Eran Palti

* [webpage](https://www.thphys.uni-heidelberg.de/~palti/)

## Selected writings

On [[string phenomenology]]:

* [[Eran Palti]], _Review of Model Building in String Theory_, talk at [String Phenomenology 2014](http://stringpheno2014.ictp.it/program.html) ([pdf](http://stringpheno2014.ictp.it/lecturenotes/Palti.pdf))

On [[intersecting D-brane models]] in [[type IIB string theory]] and [[F-theory]]:

* Anshuman Maharana, [[Eran Palti]], _Models of Particle Physics from Type IIB String Theory and F-theory: A Review_ ([arXiv:1212.0555](http://arxiv.org/abs/1212.0555))

On computation of [[Yukawa couplings]] in [[heterotic string theory]]:

* Stefan Blesneag, [[Evgeny Buchbinder]], [[Andrei Constantin]], [[Andre Lukas]], [[Eran Palti]], _Matter Field Kähler Metric in Heterotic String Theory from Localisation_ ([arXiv:1801.09645](https://arxiv.org/abs/1801.09645))

On the [[landscape of string theory vacua]] for [[heterotic string theory]] via [[heterotic line bundle models]];

* [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds_, Phys. Rev. D 84, 106005 (2011) ([arXiv:1106.4804](http://arxiv.org/abs/1106.4804))

* [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Heterotic Line Bundle Standard Models_ JHEP06(2012)113 ([arXiv:1202.1757](https://arxiv.org/abs/1202.1757))

* [[Lara Anderson]], [[Andrei Constantin]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _A Comprehensive Scan for Heterotic $SU(5)$ GUT models_, JHEP01(2014)047 ([arXiv:1307.4787](https://arxiv.org/abs/1307.4787))

* [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Heterotic standard model database_ ([web](http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/linebundlemodels/index.html.))

On [[F-theory on Spin(7)-manifolds]] and [[Witten's Dark Fantasy]]:

* [[Federico Bonetti]], [[Thomas Grimm]], [[Eran Palti]], [[Tom Pugh]], _F-Theory on $Spin(7)$ Manifolds: Weak-Coupling Limit_, JHEP02(2014)076 ([arXiv:1309.2287](https://arxiv.org/abs/1309.2287))

On [[M-theory on 8-manifolds]], [[F-theory on Spin(7)-manifolds]] and [[G-structures]]:

* Cezar Condeescu, [[Andrei Micu]], [[Eran Palti]], _M-theory Compactifications to Three Dimensions with M2-brane Potentials_, JHEP 04 (2014) 026 ([arxiv:1311.5901](https://arxiv.org/abs/1311.5901))

On the [[swampland]] and the [[weak gravity conjecture]]:

* [[Eran Palti]], _The Swampland: Introduction and Review_, lecture notes ([arXiv:1903.06239](https://arxiv.org/abs/1903.06239))

On [[string scattering amplitude|string]] [[vacuum amplitudes]]

* {#Palti} [[Eran Palti]], _The IIA/B superstring one-loop vacuum amplitude_ ([pdf](http://www.thphys.uni-heidelberg.de/~palti/Stringcourse/problemset11.pdf))

## Related $n$Lab entries

* [[string phenomenology]]

category: people


Erdal İnönü

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

## related $n$Lab entries

* [[group contraction]]

category: people

[[!redirects Erdal Inönü]]


Erez Berg

* [Research group page](https://www.weizmann.ac.il/condmat/berg/)

* [GoogleScholar page](https://scholar.google.com/citations?user=fbMOJU4AAAAJ&hl=en)

## Selected writings

On [[symmetry-protected topological order]]:

* {#PBTO09} [[Frank Pollmann]], [[Erez Berg]], [[Ari M. Turner]], [[Masaki Oshikawa]], *Symmetry protection of topological order in one-dimensional quantum spin systems*, Phys. Rev. B **85** 075125 (2012) [[arXiv:0909.4059](https://arxiv.org/abs/0909.4059), [doi:10.1103/PhysRevB.85.075125](https://doi.org/10.1103/PhysRevB.85.075125)]

category: people


Erez Urbach

* [webpage](https://www.weizmann.ac.il/physics/erez-urbach)

## Selected writings

On the [[AdS-CFT correspondence]]  for [[higher spin gauge theory]]:

* [[Ofer Aharony]], [[Shai Chester]], [[Erez Urbach]], _A Derivation of AdS/CFT for Vector Models_ ([arXiv:2011.06328](https://arxiv.org/abs/2011.06328))

category: people


Ergin Sezgin

* [webpage](http://physics.tamu.edu/directory/showpeople.php?name=Ergin%20Sezgin&userid=sezgin)

## Selected writings

On [[supergravity]]:

* [[Abdus Salam]], [[Ergin Sezgin]] (eds.), *Supergravities in Diverse Dimensions*, Elsevier & World Scientific (1990) [[doi:10.1142/0277](https://doi.org/10.1142/0277)]

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

* {#HoweSezgin97} [[Paul Howe]], [[Ergin Sezgin]], _$D=11$, $p=5$_, Phys. Lett. B **394** (1997) 62-66 [[arXiv:hep-th/9611008](https://arxiv.org/abs/hep-th/9611008), doi:10.1016/S0370-2693(96)01672-3]

* {#HoweSezginWest97} [[Paul Howe]], [[Ergin Sezgin]], [[Peter West]], _Covariant Field Equations of the M Theory Five-Brane_, Phys. Lett. B **399** (1997) 49-59 [[arXiv:hep-th/9702008](https://arxiv.org/abs/hep-th/9702008), doi:10.1016/S0370-2693(97)00257-8]

On [[D=14 supersymmetry]]:

* [[Ergin Sezgin]], _Super Yang-Mills in $(11,3)$ Dimensions_, 	Phys.Lett. B403 (1997) 265-272 ([arXiv:hep-th/9703123](https://arxiv.org/abs/hep-th/9703123))

* I. Rudychev, [[Ergin Sezgin]], P. Sundell, _Supersymmetry in Dimensions Beyond Eleven_, Nucl. Phys. Proc. Suppl. 68 (1998) 285-294 ([arXiv:hep-th/9711127](https://arxiv.org/abs/hep-th/9711127))

On the [[M2-brane]] via the [[superembedding approach]]:

* {#HoweSezgin05} [[Paul Howe]], [[Ergin Sezgin]], _The supermembrane revisited_, Class.Quant.Grav. 22 (2005) 2167-2200 ([arXiv:hep-th/0412245](https://arxiv.org/abs/hep-th/0412245))

On deriving the [[equations of motion]] for the [[M5-brane]] via [[superspace]]-methods:

* {#HoweSezginWest97} [[Paul Howe]], [[Ergin Sezgin]], [[Peter West]], _Covariant Field Equations of the M Theory Five-Brane_, Phys. Lett. B399 (1997) 49-59 ([arXiv:hep-th/9702008](https://arxiv.org/abs/hep-th/9702008))

and specifically via the [[superembedding approach]]:

* {#HoweSezgin97} [[Paul Howe]], [[Ergin Sezgin]], _$D=11$, $p=5$_, Phys.Lett. B394 (1997) 62-66 ([arXiv:hep-th/9611008](https://arxiv.org/abs/hep-th/9611008))

On [[boundary conditions]] for interacting [[M2-branes]] in regarding through the [[ABJM model]]:

* {#BPST09} [[David Berman]], [[Malcolm J. Perry]], [[Ergin Sezgin]], [[Daniel C. Thompson]], *Boundary Conditions for Interacting Membranes*, JHEP 1004:025, 2010 [[arXiv:0912.3504](https://arxiv.org/abs/0912.3504), doi:10.1007/JHEP04(2010)025]

Introducing a class of [[Lagrangian densities]] for [[D=6 N=(1,0) SCFTs]] with [[non-abelian group|non-abelian]] [[gauge group]] has been proposed in

* {#SSW11} [[Henning Samtleben]], [[Ergin Sezgin]], Robert Wimmer, _$(1,0)$ superconformal models in six dimensions_,  J. High Energ. Phys. 2011, 62 (2011) ([arXiv:1108.4060](https://arxiv.org/abs/1108.4060), doi:10.1007/JHEP12(2011)062)

reviewed in

* [[Henning Samtleben]], [[Ergin Sezgin]], Robert Wimmer, Linus Wulff, _New superconformal models in six dimensions: Gauge group and representation structure_, PoS CORFU2011 (2011) 071 ([arXiv:1204.0542](https://arxiv.org/abs/1204.0542), [spire:1102894](http://inspirehep.net/record/1102894))

On [[Freund-Rubin compactifications]] in [[11d supergravity]]:

* [[Ergin Sezgin]], _11D Supergravity on $AdS_4 \times S^7$ versus $AdS_7 \times S^4$_ ([arXiv:2003.01135](https://arxiv.org/abs/2003.01135))

On [[E11]]-[[exceptional field theory]]:

* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _On supersymmetric E11 exceptional field theory_ ([arXiv:1907.02080](https://arxiv.org/abs/1907.02080))

* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _A master exceptional field theory_ ([arXiv:2103.13411](https://arxiv.org/abs/2103.13411))

On [[higher curvature corrections]] in [[heterotic supergravity]]:

* Hao-Yuan Chang, [[Ergin Sezgin]], Yoshiaki Tanii, *Dimensional reduction of higher derivative heterotic supergravity* ([arXiv:2110.13163](https://arxiv.org/abs/2110.13163))

On [[supergravity]]:

* [[Ergin Sezgin]], *Survey of supergravities*, in *[[Handbook of Quantum Gravity]]* [[arXiv:2312.06754](https://arxiv.org/abs/2312.06754)]

On [[higher curvature corrections]] in [[supergravity]]:

* Mehmet Ozkan, Yi Pang, [[Ergin Sezgin]], *Higher Derivative Supergravities in Diverse Dimensions* [[arXiv:2401.08945](https://arxiv.org/abs/2401.08945)]

category: people


ergodic process

## Related entries

* [[probability theory]]

## References

* Wikipedia, _[Ergodic process](http://en.wikipedia.org/wiki/Ergodic_process)_

[[!redirects ergodic processes]]


ergodic theory

#Contents#
* table of contents
{:toc}

## Idea

__Ergodic theory__ studies [[dynamical system]]s in spaces with an invariant [[measure]]. In particular, the long term limit of the average of the position of a particle is studied. In [[statistical physics]], the derivation of various [[thermodynamics|thermodynamic]] principles depends much on what is called the __ergodic hypothesis__ ([wikipedia](http://en.wikipedia.org/wiki/Ergodic_hypothesis)) saying that in long the term in a system with a large number of particles the probability of passing on the constant [[energy]] surface near some particular [[state]] is independent of the state on that surface and its probability does not depend on the initial conditions (other than energy).

Ergodic reasoning is important also in construction of counterexamples in geometry and geometric group theory as well as in the construction and study of various measures, representations, random walks and so on related to algebraic objects and objects of harmonic analysis.

## References

* wikipedia [ergodic theory](http://en.wikipedia.org/wiki/Ergodic_theory), [ergodic hypothesis](http://en.wikipedia.org/wiki/Ergodic_hypothesis)

A list of references available at this course [page](http://www.math.cornell.edu/~slim/ergodic.html).

