT]]:
* [[Marco Benini]], [[Victor Carmona]], [[Alexander Schenkel]], *Strictification theorems for the homotopy time-slice axiom* [[arxiv:2208.04344](https://arxiv.org/abs/2208.04344)]
Exposition and review:
* [[Alexander Schenkel]], *Quantum field theories on Lorentzian manifolds*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#SchenkelMar2023), slides:[[Schenkel-at-QFTAndCobordism2023.pdf:file]], video:[YT](https://www.youtube.com/watch?v=YiQNAgcwZYQ)]
Formulation of the [[CS/WZW correspondence]] in [[homotopical AQFT]]:
* [[Marco Benini]], [[Alastair Grant-Stuart]], [[Alexander Schenkel]], *The linear CS/WZW bulk/boundary system in AQFT*, Annales Henri Poincaré (2023) [[arXiv:2302.06990](https://arxiv.org/abs/2302.06990)]
On the relation between [[functorial quantum field theory]] (axiomatizing the [[Schrödinger picture]] of [[quantum field theory]]) and [[algebraic quantum field theory]] (axiomatizing the [[Heisenberg picture]]):
* {#BunkMacManusSchenkel23} [[Severin Bunk]], [[James MacManus]], [[Alexander Schenkel]], *Lorentzian bordisms in algebraic quantum field theory* [[arXiv:2308.01026](https://arxiv.org/abs/2308.01026)]
## Related entries
* [[AQFT]], [[homotopical AQFT]]
* [[gauge theory]]
* [[ordinary differential cohomology]]
category: people
|
Alexander Schmeding | https://ncatlab.org/nlab/source/Alexander+Schmeding | Mathematician enjoying higher and infinite-dimensional differential geometry and its applications.
I am an associate professor at NTNU Trondheim.
Before joining NTNU, I was associate professor at Nord University in Levanger and before that at the university in Bergen. See
* [NTNU webpage](https://www.ntnu.edu/employees/alexander.schmeding)
* [Research Gate profile](https://www.researchgate.net/profile/Alexander_Schmeding)
* [Linkedin profile](https://www.linkedin.com/in/alexander-schmeding-248190141/)
## More stuff
On [[differential geometry]] with [[infinite-dimensional manifolds]]:
* [[Alexander Schmeding]], *An introduction to infinite-dimensional differential geometry*, Cambridge University Press. The open access version is available here: [CUP Open Access version](https://www.cambridge.org/core/books/an-introduction-to-infinitedimensional-differential-geometry/6483795C98EE417C0F3654F6C192C3BC)
Further, here is the [list of Errata](https://github.com/ASchmeding/An-introduction-to-infinite-dimensional-geometry)
* [Youtube](https://www.youtube.com/channel/UCTIB6ykZCrpd0trFhiSETGw) (videos on math in particular a course on infinite-dimensional geometry)
## Related $n$Lab entries
* [[bisection of a Lie groupoid]]
category: people |
Alexander Schmidt | https://ncatlab.org/nlab/source/Alexander+Schmidt | * [Web](https://www.mathi.uni-heidelberg.de/~schmidt/)
category: people |
Alexander Strohmaier | https://ncatlab.org/nlab/source/Alexander+Strohmaier |
* [personal page](http://www1.maths.leeds.ac.uk/~pmtast/)
* [Institute page](https://eps.leeds.ac.uk/maths/staff/4080/professor-alexander-strohmaier)
## Selected writings
Generalization of the [[time tube theorem]] to [[AQFT on curved spacetimes|AQFT on curved]] but [[real analytic space|real analytic]] [[spacetimes]]:
* [[Alexander Strohmaier]], *On the Local Structure of the Klein–Gordon Field on Curved Spacetimes*, Letters in Mathematical Physics **54** (2000) 249–261 [[doi:10.1023/A:1010927625112](https://doi.org/10.1023/A:1010927625112), [arXiv:math-ph/0008043](https://arxiv.org/abs/math-ph/0008043)]
* [[Alexander Strohmaier]], [[Edward Witten]], *Analytic states in quantum field theory on curved spacetimes* [[arXiv:2302.02709](https://arxiv.org/abs/2302.02709)]
* [[Alexander Strohmaier]], [[Edward Witten]], *The Timelike Tube Theorem in Curved Spacetime* [[arXiv:2303.16380](https://arxiv.org/abs/2303.16380)]
category: people |
Alexander Varchenko | https://ncatlab.org/nlab/source/Alexander+Varchenko | __Alexander Varchenko__ is a mathematician at the University of North Carolina at Chapel Hill; he was a student of [[Vladimir Arnold]]. His work is at the boundary of geometry, special functions, combinatorics, representation theory and mathematical physics.
* [list](http://www.math.unc.edu/Faculty/av/complete.htm)
* [webpage](http://www.math.unc.edu/Faculty/av)
## Selected writings
On the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of solutions to the [[Knizhnik-Zamolodchikov equation]]:
* [[Vadim Schechtman]], [[Alexander Varchenko]], *Integral representations of N-point conformal correlators in the WZW model*, Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- $[$[cds:1044951](http://cds.cern.ch/record/1044951)$]$
* [[Vadim Schechtman]], [[Alexander Varchenko]], *Hypergeometric solutions of Knizhnik-Zamolodchikov equations*, Lett. Math. Phys. **20** (1990) 279–283 $[$[doi:10.1007/BF00626523](https://doi.org/10.1007/BF00626523)$]$
* [[Vadim Schechtman]], [[Alexander Varchenko]], *Arrangements of hyperplanes and Lie algebra homology*, Inventiones mathematicae **106** 1 (1991) 139-194 $[$[dml:143938](https://eudml.org/doc/143938), [[SchechtmanVarchenko_HyperplaneArrangements.pdf:file]]$]$
On the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of [[conformal blocks]] for the [[sl(n)|$\mathfrak{sl}(n)$]] [[WZW model]]:
* {#FeiginSchechtmanVarchenko90} [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by correlators in Wess-Zumino-Witten models*, Lett Math Phys **20** (1990) 291–297 $[$[doi:10.1007/BF00626525](https://doi.org/10.1007/BF00626525)$]$
* {#FeiginSchechtmanVarchenko94} [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by hypergeometric correlators in WZW models. I*, Commun. Math. Phys. **163** (1994) 173–184 $[$[doi:10.1007/BF02101739](https://doi.org/10.1007/BF02101739)$]$
> (for [[sl(2)|$\mathfrak{sl}(2, \mathbb{C})$]])
* [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by hypergeometric correlators in WZW models. II*, Comm. Math. Phys. **170** 1 (1995) 219-247 $[$[euclid:cmp/1104272957](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-170/issue-1/On-algebraic-equations-satisfied-by-hypergeometric-correlators-in-WZW-models/cmp/1104272957.full)$]$
On [[hypergeometric functions]], the [[Knizhnik-Zamolodchikov equation]] and [[quantum groups]]:
* [[Alexander Varchenko]], *Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base*, Comm. Math. Phys. **171** 1 (1995) 99-137 $[$[arXiv:hep-th/9403102](https://arxiv.org/abs/hep-th/9403102), [doi:10.1007/BF02103772](https://doi.org/10.1007/BF02103772)$]$
* [[Alexander Varchenko]], _Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups_, Adv. Ser. in Math. Phys. __21__, World Sci. Publ. 1995. x+371 pp. ([doi:10.1142/2467](https://doi.org/10.1142/2467))
* V. Tarasov, [[Alexander Varchenko]], _Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups_, Astérisque __246__ (1997), vi+135 pp. ([arXiv:q-alg/9703044](https://arxiv.org/abs/q-alg/9703044), [numdam:AST_1997__246__R1_0](http://www.numdam.org/item/AST_1997__246__R1_0))
On [[Lie algebra weight systems]] for the [[special linear Lie algebra]] $\mathfrak{sl}(2)$:
* S. Tyurina, [[Alexander Varchenko]], *A remark on $\mathfrak{sl}_2$ approximation of the Kontsevich integral of the unknot*, Journal of Mathematical Sciences **131** (2005) 5270–5274 ([arXiv:math/0111201](https://arxiv.org/abs/math/0111201), [doi:10.1007/s10958-005-0399-1](https://doi.org/10.1007/s10958-005-0399-1))
* [[Sergei Chmutov]], [[Alexander Varchenko]], _Remarks on the Vassiliev knot invariants coming from $\mathfrak{sl}(2)$_, Topology 36 (1), 153-178, 1997 (<a href="https://doi.org/10.1016/0040-9383(95)00071-2">doi:10.1016/0040-9383(95)00071-2</a>)
Further on 1-[[twisted de Rham cohomology]] related to [[sl(2)|$\mathfrak{sl}(2)$]]-[[Verma modules]]:
* {#SlinkinVarchenko19} [[Alexey Slinkin]], [[Alexander Varchenko]], *Twisted de Rham Complex on Line and Singular Vectors in $\widehat{\mathfrak{sl}}_2$ Verma Modules*, SIGMA **15** (2019) 075 [[arXiv:1812.09791](https://arxiv.org/abs/1812.09791), [doi:10.3842/SIGMA.2019.075](https://doi.org/10.3842/SIGMA.2019.075)]
## Related $n$Lab entries
* [[hypergeometric function]]
* [[quantum group]],
* [[conformal field theory]]
* [[representation theory]]
* [[Kac-Moody Lie algebra]]
* [[singularity theory]]
[[!redirects A. Varchenko]]
category: people |
Alexander Verbovetsky | https://ncatlab.org/nlab/source/Alexander+Verbovetsky |
* [webpage](http://diffiety.ac.ru/curvita/ver.htm)
## related $n$Lab entries
* [[diffiety]]
* [[variational bicomplex]]
* [[differential operator]]
* [[jet bundle]]
category: people
|
Alexander Vilenkin | https://ncatlab.org/nlab/source/Alexander+Vilenkin |
* [webpage](http://www.phy.tufts.edu/vilenkin.html)
* [Wikipedia entry](http://en.wikipedia.org/wiki/Alexander_Vilenkin)
## Selected writings
On [[cosmic strings]], [[domain walls]], [[magnetic monopoles]], etc.:
* {#VilenkinShellard94} [[Alexander Vilenkin]], E. P. S. Shellard, *Cosmic strings and other topological defects*, Cambridge University Press (1994) [[ISBN:9780521654760](https://www.cambridge.org/ae/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/cosmic-strings-and-other-topological-defects?format=PB&isbn=9780521654760), [spire:1384873](https://inspirehep.net/literature/1384873)]
On [[false vacuum eternal inflation]]:
* {#GarrigaGuthVilenkin06} [[Jaume Garriga]], [[Alan Guth]], [[Alexander Vilenkin]], _Eternal inflation, bubble collisions, and the persistence of memory_, Phys. Rev. D76:123512, 2007 ([arXiv:hep-th/0612242](https://arxiv.org/abs/hep-th/0612242), [doi:10.1103/PhysRevD.76.123512](https://doi.org/10.1103/PhysRevD.76.123512))
## Related entries
* [[cosmic inflation]]
category: people
|
Alexander Voronov | https://ncatlab.org/nlab/source/Alexander+Voronov |
* [website](http://www.math.umn.edu/~voronov/)
## Selected writings
* [[Murray Gerstenhaber]], [[Alexander Voronov]], _Homotopy G-algebras and moduli space operad_, Internat. Math. Research Notices (1995) 141-153 ([arXiv:hep-th/9409063](https://arxiv.org/abs/hep-th/9409063))
On [[string topology]]:
* [[Ralph Cohen]], [[Alexander Voronov]], _Notes on string topology_, in: [[Ralph Cohen]], [[Kathryn Hess]], [[Alexander Voronov]], _String topology and cyclic homology_, Advanced courses in mathematics CRM Barcelona, Birkhäuser 2006 ([math.GT/05036259](http://arxiv.org/abs/math/0503625), [doi:10.1007/3-7643-7388-1](https://doi.org/10.1007/3-7643-7388-1), [pdf](http://gen.lib.rus.ec/get?md5=adde9464705ede0fea6b435edb58fbe7))
On the [[supergeometry|super]]-[[moduli space]] of [[super Riemann surfaces]] (proving it is generically not "projected" in the presence of [[Ramond punctures]], in that it does not [[retraction|retract]] onto its bosonic [[body]]):
* [[Nadia Ott]], [[Alexander A. Voronov]], *The supermoduli space of genus zero SUSY curves with Ramond punctures* [[arXiv:1910.05655](https://arxiv.org/abs/1910.05655)]
On [[mysterious duality]]/[[U-duality]] understood via [[Hypothesis H]] by [[automorphisms]] of [[iterated loop space|iterated]] [[cyclic loop spaces]] of the [[4-sphere]]:
* [[Alexander Voronov]] (joint with [[Hisham Sati]]), *Mysterious Duality*, talk at Texas Tech 2021 ([abstract pdf](https://www.math.ttu.edu/~lhoang/PureMath/abstract-2021-03-15-Voronov.pdf) [[Voronov_MysteriousAbstract.pdf:file]], [slides pdf](https://www.math.ttu.edu/~lhoang/PureMath/Slides-2021-03-15-Voronov.pdf), [[Voronov_MysteriousSlides.pdf:file]])
* [[Hisham Sati]], [[Alexander Voronov]], *Mysterious Triality and Rational Homotopy Theory* [[arXiv:2111.14810](https://arxiv.org/abs/2111.14810)]
* [[Hisham Sati]], [[Alexander Voronov]], *Mysterious Triality and M-Theory* [[arXiv:2212.13968](https://arxiv.org/abs/2212.13968)
category: people
[[!redirects Alexander A. Voronov]]
[[!redirects Sasha Voronov]]
|
Alexander Westphal | https://ncatlab.org/nlab/source/Alexander+Westphal |
* [webpage](http://www.desy.de/~westphal/)
## related $n$Lab entries
* [[moduli stabilization]]
category: people |
Alexander Zamolodchikov | https://ncatlab.org/nlab/source/Alexander+Zamolodchikov |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Alexander_Zamolodchikov)
## Selected writings
Introducing [[2d CFT|2d]] [[conformal field theory]]:
* [[Alexander Belavin]], [[Alexander Polyakov]], [[Alexander Zamolodchikov]], _Infinite conformal symmetry in two–dimensional quantum field theory_, Nuclear Physics B Volume 241, Issue 2, 23 July 1984, Pages 333-380 (<a href="https://doi.org/10.1016/0550-3213(84)90052-X">doi:10.1016/0550-3213(84)90052-X</a>)
Introducing [[parafermion]] [[2d CFT]]:
* [[Alexander B. Zamolodchikov]], [[Vladimir A. Fateev]] *Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in $Z_N$-symmetric statistical systems*, Sov. Phys. JETP **62** 2 (1985) 215-225 $[$[pdf](http://www.jetp.ras.ru/cgi-bin/dn/e_062_02_0215.pdf), [osti:5929972](https://www.osti.gov/biblio/5929972)$]$
Introducing the [[TT deformation]] in [[2d CFT]]:
* [[Alexander Zamolodchikov]], _Expectation value of composite field $T \bar T$ in two-dimensional quantum field theory_ ([arXiv:hep-th/0401146](https://arxiv.org/abs/hep-th/0401146))
## Related $n$Lab entries
* [[Knizhnik-Zamolodchikov equation]]
* [[conformal bootstrap]]
category: people
[[!redirects Alexander B. Zamolodchikov]]
|
Alexander's trick | https://ncatlab.org/nlab/source/Alexander%27s+trick |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Statement
Consider the [[mapping space]] of those [[homeomorphisms]] from the [[closed ball]] $D^{n+1}$ to itself which restrict to the [[identity]] on the [[boundary]] [[n-sphere|$n$-sphere]]. What is known as *Alexander's trick* is a construction (due to [Alexander 1923](#Alexander23)) of an explicit deformation which witnesses that this space is [[connected topological space|connected]], hence that any two such homemorphisms are connected by an [[isotopy]].
