CP^N sigma-model

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## Idea

By "$\mathbb{C}P^N$-models" one refers to ([[TQFT|topological]]) [[sigma-models]] --- often ([[topological strings|topological]]) [[string theories]] if the [[worldvolume]] is 2-dimensional --- with [[target space]] a [[complex projective space]] $\mathbb{C}P^N$ such as the [[Riemann sphere]] $\mathbb{C}P^1$.

> [NFJKK22](#NakayamaFunckeJansenKaoKühn22): "The $CP(N-1)$ models in 1+1 dimensions share many
properties with [[QCD]] in 3+1 dimensions, among them [[confinement]], [[asymptotic freedom]], [[instantons]], a [[large-N limit|$1/N$ expansion]], a topological charge, and a [[theta angle|θ-term]]. Thus, they serve as benchmark models for developing and testing both new numerical techniques and new proposed solutions to open questions of QCD. Both the $CP(N-1)$ models and QCD contain many [[nonperturbative phenomena]], which cannot be addressed
with conventional [[lattice field theory|lattice]] techniques."

## References

### $\mathbb{C}P^N$-models

The general notion of [[sigma models]] with [[target space]] a [[sphere]] or a [[complex projective space]] $\mathbb{C}P^n$:

* Golo, Perelomov, *Solution of the duality equations for the two-dimensional $SU(N)$-invariant chiral model*, Physics Letters B **79** 1–2 (1978) 112-113 [doi:10.1016/0370-2693(78)90447-1]

* [[Harald Eichenherr]], *$SU(N)$ invariant non-linear σ models*, Nuclear Physics B **146** 1 (1978) 215-223 [doi:10.1016/0550-3213(78)90439-X, errata:doi:10.1016/0550-3213(79)90287-6]

* [[Harald Eichenherr]], Michael Forger, *Higher local conservation laws for nonlinear sigma models on symmetric spaces*, Commun. Math. Phys. **82** (1981) 227–255 [[doi:10.1007/BF02099918](https://doi.org/10.1007/BF02099918)]

* [[Edward Witten]], §3 of: *Instatons, the quark model, and the $1/N$ expansion*, Nuclear Physics B **149** 2 (1979) 285-320 [doi:10.1016/0550-3213(79)90243-8

The [[lattice field theory]]-formulation:

* [[Paolo Di Vecchia]], A. Holtkamp, R. Musto, F. Nicodemi, R. Pettorini, *Lattice $CP^{N-1}$ models and their large-$N$ behaviour*, Nuclear Physics B **190** 4 (1981) 719-733 [doi:10.1016/0550-3213(81)90047-X]

and its [[supersymmetry|supersymmetric]] version:

* Christian Wozar, §6 of: *Low-dimensional supersymmetric field theories on the lattice* (1981) [[pdf](https://www.tpi.uni-jena.de/~wipf/abschlussarbeiten/wozar_diss.pdf)]

On the [[quantum cohomology]] inducd by the $\mathbb{C}P^N$-model:

* [[Sergio Cecotti]], [[Cumrun Vafa]], *Exact Results for Supersymmetric Sigma Models*, Phys. Rev. Lett. **68** (1992) 903-906 [[arXiv:hep-th/9111016](https://arxiv.org/abs/hep-th/9111016), [doi:10.1103/PhysRevLett.68.903](https://doi.org/10.1103/PhysRevLett.68.903)]

* {#CecottiVafa93} [[Sergio Cecotti]], [[Cumrun Vafa]], *On Classification of $N=2$ Supersymmetric Theories*, Comm. Math. Phys. **158** (1993) 569-644 [[arXiv:hep-th/9211097](https://arxiv.org/abs/hep-th/9211097), [doi:10.1007/BF02096804](https://doi.org/10.1007/BF02096804)]

* {#BourdeauDouglas94} [[M. F. Bourdeau]], [[Michael R. Douglas]], *Topological-Antitopological Fusion and the Large $N$ $\mathbb{C}P^N$ Model*, Nucl. Phys. B **420** (1994) 243-267 [[doi:hep-th/9312095](https://arxiv.org/abs/hep-th/9312095), doi:10.1016/0550-3213(94)90380-8]

On the [[Nicolai map]] for the $\mathbb{C}P^n$ sigma-model:

* [[Lorenzo Casarin]], [[Olaf Lechtenfeld]], [[Maximilian Rupprecht]], *Nicolai maps with four-fermion interactions* [[arXiv:2310.19946](https://arxiv.org/abs/2310.19946)]

See also:

* G. Sumbatian, E. Ievlev, A. Yung, *Large-N Solution and Effective Action of "Twisted-Mass" Deformed $\mathbb{C}P(N-1)$ Model* [[arXiv:2312.12569](https://arxiv.org/abs/2312.12569)]

### The $\mathbb{C}P^1$-model

* [[Edward Witten]], *On the structure of the topological phase of two-dimensional gravity*, Nuclear Physics B
**340** 2–3 (1990) 281-332 [doi:10.1016/0550-3213(90)90449-N]

* [[Tohru Eguchi]], Sung-Kil Yang, *The Topological $C P^1$ Model and the Large-$N$ Matrix Integral*, Mod. Phys. Lett. A **9** (1994) 2893-2902 [[arXiv:hep-th/9407134](https://arxiv.org/abs/hep-th/9407134), [doi:10.1142/S0217732394002732](https://doi.org/10.1142/S0217732394002732)]

* [[Shmuel Elitzur]], A. Forge, [[Eliezer Rabinovici]], *On effective theories of topological strings*, Nuclear Physics B **388** 1 (1992) 131-155 [doi:10.1016/0550-3213(92)90548-P]

* [[Shmuel Elitzur]], [[Yaron Oz]], [[Eliezer Rabinovici]], [[Johannes Walcher]], *Open/Closed Topological $\mathbb{C}P^1$ Sigma Model Revisited*,  J. High Energ. Phys. **2012** 101 (2012) [[arXiv:1106.2967](https://arxiv.org/abs/1106.2967)]

* [[Rajesh Gopakumar]], *What is the Simplest Gauge-String Duality?* [[arXiv:1104.2386](https://arxiv.org/abs/1104.2386)]

* [[Rajesh Gopakumar]], Roji Pius, *Correlators in the Simplest Gauge-String Duality* [[arXiv:1212.1236](https://arxiv.org/abs/1212.1236)]

* Lwazi Nkumane, *The Simplest Gauge-String Duality* (2015) [[pdf](https://core.ac.uk/download/pdf/188771529.pdf)]

* R. Banerjee, *Quantum equivalence of $O(3)$ nonlinear σ model and the $CP^1$ model: A gauge-independent Hamiltonian approach*, Phys. Rev. D **49** 2133 (1994) [[doi:10.1103/PhysRevD.49.2133](https://doi.org/10.1103/PhysRevD.49.2133)]

* {#NakayamaFunckeJansenKaoKühn22} Katsumasa Nakayama, Lena Funcke, Karl Jansen, Ying-Jer Kao, Stefan Kühn, *Phase structure of the $CP(1)$ model in the presence of a topological $\theta$-term*, Phys. Rev. D **105** 054507 (2022) [[arXiv:2107.14220](https://arxiv.org/abs/2107.14220), [doi:10.1103/PhysRevD.105.054507](https://doi.org/10.1103/PhysRevD.105.054507)]

### Flag manifold sigma-models

More generally, on sigma models with [[flag manifold]] [[target spaces]] and relation to [[Gross-Neveu models]]:

* [[Dmitri Bykov]], *Quantum flag manifold σ-models and Hermitian Ricci flow*, Commun. Math. Phys. **401** 1-32 (2023) [[arXiv:2006.14124](https://arxiv.org/abs/2006.14124), [doi:10.1007/s00220-022-04532-5](https://doi.org/10.1007/s00220-022-04532-5)]

* [[Ian Affleck]], [[Dmitri Bykov]], [[Kyle Wamer]], *Flag manifold sigma models: spin chains and integrable theories*,  Phys. Rept. **953** (2022) 1-93 [[arXiv:2101.11638](https://arxiv.org/abs/2101.11638), [doi:10.1016/j.physrep.2021.09.004](https://doi.org/10.1016/j.physrep.2021.09.004)]

See also:

* [[Dmitri Bykov]], Anton Pribytok, *Supersymmetric deformation of the $\mathbb{C}P^1$ model and its conformal limits* [[arXiv:2312.16396](https://arxiv.org/abs/2312.16396)]

[[!include quantum cohomology as Pontrjagin rings -- references]]

[[!redirects CP^n sigma-models]]

[[!redirects CP^1 sigma-model]]
[[!redirects CP^1 sigma-models]]

[[!redirects flag manifold sigma-model]]
[[!redirects flag manifold sigma-models]]


cpo

# cpos
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## Definition

The abbreviation __cpo__ stands for "complete partial order",

(beware that it does *not* generally refer to a [[partial order]] that is [[complete category|complete]] in the sense that the latter word is [[complete category|used]] in [[category theory]]: A partial order that is [[complete category|complete as a category]] is instead generally called a *[[complete lattice]]*.)

Instead, a "cpo" generally refers to either a poset that has [[suprema]] for [[directed set|directed]] subsets (a.k.a. a __dcpo__) or for [[chains]] indexed by the [[ordinal]] $\omega$ (a.k.a. an $\omega$-**cpo** --- a weaker condition).  Note also that suprema are [[colimits]] in a poset rather than [[limits]], so this is actually a [[cocomplete category|cocompleteness]] condition rather than a completeness one.

The [[morphisms]] between cpos are usually taken to be the the [[Scott-continuous function]]s ([[continuous functions]] with respect to the [[Scott topology]]).  Cpos and Scott-continuous maps form a category $CPO$.

## Applications

Cpos are important in [[domain theory]].

## References

* [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and concrete categories: the joy of cats_. [free online](http://katmat.math.uni-bremen.de/acc/acc.pdf)

[[!redirects cpo]]
[[!redirects cpos]]
[[!redirects CPO]]


CR manifold

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## Definition

A _CR manifold_ consists of a differentiable manifold $M$ together with a subbundle $L$ of the complexified tangent bundle, $L \subset TM \otimes_\mathbf{R} \mathbf{C}$ such that $[L, L ] \subset L$ and $L \cap\overline{L} =\{ 0 \} $.

### As first-order integrable $G$-structure

CR manifold structure are equivalently certain [[first-order integrable G-structures]] ([Dragomi-Tomassini 06, section 1.6](#DragomiTomassini06)), a type of [[parabolic geometry]].

### Examples

## Properties

### Relation to solutions in supergravity

A close [[analogy]] between CR geometry and [[supergravity]] [[superspacetimes]] (as both being [[torsion of a G-structure|torsion-ful]] [[integrable G-structures]]) is pointed out in ([Lott 01 exposition (4.2)](torsion+constraints+in+supergravity#Lott01)).

