AQFT

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###Context###
#### Algebraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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#### Quantum systems
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[[!include quantum systems -- contents]]
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#Contents#
* table of contents
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## Idea

**Algebraic Quantum Field Theory** or **Axiomatic Quantum Field Theory** or **AQFT** for short is a formalization of [[quantum field theory]] (and specifically full, hence [[non-perturbative quantum field theory]]) that axiomatizes the assignment of _[[algebras of observables]]_ to patches of parameter space ([[spacetime]], [[worldvolume]]) that one expects a quantum field theory to provide.

As such, the approach of AQFT is roughly dual to that of [[FQFT]], where instead _spaces of states_ are assigned to boundaries of [[cobordism]]s and propagation maps between state spaces to cobordisms themselves.

One may roughly think of AQFT as being a formalization of what in basic [[quantum mechanics]] textbooks is called the **[[Heisenberg picture]]** of quantum mechanics. On the other hand [[FQFT]] axiomatizes the _[[Schrödinger picture]]_ .

The axioms of traditional AQFT encode the properties of a [[local net]] of observables and are called the [[Haag-Kastler axioms]]. They are one of the oldest systems of axioms that seriously attempt to put [[quantum field theory]] on a solid conceptual footing.

From the [[nPOV]] we may think of a [[local net]] as a co-flabby [[presheaf|copresheaf]] of [[algebra|algebras]] on spacetime which satisfies a certain _locality_ axiom with respect to the
[[smooth Lorentzian manifold|Lorentzian structure]] of [[spacetime]]:

* **locality:** algebras assigned to spacelike separated regions commute with each other when embedded into any joint superalgebra.

This is traditionally formulated (implicitly) as a structure in ordinary [[category theory]]. More recently, with the proof of the [[cobordism hypothesis]] and the corresponding [[(∞,n)-category]]-formulation of [[FQFT]] also [[higher category theory|higher categorical]] versions of systems of local algebras of observables are being put forward and studied. Three structures are curently being studied, that are all conceptually very similar and similar to the Haag-Kastler axioms:

* [[factorization algebra]]s

* [[topological chiral homology]]

* [[blob homology]].

Initially, all three of these encoded what in physics are called _Euclidean_ quantum field theories, whereas only the notion of [[local net]] incorporated the fact that the underlying spacetime of a quantum field theory is a [[smooth Lorentzian space]]. Recent developments in the formalism of [[factorization algebra]]s have extended their theory to globally hyperbolic [[Lorentzian manifolds]].

In the context of the Haag-Kastler axioms there is a precise theorem, the [[Osterwalder-Schrader theorem]], relating the Euclidean to the Lorentzian formulation: this is the operation known as [[Wick rotation]].

Sheaves are used explicitly in:

* Roberts, John E.: [New light on the mathematical structure of algebraic field theory.](http://books.google.com/books?id=IFjzuLjE43kC&lpg=PA297&ots=5Ld1B3I45m&dq=Operator%20algebras%20and%20applications%2C%20Part%202&pg=PA523#v=onepage&q=Operator%20algebras%20and%20applications,%20Part%202&f=false)  Operator algebras and applications, Part 2 (Kingston, Ont., 1980),  pp. 523–550, Proc. Sympos. Pure Math., 38, Amer. Math. Soc., Providence, R.I., 1982.

* Roberts, John E.: [Localization in algebraic field theory](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-85/issue-1/Localization-in-algebraic-field-theory/cmp/1103921341.full). Comm. Math. Phys. 85 (1982), no. 1, 87–98.

--- much information to be filled in ---

## Axioms

* [[Wightman axioms]]

* [[Haag-Kastler axioms]]

## Theorems

* [[Reeh-Schlieder theorem]]

* [[Osterwalder-Schrader theorem]]

* [[PCT theorem]]

* [[Bisognano-Wichmann theorem]]

* [[spin-statistics theorem]]

## Properties

Generically the algebra of a relativistic AQFT turns out to be a ([[generalized the|the]]) hyperfinite type $III_1$ [[von Neumann algebra factor]]. See ([Yngvason](#Yngvason))

## Examples
 {#Examples}

Examples of AQFT [[local nets of observables]] that encode interacting quantum field theories are not easy to construct. The construction of _free_ field theories is well understood, see the references [below](#ExamplesReferences). In [[perturbation theory]] also interacting theories can be constructed, see the references [here](#ReferencesPerturbationTheory).

### Free scalar field / Klein Gordon field
 {#FreeScalarField}

A survey of the AQFT description of the [[free field theor|free]] [[scalar field]] on [[Minkowski spacetime]] is in ([Motoya, slides 11-17](#Montoya)). Discussion in more general context of [[AQFT on curved spacetimes]] in ([Brunetti-Fredenhagen, section 5.2](#BrunettiFredenhagen))

### Free fermion / Dirac field

The [[free field theory|free]] [[Dirac field]] and its deformations is discussed for instance in ([DLM, section 3.2](#DLM)), ([Dimock 83](#Dimock82)).

### Electromagnetic field

The quantized [[electromagnetic field]] is discussed for instance in ([Dimock 92](#Dimock92)).

### Proca field

([Furliani](#Furliani))

## Related concepts

* [[quantum mechanics]], [[quantum field theory]], [[perturbative quantum field theory]]

  * **AQFT**

    * [[Haag-Kastler axioms]], [[Wightman axioms]]

    * [[local net of observables]], [[field net]]

    * [[time slice axiom]], [[split property]]

    * [[quantum lattice systems]], [[string-localized quantum field]]

    * [[locally covariant AQFT]]

    * [[perturbative AQFT]]

    * [[homotopical algebraic quantum field theory]]

    * [[Haag-Ruelle scattering theory]]

  * [[FQFT]]

* [[constructive quantum field theory]]

[[!include Isbell duality - table]]

## References

### Axioms

The original article that introduced the [[Haag-Kastler axioms]] is

* {#Haag64} [[Rudolf Haag]], [[Daniel Kastler]], *An algebraic approach to quantum field theory*, Journal of Mathematical Physics, **5** (1964) 848-861 [[doi:10.1063/1.1704187](https://doi.org/10.1063/1.1704187), [spire:9124](https://inspirehep.net/literature/9124)]

following

* {#Haag59} [[Rudolf Haag]], _Discussion des "axiomes" et des propriétés asymptotiques d’une théorie des champs locales avec particules composées, Les Problèmes Mathématiques de la Théorie Quantique des Champs_, Colloque Internationaux du CNRS LXXV (Lille 1957), CNRS Paris (1959), 151.

translated to English as:

* [[Rudolf Haag]], *Discussion of the ‘axioms’ and the asymptotic properties of a local field theory with composite particles*, EPJ H 35, 243–253 (2010) ([doi:10.1140/epjh/e2010-10041-3](https://doi.org/10.1140/epjh/e2010-10041-3))

The generalization of the [[spacetime]] [[site]] from open in [[Minkowski space]] to more general and [[curvature|curved]] spacetimes (see [[AQFT on curved spacetimes]]) is due to

* {#BrunettiFredenhagen} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Quantum field theory on curved spacetimes_ [arXiv:0901.2063](http://arxiv.org/abs/0901.2063)

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Rainer Verch]], _The generally covariant locality principle -- A new paradigm for local quantum physics_ Commun. Math. Phys. 237:31-68 (2003) ([arXiv:math-ph/0112041](http://arxiv.org/abs/math-ph/0112041))

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Backgrounds_ , Proceedings of the Kompaktkurs "Quantenfeldtheorie auf gekruemmten Raumzeiten" held at Universitaet Potsdam, Germany, in 8.-12.10.2007, organized by C. Baer and K. Fredenhagen

See also _[[AQFT on curved spacetimes]]_ .

### Lecture notes and Textbooks

Introductory lecture notes:

* {#Fredenhagen03} [[Klaus Fredenhagen]], _Algebraische Quantenfeldtheorie_, lecture notes, 2003 ([[FredenhagenAQFT2003.pdf:file]])

* {#FewsterRejzner19} [[Christopher Fewster]], [[Kasia Rejzner]], _Algebraic Quantum Field Theory - an introduction_ ([arXiv:1904.04051](https://arxiv.org/abs/1904.04051))

and for just [[quantum mechanics]] in the algebraic perspective:

* {#Gleason09} [[Jonathan Gleason]], *The $C^*$-algebraic formalism of quantum mechanics* (2009) [[[Gleason09.pdf:file]], [[GleasonAlgebraic.pdf:file]]]

* {#Gleason11} [[Jonathan Gleason]], *From Classical to Quantum: The $F^\ast$-Algebraic Approach*, contribution to *[VIGRE REU 2011](http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2011.html)*, Chicago (2011) [[pdf](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf), [[GleasonFAlgebraic.pdf:file]]]

Textbook accounts:

* {#BratteliRobinson79} [[Ola Bratteli]], [[Derek W. Robinson]], *Operator Algebras and Quantum Statistical Mechanics* -- vol 1: *$C^\ast$- and $W^\ast$-Algebras. Symmetry Groups. Decomposition of States.*, Springer (1979, 1987, 2002) [[doi:10.1007/978-3-662-02520-8](https://doi.org/10.1007/978-3-662-02520-8)]

* [[Raymond F. Streater]], [[Arthur S. Wightman]], *PCT, Spin and Statistics, and All That*, Princeton University Press (1989, 2000) [[ISBN:9780691070629](https://press.princeton.edu/books/paperback/9780691070629/pct-spin-and-statistics-and-all-that), [jstor:j.ctt1cx3vcq](https://www.jstor.org/stable/j.ctt1cx3vcq)]

* [[Nikolay Bogolyubov]], A. A. Logunov, A. I. Oksak, I. T. Todorov, *General principles of quantum field theory*, Mathematical Physics and Applied Mathematics **10**, Kluwer (1990) [[doi:10.1007/978-94-009-0491-0](https://doi.org/10.1007/978-94-009-0491-0)]

* [[Rudolf Haag]], _[[Local Quantum Physics -- Fields, Particles, Algebras]]_, Texts and Monographs in Physics, Springer (1992)

* [[Huzihiro Araki]], _[[Mathematical Theory of Quantum Fields]]_ (1999)

* [[Franco Strocchi]], _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

* [[Hans Halvorson]], [[Michael Müger]], _Algebraic Quantum Field Theory_ [[arXiv:math-ph/0602036](http://arxiv.org/abs/math-ph/0602036)]

An account written by mathematicians for mathematicians:

* [[Hellmut Baumgärtel]], Manfred Wollenberg, _Causal nets of operator algebras._ Berlin: Akademie Verlag 1992 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0749.46038&format=complete))

* [[Hellmut Baumgärtel]], _Operator algebraic Methods in Quantum Field Theory. A series of lectures._ Akademie Verlag 1995 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0839.46063&format=complete))

### Reviews

Recent account of the principle of locality in AQFT from the point of view of traditional school

* [[Franco Strocchi]], _Relativistic Quantum Mechanics and Field Theory_, Found. Phys. **34** (2004) 501-527 [[arXiv:hep-th/0401143](http://arxiv.org/abs/hep-th/0401143)]

* [[Sergio Doplicher]], _The principle of locality: Effectiveness, fate, and challenges_, J. Math. Phys. __51__, 015218 (2010), [doi](http://dx.doi.org/10.1063/1.3276100)

Talk slides include

* {#Montoya} Edison Montoya, _Algebraic quantum field theory_ (2009) ([pdf](http://www.matmor.unam.mx/~robert/sem/20091021_Montoya.pdf))

More on the role of [[von Neumann algebra factors]] in AQFT

* {#Yngvason} J. Yngvason, *The role of type III factors in quantum field theory* [[arXiv:math-ph/0411058](http://arxiv.org/abs/math-ph/0411058)]

### Examples
 {#ExamplesReferences}

Construction of examples is considered for instance in

* [[Jonathan Dimock]], _Dirac quantum fields on a manifold_, Trans. Amer. Math. Soc. 269 (1982), 133-147. ([web](http://www.ams.org/journals/tran/1982-269-01/S0002-9947-1982-0637032-8/home.html))
 {#Dimock82}

