Ieke Moerdijk

**Ieke Moerdijk** is a professor of mathematics at University of Utrecht.  His main interests are in [[sheaf]] and [[topos theory]], [[descent]] and [[cohomology]], [[synthetic differential geometry]], [[operads]], [[groupoids]], [[stacks]] and [[orbifolds]], [[Lie groupoids]] and foliations, and abstract homotopy theory (Quillen [[model categories]]).

* [website](http://www.math.uu.nl/people/moerdijk/), [wikipedia](http://en.wikipedia.org/wiki/Ieke_Moerdijk), [GQT:researchers](http://www.gqt.nl/members.php)

The website of the research group is [here](http://www.math.ru.nl/~mgroth/group/group.html).

## Selected writings

On the [[categorical semantics]] of well-founded [[inductive types]] ([[W-types|$\mathcal{W}$-types]]) as given by [[initial algebra over an endofunctor|initial algebras over]] [[polynomial functor|polynomial]] [[endofunctors]] on the [[type system]]:

* {#MoerdijkPalmgren00} [[Ieke Moerdijk]], [[Erik Palmgren]], *Wellfounded trees in categories*, Annals of Pure and Applied Logic **104** 1-3 (2000) 189-218 [doi:10.1016/S0168-0072(00)00012-9]

On [[generalized Reedy categories]]:

* {#BergerMoerdijk} [[Clemens Berger]], [[Ieke Moerdijk]], *On an extension of the notion of Reedy category*, Mathematische Zeitschrift **269** (2011) 977–1004 [[arXiv:0809.3341](http://arxiv.org/abs/0809.3341), [doi:10.1007/s00209-010-0770-x](https://doi.org/10.1007/s00209-010-0770-x)]

## books with table of contents and links on the $n$Lab ##

* (with [[Saunders MacLane]]) [[Sheaves in Geometry and Logic]]

  on [[sheaf and topos theory]]

* (with [[Gonzalo Reyes]]) [[Models for Smooth Infinitesimal Analysis]]

  on [[synthetic differential geometry]]

__Other books__ by Moerdijk include

* (with [[Janez Mrčun]]) _Introduction to foliations and Lie groupoids_, Cambridge Studies in Advanced Mathematics, 91. Cambridge University Press, Cambridge, 2003. x+173 pp. ISBN: 0-521-83197-0

* _Classifying spaces and classifying topoi_, Lecture Notes in Mathematics __1616__, Springer 1995. vi+94 pp. ISBN: 3-540-60319-0

* (with [[Andre Joyal]]) _Algebraic set theory_, London Mathematical Society Lecture Note Series __220__, Cambridge University Press 1995. viii+123 pp. ISBN: 0-521-55830-1

  on [[algebraic set theory]]

[[!redirects I. Moerdijk]]

category: people


Ignatios Antoniadis

* [inspire page](https://inspirehep.net/authors/1018276?ui-citation-summary=true)

* [Wikipedia entry](https://fr.wikipedia.org/wiki/Ignatios_Antoniadis) (French)

## Selected writings

On quasi-[[phenomenology|realistic]] "flipped" [[SU(5)|$SU(5)$]]-[[GUT]],  modeled in 4d [[heterotic string theory]] and subsuming realistic [[Starobinsky model of cosmic inflation|Starobinsky-type cosmic inflation]]:

* [[Ignatios Antoniadis]], [[John Ellis]], [[John S. Hagelin]], [[Dimitri Nanopoulos]], *The flipped $SU(5) \times U(1)$ string model revamped*, Physics Letters B **231** 1–2 (1989) 65-74 [doi:10.1016/0370-2693(89)90115-9]

* {#AntoniadisNanopoulosRizos21a} [[Ignatios Antoniadis]], [[Dimitri Nanopoulos]], [[John Rizos]], *Cosmology of the string derived flipped $SU(5)$*, J. Cosm. Astrop. Phys **03** (2021) 017 [[arXiv:2011.09396](https://arxiv.org/abs/2011.09396), [doi:10.1088/1475-7516/2021/03/017](https://doi.org/10.1088/1475-7516/2021/03/017)]

* {#AntoniadisNanopoulosRizos21a} [[Ignatios Antoniadis]], [[Dimitri Nanopoulos]], [[John Rizos]], *Particle physics and cosmology of the string derived no-scale flipped $SU(5)$*,  Eur. Phys. J. C **82** (2022) 377  [[arXiv:2112.01211](https://arxiv.org/abs/2112.01211), [doi:10.1140/epjc/s10052-022-10353-6](https://doi.org/10.1140/epjc/s10052-022-10353-6)]

category: people


Igor Baković

* [website](http://www.irb.hr/users/ibakovic/)

## Selected writings

On [[fibrations of 2-categories]]:

* [[Igor Baković]], *Fibrations of bicategories* (2010) [[pdf](http://www.irb.hr/korisnici/ibakovic/groth2fib.pdf), [[Bakovic-FibrationsOfBicategories.pdf:file]]]

On the [[2-Grothendieck construction]]:

* [[Igor Baković]], *Grothendieck construction for bicategories* [[pdf](https://www2.irb.hr/korisnici/ibakovic/sgc.pdf)]

## Related entries

* [[principal 2-bundle]]

* [[Yoneda lemma for bicategories]]

* [[para construction]]

category: people

[[!redirects Igor Bakovic]]


Igor Bandos

* [webpage](https://www.ikerbasque.net/es/igor-bandos)

## Selected writings

On the [[superembedding approach]] for [[super p-branes|super $p$-branes]]/[[Green-Schwarz sigma-models]]:

* {#BPSTV95} [[Igor Bandos]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], [[Dmitry Volkov]], _Superstrings and supermembranes in the doubly supersymmetric geometrical approach_, Nucl. Phys. B **446** (1995) 79-118 [[arXiv:hep-th/9501113](https://arxiv.org/abs/hep-th/9501113)]

* [[Igor A. Bandos]], [[Dmitri P. Sorokin]], *Superembedding approach to superstrings and super-$p$-branes*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2301.10668](https://arxiv.org/abs/2301.10668)]

On [[superstrings]] and [[super membranes]] formulated in [[superspace]]:

* [[Igor Bandos]], [[Dmitrij Sorokin]], [[Dmitrij Volkov]], _On the generalized action principle for superstrings and supermembranes_, Phys. Lett. B352:269-275, 1995 ([arXiv:hep-th/9502141](https://arxiv.org/abs/hep-th/9502141))

On the [[Green-Schwarz sigma-model]] for the [[M5-brane]]:

* {#BLNPST97} [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _Covariant Action for the Super-Five-Brane of M-Theory_, Phys. Rev. Lett. 78 (1997) 4332-4334 ([arXiv:hep-th/9701149](http://arxiv.org/abs/hep-th/9701149))

* [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On the equivalence of different formulations of the M Theory five--brane_, Phys. Lett. B408 (1997) 135-141 ([arXiv:hep-th/9703127](http://arxiv.org/abs/hep-th/9703127))

On [[type II supergravity]]:

* {#BNS04} [[Igor Bandos]], [[Alexei Nurmagambetov]], [[Dmitri Sorokin]], _Various Faces of Type IIA Supergravity_, Nucl. Phys. B676 (2004) 189-228 ([arXiv:hep-th/0307153](https://arxiv.org/abs/hep-th/0307153))

On [[double supergeometry]]:

* {#Bandos15} [[Igor Bandos]], _Superstring in doubled superspace_, Physics Letters B Volume 751, 17 December 2015, Pages 408-412 ([arXiv:1507.07779](https://arxiv.org/abs/1507.07779))

On [[super-exceptional generalized geometry]]:

* {#Bandos16} [[Igor Bandos]], _Exceptional field theories, superparticles in an enlarged 11D superspace and higher spin theories_, Nucl. Phys. B925 (2017) 28-62 ([arXiv:1612.01321](https://arxiv.org/abs/1612.01321))

On [[membranes]] in [[D=4 supergravity]] ([[super 2-brane in 4d]]):

* [[Igor Bandos]], [[Fotis Farakos]] Stefano Lanza, L. Martucci,  [[Dmitri Sorokin]], _Higher Forms and Membranes in 4D Supergravities_, in: Proceedings of _[[Higher Structures in M-Theory 2018]]_, Fortschritte der Physik, Special Issue Volume 67, Issue 8-9 ([arXiv:1903.02841](https://arxiv.org/abs/1903.02841), [doi:10.1002/prop.201910020](https://doi.org/10.1002/prop.201910020))

Proposal for a non-abelian and [[supersymmetry|supersymmetric]] [[DBI-action]] for [[coincident brane|coincident]] [[D0-branes]]:

* [[Igor Bandos]], Unai D. M. Sarraga, *Complete nonlinear action for supersymmetric multiple D0-brane system*, Phys. Rev. D **106** (2022) 066004 [[doi:10.1103/PhysRevD.106.066004](https://doi.org/10.1103/PhysRevD.106.066004), [arXiv:2204.05973](https://arxiv.org/abs/2204.05973)]

* [[Igor Bandos]], Unai D. M. Sarraga, *Properties of multiple D0-brane system: 11D origin, equations of motion and their solutions* [[arXiv:2212.14829](https://arxiv.org/abs/2212.14829)]

* [[Igor A. Bandos]], [[Dmitri P. Sorokin]], *Superembedding approach to superstrings and super-$p$-branes*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2301.10668](https://arxiv.org/abs/2301.10668)]

## Related entries

* [[M5-brane]]

category: people

[[!redirects Igor A. Bandos]]


Igor Batalin

* [Scholarpedia page](http://www.scholarpedia.org/article/User:Igor_A._Batalin)

## related $n$Lab entries

* [[Batalin-Vilkovisky formalism]]

category: people


Igor Dolgachev

* [webpage](http://www.math.lsa.umich.edu/~idolga/)

## Selected writings

On basic concepts of [[quantum field theory]] and [[string theory]]:

* [[Igor Dolgachev]], *Introduction to string theory for mathematicians*, lecture at *[1999 Summer on Algebraic Geometry and Physics](https://people.sissa.it/~bruzzo/sagp99/sagp.html)* (1999) [[pdf](https://dept.math.lsa.umich.edu/~idolga/stringy.pdf), [[Dolgachev-IntroductionStringTheory.pdf:file]]]

## Related entries

* [[Polyakov action]]

category: people


Igor Frenkel

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

## Related $n$Lab entries

* [[vertex operator algebra]]

* [[Moonshine]]

category: people
[[!redirects I. Frenkel]]


Igor Khavkine

Igor Khavkine is a mathematical physicist working on [[field theory]].

* [website](http://users.math.cas.cz/~khavkine/)

## Selected publications

A review and careful re-analysis of [[covariant phase spaces]] in local Lagrangian field theory:

* {#Khavkine14a} [[Igor Khavkine]], _Covariant phase space, constraints, gauge and the Peierls formula_, International Journal of Modern Physics A 29 (2014) ([arXiv:1402.1282](http://arxiv.org/abs/1402.1282), [doi:10.1142/s0217751x14300099](http://www.worldscientific.com/doi/abs/10.1142/S0217751X14300099))

Solving the existence problem of local gauge invariant [[observables]] in [[general relativity]] by showing that while these do not exist as global functions, they do exist as a [[sheaf]] of functions on [[covariant phase space]]:

* {#Khavkine14} [[Igor Khavkine]],  _Local and gauge invariant observables in gravity_, Class. Quantum Grav. 32 185019, 2015 ([arXiv:1503.03754](https://arxiv.org/abs/1503.03754), [CQG+](http://cqgplus.com/2015/09/16/local-and-gauge-invariant-observables-in-gravity/), [pdf slides](http://www.science.unitn.it/~moretti/convegno/khavkine.pdf) for talk at [Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html) Levico Terme, Italy, September 2014)

On [[AQFT on curved spacetimes]] and its [[Hadamard states]]:

* {#KhavkineMoretti15}  [[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) [[doi:10.1007/978-3-319-21353-8](https://doi.org/10.1007/978-3-319-21353-8)]

On [[higher gauge fields]] viewed through the [[variational bicomplex]]:

* [[Igor Khavkine]], *Topology, rigid cosymmetries and linearization instability in higher gauge theories*, Annales Henri Poincaré **16** (2015) 255–288, [[arXiv:1303.2406](https://arxiv.org/abs/1303.2406), [doi:10.1007/s00023-014-0321-9](https://doi.org/10.1007/s00023-014-0321-9)]

On [[jet comonads]]:

* [[Igor Khavkine]], [[Urs Schreiber]], _[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

## Related $n$Lab entries

* [[Green hyperbolic partial differential equation]]

* [[conal manifold]]

* [[jet comonad]]

category: people


Igor Klebanov

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

## Selected writings

Inclusion of heavy [[flavor (particle physics)|flavors]] ([[strange quarks]]/[[kaons]], etc.) into the [[Skyrme model]]:

* [[Curtis Callan]], [[Igor Klebanov]], _Bound-state approach to strangeness in the Skyrme model_, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (doi10.1016/0550-3213(85)90292-5)

* [[Curtis Callan]], K. Hornbostel, [[Igor Klebanov]], _Baryon masses in the bound state approach to strangeness in the skyrme model_, Physics Letters B Volume 202, Issue 2, 3 March 1988, Pages 269-275 (doi10.1016/0370-2693(88)90022-6)

On [[black holes in string theory]] via the [[BFSS matrix model]]:

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]],  _Schwarzschild Black Holes from Matrix Theory_, Phys. Rev. Lett. 80:226-229 (1998) [[arXiv:hep-th/9709091](https://arxiv.org/abs/hep-th/9709091)]

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzchild Black Holes in Matrix Theory II_, JHEP 9801:008 (1998) [[arXiv:hep-th/9711005](https://arxiv.org/abs/hep-th/9711005)]

* [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzschild Black Holes in Various Dimensions from Matrix Theory_, Phys.Lett.B416:62-66 (1998) [[arXiv:hep-th/9709108](https://arxiv.org/abs/hep-th/9709108)]

Introducing the dictionary of the [[AdS-CFT correspondence]]:

* {#GubserKlebanovPolyakov98} [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], *Gauge theory correlators from non-critical string theory*, Physics Letters B **428** 105-114 (1998) [[hep-th/9802109](http://arxiv.org/abs/hep-th/9802109), doi:10.1016/S0370-2693(98)00377-3]

> Relations between gauge fields and strings present an old, fascinating and unanswered question. The full answer to this question is of great importance for theoretical physics. It will provide us with a theory of quark [[confinement]] by explaining the dynamics of color-electric fluxes.

