emergentism

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### Context
#### Philosophy
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[[!include philosophy - contents]]
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## Idea

*Emergentism* is a philosophical position which claims that there are properties and behaviors of entities in the world that are dependent on, but are not reducible to, the properties and behaviors of their constituent parts. This contrasts with [[reductionism]] concerning a particular phenomenon, which claims that reduction is possible there, and with *dualism*, which claims full autonomy of the higher level (see [O'Connor 2021](#oconnor)).

Emergence has long been debated in phenomena such as life and consciousness, but also regarding the chemical properties of water in relation to those of the constituent water molecules.

## Emergentism in physics

Emergentism in [[physics]], which arose in response to the position of [[reductionism]], which states that all of physics could be reduced to one basic universal theory.

In practice, emergentism is true, because even if one were to say the [[standard model of particle physics]] were to validate the reductionist paradigm, the Standard Model is so complex that particle physicists have not been able to model the dynamics of a single proton from the Standard Model, let alone the dynamics of molecules, chemical reactions, condensed matter systems, and galaxies. And furthermore, the complexity means that it is likely that even if humans could characterize such phenomena entirely based upon the Standard Model, it might be too long and complex for a single human to understand during their lifetime. This means that in practice, physicists usually have different theories for each different general phenomena; the [[quantum field theories]] used in [[condensed matter physics]] are [[effective field theories]] different from the Standard Model, and the models used in molecular chemistry and galactic physics aren't quantum field theories at all.

## See also

* [[philosophy]]

* [[reductionism]], **emergentism**

* [[pluralism]], [[monism]]

## References

* {#oconnor} Timothy O’Connor, _Emergent Properties_, The Stanford Encyclopedia of Philosophy (Winter 2021 Edition), Edward N. Zalta (ed.), [SEP page](https://plato.stanford.edu/archives/win2021/entries/properties-emergent/).

On how ontological reductionism is problematic in the philosophy of science:

* Daniel C. Dennett (1987) “True Believers." The Intentional Stance. Cambridge, MA: MIT Press.

* Jimmy Aames (2019) Patternhood and Generality: A Peircean Approach to Emergence. European Journal of Pragmatism and American Philosophy, XI (2). ([pdf](https://philsci-archive.pitt.edu/16878/1/Patternhood%20and%20Generality.pdf), [web](https://philsci-archive.pitt.edu/16878/))

A claim that [[condensed matter physics]] cannot be fully reduced to quantum mechanical theory for all the atoms of which the system consists:

* Barbara Drossel, _Strong emergence in condensed matter physics_ ([arXiv:1909.01134](https://arxiv.org/abs/1909.01134)).

On the use of mathematical tools, including [[exact sequences]], to represent emergence (termed here generative and interactive effects:

* Elie Adam, _Systems, Generativity and Interactional Effects_ ([pdf](https://elieadam.com/eadam_PhDThesis.pdf))

On zeroth and first 'homotopy posets' to measure obstructions to compositionality:

* Caterina Puca, [[Amar Hadzihasanovic]], Fabrizio Genovese, [[Bob Coecke]], _Obstructions to Compositionality_ ([arXiv:2307.14461](https://arxiv.org/abs/2307.14461)).

## External links

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


Emerton-Gee stack

#Contents#
* table of contents
{:toc}

## Idea

The _Emerton-Gee stack_ is the [[moduli stack]] of étale $(\varphi,\Gamma)$-modules. It is used as to study p-adic [[Galois modules]] $\rho:\Gal(\overline{K}/K)\to \GL_{d}(A)$, for $K$ a finite extension of $\mathbb{Q}_{p}$ and $A$ a p-adically complete $\mathbb{Z}_{p}$-algebra, as it is better behaved than the naive moduli stack of these p-adic Galois modules (which sits inside of it as a substack).

## Properties

The Emerton-Gee stack is a Noetherian formal [[algebraic stack]]. Its underlying reduced substack is an algebraic stack of finite type over $\mathbb{F}_{p}$, and is equidimensional of $[K:\mathbb{Q}_{p}]d(d-1)/2$. The irreducible components of the reduced substack are labeled by _Serre weights_ (irreducible $\overline{\mathbb{F}}_{p}$ representations of $\GL_{d}(\mathcal{O}_{K})$).

## Related concepts

* [[moduli stack of L-parameters]]

* [[Galois modules]]

* [[Galois representations]]

* [[p-adic Hodge theory]]

## References

* {#EmertonGee19} [[Matthew Emerton]], [[Toby Gee]], *Moduli stacks of étale $(\phi, \Gamma)$ modules and the existence of crystalline lifts* ([arXiv:1908.07185](https://arxiv.org/abs/1908.07185))

[[!redirects Emerton-Gee stacks]]


Emery Thomas

Paul Emery Thomas was an algebraic topologist and number theorist working at UC Berkeley.

He got his PhD in 1955 at Princeton from [[Norman Steenrod]].

His PhD students include [[Donald W. Anderson]].


Emil Artin

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

## Selected writings

Introducing the [[braid group]]:

* {#Artin25} [[Emil Artin]], *Theorie der Zöpfe*, Abh. Math. Semin. Univ. Hambg. **4**  (1925) 47–72 [[doi;10.1007/BF02950718](https://doi.org/10.1007/BF02950718)]

* [[Emil Artin]], *Theory of Braids*, Annals of Mathematics, Second Series, **48** 1 (1947) 101-126 [[doi:10.2307/1969050](https://doi.org/10.2307/1969050)]

On the [[Gamma function]]:

* {#Artin31} [[Emil Artin]], *Einführung in die Theorie der Gammafunktion*, Hamburger Mathematische Einzelschriften
l. Heft, Verlag B. G. Teubner, Leipzig (1931)

  English translation by Michael Butler: *The Gamma Function*, Holt, Rinehart and Winston (1931) [[[Artin-TheGammaFunction.pdf:file]]]

Introducing the [[Artin-Tate lemma]]:

* {#ArtinTate51} [[Emil Artin]], [[John T. Tate]], *A note on finite ring extensions*, J. Math. Soc Japan, Volume **3** (1951) 74–77 $[$[doi:10.2969/jmsj/00310074](https://doi.org/10.2969/jmsj/00310074), [pdf](https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-3/issue-1/A-Note-on-Finite-Ring-Extensions/10.2969/jmsj/00310074.pdf)$]$

On [[linear algebra]], [[projective geometry]], [[symplectic groups]] and [[orthogonal groups]]:

* [[Emil Artin]], *[[Geometric Algebra]]*, Wiley 1957 (1988) [[ISBN:978-1-118-16454-9](https://www.wiley.com/en-ae/Geometric+Algebra-p-9781118164549), Wikipedia entry, [ark:/13960/t4nk37034](https://archive.org/details/geometricalgebra033556mbp/page/n5/mode/2upa)]

## Related entries

* [[Artin stack]]

* _[[Geometric Algebra]]_

category: people

[[!redirects Artin]]


Emil Génetay-Johansen

PhD student at [[Tapio Simula]]'s *[Anyon technologies research group](https://www.swinburne.edu.au/research/centres-groups-clinics/optical-sciences-centre/our-research/anyon-technologies/)*.

## Selected writings

On [[quantum compilation]] with [[su(2)-anyon]] [[braid representation|braid gates]]:

*  [[Emil Génetay-Johansen]], [[Tapio Simula]], Section IV of: *Fibonacci anyons versus Majorana fermions -- A Monte Carlo Approach to the Compilation of Braid Circuits in $SU(2)_k$ Anyon Models*, PRX Quantum **2** 010334 (2021) [[arXiv:2008.10790](https://arxiv.org/abs/2008.10790), [doi:10.1103/PRXQuantum.2.010334](https://doi.org/10.1103/PRXQuantum.2.010334)]

category: people


Emil Martinec

* [webpage](http://hamilton.uchicago.edu/~ejm/)

## Selected writings

Introducing [[heterotic string theory]]:

* [[David Gross]], [[Jeffrey Harvey]], [[Emil Martinec]], [[Ryan Rohm]]:

  _Heterotic string theory (I). The free heterotic string_  Nucl. Phys. B 256 (1985), 253 (doi:10.1016/0550-3213(85)90394-3)

  _Heterotic string theory (II). The interacting heterotic string_ , Nucl. Phys. B 267 (1986), 75 (doi:10.1016/0550-3213(86)90146-X)

On the [[NSR superstring]] as [[2d CFT]]:

* {#FriedanMartinecShenker86} [[Daniel Friedan]], [[Emil Martinec]], [[Stephen Shenker]], _Conformal invariance, supersymmetry and string theory_, Nucl. Phys., B271 (1986) 93 (doi:10.1016/S0550-3213(86)80006-2)

On the [[AdS/CFT correspondence]]:

* [[Tom Banks]], [[Michael Douglas]], [[Gary Horowitz]], [[Emil Martinec]], _AdS Dynamics from Conformal Field Theory_ ([arXiv:hep-th/9808016](https://arxiv.org/abs/hep-th/9808016), [spire:474214](http://inspirehep.net/record/474214))

On [[black hole entropy]] via [[D1-D5 brane bound states]] as [[black holes in string theory]]:

* [[Emil Martinec]], *The branes behind black holes*, talk at *[Strings 2022](https://indico.cern.ch/event/1085701/)* [[indico:4940814](https://indico.cern.ch/event/1085701/contributions/4940814/)]

## Related $n$Lab pages

* [[string theory]]

* [[2d SCFT]]

* [[NSR superstring]]

* [[picture number]]

category: people


Emil Mottola

* [webpage](http://www.physics.fau.edu/people/faculty/mottola.php)

## related $n$Lab entries

* [[de Sitter spacetime]]

category: people


Emilio Elizalde

* [webpage](http://www.ice.csic.es/en/view_staff.php?MID=3)

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

## related $n$Lab entries

* [[zeta function regularization]]

category: people


Emilio Minichiello

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

## Selected writings

Relating the [[Dwyer-Kan loop groupoid]]-construction to the [[homotopy coherent nerve]]-construction:

* {#MinichielloRiveraZeinalian23} [[Emilio Minichiello]], [[Manuel Rivera]], [[Mahmoud Zeinalian]], *Categorical models for path spaces*, Advances in Mathematics **415** (2023) 108898 [[arXiv:2201.03046](https://arxiv.org/abs/2201.03046), [doi:10.1016/j.aim.2023.108898](https://doi.org/10.1016/j.aim.2023.108898)]

On the [[de Rham theorem]] over [[diffeological spaces]]:

* [[Emilio Minichiello]], *The Diffeological Čech-de Rham Obstruction* [[arXiv:2401.09400](https://arxiv.org/abs/2401.09400)]

category: people


Emily Nardoni

* [institite page](http://bhaumik-institute.physics.ucla.edu/postdoctoral-scholars)

## Selected writings

On [[M5-branes]] and their [[quantum anomaly]]-cancellation:

* {#BahBonettiMinasianNardoni18} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Class $\mathcal{S}$ Anomalies from M-theory Inflow_, Phys. Rev. D 99, 086020 (2019) ([arXiv:1812.04016](https://arxiv.org/abs/1812.04016))

* {#BBMN19} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Anomaly Inflow for M5-branes on Punctured Riemann Surfaces_,  J. High Energ. Phys. 2019, 123 (2019)
([arXiv:1904.07250](https://arxiv.org/abs/1904.07250))

* {#BBMN19b} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Anomalies of QFTs from M-theory and Holography_, J. High Energ. Phys. 2020, 125 (2020) ([arXiv:1910.04166](https://arxiv.org/abs/1910.04166))

On [[supersymmetry breaking]] of [[D=4 N=2 SYM]] to [[confinement|confining]] [[QCD]] with [[adjoint representation|adjoint]] [[fermions]]:

* [[Emily Nardoni]], *From $\mathcal{N} = 2$ Supersymmetry to adjoint QCD*, talk at *[[Strings 2022]]* [[indico:4940894](https://indico.cern.ch/event/1085701/contributions/4940894), [pdf](https://indico.cern.ch/event/1085701/contributions/4940894/attachments/2484206/4265102/Slides_Nardoni.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=035dfeb0-ef40-4530-8f20-01e5b890f43d)]

category:people


Emily Riehl

I am an Assistant Professor in the Department of Mathematics at [Johns Hopkins University](http://www.math.jhu.edu).

My webpage can be found [here](http://www.math.jhu.edu/~eriehl/).

## Selected writings

On [[rigidification of quasi-categories]]:

* [[Emily Riehl]], *On the structure of simplicial categories associated to quasi-categories*, Math. Proc. Camb. Phil. Soc. **150** (2011) 489-504 [[arXiv:0912.4809](https://arxiv.org/abs/0912.4809), [doi:10.1017/S0305004111000053](https://doi.org/10.1017/S0305004111000053)]

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

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

On [[Reedy model structures]] via [[weighted colimits]]:

* [[Emily Riehl]], [[Dominic Verity]], *The theory and practice of Reedy categories*, Theory and Applications of Categories **29** 9 (2014) 256-301 [[tac:29-09](http://www.tac.mta.ca/tac/volumes/29/9/29-09abs.html),[arXiv:1304.6871](https://arxiv.org/abs/1304.6871)]

An introductory [[category theory]] textbook for beginning graduate students or advanced undergraduates with an emphasis on applications of categorical concepts to a variety of areas of mathematics.

* {#RiehlCTInContext} [[Emily Riehl]], _[[Category Theory in Context]]_, Dover Publications (2017) [[pdf](http://www.math.jhu.edu/~eriehl/context.pdf)]

Textbooks on ([[simplicial homotopy theory|simplicial]]) [[homotopy theory]] and [[(∞,1)-category theory]] with emphasis on tools from [[category theory]] and [[2-category]] theory (via [[∞-cosmoi]] and the [[homotopy 2-category of (∞,1)-categories]]):

* [[Emily Riehl]], *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

* {#RiehlVerity22} [[Emily Riehl]], [[Dominic Verity]], _[[Elements of ∞-Category Theory]]_, Cambridge studies in advanced mathematics **194**, Cambridge University Press (2022) [[doi:10.1017/9781108936880](https://doi.org/10.1017/9781108936880), ISBN:978-1-108-83798-9, [pdf](https://emilyriehl.github.io/files/elements.pdf)]

On [[directed homotopy type theory]] via a form of [[simplicial type theory]]

* [[Emily Riehl]], [[Mike Shulman]], *A type theory for synthetic ∞-categories*, Higher Structures **1** 1 (2017) [[arxiv:1705.07442](https://arxiv.org/abs/1705.07442), [published article](https://higher-structures.math.cas.cz/api/files/issues/Vol1Iss1/RiehlShulman)]

Survey of [[homotopy theory]] from [[homotopical categories]] to [[(∞,1)-categories]]:

* [[Emily Riehl]], _Homotopical categories: from model categories to (∞,1)-categories_, in: [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds,) *[[Stable categories and structured ring spectra]]*,  MSRI Book Series, Cambridge University Press ([arXiv:1904.00886](https://arxiv.org/abs/1904.00886))

A new proof of the [[Strøm model structure]] using [[algebraic weak factorization systems]]:

* {#BarthelRiehl13} [[Tobias Barthel]], [[Emily Riehl]], _On the construction of functorial factorizations for model categories_, Algebr. Geom. Topol. 13 (2013) 1089-1124 ([arXiv:1204.5427](http://arxiv.org/abs/1204.5427), [doi:10.2140/agt.2013.13.1089](http://dx.doi.org/10.2140/agt.2013.13.1089))

On [[monads in computer science]]:

* {#Riehl17} [[Emily Riehl]], *A categorical view of computational effects*, talk at *[C$\circ$mp$\circ$se$\colon\!\colon$Conference](https://www.composeconference.org/2017/)* (2017) [[pdf](http://www.math.jhu.edu/~eriehl/compose.pdf), [[Riehl-ComputationalEffects.pdf:file]]]

On left-[[transferred model structures]] and [[model structures on functors]]:

* [[Marzieh Bayeh]], [[Kathryn Hess]], [[Varvara Karpova]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *Left-induced model structures and diagram categories*, in: *Women in Topology: Collaborations in Homotopy Theory*, Contemporary Mathematics **641** American Mathematical Society (2015) [[arXiv:1401.3651](http://arxiv.org/abs/1401.3651), [ISBN:978-1-4704-2495-4](https://bookstore.ams.org/conm-641)]

* {#HKRS15} [[Kathryn Hess]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *A necessary and sufficient condition for induced model structures*, J. Topology **10** 2  (2017) 324-369 [[arXiv:1509.08154](http://arxiv.org/abs/1509.08154), [doi:10.1112/topo.12011](https://doi.org/10.1112/topo.12011)]

  > (this contains an error which is corrected in [Garner, Kedziorek & Riehl 2018](transferred+model+structure#GKR18))

* {#GKR18} [[Richard Garner]], [[Magdalena Kedziorek]], [[Emily Riehl]], *Lifting accessible model structures*, J. Topology **13** 1 (2020) 59-76 [[arXiv:1802.09889](https://arxiv.org/abs/1802.09889), [doi:10.1112/topo.12123](https://doi.org/10.1112/topo.12123)]

On [[(∞,1)-category theory]] via the [[homotopy 2-category of (∞,1)-categories]] ([[formal (infinity,1)-category theory|formal $(\infty,1)$-category theory]]):

* [[Emily Riehl]], [[Dominic Verity]], _The 2-category theory of quasi-categories_, Advances in Mathematics Volume 280, 6 August 2015, Pages 549-642 ([arXiv:1306.5144](http://arxiv.org/abs/1306.5144), [doi:10.1016/j.aim.2015.04.021](https://doi.org/10.1016/j.aim.2015.04.021))

* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _Infinity category theory from scratch_, Higher Structures Vol 4, No 1 (2020) ([arXiv:1608.05314](https://arxiv.org/abs/1608.05314), [pdf](http://www.math.jhu.edu/~eriehl/scratch.pdf))