## Related entries
* [[mapping class group]]
* [[braid group]]
## References
Named after:
* {#Alexander23} [[James W. Alexander]], *On the Deformation of an $n$ Cell*, Proc Natl Acad Sci USA. **9** 12 (1923) 406–407 [[doi:10.1073%2Fpnas.9.12.406](https://doi.org/10.1073%2Fpnas.9.12.406) ]
See also:
* Wikipedia, *[Alexander's trick](https://en.wikipedia.org/wiki/Alexander%27s_trick)*
* [[Planet Math]], *[Alexander trick](https://planetmath.org/alexandertrick)*
[[!redirects Alexander trick]]
|
Alexander-Whitney map | https://ncatlab.org/nlab/source/Alexander-Whitney+map |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Let $C : sAb \to Ch_\bullet^+$ be the chains/[[Moore complex]] functor of the [[Dold-Kan correspondence]].
Let $(sAb, \otimes)$ be the standard [[monoidal category]] structure given degreewise by the [[tensor product]] on [[Ab]] and let $(Ch_\bullet^+, \otimes)$ be the standard monoidal structure on the [[category of chain complexes]].
+-- {: .un_defn}
###### Definition
For $A,B \in sAb$ two abelian [[simplicial group]]s, the **Alexander-Whitney map** is the [[natural transformation]] on [[chain complex]]es
$$
\Delta_{A,B} : C(A \otimes B) \to C(A) \otimes C(B)
$$
defined on two $n$-simplices $a \in A_n$ and $b \in B_n$ by
$$
\Delta_{A,B} : a \otimes b \mapsto \oplus_{p + q = n} (\tilde d^p a) \otimes (d^q_0 b)
\,,
$$
where the _front face map_ $\tilde d^p$ is that induced by
$$
[p] \to [p+q] : i \mapsto i
$$
and the _back face_ $d^q_0$ map is that induced by
$$
[q] \to [p+q] : i \mapsto i+p
\,.
$$
=--
+-- {: .un_defn}
###### Definition
This AW map restricts to the normalized chains complex
$$
\Delta_{A,B} : N(A \otimes B) \to N(A) \otimes N(B)
\,.
$$
=--
## Properties
\begin{proposition}
The Alexander-Whitney map is an [[oplax monoidal transformation]] that makes $C$ and $N$ into [[oplax monoidal functors]].
\end{proposition}
Beware that the AW map is _not_ [[symmetric monoidal category|symmetric]].
For details see *[[monoidal Dold-Kan correspondence]]*.
\begin{proposition}\label{EZAWDeformationRetract}
**([[Eilenberg-Zilber/Alexander-Whitney deformation retraction]])**
\linebreak
Let
* $A, B \,\in\, sAb = $ [[SimplicialAbelianGroups]]
and denote
* by $N(A), N(B) \,\in\, Ch^+_\bullet = $ [[ConnectiveChainComplexes]] their [[normalized chain complexes]],
* by $A \otimes B \,\in\, sAb$ the degreewise [[tensor product of abelian groups]],
* by $N(A) \otimes N(B)$ the [[tensor product of chain complexes]].
Then there is a [[deformation retraction]]
\begin{tikzcd}
N(A) \otimes N(B)
\ar[
rr,
bend right=20,
"\mathrm{id}"{below},
"\ "{above, name=t}
]
\ar[
rr,
phantom,
"\ "{name=s, yshift=-6pt}
]
\ar[
r,
"\nabla_{A,B}"
]
&
N( A \otimes B )
\ar[
r,
"\Delta_{A,B}"
]
&
N(A) \otimes N(B)
\ar[
from=s,
to=t,
-,
shift left=1pt
]
\ar[
from=s,
to=t,
-,
shift right=1pt
]
\end{tikzcd}
\begin{tikzcd}
N( A \otimes B )
\ar[
rr,
bend right=20,
"\mathrm{id}"{below},
"\ "{above, name=t}
]
\ar[
rr,
phantom,
"\ "{name=s, yshift=-6pt}
]
\ar[
r,
"\Delta_{A,B}"
]
&
N(A) \otimes N(B)
\ar[
r,
"\nabla_{A,B}"
]
&
N( A \otimes B )
\ar[
from=s,
to=t,
Rightarrow
]
\end{tikzcd}
where
* $\nabla_{A,B}$ is the [[Eilenberg-Zilber map]];
* $\Delta_{A,B}$ is the [[Alexander-Whitney map]].
\end{proposition}
For unnormalized chain complexes, where we have a [[homotopy equivalence]], this is the original [[Eilenberg-Zilber theorem]] ([Eilenberg & Zilber 1953](#EilenbergZilber53), [Eilenberg & MacLane 1954, Thm. 2.1](#EilenbergMacLane54)). The above [[deformation retraction]] for normalized chain complexes is [Eilenberg & MacLane 1954, Thm. 2.1a](#EilenbergMacLane54). Both are reviewed in [May 1967, Cor. 29.10](#May67). Explicit description of the [[homotopy operator]] is given in [Gonzalez-Diaz & Real 1999](#GonzalezDiazReal99)).
## Related concepts
* **Alexander-Whitney map**
* [[Eilenberg-Zilber map]]
## References
The [[Eilenberg-Zilber theorem]] is due to
* {#EilenbergZilber53} [[Samuel Eilenberg]], [[Joseph Zilber]], _On Products of Complexes_, Amer. Jour. Math. 75 (1): 200–204, (1953) ([jstor:2372629](https://www.jstor.org/stable/2372629), [doi:10.2307/2372629](https://doi.org/10.2307/2372629))
* {#EilenbergMacLane54} [[Samuel Eilenberg]], [[Saunders MacLane]], Section 2 of: *On the Groups $H(\Pi,n)$, II: Methods of Computation*, Annals of Mathematics, Second Series, Vol. 60, No. 1 (Jul., 1954), pp. 49-139 ([jstor:1969702](https://www.jstor.org/stable/1969702))
using the definition of the [[Eilenberg-Zilber map]] in:
* {#EilenbergMacLane53} [[Samuel Eilenberg]], [[Saunders MacLane]], (5.3) of: *On the groups $H(\Pi,n)$*, I*, Ann. of Math. (2) 58, (1953), 55–106. ([jstor:1969820](https://www.jstor.org/stable/1969820))
Review:
* {#May67} [[Peter May]], Section 29 of: _Simplicial objects in algebraic topology_ , Chicago Lectures in Mathematics, University of Chicago Press 1967 ([ISBN:9780226511818](https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html), [djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu), [[May_SimplicialObjectsInAlgebraicTopology.pdf:file]])
* {#GonzalezDiazReal99} Rocio Gonzalez-Diaz, Pedro Real, *A Combinatorial Method for Computing Steenrod Squares*, Journal of Pure and Applied Algebra 139 (1999) 89-108 ([arXiv:math/0110308](https://arxiv.org/abs/math/0110308))
[[!redirects Alexander-Whitney maps]]
|
Alexander-Čech duality | https://ncatlab.org/nlab/source/Alexander-%C4%8Cech+duality |
## Idea
Let $X$ be a [[finite homotopy type|finite]] [[CW complex]]
and $X \subseteq S^{n+1}$. Then there is a map
$$ X \times (S^{n+1}\backslash X) \to S^n,\,\,\,\,\,\, (x,y) \mapsto \frac{x-y}{\|x - y\|}.$$
This element determines an element
$$\delta \in H^n( X \times (S^{n+1}\backslash X)).$$
In [[topology]] there is a [[slant product]] operation, sort of _dividing_,
and in this case one can do slant product with $\delta$.
This way one obtains a map
$$
\delta_{/} : H_{i}(X)\to H^{n-i}(S^{n+1}\backslash{X}).
$$
This map is an isomorphism and it is called the __Alexander-Čech duality__ (or sometimes simply Alexander duality).
It can be considered for infinite complexes as well, but in that
case one has to change the flavour of (co)homology theories
involved. $H_i$ is then the **Steenrod-Sitnikov homology**
and $H^{n-i}$ has to be cohomology (?).
The [[Spanier-Whitehead duality]] is a generalization, where $S^{n+1}\backslash X$ is
replaced by any space $D_n(X)$, together with an element
$\delta$ such that the corresponding map
$$\delta_/ : H_i(X) \to H^{n-i}(\mathcal{D}_n X)$$
is an isomorphism. It follows that one may replace $\mathcal{D}_n X$ by its
suspension and so on, hence the stable homotopy theory is
a natural setup for this duality.
## References
Named after [[James W. Alexander]] and [[Eduard Čech]].
[[!redirects Alexander duality]]
|
Alexandr S. Mishchenko | https://ncatlab.org/nlab/source/Alexandr+S.+Mishchenko |
* [personal page](http://mech.math.msu.su/~asmish/)
* [MathNet page](https://www.mathnet.ru/eng/person8599)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alexandr_Mishchenko)
## Selected writings
On [[vector bundles]]:
* [[Glenys Luke]], [[Alexandr S. Mishchenko]], *Vector bundles and their applications*, Math. and its Appl. **447** Kluwer (1998) [[doi:10.1007/978-1-4757-6923-4](https://doi.org/10.1007/978-1-4757-6923-4), [MR99m:55019](http://www.ams.org/mathscinet-getitem?mr=99m:55019)]
On [[Hermitian K-theory]]:
* {#Mishchenko76} [[Alexandr S. Mishchenko]], *Hermitian K-Theory. The Theory of characteristic classes and methods of functional analysis*, Uspeki Mat. Nauk **31** 2 (1976) 69-134, Russ. Math. Surv. **31** 71 (1976) [[doi:10.1070/RM1976v031n02ABEH001478](https://iopscience.iop.org/article/10.1070/RM1976v031n02ABEH001478), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/mishch5.pdf)]
[[!redirects Alexandr Mishchenko]]
[[!redirects Alexander S. Mishchenko]]
[[!redirects Alexander Mishchenko]]
|
Alexandre Deur | https://ncatlab.org/nlab/source/Alexandre+Deur |
* [spire page](https://inspirehep.net/authors/1049232)
## Selected writings
On the [[running coupling constant]] in [[QCD]]:
* [[Alexandre Deur]], [[Stanley Brodsky]], [[Guy de Teramond]], *The QCD Running Coupling*, Prog. Part. Nuc. Phys. 90 1 (2016) ([arXiv:1604.08082](https://arxiv.org/abs/1604.08082), [doi:10.1016/j.ppnp.2016.04.003](https://doi.org/10.1016/j.ppnp.2016.04.003))
category: people
|
Alexandre Kirillov | https://ncatlab.org/nlab/source/Alexandre+Kirillov | **Alexandre Aleksandrovich Kirillov** (Александр Александрович Кириллов) is a professor of mathematics at the University of Pennsylvania. He invented the orbit method in [[representation theory]], developed in other flavours also by Kostant and Souriau. He was a student of [[Israel Gelfand]] and they together lead a famous journal _Functional Analysis and its Applications_.
* [wikipedia](http://en.wikipedia.org/wiki/Alexandre_Kirillov)
## Selected writings
On [[Reshetikhin-Turaev theory]] constructing [[modular functors]] from [[modular tensor categories|modular]] [[tensor categories]]:
* [[Bojko Bakalov]], [[Alexandre Kirillov]], _Lectures on tensor categories and modular functors_, University Lecture Series **21**, Amer. Math. Soc. (2001) ([ISBN:978-1-4704-2168-7](https://bookstore.ams.org/ulect-21), [webpage](http://www.math.sunysb.edu/~kirillov/tensor/tensor.html)).
[[!redirects Александр Кириллов]]
category: people |
Alexandre Miquel | https://ncatlab.org/nlab/source/Alexandre+Miquel |
* [webpage](http://perso.ens-lyon.fr/alexandre.miquel/)
category: people
|
Alexandre Vinogradov | https://ncatlab.org/nlab/source/Alexandre+Vinogradov | [[!redirects Alexander Vinogradov]]
Alexander M. Vinogradov was a Russian mathematician working in the geometric theory of [[differential equations]], the theory of [[diffieties]] with applications to [[variational calculus]].
* [webpage](https://web.archive.org/web/20180914224323/http://diffiety.ac.ru/curvita/amv.htm)
## some publications
* A. M. Vinogradov, _Cohomological analysis of partial differential equations and secondary calculus_, [bookpage](http://diffiety.ac.ru/books/mmono204.htm) Translations of Mathematical Monographs __204__, Amer. Math. Soc. 2001, 247 p.
* A. M. Vinogradov, _Local symmetries and conservation laws_, Acta Appl. Math., Vol. 2, 1984, p. 21, [MR85m:58192](http://www.ams.org/mathscinet-getitem?mr=736872), [doi](http://dx.doi.org/10.1007/BF01405491)
* A. M. Vinogradov, _Symmetries and conservation laws of partial differential equations: basic notions and results_, Acta Appl. Math., Vol. 15, 1989, p. 3. [MR91b:58282](http://www.ams.org/mathscinet-getitem?mr=1007340), [doi](http://dx.doi.org/10.1007/BF00131928)
* A. M. Vinogradov, _Scalar differential invariants, diffieties and characteristic classes_, in: Mechanics, Analysis and Geometry: 200 Years after, 379–414, [MR92e:58244](http://www.ams.org/mathscinet-getitem?mr=1098525)
* [[G. Vezzosi]], A. Vinogradov, _On higher order analogues of De Rham cohomology_, Diff. Geom. and Appl. __19__ (2003), 29-59.
* G. Vezzosi, A.M. Vinogradov, _Infinitesimal Stokes' formula for higher-order de Rham complexes_, Acta Appl. Math. __49__, N. 3 (1997), 311-329.
## related $n$Lab entries
* [[diffiety]]
* [[variational calculus]]
* [[Euler-Lagrange complex]]
category:people
[[!redirects A. Vinogradov]]
[[!redirects Alexander Vinogradov]]
[[!redirects A. M. Vinogradov]]
[[!redirects Aleksander Vinogradov]]
[[!redirects Alexandre M. Vinogradov]]
[[!redirects Aleksander M. Vinogradov]]
[[!redirects Alexander M. Vinogradov]] |
Alexandros Anastasiou | https://ncatlab.org/nlab/source/Alexandros+Anastasiou |
* [GoogleScholar page](https://scholar.google.se/citations?user=02U4jccAAAAJ)
## Selected writings
On [[Freudenthal magic square|magic squares]] of [[magic supergravities]] and [[U-duality]] via the [[classical double copy]]:
* {#ABDHN} [[Alexandros Anastasiou]], [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic pyramid of supergravities*, JHEP (2014) 178 [[arXiv:1312.6523](http://arxiv.org/abs/1312.6523)]
category: people
|
Alexandrov space | https://ncatlab.org/nlab/source/Alexandrov+space |
# Alexandrov spaces
* table of contents
{: toc}
## Idea
Alexandrov spaces are certain spaces that arise in [[differential geometry]], named after [[Alexander Alexandrov]]. They should not be confused with [[Alexandroff spaces]], which arise in [[general topology]] and are named after [[Paul Alexandroff]].