On the [[pp-wave spacetime]] [[Penrose limit]] of the [[AdS5-CFT4 correspondence]]:

* N. Itzhaki, [[Igor Klebanov]], [[Sunil Mukhi]], _PP Wave Limit and Enhanced Supersymmetry in Gauge Theories_, JHEP 0203 (2002) 048 ([arXiv:hep-th/0202153](https://arxiv.org/abs/hep-th/0202153))

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

* [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], *A semi-classical limit of the gauge/string correspondence*, Nucl. Phys. B **636** (2002) 99-114 [[arXiv:hep-th/0204051](https://arxiv.org/abs/hep-th/0204051), doi:10.1016/S0550-3213(02)00373-5]

On [[double trace operators]] in [[AdS/CFT]]:

* [[Steven Gubser]], [[Igor Klebanov]], _A universal result on central charges in the presence of double-trace deformations_, Nucl. Phys. B656 (2003) 23-36 ([arXiv:hep-th/0212138](https://arxiv.org/abs/hep-th/0212138))

On the [[confinement]]/[[mass gap]]-problem in [[holographic QCD]]:

* {#Klebanov21} [[Igor Klebanov]], *Remarks on Color Confinement*, talk at [Strings 2021](https://www.ictp-saifr.org/strings2021/) ([pdf](https://www.ictp-saifr.org/wp-content/uploads/2021/06/DiscussionConfinementKlebanov.pdf), [video](https://youtu.be/B9NI2LE7d68?t=262))

category: people


Igor Krichever

Igor Krichever (1950-2022)

* [insitute page](https://www.math.columbia.edu/~krichev/)

* [InSpire page](https://inspirehep.net/authors/1001815)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=47912)

## Selected writings

On [[Seiberg-Witten theory]] in  relation to [[integrable systems]]:

* [[Alexander Gorsky]], [[Igor Krichever]], [[Andrei Marshakov]], [[Andrei Mironov]], [[Andrey Morozov]], *Integrability and Seiberg-Witten Exact Solution*, Phys. Lett. B **355** (1995) 466-474 [[arXiv:hep-th/9505035](https://arxiv.org/abs/hep-th/9505035), doi:10.1016/0370-2693(95)00723-X]

category: people


Igor Křiž

[[!redirects Igor Kriz]]

* [website](http://www.math.lsa.umich.edu/~ikriz/)

## Selected writings

On the [[rational homotopy type]] of [[configuration spaces of points]]:

* [[Igor Kriz]], _On the Rational Homotopy Type of Configuration Spaces_, Annals of Mathematics Second Series, Vol. 139, No. 2 (Mar., 1994), pp. 227-237 ([jstor:2946581](https://www.jstor.org/stable/2946581))

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 [[equivariant complex oriented cohomology theory]] and [[equivariant formal group laws]]:

* [[Michael Cole]], [[John Greenlees]], [[Igor Kriz]], _Equivariant Formal Group Laws_, Proceedings of the LMS, Volume81, Issue2 (2000)([doi:10.1112/S0024611500012466]( https://doi.org/10.1112/S0024611500012466))

* [[Michael Cole]], [[John Greenlees]], [[Igor Kriz]], _The universality of equivariant complex bordism_, Math Z 239, 455–475 (2002) ([doi:10.1007/s002090100315](https://doi.org/10.1007/s002090100315))

On [[modular equivariant elliptic cohomology]] in [[type II string theory]]/[[F-theory]]:

* {#SatiKriz04} [[Igor Kriz]], [[Hisham Sati]], _M-theory, type IIA superstrings, and elliptic cohomology_, Adv. Theor. Math. Phys. 8 (2004), no. 2, 345–394 ([euclid:euclid.atmp/1091543172](http://projecteuclid.org/euclid.atmp/1091543172), [arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))

* {#KrizSati04b} [[Igor Kriz]], [[Hisham Sati]], _Type IIB String Theory, S-Duality, and Generalized Cohomology_, Nucl.Phys. B715 (2005) 639-664 ([arXiv:hep-th/0410293](http://arxiv.org/abs/hep-th/0410293))

* {#KrizSati05} [[Igor Kriz]], [[Hisham Sati]], _Type II string theory and modularity_, JHEP 0508 (2005) 038 ([arXiv:hep-th/0501060](http://arxiv.org/abs/hep-th/0501060))

Discussion of [[equivariant ordinary cohomology]] ([[Bredon cohomology]]) over the point but in arbitrary [[RO(G)-degree]], for [[equivariance group]] a [[dihedral group]] of [[order of a group|order]] $2p$:

* [[Igor Kriz]], [[Yunze Lu]], *On the $RO(G)$-graded coefficients of dihedral equivariant cohomology*, Mathematical Research Letters **27** 4 (2020) ([arXiv:2005.01225](https://arxiv.org/abs/2005.01225), [doi:10.4310/MRL.2020.v27.n4.a7](https://dx.doi.org/10.4310/MRL.2020.v27.n4.a7))

The [[Adams spectral sequence]] for [[Real cobordism cohomology]]:

* {#Kriz01} [[Po Hu]], [[Igor Kriz]], *Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence*, Topology **40** (2001) 317-399 [doi:10.1016/S0040-9383(99)00065-8, [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf)]

More on [[Real  cobordism cohomology]]:

* [[Po Hu]], [[Igor Kriz]], *Some Remarks on Real and Algebraic Cobordism*, K-Theory **22** (2001) 335–366 [[pdf](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/43150/10977_2004_Article_321295.pdf?sequence=1), [doi:10.1023/A:1011196901303](http://dx.doi.org/10.1023/A:1011196901303)]

* [[Po Hu]], [[Igor Kriz]], *Topological Hermitian Cobordism*, Journal of Homotopy and Related Structures, **11** (2016) 173–197 [[arXiv:1110.5608](https://arxiv.org/abs/1110.5608), [doi:10.1007/s40062-014-0100-9](https://doi.org/10.1007/s40062-014-0100-9)]

For the use of [[category theory]] in [[integration]], see

* [[Igor Kriz]], [[Aleš Pultr]], *Categorical Geometry and Integration Without Points*, Applied Categorical Structures (2014) Vol. 22, p.79-97 ([arXiv:1101.3762](https://arxiv.org/abs/1101.3762), [doi:10.1007/s10485-012-9295-2](https://doi.org/10.1007/s10485-012-9295-2))

category: people


Igor Rivin

* [webpage](https://math.temple.edu/~rivin/)

## related $n$Lab entries

* [[prime geodesic theorem]]

category: people


Igor Schnakenburg

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=229985)

## Selected writings

On [[D=11 supergravity]] (supposedly the low-energy limit of [[M-theory]]):

* [[André Miemiec]], [[Igor Schnakenburg]], _Basics of M-Theory_, Fortsch. Phys. 54 (2006) 5-72 ([arXiv:hep-th/0509137](http://arxiv.org/abs/hep-th/0509137), [doi:10.1002/prop.200510256]( https://doi.org/10.1002/prop.200510256))

category: people


Igor Shafarevich

__Igor Rostislavovich Shafarevich__ is an algebraic geometer and number theorist. He is also a pioneer in the study of infinite-dimensional algebraic groups.

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

## Selected writings

On [[algebraic geometry]]:

* {#ShafarevichVol1} [[Igor Shafarevich]], *Basic Algebraic Geometry 1 -- Varieties in Projective Space*, Springer (1977, 1994, 2013) [[pdf](http://userpage.fu-berlin.de/aconstant/Alg2/Bib/Shafarevich.pdf), [doi:10.1007/978-3-642-57908-0](https://link.springer.com/book/10.1007/978-3-642-57908-0)]

On basic [[linear algebra]], [[representation theory]] and basic [[geometry]] ([[Euclidean spaces|Euclidean]]-, [[affine spaces|affine]]-, [[projective space|projective]]-, [[hyperbolic spaces]]):

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

category: people

[[!redirects Igor R. Shafarevich]]
[[!redirects I. R. Shafarevich]]


Igor Suslov

* [webpage](http://www.kapitza.ras.ru/~suslov/)

## related $n$Lab entries

* [[asymptotic series]]

* [[Feynman perturbation series]]

* [[Yang-Mills instanton]], [[instanton in QCD]]

category: people


Igor Tyutin

[[!redirects I. V. Tyutin]]
Igor Tyutin (Russian: И́горь Ви́кторович Тю́тин) is a Russian physicist.
He is one of the originators of the [[BRST]] formalism (the T in BRST).

[Website](http://www.tamm.lpi.ru/staff/tyutin.html)

## Selected works

On what came to be known as the [[BRST complex]]:

 * [[I. V. Tyutin]], (1975), _Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism_, [arXiv:0812.0580](https://arxiv.org/abs/0812.0580).

 * [[D. M. Gitman]], [[I. V. Tyutin]], _Quantization of Fields with Constraints_, Springer, 1990.  [doi:10.1007/978-3-642-83938-2](http://dx.doi.org/10.1007/978-3-642-83938-2).


Igor V. Kanatchikov

* [InSpire page](https://inspirehep.net/authors/1003779)

* [GoogleScholar page](https://scholar.google.de/citations?user=TupRkncAAAAJ&hl=de)

## Selected writings

On [[De Donder-Weyl-Hamiltonian]]-theory ([[classical field theory]] via [[multisymplectic geometry]]):

* [[Igor V. Kanatchikov]], _De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory_, Rept. Math. Phys. **43** (1999) 157-170 [[arXiv:hep-th/9810165](http://arxiv.org/abs/hep-th/9810165), doi:10.1016/S0034-4877(99)80024-X]

and the multisumplectic analog of [[geometric prequantization]]:

* [[Igor V. Kanatchikov]], *Geometric (pre)quantization in the polysymplectic approach to field theory*, in *Differential Geometry and Its Applications*, Proc. Conf., Opava (August 2001) [[arXiv:hep-th/0112263](https://arxiv.org/abs/hep-th/0112263), [pdf](http://conferences.math.slu.cz/8icdga/PDF/309-322.pdf)]

For [[teleparallel gravity]]:

* [[Igor V. Kanatchikov]]. *The De Donder-Weyl Hamiltonian formulation of TEGR and its quantization* [[arXiv:2308.10052](https://arxiv.org/abs/2308.10052)]

[[!redirects Igor Kanatchikov ]]


IKKT matrix model

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

#Contents#
* table of contents
{:toc}

## Idea

The [[KK-compactification]] of [[D=10 super Yang-Mills theory]] all the way to the point yields a [[theory (physics)|theory]] whose [[field (physics)|fields]] are simply elements of the gauge [[Lie algebra]] $\mathfrak{g}$, hence [[matrices]] for a [[matrix Lie algebra]]. This [[model (physics)|physics]] is called the _IKKT matrix model_.

Alternatively this model can be motivated from a certain regularization of the [[worldsheet]] [[action functional]] of the [[superstring]]. This is how it was originally obtained in ([IKKT 96](#IKKT96)).

It has been argued that for $\mathfrak{g} = \mathfrak{su}(N)$ the [[special unitary Lie algebra]] and in the [[large N limit]], this model captures aspects of [[non-perturbative field theory|non-perturbative]] [[type IIB string theory]] (which would be [[F-theory]], see also at _[[M-theory]]_). Therefore this is also called the _IIB matrix model_ (in contrast to the [[BFSS matrix model]] in [[type IIA string theory]]).

Several authors have explored the possibility to lift the derivation of the IKKT model from the [[superstring]] to the [[M2-brane]]. See at _[[membrane matrix model]]_ for more on this.