* {#Riehl21} [[Emily Riehl]], *The formal theory of ∞-categories*, talk at *[Categories and Companions Symposium June 8–12, 2021](http://web.science.mq.edu.au/groups/coact/seminar/CaCS2021/)* ([video](https://youtu.be/iX5Fxen3K-U))

On [[(∞,1)-functors]] and [[(∞,1)-monads]]:

* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_, Advances in Mathematics, Volume 286, 2 January 2016, Pages 802-888 ([arXiv:1310.8279](http://arxiv.org/abs/1310.8279), [doi:10.1016/j.aim.2015.09.011](https://doi.org/10.1016/j.aim.2015.09.011))

On the [[Yoneda lemma for (∞,1)-categories]]:

* [[Emily Riehl]], [[Dominic Verity]], _Fibrations and Yoneda's lemma in an $\infty$-cosmos_, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 ([arXiv:1506.05500](https://arxiv.org/abs/1506.05500), [doi:10.1016/j.jpaa.2016.07.003](https://doi.org/10.1016/j.jpaa.2016.07.003))

On ([[discrete fibration|discrete]]) [[Grothendieck fibrations]] and [[profunctors]]:

* {#RiehlLoregian2020} [[Emily Riehl]] and [[Fosco Loregian]], _Categorical notions of fibration_, Expositiones Mathematicae **38** 4 (2020) 496-514 [[arXiv:1806.06129](https://arxiv.org/abs/1806.06129), [doi:10.1016/j.exmath.2019.02.004](https://doi.org/10.1016/j.exmath.2019.02.004)]

On the [[relation between type theory and category theory|categorical semantics]] of [[univalence|univalent]] [[homotopy type theory]] in [[(infinity,1)-topos|$\infty$-topoi]] (proof of [[Awodey's conjecture]]):

* {#Riehl22} [[Emily Riehl]], *On the $\infty$-topos semantics of homotopy type theory*, lecture at *[Logic and higher structures](https://conferences.cirm-math.fr/2689.html)* CIRM (Feb. 2022) $[$[pdf](https://emilyriehl.github.io/files/semantics.pdf), [[Riehl-InfinityToposSemantics.pdf:file]]$]$

Exposition of [[(∞,1)-category theory]] with an eye towards [[homotopy type theory]]:

* {#Riehl22} [[Emily Riehl]], *$\infty$-Category theory for undergraduates*, talk at *[[CQTS]]* (Dec. 2022) [[web](Center+for+Quantum+and+Topological+Systems#RiehlDec2022), video: [YT](https://www.youtube.com/watch?v=7g2rkiFsbXo)]

On teaching [[(∞,1)-category theory]] to undergraduate students:

* [[Emily Riehl]], *[Could $\infty$-category theory be taught to undergraduates?](https://www.ams.org/journals/notices/202305/noti2692/noti2692.html?adat=May%202023&trk=2692&galt=feature&cat=feature&pdfissue=202305&pdffile=rnoti-p727.pdf)*, Notices of the AMS (May 2023) [[published pdf](https://www.ams.org/journals/notices/202305/rnoti-p727.pdf), [arxiv:2302.07855](https://arxiv.org/abs/2302.07855)]

Formalization of the [[(infinity,1)-Yoneda lemma|$(\infty,1)$-Yoneda lemma]] via [[simplicial homotopy type theory]] (in [[Rzk]]):

* [[Nikolai Kudasov]], [[Emily Riehl]], [[Jonathan Weinberger]]. *Formalizing the $\infty$-categorical Yoneda lemma* (2023) [[arXiv:2309.08340](https://arxiv.org/abs/2309.08340)]

category: people

[[!redirects Riehl]]


Emma Previato

__Emma Previato__ was a mathematician at Boston University, a specialist in [[integrable model]]s and their relations to [[algebraic geometry]].

* [wikipedia](https://en.wikipedia.org/wiki/Emma_Previato)
* Emma Previato, _Curves in isomonodromy and isospectral deformations: [[Painleve transcendent|Painlevé]] VI as a case study_, In: Algebraic curves and their applications. Contemp. Math. __724__, 258--276, Amer. Math. Soc. 2018
* Emma Previato, _Hitchin system_, Springer Online [Enc. of Math.](http://eom.springer.de/H/h120090.htm)
* Emma Previato, Mauro Spera, _Isometric embeddings of infinite-dimensional Grassmannians_, Regul. Chaot. Dyn. __16__:356 (2011) [doi](https://doi.org/10.1134/S1560354711030099)
* Emma Previato, _Multivariable Burchnall–Chaundy theory_, Phil. Trans. R. Soc. A __366__ (2008) 1155--1177 [doi](https://doi.org/10.1098/rsta.2007.2064)
* M. J. Dupré, J. F. Glazebrook, E. Previato, _A Banach algebra version of the Sato Grassmannian and commutative rings of differential operators_ Acta Appl. Math. __92__, 241--267 (2006) [doi](https://doi.org/10.1007/s10440-006-9052-3)
* M. R. Adams, J. Harnad, E. Previato, _Isospectral Hamiltonian flows in finite and infinite dimensions. I. Generalized Moser systems and moment maps into loop algebras_, Comm. Math. Phys. __117__:3 (1988) 451--500 [doi](https://doi.org/10.1007/BF01223376)
* Emma Previato, _Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation_, Duke Math. J. 52(2): 329--377 (1985) [doi](https://doi.org/10.1215/S0012-7094-85-05218-4)

category: people


Emmanuel Dror Farjoun

Emmanuel Dror Farjoun is an [[algebraic topology|algebraic topologist]] at the Hebrew University in Jerusalem.

* [Homepage](http://www.ma.huji.ac.il/~farjoun/)

## Selected writings

On the $G$-[[Borel model structure]] and its [[Quillen equivalence]] with the [[slice model structure]] over the [[simplicial classifying space]] $\overline W G$:

* {#DDK80} [[Emmanuel Dror]], [[William Dwyer]], [[Daniel Kan]], _Equivariant maps which are self homotopy equivalences_, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 ([jstor:2043448](http://www.jstor.org/stable/2043448))

[[!redirects Emmanuel Dror]]


Emmanuel Haucourt

##Emmanuel Haucourt##

**Emmanuel Haucourt** is a researcher at the École Polytechnique in Palaiseau, near Paris, in the team of [[Eric Goubault]]. His work is in [[directed algebraic topology]] and in particular in the study of the behaviour of concurrent programs.  He has made available  notes on _Topologie Algébrique Dirigée et Concurrence_ (in both French and English), prepared for the _Master Parisien de Recherche en Informatique_.  These are available from his [web page](http://www.lix.polytechnique.fr/Labo/Emmanuel.Haucourt/).

category:people

[[!redirects E. Haucourt]]


Emmy Noether

## About

Emmy Noether (1882–1935) was a German mathematician with important results in theoretical physics (see [[Noether's theorem]]) and abstract algebra, and whose massive influence on algebra in the 20th century was spread, in part, through [[Bartel Leendert van der Waerden]]'s influential text _Moderne Algebra_, based on her ideas. [[David Hilbert]] fought to have her allowed to teach at Göttingen at a time when women had barely been allowed to attend university as students. [[Albert Einstein]] wrote, on her death,

>In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.

She got her PhD degree in 1907 from the University of Erlangen, advised by Paul Gordan.
She continued to work at the University of Erlangen until 1915 and from 1915 till 1933 at the University of Göttingen.

Noether gave a plenary talk at the 1932 International Congress of Mathematicians

In [[topology]] and [[homotopy theory]], one of the most important contributions Noether made was the [[categorification]] of the [[Betti number]]s and torsion coefficients of a [[topological space]] to [[homology]] groups, and the implicit understanding that homology was a [[functor]]. The following is the abstract of her talk at the DMV in 1925 where she pointed out this fundamental result

>Ableitung der Elementarteilertheorie aus der Gruppentheorie. Die Elementarteilertheorie gibt bekanntlich für Moduln aus ganzzahligen Linearformen eine Normalbasis von der Form $(e_1y_1, e_2y_2, ..., e_ry_r)$, wo jedes $e$ durch das folgende teilbar ist; die $e$ sind dadurch bis aufs Vorzeichen eindeutig festgelegt. Da jede Abelsche Gruppe mit endlich vielen Erzeugenden dem Restklassensystem nach einem solchen Modul isomorph ist, ist dadurch der Zerlegungssatz dieser Gruppen als direkte Summe größter zyklischer mitbewiesen. Es wird nun umgekehrt der Zerlegungssatz rein gruppentheoretisch direkt gewonnen, in Verallgemeinerung des für endliche Gruppen üblichen Beweises, und daraus durch Übergang vom Restklassensystem zum Modul selbst die Elementarteilertheorie abgeleitet. Der Gruppensatz erweist sich so als der einfachere Satz; in den Anwendungen des Gruppensatzes — z.B. Bettische und Torsionszahlen in der Topologie — ist somit ein Zurückgehen auf die Elementarteilertheorie nicht erforderlich. -Jahresbericht der Deutschen Mathematiker-Vereinigung **34** pt 2 (1926) p 104 ([source](http://www.digizeitschriften.de/dms/img/?PID=PPN37721857X_0034%7Clog44&physid=phys357#navi))

## Family

Noether is one of several mathematicians in the family with results named after them: her father Max was also a mathematician, and made contributions to classical 19th century algebraic geometry; the [[Brill-Noether theorem]] is named for Max Noether, not Emmy Noether. Her brother Fritz is also known for work in applied mathematics, and contributed to the [[Herglotz-Noether theorem]].