## References
* Yukio Otsu, _Differential geometric aspects
of Alexandrov spaces_, in: Comparison Geometry, MSRI publications 30, 1997. [pdf](http://library.msri.org/books/Book30/files/otsu.pdf)
* G. Perelman, _Elements of Morse theory on Alexandrov spaces_, SPb Math. J. 1994.
* G. Perelman, DC structure on Alexandrov spaces, [pdf](http://www.math.psu.edu/petrunin/papers/alexandrov/Cstructure.pdf)
* Yu. Burago, M. Gromov, G. Perelman, _Alexandrov spaces with curvature bounded below_, Russ. Math. Surveys 1992.
* MathOverflow [Details of Perelman's example about soul of Alexandrov space](http://mathoverflow.net/questions/22105/details-of-perelmans-example-about-soul-of-alexandrov-space), [Metrically singular Alexandrov space](http://mathoverflow.net/questions/67046/metrically-singular-alexandrov-space), [Is there Domain Invariance for Alexandrov spaces?](http://mathoverflow.net/questions/21512/is-there-domain-invariance-for-alexandrov-spaces), [Rigidity of triangle comparison in Alexandrov spaces](http://mathoverflow.net/questions/44943/rigidity-of-triangle-comparison-in-alexandrov-spaces)
[[!redirects Alexandrov space]]
[[!redirects Alexandrov spaces]]
[[!redirects Alexandrov space (differential geometry)]]
[[!redirects Alexandrov spaces (differential geometry)]]
[[!redirects Alexandrov space (in differential geometry)]]
[[!redirects Alexandrov spaces (in differential geometry)]]
[[!redirects Alexandrov space in differential geometry]]
[[!redirects Alexandrov spaces in differential geometry]]
|
Alexandru Buium | https://ncatlab.org/nlab/source/Alexandru+Buium |
* [webpage](http://www.math.unm.edu/~buium/)
###Related entries
* [[arithmetic differential geometry]]
category: people
|
Alexandru Chirvasitu | https://ncatlab.org/nlab/source/Alexandru+Chirvasitu |
* [webpage](http://math.berkeley.edu/~calex/)
category: people
|
Alexandru Dimca | https://ncatlab.org/nlab/source/Alexandru+Dimca |
* [personal page](https://math.unice.fr/~dimca/)
* [Wikipedia entry](https://math.unice.fr/~dimca/)
## Selected writings
On ([[abelian sheaf|abelian]]) [[sheaf theory]] in [[topology]], with a focus on [[constructible sheaves]] and [[perverse sheaves]]:
* [[Alexandru Dimca]], *Sheaves in Topology*, Universitext, Springer (2004) $[$[doi:10.1007/978-3-642-18868-8](https://doi.org/10.1007/978-3-642-18868-8)$]$
category: people |
Alexandru E. Stanculescu | https://ncatlab.org/nlab/source/Alexandru+E.+Stanculescu |
## Selected writings
On the [[Bousfield-Friedlander theorem]]:
* {#Stanculescu} [[Alexandru E. Stanculescu]], *Note on a theorem of Bousfield and Friedlander*, Topology and its Applications **155** 13 (2008) 1434-1438 [[arxiv:0806.4547](https://arxiv.org/abs/0806.4547), [doi:10.1016/j.topol.2008.05.003](https://doi.org/10.1016/j.topol.2008.05.003)]
On [[bifibrations]], [[factorization systems]] and [[model structures on Grothendieck fibrations]]:
* {#Stanculesu12} [[Alexandru E. Stanculescu]], *Bifibrations and weak factorisation systems*, Applied Categorical Structures **20** 1 (2012) 19–30 [<a href="https://link.springer.com/article/10.1007/s10485-009-9214-3">doi:10.1007/s10485-009-9214-3</a>]
On [[model structures on enriched categories]]:
* [[Alexandru Stanculescu]], *Constructing model categories with prescribed fibrant objects*, Theory and Applications of Categories, **29** 23 (2014) 635-653 [[arXiv:1208.6005](http://arxiv.org/abs/1208.6005), [tac:29-23](http://www.tac.mta.ca/tac/volumes/29/23/29-23abs.html)]
category: people
[[!redirects Alexandru Stanculescu]] |
Alexandru Solian | https://ncatlab.org/nlab/source/Alexandru+Solian |
## Selected writings
Early discussion of [[2-groups]] as [[Picard 2-groups]]:
* {#Solian72} [[Alexandru Solian]], *Groupe dans une catégorie*, C. R. Acad. Sc. Paris **275** (1972) [[ark:/12148/bpt6k57310477/f7](https://gallica.bnf.fr/ark:/12148/bpt6k57310477/f7.item)]
* {#Solian80} [[Alexandru Solian]], *Coherence in categorical groups*, Communications in Algebra **9** 10 (1980) 1039-1057 [[doi:10.1080/00927878108822631](https://doi.org/10.1080/00927878108822631)]
category: people
|
Alexei Abrikosov | https://ncatlab.org/nlab/source/Alexei+Abrikosov |
* <a href="https://en.wikipedia.org/wiki/Alexei_Abrikosov_(physicist)">Wikiepedia entry</a>
## Selected writings
On what came to be called [[Abrikosov vortices]] ([[vortex strings]]) in [[type II superconductors]]:
* {#Abrikosov57a} [[Alexei Abrikosov]], *On the Magnetic properties of superconductors of the second group*, Sov. Phys. JETP **5** (1957) 1174-1182; Zh. Eksp. Teor. Fiz. **32** (1957) 1442-1452 [[spire:9138](https://inspirehep.net/literature/9138), [[Abrikosov-MagneticPropertiesOfSuperconductors.pdf:file]]]
* {#Abrikosov57} [[Alexei Abrikosov]], _The magnetic properties of superconducting alloys_, Journal of Physics and Chemistry of Solids Volume 2, Issue 3, 1957, Pages 199-208 (<a href="https://doi.org/10.1016/0022-3697(57)90083-5">doi:10.1016/0022-3697(57)90083-5</a>)
category: people
[[!redirects Alexei Alexeyevich Abrikosov]]
|
Alexei Bondal | https://ncatlab.org/nlab/source/Alexei+Bondal | __Alexei Bondal__ (Алексей Игоревич Бондал, spellings also Aleksei, Alexey etc.) is a mathematician at the Steklov Institute of Russian Academy of Sciences (and at Aberdeen). He has pioneered with [[Mikhail Kapranov]] methods of replacing varieties by [[enhanced triangulated categories]] of [[coherent sheaves]] leading to early developments of [[derived algebraic geometry]], influencing early works of [[Maxim Kontsevich|Kontsevich]] and of [[Carlos Simpson|Simpson]]--[[Bertrand Toen|Toen]] school.
* [data at ras](http://www.mi.ras.ru/staff/bondal_e.html)
* math genealogy [page](http://genealogy.math.ndsu.nodak.edu/id.php?id=85435)
## Selected writings
Introducing [[Serre functors]]:
* {#BondalKapranov90} [[Alexei I. Bondal]], [[Mikhail M. Kapranov]], _Representable functors, Serre functors, and mutations_, Mathematics of the USSR-Izvestiya **35** 3 (1990) 519-541 [[doi:10.1070/IM1990v035n03ABEH000716](https://doi.org/10.1070/IM1990v035n03ABEH000716)]
## Related entries
* [[Bondal-Orlov reconstruction theorem]]
* [[triangulated categories of sheaves]]
category: people
[[!redirects Alexei Bondal]]
[[!redirects Alexei Igorevich Bondal]]
[[!redirects Alexeĭ Bondal]]
[[!redirects Alexej Bondal]]
[[!redirects Alexey Bondal]]
[[!redirects Aleksei Bondal]]
[[!redirects Alekseĭ Bondal]]
[[!redirects Aleksej Bondal]]
[[!redirects Aleksey Bondal]]
[[!redirects Алексей Бондал]]
[[!redirects Алексей Игоревич Бондал]]
[[!redirects A. Bondal]]
[[!redirects Bondal]]
[[!redirects A.I.Bondal]]
[[!redirects Alexei I. Bondal]]
|
Alexei Davydov | https://ncatlab.org/nlab/source/Alexei+Davydov |
* [website](https://www.ohio.edu/cas/davydov)
## Selected writings
{#SelectedWritings}
On [[magmoidal categories]]:
* [[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))
category: people |
Alexei Kitaev | https://ncatlab.org/nlab/source/Alexei+Kitaev |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Alexei_Kitaev)
## Selected writings
On [[quantum computation]] and [[quantum error correction]] (and stating the [[Solovay-Kitaev theorem]]):
* [[Alexei Kitaev]], *Quantum computations: algorithms and error correction*, Russian Mathematical Surveys, **52** 6 (1997) [[doi:10.1070/RM1997v052n06ABEH002155](https://iopscience.iop.org/article/10.1070/RM1997v052n06ABEH002155), <a href="https://ochicken.top/Library/Physics/Quantum_Computation_and_Quantum_Information/1997.06%20A.Yu.Kitaev_%20Quantum%20computations_%20algorithms%20and%20error%20correction%20(kitaev1997).pdf">pdf</a>]
On [[quantum circuits]] with [[mixed quantum states]]/[[density matrices]]:
* [[Dorit Aharonov]], [[Alexei Kitaev]], [[Noam Nisan]], *Quantum Circuits with Mixed States*, *Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computation (STOC)* (1998) 20-30 [[arXiv:quant-ph/9806029](https://arxiv.org/abs/quant-ph/9806029), [doi:10.1145/276698.276708](https://doi.org/10.1145/276698.276708)]
On [[quantum error correcting codes]] associated with planar bulk/boundary systems (precursor to [[holographic tensor networks]]):
* S. B. Bravyi, [[Alexei Kitaev]], *Quantum codes on a lattice with boundary* ([arXiv:quant-ph/9811052](https://arxiv.org/abs/quant-ph/9811052))
On [[computation]] in general and [[quantum computation]] in particular:
* [[Alexei Y. Kitaev]], A. H. Shen, M. N. Vyalyi, *Classical and Quantum Computation*, Graduate Studies in Mathematics **47** (2002) [[doi:10.1090/gsm/047](http://dx.doi.org/10.1090/gsm/047)]
Introducing the idea of [[topological quantum computation]] with [[anyons]]:
* [[Alexei Kitaev]], _Fault-tolerant quantum computation by anyons_, Annals Phys. 303 (2003) 2-30 ([arXiv:quant-ph/9707021](https://arxiv.org/abs/quant-ph/9707021), <a href="https://doi.org/10.1016/S0003-4916(02)00018-0">doi:10.1016/S0003-4916(02)00018-0</a>)
* [[Michael Freedman]], [[Alexei Kitaev]], [[Michael Larsen]], [[Zhenghan Wang]], _Topological quantum computation_, Bull. Amer. Math. Soc. __40__ (2003), 31-38 ([arXiv:quant-ph/0101025](https://arxiv.org/abs/quant-ph/0101025), [doi:10.1090/S0273-0979-02-00964-3](https://doi.org/10.1090/S0273-0979-02-00964-3), [pdf](http://www.ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3/S0273-0979-02-00964-3.pdf))
Introducing a model with [[Majorana zero modes]] in quantum wires ([[Kitaev spin chain]]):
* [[Alexei Kitaev]], *Unpaired Majorana fermions in quantum wires*, Physics-Uspekhi, **44** 10S (2001) 131-136 [[doi:10.1070/1063-7869/44/10S/S29](https://iopscience.iop.org/article/10.1070/1063-7869/44/10S/S29)]
and its [[interaction|interacting]] generalization:
* [[Lukasz Fidkowski]], [[Alexei Kitaev]], *The effects of interactions on the topological classification of free fermion systems*, Phys. Rev. B **81** (2010) 134509 [[arXiv:0904.2197](https://arxiv.org/abs/0904.2197), [doi:10.1103/PhysRevB.81.134509](https://doi.org/10.1103/PhysRevB.81.134509)]
Comprehensive discussion of [[anyons]]:
* {#Kitaev06} [[Alexei Kitaev]], *Anyons in an exactly solved model and beyond*, Annals of Physics **321** 1 (2006) 2-111 $[$[doi:10.1016/j.aop.2005.10.005](https://doi.org/10.1016/j.aop.2005.10.005)$]$
On [[entanglement entropy]] as an indicator of [[topological phases of matter]]:
* [[Alexei Kitaev]], [[John Preskill]], *Topological entanglement entropy*, Phys. Rev. Lett. 96 (2006) 110404 ([arXiv:hep-th/0510092](https://arxiv.org/abs/hep-th/0510092))
On potential [[experiments]] detecting [[uncertainty of fluxes]] in [[quantum electrodynamics|quantum]] [[electromagnetism]]:
* [[Alexei Kitaev]], [[Gregory W. Moore]], [[Kevin Walker]], *Noncommuting Flux Sectors in a Tabletop Experiment* [[arXiv:0706.3410](https://arxiv.org/abs/0706.3410)]
On the classification of [[gapped Hamiltonians]]/[[topological insulators]] by [[topological K-theory]]:
* [[Alexei Kitaev]], _Periodic table for topological insulators and superconductors_, talk at: L.D.Landau Memorial Conference "Advances in Theoretical Physics", June 22-26, 2008, In:AIP Conference Proceedings 1134, 22 (2009) ([arXiv:0901.2686](https://arxiv.org/abs/0901.2686), [doi:10.1063/1.3149495](https://doi.org/10.1063/1.3149495))
Introducing the notion of [[short-range entanglement]] in [[quantum materials]]:
* {#Kitaev11} [[Alexei Kitaev]], *Toward Topological Classification of Phases with Short-range Entanglement*, talk at KITP (2011) $[$[video](https://online.kitp.ucsb.edu/online/topomat11/kitaev/)$]$
* {#Kitaev13} [[Alexei Kitaev]], *On the Classification of Short-Range Entangled States*, talk at Simons Center (2013) $[$[video](https://scgp.stonybrook.edu/archives/7874)$]$
category: people
[[!redirects Alexei Y. Kitaev]]
|
Alexei Kovalev | https://ncatlab.org/nlab/source/Alexei+Kovalev |
* [webpage](https://www.dpmms.cam.ac.uk/~agk22/)
## selected writings
* {#Kovalev03} [[Alexei Kovalev]], _Twisted connected sums and special Riemannian holonomy_, J. Reine Angew. Math. 565 (2003), 125--160 ([arXiv:math/0012189](https://arxiv.org/abs/math/0012189))
(on [[compact twisted connected sum G2-manifolds]])
## related $n$Lab entries
* [[G2-manifold]]
* [[M-theory on G2-manifolds]]
category: people |
Alexei M. Tsvelik | https://ncatlab.org/nlab/source/Alexei+M.+Tsvelik |
* [GoogleScholar page](https://scholar.google.com/citations?user=Q-A96cUAAAAJ&hl=en)
* [institute page](https://www.bnl.gov/staff/atsvelik)
## Selected writings
An [[integrable model]] for [[cyclic group|$\mathbb{Z}_N$]]-[[parafermion]] [[topological order]]:
* {#Tsvelik14a} [[Alexei M. Tsvelik]], *An integrable model with parafermion zero energy modes*, Phys Rev. Lett. **113** 066401 (2014) $[$[arXiv:1404.2840](https://arxiv.org/abs/1404.2840), [doi:10.1103/PhysRevLett.113.066401](https://doi.org/10.1103/PhysRevLett.113.066401)$]$
* {#Tsvelik14b} [[Alexei M. Tsvelik]], *$\mathbf{Z}_N$ parafermion zero modes without Fractional Quantum Hall effect* $[$[arXiv:1407.4002](https://arxiv.org/abs/1407.4002)$]$
category: people
|
Alexei Morozov | https://ncatlab.org/nlab/source/Alexei+Morozov |
* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Alexey_Morozov)
* [iip page](https://www.iip.ufrn.br/eventslecturer?inf===gTEVFe)
* [InSpire page](https://inspirehep.net/authors/996770)
## Selected writings
On [[string theory]]:
* [[Alexei Morozov]], *String theory: what is it?*, Sov. Phys. Usp. **35** (1992) 671-714 [[doi:10.1070/PU1992v035n08ABEH002255](https://iopscience.iop.org/article/10.1070/PU1992v035n08ABEH002255)]
On [[topological quantum computation]] with [[Chern-Simons theory]] (e.g. with [[su(2)-anyons]]):
* [[Dmitry Melnikov]], [[Andrei Mironov]], [[Sergey Mironov]], [[Alexei Morozov]], [[Andrey Morozov]], _Towards topological quantum computer_, Nucl. Phys. B926 (2018) 491-508 ([arXiv:1703.00431](https://arxiv.org/abs/1703.00431), [doi:10.1016/j.nuclphysb.2017.11.016](https://doi.org/10.1016/j.nuclphysb.2017.11.016))
On [[Yangians]]:
* Dmitry Galakhov, [[Alexei Morozov]], Nikita Tselousov, *Towards the theory of Yangians* [[arXiv:2311.00760](https://arxiv.org/abs/2311.00760)]
category: people |