## Other Clifford-type Hypersurfaces

* [[HR manifold]]

* [[BR manifold]]

## References

The original article is

* [[Shiing-shen Chern]], [[Jürgen Moser]], _Real hypersurfaces in complex manifolds_, Acta Math. 133 (1975), 219–271 ([MathNet](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=7161&option_lang=eng))

Surveys:

* {#Jacobowitz90}  Howard J. Jacobowitz, _An Introduction to CR Structures_, Mathematical Surveys and Monographs Volume: 32; (1990) ([ISBN:978-0-8218-1533-5](https://bookstore.ams.org/surv-32), [doi:10.1090/surv/032](http://dx.doi.org/10.1090/surv/032))

* {#DragomiTomassini06} [[Sorin Dragomir]], Giuseppe Tomassini, _Differential Geometry and Analysis on CR Manifolds_, Birkhäuser, 2006 ([doi:10.1007/0-8176-4483-0](https://link.springer.com/book/10.1007/0-8176-4483-0))

See also

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

Discussion of [[orbifolds]] with CR-structure:

* [[Sorin Dragomir]], Jun Masamune, _Cauchy-Riemann orbifolds_, Tsukuba J. Math. Volume 26, Number 2 (2002), 351-386 ([euclid:tkbjm/1496164430](https://projecteuclid.org/euclid.tkbjm/1496164430))

Discussion from the point of view of [[Cartan geometry]]/[[parabolic geometry]] includes

* [[Felipe Leitner]], section I.10 of _Applications of Cartan and Tractor Calculus to Conformal and CR-Geometry_, 2007 ([pdf](http://elib.uni-stuttgart.de/opus/volltexte/2009/3922/pdf/HabilKopf.pdf))

Discussion of spherical CR manifolds locally modeled on the [[Heisenberg group]] is in:

* Robert R. Miner, _Quasiconformal equivalence of spherical manifolds_, Annales Academiae Scientiarium Fennicae, Series A. I. Mathematica, Volumen 19, 1994, 83-93 ([pdf](http://www.acadsci.fi/mathematica/Vol19/miner.pdf))

[[!redirects CR manifolds]]

[[!redirects CR-manifold]]
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[[!redirects spherical CR manifold]]
[[!redirects spherical CR manifolds]]

[[!redirects CR-orbifold]]
[[!redirects CR-orbifolds]]
[[!redirects CR orbifold]]
[[!redirects CR orbifolds]]

[[!redirects CR geometry]]
[[!redirects CR geometries]]
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CR structure

see _[[CR-manifold]]_

[[!redirects CR structures]]


Craig interpolation theorem

[[!redirects Craig's interpolation theorem]]
[[!redirects Craig interpolation lemma]]
[[!redirects interpolation theorem]]

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##Idea
The _Craig interpolation theorem_ is a basic result in the [[model theory]] of [[first-order logic]].

First proved by W. Craig for classical predicate calculus in 1957, it has been extended to [[intuitionistic logic]] by K. Schütte in 1962.
Andrew M. Pitts and [[Michael Makkai]] have proved variants in categorical logic e.g. for [[pretoposes]]. Diaconescu proved an abstract variant in _institution-independent model theory_ (cf. Augier-Barbier(2007), Diaconescu(2008)).

**Craig interpolation theorem.** Let $\varphi_1,\varphi_2$ be first-order sentences over the vocabularies $\mathcal{L}_1,\mathcal{L}_2$ such that $\varphi_1\models\varphi_2$. Then there is a sentence $\theta$ over the common vocabulary $\mathcal{L}_1\cap\mathcal{L}_2$, the _interpolant_, such that $\varphi_1\models\theta$ and $\theta\models\varphi_2$.

An important corollary is the [[Beth definability theorem]].

The theorem plays an important role in [[abstract model theory]] where the question whether the Craig interpolation property can take over the role of the [[Löwenheim-Skolem theorem]] in [[Lindström's theorems]], has been called 'one of the main open problems of the field' by Väänänen (2008, p.404).

## Related Concepts

* [[definability]]
* [[first-order logic]]
* [[Beth definability theorem]]
* [[Löwenheim-Skolem theorem]]
* [[Robinson consistency theorem]]
* [[conceptual completeness]]
* [[abstract model theory]]
* [[descent theory]]

##References

The original source is

* William Craig, _Linear Reasoning. A new form of the Herbrand-Gentzen theorem_ , JSL *22* (1957) pp.250-268.

* William Craig, _Three uses of the Herbrand-Gentzen Theorem in relating model theory and proof theory_, JSL **22** (1957) pp.269-285.

For an account in a standard reference see:

* C. C. Chang, H. J. Keisler, _Model Theory_ , North-Holland Amsterdam 1973.

A textbook account that covers the classical as well as the intuitionistic case is in:

* [[John Bell]], Moshé Machover, _A Course in Mathematical Logic_ , North-Holland Amsterdam 1977.

For a proof that uses Keisler's ultrapower theorem (Roughly: $\mathcal{A}\equiv\mathcal{B}$ iff $\mathcal{A}^I/F\cong\mathcal{B}^I/G$ for some ultrafilters $F,G$ on I) see:

* [[John Bell]], A. B. Slomson, _Models and Ultraproducts: An Introduction_ , North-Holland Amsterdam 1969.

The bible on interpolation in non-classical logic is:

* Dov Gabbay, Larisa Maksimova, _Interpolation and Definability - Modal and Intuitionistic Logics_ , Oxford UP 2005.

The following provides background on proofs of Craig's theorem and categorical logic in a gentle way:

* [[Michael Makkai]], _On Gabbay's Proof of the Craig Interpolation Theorem for Intuitionistic Predicate Logic_, Notre Dame J. Form. Logic **36** no. 3 (1995) pp.364-381. ([pdf](http://projecteuclid.org/download/pdf_1/euclid.ndjfl/1040149353))

* M. Aiguier, F. Barbier, _An institution-independent Proof of the Beth Definability Theorem_ , Studia Logica **85** no. 3 (2007) pp.333-359. ([pdf](http://perso.ecp.fr/~aiguierm/publications/communications/sl07.pdf))

* Razvan Diaconescu, _Institution-indepent Model Theory_ , Birkhäuser Basel 2008.

* Andrew M. Pitts, _Amalgamation and Interpolation in the Category of Heyting algebras_ , JPAA **29** (1983) pp.155-165.

* Andrew M. Pitts, _An Application of Open Maps to Categorical Logic_ , JPAA **29** (1983) pp.313-326.

* [[Jouko Väänänen]], _The Craig Interpolation Theorem and abstract model  theory_ , Synthese  164:401 (2008) (freely available online)


Craig Laughton

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=128370)

## Selected writings

On [[quasitoric manifold|quasitoric manifolds]], [[MSp]] and [[quaternionic oriented cohomology theory]]:

* {#Laughton08} [[Craig Laughton]], _Quasitoric manifolds and cobordism theory_, Manchester 2008 ([pdf](http://www.ma.man.ac.uk/~nige/craiglphd.pdf), [[LaughtonQuasitoric.pdf:file]])

category: people


Craig Roberts

* [personal page](https://sites.google.com/view/cdroberts/home)

* [institute page](https://inp.nju.edu.cn/Members/Staff/20191017/i43996.html)

## Selected writings

On [[confinement]] and [[hadron]]-mass generation in [[QCD]] ([[mass gap problem]]):

* {#RobertsSchmidt20} [[Craig Roberts]], [[Sebastian M. Schmidt]], _Reflections upon the Emergence of Hadronic Mass_, The European Physical Journal Special Topics volume 229, pages 3319–3340 (2020) ([arXiv:2006.08782](https://arxiv.org/abs/2006.08782), [doi:10.1140/epjst/e2020-000064-6](https://doi.org/10.1140/epjst/e2020-000064-6))

> More than 98% of visible mass is contained within [[atomic nuclei|nuclei]].  In first approximation, their atomic weights are simply the sum of the masses of all the [[neutrons]] and [[protons]] ([[nucleons]]) they contain.  Each nucleon has a [[mass]] $m_N \sim 1$ [[GeV]], i.e. approximately 2000-times the [[electron]] mass. The [[Higgs  boson]] produces the latter, but  what produces the masses of the neutron and proton?  This is the question posed above, which is pivotal to the development of modern physics: how can science explain the emergence of [[hadron|hadronic]] mass (EHM)? $[\cdots]$

> Modern science is thus encumbered with the fundamental problem of [[gluon]]  and [[quark]] [[confinement]]; and confinement is crucial because it ensures absolute stability of the [[proton]].  $[\cdots]$ Without confinement,our Universe cannot exist.

> As the 21st Century began, the Clay Mathematics Institute established seven Millennium Prize Problems [ [11](mass+gap#ReferencesMassGapProblem) ]. Each represents one of the toughest challenges in mathematics. The set contains the [[mass gap problem|problem of confinement]]; and  presenting a sound solution will win its discoverer 1,000,000 bucks. Even with  such motivation, today, almost fifty years after the discovery of [[quarks]] [12–14], no rigorous solution has been found. Confinement and EHM are inextricably linked. Consequently, as  science plans for the next thirty  years, solving the problem of EHM has become a _grand challenge_. $[\cdots]$

> In trying to match [[QCD]] with Nature, one confronts the many complexities of strong, nonlinear dynamics in relativistic quantum field theory, e.g. the loss of particle number conservation, the frame and scale dependence of the  explanations and interpretations of observable processes, and the evolving character of the relevant degrees-of-freedom. [[electroweak field|Electroweak theory]] and phenomena are essentially [[perturbative quantum field theory|perturbative]]; hence, possess little of this complexity.  Science has never before encountered an [[interaction]] such as that at work in [[QCD]]. Understanding this interaction, explaining everything of which it is capable, can potentially change the way we look at the Universe.

* [[Craig Roberts]], _On Mass and Matter_, AAPPS Bulletin volume 31, Article number: 6 (2021) ([arXiv:2101.08340](https://arxiv.org/abs/2101.08340), [doi:10.1007/s43673-021-00005-4](https://doi.org/10.1007/s43673-021-00005-4))

* Minghui Ding, [[Craig D. Roberts]], Sebastian M. Schmidt, *Emergence of Hadron Mass and Structure* $[$[arXiv:2211.07763](https://arxiv.org/abs/2211.07763)$]$

* [[Craig D. Roberts]], *Origin of the Proton Mass* $[$[arXiv:2211.09905](https://arxiv.org/abs/2211.09905)$]$

category: people

[[!redirects Craig D. Roberts]]


Craig Westerland

* [webpage](http://www.ms.unimelb.edu.au/~craigw/)

## Selected writings

On the [[J-homomorphism]]:

* {#Westerland14} [[Craig Westerland]], _Views on the J-homomorphism_ MSRI talk, 2014 ([recording](https://mathinstitutes.org/videos/videos/1464), [pdf](https://docs.google.com/viewer?url=https://www.msri.org/workshops/685/schedules/17895/documents/1978/assets/20102), [[WesterlandJHomomorphism.pdf:file]])

and in higher [[chromatic homotopy theory]]:

* [[Craig Westerland]], _A higher chromatic analogue of the image of J_, Geometry & Topology 21 (2017) 1033–1093 ([doi:10.2140/gt.2017.21.1033](https://doi.org/10.2140/gt.2017.21.1033))

On [[spherical T-duality]] in [[iterated algebraic K-theory]]:

* {#LindSatiWesterland16} [[John Lind]], [[Hisham Sati]], [[Craig Westerland]], _A higher categorical analogue of topological T-duality for sphere bundles_, Annals of K-Theory, Vol. 5 (2020), No. 1, 1–42 ([arXiv:1601.06285](http://arxiv.org/abs/1601.06285), [doi:doi:10.2140/akt.2020.5.1](https://msp.org/akt/2020/5-1/p01.xhtml))

## Related $n$Lab pages

* [[E-theory]]

* [[spherical T-duality]]

category: people


Cramer-Rao inequality

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## Overview

An [[inequality]] in [[statistics]] giving a lower bound for the  variance of a parameter estimator.