## See also

* [[conserved current]]
* [[Noether identity]]
* [[conservation law]]
* [[general relativity]]

* [[commutative algebra]]
* [[ideal]]

* [[noetherian ring]]
* [[Noetherian module]]
* [[noetherian category]]
* [[noetherian scheme]]

* [[Hilbert's basis theorem]]
* [[invariant theory]]
* [[noncommutative algebra]]

## References

Extensive information is available at

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

and

* [MacTutor biography](http://www-history.mcs.st-andrews.ac.uk/Biographies/Noether_Emmy.html).

An annotated bibliography is available on Wikipedia:

* [Emmy Noether bibliography](https://en.wikipedia.org/wiki/Emmy_Noether_bibliography)

The following mid-career paper returns to a topic related to Noether's doctoral work (under [[Paul Gordan]]), and gave a constructive approach to a problem that previously used the mere existence result ensured [[Hilbert's basis theorem]]

* [[Emmy Noether]], _Der Endlichkeitsatz der Invarianten endlicher Gruppen_, Mathematische Annalen, vol. 77, 1920, pages 89--92, (English translation by [[Colin McLarty]], at [arXiv:1503.07849](http://arxiv.org/abs/1503.07849))

category: people


empiricism

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## Idea

In the [[philosophy of science]], and particularly the [[philosophy of physics]]. _empiricism_ is the [[philosophy|philosophical]] sentiment that knowledge about the [[observable universe]] does and, ultimately, can only derive from empirical observation ([[measurement]], [[experiment]]), as opposed to from intellectual reflection, such as in [[exceptional naturalism]] and/or ([[absolute idealism|absolute]]) [[idealism]].

A variant of empiricism due to [van Fraasen 80](#vanFraasen80) came to be known as _constructive empiricism_ which (according to [SEP-CE](#SEP-CE), [here](plato.stanford.edu/entries/constructive-empiricism/#UndeConsEmpi))

> holds that science aims at truth about observable aspects of the world, but that science does not aim at truth about unobservable aspects.

A contemporary variant of empriricism appears in discussion of the "[[multiverse]]" to the extent that this rejects the existence of a fundamental [[theory (physics)|theory]] from which reality may be derived, and proclaims the necessity to simply observe fundamental fields and "constants of nature" as what they appear, giving up on theoretical explanation.

## Related concepts

* [[universal exceptionalism]]

* [[idealism]]

* [[structuralism]]

## References

See also

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

* {#SEP-CE} Stanford Encyclopedia of Philosophy, _[Constructive empiricism](https://plato.stanford.edu/entries/constructive-empiricism/)_

Authors articulating the stance of empiricism:

* [[Max Born]], _Experiment and theory in physics_, Address to the Durham Philosophical Socienty 1943, published by Dover 1956

The idea of _constructive empiricism_

* {#vanFraasen80} [[Bas van Fraassen]], _The Scientific Image_ (1980) ([philpapers:VANTSI](https://philpapers.org/rec/VANTSI),\linebreak [doi:10.1093/0198244274.001.0001/acprof-9780198244271](https://www.oxfordscholarship.com/view/10.1093/0198244274.001.0001/acprof-9780198244271))

* {#Leeds94} Stephen Leeds, _Constructive Empiricism_, Synthese Vol. 101, No. 2 (Nov., 1994), pp. 187-221 ([jstor:20117956](https://www.jstor.org/stable/20117956))

Discussion in view of contemporary issues in [[particle physics]] and [[string theory]]:

* {#Dawid05} [[Richard Dawid]], _High Energy Physics and Constructive Empiricism_ ([philsci:2243](http://philsci-archive.pitt.edu/2243/))

* {#Dawid07} [[Richard Dawid]], _Scientific realism in the age of string theory_,  Physics and Philosophy (2007) ([philpapers:DAWSRI](https://philpapers.org/rec/DAWSRI))

Further:

* [[Sabine Hossenfelder]], _Lost in Math: How Beauty Leads Physics Astray_, Basic Books 2018. German translation: _Das hässliche Universum: Warum unsere Suche nach Schönheit die Physik in die Sackgasse führt_, Fischer 2018

* [[Sabine Hossenfelder]], _Was läuft falsch in der gegenwärtigen Physik?_, talk at University of Stuttgart, 2019 ([video](https://www.youtube.com/watch?v=99hVAu1k6G8))

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###### Proposition

$$
  \label{equation1}
  x = x
$$

=--

See also (eq:equation2) in prop. \ref{Proposition2}.

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## Second chapter

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###### Proposition

$$
  \label{equation2}
  y = z
$$

=--

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empty bundle

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## Idea

In as far as a general [[bundle]] over some base [[object]] $B$ is a [[morphism]] $E \to B$ into $B$, out of some "total space" [[object]] $E$, the *empty bundle* over $B$ is the one whose total space is "[[strict initial object|empty]]", hence whose projection map is the [[empty function]].

## Definition

In [[TopologicalSpaces]] an empty bundle has the [[empty topological space]] as its total space, while in [[SimplicialSets]] the empty bundle has the the [[empty simplicial set]] as its total space, etc.:

$$
  \array{
    \varnothing
    \\
    \big\downarrow
    \\
    B
    \mathrlap{\,.}
  }
$$

Generally, one may speak of empty bundles [[internalization|internal]] to any ambient [[category]] in which the [[initial object]] is *[[strict initial object|strict]]* (e.g. any [[topos]]) in that every [[morphism]] to the [[initial object]] is an [[isomorphism]], so that

\[
  \label{InitialObjectIsEmpty}
  \exists
  \big(
    X \overset{}{\rightarrow} \varnothing
  \big)
  \;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;
  X \simeq \varnothing
  \,.
\]

## Properties

### In topological spaces
 {#InTopologicalSpaces}

In [[TopologicalSpaces]], any empty bundle

* is a [[locally trivial]] [[fiber bundle]] with [[typical fiber]] the [[empty topological space]],

  because any [[product topological space]] with the [[empty topological space]] is itself empty:

  $$
    B \times \varnothing \;\simeq\; \varnothing
  $$

* is a [[Serre fibration]] (in fact a [[Hurewicz fibration]]),

  because none of the [[commuting squares]] that one would have to [[lifting property|lift in]] actually exist, by (eq:InitialObjectIsEmpty):

  $$
    \array{
      D^{n} &\overset{ \not \exists }{\longrightarrow}& \varnothing
      \\
      \big\downarrow && \big\downarrow
      \\
      D^n \times I &\longrightarrow& B
    }
  $$

### In simplicial sets
 {#InSimplicialSets}

Similarly, in [[SimplicialSets]] every empty bundle

* is a [[Kan fibration]],

  since none of the [[commuting squares]] that one would have to [[lifting property|lift in]] actually exist, by (eq:InitialObjectIsEmpty):

  $$
    \array{
      \Lambda_k^n &\overset{ \not \exists }{\longrightarrow}& \varnothing
      \\
      \big\downarrow && \big\downarrow
      \\
      \Delta^n &\longrightarrow& B
    }
  $$

  (keeping in mind that the [0-simplex has no horns](horn#ZeroSimplexHasNoHorns), hence that all [[horns]] are [[inhabited set|inhabited]]).

## Application

### In equivariant bundle theory

Empty fiber bundles play a central role in the context of [[equivariant bundles]], where they frequently appear as [[fixed loci]] of non-empty bundles.

## Related concepts

* [[zero bundle]]

[[!include empty objects -- contents]]

[[!redirects empty bundles]]


empty category

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## Definition

The __empty category__ is [[the]] [[category]] with no [[objects]], and hence no [[morphisms]], i.e. the category whose [[class]] of objects is the [[empty set]].

This is the [[initial object]] in [[Cat]].

This is a [[groupoid]], so we may call it the __empty groupoid__.  One may similarly speak of the __empty [[n-category]]__, the __empty [[infinity-groupoid]]__, etc etc etc.

## Properties

The empty category is [[discrete category|discrete]], hence may be identified with a [[set]]: the [[empty set]].  This set is a [[subsingleton]], so we may also identify it with a [[truth value]]: the [[false]] one.

The empty category is [[initial object|initial]] in [[Cat]] (as well as [[Grpd]], [[∞Grpd]], [[Set]], etc).

## Related concepts

* [[terminal category]]

[[!include empty objects -- contents]]

[[!redirects empty category]]
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[[!redirects empty groupoids]]

[[!redirects initial category]]
[[!redirects initial categories]]


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## Idea

The [[context]] of no assumptions.

## Related concepts

[[!include empty objects -- contents]]

[[!redirects empty contexts]]


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## Definition

The _empty diagram_ in some [[category]] is the [[diagram]] indexed by the [[empty category]].

## Properties

The [[limit]] over the empty diagram is, if it exists, the [[terminal object]].