Alexei Nurmagambetov | https://ncatlab.org/nlab/source/Alexei+Nurmagambetov |
* [inSPIRE page](http://inspirehep.net/author/profile/A.J.Nurmagambetov.3)
## Selected writings
On the [[Green-Schwarz sigma-model]] for the [[M5-brane]]:
* {#BLNPST97} [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _Covariant Action for the Super-Five-Brane of M-Theory_, Phys. Rev. Lett. 78 (1997) 4332-4334 ([arXiv:hep-th/9701149](http://arxiv.org/abs/hep-th/9701149))
* [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On the equivalence of different formulations of the M Theory five--brane_, Phys. Lett. B408 (1997) 135-141 ([arXiv:hep-th/9703127](http://arxiv.org/abs/hep-th/9703127))
On [[AdS/CFT]] for the [[M5-brane]]:
* [[Alexei Nurmagambetov]], I. Y. Park, _On the M5 and the AdS7/CFT6 Correspondence_ ([arXiv:hep-th/0110192](http://arxiv.org/abs/hep-th/0110192))
On [[type II supergravity]]:
* {#BNS04} [[Igor Bandos]], [[Alexei Nurmagambetov]], [[Dmitri Sorokin]], _Various Faces of Type IIA Supergravity_, Nucl. Phys. B676 (2004) 189-228 ([arXiv:hep-th/0307153](https://arxiv.org/abs/hep-th/0307153))
## Related $n$Lab entries
* [[M5-brane]]
* [[AdS/CFT]]
[[!redirects Aleksei Nurmagambetov]]
[[!redirects Alexei J. Nurmagambetov]]
|
Alexei Oblomkov | https://ncatlab.org/nlab/source/Alexei+Oblomkov |
* [webpage](http://people.math.umass.edu/~oblomkov)
## Selected writings
On the [[Gromov-Witten/Donaldson-Thomas correspondence]]:
* [[Davesh Maulik]], [[Alexei Oblomkov]], [[Andrei Okounkov]], [[Rahul Pandharipande]], _Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds_ ([arXiv:0809.3976](https://arxiv.org/abs/0809.3976v1))
category: people |
Alexei P. Kopylov | https://ncatlab.org/nlab/source/Alexei+P.+Kopylov |
* [personal page](https://www.cs.cornell.edu/people/kopylov/)
## Selected writings
On [[affine logic]]:
* [[Alexei P. Kopylov]], *Decidability of Linear Affine Logic*, Information and Computation **164** 1 (2001) 173-198 [[doi:10.1006/inco.1999.2834](https://doi.org/10.1006/inco.1999.2834)]
category: people
[[!redirects Alexei Kopylov]] |
Alexei Pirkovskii | https://ncatlab.org/nlab/source/Alexei+Pirkovskii |
* [Institute page](https://www.hse.ru/en/staff/pirkovskii)
## Selected writings
On [[EFC-algebra|EFC-algebras]]:
* [[Alexei Pirkovskii]], *Holomorphically finitely generated algebras*, Journal of Noncommutative Geometry 9 (2015), 215–264 ([arXiv:1304.1991](https://arxiv.org/abs/1304.1991), [doi:10.4171/JNCG/192](https://www.ems-ph.org/journals/show_abstract.php?issn=1661-6952&vol=9&iss=1&rank=8)).
category: people |
Alexey O. Remizov | https://ncatlab.org/nlab/source/Alexey+O.+Remizov |
* [math-net entry](https://www.mathnet.ru/eng/person12248)
## Selected writings
On basic [[linear algebra]], [[representation theory]] and basic [[geometry]] ([[Euclidean spaces|Euclidean]]-, [[affine spaces|affine]]-, [[projective space|projective]]-, [[hyperbolic spaces]]):
* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), ISBN:9783642309939 [pdf](https://ia801507.us.archive.org/33/items/LinearAlgebraAndGeometryShafarevichIgorR.RemizovAlexey/Linear%20Algebra%20and%20Geometry%20%20%5B%20Shafarevich%2C%20Igor%20R.%2C%20Remizov%2C%20Alexey%20%5D.pdf), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]
[[!redirects Alexey Remizov]]
|
Alexey Petrov | https://ncatlab.org/nlab/source/Alexey+Petrov |
* [webpage](http://www.physics.wayne.edu/~apetrov/Welcome.html)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alexey_A._Petrov)
## Selected writings
On [[D-meson]] [[decays]]:
* Marina Artuso, Brian Meadows, [[Alexey Petrov]], _Charm Meson Decays_, Ann. Rev. Nucl. Part. Sci.58:249-291, 2008 ([arXiv:0802.2934](https://arxiv.org/abs/0802.2934))
On [[effective field theory]]:
* {#PetrovBlechman16} [[Alexey Petrov]], Andrew E Blechman, _Effective Field Theories_, World Scientific 2016 ([doi:10.1142/8619](https://doi.org/10.1142/8619))
On indirect searches for New Physics (i.e. [[phenomenology]] beyond the [[standard model of particle physics]]),
specifically via [[charm quark]] physicsl:
* Alexey Petrov, _Searching for New Physics with Charm_ ([arXiv:1003.0906](https://arxiv.org/abs/1003.0906))
## Related $n$Lab entries
* [[effective field theory]]
category: people |
Alexey Slinkin | https://ncatlab.org/nlab/source/Alexey+Slinkin |
* [GoogleScholar page](https://scholar.google.ru/citations?user=xeD0KksAAAAJ&hl=ru)
## Selected writings
On 1-[[twisted de Rham cohomology]] related to [[sl(2)|$\mathfrak{sl}(2)$]]-[[Verma modules]]:
* {#SlinkinVarchenko19} [[Alexey Slinkin]], [[Alexander Varchenko]], *Twisted de Rham Complex on Line and Singular Vectors in $\widehat{\mathfrak{sl}}_2$ Verma Modules*, SIGMA **15** (2019) 075 [[arXiv:1812.09791](https://arxiv.org/abs/1812.09791), [doi:10.3842/SIGMA.2019.075](https://doi.org/10.3842/SIGMA.2019.075)]
category: people
|
Alexis Hazell | https://ncatlab.org/nlab/source/Alexis+Hazell |
Assistant system administrator for the nLab.
[[!redirects AlexisHazell]] |
Alexis Toumi | https://ncatlab.org/nlab/source/Alexis+Toumi | [alexis.toumi.xyz](https://alexis.toumi.xyz)
## Selected writings
On [[DisCoPy]]:
* Giovanni de Felice, [[Alexis Toumi]], [[Bob Coecke]]. _DisCoPy: Monoidal Categories in Python_. Applied Category Theory, 2020. ([arXiv:2005.02975l](https://arxiv.org/abs/2005.02975l))
## Related entries
* [[DisCoPy]]
category: people |
Alexis Virelizier | https://ncatlab.org/nlab/source/Alexis+Virelizier |
* [personal page](http://math.univ-lille1.fr/~virelizi/)
## Selected writings
On [[Hopf monads]] and [[Hopf adjunctions]]:
* [[Alain Bruguières]], [[Alexis Virelizier]], *Hopf monads*, Advances in Mathematics **215** 2 (2007) 679-733 [[doi:10.1016/j.aim.2007.04.011](https://doi.org/10.1016/j.aim.2007.04.011), [arXiv:math/0604180](https://arxiv.org/abs/math/0604180)]
* [[Alain Bruguières]], [[Steve Lack]], [[Alexis Virelizier]], *Hopf monads on monoidal categories*, Adv. Math. __227__ 2 (2011) 745-800 [[arXiv:1003.1920](https://arxiv.org/abs/1003.1920), [doi:10.1016/j.aim.2011.02.008](https://doi.org/10.1016/j.aim.2011.02.008)]
category: people
|
Aleš Pultr | https://ncatlab.org/nlab/source/Ale%C5%A1+Pultr | [[!redirects Aleš Pultr]]
Aleš Pultr is a professor at Charles University.
[Website](https://kam.mff.cuni.cz/~pultr/),
[university web page](https://www.mff.cuni.cz/en/faculty/organizational-structure/people?hdl=501).
## Selected writings
A textbook on [[locales]]:
* [[Jorge Picado]], [[Aleš Pultr]], _[[Frames and Locales]]. Topology without points_, Frontiers in Mathematics, Birkhäuser (2012). |
Alfred Ewing | https://ncatlab.org/nlab/source/Alfred+Ewing |
* [Wikipedia entry](http://en.wikipedia.org/wiki/A._C._Ewing)
* [Obituary](http://www.anthonyflood.com/blanshardewing.htm), Brand Blanshard
## some writings
* _Idealism: A Critical Survey_, Barnes & Noble (1974) ([PhiPapers](http://philpapers.org/rec/EWIIAC), [Review on JSTOR](http://www.jstor.org/stable/2249864))
## related $n$Lab entries
* [[idealism]]
[[!redirects A. C. Ewing]]
[[!redirects Alfred Cyril Ewing]]
category: people |
Alfred Frölicher | https://ncatlab.org/nlab/source/Alfred+Fr%C3%B6licher |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Alfred_Fr%C3%B6licher)
## Selected writings
Introducing the [[Frölicher-Nijenhuis bracket]]:
* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms. Part I. Derivations in the graded ring of differential forms_, Indagationes Mathematicae (Proceedings) **59** (1956) 338–350 and 351–359 [<a href="https://doi.org/10.1016/s1385-7258(56)50046-7">doi:10.1016/s1385-7258(56)50046-7</a>, <a href="https://doi.org/10.1016/s1385-7258(56)50047-9">doI;10.1016/s1385-7258(56)50047-9</a>]
and its refinement for [[almost complex structures]]:
* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms. Part II. Almost-complex structures_, Indagationes Mathematicae (Proceedings) **61** (1958) 414–421 and 422-429 [<a href="https://doi.org/10.1016/S1385-7258(58)50058-4">doi:10.1016/S1385-7258(58)50058-4</a>, <a href="https://doi.org/10.1016/S1385-7258(58)50057-2">doi:10.1016/S1385-7258(58)50057-2</a>]
## Related entries
* [[Frölicher space]]
category: people
[[!redirects Alfred Frolicher]]
[[!redirects A. Frölicher]]
[[!redirects Frölicher]]
|
Alfred Gray | https://ncatlab.org/nlab/source/Alfred+Gray |
* <a href="https://en.wikipedia.org/wiki/Alfred_Gray_(mathematician)">Wikipedia entry</a>
## Selected writings
* [[Alfred Gray]], [_A Note on Manifolds Whose Holonomy Group is a Subgroup of Sp(n) $\cdot$ Sp(1)_](https://projecteuclid.org/euclid.mmj/1029000212), Michigan Math. J. Volume 16, Issue 2 (1969), 125-128.
(on [[Sp(n).Sp(1)]]-[[special holonomy]])
* {#GrayGreen70} [[Alfred Gray]], Paul S. Green, _Sphere transitive structures and the triality automorphism_, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 ([euclid:1102976640](https://projecteuclid.org/euclid.pjm/1102976640))
(on [[transitive actions]] of [[compact Lie groups]] on [[n-spheres]])
## Related $n$Lab entries
* [[Sp(n).Sp(1)]]
* [[n-sphere]]
* [[Spin(8)]]
* [[G2/SU(3) is the 6-sphere]]
* [[Spin(7)/G2 is the 7-sphere]]
* [[Spin(5)/SU(2) is the 7-sphere]]
* [[Spin(9)/Spin(7) is the 15-sphere]]
category: people |
Alfred Rieckers | https://ncatlab.org/nlab/source/Alfred+Rieckers |
* [personal page](https://homepages.uni-tuebingen.de/alfred.rieckers/)
* [InSpire page](https://inspirehep.net/authors/2270468)
## Selected writings
On [[continuous field of C-star algebras|continuous fields]] of [[Weyl algebras]] as [[continuous deformation quantizations]] of [[symplectic vector spaces]]:
* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization*, Annales Henri Poincaré **5** (2004) 327–346 [[doi:10.1007/s00023-004-0171-y](https://doi.org/10.1007/s00023-004-0171-y)]
On [[group algebras]] of ([[underlying]] [[discrete group|discrete]]) [[Heisenberg groups]] as [[strict deformation quantizations]] of [[presymplectic manifold|pre-]][[symplectic vector spaces|symplectic]] [[topological vector spaces]] by [[continuous field of C-star algebras|continuous fields of]] [[Weyl algebras]]:
* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization*, International Journal of Pure and Applied Mathematics **38** 1 (2007) [[ijpam:2007-38-1/6](https://ijpam.eu/contents/2007-38-1/6/index.html), [pdf](https://www.ijpam.eu/contents/2007-38-1/6/6.pdf)]
* [[Reinhard Honegger]], [[Alfred Rieckers]], *Heisenberg Group Algebra and Strict Weyl Quantization*, Chapter 23 in: *Photons in Fock Space and Beyond, Volume I: From Classical to Quantized Radiation Systems*, World Scientific (2015) [chapter:[doi;10.1142/9789814696586_0023](https://doi.org/10.1142/9789814696586_0023), book:[doi:10.1142/9251-vol1](https://doi.org/10.1142/9251-vol1)]
category: people
|
Alfred Schild | https://ncatlab.org/nlab/source/Alfred+Schild |
* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Alfred_Schild)
## Selected writings
Introducing [[Kerr-Schild spacetimes]]:
* [[Roy Kerr]], [[Alfred Schild]], _Some algebraically degenerate solutions of Einstein’s gravitational field equations_, Proc. Symp. Appl. Math 17, 199, 1965
category: people |
Alfred Tarski | https://ncatlab.org/nlab/source/Alfred+Tarski |
* [SEP entry](https://plato.stanford.edu/entries/tarski/)
* [wikipedia entry](http://en.wikipedia.org/wiki/Alfred_Tarski)
## Selected writings
* [[Alfred Tarski]], *Introduction to Logic and to the Methodology of Deductive Sciences* Oxford University Press (1994) $[$[doi:10.2307/2180610](https://doi.org/10.2307/2180610), ISBN 978-0-19-504472-0$]$
## Related entries
* [[Tarski universe]]
* [[Jónsson-Tarski algebra]]
* [[Jónsson-Tarski topos]]
* [[Lindenbaum-Tarski algebra]]
* [[Tarski-Seidenberg theorem]]
* [[Knaster-Tarski theorem]]
* [[correspondence theory of truth]]
* [[Euclidean geometry]]
* [[Tarski group]]
category: people
[[!redirects A. Tarski]]
[[!redirects Tarski]]
|
Alfred Whitehead | https://ncatlab.org/nlab/source/Alfred+Whitehead |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Alfred_North_Whitehead)
## Selected writings
* _[[Principia Mathematica]]_ (1910) with [[Bertrand Russell]]
## Related entries
* [[assertion]]
category: people
[[!redirects Alfred North Whitehead]]
|
Alfred Young | https://ncatlab.org/nlab/source/Alfred+Young |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alfred_Young)
## Selected writings
Introducing the [[seminormal representation]] in the [[representation theory of the symmetric group]]:
* [[Alfred Young]], *On quantitative substitutional analysis VI*, Proc. London Math. Soc. (2) **31** (1931), 253-289. Reprinted in: *The Collected Papers of Alfred Young, 1873-1940*, Mathematical Expositions 21, Univ. of Toronto Press, Toronto and Buffalo, 1977
## Related $n$Lab entries
* [[Young diagram]]
* [[Young tableaux]]
* [[semistandard Young tableaux]]
category: people
|
Alfréd Rényi | https://ncatlab.org/nlab/source/Alfr%C3%A9d+R%C3%A9nyi |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alfr%C3%A9d_R%C3%A9nyi)
## Selected writings
Introducing the notion of [[Rényi entropy]]:
* [[Alfréd Rényi]], *On Measures of Entropy and Information*, Berkeley Symposium on Mathematical Statistics and Probability, 1961: 547-561 (1961) ([euclid](https://projecteuclid.org/ebooks/berkeley-symposium-on-mathematical-statistics-and-probability/Proceedings%20of%20the%20Fourth%20Berkeley%20Symposium%20on%20Mathematical%20Statistics%20and%20Probability,%20Volume%201:%20Contributions%20to%20the%20Theory%20of%20Statistics/chapter/On%20Measures%20of%20Entropy%20and%20Information/bsmsp/1200512181))
[[!redirects Alfred Rényi]]
[[!redirects Alfréd Renyi]]
[[!redirects Alfred Renyi]]
[[!redirects A. Rényi]]
[[!redirects A. Renyi]]
category: people |
Alfsen-Shultz theorem | https://ncatlab.org/nlab/source/Alfsen-Shultz+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The theorem of ([Alfsen-Shultz 78](#AlfsenShultz78)) says that two [[C*-algebras]] have [[isomorphism|isomorphic]] [[state on an operator algebra|state spaces]] precisely if as [[Jordan algebras]] they are isomorphic.