## Related entries

* [[statistics]]

* [[information geometry]]

## Literature

* wikipedia/[Cramér–Rao bound](https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound)
* Encyclopaedia of Math, [Rao-Cramér_inequality](https://encyclopediaofmath.org/wiki/Rao-Cram%C3%A9r_inequality)

[[!redirects Cramér-Rao inequality]]
[[!redirects Rao-Cramér inequality]]


Crane-Yetter model

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## Idea ##
The Crane-Yetter model is a [[4d TQFT]], not to be confused with the [[Yetter model]].

It was defined originally as a state sum in [State-Sum Invariants of 4-Manifolds](#CraneYetterKauffman94).
One labels the elements of a triangulation with objects and morphisms of a [[ribbon category|ribbon]] [[fusion category]], usually the [[representation category]] of a [[quasitriangular Hopf algebra]], or [[quantum group]].

(The [[Yetter model]], on the contrary, is defined for a finite [[2-group]].)

## Relation to Turaev-Viro model on the boundary ##

The [[3d TQFT]] [[Turaev-Viro model]] is a [[boundary field theory]] of the Crane-Yetter model ([Barrett&Garcia-Islas&Martins 04, theorem 2](#BarrettGarciIslasMartins04)).

### As a fully extended TQFT ###
It can be understood as an [[extended TQFT|extended topological quantum field theory]], see [[4-3-2 8-7-6]].

## Relation to BF-theory and Dijkgraaf-Witten theory ##

It is expected that the classical limit of the Crane-Yetter TQFT is [[BF theory]].
See [John Baez's article](#Baez95).
A state sum model like the Crane-Yetter model, but with a classical Lie group, would be the [[Ooguri model]].
We could understand the Crane-Yetter as a [[Dijkgraaf-Witten theory]] with a quantum group instead of a classical group, whereas the Yetter model is a Dijkgraaf-Witten theory with a 2-group.
See also [this blog post](https://golem.ph.utexas.edu/category/2006/08/categorifying_the_dijkgraafwit.html).

## References

* {#CraneYetterKauffman94} [[Louis Crane]], [[Louis Kauffman]], [[David Yetter]], _State-Sum invariants of 4-manifolds I_, Journal of Knot Theory and Its Ramifications Vol. 06, No. 02, pp. 177-234 (1997)  ([arXiv:hep-th/9409167](https://arxiv.org/abs/hep-th/9409167), [pdf](http://arxiv.org/PS_cache/hep-th/pdf/9409/9409167v1.pdf), [doi:10.1142/S0218216597000145](https://doi.org/10.1142/S0218216597000145))

Relation to the [[Turaev-Viro model]]:

* {#BarrettGarciaIslasMartins04} [[John Barrett]], J. Garcia-Islas, [[João Faria Martins]], _Observables in the Turaev-Viro and Crane-Yetter models_, J. Math. Phys. 48:093508, 2007 ([arXiv:math/0411281](http://arxiv.org/abs/math/0411281))

See also:

* {#Baez95} [[John Baez]], _4-Dimensional BF Theory as a Topological Quantum Field Theory_, [arXiv:q-alg/9507006](http://arxiv.org/abs/q-alg/9507006).

[[!redirects Crane-Yetter models]]


Crans-Gray tensor product

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## Idea

The _Crans-Gray tensor product_ is a [[tensor product]] on the category of [[strict omega-category|strict omega-categories]] which is analogous to (the lax version of) the [[Gray tensor product]] for 2-categories.

This tensor product makes the category $Str\omega Cat$ of [[strict omega-category|strict omega-categories]] into a (non-symmetric) [[closed monoidal category|biclosed monoidal structure]]; thus in particular it has two internal-homs $Hom_r$ and $Hom_l$.  Both contain strict $\omega$-functors as their objects, and their $k$-cells for $k\gt 0$ are [[k-transfors]] which are lax or oplax in all dimensions (one internal-hom contains lax transfors and the other the oplax ones).

## Construction from cubical sets

One abstract way to construct the Crans-Gray tensor product is as follows.

There is an obvious [[monoidal category|monoidal structure]] on the [[cube category]] obtained from the obvious product of cellular [[geometric shapes for higher structures|cubes]] which is analogous to the cartesian product of the topological cubes $[0,1]^n$.  By [[Day convolution]] this naturally induces a [[monoidal category|monoidal structure]] on [[cubical set|cubical sets]].

Now strict $\omega$-categories, although usually defined as [[globular set|globular sets]] with extra structure, can also be regarded as [[cubical set|cubical sets]] with extra structure.  More precisely, there is a forgetful functor from $Str\omega Cat$ to $CubSet$ which is [[monadic functor|monadic]], and the monad is a [[monoidal monad]].  It follows by further general results of Day that the tensor product on $CubSet$ lifts to $Str\omega Cat$.

It is also possible to give a direct construction of this tensor product; see Crans' thesis below.

## Relation to the Gray tensor product

Strict $n$-categories for $n\lt\omega$ can of course be regarded as strict $\omega$-categories with all $k$-cells identities for $k\gt n$.   In general, the Crans-Gray tensor product of an $n$-category with an $m$-category will be an $(n+m)$-category, since the tensor product of the $n$-cube and the $m$-cube is the $(n+m)$-cube.  (Compare, for example, the fact that the [[Gray tensor product]] of two 1-categories is no longer, in general, a 1-category.)

However, we can obtain a tensor product on strict $n$-categories for any $n$ by applying the [[reflector]] (or "cotruncation") functor for the inclusion of strict $n$-categories into strict $\omega$-categories.  When $n=1$, this produces the cartesian product of categories, while when $n=2$ it reproduces the lax version of the [[Gray tensor product]].

Sjoerd Crans has also defined a tensor product *of* [[Gray-categories]], but the relationship between it and the tensor product of strict $\omega$-categories is not entirely clear, since neither $Str\omega Cat$ nor $GrayCat$ contains the other (their "intersection" is $Str3Cat$).

## Properties

The Crans-Gray tensor product makes $Str\omega Cat$ into a bi[[closed monoidal category]].  So in particular for $X, Y \in Str\omega Cat$ two strict $\omega$-categories, there is an $\omega$-[[functor category]] between them defined by

$$
   [X,Y] = Str\omega Cat( X \otimes G^\bullet, Y)
  \,,
$$

where $\otimes$ is the CG-tensor product and $G^\bullet : G \to \omega Cat$ is the  [[globular set|globular]] object of standard [[globe]]s.  The objects in $[X,Y]$ are the (strict) $\omega$-functors, while the [[k-morphism]]s are a [[lax natural transformation|lax]] sort of $k$-[[transfor]]s between these.  A dual $\omega$-functor category can be defined by $\langle X,Y\rangle = Str\omega Cat(G^{\bullet}\otimes X, Y)$; this has the same objects but its k-morphisms are oplax $k$-transfors.

+--{: .query}
Or maybe lax and oplax should be switched here?  Can someone verify?
=--

## Further Remarks

* The Crans-Gray tensor product extends the tensor product  on strict $\omega$-groupoids given by R. Brown and P.J. Higgins (see below); they construct a symmetric monoidal closed structure on these $\omega$-groupoids. It is used there to construct a monoidal closed structure on the category of [[crossed complex]]es.

* The article [Al-Agl/Brown/Steiner](#Al-Agl/Brown/Steiner) sets up an equivalence between strict globular $\omega$-categories and certain cubical $\omega$-categories with [[connection]]s, to which the monoidal closed structure in the previous paper is easily extended.

* The article [Steiner](#Steiner) gives a new construction of this tensor product in terms of augmented directed chain complexes, where he shows that it can be defined explicitly and computed efficiently on all strict ω-categories admitting a certain kind of loop-freeness assumption on their bases.  In particular, it generalizes the approaches in all earlier papers.

* There is the [[Verity-Gray tensor product]] on [[stratified simplicial set]]s (as described there). Via the [[oriental|omega-nerve]] [[strict omega-category|strict omega-categories]] form the [[complicial set]]s within all [[stratified simplicial set]]s. According to observation 60 of [Verity06](http://arxiv.org/abs/math/0604414) section 11.4 of [Verity04](http://arxiv.org/abs/math/0410412) proves that the Crans-Gray tensor product is the reflection of the Verity-Gray tensor product under this inclusion.

## References

A detailed discussion is in

* [[Sjoerd Crans]], _Pasting schemes for the monoidal biclosed structure on $\omega$-Cat_ ([ftp, gzipped PostScript](ftp://ftp.math.mcgill.ca/pub/crans/thten.ps.gz))

* {#BrownHiggins} [[Ronnie Brown]] and Phil Higgins, _Tensor products and homotopies for $\omega$-groupoids and crossed complexes_ ,  J. Pure Appl. Alg. 47 (1987) 1--33. [PDF](http://www.sciencedirect.com/science/article/pii/0022404987900995)

* F.A. Al-Agl, R. Brown and R. Steiner,  Multiple categories: the equivalence between a globular and cubical approach, Advances in Mathematics, 170 (2002) 71--118. [ArXiv](http://arxiv.org/abs/math/0007009){#Al-Agl/Brown/Steiner}

* R. Brown, P.J.Higgins, R. Sivera, _Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids_, EMS Tracts in Mathematics vol 15, 650 pages (autumn 2010 to appear). ([Nonabelian algebraic topology](http://www.groupoids.org.uk/nonab-a-t.html))

Some helpful remarks and diagrams are in

* [[Sjoerd Crans]], _A tensor product for $Gray$-categories_ ([tac](http://www.tac.mta.ca/tac/volumes/1999/n2/5-02abs.html))

which is however mainly concerned with a slightly different topic.

A collection of Crans's papers, including those on teisi, can be found here:

* [Papers of Sjoerd Crans](http://home.tiscali.nl/secrans/papers/) (the website seems down at the moment, but there is [an archived version](https://web.archive.org/web/20130514042310/http://home.tiscali.nl/secrans/papers/))

The modern version, incorporating many new primitives is given in

* [[Richard Steiner]], _Omega-categories and chain complexes_, Homology, Homotopy and Applications 6(1) (2004), 175-200 ([journal](http://www.intlpress.com/hha/v6/n1/a12/), [pdf](http://www.intlpress.com/hha/v6/n1/a12/pdf))
 {#Steiner}

A general theory of lax tensor products is in

* [[Michael Batanin]], [[Denis-Charles Cisinski]], [[Mark Weber]], _Multitensor lifting and strictly unital higher category theory_ ([arXiv:1209.2776](http://arxiv.org/abs/1209.2776))


created limit

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# Creation of limits
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## Definition

Let $F\colon C\to D$ be a [[functor]] and $J\colon I\to C$ a [[diagram]], and suppose that the composite $F \circ J$ has a [[limit]].  We say that $F$ **creates** this limit if $J$ has a limit, and $F$ both [[preserved limit|preserves]] and [[reflected limit|reflects]] limits of $J$.  The latter two conditions together mean that a [[cone]] over $J$ in $C$ is a limiting cone if and only if its image in $D$ is a limiting cone over $F\circ J$.

Of course, a functor $F$ creates a [[colimit]] if $F^{op}$ creates the corresponding limit.

If $F$ creates all limits or colimits of a given type (i.e. over a given category $I$), we simply say that $F$ creates that sort of limit (e.g. $F$ creates products, $F$ creates equalizers, etc.).

## Examples

\begin{proposition}\label{MonadicFunctorsCreateLimits}
**([[monadic functors]] [[created limit|create limits]])**
A [[monadic functor]]

1. creates all limits that exist in its [[codomain]];

1. creates all colimits that exist in its codomain and are preserved by the corresponding monad (or, equivalently, by the monadic functor itself).