The [[colimit]] over the empty diagram is, if it exists, the [[initial object]].

## Related concepts

[[!include empty objects -- contents]]

[[!redirects empty diagrams]]


empty family

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## Idea

Given any [[set]] $X$, there is a unique __empty [[family]]__ of elements of $X$.  Formally, this is given by the [[empty function]] to $X$, the unique [[function]] to $X$ from the [[empty set]].  As the empty set is [[finite set|finite]] and (a fortiori) [[countable set|countable]], this empty family counts as a [[list]] (or [[tuple]]) and a [[stream]]; in such a guise it is known as the __empty list__ (or __$0$-tuple__) or the __empty stream__ (or _empty sequence_ if the term 'sequence' is used in a sufficiently general sense).

When treating it as an element of the [[free monoid]] on $X$, the empty list may be written $()$, $*$, or $\epsilon$, perhaps with a subscript $X$ if desired.  (Thinking of all empty lists, regardless of the underlying set, as equal is like thinking of all [[empty subsets]] as equal.)

Besides empty families of elements of sets, we have the notions of the empty family of elements of a given [[proper class]], the empty family of elements of a given [[preset]], the empty family of term of a given [[type]], the empty family of [[objects]] and the empty family of [[morphisms]] of a given [[category]], and more generally the empty family of whatever you want.

## Related concepts

[[!include empty objects -- contents]]

[[!redirects empty family]]
[[!redirects empty families]]

[[!redirects empty list]]
[[!redirects empty lists]]
[[!redirects empty tuple]]
[[!redirects empty tuples]]
[[!redirects 0-tuple]]
[[!redirects 0-tuples]]
[[!redirects 0tuple]]
[[!redirects 0tuples]]
[[!redirects zero-tuple]]
[[!redirects zero-tuples]]

[[!redirects empty stream]]
[[!redirects empty streams]]
[[!redirects empty sequence]]
[[!redirects empty sequences]]


empty function

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## Definition

Given a [[set]] $X$, the __empty function__ to $X$ is a [[function]]

$$
  \varnothing
    \longrightarrow
  X
$$

[[codomain|to]] $X$ [[domain|from]] the [[empty set]].

This always exists and is unique; in other words, the empty set is an [[initial object]] in the [[category of sets]].

If regarded as a [[bundle]], the empty function is the [[empty bundle]] over its [[codomain]].

In generalization to ambient categories other that [[Sets]], an *empty morphism* would be any [[morphism]] out of a [[strict initial object]].

## Properties

The empty function to the [[empty set]] is _not_ a [[constant function]].

## Related concepts

[[!include empty objects -- contents]]

[[!redirects empty function]]
[[!redirects empty functions]]

[[!redirects empty morphism]]
[[!redirects empty morphisms]]


empty graph

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## Idea

The _empty graph_ is the [[graph]] with [[empty set]] of [[vertices]] and [[empty set]] of [[edges]].

## Properties

The empty graph is the [[initial object]] in any reasonable [[category]] of [[graphs]] (such as a [[category of simple graphs]]).

The [[free category]] on the empty graph is the [[empty category]].

## Some notions of non-empty graphs

For concreteness let us consider the topos of [[quiver|quivers]]. The _inhabited_ graphs, i.e. those satisfying $\top\vdash (\exists x)\top$ are precisely the graphs $G$ having an edge since the formula on the right is interpreted as the image of the composite $G\overset{id}{\hookrightarrow} G\to 1$ but $1$ being the loop graph and in order to validate the sequent, its loop must be contained in the image requiring an edge in the source. This notion of being inhabited is stronger than being non-empty $\neq\empty$ but weaker as _having a (global) element_ $1\to G$ which corresponds to graphs containing a loop. A _uninhabited_ graph satisfying $(\exists x)\top\vdash \bot$ is necessarily empty whence edgeless graphs with a node are internally neither inhabited nor uninhabited.

## Related concepts

[[!include empty objects -- contents]]

[[!redirects empty graphs]]


empty objects -- contents

* [[empty set]], [[empty subset]]

* [[empty type]], [[empty context]]

* [[empty family]], [[empty function]], [[empty bundle]]

* [[empty topological space]], [[empty simplicial set]]

* [[empty graph]], [[empty category]], [[empty diagram]]

* [[nothing]]


empty set

# Contents
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{: toc}

## Definition

The **empty set** $\emptyset$ is the [[set]] with no [[element]]s.

The empty set is unique in most membership-based [[foundations]], but in any case we can say 'the' empty set, since any two empty sets are isomorphic by a unique isomorphism (the [[generalized the]]).

Indeed, in any foundations (whether membership-based or structural), the empty set is characterized by a [[universal property]]: it is the [[initial object]] in the  [[category]] [[Set]] of all sets: for every set $S$, there is a unique [[function]]

$$
  \emptyset \stackrel{\exists !}{\to} S
  \,.
$$

## Properties

The empty set can be confusing, because it is a degenerate case. Indeed, it is defined as an exception: every set is [[inhabited set|inhabited]], except the empty set. Nothing belongs to the empty set, but the empty set itself is something. Mathematics students are confused that there is a function from the empty set to itself, and even grown mathematicians will misstate definitions because they forget about the empty set.

In terms of the empty set one can give sense to the [[natural number object|natural number]] expression $0^0$, the [[exponentiation]] of [[zero]] with itself, by defining the [[cartesian closed category|exponential]] $m^n$ to be the [[cardinality]] of the [[hom-set]] $\hom_{Set}([n], [m])$ where $[m]$ is an $m$-element set. Then $0^0 = 1$ because $\emptyset$ is an [[initial object]].

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

Endless debates (as seen for example at sci.math) about the meaning of $0^0$ are most easily resolved by paying attention to how exponentiation (in a [[ring]] or [[rig]] that admits [[exponentiation]]) is defined. We have just observed how $0^0 = 1$ makes good sense in the rig of natural numbers. But exponentiation for the [[real number]] field ($x^y$ for $x \gt 0$) is standardly defined by continuity considerations, and cannot be continuously extended to cover $0^0$. Of course, that has nothing to do with the empty set!
=--

## Foundational status

In [[set theory]] as a [[foundation of mathematics]], the existence of the empty set is often taken as an axiom, the __axiom of the null set__.  Actually, this axiom can be written in such a simple way that the name seems unmotivated:

+-- {: .un_defn}
###### Axiom (null set)

There exists a set.
=--

However, one can then use the axiom of separation ([[bounded separation]] is enough) to prove that there exists an empty set, and then use the [[axiom of extensionality]] to prove that this empty set is unique.

In some axiomatic frameworks, even this is unnecessary; while it makes little sense from the standpoint of [[categorial logic]], it was once traditional to take $\forall x,\; P[x] \;\Rightarrow\; \exists x,\; P[x]$ as an axiom (or theorem) of [[first-order logic]].  It is then automatic that something exists, and if one uses a [[material set theory]] of [[pure sets]], then this something must be a set.

Alternatively, since the [[axiom of infinity]] states the existence of a set (and is often phrased to state explicitly the existence of an empty set), then the axiom of the null set becomes unnecessary for another reason.

## Related concepts

[[!include empty objects -- contents]]

category: foundational axiom

[[!redirects empty set]]
[[!redirects empty sets]]

[[!redirects axiom of empty set]]
[[!redirects axiom of the empty set]]
[[!redirects axiom of null set]]
[[!redirects axiom of the null set]]


empty simplicial set

#Contents#
* table of contents
{:toc}

## Definition

The *empty simplicial set* is the [[simplicial set]] $\varnothing_\bullet$ all whose component sets are [[empty sets]]:

$$
  \underset{n \in \mathbb{N}}{\forall}
  \;\;
  \varnothing_n \;=\; \varnothing.
$$

The face- and degeneracy maps are, accordingly, each the unique [[empty function]] from the [[empty set]] to itself.

## Properties

* The empty simplicial set is a [[strict initial object]] in the [[category]] of [[SimplicialSets]].

## Related concepts

[[!include empty objects -- contents]]

[[!redirects empty simplicial sets]]


empty space

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

# The empty space
* table of contents
{: toc}

## Definition

The **empty space** is the [[topological space]] with no points.  That is, it is the [[empty set]] equipped with its unique topology.

## Properties

### General

The empty space is the ([[strict initial object|strict]]) [[initial object]] in [[TopologicalSpaces]].

It satisfies all [[separation axiom|separation]], [[compact space|compactness]], and countability conditions ([[separable space|separability]], [[first-countable space|first countability]], [[second-countable space|second-countability]]).  It is also both [[discrete space|discrete]] and [[indiscrete space|indiscrete]], a distinction it shares only with the [[point space]].

### Connectedness

Debate rages over whether the empty space is [[connected space|connected]] (and also [[path-connected space|path-connected]]).  With the common naive definitions that "a space is [[connected topological space|connected]] if it cannot be partitioned into two disjoint nonempty open subsets" and "a space is [[path-connected topological space|path-connected]] if any two points in it can be joined by a path," the empty space is trivially both connected and path-connected.