This is to some extent a [[Jordan algebra]]-analog of the [[Gelfand-Naimark theorem]].
## Related theorems
Other theorems about the foundations and [[interpretation of quantum mechanics]] include:
* [[order-theoretic structure in quantum mechanics]]
* [[Kochen-Specker theorem]]
* [[Harding-Döring-Hamhalter theorem]]
* [[Fell's theorem]]
* [[Gleason's theorem]]
* [[Wigner theorem]]
* [[Bell's theorem]]
* [[Bub-Clifton theorem]]
* [[Kadison-Singer problem]]
## References
* [[Erik Alfsen]], [[Frederic Shultz]], _A Gelfand Neumark theorem for Jordan algebras_, Advances in Math., 28 (1978), 11-56.
{#AlfsenShultz78}
* [[Erik Alfsen]], H. Hanche-Olsen, [[Frederic Shultz]], _State spaces of $C^\ast$-algebras_, Acta Math., 144 (1980), 267-305.
{#AlfsenHOShultz80}
See also
* [[Frederic Shultz]], _Dual maps of Jordan homomorphisms and ∗-homomorphisms between $C^\ast$-algebras_, Pacific J. Math. Volume 93, Number 2 (1981), 435-441 ([Euclid]( http://projecteuclid.org/euclid.pjm/1102736271))
{#Shultz81}
|
Alg | https://ncatlab.org/nlab/source/Alg |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_Alg_ is the [[category]] with [[algebra|algebras]] as [[object|objects]] and algebra homomorphisms as [[morphism]]s.
More abstractly, we can think of $Alg$ as the [[category|full subcategory]] of $Cat(Vect)$, [[internal category|internal categories]] in [[Vect]], with algebras as objects.
In case of [[Ab]], this gives us Category of Rings, namely, $Alg_Z$.
## Properties
### Relation to algebras with bimodules
Since algebras may be identified with one-object [[category|categories]] [[internalization|internal to]] vector spaces, it is sometimes useful to regard $Alg$ as a strict 2-category, namely as a full sub-2-category of the 2-category $Cat(Vect)$. In this case the 2-morphisms between morphisms of algebras come from "intertwiners": inner endomorphisms of the target algebra.
Precisely analogous statements hold for the category [[Grp]] of groups.
With $Alg$ regarded as a strict 2-category this way there is a canonical 2-functor
$$
Alg \hookrightarrow Bimod
$$
to the category [[Bimod]], which sends algebra homomorphisms $f : A \to B$ to the $A$-$B$ bimodule ${}_f B$. This exhibits $Bimod$ as a [[framed bicategory]] in the sense of Shulman.
## Related concepts
* [[Vect]]
* [[Mod]], [[2Mod]], [[nMod]]
* [[dgcAlg]]
* [[Eilenberg-Moore category]]
category: category
[[!redirects Algebras]]
[[!redirects category of algebras]]
[[!redirects categories of algebras]]
|
Alg(T) > history | https://ncatlab.org/nlab/source/Alg%28T%29+%3E+history | < [[Alg(T)]]
|
algebra | https://ncatlab.org/nlab/source/algebra |
> This page is about **algebra as a theory**. If you are looking for the term **algebra as an object** see [[associative algebra]] or [[algebra over an operad]] or the like. See below for more.
***
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
[[geometry]] $\leftarrow$ [[Isbell duality]] $\rightarrow$ **algebra**
***
#Contents#
* table of contents
{:toc}
## Idea
**Algebra** is the manipulation of symbols without (necessarily) regard for their meaning, especially in a way that can be formalized in [[cartesian logic]]. It is often seen as [[duality|dual]] to [[geometry]]. While modern algebra has ties and applications nearly everywhere in mathematics, traditionally closest ties are with the [[number theory]] and [[algebraic geometry]].
The word 'algebra' is often also used for an algebraic structure:
* often by default an [[associative unital algebra]];
* more generally a [[monoid object]];
* more generally in a different way, a [[nonassociative algebra]];
* an [[algebra over an operad]], of a [[monad]], a [[PROP]], etc;
* an [[algebra for an endofunctor]];
* a model of any [[algebraic theory]] or anything studied in [[universal algebra]];
* higher categorical analogues, many object/family versions of algebras, for example [[algebroid]]s, and [[pseudoalgebra]]s (or [[2-algebra]]s) over [[pseudomonad]]s (or [[2-monad]]s).
Various fields of mathematics or mathematical concepts can be manipulated in an algebraic or symbolic way, and such approaches or formalized subfields have names like [[categorical algebra]], [[homological algebra]], [[homotopical algebra]] and so on. Methods of combinatorics which involve much algebra, and manipulations with [[formal power series]] in particular, are called [[algebraic combinatorics]].
## Related entries
The $n$lab has a number of entries on particular algebraic structures ([[monoid]], [[semigroup]], [[group]], [[ring]], [[noetherian ring]], [[quasigroup]], [[associative algebra]], [[Lie algebra]], [[coalgebra]], [[dg-algebra]], [[bialgebra]], [[graded algebra]], [[Hopf algebra]], [[coring]], [[quasitriangular bialgebra]], [[lattice]], [[rig]], [[near-ring]], $\Omega$-[[$\Omega$-group|group]], [[field]], [[perfect field]], [[skewfield]], [[free field]], [[vector space]], [[vertex operator algebra]], [[crossed module]], [[chain complex]], [[hypermonoid]], [[hyperring]], [[hyperfield]], [[truss]], [[brace]] etc.), entries on their structural features, parts, "envelopes" or localizations ([[ideal]], [[center]], [[centralizer]], [[normal subgroup]], [[normal closure]], [[normalizer]], [[holomorph]], [[Ore set]], [[Ore localization]], [[enveloping algebra]], [[universal enveloping algebra]]) and on algebraic structures internal to other categories ([[topological group]], [[Lie group]], [[Lie groupoid]], [[algebraic group]], [[formal group]], [[dg-algebra]] etc).
There are also few pages on various invariants of algebraic objects or operations on algebraic expressions, e.g. on [[resultants]] of polynomials, [[determinant]] of a matrix, [[quasideterminant]] of a matrix with noncommutative entries.
For many algebraic structures a notion of [[action]] is defined; they embody "symmetry algebras" of some other algebraic objects. An action is expressed via a [[representation]] of one object as a subobject of a full object of another; or as a combination of the object which acts and which is acted upon (e.g. [[action groupoid]]). Objects with action are [[modules]] of the appropriate kind (possibly dualized: [[comodule]], [[contramodule]]; multiple, e.g. [[bimodule]]; or homotopized like $A_\infty$-modules). The possibilities for realizing a given algebra via symmetries of another object are systematically studied in a field called [[representation theory]].
See also
* [[commutative algebra]]
* [[counterexamples in algebra]].
[[!include Isbell duality - table]]
## References
Introductory textbooks:
* [[Carl Faith]], _Algebra: Rings, Modules, Categories I_, Grundlehren der mathematischen Wissenschaften **190**, Springer (1973) [[doi:10.1007/978-3-642-80634-6](https://doi.org/10.1007/978-3-642-80634-6)]
* [[Anthony Knapp]], *Basic Algebra*, Springer (2006) [[doi:10.1007/978-0-8176-4529-8](https://doi.org/10.1007/978-0-8176-4529-8), [pdf](https://www.math.mcgill.ca/darmon/courses/19-20/algebra2/knapp.pdf)]
* {#Aluffi09} [[Paolo Aluffi]], *Algebra: Chapter 0*, Graduate Studies in Mathematics **104**, AMS (2009) [[ISBN:978-1-4704-1168-8](https://bookstore.ams.org/gsm-104/)]
* [[Joseph A. Gallian]], *Contemporary Abstract Algebra*, Chapman and Hall/CRC (2020) [[doi:10.1201/9781003142331](https://doi.org/10.1201/9781003142331), [webpage](https://en.wikipedia.org/wiki/Joseph_Gallian), [pdf](https://ict.iitk.ac.in/wp-content/uploads/CS203-Mathematics-for-Computer-Science-III-Gallian.pdf)]
See also:
* Wikipedia, *[Algebra](http://en.wikipedia.org/wiki/Algebra)*
and see the references at *[[ring]]*, *[[module]]*, etc.
[[!redirects algebras]]
[[!redirects algebraist]] |
algebra extension | https://ncatlab.org/nlab/source/algebra+extension |
> For extension of [[morphisms]] in the sense dual to [[lift]] see at [[extension]].
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Given any kind of object $A$ in [[algebra]], such as an [[associative algebra]] or a [[group]] or a [[Lie algebra]], etc., then an _extension_ of $A$ is an [[epimorphism]]
$$
\widehat A \overset{p}{\longrightarrow} A
$$
Typically the [[kernel]] of an epimorphism will exist in the given [[category]], leading to a [[short exact sequence]]
$$
ker(p) \longrightarrow \widehat A \overset{p}{\longrightarrow} A
\,.
$$
Then one says that $\widehat A$ is an _extension by $ker(p)$_ of $A$.
## Examples
* [[group extension]]
* [[Lie algebra extension]]
* [[infinitesimal extension]]
* [[central extension]]
## Related concepts
* [[higher extension]]
[[!redirects algebra extensions]] |
algebra for a profunctor | https://ncatlab.org/nlab/source/algebra+for+a+profunctor |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
The notion of **algebra over an endo-[[profunctor]]** ($C$-$C$-[[bimodule]]) is a joint generalization of the notions [[algebra for an endofunctor]] and [[coalgebra for an endofunctor]].
## Definition
For a category $C$ and a $C$-$C$ [[bimodule]] $H : C^{op} \times C \to Set$, an **algebra** for $H$ is given by a [[functor]] $X \colon D \to C$ and an [[extranatural transformation]] $\ast \to H(X,X)$, where $\ast \colon \mathbf{1} \to Set$ is constant at the [[point]]. $X$ is called the **carrier** of the algebra. A morphism $(X, \alpha) \to (Y, \beta)$ of $H$-algebras is given by a [[natural transformation]] $\phi \colon X \Rightarrow Y$ such that $H(X,\phi) \circ \alpha = H(\phi,Y) \circ \beta$.
If $D$ is the one-object category, an algebra $(X,\alpha)$ is given by an object $X$ in $C$ and an element $\alpha \in H(X, X)$. A morphism between two algebras $(X, \alpha)$ and $(Y, \beta)$ is then a morphism $m : X \to Y$ in $C$ such that $H(X, m) (\alpha) = H(m, Y) (\beta)$, these both being elements of $H(X, Y)$.
There is an an obvious [[forgetful functor]] into $C$ from the category of algebras for $H$, which sends each algebra to its carrier and each algebra morphism to its underlying morphism in $C$; among other properties, this functor is always [[faithful]] and [[conservative functor|conservative]].
In fact, the category $Alg(H)$, together with its forgetful functor $U\colon Alg(H)\to C$, has the universal property of an [[Eilenberg-Moore object]], namely that of being the _universal_ $H$-algebra. Specifically, it is a [[terminal object]] in the category whose objects are functors $G\colon D\to C$ equipped with an [[extranatural transformation]] $\ast \to H(G-,G?)$. For such an extranatural transformation consists of, for every $d\in D$, an element $\xi_d \in H(G d,G d)$, such that for every morphism $v\colon d\to e$ in $D$, we have $H(id_d,v)(\xi_d) = H(v,id_e)(\xi_e)$. This is precisely the data of a functor $D\to Alg(H)$ lying over $C$.
## Coalgebras in Prof
One version of [[Yoneda lemma|Yoneda's lemma]] says that for a profunctor $H \colon C ⇸ C$ there is a bijection between extranatural transformations $\ast \to H$ and natural transformations $\hom_C \to H$. So there are bijections
$$
\array{
\ast \: {\ddot\to} \: H(X,X) \\
\hom_D \Rightarrow H(X,X) \\
C(1,X) \Rightarrow H \circ C(1,X)
}
$$
where the last holds by the usual properties of representable profunctors (see e.g. [[proarrow equipment]]). This exhibits each $H$-algebra on $X$ in the above sense as a $H$-[[algebra for an endomorphism|coalgebra]] in $Prof$ with carrier $C(1,X)$.
## Examples
* [[algebra for an endofunctor|Algebras]] and [[coalgebra for an endofunctor|coalgebras]] for [[endofunctor|endofunctors]] are special cases of algebras for bimodules; specifically, an algebra for an endofunctor $F$ is an algebra for the bimodule $Hom(F(-), ?)$, while a coalgebra for $F$ is an algebra for the bimodule $Hom(-, F(?))$.
