Samelson product

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[Whitehead product]]

* [[Pontrjagin product]]

## References

Original reference, proving that the [[Whitehead product]] is the [[commutator]] of the [[Pontrjagin product]]:

* [[Hans Samelson]], *A Connection Between the Whitehead and the Pontryagin Product*, American Journal of Mathematics **75** 4 (1953) 744–752  [[doi:10.2307/2372549](https://doi.org/10.2307/2372549)]

Expository accounts:

* [[Joseph Neisendorfer]], _Samelson products and exponents of homotopy groups_,  Journal of Homotopy and Related Structures 8:2 (2013), 239-277 [doi:10.1007/s40062-012-0021-4](http://dx.doi.org/10.1007/s40062-012-0021-4).

* [[Joseph Neisendorfer]].  Samelson products, Chapter 6 in: _Algebraic Methods in Unstable Homotopy Theory_, 158-220.  Cambridge University Press.  [doi:10.1017/cbo9780511691638.008](https://doi.org/10.1017/cbo9780511691638.008).

History:

* {#James71} [[Ioan Mackenzie James]], *Products between homotopy groups*,  Compositio Mathematica, **23** 3 (1971) , 329-345 [[numdam:CM_1971__23_3_329_0](http://www.numdam.org/item/?id=CM_1971__23_3_329_0)]

[[!redirects Samelson products]]


Samir Mathur

* [webpage](http://www.physics.ohio-state.edu/~mathur/)

## Selected writings

On [[multi-trace operators]] in the [[AdS/CFT correspondence]]:

* [[Eric D'Hoker]], [[Daniel Freedman]], [[Samir Mathur]], A. Matusis, [[Leonardo Rastelli]], _Extremal Correlators in the AdS/CFT Correspondence_, in: _[[The many faces of the superworld]]_ ([arXiv:hep-th/9908160](https://arxiv.org/abs/hep-th/9908160))

On [[glueballs]] in [[holographic QCD]]/[[AdS-QCD correspondence]]:

* {#BrowerMathurTan00} Richard C. Brower, [[Samir Mathur]], Chung-I Tan, _Glueball Spectrum for QCD from AdS Supergravity Duality_, Nucl. Phys. B587:249-276, 2000 ([arXiv:hep-th/0003115](https://arxiv.org/abs/hep-th/0003115))

category: people


sampling theory

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

#Contents#
* table of contents
{:toc}

## Idea

Sampling theory is a branch of [[harmonic analysis]], especially relevant for [[computation|computational]] harmonic analysis.

## Theorems

The basic result is the Kotelnikov-Whittaker-Nyquist-Shannon theorem.

## Related entries

* [[wavelet]]

* [[curvelet]]

* [[coherent state]]

* [[signal processing]]

* [[time series]]

* [[information geometry]]

## Literature

* wikipedia: [sampling (signal processing)](http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29), [Nyquist-Shannon sampling theorem](http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem), [Whittaker-Shannon interpolation formula](http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula), [signal processing](https://en.wikipedia.org/wiki/Signal_processing)
* [[Claude Shannon]], _Communication in the presence of noise_, Proc. Institute of Radio Engineers __37__:1, pp. 10–21, Jan. 1949 [IEEE reprint](http://www.stanford.edu/class/ee104/shannonpaper.pdf)
* V. A. Kotelnikov, _On the carrying capacity of the ether and wire in telecommunications_, Material for the First All-Union Conference on Questions of Communication, Izd. Red. Upr. Svyazi RKKA, Moscow, 1933 [english pdf](http://ict.open.ac.uk/classics/1.pdf)
* Kabe Moen, Hrvoje Šikić, Guido Weiss, Edward Wilson, _A panorama of sampling theory_, in: Excursions in Harmonic Analysis, vol. 1, Applied and Numerical Harmonic Analysis 2013, pp 107-123  [doi](http://dx.doi.org/10.1007/978-0-8176-8376-4_6)
* Hrvoje Šikić, Edward N. Wilson, _Lattice invariant subspaces and sampling_, Appl. Comput. Harmon. Anal. __31__ (2011), no. 1, 26–43 [MR2012i:42042](http://www.ams.org/mathscinet-getitem?mr=2795873) [doi](http://dx.doi.org/10.1016/j.acha.2010.09.006)
* Hernández, Eugenio; Šikić, Hrvoje; Weiss, Guido L.; Wilson, Edward N., _The Zak transform(s)_, in: Wavelets and multiscale analysis, 151–157, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, New York, 2011 [MR2012e:44005](http://www.ams.org/mathscinet-getitem?mr=2789161) [doi](http://dx.doi.org/10.1007/978-0-8176-8095-4_8)
* Laurent Freidel, [[nlab:Shahn Majid]], _Noncommutative harmonic analysis, sampling theory and the Duflo map in $2+1$ quantum gravity_, Classical Quantum Gravity __25__ (2008), no. 4, 045006 [MR2009f:83058](http://www.ams.org/mathscinet-getitem?mr=2388191), [doi](http://dx.doi.org/10.1088/0264-9381/25/4/045006)

category: analysis, applications, probability


Samson Abramsky

**Samson Abramsky**

* [website](http://web.comlab.ox.ac.uk/people/samson.abramsky/)

## Selected writings

On [[domain theory]]:

* [[Samson Abramsky]], [[Achim Jung]], *Domain Theory*, in: *Handbook of Logic in Computer Science* **3**, Oxford University Press (1995) [[ISBN:9780198537625](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537625?cc=de&lang=en&), [pdf](https://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf)]

On [[quantum protocols]] in [[quantum information theory]] (such as [[quantum teleportation]]) formalized as [[string diagrams]] in the [[dagger-compact category]] of [[finite dimensional vector spaces|finite-dimensional]] [[Hilbert spaces]] ([[quantum information theory via dagger-compact categories]]):

* {#AbramskyCoecke04} [[Samson Abramsky]], [[Bob Coecke]], _A categorical semantics of quantum protocols_ , Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004) [[arXiv:quant-ph/0402130](http://arxiv.org/abs/quant-ph/0402130)]

* [[Samson Abramsky]], [[Bob Coecke]], *Abstract Physical Traces*, Theory and Applications of Categories, **14** 6 (2005) 111-124. \[tac:14-06, [arXiv:0910.3144](https://arxiv.org/abs/0910.3144)\]

* {#AbramskyCoecke08} [[Samson Abramsky]], [[Bob Coecke]], *Categorical quantum mechanics*, in *[[Handbook of Quantum Logic and Quantum Structures]]*, Elsevier (2008) [[arXiv:0808.1023](http://arxiv.org/abs/0808.1023), [ISBN:9780080931661](https://www.sciencedirect.com/book/9780444528698/), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)]

Survey in contrast to [[quantum logic]]:

* {#AbramskyCoecke2007} [[Samson Abramsky]], [[Bob Coecke]], *Physics from Computer Science: a Position Statement*, [International Journal of Unconventional Computing **3** 3 (2007)](https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/) [[pdf](https://www.cs.ox.ac.uk/files/349/YORKIJUC.pdf), [ijuc-3-3-p-179-197](https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/ijuc-3-3-p-179-197/)]

On [[quantum logic]] as the [[linear logic]] [[internal language]] of [[symmetric monoidal categories]]:

* {#AbramskyDuncan05} [[Samson Abramsky]], [[Ross Duncan]], _A Categorical Quantum Logic_, Mathematical Structures in Computer Science, Volume 16, Issue 3 (2006)  pp. 469 - 489  ([arXiv:quant-ph/0512114](http://arxiv.org/abs/quant-ph/0512114), [doi:10.1017/S0960129506005275](https://doi.org/10.1017/S0960129506005275))

[[sheaf theory|Sheaf theoretic]] discussion of [[quantum contextuality]] ([[hidden variable theories]], [[Kochen-Specker theorem]]):

* {#AbramskyBrandenburger11} [[Samson Abramsky]], [[Adam Brandenburger]], _The Sheaf-Theoretic Structure Of Non-Locality and Contextuality_, New Journal of Physics **13** (2011) 113036 [[arXiv:1102.0264](https://arxiv.org/abs/1102.0264), [doi:10.1088/1367-2630/13/11/113036](https://doi.org/10.1088/1367-2630/13/11/113036)]

On [[simplicial complexes]] treated in [[homotopy type theory]]:

* Maximilian Doré, [[Samson Abramsky]], _Towards Simplicial Complexes in Homotopy Type Theory_ ([pdf](https://hott-uf.github.io/2020/HoTTUF_2020_paper_11.pdf))

category: people


Samson Sanebildze

[[!redirects Sanebildze]]

* [Webpage](http://www.rmi.ge/~sane/)

##List of publications etc.

* [Publications](http://www.rmi.ge/~sane/CVSSS.htm)

category:people

[[!redirects S. Saneblidze]]


Samuel A. Goudsmit

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

## Selected writings

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

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

Historical recollection:

* [[Samuel A. Goudsmit]], *The discovery of the electron spin*, in: *Foundations of Modern EPR*, World Scientific (1998) [[pdf](https://www.ifsc.usp.br/~laf/quantica1/goudsmit_discovery_of_spin.pdf), [webpage](https://lorentz.leidenuniv.nl/history/spin/goudsmit.html), [doi:10.1142/3624](https://doi.org/10.1142/3624)]

category: people

[[!redirects Samuel Goudsmit]]


Samuel Adrian Antz

* [MathOverflow page](https://mathoverflow.net/users/479945/samuel-adrian-antz)

* [Physics.SE page](https://physics.stackexchange.com/users/332309/samuel-adrian-antz)

category: people

[[!redirects Samuel A. Antz]]
[[!redirects Samuel Antz]]


Samuel Clarke

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

category: people


Samuel compactification

The _Samuel compactification_ of a [[uniform space]] is introduced in

* P. Samuel, _Ultrafilters and compactifications of uniform spaces_, Trans. Amer. Math. Soc. __64__ (1948) 100-132 [MR0025717](http://www.ams.org/mathscinet-getitem?mr=0025717) [link](http://www.ams.org/tran/1948-064-01/S0002-9947-1948-0025717-6)

An abstract study of localization phenomena which includes Samuel compactification is in

* [[Joachim Lambek]], B. A. Rattray, _Localization at injectives in complete categories_, Proc. Amer. Math. Soc. __41__ (1973) 1-9 [MR414651](http://www.ams.org/mathscinet-getitem?mr=414651) [doi](http://dx.doi.org/10.1090/S0002-9939-1973-0414651-5)

category: topology


Samuel Dobson

Mathematics PhD student at the University of Auckland in New Zealand. Computer Science and Physics major, studying cryptography.


Samuel Eilenberg

With [[Saunders Mac Lane]], 'Sammy' Eilenberg was one of the 'fathers' of [[category theory]]. In their 1945 paper *[[General Theory of Natural Equivalences]]* they introduced [[category theory]] with the notions of *[[categories]]*, *[[functors]]* and *[[natural transformations]]*.

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

## Selected writings

On [[group extensions]] and [[group homology]] via [[Ext]]/[[Tor]]-[[functors]]:

* {#EilenbergMacLane42} [[Samuel Eilenberg]], [[Saunders MacLane]], *Group Extensions and Homology*, Annals of Mathematics **43** 4 (1942) 757-831 [[doi:10.2307/1968966](https://doi.org/10.2307/1968966), [jstor:1968966](https://www.jstor.org/stable/1968966)]

Introducing [[category theory]]:

* {#EilenbergMacLane45} [[Samuel Eilenberg]], [[Saunders MacLane]], _[[General Theory of Natural Equivalences]]_,  Transactions of the American Mathematical Society **58** 2 (1945) 231-294 [[doi:10.1090/S0002-9947-1945-0013131-6](https://doi.org/10.1090/S0002-9947-1945-0013131-6), [jstor:1990284](http://www.jstor.org/stable/1990284)]

Introducing the *[[Eilenberg-Steenrod axioms]]* for [[ordinary cohomology]]:

* [[Samuel Eilenberg]], [[Norman E. Steenrod]], _Axiomatic Approach to Homology Theory_, Proceedings of the National Academy of Sciences **31** 4 (1945) 117–120  [[doi:10.1073/pnas.31.4.117](http://dx.doi.org/10.1073/pnas.31.4.117)]

Introducing abstract [[group cohomology]]:

* [[Samuel Eilenberg]], [[Saunders MacLane]], *Cohomology Theory in Abstract Groups. I*, Annals of Mathematics, Second Series **48** 1 (1947) 51-78 [[doi:10.2307/1969215](https://doi.org/10.2307/1969215)]

Introducing [[simplicial sets]] and [[semi-simplicial sets]] (see the historical remarks [here](semi-simplicial+set#History)):

* {#EilenbergZilber50} [[Samuel Eilenberg]], [[Joseph Zilber]], *Semi-Simplicial Complexes and Singular Homology*, *Annals of Mathematics* vol. 51 no. 3 (1950), 499--513 ([jstor:1969364](https://www.jstor.org/stable/1969364))

On [[algebraic topology]]:

* [[Samuel Eilenberg]], [[Norman Steenrod]], _Foundations of Algebraic Topology_, Princeton University Press 1952 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/eilestee.pdf), [ISBN:9780691653297](https://press.princeton.edu/books/hardcover/9780691653297/foundations-of-algebraic-topology))

