Michael Johnson

* [website](http://web.science.mq.edu.au/~mike/)

* [ORC-ID](https://orcid.org/0000-0001-9858-5542)

### Selected writings

On the [[nerve]] of [[n-categories]]:

* [[Michael Johnson]] and [[R. F. C. Walters]], _On the nerve of an n-category_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 28, No. 4, 1987, pp. 257-282. [[numdam](http://www.numdam.org/item/CTGDC_1987__28_4_257_0/)]

On the [[combinatorics]] of n-categorical [[pasting]]:

* [[Michael Johnson]], _Pasting Diagrams in n-Categories with Applications to Coherence Theorems and Categories of Paths_, PhD thesis, University of Sydney (1987) [[pdf](http://web.science.mq.edu.au/~mike/papers/0.pdf)]

* [[Michael Johnson]], _The combinatorics of n-categorical pasting_, Journal of Pure and Applied Algebra **62** 3 (1989) 211-225 [doi:10.1016/0022-4049(89)90136-9]

On representations of [[universal algebras|universal algebra]] by [[sheaves]] of [[quotient algebras]]:

* [[Michael Johnson]] and Shu-Hao Sun, _Remarks on representations of universal algebras by sheaves of quotient algebras_, Proceedings of the 1991 Summer Category Theory Meeting (1992) [pdf]

On well-behaved [[lenses (in computer science)]] as the [[coalgebra over a comonad|coalgebras]] over the [[costate comonad]]:

* {#GibbonsJohnson12} [[Jeremy Gibbons]], [[Michael Johnson]], *Relating Algebraic and Coalgebraic Descriptions of Lenses*, Electronic Communications of the EASST **49** (2012) [[doi:10.14279/tuj.eceasst.49.726](http://dx.doi.org/10.14279/tuj.eceasst.49.726), [pdf](http://www.cs.ox.ac.uk/jeremy.gibbons/publications/colens.pdf)]

On [[lenses (in computer science)]]:

* {#JohnsonRosebrughWood10} [[Michael Johnson]], [[Robert Rosebrugh]], [[Richard Wood]], _Algebras and Update Strategies_, Journal of Universal Computer Science **16** (2010) [[doi:10.3217/jucs-016-05-0729](http://dx.doi.org/10.3217/jucs-016-05-0729)]

* [[Michael Johnson]], [[Robert Rosebrugh]], and  [[Richard J. Wood]], *Lenses, fibrations and universal translations* Math. Structures Comput. Sci. **22** 1 (2012) 25–42 [[doi:10.1017/S0960129511000442](https://doi.org/10.1017/S0960129511000442)]

* {#JohnsonRosebrugh16} [[Michael Johnson]], [[Robert Rosebrugh]], *Unifying Set-Based, Delta-Based and Edit-Based Lenses*, CEUR Workshop Proceedings **1571** (2016) [[pdf](http://ceur-ws.org/Vol-1571/paper_13.pdf)]

* [[Michael Johnson]], [[Robert Rosebrugh]], *The more legs the merrier: A new composition for symmetric (multi-)lenses*, EPTCS **333** (2021) 92-107 [[arXiv:2101.10482](https://arxiv.org/abs/2101.10482), [doi:10.4204/EPTCS.333.7](https://doi.org/10.4204/EPTCS.333.7)]

category: people


Michael Kapovich

* [webpage](https://www.math.ucdavis.edu/~kapovich/)

## Selected writings

On [[hyperbolic manifolds]] (and [[orbifolds]])

* [[Michael Kapovich]], _Hyperbolic Manifolds and Discrete Groups_,  Modern Birkhäuser Classics, Birkhäuser 2008 ([doi:10.1007/978-0-8176-4913-5](https://link.springer.com/book/10.1007/978-0-8176-4913-5))

On [[geometric group theory]]:

* [[Cornelia Druţu]], [[Michael Kapovich]] (appendix by [[Bogdan Nica]]), *Geometric group theory*, Colloquium Publications **63**, AMS 2018 ([ISBN:978-1-4704-1104-6](https://bookstore.ams.org/coll-63/) [pdf](https://courses.maths.ox.ac.uk/node/view_material/35649))

category: people


Michael Larsen

* [webpage](http://mlarsen.math.indiana.edu/~larsen/)

## Selected writings

On [[topological quantum computation]] with [[anyons]]:

* [[Michael Freedman]], [[Alexei Kitaev]], [[Michael Larsen]], [[Zhenghan Wang]], _Topological quantum computation_,  Bull. Amer. Math. Soc. __40__ (2003), 31-38 ([arXiv:quant-ph/0101025](https://arxiv.org/abs/quant-ph/0101025), [doi:10.1090/S0273-0979-02-00964-3](https://doi.org/10.1090/S0273-0979-02-00964-3), [pdf](http://www.ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3/S0273-0979-02-00964-3.pdf))

On [[braiding]] of [[su(2)-anyons]] providing universal [[quantum gates]] for [[topological quantum computation]]:

* {#FreedmanLarsenWang00} [[Michael Freedman]], [[Michael Larsen]], [[Zhenghan Wang]], _A modular functor which is universal for quantum computation_, Communications in Mathematical Physics **227** (2002) 605–622 [[arXiv:quant-ph/0001108](http://arxiv.org/abs/quant-ph/0001108), [doi:10.1007/s002200200645](https://doi.org/10.1007/s002200200645)]

category: people


Michael Lesnick

* [personal page](https://www.albany.edu/~ml644186/)

## Selected writings

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

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

* [[Ulrich Bauer]], [[Michael Lesnick]], *Induced Matchings and the Algebraic Stability of Persistence Barcodes*, Journal of Computational Geometry **6** 2 (2015) 162-191 $[$[arXiv:1311.3681](https://arxiv.org/abs/1311.3681), [doi:10.20382/jocg.v6i2a9](https://doi.org/10.20382/jocg.v6i2a9)$]$

On [[persistent homotopy theory]] and its [[stability of persistence diagrams|stability theorem]]:

* {#BlumbergLesnick17} [[Andrew J. Blumberg]], [[Michael Lesnick]], *Universality of the Homotopy Interleaving Distance* $[$[arXiv:1705.01690](https://arxiv.org/abs/1705.01690)$]$

category: people


Michael Levin

> This page is about the [[mathematical physics|mathematical physicist]] Michael levin, while there is also a [[biology|biologist]] of that name, see the entry *[[Michael Levin (biologist)]]*. (Both authors publish on [[quantum systems]].)

***

* [Institute page](https://physics.uchicago.edu/people/profile/michael-levin)

## Selected writings

Introducing [[string-net models]] and the [[Levin-Wen model]]:

* [[Michael Levin]], [[Xiao-Gang Wen]], _String-net condensation: A physical mechanism for topological phases_, Phys.Rev. B71 (2005) 045110 ([arXiv:cond-mat/0404617](http://arxiv.org/abs/cond-mat/0404617))

Introducing [[topological entanglement entropy]] in view of [[string-net models]]:

* [[Michael Levin]], [[Xiao-Gang Wen]], *Detecting topological order in a ground state wave function*, Phys. Rev. Lett., 96, 110405 (2006) ([arXiv:cond-mat/0510613](https://arxiv.org/abs/cond-mat/0510613))

On [[string-net models]] via [[tensor networks]]:

* [[Zheng-Cheng Gu]], [[Michael Levin]], [[Brian Swingle]], [[Xiao-Gang Wen]], *Tensor-product representations for string-net condensed states*, Phys. Rev. B **79** (2009) 085118 $[$[doi:10.1103/PhysRevB.79.085118](https://doi.org/10.1103/PhysRevB.79.085118)$]$

On [[braid group]]-effects in 2d [[symmetry protected topological phases]]:

* [[Michael Levin]], [[Zheng-Cheng Gu]], *Braiding statistics approach to symmetry-protected topological phases*, Phys. Rev. B **86** (2012) 115109 ([doi:10.1103/PhysRevB.86.115109](https://doi.org/10.1103/PhysRevB.86.115109))

On [[loop braid group]]-[[representations]] in [[model (in theoretical physics)|models]] of [[solid state physics]]:

* [[Chenjie Wang]], [[Michael Levin]], *Braiding Statistics of Loop Excitations in Three Dimensions*, Phys. Rev. Lett. **113** (2014) 080403 ([doi:10.1103/PhysRevLett.113.080403](https://doi.org/10.1103/PhysRevLett.113.080403))

On [[string-net models]]:

* Chien-Hung Lin, [[Michael Levin]], Fiona J. Burnell, *Generalized string-net models: A thorough exposition*, Phys. Rev. B **103** (2021) 195155  [[arXiv:2012.14424](https://arxiv.org/abs/2012.14424), [doi:10.1103/PhysRevB.103.195155](https://doi.org/10.1103/PhysRevB.103.195155)]

category:people

[[!redirects Michael Levin (physicist)]]


Michael Levin (biologist)

> This page is about the [[biology|biologist]] Michael Levin, while there is also a [[mathematical physics|mathematical physicist]] of that name, see the entry *[[Michael Levin]]*. (Both authors publish on [[quantum systems]].)

***

* [Institute page](https://as.tufts.edu/biology/people/faculty/michael-levin) at Tufts University.

##Selected writings##

On [[quantum biology]], arguing that cellular [[information theory|information processing]] must employ [[quantum coherence]] as a resource for [[reversible computation|reversibility]] in order to maintain a biologically reasonable [[free energy]] budget:

* [[Chris Fields]], [[Michael Levin (biologist)|Michael Levin]], _Metabolic limits on classical information processing by biological cells_ Biosystems **209** (2021) 104513 [[doi:10.1016/j.biosystems.2021.104513](https://doi.org/10.1016/j.biosystems.2021.104513)]

category: people


Michael Lindgren

* [webpage](https://www.kth.se/en/2.82512/kalender/examensarbete/michael-lindgren-dependent-products-as-relative-adjoints-1.1113011?date=2021-10-29&orgdate=2021-09-04&length=1&orglength=119)

## Selected writings

On the [[categorical semantics]] of [[dependent product types]] as *[[relative adjoint functor|relative]]* [[right adjoints]] to [[context extension]] in [[comprehension categories]]:

* [[Michael Lindgren]], *Dependent products as relative adjoints*, Stockholm (2021) [[[Lindgren-DependentProductsAsRelativeAdjoints.pdf:file]]]

## Related entries

* [[UniMath project]]

category: people


Michael M. Wolf

* [institute page](https://www.math.cit.tum.de/en/math/people/professors/wolf-michael/)

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

## Selected writings

On [[extremal quantum channels]]:

* [[Christian B. Mendl]], [[Michael M. Wolf]], *Unital Quantum Channels -- Convex Structure and Revivals of Birkhoff's Theorem*, Commun. Math. Phys. **289**  (2009) 1057-1096 [[arXiv:0806.2820](http://arxiv.org/abs/0806.2820)]

category: people

[[!redirects Michael Wolf]]


Michael Makkai

__Michael Makkai__ (Hungarian original __Mihály Makkai__) is a Canadian mathematician born in Hungary; he studied in Budapest and Warsaw. He is the inventor of [[anafunctors]] and [[FOLDS]], which are both ways of doing [[category theory]] and [[higher category theory]] without violating the [[principle of equivalence]] or requiring the [[axiom of choice]].  Makkai is a pioneer in the [[category theory|category-theoretic]] approach to [[logic]] (which is mostly the approach that you see here on the $n$-Lab).

* [homepage](http://www.math.mcgill.ca/makkai)

* [Mathematics Genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=62420)

* _Models, logics, and higher-dimensional categories  A tribute to the work of Mihály Makkai_ Proceedings of the meeting held at the Université de Montréal, June 18–20, 2009. Edited by Bradd Hart, Thomas G. Kucera, Anand Pillay, Philip J. Scott and Robert [[R.A.G. Seely]]. CRM Proceedings & Lecture Notes __53__,  Amer. Math. Soc. 2011. x+426 pp.