* [[Jonathan Dimock]], _Quantized electromagnetic field on a manifdold_, Reviews in mathematical physics, Volume 4, Issue 2 (1992) ([web](http://www.worldscinet.com/rmp/04/0402/S0129055X92000078.html))
 {#Dimock92}

* Edward Furliani, _Quantization of massive vector fields in curved space–time_, J. Math. Phys. 40, 2611 (1999) ([web](http://jmp.aip.org/resource/1/jmapaq/v40/i6/p2611_s1?isAuthorized=no))
 {#Furliani}

General discussion of AQFT quantization of [[free quantum fields]] is in

* [[Christian Bär]], N. Ginoux, [[Frank Pfäffle]], _Wave Equations on Lorentzian Manifolds and Quantization_, (EMS, 2007) ([arXiv:0806.1036](http://arxiv.org/abs/0806.1036))

* [[Christian Bär]], N. Ginoux, _Classical and quantum fields on lorentzian manifolds_ (2011) ([arXiv:1104.1158](http://arxiv.org/abs/1104.1158))

Examples of [[non-perturbative quantum field theory|non-perturbative]] [[interacting quantum field theory|interacting]] [[scalar field theory]] in _any_ [[spacetime]] [[dimension]] (in particular in $d \geq 4$) are claimed in

* {#BuchholtzFredenhagen20} [[Detlev Buchholz]], [[Klaus Fredenhagen]], _A $C^\ast$-algebraic approach to interacting quantum field theories_,  Commun. Math. Phys. **377** (2020) 947–969  [[arXiv:1902.06062](https://arxiv.org/abs/1902.06062), [doi:10.1007/s00220-020-03700-9](https://doi.org/10.1007/s00220-020-03700-9)]

### Local gauge theory
 {#LocalGaugeTheory}

Discussion of aspects of [[gauge theory]] includes

* Fabio Ciolli, [[Giuseppe Ruzzi]], Ezio Vasselli, _Causal posets, loops and the construction of nets of local algebras for QFT_ ([arXiv:1109.4824](http://arxiv.org/abs/1109.4824))

* Fabio Ciolli, [[Giuseppe Ruzzi]], Ezio Vasselli, _QED Representation for the Net of Causal Loops_ ([arXiv:1305.7059](http://arxiv.org/abs/1305.7059))

* [[Giuseppe Ruzzi]], _Nets of local algebras and gauge theories_, 2014 ([pdf slides](http://www.aqft14.eu/wp-content/uploads/2014/05/Ruzzi.pdf))

Construction and axiomatization of gauge field AQFT via [[homotopy theory]] and [[homotopical algebra]] (see also at _[[field bundle]]_) is being developed in

* {#BDS} [[Marco Benini]], [[Claudio Dappiaggi]], [[Alexander Schenkel]], _Quantized Abelian principal connections on Lorentzian manifolds_, Communications in Mathematical Physics 2013 ([arXiv:1303.2515](http://arxiv.org/abs/1303.2515))

* {#BeniniSchenkelSzabo15} [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Homotopy colimits and global observables in Abelian gauge theory_ ([arXiv:1503.08839](http://arxiv.org/abs/1503.08839))

* {#BeniniSchenkel16} [[Marco Benini]], [[Alexander Schenkel]], _Quantum field theories on categories fibered in groupoids_ ([arXiv:1610.06071](https://arxiv.org/abs/1610.06071))

The issue of the tension between local gauge invariance and locality and the need to pass to [[stacks]]/[[higher geometry]] is made explicit in

* {#Schenkel14} [[Alexander Schenkel]], _On the problem of gauge theories
in locally covariant QFT_, talk at _[Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html)_ Trento, 2014 ([[SchenkelTrento2014.pdf:file]]) (with further emphasis on this point in the companion talk [Schreiber 14](field+bundle#Schreiber14))

Further development of this [[homotopical algebraic quantum field theory]] includes

* {#BeniniSchenkel16} [[Marco Benini]], [[Alexander Schenkel]], _Quantum field theories on categories fibered in groupoids_ ([arXiv:1610.06071](https://arxiv.org/abs/1610.06071))

### Perturbation theory and renormalization
 {#ReferencesPerturbationTheory}

[[perturbation theory|Perturbation theory]] and [[renormalization]] in the context of AQFT and  is discussed in the following articles.

The observation that in [[perturbation theory]] the [[renormalization|Stückelberg-Bogoliubov-Epstein-Glaser]] local [[S-matrix|S-matrices]] yield a [[local net of observables]] was first made in

* V. Il'in, D. Slavnov, _Observable algebras in the S-matrix approach_ Theor. Math. Phys. **36** , 32 (1978)

which was however mostly ignored and forgotten. It is taken up again in

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

(a quick survey is in section 8, details are in section 2).

Further developments along these lines are in

* [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic quantum field theory and deformation quantization_, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) ([arXiv:hep-th/0101079](http://xxx.uni-augsburg.de/abs/hep-th/0101079))

(relation to [[deformation quantization]])

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

* [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_ Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](http://arxiv.org/abs/0901.2038))

(relation to [[renormalization]])

* [[Michael Dütsch]], [[Klaus Fredenhagen]], _A local (perturbative) construction of observables in gauge theores: the example of qed_ , Commun. Math. Phys. 203 (1999), no.1, 71-105,  ([arXiv:hep-th/9807078](http://xxx.uni-augsburg.de/abs/hep-th/9807078)).

(relation to [[gauge theory]] and [[QED]])

Lecture notes are in

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

* [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Perturbative Construction of Models of Algebraic Quantum Field Theory_ ([arXiv:1503.07814](https://arxiv.org/abs/1503.07814))

and a textbook acount is in

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

### Further developments

* {#DLM} [[Claudio Dappiaggi]], [[Gandalf Lechner]], E. Morfa-Morales, _Deformations of quantum field theories on spacetimes with Killing vector fields_, Commun.Math.Phys.305:99-130, (2011), ([arXiv:1006.3548](http://arxiv.org/abs/1006.3548))

Relation to [[factorization algebras]]:

* [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], _Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifolds_, Communications in Mathematical Physics volume 377, pages 971–997 (2020) ([arXiv:1903.03396v2](https://arxiv.org/abs/1903.03396), [doi:10.1007/s00220-019-03561-x](https://doi.org/10.1007/s00220-019-03561-x))

Relation to [[smooth stacks]]:

* [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], *Smooth 1-dimensional algebraic quantum field theories* ([arXiv:2010.13808](https://arxiv.org/abs/2010.13808))

### Relation to holographic entanglement entropy

Discussion of [[local nets of observables]] in [[AQFT]] as the natural language for grasping [[holographic entanglement entropy]]:

* [[Edward Witten]], _Notes on Some Entanglement Properties of Quantum Field Theory_, Rev. Mod. Phys. 90, 45003 (2018) ([arXiv:1803.04993](https://arxiv.org/abs/1803.04993))

* [[Thomas Faulkner]], _The holographic map as a conditional expectation_ ([arXiv:2008.04810](https://arxiv.org/abs/2008.04810))

[[!include relation between algebraic and functorial field theory -- references]]

[[!redirects algebraic quantum field theory]]
[[!redirects algebraic quantum field theories]]


AQFT and operator algebra contents

**[[algebraic quantum field theory]]** ([[perturbative AQFT|perturbative]], [[AQFT on curved spacetime|on curved spacetimes]], [[homotopical algebraic quantum field theory|homotopical]])

[[A first idea of quantum field theory|Introduction]]

## Concepts

**[[field theory]]**:

* [[classical field theory|classical]], [[prequantum field theory|pre-quantum]], [[quantum field theory|quantum]], [[perturbative quantum field theory|perturbative quantum]]

* [[relativistic field theory|relativistic]], [[Euclidean field theory|Euclidean]], [[thermal quantum field theory|thermal]]

**[[Lagrangian field theory]]**

* [[field (physics)]]

  * [[field bundle]]

  * [[field history]]

  * [[space of field histories]]

* [[Lagrangian density]]

  * [[Euler-Lagrange form]], [[presymplectic current]]

  * [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]]

* [[locally variational field theory]]

* [[covariant phase space]]

  * [[Peierls-Poisson bracket]]

    * [[advanced and retarded propagator]],

    * [[causal propagator]]

**[[quantization]]**

* [[geometric quantization]] [[geometric quantization of symplectic groupoids|of symplectic groupoids]]

* [[algebraic deformation quantization]], [[star algebra]]

**[[quantum mechanical system]]**, **[[quantum probability]]**

* [[subsystem]]

* [[observables]]

  * [[field observables]]

  * [[local observables]]

  * [[polynomial observables]]

  * [[microcausal observables]]

  * [[operator algebra]], [[C*-algebra]], [[von Neumann algebra]]

* [[local net of observables]]

  * [[causal locality]]

  * [[Haag-Kastler axioms]]

  * [[Wightman axioms]]

  * [[field net]]

  * [[conformal net]]

* [[state on a star-algebra]], [[expectation value]]

  * [[pure state]]

    [[wave function]]

    [[collapse of the wave function]]/[[conditional expectation value]]

  * [[mixed state]], [[density matrix]]

  * [[space of quantum states]]

  * [[vacuum state]]

    * [[quasi-free state]],

    * [[Hadamard state]]

    * [[Wightman propagator]]

* [[picture of quantum mechanics]]

**[[free field]] [[quantization]]**

* [[star algebra]], [[Moyal deformation quantization]]

* [[Wick algebra]]

  * [[canonical commutation relations]], [[Weyl relations]]

  * [[normal ordered product]]

* [[Fock space]]

**[[gauge theories]]**

* [[gauge symmetry]]

* [[BRST complex]], [[BV-BRST formalism]]

* [[local BV-BRST complex]]

* [[BV-operator]]

* [[quantum master equation]]

* [[master Ward identity]]

* [[gauge anomaly]]

**[[interacting field theory|interacting field]] [[quantization]]**

* [[causal perturbation theory]], [[perturbative AQFT]]

* [[interaction]]

* [[S-matrix]], [[scattering amplitude]]

  * [[causal additivity]]

  * [[time-ordered product]], [[Feynman propagator]]

  * [[Feynman diagram]], [[Feynman perturbation series]]

  * [[effective action]]

  * [[vacuum stability]]

* [[interacting field algebra]]

  * [[Bogoliubov's formula]]

  * [[quantum Møller operator]]

* [[adiabatic limit]]

  * [[infrared divergence]]

  * [[interacting vacuum]]

**[[renormalization]]**

* [[renormalization scheme|("re-")normalization scheme]]

  * [[extension of distributions]]

  * [[renormalization condition|("re"-)normalization condition]]

  * [[quantum anomaly]]

* [[renormalization group]]

  * [[interaction vertex redefinition]]

  * [[Stückelberg-Petermann renormalization group]]

  * [[renormalization group flow]]/[[running coupling constants]]

* [[effective quantum field theory]]

  * [[UV cutoff]]

  * [[counterterms]]

  * [[relative effective action]]

  * [[Wilsonian RG]], [[Polchinski flow equation]]

## Theorems
 {#Theorems}

### States and observables

* [[order-theoretic structure in quantum mechanics]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Kochen-Specker theorem]]

* [[Bell's theorem]]

* [[Fell's theorem]]

* [[Gleason's theorem]]

* [[Wigner theorem]]

* [[Bub-Clifton theorem]]

* [[Kadison-Singer problem]]

### Operator algebra

* [[Wick's theorem]]

* [[GNS construction]]

  * [[cyclic vector]], [[separating vector]]

* [[modular theory]]

* [[Fell's theorem]]

* [[Stone-von Neumann theorem]]

* [[Haag's theorem]]

### Local QFT

* [[Reeh-Schlieder theorem]]

* [[Bisognano-Wichmann theorem]]

* [[PCT theorem]]

* [[spin-statistics theorem]]

* [[DHR superselection theory]]

* [[Osterwalder-Schrader theorem]] ([[Wick rotation]])

### Perturbative QFT

* [[Schwinger-Dyson equation]]

* [[main theorem of perturbative renormalization]]


AQFT on curved spacetimes

+-- {: .rightHandSide}
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###Context###
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
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#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Where the [[Haag-Kastler axioms]] formulate [[quantum field theory]] on [[Minkowski spacetime]], known as _[[algebraic quantum field theory]]_ (AQFT) there is a generalization of these axioms to [[curved spacetimes]] ([Brunetti-Fredenhagen 01](#BrunettiFredenhagen01)), also known as _locally covariant algebraic quantum field theory_. For the case of [[perturbative quantum field theory]] this is _[[locally covariant perturbative quantum field theory]]_, see there for more.