Introducing the [[Eilenberg-Zilber theorem]]:

* [[Samuel Eilenberg]], [[Joseph Zilber]], _On Products of Complexes_, Amer. Jour. Math. 75 (1): 200–204, (1953) ([jstor:2372629](https://www.jstor.org/stable/2372629))

Introducing [[Eilenberg-MacLane spaces]]:

* {#EilenbergMacLane53I} [[Samuel Eilenberg]], [[Saunders Mac Lane]], _On the Groups $H(\Pi,n)$, I_, Annals of Mathematics Second Series, Vol. 58, No. 1 (Jul., 1953), pp. 55-106 ([jstor:1969820](https://www.jstor.org/stable/1969820))

* {#EilenbergMacLane54II} [[Samuel Eilenberg]], [[Saunders Mac Lane]], _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))

* {#EilenbergMacLane54III} [[Samuel Eilenberg]], [[Saunders Mac Lane]], _On the Groups $H(\Pi,n)$, III: Operations and Obstructions_, Annals of Mathematics Second Series, Vol. 60, No. 3 (Nov., 1954), pp. 513-557 ([jstor:1969849](https://www.jstor.org/stable/1969849))

On [[separable algebras]] and [[Frobenius algebras]]:

* {#EilenbergNakayama55} [[Samuel Eilenberg]], [[Tadasi Nakayama]], *On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings*, _Nagoya Math. J._ **9** (1955) 1-16  [[euclid](https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-9/issue-none/On-the-dimension-of-modules-and-algebras-II-Frobenius-algebras/nmj/1118799677.pdf)]

On [[homological algebra]]:

* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], *[[Homological Algebra]]*, Princeton Univ. Press (1956), Princeton Mathematical Series **19** (1999) [[ISBN:9780691049915](https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19), [doi:10.1515/9781400883844](https://doi.org/10.1515/9781400883844), [pdf](http://www.math.stonybrook.edu/~mmovshev/BOOKS/homologicalalgeb033541mbp.pdf)]

On [[homology of loop spaces]]:

* [[Samuel Eilenberg]], [[John Moore]], _Homology and fibrations_, Comment. Math. Helv., 40 (1966), pp. 199-236 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/eilemoor.pdf))

Introducing the notion of [[closed categories]]:

* {#EilenbergKelly66} [[Samuel Eilenberg]], [[G. Max Kelly]], *Closed Categories*, pp. 421-562 in:  [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

On [[categorical algebra]]:

* [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

## Related entries

* [[category theory]]

* [[Eilenberg-MacLane space]]

* [[history of cohomology with local coefficients]]

category: people

[[!redirects Sammy Eilenberg]]
[[!redirects Eilenberg]]
[[!redirects S. Eilenberg]]


Samuel Gitler

Sam Gitler was a Mexican [[algebraic topology|algebraic topologist]]. He was very important not only for his contribution to [[Yang-Mills theory]] and the [[Brown-Gitler spectrum]], but also for his creation, with [[José Ádem]] of the school of algebraic topology in Mexico.

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

## Selected writings

On [[cohomology operations]] in [[twisted ordinary cohomology]]:

* [[Samuel Gitler]], _Cohomology Operations with Local Coefficients_, American Journal of Mathematics Vol. 85, No. 2 (Apr., 1963), pp. 156-188 ([jstor:2373208](https://www.jstor.org/stable/2373208))

On [[algebraic topology]] and [[homotopy theory]]:

* [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]],  _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

On the [[ordinary homology]] of [[loop spaces]] of [[configuration spaces of points]] and their relation to [[Vassiliev invariants]]:

* [[Fred Cohen]], [[Samuel Gitler]], _Loop spaces of configuration spaces, braid-like groups, and knots_, In: Jaume Aguadé, Carles Broto, [[Carles Casacuberta]]  (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel ([doi:10.1007/978-3-0348-8312-2_7](https://doi.org/10.1007/978-3-0348-8312-2_7))

* [[Fred Cohen]], [[Samuel Gitler]], _On loop spaces of configuration spaces_, Trans. Amer. Math. Soc. __354__ (2002), no. 5, 1705–1748, ([jstor:2693715](https://www.jstor.org/stable/2693715), [MR2002m:55020](http://www.ams.org/mathscinet-getitem?mr=1881013))

## Related $n$Lab entries

* [[algebraic topology]]

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

* [[Brown-Gitler spectrum]]

category:people

[[!redirects Sam Gitler]]


Samuel Isaacson

* [website](http://www.ma.utexas.edu/users/isaacson/)

category: people


Samuel J. Lomonaco

* [personal page](https://www.csee.umbc.edu/~lomonaco/)

## Selected writings

On [[Lord Kelvin]]'s ideas "[[On Vortex Atoms]]" in view of [[classical field theory|classical]] [[electromagnetism]]:

* [[Samuel Lomonaco]], _The modern legacies of Thomson's atomic vortex theory in classical electrodynamics_, in: [[Louis Kauffman]] (ed.) _The Interface of Knots and Physics_, Proceedings of Symposia in Applied Mathematics, Volume 51 (1996) ([pdf](http://www.csee.umbc.edu/~lomonaco/kelvin/kelvin23.pdf), [doi:10.1090/psapm/051](https://doi.org/10.1090/psapm/051))

On [[quantum computation]]:

* [[Samuel J. Lomonaco]] (ed.), *Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium*, Proceedings of Symposia in Applied Mathematics **58**, AMS (2002) [[doi:10.1090/psapm/058](https://doi.org/10.1090/psapm/058)]

On [[topological quantum computation]] with [[anyons]]/[[braid group statistics]] ([[braid group representations]]):

* [[Louis H. Kauffman]], [[Samuel J. Lomonaco]], *Braiding Operators are Universal Quantum Gates*,  New Journal of Physics, Volume 6, January 2004 ([arXiv:quant-ph/0401090](https://arxiv.org/abs/quant-ph/0401090), [doi:10.1088/1367-2630/6/1/134](https://iopscience.iop.org/article/10.1088/1367-2630/6/1/134))

* [[Samuel J. Lomonaco]], [[Louis Kauffman]], _Topological Quantum Computing and the Jones Polynomial_, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (12 May 2006) ([arXiv:quant-ph/0605004](https://arxiv.org/abs/quant-ph/0605004))

* [[Louis H. Kauffman]], [[Samuel J. Lomonaco]], *Topological quantum computing and $SU(2)$ braid group representations*, Proceedings Volume 6976, Quantum Information and Computation VI; 69760M (2008) ([doi:10.1117/12.778068](https://doi.org/10.1117/12.778068), [rg:228451452](https://www.researchgate.net/publication/228451452))

On [[quantum computation]] and more generally on [[quantum systems]]:

* [[Goong Chen]], [[Louis Kauffman]], [[Samuel J. Lomonaco]] (eds.), *Mathematics of Quantum Computation and Quantum Technology*, Chapman and Hall/CRC (2007) (ISBN:9780367388614, [doi:10.1201/9781584889007](https://doi.org/10.1201/9781584889007))

category: people

[[!redirects Samuel Lomonaco]]


Samuel L. Braunstein

Sam Braunstein is a professor in the department of Computer Science of the University of York. Research interests include: quantum and non-standard computation; algorithm and hardware design; quantum information; quantum communication and foundations.

* [Webpage](https://www-users.cs.york.ac.uk/~schmuel/)

## Selected writings

Introducing the statement of the "[[no-deleting theorem]]" in [[quantum physics]]/[[quantum information theory]]:

* [[Arun Kumar Pati]], [[Samuel L. Braunstein]], *Impossibility of deleting an unknown quantum state*, Nature **404** (2000) 164-165 [[doi:10.1038/404130b0](https://doi.org/10.1038/404130b0), [arXiv:quant-ph/9911090](https://arxiv.org/abs/quant-ph/9911090)]

On [[quantum information theory]] with continuous variables and the corresponding [[quantum error correction]]:

* [[Samuel L. Braunstein]], Peter van Loock, *Quantum information with continuous variables*, Rev. Mod. Phys. **77** 2 (2005) 513 [[arXiv:quant-ph/0410100](https://arxiv.org/abs/quant-ph/0410100), [doi:10.1103/RevModPhys.77.513](https://doi.org/10.1103/RevModPhys.77.513)]

* [[Samuel L. Braunstein]], Arun K. Pati, *Quantum Information with Continuous Variables*, Springer (2003) [[doi:10.1007/978-94-015-1258-9](https://doi.org/10.1007/978-94-015-1258-9)]

On the [[black hole firewall problem]]:

* [[Samuel L. Braunstein]]; S. Pirandola, K. Życzkowski, *Better Late than Never: Information Retrieval from Black Holes* (2013) [[arXiv:0907.1190](http://arxiv.org/abs/0907.1190)]

category: people

[[!redirects Samuel Braunstein]]


Samuel Lerbet

* [ResearchGate page](https://www.researchgate.net/profile/Samuel-Lerbet)

## Selected writings

On ([[stable cohomotopy|stable]]) [[motivic cohomology|motivic]] [[Cohomotopy]] of [[schemes]] (as  [[motivic homotopy theory|motivic homotopy classes]] of maps into [[motivic sphere|motivic]] [[Tate spheres]]):

* [[Samuel Lerbet]], *Motivic stable cohomotopy and unimodular rows*, Advances in Mathematics **436** 109415 (2024) [[arXiv:2206.11688](https://arxiv.org/abs/2206.11688), [doi:10.1016/j.aim.2023.109415](https://doi.org/10.1016/j.aim.2023.109415)]

category: people


Samuel Mimram

Samuel Mimram is a professor in (and head of) the Cosynus team at the LIX laboratory of École Polytechnique. Previously he was a post-doc in the _Modélisation et Analyse de Systèmes en Interaction_ team at CEA Saclay, France and before that a post-doc  in the [PPS group](http://www.pps.jussieu.fr/) at the _Université Paris Diderot - Paris 7_.

* [webpage](http://www.lix.polytechnique.fr/Labo/Samuel.Mimram/)

##Selected Preprints and publications

On [[rewriting]] theory:

* [[Samuel Mimram]], _Towards Higher dimensional rewriting theory_, [semanticscholar](https://pdfs.semanticscholar.org/a887/10b141afeb7bcee157e56beff635dcfb4350.pdf?_ga=2.155145712.991874601.1564010891-2034645618.1564010891)

* [[Samuel Mimram]], _Towards 3-dimensional rewriting theory_, [arxiv](https://arxiv.org/pdf/1403.4094.pdf)

On [[polynomial functors]]:

* {#FMLS} [[Eric Finster]], [[Samuel Mimram]], Maxime Lucas, Thomas Seiller, _A Cartesian Bicategory of Polynomial Functors in Homotopy Type Theory_ ([arXiv:2112.14050](https://arxiv.org/abs/2112.14050)).


Samuel Monnier

* [Mathematics department, University of Geneva](http://www.unige.ch/math/en/people/enseigne/)

category: people


Sanath Devalapurkar

- [webpage](http://sites.google.com/site/homotopytheory/)


Sand's Box

This box belongs to the sand!

It has a [[Sand's Box|self-link]].

[[!redirects Sand's Box]]
[[!redirects Sand's Box]]


Sandbox

+-- {: .rightHandSide}
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###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

* A *functor*[^ety] is a [[homomorphism]] of [[categories]].

* A functor between [[small categories]] is a [[homomorphism]] of the [[underlying]] [[graphs]] that respects the [[composition]] of [[edges]].

So, for $C$, $D$ two [[categories]], a functor $F \colon C \to D$ consists of

1. a component-[[function]] of the [[classes]] of [[objects]];

   $F_0 \colon Obj(C) \to Obj(D)$

* a component-[[function]] of [[sets]] of [[morphisms]]

  $F_1 \colon Mor(C) \to Mor(D)$

  or equivalently:

  for each [[pair]] $x,y \in Obj(C)$ of [[objects]], a component-[[function]]

  \[
    \label{ComponentFunctionOnHomSets}
    F_{x,y} \;\colon\; C(x,y) \xrightarrow{\;\;} D(F(x), F(y))
  \]

  between [[hom-sets]];

such that:

* it respects [[source]] and [[target]] of [[morphisms]]: $F_1$ coincides with $F_0$ on source and target objects;

* it respects [[identity morphisms]]: $F(id_X) = id_{F(X)}$;

* it respects [[composition]]: the image of the composite of two morphisms under $F$ is the composite of their images.

These last two properties are the decisive ones of a functor; they are called the **functoriality conditions**.  These are a direct generalization of the notion of [[homomorphism]] (of [[monoids]], [[groups]], [[algebras]], etc.) to the case that there are more objects. As a slogan:

The notion of functor is a [[horizontal categorification]] of that of [[homomorphism]].

## Definition

### External definition

A **functor** $F$ from a [[category]] $C$ to a category $D$ is a map sending each [[object]] $x \in C$ to an object $F(x) \in D$ and each [[morphism]] $f : x \to y$ in $C$ to morphism $F(f) : F(x) \to F(y)$ in $D$, such that

* $F$ preserves [[composition]]: $F(g\circ f) = F(g)\circ F(f)$ whenever the left-hand side is well-defined,

* $F$ preserves [[identity morphisms]]: for each object $x \in C$, $F(1_x) = 1_{F(x)}$.