* A [[natural transformation]] between functors $F$ and $G$ from $C$ to $D$ is a [[section]] of the forgetful functor into $C$ from the category of algebras for the $C-C$ bimodule $Hom_D(F(-), G(?))$. That is, it gives every object of $C$ the structure of an algebra for $Hom_D(F(-), G(?))$ in such a way as that every morphism of $C$ has the property of being an algebra morphism between the algebras on its domain and codomain.
* A [[natural numbers object]] (in the weak, unparametrized sense) in a category $C$ with terminal object $1$ is an initial object in the category of algebras for the bimodule $Hom_C(1, ?) \times Hom_C(-, ?)$. If $C$ has binary coproducts, then this is of course the same as an initial algebra for the endofunctor $1+(-)$.
## Related concepts
* [[algebra for an endomorphism]]
* [[algebra over a monad]], [[algebra over an endofunctor]], [[coalgebra over an endofunctor]], [[dialgebra]]
[[!redirects algebra for an endo-profunctor]]
[[!redirects algebra for an endoprofunctor]]
[[!redirects algebra for a bimodule]]
[[!redirects algebra for a C-C bimodule]]
[[!redirects algebra over a profunctor]] |
algebra for an endofunctor | https://ncatlab.org/nlab/source/algebra+for+an+endofunctor |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
An _algebra over an endofunctor_ is like an [[algebra over a monad]], but without a notion of [[associativity]] (given that a plain [[endofunctor]] is not equipped with a multiplication-operation that would make it a [[monad]]).
## Definition
+-- {: .num_defn}
###### Definition
For a [[category]] $C$ and [[endofunctor]] $F$, an __algebra__ (or __module__) of $F$ is an [[object]] $X$ in $C$ and a [[morphism]] $\alpha\colon F(X) \to X$. ($X$ is called the __carrier__ of the algebra)
A [[homomorphism]] between two algebras $(X, \alpha)$ and $(Y, \beta)$ of $F$ is a morphism $m\colon X \to Y$ in $C$ such that the following [[commuting diagram|square commutes]]:
$$
\array{
F(X)
&
\overset{F(m)}{\longrightarrow}
&
F(Y)
\\
\mathllap{{}^{\alpha}}\big\downarrow
&&
\big\downarrow\mathrlap{{}^{\beta}}
\\
X
&
\underset{m}{\longrightarrow} & Y
}
\,.
$$
[[composition|Composition]] of such homomorphisms of algebras is given by composition of the [[underlying]] morphisms in $C$. This yields the [[category]] of $F$-algebras, which comes with a [[forgetful functor]] to $C$.
=--
+-- {: .num_remark}
###### Remark
The dual concept is a [[coalgebra for an endofunctor]]. Both algebras and coalgebras for endofunctors on $C$ are special cases of [[algebra for a C-C bimodule|algebras for bimodules]].
=--
If $F$ is a [[pointed endofunctor]] with point $\eta : Id \to F$, then by an **algebra** for $F$ one usually means a *pointed algebra*, i.e. one such that $\alpha \circ \eta_X = id_X$.
## Properties
### Relation to algebras over a monad
To a [[category theory|category theorist]], [[algebras over a monad]] may be more familiar than algebras over just an endofunctor. In fact, when $C$ and $F$ are well-behaved, then algebras over an endofunctor $F$ are equivalent to algebras over a certain monad, the [[algebraically-free monad]] generated by $F$ ([Pirog](#Maciej), [Gambino-Hyland 04, section 6](#GambinoHyland04)).
This is analogous to the relationship between an [[action]] $M \times B \to B$ of a [[monoid]] $M$ and a [[binary function]] $A \times B \to B$ (an action of a [[set]]): such a function is the same thing as an action of the [[free monoid]] $A^*$ on $B$.
Returning to the endofunctor case, the general statement is:
+-- {: .num_prop}
###### Proposition
The [[category]] of algebras of the endofunctor $F\colon \mathcal{C} \to \mathcal{C}$ is [[equivalence of categories|equivalent]] to the category of [[algebra over a monad|algebras]] of the [[algebraically-free monad]] on $F$, should such exist.
=--
Actually, this proposition is merely a definition of the term "algebraically-free monad". If $F$ has an algebraically-free monad, denoted say $F^*$, then in particular the forgetful functor $F Alg \to C$ has a [[left adjoint]], and $F^*$ is the monad on $C$ generated by this [[adjunction]]. Conversely, if such a left adjoint exists, then the monad it generates is algebraically-free on $F$; for the straightforward proof, see for instance ([Pirog](#Maciej)). An explicit construction of the algebraically free monad in terms of [inductive types](#initalg) is given below.
Algebraically-free monads exist in particular when $C$ is a [[locally presentable category]] and $F$ is an [[accessible functor]]; see [[transfinite construction of free algebras]].
+-- {: .num_remark}
###### Remark
It turns out that an algebraically-free monad on $F$ is also [[free object|free]] in the sense that it receives a universal arrow from $F$ relative to the [[forgetful functor]] from [[monads]] to [[endofunctors]]. The converse, however, is not necessarily true: a free monad in this sense need not be algebraically-free. It is true when $C$ is [[complete category|complete]], however.
=--
Entirely analogous facts are true for pointed algebras over pointed endofunctors.
## Relationship to inductive types {#initalg}
The [[initial algebra of an endofunctor]] provides [[categorical semantics]] for [[inductive types]].
The construction of an algebraically free monad may be cast in the language of such initial algebras. Suppose $C$ is a category with coproducts and $F: C \to C$ is an endofunctor. Let $F$-$alg$ be the category of $F$-algebras, and let $U: F\text{-}alg \to C$ be the usual forgetful functor. A [[left adjoint]] to $U$ then takes an object $d$ of $C$ to the initial algebra $\Phi(d)$ of the endofunctor $c \mapsto d + F(c)$, provided this initial algebra exists. For, by the usual comma category description (see for example [[adjoint functor theorem]]), $\Phi(d)$ is the [[initial object]] of the category $(d \downarrow U)$. However, an object of $(d \downarrow U)$ is a triple $(c, \alpha: F(c) \to c, \beta: d \to c)$, equivalently a pair $(c, \gamma: d + F(c) \to c)$, equivalently an algebra of $c \mapsto d + F(c)$. Hence an initial object of $(d \downarrow U)$ is an initial algebra of an endofunctor.
The [[monad]] structure of the algebraically free monad $F^\ast = U\Phi$ may be straightforwardly extracted from this initial algebra description. This is made explicit in [Pirog](#Maciej). For example, to describe the multiplication $\mu: F^\ast F^\ast \to F^\ast$, let $d$ be an object; then $F^\ast d$ has an algebra structure $[i, \theta]: d + F(F^\ast d) \to F^\ast d$. It therefore also has a structure of algebra over the endofunctor $c \mapsto F^\ast d + F(c)$, namely $[1, \theta]: F^\ast d + F(F^\ast d) \to F^\ast d$. But since $F^\ast F^\ast d$ is the initial algebra for the monad $c \mapsto F^\ast d + F(c)$, we obtain a unique algebra map $F^\ast F^\ast d \to F^\ast d$. This is the component $\mu_d$ of the monad multiplication.
## Related concepts
* [[free monad]]
* [[endofunctor]], [[pointed endofunctor]]
* [[algebra over a monad]], [[algebra over a profunctor]], [[coalgebra over an endofunctor]]
* [[algebraically compact category]]
## References
A textbook account of the basic theory is in [chapter 10](http://www.andrew.cmu.edu/course/80-413-713/notes/chap10.pdf) of
* [[Steve Awodey]], _Category theory_ lecture notes (2011) ([web](http://www.andrew.cmu.edu/course/80-413-713/))
The relation to [[free monads]] is discussed in
* [Maciej Pirog](http://maciejcs.wordpress.com/), _[Free monads and their algebras](http://maciejcs.wordpress.com/2012/04/17/free-monads-and-their-algebras/)_
{#Maciej}
* [[Nicola Gambino]], [[Martin Hyland]], _Wellfounded trees and dependent polynomial functors_. In Types for proofs and programs, volume 3085 of Lecture Notes in Comput. Sci., pages 210–225. Springer-Verlag, Berlin, 2004 ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.98.4534))
{#GambinoHyland04}
[[!redirects algebra of a functor]]
[[!redirects algebra of an endofunctor]]
[[!redirects algebra for a functor]]
[[!redirects algebra for an endofunctor]]
[[!redirects algebra over a functor]]
[[!redirects algebra over an endofunctor]]
[[!redirects algebras of a functor]]
[[!redirects algebras of an endofunctor]]
[[!redirects algebras for a functor]]
[[!redirects algebras for an endofunctor]]
[[!redirects algebras over a functor]]
[[!redirects algebras over an endofunctor]]
[[!redirects algebras of functors]]
[[!redirects algebras of endofunctors]]
[[!redirects algebras for functors]]
[[!redirects algebras for endofunctors]]
[[!redirects algebras over functors]]
[[!redirects algebras over endofunctors]]
[[!redirects F-algebra]]
[[!redirects algebra of a pointed endofunctor]]
[[!redirects algebra for a pointed endofunctor]]
[[!redirects algebra over a pointed endofunctor]]
[[!redirects algebras of a pointed endofunctor]]
[[!redirects algebras for a pointed endofunctor]]
[[!redirects algebras over a pointed endofunctor]]
[[!redirects algebras of pointed endofunctors]]
[[!redirects algebras for pointed endofunctors]]
[[!redirects algebras over pointed endofunctors]]
|
algebra for an endomorphism | https://ncatlab.org/nlab/source/algebra+for+an+endomorphism | [[!redirects algebra over an endomorphism]]
## Definition
If $K$ is a [[bicategory]] and $f \colon a \to a$ is an [[endomorphism]] in $K$, then a (**left**) $f$-**algebra** or $f$-**module** is given by a 1-cell $x \colon b \to a$ together with a 2-cell $\lambda \colon f x \Rightarrow x$.
One can also define right modules/algebras, comodules/coalgebras and bimodules as for [[module over a monad|monads]].
## Examples
If $K$ is $Cat$, an [[algebra for an endofunctor]] $F \colon C \to C$ is the same thing as an $F$-algebra $A \colon \ast \to C$ in the sense above.
Every [[module over a monad]] $(t, \eta, \mu)$ is an algebra over the underlying endomorphism $t$.
An [[algebra for a profunctor]] (q.v.) $H \colon C ⇸ C$ on $X \colon D \to C$ is essentially the same as a $H$-coalgebra $C(1,X) \Rightarrow H \circ C(1,X)$ in $Prof$, the bicategory of categories and [[profunctors]].
|
algebra modality | https://ncatlab.org/nlab/source/algebra+modality | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of algebra modality is used to define [[differential categories|codifferential categories]]. However, it could be use in other types of [[doctrines|categorical doctrines]].
## Definition
\begin{definition}
An [[algebra modality]] in a [[symmetric monoidal category]] $(\mathcal{C}, \otimes, I)$ is given by a [[monad]] $(S,m,u)$ and two [[natural transformations]] $\eta:I \rightarrow S(A)$ and $\nabla:S(A) \otimes S(A) \rightarrow S(A)$ such that for every $A \in \mathcal{C}$, $(S(A), \nabla, \eta)$ is a [[commutative monoid]] in $(\mathcal{C},\otimes,I)$ and this diagram commutes:
\begin{tikzcd}
S(S(A)) \otimes S(S(A)) \arrow[dd, "m \otimes m"'] \arrow[rr, "\nabla"] & & S(S(A)) \arrow[dd, "m"] \\
& & \\
S(A) \otimes S(A) \arrow[rr, "\nabla"'] & & S(A)
\end{tikzcd}
\end{definition}
## Properties
\begin{proposition}
If $R$ is a commutative rig, then the symmetric algebra defines an algebra modality in $Mod_{R}$.
* $Sym(A)$ is the [[symmetric algebra]] of the module $A$.
* We have $\nabla_{A}:Sym(A) \otimes Sym(A) \rightarrow Sym(A)$.
* We have $\eta_{A}:A \rightarrow Sym(A)$.
* The unit $A \rightarrow Sym(A)$ of the monad is just the injection $x \mapsto x$.
* The multiplication $Sym(Sym(A)) \rightarrow Sym(A)$ of the monad is given on pure tensors by $(x_{1}^{(1)} \otimes_{s} ... \otimes_{s} x_{n_{1}}^{(1)}) \boxtimes_{s} ... \boxtimes_{s} (x_{1}^{(p)} \otimes_{s} ... \otimes_{s} x_{n_{p}}^{(p)}) \mapsto x_{1}^{(1)} \otimes_{s} ... \otimes_{s} x_{n_{1}}^{(1)} \otimes ... \otimes x_{1}^{(p)} \otimes_{s} ... \otimes_{s} x_{n_{p}}^{(p)}$. It is a kind of composition of polynomials.
\end{proposition}
\begin{proposition}
[[rigs|Commutative rigs]] are exactly the [[algebra over a monad|algebras]] (over a monad) over the algebra modality $Sym$ in the category of commutative monoids.
\end{proposition}
## References
Algebra modalities are part of the definition of a codifferential category ie. a category $\mathcal{C}$ such that $\mathcal{C}^{op}$ is a differential category.
Differential categories were introduced in:
* {#BluteCocketSeely06} [[R. F. Blute]], [[J. R. B. Cockett]], and [[R. A. G. Seely]]: _Differential categories_. Math. Struct. Comput. Sci. 16(06), 1049–1083 (2006) ([doi:10.1017/S0960129506005676](https://doi.org/10.1017/S0960129506005676))
and revised in:
* [[Richard Blute]], [[Robin Cockett]], [[Jean-Simon Lemay]] and [[Robert Seely]], _Differential categories revisited,_ Appl. Categ. Struct. 28, 171-235 (2020). ([arXiv:1806.04804](https://arxiv.org/abs/1806.04804), [doi:10.1007/s10485-019-09572-y](https://doi.org/10.1007/s10485-019-09572-y)) |
algebra of an endofunctor > history | https://ncatlab.org/nlab/source/algebra+of+an+endofunctor+%3E+history | < [[algebra of an endofunctor]]
[[!redirects algebra of an endofunctor -- history]] |
algebra of functions | https://ncatlab.org/nlab/source/algebra+of+functions |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
### Algebra of function on a set
For $R$ a [[ring]] and $S$ a [[set]], the set of [[functions]] $S \to R$ (to the underlying set of $R$) is itself naturally an [[associative algebra]] over $R$, where addition and multiplication is given pointwise in $S$ by addition and multiplication in $R$: for $f_1, f_2 \colon S \to R$ their sum is the function
$$
(f_1 + f_2) \colon s \mapsto f_1(s) + f_2(s) \in R
\,,
$$
their product is the function
$$
(f_1 \cdot f_2) \colon s \mapsto f_1(s) f_2(s) \in R
$$
and the [[ring]] inclusion $R \to [S,R]$ is given by sending $r \in R$ to the constant function with value $r$.
### Algebra of functions on an $\infty$-stack
More generally, in the context of [[(∞,1)-topos theory]] and [[higher algebra]], there is a notion of [[function algebras on ∞-stacks]].
## Properties
### Relation to free modules
If $S$ is a [[finite set]] or else if one restricts to functions that are non-vanishing only for finitely many elements in $S$, then the algebra of functions with values in $R$ also forms the [[free module]] over $R$ generated by $S$.