## Selected writings

On [[accessible categories]] and [[categorical semantics|categorical]] [[model theory]] via [[sketches]]:

* {#MakkaiPare} [[Michael Makkai]], [[Robert Paré]],  _Accessible categories: The foundations of categorical model theory_,  Contemporary Mathematics 104. American Mathematical Society, Rhode Island (1989) ([ISBN:978-0-8218-7692-3](https://bookstore.ams.org/conm-104))

Introducing [[cellular categories]]:

* {#MakkaiRosický14} [[M. Makkai]], [[J. Rosický]], _Cellular categories_, J. Pure Appl. Alg. **218** (2014) 1652-1664 [[arXiv:1304.7572](https://arxiv.org/abs/1304.7572), [doi:10.1016/j.jpaa.2014.01.005](https://doi.org/10.1016/j.jpaa.2014.01.005)]

## Related $n$Lab entries

* [[FOLDS]]

* [[opetopic omega-category]]

* [[Makkai duality]]

* [[conceptual completeness]]

category: people

[[!redirects Mihály Makkai]]
[[!redirects M. Makkai]]
[[!redirects Mihaly Makkai]]


Michael Mandell

* [website](http://mypage.iu.edu/~mmandell/)

## Selected writings

On [[higher algebra]] ([[brave new algebra]]) in [[stable homotopy theory]], i.e. on [[ring spectra]], [[module spectra]] etc.:

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], _[[Rings, modules and algebras in stable homotopy theory]]_, Mathematical Surveys and Monographs Volume 47, AMS 1997 ([ISBN:978-0-8218-4303-1](https://bookstore.ams.org/surv-47-s), [pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))

On the [[equivariant stable homotopy theory]] of [[cyclotomic spectra]]:

* {#BlumbergMandell13} [[Andrew Blumberg]], [[Michael Mandell]], _The homotopy theory of cyclotomic spectra_, Geom. Topol. **19** (2015) 3105-3147 [[arXiv:1303.1694](http://arxiv.org/abs/1303.1694), [doi:10.2140/gt.2015.19.3105](https://doi.org/10.2140/gt.2015.19.3105)]

category: people

[[!redirects Mike Mandell]]


Michael Mendler

* [webpage](https://www.uni-bamberg.de/gdi/team/michael-mendler/)

## Related entries

* [[computational type theory]]

category: people


Michael Mertens

* [website](http://www.mi.uni-koeln.de/~mmertens/)

category: people


Michael Mislove

* [institute page](https://mislove.wp.tulane.edu/)

* [GoogleScholar page](https://scholar.google.com/citations?user=tMa-iJ8AAAAJ&hl=en)

## Selected writings

On the [[convex powerset of distributions monad]] in [[domain theory]]:

* [[Michael W. Mislove]], *Nondeterminism and probabilistic choice: Obeying the laws*, in: Proc. CONCUR 2000, Lecture Notes in Computer Science **1877**, Springer (2000) 350–364 [[doi:10.1007/3-540-44618-4\_26](https://doi.org/10.1007/3-540-44618-4_26)].

On [[domain theory]] and [[Baire lattices]]:

 * G. Gierz, [[Karl H. Hofmann]], K. Keimel, J. D. Lawson, [[Michael W. Mislove]], [[Dana S. Scott]], *Continuous Lattices and Domains* in *Encyclopedia of Mathematics and its Applications* 93, Cambridge University Press (2003)  [[doi:10.1017/CBO9780511542725](https://doi.org/10.1017/CBO9780511542725)]

Introducing a notion of [[quantum CPOs]] for [[quantum computation]] (via [[quantum sets]] carrying [[quantum relations]]):

* [[Andre Kornell]], [[Bert Lindenhovius]], [[Michael Mislove]], *Quantum CPOs*, EPTCS **340** (2021) 174-187 [[arXiv:2109.02196](https://arxiv.org/abs/2109.02196), [doi:10.4204/EPTCS.340.9](https://doi.org/10.4204/EPTCS.340.9)]

category: people

[[!redirects Michael W. Mislove]]


Michael Murray

* [website](http://www.maths.adelaide.edu.au/michael.murray/)

* [publications](http://www.maths.adelaide.edu.au/michael.murray/publications.html)

## Selected writings

On [[bundle gerbes]]:

* [[Michael Murray]], _Bundle gerbes_, J. Lond. Math. Soc. **54** (1996) pp. 403-416 ([arXiv:dg-ga/9407015](http://arxiv.org/abs/dg-ga/9407015))

On identifying the [[moduli space of Yang-Mills monopoles]] for [[gauge group]] any [[classical Lie group]] with a space of [[rational maps]], namely [[holomorphic maps]] from the [[Riemann sphere]] to a [[flag variety]]:

* [[Jacques Hurtubise]], [[Michael K. Murray]], *On the construction of monopoles for the classical groups*,  Comm. Math. Phys. 122(1): 35-89 (1989) ([euclid:cmp/1104178316](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-122/issue-1/On-the-construction-of-monopoles-for-the-classical-groups/cmp/1104178316.full))

* [[Michael Murray]], *Stratifying monopoles and rational maps*, Commun. Math. Phys. 125, 661–674 (1989) ([doi:10.1007/BF01228347](https://doi.org/10.1007/BF01228347))

* [[Jacques Hurtubise]], [[Michael K. Murray]], *Monopoles and their spectral data*, Comm. Math. Phys. 133(3): 487-508 (1990) ([euclid:cmp/1104201504](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-133/issue-3/Monopoles-and-their-spectral-data/cmp/1104201504.full))

Discussion of [[twisted cohomology|twists]] of [[KR-theory]] by [[HZR-theory]] in degree 3 via [[bundle gerbes]] ([[Jandl gerbes]]) suitable for classifying [[D-brane charge]] on [[orientifolds]]:

* [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Real bundle gerbes, orientifolds and twisted KR-homology_, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number  ([arXiv:1608.06466](http://arxiv.org/abs/1608.06466), [doi:10.4310/ATMP.2019.v23.n8.a5](https://dx.doi.org/10.4310/ATMP.2019.v23.n8.a5))

* [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Sign choices for orientifolds_ ([arXiv:1905.06041](https://arxiv.org/abs/1905.06041))

## Related entries

* [[bundle gerbe]]

* [[Chern-Simons 2-gerbe]]

category: people

[[!redirects Michael K. Murray]]
[[!redirects Michael K Murray]]


Michael Müger

* [website](http://www.math.ru.nl/~mueger/)

## Selected writings

On [[Drinfeld centers]] and [[von Neumann algebra factors]]:

* {#Mueger03} [[Michael Mueger]], *From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors*, J. Pure Appl. Alg. 180, 159-219 (2003) ([arXiv:math/0111205](https://arxiv.org/abs/math/0111205))

On [[general topology|general]] [[topology]]

* {#Mueger18} [[Michael Müger]], _Topology for the working mathematician_, Nijmegen 2018 ([pdf](https://www.math.ru.nl/~mueger/topology.pdf))

## Related $n$Lab entries

* [[spherical category]]

* [[modular tensor category]]

* [[conformal field theory]]

category: people

[[!redirects Michael Mueger]]


Michael Oberguggenberger

* [webpage](https://www.uibk.ac.at/techmath/michael/)

## related $n$Lab entries

* [[product of distributions]]

category: people


Michael P. Frank

* [personal page](https://web1.eng.famu.fsu.edu/~mpf/)

## References

On [[reversible computation]] in view of [[quantum computation]]:

* {#Frank03} [[Michael P. Frank]], *Reversible Computing -- Quantum Computing’s Practical Cousin*, Simons Conference Lecture, Stony Brook (2003) [[pdf](http://insti.physics.sunysb.edu/conf/simons-qcomputation/talks/frank.pdf), [[Frank-ReversibleComputing.pdf:file]]]

* [[Michael P. Frank]], Karpur Shukla, *Quantum Foundations of Classical Reversible Computing*, Entropy **23** 6 (2021) 701  [[arXiv:2105.00065](https://arxiv.org/abs/2105.00065), [doi:10.3390/e23060701](https://doi.org/10.3390/e23060701)]

category: people

[[!redirects Michael Frank]]


Michael Penkava

* [webpage](https://people.uwec.edu/penkavmr/penkavmr.htm)

## related $n$Lab entries

* [[polynomial Poisson algebra]]

* [[formal deformation quantization]]

category: people


Michael Peskin

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

## Selected writings

On [[perturbative quantum field theory]]:

* [[Michael Peskin]], [[Daniel Schroeder]], _An Introduction to Quantum Field Theory_, 1995 ([spire:407703](https://inspirehep.net/literature/407703), [ISBN 9780201503975](https://www.routledge.com/An-Introduction-To-Quantum-Field-Theory/Peskin-Schroeder/p/book/9780201503975))

On precision theory of the [[electroweak field|electroweak interaction]]:

* [[Michael Peskin]], _Precision Theory of Electroweak Interactions_ ([arXiv:2003.05433](https://arxiv.org/abs/2003.05433))

## Related $n$Lab entries

* [[quantum field theory]]

* [[GUT]]

category: people


Michael R. Mather

> Not to be confused with [[John Norman Mather]] of [[Mather's stability theorem]].

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=19528)

Michael R. Mather was a mathematician at the University of Toronto.
He got his PhD in 1966 from the University of Cambridge, advised by [[David Epstein]].

## Selected writings

On [[partitions of unity]]:

* {#Mather65} [[Michael R. Mather]], _Paracompactness and partitions of unity_, PhD thesis, Cambridge (1965).

category: people

[[!redirects Michael Mather]]


Michael Rapoport

* [webpage](http://www.math.uni-bonn.de/ag/alggeom/rapoport)

## Selected writings

On [[elliptic curves]] over general [[commutative ring|commutative]] [[ground rings]] (in [[arithmetic geometry]]):

* [[Pierre Deligne]], [[Michael Rapoport]], _Les Schémas de Modules de Courbes Elliptiques_ In: Deligne P., Kuijk W. (eds) _Modular Functions of One Variable II_, Lecture Notes in Mathematics, vol 349. Springer (1973) ([doi:10.1007/978-3-540-37855-6_4](https://doi.org/10.1007/978-3-540-37855-6_4))

On [[Deligne cohomology]] and [[Beilinson's conjectures]] on [[special values of L-functions]]:

* [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_  Perspectives in Mathematics, Volume 4, Academic Press, Inc. 1988 (ISBN:978-0-12-581120-0)

category: people


Michael Rathjen

Michael Rathjen is professor for [[mathematical logic]] at Leeds.

* [webpage](http://www1.maths.leeds.ac.uk/~rathjen/rathjen.html)

* Research grant _[Homotopical Inductive Types](http://www2.macs.hw.ac.uk/~cm389/hexmaps/2014/03/epsrc-ict-50/grants/EP-K023128-1.php)_ on [[higher inductive types]]

## Selected writings

On [[constructive set theory]]:

* {#AczelRathjen01} [[Peter Aczel]], [[Michael Rathjen]], _Notes on Constructive Set theory_, 2001 ([pdf](https://events.math.unipd.it/3wftop/pdf/AczelRathjen.pdf), [[AczelRathjenCST.pdf:file]])

On [[constructible universes]]:

* [[Richard Matthews]], [[Michael Rathjen]], _Constructing the Constructible Universe Constructively_ [[arxiv:2206.08283](https://arxiv.org/abs/2206.08283)]

On [[Lifschitz realizability]]:

* [[Michael Rathjen]], [[Andrew Swan]], *Lifschitz Realizability as a Topological Construction*. The Journal of Symbolic Logic, Volume 85, Issue 4, December 2020, pp. 1342 - 1375. [[doi:10.1017/jsl.2021.1](https://doi.org/10.1017/jsl.2021.1), [arXiv:1806.10047](https://arxiv.org/abs/1806.10047)]

* [[Ray-Ming Chen]], [[Michael Rathjen]], *Lifschitz realizability for intuitionistic Zermelo-Fraenkel set theory*. Archive for Mathematical Logic, Volume 51, pages 789–818, (2012). [[doi:10.1007/s00153-012-0299-2](https://doi.org/10.1007/s00153-012-0299-2), [pdf](https://www1.maths.leeds.ac.uk/~rathjen/Lifschitz.pdf)]

On the [[limited principle of omniscience]] in [[CZF]]:

* {#Rathjen13} [[Michael Rathjen]], *Constructive Zermelo-Fraenkel set theory and the limited principle of omniscience*. [[arXiv:1302.3037](https://arxiv.org/abs/1302.3037)]

## Related entries

* [[constructive set theory]]

* [[axiom of multiple choice]]

* [[type of types]]

* [[WISC]]

* [[regular extension axiom]]

* [[small violations of choice]]

* [[countable ordinal]]

* [[presentation axiom]]

* [[higher inductive type]]

* [[taboo]]

category: people


Michael Ratz

* [isntitute page](https://www.faculty.uci.edu/profile/?facultyId=6279)