Discussion of [[cohomology]]-considerations in ergodic theory is in

* Anatole Katok, E. A. Robinson, Jr., _Cocycles, cohomology and combinatorial constructions in ergodic theory_, Proceedings of Symposia in Pure Mathematics Volume 00, 2001 ([pdf](http://www.math.psu.edu/katok_a/pub/KR.pdf))


Erhard Scheibe

* [Wikipedia page](https://de.wikipedia.org/wiki/Erhard_Scheibe) (de)

* [institute page](https://www.uni-heidelberg.de/fakultaeten/philosophie/philsem/personal/scheibe.html)

* Brigitte Falkenburg: *The Philosopher of the Physicists: The Legacy of Erhard Scheibe*, Journal for General Philosophy of Science **42** (2011) 1–15 [[doi:10.1007/s10838-011-9158-9](https://doi.org/10.1007/s10838-011-9158-9)]

## Selected writings

On the [[axiom|axiomatization]], [[quantum logic|logic]] and [[interpretation of quantum mechanics|interpretation]] of [[quantum mechanics]]:

* {#Scheibe73} [[Erhard Scheibe]], _The logical analysis of quantum mechanics_, Pergamon Press, Oxford (1973)

## Related entries

* [[Bohr topos]]

* [[quantum measurement]]

* [[wavefunction collapse]]

category: people


Eric > history

< [[Eric]]
[[!redirects Eric -- history]]


Eric Bergshoeff

* [webpage](http://thep.housing.rug.nl/people/eric-bergshoeff)

## Selected writings

On [[double dimensional reduction]] of the [[M5-brane]] to the [[D4-brane]]:

* [[Eric Bergshoeff]], [[Mees de Roo]], [[Tomas Ortin]], *The Eleven-dimensional Five-brane*, Phys. Lett. B **386** (1996) 85-90 [[arXiv:hep-th/9606118](https://arxiv.org/abs/hep-th/9606118), doi:10.1016/0370-2693(96)00913-6]

On [[brane intersection]] laws of [[M-branes]]:

* [[Eric Bergshoeff]], [[Mees de Roo]], [[Eduardo Eyras]], [[Bert Janssen]], [[Jan Pieter van der Schaar]], _Intersections involving waves and monopoles in eleven dimensions_, Class. Quantum Grav. **14** (1997) 2757 [[doi:0264-9381/14/10/005](http://iopscience.iop.org/0264-9381/14/10/005)]

* [[Eric Bergshoeff]], [[Joaquim Gomis]], [[Paul Townsend]], _M-brane intersections from worldvolume superalgebras_, Phys.Lett. B421 (1998) 109-118 ([arXiv:hep-th/9711043](http://arxiv.org/abs/hep-th/9711043))

On [[black brane|black]]$\,$[[D8-brane]]-solutions in [[massive type IIA supergravity]]/[[massive type IIA string theory]]:

* {#BergshoessdeRooGreenPapadopoulosTownsend96} [[Eric Bergshoeff]], [[Mees de Roo]], [[Michael Green]], [[George Papadopoulos]], [[Paul Townsend]], _Duality of Type II 7-branes and 8-branes_, Nucl. Phys. B470:113-135, 1996 ([arXiv:hep-th/9601150](https://arxiv.org/abs/hep-th/9601150))

On [[massive type IIA string theory]] and its embedding in [[M-theory]] via [[M9-branes]]:

* [[Eric Bergshoeff]], [[Yolanda Lozano]], [[Tomas Ortin]], *Massive Branes*, Nucl. Phys. B **518** (1998) 363-423 [[arXiv:hep-th/9712115](https://arxiv.org/abs/hep-th/9712115), doi:10.1016/S0550-3213(98)00045-5]

On the [[KK-monopole]] in massive [[D=11 supergravity]], [[KK-reduction|reducing]] to the [[D6-brane]] in [[massive type IIA string theory]]:

* [[Eric Bergshoeff]], [[Eduardo Eyras]], [[Yolanda Lozano]], *The massive Kaluza-Klein monopole*, Physics Letters B
**430** 1–2 (1998) 77-86 [doi:10.1016/S0370-2693(98)00501-2, [arXiv:hep-th/9802199](https://arxiv.org/abs/hep-th/9802199)]

More on [[M9-branes]]:

* [[Eric Bergshoeff]], [[Jan Pieter van der Schaar]], *On M-9-branes*, Class. Quant. Grav. **16** (1999) 23-39 [[arXiv:hep-th/9806069](http://arxiv.org/abs/hep-th/9806069), [doi:10.1088/0264-9381/16/1/002](https://doi.org/10.1088/0264-9381/16/1/002)]

Formulation of ([[Lagrangian densities]] for) [[type II supergravity]] with "democratic"/"[[pregeometric RR-field|pregeometric]]" [[RR-fields]] subject to [[self-dual higher gauge theory|self-duality]]:

* [[Eric Bergshoeff]], [[Renata Kallosh]], [[Tomas Ortin]], [[Diederik Roest]], [[Antoine Van Proeyen]], *New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls*, Class. Quant. Grav. **18** (2001) 3359-3382 [[arXiv:hep-th/0103233](https://arxiv.org/abs/hep-th/0103233), [doi:10.1088/0264-9381/18/17/303](https://doi.org/10.1088/0264-9381/18/17/303)]

On the [[non-abelian DBI action]]:

* [[Eric Bergshoeff]], [[Adel Bilal]], [[Mees de Roo]] and A. Sevrin, _Supersymmetric non-abelian Born-Infeld revisited_, JHEP 0107, 029 (2001) ([arXiv:hep-th/0105274](https://arxiv.org/abs/hep-th/0105274))

On open [[M5-branes]] [[intersecting brane|intersecting]] [[M9-branes]] in a [[Yang monopole]]:

* [[Eric Bergshoeff]], [[Gary Gibbons]], [[Paul Townsend]], _Open M5-branes_, Phys. Rev. Lett. 97:231601 2006 ([arXiv:hep-th/0607193](http://arxiv.org/abs/hep-th/0607193), [doi:10.1103/PhysRevLett.97.231601](https://doi.org/10.1103/PhysRevLett.97.231601))

On [[defect branes]]:

* {#BershoeffOrtinRiccioni11} [[Eric Bergshoeff]], [[Tomas Ortin]], [[Fabio Riccioni]], _Defect Branes_, Nuclear Physics B **856** 2 (2012) 210-227 [[arXiv:1109.4484](https://arxiv.org/abs/1109.4484), [doi:10.1016/j.nuclphysb.2011.10.037](https://doi.org/10.1016/j.nuclphysb.2011.10.037)]

On [[black branes]]/[[solitonic branes|solitonic]] [[exotic branes]] and their [[double dimensional reduction]]:

* [[Eric A. Bergshoeff]], [[Fabio Riccioni]], *Solitonic branes and wrapping rules*, Phys. Part. Nuclei **43** (2012) 557–561 [[doi:10.1134/S106377961205005X](https://doi.org/10.1134/S106377961205005X)]

category: people

[[!redirects Eric A. Bergshoeff]]


Eric D'Hoker

* [webpage](http://www.physics.ucla.edu/~dhoker/index.htm)

## Selected writings

On the [[Skyrme model]] including [[leptons]]:

* [[Eric D'Hoker]], Edward Farhi, _The decay of the skyrmion_, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 86-90 (doi:10.1016/0370-2693(84)90991-2)

Exposition of the [[Skyrme model]]:

* [[Eric D'Hoker]], Edward Farhi, _The Proton as a Topological Twist: A model of elementary particles, without reference to quarks, contains topological structures whose properties match experimental observations remarkably well_, American Scientist Vol. 73, No. 6 (November-December 1985), pp. 533-540 ([jstor:27853483](https://www.jstor.org/stable/27853483))

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

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

On [[black brane|black]] [[M2-M5 brane bound states]]:

* [[Eric D'Hoker]], John Estes, Michael Gutperle, Darya Krym, _Exact Half-BPS Flux Solutions in M-theory I, Local Solutions_, JHEP 0808:028, 2008 ([arXiv:0806.0605](https://arxiv.org/abs/0806.0605))

Discussion of [[BTZ black hole|BTZ]] [[black hole entropy]] and more generally of [[holographic entanglement entropy]] in [[3d quantum gravity]]/[[AdS3/CFT2]] via [[Wilson line observables]] in [[Chern-Simons theory]]:

* [[Eric D'Hoker]], [[Per Kraus]], _Gravitational Wilson lines in $AdS_3$_ ([arXiv:1912.02750](https://arxiv.org/abs/1912.02750))

On [[2-loop]] [[string scattering amplitudes]] of the [[superstring]]:

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings I, Main Formulas_, Phys. Lett. B529:241-255, 2002 ([arXiv:hep-th/0110247](http://arxiv.org/abs/hep-th/0110247))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings II, The Chiral Measure on Moduli Space_, Nucl. Phys. B636:3-60, 2002 ([arXiv:hep-th/0110283](http://arxiv.org/abs/hep-th/0110283))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings III, Slice Independence and Absence of Ambiguities_, Nucl. Phys. B636:61-79, 2002 ([arXiv:hep-th/0111016](http://arxiv.org/abs/hep-th/0111016))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings IV, The Cosmological Constant and Modular Forms_, Nucl. Phys. B639:129-181, 2002 ([arXiv:hep-th/0111040](http://arxiv.org/abs/hep-th/0111040))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings V: Gauge Slice Independence of the N-Point Function_, Nucl. Phys. B715:91-119, 2005 ([arXiv:hep-th/0501196](http://arxiv.org/abs/hep-th/0501196))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings VI: Non-Renormalization Theorems and the 4-Point Function_,  	Nucl. Phys. B715:3-90, 2005 ([arXiv:hep-th/0501197](http://arxiv.org/abs/hep-th/0501197))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings VII, Cohomology of Chiral Amplitudes_, Nucl. Phys. B804:421-506, 2008 ([arXiv:0711.4314](http://arxiv.org/abs/0711.4314))

* [[Eric D'Hoker]], [[Carlos Mafra]], [[Boris Pioline]], [[Oliver Schlotterer]], _Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors_ ([arXiv:2006.05270](https://arxiv.org/abs/2006.05270))

* [[Eric D'Hoker]], [[Carlos Mafra]], [[Boris Pioline]], [[Oliver Schlotterer]], _Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality_ ([doi:2008.08687](https://arxiv.org/abs/2008.08687))

On [[modular forms]] in relation to [[string theory]]:

* [[Eric D'Hoker]], [[Justin Kaidi]], *Lectures on modular forms and strings* [[arXiv:2208.07242](https://arxiv.org/abs/2208.07242)]

## Related entries

* [[string theory]]

* [[string scattering amplitude]]

category: people


Eric Finster

* [webpage](http://ericfinster.github.io/)

## Selected writings

Formalization of (the proof of) the [[Blakers-Massey theorem]] in [[homotopy type theory]]:

* {#FFLL16} [[Kuen-Bang Hou]] ([[Favonia]]), [[Eric Finster]], [[Dan Licata]], [[Peter LeFanu Lumsdaine]], *A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory*, LICS '16 (2016) 565–574  [[arXiv.1605.03227](https://arxiv.org/abs/1605.03227), [doi:10.1145/2933575.2934545](https://doi.org/10.1145/2933575.2934545)]

Proving the [[Blakers-Massey theorem]] in any [[(infinity,1)-topos|$(\infty,1)$-topos]] and with the [[(n-connected, n-truncated) factorization system]] allowed to be replaced by more general [[modal homotopy type theory|modalities]]:

* {#AnelBiedermanFinsterJoyal17a} [[Mathieu Anel]], [[Georg Biedermann]], [[Eric Finster]], [[André Joyal]], _A Generalized Blakers-Massey Theorem_, Journal of Topology **13** 4 (2020) 1521-1553 $[$[arXiv:1703.09050](https://arxiv.org/abs/1703.09050), [doi:10.1112/topo.12163](https://doi.org/10.1112/topo.12163)$]$

On combining [[homotopy type theory]] with [[opetopic type theory]] (for [[higher algebra]]):

* {#AlliouxFinsterSozeau21} [[Antoine Allioux]], [[Eric Finster]], [[Matthieu Sozeau]], _Types are internal infinity-groupoids_, 2021 ([hal:03133144](https://hal.inria.fr/hal-03133144), [pdf](https://hal.inria.fr/hal-03133144/document))

reviewed in:

* {#Finster18} [[Eric Finster]], _[[Towards Higher Universal Algebra in Type Theory|Towards Higher Universal Algebra in Type Theory]]_, [Homotopy Type Theory Electronic Seminar](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html) 2018 ([recording](https://www.youtube.com/watch?v=hlCVHVtAlqQ), [[Agda]] [code](https://github.com/ericfinster/higher-alg))

* [[Eric Finster]], *The $(\infty,1)$-category of Types*, talk at [[CQTS]] (Dec 2022) [slides: [[Finster-CQTS2022.pdf:file]], video: [rec](https://nyu.zoom.us/rec/share/MFJT2B2SX8XUsB0n4z-hwMKb-mxyenQhE5HgBC72qIdHb-ixm7FqQ_KN8hDa0YWl.tEp5kHPzkef1ZDHU)]

  > Abstract: A major outstanding difficulty in [[homotopy type theory|Homotopy Type Theory]] is the description of [[coherence law|coherent]] [[nLab:higher algebra|higher algebraic structures]]. As an example, we know that the algebraic structure possessed by the collection of types and functions between them is *not* a traditional 1-category, but rather an [[(infinity,1)-category|(∞,1)-category]]. In this talk, I will describe how the addition of a finite collection of additional definitional equalities designed to render the notion of "[[opetopic type theory|opetopic type]]" definable in fact allows one to construct the [[(∞,1)-category]] structure on the [[universe of types]].

On [[homotopy type theory|homotopy]] [[dependent linear type theory]] of dependent [[stable homotopy types]] with [[categorical semantics]] in [[parametrized spectra]]:

* {#RileyFinsterLicata21} [[Mitchell Riley]], [[Eric Finster]], [[Daniel Licata]], *Synthetic Spectra via a Monadic and Comonadic Modality* ([arXiv:2102.04099](https://arxiv.org/abs/2102.04099))

On [[polynomial functors]]:

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

On [[weak omega-category|weak $\omega$-categories]] via [[computads]] construed as [[inductive types]]:

* [[Christopher J. Dean]], [[Eric Finster]], [[Ioannis Markakis]], [[David Reutter]], [[Jamie Vicary]], *Computads for weak $\omega$-categories as an inductive type* [[arXiv:2208.08719](https://arxiv.org/abs/2208.08719)]

## Related entries

* [[opetopic type theory]]

* [[homotopytypetheory:Eric Finster, Towards Higher Universal Algebra in Type Theory]]

category: people


Eric Forgy

[[eric.jpg:pic]]

* [[ericforgy:HomePage|Personal nLab Area]]
* [[ericforgy:Eric Forgy|Short bio]]

[[An Exercise in Groupoidification]]

category: people

[[!redirects Eric]]
[[!redirects nlabmeta:Eric Forgy]]


Eric Friedlander

Eric Friedlander is an American mathematician, who has worked extensively in algebraic K-theory, and its relationship with algebraic geometry.