(This falls short of being a theory of [[quantum gravity]], instead it describes [[quantum field theory]] on classical [[background field]] configurations of [[gravity]].)

This is the mathematically rigorous framework for studying subjects such as the [[cosmological constant]] (see [there](cosmological+constant#InPerturbativeQuantumGravity)), [[Hawking raditation]] or the [[cosmic microwave background]] ([Fredenhagen-Hack 13](#FredenhagenHack13)).

## Applications

### Vacuum energy and Cosmological constant

The [[renormalization]] freedom in [[perturbative QFT|perturbative]] [[quantization]] of [[gravity]] ([[perturbative quantum gravity]]) induces freedom in the choice of [[vacuum expectation value]] of the [[stress-energy tensor]] and hence in the [[cosmological constant]].

Review includes ([Hack 15, section 3.2.1](#Hack15)).

For more see at _[[cosmological constant]]_ [here](cosmological+constant#InPerturbativeQuantumGravity).

## Related concepts

* [[general covariance]]

* [[causal structure]]

## References

To some extent the problem of AQFT on curved spacetime was formulated in

* {#Dyson72} [[Freeman Dyson]], *Missed opportunities*, Bulletin of the AMS, Volume 78, Number 5, September 1972 ([pdf](https://www.math.uh.edu/~tomforde/Articles/Missed-Opportunities-Dyson.pdf))

  > $\,$

  > $[$ the [[Haag-Kastler axioms]] $]$  taken  together  with  the  axioms  defining  a  [[C*-algebra]]  are  a  distillation  into  abstract  mathematical  language  of  all  the   general  truths  that  we  have  learned  about  the  physics  of  microscopic   systems  during  the  last  50 years. They  describe a  mathematical  structure of  great  elegance  whose  properties  correspond  in  many  respects  to  the facts  of  experimental  physics.  In  some  sense,  the  axioms  represent  the most  serious  attempt  that  has  yet  been  made  to  define  precisely  what physicists mean by the words "observability, causality, locality, relativistic invariance," which  they are constantly  using  or  abusing  in their  everyday speech. $[$...$]$ I therefore  propose as an  outstanding opportunity  still open to the pure mathematicians, to  create a mathematical structure preserving the  main features of the Haag-Kastler axioms but possessing E-invariance instead of P-invariance.

$P$ here denotes the [[Poincaré group]], while $E$ denotes what Dyson calls the 'Einstein group', which is now called the [[diffeomorphism group]].

General accounts of (perturbative, algebraic) quantum field theory on curved spacetimes include

* N. Birrell, P.  Davies, _Quantum  Fields in Curved Space_, Cambridge:  Cambridge University Press, 1982

* [[Robert Wald]], _Quantum field theory in curved spacetime and black hole thermodynamics_. Univ. of Chicago Press 1994 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0842.53052&format=complete)).

* {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], _Quantum fields in curved spacetime_, Physics Reports Volume 574, 16 April 2015, Pages 1-35 ([arXiv:1401.2026](https://arxiv.org/abs/1401.2026), [doi:10.1016/j.physrep.2015.02.001](https://doi.org/10.1016/j.physrep.2015.02.001))

* [[Christopher Fewster]], [[Rainer Verch]], _Algebraic quantum field theory in curved spacetimes_ ([arXiv:1504.00586](https://arxiv.org/abs/1504.00586))

See also:

* [[Bernard S. Kay]], *Quantum Field Theory in Curved Spacetime*, Encyclopaedia of Mathematical Physics (2023) [[arXiv:2308.14517](https://arxiv.org/abs/2308.14517)]

Foundations for [[perturbative quantum field theory]] on curved spacetimes in terms of [[causal perturbation theory]] were laid in

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

* {#HollandsWald01} [[Stefan Hollands]], [[Robert Wald]], _Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime_, Commun. Math. Phys. 223:289-326,2001 ([arXiv:gr-qc/0103074](https://arxiv.org/abs/gr-qc/0103074))

* {#HollandsWald02} [[Stefan Hollands]], [[Robert Wald]], _On the Renormalization Group in Curved Spacetime_, Commun. Math. Phys. 237 (2003) 123-160 ([arXiv:gr-qc/0209029](https://arxiv.org/abs/gr-qc/0209029))

* {#HollandsWald04} [[Stefan Hollands]], [[Robert Wald]], _Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes_, Rev.Math.Phys. 17 (2005) 227-312 ([arXiv:gr-qc/0404074](https://arxiv.org/abs/gr-qc/0404074))

The [[AQFT]]-style axiomatization via [[local nets]] on a category of [[Lorentzian manifolds]] ([[locally covariant perturbative quantum field theory]]) is due to:

* {#BrunettiFredenhagen01} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Rainer Verch]], _The generally covariant locality principle -- A new paradigm for local quantum physics_, Commun. Math. Phys. 237:31-68 (2003) ([arXiv:math-ph/0112041](http://arxiv.org/abs/math-ph/0112041))

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Backgrounds_ , Proceedings of the Kompaktkurs "Quantenfeldtheorie auf gekruemmten Raumzeiten" held at Universitaet Potsdam, Germany, in 8.-12.10.2007, organized by C. Baer and K. Fredenhagen ([arXiv:0901.2063](http://arxiv.org/abs/0901.2063))

Reviews with emphasis on the AQFT-local-nets point of view:

* [[Robert Wald]], _The Formulation of Quantum Field Theory in Curved Spacetime_ ([arXiv:0907.0416](https://arxiv.org/abs/0907.0416))

* [[Robert  Wald]], _The History and Present Status of Quantum Field Theory in Curved Spacetime_ ([arXiv:gr-qc/0608018](https://arxiv.org/abs/gr-qc/0608018))

* [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _QFT on curved spacetimes: axiomatic framework and examples_ ([arXiv:1412.5125](http://arxiv.org/abs/1412.5125))

* Níckolas de Aguiar Alves, *Nonperturbative Aspects of Quantum Field Theory in Curved Spacetime* [[arXiv:2305.17453](https://arxiv.org/abs/2305.17453)]

On the [[perturbative algebraic quantum field theory|locally covariant pAQFT approach]] to effective [[quantum gravity]] and applications to experiment:

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Kasia Rejzner]], *Locally covariant approach to effective quantum gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07800](https://arxiv.org/abs/2212.07800)]

There is also a complementary approach via [[OPEs]]:

* [[Stefan Hollands]], [[Robert Wald]], _Axiomatic quantum field theory in curved spacetime_, Commun. Math. Phys. 293:85-125, 2010 ([arXiv:0803.2003](https://arxiv.org/abs/0803.2003))

On the application of [[microlocal analysis]]:

* Alexander Strohmaier, [[Rainer Verch]], Manfred Wollenberg: _Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems_ ([arXiv] (http://xxx.uni-augsburg.de/abs/math-ph/0202003)).

Discussion of [[renormalization]] in AQFT on curved spacetimes includes

* {#KhavkineMoretti16} [[Igor Khavkine]], [[Valter Moretti]], _Analytic Dependence is an Unnecessary Requirement in Renormalization of Locally Covariant QFT_, Communications in Mathematical Physics, March 2016 ([arXiv:1411.1302](http://arxiv.org/abs/1411.1302), [publisher](http://link.springer.com/article/10.1007%2Fs00220-016-2618-7))

Discussion of the [[cosmology]] in the context of AQFT on curved spacetimes includes

* {#FredenhagenHack13} [[Klaus Fredenhagen]], [[Thomas-Paul Hack]], _Quantum field theory on curved spacetime and the standard cosmological model_ ([arXiv:1308.6773](http://arxiv.org/abs/1308.6773))

* {#BrunettiFredenhagenHackPinamontoRejzner16} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Thomas-Paul Hack]], [[Nicola Pinamonti]], [[Katarzyna Rejzner]], _Cosmological perturbation theory and quantum gravity_ ([arXiv:1605.02573](https://arxiv.org/abs/1605.02573))

* {#Hack15} [[Thomas-Paul Hack]], _Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes_, Springer 2016 ([arXiv:1506.01869](https://arxiv.org/abs/1506.01869), [doi:10.1007/978-3-319-21894-6](https://doi.org/10.1007/978-3-319-21894-6))

Relation to [[2d CFT]]:

* [[Marco Benini]], [[Luca Giorgetti]], [[Alexander Schenkel]], *A skeletal model for 2d conformal AQFTs* ([arXiv:2111.01837](https://arxiv.org/abs/2111.01837))

[[!redirects AQFT on curved spacetime]]

[[!redirects quantum field theory on curved spacetime]]

[[!redirects AQFT on curved backgrounds]]

[[!redirects locally covariant algebraic quantum field theory]]
[[!redirects locally covariant AQFT]]

[[!redirects quantum field theory on curved spacetime]]
[[!redirects quantum field theory on curved spacetimes]]

[[!redirects quantum field theories on curved spacetime]]
[[!redirects quantum field theories on curved spacetimes]]

[[!redirects QFT on curved spacetime]]
[[!redirects QFT on curved spacetimes]]

[[!redirects QFTs on curved spacetime]]
[[!redirects QFTs on curved spacetimes]]


Arakelov geometry

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

#Contents#
* table of contents
{:toc}

## Motivation

The aim of _Arakelov geometry_ is to extend [[intersection theory]] to the case of [[algebraic curves]] over $Spec(\mathbb{Z})$, hence in [[arithmetic geometry]].

Arakelov complemented the algebraic geometry at finite primes with a holomorphic piece at a [[place at infinity]]. Then using [[complex analytic geometry]] and [[Green's functions]] he defined the [[intersection numbers]] using the complementary piece at infinity.

## Definitions

... e.g. ([Durov 07](#Durov07))

## Properties

### Arithmetic Riemann-Roch theorem

* [[arithmetic Riemann-Roch theorem]]

## Related concepts

* [[Mordell conjecture]]

* [[arithmetic Chow group]]

* [[Vojta's conjecture]]

* [[global analytic geometry]]

* [[function field analogy]]

## References

### Introductions and surveys
 {#IntroductionsAndSurveys}

* Wikipedia: [Arakelov theory](http://en.wikipedia.org/wiki/Arakelov_theory)

* {#Lang88} [[Serge Lang]], _Introduction to Arakelov theory_ Springer-Verlag, New York, 1988.

* {#SABK91} [[Christophe Soulé]], D. Abramovich, , J.-F. Burnol, J. Kramer, _Lectures on Arakelov Geometry_, Cambridge University Press 1991

* {#deJong04} [[Robin de Jong]], _Explicit Arakelov geometry_, PhD thesis 2004 ([pdf](http://www.math.leidenuniv.nl/~rdejong/publications/thesis.pdf))

* Alberto Camara, _Notes on Arakelov theory_, 2011 ([[CamaraOnArakelov11.pdf:file]])

### Original articles
 {#ReferencesOriginal}

The theory originates in

* [[Suren Arakelov]], _Intersection theory of divisors on an arithmetic surface_, Math. USSR Izv. 8 (6): 1167–1180, 1974, [doi](http://dx.doi.org/10.1070%2FIM1974v008n06ABEH002141); _Theory of intersections on an arithmetic surface_, Proc. ICM Vancouver 1975, vol. 1, 405–408, Amer. Math. Soc. 1975, [djvu](http://www.mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0405.0408.ocr.djvu), [pdf](http://www.mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0405.0408.ocr.pdf)

After Arakelov there were main improvements by Faltings and Gillet and Soulé.