\end{proposition}
(e.g. [MacLane 71, Exercise IV.2.2 (p. 138)](#MacLane71))

Creation of a particular sort of [[split coequalizer]] figures prominently in Beck's [[monadicity theorem]].

## Terminological remarks

### Creation of non-existing limits

It seems that the notion of "creating a limit" is used most frequently when the limits exist in the codomain.  One may want to extend the terminology to cases when such limits don't exist, which would require making a choice about whether a non-existing limit should be regarded as "created".

In [[Categories Work]] the convention is that a functor creates *all* limits that do not exist in its codomain.  In this case, the more generally applicable definition could be stated as "$F$ creates limits for $J$ if $J$ has a limit whenever $F\circ J$ has a limit, and *in that case* limits of $J$ are preserved and reflected by $F$."  (But see below for an additional difference with [[Categories Work]].)

On the other hand, one might argue that it doesn't make sense to regard a limit that exists in the domain as being "created by the functor" if the limit in the codomain doesn't even exist.  In this case the more generally applicable definition could be stated as "$F$ creates limits for $J$ if $J$ has a limit whenever $F\circ J$ has a limit, and *furthermore in all cases* limits of $J$ are preserved and reflected by $F$."

Finally, one might even argue that based on the meaning of the English word "created", only something that exists can be created at all.  In this case the more generally applicable definition could be stated as "$F$ creates limits for $J$ if $J$ and $F\circ J$ both have limits, and furthermore limits of $J$ are preserved and reflected by $F$."

### Strictness

The definitions given above are all "up to isomorphism", i.e. they satisfy the [[principle of equivalence]].  The definition in [[Categories Work]] is additionally *strict*: it requires that for every limiting cone $L$ over $F J$ in $D$ there exists a *unique* cone $L'$ over $J$ which is mapped *exactly* to $L$, and this $L'$ is a limit of $J$.  This is used in stating the version of the [[monadicity theorem]] that characterizes the [[category of algebras for a monad]] up to [[isomorphism]] rather than [[equivalence of categories]]. For [[amnestic functor|amnestic]] [[isofibrations]] the strict and the non-strict notion are equivalent.

## Remarks

[Kissinger](https://raw.githubusercontent.com/punkdit/categories/26b751bbcbe6765fe447e805629ff6416f4c38b1/gmane/science/mathematics/categories/6644) suggested a concise way to state creation/preservation/etc. of limits.  However, there is [some dispute](https://nforum.ncatlab.org/discussion/7024/lifted-limit/?Focus=56765#Comment_56765) about its correctness.

## Related pages

* [[preserved limit]]

* [[reflected limit]]

* **created limit**

* [[lifted limit]]

## References

* {#MacLane71} [[Saunders Mac Lane]], Definition V.1 in: _[[Categories for the Working Mathematician]]_ (1971)

* [[Jiří Adámek]], [[Horst Herrlich]], [[George E. Strecker]], Definition 13.17(2) in: _[[Abstract and Concrete Categories]]_.

* [[Emily Riehl]], §3.3 in: *[[Category Theory in Context]]*, Dover Publications (2017) [[pdf](https://math.jhu.edu/~eriehl/context.pdf), [book website](https://math.jhu.edu/~eriehl/context/)]

[[!redirects created limits]]
[[!redirects creation of limits]]
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[[!redirects created colimits]]
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[[!redirects creates colimits]]

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CRing

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#### Category theory
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#Contents#
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## Definition ##

$CRing$ is the category of commutative [[ring]]s and ring homomorphisms.

A commutative ring is a commutative [[monoid object]] in [[Ab]], so $CRing = CMon(Ab)$ is the [[category of commutative monoids]] in [[abelian groups]].

The [[opposite category]] $CRing^{op}$ is the category of [[affine schemes]].

## Properties

### Epi/Monomorphisms

Every [[surjective function|surjective]] [[homomorphism]] of commutative rings is an [[epimorphism]] in $CRing$, but not every epimorphism is surjective.

A counterexample is the defining inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ of the ring of [[integers]] into the ring of [[rational numbers]]. This is an injective epimorphism of rings.

For more see for instance at [[Stacks Project]], _[10.106 Epimorphisms of rings](http://stacks.math.columbia.edu/tag/04VM)_.

### Cocartesian co-monoidal structure
 {#CocartesianComnonoidalStructure}

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

The [[coproduct]] in $CRing$ is given by the underlying [[tensor product of abelian groups]], equipped with its canonically induced commutative ring structure.

=--

By [this general proposition](category+of+monoids#PushoutsInCMonGivenByTensorProduct) discussed at _[[category of commutative monoids]]_.

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

Prop. \ref{CoproductIsTensorProduct} means that tensor product of commutative rings exhibits [[cartesian monoidal category]] structure on the [[opposite category]] $CRing^{op}$.

=--

## Generalizations ##

* The [[slice category]] of $CRing$ under a ring $R$ is the category $R$[[CAlg]] of commutative [[associative algebra]]s over $R$.

* There is a "smooth" version of $CRing^{op}$: the category of [[smooth locus|smooth loci]].

* There is a [[higher category theory]] version of $CRing$: the $(\infty,1)$-[[(∞,1)-category|category]] of $E_\infty$-[[E-∞ ring|rings]].

## Related categories

* [[Ring]]

* [[Mod]]

## References

* [[Stacks Project]], _[10.106 Epimorphisms of rings](http://stacks.math.columbia.edu/tag/04VM)_

category: category

[[!redirects category of commutative rings]]
[[!redirects categories of commutative rings]]

[[!redirects CRings]]
[[!redirects Cring]]
[[!redirects Crings]]
[[!redirects CommutativeRing]]
[[!redirects CommutativeRings]]


Cristian Ortiz

* [webpage](http://people.ufpr.br/~cristian.ortiz/Cristian_Ortiz_Homepage/Welcome.html)

category: people


critical point

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#Contents#
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## Definition

For $f : X \to Y$ a [[smooth function]], a **critical point** is a point $x \in X$ at which the [[derivative]] $d f : T X \to T Y$ has [[rank]] strictly less than the [[dimension]] of $Y$.

The collection of all critical points is also called the **critical locus** of $f$.

The $f$-image of a critical point is known as a __[[critical value]]__.
A point in $Y$ that is not a [[critical value]] is known as a __[[regular value]]__.

A formalization of this in [[cohesive (infinity,1)-topos|cohesive geometry]] is at _[cohesive (infinity,1)-topos -- infinitesimal cohesion -- critical locus](cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#CriticalLocus)_.

## Applications

* Critical loci are used to study [[topology]] in terms of [[Morse theory]].

* Critical loci of [[functionals]] on [[jet bundles]] are studied in [[variational calculus]]. See also at _[[shell]]_.

## Related concepts

* [[Morse–Sard theorem]]

* [[extremum]]

* [[locus]]

* [[derived critical locus]]

## Rereferences

* {#Thom54} [[René Thom]], p. 18 of: _[[Quelques propriétés globales des variétés différentiables]]_, Comment. Math. Helv. 28, (1954). 17-86 ([doi:10.1007/BF02566923](https://doi.org/10.1007/BF02566923), [dml:139072](https://eudml.org/doc/139072), [digiz:GDZPPN002056259](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf))

[[!redirects critical locus]]

[[!redirects critical points]]
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criticism of string theory

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## Idea

The [[theory (physics)|theory]] of fundamental [[physics]] called _[[string theory]]_ has been the topic of many discussions, scientific discussion but also much discussion about the sociology of the higher energy physics community, concerned with the problem of how to do theoretical high energy physics "beyond the [[standard model of particle physics]]". While during the 1980s and 1990s the "public perception" of string theory, as expressed by mass media coverage and by scientific funding, had been dominantly enthusiastic with tendentially hysterical hopeful expectations, somewhere in the 2000s the "public perception" in mass media and blogs has gradually turned towards a gloomy attitude with tendentially hysterical pessimistic expectations.

This page is to collect pointers to some of the criticisms of the theory and/or the way it is treated in the higher energy physics community, as expressed in these discussions.

For general non-technical background information see also at _[[string theory FAQ]]_.

## References

The two popular books whose appearance in hindsight mark the change of the nature in the "public perception" of string theory are

* [[Peter Woit]], [Not Even Wrong](http://www.amazon.com/gp/product/0465092756) and [blog](http://www.math.columbia.edu/~woit/wordpress/)

* [[Lee Smolin]], [The Trouble with Physics](http://www.thetroublewithphysics.com)

More technical critical discussion is for instance in

* [[Carlo Rovelli]], _A critical look at strings_, [arxiv/1108.0868](http://lanl.arxiv.org/abs/1108.0868)

See also

* Amelino Camelia.

* Special issue of _Foundations of physics_ on superstring theory [Forty Years of String Theory - Reflecting on the Foundations](http://www.springer.com/physics/history+%26+philosophical+foundations+of+physics/journal/10701)
* Glimm, Jaffe, Witten, Thurston etc. about the idea of Experimental and theoretical mathematics, Bull. AMS -discussions prompted by Arthur Jaffe, Frank Quinn, _"Theoretical mathematics": Toward a cultural synthesis of mathematics and theoretical physics_ , Bull. Amer. Math. Soc. (N.S.) __29__ (1993), no. 1, 1--13; [MR94h:00007](http://www.ams.org/mathscinet-getitem?mr=1202292),[math.HO/9307227](http://arxiv.org/abs/math.HO/9307227), [doi](http://dx.doi.org/10.1090/S0273-0979-1993-00413-0)
* [Струнная клиника](http://stringworld.ru/strunnaya-klinika-neadekvatnye-veb-resursy-po-teorii-strun.html): неадекватные веб-ресурсы по теории струн (critics of string critics at [stringworld.ru](http://stringworld.ru))
* $n$Forum about this page [here](http://nforum.mathforge.org/discussion/2985/critics-of-string-theory)

(under construction)

[[!redirects string critique]]
[[!redirects critics of strings]]
[[!redirects critics of superstring theory]]
[[!redirects critique of string theory]]

[[!redirects critics of string theory]]


Critique of Pure Reason

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* [[Kant]]

  _Kritik der Reinen Vernunft_ (Critique of Pure Reason), abbreviated KrV.

  * [first edition 1781](#1stEd). Usually and in this article denoted by "A", e.g. "A111" refers to page 111 of the original first edition. But the Akademieausgabe uses A¹.

  * [second edition 1787](#2ndEd). Usually and in this article denoted by "B". Akademieausgabe uses A².

  The second edition was "improved here and there"; some passages have been replaced entirely, like [in the Transcendental Induction](#deductionA).

#Contents#
* table of contents
{:toc}

## __Introduction__ and Prefaces

## __I. Transcendental Doctrine of Elements.__ __1st Part.__ Transcendental Aesthetics

> The science of all the principles of sensibility a priori, I call transcendental aesthetic.* There must, then, be such a science forming the first part of the transcendental doctrine of elements, in contradistinction to that part which contains the principles of pure thought, and which is called transcendental logic.

### __First Secton.__ On Space.

For Kant space is not a concept but an intuition, more precisely an intuition, "which serves for the foundation of all external intuitions". Kant develops this view in the following steps (A[23](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_023.png)-[24](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_025.png)/B38-40)

>1. Space is not a conception which has been derived from outward experiences. For, in order that certain sensations may relate to something without me (that is, to something which occupies a different part of space from that in which I am); in like manner, in order that I may represent them not merely as without, of, and near to each other, but also in separate places, the representation of space must already exist as a foundation. ...

>2. Space then is a necessary representation a priori, which serves for the foundation of all external intuitions. We never can imagine or make a representation to ourselves of the non-existence of space, though we may easily enough think that no objects are found in it. ...