### Hadamard product
If $S$ is a [[set]] and $R$ is a [[commutative ring]], then the pointwise multiplication on the function algebra $S \to R$ is the [[Hadamard product]] on the function algebra.
### Duality between algebra and geometry
Sending [[spaces]] to their suitable algebras of functions constitutes a basic [[duality]] operation that relates [[geometry]] and [[algebra]]. For more on this see at _[[Isbell duality]]_.
## Related concepts
* [[pullback of functions]]
* [[composition ring]]
[[!redirects algebras of functions]]
[[!redirects function algebra]]
[[!redirects function algebras]]
[[!redirects function ring]]
[[!redirects function rings]]
[[!redirects ring of functions]]
[[!redirects rings of functions]]
|
algebra over a Lawvere theory | https://ncatlab.org/nlab/source/algebra+over+a+Lawvere+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Categorical algebra
+--{: .hide}
[[!include categorical algebra -- contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A [[Lawvere theory]] is encoded in its [[syntactic category]] $T$, a category with finite products such that all objects are finite products of a given object.
An **algebra over a Lawvere theory $T$**, or **$T$-algebra** for short, is a [[model]] for this [[algebraic theory]]: it is a product-preserving [[functor]]
$$
A : T \to Set
\,.
$$
The [[category]] of $T$-algebras is the [[full subcategory]] of the [[functor category]] on the product-preserving functors
$$
T Alg := [T,Set]_\times \subset [T,Set]
\,.
$$
For more discussion, properties and examples see for the moment [[Lawvere theory]].
## Properties
+-- {: .un_prop}
###### Proposition
The category $T Alg$ has all [[limit]]s and these are computed objectwise, hence the embedding $T Alg \to [T,Set]$ preserves these limits.
=--
+-- {: .un_prop}
###### Proposition
$T Alg$ is a [[reflective subcategory]] of $[T, Set]$:
$$
T Alg \stackrel{\leftarrow}{\hookrightarrow} [T,Set]
\,.
$$
=--
+-- {: .proof}
###### Proof
With the above this follows using the [[adjoint functor theorem]].
=--
+-- {: .un_corollary}
###### Corollary
The category $T Alg$ has all [[colimit]]s.
=--
for more see [[Lawvere theory]] for the moment
## Examples
* [[group]]s
* [[ring]]s
* $k$-[[associative algebra]]s
* [[smooth algebra]]s
## Related concepts
* [[algebra over a monad]]
[[∞-algebra over an (∞,1)-monad]]
* **algebra over an algebraic theory**
[[∞-algebra over an (∞,1)-algebraic theory]]
* [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]
* [[algebra over an operad]]
[[∞-algebra over an (∞,1)-operad]]
* [[model structure on algebras over an operad]]
[[!redirects algebras over a Lawvere theory]]
[[!redirects algebra over an algebraic theory]]
[[!redirects algebras over an algebraic theory]]
[[!redirects algebra for an algebraic theory]]
[[!redirects algebras for an algebraic theory]]
[[!redirects T-algebra]] |
algebra over a monad | https://ncatlab.org/nlab/source/algebra+over+a+monad | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
There are different, related ways in which one could view the notion of *algebra over a [[monad]]*:
* If one views monads as a generalization of [[algebraic theories]] (see [[algebraic theory#RelationToMonads|algebraic theory - relation to monads]]), an algebra over a monad is the corresponding generalization of an [[algebra over a Lawvere theory|algebra over a theory]]. In particular, if one views a monad as a way of prescribing particular [[operations]], an algebra is a context where those specified formal expressions can be evaluated to an actual result.
* From the 2-dimensional point of view, an algebra over a monad is a special case of a [[module over a monad]] for the bicategory [[Cat]], where the arrow is from the [[terminal category]].
* If one views monads as a way to model computational effects (see [[monad in computer science]]), an algebra is a context in which the extra effects can be reincorporated into the main data.
Algebras over a monad are usually objects equipped with extra [[structure]], not just [[properties]]. (They can also be seen as [[algebra over an endofunctor|algebras over the underlying endofunctor]], satisfying extra compatibility [[properties]].)
The corresponding [[formal duality|dual]] notion is that of *[[coalgebra over a comonad]]*.
## Definition
{#Definition}
Let $(T,\eta, \mu)$ be a [[monad]] on a [[category]] $\mathcal{C}$.
### Algebras
{#Algebras}
\begin{definition}
An **algebra over $T$** (or just **$T$-algebra** or **$T$-module**) consists of:
1. an [[object]] $A$ of $\mathcal{C}$,
1. a [[morphism]] $a \colon T (A) \longrightarrow A$ of $\mathcal{C}$,
such that the following [[commuting diagram|diagrams commute]] in $\mathcal{C}$ (cf. the definition of *[[module object]]* [here](module+object#MonoidsInMonoidalCategory)):
\[
\label{UnitProperty}
\text{Unit Property:}
\]
\begin{tikzcd}
A
\ar{r}{\eta}
\ar[swap]{dr}{\mathrm{id}}
&
T(A)
\ar{d}{a}
\\
&
A
\end{tikzcd}
\linebreak
\[
\label{ActionProperty}
\text{Action Property:}
\]
\begin{tikzcd}
T\big(T(A)\big)
\ar{r}{T(a)}
\ar{d}{\mu}
&
TA
\ar{d}{a}
\\
TA
\ar{r}{a}
&
A
\end{tikzcd}
\end{definition}
\begin{remark}
The diagram (eq:UnitProperty) is also sometimes called the _unit triangle_, and the diagram (eq:ActionProperty) is also called the _multiplication square_ or _algebra square_.
\end{remark}
### Homomorphisms
Let $(A,a)$ and $(B,b)$ be $T$-algebra. A **[[homomorphism]] of $T$-algebras** is a [[morphism]] $f \colon A \to B$ of $\mathcal{C}$ which makes the following [[commuting diagram|diagram commute]].
\begin{tikzcd}
TA
\ar{r}{Tf}
\ar{d}{a}
&
TB
\ar{d}{b}
\\
A
\ar{r}{f}
&
B
\end{tikzcd}
The [[category]] formed by $T$-algebras and their homomorphisms is known as the **[[Eilenberg-Moore category]]** of $T$ and often denoted by $\mathcal{C}^T$.
### Free algebras
{#FreeAlgebras}
Given a [[monad]] $(T,\mu,\eta)$ on a [[category]] $\mathcal{C}$, then for every [[object]] $X$ of $\mathcal{C}$, the object $T X$ is canonically equipped with a $T$-algebra [[structure]], given by the multiplication map $\mu$ of the monad. The relevant diagrams commute by the monad axioms.
$T$-Algebras of this sort are called **[[free construction|free]] $T$-algebras**.
Given any [[morphism]] $\phi \colon X \to Y$ of $\mathcal{C}$, the [[morphism]] $T \phi \colon T X \longrightarrow T Y$ is evidently a [[homomorphism]] of $T$-algebras, by [[natural transformation|naturality]] of $\mu$. But not every homomorphism of $T$-algebras between the free $T$-algebras $T X$ and $T Y$ arises this way, in general.
However, for any morphism of the form
$$
f \colon X \longrightarrow T Y
$$
in $\mathcal{C}$ (called a $T$-*[[Kleisli morphism]]*), the induced morphism
\[
\label{AlgebraMapInducedByKleisliMorphism}
\mu_{Y} \circ T f
\;\colon\;
T X
\xrightarrow{\;\; T f \;\;}
T T Y
\xrightarrow{\;\; \mu_Y \;\;}
TY
\]
*is* a [[homomorphism]] of $T$-algebras between these free $T$-algebras, as one verifies again using the [[natural transformation|naturality]] of $\mu$. Now, all homomorphisms of $T$-algebras between the free algebras $T X$ and $T Y$ *do* arise this way.
Moreover, given in addition a morphism in $\mathcal{C}$ of the form $g \colon Y \longrightarrow T Z$,
then, under this association, the [[composition]] of the corresponding $T$-algebra morphisms (eq:AlgebraMapInducedByKleisliMorphism) of $f$ and $g$
equals the $T$-algebra homomorphism corresponding to their *[[Kleisli composition|Kleisli composite]]*, defined by
$$
g \circ_T f
\;\coloneqq\;
X
\xrightarrow{ f }
T Y
\xrightarrow{ T g }
T T Z
\xrightarrow{ \mu_Z }
T Z
\,.
$$
The [[category]] of [[Kleisli morphisms]] equipped with this [[Kleisli composition]] is called the *[[Kleisli category]]* and is [[equivalence of categories|equivalent]] to the [[full subcategory]] of $T$-algebras on the free $T$-algebras (see [there](Kleisli+category#InTermsOfKleisliMorphisms) for more).
### Tensor product
In the case of a [[commutative monad]] $T$ , one can define a [[tensor product of algebras over a commutative monad|tensor product of monad algebras]], see there for more.
## Examples
Many monads are named after their (free) algebras:
* The algebras of the _free monoid monad_ on [[Set]] are [[monoids]], and the morphisms of algebras the monoid homomorphisms.
* The algebras of the _free commutative monoid monad_ on [[Set]] are [[commutative monoids]], and their morphisms the monoid homomorphisms between them.
* The algebras of the _free group monad_ on [[Set]] are groups, and their morphisms are the group homomorphisms.
* ...and so on.
In these cases, the notion of _free group_, _free monoid_, et cetera coincide with the notion of free algebra given above.
* Given a [[monoid]] or [[group]] $M$, the algebras of the $M$-[[action monad]] on [[Set]] are the $M$-sets, i.e. sets equipped with an [[action]] of $M$. The morphisms are the [[equivariant maps]].
* The example above generalizes to [[action monads]] given by [[monoid objects]] in a general [[monoidal category]]. Famous examples of this construction in mathematics are smooth actions of [[Lie groups]] on [[manifolds]] and actions of [[rings]] on their [[modules]].
* The algebras of the [[maybe monad]] $(-)_*\colon Set \to Set$, which adds a disjoint point, are the pointed sets.
* The algebras of the [[power set]] monad are the sup-[[semilattices]].
* The algebras of the [[distribution monad]] are [[convex spaces]], and more generally algebras of [[probability monads]] correspond to generalized [[convex spaces]] or [[conical spaces]] (see [[probability monad#algebras_expectation_values|probability monad - algebras]]).
## Generalizations
An algebra over a monad is a special case of a [[module over a monad]] in a [[bicategory]]. See there for more information.
The Eilenberg-Moore and Kleisli categories are also special cases of more general 2-dimensional [[universal constructions]], namely the [[Eilenberg-Moore object]] and the [[Kleisli object]]. See those pages for more information.
## Related concepts
* **algebra over a monad**, [[module over a monad]], [[algebra over an endofunctor]], [[coalgebra over an endofunctor]], [[algebra over a profunctor]]
[[∞-algebra over an (∞,1)-monad]]
* [[model structure on algebras over a monad]]
* [[algebra over an algebraic theory]]
[[∞-algebra over an (∞,1)-algebraic theory]]
* [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]
* [[algebra over an operad]]
[[∞-algebra over an (∞,1)-operad]]
* [[model structure on algebras over an operad]]
* [[Eilenberg-Moore category]], [[Kleisli category]], [[Eilenberg-Moore object]], [[Kleisli object]]
## References
### General
> See the *[References](monad#References)* at *[[monad]]*, such as:
* [[Saunders MacLane]], §VI.2 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]
An introduction to the basic ideas, which gives some intuition for newcomers, can be found in
* [[Paolo Perrone]], Chapter 5 of: _Notes on Category Theory with examples from basic mathematics_, [[arXiv:1912.10642](http://arxiv.org/abs/1912.10642)]
### Kleisli/extension system-style
{#ReferencesKleisliStyle}
For monads presented in "[[extension system]]"/"[[Kleisli triple]]"-form (the way traditionally used for [[monads in computer science]] -- i.e. in terms of a "bind"-operation taking [[Kleisli maps]] to actual [[morphisms]], not explicitly referring to the monad product) there is the corresponding "Kleisli-triple style" or "Mendler style" [[Uustalu (2021), p. 4](#Uustalu21Lecture2)] for presenting the algebra/module-structures for these monads:
* {#MarmolejoWood10} [[F. Marmolejo]], [[Richard J. Wood]], Def. 3.1 in: *Monads as extension systems -- no iteration is necessary* [[TAC]] **24** 4 (2010) 84-113 [[tac:24-04](http://www.tac.mta.ca/tac/volumes/24/4/24-04abs.html)]
* [[Thorsten Altenkirch]], [[James Chapman]], [[Tarmo Uustalu]], Def. 2.11 in: *Monads need not be endofunctors*, Logical Methods in Computer Science **11** 1:3 (2015) 1–40 [[arXiv:1412.7148](https://arxiv.org/abs/1412.7148), [pdf](http://www.cs.nott.ac.uk/~txa/publ/jrelmon.pdf), <a href="https://doi.org/10.2168/LMCS-11(1:3)2015">doi:10.2168/LMCS-11(1:3)2015</a>]
> (stated in the generality of [[relative monads]])
* {#Uustalu21Lecture2} [[Tarmo Uustalu]], p. 4 of: *Monads and Interaction Lecture 2* lecture notes for [MGS 2021](https://staffwww.dcs.shef.ac.uk/people/G.Struth/mgs21.html) (2021) [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs2.pdf), [[Uustalu-Monads2.pdf:file]]]
See also
* [[R. F. C. Walters]], Chapter I of: *A categorical approach to universal algebra*, Ph.D. Thesis (1970) [[anu:1885/133321](https://openresearch-repository.anu.edu.au/handle/1885/133321)]
[[!redirects algebra for a monad]]
[[!redirects algebra over a monad]]
[[!redirects algebras of a monad]]
[[!redirects algebras for a monad]]
[[!redirects algebras over a monad]]
[[!redirects algebras for monads]]
[[!redirects algebras over monads]]
[[!redirects Eilenberg-Moore algebra]]
[[!redirects Eilenberg-Moore algebras]] |
algebra over an operad | https://ncatlab.org/nlab/source/algebra+over+an+operad |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
An [[operad]] is a structure whose elements are _formal operations_, closed under the operation of plugging some formal operations into others. An **algebra over an operad** is a structure in which the formal operations are interpreted as actual operations on an object, via a suitable [[action]].
Accordingly, there is a notion of [[module over an algebra over an operad]].
## Definition
Let $M$ be a [[closed monoidal category|closed symmetric monoidal category]] with monoidal unit $I$, and let $X$ be any object. There is a canonical or tautological [[operad]] $Op(X)$ whose $n^{th}$ component is the internal hom $M(X^{\otimes n}, X)$; the operad identity is the map
$$1_X: I \to M(X, X)$$
and the operad multiplication is given by the composite
$$\array{
M(X^{\otimes k}, X) \otimes M(X^{\otimes n_1}, X) \otimes \ldots \otimes M(X^{\otimes n_k}, X) & \stackrel{1 \otimes func_\otimes}{\to} & M(X^{\otimes k}, X) \otimes M(X^{\otimes n_1 + \ldots + n_k}, X^{\otimes k}) \\
& \stackrel{comp}{\to} & M(X^{\otimes n_1 + \ldots + n_k}, X)
}$$
Let $O$ be any operad in $M$. An **algebra over** $O$ is an object $X$ equipped with an operad map $\xi: O \to Op(X)$. Alternatively, the data of an $O$-algebra is given by a sequence of maps
$$O(k) \otimes X^{\otimes k} \to X$$
which specifies an action of $O$ via finitary operations on $X$, with compatibility conditions between the operad multiplication and the structure of plugging in $k$ finitary operations on $X$ into a $k$-ary operation (and compatibility with actions by permutations).