* [[Gerd Faltings]], _Calculus on arithmetic surfaces_, Ann. of Math. (2) 119 (1984), no. 2, 387–424, [MR86e:14009](http://www.ams.org/mathscinet-getitem?mr=740897), [doi](http://dx.doi.org/10.2307/2007043); _Arakelov's theorem for abelian varieties_, Invent. Math. __73__ (1983), no. 3, 337–347, [MR85m:14061](http://www.ams.org/mathscinet-getitem?mr=718934), [doi](http://dx.doi.org/10.1007/BF01388431)

The [[arithmetic Riemann-Roch theorem]] is due to

* [[Henri Gillet]], [[Christophe Soulé]], _An arithmetic Riemann–Roch Theorem_, Invent. Math. __110__: 473–543, 1992, [doi](http://dx.doi.org/10.1007/BF01231343)

* Shou-Wu Zhang, _Small points and Arakelov theory_, Proc. ICM 1998, vol. 2, [djvu](http://www.mathunion.org/ICM/ICM1998.2/Main/03/Zhang.MAN.ocr.djvu), [pdf](http://www.mathunion.org/ICM/ICM1998.2/Main/03/Zhang.MAN.ocr.pdf)

In a recent Bonn thesis under Faltings' supervision,

* {#Durov07} [[Nikolai Durov]], _A new approach to Arakelov geometry_, [arxiv/0704.2030](http://arxiv.org/abs/0704.2030)

a completely algebraic replacement (using generalized [[schemes]] whose local models are spectra of commutative [[algebraic monad]]s) for the original mixed approach is proposed; it is not known if that approach can be closely and precisely compared with the traditional.

See also

* [[Jean-Benoit Bost]], [[Klaus Künnemann]], _Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers_ ([arXiv:math/0701343](http://arxiv.org/abs/math/0701343))

* [[Jean-Benoit Bost]], [[Klaus Künnemann]], _Hermitian vector bundles and extension groups on arithmetic schemes. II. The arithmetic Atiyah extension_ ([arXiv:0807.4374](http://arxiv.org/abs/0807.4374))

### Relation to AdS3/CFT2

Relation of [[AdS3/CFT2]] to [[hyperbolic geometry]] and [[Arakelov geometry]] of [[algebraic curves]]:

* [[Yuri Manin]], [[Matilde Marcolli]], _Holography principle and arithmetic of algebraic curves_, Adv. Theor. Math. Phys. 5 (2002) 617-650 ([arXiv:hep-th/0201036](https://arxiv.org/abs/hep-th/0201036)

[[!redirects Arakelov theory]]


Araminta Amabel

* [personal page](https://math.mit.edu/~araminta/)

## Selected writings

On [[Whitehead-generalized cohomology theory|Whitehead-generalized]] [[differential cohomology]] via [[sheaves of spectra]]:

* {#AmabelDebrayHaine21} [[Araminta Amabel]], [[Arun Debray]], [[Peter J. Haine]] (eds.), _Differential Cohomology: Categories, Characteristic Classes, and Connections_. Based on [Fall 2019 talks at MIT's Juvitop seminar](https://math.mit.edu/juvitop/pastseminars/2019_Fall.html) by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. ([arXiv:2109.12250](https://arxiv.org/abs/2109.12250))

On [[factorization algebras]] in relation to [[functorial field theory]]:

* [[Araminta Amabel]], *Notes on Factorization Algebras and TQFTs* [[arXiv:2307.01306](https://arxiv.org/abs/2307.01306)]

category: people


Aravind Asok

* [institute page](https://dornsife.usc.edu/aravind-asok/)

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

## Selected writings

On [[motivic homotopy theory]]

and [[h-cobordism]]:

* {#AsokMorel} [[Aravind Asok]], [[Fabien Morel]], _Smooth varieties up to $\mathbb{A}^1$-homotopy and algebraic h-cobordisms_, Adv. Math. **227** (5) (2011) 1990-2058 [[arXiv:0810.0324](http://arxiv.org/abs/0810.0324), [doi:10.1016/j.aim.2011.04.009](https://doi.org/10.1016/j.aim.2011.04.009)]

and [[representable functor|representability]]:

* [[Aravind Asok]], [[Marc Hoyois]], [[Matthias Wendt]], *Affine representability results in $\mathbb{A}^1$-homotopy theory I: Vector bundles*, Duke Math. J. **166** 10 (2017) 1923-1953 [[arXiv:1506.07093](http://arxiv.org/abs/1506.07093), [doi:10.1215/00127094-0000014X](https://doi.org/10.1215/00127094-0000014X)]

* [[Aravind Asok]], [[Marc Hoyois]], [[Matthias Wendt]], _Affine representability results in $\mathbb{A} ^1$-homotopy theory II: Principal  bundles and homogeneous spaces_, Geom. Topol. **22** (2018) 1181-1225 [[arXiv:1507.08020](http://arxiv.org/abs/1507.08020), [doi:10.2140/gt.2018.22.1181](https://doi.org/10.2140/gt.2018.22.1181)]

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

* [[Aravind Asok]], [[Jean Fasel]], [[Mrinal Kanti Das]], *Euler class groups and motivic stable cohomotopy*, Journal of the EMS **24** 8 (2022) 2775–2822 [[arXiv:1601.05723](https://arxiv.org/abs/1601.05723), [doi:10.4171/jems/1156](https://doi.org/10.4171/jems/1156)]

category: people


arc space

#Contents#
* table of contents
{:toc}

## Idea

While for finite $m$, $m$-jets of a scheme of finite type (over an algebraically closed field of characteristic $0$) are represented by a scheme, the $\infty$-[[jet scheme]], the (inverse) limit of $m$-jet schemes is not
of finite type; this is the arc space.

## Motivation

The arc space (and the jet schemes) of a variety $X$ gives information about the singular locus $X_{sing}$.

## Definition

Let $k$ be the algebraically closed field, $Sch/k$ the category of schemes over $k$ and $X$ an object in $Sch/k$. The presheaf
$$
(Sch/k)^{op}\to Set\,\,\,\,\,\,\,\,\,\,
 Y\mapsto (Sch/k) (Y\times_k k[t]/t^{m+1},X)
$$
is representable by a $k$-scheme of finite type $X_m$ the $m$-jet scheme.
For $s\geq 1$, the canonical maps $k[t]/t^{m+1}\to k[t]^{m+s+1}$ induces maps
$(Sch/k) (Y\times_k k[t]/t^{m+s+1},X)\to (Sch/k)(Y\times_k k[t]/t^{m+1},X)$,
what is
$(Sch/k) (Y,X_{m+1})\to (Sch/k) (Y, X_m)$ hence also on representing objects
$X_{m+1}\to X_m$. The limit is the __arc space__ $X_\infty = lim_m X_m$ of $X$
and it comes along with natural projections $X_\infty\to X_m\to X$ (under some
assumptions each of the maps is locally trivial).

## Properties

If $X$ is a scheme of finite type over $k$ then there is a bijection

$$
(Sch/k) (Y,X_\infty) \cong (ind-Sch/k) (Y\hat\times_{Spec k} Spec k[[t]],X)
$$

natural in $Y$ in $Sch/k$, where $Y\hat\times_k k[[t]]$ is the formal completion of
$Y$ along subscheme $Y\times_{Spec k} \{0\}$.1

## Related entries

* [[singularity]]

* [[loop space]]

* [[jet space]]

*  [[motivic integration]]

* [[Greenberg scheme]]

## Literature

Early ideas appeared in

* J. Nash Jr., _Arc structure of singularities_, Duke Math. J.,
81 (1995), 31–38.

and its appearance in motivic integration stems from

* [[M. Kontsevich]], lecture on motivic integration, Orsay, December 7, 1995.

For basic lectures see

* M. Mustaţǎ, _Spaces of arcs in birational geometry_, [pdf](http://www.math.lsa.umich.edu/~mmustata/lectures_arcs.pdf)
* M. Popa, 571 Ch. 5. _Jet schemes and arc spaces_, [pdf](http://homepages.math.uic.edu/~mpopa/571/chapter5.pdf)

Surveys:

* Jan Denef, Francois Loeser, _Geometry on arc spaces of algebraic varieties_, Proceedings of 3rd ECM, Barcelona, July 10-14, 2000, [math.AG/0006050](http://arxiv.org/abs/math/0006050)

* L. Ein, M. Mustaţǎ, _Jet schemes and singularities_, Algebraic geometry-
Seattle 2005, 505–546, Proc. Sympos. Pure Math. 80, Part 2, Amer. Math. Soc.,
Providence, RI, 2009 [MR2483946](http://www.ams.org/mathscinet-getitem?mr=2483946)

* Tommaso de Fernex, _The space of arcs of an algebraic variety_, [arxiv/1604.02728](http://arxiv.org/abs/1604.02728)

On connections to combinatorics and representation theory:

* Clemens Bruschek, Hussein Mourtada, Jan Schepers, _Arc spaces and the Rogers–Ramanujan identities_, The Ramanujan Journal 30:1 (2013) 9-38

Other papers

* J. Denef, F. Loeser, _Germs of arcs on singular algebraic varieties and motivic
integration_, Invent. Math. 135 (1999), 201–232.
* S Ishii, J Kollár, _The Nash problem on arc families of singularities, _
Duke Math. J., 120 (2003) 601–620 [math.AG/0207171](http://arxiv.org/abs/math/0207171)

* Shihoko Ishii, _The arc space of a toric variety_, [doi](http://dx.doi.org/10.1016/j.jalgebra.2003.12.015) [arxiv/0312324](http://arxiv.org/abs/math/0312324)

* L Ein, R Lazarsfeld, M Mustaţǎ, _Contact loci in arc spaces_, Comput. Math. and [math.AG/0303268](http://arxiv.org/abs/math/0303268)

* M Mustaţǎ, _Jet schemes of locally complete intersection canonical singularities_, with an appendix by David Eisenbud and Edward Frenkel, Invent. Math., 145 (2001) 397–424; _Singularities of pairs via jet schemes_, J. Amer. Math. Soc., 15 (2002) 599–615

* Cobo Pablos, H. and González Pérez, Pedro Daniel (2012) Motivic Poincaré series, toric singularities and logarithmic Jacobian ideals. Journal of algebraic geometry, 21 (3) 495-529 [pdf](http://www.ams.org/journals/jag/2012-21-03/S1056-3911-2011-00567-5/S1056-3911-2011-00567-5.pdf)

* Dave Anderson, Alan Stapledon, _Arc spaces and equivariant cohomology_, Transformation Groups 18:4 (2013) 931-969
* J. Nicaise, _Arcs and resolution of singularities_, Manuscr. Math. 116: pp. 297-322 (2005)

* W. Veys, _Arc spaces, motivic integration and stringy invariants_, in: Singularity theory and its applications, Adv. Stud. Pure Math. 43, Math. Soc. Japan, Tokyo (2006) 529-572

See also Corollary 4.4 in

* [[Bhargav Bhatt]], _Algebraization and Tannaka duality_,
Cambridge J. Math. __4__: 4 (2016)
 403-461 [doi](https://doi.org/10.4310/CJM.2016.v4.n4.a1)
[arXiv:404.7483](https://arxiv.org/abs/1404.7483)

A formal version (ind-scheme) of free loop space for a complex algebraic variety containing the Kontsevich-Denef-Loeser arc scheme is studied in

* [[Mikhail Kapranov]], [[Eric Vasserot]], _Vertex algebras and formal loop space_, Publ. Math., Inst. Hautes Étud. Sci. 100 (2004) 209--269 (2004) [doi](https://doi.org/10.1007/s10240-004-0023-9) [arXiv:math/0107143](https://arxiv.org/pdf/math/0107143.pdf)

[[!redirects arc spaces]]
[[!redirects arc schemes]]
[[!redirects arc scheme]]


arccos

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### Context
#### Trigonometry
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[[!include trigonometry -- contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

In [[trigonometry]] the arccosine or "inverse cosine" is one of the [[inverse trigonometric functions]].

## Definition

The arccosine is a [[section]] (or [[right inverse]]) to the cosine function [[cos]], which in turn is one of the [[trigonometric functions]]. More precisely, it is the unique [[continuous function|continuous]] section of the [[epimorphism]] in the [[(epi, mono) factorization]] of the cosine such that its value at $0$ is $\pi/2$ (see [[pi]]).