>3. Space is no discursive, or as we say, general conception of the relations of things, but a pure intuition. For, in the first place, we can only represent to ourselves one space, and, when we talk of divers spaces, we mean only parts of one and the same space. Moreover, these parts cannot antecede this one all-embracing space, as the component parts from which the aggregate can be made up, but can be cogitated only as existing in it. Space is essentially one, and multiplicity in it, consequently the general notion of spaces, of this or that space, depends solely upon limitations. ...

The intuition (in the ordinary sense) Kant has concerning space is similar to a number theorist who considers "the" [[natural number|natural numbers]] as the object of her studies, though she certainly has limitations in specifying what she means. Indeed, common way to do so have limitation: for instance, one can not uniquely determine the natural numbers by a list of axioms, as [[incompleteness theorem|Gödel's fist incompleteness theorem]] demonstrates, nor is it satisfying to describe the natural numbers by a concrete model, because there are several ways to do so.

Due to his intuition that there is "one" space, Kant discards the first two out of the three possibilities he discusses.
Since Kant this intuition has been dropped mostly by geometers in favor of the first introducing pioneering concepts like [[On the hypotheses which underlie geometry|Riemannian manifolds in 1854]]. In his very early work [[Bertrand Russell]] tried to adapt Kant's work to this new view (see [[An Essay on the Foundations of Geometry]]). Later Kant's idea appeared again in the framework of [[noncommutative geometry]] ([Petitot 2009](#Petitot09)).

Actually, all three possibilities discussed by Kant appear in modern mathematics:

1. the "discursive, or ..., general conception" is just the most common way to describe a space by a list of axioms, e.g. a [[topological space]] or a [[metric space]].

1. a space that "antecede this one all-embracing space, as the component parts from which the aggregate can be made up" (German: "vor dem einigen allbefassenden Raume gleichsam als dessen Bestandtheile (daraus seine Zusammensetzung möglich sei) vorhergehen"). A modern concept that matches this description quite well is the concept of a [[classifying space]], that does not contain each member of a certain class of spaces as a subspace, but from which "antecede" each such space by [[pullback]].

1. "Space is essentially one, and multiplicity in it, consequently the general notion of spaces, of this or that space, depends solely upon limitations." (German: "Er [Der Raum] ist wesentlich einig, das Mannigfaltige  in ihm, mithin auch der allgemeine Begriff von Räumen überhaupt beruht lediglich auf Einschränkungen.") A situation like this is found if any space in a geometrical theory can be considered as a subspace of one fix space. Good examples of such situations are given, as mentioned, by noncommutative geometry ([Petitot 2009](#Petitot09)), which is concerned with subspaces of the space of bounded operators on a [[separable space|separable]] [[Hilbert space]], or by [[descriptive set theory]] that can be understood as the study of subsets of the real line.

  Making such comparisons maybe a word of caution is in order to note that what Kant meant by the intuition of e.g. "space" is certainly not exactly any of the objects described above. But Kant's "formal intuition" ("formale Anschauung"), which he mentions later in the Transcendental Deduction as a footnote in B §26, comes probably quite close.

### __Second Section.__ On Time.

"Time is nothing other than the form of inner sense, i.e., of the intuition of our self and our inner state." ([A33](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_033.png)/B50)

## __I. Transcendental Doctrine of Elements.__ __2nd Part.__ Transcendental Logic

### _Introduction._ __The Idea of a Transcendental Logic.__

In the development of Kant's distinction between general logic and transcendental logic his _[[Attempt to Introduce the Concept of Negative Quantities into Philosophy]]_ from 1763 constitutes a crucial step.

###### _I._ On logic in general. ([A50](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_050.png)-5/B74-9)

In this paragraph, the scope of logic as a science is clarified, distinguishing it from aesthetics and structuring its subdisciplines, to then be able to show the place of transcendental logic among them in the following paragraph.
To do so, Kant firstly provides a plethora of terminology (“receptivity of impressions”, “spontaneity of concepts”, “empirical”, “sensibility”, “receptivity”), to then distinguish between the science of aesthetics and logic, the former concerning the rules of sensibility and the latter the rules of thinking. ([A50](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_050.png)-2/B74-6).

Secondly ([A52](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_052.png)/B76), Kant divides the science of logic into __general logic__ (synonymous with __elementary logic__) which contains the necessary rules of thinking, regardless of its object and __particular logic__ which contains the necessary rules of thinking about certain kinds of objects. Such a set of rules is called "organon of this or that science". In modern language this is the [[methodology]] of a [[science|scientific discipline]].
Finally ([A53](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_053.png)-5/B77-9), general logic is then again subdivided into __pure logic__---which "abstract[s] from all empirical conditions under which our understanding is exercised"---and __applied logic__---which deals with the conditions of thinking “under the subjective empirical conditions that psychology teaches us”, which are only given a posteriori, and which can all be given only empirically.

###### _II._ On __transcendental logic__. ([A55](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_055.png)-7/B79-82)

In this paragraph, the concept of transcendental logic as a particular kind of applied logic is established, contrasting it to pure logic. Pure logic abstracts from all contents of cognition [Erkenntnis] and is only concerned with the form of thinking in general. Transcendental logic contains the “rules of pure thinking of an object” ([A55](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_055.png)/B80).
Kant then notes the difference between cognition a priori and transcendental cognition. The former is merely any cognition obtained without recourse to intuition. The latter is cognition concerning the possibility or use of cognition a priori. ([A56](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_056.png)-7/B80-1)
Hence, transcendental aesthetics would describe the pure forms of intuition, which can then be drawn on to justify e.g. the possibility of geometric cognition a priori. Analogously there would need to be a transcendental logic, which would be __the science of the possibilities and rules of thinking about objects a priori__. ([A56](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_056.png)-7/B81-2)

###### _III._ On the division of general logic into __analytic__ and __dialectic__. ([A57](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_057.png)-62/B82-6)

In this paragraph, general logic is established as a negative criterion of truth, while arguing against its capability to serve as a positive (sufficient) criterion.
To show, that general pure logic is only capable of providing a negative criterion of truth, Kant firstly supposes a definition of truth which is: Truth “is the agreement of cognition with its object” ([A58](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_058.png)/B82).
Now, using this supposition as a premise, he attempts to prove the self-contradictory nature of a “certain and general criterion of truth of any cognition” ([A58](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_058.png)/B82) according to the following schema:

P1
: Truth “is the agreement of cognition with its object” ([A58](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_058.png)/B82), hence “a cognition is false if it does not agree with the object to which it is related even if it contains something that could well be valid of other objects”

P2
: “[A] general criterion of truth [found in general logic] would be that which was valid of all cognitions without any distinction among their objects” ([A58](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_058.png)/B83) since general logic abstracts from all objects and only concerns the pure form of thinking.

C
: A general criterion of truth found in general logic is self-contradictory. ([A59](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_059.png)/B83)

Nevertheless, since “[g]eneral logic analyses the entire formal business of the understanding and reason into its elements, and presents these as principles of all logical assessment of our cognition. This part of logic can therefore be called an __analytic__ [our emphasis], and is on that very account at least the negative touchstone of truth.” ([A60](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_060.png)/B84)
On the other hand, "general logic, which is merely a __canon__ for judging, has been used as if it were an __organon__ for the actual production of at least the semblance of objective assertions, and thus in fact it has thereby been misused. Now general logic, as a putative organon, is called __dialectic__." ([A61](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_061.png)/B85)

###### _IV._ __On the division of transcendental logic__ into the __transcendental analytic__ and __dialectic.__ ([A62](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_062.png)-4/B87-8)

The purpose of this paragraph is to narrow down the scope of transcendental logic, applying the division established in the last paragraph. From [A62](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_062.png)-3/B87:

> The use of this pure cognition [described in transcendental logic], however, depends on this as its condition: that objects are given to us in intuition, to which it can be applied. For without intuition all of our cognition would lack objects and therefore remain completely empty. The part of transcendental logic, therefore, that expounds the elements of the pure cognition of the understanding and the principles without which no object can be thought at all, is the __transcendental analytic and at the same time a logic of truth__  [our emphasis] For no cognition can contradict it without at the same time losing all content, i.e., all relation to any object ...

One may summarize, that the transcendental logic, in its analytic part, provides the necessary criterion of truth, not only regarding its logical form, which is already accomplished by general pure logic, but also regarding the necessary form which truth has to have as it necessarily relates to objects. Kant continues ([A63](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_063.png)/B88):

> [Transcendental logic] should properly be only a canon for the assessment of empirical use [of pure cognitions of the understanding], it is misused if one […] dares to synthetically judge, assert, and decide about objects in general with the pure understanding alone. The use of the pure understanding would in this case therefore be dialectical. __The second part of the transcendental logic must therefore be a critique of this dialectical illusion__ [our emphasis].

### __First Division.__ The Transcendental Analytic

#### __First Book.__ The Analytik of Concepts

##### _First Chapter._ On the Transcendental Clue [Leitfaden] to the Discovery of all Pure Concepts of the Understanding

###### _Second Section._ On the logical function of the understanding in judgments.

|        |Table of Judgements|               |
|--------|-------------------|---------------|
|  $\,$  | __1. Quantity of judgments__
 [[universal quantifier|Universal]]
 [[existential quantifier|Particular]]
  Singular | $\,$|
| __2. Quality__
 Affirmative
 [[negation|Negative]]
 [[infinite judgement|Infinite]] | $\,$ | __3. Relation__
 Categorical
 [[hypothetical judgement|Hypothetical]]
 [[disjunction|Disjunctive]]|
|   | __4. [[modality|Modality]]__
 [[possibility|Problematic]]
 Assertoric
 [[necessity|Apodictic]] |   |

Note that at least in ([Jäsche Logik, 1800](#Jaesche)) the negative and infinite judgements collapse in pure logic:

> Nach dem Principium der Ausschließung jedes Dritten (exclusi tertii) ist die Sphäre eines Begriffs relativ auf eine andre entweder ausschließend oder einschließend. - Da nun die Logik bloß mit der Form des Urteils, nicht mit den Begriffen ihrem Inhalte nach, es zu tun hat: so ist die Unterscheidung der unendlichen von den negativen Urteilen nicht zu dieser Wissenschaft gehörig.