* [InSpire page](https://inspirehep.net/authors/992219?ui-citation-summary=true)

## Selected writings

On [[heterotic string theory|heterotic]] [[string phenomenology]]:

* Wilfried Buchmuller, Koichi Hamaguchi, Oleg Lebedev, [[Michael Ratz]], _Supersymmetric Standard Model from the Heterotic String_,  Phys. Rev. Lett. **96** 121602 (2006) (\[doi:v[arXiv:hep-ph/0511035](https://arxiv.org/abs/hep-ph/0511035)\]

* [[Saul Ramos-Sanchez]], [[Michael Ratz]], *Heterotic Orbifold Models*, in *[[Handbook of Quantum Gravity]]*, Springer (2024) [[arXiv:2401.03125](https://arxiv.org/abs/2401.03125), [doi:10.1007/978-981-19-3079-9](https://doi.org/10.1007/978-981-19-3079-9)]

  > (on [[orbifolds]])

category: people


Michael Reed

* [personal page](https://services.math.duke.edu/~reed/)

* [Wikipedia page](https://en.wikipedia.org/wiki/Michael_C._Reed)

## Selected writings

On [[mathematical physics]]:

* [[Michael Reed]] and [[Barry Simon]]: *[[Methods of Modern Mathematical Physics]]*, Academic Press (1978)

category: people


Michael Rios

* [webpage](https://quantumgravityresearch.org/michael-rios)

## Selected writings

On generalizations of [[exceptional structures]], including [[E8]], [[octonions]] and the [[exceptional Jordan algebra]]:

* [[Piero Truini]], [[Michael Rios]], [[Alessio Marrani]], *The Magic Star of Exceptional Periodicity*, J. Phys.: Conf. Ser. **1194** (2019) 012106 [[arXiv:1711.07881](https://arxiv.org/abs/1711.07881), [doi:10.1088/1742-6596/1194/1/012106](https://iopscience.iop.org/article/10.1088/1742-6596/1194/1/012106)]

On [[D=14 supersymmetry]]:

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], _The Geometry of Exceptional Super Yang-Mills Theories_, Phys. Rev. D 99, 046004 (2019) ([arXiv:1811.06101](https://arxiv.org/abs/1811.06101))

On [[12-dimensional supergravity]], [[D=14 supersymmetry]] et al. and further indications that [[M-theory]] in 10+1 dimensions may be understood as the [[KK-compactification]] on Cayley-plane [[fibers]] of some kind of [[bosonic M-theory]] in 26+1 dimensions:

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], _The Geometry of Exceptional Super Yang-Mills Theories_, Phys. Rev. D **99** (2019) 046004  [[arXiv:1811.06101](https://arxiv.org/abs/1811.06101), [doi:10.1103/PhysRevD.99.046004](https://doi.org/10.1103/PhysRevD.99.046004)]

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], *Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory*, Phys. Lett. B **808** (2020) 135674 [[arXiv:1906.10709](https://arxiv.org/abs/1906.10709), [doi:10.1016/j.physletb.2020.135674](https://doi.org/10.1016/j.physletb.2020.135674)]

Relation to the [[Monster group]], [[Moonshine]] and the [[Monster vertex operator algebra]]:

* [[Alessio Marrani]], [[Michael Rios]], [[David Chester]], *Monstrous M-theory*, Symmetry **15** 2 (2023) 490; [[doi:10.3390/sym15020490](https://doi.org/10.3390/sym15020490), [arXiv:2008.06742](https://arxiv.org/abs/2008.06742)]

category: people


Michael Roy

* [personal page](https://euclid.colorado.edu/~roymd/)

* [institute page](https://www.colorado.edu/math/michael-roy)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=39506)

## Selected writings

* {#Roy97} [[Michael Roy]], *The topos of ball complexes*, PhD thesis (1997), reprinted as: TAC Reprints **28** (2021) 1-62 [[tac:tr28](http://www.tac.mta.ca/tac/reprints/articles/28/tr28abs.html)]

On [[level of a topos|topos level]] in [[simplicial sets]] and [[cubical sets]]:

* {#KRRZ11} C. Kennett, [[Emily Riehl]], [[Michael Roy]], M. Zaks, *Levels in the toposes of simplicial sets and cubical sets*, JPAA **215**  5 (2011)  949-961 [[arXiv:1003.5944](http://arxiv.org/abs/1003.5944), [doi:j.jpaa.2010.07.002](https://doi.org/10.1016/j.jpaa.2010.07.002)]

## Related entries

* [[Aufhebung]]

category: people


Michael Schlessinger

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

## related $n$Lab entries

* [[deformation theory]]

category: people

[[!redirects Mike Schlessinger]]


Michael Sipser

Michael Sipper is a professor of Mathematics and member of the Computer Science and Artificial Intelligence Laboratory at MIT.

* [webpage](https://math.mit.edu/~sipser/)

## Selected writings
 {#SelectedWritings}

Textbook covering [[automaton| automata theory]], [[computability theory]], and [[complexity theory]]:

* [[Michael Sipser]], *Introduction to the Theory Of Computation*, 3rd ed: Cengage Learning (2012) [ISBN:978-1-133-18779-0,[pdf](https://www.mog.dog/files/SP2019/Sipser_Introduction.to.the.Theory.of.Computation.3E.pdf), [pdf](https://fuuu.be/polytech/INFOF408/Introduction-To-The-Theory-Of-Computation-Michael-Sipser.pdf)]

category: people


Michael Spivak

* Michael Spivak, *Physics for Mathematicians, Mechanics I*, xvi + 733 pages, Publish or Perish 2010 [contents](http://olivier.thill.perso.neuf.fr/books/bospphma.htm), [amazon book page](http://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322)
* Michael Spivak, _A comprehensive introduction to differential geometry_, Brandeis University, 1969; 5 vols.; review by V. Guillemin, Bull. AMS __79__ (1973), 303-306  [page](http://www.ams.org/journals/bull/1973-79-02/S0002-9904-1973-13149-0), [doi](http://dx.doi.org/10.1090/S0002-9904-1973-13149-0 )

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

## Related entries

* [[differential geometry]]

category: people


Michael Teper

* [personal page](http://www-thphys.physics.ox.ac.uk/people/MikeTeper/)

* [institute page](https://www.physics.ox.ac.uk/our-people/teper)

## Selected talks

Introduction to seeing [[confinement]] in [[lattice gauge field theory]]:

* [[Michael Teper]], *Lattice Gauge Theory and Confinement -- an Introduction*, talk at *[[Prospects in Theoretical Physics 2023 -- Understanding Confinement]]*, IAS (2023)  [[part I](https://www.ias.edu/video/lattice-gauge-theory-and-confinement-introduction), [II](https://www.ias.edu/video/lattice-gauge-theory-and-confinement-introduction-ii)]

category: people


Michael Völkl

* [institute page](http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/personen/wissbang.html)

## Related $n$Lab pages

* [[differential cohomology diagram]]

category: people

[[!redirects Michael Voelkl]]


Michael Warren

[[!redirects Michael A. Warren]]

* [website](http://www.math.ias.edu/~mwarren/)

## Selected writings

Early discussion of what came to be known as [[homotopy type theory]]:

* {#Warren08} [[Michael Warren]], *Homotopy theoretic aspects of constructive type theory*, PhD thesis (2008) [[pdf](http://www.andrew.cmu.edu/user/awodey/students/warren.pdf), [[Warren-HomotopyTypeTheory.pdf:file]]]

Proof that every [[simplicial model category]] in which the [[cofibrations]] are [[monomorphisms]] provides a sound [[model]] for [[intensional type theory|intensional]] [[Martin-Löf type theory]] (i.e. including the [[identity types]] used in [[homotopy type theory]]):

* {#AwodeyWarren07} [[Steve Awodey]], [[Michael Warren]], *Homotopy theoretic models of identity type*,  Mathematical Proceedings of the Cambridge Philosophical Society **146** 1 (2009) [[arXiv:0709.0248](http://arxiv.org/abs/0709.0248), [doi:10.1017/S0305004108001783](https://doi.org/10.1017/S0305004108001783)]

On the [[categorical semantics of dependent type theory]] via [[comprehension categories]]:

* {#LumsdaineWarren13} [[Peter LeFanu Lumsdaine]], [[Michael Warren]], *The local universes model: An overlooked coherence construction for dependent type theory*, ACM Trans. Comput. Log., **16** 3 no 23 (2015) 1-31 [[arXiv:1411.1736](https://arxiv.org/abs/1411.1736), [doi:10.1145/2754931](https://doi.org/10.1145/2754931)]

## Related entries

* [[homotopy type theory]]

category: people


Michael Weiss

* [webpage](http://www.math.uni-muenster.de/u/michael.weiss/)

## Selected writings

On [[manifold calculus]] via [[(infinity,1)-sheaves]] on the [[Weiss topology]]:

* {#BriWei} [[Pedro Boavida de Brito]], [[Michael Weiss]], _Manifold calculus and homotopy sheaves_, Homology, Homotopy and Applications, vol. 15(2), 2013, pp.361–383 ([arXiv:1202.1305](http://arxiv.org/abs/1202.1305))

## Related $n$Lab entries

* [[orthogonal calculus]]

* [[Weiss topology]]

* [[configuration category]]

category: people

[[!redirects Michael S. Weiss]]


Michael Westmoreland

* [personal page](https://personal.denison.edu/~westmoreland/)

## Selected writings

On [[quantum systems]] and [[quantum information theory]]:

* [[Benjamin Schumacher]], [[Michael Westmoreland]], *Quantum Processes, Systems, and Information*, Cambridge University Press (2010) [[doi:10.1017/CBO9780511814006](https://doi.org/10.1017/CBO9780511814006)]

category: people


Michael Wiemeler

* [Institute page](https://www.uni-muenster.de/FB10/Service/show_perspage.shtml?id=1305)

## Selected writings

On [[equivariant bordism homology theory]] and applicationto [[differential geometry]]:

* [[Michael Wiemeler]], *Equivariant bordism and applications in Differential Geometry*, talk at Math Coll. Augsburg (Jan 2016) [[pdf](https://www.uni-muenster.de/IVV5WS/WebHop/user/wiemelem/slides/kol_augsburg.pdf), [[Wiemeler-EquivariantBordism.pdf:file]]]

category: people


Michael Wooldridge

* [Home page](http://www.cs.ox.ac.uk/people/michael.wooldridge/)

category:people


Michael Wright

runs the _[[Archive for Mathematical Sciences & Philosophy]]_

category: people


Michael's theorem

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

#Contents#
* table of contents
{:toc}

## Idea

There are several theorems of [[Ernest Michael]] from the 1950s about [[paracompactness]]. There is also Michael's 1972 characterization of paracompact [[locally compact spaces]] under certain class of quotient maps.

## Statement

+-- {: .num_prop #RegularSpacesWherOpenCoveringIsRefinedbyCountableLocallyFiniteConnectionsofOpensAreParacompact}
###### Proposition
**(detection of paracompactness, [Michael 53, theorem 1](#Michael53))**

Let $X$ be a [[topological space]] such that

1. $X$ is [[regular topological space|regular]];

1. every [[open cover]] of $X$ has a [[refinement]] by a union of a [[countable set]] of

   [[locally finite sets of subsets|locally finite]] sets of open subsets (not necessarily covering).

Then $X$ is [[paracompact topological space]].

=--

Prop. \ref{RegularSpacesWherOpenCoveringIsRefinedbyCountableLocallyFiniteConnectionsofOpensAreParacompact} immediately implies that

* [[second-countable regular spaces are paracompact]],

* [[locally compact and sigma-compact spaces are paracompact]].

+-- {: .num_prop }
###### Proposition
**(on the closed image of a paracompact space, [Michael 57, corollary 1](#Michael57))**

The [[image]] of a [[paracompact Hausdorff space]] under a [[closed maps|closed]] [[continuous function]] is also paracompact Hausdorff.

=--

+-- {: .num_prop }
###### Proposition
**(Michael [[selection theorem]])**

A [[lower semicontinuous map]] from a [[paracompact topological space]] $X$ to a [[Banach space]] $E$ with convex closed values has a continuous subrelation which is a function. If this is true for a given topological space $Y$ instead of $E$ and all such functions and codomains $E$, then $Y$ is paracompact.