In ([Kim-Nishimura-Tsuchiya 12](#KimNishimuraTsuchiya12)) it is claimed that computer simulation of the IKKT matrix model, regarded as non-perturbative [[type IIB string theory]], shows a spontaneous emerging spacetime of macroscopic dimension 3+1, with 6 microscopic dimensions. (A similar claim results from a very different argument: the _[[Brandenberger-Vafa mechanism]]_.)

## Related concepts

[[!include brane matrix models -- content]]

* [[lattice gauge theory]]

## References
 {#References}

### General

First inkling of [[matrix models]] from the [[large N limit]] of [[QCD]]:

* {#EguchiKawai82} [[Tohru Eguchi]], [[Hikaru Kawai]], _Reduction of Dynamical Degrees of Freedom in the Large-$N$ Gauge Theory_, Phys. Rev. Lett. 48, 1063 (1982) ([spire:176459](http://inspirehep.net/record/176459), [doi:10.1103/PhysRevLett.48.1063](https://doi.org/10.1103/PhysRevLett.48.1063))

* A. Gonzalez-Arroyo, M. Okawa, _A twisted model for large $N$ lattice gauge theory_, Physics Letters B Volume 120, Issues 1–3, 6 January 1983, Pages 174-178 (doi:10.1016/0370-2693(83)90647-0)

* A. Gonzalez-Arroyo, M. Okawa, _Twisted-Eguchi-Kawai model: A reduced model for large-
$N$ lattice gauge theory_, Phys. Rev. D 27, 2397 (1983) ([doi:10.1103/PhysRevD.27.2397](https://doi.org/10.1103/PhysRevD.27.2397))

The original articles on the IKKT matrix model:

* {#IKKT96} [[Noboyuki Ishibashi]], [[Hikaru Kawai]], Y. Kitazawa, A. Tsuchiya, _A Large-$N$ Reduced Model as Superstring_, Nucl. Phys. B498 (1997) 467-491 ([arXiv:hep-th/9612115](http://arxiv.org/abs/hep-th/9612115), doi:10.1016/S0550-3213(97)00290-3)

* H. Aoki, S. Iso, [[Hikaru Kawai]], Y. Kitazawa, T. Tada, A. Tsuchiya, _IIB Matrix Model_, Prog.Theor.Phys.Suppl.134:47-83,1999 ([arXiv:hep-th/9908038](http://arxiv.org/abs/hep-th/9908038))

See also

* [[Mikhail Movshev]], [[Albert Schwarz]], _On maximally supersymmetric Yang-Mills theories_, Nucl.Phys. B681 (2004) 324-350 ([arXiv:hep-th/0311132](https://arxiv.org/abs/hep-th/0311132))

Review of the grand perspective with emphasis on [[noncommutative geometry]] ([[fuzzy spheres]]):

* {#Steinacker10} [[Harold Steinacker]], *Emergent Geometry and Gravity from Matrix Models: an Introduction*, Class. Quant. Grav. **27** (2010) 133001  [[arXiv:1003.4134](https://arxiv.org/abs/1003.4134)]

* [[Harold Steinacker]], _Non-commutative geometry and matrix models_, PoS QGQGS2011 (2011) 004 [[arXiv:1109.5521](https://arxiv.org/abs/1109.5521)]

* {#Ydri18} [[Badis Ydri]], _Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory_ ([arXiv:1708.00734](https://arxiv.org/abs/1708.00734)), published as: _Matrix Models of String Theory_, IOP 2018 ([ISBN:978-0-7503-1726-9](https://iopscience.iop.org/book/978-0-7503-1726-9))

See also at _[[D-brane geometry]]_.

Derivation from [[open string field theory]] is discussed in

* [[Taejin Lee]], _Covariant Open String Field Theory on Multiple D$p$-Branes_ ([arXiv:1703.06402](https://arxiv.org/abs/1703.06402))

* [[Harold C. Steinacker]], [[Tung Tran]], *Soft limit of higher-spin interactions in the IKKT model* [[arXiv:2311.14163](https://arxiv.org/abs/2311.14163)]

* [[Harold C. Steinacker]], [[Tung Tran]], *Spinorial description for Lorentzian $\mathfrak{hs}$-IKKT* [[arXiv:2312.16110](https://arxiv.org/abs/2312.16110)]

Arguments that full [[Yang-Mills theory]] generalized to [[noncommutative geometry]] is recovered as the [[perturbation theory]] around classical solutions of the IKKT model are in

* H. Aoki, [[Noboyuki Ishibashi]], S. Iso, [[Hikaru Kawai]], Y. Kitazawa, T. Tada, _Noncommutative Yang-Mills in IIB Matrix Model_, Nucl.Phys. B565 (2000) 176-192 ([arXiv:hep-th/9908141](http://arxiv.org/abs/hep-th/9908141))

* Tatsuo Azeyanagi, Masanori Hanada, Tomoyoshi Hirata, _On Matrix Model Formulations of Noncommutative Yang-Mills Theories_, Phys.Rev.D78:105017,2008 ([arXiv:0806.3252](http://arxiv.org/abs/0806.3252))

Emergence of [[higher spin gauge theory|higher spin]] [[gravity]] from the [[IKKT matrix model]]:

* [[Harold Steinacker]], *Higher-spin gravity and torsion on quantized space-time in matrix models*, Journal of High Energy Physics **2020** 111 (2020) [[arXiv:2002.02742](https://arxiv.org/abs/2002.02742), doi:10.1007/JHEP04(2020)111]

* [[Harold Steinacker]], *Gravity as a Quantum Effect on Quantum Space-Time*, Phys.Lett. B **827** (2022) 136946 [[arXiv:2110.03936](https://arxiv.org/abs/2110.03936), [doi:10.1016/j.physletb.2022.136946](https://doi.org/10.1016/j.physletb.2022.136946)]

* [[Yuhma Asano]], [[Harold Steinacker]], *Spherically symmetric solutions of higher-spin gravity in the IKKT matrix model*, Nuclear Physics B **980** (2022) 115843 [[arXiv:2112.08204](https://arxiv.org/abs/2112.08204), [doi:10.1016/j.nuclphysb.2022.115843](https://doi.org/10.1016/j.nuclphysb.2022.115843)]

* [[Harold Steinacker]], [[Tung Tran]], *Spinorial higher-spin gauge theory from IKKT in Euclidean and Minkowski signatures* [[arXiv:2305.19351](https://arxiv.org/abs/2305.19351)]

Arguments that [[closed string field theory]] arises from the [[dynamics]] of [[Wilson loops]] in the IKKT model are in

* M. Fukuma, [[Hikaru Kawai]], Y. Kitazawa, A. Tsuchiya, _String Field Theory from IIB Matrix Model_, Nucl.Phys.B510:158-174,1998 ([arXiv:hep-th/9705128](http://arxiv.org/abs/hep-th/9705128))

* Daiji Ennyu, Hiroshi Kawabe, Naohito Nakazawa, _Note on a Closed String Field Theory from Bosonic IIB Matrix Model_, JHEP 0301 (2003) 025 ([arXiv:hep-th/0212044](http://arxiv.org/abs/hep-th/0212044))

Possibilities of generalizing the IKKT model from [[Lie algebras]] to [[Lie 2-algebras]] in some [[membrane matrix model]] are explored in

* [[Patricia Ritter]], [[Christian Saemann]], _Lie 2-algebra models_ ([arXiv:1308.4892](http://arxiv.org/abs/1308.4892))

Discussion of [[standard model of particle physics|standard model]] [[phenomenology]] within the IKKT model includes

* [[Harold Steinacker]], Jochen Zahn, _An extended standard model and its Higgs geometry from the matrix model_, PTEP 2014 (2014) 8, 083B03 ([arXiv:1401.2020](http://arxiv.org/abs/1401.2020))

See also

* A. Stern, Chuang Xu, _Signature change in matrix model solutions_ ([arXiv:1808.07963](https://arxiv.org/abs/1808.07963))

* [[Frans R. Klinkhamer]], _On the emergence of an expanding universe from a Lorentzian matrix model_, Prog. Theor. Exp. Phys. 2020, 103B03 (2020) ([arXiv:1912.12229](https://arxiv.org/abs/1912.12229))

* [[Frans R. Klinkhamer]], _IIB matrix model: Emergent spacetime from the master field_ ([arXiv:2007.08485](https://arxiv.org/abs/2007.08485))

* [[Frans R. Klinkhamer]], _IIB matrix model: Extracting spacetime points_ ([arXiv:2008.01058](https://arxiv.org/abs/2008.01058))

* [[Frans R. Klinkhamer]], _Regularized big bang and IIB matrix model_ ([arXiv:2009.06525](https://arxiv.org/abs/2009.06525))

* [[Frans R. Klinkhamer]], _M-theory and the birth of the Universe_ ([arXiv:2102.11202](https://arxiv.org/abs/2102.11202))

* [[Frans R. Klinkhamer]], *A first look at the master-field equation of the IIB matrix model* ([arXiv:2105.05831](https://arxiv.org/abs/2105.05831))

* Emmanuele Battista, [[Harold C. Steinacker]], *One-loop effective action of the IKKT model for cosmological backgrounds* [[arXiv:2310.11126](https://arxiv.org/abs/2310.11126)]

* Samuel Laliberte, *Effective mass and symmetry breaking in the IKKT matrix model from compactification* [[arXiv:2401.16401](https://arxiv.org/abs/2401.16401)]

### Computer simulation
 {#ReferencesComputerSimulation}

There are claims that numerical computer simulations (as in [[lattice gauge theory]], see the references [there](lattice+gauge+theory#ForSuperYangMills)) show that the IKKT matrix model  predicts a spontanously generated spacetime where exactly 3+1 dimensions become macroscopic (hence effectively predicts _[[moduli stabilization]]_ in spintaneous [[KK-compactification]] of [[M-theory]] to $D = 3+1$ macroscopic dimensions ):

* {#KimNishimuraTsuchiya12} S.-W. Kim, J. Nishimura, and A. Tsuchiya, _Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions_, Phys. Rev. Lett. 108, 011601 (2012), ([arXiv:1108.1540](https://arxiv.org/abs/1108.1540)).

* S.-W. Kim, J. Nishimura, and A. Tsuchiya, _Late time behaviors of the expanding universe in the IIB matrix model_, JHEP 10, 147 (2012), ([arXiv:1208.0711](https://arxiv.org/abs/1208.0711)).

* Yuta Ito, Jun Nishimura, Asato Tsuchiya, _Large-scale computation of the exponentially expanding universe in a simplified Lorentzian type IIB matrix model_ ([arXiv:1512.01923](https://arxiv.org/abs/1512.01923))

* {#AHINT19} Toshihiro Aoki, Mitsuaki Hirasawa, Yuta Ito, Jun Nishimura, Asato Tsuchiya, _On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model_ ([arXiv:1904.05914](https://arxiv.org/abs/1904.05914))

* Kohta Hatakeyama, Akira Matsumoto, Jun Nishimura, Asato Tsuchiya, Atis Yosprakob, _The emergence of expanding space-time and intersecting D-branes from classical solutions in the Lorentzian type IIB matrix model_ ([arxiv:1911.08132](https://arxiv.org/abs/1911.08132))

* Konstantinos N. Anagnostopoulos, Takehiro Azuma, Yuta Ito, Jun Nishimura, Toshiyuki Okubo, Stratos Kovalkov Papadoudis, _Complex Langevin analysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model_ ([arXiv:2002.07410](https://arxiv.org/abs/2002.07410))

Status report:

* *Progress in the numerical studies of the type IIB matrix model* [[arXiv:2210.17537](https://arxiv.org/abs/2210.17537)]

Further:

* *The emergence of expanding space-time in the Lorentzian type IIB matrix model with a novel regularization* [[arXiv:2307.01681](https://arxiv.org/abs/2307.01681)]

[[!redirects IIB matrix model]]

[[!redirects IKKT model]]


Ilan Barnea

Ilan Barnea is a postdoc at the University of Muenster, Germany, in the research team of Prof. [[Michael Weiss]]. He is interested in abstract homotopy theory, category theory and their applications.

##Papers and preprints

* I. Barnea and [[T. M. Schlank]], 2011, _A Projective Model Structure on
Pro Simplicial Sheaves, and the Relative Etale Homotopy Type_, [arXiv:1109.5477](http://arxiv.org/abs/1109.5477)

*  I. Barnea and [[T. M. Schlank]], 2013, _Functorial Factorizations in Pro
Categories_, [arXiv:1305.4607](http://arxiv.org/abs/1305.4607).

* I. Barnea and [[T. M. Schlank]], (2014), _Model Structures on Ind Categories and the Accessibility Rank of Weak Equivalences_, [arXiv:1407.1817](http://arxiv.org/abs/1407.1817)

* [[I. Barnea]], [[M. Joachim]], [[Snigdhayan Mahanta|S. Mahanta]], (2015). _Model structure on projective systems of C∗-algebras and bivariant homology theories_, [arXiv:1508.04283](http://arxiv.org/abs/1508.04283)

category:people
[[!redirects I. Barnea]]
[[!redirects ilan barnea]]


ILH manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

An _ILH manifold_ -- for a (possibly [[infinite-dimensional manifold|infinite dimensional]]) [[manifold]] modelled on an _[[inverse limit]] of [[Hilbert spaces]]_ -- is a [[topological manifold]] equipped with a kind of [[smooth structure]] that is more general than that of a [[Hilbert manifold]] but less general than that of a [[Fréchet manifold]].