###### _Third Section._ On the pure concepts of the understanding or categories.

"... there arise exactly as many pure concepts of the understanding ... as there were logical functions of all possible judgments in the previous table:" ([A79](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_079.png)/B105 f.)

|        |Table of Categories|               |
|--------|-------------------|---------------|
|  $\,$  | __1. Of Quantity__
 Unity
 Plurality
 Totality | $\,$|
| __2. Of Quality__
 Reality
 $\qquad$
 [[negation|Negation]]
 $\qquad$
 Limitation
 $\qquad$  | $\,$ | __3. Of Relation__
 Of Inherence and Subsistence
 $\qquad$  (*substantia et accidens*)
 Of Causality and Dependence
 $\qquad$ (cause and effect)
 Of Community (reciprocity
 $\qquad$ between agent and patient) |
|   | __4. Of [[modality|Modality]]__
 [[possibility|Possibility]]
 Existence - Non-existence
 Necessity - Contingency |   |

##### _Second Chapter._ On the Deduction of the Pure Concepts of the Understanding [Transcendental Deduction]

Kant explains the transcendental deduction as follows:

> I therefore call the explanation of the way in which concepts can relate to objects a priori their __transcendental deduction__. (B117)

Kant begins by describing the way in which the manifold of intuition is formed:

> Yet the __combination__ (conjunctio) of a manifold in general can never come to us through the senses, and therefore cannot already be contained in the pure form of sensible intuition; for it is an act of the spontaneity of the power of representation. (B129)

He justifies the transcendental unity in the following sentence:

> I am therefore conscious of the identical self in regard to the manifold of the representations that are given to me in an intuition because I call them all together __my__ representations, which constitute __one__. (B135)

Presupposing that the human understanding only operates with concepts  rather than with intuitions, Kant defines "understanding" in the following way:

> __Understanding__ is, generally speaking, the faculty of __cognitions__. (B137)

To advance his argument, Kant then needs the synthetic unity, because:

> The synthetic unity of consciousness is ... an objective condition of all cognition, ... (B138)

Now a question regarding the purpose of the categories arises since for Kant

> [the Categories] are only rules for an understanding whose entire capacity consists in thinking, i.e., in the action of bringing the synthesis of the manifold that is given to it in intuition from elsewhere to the unity of apperception, which therefore __cognizes__ nothing at all by itself, but only combines and orders the material for cognition, the intuition, which must be given to it through the object. (B145)

While "Space and time are valid, as conditions of the possibility of how objects can be given to us, no further than for objects of the senses, hence only for experience." (B148) the categories are rules regarding how understanding may occur.

The following quote enlightens the connection between natural laws, which Kant describes as derivable from nature, and the categories, which are conditions of possible experience:

> Particular laws, because they concern empirically determined appearances, __cannot__ be __completely derived__ from the categories, although they all stand under them. (B165)

In the end Kant arrives at the result that "__no a _priori___ cognition is possible for us except solely of objects of possible experience." (B166)

######{#deductionA} _Second Section_ __[A]__. On the a priori grounds for the possibility of experience. [A deduction]

In the first edition Kant provides a more thorough description on how the formation of concepts in our mind works. This description was left out in the corresponding chapter of the second edition ("B deduction"). Concepts are provided by a _threefold synthesis_:

1. _The synthesis of_ __apprehension in the intuition.__
All representations belong "as modifications of the mind ... to inner sense, and as such all of our cognitions are in the end subjected to the formal condition of inner sense, namely time" ([A99](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_099.png)).
Such representations may be though of as states of the mind. One may be inclined to construe them as objects in a category as "Every intuition [Anschauung] contains a manifold in itself" ([A98/9](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_098.png))

2. _The synthesis of_ __reproduction in the imagination.__
"It is ... a merely empirical law in accordance with which representations that have often followed or accompanied one another are finally associated with each other and thereby placed in a connection in accordance with which, even without the presence of the object, one of these representations brings about a transition of the mind to the other in accordance with a constant rule [beständige Regel]. This law of reproduction, however,
presupposes that the appearances themselves are actually subject to
such a rule ..." ([A100](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_100.png))

3. _The synthesis of_ __recognition in the concept.__
"Without consciousness that that which we think is the very same as what we thought a moment before, all reproduction in the series of representations would be in vain. ... The word "concept" itself could already lead us to this remark. For it is this one consciousness that unifies the manifold that has been successively intuited, and then also reproduced, into one representation." ([A103](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_103.png)) The crucial point is this __unity of consciousness__.

So a _[constant] rule_ appears as a relationship between two states of consciousness while _rules_ appear as a connection of a concept to such states. Kant speaks also of a __unity of rule__ that "determines every manifold".
As for a modern formalization the former would could be interpreted as a morphism and the letter reminds one of a diagram that has the concept as its cone. When taking sets as object the rules should be some kind of [[relation]]. But it is not clear from Kant's formulation if the class of morphisms is restricted in some way, e.g. to functions in one direction.

Kant continues to state that these three step process also happens for a priori concepts.
This is to say for our cognition (Erkenntnis) thereof.
But this unity of the consciousness means that for each cognition  a concept is assigned to the intuition in some way, i.e.
that there is a corresponding "rule of intuitions"
($cognition \coloneqq (concept \to Regel\, der\, Anschauungen) $ ) ([A106](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_106.png)).
An instance of this is e.g. "object in general" ("Objekt überhaupt").
Every instance comes with a necessity that there is such a concept and a rule of intuitions.
But a necessity can only origin from some "transcendental condition", that Kant also calls __transcendental apperception__.
These are the pure concepts of the understanding we sought.

As example Kant provides: _a triangle as an object by being conscious of the composition of three straight lines
in accordance with a rule according to which such an intuition can always be exhibited_.
Moreover
$$
  cognition\, of\, outer\, appearances
= \left(body \to \substack{the\, unity\, of\, the\, manifold\\ that\, is\, thought\, through\, [the\, concept\, of\, body]}\right).
$$
Further one should remark that a priori concepts are distinguished by their numerical unity. Kant elaborates on this distinction [in the next book](#identity_difference).

#### Achourioti and van Lambalgen's formalization

Starting from the description of the [A deduction](#deductionA) above Achourioti and van Lambalgen ([2011/2012](#AchouriotiVanLambalgen)) suggested a formalization of Kant's logic in terms of [[geometric logic]].
The idea is to assign each moment of time, i.e. impression, a [[model]]. Then a directed system of such models represents the syntheses of apprehension and reproduction. Taking the limit along this directed system ("synthesis of recognition") gives the concept. Geometric implications come into play as they are exactly the formulas whose truth is stable under taking the limit.

Although the natural category for interpreting [[geometric logic]] is a [[geometric category]] (e.g. a [[topos]]), Achourioti and van Lambalgen work only in [[Set]] for technical convenience. From a philosophical point of view they remark that the underlying category should have additional structure. They elaborate on a specific finite set model of their interpretation, for which they can show a completeness theorem. We will state their main results with small reformulations.

...

##### Dictionary

| Kant | [[mathematical logic]] and [[model theory]] |
|------|---------------------------------------------|
| time | directed index category |
| a manifold of representation | object in the underlying category |
| synthesis of apprehension | objects of the index category of a directed system (inside the underlying category) |
| synthesis of reproduction | morphisms of the index category of a directed system (inside the underlying category) |
| synthesis of recognition | limit along the directed system |

#### __Second Book.__ The Analytic of Principles

##### _Appendix._ On the amphiboly of the concepts of reflection through the confusion of the empirical use of the understanding with the transcendental.

{#identity_difference}
> 1. __Identity__ and __difference__. If an object is presented to us several times, but always with the same inner determinations (_qualitas et quantitas_), then it is always exactly the same if it counts as an object of pure understanding, not many but only one thing (_numerica identitas_); but if it is appearance, then the issue is not the comparison of concepts, but rather, however identical everything may be in regard to that, the difference of the places of these appearances at the same time is still an adequate ground for the __numerical difference__ of the object (of the senses) itself. ([A263](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_263.png)/B319)

## __II. Transcendental Doctrine of Methods.__

## On Aristotelian logic

Regarding [[Aristotelian logic]]:

> _Concepts [Begriffe] ... serve as predicates of possible judgements._ ([A69](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_069.png)/B94)

Compare to the identification of concepts with the [[types]] in the sense of [[type theory]], see the references [here](Science+of+Logic#ReferencesConceptLogicAsTypeTheory).

...

## Related entries

* [[category (philosophy)]]

* [[transcendental grammar]]

* [[Stephan Wannowski]]

* [[Aristotelian logic]]

* [[idealism]]

* [[Science of Logic]]

## References

Publications of Kant's original work include:

* Cambridge University Press, First and second editions, English translation. _Critique of Pure Reason_ (1998) (The Cambridge Edition of the Works of Immanuel Kant) (P. Guyer & A. Wood, Eds.).

* {#1stEd}Wikisource, [First edition, German](https://de.wikisource.org/wiki/Critik_der_reinen_Vernunft_%281781%29) (1781) and [several English translations](https://en.wikisource.org/wiki/Critique_of_Pure_Reason)

* {#2ndEd}Wikisource, [Second edition, English](https://en.wikisource.org/wiki/Critique_of_Pure_Reason_%28Meiklejohn%29) (1855) translated by John Miller Dow Meiklejohn.

* [english translation](http://www.gutenberg.org/files/4280/4280-h/4280-h.htm) by [J. M. D. Meiklejohn](https://en.wikipedia.org/wiki/John_Meiklejohn)

* [comparison of first and second editions](http://www.zbk-online.de/texte/A0368.htm) (German), Bär, Jochen A. (ed.) (2010 ff.): _Zentralbegriffe der klassisch-romantischen „Kunstperiode“ (1760–1840). Wörterbuch zur Literatur- und Kunstreflexion der Goethezeit._ [http://www.zbk-online.de](http://www.zbk-online.de).

Related primary literature from Kant comprises

* {#Jaesche} G. B. Jäsche (ed.), _Immanuel Kants Logik ein Handbuch zu Vorlesungen_, pp. 419-582 in Kant, _Schriften zur Metaphysik und Logik 2_, Suhrkamp Frankfurt a. M. 1985[1800].

Secondary literature includes:

* Wikipedia, _[Critique of Pure Reason](https://en.wikipedia.org/wiki/Critique_of_Pure_Reason)_

* {#Petitot09}Jean Petitot, _[Noncommutative Geometry and Transcendental Physics](http://topologicalmedialab.net/xinwei/classes/readings/Petitot/Petitot_NoncommutativeGeometry.pdf)_, pp. 415-455.
In: Bitbol, Michel, Kerszberg, Pierre, Petitot, Jean (Eds.), _Constituting Objectivity: Transcendental Perspectives on Modern Physics_ Springer (2009)

{#AchouriotiVanLambalgen}Achourioti and van Lambalgen's approach appeared in

* Achourioti and van Lambalgen, _A formalization of Kant's transcendental logic_. (June 2011). Review of Symbolic Logic 4 (2):254-289.

* Achourioti and van Lambalgen, _The Completeness of Kant's Table of Judgments & Consequences for Philosophy of Mathematics_. Talk given at the Munich Center for Mathematical Philosophy (November 2011). [Video](https://www.youtube.com/watch?v=gPAZcxAGRVs).

Later the authors elaborated on the table of judgements in

* Achourioti and van Lambalgen, _Kant’s Logic Revisited_ (January 2016). In: Proceedings of PhML 2012, Saint Petersburg. Also published (2017) in: The IfCoLog Journal of Logics and their Applications  4(4), 845-865.

For criticism of Kant's work on the categories, see

* [[Gilbert Ryle]], _Categories_, Proceedings of the Aristotelian Society, New Series, Vol. 38 (1937 - 1938), pp. 189-206.

> Kant's line of approach was, in principle, much more enlightened than Aristotle's had been. Unfortunately his execution was hopelessly misguided.

category: reference

[[!redirects Kritik der Reinen Vernunft]]

[[!redirects KrV]]


Croatian Black Hole School

The __Croatian Black Hole School__ is a school inspired by the traditional Modave school in Belgiumč it took place on 21--25 June 2010 at [Trpanj](http://hr.wikipedia.org/wiki/Trpanj), on the peninsula Pelješac in southern Croatia. The lectures and housing for most people were organized at Pansion Loviška whose personel was very kind and helpful. The organizer was [[Jarah Evslin]] and the main topic is the physics of [[black holes]], both theoretical and observational.

The official web page is [http://antimodave.jimdo.com](http://antimodave.jimdo.com).

Lecture series:

* Chethan Krishnan (SISSA): Quantum field theory and black holes

* [[Gaston Giribet]] (Buenos Aires):  Black hole physics and AdS3/CFT2 correspondence. (This includes Black hole physics in three-dimensional massive gravity, applications to four-dimensions (i.e. Kerr/CFT correspondence) and other topics (e.g. Lifshitz black holes).)