=--

## Related entries

* [[selection theorem]]

## References

The original articles are the following:

* {#Michael53} [[Ernest Michael]], _A note on paracompact spaces_, Proc. Amer. Math. Soc. 1953 ([jstor](http://www.jstor.org/stable/2032419), [pdf](http://www.renyi.hu/~descript/papers/Michael_parakompakt.pdf))

* {#Michael57} [[Ernest Michael]], _Another note on paracompactness_, 1957, ([pdf](krein.unica.it/~cornelis/private/PDF/AMS/PAMS_8_822.pdf), [pdf](http://www.ams.org/journals/proc/1957-008-04/S0002-9939-1957-0087079-9/S0002-9939-1957-0087079-9.pdf))

* [[Ernest Michael]], _Continuous selections. I_, Annals of Math. __63__ (2): 361–382 ([doi:10.2307/1969615](http://dx.doi.org/10.2307/1969615), MR0077107)

* [[Ernest Michael]], _A quintuple quotient test_, Gen. Topol. Appl. __2__ (1972) 91–138, [link](www.sciencedirect.com/science/article/pii/0016660X729004021972)

* discussion at A. Mathew's [blog](http://amathew.wordpress.com/2010/08/19/a-theorem-of-michael-on-paracompactness) and also on an application to metric spaces [here](http://amathew.wordpress.com/2010/08/19/a-metric-space-is-paracompact)

See also:

* R. Engelking, _General topology_

* [[Kenneth Kunen]], _Paracompactness of box products of compact spaces_, Trans. Amer. Math. Soc. __240__, (1978) 307-316, [jstor](http://www.jstor.org/stable/1998822)

* [[Akhil Mathew]], _[A theorem of Michael on paracompactness](https://amathew.wordpress.com/2010/08/19/a-theorem-of-michael-on-paracompactness/)_

* wikipedia [Michael selection theorem](http://en.wikipedia.org/wiki/Michael_selection_theorem)

[[!redirects Michael theorem]]

[[!redirects Michael's theorems]]


Michal Adamaszek

* [personal page](https://www.mimuw.edu.pl/~aszek/)

## Selected writings

On [[homotopy of rational maps]] between [[real projective spaces]]:

* {#AdamaszekKozlowskiYamaguchi08} [[Michal Adamaszek]], [[Andrzej Kozlowski]], [[Kohhei Yamaguchi]], *Spaces of algebraic and continuous maps between real algebraic varieties*, Quart. J. Math. 62 (2011), 771–790 ([arXiv:0809.4893](https://arxiv.org/abs/0809.4893), [doi:10.1093/qmath/haq029](https://doi.org/10.1093/qmath/haq029))

category: people


Michal Horodecki

* [Wikipedia entry](https://en.wikipedia.org/wiki/Micha%C5%82_Horodecki)

* [InSpire page](https://inspirehep.net/authors/1956876)

## Selected writings

Discussion of "[[noisy quantum operations]]" in the particular sense of [[quantum channels]] that admit an [environmental representation](quantum+channel#QuantumChannelsAndDecoherence) where the environment/bath system $\mathscr{B}$ is in its *maximally* [[mixed quantum state]]:

* {#HorodeckiHorodeckiOppenheim03} [[Michal Horodecki]], [[Pawel Horodecki]], [[Jonathan Oppenheim]], *Reversible transformations from pure to mixed states, and the unique measure of information*, Phys. Rev. A 67 062104 (2003) [[doi:10.1103/PhysRevA.67.062104](https://doi.org/10.1103/PhysRevA.67.062104), [arXiv;quant-ph/0212019](https://arxiv.org/abs/quant-ph/0212019)]

category: people


Michal Malinský

* [webpage](http://www-ucjf.troja.mff.cuni.cz/~malinsky/)

## Selected writings

On [[grand unified theories]]:

* {#BLM09} Stefano Bertolini, Luca Di Luzio, [[Michal Malinský]], _Intermediate mass scales in the non-supersymmetric SO(10) grand unification: a reappraisal_, Phys. Rev. D80:015013, 2009 ([arXiv:0903.4049](https://arxiv.org/abs/0903.4049))

* {#KM14} Helena Kolešová, [[Michal Malinský]], _Proton lifetime in the minimal $SO(10)$ GUT and its implications for the LHC_, Phys. Rev. D 90, 115001 (2014) ([arXiv:1409.4961](http://arxiv.org/abs/1409.4961))

* {#Malinsky09} [[Michal Malinský]], _35 years of GUTs - where do we stand?_, 2009 ([pdf](https://www.mpi-hd.mpg.de/lin/seminar_theory/talks/Malinsky.pdf))

On [[naturalness]]:

* Matěj Hudec, [[Michal Malinský]], _Hierarchy and decoupling_ ([arXiv:1902.04470](https://arxiv.org/abs/1902.04470))

On [[flavour anomalies]]:

* [[Michal Malinský]], _Lepton non-universality in $B$-decays in the minimal leptoquark gauge model_ ([arXiv:1906.09174](https://arxiv.org/abs/1906.09174))

## Related $n$Lab entries

* [[grand unified theory]]

* [[flavour anomalies]]

category: people

[[!redirects Michal Malinsky]]


Michal Marvan

* [webpage](http://www.slu.cz/math/cz/lide/marvan-michal/)

## related $n$Lab entries

* [[jet bundle]]

* [[differential operator]]

category: people


Michal Przybylek

* [personal page](https://www.mimuw.edu.pl/~mrp/)

## Selected writings

On the [[pasting law for pullbacks]]:

* {#Przybylek} [[Michal Przybylek]], _The other pullback lemma_ (2013) [[arXiv:1311.2974](https://arxiv.org/abs/1311.2974)]

category: people


Michal-Bastiani smooth map

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

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

## Idea

One can ask whether a nonlinear map between locally convex topological vector spaces is smooth according to various definitions. The definition due to Michal, developed by [[Andree Ehresmann|Bastiani]], is a very general such notion.

## Definition

A map $\Phi:E\rightarrow F$ from a [[locally convex topological vector space]] (lctvs) $E$ into another lctvs $F$ is said to be **Michal-Bastiani smooth** (or **MB-smooth**) if its [[directional derivatives]] of order $k$,

$$
D^k\Phi[x](y_1,\ldots,y_k) = \left.\frac{\partial^k}{\partial\lambda_1\cdots\partial\lambda_k}\right|_{\lambda_1=\cdots=\lambda_k=0}\Phi\left(x+\sum^k_{j=1}\lambda_j y_j\right)
$$

exist and the maps $D^k\Phi:E\times E^k\rightarrow F$ are [[jointly continuous map|jointly continuous]] for all $k\in\mathbb{N}$. The notion is due to Michal ([Michal 38](#Michal38),[Michal 40](#Michal40)), and was further developed by ([Bastiani 64](#Bastiani64)).

This notion of smoothness is the one used in [[Milnor]]'s treatment of infinite-dimensional [[Lie groups]] ([Milnor 84](#Milnor)) and Hamilton's exposé of the Nash-Moser [[inverse function theorem]] ([Hamilton 92](#Hamilton)). A comparison to other notions of smoothness for topological vector spaces with various properties is in ([Keller 74](#Keller)).

## Properties

MB-smooth maps have the following properties.

* Continuous [[linear maps]] are MB-smooth;

* MB-smooth maps are [[continuous map|continuous]];

* MB-smooth maps are smooth in the sense of [[convenient vector spaces]], since the [[chain rule]] holds;

* Maps between [[Fréchet spaces]] are MB-smooth if and only if they are conveniently smooth.

Note that a map that is smooth in the sense of a convenient vector space (equivalently, a smooth map between the corresponding [[diffeological spaces]]) is not necessarily continuous, so not all convenient smooth maps between lctvs are MB-smooth. Explicit examples have been given in ([Glöckner 06](#Glockner06)). It is even the case that one can have a convenient _isomorphism_ that is not MB-smooth, so the faithful and non-full functor
$$
lctvs_{MB} \to lctvs_{convenient} \hookrightarrow DiffeologicalSpace
$$
is not even injective on isomorphism classes. The following example was supplied in ([TaQ 15](#TaQ15)):

Let $E$ be the vector space of all real (two-sided) sequences $x=\langle\sp x_i\mid i\in\mathbb{Z}\rangle$ for which $x_i=0$ for $i\ll 0$, topologized so that we get a linear homeomorphism $E\to\mathbb{R}^\mathbb{N}\times\mathbb{R}^{(\mathbb{N})}$ defined by $x\mapsto(u,v)$ where $u_i=x_{i-1}$ and $v_i=x_{-i}$ for $i\gt 0}$.

Then define the bijection $f:E\to E$ by $x\mapsto y$ where $y_i=x_i$ for $i\neq 0$ and $y_0=x_0+\sum_{i\in\mathbb{N}}(x_i\cdot x_{-i})$.
The inverse is given by $y\mapsto x$ where $y_i=x_i$ for $i\neq 0$ and $x_0=y_0-\sum_{i\in\mathbb{N}}(y_i\cdot y_{-i})$.
The duality map $\mathbb{R}^\mathbb{N}\times\mathbb{R}^{(\mathbb{N})}\to\mathbb{R}$, $x\mapsto \sum_{i\in\mathbb{N}}(x_i\cdot x_{-i})$, is discontinuous but bornological, and hence a conveniently smooth bilinear map. Thus $f$ is discontinuous and so not MB-smooth.

## References

* {#Michal38} Aristotle Demetrius Michal, _Differential calculus in linear topological spaces_, Proc. Nat. Acad. Sci. USA **24** (1938), 340-342 ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1077109/pdf/pnas01796-0040.pdf))

* {#Michal40} Aristotle Demetrius Michal, _Differential of  functions with arguments and values in  topological  abelian groups_, Proc. Nat. Acad. Sci. USA **26** (1940), 356–359. ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078188/pdf/pnas01616-0038.pdf))

* {#Bastiani64} [[Andree Bastiani]], _Applications différentiables et variétés différentiables de dimension infinie_, J. Anal. Math. **13** (1964), 1–114. ([article](http://dx.doi.org/10.1007/BF02786619), paywalled)

* {#Milnor} [[John Milnor]], _Remarks on Infinite-Dimensional Lie Groups_, in: B. DeWitt, R. Stora, eds., Les Houches Session XL, _Relativity, Groups and Topology II_ (North-Holland, 1984), pp. 1007-1057

* {#Hamilton} [[Richard Hamilton]], _The Inverse Function Theorem of Nash and Moser_. Bull. Amer. Math. Soc. **7** (1982) 65-222.

* {#Keller} Keller, H. H., _Differential Calculus in Locally Convex Spaces_, Lecture Notes in Mathematics 417, Springer-Verlag, 1974 ([Springerlink](http://dx.doi.org/10.1007/BFb0070564), paywalled)

* {#Glockner06} [[Helge Glöckner]], _Discontinuous non-linear mappings on locally convex direct limits_, Publ. Math. Debrecen 68 (2006) 1-13, [arXiv:math/0503387](http://arxiv.org/abs/math/0503387).

* {#TaQ15} TaQ ([user 12643](http://mathoverflow.net/users/12643/taq)), Answer to _Do locally convex topological vector spaces embed into diffeological spaces?_, MathOverflow, <http://mathoverflow.net/q/209470> (version: 2015-06-16)

[[!redirects Michal-Bastiani smooth maps]]
[[!redirects MB-smooth maps]]
[[!redirects MB-smooth map]]


Michel André

Michel André (26 March 1936 – 9 July 2009) was a Swiss mathematician, specializing in non-commutative algebra and its applications to topology.

* [Wikipedia page](https://en.wikipedia.org/wiki/Michel_Andr%C3%A9_(mathematician))

##Relevant links;

* [[André-Quillen cohomology]]

##Selected publications:

* Michel André, _Méthode simpliciale en algèbre homologique et algèbre commutative_, Springer Lecture Notes in Mathematics, Vol 32, 1967.

* Michel André, _Homologie des algèbres commutatives_ Grundlehren der mathematischen Wissenschaften, Band 206. Springer. 1974

category:people


Michel Brion

* [webpage](http://www-fourier.ujf-grenoble.fr/~mbrion/)

category: people


Michel Cahen

* [webpage](http://homepages.ulb.ac.be/~mcahen/)

## related $n$Lab entries

* [[metaplectic group]]

category: people


Michel Coste

**Michel Coste** is a French mathematician who works on [[real algebraic geometry]] and [[o-minimal structures]].

* [homepage](http://perso.univ-rennes1.fr/michel.coste/index.html)
* [list of publications](http://perso.univ-rennes1.fr/michel.coste/articles.html)

category:people


Michel Demazure

Michel Demazure was a French mathematician, student of [[Alexander Grothendieck]], and member of the [[Bourbaki]] collective.