## Related concepts

* [[cos]]

* [[arctan]]

* [[pi]]

## References

See also

* Wikipedia, _Inverse trigonometric functions_

[[!redirects arccosine]]


Archana Morye

* [webpage](http://mathstat.uohyd.ernet.in/people/profile/archana-s-morye)

category: people


archetypal example

An __archetypal example__ of a notion in mathematics is a class of examples such that every instance of the notion in some sense reduces to it: by being isomorphic to it, equivalent in some other sense, or at least that the archetypal example has all essential features (for the problem at hand) found in general case.

It is often the case that the fact that some example is in fact archetypal is an insight which comes only after nontrivial study of the subject. Its special role is often a conjecture in the development of the subject. The terminology is not meant so much to label the subclasses of objects, as to emphasise on the exhaustive status of some examples in studying the general case of the notion. It is often used not only for objects but often for procedures, algorithms and alike.

Pedagogically it is often good to introduce some definition by introducing the archetypal example first. However the general definition often has appeal of often more invariant definition, and allows possibly some new but nonsubstantial examples.

## Examples

Each sheaf is isomorphic to a sheaf of sections of some etale space. Hence sheaves of sections of etale spaces are the archetypal example of a sheaf.

In the axiomatic (synthetic) theory of projective spaces, a well known results is that every projective space of dimension different from $2$ is isomorphic to a projective space defined in the algebraic way from a vector space over a division ring. Hence in dimension $n$ different from $2$ (as well as e.g. all finite planes) the algebraic construction of a projective space from the vector space $k^n$ (where $k$ is the ground division ring) amounts to the archetypal example of projective space in synthetic sense.

All [[homogeneous spaces]] for Hausdorff paracompact topological groups are _isomorphic_ to [[coset space]]s of that group. Hence the coset spaces are the archetypal example of a homogeneous space, provided we take rather weak conditions on a topological group.

Exponential function can be defined for finite matrices (over say real numbers) by the exponential power series. In practice, one uses similarity transformations to reduce the matrix to a similar Jordan form matrix, calculates the exponential for that case and then by the inverse similarity transformation obtains the answer. The calculation for the Jordan forms reduces to calculating for each Jordan block separately and then using the rule that for every block matrix one calculates the exponential and inserts it. In particular for the diagonal matrix one can just take the exponentials of the diagonal entries. Thus, the two examples of the calculation of the exponential function, that is calculating it for the block matrix and for the single Jordan block, are the acrhetypal examples, at least provided we consider the similarity transformation nonessential.

An arbitrary full subcategory of the category of all modules over a ring is an archetypal example of an [[abelian category]] by the Freyd-Mitchell embedding theorem. The theorem has conditions of set versus class size which are nonessential for almost all practical purposes in mathematics. Hence we can consider categories of modules over rings as the archetypal examples of abelian categories. This motivated many techniques, for example the method of elements in the study of abelian categories. According to some mathematicians the embedding theorem has its usefulness also used in converse sense. Namely, finding the abelian category proof of some fact on categories of modules is often beneficial as it may bring clarity and it may surface deeper essential ideas (and sometimes even lead to simplifications). Why the embedding theorem helps and not just the fact that the category of modules is an abelian category:
if it were not an almost general example, then we would be unsure that if the internal proof even exists, hence we would less likely work hard on it and find it.

## Non-examples

While [[vector space]] is a basic example of a [[module]] (namely, when the underlying ring is a [[field]] or [[division ring]]) it is not an archetypal example. Indeed, most interesting phenomena in modules become trivial or nonexistent in the case of vector spaces. For example, in every dimension there is only one isomorphism class of vector spaces, while this is far from so in the case of general modules. Similarly, every vector space is semisimple (and there is only one isomorphism class of simple vector spaces) while for many rings there are nonsemisimple modules.

[[!redirects archetypal examples]]


Archibald Richardson

[[!redirects Richardson]]
Archibald Richardson (1881-1954) was a British algebraist, who was a professor of mathematics at Swansea  from 1920. He collaborated with [[D. E. Littlewood]] on invariants and the theory of group representations. They introduced the immanant of a matrix, studied [[Schur functions]] and developed the [[Littlewood-Richardson rule]] for their multiplication.

* [St. Andrews history entry](mathshistory.st-andrews.ac.uk/Biographies/Richardson_Archibald/)

category: people


archimedean difference protoring

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#### Algebra
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[[!include higher algebra - contents]]
=--
=--
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# Contents
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## Definition ##

A [[protoring]] $M$ is an **archimedean difference protoring** it is both an [[archimedean protoring]] and a [[difference protoring]].

## Examples ##

* Every [[archimedean integral domain]] is an archimedean difference protoring.

* In [[impredicative mathematics]], the [[Dedekind real numbers]] are the [[terminal object|terminal]] archimedean difference protoring.

## See also ##

* [[protoring]]

* [[archimedean protoring]]

* [[difference protoring]]

* [[archimedean integral domain]]

* [[streak]]

## References

* [[Davorin Lešnik]], *Synthetic Topology and Constructive Metric Spaces*, ([arxiv:2104.10399](https://arxiv.org/abs/2104.10399))

[[!redirects archimedean difference protorings]]
[[!redirects Archimedean difference protoring]]
[[!redirects Archimedean difference protorings]]


Archimedean field

## Disambiguation

The term *Archimedean field* may refer to either:

* [[Archimedean ordered fields]]

* [[Archimedean valued fields]]

category: disambiguation

[[!redirects archimedean field]]
[[!redirects archimedean fields]]
[[!redirects non-archimedean field]]
[[!redirects non-archimedean fields]]
[[!redirects nonarchimedean field]]
[[!redirects nonarchimedean fields]]

[[!redirects Archimedean field]]
[[!redirects Archimedean fields]]
[[!redirects non-Archimedean field]]
[[!redirects non-Archimedean fields]]


archimedean group

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* table of contents
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## Definition

An _archimedean group_ is a [[strict total order|strictly]] [[ordered group]] which satisfies the [[Archimedean property]], in which every positive element is bounded above by a [[natural number]].

So an archimedean group has no infinite elements (and thus no non-zero [[infinitesimal object|infinitesimal]] elements).

## Properties

* Every archimedean group is an [[abelian group]] and has no bounded [[cyclic group|cyclic]] [[subgroups]]. Every archimedean group admits an [[embedding]]  into the group of [[real numbers]].

* Every archimedean group is a [[flat module]] and a torsion-free group.

* Every Dedekind complete archimedean group is isomorphic to the [[integers]], if the group is not dense, or the [[Dedekind real numbers]], if the group is dense.

## Examples

Archimedean groups include

* [[integers]]

* [[half integers]]

* [[rational numbers]]

* [[dyadic rationals]]

* [[decimal rationals]]

* [[real numbers]]

Non-archimedean groups include

* [[p-adic integers]]

* [[p-adic numbers]]

## See also

* [[archimedean property]]

* [[archimedean protoring]]

## External links

* Wikipedia, _Archimedean group_

[[!redirects archimedean group]]
[[!redirects archimedean groups]]
[[!redirects non-archimedean group]]
[[!redirects non-archimedean groups]]
[[!redirects nonarchimedean group]]
[[!redirects nonarchimedean groups]]

[[!redirects Archimedean group]]
[[!redirects Archimedean groups]]
[[!redirects non-Archimedean group]]
[[!redirects non-Archimedean groups]]


Archimedean ordered Artinian local ring

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#### Differential geometry
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## Idea

Since every [[ordered Artinian local ring]] $R$ has [[characteristic zero]], the positive integers $\mathbb{Z}_+$ are a subset of $R$, with [[injection]] $i:\mathbb{Z}_+ \hookrightarrow R$. An **Archimedean ordered Artinian local ring** is an [[ordered Artinian local ring]] which satisfies the [[archimedean property]]: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$.

Archimedean ordered Artinian local rings are important because they are the [[ordered local rings]] with [[nilpotent]] [[infinitesimals]] but no [[infinite elements]] or invertible infinitesimals, and thus play an important role in synthetic approaches to [[analysis]] and [[differential geometry]].

[[Archimedean ordered fields]] are the Archimedean ordered Artinian local rings in which the [[nilradical]] is the [[trivial|trivial ideal]], or equivalently, in which every non-invertible element is equal to zero.

## Properties

### Purely real and purely infinitesimal elements

Suppose that $K$ is an [[Archimedean ordered field]] and $A$ is an Archimedean ordered Artinian local $K$-algebra. Since $A$ is a [[local ring]], the quotient of $A$ by its ideal of non-invertible elements $D$ is the [[residue field]] $K$ itself, and the canonical function used in defining the quotient is the function $\Re:A \to K$ which takes a number $a \in A$ to its purely real component $\Re(a) \in K$. Since $A$ is an ordered $K$-algebra, there is a [[strictly monotone]] [[ring homomorphism]] $h:K \to A$. An element $a \in A$ is **purely real** if $h(\Re(a)) = a$, and an element $a \in A$ is **purely [[infinitesimal]]** if it is in the [[fiber]] of $\Re$ at $0 \in K$. Zero is the only element in $A$ which is both purely real and purely infinitesimal.

### Prelattice structure

Now, suppose that the [[Archimedean ordered field]] $K$ has [[lattice]] structure $\min:K \times K \to K$ and $\max:K \times K \to K$. Then the Archimedean ordered local $K$-algebra $A$ has a [[prelattice]] structure given by functions $(-)\wedge(-):A \times A \to A$ and $(-)\vee(-):A \times A \to A$, defined by
$$a \wedge b \coloneqq h(\min(\Re(a), \Re(b)))$$
and
$$a \vee b \coloneqq h(\max(\Re(a), \Re(b)))$$

### Pseudometric and seminorm structure

The [[distance]] function on $A$ is given by the function $\rho:A \times A \to K$, defined as

$$\rho(a, b) \coloneqq \max(\Re(a), \Re(b)) - \min(\Re(a), \Re(b))$$

and the [[absolute value]] on $A$ is given by the function $\vert-\vert:A \times A \to K$, defined as

$$\vert a \vert \coloneqq \rho(a, 0)$$

The distance function and absolute value are [[pseudometrics]] and multiplicative [[seminorms]], because every [[Archimedean ordered field]] $K$ embeds into the [[real numbers]] $\mathbb{R}$, and since $\min(a, b) \leq \max(a, b)$, the pseudometric and seminorm are always non-negative. This also implies that in every Archimedean ordered field with lattice structure, the pseudometric defined above is a metric.

### Smooth and differentiable function structure

Suppose that $K$ is an [[Archimedean ordered field]] with lattice structure, and $A$ is an [[Archimedean ordered local ring|Archimedean ordered]] [[Artinian local algebra|Artinian local $K$-algebra]]. Then [[continuous functions]], [[differentiable functions]], and [[smooth functions]] are each definable on $K$ using the algebraic, order, and metric structure on $K$.

The ring homomorphism $h:K \to A$ preserves [[smooth functions]]: given a natural number $n \in \mathbb{N}$ and a purely infinitesimal element $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$, then for every [[smooth function]] $f \in C^\omega(K)$, there is a function $f_A:A \to A$ such that for all elements $x \in K$, $f_A(h(x)) = h(f(x))$ and
$$f_A(h(x) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}(x)\right) \epsilon^i$$

If we restrict to Archimedean ordered Artinian local $K$-algebras $A$ where every element of the nilradical $D$ is a nilsquare element, where for all $\epsilon \in D$, $\epsilon^2 = 0$, then the ring homomorphism $h:K \to A$ preserves [[differentiable functions]]; for every [[differentiable function]] $f:K \to K$ with given [[derivative]] $f':K \to K$, there is a function $f_A:A \to A$ such that for all elements $x \in K$ and nilpotent elements $\epsilon \in D$, $f_A(h(x)) = h(f(x))$ and
$$f_A(h(x) + \epsilon) = h(f(x)) + h(f'(x)) \epsilon$$

In particular, every [[polynomial function]] $p:K \to K$ lifts to a polynomial function $p_A:A \to A$.

Alternatively, one could use this property to define differentiable and smooth functions in $K$, such as the [[exponential function]], [[natural logarithm]], [[sine function]], and [[cosine function]].