Or equivalently, since compositions $g f = g\circ f$ (commuting triangles) and identities $1_x$ (commuting loops) are both simple commuting diagrams, we can combine the above conditions to the single statement

* $F$ preserves [[commuting diagrams]].

Given [[morphisms]] $f \colon X\to Y$, $g \colon Y\to Z$, and $h \colon X\to Z$, declaring the triangle commutes amounts to declaring
$$h = g\circ f.$$
In this case, for $F:C\to D$ to preserve the commutative triangle means
$$F(h) = F(g)\circ F(f)$$
as depicted below

Preserving commuting triangles means $F$ preserves compositions.

Given morphisms $f:X\to Y$, $g:Y\to Z$, and $h:Z\to X$, declaring the loop commutes amounts to declaring
$$1_X = h\circ g\circ f.$$
In this case, for $F:C\to D$ to preserve the commutative loop means
$$F(1_X) = F(h)\circ F(g)\circ F(f)$$
as depicted below

However, it means more than that. Since any commutative loop is equal to the identity morphism, we must also have
$$1_{F(X)} = F(h)\circ F(g)\circ F(f)$$
implying
$$F(1_X) = 1_{F(X)}.$$

Preserving commuting loops means $F$ preserves identity morphisms.

Another equivalent way to say this is that a functor $F : C \to D$ is precisely a morphism of [[simplicial sets]] $N(F) : N(C) \to N(D)$ between the [[nerves]] of these categories

* the [[objects]] of $C$ and $D$ are the 0-cells of $N(C)$ and $N(D)$, so $N(F)_0 : N(F)_0 \to N(F)_0$ maps objects of $C$ to objects of $D$;

* the [[morphisms]] of $C$ and $D$ are the 1-cells of $N(C)$ and $N(D)$, so $N(F)_1 : N(F)_1 \to N(F)_1$ maps morphisms of $C$ to objects of $D$;

* the identity morphisms of $C$ and $D$ are the degenerate 1-cells of $N(C)$ and $N(D)$, so the fact that $N(F)$ respects degeneracy maps means that $F$ respects identities;

* the commuting triangles of $C$ and $D$ are the 2-cells of $N(C)$ and $N(D)$, so the fact that $N(F)$ maps 2-cells to 2-cells means that it respects commuting triangles, hence that it respects composition.

See [[nerve]] for more details on this.

The functors between two categories $C$ and $D$ form themselves a category, the [[functor category]] $[C,D]$, whose morphisms are [[natural transformations]]. Equipped with these functor categories as [[hom-objects]], we have a $2$-[[2-category|category]] [[Cat]] of categories, functors and natural transformations.  In other words, functors are [[morphisms]] in $Cat$.

### Internal definition {#InternalDefinition}

If $C$ and $D$ are [[internal categories]] in some ambient category $A$, then an __internal functor__ $F : C \to D$ is

* a morphism of objects $F_0 : C_0 \to D_0$ in $A$;

* a morphisms of morphisms $F_1 : C_1 \to D_1$ in $A$;

* such that the following diagrams commute

  * respect for the source map:
  $
 \array{
  C_1 &\stackrel{f_1}{\to}& D_1
  \\
  \downarrow^s && \downarrow^s
  \\
  C_0
  &\stackrel{f_0}{\to}&
  D_0
 }
$;

   * respect for the target map:
  $
 \array{
  C_1 &\stackrel{f_1}{\to}& D_1
  \\
  \downarrow^t && \downarrow^t
  \\
  C_0
  &\stackrel{f_0}{\to}&
  D_0
 }
$;

   * respect for identities
  $
 \array{
   C_0 &\stackrel{f_0}{\to}&
   D_0
   \\
   \downarrow^i && \downarrow^i
   \\
   C_1 &\stackrel{f_1}{\to}& D_1
 }
$;

   * respect for composition
  $
  \array{
    C_1 \times_{t,s}
   C_1 &\stackrel{f_1\times_{t,s} f_1}{\to}&
   D_1 \times_{t,s} D_1
  \\
   \downarrow^{\circ} && \downarrow^{\circ}
  \\
  C_1 &\stackrel{f_1}{\to}& D_1
  }
$.

This reproduces the external definition of functors above for [[small categories]], which are categories [[internalization|internal to]] [[Set]]

In many cases, this notion is too restrictive, and we should use internal [[anafunctors]] instead.

### Enriched definition

In [[enriched category theory]] a functor maps not [[hom-sets]] but the given [[hom-objects]] to each other, in a way that respects their composition. This is described at

* [[enriched functor]].

### Profunctors

A generalization of the notion of [[enriched functor]] is the notion of [[profunctor]].

### Higher categorical functors

In [[higher category theory]] there are corresponding higher notions of functor, such as

* [[2-functor]]

  * [[strict 2-functor]]

  * [[pseudofunctor]]

  * [[lax functor]]

* [[(∞,1)-functor]]

* See also an informal discussion about an [[experimental alternative definition of functor]].

### In homotopy type theory

Note: the [[HoTT book]] calls a [[category]] a "precategory" and a [[univalent category]] a "category", but here we shall refer to the standard terminology of "category" and "univalent category" respectively.

The definition of functor in homotopy type theory is a straightforward translation of the ordinary one.  However, the notion of [[univalent category]] allows us to construct some such functors that in classical mathematics would require either the [[axiom of choice]] or the use of [[anafunctors]].

 {#Definition}

Let $A$ and $B$ be [[categories]]. Informally, a **functor** $F : A \to B$ consists of

* A function $F_0 : A_0 \to B_0$
* For each $a,b:A$, a function $F_{a,b}:hom_A(a,b) \to hom_B(F a,F b)$, generally also denoted $F$.
* For each $a:A$, we have $F(1_a)=1_{F a}$.
* For each $a,b,c: A$ and $f:hom_A(a,b)$ and $g:hom_A(b,c)$, we have

$$F(g \circ f) = F g \circ F f$$

Formally, the type of functors from $A$ to $B$ is

$$ Func(A,B) \coloneqq
\sum_{F_0:A_0\to B_0} \sum_{F:\prod_{a,b:A} hom_A(a,b) \to \hom_B(F a,F b)} \Big(\prod_{a:A} F(1_a) = 1_{F a}\Big) \times \Big( \prod_{a,b,c:A} \prod_{f:\hom_A(a,b)} \prod_{g:\hom_A(b,c)} F(g \circ f) = F g \circ F f\Big) $$

A formal definition in [[Coq]] can be found in [Ahrens-Kapulkin-Shulman 13](#AhrensKapulkinShulman13).

## Properties

These properties come from the [[HoTT book]].

By induction on [[identity]], a functor also preserves $idtoiso$ (See [[category]]).

### Composition of functors ###

For functors $F:A\to B$ and $G:B \to C$, their composite $G \circ F : A \to C$ is given by

* The composite $(G_0 \circ F_0): A_0 \to C_0$
* For each $a,b:A$, the composite
$$(G_{F a, F b} \circ F_{a,b}):hom_A(a,b)\to hom_C(G F a, G F b)$$

### Lemma 9.2.9 in HoTT book###
Composition of functors is associative $H(G F)=(H G)F$.

**Proof:** Since composition of functions is associative, this follows immediately for the actions on objects and on homs. And since hom-sets are sets, the rest of the data is automatic. $\square$

### Lemma 9.2.10 in HoTT book###
Lemma 9.2.9 is coherent, i.e. the following pentagon of equalities commutes:

$$
   \array{
     && (K H)(G F)
     \\
     & \nearrow
     &&
     \searrow
     \\
     ((K H) G) F
     && &&
     K (H (G F))
     \\
     \downarrow
     && &&
     \uparrow
     \\
     (K(H G)) F
     && \longrightarrow &&
     K( (H G) F)
   }
$$

## Types of functors

Specific types of functors are important in applications. See for instance

* [[essentially surjective functor]]

* [[full functor]]

* [[faithful functor]]

* [[full and faithful functor]]

And for more background on this see [[stuff, structure, property]].

##Examples

### Morphisms of monoids and groups

For $A,B$ [[monoids]] or $G, H$ [[groups]], let $\mathbf{B}A, \mathbf{B}B$, $\mathbf{B}G$, $\mathbf{B}H$ be the corresponding one-object [[category|categories]] (as described at [[delooping]]). Then functors

$$
  \mathbf{B}A \to \mathbf{B}B
$$

are canonically in bijection with monoid homomorphisms $A \to B$ and accordingly functors

$$
  \mathbf{B}G \to \mathbf{B}H
$$

are canonically in bijection with group homomorphisms $G \to H$.

### Representations

With $\mathbf{B}G$ as above, functors on $\mathbf{B}G$ with values in [[Vect]] are the same as linear [[representations]] of the [[group]] $G$. In fact, we have a canonical isomorphism of categories

$$
  Funct(\mathbf{B}G, Vect) \simeq Rep(G)
$$

of the [[functor category]] with the representation category.

### Linear Maps

Let $\mathbf{B} End(U)$ and $\mathbf{B} End(V)$ be one-object categories whose objects are each finite-dimensional vector space and whose morphisms are all of the linear [[endomorphisms]] on that space, i.e. one-object [[full subcategory|full subcategories]] of $Fin Vect$.

#### Left Inverses

If the linear map $F:U\to V$ has a left inverse, i.e.

$$F^*\circ F = 1_U,$$

where $F^*$ is the [[preimage]], then we can construct a functor

$$F_*:\mathbf{B} End(U)\to\mathbf{B} End(V)$$

by defining its action on objects by

$$F_*(U) = F_* U$$

where $F_*$ on the right-hand side is the [[image]] and its action on endomorphisms by

$$F_*(f) = F f F^*.$$

Composition follows immediately

$$F_*(f g) = F f g F^* = F f F^* F g F^* = F_*(f) F_*(g).$$

Identity morphisms are preserved since for any vector $y$ in $F_*U\subset V$, then

$$y = F x$$

for some vector $x$ in $U$ and we have

$$F_*(1_U)y = F\circ F^* y = F\circ F^*\circ F x = F x = y$$

so that

$$F_*(1_U) = 1_{F_*(U)}$$

as required. Hence, $F_*$ is a functor.

#### Right Inverses

If the linear map $F:U\to V$ has a right inverse, i.e.

$$F\circ F^* = 1_V,$$

where $F^*$ is the preimage, then we can construct a functor

$$F^*:\mathbf{B} End(V)\to\mathbf{B} End(U)$$

by defining its action on objects by

$$F^*(V) = F^*V$$

where $F^*$ on the right-hand side is preimage, and its action on endomorphisms by

$$F^*(f) = F^* f F.$$

Composition follows immediately

$$F^*(f g) = F^* f g F = F^* f F F^* g F = F^*(f) F^*(g).$$

Identity morphisms are preserved since for any vector $x$ in $F^*V\subset U$, then

$$x = F^*y$$

for some vector $y$ in $V$ and we have

$$F^*(1_V)x = F^*\circ F x = F^*\circ F \circ F^* y = F^* y = x$$

so that

$$F^*(1_V) = 1_{F^*(V)}$$

as required. Hence, $F^*$ is a functor.

### Presheaves

Functors $F : C \to Set$ with values in [[Set]] are also called [[presheaf|presheaves]]. As such one calls them presheaves on the [[opposite category]] $C^{op}$ of $C$. See [[presheaf]] for more on this.

### Functors and generalized elements {#OnGeneralizedElements}

For $C$ a [[category]], and $X \in C$ an [[object]], and $U$ any other object, a [[morphism]] $x : U \to X$ may be regarded as a [[generalized element]] of $X$, written $x \in X$ (For this language applied to the category [[Set]] of sets see [[ETCS]]. For the general case see [[type theory]]).

The _[[set]]_ of generalized elements of an object $X \in C$ is thus the union of [[hom-sets]] $\coprod_{U \in C} Hom_C(U,X)$.

While a [[morphism]] $f : X \to Y$ in an  arbitrary category $C$ need not at all come to us as a function of sets, it always induces a function of sets of _generalized elements_ : it sends the generalized element $x : U \to X$ of $X$ to the generalized element

$$
  f(x) : U \stackrel{x}{\to} X \stackrel{f}{\to} Y
$$

of $Y$, using the composition of the morphism $f$ with the morphism $x$ in $C$.

In terms of this notation, the functoriality condition on a functor $F : C \to D$, which is

$$
  F(U \stackrel{x}{\to} X \stackrel{f}{\to} Y)
  =
  F(U) \stackrel{F(x)}{\to} F(X) \stackrel{F(f)}{\to} F(Y)
$$

appears as

$$
  F(f(x)) = F(f)(F(x))
  \,.
$$

This can be illustrated in the following diagram

which provides an alternative expression of the functoriality condition as simply a statement that commuting diagrams in $C$ map to commuting diagrams in $D$.

## Related concepts

* [[endofunctor]]

* [[bifunctor]]

* [[profunctor]]

* [[multifunctor]]

* [[function]]

* **functor**

  * [[endofunctor]]

  * [[polynomial functor]]

* [[2-functor]] / [[pseudofunctor]]

  * [[monoidal functor]]

* [[n-functor]]

* [[(∞,1)-functor]]

* [[(∞,n)-functor]]

[[!include properties of functors -- contents]]

## References

Textbook accounts:

* [[Saunders MacLane]], §I.3 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], Section 1.2 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory* [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

See also the references at:

* [[category theory#references|category theory - references]].