* [website](http://www.math.northwestern.edu/~eric/)

## Selected writings

* E. M. Friedlander, 1982, _Étale homotopy of simplicial schemes_ , volume 104 of Annals of Mathematics Studies , Princeton University Press, Princeton, N.J.

*  E. M. Friedlander, _Fibrations in étale homotopy theory [(numdam)](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1973__42_/PMIHES_1973__42__5_0/PMIHES_1973__42__5_0.pdf)_, Publ. Math.
Inst. des Haut. Études Scient., 42, (1973), 5 – 46.

Establishing the [[Bousfield-Friedlander model structure]], in particular the [[stable model category|stable]] [[model structure on topological sequential spectra]]:

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf), [[BousfieldFriedlanderSpectra.pdf:file]])

## Related $n$Lab entries

* [[Bousfield-Friedlander model structure]]

* [[Bousfield-Friedlander theorem]]

* [[proper model category]]

* [[étale homotopy]]

* [[Nisnevich site]]

* [[motivic cohomology]]

category: people

[[!redirects E. M. Friedlander]]
[[!redirects Eric M. Friedlander]]


Eric Gimon

## Selected writings

On [[orientifolds]], [[O-planes]] and [[RR-field tadpole cancellation]]:

* {#GimonPolchinski96} [[Eric Gimon]], [[Joseph Polchinski]], _Consistency Conditions for Orientifolds and D-Manifolds_, Phys.Rev.D54:1667-1676, 1996 ([arXiv:hep-th/9601038](https://arxiv.org/abs/hep-th/9601038))

  (specifically for [[D5-D9 brane bound states]] [[duality between heterotic and type I string theory|dual]] to [[small instantons]] in [[heterotic string theory]])

* {#GimonJohnson96} [[Eric Gimon]], [[Clifford Johnson]], _K3 Orientifolds_, Nucl. Phys. B477: 715-745, 1996 ([arXiv:hep-th/9604129](https://arxiv.org/abs/hep-th/9604129))

* {#BergmanGimonSugimoto01} [[Oren Bergman]], [[Eric Gimon]], [[Shigeki Sugimoto]], _Orientifolds, RR Torsion, and K-theory_, JHEP 0105:047, 2001 ([arXiv:hep-th/0103183](https://arxiv.org/abs/hep-th/0103183))

and their lift to [[M-theory]] ([[MO5]], [[MO9]]):

* {#Gimon98} [[Eric Gimon]], _On the M-theory Interpretation of Orientifold Planes_ ([arXiv:hep-th/9806226](https://arxiv.org/abs/hep-th/9806226), [spire:472499](http://inspirehep.net/record/472499))

Understanding [[solitonic brane|solitonic]] (non-[[singular brane|singular]]) [[D-branes]] and their [[T-duality]] in terms of the [[D-brane charge quantization in K-theory]]:

* {#BergmanGimonHořava99} [[Oren Bergman]], [[Eric G. Gimon]], [[Petr Hořava]], *Brane Transfer Operations and T-Duality of Non-BPS States*, JHEP 9904 (1999) 010 \[doi:10.1088/1126-6708/1999/04/010, [arXiv:hep-th/9902160](https://arxiv.org/abs/hep-th/9902160)\]

category: people

[[!redirects Eric G. Gimon]]


Eric Hoffbeck

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

## related $n$Lab entries

* [[shuffle]], [[shuffle of trees]]

category: people


Eric Hogle

* [personal page](https://connect.gonzaga.edu/hogle)

* [institute page](https://www.gonzaga.edu/academics/faculty-listing/detail/hogle)

## Selected writings

On [[Kronholm's freeness theorem]] in [[cyclic group of order 2|$\mathbb{Z}/2$]]-[[equivariant cohomology|equivariant]] [[Bredon cohomology]]:

* [[Eric Hogle]], [[Clover May]], *The freeness theorem for equivariant cohomology of $Rep(C_2)$-complexes*, Topology and its Applications **285** 1 (2020) 107413 [[arXiv:2005.07300](https://arxiv.org/abs/2005.07300), [doi:10.1016/j.topol.2020.107413](https://doi.org/10.1016/j.topol.2020.107413)]

category: people


Eric J. Moore

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=16127)

## Selected writings

On [[strict (2,1)-categories]] in the guise of [[Grpd]]-[[enriched categories]], motivated from the [[homotopy 2-category]] of [[topological spaces]]:

* [[Peter H. H. Fantham]], [[Eric J. Moore]], *Groupoid Enriched Categories and Homotopy Theory*, Canadian Journal of Mathematics **35** 3 (1983) 385-416 ([doi:10.4153/CJM-1983-022-8](https://doi.org/10.4153/CJM-1983-022-8))

category: people

[[!redirects Eric Moore]]

[[!redirects E.J. Moore]]


Eric Laenen

* [webpage](http://www.nikhef.nl/~t45/wp/wordpress/)

category: people


Eric Oliver Paquette

## Selected writings

On [[quantum measurement]] formulated in [[finite quantum mechanics in terms of dagger-compact categories ]] in terms of [[Frobenius algebras]] and the [[quantum reader monad]]:

* [[Bob Coecke]], [[Eric Oliver Paquette]], *POVMs and Naimark's theorem without sums*, Electronic Notes in Theoretical Computer Science **210** (2008) 15-31 [[arXiv:quant-ph/0608072](https://arxiv.org/abs/quant-ph/0608072), [doi:10.1016/j.entcs.2008.04.015](https://doi.org/10.1016/j.entcs.2008.04.015)]

* [[Bob Coecke]], [[Eric Paquette]], [[Dusko Pavlovic]], *Classical and quantum structures* (2008) [[pdf](http://www.comlab.ox.ac.uk/files/627/RR-08-02.pdf), [[Coecke-Paquette-Pavlovic-CQS.pdf:file]]]

* [[Bob Coecke]], [[Eric Oliver Paquette]], [[Duško Pavlović]], *Classical and quantum structuralism* [[arXiv:0904.1997](https://arxiv.org/abs/0904.1997)]

On [[quantum information theory via dagger-compact categories]]:

* [[Bob Coecke]], [[Eric Oliver Paquette]], *Categories for the practising physicist*, in: *New Structures for Physics*, Lecture Notes in Physics **813**, Springer (2010) [[arXiv:0905.3010](http://arxiv.org/abs/0905.3010), [doi:10.1007/978-3-642-12821-9_3](https://doi.org/10.1007/978-3-642-12821-9_3)]

## Related entries

* [[quantum information theory via dagger-compact categories]]

  * [[classical structure]]

category: people

[[!redirects Eric Paquette]]


Eric Perlmutter

* [webpage](https://www.perimeterinstitute.ca/people/eric-perlmutter)

## Selected writings

Discussion of [[BTZ black hole|BTZ]] [[black hole entropy]] and more generally of [[holographic entanglement entropy]] in [[3d quantum gravity]]/[[AdS3/CFT2]] via [[Wilson line observables]] in [[Chern-Simons theory]]:

* Alejandra Castro, Stephane Detournay, Nabil Iqbal, [[Eric Perlmutter]], _Holographic entanglement entropy and gravitational anomalies_, JHEP 07 (2014) 114 ([arXiv:1405.2792](https://arxiv.org/abs/1405.2792))

On probing [[M-theory]] via [[AdS/CFT]] and the [[conformal bootstrap]]:

* {#ChesterPerlmutter18} [[Shai Chester]], [[Eric Perlmutter]], _M-Theory Reconstruction from $(2,0)$ CFT and the Chiral Algebra Conjecture_ ([arXiv:1805.00892](https://arxiv.org/abs/1805.00892))

On [[string scattering amplitudes]] via [[AdS/CFT]] and the [[conformal bootstrap]]:

* {#ABP18} [[Luis Alday]], [[Agnese Bissi]], [[Eric Perlmutter]], _Genus-One String Amplitudes from Conformal Field Theory_, JHEP06(2019) 010 ([arXiv:1809.10670](https://arxiv.org/abs/1809.10670))

On [[AdS3/CFT2]] and relating to [[random matrix theory]]:

* Gabriele Di Ubaldo, [[Eric Perlmutter]], *$AdS_3/RMT_2$ Duality* [[arXiv:2307.03707](https://arxiv.org/abs/2307.03707)]

category: people


Eric Rowell

* [webpage](http://www.math.tamu.edu/~rowell/)

## Selected writings

On [[modular tensor categories]]:

* [[Eric Rowell]], [[Richard Stong]], [[Zhenghan Wang]], *On classification of modular tensor categories*, Comm. Math. Phys. 292 (2009) no. 2, 343--389 ([arXiv:0712.1377](https://arxiv.org/abs/0712.1377))

On [[braid group representations]] (specifically the [[Burau representation]] and its relation to the [[Alexander polynomial]]) and their application to [[topological quantum computation]]:

* [[Colleen Delaney]], [[Eric C. Rowell]], [[Zhenghan Wang]], *Local unitary representations of the braid group and their applications to quantum computing*, Revista Colombiana de Matemáticas(2017), 50 (2):211 ([arXiv:1604.06429](https://arxiv.org/abs/1604.06429), [doi:10.15446/recolma.v50n2.62211](http://dx.doi.org/10.15446/recolma.v50n2.62211))

On [[topological quantum computation]] by [[braid representation|braiding]] of [[anyons]] (via [[Chern-Simons theory]]):

* [[Eric C. Rowell]], *An Invitation to the Mathematics of Topological Quantum Computation*, J. Phys.: Conf. Ser. 698 (2016) 012012 ([doi:10.1088/1742-6596/698/1/012012](https://iopscience.iop.org/article/10.1088/1742-6596/698/1/012012))

* [[Eric Rowell]], [[Zhenghan Wang]], _Mathematics of Topological Quantum Computing_, Bull. Amer. Math. Soc. 55 (2018), 183-238 ([arXiv:1705.06206](https://arxiv.org/abs/1705.06206), [doi:10.1090/bull/1605](https://doi.org/10.1090/bull/1605))

* [[Eric C. Rowell]], *Braids, Motions and Topological Quantum Computing* [[arXiv:2208.11762](https://arxiv.org/abs/2208.11762)]

## Related entries

* [[modular tensor category]]

category: people

[[!redirects Eric C. Rowell]]


Eric Schechter

[[!redirects Erich Schechter]]

* [webpage](https://math.vanderbilt.edu/schectex/)

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

## Writings

* _[[Handbook of Analysis and its Foundations]]_

## related $n$Lab entries

* [[analysis]]

category: people


Eric Sharpe

* [website](http://www.phys.vt.edu/~ersharpe/)

Eric Sharpe works on [[theoretical physics]] related to [[string theory]] with an emphasis on structural aspects and mathematical formalization. He got his Phd under [[Edward Witten]].

Sharpe has early on realized the role of [[higher differential geometry]] in the form of [[stacks]] and [[gerbes]] in [[string theory]]. He identified string theorists's  "[[discrete torsion]]" with the [[equivariant cohomology]] of [[bundle gerbes]] on [[orbifolds]] ([Sharpe 99](#Sharpe99), [Sharpe 01](#Sharpe01)) and he was the first to consider the possibility that  [[target spaces]] for [[string]] [[sigma-models]] may themselves be [[stacks]]/[[gerbes]] ([Pantev-Sharpe 05a](#PantevSharpe05a), [Pantev-Sharpe 05b](#PantevSharpe05b), [Hellerman-Sharpe 10](#HellermanSharpe10)).

More recently he has been investigating [[quantum sheaf cohomology]].

In ([Sharpe 15](#Sharpe15)) he lays out a general perspective of the role of [[2-group]] and generally [[n-group]] ([[infinity-group]]) global [[higher gauge symmetry]] in [[quantum field theory]] and [[string theory]].