* [John Wang](http://www.niagara.edu/chemistry-faculty/showfaculty/133) (Niagara Univ and Buffalo) Introduction into the physics of black holes

* [Dieter Van den Bleeken](http://www.physics.rutgers.edu/people/pips/vandenBleeken.html) (Rutgers): BPS black holes and the attractor mechanism

* [[Mario Pasquato]] (Pisa): Intermediate mass black holes - observational challenges

* [Malcolm Fairbairn](http://www.kcl.ac.uk/schools/pse/physics/people/academic/fairbairn) (King's College):  Cosmology, astrophysics and astrophysical black goles

* Holger Nielsen: discussion sessions -- Black hole information paradox

Key words and related entries: [[AdS-CFT|AdS3/CFT2]], Penrose diagram, Hawking radiation, [[asymptotic isometry]], [[intermediate mass black hole]], [[cosmology]], [[string theory]], horizon, information paradox, [[supersymmetry]], [[supergravity]], [[observing black holes]], [[quantum gravity]]

[[!redirects Croatian Black Hole School]]
[[!redirects Croatian black hole school]]

category: reference


cross product

# Cross products
* table of contents
{: toc}

## Idea

In general, the term 'cross product' is used for any operation denoted by the symbol '$\times$', such as [[cartesian product]], [[direct product]], [[Tychonoff product]], or (subsuming all of these) [[product|product in a category]].

However, there is another completely different context, used in the elementary analysis of [[vector spaces]], and that is what we discuss here.  Originally isolated from the multiplication operation in [[quaternions]] as a binary operation on $\mathbb{R}^3 \simeq Im(\mathbb{H})$, the cross product now has generalisations to other arities, other dimensions, and other ground fields.

## Definitions

### Classical

The **classical cross product** on the [[Cartesian space]] $\mathbb{R}^3$ is the [[bilinear function]]

$$
  (-) \times (-)\colon \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3
$$

given by

$$ (a,b,c) \times (d,e,f) = (b f - c e, c d - a f, a e - b d) .$$

This operation is invariant under [[orthogonal group|orthogonal transformations]] and there is nothing special here about the [[real numbers]], so given any $3$-dimensional [[orientation|oriented]] [[inner product space]] $V$ over any [[field]], we have a bilinear cross product

$$
  (-) \times (-)\colon V \times V \to V
$$

given, upon choosing any oriented orthonormal basis for $V$, by the formula above.  We can even do this over any [[free module]] of rank $3$ over any [[commutative ring]] (or arguably a non-commutative [[ring]], but not a [[rig]]) equipped (as may always be done) with an inner product that admits an orthonormal basis.

### Binary {#binary}

We have already, trivially, generalized the cross product to other ground fields.  One way to generalise it to other dimensions is to identify characteristic features as a bilinear operation and see what operations in other dimensions have these.

In this vein, a __binary cross product__ on any [[inner product space]] $V$ is a [[bilinear function]]

$$
  (-) \times (-)\colon V \times V \to V
$$

such that for all $x, y\in V$ we have

1. Alternation: $x \times x = 0$.

2. Orthgonality: $x \times y$ is orthogonal to both $x$ and $y$; that is, $x \cdot (x \times y) = (x \times y) \cdot y = 0$.

3. Area: ${\|x \times y\|} = {\|x\|} {\|y\|}$ if $x, y$ are orthogonal.

From these, we can prove a more general formula for ${\|x \times y\|}$:

$$ {\|x \times y\|}^2 = {\|x\|}^2 {\|y\|}^2 - (x \cdot y)^2 ,$$

or equivalently

$$ {\|x \times y\|} = {\|x\|} {\|y\|} {|\sin\angle(x,y)|} .$$

(Using the [[polarization identity]] to express $x \cdot y$ in terms of $\|x\|$, $\|y\|$, and either $\|x + y\|$ or $\|x - y\|$, this is the double of Hero's Formula for the area of a triangle.)

Conversely, using this more general area formula, we can prove both the restricted area formula and alternation, so that only orthogonality is needed as a separate axiom.

We then have over the [[real numbers]]:

* In [[dimension]] $0$ or $1$, there is one binary cross product: $x \times y \coloneqq 0$.

* In dimension $3$, there are two binary cross products, one for each orientation, matching the classical cross product.

* In dimension $7$, there are uncountably many binary cross products.  Even fixing an orthonormal basis and requiring compatibility with this (in that the cross product of any two distinct basis vectors must be another basis vector or its opposite), there are still 480 distinct cross products.

* In any other finite dimension, there are no binary cross products at all.

These cross products exist over any base field, but as far as I know there may be additional cross products over some fields in dimensions greater than $1$.  (The claim that there are uncountably many cross products in $7$ dimensions becomes the claim that the [[algebraic variety]] of these inner products has dimension greater than $0$; even in the real case, it would be worthwhile to describe this variety in more detail.)  If we pick an orthonormal basis and demand compatibility with it, then the classification above is complete over any field (as the problem becomes essentially combinatorial), except that some of these cross products will be identified in characteristic $2$.

Binary cross products are closely related to [[normed division algebras]] (NDAs).  Given a normed division algebra $A$, the imaginary hyperplane $Im(A)$ inherits an inner product from $A$ and gains a cross product as
$$ x \times y \coloneqq Im(x y) = \frac{1}{2}[x,y] = x y + x \cdot y .$$
Conversely, given an inner product space $V$ with a binary cross product, the orthogonal [[direct sum]] $K \oplus V$ becomes a NDA as
$$ (a,x) (b,y) = (a b - x \cdot y, a y + b x + x \times y) ,$$
where $K$ is the ground field.  (Compare the relationship between complex $*$-algebras and [[Jordan--Lie algebras]], where $A$ is $V \oplus V$ instead of $K \oplus V$, which we again think of as consisting of real and imaginary parts and where we again have two multiplication operations on $V$.)

By [[Hurwitz theorem|Hurwitz's theorem]], the only finite-dimensional NDAs over $\mathbb{R}$ are $\mathbb{R}$ itself (the [[real numbers]]), $\mathbb{C}$ (the [[complex numbers]]), $\mathbb{H}$ (the [[quaternions]]), and $\mathbb{O}$ (the [[octonions]]).  Thus the limited possibilities for binary cross products are determined by the limited possibilities for NDA structures.

By one of the deeper strands of mathematics, this classification of something as innocent-looking as cross-products is closely related not just to the existence of [[normed division algebras]] over the [[real numbers]], but also to all of the following: [[parallelizable manifold|parallelizable]] [[n-spheres]], the existence of [[real spin representations]] (see also at _[[supersymmetry and division algebras]]_), and the [[homotopy groups of spheres]] of [[Hopf invariant one]]; see [there](Hopf+invariant+one#RelationToHSpaceStructureAndNormedDivisionAlgebras).

### $(n-1)$-ary {#counary}

Given an [[orientation|oriented]] [[inner product space]] $V$ of finite [[dimension]] $n$, we can define the signed [[volume]] of an $n$-[[tuple]] of vectors.  (See also [[volume form]].)  This allows us to characterise an $(n - 1)$-ary or __co-unary cross product__ of $n - 1$ vectors as a multilinear operation
$$ ⨉\colon V^{n - 1} \to V $$
such that
$$ v_0 \cdot ⨉(v_1,\ldots,v_{n-1}) = vol(v_0,v_1,\ldots,v_{n-1}) $$
always.  There is exactly one such cross product on any such $V$ (so two if we start with an unoriented inner product space).

In $3$ dimensions, this also recovers the classical cross product.  In $2$ dimensions, this produces a __unary cross product__ given by ${\times}(a,b) = (b,-a)$; in a counterclockwise-oriented plane, it rotates a vector clockwise through a right angle.  In $1$ dimension, this is a nullary operation (a constant) whose value is the positive normal vector $1$.

### Vector-valued {#vectorvalued}

Generalizing all of the above, let a __vector-valued cross product__ on any [[inner product space]] $V$ be a multilinear function
$$ ⨉\colon V^k \to V $$
for some [[natural number]] $k$ (called the [[arity]]) such that:

1. Alternation: $⨉(v_1,\ldots,v_k) = 0$ if $v_i = v_j$ for some $i \ne j$.

2. Orthgonality: $⨉(v_1,\ldots,v_k)$ is orthogonal to each $v_i$.

3. Area: ${\|⨉(v_1,\ldots,v_k)\|} = \prod_i {\|v_i\|}$ if the $v_i$ are mutually orthogonal.

We can again extend (3) to get the magnitude of the cross product of any $k$ vectors; its square is the [[determinant]] of the matrix whose $(i,j)$th entry is $v_i \cdot v_j$ (the [[Gram determinant]]), and then (1) again follows.

Then for an inner product space $V$ over $\mathbb{R}$ of finite dimension $n$, we have:

* If $k = 0$, then there is one cross product for each [[unit vector]] of $V$: none when $n = 0$, two (one for each orientation) when $n = 1$, and uncountably many when $n \geq 2$ (of which $2n$ are compatible with a given basis).

* If $k = 1$, then there is one cross product for each way of writing $\mathbb{R}^n$ as as $n/2$-fold [[internal direct sum]] of $2$-dimensional subspaces and orienting each subspace: one (always $0$) when $n = 0$, two (one for each orientation) when $n = 2$, uncountably many when $n \geq 4$ is even (of which $2^{n/2}(n-1)!!$ are compatible with a given basis), and none when $n$ is odd.

* If $k = 2$, then the possibilities are as listed [above](#binary) for binary cross products: one (always $0$) when $n = 0, 1$, two (one for each orientation) when $n = 3$, uncountably many (related to the octonions, and of which $480$ are compatible with a given basis) when $n = 7$, and none when $n \ne 0, 1, 3, 7$.

* If $k = 3$, then there is one cross product (always $0$) when $n = 0, 1, 2$, two cross products (one for each orientation) when $n = 4$, uncountably many cross products (related to the octonions) when $n = 8$, and none when $n \ne 0, 1, 2, 4, 8$.

* If $4 \leq k \lt n - 1$, then there are no cross products.

* If $k = n - 1$, then there are two cross products, one for each orientation, as described [above](#counary) for $(n-1)$-ary cross products.

* If $k = n$, then there are no cross products.

* If $k \geq n + 1$, then there is one cross product (always $0$).

Or organized by dimension ($n$) rather than arity ($k$):

* If $n = 0$, then there is no cross product when $k = 0$ and one cross product (always zero) when $k \geq 1$.

* If $n = 1$, then there are two cross products (one for each unit vector, or equivalently for each orientation) when $k = 0$, none when $k = 1$, and one (always zero) when $k \geq 2$.

* If $n = 2$, then there are uncountably many cross products (one for each unit vector, of which four are compatible with a given basis) when $k = 0$, two (one for each orientation) when $k = 1$, none when $k = 2$, and one (always zero) when $k \geq 3$.

* If $n = 3$, then there are uncountably many cross products (one for each unit vector, of which six are compatible with a given basis) when $k = 0$, none when $k = 1$, two (one for each orientation) when $k = 2$, none when $k = 3$, and one (always zero) when $k \geq 4$.

* If $n = 4$, then there are uncountably many cross products (one for each unit vector, of which eight are compatible with a given basis) when $k = 0$, uncountably many (four for each $2$-dimensional subspace, and of which twelve are compatible with a given basis) when $k = 1$, none when $k = 2$, two (one for each orientation) when $k = 3$, none when $k = 4$, and one (always zero) when $k \geq 5$.