An _algebra over an [[operad]]_ can equivalently be defined as a [[category over an operad]] which has a single [[object]].
If $M$ is cocomplete, then an operad in $M$ may be defined as a monoid in the symmetric monoidal category $(M^{\mathbb{P}^{op}}, \circ)$ of permutation representations in $M$, aka [[species]] in $M$, with respect to the substitution product $\circ$. There is an [[actegory]] structure
$M^{\mathbb{P}^{op}} \times M \to M$ which arises by restriction of the monoidal product $\circ$ if we consider $M$ as fully embedded in $M^{\mathbb{P}^{op}}$:
$$i: M \to M^{\mathbb{P}^{op}}: X \mapsto (n \mapsto \delta_{n 0} \cdot X)$$
(interpret $X$ as concentrated in the 0-ary or "constants" component), so that an operad $O$ induces a [[monad]] $\hat{O}$ on $M$ via the actegory structure. As a functor, the monad may be defined by a [[coend]] formula
$$\hat{O}(X) = \int^{k \in \mathbb{P}} O(k) \otimes X^{\otimes k}$$
An $O$-algebra is the same thing as an algebra over the monad $\hat{O}$.
**Remark** If $C$ is the [[symmetric monoidal category|symmetric monoidal]] [[enriched category|enriching category]], $O$ the $C$-enriched operad in question, and $A \in Obj(C)$ is the single [[hom-object]] of the [[category over an operad|O-category]] with single object, it makes sense to write $\mathbf{B}A$ for that $O$-category. Compare the discussion at [[monoid]] and [[group]], which are special cases of this.
## Examples
### Over single-coloured operads
* an [[associative algebra]] is an algebra over the [[associative operad]].
* an [[A-infinity algebra]] is an algebra over a cofibrant [[resolution]] of $Assoc$.
* a [[commutative algebra]] is an algebra over the [[commutative operad]].
* an [[E-infinity algebra]] is an algebra over a cofibrant [[resolution]] of the commutative operad
* etc.
### Over coloured operads
* There is a [[coloured operad]] $Mod_P$ whose algebras are pairs consisting of a $P$-algebra $A$ and a [[module]] over $A$;
* For a single-coloured operad $P$ there is a coloured operad $P^1$ whose algebras are triples consisting of two $P$ algebras and a [[morphism]] $A_1 \to A_2$ between them.
* Let $C$ be a set. There is a $C$-coloured operad whose algebras are $V$-[[enriched categories]] with $C$ as their set of objects.
## Literature
### Related $n$Lab entries
* [[algebra over a monad]]
[[∞-algebra over an (∞,1)-monad]]
* [[algebra over an algebraic theory]]
[[∞-algebra over an (∞,1)-algebraic theory]]
* [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]
* **algebra over an operad**
[[∞-algebra over an (∞,1)-operad]]
* [[model structure on algebras over an operad]]
### Generalizations
* S. N. Tronin, _Algebras over multicategories_, Russ Math. (2016) 60: 52. [doi](http://dx.doi.org/10.3103/S1066369X16020092); Rus. original: С. Н. Тронин, _Об алгебрах над мультикатегориями_, Изв. вузов. Матем., 2016, № 2, 62–74
[[!redirects algebra over operad]]
[[!redirects algebra for an operad]]
[[!redirects algebra of an operad]]
[[!redirects algebras over an operad]]
[[!redirects algebras over operads]]
[[!redirects algebras over operad]]
[[!redirects algebras for an operad]]
[[!redirects algebras of an operad]] |
algebra spectrum | https://ncatlab.org/nlab/source/algebra+spectrum |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _algebra spectrum_ or _[[A-∞ algebra]]-spectrum_ over a [[ring spectrum]] is the analog in the [[higher algebra]] of [[stable homotopy theory]] of an [[associative algebra]] over a [[ring]] on ordinary [[algebra]].
## Definition
Abstractly, an _$A_\infty$-algebra spectrum_ over an [[E-∞ ring spectrum]] $R$ is [[algebra in an (∞,1)-category]] in the [[stable (∞,1)-category]] of $R$-[[module spectra]].
Concretely this [[(∞,1)-category]] is [[presentable (∞,1)-category|presented]] by the [[model structure on monoids]] in the [[monoidal model category|monoidal]] $R$-modules in the [[model structure on symmetric spectra]].
(...)
## Properties
### Stable monoidal Dold-Kan correspondence
Let $R := H \mathbb{Z}$ be the [[Eilenberg-MacLane spectrum]] for the [[integer]]s.
+-- {: .num_prop}
###### Proposition
There is a zig-zag of [[monoidal Quillen adjunction|lax monoidal]] [[Quillen equivalences]]
$$
H \mathbb{Z} Mod
\stackrel{\overset{Z}{\longrightarrow}}{\underset{U}{\leftarrow}}
Sp^\Sigma(sAb)
\stackrel{\overset{L}{\leftarrow}}{\underset{\phi^* N}{\longrightarrow}}
Sp^\Sigma(Ch_+)
\stackrel{\overset{D}{\longrightarrow}}{\underset{R}{\leftarrow}}
Ch_\bullet
\,,
$$
between [[monoidal model categories]] satisfying the [[monoid axiom in a monoidal model category]]:
* the model structure for $H \mathbb{Z}$-[[module spectra]];
* the [[model structure on symmetric spectrum objects]] in [[simplicial abelian group]]s and in [[chain complex]]es;
* and the [[model structure on chain complexes]] (unbounded).
This induces a [[Quillen equivalence]] between the corresponding [[model structures on monoids]] in these [[monoidal category|monoidal categories]], which on the left is the model structure on $H \mathbb{Z}$-algebra spectra and on the right the [[model structure on dg-algebras]]:
$$
H \mathbb{Z} Alg \simeq dgAlg_\mathbb{Z}
\,.
$$
=--
This is due to ([Shipley](#Shipley)). The corresponding [[equivalence of (∞,1)-categories]] for $R$ a commutative rings with the intrinsically defined [[(∞,1)-category]] of [[En-algebra|E1-algebra]] objects on the left appears as ([Lurie, prop. 7.1.4.6](#Lurie)).
+-- {: .num_remark}
###### Remark
This is a stable version of the [[monoidal Dold-Kan correspondence]]. See there for more details.
=--
## Related concepts
* [[ring spectrum]], [[module spectrum]]
* [[symmetric algebra spectrum]]
* [[A-∞ algebra]], [[algebra in an (∞,1)-category]]
## References
An account in terms of [[(∞,1)-category theory]] is in section 7.1.4 of
* [[Jacob Lurie]], _[[Higher Algebra]]_
The equivalence of $H \mathbb{Z}$-algebra spectra with [[dg-algebra]]s is due to
* [[Brooke Shipley]], _$H \mathbb{Z}$-algebra spectra are differential graded algebras_ , Amer. Jour. of Math. 129 (2007) 351-379. ([arXiv:math/0209215](http://arxiv.org/abs/math/0209215))
{#Shipley}
Eilenberg-MacLane spectra $H R$ for $R$ itself a [[dg-algebra]] are discussed in
* [[Daniel Dugger]], [[Brooke Shipley]], _Topological equivalences for differential graded algebras_ ([arXiv:math/0604259](http://arxiv.org/abs/math/0604259))
See also the references at [[stable homotopy theory]].
[[!redirects algebra spectra]]
|
algebrad | https://ncatlab.org/nlab/source/algebrad | The notion of **algebrad** is due to
* [[Nikolai Durov]], _Classifying vectoids and generalisations of operads_, [arxiv/1105.3114](http://arxiv.org/abs/1105.3114), the translation of "Классифицирующие вектоиды и классы операд", Trudy MIAN, vol. 273
based on the earlier conference talk:
* _Classifying vectoids and generalizations of operads_ , The International Conference "Contemporary Mathematics", June 12, 2009, [link](http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=34084)
A **vectoid** is a finitely complete and [[cocomplete category]] $C$ with a [[small set]] of [[generators]], where all [[epimorphisms]] are universally [[effective epimorphism|effective]] and where the following "completeness/totality" axiom holds: every [[functor]] $F : C^{op} \to Set$ commuting with all [[colimit]]s is [[representable functor|representable]].
The concept of vectoid simultaneously generalizes [[topoi]] and [[abelian categories]] of $O$-[[module]]s for [[ringed topos|ringed topoi]]: intuitively it is roughly to the category of $O$-modules for a ringed topos $(X,O)$ what a [[generalized ring]] (algebraic monad in Set) is to a ring.
There are monoidal, symmetric monoidal and usual variant of vectoids; for monoidal versions one needs to impose a cocontinuity of the tensor product in each argument,
Vectoids are organized in a 2-category $Vectoid$ of vectoids. The name vectoid because of some analogies of that 2-category with the category of vector spaces, including a universal property of an external tensor product between vectoids which is similar to the universal property of the tensor product for vector spaces; where (bi)cocontinuous functors are analogous to (bi)cocontinuous maps. A monad in the 2-category of vectoids, that is a monoid with respect to the composition product, is called an **algebrad**.
A main source of algebrads are classifying vectoids; classifying vectoid come from the problem of representing $Cat$-valued 2-presheaves on $Vectoid$. One looks at algebrads whose underlying 1-cell is an endomorphism of a classifying vectoid. Examples of the algebrads of that kind are symmetric [[operad]]s (which come from the classifier of objects), algebraic [[monad]]s (from the classifier of (cocommutative) [[coalgebra]]s) and a new type from the classifier of algebras. While the moduli spaces of algebras are hard to construct, the classifying vectoids are in these examples constructed with relatively little pain.
Abstract (in Russian, from [link](http://www.pdmi.ras.ru/EIMI/2009/cm/abs.html)):
Н. В. Дуров Классифицирующие вектоиды и обобщения операд
Первая треть доклада посвящена изложению основ теории вектоидов, представляющих собой совместное обобщение понятия топоса, окольцованного топологического пространства и обобщенного кольца. Далее будет обсуждаться задача построения "классифицирующего вектоида" или "пространства модулей" для различных алгебраических структур, и будет приведена простая комбинаторная конструкция таких вектоидов в простейших случаях (классификатор объектов, алгебр и коалгебр). Оказывается, что моноиды в категории эндоморфизмов классифицирующего вектоида представляют собой естественное обобщение понятие операды, зависящее от выбора классифицирующего вектоида. Например, классические операды получаются из классификатора объектов, а алгебраические монады - из классификатора коалгебр. Случай классификатора алгебр представляется не менее естественным, хотя соответствующее обобщение операды, по всей видимости, является новым. Изложению его основных свойств и будет посвящена заключительная часть доклада.
English: We start with a brief discussion of "vectoids", which are a common generalization of topoi, ringed spaces and generalized rings. After that we list several "classifying vectoid" or "moduli space" construction problems for different algebraic structures, and present a straightforward combinatorial construction of such classifying vectoids for the simplest cases (such as the classification of objects, algebras and coalgebras). Once a classifying vectoid is constructed, one can study monoids with respect to the composition in the category of its endomorphisms; such monoids turn out to be a natural generalization of the notion of an operad. Each choice of a classifying vectoid leads to its own kind of generalization, for example, the classifying vectoid of objects leads to (classical) operads, and that of coalgebras - to algebraic monads. The case of classifying vectoid of algebras seems to be as natural as these two other cases; however, corresponding generalization of operads, which we call "algebrads", appears to be new. Therefore, an elementary description of algebrads will be given.
(ZS: it seems that the time constraint unfortunately caused some cuts in the last part of the plan of the abstract)
(for now this entry also redirects vectoid)
+--{.query}
[[Todd Trimble|Todd]]: "Commutes with colimits" must really mean: $F: A^{op} \to Set$ takes colimits in $A$ to limits in $Set$, and the axiom is that such continuous functors are representable. This reminds me of notions of totality in category theory.
[[Mike Shulman]]: Yes, that exact condition has been studied by category theorists under the name of a "compact" category. That's a terrible name, of course, so even the odd-sounding (to me) "vectoid" is better. I think the original reference is Isbell's paper [Small subcategories and completeness](http://www.ams.org/mathscinet-getitem?mr=0224670), and one later one is [Compact and hypercomplete categories](http://www.ams.org/mathscinet-getitem?mr=614376) by Börger, Tholen, Wischnewsky, and Wolff. The property is implied by totality (= the Yoneda embedding has a left adjoint), and implies hypercompleteness (= admits every limit which it could conceivably admit, subject to local smallness).
=--
## Related entries
* [[2-ring]]
[[!redirects vectoid]]
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algebraic analysis | https://ncatlab.org/nlab/source/algebraic+analysis | **Algebraic analysis** is a program introduced by [[Mikio Sato]] from around 1958, based on the idea that the study of differential equations should be done in a coordinate-free manner, and operations should follow general nonsense geometric and algebraic constructions. One of the first steps was the introduction of the concept of [[D-module]], and of a [[holonomic D-module]], [[hyperfunction]]s (as a sheaf theoretic approach to distribution theory), then relying on homological algebra in [[derived categories]] (it seems that Sato introduced them independently from Grothendieck-Verdier, without publication at the time). Study of nonlinear and nonholonomic system was supposed to reduce on study of holonomic systems on more complicated spaces, e.g. on the cartesian square of the original space and so on. This depends on subtle properties of the study of singularities and other aspects and works only in some generality, with ongoing progress. Singularity theory and the lagrangian geometry are very important aspects of the algebraic analysis. Later Sato introduced [[microlocal analysis|microlocalization]] and his program joined young [[Masaki Kashiwara]] around 1968. Numerous connections to mathematical physics (e.g. [[holonomic quantum field]]s, [[integrable systems]]) and [[Hodge theory]] gradually entered into the program.
It seems that the vision of this program fits well with [[nPOV]]. See also [[D-geometry]].
* interview with [[Mikio Sato]] in [Notices AMS](http://www.ams.org/notices/200702/fea-sato-2.pdf)
* [[Masaki Kashiwara]], Takahiro Kawai, Tatsuo Kimura, _Foundations of algebraic analysis_, Transl. from Japanese by Goro Kato. Princeton Mathematical Series __37__, 1986. xii+255 pp.
[MR87m:58156](http://www.ams.org/mathscinet-getitem?mr=855641); [[J.-L. Brylinski]], Book Review: Foundations of algebraic analysis. Bull. Amer. Math. Soc. (N.S.) __18__ (1988), no. 1, 104–108, [doi](http://dx.doi.org/10.1090/S0273-0979-1988-15622-4)
* [[Tadao Oda]], *Introduction to algebraic analysis on complex manifolds*, Algebraic varieties and analytic varieties (Tokyo, 1981), 29–48, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam (1983) $[$[doi:10.2969/aspm/00110029](https://doi.org/10.2969/aspm/00110029), [MR85a:14010](http://www.ams.org/mathscinet-getitem?mr=715644)$]$
A short but quite complete overview of algebraic analysis can be found in:
* [[Pierre Schapira]] Triangulated categories for the analysts. In "Triangulated categories" London Math. Soc. LNS 375 Cambridge University Press, pp 371-389 (2010) [pdf](https://webusers.imj-prg.fr/~pierre.schapira/ResPapers/Tricat.pdf); |