## Selected writings

On [[discrete torsion]]:

* [[Eric Sharpe]], _Discrete torsion_, Phys.Rev. D68 (2003) 126003 ([arXiv:hep-th/0008154](https://arxiv.org/abs/hep-th/0008154))

* {#Sharpe99} [[Eric Sharpe]], _Discrete Torsion and Gerbes I_ ([arXiv:hep-th/9909108](http://arxiv.org/abs/hep-th/9909108))

  _Discrete Torsion and Gerbes II_ ([arXiv:hep-th/9909120](http://arxiv.org/abs/hep-th/9909120))

On [[stacks]] as [[target spaces]] for [[sigma-models]]:

* {#Sharpe01} [[Eric Sharpe]],  _Discrete Torsion, Quotient Stacks, and String Orbifolds_ ([arXiv:math/0110156](http://arxiv.org/abs/math/0110156))

* {#PantevSharpe05a} [[Tony Pantev]], [[Eric Sharpe]], _String compactifications on Calabi-Yau stacks_, Nucl.Phys. B733 (2006) 233-296, ([arXiv:hep-th/0502044](http://arxiv.org/abs/hep-th/0502044))

* {#PantevSharpe05b} [[Tony Pantev]], [[Eric Sharpe]], _Gauged linear sigma-models for gerbes (and other toric stacks)_, ([arXiv:hep-th/0502053](http://arxiv.org/abs/hep-th/0502053))

* {#HellermanSharpe10} S. Hellerman, [[Eric Sharpe]], _Sums over topological sectors and quantization of Fayet-Iliopoulos parameters_, ([arXiv:1012.5999](http://arxiv.org/abs/1012.5999))

* {#Sharpe15} Eric Sharpe, _Notes on generalized global symmetries in QFT_ ([arXiv:1508.04770](http://arxiv.org/abs/1508.04770))

On [[heterotic string]] [[perturbative string theory vacua|vacua]]:

* [[Eric Sharpe]], _Recent developments in heterotic compactifications_, AMS/IP Stud. Adv. Math. 44 (2008) 209-230 ([arXiv:0801.4080](https://arxiv.org/abs/0801.4080))

On [[elliptic genera]] of [[Landau-Ginzburg models]]:

* [[Matthew Ando]], [[Eric Sharpe]], _Elliptic genera of Landau-Ginzburg models over nontrivial spaces_, Adv. Theor. Math. Phys. 16 (2012) 1087-1144 ([arXiv:0905.1285](https://arxiv.org/abs/0905.1285))

On [[quantum sheaf cohomology]]:

* [[Ron Donagi]], [[Josh Guffin]], [[Sheldon Katz]], [[Eric Sharpe]], *A Mathematical Theory of Quantum Sheaf Cohomology*, Asian J. Math. **18** (2014) 387-418 [[arXiv:1110.3751](https://arxiv.org/abs/1110.3751), [doi:10.4310/AJM.2014.v18.n3.a1](https://doi.org/10.4310/AJM.2014.v18.n3.a1)]

On [[duality in string theory]] in view of [[category theory]] and [[higher geometry]]:

* {#Sharpe19} [[Eric Sharpe]], _Categorical Equivalence and the Renormalization Group_, Proceedings of LMS/EPSRC Symposium _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_, Fortschritte der Physik 2019 ([arXiv:1903.02880](https://arxiv.org/abs/1903.02880))

Discussion of the [[Hilbert schemes of points]] of [[ADE-singularities]]:

* [[Ron Donagi]], [[Sheldon Katz]], [[Eric Sharpe]], _Spectra of D-branes with Higgs vevs_, Adv.Theor.Math.Phys. 8 (2005) 813-859 ([arXiv:hep-th/0309270](https://arxiv.org/abs/hep-th/0309270))

On [[orbifolds]] by [[2-groups]] in view of [[sigma-models]] inspired from [[string theory]]:

* [[Alonso Perez-Lona]], [[Eric Sharpe]], *Three-dimensional orbifolds by 2-groups* [[arXiv:2303.16220](https://arxiv.org/abs/2303.16220)]

On [[quantum K-theory rings]]:

* W. Gu, [[Leonardo C. Mihalcea]], [[Eric Sharpe]], H. Zou, *Quantum K theory of symplectic Grassmannians* [[arXiv:2008.04909](https://arxiv.org/abs/2008.04909)

On [[gauged linear sigma-models]]:

* [[Eric Sharpe]], *A survey of recent developments in GLSMs* [[arxiv:2401.11637](https://arxiv.org/abs/2401.11637)]

## Selected talks

* [[Eric Sharpe]], _Sigma models on gerbes_, talk at _[Higher Structures in M-Theory, Durham, August 2018](http://www.maths.dur.ac.uk/lms/109/index.html)_ ([pdf slides](http://www.maths.dur.ac.uk/lms/109/talks/1864sharpe.pdf), [video recording](http://www.maths.dur.ac.uk/lms/109/movies/1864shar.mp4))

category: people


Eric Vasserot

* [webpage](https://webusers.imj-prg.fr/~eric.vasserot/)

## related $n$Lab entries

* [[superscheme]]

category: people


Eric Zaslow

* [webpage](http://www.math.northwestern.edu/~zaslow/)

## Selected writings

On [[quantum cohomology rings]] for [[orbifolds]] of [[Riemann sphere|$\mathbb{C}P^1$]]:

* [[Eric Zaslow]], *Topological Orbifold Models and Quantum Cohomology Rings*, Commun. Math. Phys. **156** (1993) 301-331 [[arXiv:hep-th/9211119](https://arxiv.org/abs/hep-th/9211119), [doi:10.1007/BF02098485](https://doi.org/10.1007/BF02098485)]

## Related entries

* [[topological string]]

* [[mirror symmetry]], [[T-duality]]

category: people


Erich Hecke

[Wikipedia](https://en.wikipedia.org/wiki/Erich_Hecke)

## Selected writings

On [[modular forms]]:

* [[Erich Hecke]], *Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I.*, Mathematische Annalen **114** (1937)  1–28 [[doi:10.1007/BF01594160](https://doi.org/10.1007/BF01594160)]

* [[Erich Hecke]], *Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II.*, Mathematische Annalen (in German), **114** (1937) 316–351 [[doi:10.1007/BF01594180](https://doi.org/10.1007/BF01594180)]

category: people


Erich Jantsch

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

## related $n$Lab entries

* [[dissipative system]]

* [[autopoiesis]]

## writings

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

category: people


Erich Joos

* [personal page](http://www.erichjoos.de/)

* [www.decoherence.de](http://www.decoherence.de)

## Selected writings

Introducing the idea of dynamical [[quantum decoherence]]:

* [[Erich Joos]], [[H. Dieter Zeh]], *The emergence of classical properties through interaction with the environment*, Z. Physik B -- Condensed Matter **59** (1985) 223–243 [[doi:10.1007/BF01725541](https://doi.org/10.1007/BF01725541)]

Review:

* Claus Kiefer, [[Erich Joos]], *Decoherence: Concepts and Examples*, in: *Quantum Future From Volta and Como to the Present and Beyond*, Lecture Notes in Physics *517** (1999) [[arXiv:quant-ph/9803052](https://arxiv.org/abs/quant-ph/9803052), [doi:10.1007/BFb0105342](https://doi.org/10.1007/BFb0105342)]

category: people


Erik Alfsen

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

## Related entries

* [[Alfsen-Shultz theorem]]

category: people


Erik Palmgren

Erik Palmgren (1963–2019) was a Swedish [[category theory|category theorist]].

* [home page](https://staff.math.su.se/palmgren/) (broken) [archived](https://web.archive.org/web/20230325203454/https://staff.math.su.se/palmgren/).

* [obituary by Peter LeFanu Lumsdaine](https://www.math.su.se/om-oss/nyheter/erik-palmgren-1963-2019-1.463835) (broken) [archived](https://web.archive.org/web/20200828141636/https://www.math.su.se/om-oss/nyheter/erik-palmgren-1963-2019-1.463835).

## Selected writings

On [[type universes]]:

* {#Palmgren} [[Erik Palmgren]], _On Universes in Type Theory_, in *Twenty-Five Years of Constructive Type Theory*, Oxford University Press (1998) 191–204 [[pdf](http://www2.math.uu.se/~palmgren/universe.pdf), [doi:10.1093/oso/9780198501275.003.0012](https://doi.org/10.1093/oso/9780198501275.003.0012)]

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 the [[locally cartesian closed category|local cartesian closure]] of the [[2-category]] of groupoids (and the failure of this property for the [[1-category]] version):

* [[Erik Palmgren]], *Groupoids and local cartesian closure* (2003) [[[Palmgren-GroupoidsAndLCC.pdf:file]]]

On [[set theory]] in [[constructive mathematics]] via [[Bishop sets]] and constructive [[ETCS]]:

* {#Palmgren05} [[Erik Palmgren]],  _Bishop's set theory_ (2005) [[pdf](http://www.cse.chalmers.se/research/group/logic/TypesSS05/Extra/palmgren.pdf), [[Palmgren-BishopSetTheory.pdf:file]]]

* [[Erik Palmgren]], _Constructivist and Structuralist Foundations: Bishop's and Lawvere's Theories of Sets_, Annals of Pure and Applied Logic **163** 10 (2012) 1384-1399 (2009) [[doi:10.1016/j.apal.2012.01.011](https://doi.org/10.1016/j.apal.2012.01.011), [arXiv:1201.6272](https://arxiv.org/abs/1201.6272)]

* [[Erik Palmgren]], *Bishop-style constructive mathematics in type theory -- a tutorial* (2013) [[pdf](https://staff.math.su.se/palmgren/Palmgren_Nis.pdf), [[Palmgren-BishopConstructiveMathematics.pdf:file]]]

On [[setoids]]:

* {#PalmgrenWilander14} [[Erik Palmgren]], [[Olov Wilander]], *Constructing categories and setoids of setoids in type theory*, Logical Methods in Computer Science, **10** 3 (2014) lmcs:964 [[arXiv:1408.1364](https://arxiv.org/abs/1408.1364), doi:10.2168/LMCS-10(3:25)2014]

On [[FOLDS]]:

* [[Erik Palmgren]], _Categories with families, FOLDS and logic enriched type theory_ (2016) [[arXiv:1605.01586](https://arxiv.org/abs/1605.01586), [diva2:926427](http://su.diva-portal.org/smash/record.jsf?pid=diva2%3A926427)]

## Related entries

* [[localic homotopy theory]]

* [[Bishop set]]

category: people


Erik Plauschinn

* [personal page](https://webspace.science.uu.nl/~plaus101/)

## Selected writings

On [[conformal field theory]] with applications to [[string theory]]:

* [[Ralph Blumenhagen]], [[Erik Plauschinn]], *Introduction to Conformal Field Theory -- With Applications to String Theory*, Lecture Notes in Physics **779**, Springer (2009) [[doi:10.1007/978-3-642-00450-6](https://doi.org/10.1007/978-3-642-00450-6)]

category: people


Erik van Erp

* [webpage](http://www.math.dartmouth.edu/~vanerp/index.html)

## Selected writings

On [[T-duality]] and [[Langlands dual groups]]:

* {#DaenzerErp} [[Calder Daenzer]], [[Erik Van Erp]], *T-Duality for Langlands Dual Groups*, Advances in Theoretical and Mathematical Physics **18** 6 (2014) 1267–1285 [[arXiv:1211.0763](http://arxiv.org/abs/1211.0763), [doi:10.4310/ATMP.2014.v18.n6.a2](https://dx.doi.org/10.4310/ATMP.2014.v18.n6.a2)]

## Related entries

* [[T-duality]]

category: people

[[!redirects Erik Van Erp]]


Erik Van Erp > history

see *[[Erik van Erp]]*


Erik Verlinde

* [webpage](http://staff.science.uva.nl/~erikv/)

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

## Selected writings

Introducing the [[Verlinde formula]] in [[2d CFT]]:

* [[Erik Verlinde]], *Fusion rules and modular transformations in 2D conformal field theory*, Nuclear Physics B **300** (1988) 360-376 $[$doi:10.1016/0550-3213(88)90603-7$]$

On [[AdS3-CFT2]] for [[D1/D5 brane bound states]] and [[black hole entropy]] [[black holes in string theory|in string theory]]:

* [[Robbert Dijkgraaf]], [[Juan Maldacena]], [[Gregory Moore]], [[Erik Verlinde]], _A Black Hole Farey Tail_ ([arXiv:hep-th/0005003](https://arxiv.org/abs/hep-th/0005003), [spire:526744](http://inspirehep.net/record/526744))

Early proposal relating [[black hole entropy]] to [[quantum error correcting codes]]:

* {#VerlindeVerlinde12} [[Erik Verlinde]], [[Herman Verlinde]], *Black Hole Entanglement and Quantum Error Correction*, J. High Energ. Phys. 2013, 107 (2013) ([arXiv:1211.6913](https://arxiv.org/abs/1211.6913),  doi:10.1007/JHEP10(2013)107)

On [[ER = EPR]]:

* [[Erik Verlinde]], [[Herman Verlinde]], *A Conversation on ER = EPR* [[arXiv:2212.09389](https://arxiv.org/abs/2212.09389)]

## Related $n$Lab entries

* [[Verlinde algebra]]

category: people


Erlangen program

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The _Erlangen program_ (in German, _Erlanger Programm_ ) is a project, begun by [[Felix Klein]] at Erlangen in the 19th century ([Klein 1872](#Klein1872)), to study [[geometry]] via [[symmetry]] [[groups]] of "geometric shapes", hence from the perspective of [[group theory]]. The idea is to take the elementary building blocks of geometry to be not just [[Euclidean spaces]] but more generally [[homogeneous spaces]] $G/H$. These have then also been called _[[Klein geometries]]_ .

In ([Klein 1872, section 1](#Klein1872)) the theme of the program is described as follows:

> Given a manifold and a transformation group acting on it, to investigate those properties of figures [Gebilde] on that manifold which are invariant under [all] transformations of that group.

In modern language this means to consider a group of [[homeomorphisms]] ([[diffeomorphisms]]) [[action|acting]] on a ([[smooth manifold|smooth]]) [[manifold]] together with its [[stabilizer subgroup]] of any prescribed [[submanifold]]. (The concept of [[Lie group]] only emerged at that time, in fact Klein and [[Sophus Lie]] were in close contact, see [Birkhoff-Bennett](#BirkhoffBennett).)

The group of all such transformations

> by which the geometric properties of configurations in space remain entirely unchanged

is called the "Hauptgruppe", translated to "principal group".