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

* [math genealogy](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=76208)

[[!redirects Demazure]]
[[!redirects M. Demazure]]


Michel Dubois-Violette

__Michel Dubois-Violette__ is a mathematical physicist in Paris. His more recent contributions involve noncommutative geometry (where he often collaborated with [[Alain Connes]]), homological algebra including the study of Koszul duality and noncommutative differential forms.

* Alain Connes, Michel Dubois-Violette, _Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples_,
Commun. in Math. Physics __230__:3 (2002) 539--579 [math.QA/0107070](http://arxiv.org/abs/math/0107070) [doi](http://dx.doi.org/10.1007/s00220-002-0715-2)

* M. Dubois-Violette, P. Michor, _Connections on central bimodules in noncommutative differential geometry_, J. Geom. Phys. __20__ (1996) 218
* [[Michel Dubois-Violette|M. Dubois-Violette]], J. Madore, T. Masson, J. Mourad, _On curvature in noncommutative geometry_, [q-alg/9512004](http://arxiv.org/abs/q-alg/9512004)

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

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

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

category: people


Michel Duflo

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

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=79929)

## Selected writings

Introducing the [[Duflo isomorphism]]:

* [[Michel Duflo]], *Opérateurs différentiels bi-invariants sur un groupe de Lie*, Ann. Sci. École Norm. Sup. (4) 10 (1977), 265–288  ([numdam:ASENS_1977_4_10_2_265_0](http://www.numdam.org/item/ASENS_1977_4_10_2_265_0/), [MR56:3188](http://www.ams.org/mathscinet-getitem?mr=56:3188))

On [[equivariant de Rham cohomology]] and the [[equivariant Chern character]]:

* [[Michel Duflo]], [[Michèle Vergne]], *Cohomologie équivariante et descente *, Astérisque, no. 215 (1993) ([numdam:AST_1993__215__5_0](http://www.numdam.org/book-part/AST_1993__215__5_0))

category: people


Michel Egeileh

* [webpage](http://www.aub.edu.lb/fas/math/Pages/egeileh.aspx)

## related $n$Lab entries

* [[super-Cartan geometry]]

* [[torsion constraints of supergravity]]

category: people


Michel Hilsum

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

category: people


Michel Kervaire

Michel André Kervaire (26 April 1927 – 19 November 2007) was a French mathematician who made significant contributions to topology and algebra.

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

*  [Obituary](http://smf4.emath.fr/Publications/Gazette/2008/116/smf_gazette_116_77-82.pdf)

## Selected writings

On [[Pontryagin's theorem]] (the unstable and framed version of the [[Pontryagin-Thom construction]]) and its application to the computation of [[stable homotopy groups of spheres]]:

* [[Michel Kervaire]], _La méthode de Pontryagin pour la classification des applications sur une sphère_, in: E. Vesentini (ed.), _Topologia Differenziale_, CIME Summer Schools, vol. 26, Springer 2011 ([doi:10.1007/978-3-642-10988-1_3](https://doi.org/10.1007/978-3-642-10988-1_3))

## Related $n$Lab entries

* [[Kervaire invariant]]

* [[Arf-Kervaire invariant problem]]

[[!redirects Kervaire]]
category:people


Michel Lazard

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

## Selected writings

On that came to be known as the *[[Lazard ring]]* and *[[Lazard's theorem]]*:

* [[Michel Lazard]], *Sur les groupes de Lie Formels à un Paramètre*, Bull. Soc. France **83** (1955) [[numdam:BSMF_1955__83__251_0](http://www.numdam.org/item/?id=BSMF_1955__83__251_0)]

## Related entries

* [[Lazard's theorem]]

category: people


Michel Raynaud

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

category: people


Michel Van den Bergh

__Michel Van den Bergh__ is a Belgian algebraist and algebraic geometer.

* [homepage](http://hardy.uhasselt.be/personal/vdbergh)
* [wikipedia](https://en.wikipedia.org/wiki/Michel_Van_Den_Bergh)

* [publications](http://hardy.uhasselt.be/personal/vdbergh/Publications/michel_pub.html)

## Selected writings

Introducing the notion of [[separable functors]]:

* {#NăstăsescuVanDen BerghVanOystaeyen89} [[Constantin Năstăsescu]], [[Michel van den Bergh]], [[F. van Oystaeyen]], _Separable functors applied to graded rings_, J. Algebra __123__ (1989) 397-413 [doi:10.1016/0021-8693(89)90053-7]

category: people

[[!redirects M. Van den Bergh]]
[[!redirects Michel van den Bergh]]
[[!redirects Van den Bergh]]


Michel Vaquié

* [web](http://www.math.univ-toulouse.fr/~vaquie/)

## Selected writings

Discussion of [[opposite categories]] of [[commutative monoids in a symmetric monoidal category]] regarded as categories of generalized [[affine schemes]] (and [[Spec(Z)]]):

* [[Bertrand Toën]], [[Michel Vaquié]], *Au-dessous de $Spec \mathbb{Z}$*, Journal of K-Theory **3** 3 (2009) 437-500 [[doi:10.1017/is008004027jkt048](https://doi.org/10.1017/is008004027jkt048)]

category: people

[[!redirects Michel Vaquie]]
[[!redirects M. Vaquie]]
[[!redirects Vaquie]]
[[!redirects M. Vaquié]]
[[!redirects Vaquié]]


Michel Zisman

* Mathematics Genealogy [entry](http://genealogy.math.ndsu.nodak.edu/id.php?id=154785)

## Book

* with [[Pierre Gabriel]], _[[Calculus of fractions and homotopy theory]]_

## References

* _Comparaison de deux diagonales pour un ensemble bisimplicial_, [[Journal of Homotopy and Related Structures]], Dec 2014, doi:[10.1007/s40062-014-0095-2](http://dx.doi.org/10.1007/s40062-014-0095-2) (paywall)

[[!redirects M. Zisman]]
category: people


Michele Del Zotto

* [webpage](https://katalog.uu.se/profile/?id=N20-103)

## Selected writings

On [[M-theory on S1/G_HW times H/G_ADE]] and [[D=6 N=(1,0) SCFT]]:

* {#DHTV14} [[Michele Del Zotto]], [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], _6d Conformal Matter_, JHEP02(2015)054 [[arXiv:1407.6359](https://arxiv.org/abs/1407.6359)]

On [[2-group]] [[higher gauge theory|higher symmetry]]:

* [[Michele Del Zotto]] and [[Kantaro Ohmori]], _2-Group Symmetries of 6D Little String Theories and T-Duality_, Annales Henri Poincaré volume 22, pp. 2451–2474 (2021) ([doi:10.1007/s00023-021-01018-3)](https://doi.org/10.1007/s00023-021-01018-3), [arXiv:2009.03489](https://arxiv.org/abs/2009.03489))

On [[little string theory]] in [[heterotic string theory]] via [[F-theory]]-methods:

* [[Michele Del Zotto]], Muyang Liu, Paul-Konstantin Oehlmann, *6D Heterotic Little String Theories and F-theory Geometry: An Introduction* [[arXiv:2303.13502](https://arxiv.org/abs/2303.13502)]

On [[QFT with defects]]:

* [[Nils Carqueville]], [[Michele Del Zotto]], [[Ingo Runkel]], *Topological defects*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2311.02449](https://arxiv.org/abs/2311.02449)]

On the [[Coulomb branch]] of [[D=3 N=4 SYM]] from [[heterotic string theory]] on [[ADE-singularities]]:

* [[Michele Del Zotto]], [[Marco Fazzi]], Suvendu Giri, *A new vista on the Heterotic Moduli Space from Six and Three Dimensions* [[arXiv:2307.10356](https://arxiv.org/abs/2307.10356)]

category: people


Michele Mosca

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

* [institute page](https://uwaterloo.ca/institute-for-quantum-computing/profiles/michele-mosca)

## Selected writings

On [[quantum computation]]:

* [[Phillip Kaye]], [[Raymond Laflamme]], [[Michele Mosca]], *An Introduction to Quantum Computing*, Oxford University Press (2007) [[pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149125645.pdf), [ISBN:9780198570493](https://global.oup.com/academic/product/an-introduction-to-quantum-computing-9780198570493?cc=at&lang=en&)]

category: people


Michele Pagani

Michele Pagani is a proof theorist working at [[IRIF]].

* [webpage](https://www.irif.fr/~michele/index.html)

## Selected writings
 {#SelectedWritings}

On [[strong normalization]]:

* {#PaganiTortora09} [[Michele Pagani]], [[Lorenzo Tortora de Falco]], *Strong Normalization Property for Second Order Linear Logic* (2009) [[pdf](https://www.irif.fr/~michele/snllTCS-1.pdf)]

category: people


Michele Schiavina

* [webpage](https://math.berkeley.edu/people/faculty/michele-schiavina)

## Selected writings

On the [[phase space]] of [[first-order formulation of gravity|first-order]] [[gravity]] ([[general relativity]]) in a local formulation that allows to decompose spacetime into pieces and then re-glue ([[BV-BFV formalism]]):

* [[Michele Schiavina]], _BV-BFV approach to general relativity_ (2015) [[pdf](http://user.math.uzh.ch/cattaneo/schiavina.pdf), [[Schiavina-Relativity.pdf:file]]]

* {#CattaneoSchiavina15} [[Alberto Cattaneo]], [[Michele Schiavina]], _BV-BFV approach to General Relativity, Einstein-Hilbert action_, J. Math. Phys. **57** 023515 (2016) [[arXiv:1509.05762](https://arxiv.org/abs/1509.05762), [doi:10.1063/1.4941410](https://doi.org/10.1063/1.4941410)]

* {#CattaneoSchiavina17a} [[Alberto Cattaneo]], [[Michele Schiavina]], _The reduced phase space of Palatini-Cartan-Holst theory_, Ann. Henri Poincaré **20** (2019) 445 [[arXiv:1707.05351](https://arxiv.org/abs/1707.05351), [doi:10.1007/s00023-018-0733-z](https://doi.org/10.1007/s00023-018-0733-z)]

* {#CattaneoSchiavina17b} [[Alberto Cattaneo]], [[Michele Schiavina]], *BV-BFV approach to General Relativity: Palatini-Cartan-Holst action*, Adv. Theor. Math. Phys. **23**  (2019) 1801-1835 [[arXiv:1707.06328](https://arxiv.org/abs/1707.06328), [doi:10.4310/ATMP.2019.v23.n8.a3](https://doi.org/10.4310/ATMP.2019.v23.n8.a3)]

* [[Alberto S. Cattaneo]], *Phase space for gravity with boundaries*,  Encyclopedia of Mathematical Physics (2023) [[arXiv:2307.04666](https://arxiv.org/abs/2307.04666)]

and relation to [[BF-theory]]:

* [[Alberto S. Cattaneo]], [[Leon Menger]], [[Michele Schiavina]], *Gravity with torsion as deformed BF theory* [[arXiv:2310.01877](https://arxiv.org/abs/2310.01877)]

Discussion of the [[Nambu-Goto action]] and [[Polyakov action]] on [[worldsheets]] with [[manifold with boundary|boundary]] (i.e. in the generality of [[open strings]]) and cast in [[BV-BRST formalism]]:

* {#MartinoliSchiavina22} S. Martinoli, [[Michele Schiavina]], *BV analysis of Polyakov and Nambu–Goto theories with boundary*,  Lett. Math. Phys. **112** 35 (2022) [[doi:10.1007/s11005-022-01526-1](https://doi.org/10.1007/s11005-022-01526-1), [arXiv:2106.02983](https://arxiv.org/abs/2106.02983)]

On the [[relativistic particle]], and related systems, in [[BV-BRST formalism]]:

* [[Alberto Cattaneo]], [[Michele Schiavina]], _On time_, Lett. Math. Phys. **107** (2017) 375-408 [[doi:10.1007/s11005-016-0907-x](https://doi.org/10.1007/s11005-016-0907-x), [arXiv:1607.02412](http://arxiv.org/abs/1607.02412)]

On [[BV quantization]]:

* [[Alberto S. Cattaneo]], [[Pavel Mnev]], [[Michele Schiavina]], *BV Quantization*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2307.07761](https://arxiv.org/abs/2307.07761)]

category: people


Micheline Vigué-Poirrier

* [webpage](http://www.math.univ-paris13.fr/~vigue/)

## Selected writings

On [[Sullivan models for free loop spaces]]:

* {#VigueSullivan76} [[Micheline Vigué-Poirrier]], [[Dennis Sullivan]], _The homology theory of the closed geodesic problem_, J. Differential Geometry 11 (1976) 633-644.