* [[André Joyal]]'s CatLab: _[[joyalscatlab:Functors]]_

For functors in homotopy type theory

* {#AhrensKapulkinShulman13} [[nLab:Benedikt Ahrens]], [[nLab:Chris Kapulkin]], [[nLab:Michael Shulman]], section 4 of _Univalent categories and the Rezk completion_, Mathematical Structures in Computer Science 25.5 (2015): 1010-1039 ([arXiv:1303.0584](http://arxiv.org/abs/1303.0584))

* {#HoTTBook} [[nLab:UF-IAS-2012|Univalent Foundations Project]], section 9.2 of _[[HoTT Book|Homotopy Type Theory – Univalent Foundations of Mathematics]]_, IAS 2013

[[Coq]] code formalizing the concept of functors includes the following:

* _[functors_transformations.v](https://github.com/benediktahrens/rezk_completion/blob/master/functors_transformations.v)_

Formalization in [[cubicalAgda]]:

* [[1lab]], *[Functors](https://1lab.dev/Cat.Functor.Base.html#functors)*

[[!redirects functors]]
[[!redirects covariant functor]]
[[!redirects covariant functors]]

[[!redirects internal functor]]
[[!redirects internal functors]]

[[!redirects functorial]]


Sandbox > 905234

< [[Sandbox]]


Sandbox > SVG > history


Sandbox > test

< [[Sandbox]]
[[!redirects Sandbox/test]]
[[!redirects Sandbox -- test]]


Sandbox/Set

# The category of sets
* table of contents
{: toc}

## Definition

__$Set$__ is the (or a) [[category]] with [[sets]] as [[objects]] and [[functions]] between sets as [[morphisms]].

## Properties

This category has many marvelous properties, which make it a common choice for serving as a '[[foundations|foundation]]' of mathematics.  For instance:

* It is a [[well-pointed category|well-pointed]] {:yes} [[topos]]{:},

  So in particular it is [[locally cartesian closed category|locally cartesian closed]].

* It is [[locally small category|locally small]].

* It is [[complete category|complete]] and [[cocomplete category|cocomplete]], and therefore $\infty$-[[extensive category|extensive]] (as is any cocomplete topos).

At least assuming [[classical logic]], these properties suffice to characterize $Set$ uniquely up to equivalence among all categories; see [[cocomplete well-pointed topos]].  Note, however, that the definitions of "locally small" and "(co)complete" presuppose a notion of _small_ and therefore a knowledge of what a _set_ (as opposed to a [[proper class]]) is.

As a topos, $Set$ is also characterized by the fact that

* $Set$ is the [[terminal object]] in the [[category]] of [[Grothendieck topos]]es and [[geometric morphism]]s. The terminal morphism $\Gamma\colon E \to Set$ from any other topos $E$ is the [[global section]]s functor.

It is usually assumed that $Set$ satisfies the [[axiom of choice]] and has a [[natural numbers object]].  In Lawvere's theory [[ETCS]], which can serve as a foundation for much of mathematics, $Set$ is asserted to be a well-pointed topos that satisfies the axiom of choice and has a natural numbers object.  It follows that it is automatically "locally small" and "complete and cocomplete" relative to the notion of "smallness" defined in terms of itself (actually, this is true for any topos).

Conversely, $\Set$ in [[constructive mathematics]] cannot satisfy the axiom of choice (since this implies [[excluded middle]]), although constructivists might accept [[COSHEP]] (that $Set$ has [[projective object|enough projectives]]).  In [[predicative mathematics]], $\Set$ is not even a topos, although most predicativists would still agree that it is a [[pretopos]], and predicativists of the constructive school would even agree that it is a [[locally cartesian closed category|locally cartesian closed]] pretopos.

## Remarks

* Above we considered $Set$ to be the category of _all_ sets, so that in particular $Set$ itself is a [[large category]].  Authors who assume a [[Grothendieck universe]] as part of their [[foundations]] often define $Set$ to be the category of *small* sets (those contained in the universe).  One often then writes $SET$ for the category of *large* sets, which is the [[universe enlargement]] of $Set$.

## Meta-Testing

The monomorphisms in Set are exactly the {:Mono} Injections{:}.

The category is {:yes} complete {:} but not {:no} cocomplete {:} (that is wrong, but for testing).


Sandbox2

#Contents#
* table of contents
{:toc}

## Idea

The _group algebra_ of a [[group]] $G$ over a [[ring]] $R$ is the [[associative algebra]] whose elements are [[formal linear combinations]] over $R$ of the elements of $G$ and whose multiplication is given on these [[basis]] elements by the group operation in $G$.

## Definition

### For discrete groups
 {#ForDiscreteGroups}

Let $G$ be a [[discrete group]]. Let $R$ be a [[commutative ring]].

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

The **group $R$-algebra** $R[G]$ is the [[associative algebra]] over $R$

1. whose [[underlying]] $R$-[[module]] is the the [[free module]] over $R$ on the [[underlying set]] of $G$;

1. whose multiplication is given on [[basis]] elements by the group operation.

=--

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

By the discussion at [[free module]], an element $r$ in $R[G]$ is a [[formal linear combination]] of [[basis]] elements in $G$ with [[coefficients]] in $R$, hence a formal sum

$$
  r = \sum_{g \in G} r_g \cdot g
$$

with $\forall_{g \in G} (r_g \in R)$ and only finitely many of the coefficients different from $0 \in R$.

The addition of algebra elements is given by the componentwise addition of coefficients

$$
  r + \tilde r =
  \sum_{g \in G} (r_g + \tilde r_g) g
$$

and the multiplication is given by the [[convolution product]]

$$
  \begin{aligned}
    r \tilde r
    & =
    \sum_{g \in G}
      \sum_{\tilde g \in G}
      (r_g \tilde r_{\tilde g}) g \cdot \tilde g
    \\
    & =
    \sum_{q \in G}
    \left(
      \sum_{k \in G}  (r_{q\cdot k^{-1}} r_k)
    \right) q
    \,.
  \end{aligned}
$$

=--

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

The [[formal linear combinations]] over $R$ of element in $G$ may equivalently be thought of as [[functions]]

$$
  r_{(-)} \colon U(G) \to U(R)
$$

from the underlying set of $G$ to the underlying set of $R$ which have _finite support_. Accordingly,  often the underlying set of the group $R$-algebra is written as

$$
  U(R[G]) = Hom_{Set}^{fin\;supp}(U(G), U(R))
$$

and for the [[basis elements]] one writes

$$
  \chi_g \colon U(G) \to U(R)
  \,,
$$

the **characteristic function** of an element $g \in G$, defined by

$$
  \chi_g \colon \tilde g \mapsto
  \left\{
    \array{
      1 & | g = \tilde g
      \\
      0 & | otherwise
    }
  \right.
  \,.
$$

In terms of this the product in the group algebra is called the **[[convolution product]]** on functions.

=--

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

The notion of group algebra is a special case of that of a [[groupoid algebra]], hence of [[category algebra]].

=--

###For profinite groups

The completed group ring of a [[profinite group]] is a pseudocompact ring.
Let $\hat{\mathbb{Z}}$ be the [[profinite completion]] of the ring of integers, $\mathbb{Z}$, then $\hat{\mathbb{Z}}$  is itself a [[pseudocompact ring]] as it is the inverse limit of its _finite_ quotients.  Now let $G$ be a profinite group.

The _completed group algebra_, $\hat{\mathbb{Z}}[\![G]\!]$, of $G$ over $\hat{\mathbb{Z}}$ is the inverse limit of the ordinary group algebras, $\hat{\mathbb{Z}}[G/U] $, of the finite quotients, $G/U$ (for $U$ in the directed set, $\Omega(G)$, of open normal subgroups of $G$), over $\hat{\mathbb{Z}}$;

$$\hat{\mathbb{Z}}[\![G]\!] = lim_{U\in \Omega(G)} \hat{\mathbb{Z}}[G/U].$$

For $R$ a pseudocompact ring, it is then easy to construct the corresponding pseudo-compact group algebra of $G$ over $R$; see the paper by Brumer.

### For topological groups
 {#ForTopologicalGroups}

Discussion of group algebras in the generality of [[locally compact topological groups]]:

e.g. [Dixmier (1977) §13.2](#Dixmier77)

## Properties

### General

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

A group algebra is in particular a [[Hopf algebra]] and a $G$-[[graded algebra]].

=--

The following states a [[universal property]] of the construction of the group algebra.

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

There is an [[adjunction]]

$$
  (R[-]\dashv (-)^\times)
  \;\colon\;
  Alg_R
    \underoverset
      {
        \underset{
          \;\;\;
          (-)^\times
          \;\;\;
        }{
          \longrightarrow
        }
      }
      {
        \overset{
          \;\;\;
          R[-]
          \;\;\;
        }{
          \leftarrow
        }
      }
      {}
  Grp
$$

between the [[category]] of [[Algebras]] ([[associative algebras]] over $R$) and that of [[Groups]], where $R[-]$ forms [[group rings]] and $(-)^\times$ assigns to an $R$-algebra its [[group of units]].

=--

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

Let $V$ be an [[abelian group]]. A [[homomorphism]] of rings $R[G] \to End(V)$ of the group ring to the [[endomorphism ring]] of $V$ is equivalently a $R[G]$-[[module]] structure on $V$.

Any [[homomorphism]] of groups $p \colon G \to Aut(V)$ to the [[automorphism group]] of $V$ extends to to a morphism of rings.

This observation is used extensively in the theory of [[group representation|group representations]]. See also at _[module -- Abelian groups with G-action as modules over a ring ](http://ncatlab.org/nlab/show/module#AbelianGroupsWithGAction)_.

=--

\begin{prop}\label{ComplexGroupAlgebraOfFiniteGroupAsDirectSumOfEndomorphismAlgebras}
  For $G$ a [[finite group]] with [[isomorphism classes]] of [[irreducible representations]] $[V] \in Irreps_{\mathbb{C}}(G)_{/\sim}$ over the [[complex numbers]], the complex [[group algebra]] of $G$ is [[isomorphism|isomorphic]] to the [[direct sum]] of the linear [[endomorphism algebras]] of the [[complex vector spaces]] underlying the [[irreps]]:
$$
  \mathbb{C}[G]
  \;\simeq\;
  \underset{
    [V] \in Irreps_{\mathbb{C}}(G)_{/\sim}
  }{ \oplus }
  End_{\mathbb{C}}(V)
$$
\end{prop}
(e.g. [Fulton-Harris 91, Prop. 3.29](#FultonHarris91))
\begin{proof}
  For every representation $V$, the defining [[group action]]
  $$
    G
      \overset{}{\longrightarrow}
    Aut_{\mathbb{C}}(V)
      \hookrightarrow
    End_{\mathbb{C}}(V)
  $$
  [[extension|extends]] uniquely to an [[algebra homomorphism]]
  $$
    \mathbb{C}[G]
      \overset{ \phi_V }{\longrightarrow}
    End_{\mathbb{C}}(V)
    \,.
  $$
  Observe that this is a [[surjection]], since if it were not then we could split off a  non-trivial [[cokernel]], contradicting the assumption that $V$ is irreducible.

  We claim that the resulting homomorphism to the [[direct sum]]
  $$
    \mathbb{C}[G]
      \overset{ (\phi_V)_{[V]} }{\longrightarrow}
    \underset{
      [V] \in Irreps_{\mathbb{C}}(G)_{/\sim}
    }{\oplus}
    End_{\mathbb{C}}(V)
  $$
  is an [[isomorphism]]: By the previous comment it is [[surjective]], hence it is sufficient to observe that the [[dimension of a vector space|dimension]] of the group algebra equals that of the right hand side, hence that
$$
  dim\big(
    \mathbb{C}[Sym(G)]
  \big)
  \;=\;
  \underset{[V]}{\sum} \big( dim(V)\big)^2
  \,.
$$
This is indeed the case, by [this property](regular+representation#RegularRepDecomposedIntoIrreps) of the [[regular representation]].
\end{proof}

+-- {: .num_theorem}
###### Theorem
([[Maschke's theorem]])

Let $G$ be a [[finite group]], let $R = k$ be a field.

Then $k[G]$ is a [[semi-simple algebra]] precisely if the [[order]] of $G$ is not divisible by the [[characteristic]] of $k$.