However, in some ways these definitions are too naive.  The question of whether the empty set is (path-)connected is analogous in many ways to the question of whether $1$ is [[prime number|prime]].  The above definitions are then analogous to saying that "a natural number $p$ is prime if any factor of it is either equal to $1$ or to $p$," according to which $1$ is prime---but there are better definitions that exclude $1$.

For instance, we may say that "$p$ is prime if it has exactly two factors, itself and $1$;" with this definition $1$ is not prime, since it has exactly _one_ factor.   Likewise, we may say that a space is (path-)connected if it has exactly one (path-)[[component]]; with this definition the empty space is not connected, since it has exactly _zero_ components.  (Lest you question that last statement, note that the correct definition of a (path-)component of a space $X$ is as an [[equivalence class]] of points of $X$ under some [[equivalence relation]].  There is a unique equivalence relation on the empty set, and it has _zero_ equivalence classes.)

Here are some other reasons why the empty space should not be considered (path-)connected:

* {#IfItwerePathConnectedDecompositionWouldFail} If the empty space were (path-)connected, unique decomposition into (path-)connected components would fail: $X \sqcup Y = \emptyset \sqcup X \sqcup Y = \dots$.  This is analogous to how if $1$ were a prime, then unique factorization into primes would fail: $6 = 2 \cdot 3 = 1 \cdot 2 \cdot 3 = 1 \cdot 1 \cdot 2 \cdot 3 = \dots$.

* In [[homotopy theory]], one defines a space $X$ to be $k$-connected if $\pi_i(X)$ is trivial (that is, has exactly one element) for $i \le k$.  When $k =0$ this says that $\pi_0(X)$ should have exactly one component---that is, that $X$ should be path-connected.  (Actually, this definition really only makes sense if we phrase it in terms of homotopy [[groupoids]]; homotopy _groups_ are only defined once we choose a basepoint, which is clearly impossible for the empty space.)

* Category-theoretically, one may say that a space $X$ is connected if the [[functor]] $hom(X,-)$ preserves [[coproducts]].  Since $\hom(\emptyset,-)$ is constant at the point, it certainly does not preserve coproducts.

* The statement that a product $X\times Y$ is connected if and only if its components $X$ and $Y$ are connected fails if the empty set is regarded as connected.

* A general result, e.g., in the theory of [[combinatorial species]], is that the [[logarithm]] of [[exponential]] [[generating functions]] of some type of objects should be the exponential generating function for the connected objects of that type. Since this logarithm has no constant term, this suggests the empty object should not count as connected. This result is also known in the physics literature as the linked-cluster theorem (see [this Prop.](geometry+of+physics+--+perturbative+quantum+field+theory#LogarithmEffectiveAction) at _[[geometry of physics -- perturbative quantum field theory]]_).

See [[too simple to be simple]] for general theory.

## Related concepts

[[!include empty objects -- contents]]

* [[Sierpinski space]]

* [[discrete space]]

[[!include universal constructions of topological spaces -- table]]

[[!redirects empty space]]
[[!redirects empty spaces]]

[[!redirects empty topological space]]
[[!redirects empty topological spaces]]


empty subset

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[set]] $A$, the __empty subset__ of $A$, denoted $\empty_A$, is the [[subset]] of $A$ defined by the property that, for every [[element]] $x$ of $A$, it is [[false]] that $x$ belongs to $\empty_A$.

The underlying set (or _shadow_) of any empty subset is [[the]] [[empty set]].  That is, if we interpret $\empty_A$ as an [[injective function]] $S \hookrightarrow A$, then the [[source]] $S$ of this function is the empty set.

In the usual framework of [[material set theory]], every empty subset is identical to the empty set.  For this reason, it is common to write simply $\empty$ instead of $\empty_A$.  Even from a [[structural set theory|structural]] perspective, this is an abuse of language that is unlikely to cause any confusion.

In the context of [[topology]], we often speak of the __empty subspace__.  In point-set topology, this is indeed an empty subset of the set of points, but in point-free topology, a space is not necessarily the [[empty space]] just because it has no points, and the empty subspace is similarly subtle.

## Related concepts

[[!include empty objects -- contents]]

[[!redirects empty subset]]
[[!redirects empty subsets]]

[[!redirects empty subspace]]
[[!redirects empty subspaces]]


empty theory

The **empty theory** is the [[theory]] with no sorts (and therefore no functions or relations) and no axioms.  This can be formulated in any [[doctrine]]: there is an empty algebraic theory, an empty essentially algebraic theory, an empty first-order theory, and so on.

Each empty theory has a unique model in any category of the sort appropriate to the doctrine.  Its classifying category is the initial object of the category of such categories.  For instance, the [[classifying topos]] of the empty geometric theory is the terminal (since geometric morphisms of toposes are backwards from the "logical" direction) Grothendieck topos [[Set]].


empty topological space > history

> see _[[empty space]]_


empty type

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

# Contents
* table of contents
{: toc}

## Idea

In [[type theory]] the _empty type_ is the [[type]] with no [[term]].

In a [[model]] by [[categorical semantics]] (cf. [[relation between type theory and category theory]]), this is an [[initial object]]. In [[set theory]], it is an [[empty set]]. In [[logic]], especially via the [[propositions as types]] interpretation of type theory, it represents [[falsehood]]; and constructing a [[term]] of an empty type represents a [[contradiction]]; thus [[functions]] into the empty type are regarded as the [[negation]] of their [[domain]] [[proposition]].

## Definition

Like all [[type formation|type constructors]] in [[type theory]], to characterize the empty type we must specify how to build it, how to [[term introduction|construct elements]] of it, how to [[term elimination|use such elements]], and the [[computation rules]].

To start with, since the empty type is not [[dependent type|dependent]], its [[type formation]] rule just says that it exists:

$$\frac{ }{\emptyset\colon Type}$$

### As a positive type

The empty type is most naturally presented as a [[positive type]], so that the [[term introduction|constructor rules]] are primary.  However, since the empty type is supposed to contain no elements, there *are* no constructor rules.

In fact one may understand the positively defined empty type as that *[[inductive type]]* which is given by *no constructors*.

Therefore, in [[programming languages]] supporting a [[calculus of constructions]], such as *[[Coq]]*,  the empty type may be defined by the following [[syntax]] for [[inductive type|inductive]] [[data types]] using literally an [[empty set|empty]] [[string (computer science)|string]] of constructors on the right:

    Inductive empty : Type :=

#### Recursion principle
 {#RecursionPrinciple}

We start by considering the [[recursion principle]] of the empty type, hence its un-[[dependent type|dependent]] [[elimination rule]]:

The [[term elimination|eliminator rules]] are derived from the constructor rules in the usual way: to use a term $x \,\colon\, \varnothing$, given any $D \,\colon\, Type$,
it suffices to specify what should be done for all the ways that $x \,\colon\, \varnothing$ could have been constructed. But for the empty type there are [[empty set|no such]] ways and hence -- *assuming* $x \,\colon\, \varnothing$ -- we obtain a term of any type $D$ under no further conditions:

$$
  \frac{
    D \,\colon\, Type
  }
  {
    x \,\colon\, \varnothing
    \;\; \vdash \;\;
    ind_D(x) \,\colon\, D
  }
$$

Some ways to read this recursion principle:

* Understood as a statement in [[propositional logic]] (regarding *[[propositions as types]]*), this says that from a [[false]] assumption every [[proposition]] $D$ is [[implication|implied]] (*[[ex falso quodlibet]]*):

$$
  False \;\Rightarrow\; D
  \,.
$$

* Understood in the [[categorical semantics]] (cf. the [[relation between type theory and category theory]]) this translates to the first part of the characterization of an [[initial object]], namely the existence of [[morphisms]] into any [[codomain]] [[object]]:

$$
  \varnothing \xrightarrow{\exists} D
  \,.
$$

What is not manifest from the above rule is the (essential) uniqueness of these morphisms; on that point see also the [eta conversion rule](#EtaConversion), below.

#### Induction principle
 {#InductionPrinciple}

In [[dependent type theory]] the elimination rule of an [[inductive type]] is a dependent [[induction principle]] which involves any $\varnothing$-[[dependent type]].

{#InferenceRules}  Applying the general rules for [[inductive types]] to the case of no generators, one obtains the following [[inference rules]]:

**[[type formation rule]]:**

$$
  \frac{
  }{
    \mathclap{\phantom{\vert^{\vert}}}
    \varnothing \,\colon\, Type
  }
$$

\linebreak

**[[term introduction rule]]:**

$$
\text{--- none ---}
$$

\linebreak

**[[term elimination rule]]:**

$$
  \frac{
    x \,\colon\, \varnothing
    \;\vdash\;\;
    D(x) \,:\, Type
  }{
    \mathclap{\phantom{\vert^{\vert}}}
    ind_{(D)}
    \,\colon\,
    \underset{x \colon \varnothing}{\prod}
    D(x)
  }
$$

\linebreak

**[[computation rule]]:**

$$
\text{--- none ---}
$$

Notice that the induction principle entails the recursion principle [above](#RecursionPrinciple) (cf. e.g. [MHE, §2.6](#MHE)), as for any inductive type:

$$
  \frac{
  \frac{
    D \,\colon\, Type
  }
  {
  \big(
    (x \,\colon\, \varnothing)
    \mapsto
    D
  \big)
  \,\colon\,
  (\varnothing \to Type)
  }
  }{
  ind_{ (x \mapsto D) }
  \,\colon\,
  (x \colon \varnothing) \to D
  }
$$

#### Eta-conversion
 {#EtaConversion}

Notice that there is no [[beta-reduction]] rule for the positive empty type, since there are no constructors to compose with the eliminator.