One could also work with [[partial functions]] instead. Given a predicate $P$ on the real numbers $\mathbb{R}$, let $I$ denote the set of all elements in $\mathbb{R}$ for which $P$ holds. A [[partial function]] $f:\mathbb{R} \to \mathbb{R}$ is equivalently a function $f:I \to \mathbb{R}$ for any such predicate $P$ and set $I$.

A function $f:I \to \mathbb{R}$ is **smooth at a subset** $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R}$ with $(D^0 j)(a) = a$ for all $a \in S$, such that for all Archimedean ordered Artinian local $\mathbb{R}$-algebras $A$ with ring homomorphism $h_A:\mathbb{R} \to A$, natural numbers $n \in \mathbb{N}$, and purely infinitesimal elements $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$
$$f_A(h_A(j(a)) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h_A\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i$$

A function $f:I \to \mathbb{R}$ is **smooth at an element** $a \in I$ if it is smooth at the [[singleton subset]] $\{a\}$, and a function $f:I \to \mathbb{R}$ is **smooth** if it is smooth at the [[improper subset]] of $I$.

A function $f:I \to \mathbb{R}$ is **differentiable at a subset** $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d f}{d x}:S \to \mathbb{R}$ such that for all Archimedean ordered Artinian local $K$-algebras $A$ with ring homomorphism $h:K \to A$ such that for all $\epsilon \in D$, $\epsilon^2 = 0$, for all nilpotent elements $\epsilon \in D$,
$$f_A(h(j(a)) + \epsilon) = h(j(a)) + h\left(\frac{d f}{d x}(a)\right) \epsilon$$

A function $f:I \to \mathbb{R}$ is **differentiable at an element** $a \in I$ if it is differentiable at the [[singleton subset]] $\{a\}$, and a function $f:I \to \mathbb{R}$ is **differentiable** if it is differentiable at the [[improper subset]] of $I$.

### Square roots and Euclidean pseudometric structure

Now, assume that $K$ is an [[Euclidean field]] as well, in addition to being an [[Archimedean ordered field]]. While $K$ has a [[principal square root function]] $\sqrt{-}:[0, \infty) \to [0, \infty)$, not every Archimedean ordered local $K$-algebra $A$ has a principal square root function $\sqrt{-}:[0, \infty) \to [0, \infty)$, because purely infinitesimal elements in $A$ are not guaranteed to have [[square roots]]. An Archimedean ordered Artinian local $K$-algebra is **Euclidean** if every nilpotent element has a square root.

However, given an Archimedean ordered Artinian local $K$-algebra $A$, every [[rank]] $n$ $A$-module $V$ with basis $v:\mathrm{Fin}(n) \to V$ has a [[Euclidean pseudometric]] $\rho_V:V \times V \to K$, given by

$$\rho_V(a, b) \coloneqq \sqrt{\sum_{i \in \mathrm{Fin}(n)} \rho(a_i, b_i)^2}$$
for module elements $a \in V$ and $b \in V$ and scalars $a_i \in A$ and $b_i \in A$ for index $i \in \mathrm{Fin}(n)$, where
$$a = \sum_{i \in \mathrm{Fin}(n)} a_i v_i \quad b = \sum_{i \in \mathrm{Fin}(n)} b_i v_i$$

If $A$ is an [[ordered field]], then the Euclidean pseudometric on $V$ is a [[metric]].

## See also

* [[Archimedean ordered field]]
* [[Archimedean ordered local ring]]
* [[ordered Artinian local ring]]

[[!redirects Archimedean ordered Artinian local ring]]
[[!redirects Archimedean ordered Artinian local rings]]

[[!redirects Archimedean ordered local Artinian ring]]
[[!redirects Archimedean ordered local Artinian rings]]

[[!redirects Archimedean ordered Artinian local algebra]]
[[!redirects Archimedean ordered Artinian local algebras]]

[[!redirects Archimedean ordered local Artinian algebra]]
[[!redirects Archimedean ordered local Artinian algebras]]

[[!redirects Archimedean ordered Artin local ring]]
[[!redirects Archimedean ordered Artin local rings]]

[[!redirects Archimedean ordered local Artin ring]]
[[!redirects Archimedean ordered local Artin rings]]

[[!redirects Archimedean ordered Artin local algebra]]
[[!redirects Archimedean ordered Artin local algebras]]

[[!redirects Archimedean ordered local Artin algebra]]
[[!redirects Archimedean ordered local Artin algebras]]


Archimedean ordered field

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## Idea

An _Archimedean ordered field_ is an [[ordered field]] that satisfies the [[archimedean property]].

## Definition

### Using field homomorphisms from the rationals

The [[rational numbers]] are the [[initial]] [[ordered field]], so for every ordered field $F$ there is a field homomorphism $h:\mathbb{Q}\to F$. Since every field homomorphism between ordered fields is an [[injection]], the rational numbers $\mathbb{Q}$ is a [[subset]] of the ordered field $F$, and we can suppress the field homomorphism via [[coercion]], such that given $q \in \mathbb{Q}$ one can derive that $q \in F$. Thus, $F$ is _Archimedean_ if for all elements $x \in F$ and $y \in F$, if $x \lt y$, then there exists a rational number $q\in \mathbb{Q}$ such that $x \lt q$ and $q \lt y$.

### Using Dedekind cut-like conditions

A [[real number]] in an [[ordered field]] $F$ is an element $x \in F$ which satisfies the [[Dedekind cut]] axioms:

1. there exists a rational number $q \in \mathbb{Q}$ such that $q \lt x$
1. there exists a rational number $r \in \mathbb{Q}$ such that $x \lt r$
1. for all rational numbers $q \in \mathbb{Q}$ and $q' \in \mathbb{Q}$, $q \lt q'$ and $q' \lt x$ implies that $q \lt x$
1. for all rational numbers $r \in \mathbb{Q}$ and $r' \in \mathbb{Q}$, $x \lt r'$ and $r' \lt r$ implies that $x \lt r$
1. for all rational numbers $q \in \mathbb{Q}$, $q \lt x$ implies that there exists a rational number $q' \in \mathbb{Q}$, such that $q \lt q'$ and $q' \lt x$
1. for all rational numbers $r \in \mathbb{Q}$, $x \lt r$ implies that there exists a rational number $r' \in \mathbb{Q}$, such that $x \lt r'$ and $r' \lt r$
1. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt x$ and $x \lt r$ implies that $q \lt r$
1. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt r$ implies that $q \lt x$ or $x \lt r$

We have the following results:

* The first and second conditions say that every element $x \in F$ is bounded below and above by rational numbers, and thus strictly not an [[infinite]] element. This also implies that there are no infinitesimal elements, because there are no element $x \in F$ whose multiplicative inverse is an infinite element.

* The fifth and sixth conditions independently imply that every element $x \in F$ is strictly not an [[infinitesimal]] element.

These four conditions together imply the [[archimedean property]] for the ordered field $F$.

* The third, fourth, and seventh conditions are always true for all elements $x \in F$ because of transitivity of the [[strict order]] relation.

* Finally, the eighth condition says that every element $x \in F$ is located, and is true for all elements $x \in F$ because the [[pushout]] of the [[open intervals]] $(q, \infty)$ and $(-\infty, r)$ with canonical inclusions $(q, r) \to (q, \infty)$ and $(q, r) \to (-\infty, r)$ is equivalent to $F$ itself.

Thus, an **Archimedean ordered field** is an ordered field $F$ where every element $x \in F$ is a real number.

Note that this definition is not the same as saying that $F$ contains every [[real number]] - the latter definition results in the [[Dedekind real numbers]], which is the union of all Archimedean ordered fields and the [[terminal object|terminal]] Archimedean ordered field.

## Properties

Every Archimedean ordered field is a [[dense linear order]]. This means that the [[Dedekind completion]] of every Archimedean ordered field is the field of all [[real numbers]].

### Dedekind cuts

Every element $x \in F$ in an Archimedean ordered field satisfies the axioms of [[Dedekind cuts]]:

1. there exists a rational number $q \in \mathbb{Q}$ such that $q \lt x$
1. there exists a rational number $r \in \mathbb{Q}$ such that $x \lt r$
1. for all rational numbers $q \in \mathbb{Q}$ and $q' \in \mathbb{Q}$, $q \lt q'$ and $q' \lt x$ implies that $q \lt x$
1. for all rational numbers $r \in \mathbb{Q}$ and $r' \in \mathbb{Q}$, $x \lt r'$ and $r' \lt r$ implies that $x \lt r$
1. for all rational numbers $q \in \mathbb{Q}$, $q \lt x$ implies that there exists a rational number $q' \in \mathbb{Q}$, such that $q \lt q'$ and $q' \lt x$
1. for all rational numbers $r \in \mathbb{Q}$, $x \lt r$ implies that there exists a rational number $r' \in \mathbb{Q}$, such that $x \lt r'$ and $r' \lt r$
1. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt x$ and $x \lt r$ implies that $q \lt r$
1. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt r$ implies that $q \lt x$ or $x \lt r$

We have the following results:

* The first condition is always true because for all $x \in F$, we have $x - 1 \in F$, and by the Archimedean principle there exists a rational number $q \in \mathbb{Q}$ such that $x - 1 \lt q \lt x$.

* The second condition is always true because for all $x \in F$, we have $x + 1 \in F$, and by the Archimedean principle there exists a rational number $r \in \mathbb{Q}$ such that $x \lt r \lt x + 1$.

* The fifth condition is always true because for all $x \in F$ and $q \in \mathbb{Q}$, if $q \lt x$, then by the Archimedean principle there exists a rational number $q' \in \mathbb{Q}$ such that $q \lt q' \lt x$.

* The sixth condition is always true because for all $x \in F$ and $r \in \mathbb{Q}$, if $x \lt r$, then by the Archimedean principle there exists a rational number $q' \in \mathbb{Q}$ such that $x \lt r' \lt r$.

* The third, fourth, and seventh conditions are always true for all elements $x \in F$ because of transitivity of the [[strict order]] relation.

* Finally, the eighth condition says that every element $x \in F$ is located, and is true for all elements $x \in F$ because the [[union]] $(q, \infty) \cup (-\infty, r)$ of the [[open intervals]] $(q, \infty)$ and $(-\infty, r)$ is the [[improper subset]] of $F$.

### Continuous and differentiable structure

Every Archimedean ordered field is a [[differentiable space]]:

#### Pointwise continuous functions

Let $F$ be an Archimedean ordered field. A function $f:F \to F$ is __continuous at a point__ $c \in F$ if

$$isContinuousAt(f, c) \coloneqq \forall \epsilon \in (0, \infty). \forall x \in F. \exists \delta \in (0, \infty). (\vert x - c \vert \lt \delta) \implies (\vert f(x) - f(c) \vert \lt \epsilon)$$

$f$ is __pointwise continuous__ in $F$ if it is continuous at all points $c$:
$$isPointwiseContinuous(f) \coloneqq \forall c \in F. isContinuousAt(f, c)$$

The set of all pointwise continuous functions is defined as

$$C^0(F) \coloneqq \{f \in F \to F \vert isPointwiseContinuous(f)\}$$

#### Pointwise differentiable functions

Let $F$ be an Archimedean ordered field. A function $f:F \to F$ is __differentiable at a point__ $c \in F$ if

$$isDifferentiableAt(f, c) \coloneqq isContinuousAt(f, c) \times \exists L \in F. \forall \epsilon \in (0, \infty). \forall x \in F. \exists \delta \in (0, \infty). \forall h \in (-\delta, 0) \cup (0, \delta). \left| \frac{f(c + h) - f(c)}{h} - L \right| \lt \epsilon$$

$f$ is __pointwise differentiable__ in $F$ if it is differentiable at all points $c$:
$$isPointwiseDifferentiable(f) \coloneqq \forall c \in F. isDifferentiableAt(f, c)$$

The set of all pointwise differentiable functions is defined as

$$D^0(F) \coloneqq \{f \in F \to F \vert isPointwiseDifferentiable(f)\}$$

## Category of Archimedean ordered fields

The **category of Archimedean ordered fields** is the [[category]] whose [[objects]] are Archimedean ordered fields and whose [[morphisms]] are [[strictly monotonic]] field [[homomorphisms]] between Archimedean ordered fields.