## Related concepts

* [[Banach manifold]]

* [[Hilbert manifold]]

* [[convenient manifold]]

* [[Fréchet manifold]]

## References

* H. Omori, _Infinite dimensional Lie transformation groups_, Springer lecture notes in mathematics 427 (1974)

* H. Omori, _Infinite dimensional Lie groups_, translations of Mathematical monographs, 158

[[!redirects IHL manifold]]
[[!redirects IHL manifolds]]
[[!redirects IHL-manifold]]
[[!redirects IHL-manifolds]]

[[!redirects ILH manifold]]
[[!redirects ILH manifolds]]
[[!redirects ILH-manifold]]
[[!redirects ILH-manifolds]]

[[!redirects ILH space]]
[[!redirects ILH spaces]]


Ilia Pirashvili

## related $n$Lab entries

* [[fundamental groupoid]]

category: people


Ilias Amrani

* [webpage](https://www.muni.cz/people/116793)

## related $n$Lab entries

* [[model structure on dg-algebras]]

category: people


Ilka Agricola

* [webpage](http://www.mathematik.uni-marburg.de/~agricola/)

category: people


Ilka Brunner

* [Institute page](https://www.theorie.physik.uni-muenchen.de/lsluest/members/asc/professors/brunner_ilka/index.html)

## Selected writings

On [[boundary conformal field theory]]:

* [[Ilka Brunner]], [[Michael Douglas]], Albion Lawrence, Christian Romelsberger, _D-branes on the Quintic_, JHEP 0008 (2000) 015 ([arXiv:hep-th/9906200](https://arxiv.org/abs/hep-th/9906200))

On [[Landau-Ginzburg models]]:

* [[Ilka Brunner]], [[Daniel Roggenkamp]], _B-type defects in Landau-Ginzburg models_, JHEP 0708 (2007) 093, ([arXiv:0707.0922](http://arxiv.org/abs/0707.0922))

The graded pivotal bicategory of B-twisted affine LG models is studied in detail in

* Nils Carqueville, [[Daniel Murfet]], _Adjunctions and defects in Landau-Ginzburg models_, Advances in Mathematics, Volume 289 (2016), 480-566, ([arXiv:1208.1481](http://arxiv.org/abs/1208.1481))

Orbifolds of defects are studied in

* [[Ilka Brunner]], [[Daniel Roggenkamp]], _Defects and Bulk Perturbations of Boundary Landau-Ginzburg Orbifolds_, JHEP 0804 (2008) 001, ([arXiv:0712.0188](http://arxiv.org/abs/0712.0188))

* Nils Carqueville, Ingo Runkel, _Orbifold completion of defect bicategories_, ([arXiv:1210.6363](http://arxiv.org/abs/1210.6363))

* [[Ilka Brunner]], Nils Carqueville, Daniel Plencner, _Orbifolds and topological defects_, Comm. Math. Phys. 332 (2014), 669-712, ([arXiv:1307.3141](http://arxiv.org/abs/1307.3141))

* [[Ilka Brunner]], Nils Carqueville, Daniel Plencner, _Discrete torsion defects_, Comm. Math. Phys. 337 (2015), 429-453, ([arXiv:1404.7497](http://arxiv.org/abs/1404.7497))

On [[permutation D-branes]]:

* [[Ilka Brunner]], [[Matthias Gaberdiel]], _Matrix factorisations and permutation branes_, JHEP 0507:012, 2005 ([arXiv:hep-th/0503207](https://xxx.lanl.gov/abs/hep-th/0503207))

On [[M-theory on S1/G_HW times H/G_ADE]]:

* {#BrunnerKarch97} [[Ilka Brunner]], [[Andreas Karch]], _Branes at Orbifolds versus Hanany Witten in Six Dimensions_, JHEP 9803:003, 1998 ([arXiv:hep-th/9712143](https://arxiv.org/abs/hep-th/9712143))

On [[torsion subgroup|torsion]] in the [[D-brane charge quantization in topological K-theory]] related to [[non-geometric vacua]]:

* [[Ilka Brunner]], [[Jacques Distler]], *Torsion D-Branes in Nongeometrical Phases*, Adv. Theor. Math. Phys. **5** (2002) 265-309 \[doi:10.4310/ATMP.2001.v5.n2.a3, [arXiv:hep-th/0102018](https://arxiv.org/abs/hep-th/0102018)\]

* [[Ilka Brunner]], [[Jacques Distler]], Rahul Mahajan, *Return of the Torsion D-Branes*, Adv. Theor. Math. Phys. **5** (2002) 311-352 \[doi:10.4310/ATMP.2001.v5.n2.a4, [arXiv:hep-th/0106262](https://arxiv.org/abs/hep-th/0106262)\]

Discussion of [[Rozansky-Witten theory|Rozansky-Witten]] [[defect QFT]] as an [[extended TQFT]] via the [[cobordism hypothesis]]:

* [[Ilka Brunner]], [[Nils Carqueville]], [[Daniel Roggenkamp]], *Truncated affine Rozansky--Witten models as extended TQFTs*,  Commun. Math. Phys. (2023) [[arXiv:2201.03284](https://arxiv.org/abs/2201.03284), [doi:10.1007/s00220-022-04614-4](https://doi.org/10.1007/s00220-022-04614-4)]

category: people


Ilya L. Shapiro

> beware that there is also mathematician *[[Ilya Shapiro]]*

***

* [GoogleScholar page](https://scholar.google.com.br/citations?user=V1A4RrIAAAAJ&hl=en)

## Selected writings

On [[quantum gravity]]:

* C. Bambi, L. Modesto, [[Ilya Shapiro]]: *[[Handbook of Quantum Gravity]]*, Springer (expected in 2023)

On [[perturbative quantum gravity]]:

* [[Ilya L. Shapiro]], *The background information about perturbative quantum gravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.12319](https://arxiv.org/abs/2210.12319)]

On [[gauge invariance|gauge invariant]] [[renormalization]] of [[perturbative quantum gravity]]:

* [[Petr M. Lavrov]], [[Ilya L. Shapiro]], *Gauge invariant renormalizability of quantum gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.09271](https://arxiv.org/abs/2210.09271)]

category: people


Ilya Prigogine

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

## related $n$Lab entries

* [[dissipative system]], [[autopoiesis]]

* _[[The Self-Organizing Universe]]_

category: people


Ilya Shapiro

> beware that there is also *[[Ilya L. Shapiro]]*.

***

* [GoogleScholar page](https://scholar.google.com/citations?user=fMr8lfIAAAAJ)

## Selected writings

On 1-[[twisted de Rham cohomology]]:

* [[Albert Schwarz]], [[Ilya Shapiro]], *Twisted de Rham cohomology, homological definition of the integral and "Physics over a ring"*, Nucl. Phys. B **809** (2009) 547-560 [[arXiv:0809.0086](https://arxiv.org/abs/0809.0086), [doi:10.1016/j.nuclphysb.2008.10.005](https://doi.org/10.1016/j.nuclphysb.2008.10.005)]

category: people


image

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The **image** of a [[function]] $f\colon A\to B$ between [[sets]] is the [[subset]] of $B$ consisting of all those elements $b\in B$ that are of the form $f(a)$ for some $a\in A$.  This notion can be generalized from [[Set]] to other categories, as follows.

To discuss images in a category $C$, we must first fix a notion of [[subobject]] or [[embedding]] in $C$.  (Sometimes we want these to be the [[monomorphisms]], but sometimes we want the [[regular monomorphisms]] instead.)  Then [[generalized the|the]] **image**  of a [[morphism]] $f\colon A\to B$ in $C$ is a universal factorization of $f$ into a composite $A \to im(f) \to B$ such that $im(f)\to B$ is a subobject, of the specified sort.

Note that in this generality, a given morphism may or may not have an image, although if it does, it is unique up to isomorphism by universality.  In many cases, images can be constructed out of [[limits]] and [[colimits]] in the ambient category.  In particular, in a [[regular category]], images (relative to all monomorphisms) can be constructed as the [[quotient object]] of a [[kernel pair]].

## Definition

Let $C$ be a [[category]], let $M\subset Mono(C)$ be a subclass of the monomorphisms in $C$, and let $f: c \to d$ be a [[morphism]] in $C$.  The ($M$)-**image** of $f$ is the smallest $M$-[[subobject]] $im(f) \hookrightarrow d$ through which $f$ factors (if it exists). The factorizing morphism $c \to im(f)$ is sometimes called the **[[corestriction]]** of $f$ (or **coastriction**, see [mathoverflow](http://mathoverflow.net/questions/29911/whats-the-notation-for-a-function-restricted-to-a-subset-of-the-codomain/65813#65813)):

In other words, it is a factorization $c \overset{e}{\to} im(f) \overset{m}{\to} d$ of $f$ (i.e. $f = m e$) such that $m\in M$, and given any other factorization $f = m' e'$ with $m'\in M$, we have $m \subseteq m'$ as subobjects of $C$ (i.e. $m$ factors through $m'$, $m = m' k$ for some $k$).  Such a factorization is unique up to unique isomorphism, if it exists.

(For instance if $M$ is the class of [[regular monomorphisms]], then the $M$-image is the _[[regular image]]_. See [below](#AsEqualizer) for more.)

This can be phrased equivalently as follows.  Let $C/d$ be the [[slice category]] of $C$ over $d$, and let $M/d$ be its full subcategory whose objects are $M$-morphisms into $d$.  If all images exist in $C$, then taking the image of a map $f: c \to d$ provides a [[left adjoint]]

$$C/d \to M/d$$

to the inclusion $M/d \hookrightarrow C/d$.  More generally, an image of a single morphism $f\colon c\to d$ is a [[universal arrow]] from $f$ to this inclusion.

## Examples

* In [[Set]], for $M=$ monomorphisms = [[injections]], this reproduces the ordinary notion of image.

* More generally in any [[topos]] the [[(epi,mono) factorization system]] gives the factorization through the image $f \colon A \overset{epi}{\to} im(f) \overset{mono}{\to} B$.

* Similarly in an [[abelian category]] ([by definition](abelian+category#AbelianCategory)) [[(epi,mono) factorization system|(epi,mono) factorization]] gives factorization through the image.

* In algebraic categories such as [[Grp]], for $M=$ monomorphisms, this also reproduces the ordinary notions of image.

* In [[Top]], for $M=$ [[subspace]] inclusions, the $M$-image is the set-theoretic image topologized as a [[subspace]] of the [[target|codomain]], see [[subspace topology]] and more comprehensive [Introduction to topology](https://ncatlab.org/nlab/show/Introduction+to+Topology+--+1#subspaces). On the other hand, for $M=$ injective continuous maps, the $M$-image is the set-theoretic image topologized as a [[quotient space]] of the [[source|domain]].

* A [[regular category]] can be defined as a [[finitely complete category]] in which all images exist for $M=$ monomorphisms, and such images are moreover stable under [[pullback]].  In particular, this includes any [[topos]].

* In [[Cat]] (considered as a [[1-category]] of [[strict categories]]), the image of a [[functor]] $F\colon A\to B$ is the smallest [[subcategory]] of $B$ which contains images through $F$ of all morphisms in $A$.  Note that some of the morphisms in the image may not be images of any morphism in $A$; all morphisms in the image of $F$ are [[composite|compositions]] in $B$ of $B$-composable sequences of images of morphisms in $A$, but these themselves do not necessarily form $A$-composable sequences of morphisms in $A$.

  {#RepleteImages} See also at at *[[replete subcategory]]* the section *[Repletion and replete images](replete+subcategory#RepletionAndRepleteImages)*.

  Often it is desireable to instead treat $Cat$ as a [[2-category]], in which case one can use a more 2-categorical notion of image.  See, for instance, *[[full image]]* and *[[essential image]]*.

## Properties

### Basic properties

+-- {: .num_prop #PreservationOfUnionsAndIntersectionsOfSets}
###### Proposition
**([[images preserve unions]] but not in general [[intersections]])**

Let $f \colon X \longrightarrow Y$ be a [[function]] between [[sets]]. Let $\{S_i \subset X\}_{i \in I}$ be a set of [[subsets]] of $X$. Then.

1. $im_f\left( \underset{i \in I}{\cup}  S_i\right) = \left(\underset{i \in I}{\cup} im_f(S_i)\right)$

1. $im_f\left( \underset{i \in I}{\cap}  S_i\right) \subset \left(\underset{i \in I}{\cap} im_f(S_i)\right)$

The inclusion in the second item is in general proper. If $f$ is an [[injective function]] and $I$ is [[inhabited set|inhabited]] then this is a [[bijection]]:

* $(f\,\text{injective}) \Rightarrow \left(im_f\left( \underset{i \in I}{\cap}  S_i\right) = \left(\underset{i \in I}{\cap} im_f(S_i)\right)\right)$

=--

For details see at _[[interactions of images and pre-images with unions and intersections]]_.