However, one may consider an [[eta-conversion]] rule, which says that for any term $e \,\colon\, \varnothing \;\;\vdash\;\; c \,\colon\, C$ in a context including a term of type $\emptyset$, we have

$$
  abort_C(e)
  \;\;\;
    \leftrightarrow_\eta
  \;\;\;
  c
  \,.
$$

As with the [[eta-conversion|$\eta$-conversion]] rule for the negative presentation of the [[unit type]], this is ill-defined as a *reduction* (since we cannot determine $c$ from $abort_C(e)$), but makes sense as an *expansion*.
   .
Notice that [[Coq]] implements the [[beta reduction]] rule, but not the [[eta conversion]] (although eta equivalence is provable for the inductively defined [[identity types]], using the dependent eliminator mentioned above).

### As a negative type

As for binary [[sum types]], it is possible to present the empty type as a [[negative type]] as well, but only if we allow [[sequents]] with multiple conclusions.  This is common in [[sequent calculus]] presentations of [[classical logic]], but not as common in type theory and almost unheard of in [[dependent type theory]].

The two definitions are provably equivalent, but only using the [[contraction rule]] and the [[weakening rule]].  Thus, in [[linear logic]] they become distinct; the positive empty type is "zero" $\mathbf{0}$ and the negative one is "bottom" $\bot$.

### Using a type of propositions

Suppose that the [[dependent type theory]] has a [[type of propositions]] $\mathrm{Prop}$, such as the one derived from a type universe $U$ - $\sum_{A:U} \mathrm{isProp}(A)$. Then the empty type is defined as the [[dependent function type]]

$$\emptyset \equiv \prod_{P:\mathrm{Prop}} P$$

whose elements are witnesses that all propositions are [[pointed types]] (and thus [[contractible types]]).

By [[weak function extensionality]], the empty type is a proposition, and if it has an element, then every proposition in $\mathrm{Prop}$ has an element and is contractible.

## Properties

### Equivalence of the definitions of the empty type

\begin{theorem}
The inductive definition of the empty type and the definition of the empty type in terms of dependent product types and the type of propositions are equivalent to each other.
\end{theorem}

\begin{proof}
Since the empty type is a proposition, it is equivalent to its own [[propositional truncation]] $\emptyset \simeq [\emptyset]$.

Meanwhile, the [[propositional truncation]] of a type $A$ is equivalent to the dependent product type

$$[A] \simeq \prod_{P:\mathrm{Prop}} (A \to P) \to P$$

Substituting the empty type for $A$, we have

$$[\emptyset] \simeq \prod_{P:\mathrm{Prop}} (\emptyset \to P) \to P$$

By the recursion principle of the empty type, for every type $P$, the type $\emptyset \to P$ is contractible, and for every contractible type $I$, the type $I \to P$ is equivalent to $P$. Thus, we have equivalences of types

$$\emptyset \simeq [\emptyset] \simeq \left(\prod_{P:\mathrm{Prop}} (\emptyset \to P) \to P\right) \simeq \left(\prod_{P:\mathrm{Prop}} P\right)$$
\end{proof}

### The empty type as a univalent universe

The empty type can be represented as a univalent universe. We [[inductive definition|inductively define]] the type family $x:\mathbb{0} \vdash \mathrm{El}_\mathbb{0}(x) \; \mathrm{type}$ by defining
$$\mathrm{El}_\mathbb{0}(*) \coloneqq \mathbb{0}$$
The extensionality principle for the unit type then is simply the [[univalence axiom]]:
$$\mathrm{ext}_\mathbb{0}:\prod_{x:\mathbb{0}} \prod_{y:\mathbb{0}} (x =_\mathbb{0} y) \simeq (\mathrm{El}_\mathbb{0}(x) \simeq \mathrm{El}_\mathbb{0}(y))$$
The empty type $\mathbb{0}$ represents the empty universe, the universe with no elements up to identification and thus no small types. Thus, the univalence axiom is trivially true.

## Related concepts

[[!include empty objects -- contents]]

* [[falsehood]], [[ex falso quodlibet]]

* [[sum type]]

* [[unit type]], [[contractible type]]

## References

* {#UFP13} [[Univalent Foundations Project]], §1.7 and §A.28 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* [[Egbert Rijke]], Def. 3.2.1 in: *Inductive types and the universe*, Lecture 3 in: *[Introduction to Homotopy Type Theory 2018](https://www.andrew.cmu.edu/user/erijke/hott/)* [[pdf](https://www.andrew.cmu.edu/user/erijke/hott/inductive.pdf)]

The definition of the empty type from the type of propositions and dependent product types can be found in:

* Madeleine Birchfield, *Constructing coproduct types and boolean types from universes*, MathOverflow ([web](https://mathoverflow.net/questions/457904/constructing-coproduct-types-and-boolean-types-from-universes))

In [[Agda]]:

* {#MHE} [[Martín Hötzel Escardó]], [§2.6](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#emptytype) in *[Introduction to Univalent Foundations of Mathematics with Agda](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/)* (2019-2022)

See also:

* Wikipedia, *[Principle of explosion](https://en.wikipedia.org/wiki/Principle_of_explosion)*

[[!redirects bottom type]]
[[!redirects bottom types]]

[[!redirects empty type]]
[[!redirects empty typpe]]
[[!redirects empty types]]


empty166


empty167


empty168


empty169

category: empty


emptypage


En-algebra

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

#Contents#
* table of contents
{:toc}

## Definition

An **$E_n$-algebra** is an [[∞-algebra over an (∞,1)-operad|∞-algebra]] over the [[E-k operad]].

## Special cases

### $E_1$-algebras

$E_1$-algebras are often called [[A-∞ algebras]]. See also [[algebra in an (∞,1)-category]].

An $E_1$ algebra in the [[symmetric monoidal (∞,1)-category]] [[Spec]] of [[spectra]] is a [[ring spectrum]].

### $E_2$-algebras

The [[homology]] of an $E_2$-algebra in [[chain complex]]es is a [[Gerstenhaber algebra]].

### $E_\infty$-algebra

See [[E-∞ algebra]].

## Examples

* [[universal enveloping E-n algebra]]

## Properties

### Relation to Poisson $n$-algebras

The [[homology]] of an $E_n$-algebra for $n \geq 2$ is a
[[Poisson n-algebra]].

Moreover, in [[chain complexes]] over a [[field]] of [[characteristic]] 0 the [[E-n operad]] is [[formal dg-algebra|formal]], hence equivalent to its homology, and so in this context $E_n$-algebras are equivalent to [[Poisson n-algebras]].

See there for more.

## Related concepts

[[!include k-monoidal table]]

* [[May recognition theorem]]

* [[monoidal groupoid]], [[braided monoidal groupoid]], [[symmetric monoidal groupoid]]

* [[2-group]], [[braided 2-group]], [[symmetric 2-group]]

* [[k-tuply monoidal n-groupoid]]

* [[k-tuply monoidal (n,r)-category]]

* [[k-tuply groupal n-groupoid]]

* [[A-n algebra]]

* [[nonunital Ek-algebra]]

## References

Section 5 of

* [[Jacob Lurie]], _[[Higher Algebra]]_,

some summary of which is at _[[Ek-Algebras]]_.

Discussion of [[derived noncommutative geometry]] over [[formal duals]] of $E_n$-algebras is in

* [[John Francis]], _Derived algebraic geometry over $\mathcal{E}_n$-Rings_ ([pdf](http://math.northwestern.edu/~jnkf/writ/thezrev.pdf))

* [[John Francis]], _The tangent complex and Hochschild cohomology of $\mathcal{E}_n$-rings_ ([pdf](http://math.northwestern.edu/~jnkf/writ/cotangentcomplex.pdf))

[[!redirects En-algebras]]

[[!redirects E1-algebra]]
[[!redirects E1-algebras]]

[[!redirects E2-algebra]]
[[!redirects E2-algebras]]

[[!redirects E3-algebra]]
[[!redirects E3-algebras]]

[[!redirects E4-algebra]]
[[!redirects E4-algebras]]

[[!redirects E5-algebra]]
[[!redirects E5-algebras]]

[[!redirects E2-space]]
[[!redirects E2-spaces]]

[[!redirects E3-space]]
[[!redirects E3-spaces]]

[[!redirects E4-space]]
[[!redirects E4-spaces]]

[[!redirects E5-space]]
[[!redirects E5-spaces]]

[[!redirects Ek-algebra]]

[[!redirects E-k algebra]]
[[!redirects E-k algebras]]

[[!redirects E-n algebra]]
[[!redirects E-n algebras]]

[[!redirects Ek-algebra]]
[[!redirects Ek-algebras]]


encode-decode method

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

A proof technique used in [[homotopy type theory]] and [[synthetic homotopy theory]].