The category of Archimedean ordered fields is a [[thin category]]. It is also a [[skeletal category]] and a [[gaunt category]], and impredicatively is the subset of the [[power set]] of [[real numbers]] which consists of all the Archimedean ordered [[subfields]] of the real numbers.

The [[initial object]] in the category of Archimedean ordered fields is the [[rational numbers]] and the [[terminal object]] in the category of Archimedean ordered fields is the ([[Dedekind real number|Dedekind]]) [[real numbers]].

## Examples

Archimedean ordered fields include

* [[rational numbers]]
* [[real closed fields]]
* [[real numbers]]

In [[constructive mathematics]], one has the different notions of real numbers

* [[Cauchy real numbers]] (terminal Archimedean ordered field where every element merely has a [[locator]])
* [[HoTT book real numbers]] (initial [[Cauchy complete]] Archimedean ordered field)
* [[Dedekind real numbers]] (terminal Archimedean ordered field, also initial [[Dedekind complete]] Archimedean ordered field)

These notions of real numbers are the same if every Dedekind real number merely has a [[locator]], so that the Cauchy real numbers are Dedekind complete. Both [[excluded middle]] and [[countable choice]] imply that every Dedekind real number has a [[locator]].

Non-Archimedean ordered fields include

* [[finite fields]]

* [[complex numbers]]

* [[p-adic numbers]]

* [[surreal numbers]]

## Related concepts

* [[archimedean valued field]]

* [[archimedean integral domain]]

* [[archimedean group]]

* [[Archimedean ordered Artinian local ring]]

* [[Archimedean ordered reduced local ring]]

* [[Archimedean ordered local integral domain]]

* [[differentiable space]]

## References

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

The definition of the Archimedean property for an ordered field is given in section 4.3 of

* Auke B. 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)$]$

The [[real numbers]] are defined as [[generalized the|the]] [[terminal object|terminal]] [[Archimedean ordered field]] and the [[complete space|complete]] [[Archimedean ordered field]] in:

* {#Richman08} [[Fred Richman]], *Real numbers and other completions*, Mathematical Logic Quarterly **54** 1 (2008) 98-108 [[doi:10.1002/malq.200710024](https://onlinelibrary.wiley.com/doi/10.1002/malq.200710024)]

[[!redirects archimedean ordered field]]
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[[!redirects category of archimedean ordered fields]]
[[!redirects category of Archimedean ordered fields]]


Archimedean ordered integral domain

# Contents
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## Idea

An _Archimedean ordered integral domain_ is an [[ordered integral domain]] that satisfies the [[Archimedean property]].

## Properties

Every Archimedean ordered integral domain extension of the integers $\mathbb{Z}[x]$ is a [[dense linear order]].

Since $\mathbb{Z}[x]$ is Archimedean, it does not have either infinite or infinitesimal elements, which means there exists an integer $a$ and natural number $b$ such that $0 \lt x - a \lt 1$ and $1 \lt b(x - a)$. $0 \lt x - a \lt 1$ implies $0 \lt (d - c)(x - a) \lt d - c$ and $c \lt (d - c)(x - a) + c \lt d$ for all $c$ and $d$ in $\mathbb{Z}[x]$. Let $y = (d - c)(x - a) + c$. Since there exists an element $y$ such that $c \lt y \lt d$ for all $c$ and $d$, $\mathbb{Z}[x]$ is a [[dense linear order]].

This means that the [[Dedekind completion]] of every Archimedean ordered integral domain extension of the integers is the integral domain of [[real numbers]].

## Examples

Archimedean ordered integral domains include

* [[integers]]
* [[rational numbers]]
* [[decimal rational numbers]]
* [[dyadic rational numbers]]
* [[real numbers]]

Non-Archimedean ordered integral domains include

* [[p-adic integers]].

## Related concepts

* [[Archimedean ordered field]]

* [[Archimedean group]]

* [[archimedean difference protoring]]

* [[enriched set]]

[[!redirects archimedean ordered integral domain]]
[[!redirects archimedean ordered integral domains]]

[[!redirects Archimedean ordered integral domain]]
[[!redirects Archimedean ordered integral domains]]

[[!redirects archimedean integral domain]]
[[!redirects archimedean integral domains]]

[[!redirects Archimedean integral domain]]
[[!redirects Archimedean integral domains]]


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## Idea

Since every [[ordered local integral domain]] $R$ has [[characteristic zero]], the positive integers $\mathbb{Z}_+$ are a subset of $R$, with [[injection]] $i:\mathbb{Z}_+ \hookrightarrow R$. An **Archimedean ordered local integral domain** is an [[ordered local integral domain]] which satisfies the [[archimedean property]]: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$.

Archimedean ordered local integral domains are important because they are the [[ordered local rings]] with [[infinitesimals]] but no [[infinite elements]], infinitesimals for which the zero-product property fails, or invertible infinitesimals. These play an important role in some approaches to [[analysis]] and [[differential geometry]], such as the approaches where the ring of [[formal power series]] with $n$ commuting variables on the [[real numbers]] is used to model [[infinitesimals]] in the real line.

[[Archimedean ordered fields]] are the Archimedean ordered local integral domains in which every non-invertible element is equal to zero. However, unlike Archimedean ordered fields, Archimedean ordered local integral domains can be defined inside of an arbitrary [[arithmetic pretopos]] in [[coherent logic]].

## See also

* [[Archimedean ordered field]]
* [[Archimedean ordered local ring]]
* [[Archimedean ordered reduced local ring]]
* [[ordered local integral domain]]
* [[formal power series]]

[[!redirects Archimedean ordered local integral domain]]
[[!redirects Archimedean ordered local integral domains]]


Archimedean ordered local ring

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\tableofcontents

## Definition

Since every [[ordered local ring]] $R$ has [[characteristic zero]], the positive integers $\mathbb{Z}_+$ are a subset of $R$, with [[injection]] $i:\mathbb{Z}_+ \hookrightarrow R$. An **Archimedean ordered local ring** is an [[ordered local ring]] which satisfies the [[archimedean property]]: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$.

Unlike [[Archimedean ordered fields]], which require arithmetic [[Heyting pretoposes]], Archimedean ordered local rings are definable in any [[arithmetic pretopos]].

## In analysis

Archimedean ordered local rings are important for modeling notions of [[infinitesimals]]. These include the dual numbers, which represent nilsquare infinitesimals and are used to synthetically define [[differentiable functions]] in the real numbers, Archimedean ordered Weil rings, which represent nilpotent infintiesimals and are used to synthetically define [[smooth functions]] in the real numbers, as well as [[formal power series]] on the [[ground ring]], which represent infinitesimals which are not nilpotent and are used to synthetically define [[analytic functions]] in the real numbers.

### Kock-Lawvere axiom

An Archimedean ordered local ring $R$ satisfies the *[[Kock-Lawvere axiom]]* if and only if given any Weil $R$-algebra $W$, the canonical function from $W$ to $R^{\mathrm{spec}_R^W}$ the [[function algebra]] with [[domain]] the [[formal spectrum]] of $W$ and codomain $R$ is an $R$-algebra [[isomorphism]].

## See also

* [[Archimedean ordered field]]
* [[ordered local ring]]
* [[Archimedean ordered Artinian local ring]]
* [[Archimedean ordered reduced local ring]]
* [[Archimedean ordered local integral domain]]

## References

* [[Ieke Moerdijk]], [[Gonzalo E. Reyes]]: **Models for Smooth Infinitesimal Analysis**, Springer (1991), [doi:10.1007/978-1-4757-4143-8](https://link.springer.com/book/10.1007/978-1-4757-4143-8)

[[!redirects Archimedean ordered local ring]]
[[!redirects Archimedean ordered local rings]]

[[!redirects archimedean ordered local ring]]
[[!redirects archimedean ordered local rings]]

[[!redirects Archimedean local ring]]
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Archimedean ordered reduced local ring

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## Idea

Since every [[ordered reduced local ring]] $R$ has [[characteristic zero]], the positive integers $\mathbb{Z}_+$ are a subset of $R$, with [[injection]] $i:\mathbb{Z}_+ \hookrightarrow R$. An **Archimedean ordered reduced local ring** is an [[ordered reduced local ring]] which satisfies the [[archimedean property]]: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$.

An important class of Archimedean ordered reduced local rings are [[Archimedean ordered local integral domains]], which are used in [[differential geometry]] and [[analysis]] to define [[analytic functions]] in the same way that [[Archimedean ordered Artinian local rings]] are used to define [[smooth functions]]. [[Archimedean ordered fields]] are the Archimedean ordered reduced local rings in which every non-invertible element is nilpotent, or equivalently, in which every non-invertible element is equal to zero.

## See also

* [[Archimedean ordered field]]
* [[Archimedean ordered local ring]]
* [[Archimedean ordered local integral domain]]
* [[ordered reduced local ring]]
* [[formal power series]]

[[!redirects Archimedean ordered reduced local ring]]
[[!redirects Archimedean ordered reduced local rings]]

[[!redirects Archimedean ordered reduced local algebra]]
[[!redirects Archimedean ordered reduced local algebras]]


Archimedean property

\tableofcontents

## Idea

The Archimedean property states that every positive element in a [[strictly weakly ordered]] [[cancellative monoid|cancellative]] [[commutative monoid]] is bounded above by a [[natural number]].

So an object satisfying the Archimedean property has no infinite elements. If the [[strict weak order]] is additionally a [[connected relation]] and thus a [[pseudo-order]], every [[infinitesimal]] element is equal to zero.

## Definition

### In general

Let $(\mathbb{N}^+,1:\mathbb{N}^+,s:\mathbb{N}^+\to \mathbb{N}^+)$ be the set of positive integers.

Let $(A,\lt, +, 0)$ be a [[strictly ordered]] [[cancellative monoid|cancellative]] [[commutative monoid]]. The positive integers are embedded into the [[function algebra|function monoid]] $A \to A$; there is an injection $inj:\mathbb{N}^+\to (A \to A)$ such that $inj(1) = id_A$ and $inj(s(n)) = inj(n) + id_A$ for all $n:\mathbb{N}^+$.

The __archimedean property__ states that for every $a,b:A$ such that $0 \lt a$ and $0 \lt b$, then there exist $n:\mathbb{N}^+$ such that $a \lt inj(n)(b)$.

By [[uncurrying]] $inj$ one gets an [[action]] $act: (\mathbb{N}^+\times A) \to A$ such that $act(1,a) = a$ and $act(s(n),a) = act(n,a) + a$ for all $n:\mathbb{N}^+$ and $a:A$. The archimedean property then states that for all $a,b:A$ such that $0 \lt a$ and $0 \lt b$, there exist $n:\mathbb{N}^+$ such that $a \lt act(n,b)$.

### For ordered fields

For [[ordered fields]] $F$, since $\mathbb{Q}$ is the initial ordered field, there exists a unique ordered field homomorphism $i:\mathbb{Q} \to F$ for all ordered fields $F$. Then the **archimedean property** for $F$ states that the rational numbers are [[dense]] in $F$:

* for all $x \in F$ and $y \in F$ such that $x \lt y$, there exists a rational number $q:\mathbb{Q}$ such that $x \lt i(q) \lt y$.

## Archimedean structure

In [[dependent type theory]], both definitions of the usual archimedean property above use the phrase "there exists...". This existence in the definitions is mere existence; i.e. using the [[existential quantifier]] rather than the [[dependent sum type]]. The untruncated version of the archimedean property using dependent sum types can be called **archimedean structure**.