* If $n = 5$, then there are uncountably many cross products (one for each unit vector, of which ten are compatible with a given basis) when $k = 0$, none when $k = 1, 2, 3$, two (one for each orientation) when $k = 4$, none when $k = 5$, and one (always zero) when $k \geq 6$.

* If $n = 6$, then there are uncountably many cross products (one for each unit vector, of which twelve are compatible with a given basis) when $k = 0$, uncountably many (related to the $2$-dimensional decompositions, and of which $120$ are compatible with a given basis) when $k = 1$, none when $k = 2, 3, 4$, two (one for each orientation) when $k = 5$, none when $k = 6$, and one (always zero) when $k \geq 7$.

* If $n = 7$, then there are uncountably many cross products (one for each unit vector, of which $14$ are compatible with a given basis) when $k = 0$, none when $k = 1$, uncountably many (related to the octonions, and of which $480$ are compatible with a given basis) when $k = 2$, none when $k = 3, 4, 5$, two (one for each orientation) when $k = 6$, none when $k = 7$, and one (always zero) when $k \geq 8$.

* If $n = 8$, then there are uncountably many cross products (one for each unit vector, of which $16$ are compatible with a given basis) when $k = 0$, uncountably many (related to the $2$-dimensional decompositions, and of which $1680$ are compatible with a given basis) when $k = 1$, none when $k = 2, 3$, uncountably many (related to the octonions) when $k = 4$, none when $k = 5, 6$, two (one for each orientation) when $k = 7$, none when $k = 8$, and one (always zero) when $k \geq 9$.

* If $n \geq 9$ is odd, then there are uncountably many cross products (one for each unit vector, of which $2n$ are compatible with a given basis) when $k = 0$, none when $1 \leq k \lt n - 1$, two (one for each orientation) when $k = n - 1$, none when $k = n$, and one (always zero) when $k \geq n + 1$.

* If $n \geq 10$ is even, then there are uncountably many cross products (one for each unit vector, of which $2n$ are compatible with a given basis) when $k = 0$, uncountably many (related to the $2$-dimensional decompositions, and of which $2^{n/2}(n-1)!!$ are compatible with a given orientation) when $k = 1$, none when $2 \leq k \lt n - 1$, two (one for each orientation) when $k = n - 1$, none when $k = n$, and one (always zero) when $k \geq n + 1$.

### Tensor-valued {#tensorvalued}

Fixing a field $K$, let [[Vect]] be $Vect_K$, the [[symmetric monoidal category]] of vector spaces over $K$ (with the usual [[tensor product]]), and let $T$ be any [[symmetric monoidal functor]] from $Vect$ to itself.  Note that any inner product $g\colon V \otimes V \to K$ extends to a (possibly degenerate) inner product $T(g)\colon T(V) \otimes T(V) \to K$; similarly, any element $x$ of $V$, thought of as a linear map $x\colon K \to V$, gives rise to an element $T(x)$ of $T(V)$.  The vector-valued cross products [above](#vectorvalued) use the [[identity functor]] for $T$, but other possible choices for $T$ are $V \mapsto V \otimes V$, $V \mapsto \Lambda^2 V$, and the constant functor $V \mapsto K$.  We could also take $Vect$ to be a [[full subcategory]] of $Vect_K$ closed under the tensor product, such as $Fin Vect_K$; this may allow more possibilities for $T$ in exchange for fewer possibilities for $V$.

Given an inner-product space $V$ and a symmetric monoidal functor $T$, a __$T$-valued cross product__ on $V$ is a multilinear function
$$ ⨉\colon V^k \to T(V) $$
for some natural arity $k$ such that:

1. Alternation: $⨉(v_1,\ldots,v_k) = 0$ if $v_i = v_j$ for some $i \ne j$.

2. Orthogonality: $⨉(v_1,\ldots,v_k)$ is orthogonal (in $T(V)$) to each $T(v_i)$.

3. Area: ${\|⨉(v_1,\ldots,v_k)\|} = \prod_i {\|v_i\|}$ if the $v_i$ are mutually orthogonal (in $V$).

Again it follows that in any case ${\|⨉(v_1,\ldots,v_k)\|}$ is a square root of the Gram determinant, and again this implies both (1) and (3).

I do not know a full list of these, but one important example is the __scalar-valued binary cross product__ in $2$ dimensions:
$$ (a, b) \times (c, d) = a d - b c .$$
Actually, this scalar-valued cross product $x \times y$ is simply the dot product $x \cdot {\times}y$, where ${\times}y$ is the [unary](#counary) cross product in $2$ dimensions.

More generally, in any number $n \geq 2$ of dimensions, there is a __multivector-valued binary cross product__ (actually one for each orientation) whose values are $(n-2)$-[[multivector|vectors]]; this includes the scalar-valued cross product when $n = 2$ and the classical cross product when $n = 3$, but gets more complicated for larger values of $n$.  In particular, for $n = 4$, we have a [[bivector]]-valued cross product, which is the [[Hodge dual]] of the [[exterior product]].

Or generalizing the scalar-valued binary cross product in a different way, the [[volume]] form on an $n$-dimensional inner-product space is a __scalar-valued co-nullary cross product__ (so $n$-ary).  Notice that there are two of these, one for each orientation, and each of these is the dot product with one of the vector-valued $(n-1)$-ary cross products.  There are no other scalar-valued cross products, except for the identically zero products of arity $k \gt n$ and the two nullary products given by the unit-norm scalars (so just $1$ and $-1$ over the real numbers) for arbitrary $n$.

## Relationships

### Exterior products

The cross product is also called 'outer product', and both of these terms are sometimes also used for the [[exterior product]].  In its most basic form, the exterior product of two vectors $u,v$ is a [[bivector]] $u \wedge v$.  But note that this is *not* a bivector-valued cross product by the definition [above](#tensorvalued), since it lacks orthogonality (and indeed has nothing to do with the inner product).

In $3$ dimensions, given an inner product and an orientation, we can use the [[Hodge dual]] to turn the exterior product into a vector, and this is the classical cross product once more.  In $2$ dimensions, using the same structure, we can turn the bivector into a [[scalar]]; this recovers the scalar-valued binary cross product [above](#tensorvalued).  In general, this produces the binary $(n-2)$-vector-valued cross product.

Using only the inner product but not the orientation, we get (respectively) a [[pseudovector]] (sometimes called an axial vector) or more generally a [[pseudoscalar]] or other [[pseudotensor]]; this perspective is common in [[geometric algebra]].  In general in dimension $n$, a bivector becomes an $(n-2)$-pseudovector, but this is not usually a simplification.

In classical applications of the cross product, often not all of the structure is needed, and the exterior product is really the fundamental concept.

### Curls

If $M$ is a [[Riemannian manifold]], then the [[tangent space]] at each point is an [[inner product space]], so it may be possible to smoothly assign a $k$-ary cross product to these spaces.  If this is done, then we can take the [[curl]] of a $(k-1)$-[[multivector|vector]] [[vector field|field]] as follows:

*  Use the metric to turn the $(k-1)$-vector field into a $(k-1)$-[[differential form|form]].
*  Take the [[exterior differential]] to get a $k$-form.
*  Use the metric again to get a $k$-vector field.
*  Apply the cross product to get a [[vector field]] (or more generally a [[tensor field]] for a tensor-valued cross product).

This vector field is the __curl__ of the original $(k-1)$-vector field.  This justifies the notation $\Del \times X$ for the curl.

(It\'s important that the cross product is [[alternating form|alternating]] and [[multilinear map|multilinear]], so that it makes sense to apply it to a $k$-vector rather than to $k$ individual vectors.)

When $k = 2$ and $n = 3$, there is one smooth choice of cross product for each [[orientation]] of $M$, and we recover the classical notion of curl.

When $k = 1$ and $n = 2$, we may also consider the scalar-valued curl, using the scalar-valued binary cross product described [above](#tensorvalued).  The scalar-valued curl of a vector field $X$ is the same as the [[divergence]] of the rotated vector field ${\times}X$ (using the [unary](#counary) cross product in $2$ dimensions); that is, $\Del \times X = \Del \cdot {\times}X$.

In modern mathematics, of course, we usually think of the exterior differential of differential forms as the fundamental concept, and only turn these forms into vector fields and the like under certain circumstances.

## Related concepts

* [[associative 3-form]]

## References

* The English Wikipedia article _[Seven-dimensional cross product](https://en.wikipedia.org/wiki/Seven-dimensional_cross_product)_ actually covers all non-zero vector-valued cross products in finite dimensions over the real numbers.

[[!redirects cross product]]
[[!redirects cross products]]
[[!redirects vector product]]
[[!redirects vector products]]
[[!redirects vector cross product]]
[[!redirects vector cross products]]

[[!redirects scalar cross product]]
[[!redirects scalar cross products]]


cross ratio

Any triple of distinct points in a [[projective line]] over a [[field]] $K$ may be transformed by a [[projective transformation]] (an element of [[projective general linear group|PGL(2,K)]], in the case of $K = \mathbf{C}$ (the [[complex numbers]]) known as a [[Möbius transformation]]) to another given triple of distinct points (i.e., the group $PGL(2,K)$ is 3-transitive on the projective line). Therefore no projective invariant can be attached to a triple of points and if three points are the starting parameters for some problem in projective geometry we may and typically do give them fixed values of $0,1,\infty$. The unique element in $PGL_2(K)$ that takes $(A, B, C)$ to $(\infty, 0, 1)$ may be described as the [[fractional linear transformation]]

$$p_{A, B, C}: z \mapsto \frac{z - B}{z - A} \cdot \frac{C - A}{C - B}$$

The simplest projective invariant can be attached to a quadruple of points and it is the cross ratio. It can be expressed in various ways, e.g.,

$$
(A,B,C,D) \coloneqq \frac{A - C}{A - D}\frac{B - D}{B - C}
$$

which is the value $p_{A, B, C}(D)$ using the fractional linear transformation above.

If the cross ratio for $(A,B,C,D)$ is $\lambda$ and the order of the four points is changed than the cross ratio takes one of the following values, depending on the permutation:

$$
\lambda, \frac{1}\lambda, 1-\lambda, \frac{1}{1-\lambda}, \frac{\lambda - 1}\lambda, \frac\lambda{\lambda-1}
$$

An elementary introduction can be found in

* Lucienne Félix, The modern aspect of mathematics, (Rus. transl.: Элементарная математика в современном изложении, 1967)

and a quick overviews in

* F. Labourie, What is a Cross Ratio, Notices of the AMS, 55 (2008), no. 10, pp.1234–1235 [pdf](https://www.math.u-psud.fr/~labourie/preprints/pdf/whatis.pdf)
* wikipedia [cross-ratio](https://en.wikipedia.org/wiki/Cross-ratio), [Möbius transformation](https://en.wikipedia.org/wiki/M%C3%B6bius_transformation)

There is a noncommutative version

* Vladimir Retakh, _Noncommutative cross-ratios_, Journal of Geometry and Physics __82__ (2014) 13-17 [arxiv/1401.5770](https://arxiv.org/abs/1401.5770) [doi](https://doi.org/10.1016/j.geomphys.2014.04.001)

category: geometry


cross-composition product

**Cross-composition products** (including cross-composition products of [[funcoid]]s, [[reloid]]s, [[multifuncoid]]s, [[multireloid]]s) arise in the research of [[Victor Porton]].

A **cross-composition product** is a [[pointfree funcoid]].

TODO: Write more here.

[[!redirects cross-composition product]]
[[!redirects cross-composition products]]