### Relation to factorization systems

Suppose that $M$ is closed under composition, and that $f = m e$ is an image factorization relative to $M$.  Then $e$ has the property that if $e = n g$ for some $n\in M$, then $n$ is an isomorphism --- for then we would have $f = (m n) g$ and so by universality of images, $m$ would factor through $m n$.  In particular, if $M$ is the class of all monomorphisms and $C$ has [[equalizers]], then $e$ is an [[extremal epimorphism]].

If $C$ has [[pullbacks]] and $M$ is closed under pullbacks, then we can say more: $e$ is [[orthogonality|orthogonal]] to $M$.  For if
$$\array{a & \overset{h}{\to} & b\\
  ^e\downarrow && \downarrow^n\\
  c  & \underset{k}{\to} & d}$$
is a commutative square with $n\in M$, then the pullback $k^*n$ is an $M$-morphism through which $e$ factors.  Hence $k^*n$ must be an isomorphism, and so the square admits a diagonal filler, which is unique since $n\in M$ is monic.  It follows that if all $M$-images exist in $C$, then $M$ is the right class of an [[orthogonal factorization system]], and $M$-images are precisely the factorizations in this OFS.

Conversely, it is easy to see that if $(E,M)$ is an OFS on a category $C$, then all $M$-images exist and are given by the factorizations of the OFS.  Therefore, to give a notion of image is more or less equivalent to giving an orthogonal factorization system.

**Duality**

Note that the notion of factorization system is self-dual.  Therefore, if $(E,M)$ is a factorization system and $c \overset{e}{\to} a \overset{m}{\to} d$ is an $(E,M)$-factorization of $f\colon c\to d$, then not only is $m$ the $M$-image of $f$ (the least $M$-subobject through which $f$ factors), but dually $e$ is also the **$E$-coimage** of $f$, i.e. the greatest $E$-quotient through which $f$ factors.

However, see below for additional remarks on the usage of the terms "image" and "coimage."

### Construction via limits
 {#AsEqualizer}

Suppose that the category $C$ admits [[finite limits]] and [[finite colimits]], and that $M=RegMono$ consists of the [[regular monomorphisms]].  Then the $M$-image of a morphism $f : c \to d$ may be constructed as

$$
  Im f \simeq lim (d \rightrightarrows d \sqcup_c d)
  \,,
$$

where $d \sqcup_c d$ denotes the [[pushout]]

$$
  \array{
    c &\stackrel{f}{\to}& d
    \\
    \downarrow^{f} && \downarrow
    \\
    d &\to& d \sqcup_c d
  }
  \,.
$$

In other words, the **[[regular image]]** is the [[equalizer]] of the [[cokernel pair]].  To see that this is in fact the $RegMono$-image, we first note that it is of course a regular monomorphism by definition, and then invoke the fact that in a category with [[finite limits]] and colimits, a monomorphism is regular if and only if it is the equalizer of its cokernel pair.

Dually, the **regular coimage** of a morphism is the [[coequalizer]] of its [[kernel pair]].  In [[Set]] (and more generally in any [[topos]]) these two constructions coincide, but in general they are distinct.  For example, in [[Top]] the regular image is the set-theoretic image topologized as a subspace of the [[target|codomain]], while the regular coimage is the set-theoretic image topologized as a quotient space of the [[source|domain]].

Note that some authors drop the "regular" and simply call these constructions the **image** and **coimage** respectively.  This can be confusing, however, since in many cases (such as in any [[regular category]]) the *regular coimage* coincides with the $M$-image for $M=Mono$ the class of all monomorphisms, which it is also natural to simply call the *image*.

### Comparison of regular images and coimages

Suppose that $M_1$ and $M_2$ are two classes with $M_1\subseteq M_2$.  If $f$ has both an $M_1$-image $im_1(f)$ and an $M_2$-image $im_2(f)$, then by universality, the latter must factor through the former.  The correspondence between images and factorization systems also extends to pairs; see [[ternary factorization system]].

As a special case of this, we have:

+-- {: .num_lemma}
###### Lemma
If $C$ has finite limits and colimits, then there is a unique map
$$
 u :  coim f \to im f
$$
from its regular coimage to its regular image such that
$$
  f = (c \to coim f \stackrel{u}{\to} im f \to d)
  \,.
$$
=--

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

Because $f$ coequalizes $c \times_d c \rightrightarrows c$, a morphism $h$ in

$$
  \array{
    c
      &\times_d c \rightrightarrows&
    c
    &\stackrel{f}{\to}&
    d
    &\rightrightarrows&
    d \sqcup_{c} d
    \\
   && {}^{\;}\downarrow^{epi} &{}^{h}\nearrow& {}^{\;}\uparrow^{mono}
   \\
    && coim f && im f
  }
$$

exists uniquely.

Because $c \to coim f$ is [[epimorphism|epi]] it follows that $h$ equalizes $d \rightrightarrows d \sqcup_c d$ and hence $u$ in the diagram

$$
  \array{
    c
      &\times_d c \rightrightarrows&
    c
    &\stackrel{f}{\to}&
    d
    &\rightrightarrows&
    d \sqcup_{c} d
    \\
   && {}^{\;}\downarrow^{epi} &{}^{h}\nearrow& {}^{\;}\uparrow^{mono}
   \\
    && coim f &\stackrel{u}{\to}& im f
  }
$$

exists uniquely.

=--

If this map $u$ is an [[isomorphism]], then $f$ is sometimes called a [[strict morphism]].  In particular, if $C$ has finite limits and colimits and every morphism is a strict morphism, then the regular image and regular coimage factorizations coincide and give an epi-mono factorization system.

## In higher category theory

In [[higher category theory]] there are generalizations of the notion of image, such as these:

* [[essential image]] (of a [[functor]])
* [[homotopy image]].

However, it is not clear that either serves as the proper categorification of the notion described above.

There are several properties we might want a 'higher image' to have. For example, in an $2$-category, we might want isomorphic 1-cells to have equivalent images. In **Cat**, we might want the image of a functor between discrete categories to be its image as a function.  One fruitful direction is to study a [[factorization system in a 2-category]].

### In $(\infty,1)$-category theory {#InfImage}

A **(regular) $(\infty,1)$-image** of a morphism $f : c \to d$ in an [[(∞,1)-category]] with [[(∞,1)-limit|(∞,1)-limits]] and -colimits should be defined to be the [[(∞,1)-limit]] over the [[Čech nerve|Cech co-nerve]] of $f$:

$$
  im f := \lim_{\leftarrow}
  \left(
    d \rightrightarrows d \coprod_c d
    \stackrel{\to}{\rightrightarrows}
    d \coprod_c d \coprod_c d
    \stackrel{\to}{\stackrel{\to}{\rightrightarrows}}
    \cdots
  \right)
  \,.
$$

Notice that

* this reduces to the above [equalizer definition](#AsEqualizer) in the case that the ambient $(\infty,1)$-category is just an ordinary category;

* this implies that the inclusion $im f \to d$ is a [[regular monomorphism]] in the $(\infty,1)$-category sense (described [here](http://ncatlab.org/nlab/show/regular+monomorphism#Infty1Version)).

For more see _[[n-image]]_.

#### Examples {#InfImageExamples}

Applied to the $(\infty,1)$-category [[∞Grpd]] this gives a notion of image of [[(∞,1)-functor|(∞,1)-functors]] between [[∞-groupoid|∞-groupoids]] and hence a notion of image of [[functor|functors]] between [[groupoid|groupoids]], [[2-functor|2-functors]] between [[2-groupoid|2-groupoids]], etc.

## Related concepts

* [[preimage]]

* [[coimage]]

* [[inverse image]], [[direct image]]

* [[kernel]], [[cokernel]]

* [[continuous images of compact spaces are compact]]

* [[value]]

[[!redirects images]]

[[!redirects regular image]]
[[!redirects regular images]]

[[!redirects regular coimage]]
[[!redirects regular coimages]]

[[!redirects image factorization]]
[[!redirects image factorizations]]


image of a rational map

As [[rational map]] $f: X \dashrightarrow Y$ among projective [[algebraic varieties]] is not defined everywhere there is also a nontrivial definition of an image of a [[rational map]]: it is defined by help of a graph of a rational map.

For this embed $Y$ in a [[projective space]] $P^n = P^n_k$ of dimension $n$; now the image of $f$ is the image of the composition still denoted $f: X\dashrightarrow P^n$. It is defined as a regular map on some open dense subset $f_U: U\to P^n$. Let the [[graph of a function|set theoretical graph]] be $setgraph f_U \subset U\times P^n$; then define the **graph of the rational map** $f$ as the closure of the $setgraph f_U$ within $X\times P^n$. This does not depend on the choice of the dense open subset $U$ and agrees with the usual graph when $f$ is regular.

Notice that not every $k$-point in the image of a rational map is actually set theoretically in image of the underlying map of sets of $k$-points.

A rational map $f: X\dashrightarrow Y$ is __dominant__ if its image as a rational map is the whole of $Y$.


image of J -- table

| | $n$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|-|-----|---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|
| [[Whitehead tower]] of [[orthogonal group]] |  | [[orientation]]  | [[spin group]]  |  | [[string group]]  |  |  |  | [[fivebrane group]] |  |  |  | [[ninebrane group]] |  |  |  |  | |
| higher versions |  | [[special orthogonal group]]  | [[spin group]]  |  | [[string 2-group]]  |  |  |  | [[fivebrane 6-group]] |  |  |  | [[ninebrane 10-group]] |  |  |  |  | |
| [[homotopy groups]] of [[stable orthogonal group]] | $\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$  |  $\mathbb{Z}_2$ | 0  | $\mathbb{Z}$  | 0  | 0 | 0 | $\mathbb{Z}$  | $\mathbb{Z}_2$ |
| [[stable homotopy groups of spheres]] | $\pi_n(\mathbb{S})$ | $\mathbb{Z}$ | $\mathbb{Z}_2$  | $\mathbb{Z}_2$ | $\mathbb{Z}_{24}$ | 0  | 0 | $\mathbb{Z}_2$ | $\mathbb{Z}_{240}$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_6$  | $\mathbb{Z}_{504}$ | 0 | $\mathbb{Z}_3$  | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$  | $\mathbb{Z}_{480} \oplus \mathbb{Z}_2$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2 $ |
| [[image of J-homomorphism]] | $im(\pi_n(J))$ |  0 | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}_{24}$ | 0 | 0 | 0 | $\mathbb{Z}_{240}$ | $\mathbb{Z}_2$  | $\mathbb{Z}_2$  | 0  | $\mathbb{Z}_{504}$  |  0 |  0 |  0 | $\mathbb{Z}_{480}$ | $\mathbb{Z}_2$ |


imaginary number

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#### Arithmetic
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[[!include arithmetic geometry - contents]]
=--
=--
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# Contents
* table of contents
{: toc}

## Definitions

A [[complex number]] $z$ is __purely imaginary__ if its [[real part]] $\Re{z}$ is [[zero]], hence if under [[complex conjugation]] we have $\overline{z} = - z$.  The complex number $z$ is __imaginary__ if it is not [[real number|real]], in other words if $z \ne \Re{z}$, or equivalently if its [[imaginary part]] $\Im{z}$ is nonzero, or if $ z \ne \overline{z}$.  (In [[constructive mathematics]], we mean [[apartness]] here.)

The same terminology applies to [[quaternions]], [[octonions]], and other $*$-[[star-algebra|algebras]] of [[hypercomplex numbers]]. (In yet other $*$-algebras, such as in [[matrix algebras]], one tends to say "skew-[[hermitian matrix|hermitian]]" instead of "purely imaginary". There is no clear analogue of "imaginary" here since $z \notin \mathbb{R}$ and $z \ne \overline{z}$ diverge.)

Beware that often one says just "imaginary" for "purely imaginary".  (For example, $2 + 3\mathrm{i}$ is imaginary but not purely imaginary; while $0$ is the unique purely imaginary number that is not imaginary.)  This may be because the imaginary numbers, as is typical for things defined by an [[inequality]], do not form an interesting collection as a whole (for example, they are not even closed under addition).  Compare [[irrational number]].

The purely imaginary complex numbers, on the other hand, form the [[Lie algebra]] $\mathfrak{u}(1)$.  Often one substitutes $\mathbb{R}$ (the algebra of real numbers), which is simpler, when one only cares about this Lie algebra up to [[isomorphism]].  However, using $\mathrm{i}\mathbb{R}$ (the algebra of purely imaginary numbers) makes $\mathfrak{u}(1)$ fit with the [[matrix]] formulas used in higher dimensions.