### In general

Archimedean structure for [[strictly ordered]] [[cancellative monoid|cancellative]] [[commutative monoids]] $(A,\lt, +, 0)$ using the positive natural numbers says that

* for every $x:A$, $y:A$ such that $0 \lt x$ and $0 \lt y$, then there exist as structure a positive natural number $n:\mathbb{N}^+$ such that $x \lt \mathrm{inj}(n)(y)$.

$$\prod_{x:A} \prod_{y:A} (0 \lt x) \times (0 \lt y) \to \sum_{n:\mathbb{N}^+} x \lt \mathrm{inj}(n)(y)$$

Defining the type of positive elements in $A$ as $A^+ \coloneqq \sum_{x:A} 0 \lt x$, by [[uncurrying]] and using the [[type theoretic axiom of choice]], one gets the equivalent statement

$$\prod_{x:A^+} \prod_{y:A^+} \sum_{n:\mathbb{N}^+} x \lt \mathrm{inj}(n)(y)$$

which by the type theoretic axiom of choice is the same as saying that

* there exists as structure a function $M:(A^+ \times A^+) \to \mathbb{N}^+$ such that for every $x:A^+$ and $y:A^+$,  $x \lt \mathrm{inj}(M(x, y))(y)$

$$\sum_{M:(A^+ \times A^+) \to \mathbb{N}^+} \prod_{x:A^+} \prod_{y:A^+} x \lt \mathrm{inj}(M(x, y))(y)$$

These functions can be called the **archimedean modulus** in parallel to the [[modulus of convergence]] and [[modulus of continuity]] which are defined for the untruncated versions of the definitions of [[Cauchy sequence]] and [[continuous function]].

### For ordered fields

Similarly, archimedean structure for ordered fields states that

* for all $x:F$ and $y:F$ such that $x \lt y$, there exists as structure a rational number $q:\mathbb{Q}$ such that $x \lt i(q) \lt y$.

$$\prod_{x:F} \prod_{y:F} x \lt y \to \sum_{q:\mathbb{Q}} (x \lt i(q)) \times (i(q) \lt y)$$

One could construct archimedean structure for any Archimedean ordered field where every element has a [[locator]]. See lemma 6.7.3 of [Booij20](#Booij20).

In particular, one could construct archimedean structure for any Archimedean ordered field satisfying [[trichotomy]], that for all $x:F$ and $y:F$, exactly one of $x \gt y$, $x \lt y$, $x = y$ is true.

Suppose the Archimedean ordered field $F$ satisfies [[trichotomy]]. Then, the [[ceiling]] $x \mapsto \lceil x \rceil$ and [[floor]] $x \mapsto \lfloor x \rfloor$ are well defined functions over the entire domain of $F$. By definition of the ceiling and the fact that $x \lt y$, we have that

$$\left\lceil\frac{2}{y - x}\right\rceil \geq \frac{2}{y - x}$$
and thus
$$\left\lceil\frac{2}{y - x}\right\rceil (y - x) = \left\lceil\frac{2}{y - x}\right\rceil y - \left\lceil\frac{2}{y - x}\right\rceil x \geq 2$$
so we have
$$\left\lceil\frac{2}{y - x}\right\rceil x \lt \frac{\left\lceil \left\lceil\frac{2}{y - x}\right\rceil x \right\rceil + \left\lfloor \left\lceil\frac{2}{y - x}\right\rceil y \right\rfloor}{2} \lt \left\lceil\frac{2}{y - x}\right\rceil y$$
Since $x \lt y$, $0 \lt \left\lceil\frac{2}{y - x}\right\rceil$, and so it is possible to divide all sides by $\left\lceil\frac{2}{y - x}\right\rceil$ and keep the elements in the inequality in order.

$$x \lt \frac{\left\lceil \left\lceil\frac{2}{y - x}\right\rceil x \right\rceil + \left\lfloor \left\lceil\frac{2}{y - x}\right\rceil y \right\rfloor}{2 \left\lceil\frac{2}{y - x}\right\rceil} \lt y$$

Thus, if the Archimedean ordered field satisfies [[trichotomy]], then the Archimedean ordered field has Archimedean structure. In [[classical mathematics]], as well as in [[constructive mathematics]] in which the [[analytic LPO]] for the [[terminal object|terminal]] Archimedean ordered field is true, every Archimedean ordered field has Archimedean structure.

## Examples

* [[Archimedean group]]
* [[Archimedean ordered field]]
* [[Archimedean ordered integral domain]]
* [[Archimedean ordered local ring]]

## References

* Wikipedia, _Archimedean property_

The alternative definition of the Archimedean property for ordered fields using the rational numbers is defined in section 11.2.1 of

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

and in section 4.3 of

* {#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)$]$

[[!redirects archimedean property]]
[[!redirects Archimedean property]]

[[!redirects archimedean structure]]
[[!redirects Archimedean structure]]


archimedean protoring

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## Definition ##

A [[protoring]] $M$ is an **archimedean protoring** if for all terms $a, b, c, d \in M$, $c \lt d$ implies that there exists a positive [[natural number]] $n \in \mathbb{N}_+$ such that

$$b + \sum_{i=1}^n c \lt a + \sum_{i=1}^n d$$

where

$$\sum_{i=1}^{(-)} (-):\mathbb{N}_+ \times M \to M$$

is the canonical left non-unital $\mathbb{N}_+$-[[action]] for [[commutative semigroups]] defined inductively by

$$\sum_{i=1}^{1} c \coloneqq c$$

$$\sum_{i=1}^{n + 1} c \coloneqq c + \sum_{i=1}^{n}$$

## See also ##

* [[archimedean group]]

* [[archimedean integral domain]]

* [[archimedean difference protoring]]

* [[protoring]]

## Examples ##

* Every [[archimedean integral domain]] is a archimedean protoring.

## References

* Davorin Lešnik, Synthetic Topology and Constructive Metric Spaces, ([arxiv:2104.10399](https://arxiv.org/abs/2104.10399))

[[!redirects archimedean protorings]]
[[!redirects Archimedean protoring]]
[[!redirects Archimedean protorings]]


archimedean valued field

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## Definition

A (complete) _Archimedean valued field_ is a [[field]] equipped with an [[archimedean absolute value]] (and [[complete field|complete]] with respect to it).

A non-Archimedean valued field is one that is not, hence one whose [[norm]] satisfies the [[ultrametric]] [[triangle inequality]].

## Properties

One of [[Ostrowski's theorems]] says that for $k$ a field [[complete field|complete]] with respect to an [[absolute value]] ${\vert - \vert}$ either the absolute value is archimedean valued, in which case $k$ is either the field of [[real numbers]] or of [[complex numbers]], or the absolute value is non-archimedean.

### Non-Archimedean valued fields

For $k$ a non-Archimedean valued field for some non-Archimedean [[absolute value]] ${\vert -\vert}$ one defines

* its **ring of integers** to be

  $$
    k^\circ := \{x \in k \,|\, {\vert x\vert} \leq 1\}
    \,.
  $$

This is a [[local ring]] with maximal ideal

$$
  k^{\circ\circ} := \{x \in k \,|\, {\vert x\vert} \lt 1\}
  \,.
$$

* The **residue field** of $k$ is the [[quotient]]

  $$
    \tilde k := k^\circ / k^{\circ \circ}
    \,.
  $$

## Examples

Archimedean valued fields include

* [[real numbers]]
* [[complex numbers]]

Non-Archimedean valued fields include

* [[p-adic numbers]].

## Related concepts

* [[archimedean field]]

* [[absolute value]]

* [[non-Archimedean analytic geometry]]

[[!redirects archimedean valued field]]
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[[!redirects archimedean-valued fields]]
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[[!redirects archimedean valuation]]
[[!redirects archimedean valuations]]

[[!redirects non-archimedean valued field]]
[[!redirects non-archimedean valued fields]]
[[!redirects non-Archimedean valued field]]
[[!redirects non-Archimedean valued fields]]
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Archimedes

* [Wikiedpia entry](https://en.wikipedia.org/wiki/Archimedes)

## related $n$Lab entries

* [[pi]]

category: people


The "Archive for Mathematical Sciences & Philosophy"

* [webpage](http://www.archmathsci.org)

run by [[Michael Wright]] claims to have a considerable collection of recordings (video, also pure audio) of talks on [[topos theory]] and [[category theory]] , such as notably talks and private discussion by [[William Lawvere]].

But few or none of these are publically available.

Michael Wright actively keeps traveling around to attend conferences and other meetings on topos theory and category theory, video recording talks. He says he is looking for funding that would eventually allow to make his archived recordings publically available.

category: reference

[[!redirects Archive for Mathematical Sciences and Philosophy]]


Archive of Formal Proofs

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## Idea

The _Archive of Formal Proofs_ (*AFP*) is an online library of [[formal proofs]], examples, and projects formalized in the [[proof assistant]] [[Isabelle]].

In contrast to other current projects, AFP and Isabelle are based on classical [[set theory|set theoretic]] [[foundation of mathematics|foundations]].
It already comprises a large amount of concepts like [smooth manifolds](https://www.isa-afp.org/entries/Smooth_Manifolds.html), [Markov models](https://www.isa-afp.org/entries/Markov_Models.html), [Monoidal Categories](https://www.isa-afp.org/entries/MonoidalCategory.html), or [ordinals and cardinals](https://www.isa-afp.org/entries/Ordinals_and_Cardinals.html) to mention a few.

## Related projects

* [[Xena project]]

* [[ForMath project]]

* [[The QED project]]

* [[UniMath project]]

## References

* [project page](https://www.isa-afp.org/)

[[!redirects AFP]]


arcsin

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## Idea

In [[trigonometry]] the arcsine or "inverse sine" is one of the [[inverse trigonometric functions]].

## Definition

The arcsine is a [[section]] (or [[right inverse]]) to the sine function [[sin]], which in turn is one of the [[trigonometric functions]]. More precisely, it is the unique [[continuous function|continuous]] section of the [[epimorphism]] in the [[(epi, mono) factorization]] of the sine such that its value at $0$ is $0$ (see [[pi]]).

## Related concepts

* [[sin]]

* [[arctan]]

* [[pi]]

## References

See also

* Wikipedia, _Inverse trigonometric functions_

[[!redirects arcsine]]


arctangent

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## Idea

In [[trigonometry]], the _arctangent_ or _inverse tangent_ is one of the basic [[trigonometric functions]].  It is the unique [[continuous function|continuous]] [[section]] of the [[tangent function]] whose value at $0 \in \mathbb{R}$ is $0$, and it is an [[analytic function]] on the [[real line]].

## Related concepts

* [[trigonometric identity]]

## References

* Wikipedia, _[Inverse trigonometric function](https://en.wikipedia.org/wiki/Inverse_trigonometric_functions)_

[[!redirects inverse tangent function]]
[[!redirects inverse tangent functions]]

[[!redirects arctangent]]
[[!redirects arctangents]]

[[!redirects arctan]]
[[!redirects artan]]
[[!redirects atan]]
[[!redirects tan⁻¹]]


area

#Contents#
* table of contents
{:toc}

## Idea

The [[volume]] of a [[surface]].

## Definition

### Of polygons

Let $\mathrm{Polygons}$ be the set of all [[polygons]] in the [[Euclidean plane]] $\mathbb{R}^2$. Then the area is a function $A:\mathrm{Polygons} \to \mathbb{R}$ such that for all polygons $P \in \mathrm{Polygons}$,

* $A$ is invariant under [[translations]]:

* Given a [[linear transformation]] $L$ and a polygon $P$, $A(L P) = \det(L) A(P)$

* Given two vertices $p$ and $q$ of $P$, ...

### In terms of Jordan content

Given a [[large set]] $M$ of [[Jordan content|Jordan-measurable]] subsets of $\mathbb{R}^2$ bounded by a [[Jordan curve]] called shapes, the **area** of a shape $S \in M$ is the [[Jordan content]] of $S$.

## Related concepts

* [[Jordan content]]

* [[length]]

* [[volume]]

* [[surface]]

* [[plane]]

## See also

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

## References

* [[Frank Quinn]], *Proof Projects for Teachers* ([pdf](https://personal.math.vt.edu/fquinn/education/pfs4teachers0.pdf))

* Apostol, Tom (1967). Calculus. Vol. I: One-Variable Calculus, with an Introduction to Linear Algebra. pp. 58–59. ISBN 9780471000051.

* Moise, Edwin (1963). Elementary Geometry from an Advanced Standpoint. Addison-Wesley Pub. Co. ([web](https://archive.org/details/elementarygeomet0000mois))

[[!redirects areas]]