A few lines below in ([Klein 1872, section 1](#Klein1872)) is the converse statement

> Given a manifold, and a transformation group acting on it, to study its invariants.

Hence to find the figures which are left invariant by a given group [[action]].

Then in ([Klein 1872, end of section 5](#Klein1872)) it says:

> Suppose in space some group or other, the principal group for instance, be given. Let us then select a single configuration, say a point, or a straight line, or even an ellipsoid, etc., and apply to it all the transformations of the principal group. We thus obtain an infinite manifoldness with a number of dimensions in general equal to the number of arbitrary parameters contained in the group, but reducing in special cases, namely, when the configuration originally selected has the property of being transformed into itself by an infinite number of the transformations of the group. Every manifoldness generated in this way may be called, with reference to the generating group, a body.

This means in modern language, that if $G$ is the given group acting on a given space $X$, and if $S \hookrightarrow X$ is a given subspace (a configuration), then the "body" ("Körper") generated by this is the [[coset]]

$$
  G/Stab_G(S)
$$

of $G$ by the [[stabilizer subgroup]] of that configuration. In the case that $S$ is a point we would now call this the [[orbit]] of $S$. See also there at _[Stabilizer of shapes -- Klein geometry](stabilizer%20group#KleinGeometry)_.

The text goes on to argue that spaces of this form $G/Stab_G(S)$ are of fundamental importance:

> If now we desire to base our investigations upon the group, selecting at the same time certain definite configurations as space-elements, and if we wish to represent uniformly things which are of like characteristics, we must evidently choose our space-elements in such a way that their manifoldness either is itself a body or can be decomposed into bodies.

Following this, such [[coset spaces]] $G/H$ have come to also be called _[[Klein geometries]]_.

When it was proposed, the Erlangen program served to unify various different kinds of geometry, discovered and studied at that time, into a common framwork. On the other hand, many kinds of geometries without global symmetries are not [[Klein geometries]], notably [[Riemannian geometry]] is (in general) not. But a ([[pseudo-Riemannian manifold|pseudo]]) [[Riemannian manifold]] is _locally_ (tangentially) modeled on [[Euclidean space]] ([[Minkowski spacetime]]) and this local model space _is_ a Klein geometry. The generalization of Klein geometry to such local situations is _[[Cartan geometry]]_, see [below](#CartanGeometry).

###In homotopy type theory

In [[homotopy type theory]] the idea of a group of symmetries preserving a figure inside the larger group of symmetries acting on what the figure is inscribed in is represented by any map of the form:

$$
B H \to B G
$$

The [[homotopy fiber]] of such a map is the Klein space $G/H$.

## Refinements and generalizations

### From local to global geometry -- Cartan geometry
 {#CartanGeometry}

While many types of geometries (such as [[Riemannian geometry]]) are not in general [[Klein geometries]], they are _locally_ like Klein geometries. This generalization of Klein geometry is known as _[[Cartan geometry]]_.

 local model                |   global geometry
----------------------------|-----------------
 [[Klein geometry]]         |   [[Cartan geometry]]
 [[Klein 2-geometry]]       |   [[Cartan 2-geometry]]
 [[higher Klein geometry]]  |   [[higher Cartan geometry]]

In [[physics]] terminology this corresponds to "locally gauging" the [[symmetry group]]. For instance for $H \hookrightarrow G$ the inclusion of the [[Lorentz group]] into the [[Poincare group]], then the corresponding [[Klein geometry]] is just [[Minkowski spacetime]], but a corresponding [[Cartan geometry]] is any [[pseudo-Riemannian manifold]], i.e. a [[spacetime]] characterized in the [[first-order formulation of gravity]].

[[!include local and global geometry - table]]

### Higher Klein geometry

Aspects of Klein geometry may be generalized from groups to [[groupoids]] and even [[category|categories]] or $\infty$-[[infinity-groupoid|groupoids]]. See at _[[higher Klein geometry]]_.

### Logicality and invariance

Logicians have attempted to demonstrate that specifically _logical_ constructions are those invariant under the largest group of transformations, in the sense of the Erlangen program. See [[logicality and invariance]].

## Related concepts

* [[symmetry]]

* [[homogeneous space]], [[coset space]]

* [[crystallographic group]]

## References

### Klein's original and historical context

The original article:

* {#Klein1872} [[Felix Klein]], _[[Vergleichende Betrachtungen über neuere geometrische Forschungen]]_ (1872) Mathematische Annalen volume 43, pages 63–100 1893  ([doi:10.1007/BF01446615](https://doi.org/10.1007/BF01446615))

  English translation by M. W. Haskell:

  _A comparative review of recent researches in geometry_, Bull. New York Math. Soc. 2, (1892-1893), 215-249. ([euclid:1183407629](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-2/issue-10/A-comparative-review-of-recent-researches-in-geometry/bams/1183407629.full), LaTeX version retyped by Nitin C. Rughoonauth: [arXiv:0807.3161](https://arxiv.org/abs/0807.3161))

Historical context and related material:

* {#BirkhoffBennett} Garrett Birkhoff, M. K. Bennett, _Felix Klein and His "Erlanger Program"_, in Aspray & Kitcher (eds.), *History and Philosophy of Modern Mathematics*, University of Minnesota Press (1988) 145-176 $[$[jstor:10.5749/j.cttttp0k.9](https://www.jstor.org/stable/10.5749/j.cttttp0k.9), [pdf](https://conservancy.umn.edu/bitstream/handle/11299/185660/11_06Birkhoff.pdf?sequence=1), BirkhoffBennett-KleinAndErlangerProgram.pdf_file$]$

* Jeremy Gray, _Felix Klein's Erlangen programme_, in I. Grattan-Guinness (ed.),  _Landmark Writings in Western Mathematics 1640-1940_, Elsevier (2005) 544-552 $[$[doi:10.1016/B978-0-444-50871-3.X5080-3](https://doi.org/10.1016/B978-0-444-50871-3.X5080-3)$]$

* Vladimir Kisil, _Erlangen Programme at Large: An Overview_ ([arXiv:1106.1686](http://arxiv.org/abs/1106.1686))

* [[John Baez]], [webpage on the Erlangen program](http://math.ucr.edu/home/baez/erlangen)

### Related commentary

* {#Weyl38} [[Hermann Weyl]], *Symmetry*, Journal of the Washington Academy of Sciences, **28** 6 (1938)  253-271 $[$[jstor:24530200](https://www.jstor.org/stable/24530200)$]$

* [[Ernst Cassirer]], _The concept of Group and The Theory of Perception_, Philosophy and Phenomenological Research **5** 1 (1944) 1-36 $[$[doi:10.2307/2102891](https://doi.org/10.2307/2102891)$]$

* {#GruenbaumShephard10} Branko Grünbaum, G. C. Shephard: *A hierarchy of classification methods for patterns*,  Zeitschrift für Kristallographie - Crystalline Materials, **154**  (2010) 163-187 $[$[doi:10.1524/zkri.1981.154.14.165](https://doi.org/10.1524/zkri.1981.154.14.165), [sci-hub:10.1524/zkri.1981.154.3-4.163](https://sci-hub.hkvisa.net/10.1524/zkri.1981.154.3-4.163)$]$

[[!redirects Erlangen programme]]

[[!redirects Erlanger program]]


Ernest Manes

Ernie Manes wrote one of the well-known books on [[algebraic theory|algebraic theories]].  He was a long term collaborator of [[Michael Arbib]].

* [MathematicsGenealogy entry](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=38925)

## Selected writings

On [[algebraic theories]] (such as via [[monads]] and along the lines introducing their formulation as "[[extension systems]]"):

* {#Manes76} [[Ernest G. Manes]], *Algebraic Theories*, Springer (1976) [[doi:10.1007/978-1-4612-9860-1](https://doi.org/10.1007/978-1-4612-9860-1)]

On [[monads]] in [[universal algebra]] and ([[cohomology|co-]])[[homology]]-theory:

* [[H. Applegate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]: _[[Seminar on Triples and Categorical Homology Theory]]_, ETH 1966/67, edited by [[Beno Eckmann]] and [[Myles Tierney]], **[[LNM 80]]**, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

On some aspects of [[algebras over a monad]]:

* [[Ernest Manes]], *A triple miscellany: Some aspects of the theory of algebra over a triple*, PhD thesis, Wesleyan University, 1967. [[pdf](https://github.com/CategoryTheoryArchive/archive/blob/main/resources/1967_manes_triple-miscellany.pdf)]

On [[compact]] algebras:

* [[Ernest Manes]], *A triple theoretic construction of compact algebras*, Lecture Notes in Mathematics **80**, 1969. [[doi:10.1007/BFb0083083](https://doi.org/10.1007/BFb0083083)]

category:people

[[!redirects Ernest G. Manes]]
[[!redirects Ernie Manes]]
[[!redirects E. Manes]]


Ernest Michael

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

## related $n$Lab entries

* [[Michael's theorem]]

* [[paracompact topological space]]

category: people


Ernesto Lupercio

* [publications](http://en.scientificcommons.org/ernesto_lupercio)

## Selected writings

Introducing [[twisted K-theory|twisted]] [[orbifold K-theory]] presented by
[[twisted bundles|twisted]] [[equivariant vector bundles]] (including their form that came to be called called *[[bundle gerbe modules]]*):

* {#LupercioUribe01} [[Ernesto Lupercio]], [[Bernardo Uribe]], *Gerbes over Orbifolds and Twisted K-theory*, Comm. Math. Phys. 245(3): 449-489.  ([arXiv:math/0105039](http://arxiv.org/abs/math/0105039), [doi:10.1007/s00220-003-1035-x](https://doi.org/10.1007/s00220-003-1035-x))

On [[equivariant ordinary differential cohomology]]:

* [[Ernesto Lupercio]], [[Bernardo Uribe]], _Deligne Cohomology for Orbifolds, discrete torsion and B-fields_, in: _Geometric and topological methods for quantum field theory_ Proceedings, Summer School, Villa de Leyva, Colombia, July 9-27, 2001 ([arXiv:hep-th/0201184](https://arxiv.org/abs/hep-th/0201184), [spire:582101](https://inspirehep.net/literature/582101))

On [[transgression]] and [[higher holonomy]] in [[Deligne cohomology]] (for [[bundle gerbes with connection]]) over [[orbifolds]]:

* [[Ernesto Lupercio]], [[Bernardo Uribe]], *Holonomy for Gerbes over Orbifolds*, J. Geom.Phys. **56** (2006) 1534-1560 [[arXiv:math/0307114](https://arxiv.org/abs/math/0307114), [doi:10.1016/j.geomphys.2005.08.006](https://doi.org/10.1016/j.geomphys.2005.08.006)]

Exposition of [[noncommutative geometry]]:

* [[Ernesto Lupercio]], _Non-commutative Geometry Indomitable_ ([arXiv:2008.08529](https://arxiv.org/abs/2008.08529))

category: people


Ernst Binz

* [institute page](https://binz.math.uni-mannheim.de/de/kontakt/index.html)

* [GoogleScholar page](https://inspirehep.net/authors/2270452)

## Selected writings

On [[continuous field of C-star algebras|continuous fields]] of [[Weyl algebras]] as [[continuous deformation quantizations]] of [[symplectic vector spaces]]:

* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization*, Annales Henri Poincaré **5** (2004) 327–346 [[doi:10.1007/s00023-004-0171-y](https://doi.org/10.1007/s00023-004-0171-y)]

On [[group algebras]] of ([[underlying]] [[discrete group|discrete]]) [[Heisenberg groups]] as [[strict deformation quantizations]] of [[presymplectic manifold|pre-]][[symplectic vector spaces|symplectic]] [[topological vector spaces]] by [[continuous field of C-star algebras|continuous fields of]] [[Weyl algebras]]:

* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization*, International Journal of Pure and Applied Mathematics **38** 1 (2007) [[ijpam:2007-38-1/6](https://ijpam.eu/contents/2007-38-1/6/index.html), [pdf](https://www.ijpam.eu/contents/2007-38-1/6/6.pdf)]

On [[Heisenberg groups]], [[Heisenberg Lie algebras]] and [[strict deformation quantization]]:

* [[Ernst Binz]], Sonja Pods, Ch 1 in: *The geometry of Heisenberg groups --- With Applications in Signal Theory, Optics, Quantization, and Field Quantization*, Mathematical Surveys and Monographs **151**, American Mathematical Society (2008) [[ams:surv-151](https://bookstore.ams.org/surv-151)]

category: people


Ernst Cassirer

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

#Contents#
* table of contents
{:toc}

**Ernst Cassirer** (1874-1945) was a German philosopher, famous for many works in the history of philosophy and for developing a theory of *symbolic forms* -- language, myth, religion, art, history, science. As one of the main exponents of the _Marburg school of neo-Kantianism_ and as a specialist in the philosophy of physics and mathematics who was in constant dialogue with the scientists of his time his work builds a bridge between classical (German) philosophy and more modern threads of analytical philosophy.

