calibration

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## Idea

A degree-$p$ _calibration_ of an oriented [[Riemannian manifold]] $(X,g)$ is a [[differential p-form]] $\omega \in \Omega^p(X)$ with the property that

1. it is [[closed differential form|closed]] $d \omega = 0$;

1. evaluated on any oriented $p$-dimensional subspace of any [[tangent space]] of $X$, it is less than or equal to the induced degree-$p$ [[volume form]], with [[equality]] for at least one choice of subspace.

A Riemannian manifold equipped with such a calibration is also called a _calibrated geometry_ ([Harvey-Lawson 82](#HarveyLawson82)) or similar.

A _calibrated submanifold_ of a manifold with calibration is an oriented [[submanifold]] such that restricted to each of its tangent spaces $\omega$ _equals_ the induced volume form of the submanifold there.

## Properties

### Minimal volume submanifolds

Any calibrated submanifold $\Sigma \hookrightarrow X$ minimizes [[volume]] in its [[homology]] class.

For Let $\tilde \Sigma \hookrightarrow X$ be a homologous [[submanifold]]. Then [[Stokes theorem]] together with the condition that $d \phi = 0$ implies that the [[integration of differential forms]] of $\phi$ over $\Sigma$ equals that over $\tilde \Sigma$. The defining conditions on calibrations and on calibrated submanifolds then imply the [[inequality]]

$$
  vol(\Sigma)
  \stackrel{cal\,subm}{=}
  \int_\Sigma \phi
  \stackrel{Stokes}{=}
  \int_{\tilde \Sigma} \phi
  \stackrel{calib}{\leq}
  \int_{\tilde \Sigma} d vol
  =
  vol(\tilde \Sigma)
  \,.
$$

### Calibrations from spinors
 {#CalibrationsFromSpinors}

> under construction

For suitable $n$ and $p$, and given a real [[spin representation]] of $Spin(n)$,
then the [[Cartesian space]] $\mathbb{R}^n$ with its canonical Riemannian structure
becomes $p$-calibrated with the calibration form being

$$
  \omega_{\epsilon}
  \coloneqq
  (\overline{\epsilon} \Gamma_{a_1 a_2 \cdots a_p} \epsilon) \, e^{a_1} \wedge \cdots \wedge e^{a_p}
$$

where

1. $\{e^a\}$ denotes the canonical [[linear basis]] of [[differential 1-forms]];

1. $\epsilon$ is a non-vanishing [[spinor]];

1. $\overline{\epsilon} \Gamma_{a_1 a_2 \cdots a_p} \epsilon$ is the [canonical bilinear pairing](spin+representation#SpinorBilinearForms) which in components is given by evaluating $\epsilon$ in the [[quadratic form]] given by multiplying the skew-symmetrized product of $p$ of the representation matrices $\Gamma^a$ of the [[Clifford algebra]] with the [[charge conjugation matrix]] $C$.

(e.g. [Dadok-Harvey 93](#DadokHarvey93)).

For instance for $n = 7$ and $p = 3$ then this gives the [[associative 3-form]] calibration.

More generally for $X$ an $n$-dimensional [[Riemannian manifold]] with a [[covariantly constant spinor]] $\epsilon$, then under suitable conditions applying this construction in each [[tangent space]] gives a calibration.

## Examples

* The globalization of the [[associative 3-form]] of a [[G2-manifold]] is a calibration. A calibrated submanifold in this case is also called an [[associative submanifold]].

* The [[Cayley 4-form]] on [[Spin(7)-manifolds]].

## References

### General

The original articles are

* {#HarveyLawson82} [[Reese Harvey]], [[H. Blaine Lawson]], _Calibrated geometries_, Acta Mathematica July 1982, Volume 148, Issue 1, pp 47-157

* [[Reese Harvey]], _Calibrated geometries_, Proceeding of the ICM 1983 ([pdf](http://www.mathunion.org/ICM/ICM1983.1/Main/icm1983.1.0797.0808.ocr.pdf))

The relation to [[Killing spinors]] goes back to

* [[Reese Harvey]], _Spinors and Calibrations_, Academic Press, 1990 ([publisher](http://www.elsevier.com/books/spinors-and-calibrations/harvey/978-0-12-329650-4))

* {#DadokHarvey93} [[Jiri Dadok]], [[Reese Harvey]], _Calibrations and spinors_, Acta Mathematica 1993, Volume 170, Issue 1, pp 83-120

See also

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

* Jason Dean Lotay, _Calibrated submanifolds and the Exceptional geometries_, 2005 ([pdf](https://people.maths.ox.ac.uk/joyce/theses/LotayDPhil.pdf))

### Application in string theory

Discussion in [[string theory]]/[[M-theory]] includes the following.