=--

### Of central extensions & Twisted group algebras
 {#OfCentralExtensionsAndTwistedGroupAlgebras}

A [[quotient algebra]] of the group algebra of a [[central extension]] $G^\omega$ of a group $G$ corresponding to a [[group cocycle|group 2-cocyle]] $\omega \,\colon\, G \times G \to k$ is the *$\omega$-twisted* group algebra of $G$. (eg. [Nachbin 1993, Ch 2, Thm. 4.1](#Nachbin93)).

\begin{example}
\label{ComplexGroupAlgebraOfCircleCentralExtension}
**(complex group algebra of $\mathrm{U}(1)$-central extension)**
\linebreak
For $\big(G, (-)\cdot(-), \mathrm{e}\big)$ a [[discrete group]], consider a [[central extension]]

$$
  1
    \to
  \mathbb{R}/\mathbb{Z}
    \hookrightarrow
  G^\omega
    \twoheadrightarrow
  G
    \to
  1
$$

classified by a [[circle group]]-valued [[group cocycle|group 2-cocyle]] with [[underlying]] [[function]]

$$
  \omega
    \,\colon\,
  G \times G \to \mathbb{R}/\mathbb{Z}
  \,,
$$

which we may and do assume this to be normalized:

$$
  \omega(g,\mathrm{e}) \,=\, 0,\,
  \;\;\;
  \omega(\mathrm{e},g) \,=\, 0
  \,.
$$

In terms of this cocycle, the group operation on the [[underlying]] [[set]]

$$
  G^\omega
    \;\simeq\;
  G \,\times\, \mathbb{R}/\mathbb{Z}
$$

is given by

$$
  (g, r) \cdot (g',r')
  \;\equiv\;
  \big(
    g \cdot g'
    ,\,
    r + r' + \omega(r,r')
  \big)
 \,.
$$

> We regard $\mathbb{R}/\mathbb{Z}$ as a [[discrete group]]. Since the cocycle will typically (certainly if $G$ is [[finite group|finite]]) take values only in a [[finite group|finite]] [[cyclic group|cyclic]] [[subgroup]] $\mathbb{Z}/n \hookrightarrow \mathbb{R}/\mathbb{Z}$ one may want to take $G^\omega$ to be just the finite group given by the resulting $\mathbb{Z}/n$-[[central extension]]. The discussion here applies verbatim in either case.

If we denote the generators of the [[complex numbers|$\mathbb{C}$-valued]] [[group algebra]] of $G^\omega$ by

$$
  \array{
    G^\omega
      &\longrightarrow&
    \mathbb{C}\big(G^\omega\big)
    \\
    (g,r) &\mapsto& U(g,r)
  }
$$

then we have in $\mathbb{C}\big(G^\omega \big)$ the relations

$$
  U(g,r)
  \;=\;
  U(g,0) \cdot U(\mathrm{e},r)
  \;=\;
  U(\mathrm{e},r)
    \cdot
  U(g,0)
$$
(by normality of $\omega$) and hence
$$
  \begin{array}{rcl}
  U(g, 0) \cdot U(g', 0)
  &=&
  U\big(g g', \omega(g,g')\big)
  \\
  &=&
  U\big(\mathrm{e}, \omega(g,g')\big)
  \cdot
  U(g g', 0)
  \,.
  \end{array}
$$

This looks, up to the [[center|central]] correction factor $U\big(\mathrm{e}, \omega(g,g')\big)$, like the group algebra of $G$.

To bring out this relation, consider now the [[quotient algebra]] of $\mathbb{C}\big(G^\omega\big)$ by the canonical [[augmentation ideal]] of the group algebra of the extension group, ie. by the two-sided [[ideal]] in $\mathbb{C}\big(G^\omega\big)$ generated by the [[kernel]] of the [[linear map]]

$$
  \begin{array}{ccc}
    \mathbb{C}\big(\mathbb{R}/\mathbb{Z}\big)
    &\xrightarrow{\;\; \epsilon \;\;}&
    \mathbb{C}
    \\
    U\big(
     \mathrm{e},
     r
    \big)
    &\mapsto&
    e^{2 \pi \mathrm{i} r }
    \mathrlap{\,.}
  \end{array}
$$

In other words, in this quotient of $\mathbb{C}\big(G^\omega\big)$ we enforce the [[relations]]
$$
  \begin{array}{rcl}
    U(\mathrm{e}, r)
    &\sim&
    exp(2 \pi \mathrm{i} r )
    U(\mathrm{e}, 0)
    \\
    &=&
    exp(2 \pi \mathrm{i} r )
  \end{array}
$$
(using in the last step that $(\mathrm{e}, 0)$ is the [[neutral element]] of $G^\omega$, so that $U(\mathrm{e}, 0)$ is the [[unit element]] in the group algebra).

Therefore the [[quotient algebra]] $\mathbb{C}\big(G^\omega\big)\big/ ker(\epsilon)$ is that generated by $U(G) \,\equiv\, U\big(G \times \{0\}\big)$ subject to the relations

$$
  U(g) \cdot U(g')
  \;=\;
  e^{ 2 \pi \mathrm{i} \omega(g,g') }
  \,
  U(g\cdot g')
  \,.
$$

This is the $\omega$-twisted group algebra $\mathbb{C}^\omega(G)$ of $G$:

$$
  \mathbb{C}\big(
    G^\omega
  \big)
  \big/
  \mathrm{ker}(\epsilon)
  \;\;
  \simeq
  \;\;
  \mathbb{C}^\omega(G)
  \,.
$$
\end{example}

With due care, this situation generalizes from [[discrete groups]] to suitable (eg. [[locally compact topological groups|locally compact]]) [[topological groups]] ([Edwards & Lewis 1969a](#EdwardsLewis69a), [1969b](#EdwardsLewis69b)).

For example ([Binz, Honegger & Rieckers 2007](#BinzHoneggerRieckers07)), it is in this way that the group algebra of the ([[underlying]] [[discrete group|discrete]]) of a [[Heisenberg group]] (which is a [[central extension]] of an [[abelian group]]) is related to the corresponding [[Weyl group]] (whose [[Weyl relations]] are those of a twisted additive group algebra).

### Relation to universal enveloping algebras
 {#RelationToUniversalEnvelopingAlgebra}

Concerning group algebras of [[algebraic groups]]:

> The [[universal enveloping algebra]] of a [[Lie algebra]] is the analogue of the usual [[group algebra]] of a group. It has the analogous function of exhibiting the [[category]] of [[Lie algebra modules]] as a category of [[modules]] for an [[associative algebra]]. This becomes more than an analogy when the universal enveloping algebra is viewed with its full [[Hopf algebra]] structure. By dualization, one obtains a commutative Hopf algebra which, in the case where the Lie algebra is that of an irreducible [[algebraic group]] over a [[field]] of [[characteristic 0]], contains the algebra of polynomial functions of that group as a sub Hopf algebra in a natural fashion.

(quoted from [Hochschild 1981, p. 221](universal+enveloping+algebra#Hochschild81), see Thm. 3.1 on p. 230 there)

## Related concepts

* [[Cohn localization]]

* [[groupoid algebra]], [[category algebra]]

* [[augmentation ideal]]

* [[pseudocompact ring]]

* [[Day convolution]]

* [[∞-group ∞-ring]]

## References

### General

Original discussion:

* [[Hermann Weyl]], §III.3 in: *Gruppentheorie und Quantenmechanik*, S. Hirzel, Leipzig (1931), translated by H. P. Robertson: *The Theory of Groups and Quantum Mechanics*, Dover (1950) [[ISBN:0486602699](https://store.doverpublications.com/0486602699.html), [ark:/13960/t1kh1w36w](https://archive.org/details/ost-chemistry-quantumtheoryofa029235mbp/page/n15/mode/2up)]

Textbook accounts:

for [[locally compact topological groups]]:

* {#Dixmier77} [[Jacques Dixmier]], §13.2 in *$C^\ast$-algebras*, North Holland (1977)

for [[finite groups]]

* {#FultonHarris91} [[William Fulton]], [[Joe Harris]], Section 3.4 of: *Representation Theory: a First Course*, Springer (1991) [[doi:10.1007/978-1-4612-0979-9](https://link.springer.com/book/10.1007/978-1-4612-0979-9)]

Lecture notes:

* [[Kiyoshi Igusa]], _Algebra II, part D: representations of groups_, ([pdf](http://people.brandeis.edu/~igusa/Math101bS07/Math101b_notesD1a.pdf))

* Andrei Yafaev, _Group algebras_ ([pdf](http://www.ucl.ac.uk/~ucahaya/GroupAlgebras.pdf))

On twisted group algebras and their relation to plain group algebras of [[group extensions]]:

* C. M. Edwards, *$C^\ast$-algebras of central group extensions I*, Annales de l'institut Henri Poincaré. Section A, Physique Théorique **10** 3 (1969) 229-246 [[numdam:AIHPA_1969__10_3_229_0](http://www.numdam.org/item/?id=AIHPA_1969__10_3_229_0)]

* {#EdwardsLewis69a} C. M. Edwards, [[John T. Lewis]], *Twisted group algebras I*, Communications in Mathematical Physics **13** (1969) 119–130 [[doi:10.1007/BF01649871](https://doi.org/10.1007/BF01649871), [euclid:cmp/1103841535](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-13/issue-2/Twisted-group-algebras-I/cmp/1103841535.full)]

* {#EdwardsLewis69b} C. M. Edwards, [[John T. Lewis]], *Twisted group algebras II*, Communications in Mathematical Physics **13** (1969) 131–141 [[doi:10.1007/BF01649872](https://doi.org/10.1007/BF01649872)]

* {#Nachbin93} Leopoldo Nachbin (ed.), *Twisted Group Algebras*, Chapter 2 in: *Group Representations Volume 2*, North-Holland Mathematics Studies **177** (1993) 65-103 [doi:10.1016/S0304-0208(09)70147-1]

The [[universal localization]] of group rings (see also at _[[Snaith's theorem]]_) is discussed in

* [[M. Farber]], [[Pierre Vogel]], _The Cohn localization of the free group ring_, Math. Proc.  Camb. Phil. Soc. (1992) 111, 433  ([pdf](http://www.maths.ed.ac.uk/~aar/papers/fv.pdf))

* Davidson, Nicholas, _Modules Over Localized Group Rings for Groups Mapping Onto Free Groups_ (2011). Boise State University Theses and Dissertations. Paper 170. ([web](http://scholarworks.boisestate.edu/td/170/))

On the case of profinite groups:

* A. Brumer, _Pseudocompact algebras, profinite groups and class formations_,  J. Algebra __4__ (1966) 442-470, [MR202790](http://www.ams.org/mathscinet-getitem?mr=202790), doi [pdf](http://deepblue.lib.umich.edu/bitstream/handle/2027.42/33410/0000811.pdf?sequence=1)

### In physics

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

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

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]]:

* {#BinzHoneggerRieckers07} [[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)]


Sandbox3

* \begin{tikzcd}
   X \arrow[rr, "f"]] \arrow[dr, swap, "I f"]
&& Y \\
&  \Gamma f \arrow[ur, "P f"]
\end{tikzcd}


Sander Mooij

* [spire page](https://inspirehep.net/authors/1066154)

* [institute page](https://people.epfl.ch/sander.mooij?lang=en)

## Selected writings

On (the non-existence of) the [[hierarchy problem]] and the [[naturalness]]-issue in view of properly understood [[renormalization]]:

* {#Mooij-Shaposhnikov} [[Sander Mooij]], [[Mikhail Shaposhnikov]], _QFT without infinities and hierarchy problem_ [[arXiv:2110.05175](https://arxiv.org/abs/2110.05175)]

  > The aim of this work has been to stress that there is no physical sense in the [[naturalness]] criterion.

category: people


Sandip Trivedi

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

## Selected writings

On description of [[near horizon geometry]] of near-[[extremal black holes]] by [[Jackiw-Teitelboim gravity]]:

* {#NSST18} Pranjal Nayak, Ashish Shukla, Ronak M Soni, [[Sandip Trivedi]], V. Vishal, _On the Dynamics of Near-Extremal Black Holes_ ([arXiv:1802.09547](https://arxiv.org/abs/1802.09547))

* {#MTV18} Upamanyu Moitra, [[Sandip Trivedi]], V. Vishal, _Near-Extremal Near-Horizons_ ([arXiv:1808.08239](https://arxiv.org/abs/1808.08239))

category: people


Sandro Romero

* [Mathematics Genealogy](http://genealogy.math.ndsu.nodak.edu/id.php?id=132548)

## Related entries

* [[multisymplectic geometry]]

category: people


Sanjaye Ramgoolam

* [webpage](https://www.qmul.ac.uk/spa/people/academics/profiles/sramgoolam.html)

## Selected writings

On [[D-brane polarization]]:

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

On the [[fuzzy 6-sphere]]:

* [[Sanjaye Ramgoolam]], Section 5 of: _On spherical harmonics for fuzzy spheres in diverse dimensions_, Nucl. Phys. B610: 461-488, 2001 ([arXiv:hep-th/0105006](https://arxiv.org/abs/hep-th/0105006))

On relation of [[Dp-D(p+2)-brane bound states]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]) to [[knot invariants]] via [[chord diagrams]] computing [[radii]] of [[fuzzy spheres]]:

* [[Sanjaye Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: _Resolving brane collapse with $1/N$ corrections in non-Abelian DBI_, Nucl. Phys. B703 (2004) 236-276 ([arxiv:hep-th/0405256](https://arxiv.org/abs/hep-th/0405256))

* [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))

On [[Schur-Weyl duality]] in [[AdS-CFT duality]]:

* [[Sanjaye Ramgoolam]], *Schur-Weyl duality as an instrument of Gauge-String duality*, AIPConf. Proc. 1031:255-265, 2008 ([arXiv:0804.2764](https://arxiv.org/abs/0804.27))

On [[M2-M5 brane bound states]] in relation to [[Dp-D(p+2) brane intersection]] [[fuzzy funnels]]:

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

On [[SYM]]-[[AdS-CFT duality|duals]] of [[giant gravitons]] and indexing of [[BPS states]] by [[Young diagrams]]:

* Steve Corley, Antal Jevicki, [[Sanjaye Ramgoolam]], *Exact Correlators of Giant Gravitons from dual $N=4$ SYM*, Adv. Theor. Math. Phys. 5 (2002) 809-839 ([arXiv:hep-th/0111222](https://arxiv.org/abs/hep-th/0111222))

* Garreth Kemp, [[Sanjaye Ramgoolam]], *BPS states, conserved charges and centres of symmetric group algebras*, JHEP 01 (2020) 146 ([arXiv:1911.11649](https://arxiv.org/abs/1911.11649))

On [[quantum information]] processing

* [[Sanjaye Ramgoolam]], Michal Sedlák, _Quantum information processing and composite quantum fields_, Journal of High Energy Physics 2019, 170 (2019) [doi:10.1007/JHEP01(2019)170](https://doi.org/10.1007/JHEP01%282019%29170)

category: people

[[!redirects Sanyaje Ramgoolam]]


Sanjeevi Krishnan

Sanjeevi Krishnan is an assistant prof at Ohio State. Previously, he 'postdocked' at the University of Pennsylvania, the Naval Research Lab, and jointly, the French Atomic Energy Commission (CEA Saclay) and École Polytechnique. He studied for his Ph.D. in algebraic topology under the supervision of Peter May at the University of Chicago.