## Related entries

* [[Hermann Cohen]]
* [[Albert Einstein]]

## Links

* [Stanford Encyclopedia of Philosophy entry](http://plato.stanford.edu/entries/cassirer/)
* [Wikipedia entry (German)](http://de.wikipedia.org/wiki/Ernst_Cassirer)
* [Wikipedia entry](http://en.wikipedia.org/wiki/Ernst_Cassirer)

## References

* E. Cassirer,  _The Concept of Group and the Theory of Perception_ , Phil. Phen. Res. **V** (1944) pp. 1-35.

* E. Cassirer, _Structuralism in modern linguistics_ , Word **1** (1945) pp.99-120.

* E. Cassirer, _Reflections on the Concept of Group and the Theory of Perception_ , pp.271-291 in  D. Ph. Verene (ed.), _Symbol, Myth, and Culture: Essays and Lectures of Ernst Cassirer 1935-1945_ , London 1979.

* E. Cassirer, _Zur Modernen Physik_ , Wissenschaftliche Buchgesellschaft Darmstadt 1957.

* J. Heis, _Ernst Cassirer's neo-Kantian Philosophy of Geometry_ , Brit. J. Hist. Phil. **19** no.4 (2011)  pp.759–794.

category:people


Ernst Kummer

* [Wikipedia enty](http://en.wikipedia.org/wiki/Ernst_Kummer)

## related $n$Lab entries

* [[Kummer theory]]

* [[Kummer exact sequence]]

category: people

[[!redirects Ernst Eduard Kummer]]


Ernst Leonard Lindelöf

* [Wikipedia entry](https://en.wikipedia.org/wiki/Ernst_Leonard_Lindel%C3%B6f)

## related $n$Lab enties

* [[Lindelöf topological space]]

category: people

[[!redirects Ernst Lindelöf]]

[[!redirects Ernst Lindeloef]]
[[!redirects Ernst Leonard Lindeloef]]

[[!redirects Ernst Lindelof]]
[[!redirects Ernst Leonard Lindelof]]


Ernst Specker

**Ernst Specker** (1920-2011) was a Swiss mathematician. He was professor of mathematics at ETH Zürich from 1955-1987.

His approach to [[set theory]] proved instrumental for the [[topos|topos-theoretic]] approach to set theory.

* [Wikipedia entry](https://de.wikipedia.org/wiki/Ernst_Specker)

* [Detailed biography](http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Specker.html)

* E. Engeler, N. Hungerbühler, J. A. Makowsky, _Rembering Ernst Specker (1920-2011)_ , Elem. Math. **67** (2012) pp.1 – 27. ([pdf](http://www.math.ethz.ch/~engeler/Specker_Nachruf.pdf))

## Selected writings:

On axiomatic [[set theory]]:

* E. Specker, _Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom)_ , Z. Math. Logik Grundlagen
Math. **3** (1957) pp.173–210.

* E. Specker, _Die Entwicklung der axiomatischen Mengenlehre_ , JBer. DMV **81** (1978) pp.13-21. ([Jber. DMV](http://dml.math.uni-bielefeld.de/JB_DMV/JB_DMV_081_1.pdf), 6.21 MB)

Introducing the [[Kochen-Specker theorem]]:

* {#KochenSpecker67} [[Simon Kochen]], [[Ernst Specker]], _The problem of hidden variables in quantum mechanics_, Journal of Mathematics and Mechanics **17** (1967) 59-87 [[pdf](http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1968/17/17004/pdf), [jstor:24902153](https://www.jstor.org/stable/24902153), [doi:10.1007/978-94-010-1795-4_17](https://doi.org/10.1007/978-94-010-1795-4_17)]

## Related entries

* [[Kochen-Specker theorem]]

* [[elementary theory of the category of sets]]

* [[constructive mathematics]]

* [[Timeline of category theory and related mathematics]]

category: people


Ernst Steinitz

Ernst Steinitz (1871-1928) was a mathematician.
Nowadays, he is remembered primarily through two, unrelated, theorems:

* Steinitz's theorem on realizability of [[three-connected|3-connected]] [[planar graphs]] by 1-skeleta of 3-dimensional polyhedra,

* Steinitz's exchange lemma for sets of vectors.

## related $n$Lab entries

* [[polyhedron]]
* [[basis of a vector space]]

## References

* [English Wikipedia](https://en.wikipedia.org/wiki/Steinitz%27s_theorem) on Steinitz's theorem on skeleta of polyhedra

category: people


Ernst Stückelberg

[[!redirects Ernst Stueckelberg]]

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

## related $n$Lab entries

* [[causal perturbation theory]]

* [[Stückelberg-Petermann renormalization group]]

category: people


Ernst Witt

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

category: people


Ernst Zermelo

Ernst Zermelo (1871–1953) was a German mathematician best known for his work in the foundations of mathematics, in particular for his axiomatisation of [[set theory]]. The Zermelo–Fraenkel axioms of [[ZFC]] are named after him.

* [wikipedia entry](http://de.wikipedia.org/wiki/Ernst_Zermelo)

category: people

[[!redirects Zermelo]]


Ernst-Erich Doberkat

* [webpage](http://www.cs.tu-dortmund.de/nps/de/Home/Personen/D/Doberkat__Ernst-Erich.html)

## Selected writings

* [[Ernst-Erich Doberkat]], _Stochastic Relations: Foundations for Markov Transition Systems_ CRC Press (2007) [Google Books preview](https://books.google.com.au/books?id=_OmI_xrYqawC&printsec=frontcover#v=onepage&q&f=false)

On the [[Giry monad]]:

* {#Doberkat03} [[Ernst-Erich Doberkat]], _Characterizing the Eilenberg-Moore algebras for a monad of stochastic relations_ ([pdf](https://eldorado.tu-dortmund.de/bitstream/2003/2717/1/147.pdf))

On [[measure theory]]:

* [[Ernst-Erich Doberkat]], _Measures and all that --- A Tutorial_ ([arxiv:1409.2662](http://arxiv.org/abs/1409.2662))

category: people


Ernst-Wilhelm Zink

* [Institute page](https://www.mathematik.hu-berlin.de/en/people/chairs/1919)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=51063)

## Related concepts

On [[L-parameters]] in the [[local Langlands correspondence]]

* {#SilbergerZink14} [[Allan J. Silberger]], [[Ernst-Wilhelm Zink]], Section 3 of: *Langlands Classification for L-Parameters*, Journal of Algebra Volume 511, 1 October 2018, Pages 299-357 ([arXiv:1407.6494](https://arxiv.org/abs/1407.6494), [doi:10.1016/j.jalgebra.2018.06.012](https://doi.org/10.1016/j.jalgebra.2018.06.012))

Talk notes:

* [[Ernst-Wilhelm Zink]], with [[Allan J. Silberger]], *Langlands classification for L-parameters* ([pdf](https://www.math.hu-berlin.de/~zyska/zink/Lvortrag1.pdf))

category: people


Errett Bishop

# Errett Bishop

Errett Bishop (1928--1983) was responsible for reviving interest in [[constructive mathematics]] as a branch of [[mathematics]].  Before this, he had a career in areas of classical (i.e., non-constructive) [[analysis]], including [[approximation theory]], [[several complex variables]], [[function algebras]], and other fields.

See [[Bishop's constructive mathematics]] for a detailed discussion of the system he developed.  Bishop\'s important works in constructive mathematics include:

## Selected writings

On [[constructive mathematics]] in general and [[constructive analysis]] in particular

* [[Errett Bishop]], [[Douglas Bridges]] *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

On [[constructive mathematics]]:

* "Schizophrenia in contemporary mathematics" (1973), a philosophical attack on nonconstructive mathematics;

  see also: "The crisis in contemporary mathematics", Historia Mathematica **2** 4 (1975) 507-517 [[doi.org/10.1016/0315-0860(75)90113-5](https://doi.org/10.1016/0315-0860(75)90113-5)]

On constructive [[measure theory]]:

* _Constructive Measure Theory_ (1972, with Henry Cheng), featuring a better [[measure theory]] than in _FCA_;

category: people


error correcting code

#Contents#
* table of contents
{:toc}

## Idea

In [[coding theory]], an *error correcting code* is a means to encode data in a way that is robust against errors ([[noise]]).

Very broadly, for $L$ a [[finite set]] playing the role of a [[space of states]] that is to be saved/communicated/analyzed, an error correcting code for $L$ is an [[injection]] $L \overset{\;\;\;}{\hookrightarrow} P$ into a larger set. The idea is that noise/errors move the image of $L$ within $P$, but if $P$ is large enough and the embedding chosen well enough, then a sufficiently small number of errors stays within a small neighbourhood of $L$ in $P$ that allows to [[retraction|retract]] back to $L$.

The simplest example is the *repetition code*, where the inclusion is the [[diagonal]] on the $n$-fold [[Cartesian product]]

$$
  \array{
  L
  &
    \overset{diag}{\hookrightarrow}
  &
  P \coloneqq \underset{n \; factors}{\underbrace{L \times \cdots \times L}}
  \\
  \ell &\mapsto& (\ell, \cdots, \ell)
  }
  \,.
$$

This code "protects against $n/2-1$ errors" in an evident sense.

Much attention in [[coding theory]] is instead on the special class of *[[linear codes]]*, where $L$ and $P$ carry the structure of [[vector spaces]] (necessarily over a [[finite field]] if they are [[finite sets]] of relevance in practice) and where the inclusion $L \hookrightarrow P$ is a [[linear map]].

## Examples

* [[linear code]]

  * [[Hamming code]]

  * [[binary linear code]]

    * [[binary Golay code]]

* [[quantum error correcting code]]

  * [[bit flip code]]

  * [[stabilizer code]]

  * [[HaPPY code]]

  * [[Majorana dimer code]]

## References

### General

See also the references at *[[coding theory]]* and *[[linear code]]*.

* [[Victor V. Albert]] et al.,  *[errorcorrectionzoo.org](https://errorcorrectionzoo.org)*

An observation on classical codes preconceiving aspects of [[holographic tensor network]] [[quantum error correcting codes]]:

* [[Beni Yoshida]], *Information storage capacity of discrete spin systems*, Annals of Physics 338, 134 (2013) ([arXiv:1111.3275](https://arxiv.org/abs/1111.3275))

### Relation to 2d CFT

Construction of chiral [[2d SCFTs]] from error-correcting codes:

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Holomorphic SCFTs with small index*, Canadian Journal of Mathematics , **74** 2 (2022) 573-601  [[arXiv:1811.00589](https://arxiv.org/abs/1811.00589), [doi:10.4153/S0008414X2100002X](https://doi.org/10.4153/S0008414X2100002X)]

On their [[elliptic genera]]

* Kohki Kawabata, Shinichiro Yahagi, *Elliptic genera from classical error-correcting codes* [[arXiv:2308.12592](https://arxiv.org/abs/2308.12592)]

[[!redirects error correcting codes]]
[[!redirects error-correcting code]]
[[!redirects error-correcting codes]]

[[!redirects error correction]]
[[!redirects error corrections]]
[[!redirects error-correction]]
[[!redirects error-corrections]]


error correction code

[[!redirects error correction]]

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### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* in [[quantum computation]]: [[quantum error correction]]

## See also

* Wikipedia, *[Error correction code](https://en.wikipedia.org/wiki/Error_correction_code)*

[[!redirects error corrections]]


Erwin Schrödinger

* [Wikipedia entry](http://de.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger)

## Selected writings

Original article on [[quantum mechanics]]:

* [[Erwin Schrödinger]], *Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen*, Annalen der Physil **384** 8 (1926) 734-756 $[$[doi:10.1002/andp.19263840804]( https://doi.org/10.1002/andp.19263840804)$]$

Early discussion of [[composite quantum systems]] and their [[quantum entanglement]] (and introducing this terminology):

* [[Erwin Schrödinger]], *Discussion of Probability Relations between Separated Systems*, Mathematical Proceedings of the Cambridge Philosophical Society, **31** 4 (1935) 555-563 [[doi:10.1017/S0305004100013554](https://doi.org/10.1017/S0305004100013554)]

Early discussion of [[quantum biology]]:

* [[Erwin Schrödinger]], *What is life?*, Cambridge (1948) [cover:[web](https://zbphysik.univie.ac.at/ausstellung/schroedinger/dokumenteOG/whatislife1.htm), retyped:[pdf](https://www.arvindguptatoys.com/arvindgupta/whatislife-schrodinger.pdf), [Wikipedia entry](https://en.wikipedia.org/wiki/What_Is_Life%3F)]

## Related entries

* [[quantum mechanics]]

* [[Schrödinger equation]]

category: people

[[!redirects Erwind Schrodinger]]
[[!redirects Erwin Schroedinger]]


Escardo-Simpson real number

\tableofcontents

## Idea

A notion of the real numbers which is constructed as an [[initial object|initial]] [[interval object]].