* [[Gary Gibbons]], [[George Papadopoulos]], _Calibrations and Intersecting Branes_ ([arXiv:hep-th/9803163](http://arxiv.org/abs/hep-th/9803163))

* [[Jerome Gauntlett]], [[Neil Lambert]], [[Peter West]], _Branes and Calibrated Geometries_, Commun.Math.Phys. 202 (1999) 571-592 ([arXiv:hep-th/9803216](http://arxiv.org/abs/hep-th/9803216))

* [[George Papadopoulos]], [[Jan Gutowski]], _AdS Calibrations_, Phys.Lett.B462:81-88,1999 ([arXiv:hep-th/9902034](http://arxiv.org/abs/hep-th/9902034))

* [[Jan Gutowski]], [[George Papadopoulos]], [[Paul Townsend]], _Supersymmetry and generalized calibrations_, Phys.Rev.D60:106006, 1999 ([arXiv:hep-th/9905156](http://arxiv.org/abs/hep-th/9905156))

* [[Jan Gutowski]], S. Ivanov, [[George Papadopoulos]], _Deformations of generalized calibrations and compact non-Kahler manifolds with vanishing first Chern class_, Asian Journal of Mathematics 7 (2003), 39-80 ([arXiv:0205012](http://arxiv.org/abs/math/0205012))

[[!redirects calibrations]]

[[!redirects calibrated geometry]]
[[!redirects calibrated geometries]]

[[!redirects calibrated manifold]]
[[!redirects calibrated manifolds]]

[[!redirects calibrated submanifold]]
[[!redirects calibrated submanifolds]]

[[!redirects manifold with calibration]]
[[!redirects manifolds with calibration]]
[[!redirects manifold with calibrations]]
[[!redirects manifolds with calibrations]]


Calin Lazaroiu

* [Home page](https://cgp.ibs.re.kr/~calin/)

* [Mathematics Genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=99921)

## Selected writings

On [[M-theory on 8-manifolds]] in terms of [[G-structures]]:

* [[Elena Babalic]], [[Calin Lazaroiu]], _Singular foliations for M-theory compactification_, JHEP 03 (2015) 116 ([arXiv:1411.3497](https://arxiv.org/abs/1411.3497))

* [[Elena Babalic]], [[Calin Lazaroiu]], _Foliated eight-manifolds for M-theory compactification_, JHEP 01 (2015) 140 ([arXiv:1411.3148](https://arxiv.org/abs/1411.3148))

* [[Elena Babalic]], [[Calin Lazaroiu]], _The landscape of $G$-structures in eight-manifold compactifications of M-theory_, JHEP 11 (2015) 007 ([arXiv:1505.02270](https://arxiv.org/abs/1505.02270))

* [[Elena Babalic]], [[Calin Lazaroiu]], _Internal circle uplifts, transversality and stratified $G$-structures_, JHEP 11 (2015) 174 ([arXiv:1505.05238](https://arxiv.org/abs/1505.05238))

On [[D-branes]] under the [[flop transition]]:

* [[Brian Greene]], [[Calin Lazaroiu]], _Collapsing D-Branes in Calabi-Yau Moduli Space: I_, Nucl.Phys. B604 (2001) 181-255 ([arXiv:hep-th/0001025](https://arxiv.org/abs/hep-th/0001025))

* [[Calin Lazaroiu]], _Collapsing D-branes in one-parameter models and small/large radius duality_, Nucl.Phys. B605 (2001) 159-191 ([arXiv:hep-th/0002004](https://arxiv.org/abs/hep-th/0002004))

* [[Calin Lazaroiu]], _On the non-commutative geometry of topological D-branes_, JHEP 0511 (2005) 032, [arXiv:hep-th/0507222](https://arxiv.org/abs/hep-th/0507222)