* {#VigueBurghelea85} [[Micheline Vigué-Poirrier]], Dan Burghelea, _A model for cyclic homology and algebraic K-theory of 1-connected topological spaces_, J. Differential Geom. Volume 22, Number 2 (1985), 243-253 ([Euclid](https://projecteuclid.org/euclid.jdg/1214439821))

More generally on [[rational models of mapping spaces]]:

* [[Micheline Vigué-Poirrier]], _Rational formality of function spaces_, Journal of Homotopy and Related Structures 2.1 (2007): 99-108 ([arXiv:0706.2977](https://arxiv.org/abs/0706.2977))

## Related $n$Lab entries

* [[Sullivan model of free loop space]]

* [[rational models of mapping spaces]]]

category: people

[[!redirects Micheline Vigue-Poirrier]]
[[!redirects Micheline Vigué]]
[[!redirects Micheline Vigue]]


Michelle Wachs

* [personal page](https://www.math.miami.edu/~wachs/)

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

## Selected writings

On something similar to (but different from) [[n-Lie algebras]]:

* [[Phil Hanlon]], [[Michelle Wachs]], _On Lie $k$-Algebras_, Advances in Mathematics Volume 113, Issue 2, July 1995, Pages 206–236 ([doi:10.1006/aima.1995.1038](https://doi.org/10.1006/aima.1995.1038))

category: people


Michelson-Morley experiment

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[experiment]]

* [[null result]]

* [[speed of light]]

* [[special relativity]], [[general relativity]]

## References

See also

* Wikipedia, _[Michelson-Morley experiment](https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment)_


Michiel Hazewinkel

[[!redirects Michael Hazewinkel]]

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

## Selected publications

* [[Hazewinkel, Witt vectors]]

* [[Encyclopedia of Mathematics]]

## Related entries

* [[Lambda-ring]]

* [[ring of Witt vectors]]

* [[symmetric function]]

category: people


Michio Jimbo

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

## Selected writings

On [[holonomic quantum fields]]:

* [[Tetsuji Miwa]], [[Michio Jimbo]], *Introduction to holonomic quantum fields*, pp. 28--36 in: _The Riemann problem, complete integrability and arithmetic applications_, Lec. Notes in Math. **925**, Springer (1982) $[$[doi:10.1007/BFb0093497](http://dx.doi.org/10.1007/BFb0093497)$]$

* {#JMSM05} [[Michio Jimbo]], [[Tetsuji Miwa]], [[Mikio Sato]], Yasuko Môri, *Holonomic Quantum Fields — The unanticipated link between deformation theory of differential equations and quantum fields —*,  In: [[Konrad Osterwalder|K. Osterwalder]] (ed.), *Mathematical Problems in Theoretical Physics*, Lecture Notes in Physics, **116** Springer (2005) $[$[doi:10.1007/3-540-09964-6_310](https://doi.org/10.1007/3-540-09964-6_310)$]$

On the [[quantum Yang-Baxter equation|quantum]] [[Yang-Baxter equation]]:

* [[Michio Jimbo]], *Introduction to the Yang-Baxter Equation*, Int. J. Modern Physics A **4** 15 (1989) 3759-3777 [[doi:10.1142/S0217751X89001503](https://doi.org/10.1142/S0217751X89001503)]

  reprinted in: *Braid Group, Knot Theory and Statistical Mechanics*, Advanced Series in Mathematical Physics **9**, World Scientific (1991) [[doi:10.1142/0796](https://doi.org/10.1142/0796)]

On the [hypergeometric construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of solutions to the [[Knizhnik-Zamolodchikov equation]]:

* [[Etsuro Date]], [[Michio Jimbo]], [[Atsushi Matsuo]], [[Tetsuji Miwa]], *Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation*,  International Journal of Modern Physics B **04** 05 (1990) 1049-1057   [[doi:10.1142/S0217979290000528](https://doi.org/10.1142/S0217979290000528)]

On [[solitons]]:

* [[Michio Jimbo]], [[Tetsuji Miwa]], [[Etsuro Date]], (transl. from Japanese by [[Miles Reid|M. Reid]]), *Solitons: differential equations, symmetries and infinite dimensional algebras*,  Cambridge Tracts in Mathematics **135** (2012) [[ISBN:9781107404199](https://www.cambridge.org/us/universitypress/subjects/mathematics/geometry-and-topology/solitons-differential-equations-symmetries-and-infinite-dimensional-algebras)]

category: people


Michèle Audin

__Michèle Audin__ is a French mathematician. Her main subject is [[symplectic geometry]] and its connections to mathematical physics including integrable systems and theoretical mechanics.

* [wikipedia](https://en.wikipedia.org/wiki/Mich%C3%A8le_Audin)

* Michèle Audin, _Hamiltonian systems and their integrability_, SMF/AMS Texts and Monographs __15__ (2008)
* Michele Audin (editor), _[[Matériaux pour l’histoire des mathématiques au XXe siècle]]_ - Actes du colloque à la mémoire de Jean Dieudonné (Nice 1996) Séminaires & Congrès. Numèro 3. Paris (Socièté Mathématique de France). 1998. 283 pp.

category: people

[[!redirects M. Audin]]


Michèle Friend

* [website](https://philosophy.columbian.gwu.edu/michele-friend)

## Selected publications

* [[Michèle Friend]], *Varieties of Pluralism and Objectivity in Mathematics*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[doi:10.1007/978-3-030-15655-8_15](https://doi.org/10.1007/978-3-030-15655-8_15)]

* [[Michèle Friend]], *Pluralism in Mathematics: A New Position in Philosophy of Mathematics*, Logic, Epistemology and the Unity of Science, Springer, 2014.  [ISBN 978-94-007-7057-7, [doi:10.1007/978-94-007-7058-4](https://doi.org/10.1007/978-94-007-7058-4), [excerpt](https://content.e-bookshelf.de/media/reading/L-2680820-cee701bdbb.pdf)]

## Related concepts

* [[pluralism]]

* [[philosophy of mathematics]]

* [[foundations of mathematics]]

category: people


Michèle Giry

## Selected writings

On the [[monad]] that came to be called the [[Giry monad]]:

* {#Giry80} [[Michèle Giry]], _A categorical approach to probability theory_, Categorical aspects of topology and analysis (Ottawa, Ont., 1980), pp. 68–85, Lecture Notes in Math. **915** Springer 1982 ([doi:10.1007/BFb0092872](https://doi.org/10.1007/BFb0092872))

category: people


Michèle Raynaud

* [Wikipedia entry](https://en.wikipedia.org/wiki/Mich%C3%A8le_Raynaud)

## Selected writings

On [[étale fundamental groups]] in [[algebraic geometry]]:

* [[Alexander Grothendieck]], [[Michèle Raynaud]], *Revêtements étales et groupe fondamental* ([[SGA 1]]),  Lecture Notes in Mathematics **224**, Springer 1971 ([arXiv:math/0206203](https://arxiv.org/abs/math/0206203), [doi:10.1007/BFb0058656](https://link.springer.com/book/10.1007/BFb0058656))

category: people


Michèle Vergne

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

## Selected writings

On [[heat kernels]], [[Dirac operators]] and [[index theory]]:

* [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], *Heat Kernels and Dirac Operators*, Springer 2004 ([ISBN:9783540200628](https://www.springer.com/gp/book/9783540200628))

On [[equivariant de Rham cohomology]] and the [[equivariant Chern character]]:

* [[Michel Duflo]], [[Michèle Vergne]], *Cohomologie équivariante et descente *, Astérisque, no. 215 (1993) ([numdam:AST_1993__215__5_0](http://www.numdam.org/book-part/AST_1993__215__5_0))

On [[equivariant elliptic cohomology]]:

* [[Michèle Vergne]], *Bouquets revisited and equivariant elliptic cohomology*, International Journal of Mathematics 2021 ([arXiv:2005.00312](https://arxiv.org/abs/2005.00312), [doi:10.1142/S0129167X21400127](https://doi.org/10.1142/S0129167X21400127))

category: people

[[!redirects Michele Vergne]]


microbundle

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

#Contents#
* table of contents
{:toc}

## Idea

A microbundle is something like an approximation to the notion of [[vector bundle]]: a locally trivial [[bundle]] $E \to X$ of topological spaces that has a [[section]]. Indeed, as observed by Milnor,  every vector bundle gives an example of a microbundle (for a modern treatment see [Kupers18, Example 27.2.3](#Kupers18) or Lurie's course, [Topics in Geometric Topology, Lecture 10](http://www.math.harvard.edu/~lurie/937notes/937Lecture10.pdf)).

## Definition

### Microbundles

A real __microbundle of dimension $n$__ is a 4-tuple $\xi = (E,p,B,i)$ where

* $E$ is a [[topological space]] (the _total space_ of $\xi$),

* $B$ is a topological space (the _base space_ of $\xi$,

* and $p:E\to B$ a continuous map (projection),

* $i:B\hookrightarrow E$ another continuous map (inclusion of base space)

such that

* $i$ is a [[section]] of $p$, i.e. $p\circ i = id_B$

* the _local triviality_ condition holds:

  for all $b\in B$, there are neighborhoods $U\ni b$ and $V\ni i(b)$ and a homeomorphism $h:U\times R^n\to V\cap p^{-1}(U)$ such that $p(h(u,v))=u$ and $h(u,0)=i(u)$ for all $u\in U$. The open subspace $i(B)$ is called the _zero section_ of $\xi$.

### Morphisms of microbundles

A __[[morphism]] of microbundles__ $\phi:\xi\to\xi'$ is a [[germ]] of maps from neighborhoods of the zero section of $\xi$ to $\xi'$, which commutes with projections and inclusions, with composition defined for representatives as composition of functions on smaller neighborhoods.

In particular, an __[[isomorphism]] of microbundles__ can be represented by a [[homeomorphism]] from a neighborhood $V$ of the zero section in $\xi$ to a neighborhood $V'$ of the zero section in $\xi'$ commuting with projections and inclusions of the zero sections.

## Examples

### Tangent microbundle

The main example is the __tangent microbundle__ $(M\times M,p_1,M,i)$ of a topological [[manifold]] $M$ where $p_1:M\times M\to M$ is the projection onto the first factor. If $(U,f)$ is a chart of the manifold $M$ around point $x\in M$ (where $x\in U\subset M$ and $f:U\to R^n$ is a homeomorphism with $h(x)=0$) then define $h:U\times R^n\to U\times U$ by $h(u,v)=(u,f^{-1}(f(v)-u))$.

If $M$ is a smooth manifold, then the tangent microbundle is equivalent to the tangent bundle ([Kupers18, Example 27.2.3](#Kupers18)).

## References and Literature

The original reference is

* [[John Milnor]], _Microbundles.  Part I_, Topology 3 (1964), Supplement 1, 53–80. doi: https://doi.org/10.1016/0040-9383(64)90005-9, [PDF](http://www.maths.ed.ac.uk/~aar/papers/micro001.pdf).

Classic treatments of their elementary theory include:

* [[N. H. Kuiper]] and [[Richard Lashof|R. K. Lashof]], _Microbundles and Bundles I. Elementary Theory_, Invent. Math., 1, (1966), 1 – 17.

* [[N. H. Kuiper]] and [[Richard Lashof|R. K. Lashof]], _Microbundles and Bundles II. Semisimplicial Theory_, Invent. Math., 1, (1966), 243 – 259.

Useful references are for instance

* [[Jacob Lurie]], Spring 2009, [Topics in Geometric Topology](http://www.math.harvard.edu/~lurie/937.html)

* S. Buoncristiano, 2003, [Fragments of geometric topology from the sixties](http://www.emis.de/journals/GT/gtmcontents6.html).