## Related concepts

* [[imaginary unit]]

* [[imaginary part]], [[real part]]

[[!redirects imaginary number]]
[[!redirects imaginary numbers]]

[[!redirects purely imaginary number]]
[[!redirects purely imaginary numbers]]


imaginary unit

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The [[complex numbers]] $+i$ and $-i$, i.e. the two [[solutions]] $x$ to the [[equation]] $x^2 = -1$, are the _imaginary units_.

A [[complex number]] of the form $y i$ with $y \in \mathbb{R}$ a [[real number]] is called a _[[purely imaginary number]]_.

## References

* Wikipedia, _[Imaginary unit](https://en.wikipedia.org/wiki/Imaginary_unit)_

[[!redirects imaginary units]]


Imma Gálvez-Carrillo

# Imma Gálvez-Carrillo

* [website](https://web.mat.upc.edu/m.immaculada.galvez/)

## related $n$Lab pages

* [[homotopy BV-algebra]]
* [[2-Segal space]]

## Publications

* Imma Gálvez-Carrillo, [[Andy Tonks]], [[Bruno Vallette]], _Homotopy Batalin-Vilkovisky algebras_ ([arXiv:0907.2246](http://arxiv.org/abs/0907.2246))

* Imma Gálvez, [[Vassily Gorbounov]], Andrew Tonks, _Homotopy Gerstenhaber structures and vertex algebras_, [math/0611231.QA](http://arxiv.org/abs/math/0611231)

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

* Imma Gálvez-Carrillo, [[Andy Tonks]], [[Bruno Valette]], _Homotopy Batalin-Vilkovisky algebras_, J. Noncomm. Geom. __6__:3, 539--602 [arXiv:0907.2246](https://arxiv.org/abs/0907.2246) [doi](https://doi.org/10.4171/JNCG/99)

category: people

[[!redirects Imma Gálvez-Carrillo]]
[[!redirects Imma Gálvez]]
[[!redirects Imma Galvez-Carrillo]]
[[!redirects Imma Galvez]]
[[!redirects Imma Carrillo]]


immanant

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#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

This is a joint generalization of [[determinants]] and [[permanent|permanents]].

If $A = (a_{i,j}) \in \mathcal{M}_{n,p}(R)$ where $R$ is a [[commutative ring]] and if $\lambda \vdash n$ is a [[partition]] of $n$, we define:

$$
Imm_{\lambda}(A) = \underset{\sigma \in \mathfrak{S}_{n}}{\sum}\chi_{\lambda}(\sigma)a_{i,\sigma(i)}
$$

where $\chi_{\lambda}$ is the [[character]] of the [[irreducible representation]] of the [[symmetric group]] $\mathfrak{S}_{n}$ associated to $\lambda$.

## Related entries

* [[determinant]]
* [[permanent]]

## References

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


Immanuel Kant

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### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

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

## Quotes

On [[mathematics]] (later quoted by [[David Hilbert]] in *[[Naturerkennen und Logik]]* (1930), see [there](Naturerkennen+und+Logik#TheInstrumentThatMediates)):

> {#IchBehaupteAber} Ich behaupte aber, daß in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist.

> I maintain that in each particular natural science there is only as much true science as there is mathematics.

> [From *[Metaphysische Anfangsgründe der Naturwissenschaft](https://de.wikipedia.org/wiki/Metaphysische_Anfangsgr%C3%BCnde_der_Naturwissenschaft) (1786), [Vorrede](http://www.zeno.org/Philosophie/M/Kant,+Immanuel/Metaphysische+Anfangsgr%C3%BCnde+der+Naturwissenschaft/Vorrede)*]

## Related entries

* [[Critique of Pure Reason]]

* [[Attempt to Introduce the Concept of Negative Quantities into Philosophy|Concept of Negative Quantities]]

* [[transcendental ideal]]

* [[infinite judgement]]

* [[construction in philosophy]]

* [[transcendental grammar]]

* [[Stephan Wannowski]]

* [[category (philosophy)]]

* [[Aufhebung]]

* [[Johann Fichte|Johann Gottlieb Fichte]]

* [[Salomon Maimon]]

* [[Hermann Cohen]]

* [[Ernst Cassirer]]

## References

* [[Georg Hegel]], _[[Lectures on the History of Philosophy]] -- [Kant](Lectures+on+the+History+of+Philosophy#Kant)_

* Bitbol, Michel, Kerszberg, Pierre, Petitot, Jean (Eds.), _Constituting Objectivity: Transcendental Perspectives on Modern Physics_ Springer (2009)
  * Jean Petitot, _[Noncommutative Geometry and Transcendental Physics](http://topologicalmedialab.net/xinwei/classes/readings/Petitot/Petitot_NoncommutativeGeometry.pdf)_, pp. 415-455

[[!redirects Kant]]

category: people


immersion

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###Context###
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--

There are several concepts of an immersion, depending on the category of spaces. The most basic ones are

* [[closed immersion of schemes]]
* [[open immersion of schemes]]
* [[immersion of smooth manifolds]]

[[!redirects immersions]]

[[!redirects smooth immersion]]
[[!redirects smooth immersions]]


immersion of smooth manifolds

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=--
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $f \colon X \to Y$ be a [[differentiable function]]
between [[smooth manifolds]] (or just [[differentiable manifolds]]) of [[finite number|finite]] [[dimension]].

We denote by $T X$,  $T Y$ the [[tangent bundles]] and by $(-)\times_Y (-)$ the [[fiber product]] of [[differentiable functions]] into $Y$. In particular, $f^\ast T Y \,=\, X \times_Y T Y$ is the [[pullback bundle]] of $T Y$ along $f$ to a [[real vector bundle]] over $X$.

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

The [[differentiable function]] $f : X \to Y$ is called an **immersion** precisely if the canonical [[map]]

$$
  T X \longrightarrow X \times_Y T Y \eqqcolon f^* T Y
$$

is a [[monomorphism]].

=--

This map is the one induced from the [[universal property]] of the [[pullback]] by the [[commuting diagram]]

$$
  \array{
    T X &\overset{d f}{\longrightarrow}& T Y
    \\
    \big\downarrow
      &&
    \big\downarrow
    \\
    X &\underset{f}{\longrightarrow}&  Y
  }
$$

given by the [[differential]] of $f$ going between the [[tangent bundles]].

Equivalently this means the following:

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

The function $f : X \to Y$ is an immersion precisely if for every point $x \in X$ the [[differential]]

$$
  d f|_x \;\colon\; T_x X \to T_{f(x)} Y
$$

between the [[tangent space]] of $X$ at $x$ and the tangent space of $Y$ at $f(y)$ is an [[injection]].

=--

+-- {: .num_defn #EmbeddingOfSmoothManifolds}
###### Definition
**([[embedding of smooth manifolds]])**

An immersion whose underlying [[continuous function]] is an [[embedding of topological spaces]] is called an an _[[embedding of smooth manifolds]]_.

=--

+-- {: .num_example #ImmersionsThatAreNotEmbeddings}
###### Example
**(immersions that are not embeddings)**

Consider an immersion $f \;\colon\; (a,b) \to \mathbb{R}^2$ of an [[open interval]] into the [[Euclidean plane]] (or the [[2-sphere]]) as shown on the right. This is not an [[embedding of smooth manifolds]]: around the points where the image crosses itself, the function is not even [[injective]], but even
at the points where it just touches itself, the pre-images under $f$ of open subsets of $\mathbb{R}^2$ do not exhaust the open subsets of $(a,b)$, hence do not yield the [[subspace topology]].

Concretely, consider the function $\big($[[sin]]$(2-),$ [[sin]]$(-)\big) \;\colon\; (-\pi, \pi) \longrightarrow \mathbb{R}^2$. While this is an immersion and [[injective]], it fails to be an [[embedding of smooth manifolds|embedding]] due to the points at $t = \pm \pi$ "touching" the point at $t = 0$.

> figure from [Lee (2012, Fig. 4.3)](#Lee12)

=--

\linebreak

## Properties

### Relation to embeddings

An immersion $f : X \to Y$ is precisely a _local [[embedding of smooth manifolds|embeddings]]_: for every point $x \in X$ there is an [[open neighbourhood]] $x \in U \subset X$ such that $f|_U : U \to Y$ is an [[embedding of smooth manifolds]].

### Relation to formal immersions

The related concept of [[formal immersion of smooth manifolds]], defined as an injective bundle morphism $T M \to T N$ between tangent bundles, is in some ways easier to study, in the sense that the collection of all such formal immersions, $Imm^f(M, N)$, is simpler to analyze. Then under some conditions on $M$ and $N$ (see there), it is the case that the map $Imm(M, N) \to \Imm^f(M, N)$ is a [[weak homotopy equivalence]]. This is a case of the [[h-principle]].

### Characterization in differential cohesion
 {#InDifferentialCohesion}

A [[smooth function]] $f : X \to Y$ between [[smooth manifolds]] is canonically regarded as a morphism in the [[cohesive (∞,1)-topos]] [[SynthDiff∞Grpd]]. With respect to the canonical [[infinitesimal cohesion|infinitesimal neighbourhood inclusion]] $i : $ [[Smooth∞Grpd]] $\hookrightarrow$ [[SynthDiff∞Grpd]] there is a notion of [[formally unramified morphism]] in $SynthDiff\infty Grpd$.

$f$ is an immersion precisely if it is formally unramified with respect to this infinitesimal cohesion.

See the discussion at [[SynthDiff∞Grpd]] for details.

## Variants

The [[algebraic geometry]] analogue of a submersion is a [[smooth morphism of schemes|smooth morphism]].

The analogue between arbitrary [[topological spaces]] (not manifolds) is simply an [[open map]]. There is also [[topological submersion]], of which there are two versions.

## Related concepts

* **immersion**, [[formal immersion of smooth manifolds]], [[submersion]], [[local diffeomorphism]]

* [[embedding of smooth manifolds]]

* [[regular value]]

## References

* {#Lee12} [[John Lee]], Chapter 4 of: *Introduction to Smooth Manifolds*, Springer (2012) [[doi:10.1007/978-1-4419-9982-5](https://doi.org/10.1007/978-1-4419-9982-5), [book webpage](https://sites.math.washington.edu/~lee/Books/ISM/), [pdf](https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth_manifolds.pdf)]

[[!redirects immersion of differentiable manifolds]]
[[!redirects immersions of differentiable manifolds]]


imperative programming

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### Context
#### Computing
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[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[computer science]], under *imperative programming* one understands a [[programming language]]-style in which a program largely consists of

1. a succession of *commands* (whence "imperative")

1. for (in particular) accessing and modifying the *global state* of the computing environment

   (such as the [[RAM]] and [[I/O]]-devices).

A typical example of imperative [[pseudocode]] would be of the form

> `n <- Read`

> `n++`

> `Write n`

being the sequence of commands to, in this case:

1. read a (numerical) datum (from some input device) and store it in a global variable `n`

1. increment `n`

1. write the value of `n` (to some output device).

Low-level languages are often imperative, reflecting the actual operation of the underlying computing machine; and medium-level imperative languages may be felt to have a transparent flow logic.

In its naive form the access and modification of global state variables means that imperative programs are hard to [[software verification|verify]] (since there is no guarantee that the global state is in the assumed form). This is generally in contrast to *pure* [[functional programming languages]], which enforce that every procedure be a *pure function* in that its output and side-effect depends deterministically on its input (instead of also on the value of some global variables).

On the other hand, much of the features of imperative programs with access to a global state can be emulated/encapsulated in pure [[functional programming languages]] via the scheme of [[monad in computer science|monadic data types]] (see there for more). With the [[syntactic sugar]] of [do-notation](monad+in+computer+science#DoNotation) [[Launchbury 1993 §3.3](#Launchbury93)] this allows purely functional languages to look essentially like imperative code while remaining [[software verification|verifiable]].

## Related concepts

* [[functional programming]]

* [[object-oriented programming]]

* [[monad (in computer science)]]

## References

See also:

* Wikipedia, _[Imperative programming](https://en.wikipedia.org/wiki/Imperative_programming)_

The origin of [do-notation](monad+in+computer+science#DoNotation) for emulating imperative programming in [[functional programming languages]] via [[monad in computer science|monadic data types]]:

* {#Launchbury93} [[John Launchbury]], §3.3 in: *Lazy imperative programming*, Proceedings of *ACM Sigplan Workshop on State in Programming Languages*, Copenhagen (1993) [[pdf](https://launchbury.files.wordpress.com/2019/01/lazy-imperative-programming.pdf), [[Launchbury-LazyImperative.pdf:file]]]

Conversely, building features of [[functional programming languages]] into imperative languages:

* David R. Cok, *Reasoning about functional programming in Java and C++*, ISSTA '18: Companion Proceedings for the ISSTA/ECOOP 2018 WorkshopsJuly 2018 Pages 37–39 ([doi:10.1145/3236454.3236483](https://doi.org/10.1145/3236454.3236483))

[[!redirects imperative programming language]]
[[!redirects imperative programming languages]]

[[!redirects imperative program]]
[[!redirects imperative programs]]


implication

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Implication
* table of contents
{: toc}

## Definitions

An _implication_ may be either an _entailment_ or a _conditional_ statement; these are closely related but not quite the same thing.