On [[duality-symmetric abelian Yang-Mills theory]] ("[[premetric electromagnetism]]") in the generality allowing "[[U-duality]]-twists" among several abelian gauge fields, motivated by application to [[D=4 supergravity]]:

* [[Calin Lazaroiu]], [[Carlos S. Shahbazi]], *The duality covariant geometry and DSZ quantization of abelian gauge theory*, Advances in Theoretical and Mathematical Physics **26** (2022) 2213–2312 [[arXiv:2101.07236](https://arxiv.org/abs/2101.07236), [doi:10.4310/ATMP.2022.v26.n7.a5](https://dx.doi.org/10.4310/ATMP.2022.v26.n7.a5)]

* [[Calin Lazaroiu]], [[Carlos S. Shahbazi]], *The geometry and DSZ quantization of four-dimensional supergravity*, Letters in Mathematical Physics  **113** 4 (2023) [[arXiv:2101.07778](https://arxiv.org/abs/2101.07778), [doi:10.1007/s11005-022-01626-y](https://doi.org/10.1007/s11005-022-01626-y)]

## Related entries

* [[flop transition]]

category: people


Calkin algebra

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## Idea

The __Calkin algebra__ is the [[quotient]] $B(H)/K(H)$ of the [[C*-algebra]] of [[bounded operator|bounded]] [[linear operators]] on an infinite-dimensional separable complex [[Hilbert space]] $H$ by the closed ideal of [[compact operators]].
It is widely used in [[index theory]] and [[operator algebras]].

It is sometimes viewed as a noncommutative analogue of the space of continuous functions on $\beta(\omega) \,\backslash\, \omega$ (where $\beta$ means taking the [[Stone-Čech compactification]]).

## Properties

### General

Some results invoking the Calkin algebra rely on extra [[axioms]] in [[set theory]], see for instance ([Weaver 2007](#Weaver)) for a sample of such results.

### Automorphisms

[[Todorcevic's Axiom]] (TA, also called the _Open Coloring Axiom_) implies that all [[automorphisms]] of the Calkin algebra are [[inner automorphism|inner]] ([Farah 2011a](#Farah11a)), whereas assuming the [[continuum hypothesis]], one can construct an [[outer automorphism]] ([Phillips-Weaver 2007](#PhillipsWeaver)). Note that TA contradicts CH ([Rinot 2013](#Rinot13)), so these two options are mutually exclusive.

There are also some analogues of the above results where the separable Hilbert space is replaced by an arbitrary infinite-dimensional complex Hilbert space. For instance, the [[Proper Forcing Axiom]] (PFA) implies that for every Calkin algebra, all automorphisms are inner ([Farah 2011b](#Farah11b)).

Another result ([Farah and Hirshberg 2016](#FarahHirshberg16)) of Farah and Hirshberg states that the Calkin algebra of a separable Hilbert space is not countably [[homogeneous structure|homogeneous]]; in particular, it cannot be realized as a [[Fraïssé limit]] of [[continuous logic|metric structures]].

## Related entries

* [[Brown-Douglas-Fillmore theory]]

## References

* {#PhillipsWeaver} N. Christopher Phillips, [[Nik Weaver]], _The Calkin algebra has outer automorphisms_, Duke Math. J. **139** 1 (2007) 185-202 [[arXiv:math/0606594](https://arxiv.org/abs/math/0606594), [doi:10.1215/S0012-7094-07-13915-2](https://projecteuclid.org/journals/duke-mathematical-journal/volume-139/issue-1/The-Calkin-algebra-has-outer-automorphisms/10.1215/S0012-7094-07-13915-2.short)]

* {#Farah11a} Ilijas Farah, _All automorphisms of the Calkin algebra are inner_, Annals of Mathematics 173 (2011), 619-661, doi: 10.4007/annals.2011.173.2.1 [arXiv:0705.3085](https://arxiv.org/abs/0705.3085)
* {#Farah11b} Ilijas Farah, _All automorphisms of all Calkin algebras_, Math. Res. Lett. 18 (2011), no. 3, 489-503 [arXiv:1007.4034](https://arxiv.org/abs/1007.4034)