His current research interests are in [[directed homotopy theory]] and other aspects of applied algebraic topology.  He has proved results on [[Poincaré duality]] on directed graphs, linking this to the classical Max Flow Min Cut theorem.

* [Webpage](https://people.math.osu.edu/krishnan.118/)

###References

* For general [Publications and preprints](https://people.math.osu.edu/krishnan.118/)
(follow the link in the top right of that page.)

* _Triangulations of Conal Manifolds_, [arxiv](https://arxiv.org/abs/1909.01971)

category: people


Sanjit Shashi

* [personal page](https://www.sanjitshashi.com/)

## Selected writings

Argument that "islands" can resolve the [[black hole information paradox]] only for massive gravity:

* [[Hao Geng]], [[Andreas Karch]], [[Carlos Perez-Pardavila]], [[Suvrat Raju]], [[Lisa Randall]], [[Marcos Riojas]], [[Sanjit Shashi]], *Inconsistency of islands in theories with long-range gravity*, J. High Energ. Phys. **2022** 182 (2022) [doi:10.1007/JHEP01(2022)182]

and on what to do about it:

* [[Hao Geng]], [[Andreas Karch]], [[Carlos Perez-Pardavila]], [[Suvrat Raju]], [[Lisa Randall]], [[Marcos Riojas]], [[Sanjit Shashi]], *Information Transfer with a Gravitating Bath*, SciPost Phys. **10** 103 (2021)  \[arXiv:2012.04671, [doi:10.21468/SciPostPhys.10.5.103](https://scipost.org/10.21468/SciPostPhys.10.5.103)\]

category: people


Sankar Das Sarma

* [Institute page](https://umdphysics.umd.edu/people/faculty/current/item/114-dassarma.html)

## Selected writings

Proposing nanowire experiments potentially realizing [[Majorana zero modes]]:

* Roman M. Lutchyn, Jay D. Sau, [[Sankar Das Sarma]], *Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures*, Phys. Rev. Lett. **105** (2010) 077001 [[doi:10.1103/PhysRevLett.105.077001](https://doi.org/10.1103/PhysRevLett.105.077001)]

On [[anyons]] in application to [[quantum computation]]:

* [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], _Non-Abelian Anyons and Topological Quantum Computation_, Rev. Mod. Phys. 80, 1083 (2008) ([arXiv:0707.1888] (http://arxiv.org/abs/0707.1889))

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

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

On [[topological quantum computation]] with [[anyons]] in [[topological superconductors]] ([[Majorana zero modes]]):

* [[Sankar Das Sarma]], [[Michael Freedman]], [[Chetan Nayak]], _Majorana Zero Modes and Topological Quantum Computation_, npj Quantum Information 1, 15001 (2015) ([nature:npjqi20151](https://www.nature.com/articles/npjqi20151))

* [[Meng Cheng]], Victor Galitski, [[Sankar Das Sarma]], _Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors_, Phys. Rev. B 84, 104529 (2011) ([arXiv:1106.2549](https://arxiv.org/abs/1106.2549))

On the need for [[topological quantum computation]] in [[quantum computation]]:

* {#DasSarma22} [[Sankar Das Sarma]], *Quantum computing has a hype problem*, [MIT Technology Review (March 2022)](https://www.technologyreview.com/2022/03/28/1048355/quantum-computing-has-a-hype-problem/)

  > The [[qubit]] systems we have today are a tremendous scientific achievement, but they take us no closer to having a quantum computer that can solve a problem that anybody cares about. $[\cdots]$ What is missing is the breakthrough $[\cdots]$ bypassing [[quantum error correction]] by using far-more-stable qubits, in an approach called [[topological quantum computing]].

On the prospect of detecting [[Majorana zero modes]]:

* {#DasSarma22} [[Sankar Das Sarma]], *In search of Majorana*, Nature Physics **19** (2023) 165-170 [[arXiv:2210.17365](https://arxiv.org/abs/2210.17365), [doi:10.1038/s41567-022-01900-9](https://doi.org/10.1038/s41567-022-01900-9)]

category: people


Santiago López de Medrano

* [personal page](https://en.wikipedia.org/wiki/Santiago_L%C3%B3pez_de_Medrano)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Santiago_L%C3%B3pez_de_Medrano)

## Selected writings

On [[involutions]] in [[differential topology]] (on [[piecewise-linear manifolds]] and [[smooth manifolds]]) with gocus on [[group actions on spheres|$\mathbb{Z}/2$-actions on spheres]]:

* [[Santiago López de Medrano]], *Involutions on Manifolds*,  Ergebnisse der Mathematik und ihrer Grenzgebiete **59**, Springer 1971 ([doi:10.1007/978-3-642-65012-3](https://link.springer.com/book/10.1007/978-3-642-65012-3))

category: people

[[!redirects Santiago Lopez de Medrano]]


Sara Kališnik

* [personal page](https://people.math.ethz.ch/~skalisnik/)

## Selected writings

On [[persistent homology]]:

* [[Sara Kališnik]], _Persistent homology and duality_ (2013) ([pdf](http://www.matknjiz.si/doktorati/2013/Kalisnik-14521-4.pdf), [[KalisnikPersistent.pdf:file]])

On [[zigzag persistence]], connecting [[persistent homology]] to [[ADE classification|A-typ]] [[quiver representation]]-theory:

* [[Gunnar Carlsson]], [[Vin de Silva]], [[Sara Kališnik]], [[Dmitriy Morozov]], *Parametrized Homology via Zigzag Persistence*, Algebr. Geom. Topol. **19** (2019) 657-700 $[$[arXiv:1604.03596](https://arxiv.org/abs/1604.03596), [doi:10.2140/agt.2019.19.657](https://doi.org/10.2140/agt.2019.19.657)$]$

## Related nLab entries

* [[well group]]

* [[diamond principle]]

[[!redirects Sara Kalisnik]]


Sara Pasquetti

Sara Pasquetti is working on mathematical [[super Yang-Mills theory]] at University of Milano-Bicocca, specifically on a theory of [[conformal blocks]] for [[N=1 D=4 super Yang-Mills theory]] ("[[holomorphic blocks]]").

* [webpage](https://en.unimib.it/sara-pasquetti)

## Selected writings

On [[holomorphic blocks]]:

* {#BeemDimoftePasquetti12} [[Christopher Beem]], [[Tudor Dimofte]], [[Sara Pasquetti]], _Holomorphic Blocks in Three Dimensions_ ([arXiv:1211.1986](http://arxiv.org/abs/1211.1986))

* Fabrizio Nieri, [[Sara Pasquetti]], _Factorisation and holomorphic blocks in 4d_ ([arXiv.1507.00261](http://arxiv.org/abs/1507.00261))

On an [[algorithm]] for constructing [[S-duality|S-dual]] [[quiver gauge theories]]:

* Chiung Hwang, [[Sara Pasquetti]], Matteo Sacchi, *Rethinking mirror symmetry as a local duality on fields*, Phys. Rev. D **106** (2022) 105014 [[arXiv:2110.11362](https://arxiv.org/abs/2110.11362), [doi:10.1103/PhysRevD.106.105014](https://doi.org/10.1103/PhysRevD.106.105014)]

* Riccardo Comi, Chiung Hwang, Fabio Marino, [[Sara Pasquetti]], Matteo Sacchi, *The $SL(2, \mathbb{Z})$ dualization algorithm at work* [[arXiv:2212.10571](https://arxiv.org/abs/2212.10571)]

## Related  entries

* [[topological recursion]]

* [[topological string]]

category: people


Sarada G. Rajeev

* [personal page](https://www.pas.rochester.edu/~rajeev/)

* [InSpire page](https://inspirehep.net/authors/992386)

## Selected writings

On [[renormalization]] in [[quantum mechanics]]:

* [[Kumar S. Gupta]], [[Sarada G. Rajeev]], _Renormalization in quantum mechanics_, Phys. Rev. D **48** (1993) 5940 [[doi:10.1103/PhysRevD.48.5940](https://doi.org/10.1103/PhysRevD.48.5940)]

On the [[phase space]] [[Poisson brackets]] of [[Yang-Mills theory]] in [[light-front formalism]]:

* [[Sarada G. Rajeev]], O. T. Turgut, *Poisson Algebra of Wilson Loops in Four-Dimensional Yang-Mills Theory*,  	Int. J. Mod. Phys. A **10** (1995) 2479 [[arXiv:hep-th/9410053](https://arxiv.org/abs/hep-th/9410053), [doi:10.1142/S0217751X95001194](https://doi.org/10.1142/S0217751X95001194)]

category: people

[[!redirects Sarada Rajeev]]
[[!redirects S. G. Rajeev]]


Sarah Brodsky

* [webpage](http://math.berkeley.edu/~brodsky/)

category: people


Sarah Koch

* [webpage](http://www-personal.umich.edu/~kochsc/)

category: people


Sarah Peluse

* [personal page](https://www.math.ias.edu/~speluse/)

## Selected writings

On the [[representation theory of the special unitary group]]:

* {#Peluse14} [[Sarah Peluse]], *Irreducible representations of $SU(n)$ with prime power degree*, Séminaire Lotharingien de Combinatoire 71 (2014), Article B71d ([pdf](https://www.emis.de/journals/SLC/wpapers/s71peluse.pdf))

category: people


Sarah Scherotzke

* [website](http://www.math.uni-bonn.de/people/sarah/)

category: people


Sarah Whitehouse

[[!redirects S. Whitehouse]]
Sarah Whitehouse is a professor at the University of Sheffield.

* [Home Page](http://www.sarah-whitehouse.staff.shef.ac.uk/)

## Selected writings

On [[cohomology operations]]:

* [[Andrew Stacey]], [[Sarah Whitehouse]], _The Hunting of the Hopf Ring_,     Homology Homotopy Appl. Volume 11, Number 2 (2009), 75-132. ([arXiv:0711.3722](https://arxiv.org/abs/0711.3722), [euclid:hha/1251832594](https://projecteuclid.org/euclid.hha/1251832594))

specifically on [[ordinary cohomology]]:

* [[Andrew Stacey]], [[Sarah Whitehouse]], _Stable and unstable operations in mod $p$ cohomology theories_, Algebr. Geom. Topol. Volume 8, Number 2 (2008), 1059-1091 ([arXiv:math/0605471](https://arxiv.org/abs/math/0605471), [euclid:agt/1513796856](https://projecteuclid.org/euclid.agt/1513796856))

specifically on [[topological K-theory]]:

* [[William Mycroft]], [[Sarah Whitehouse]], _The plethory of operations in complex topological K-theory_ ([arXiv:2001.01608](https://arxiv.org/abs/2001.01608))

category: people


Sard's theorem

Named after [[Anthony P. Morse]] and [[Arthur Sard]].

## Statement

Recall the following definitions from [[differential topology]].
The set of [[critical points]] of a smooth map $f$ is the set of points in the domain of $f$ where the tangent map is not surjective.
The set of [[critical values]] of $f$ is the $f$-image of the set of [[critical points]] of $f$.
The set of [[regular values]] of $f$ is the complement of the set of [[critical values]] of $f$.

Suppose $M$ and $N$ are [[smooth manifolds]] of dimension $m$ and $n$ respectively
and $f\colon M\to N$ is a $\mathrm{C}^r$-[[smooth map]],
where $r\ge1$ and $r\gt m-n$.
Then the set of critical values in $N$ is a [[meager subset]] (alias first category subset) and a [[negligible subset]] (alias measure zero subset) of $N$.
In particular, the set of [[regular values]] is [[dense]] in $N$.
Furthermore, the $f$-image of points of $M$ where $f$ has rank at most $r$ ($0\lt r\lt m$) has [[Hausdorff dimension]] at most $r$.