1. __Entailment__ is a [[preorder]] on [[propositions]] within a given [[context]] in a given [[logic]].

   We say that $p$ entails $q$ _[[syntax|syntactically]]_, written as a [[sequent]] $p \vdash q$, if $q$ can be [[proof|proved]] from the assumption $p$.

   We say that $p$ entails $q$ _[[semantics|semantically]]_, written $p \vDash q$, if $q$ holds in every [[model]] in which $p$ holds.

   (These relations are often equivalent, by various [[soundness theorem|soundness]] and [[completeness theorem|completeness]] theorems.)

1. A __conditional__ statement is the result of a [[binary operation]] on [[propositions]] within a given [[context]] in a given [[logic]].  If $p$ and $q$ are propositions in some context, then so is the conditional statement $p \to q$, at least if the logic has a notion of conditional.

Notice that $p$, $q$, and $p \to q$ are all statements in the _object_ language (the language that we are talking about), whereas the [[hypothetical judgements]] $p \vdash q$ and $p \vDash q$ are statements in the _[[metalanguage]]_ (the language that we are using to talk about the [[object language]]).

### Relations between the definitions

Depending on what logic one is using, $p \to q$ might be anything, but it\'s probably not fair to consider it a conditional statement unless it is related to entailment as follows:
+-- {: .standout}
If, in some context, $p$ entails $q$ (either syntactically or semantically), then $p \to q$ is a [[theorem]] (syntactically) or a [[tautology]] (semantically) in that context, and conversely.
=--
In particular, this holds for [[classical logic]] and [[intuitionistic logic]].

You can think of entailment as being an [[external hom]] (taking values in the poset of [[truth values]]) and the conditional as being an [[internal hom]] (taking values in the poset of [[propositions]]).  In particular, we expect these to be related as in a [[closed category]]:

*  $ q \to r \vdash (p \to q) \to (p \to r) $,
*  $ p \equiv \top \to p $,
*  $ \top \vdash p \to p $,

where $\top$ is an appropriate constant statement (often satisfying $p \vdash \top$, although not always, as in [[linear logic]] with $\multimap$ for $\to$ and $1$ for $\top$).

Most kinds of logic used in practice have a notion of entailment from a [[list]] of multiple premises; then we expect entailment and the conditional to be related as in a closed [[multicategory]].

Just as we may identify the internal and external hom in [[Set]], so we may identify the entailment and conditional of [[truth values]].  In the $n$Lab, we tend to write this as $\Rightarrow$, a symbol that is variously used by other authors in place of $\vdash$, $\vDash$, and $\rightarrow$.

## In various formalizations

### In Heyting algebras

Although [[Heyting algebras]] were first developed as a way to discuss [[intuitionistic logic]], they appear in other contexts; but their characterstic feature is that they have an operation analogous to the conditional operation in logic, usually called __Heyting implication__ and denoted $\rightarrow$ or $\Rightarrow$.  If you use $\to$ and replace $\vdash$ above with the Heyting algebra\'s partial order $\leq$, then everything above applies.

### In natural deduction

In [[natural deduction]] the [[inference rules]] for implication are given as

$$\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma \vdash P \to Q \; \mathrm{prop}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma, P \; \mathrm{true} \vdash Q \; \mathrm{true}}{\Gamma \vdash P \to Q \; \mathrm{true}} \qquad \frac{\Gamma \vdash P \to Q \; \mathrm{true}}{\Gamma, P \; \mathrm{true} \vdash Q \; \mathrm{true}}$$

### In type theory

In [[type theory]]

* a _conditional statement_ is, under [[propositions-as-types]] a [[function type]] $p \to q$ (or the [[bracket type]] thereof).

* an _entailment_ is a [[hypothetical judgement]] or [[sequent]].

## Related concepts

* [[function type]]

* [[hypothetical judgement]]

* [[linear implication]]

* [[contrapositive]]

[[!include logic symbols -- table]]

[[!redirects implication]]
[[!redirects implications]]

[[!redirects entailment]]
[[!redirects entailments]]

[[!redirects conditional]]
[[!redirects conditionals]]
[[!redirects conditional statement]]
[[!redirects conditional statements]]

[[!redirects Heyting implication]]
[[!redirects Heyting implications]]


implicative algebra

# Contents
* table of contents
{:toc}

## Idea

An *implicative algebra* is an [[implicative structure]] equipped with a [[filter]]-like [[subset]] called a *separator*.

Recall an implicative structure is a generalized [[complete Heyting algebra]] in which the implication $\to$ is only required to be right continuous and not necessarily [[right adjoint]] to the [[meet]].

The separator is defined in such a way as to induce a quotient which makes the implicative structure into a bona-fide Heyting algebra. It is in this sense that a separator is like a (ultra)filter, whose primary purpose is to be used to define quotients which usually restore some property.

Given the completeness result in [(Miquel'20b)](#Miquel20b), we can say implicative algebras are to ($\mathbf{Set}$-)triposes what [[complete Heyting algebra|complete Heyting algebras]] (equivalently, [[locale|locales]]) are to [[Grothendieck topos|Grothendieck toposes]].

## Definition

Let $(\mathcal{A}, \leq, \to)$ be an [[implicative structure]].

\begin{definition}
A **separator** of $\mathcal{A}$ is a subset $S \subseteq \mathcal{A}$ such that

1. $S$ is upwards closed: for all $a \leq b \in \mathcal{A}$, $a \in S$ implies $b \in S$.
2. $S$ is closed under 'modus ponens': for all $a, b \in \mathcal{A}$, $a \in S$ and $a \to b \in S$ implies $b \in S$.
3. $S$ contains the following elements:
\[
K = \bigwedge_{a,b \in \mathcal{A}} (a \to b \to a),
\]
\[
S = \bigwedge_{a,b,c \in \mathcal{A}} ((a \to b \to c) \to (a \to b) \to a \to c)
\]

\end{definition}

\begin{remark}
In the case $\mathcal{A}$ is actually an [[Heyting algebra]], separators coincide with filters.
\end{remark}

\begin{proposition}
Let $S$ be a separator of $\mathcal{A}$. Then we can define a relation on $\mathcal{A}$:
\[
a \vdash_S b \quad \text{iff}\quad (a \to b) \in S.
\]
This relation is a [[preorder]], and its [[posetal reflection]] is an Heyting algebra with implication given by:
\[
[a] \Rightarrow [b] := [a \to b].
\]
\end{proposition}
\begin{proof}
This is Proposition 3.2.1 in [(Miquel'20)](#Miquel20).
\end{proof}

The Heyting algebra obtained from an implicative algebra is called its **induced Heyting algebra**, denoted $(\mathcal{A}/S, \leq_S)$. It is proven in [(Miquel'20)](#Miquel20) that this operation is functorial .

\begin{remark}
When $\mathcal{A}$ is an Heyting algebra to begin with, $(\mathcal{A}/S, \leq_S)$ is the quotient induced by $S$, seen as a filter.
\end{remark}

## See also

* [[filter]]
* [[complete Heyting algebra]]
* [[realizability]]
* [[forcing]]
* [[tripos]], in particular [[implicative tripos]]

## References
* {#Miquel20} Alexandre Miquel, _Implicative algebras: a new foundation for realizability and forcing_, Mathematical Structures in Computer Science 30 (5): 458–510. [(doi)](https://doi.org/10.1017/S0960129520000079).

* {#Miquel20b} Alexandre Miquel. 2020. _Implicative Algebras II: Completeness w.r.t. Set-Based Triposes_.
 [(arxiv)](https://arxiv.org/abs/2011.09085)


implicative structure

# Contents
* table of contents
{:toc}

## Idea

An implicative structure is an ample generalization of [[complete Heyting algebra|complete Heyting algebras]] introduced by Alexandre Miquel in [(Miquel '20a)](#Miquel20a).

Implicative structures are used to define [[implicative algebra|implicative algebras]], with are implicative structures equipped with a 'separator', which generalizes a filter for an Heyting algebra. From an implicative algebra, one can then construct an [[implicative tripos]], a construction which is the real motivation behind implicative structures and algebras.

These triposes end up generalizing many others, constructed from [[complete Heyting algebra|complete Heyting algebras]], [[partial combinatory algebras]], ordered combinatory algebras, and Krivine structures, thus unifying the methodology of realizability theory. More strikingly, Miquel proved all triposes over $\mathbf{Set}$ are implicative triposes [(Miquel'20b)](#Miquel20b).

## Definition
\begin{definition}
An **implicative structure** is a [[complete lattice|complete meet semilattice]] $(\mathcal{A}, \leq)$ equipped with a right-continuous monotone operation $\to : \mathcal{A}^{op} \times \mathcal{A} \to \mathcal{A}$.
\end{definition}

Explicitly, this means $\mathcal{A}$ is a [[poset]] with all meets, and $\to$ satisfies two axioms:

1. for all $a' \leq a, b \leq b' \in \mathcal{A}$, $(a \to b) \leq (a' \to b')$,
2. for all $a \in \mathcal{A}, B \subseteq \mathcal{A}$, $a \to \bigwedge_{b \in B} b = \bigwedge_{b \in B} (a \to b)$

We don't assume $\to$ to also be left-continuous (i.e. commute with $\bigvee$ in the first argument, which comes out as a $\bigwedge$ because of the antimonotonicity), though in most examples $\to$ satisfies this additional property.

\begin{remark}

Two observations follow immediately from this definition:

1. A complete semi-lattice is, at the end of the day, the same thing as a complete lattice since $\bigvee A = \bigwedge (\mathcal{A}\setminus A)$, where $A \subseteq \mathcal{A}$.
2. By the [[adjoint functor theorem]] for posets, $a \to -$ has a left adjoint $- \otimes a : \mathcal{A} \to \mathcal{A}$. In this sense, implicative structures generalize Heyting algebras by weakening the requirement that $- \otimes a = - \wedge a$. In fact, we have no requirement whatsoever on $- \otimes a$, which makes $\to$ a much weaker operation than $\Rightarrow$, at least on paper.

\end{remark}

Hence an alternative definition of implicative structure is:

\begin{definition}
An implicative structure is a complete lattice $\mathcal{A}$ equipped with a left-continuous, monotone operation $\otimes : \mathcal{A} \times \mathcal{A} \to \mathcal{A}$.
\end{definition}

In this form, it is easy to see this as broadly generalizing [[poset|posetal]] [[Benabou cosmoi]], more commonly known as [[quantale|quantales]].
In fact implicative structures are quantales, but for the fact that $\otimes$ doesn't satisfy the axioms of a monoidal operation on $\mathcal{A}$.

This is interesting since cosmoi are 'good enrichment bases' for *categories*, while implicative structures are 'good enrichment bases' for *sets*.

## See also

* [[implicative algebra]]
* [[realizability]]
* [[forcing]]
* [[tripos]], in particular [[implicative tripos]]

## References
* {#Miquel20a} Alexandre Miquel, _Implicative algebras: a new foundation for realizability and forcing_, Mathematical Structures in Computer Science 30 (5): 458–510. [(doi)](https://doi.org/10.1017/S0960129520000079).
* {#Miquel20b} Alexandre Miquel. 2020. _Implicative Algebras II: Completeness w.r.t. Set-Based Triposes_. arXiv. https://doi.org/10.48550/arXiv.2011.09085.
 [(arxiv)](https://arxiv.org/abs/2011.09085)


implicative tripos

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### Context
#### Type theory
+-- {: .hide}
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#### $(0,1)$-Category theory
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[[!include (0,1)-category theory - contents]]
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# Contents
* table of contents
{:toc}

## Idea
An implicative tripos is a [[tripos]] built out from an [[implicative algebra]]. It thus captures the particular logic encoded within such a structure.

Implicative triposes are triposes of $H$-valued sets for the Heyting algebra $H$ induced by the implicative algebra one started with (see [[implicative algebra]]).

In [(Miquel'20b)](#Miquel20b), it is proven *all triposes on $\mathbf{Set}$ arise in this way*, which means implicative algebras play, for ($\mathbf{Set}$-)triposes, the same role [[complete Heyting algebra|complete Heyting algebras]] (thus [[locale|locales]]) play for [[Grothendieck topos|Grothendieck toposes]].

## Construction

(...)

## See also
* [[implicative algebra]], [[implicative structure]]
* [[tripos]]

## References
* {#Miquel20a} Alexandre Miquel, _Implicative algebras: a new foundation for realizability and forcing_, Mathematical Structures in Computer Science 30 (5): 458–510. [(doi)](https://doi.org/10.1017/S0960129520000079).