* {#FarahHirshberg16} Ilijas Farah and Ilan Hirshberg, _The Calkin algebra is not countably homogeneous_, Proc. AMS, to appear. [arXiv:1506.07455](https://arxiv.org/abs/1506.07455)

* {#Rinot13} Assaf Rinot, _Open coloring and the cardinal invariant $\mathfrak{b}$_, (2013) [blog post](http://blog.assafrinot.com/?p=3243)

* {#Weaver07} [[Nik Weaver]], _Set theory and $C^\ast$-algebras_, The Bulletin of Symbolic Logic Vol. 13, No. 1 (Mar., 2007), pp. 1-20, [arxiv:math/060198.LO](https://arxiv.org/abs/math/0604198)

category: operator algebras


call-by-push-value

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# Contents
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## Idea

Call-by-push-value is a [[type theory]] and [[programming language]] paradigm which has full embeddings of [[call-by-value]] and [[call-by-name]] lambda-calculi, including their [[operational semantics]], [[equational theories]] and [[denotational semantics|denotational]] models.

Its [[semantics]] decomposes the semantics of [[monad (in computer science)|effectful languages]] using a [[strong monad]] into a [[strong adjunction]]. Then the embeddings of call-by-value and call-by-name correspond to the construction of [[Kleisli category|Kleisli and co-Kleisli categories]] from an adjunction.

## Syntax

### Judgmental Structure

CBPV has two "kinds" of [[types]]: value types --- written $A,A',A_i$ --- and computation types --- written $B,B',B_i$. As originally presented. there are 3 term [[judgments]] of CBPV: Values, Terms, and Stacks.

The value judgment is an ordinary simple type theory. A context $\Gamma$ is a sequence of value typed variables $x_1:A_1,\ldots$ and the judgment $\Gamma \vdash V : A$ admits symmetry, contraction, weakening and the corresponding substitution principle: if for every $x_i:A_i$ in $\Gamma$, $\Gamma' \vdash \gamma(x_i) : A_i$, then there is a value $\Gamma' \vdash V[\gamma] : A$.

The term judgment has again a value typed context $\Gamma$, but the output type is a computation type $\Gamma \vdash M : B$. Again it admits symmetry, contraction and weakening, and admits a substitution principle on its inputs, if we have a $\gamma$ as above, then there is a term $\Gamma' \vdash M[\gamma] : B$.

Finally the stack judgment has a value context $\Gamma$ as input, but also a computation type as input and output: $\Gamma | B \vdash S : B'$. Stacks have an admissible substitution operation $S[\gamma]$ as above. Stacks also have a composition operation: if $\Gamma | B \vdash S : B'$ and $\Gamma | B' \vdash S' : B''$ then $\Gamma | B \vdash S'[S] : B''$, which is associative and has a unit: the empty stack $\Gamma | B \vdash \bullet : B$. Finally, stacks have an action on terms: if $\Gamma \vdash M : B$ and $\Gamma | B \vdash S : B'$ then $\Gamma \vdash S[M] : B'$, which is associative with $\bullet$ as identity.

We can get a slightly simpler presentation by combining the stack and term judgments into a single term judgment with a [[stoup]]: a context that is either a single type $B$ or empty. Then a term as before is a term typed with the empty stoup and a stack is a term typed with a full stoup. This has the benefit of combining the action of stacks on terms and the composition of stacks as one operation. Below we write a stoup as $\Delta$, and if it is empty we don't write it at all.

### Type Structure

Call-by-push-value is characterized not just by its judgmental structure, but by a specific *choice* of which connectives to use. Except for the shifts $F,U$, all value type connectives are left adjoints, and all computation type connectives are right adjoints.

For this reason, some connectives are definable from the judgments but excluded because they violate this discipline. These are a tensor product $A \otimes B$ which is a computation type, a unit $I$ computation type, and a linear function space $B \multimap B'$ which is a computation type. When these are added, the system is called the [Enriched Effect Calculus](#EEC) (modulo some superficial syntactic differences).