## Related concepts

* [[real number]]

* [[unit interval]]

* [[HoTT book real numbers]]

## References

* [[Martín Escardó]] and [[Alex Simpson]]. *A universal characterization of the closed Euclidean interval.* In 16th Annual IEEE Symposium on Logic in Computer Science, Boston, Massachusetts, USA, June 16-19, 2001, Proceedings, pages 115–125. IEEE Computer Society, 2001. ([doi:10.1109/LICS.2001.932488](https://doi.org/10.1109/LICS.2001.932488), [pdf](http://www.cs.bham.ac.uk/~mhe/papers/interval.pdf))

* {#UFP13} [[Univalent Foundations Project]], *[[Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* {#Booij20} [[Auke Booij]], *Analysis in Univalent Type Theory* (2020) [[etheses:10411](http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf), [[Booij-AnalysisInUF.pdf:file]]]

[[!redirects Escardo-Simpson real number]]
[[!redirects Escardo-Simpson real numbers]]


Esfandir Haghverdi

* [webpage](http://xavier.informatics.indiana.edu/~ehaghver/)

## Related entries

* [[Geometry of Interaction]]

* [[linear logic]]

category: people


ESI

Erwin Schroedinger Institute for Mathematical Physics, Vienna

* [web](http://www.esi.ac.at)


eso morphism

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#### 2-category theory
+--{: .hide}
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# Eso morphisms
* table of contents
{: toc}

## Definition

In any [[2-category]] $K$, a morphism $f:A\to B$ is called **eso**, or **strong 1-epic**, if for any [[fully faithful morphism]] $m:C\to D$, the following square is a ([[2-limit|2-categorical]]) [[pullback]] in [[Cat]]:
$$\array{K(B,C) & \to & K(B,D)\\
\downarrow & & \downarrow \\
K(A,C) & \to & K(A,D)}$$
This can be rephrased in elementary terms, without the need for a category $Cat$ in which the hom-categories of $K$ live.

One easily checks that when $K=$ [[Cat]], a functor $f$ is eso if and only if it is [[essentially surjective functor|essentially surjective on objects]] in the usual sense.  (This requires either the [[axiom of choice]] or the use of [[anafunctor]]s in defining $Cat$.)

## Remarks

* If $K$ has finite limits, then $f:A\to B$ is eso if and only if whenever $f\cong m g$ where $m$ is ff, then $m$ is an [[equivalence]].

* Any [[coinserter]], co-[[isoinserter]], [[coinverter]], [[coequifier]], or (lax or oplax) [[codescent object]] is eso.

* If $K$ has finite limits and $f:A\to B$ is eso, then for any $Z$ the functor $K(B,Z)\to K(A,Z)$ is [[faithful functor|faithful]] and [[conservative functor|conservative]].

* If $K$ is a 1-category with finite limits, regarded as a 2-category with only identity 2-cells, then a morphism in $K$ is eso if and only if it is an [[extremal epimorphism]] (equivalently, a [[strong epimorphism]]).

[[!redirects eso morphism]]
[[!redirects eso morphisms]]
[[!redirects strong 1-epimorphism]]
[[!redirects strong 1-epimorphisms]]
[[!redirects strong 1-epic morphism]]
[[!redirects strong 1-epic morphisms]]
[[!redirects strong 1-epic]]
[[!redirects strong 1-epics]]
[[!redirects strong 1-epi]]
[[!redirects strong 1-epis]]


Esquisse d'un programme

##Subtitle: a sketch of a programme

This was a document produced by [[Alexander Grothendieck]] that outline a series of interrelated research projects, (see the [Wikipedia page](http://en.wikipedia.org/wiki/Esquisse_d%27un_Programme)).

###Summary  of Grothendieck's programme

(English translation with original in brackets.)

("Sommaire")

1. The Proposal and enterpise ("Envoi").

1.     "Teichmüller's Lego-game and the [[Galois group]] of Q over Q" ("Un jeu de "Lego-Teichmüller" et le groupe de Galois de Q sur Q").

       (on the relation between the [[absolute Galois group]] and the [[Grothendieck-Teichmüller group]])

1.    Number fields associated with "[[children's drawings]]". ("Corps de nombres associés à un [[dessin d'enfant]]").

1.     Regular polyhedra over finite fields ("Polyèdres réguliers sur les corps finis").

1.    Denunciation of so-called "general" topology, and heuristic reflections
towards a so-called "tame" topology.
 ("Haro sur la topologie dite 'générale', et réflexions heuristiques vers une topologie dite 'modérée").

1.     Differentiable theories and moderated theories ("Théories différentiables" (à la Nash) et "théories modérées").

1.    [[Pursuing Stacks]] ("À la Poursuite des Champs"}.

1.     Two-dimensional geometry ("Digressions de géométrie bidimensionnelle".

1.   Summary of proposed studies ("Bilan d'une activité enseignante").

1.    Epilogue.

   Notes

##References

* A copy of the original version is [here](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/EsquisseFr.pdf).

* An English translation is [here](http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/EsquisseEng.pdf).

* [MO discussion](http://mathoverflow.net/q/24900/381)

* Notes Anabéliennes de A. Grothendieck [link](https://github.com/carmonamateo/NotesAnabeliennes)

[[!redirects Esquisse d’un programme]]


Esquisse d’un programme

"Esquisse d'un Programme" is a proposal for his continuing research made by the [[Alexander Grothendieck]] in 1984.

* [pdf file in French](https://wwwmath.uni-muenster.de/reine/u/topos/AG-Leray/Operads/Grothendieck-Esquisse%20d'un%20programme.pdf)

* [English language version](http://www.landsburg.com/grothendieck/EsquisseEng.pdf)


Esquisses mathématiques

_Esquisses Mathématiques_ is a series of Paris-Amiens University reports published from 1970 to 1983 which contains more or less preliminary works, in particular theses of members of the research team _Théorie et Applications des Catégories_ (Paris-Amiens), directed by [[Charles Ehresmann]] and [[Andrée Ehresmann]].

Most of them are still available and can be obtained by sending an e-mail to [[Andrée Charles Ehresmann]].

A list of titles is on the page linked below:

* [link](https://mes-ehres.fr/CTGD/Ctgdc.htm)

## Related pages

* [[Comptes rendus hebdomadaires des séances de l'Académie des sciences]]

category: reference


essential fiber

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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The notion of *essential fiber* of a [[functor]] is an enhancement of the naive notion of *[[fiber]]* which, for [[functors]], would violate the [[principle of equivalence]].  It is a [[category theory|category-theoretic]] version of a [[homotopy fiber]].

The essential fiber of a functor $p:E\to B$ over an [[object]] $b \in B$ can be thought of as the [[category]] of ways that $b$ can arise from applying $p$ to some object in $E$.

## Definition

\begin{definition}
\label{EssentialFiber}
For $p \colon E\to B$ be a [[functor]] and $b\in B$ an [[object]]  the **essential fiber** of $p$ over $b$ is the [[category]] whose:

* [[objects]] are [[pairs]] $(e,\phi)$ where $e\in E$ is an object and $\phi\colon p(e)\cong b$ is an [[isomorphism]],

* [[morphisms]] $(e,\phi)\to (e',\phi')$ are morphisms $f\colon e\to e'$ in $E$ such that $\phi' \circ p(f) = \phi$,

* [[composition]] operation is the evident one.

\end{definition}

\begin{remark}
The notion of essential fiber in Def. \ref{EssentialFiber} can be identified with that of the *[[pseudopullback]]* or "[[isocomma object]]" of $p$ along the functor $b\colon 1\to B$ from the [[terminal category]] which picks out the object $b$:

$$
  \array{
    E \times_B 1
    &\longrightarrow&
    E
    \\
    \Big\downarrow
    &&
    \Big\downarrow
    \mathrlap{{}_p}
    \\
    1 \!\!\!
    &\underset
      {b}
      {\longrightarrow}
    &
    B
  }
$$

Moreover, the notion can also be identified with a [[homotopy fiber]] in the [[canonical model structure]] on [[Cat]].

Finally, with [[groupoids]] identified as [[homotopy 1-types]], the essential fiber of a functor between groupoids and thought of as [[infinity-groupoids|$\infty$-groupoids]] actually coincides with its *[[homotopy fiber]]* in the classical sense of [[homotopy theory]] (well-defined up to [[weak homotopy equivalence]]).
\end{remark}

## Relationship to fibrations

If $p$ is an [[isofibration]], then any of its essential fibers (Def. \ref{EssentialFiber}) is [[equivalence of categories|equivalent]] to the corresponding strict [[fiber]].  This includes the case when $p$ is a [[Grothendieck fibration]].

On the other hand, when $p$ is a [[Street fibration]] (the version of [[Grothendieck fibration]] which respects the [[principle of equivalence]]), then essential fibers do *not* coincide with strict fibers, and essential fibers are the more useful notion.  In particular, the [correspondence](Grothendieck+construction#EquivalenceBetweenFiberedCategoriesAndPsuedofunctors) between [[Grothendieck fibrations]] and [[pseudofunctors]] only goes through for Street fibrations if one defines the pseudofunctor using essential fibers.

## Properties

Some properties of a functor are reflected in properties of its essential fibers (Def. \ref{EssentialFiber}).   A good intuition is that the more a functor resembles an [[injective function]], the simpler its essential fibers are.

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

A functor $f \colon A \to B$ is [[conservative functor|conservative]] if and only if all its essential fibers are [[groupoids]].
=--

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

If the functor $p \colon E \to B$ is [[faithful]], all its essential fibers are [[preorders]].  (The converse is not true.)
=--

and thus:

+-- {: .num_prop #essential_fibers_faithful_and_conservative_1}
###### Proposition

If the functor $p \colon E \to B$ is [[faithful functor|faithful]] and [[conservative functor|conservative]] then all its essential fibers are [[equivalence of categories|equivalent]] to [[discrete categories]].
=--

\begin{example}
There typically is a nontrivial [[preorder]] of ways that a [[set]] can arise as the [[underlying]] set of a [[topological space]], because the [[forgetful functor]] from [[Top|$\mathrm{Top}$]] to [[Set|$\mathrm{Set}$]] is [[faithful functor|faithful]] but not [[conservative functor|conservative]].

On the other hand, there is a mere set (i.e., a [[discrete category]]) of ways that a set can arise as the [[underlying]] set of a [[group]], because the [[forgetful functor]] from [[Grp|$\mathrm{Grp}$]] to [[Set|$\mathrm{Set}$]], being [[monadic]], is both faithful and conservative.
\end{example}

\begin{remark}
The [[automorphism group]] of $b \in B$ always [[action|acts]] on the essential fiber of $b$.

For example, on objects, $\alpha \in \mathrm{Aut}_B(b)$ acts to send $(e,\phi)$ to $(e, \alpha \circ \phi)$.
\end{remark}

When the essential fiber is essentially a set as in proposition \ref{essential_fibers_faithful_and_conservative_1}, this allows us to describe the essential fiber as a [[union]] of [[orbits]]:

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

If the functor $p \colon E \to B$ is [[faithful functor|faithful]] and [[conservative functor|conservative]], the essential fiber over $b \in B$ is [[equivalence of categories|equivalent]] to the [[discrete category]] on the set
$$
  \coprod_e
  \big(
  \mathrm{Aut}_B (b)
  /
  \mathrm{Aut}_E (e)
  \big)
  \,
$$
where $e \in E$ ranges over one representative of each [[isomorphism class]] in $E$ whose [[image]] is the isomorphism class of $b$.
=--

When $p \colon E \to B$ is not only faithful and conservative but also [[injective]] on [[isomorphism classes]], there is at most one [[isomorphism class]] in $E$ whose [[image]] is the isomorphism class of $b$.  Thus the [[coproduct]] in proposition \ref{essential_fibers_faithful_and_conservative} has at most one summand, and the automorphism group of $b$ acts [[transitive action|transitively]] on the relevant set:

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

If the functor $p \colon E \to B$ is [[faithful]], [[conservative functor|conservative]] and it is [[injective]] on [[isomorphism classes]], then for any $e \in E$, the essential fiber over $b = f(e)$ is [[equivalence of categories|equivalent]] to the [[discrete category]] on the set
$$
  \mathrm{Aut}_B (b)
  /
  \mathrm{Aut}_E (e)
  \,.
$$
Thus $\mathrm{Aut}_B (b)$ [[action|acts]] [[transitive action|transitively]] on this set.
=--

\begin{example}
The [[forgetful functor]] from [[complex vector spaces]] to [[real vector spaces]] is faithful, conservative and injective on isomorphism classes.  The essential fiber over a given real vector space is the set of [[complex structures]] on this vector space, and if this vector space is $\mathbb{R}^{2n}$, proposition \ref{essential_fibers_faithful_conservative_injective} implies that this set of complex structures is isomorphic to (the [[underlying]] set of) the [[coset space]]
$$
  \mathrm{GL}(2n,\mathbb{R})/\mathrm{GL}(n,\mathbb{C})
\,.
$$
\end{example}

## Related entries

* [[fiber]]

* [[homotopy fiber]]

* [[fibered category]]

## References

The above results on essential fibers were proved in this discussion:

* Category Theory Community Server: *[Homotopy fibers](https://categorytheory.zulipchat.com/#narrow/stream/266967-general.3A-mathematics/topic/homotopy.20fibers/near/353365094)* (2023)

[[!redirects essential fibre]]
[[!redirects essential fibers]]
[[!redirects essential fibres]]


essential geometric morphism

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###Context###
#### Topos Theory
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[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[geometric morphism]] $f : E \to F$ between [[topos]]es is a [[functor]] of the underlying categories that is consistent with the interpretation of $E$ and $F$ as generalized [[topological space]]s.