## See also ##

* [[identity type]]
* [[inductive type]], [[higher inductive type]], [[quotient inductive type]]
* [[homotopy type theory]]
* [[synthetic homotopy theory]]

## References ##

* {#UBP13} [[Univalent Foundations Project]], Sections 2.12, 2.13, 7.3, 8.1, 8.6, 8.7, 8.9, 9.9, 10.1, and 11.3 of: *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

[[!redirects encode-decode]]


Encyclopaedia of Mathematics

[[!redirects eom]]

* [Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Main_Page) (eom)

is an online encyclopedia on *[[mathematics]]*.

Russian original:

* Математическая энциклопедия.  Под редакцией [[И. М. Виноградов|И. М. Виноградовa]].  Издательство «Советская энциклопедия».  Том 1: 1977, Том 2: 1979, Том 3: 1982, Том 4: 1984, Том 5: 1985.

English translation (enhanced with comments):

* Encyclopaedia of Mathematics.  Edited by [[Michiel Hazewinkel]].
D. Reidel Publishing Co. (starting from Volume 2: Kluwer Academic Publishers), Dordrecht,
* Volume 1 (1988. x+488 pp. ISBN: 1-55608-000-X);
* Volume 2 (1988. x+508 pp. ISBN: 1-55608-001-8);
* Volume 3 (1989. x+491 pp. ISBN: 1-55608-002-6);
* Volume 4 (1989. x+516 pp. ISBN: 1-55608-003-4);
* Volume 5 (1990. x+534 pp. ISBN: 1-55608-004-2);
* Volume 6 (1990. ix+546 pp. ISBN: 1-55608-005-0);
* Volume 7 (1991. ix+506 pp. ISBN: 1-55608-006-9);
* Volume 8 (1992. ix+540 pp. ISBN: 1-55608-007-7);
* Volume 9 (1993. x+536 pp. ISBN: 1-55608-008-5);
* Volume 10 (1994. xiv+732 pp. ISBN: 1-55608-009-3).

Paperback edition:

* Kluwer Academic Publishers, Dordrecht, 1995. Vol. 1: x+919 pp.; Vol. 2: ii+963 pp.; Vol. 3: ii+950 pp.; Vol. 4: ii+929 pp.; Vol. 5: ii+906 pp.; Vol. 6: iv+732. ISBN: 1-55608-010-7

Supplements:

* Vol. I. Edited by M. Hazewinkel. Kluwer Academic Publishers, Dordrecht, 1997. viii+587 pp. ISBN: 0-7923-4709-9

* Vol. II. Edited by M. Hazewinkel. Kluwer Academic Publishers, Dordrecht, 2000. vi+631 pp. ISBN: 0-7923-6114-8

* Vol. III. Edited by M. Hazewinkel. Kluwer Academic Publishers, Dordrecht, 2001. viii+557 pp. ISBN: 1-4020-0198-3

Electronic edition:

* Encyclopaedia of mathematics on CD-ROM.  Network version. Revised from the 1988–1997 English editions. Kluwer Academic Publishers, Dordrecht, 1997. Booklet (15 pp.) + 1 CD-ROM (Windows and UNIX). ISBN: 0-7923-4805-2

Online:

* [doi](https://doi.org/10.1007/978-94-009-5991-0).

Encyclopaedia of Mathematics is an enhanced English translation of the Soviet version in Russian (5 volumes), which was a collective effort.  Most of the entries had originally a single author, who signed it as an article; most articles had a very useful bibliography. The English edition was enhanced with editorial additions to the original passages (paragraphs cleanly separated from the original entries), as well as new articles.

## Related entries

* [[Online Resources]]

category: reference

[[!redirects Springer Online Encyclopaedia of Mathematics]]

[[!redirects Encyclopedia of Mathematics]]


This page compiles pointers to material contained in the (upcoming)

\linebreak

* **Encyclopedia of Mathematical Physics**, 2nd edition

  \linebreak

  Elsevier (expected Oct 2024)

  (cf. 1st ed. leaflet: [pdf](https://www.math.ucdavis.edu/~temple/!!!PubsForWeb/cv70.pdf))

on [[mathematical physics]].

\linebreak

## Contents

> (in no particular order, for the time being)

On the [[phase space]] of [[Einstein gravity]] on [[manifolds with boundary]]:

* [[Alberto S. Cattaneo]], *Phase space for gravity with boundaries*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]* (2023) [[arXiv:2307.04666](https://arxiv.org/abs/2307.04666)]

On [[celestial holography]]:

* [[Sabrina Pasterski]], *A Chapter on Celestial Holography*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, [[arXiv:2310.04932](https://arxiv.org/abs/2310.04932)]

On [[principal infinity-bundles|principal $\infty$-bundles]]:

* [[Severin Bunk]], *$\infty$-Bundles*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2308.04196](https://arxiv.org/abs/2308.04196)]

On the [[FRS theorem]] classifying [[rational CFT|rational]] [[2d conformal field theory]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], *Algebraic structures in two-dimensional conformal field theory*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

On [[BV quantization]]:

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

On [[shifted symplectic structures]]:

* [[Damien Calaque]], *Derived symplectic geometry*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2308.04210](https://arxiv.org/abs/2308.04210)]

On [[KK-compactification]] in [[string theory]]:

* [[Michael R. Douglas]], [[Liam McAllister]], *Compactification of Superstring Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2310.20118](https://arxiv.org/abs/2310.20118)]

On [[defect QFT]]:

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

On [[higher topos theory]] in physics:

* [[Urs Schreiber]], *[[schreiber:Higher Topos Theory in Physics|Higher Topos Theory in Physics]]* [[arXiv:2311.11026](https://arxiv.org/abs/2311.11026)]

On [[higher geometry]] in physics:

* [[Luigi Alfonsi]], *Higher geometry in physics*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2312.07308](https://arxiv.org/abs/2312.07308)]

On [[3d quantum gravity]]:

* [[Steve Carlip]], *Quantum Gravity in 2+1 Dimensions*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.12596](https://arxiv.org/abs/2312.12596)]

On [[string structures]] seen as a kind of [[spin structures]] after [[transgression]] to [[loop space]] (cf. [[stringor bundle]]*):

* [[Konrad Waldorf]], *String structures and loop spaces*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.12998](https://arxiv.org/abs/2312.12998)]

On [[Donaldson-Witten theory]]:

* [[Jan Manschot]], *Four-Manifold Invariants and Donaldson-Witten Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.14709](https://arxiv.org/abs/2312.14709)]

On [[differential cohomology]]:

* [[Arun Debray]], *Differential Cohomology*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.14338](https://arxiv.org/abs/2312.14338)]

On [[twisted cohomology]] (with focus on [[twisted ordinary cohomology]] and [[twisted K-theory]]):

* [[Jonathan Rosenberg]], *Twisted cohomology*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.03966](https://arxiv.org/abs/2401.03966)]

On [[higher gauge theory]]:

* [[Leron Borsten]], [[Mehran Jalali Farahani]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Jiří Nárožný]], [[Dominik Rist]], [[Christian Saemann]], [[Martin Wolf]], *Higher Gauge Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.05275](https://arxiv.org/abs/2401.05275)]

On [[quantum gravity]] via [[causal dynamical triangulation]]:

* J. Ambjørn, R. Loll, *Causal Dynamical Triangulations: Gateway to Nonperturbative Quantum Gravity*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.09399](https://arxiv.org/abs/2401.09399)]

(...)

category: reference


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

This page collect material related to

* [[Georg Hegel]],

  _Encyclopedia of the philosophical Sciences_

  1817

which gives a relatively brief survey of _Hegel's system_

1. starting in the first part (widely known as _The Shorter Logic_) with a summary of the _[[Science of Logic]]_ ([[ontology]], [[metaphysics]]);

1. proceeding in the second part with a _Philosophy of Nature_ ([[physics]]);

1. and ending in the third part with a _Philosophy of the Spirit_ which in turn

   1. contains a summary of the _[[Phenomenology of Spirit]]_ (in its first section, B)

   1. and overlaps with (the preface of) the [[Lectures on the Philosophy of History]] (second section, C)

   1. and with the [[Lectures on the Philosophy of Religion]] (third section, b and c).

Hegel defines these three parts in the Introduction (§ 18) as:

   1. Logic: the science of the Idea in and for itself.

   1. The Philosophy of Nature: the science of the Idea in its otherness.

   1. The Philosophy of Mind: the science of the Idea come back to itself out of that otherness.

## References

* Wikipedia, _[Encyclopedia of the Philosophical Sciences](http://en.wikipedia.org/wiki/Encyclopedia_of_the_Philosophical_Sciences)_

category: reference

[[!redirects Enzyklopädie der Philosophischen Wissenschaften im Grundrisse]]


Encyclopedic Dictionary of Mathematics

* Wikipedia, _[Encyclopedic Dictionary of Mathematics](https://en.wikipedia.org/wiki/Encyclopedic_Dictionary_of_Mathematics)_