#### Shifts

The shift types express the adjunction between values and stacks.
The type $UB$ is a value type that is pronounced as "thunk of $B$", and can be thought of as the type of [[closures]] that when called behave as $B$.
It is a value type, but a right adjoint, with invertible introduction rule

$$ \frac { \Gamma \vdash M : B }
         { \Gamma \vdash thunk M : UB } $$

and elimination rule:

$$ \frac { \Gamma \vdash V : UB }
         { \Gamma \vdash force V : B } $$

With $\beta$ rule $force thunk M = M$ and $\eta$ rule $V = thunk force V$.

The type $FA$ is a computation type that is the type of "returners of $A$s".
It is a computation type, but a left adjoint, with invertible elimination rule that enables the sequencing of effects:

$$ \frac { \Gamma | \Delta \vdash M : F A \qquad \Gamma, x : A \vdash N : B}
         { \Gamma | \Delta \vdash M to x. N : B } $$

with introduction rule:

$$ \frac {\Gamma \vdash V : A } { \Gamma \vdash return V : F A } $$

with $\beta$ rule $(return V) to x. N = N[V/x]$ and $\eta$ rule that for any stack $\Gamma | F A \vdash S : B$, we have $S = \bullet to x. S[return x]$.

## As an Adjoint Logic

The semantics of Call-by-push-value was originally presented using fibred categories, but we can get a presentation of its semantics in a more multi-categorical flavor by expressing it as an [[adjoint logic]] from a specific mode theory in the sense of [LSR](#LSR).

As a summary, the mode theory of CBPV is the theory of a cartesian monoid acting on a pointed object.
In more detail, we have two modes: $v$ for values and $c$ for computations. Then the $v$ mode is axiomatized as a cartesian monoid $(v,\times,1)$, which gives the value judgment. To axiomatize the term and stack judgments, we add an asymmetric tensor product

$$a:v,b:c \vdash a\otimes b : c$$

to represent the combined context $\Gamma | B$ and also a "point" for the computation types:

$$\vdash i : c$$

which represents the empty stoup.

To correctly model the substitution structure, we add equations that say that the tensor product constitutes an action of the monoid, i.e., associativity and unitality:

$$(a \times a') \otimes b = a \otimes (a' \otimes b) \qquad 1 \otimes b = b$$

Then for example, the $F$ type above is $F_{x \otimes i}$ and the $U$ type is $U_{x. x \otimes i}$.

Furthermore, the mode theory also gives the "missing" types of the Enriched Effect Calculus. The tensor product is $F_{x \otimes y}$, the unit is $F_{i}$ and the linear function space is $U_{x. x\otimes y}$.

Note that this mode theory is equivalent to the one given for a strong monad in [LSR](#LSR), which instead of the point $i$ adds a morphism $a : v \vdash g(a) : c$, and requires this be a homomorphism of $v$-actions in that

$$ g(a \times b) = a \otimes g(b)$$

However, this is equivalent to our above presentation because for any $a:v$, we have $a = a \times 1$, so $g(a) = g(a \times 1) = a\otimes g(1)$, so $g$ is completely determined by where it takes $1$, so we can axiomatize that directly as the point $i$.

## Related

* The linear term judgment in [[linear-non-linear logic]] can be seen as extending the call-by-push-value stack judgment to allow for "multi-hole" stacks.

## References

* [[Paul-Blain Levy]], [_Call-by-Push-Value: A Subsuming Paradigm_, TLCA 1999](https://link.springer.com/chapter/10.1007/3-540-48959-2_17)

* [[Paul-Blain Levy]], _Adjunction Models for Call-by-push-value with Stacks_, [CTCS version](https://www.cs.bham.ac.uk/~pbl/papers/ctcs02journal.pdf), [EPTCS journal version](https://www.sciencedirect.com/science/article/pii/S1571066104805681), 2003

* {#EEC} Jeff Egger, Rasmus Ejlers Møgelberg and Alex Simpson, [The Enriched Effect Calculus: Syntax and Semantics](https://pdfs.semanticscholar.org/2f3f/21259ffaa5cc56bb280914a028cfed3454e2.pdf), Journal of Logic and Computation, Volume 24.