* {#Miquel20b} Alexandre Miquel. 2020. _Implicative Algebras II: Completeness w.r.t. Set-Based Triposes_. arXiv. https://doi.org/10.48550/arXiv.2011.09085.
 [(arxiv)](https://arxiv.org/abs/2011.09085)


implicit function theorem

# Implicit function theorems
* table of contents
{: toc}

## Idea

Implicit function theorems give sufficient conditions for the existence of a [[differentiable map|differentiable]] [[inverse]] of a [[germ]] $f_p$ of a [[differentiable map]] $f\colon M \to N$ of [[smooth manifold]]s at a point $p$. The invertibility is trivially equivalent to the statement that the germ is a [[local diffeomorphism]] of some [[neighborhood]] of $p$ to some neighborhood of $f(p)$. If it is invertible, then we can consider the [[tangent map]] $T_p f\colon T_p M \to T_{f(p)}N$. If $f$ is locally invertible with differentiable inverse, then for all $y$ in some neighborhood of $y$ the [[functor]]iality of $T$ implies that $Id_{T_y} = T_y (f^{-1} \circ f) = T_{f(y)} f^{-1} \circ T_y f$ and alike for $f \circ f^{-1}$ at $f(y)$, demonstrating that $T_y f$ must then be invertible. The [[inverse function theorem]] says that the invertibility of $T_p f$ is in fact sufficient for the invertibility of the germ, which is then automatically differentiable.

## In $\mathbf{R}^n$

Let $U \subset \mathbf{R}^n$ be an [[open set]] in a [[cartesian space]], $a\in U$, $f\colon U \to \mathbf{R}^n$ a [[continuously differentiable map|map]] of class $C^1$ and $det\left(\frac{\partial f_i}{\partial x_j}(a)\right)\neq 0$. Then there are open sets $V \ni a$, $W \ni f(a)$, $V \subset U$ such that $f|_V\colon V \to W$ is a [[diffeomorphism]] and for all $y \in W$ and $(f^{-1})'(y) = (f'[f^{-1}(y)])^{-1}$.

## Local statement on manifolds

This is the theorem stated in the Idea section; the differentiable germ is assumed to be of class $C^1$ ([[continuously differentiable map|continuously differentiable]]). The statement is local, so one can consider it in [[chart]]s, hence the proof reduces to the case of $\mathbf{R}^n$.

## Global statement on manifolds

Let $f\colon M \to N$ be a [[smooth map]] of smooth manifolds. A point $q \in N$ is a __regular value__ of $f$ if for every point $p \in f^{-1}(q)$ the [[differential]] $T_p f\colon T_p M \to T_q N$ is an [[epimorphism]]. The implicit function theorem asserts that $Q = f^{-1}(p)$ is a smooth [[submanifold]] of $M$ and the [[tangent bundle]] $T M|_N$ globally splits as $T M|_N \cong T N \oplus \mathbf{R}^n$ where $n = dim N$.

More generally, if $W \subset N$ is a submanifold, we say that the map $f$ is __transversal along__ $W$ if for every point $x\in f^{-1}(W)$ there is an equality
$$
T_{f(x)} N = T_{f(x)}W + (T_x f)(T_x X)
.$$
In particular, $f$ is transveral along every regular value $p \in N$. The implicit function theorem asserts that the [[preimage]] $f^{-1}(W)$ is a smooth submanifold of $M$, the [[normal bundle]] $\nu(f^{-1}(W) \subset M)$ is isomorphic to $f^*(\nu(W\subset N))$, and the differential $T f$ exhibits the fiberwise isomorphism $\nu(f^{-1}(W)\subset M)\to \nu(W\subset N)$.

## References

* L. H. Loomis, [[S. Sternberg]], _Advanced calculus_, 1968, 1990 (3.11 in 1990 edition)

* S. Lang, _Analysis I_

Various applications and related theorems can be found in chapter 5: _Local and tangential properties_ of

* T. Bröcker, K. Jänich, C. B. Thomas, M. J. Thomas, _Introduction to differentiable topology_, 1982 (translated from German 1973 edition; $\exists$ also 1990 German 2nd edition)

An invariant global statement on manifolds is at page 44 of

* А. С. Мищенко, Векторные расслоенния и их применения, Moscow, Nauka 1984

Elementary course notes of the case in $\mathbf{R}^n$ (mainly lots of examples):

* Frank Jones, _Implicit function theorem_, [pdf](http://www.owlnet.rice.edu/~fjones/chap6.pdf)

On the implicit function theorem in the generality of [[Fréchet manifolds]]:

* [[Richard S. Hamilton]], *The inverse function theorem of Nash and Moser*, Bull. Amer. Math. Soc. **7** (1982) 65-222 [[doi:1982-07-01/S0273-0979-1982-15004-2](https://www.ams.org/journals/bull/1982-07-01/S0273-0979-1982-15004-2)]

[[!redirects implicit function theorem]]
[[!redirects implicit function theorems]]


+-- {: .rightHandSide}
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### Context
#### $(\infty,1)$-Category theory
+-- {: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

Work on $\infty$-[[infinity-category|category]] and $(\infty,1)$-[[(infinity,1)-category|category]] theory often uses the **implicit $\infty$-category theory convention**, leaving out the "$\infty$" or "$(\infty,1)$" symbol in notation and terminology.

In this convention

* _category_ means _[[(∞,1)-category]]_, and so $Cat$ means _[[(∞,1)Cat]]_

* _topos_ means _[[(∞,1)-topos]]_, and so $Topos$ means _[[(∞,1)Topos]]_

* _groupoid_ means _[[∞-groupoid]]_, and so $Grpd$ means _[[∞Grpd]]_

* _$n$-category_ means _[[(∞,n)-category]]_

* _functor_ means _[[(∞,1)-functor]]_ or _[[(∞,n)-functor]]_ for some $n \in \mathbb{N}$, depending on the codomain;

* _presheaf_ means _[[(∞,1)-presheaf]]_,

* _sheaf_ means _[[(∞,1)-sheaf]]_ , a.k.a. _[[∞-stack]]_

* _limit_ means _[[(∞,1)-limit]]_

* _algebraic theory_ means _[[(∞,1)-algebraic theory]]_

* _algebra over an algebraic theory_ means _[[∞-algebra over an (∞,1)-algebraic theory]]_ .

Etc.

This convention is used for example by [[Jacob Lurie]] and often on [[nLab]] entries that discuss higher category theory.

(Note that this convention *never* applies to the *names* of pages in the Lab; see [Naming conventions](/nlab/show/HowTo#naming).  That also means that when writing pages using this convention, one must be sure to use the appropriate "pipelinks," such as `[[(infinity,1)-topos|topos]]`.)

[[!redirects implicit infinity-category theory convention]]
[[!redirects implicit ∞-category theory convention]]
[[!redirects implicit (infinity,1)-category theory convention]]
[[!redirects implicit (∞,1)-category theory convention]]
[[!redirects implicit infinity-category convention]]
[[!redirects implicit ∞-category convention]]
[[!redirects implicit (infinity,1)-category convention]]
[[!redirects implicit (∞,1)-category convention]]


imprecise probability

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#### Imprecise Probability theory
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=--

#Contents#
* table of contents
{:toc}

## Idea

Imprecise probability is a generalization of the classical probability theory that allows for the representation of uncertainty in situations where precise probabilities are not available or not suitable. The concept is useful for modeling and reasoning with incomplete or ambiguous information. Imprecise probability encompasses various formalisms, such as credal sets, lower previsions, and belief functions, among others.

Imprecise probability can be formalized in different ways,some of which include

* Credal Sets: A credal set is a closed convex set of probability distributions, representing a range of possible distributions that may accurately represent an agent's uncertainty. Credal sets were introduced by Levi (1980) and further developed by Walley (1991). Most other formalizations of imprecise probability can be seen as special cases of credal sets embbed naturally in them.

* Belief Functions: Belief functions, also known as Dempster-Shafer theory or evidence theory, are a generalization of probability theory that allows for representing partial beliefs. They are based on the concept of mass functions, which assign a weight to each subset of a finite set of alternatives, subject to certain consistency conditions.

## Related concepts

* [[infra-Bayesianism]]

## References

* Walley, P. (1991). Statistical reasoning with imprecise probabilities. London: Chapman and Hall.

* Augustin, T., Coolen, F., de Cooman, G., & Troffaes, M. C. M. (2014). Introduction to imprecise probabilities. Wiley.

* Levi, I. (1980). The Enterprise of Knowledge: An Essay on Knowledge, Credal Probability, and Chance. MIT Press.

* Shafer, G. (1976). A Mathematical Theory of Evidence. Princeton University Press.


improper filter

# The improper filter
* table of contents
{: toc}

## Idea

A [[filter]] is __improper__ if it is the whole thing ... whatever thing the filters are filters in.

## Definitions

+-- {: .num_defn}
###### Definition (improper filter in a lattice or other proset)

If $L$ is a [[lattice]] or even a [[proset]], then $L$ is an [[filter]] in itself, the __improper filter__.
=--

+-- {: .num_defn}
###### Definition (improper filter on a set)

If $S$ is a [[set]], then the __improper filter__ on $S$ is the [[power set]] of $S$.
=--

Of course, the improper filter *on* $S$ is simply the improper filter *in* the power set of $S$.

## Properties

The improper filter contrasts with [[proper filters]] (all of the other filters).

The improper filter does not count as a [[prime filter]] or an [[ultrafilter]], because it is [[too simple to be simple]], although it may satisfy careless or naïve definitions of those concepts (which must be required to be proper).

Conversely, sometimes one defines 'filter' to exclude the improper filter, especially in [[analysis]] and [[topology]]; however, this makes the set of all filters less nice to work with.

[[!redirects improper filter]]
[[!redirects improper filters]]


improper ideal

# The improper ideal
* table of contents
{: toc}

## Idea

An [[ideal]] is __improper__ if it is the whole thing ... whatever thing the ideals are ideals in.

## Definitions

As there are different (but related) kinds of ideal, there are different (but very similar) kinds of improper ideal.

+-- {: .num_defn}
###### Definition (improper ideal of a ring or other rig)

If $R$ is a [[ring]] or even a [[rig]], then $R$ is a two-sided [[ideal]] of itself, the __improper ideal__.
=--

+-- {: .num_defn}
###### Definition (improper ideal of a lattice or other proset)

If $L$ is a [[lattice]] or even a [[proset]], then $L$ is an [[ideal]] of itself, the __improper ideal__.
=--

## Properties

The improper ideal contrasts with [[proper ideals]] (all of the other ideals).

The improper ideal does not count as a [[prime ideal]] or a [[maximal ideal]], because it is [[too simple to be simple]], although it may satisfy careless or naïve definitions of those concepts (which must be required to be proper).

Conversely, sometimes one defines 'ideal' to exclude the improper ideal, but this makes the set of all ideals less nice to work with.

[[!redirects improper ideal]]
[[!redirects improper ideals]]


improper subset

\tableofcontents

## Idea

Let $S$ be a [[set]].  Then the __improper subset__ of $S$ is $S$ itself, viewed as a [[subset]] of itself.

As the subsets of $S$ correspond to the [[predicates]] on $S$ and to the [[unary relations]] on $S$, so the improper subset corresponds to the predicate that is always [[true]] and the relation which always holds.

As the subsets of $S$ correspond to the [[injections]] to $S$ (up to [[isomorphism]] in the [[slice category]] $Set/S$), so the improper subset corresponds to the [[identity function]] $\id_S\colon S \to S$ (or to any [[bijection]] to $S$, since these are all isomorphic in $Set/S$).

The improper subset is the [[top element]] of the [[power set]] $\mathcal{P}S$, viewed as a [[lattice]].

The improper subset is so called because it is not a [[proper subset]]; by [[excluded middle]], it is the only subset that is not proper.

## Generalizations

The notion of improper subset could be generalized from [[Set]] to any category $C$ as the notion of **improper subobject**.

If a [[subobject]] of an [[object]] $A \in \mathrm{Ob}(C)$ is an [[isomorphism class]] of [[monomorphisms]] into $A$, then an **improper subobject** is of an object $A \in \mathrm{Ob}(C)$ is an [[isomorphism class]] of [[isomorphisms]] into $A$.

## See also

* [[monomorphism]], [[isomorphism]]

* [[subset]], [[proper subset]], [[improper subset]]

* [[subobject]], [[proper subobject]], [[improper subobject]]

* [[filter]], [[proper filter]], [[improper filter]]

[[!redirects improper subset]]
[[!redirects improper subsets]]

[[!redirects improper subobject]]
[[!redirects improper subobjects]]


Imre Simon

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

## Selected writings

On [[tropical semirings]]:

* [[Imre Simon]], (1978), _Limited Subsets of a Free Monoid_, in Proc. 19th Annual Symposium on Foundations of Computer Science, Piscataway, N.J.,
Institute of Electrical and Electronics Engineers, 143–150 ([doi:10.1109/SFCS.1978.21](https://doi.org/10.1109/SFCS.1978.21))

category:people


IMU

__International Mathematical Union__ (abbreviation: IMU) is the principal international organization of mathematicians; it includes many national mathematical societies as collective members. IMU organizes the International Congress of Mathematicians ([[ICM]]) every four years and awards the Fields Medal
([wikipedia](http://en.wikipedia.org/wiki/Fields_Medal), [F.M. at IMU](http://www.mathunion.org/general/prizes/fields/details)).

* [www.mathunion.org](http://www.mathunion.org)

[[!redirects International Mathematical Union]]