* {#Kupers18} [[Alexander Kupers]], _Lectures on diffeomorphisms groups of manifolds_, ([pdf](http://www.math.harvard.edu/~kupers/teaching/272x/book.pdf))

[[!redirects tangent microbundle]]
[[!redirects tangent microbundles]]
[[!redirects microbundles]]


microcausal polynomial observable

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

#Contents#
* table of contents
{:toc}

## Idea

The _microcausal functionals_ on the space $C^\infty(X)$ of [[smooth functions]] on a [[globally hyperbolic spacetime]] $(X,e)$ are those which come from [[compactly supported distributions]] on some [[Cartesian product]] of copies of $X$ such that the [[wave front set]] of the distributions excludes those covectors to a point in $X^n$ all whose components are in the [[closed future cone]] or all whose components are in the [[closed past cone]]

These functionals underly the [[Wick algebra]] of [[free field theories]]. The condition on the wave front is such that the [[product of distributions]] with a [[Hadamard distribution]] is well defined, so that the coresponding [[Moyal star product]] is well defined, which gives the [[Wick algebra]]. At the same time the condition includes [[local observables]] and hence in particular the usual ([[adiabatic switching|adiabatically switched]]) point-[[interaction]] terms, such as of [[phi^4 theory]].

[[!include perturbative observables -- table]]

## Definition

+-- {: .num_defn #PolynomialObservable}
###### Definition
**([[polynomial observable]])**

Let $E \overset{fb}{\to}$ be [[field bundle]] which is a [[vector bundle]]. An [[off-shell]] _[[polynomial observable]]_ is a [[smooth function]]

$$
  A
  \;\colon\;
  \Gamma_\Sigma(E)
    \longrightarrow
  \mathbb{C}
$$

on the [[on-shell]] [[space of sections]] of the [[field bundle]] $E \overset{fb}{\to} \Sigma$ (space of field histories) which may be expressed as

$$
  A(\Phi)
  \;=\;
  \alpha^{(0)}
  +
  \int_\Sigma
    \alpha^{(1)}_a(x) \Phi^a(x)
   \,
  dvol_\Sigma(x)
  +
  \int_\Sigma \int_\Sigma
   \alpha^{(2)}_{a_1 a_2}(x_1, x_2) \Phi^{a_1}(x_1) \Phi^{a_2}(x_2)
  \,dvol_\Sigma(x_1)
  \, dvol_\Sigma(x_2)
  +
  \cdots
  \,,
$$

where

$$
  \alpha^{(k)} \in \Gamma'_{\Sigma^k}\left((E^\ast)^{\boxtimes^k_{sym}} \right)
$$

is a [[compactly supported distribution]] [[distribution of two variables|of k variables]] on the $k$-fold graded-symmetric [[external tensor product of vector bundles]] of the [[field bundle]] with itself.

Write

$$
  PolyObs(E) \hookrightarrow Obs(E)
$$

for the [[subspace]] of off-shell polynomial observables onside all off-shell [[observables]].

Let moreover $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] whose [[equations of motion]] are [[Green hyperbolic differential equations]]. Then an _[[on-shell]] polynomial observable_ is the [[restriction]] of an off-shell polynomial observable along the inclusion of the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E)$. Write

$$
  PolyObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})
$$

for the subspace of all on-shell polynomial observables inside all on-shell [[observables]].

By [this prop.](Green+hyperbolic+partial+differential+equation#DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions) restriction yields an [[isomorphism]] between polynomial on-shell observables and polynomial off-shell observables modulo the image of the [[differential operator]] $P$:

$$
  PolyObs(E,\mathbf{L})
    \underoverset{\simeq}{\text{restriction}}{\longleftarrow}
  PolyObs(E)/im(P)
  \,.
$$

=--

+-- {: .num_defn #MicrocausalObservable}
###### Definition
**([[microcausal observable]])**

For $\Sigma$ a [[spacetime]], hence a [[Lorentzian manifold]] with [[time orientation]], then a _[[microcausal observable]]_ is a [[polynomial observable]] (def. \ref{PolynomialObservable}) such that each [[coefficient]] $\alpha^{(k)}$ has [[wave front set]] excluding those points  where all $k$ [[wave vectors]] are in the [[closed future cone]] or all in the [[closed past cone]].

=--

## Examples

+-- {: .num_example }
###### Example
**([[non-singular distribution|non-singular observables]] are [[microcausal observables|microcausal]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]].

Then a [[regular observable]], hence a [[polynomial observable]] ([this def.](A+first+idea+of+quantum+field+theory+--+Observables#PolynomialObservables)) whose [[distribution|distributional]] [[coefficients]] $\alpha_{a_1 \cdots a_k}$ (eq:RecalledForWickAlgebraExpansionOfPolynomialObservables) are [[non-singular distributions]] is a [[microcausal observable]] (def. \ref{MicrocausalFunctionals}).

This is simply because the [[wave front set]] of [[non-singular distributions]] is [[empty set|empty]] (by definition, via the [[Paley-Wiener-Schwartz theorem]], [this prop.](Schwartz+theorem#DecayPropertyOfFourierTransformOfCompactlySupportedFunctions)).

=--

+-- {: .num_example}
###### Example
**(compactly averaged point evaluations are microcausal)**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]]. Assume the [[field bundle]] $E$ is a [[trivial vector bundle]] with linear [[fiber]] coordinates $(\phi^a)$.

Let $g \in C^\infty_c(X)$ be a [[bump function]], then for $n \in \mathbb{N}$ the polynomial observables ([this def.](A+first+idea+of+quantum+field+theory+--+Observables#PolynomialObservables)) of the form

$$
  \array{
    \Gamma_\Sigma(E) &\overset{}{\longrightarrow}& \mathbb{C}
    \\
    \Phi
      &\mapsto&
    \int_X g(x)
      \tilde \alpha_{a_1 \cdots a_k}(x)
      \Phi^{a_1}(x) \cdots \Phi^(a_k)\,  dvol_\Sigma(x)
  }
$$

are [[microcausal observable|microcausal]] (def. \ref{MicrocausalFunctionals}).

If here we think of $\phi(x)^n$ as a point-[[interaction]] term (as for instance in [[phi^4 theory]]) then $g$ is to be thought of as an "[[adiabatic switching|adiabatically switched]]" [[coupling constant]]. These are the relevant [[interaction terms]] to be quantized via [[causal perturbation theory]].

=--

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

For notational convenience, consider the case of the [[scalar field]] with $k = 2$; the general case is directly analogous. Then the [[local observable]] coming from $\phi^2$ (a [[phi^n interaction]]-term), has, regarded as a [[polynomial observable]], the [[delta distribution]] $\delta(x_1-x_2)$ as [[coefficient]] in degree 2:

$$
  \begin{aligned}
    A(\Phi)
    & =
    \underset{\Sigma}{\int} g(x) (\Phi(x))^2 \,dvol_\Sigma(x)
    \\
    & =
    \underset{\Sigma \times \Sigma}{\int}
      \underset{ = \alpha^{(2)}}{
      \underbrace{
        g(x_1)
        \delta(x_1 - x_2)
      }}
      \,
     \Phi(x_1) \Phi(x_2) \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2)
  \end{aligned}
  \,.
$$

Now for $(x_1, x_2) \in \Sigma \times \Sigma$ and $\mathbb{R}^{2n} \simeq U \subset X \times X$ a [[chart]] around this point,  the [[Fourier transform of distributions]] of $g \cdot \delta(-,-)$ restricted to this chart is proportional to the Fourier transform $\hat g$ of $g$ evaluated at the sum of the two covectors:

$$
  \begin{aligned}
    (k_1, k_2)
      & \mapsto
      \underset{\mathbb{R}^{2n}}{\int}
      g(x_1)
      \delta(x_1, x_2)
      e^{i (k_1 \cdot x_1 + k_2 \cdot x_2 )}
      \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2)
    \\
    & \propto \hat g(k_1 + k_2)
   \end{aligned}
  \,.
$$

Since $g$ is a plain [[bump function]], its [[Fourier transform]] $\hat g$ is quickly decaying (according to [this inequality](compactly+supported+distribution#eq:DecayEstimateForFourierTransformOfNonSingularDistribution))  along $k_1 + k_2$ ([this prop.](wavefront+set#EmptyWaveFronSetCorrespondsToOrdinaryFunction)), as long as $k_1 + k_2 \neq 0$. Only on the cone $k_1 + k_2 = 0$ the Fourier transform is constant, and hence in particular not decaying.

{#WaveFrontOfdeltaxy} This means that the wave front set consists of the elements of the form $(x, (k, -k))$ with $k \neq 0$. Since $k$ and $-k$ are both in the [[closed future cone]] or both in the [[closed past cone]] precisely if $k = 0$, this situation is excluded in the wave front set and hence the distribution $g \cdot \delta(-,-)$ is [[microcausal observable|microcausal]].

> (graphics grabbed from [Khavkine-Moretti 14, p. 45](#KhavkineMoretti14))

=--

This shows that microcausality in this case is related to conservation of momentum in the point interaction.

More generally:

+-- {: .num_example #CompactlySupportedPolynomialLocalDensities}
###### Example
**(polynomial [[local observables]] are microcausal)**

Write

$$
  \Omega_{poly}^{h,v}(E)
$$

for the space of differential forms on the [[jet bundle]]
of the [[field bundle]] $E$ which locally are [[polynomials]]
in the field variables.

$$
  \mathcal{F}_{loc}
    \; \subset \;
  C^\infty_c(\Sigma) \underset{\Omega_{poly}^{0,0}(E)}{\otimes} \Omega_{poly}^{d,0}(E)
$$

for the subspace of [[horizontal differential forms]] of degree $d$ on the [[jet bundle]] ([[local Lagrangian densities]])
of those which are [[compact support|compactly supported]] with respect to $\Sigma$ ([[local observables]]) and [[polynomial]] with respect to the
field variables.

Every $L \in \mathcal{F}_{loc}$ induces a functional

$$
   \Gamma_\Sigma(E) \longrightarrow \mathbb{R}
$$

by [[integration of differential forms|integration]] of the [[pullback of differential forms|pullback]]
of $L$ along the [[jet prolongation]] of a given [[section]]:

$$
  \phi \mapsto \int_{\Sigma} j^\infty(\phi)^\ast L
  \,.
$$

These functionals happen to be [[microcausal functional|microcausal]], so that there is an inclusion

$$
  \mathcal{F}_{loc} \hookrightarrow \mathcal{F}_{mc}
$$

into the space of microcausal functionals (e.g. [Fredenhagen-Rejzner 12, p. 21](#FredenhagenRejzner12)). In fact this is a [[dense subspace]] inclusion (e.g. [Fredenhagen-Rejzner 12, p. 23](#FredenhagenRejzner12))

=--

## Properties

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

Write $\mathcal{F}_{reg} \subset \mathcal{F}_{mc}$ for that subalgebra of the algebra of microcausal functionals whose [[coefficients]] are [[non-singular distributions]].

Let

$$
  \langle -\rangle
  \;\colon\;
  \mathcal{F}_{reg}
  \longrightarrow
  \mathbb{C}
$$

be a [[state on a star-algebra|state]] on regular observables which is [[quasi-free Hadamard state|quasi-free Hadamard]]. Then this uniquely [[extensions|extends]] to a state on microcausal functionsal

$$
  \array{
    \mathcal{F}_{reg}
      &\overset{\langle -\rangle}{\longrightarrow}&
    \mathbb{C}
    \\
    \downarrow & \nearrow_{\mathrlap{\exists ! \langle -\rangle}}
    \\
    \mathcal{F}_{mc}
  }
$$

=--

([Hollands-Ruan 01, remark 1 on p. 12](#HollandRuan01), implied by [Brunetti-Fredenhagen 00](#BrunettiFredenhagen00), [Hollands-Wald 01](#HollandsWald01), a special case of [Hollands-Ruan 01, theorem III.1 (ii)](#HollandRuan01))

## Related concepts

[[!include states and observables -- content]]

## References

### Original articles

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661, 2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))

* {#DuetschFredenhagen01a} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion_, Commun.Math.Phys. 219 (2001) 5-30 ([arXiv:hep-th/0001129](https://arxiv.org/abs/hep-th/0001129))

* {#HollandsWald01} [[Stefan Hollands]], [[Robert Wald]], _Local Wick polynomials and time ordered products of quantum fields in curved spacetime_, Commun. Math. Phys., Commun.Math.Phys.223:289-326,2001 ([arXiv:gr-qc/0103074](https://arxiv.org/abs/gr-qc/0103074))

* {#DutschFredenhagen01b} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic field theory, and deformation quantization_, in [[Roberto Longo]] (ed.), _Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects_, volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001
([arXiv:hep-th/0101079](https://arxiv.org/abs/hep-th/0101079))

* {#HollandRuan01} [[Stefan Hollands]], Weihua Ruan, _The State Space of Perturbative Quantum Field Theory in Curved Spacetimes_, Annales Henri Poincare 3 (2002) 635-657 ([arXiv:gr-qc/0108032](https://arxiv.org/abs/gr-qc/0108032))

* {#BrunettiFredenhagenRibeiro12} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Pedro Lauridsen Ribeiro]], section 4.1 of _Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields_ ([arXiv:1209.2148](https://arxiv.org/abs/1209.2148))

### Review

* {#FredenhagenRejzner12} [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Perturbative algebraic quantum field theory_, In _Mathematical Aspects of Quantum Field Theories_, Springer 2016 ([arXiv:1208.1428](https://arxiv.org/abs/1208.1428))

* {#KhavkineMoretti14} [[Igor Khavkine]], [[Valter Moretti]], _Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction_, Chapter 5 in [[Romeo Brunetti]] et al. (eds.) _Advances in Algebraic Quantum Field Theory_, Springer, 2015 ([arXiv:1412.5945](https://arxiv.org/abs/1412.5945))

* [[Katarzyna Rejzner]], section 4.4.1 of _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

* {#Duetsch18} [[Michael Dütsch]], def. 1.2 and equation (2.47) in _[[From classical field theory to perturbative quantum field theory]]_, 2018

[[!redirects microcausal polynomial observables]]

[[!redirects microcausal observable]]
[[!redirects microcausal observables]]

[[!redirects microcausal functional]]
[[!redirects microcausal functionals]]


microcosm principle

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

#Contents#
* table of contents
{:toc}

## Idea

In [[higher algebra]]/[[higher category theory]] one can define (generalized) [[algebra|algebraic structures]] [[internalization|internal]] to [[categories]] which themselves are equipped with certain algebraic structure, in fact with the _same kind_ of algebraic structure. In ([Baez-Dolan 97](#BaezDolan)) this has been called the _microcosm principle_.