* {#LSR} [[Daniel Licata]], [[Mike Shulman]], and [[Mitchell Riley]], _A Fibrational Framework for Substructural and Modal Logics (extended version)_, in Proceedings of 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) ([doi: 10.4230/LIPIcs.FSCD.2017.25](http://drops.dagstuhl.de/opus/volltexte/2017/7740/), [pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/lsr17multi-ex.pdf))


Calogero model

# Overview

_Calogero_ models comprise several [[integrable system|integrable systems]] in classical mechanics and their relatives; they are describing interacting systems of $n$ particles on a circle or on a line. Basic variants include Calogero-Moser system, Calogero-Sutherland system and "relativistic" [[Ruijsenaars model|Ruijsenaars models]]. Calogero-Moser system was a historical source of the study of [[Dunkl operator|Dunkl operators]] and Cherednik's [[double Hecke algebra|double Hecke algebras]]. The Calogero-Moser system can also be related to the rational $n$-soliton solutions of rational KP equation; the elucidation of the relation between the soliton solutions and solutions of the Calogero-Moser system is rather deep and is called __Calogero-Moser correspondence__. Its modern formulation involves [[noncommutative algebraic geometry]].

Some special functions come out of analysis of Calogero models, like [[Jack polynomial|Jack polynomials]].

# Literature

* Pavel Etingof, _Lectures on Calogero-Moser systems_, [pdf](http://www-math.mit.edu/~etingof/zlecnew.pdf)
* [[D. Kazhdan]], [[B. Kostant]], [[S. Sternberg]], _Hamiltonian group actions and dynamical systems of Calogero type_, Comm. Pure Appl. Math. __31__ (1978), 481-508, [MR478225](http://www.ams.org/mathscinet-getitem?mr=478225), [doi](http://dx.doi.org/10.1002/cpa.3160310405)
* H. Airault, H. P. McKean, J. Moser, _Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem_, Comm. Pure Appl. Math. __30__ (1977), no. 1, 95--148; [MR0649926](https://mathscinet.ams.org/mathscinet-getitem?mr=649926), [doi](https://doi.org/10.1002/cpa.3160300106)
* E. Mukhin, V. Tarasov, A. Varchenko, _KZ characteristic variety as the zero set of classical Calogero-Moser Hamiltonians_, [arxiv/1201.3990](https://arxiv.org/abs/1201.3990)
* Sergio Andraus, Makoto Katori, Seiji Miyashita, _Calogero-Moser systems as a diffusion-scaling transform of Dunkl processes on the line_, [arxiv/1211.6561](https://arxiv.org/abs/1211.6561)
* scholarpedia [Calogero-Moser system](http://www.scholarpedia.org/article/Calogero-Moser_system)
* [[Yuri Berest]], Oleg Chalykh, _Deformed Calogero--Moser operators and ideals of rational Cherednik algebras_, Commun. Math. Phys. 400, 133--178 (2023) [doi](https://doi.org/10.1007/s00220-022-04595-4) [arXiv:2002.08691](https://arxiv.org/pdf/2002.08691)

On Calogero-Moser correspondence:

* I. M. Kričever, _Rational solutions of the Kadomcev-Petviašvili equation and the integrable systems of $N$ particles on a line_, Funkcional. Anal. i Priložen. __12__ (1978), no. 1, 76–78, [MR488139](https://mathscinet.ams.org/mathscinet-getitem?mr=488139)
* Takahiro Shiota, _Calogero-Moser hierarchy and KP hierarchy_, J. Math. Phys. __35__ (1994), no. 11, 5844--5849, [MR95i:58095](https://mathscinet.ams.org/mathscinet-getitem?mr=1299922)
* [[George Wilson]], _Collisions of Calogero-Moser particles and an adelic Grassmannian_, With an appendix by I. G. Macdonald. Invent. Math. __133__ (1998), no. 1, 1--41, [MR99f:58107](https://mathscinet.ams.org/mathscinet-getitem?mr=1626461), [doi](https://doi.org/10.1007/s002220050237)
* [[Yuri Berest]], [[George Wilson]], _Ideal classes of the Weyl algebra and noncommutative projective geometry_ (with an Appendix by [[M. Van den Bergh]]), Internat. Math. Res. Notices __26__ (2002), 1347--1396.
* [[Yuri Berest]], [[George Wilson]], _Mad subalgebras of rings of differential operators on curves_, Advances in Math. __212__ no. 1 (2007), 163--190.
* [[Victor Ginzburg]], _Non-commutative symplectic geometry, quiver varieties, and operads_, Math. Res. Lett. 8 (2001), 377-400.
* [[Yuri Berest]], Oleg Chalykh, _$A_\infty$-modules and Calogero-Moser spaces_, J. Reine Angew. Math. __607__ (2007), 69--112, [MR2009f:16019](https://mathscinet.ams.org/mathscinet-getitem?mr=2338121), [doi](https://doi.org/10.1515/CRELLE.2007.046)
[[!redirects Calogero models]]
[[!redirects Calogero-Moser model]]
[[!redirects Calogero-Moser system]]
[[!redirects Calogero-Sutherland model]]
[[!redirects Calogero integrable system]]