If $F = Set = Sh(*)$ is the terminal [[sheaf topos]], then $E \to Set$ is essential if $E$ is a **[[locally connected topos]]**. In general, $f$ being essential is a necessary (but not sufficient) condition to ensure that $f$ behaves like a map of topological spaces whose [[fiber]]s are locally connected: that it is a [[locally connected geometric morphism]].

## Definition

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

Given a [[geometric morphism]] $(f^* \dashv f_*) : E \to F$, it is an **essential geometric morphism** if the [[inverse image]] functor $f^*$ has not only the [[right adjoint]] $f_*$, but also a [[left adjoint]] $f_!$:

$$
  (f_! \dashv f^* \dashv f_*)
  \;\;\; : \;\;\;
  E
    \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^*}{\longleftarrow}}{\underset{f_*}{\longrightarrow}}}
  F
  \,.
$$

=--

A [[point of a topos]] $x : Set \to E$ which is given by an essential geometric morphism is called an **essential point** of $E$.

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

There are various further conditions that can be imposed on a geometric morphism:

* If $f_!$ can be made into an $E$-[[indexed functor]] and $f^*$ satisfies some extra conditions, the geometric morphism $f$ is a [[locally connected geometric morphism]] (see there for details).

* If $f_!$ preserves finite [[products]] then $f$ is called **connected surjective**.

* If in addition to the above $f$ is a [[local geometric morphism]] in that there is a further functor $f^! : F \to E$ which is  [[right adjoint]] $(f_* \dashv f^!)$ and [[full and faithful functor|full and faithful]] then the geometric morphism $f$ is called **[[cohesive topos|cohesive]]**.

=--

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

Since for a [[Grothendieck topos]] $E$ the [[inverse image]] $x^*$ of an essential point is of the form $Hom_E(P,-)$ where $P\ncong\emptyset$ is [[projective object|projective]] and [[connected]], objects satisfying these three conditions are sometimes called _essential objects_ (cf. [Johnstone 1977, p.255](#JTT77)).

=--

## Properties
 {#Properties}

### A criterion for Grothendieck toposes

The inverse image $f^\ast$ of an essential geometric morphism preserves small limits since it is a right adjoint. Hence, this provides a minimal requirement to satisfy for a general geometric morphism $f^\ast\dashv f_\ast$ in order to qualify for being essential. In case, the toposes involved are [[Grothendieck toposes]] this condition is not only necessary but sufficient.

+-- {: .num_prop}
###### Proposition
Let $f^\ast\dashv f_\ast:\mathcal{E}\to\mathcal{F}$ a geometric morphism between Grothendieck toposes. Then $f$ is essential iff $f^\ast$ preserves small limits iff $f^\ast$ preserves small products.
=--

+-- {: .proof}
###### Proof
Grothendieck toposes are [[locally presentable category|locally presentable]] and $f^\ast$ has rank i.e. it preserves $\alpha$-filtered colimits for some [[regular cardinal]] $\alpha$ since it is a left adjoint. But by ([[Handbook of Categorical Algebra|Borceux vol. 2]]  Thm.5.5.7, p.275) a functor between two locally presentable categories has a left adjoint precisely if it has rank and preserves small limits.

Since limits can be constructed from products and equalizers and $f^\ast$ preserves the latter, it preserves small limits precisely when it preserves small products.
=--

### Relation to morphisms of (co)sites
 {#RelationToSiteMorphisms}

For $C$ and $D$ [[small categories]] write $[C,Set]$ and $[D,Set]$ for the corresponding [[copresheaf]] [[presheaf topos|toposes]]. (If we think of the [[opposite categories]] $C^{op}$ and $D^{op}$ as [[sites]] equipped with the trivial [[coverage]], then these are the corresponding [[sheaf toposes]].)

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

This construction extends to a [[2-functor]]

$$
  [-,Set] : Cat_{small}^{co} \to Topos_{ess}
$$

from the [[2-category]] [[Cat]]${}_{small}$ with [[2-morphism]]s reversed) to the sub-2-category of [[Topos]] on essential geometric morphisms, where a [[functor]] $f : C \to D$ is sent to the essential geometric morphism

$$
  (f_! \dashv f^* \dashv f_*) :
  [C,Set]
    \stackrel{\overset{f_! := Lan_f}{\to}}{\stackrel{\overset{f^* := (-) \circ f}{\leftarrow}}{\underset{f_* := Ran_f}{\to}}}
  [D,Set]
  \,,
$$

where $Lan_f$ and $Ran_f$ denote the left and right [[Kan extension]] along $f$, respectively.

=--

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

This 2-functor is a [[full and faithful 2-functor]] when restricted to [[Cauchy complete categories]]:

$$
  [-, Set] : Cat^co_{CauchyComp} \hookrightarrow Topos_{ess}
  \,.
$$

For all small categories $C,D$ we have an [[equivalence of categories]]

$$
  Func(\overline{C},\overline{D})^{op}
  \simeq
  Topos_{ess}([C,Set], [D,Set])
$$

between the [[opposite category]] of the [[functor category]] between the [[Cauchy completion]]s of $C$ and $D$ and the the category of essential geometric morphisms between the copresheaf toposes and [[geometric transformation]]s between them.

=--

In particular, since every [[poset]] -- when regarded as a [[category]] -- is Cauchy complete, we have

+-- {: .num_cor}
###### Corollary

The [[2-functor]]

$$
  [-,Set] : Poset \to Topos_{ess}
$$

is a [[full and faithful 2-functor]].

=--

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

Sometimes it is useful to decompose this statement as follows.

There is a functor

$$
  Alex : Poset \to Locale
$$

which assigns to each [[poset]] a [[locale]] called its [[Alexandroff locale]]. By a theorem discussed there, a morphisms of locales $f : X \to Y$ is in the image of this functor precisely if its inverse image morphism $f^* Op(Y) \to Op(X)$ of [[frame]]s has a [[left adjoint]] in the [[2-category]] [[Locale]].

Moreover, for any poset $P$ the [[sheaf topos]] over $Alex P$ is naturally equivalent to $[P,Set]$

$$
  [-,Set] \simeq Sh \circ Alex
  \,.
$$

With this, the fact that $[-,Set] : Poset \to Topos$ hits precisely the essential geometric morphisms follows with the basic fact about [[localic reflection]], which says that $Sh : Locale \to Topos$ is a [[full and faithful 2-functor]].

=--

### Some morphism calculus

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

Let $f : E \to F$ be an essential geometric morphism.

For every $\phi : X \to f^* f_* A$ in $E$ the diagram

$$
  \array{
    X &\stackrel{\phi}{\to}& f^* f_* A
    \\
    \downarrow && \downarrow
    \\
    f^* f_! X &\stackrel{}{\to}& A
  }
$$

commutes, where the vertical morphisms are unit and counit, respectively, and where the bottom horizontal morphism is the [[adjunct]] of $\phi$ under the composite adjunction $(f^* f_! \dashv f^* f_*)$.

=--

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

The morphism $\phi : X \to f^* f_* A$ is the component of a [[natural transformation]]

$$
  \array{
    *&&\overset{X}{\to}&& E
    \\
    {}^{\mathllap{A}}\downarrow
    &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow
    \\
    E &\underset{f_*}{\to}&F
  }
  \,.
$$

The composite $X \stackrel{\phi}{\to}  f^* f_* A \to A$ is the component of this composed with the counit $f^* f_* \Rightarrow Id$.

We may insert the 2-identity given by the zig-zag law

$$
  \cdots \;\;\; = \;\;\;
  \array{
    *&&\overset{X}{\to}&& E && = && E
    \\
    {}^{\mathllap{A}}\downarrow
    &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow &\Downarrow& \searrow^{f_!}
    &\Downarrow& \nearrow_{\mathrlap{f^*}}
    \\
    E &\underset{f_*}{\to}&F &&=&& F
  }
  \,.
$$

Composing this with the counit $f^* f_* \Rightarrow Id$ produces the transformation whose component is manifestly the morphism $X \to f^* f_! X \to A$.

=--

## Examples

### Logical functors and etale geometric morphisms

A [[logical functor]] $E\to F$ with a [[left adjoint]] has automatically also a [[right adjoint]] whence is the [[inverse image]] part of an essential geometric morphism $F\to E$.

A particularly important instance of this situation is the following:

For any morphism $f\colon A\to B$ in a topos $E$, the induced geometric morphism $f\colon E/A \to E/B$ of [[overcategory]] [[topos]]es is essential. Here, the logical functor is given by the pullback functor $f^*:E/B\to E/A$ of course.

In the special case $B = *$ the [[terminal object]], the essential geometric morphism

$$
  \pi : E/A \to E
$$

is also called an [[etale geometric morphism]].

Conversely, (almost) any essential geometric morphism $\pi$ whose [[inverse image]] $\pi^*$ is a [[logical functor]] has the form of an [[etale geometric morphism]]:

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

If the [[left adjoint]] $\pi_!$ of a logical functor $\pi^*:E\to F$ furthermore preserves equalizers, then the corresponding essential geometric morphism is up to equivalence an [[etale geometric morphism]] $\pi : F\simeq E/A \to E$ for an object $A$ in $E$ that is determined up to isomorphism.

=--

For a proof see e.g. Johnstone [(1977, p.37)](#JTT77).

### Locally connected toposes

A [[locally connected topos]] $E$ is one where the [[global section]] [[geometric morphism]] $\Gamma : E \to Set$ is essential.

$$
  (f_! \dashv f^* \dashv f_*) \;\;\; : \;\;\;
  E
    \stackrel{\overset{\Pi_0}{\longrightarrow}}{\stackrel{\overset{LConst}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}}}
  Set
  \,.
$$

In this case, the functor $\Gamma_! = \Pi_0 : E \to Set$ sends each object to its set of connected components.  More on this situation is at [[homotopy groups in an (∞,1)-topos]].

Note, though that if $p\colon E\to S$ is an arbitrary geometric morphism through which we regard $E$ as an $S$-topos, i.e. a topos "in the world of $S$," the condition for $E$ to be locally connected as an $S$-topos is not just that $p$ is essential, but that the left adjoint $p_!$ can be made into an $S$-[[indexed functor]] (which is automatically true for $p^*$ and $p_*$).  This is automatically the case for $Set$-toposes (at least, when our [[foundation]] is [[material set theory]]---and if our foundation is [[structural set theory]], then our large categories and functors all need to be assumed to be $Set$-indexed anyway). For more see [[locally connected geometric morphism]].

### Tiny objects

The [[tiny object]]s of a [[presheaf topos]] $[C,Set]$ are precisely the essential points $Set \to [C,Set]$. See [[tiny object]] for details.

## Related concepts

* [[essential subtopos]] / [[level of a topos]]

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

* [[connected topos]] / [[∞-connected (∞,1)-topos]]

* [[totally connected topos]] / [[totally connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

## References

Like many other things, it all started as an exercise in

* {#SGA4}[[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (exposé IV, ex.7.6, pp.414-417)

Speaking of exercises, consider the results of [[Jan Erik Roos|Roos]] on essential points reported in exercise 7.3 of

* {#JTT77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Dover reprint 2014, pp.254f)

The case of sheaves valued in [[FinSet]] is considered in

* J. Haigh, _Essential geometric morphisms between toposes of finite sets_ , Math. Proc. Phil. Soc. **87** (1980) pp.21-24.

The standard reference for _essential localizations_ [^further], aka [[level of a topos|levels]], is

* {#KL89}[[G. M. Kelly]], [[F. W. Lawvere]], _On the Complete Lattice of Essential Localizations_ , Bull. Soc. Math. de Belgique **XLI** (1989) 289-319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

[^further]: See at [[Aufhebung]] for further references on essential localizations.

For more on this see also

* {#Lima15} [[Guilherme Frederico Lima]], _From Essential Inclusions to Local Geometric Morphisms_, talk at [Topos à l'IHES](https://indico.math.cnrs.fr/event/747/), November 2015, Paris ([video](https://www.youtube.com/watch?v=YsoGN91Rh_s))

The definition of essential geometric morphisms appears before Lemma A.4.1.5 in

* [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. I_ , Oxford UP 2002.

Connected surjective and local geometric morphisms are discussed in

* [[Peter Johnstone]], [[Ieke Moerdijk]], _Local maps of toposes_ , Proceedings London Mathematical Society **58** (1989), pp.281-305.

Further refinements are in

* [[Bill Lawvere]], _Axiomatic cohesion_ Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))

[[!redirects essential geometric morphisms]]


essential ideal

Let $C$ be some [[category]] of [[modules]] or [[bimodules]] (say over a [[ring]], algebra, or operator algebra). Then an [[subobject]] $M\subset N$ is an __essential submodule__ of $N$ (or, we say that $N$ is an __essential extension__ of $M$) if $M$ has nonzero intersection ([[pullback]] in more abstract situations) with any nonzero subobject of $N$ (or equivalently, $M$ has zero intersection with only the [[zero subobject]] of $N$).

In particular, one applies this terminology to [[ideals]], i.e. submodules (or subbimodules in the $2$-sided case) of a ring, algebra, or operator algebra itself. Hence we talk about __essential ideals__. For essential extensions, one considers [[extensions]] of algebras, where 'essential' still refers to non-intersection with submodules rather than with [[subalgebras]].

category: algebra, operator algebras

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