> **Microcosm principle**: _Certain [[algebra|algebraic structures]] can be defined [[internalization|in]] any category equipped with a [[categorification|categorified version]] of the same structure._

A standard example is the notion of _[[monoid object]]_ which make sense in a _[[monoidal category]]_, and the notion of _[[commutative monoid object]]_ which makes sense in a _[[symmetric monoidal category]]_.

Moreover, in many cases the "internal" version of the structure can be identified with [[lax morphisms]] of categorified structures with [[terminal object|terminal]] domain.  For instance, a monoid in a monoidal category is equivalently a [[lax monoidal functor]] with terminal domain.  More generally, the categorified structure can often be generalized to a "virtual" ([[generalized multicategory]]-like) structure.

The term "microcosm principle" arises from the idea that the monoid object (the 'microcosm') can only thrive in a category (a 'macrocosm') that resembles itself.

## Examples

* [[monoid objects]] can be defined in [[monoidal categories]] (categorified monoids), or more generally [[multicategories]] (virtual monoidal categories).
* [[commutative monoid objects]] can be defined in [[symmetric monoidal categories]], or more generally in [[braided monoidal categories]].
* Categorifying these examples, [[monoidal categories]] are instances of [[pseudomonoids]], which can be defined in any [[monoidal bicategory]].  There are also versions for braided, symmetric, closed, compact closed, and so on.
* [[trace in a bicategory|traces]] can be defined in bicategories equipped with a [[shadow]] (a categorified trace).
* [[categories]] are instances of [[monads]], which can be defined in any [[bicategory]] or more generally any [[double category]] (both a kind of categorified category) or [[virtual double category]].  This generalization includes [[internal categories]] and [[enriched categories]].
* [[enriched categories]] are also objects of the free cocompletion of an [[enriched bicategory]] under a kind of [[Kleisli object]].
* [[generalized multicategories]] are naturally defined internal to a [[virtual double category]], which is itself a kind of generalized multicategory.
* [[double categories]] are instances of [[intermonads]], which can be defined in any [[intercategory]] (a categorified double category).
* [[duoids]] (objects with two monoidal structures) can be defined in any [[duoidal category]] (a category with two monoidal structures).
* [[Frobenius algebras]] can be defined in any [[star-autonomous category]], and cocomplete $\ast$-autonomous posets are Frobenius algebras in the $\ast$-autonomous category [[Sup]], while [[promonoidal category|pro]]-$\ast$-autonomous categories are [[Frobenius pseudomonoids]] in the $\ast$-autonomous (indeed, compact closed) bicategory [[Prof]].
* The prototypical example of a [[2-fibration]] consists of internal [[fibrations in a 2-category]].
* [[type theories]] can be defined inside a "2-type theory"; see [[adjoint type theory]] and [[logical framework]].

## Formalizations

The microcosm principle is a general heuristic, but in some contexts, general versions of it can be defined and proven formally.  These do not generally include all examples, but often include many of them.

### For $n$-categorical operad algebras

[Baez and Dolan](#BaezDolan) gave the following precise definition: for any [[operad]] $\mathcal{O}$, an $n$-categorical $\mathcal{O}$-algebra internal to an $(n+1)$-categorical $\mathcal{O}$-algebra $A$ is a [[lax morphism]] from the [[terminal object|terminal]] $(n+1)$-categorical $\mathcal{O}$-algebra to $A.$

### For categorical algebras over cartesian monads

[Batanin](#Batanin) gave the following definition: for a [[cartesian monad]] $T$, an internal $T$-algebra in an algebra $A$ for the extension of $T$ to [[internal categories]] is a lax $T$-morphism from the [[terminal object|terminal]] such algebra to $A$.  More generally, $A$ can be an algebra for some other monad, for instance which builds in some commutativity.

### For categorical algebras over Lawvere theories

[Hasuo, Jacobs, and Sokolova](#HJS) gave the following very similar definition: for a [[Lawvere theory]] $L$, an internal $L$-algebra in a categorical $L$-algebra (an $L$-algebra in [[Cat]]) is a lax $L$-morphism from the [[terminal object|terminal]] such algebra to $A$.  They used this to prove a general result about the liftings of $L$-algebra structures to coalgebras.

### For algebras over (∞,1)-operads
 {#ForAlgebrasOverInfinityOperads}

Another such formalization was given (independently) in ([Lurie](##Lurie)) in the context of [[higher algebra]] in [[homotopy theory]]/[[(∞,1)-category theory]]:

given an [[(∞,1)-operad]] $\mathcal{O}$ we have:

1. an [[(∞,1)-category]] $\mathcal{C}$ equipped with algebraic structure as encoded by $\mathcal{O}$ -- an $\mathcal{O}$-[[monoidal (∞,1)-category]] -- is equivalently encoded by a [[coCartesian fibration of (∞,1)-operads]] $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$;

1. an [[(∞,1)-algebra over an (∞,1)-operad]] over $\mathcal{O}$ can be defined internal to such a $\mathcal{O}$-monoidal $(\infty,1)$-category  $\mathcal{C}$ and is equivalently given by a [[section]] of this map.

See at _[[(∞,1)-algebra over an (∞,1)-operad]]_ for examples and further details.

## Reference

The term "microcosm principle" was coined in

* [[John Baez]], [[James Dolan]], _Higher-Dimensional Algebra III_ ([arXiv:q-alg/9702014](http://arxiv.org/abs/q-alg/9702014))
 {#BaezDolan}

Discussion is in

* [The Microcosm Principle](http://golem.ph.utexas.edu/category/2008/12/the_microcosm_principle.html)

Other formalizations can be found in

* [[Michael Batanin]], _The Eckman-Hilton argument and higher operads_, [arxiv:0207281](https://arxiv.org/abs/math/0207281)
 {#Batanin}

* Ichiro Hasuo, [[Bart Jacobs]], and Ana Sokolova. (2008)  _The Microcosm Principle and Concurrency in Coalgebra_.  In: Amadio R. (eds) _Foundations of Software Science and Computational Structures_. FoSSaCS 2008. Lecture Notes in Computer Science, vol 4962. Springer, Berlin, Heidelberg. [doi:10.1007/978-3-540-78499-9_18](https://doi.org/10.1007/978-3-540-78499-9_18), [paper](http://cs.uni-salzburg.at/~anas/papers/HasuoJS.pdf), [slides](http://cs.uni-salzburg.at/~anas/papers/MicrocosmPresentation.pdf), [more slides](http://cs.uni-salzburg.at/~anas/papers/Ohrid2008ana.pdf)
 {#HJS}

* Ichiro Hasuo, Chris Heunen, Bart Jacobs, Ana Sokolova, _Coalgebraic Components in a Many-Sorted Microcosm_,  International Conference on Algebra and Coalgebra in Computer Science CALCO 2009: Algebra and Coalgebra in Computer Science pp 64-80, ([pdf](http://homepages.inf.ed.ac.uk/cheunen/publications/2009/component/component.pdf))

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}


microflexible sheaf

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

#Contents#
* table of contents
{:toc}

## Definition

Denote by $Emb_n$ the [[site]] of $n$-dimensional [[smooth manifolds]] and [[open embeddings]].

An [[(∞,1)-sheaf]] $F\colon Emb_n^op\to Top$ of [[topological spaces]] is __microflexible__ if for any closed inclusion $K\to K'$ of [[compact spaces]],
the induced map $F(K')\to F(K)$ is a [[Serre microfibration]].

An [[(∞,1)-sheaf]] $F\colon Emb_n^op\to Top$ of [[topological spaces]] is __flexible__ if for any closed inclusion $K\to K'$ of [[compact spaces]],
the induced map $F(K')\to F(K)$ is a [[Serre fibration]].

## Gromov's theorem

Given an [[open manifold]] $M$,
the inclusion of microflexible sheaves into flexible sheaves
on the [[slice site]] site $Emb_n/M$ is an [[equivalence of (∞,1)-categories]].

## Related concepts

* [[h-principle]]

* [[Serre microfibration]]

## Literature

The original reference is

* [[M. L. Gromov]], _Stable mappings of foliations into manifolds_, Mathematics of the USSR-Izvestiya 3:4 (1969), 671-694.  [doi](http://dx.doi.org/10.1070/im1969v003n04abeh000796).

The canonical reference is Section 2.2.1 of

* [[Mikhael Gromov]], _Partial differential relations_, 1986, [doi](http://dx.doi.org/10.1007/978-3-662-02267-2).

See also

* [[John Francis]], *The H-principle for microflexible sheaves*, 2010 ([pdf](http://math.northwestern.edu/~jnkf/classes/hprin/20microflexible.pdf))

* [[Alexander Kupers]], Section 2 of: *Three applications of delooping to H-principles*, Geom Dedicata **202** (2019) 103–151  ([arXiv:1701.06788](https://arxiv.org/abs/1701.06788), [doi:10.1007/s10711-018-0405-7](https://doi.org/10.1007/s10711-018-0405-7))

[[!redirects microflexible sheaves]]


microformal morphism

* Theodore Th. Voronov, _Microformal geometry and homotopy algebras_, [arxiv/1411.6720](https://arxiv.org/abs/1411.6720)

> We enlarge the category of smooth maps of (super)manifolds by introducing a notion of 'microformal morphism, starting from the classical notion of canonical relation and using formal power expansions in the cotangent directions. The result is a 'formal category'; that is, the composition of microformal morphisms is defined by a formal power series. Microformal morphisms act on functions by pullbacks which are in general nonlinear transformations. More precisely, they are formal mappings of formal manifolds of even functions ("bosonic fields"). Their derivatives at each point are algebra homomorphisms. This suggests a notion of "nonlinear homomorphisms" of algebras and an extension of the classic algebra/geometry duality. There is a parallel "fermionic" version of these constructions.
Using microformal geometry, we obtain a general construction of L∞-morphisms for functions on homotopy Poisson (P?) or homotopy Schouten (S?) manifolds. Further, we show that the familiar adjoint of a linear operator on vector spaces or vector bundles can be generalized to the nonlinear situation as a microformal morphism. We apply this to L∞-algebroids and show that an L∞-morphism of L∞-algebroids induces an L∞-morphism of the homotopy Lie--Poisson algebras of functions on the dual vector bundles. We apply that to higher Koszul brackets on forms and triangular L∞-bialgebroids. We also obtain a quantum version (for the bosonic case) related with the above as the Schroedinger equation is related with the Hamilton--Jacobi equation. We show that nonlinear pullbacks by microformal morphisms are the limits when ℏ→0 of "quantum pullbacks", which are Fourier integral operators of special type.

Related entries: [[cotangent bundle]], [[Poisson manifold]], [[microlocal analysis]], [[Weinstein symplectic category]]