caloron

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## Idea

Calorons are topologically non-trivial [[field (physics)|field]]-configurations in [[Yang-Mills theory]] at [[thermal field theory|positive temperature]] $T \gt 0$. They correspond to what at vanishing [[temperature]] $T =  0$ are called [[instantons]], specifically [[BPST instantons]] (but there is no continuous limit relating the two).

While the physical reality of [[instantons]] is at best subtle (they are interpreted as witnessing [[quantum tunneling]] between actual [[vacua]]) calorons are meant to correspond to actual [[vacua]] of [[Yang-Mills theory]] at [[thermal field theory|positive temperature]], reflecting the general statement that [[Wick rotation]] has good physical meaning at positive temperature.

More in detail:

Upon [[Wick rotation]], $G$-[[Yang-Mills theory]] on [[Minkowski spacetime]] $\mathbb{R}^{3,1}$ in a [[KMS state]] of [[positive number|positive]] [[temperature]] $T$ is expressed by [[Euclidean field theory]] on the [[Riemannian manifold]] $\mathbb{R}^3 \times S^1_{\beta}$, where the first factor is 3-dimensional [[Euclidean space]] and the second factor is a [[circle]] of [[length]] $\beta \coloneqq 1/T$ (proportional to) the inverse [[temperature]].

A _caloron_ is a [[gauge field]]-configuration in this Euclidean Yang-Mills theory on $\mathbb{R}^3 \times S^1_\beta$ [[vanishing at infinity]]. By the [[clutching construction]] the topological class of these bundles (their [[second Chern class]] for [[gauge group]] a  [[simple Lie group]]) is given by [[homotopy classes]] of maps

$$
  S^2 \times S^1 \longrightarrow G
$$

hence

$$
  S^3 \times S^1 \longrightarrow B G
$$

For the case $G = $ [[special unitary group|SU(2)]] $\simeq S^3$ and via the usual math/physics translation of terminology ([here](BPST-instanton#FromTheMathsToThePhysicsStory)) this was first considered in [Harrington-Shepard 77, p. 3](#HarringtonShepard77), and baptized a "caloron"-configuration in [Harrington-Shepard 78, p. 1](#HarringtonShepard78).

A further refinement of the construction included the nonzero [[vacuum expectation value]] $\langle (A_4)^3\rangle$ of the "time" component of the [[vector potential]], called the _KvBLL caloron_ ([Kraan-van Baal 98a](#KraanVanBaal98a), [Kraan-van Baal 98b](#KraanVanBaal98b), [Lee-Lu 98](#LeeLu98)).

This solution revealed the substructure: it  gets  split  into $N_c$ (number  of  colors)  separate  (anti)self- dual  3d  solitons  with  nonzero  (Euclidean)  electric  and  magnetic charges (see e.g. [Larsen-Shuryak 14](#LarsenShuryak14)).

## Properties

### Relation to confinement
 {#RelationToConfinement}

As part of a possible solution to the [[confinement]]/[[mass gap]]-problem:

[Greensite 11, section 8.5](#Greensite11):

> it is natural to wonder if [[confinement]] could be derived from some [[semiclassical approximation|semiclassical]] treatment of [[Yang–Mills theory]] based on the [[instanton]] solutions of [[nonabelian group|non-abelian]] [[gauge theories]]. The [[BPST instanton|standard]] [[instantons]], introduced by Belavin et al. ([40](BPST-instanton#BelavinPolyakovSchwartzTyupkin75)), do not seem to work; their [[field strengths]] fall off too rapidly to produce the desired  magnetic  disorder in the vacuum.