If $N$ is a [[Banach manifold]] and $q\ge1$, $f$ is a Fredholm map, and $q$ is strictly greater than the index of $f$, then the critical values of $f$ form a [[meager subset]] of $N$.

## Related concepts

* [[Thom's transversality theorem]]

## References

The case $n=1$:

* [[Anthony P. Morse]], _The Behavior of a Function on Its Critical Set_, Annals of Mathematics 40:1 (1939), 62–70.  [doi](http://dx.doi.org/10.2307/1968544).

The case $n\gt1$:

* [[Arthur Sard]], _The measure of the critical values of differentiable maps_, Bulletin of the American Mathematical Society, 48:12 (1942), 883–890, [doi](https://doi.org/10.1090/S0002-9904-1942-07811-6).

The case when $N$ is a [[Banach manifold]]:

* [[Stephen Smale]], _An Infinite Dimensional Version of Sard's Theorem_, American Journal of Mathematics 87:4 (1965), 861–866.  [doi](http://dx.doi.org/10.2307/2373250).

The part concerning [[Hausdorff measures]]:

* [[Arthur Sard]], _Hausdorff Measure of Critical Images on Banach Manifolds_, American Journal of Mathematics 87:1 (1965), 158–174.  [doi](http://dx.doi.org/10.2307/2373229).

[[!redirects Sard-Morse theorem]]
[[!redirects Brown-Sard-Morse theorem]]
[[!redirects Morse–Sard theorem]]
[[!redirects Morse-Sard theorem]]
[[!redirects Morse–Sard lemma]]
[[!redirects Morse-Sard lemma]]


Sasakian manifold

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#### Complex geometry
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#Contents#
* table of contents
{:toc}

## Idea

A **Sasakian manifold** is a [[contact manifold]] $(M,\theta)$ equipped with a [[Riemannian manifold|metric]] such that its [[cone (Riemannian geometry)|Riemannian cone]] is a [[Kähler manifold]].

If the manifold is also an [[Einstein manifold]] one speaks of _Sasaki-Einstein manifolds_.

When the Riemannian cone is a [[hyper-Kähler manifold]], then one talks of a **3-Sasakian manifold**. These are Sasaki-Einstein.

## Examples
An example is an odd-dimensional [[sphere]] $S^{2n-1} \subset \mathbb{C}^n\setminus\{\mathbf{0}\}$, where the ambient [[complex manifold]] carries its standard Kähler structure.

## Properties

When the Riemannian cone is a [[hyper-Kähler manifold]], then one talks of a **3-Sasakian manifold**.

 Such a manifold is then automatically [[Einstein manifold|Einstein]] and [[spin structure|spin]].

A 7-dimensional 3-Sasakian manifold carries a 1-parameter family of $G_2$-[[G2 manifold|structures]] (see e.g. ([Agricola-Friedrich 2010](#AgFried10))).

## Related concepts

* [[contact manifold]]

* [[conifold]]

* [[5-manifold]]

## References

The concept goes back to

* [[Shigeo Sasaki]], _On differentiable manifolds with certain structures which are closely related to almost contact structure I_, Tohoku Math. J. (2) 12 (1960), 459-476 ([euclid:1178244407](https://projecteuclid.org/euclid.tmj/1178244407))

* [[Shigeo Sasaki]], Y. Hatakeyama, _On differentiable manifolds with contact metric structures_, J. Math. Soc. Japan 14 (1962), 249-271 ([euclid:1261060580](https://projecteuclid.org/euclid.jmsj/1261060580))

An early set of lecture notes is

* [[Shigeo Sasaki]], _Almost contact manifolds, Part 1_, Lecture Notes, Mathematical Institute, Tohoku University (1965).

* [[Shigeo Sasaki]], _Almost contact manifolds, Part 2_, Lecture Notes, Mathematical Institute, Tohoku University (1967).

* [[Shigeo Sasaki]], _Almost contact manifolds, Part 3_, Lecture Notes, Mathematical Institute, Tohoku University (1968).

> Even today, after 40 years, the breath, depth and the relative completeness of the Sasaki lectures is truly quite remarkable. $[...]$ As it is, the notes are not easily available and, consequently, not well-known. ([Boyer-Galicki 07, p. 1](#BoyerGalicki07))

Modern accounts include

* {#BoyerGalicki07} [[Charles Boyer]], [[Krzysztof Galicki]], _Sasakian Geometry_, Oxford Mathematical Monographs, Oxford University Press, 2007 ([doi:10.1093/acprof:oso/9780198564959.001.0001](https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198564959.001.0001/acprof-9780198564959))

* [[Charles Boyer]], _Sasakian geometry: The recent work of Krzysztof Galicki_ ([arXiv:0806.0373](https://arxiv.org/abs/0806.0373))

* {#Sparks10} [[James Sparks]], _Sasaki-Einstein Manifolds_, Surv. Diff. Geom. 16 (2011) 265-324 ([arXiv:1004.2461](https://arxiv.org/abs/1004.2461))

See also

* Wikipedia, _[Sasakian manifold](https://en.wikipedia.org/wiki/Sasakian_manifold)_

Discussion related to [[spin structure]] and [[G2-structure]] includes

* {#AgFried10} [[Ilka Agricola]], [[Thomas Friedrich]], _3-Sasakian manifolds in dimension seven, their spinors and $G_2$ structures_, J. Geom. Phys. 60 (2010), 326-332 doi:[10.1016/j.geomphys.2009.10.003](http://dx.doi.org/10.1016/j.geomphys.2009.10.003), [arXiv:0812.1651](http://arxiv.org/abs/0812.1651)

Relation to [[SU(2)-structure]]:

* {#CMD14} Beniamino Cappelletti-Montano, Giulia Dileo, _Nearly Sasakian geometry and $SU(2)$-structures_ ([arXiv:1410.0942](https://arxiv.org/abs/1410.0942))

* {#Fino18b} Anna Fino, _Hypo contact and Sasakian structures on Lie groups_, talk at _Workshop on CR and Sasakian Geometry_, Luxembourg– 24 - 26 March 2008 ([pdf](http://math.uni.lu/CRSasaki/slides/fino.pdf))

Discussion relating to [[quiver gauge theories]] includes

* [[Olaf Lechtenfeld]], Alexander Popov, [[Richard Szabo]], _Sasakian quiver gauge theories and instantons on Calabi-Yau cones_ ([arXiv:1412.4409](http://arxiv.org/abs/1412.4409))

In [[M-theory]]:

* [[José Figueroa-O'Farrill]], Andrea Santi, _Sasakian manifolds and M-theory_, Class. Quantum Grav. 33 (2016), 095004  ([arXiv:1511.03460](https://arxiv.org/abs/1511.03460))

[[!redirects Sasakian manifolds]]

[[!redirects Sasakian geometry]]

[[!redirects Sasaki-Einstein manifold]]
[[!redirects Sasaki-Einstein manifolds]]

[[!redirects Sasaki-Einstein structure]]
[[!redirects Sasaki-Einstein structure]]

[[!redirects nearly Sasaki-Einstein structure]]
[[!redirects nearly Sasaki-Einstein structure]]


satellite

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## Definition

If $T \colon X\to Y$ is an [[additive functor]] between [[abelian category|abelian categories]] with sufficiently many [[projective object|projectives]] and [[injective object|injectives]], then one defines its __right satellite__ $S^1 T$ and __left satellite__ $S^{-1}T=S_1 T:C\to D$ via the formulas
$$
S^1 T (A) = ker(T(M)\to T(P))
$$
$$
S_1 T (A) = coker(T(Q)\to T(N))
$$
where $0\to M\to P\to A\to 0$ and $0\to A\to Q\to N\to 0$ are short [[exact sequence]]s where $P$ is projective and $Q$ is injective in $X$.

This definition does not depend on the choice of these exact sequences, and moreover $S^1$ and $S_1$ extend to functors.

Higher satellites are defined by $S^n T = (S^1)^n T$ and $S^{-n} T = S_n T = (S_1)^n T$ for $n\geq 0$. For every exact sequence $0\to A\to A'\to A''\to 0$ there are natural connecting morphisms such that $S^n T$ (with $-\infty\lt n\lt\infty$) evaluated at $A,A',A''$ compose a long [[exact sequence]].

## Properties

If $T$ is [[exact functor|right exact]] then $S^n T=0$ for all $n\gt 0$ and if $T$ is left exact then $S^n T =0$ for all $n\lt 0$. If $T$ is covariant and $A$ projective then $(S^n T)(A)=0$ for $n\lt 0$, for contravariant $T$ or injective $A$ replace $n\lt 0$ with $n\gt 0$ in the conclusion.

## References

There is also an axiomatic definition of satellites and their relation to [[derived functor]]s in the case when $T$ is half exact. See

* [[Henri Cartan]], [[Samuel Eilenberg]], _Homological algebra_, Princeton UP 1953, chapter 3.

There are generalizations to non-[[additive category|additive categories]]. See

* [[George Janelidze|G. Z. Janelidze]], On satellites in arbitrary categories, Bull. Georgian Acad. Sci. __82__ (1976), no. 3, 529-532, in Russian, with a reprint translated in English at [arXiv:0809.1504](http://front.math.ucdavis.edu/0809.1504).

for early work; and also in a bit different setup recent

* [[Alexander Rosenberg|A. Rosenberg]], [Homological algebra of noncommutative 'spaces' I](http://www.mpim-bonn.mpg.de/preprints/send?bid=3623), 199 pages, preprint Max Planck, Bonn: MPIM2008-91.

or for a shorter presentation, Ch. 3 or so of the survey

* [[Alexander Rosenberg|A. Rosenberg]], _Topics in noncommutative algebraic geometry, homological algebra and K-theory_, preprint MPIM Bonn 2008-57 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3589)

[[!redirects satellites]]


Sato Grassmannian

[[Mikio Sato]] introduced an infinite-dimensional Grassmannian in relation to the [[integrable system]]s. It gives a standard way to describe the $\tau$-[[tau-function|function]]. Constructed also by [[Graeme Segal]] and Wilson so it is often called Sato-Segal-Wilson Grassmannian.

* [[Mikio Sato|M. Sato]], _The KP hierarchy and infinite dimensional Grassmann manifolds_, Proc. Sympos. Pure Math. 49 Part 1. (pp. 51–66) Amer. Math. Soc. 1989.
* [[Graeme Segal]], [[George Wilson]], _Loop groups and equations of KdV type_, Publ. Math. IHÉS __61__, 5--65 (1985) [numdam](http://www.numdam.org/item/PMIHES_1985__61__5_0) [MR87b:58039](https://mathscinet.ams.org/mathscinet-getitem?mr=87b:58039) [Zbl0592.35112](https://zbmath.org/?q=an:0592.35112)

Textbooks include

* [[Andrew Pressley]], [[Graeme Segal]], _Loop groups_, Clarendon Press 1989

* T. Miwa, M. Jimbo, E. Date, _Solitons: differential equations, symmetries and infinite dimensional algebras_ (translated from Japanese by Miles Reid) Cambridge Tracts in Math. 135, 120 pp.

Other references

* [[Mattia Cafasso]], Chao-Zhong Wu, _Tau functions and the limit of block Toeplitz determinants_ [arxiv/1404.5149](https://arxiv.org/abs/1404.5149)

> A classical way to introduce tau functions for integrable hierarchies of solitonic equationsis by means of the Sato–Segal–Wilson infinite-dimensional Grassmannian. Every point in the Grassmannian is naturally related to a [[Riemann–Hilbert problem]] on the unit circle, for which Bertola proposed a tau function that generalizes the Jimbo–Miwa–Ueno tau function for isomonodromic deformation problems. In this paper, we prove that the Sato–Segal–Wilson tau function and the (generalized) Jimbo–Miwa–Ueno isomonodromy tau function coincide under a very general setting, by identifying each of them to the large-size limit of a block Toeplitz determinant. As an application, we give a new definition of tau function for Drinfeld–Sokolov hierarchies (and their generalizations) by means of infinite-dimensional Grassmannians, and clarify their relation with other tau functions given in the literature.

* Dinakar Muthiah, Alex Weekes, Oded Yacobi, _The equations defining affine Grassmannians in type A and a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman_ [arxiv/1708.06076](https://arxiv.org/abs/1708.06076)

> The affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a [[Plücker embedding]] into the projectivization of Fermion Fock space...

* Maurice J. Dupré, James F. Glazebrook, [[Emma Previato]], _A Banach algebra version of the Sato Grassmannian and commutative rings of differential operators_ Acta Appl Math (2006) 92: 241 [doi](https://doi.org/10.1007/s10440-006-9052-3); _On Banach bundles and operator-valued Baker functions_, [pdf](http://mittag-leffler.se/sites/default/files/IML-0607-12.pdf)

* [[Emma Previato]], Mauro Spera, _Isometric embeddings of infinite-dimensional Grassmannians_, Regul. Chaot. Dyn. (2011) 16: 356 [doi](https://doi.org/10.1134/S1560354711030099)
* M. J. Dupré, J. F. Glazebrook, E. Previato, _A Banach algebra version of the Sato Grassmannian and commutative rings of differential operators_ Acta Appl. Math. __92__, 241--267 (2006) [doi](https://doi.org/10.1007/s10440-006-9052-3)

[[!redirects Segal-Wilson Grassmannian]]
[[!redirects Sato-Segal-Wilson Grassmannian]]
[[!redirects Sato Grassmanian]]


Satoshi Kobayashi

* [researchmap page](https://researchmap.jp/ksu-kbys?lang=en)

## Selected writings

On [[modal operators]] in [[S4 modal logic|S4]]/[[S5 modal logic|S5]] [[modal logic]] regarded as ([[comonad|co]]-)[[monads]]:

* {#Kobayashi97} [[Satoshi Kobayashi]], *Monad as modality*, Theoretical Computer Science **175** 1 (1997) 29-74 [doi:10.1016/S0304-3975(96)00169-7]

category: people


Satoshi Tomoda

* [Institute page](https://www.okanagan.bc.ca/science/math-and-statistics)

* [Personal page](http://people.okanagan.bc.ca/STomoda/)

## Selected writings

On [[fundamental groups]] of [[3-manifolds]] and the [[group cohomology]] of [[finite rotation groups]]:

* {#TomodaZvengrowski08} [[Satoshi Tomoda]], [[Peter Zvengrowski]], *Remarks on the cohomology of finite fundamental groups of 3-manifolds*, Geom. Topol. Monogr. 14 (2008) 519-556 ([arXiv:0904.1876](https://arxiv.org/abs/0904.1876))

category: people


saturated class of limits

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#### Limits and colimits
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# Saturated classes of limits
* table of contents
{:toc}

## Idea

A class $\mathcal{X}$ of [[limits]] is *saturated* if it is closed under the "construction" of other limits out of limits in $\mathcal{X}$.

## Definition

Let $V$ be a [[Bénabou cosmos]], and let $\mathcal{X}$ be a [[class]] of $V$-functors $\Phi\colon D\to V$ where $D$ is a [[small category|small]] $V$-category.  Note that the domain $D$ may be different for different elements of $\mathcal{X}$.

Any such $\Phi\colon D\to V$ can serve as a *weight* for defining the [[weighted limit]] of a $V$-functor $D\to C$, for any $V$-category $C$.  We say that a $V$-category is **$\mathcal{X}$-complete** if it admits all such limits, for all $\Phi\in\mathcal{X}$, and that a $V$-functor between $\mathcal{X}$-complete $V$-categories is **$\mathcal{X}$-continuous** if it preserves all such limits.

The **saturation** of $\mathcal{X}$ is the class of all weights $\Phi\colon D\to V$ (with $D$ small) such that

1. Any $\mathcal{X}$-complete $V$-category admits $\Phi$-weighted limits, and
1. Any $\mathcal{X}$-continuous $V$-functor preserves $\Phi$-weighted limits.

It is an open question whether the second condition is implied by the first in general.

Finally, $\mathcal{X}$ is **saturated** if it is its own saturation.

### The conical case

When $V=Set$, we frequently discuss only *conical* limits, i.e. limits whose weight $\Phi\colon D\to Set$ is the constant functor $\Delta_D 1$ at the terminal set.  These give the classical notion of [[limit]] in a [[category]].

In this case, we may consider instead classes $\mathcal{J}$ of small categories; we write $\Delta_{\mathcal{J}}$ for the class of weights $\{ \Delta_C 1 | C \in \mathcal{J}\}$.  We say that a category $D$ lies in the **saturation** of $\mathcal{J}$ if the weight $\Delta_D 1$ lies in the saturation of $\Delta_{\mathcal{J}}$, and that $\mathcal{J}$ is **saturated** if it is its own saturation.

Note that in practically all cases, the saturation of $\Delta_{\mathcal{J}}$ will contain weights that are not of the form $\Delta_D 1$.  Moreover, even when $V=Set$ there are nontrivial saturated classes of *weights* that do not contain any nontrivial *conical* weights, such as the saturation of the weight for "cartesian squares" $A\times A$.

However, for conical weights the answer to the above open question is known to be affirmative.  On the one hand, if $\mathcal{X}$ is a class of $Set$-weights such that every $\mathcal{X}$-complete category is also $\Delta_D 1$-complete, then every $\mathcal{X}$-continuous functor is also $\Delta_D 1$-continuous.  See [AK](#AK) for a proof of this.

On the other hand, if $\mathcal{J}$ is a class of $Set$-categories and $\Phi$ is a $Set$-weight such that every $\Delta_{\mathcal{J}}$-complete category is also $\Phi$-complete, then every $\Delta_{\mathcal{J}}$-continuous functor is also $\Phi$-continuous.  In fact, this is still true if instead of $\Delta_{\mathcal{J}}$ we consider a class of weights all of which take only nonempty sets as values.  See [KP](#KP) for a proof of this.

## Characterization

The main theorem of [AK](#AK) (which introduced the notion under the name "closure") is the following.

+-- {: .un_theorem}
###### Theorem
$\Phi\colon D\to V$ lies in the saturation of $\mathcal{X}$ if and only if it lies in the closure of the [[representable functor|representables]] under $\mathcal{X}$-weighted colimits in $[D,V]$.
=--

## Examples

The following examples are all for $V=Set$, restricted to the conical case.

* The class of small [[products]] is not saturated; its saturation includes all $\Delta_D 1$ where $D$ has local initial objects. Similarly, the class of finite products is not saturated; the saturation of this coincides with the saturation of the finite class containing only [[terminal objects]] and binary products.

* The class of [[L-finite limits]] is saturated with respect to conical weights (but not in the sense of the definition above); it is the saturation of the class of [[finite limits]].  It is also the saturation of the finite class containing only terminal objects and [[pullbacks]], and the saturation of the class containing only finite products and [[equalizers]].

* The class of [[connected limits]] is saturated with respect to conical weights. It is the saturation of the class consisting of [[wide pullbacks]] and equalizers.  Similarly, the class of L-finite connected limits is the saturation of the finite class of pullbacks and equalizers.  See also [[pullback]] and [[wide pullback]] for their saturations.

There are also interesting examples for other $V$.

* When $V=Cat$, the classes of [[PIE-limits]] and [[flexible limits]] are saturated.  The former is, essentially by definition, the saturation of the class containing products, [[inserters]], and [[equifiers]].  The latter can be proven to be the saturation of the class containing products, inserters, equifiers, and [[splitting of idempotents]].

* When $V=F$ is the category of [[fully faithful functors]], so that a $V$-category is an [[F-category]], the class of $w$-[[rigged weight|rigged weights]] is saturated (for any of $w=p$, $l$, or $c$ denoting pseudo, lax, or colax).

* For any $V$, the class of [[absolute colimits]] is saturated.  When $V=Set$, this is the saturation of the [[split idempotent|splitting of idempotents]].

It is also worth mentioning some non-examples.

* For $V=Set$, the class of [[finite limits]] is not saturated; its saturation is the class of [[L-finite limits]].

* For $V=Cat$, the class of [[strict pseudo-limits]] is not saturated; it does not even contain the representables.  (The same is true for strict lax limits.)  It is unclear precisely what its saturation looks like.

## References

* Albert and [[Max Kelly|Kelly]], "The closure of a class of colimits", J. Pure. App. Alg. 51 (1988), 1--17
 {#AK}

* [[Max Kelly]] and [[Robert Paré]], "A note on the Albert-Kelly paper 'The closure of a class of colimits'", JPAA 51 (1988), 19--25
 {#KP}

[[!redirects saturated class of limits]]
[[!redirects saturated classes of limits]]
[[!redirects saturated class of colimits]]
[[!redirects saturated classes of colimits]]
[[!redirects saturated class of weights]]
[[!redirects saturated classes of weights]]
[[!redirects saturation of a class of weights]]
[[!redirects saturation of a class of limits]]
[[!redirects saturation of a class of colimits]]


saturated class of maps

In the context of [[cofibrantly generated model category]] theory one says that a collection $I$ of [[morphism]]s in a [[category]] $C$ with [[colimit]]s is _saturated_ or _weakly saturated_ if it is closed under the operations of forming

1. [[pushouts]]

1. [[transfinite composition]]

1. [[retracts]] (in the [[arrow category]] $Arr(C)$)

[[!redirects saturated classes of maps]]

[[!redirects saturated class of morphisms]]
[[!redirects saturated classes of morphisms]]

[[!redirects weakly saturated class of morphisms]]
[[!redirects weakly saturated classes of morphisms]]

[[!redirects weakly saturated class of maps]]
[[!redirects weakly saturated classes of maps]]

[[!redirects weak saturation]]


saturated set

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=--
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=--

## Definition

A [[subset]] $S$ of a [[topological space]] $X$ is called **saturated** if it is the intersection of its [[open neighborhoods]].

## Properties

* If $X$ is [[T1]], every subset of $X$ is saturated. In particular, every [[singleton]] subset of a [[T1]] space is the [[intersection]] of its [[open neighborhoods]].

* $X$ is not [[T1]] precisely if there exists a non-saturated singleton.

## See also

* [[open set]], [[closed set]]
* [[neighborhood filter]]


saturated subset

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# Contents
* table of contents
{: toc}

## Definition

The following terminology is used in [[topology]]:

+-- {: .num_defn #SubsetSaturated}
###### Definition
**(saturated subset)**

Let $f \;\colon\; X \longrightarrow Y$ be a [[function]] of [[sets]].
Then a [[subset]] $S \subset X$ is called an _$f$-[[saturated subset]]_ (or just _saturated subset_, if $f$ is understood)
if $S$ is the [[pre-image]] of its [[image]]:

$$
  \left(S \subset X \,\, f\text{-saturated} \right)
    \,\Leftrightarrow\,
  \left(
    S = f^{-1}(f(S))
  \right)
  \,.
$$

If $S$ is not necessarily $f$-saturated, then $f^{-f}(f(S))$ is called its _saturation_. Notice that $S \subset f^{-1}(f(S))$.

=--

## Properties

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

Let $f \colon X \longrightarrow Y$ be a [[function]]. Then a [[subset]] $S \subset X$
is $f$-saturated (def. \ref{SubsetSaturated}) precisely if its [[complement]]
$X \backslash S$ is so.

=--

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

A [[continuous function]]

$$
  f \;\colon\; (X, \tau_X) \longrightarrow (Y,\tau_Y)
$$

whose underlying function $f \colon X \longrightarrow Y$ is [[surjective function|surjective]]
exhibits $\tau_Y$ as the corresponding [[quotient topological space|quotient topology]]
precisely if $f$ sends open and $f$-[[saturated subsets]] in $X$ (def. \ref{SubsetSaturated}) to open subsets of $Y$.
By lemma \ref{ComplementOfSaturatedSubsetIsSaturated} this is the case precisely if it sends
closed and $f$-saturated subsets to closed subsets.

=--

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

Let

1. $f \;\colon\; (X, \tau_X) \longrightarrow (Y, \tau_Y)$ be a [[closed map]].

1. $C \subset X$ be a [[closed subset]] which is $f$-[[saturated subset|saturated]];

1. $U \supset C$ an [[open subset]] containing $C$

then there exists a smaller open subset $V$ still containing $C$

$$
  U \supset V \supset C
$$

and such that $V$ is $f$-[[saturated subset|saturated]].

=--

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

We claim that

$$
  V \coloneqq X \backslash \left(  f^{-1}\left( f\left(  X \backslash U \right)  \right) \right)
$$

has the desired properties. To see this, observe first that

1. the [[complement]] $X \backslash U$ is closed, since $U$ is assumed to be open;

1. hence the image $f(X\backslash U)$ is closed, since $f$ is assumed to be a closed map;

1. hence the pre-image $f^{-1}\left( f\left(  X \backslash U \right)\right)$ is closed, since $f$ is continuous, therefore its complement $V$ is indeed open;

1. this pre-image $f^{-1}\left( f\left(  X \backslash U \right) \right)$ is saturated, as all pre-images clearly are,
   and hence also its complement $V$ is saturated, by lemma \ref{ComplementOfSaturatedSubsetIsSaturated}.

Therefore it now only remains to see that $U \supset V \supset C$.

The inclusion $U \supset V$ means equivalently that $f^{-1}\left( f\left(  X \backslash U \right)\right) \supset X \backslash U$,
which is clearly the case.

The inclusion $V \supset C$ meas that $f^{-1}\left( f\left(  X \backslash U \right) \right) \,\cap \, C = \emptyset$.
Since $C$ is saturated by assumption, this means that $f^{-1}\left( f\left(  X \backslash U \right)\right) \,\cap \, f^{-1}(f(C)) = \emptyset$.
This in turn holds precisely if $f\left(  X \backslash U \right) \,\cap \, f(C)  = \emptyset$.
Since $C$ is saturated, this holds precisely if $X \backslash U \cap C = \emptyset$, and this is true by the assumption
that $U \supset C$.

=--

## Applications

* [[quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff]]

[[!redirects saturated subset]]
[[!redirects saturated subsets]]


Saul Glasman

* [personal page](http://www.math.ias.edu/~sglasman/) (down)

* [MathGenealogy page](https://genealogy.math.ndsu.nodak.edu/id.php?id=191840)

## Related entries

* [[cyclotomic spectrum]]

* [[spectral Mackey functor]]

  * [[equivariant stable homotopy theory]]

  * [[Goodwillie calculus]]

category: people