> In  recent  years,  however, it has been realized that instanton solutions at [[thermal field theory|finite temperature]], known as _calorons_, might do the job. These caloron solutions were introduced independently by Kraan and van Baal  ([41](#KraanVanBaal98), [42](#KraanVanBaal98b)) and Lee and Lu  ([43](#LeeLu98)) (KvBLL), and they have the remarkable property of containing [[monopole]] constituents which may, depending on the type of caloron, be widely separated.

> $[...]$

> The caloron idea is probably the most promising current version of [[monopole]] [[confinement]] in pure non-abelian gauge theories, but it is  basically  (in certain [[gauge fixing|gauges]]) a superposition of [[monopoles]] with spherically  symmetric abelian fields, and this leads to the same questions raised in connection with monopole Coulomb gases.

### Relation to heterotic string theory

The _gauge solution_ of the [[NS5-brane]] in [[heterotic string theory]] contains a [[gauge field]] on the transverse space $\mathbb{R}^4$ which is a [[BPST-instanton]]. Then, if we consider an array of [[NS5-brane]]s at fixed distance along the circle of a compactified transverse space $\mathbb{R}^3\times S^1$, we obtain a Harrington-Shepard [[caloron]] (see e.g. [SasYat16, pag.12](#Sasaki-Yata16)). If we send the radius to zero $R\rightarrow 0$ we obtain a smeared [[NS5-brane]] in [[heterotic string theory]].

## History

The paper [Harrington-Shepard 1978](#Harrington-Shepard_78) considers the contribution from periodic solutions at _positive, finite_ temperature to the [[Yang-Mills equations]] to the action of a 'Yang-Mills gas': an equilibrium ensemble of YM fields. They give explicit solutions to the classical equations of motion 'of charge 1' i.e. representing a generator in the [[homotopy group]] classifying topologically inequivalent solutions.

[[Werner Nahm|Nahm]] ([Nahm 1984](#Nahm_84)) continued the study of calorons, linking them with self-dual [[monopoles]] and the [[ADHM construction]].

## Related concepts

* [[instanton]], [[soliton]], [[vortex]]

* [[thermal field theory]]

* [[caloron correspondence]]

* [[Nahm transform]]

## References

### General

The concept was introduced in

* {#HarringtonShepard77} Barry J. Harrington, Harvey K. Shepard, _Euclidean solutions and finite temperature gauge theory_, Nuclear Physics B Volume 124, Issue 4, 27 June 1977, Pages 409-412 (doi:10.1016/0550-3213(77)90413-8)

* {#HarringtonShepard78} Barry J. Harrington, Harvey K. Shepard, _Periodic Euclidean solutions and the finite-temperature Yang-Mills gas_, Phys. Rev. D 17, 2122 – April 1978 ([doi:10.1103/PhysRevD.17.2122](https://doi.org/10.1103/PhysRevD.17.2122))

Further development includes

* {#Nahm_84} W. Nahm, _Self-dual monopoles and calorons_, in Group Theoretical Methods in Physics, Ed. G. Denardo, G. Ghirardi, and T. Weber,  Lecture Notes in Physics **201** (1984) pp 189-200. doi:[10.1007/BFb0016145](http://dx.doi.org/10.1007/BFb0016145)

* {#KraanVanBaal98a} Thomas C. Kraan, Pierre van Baal, _Periodic Instantons with non-trivial Holonomy_, Nucl.Phys. B533 (1998) 627-659 ([arXiv:hep-th/9805168](https://arxiv.org/abs/hep-th/9805168))

* {#KraanVanBaal98b} Thomas C. Kraan, Pierre van Baal, _Exact T-duality between Calorons and Taub-NUT spaces_, Phys.Lett.B428:268-276, 1998 ([arXiv:hep-th/9802049](https://arxiv.org/abs/hep-th/9802049))

* {#LeeLu98} Kimyeong Lee, Changhai Lu, _$SU(2)$ Calorons and Magnetic Monopoles_, Phys. Rev. D 58, 025011 (1998) ([arXiv:hep-th/9802108](https://arxiv.org/abs/hep-th/9802108))

See also

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

Discussion in [[lattice gauge theory]]:

* P. Gerhold, E.‐M. Ilgenfritz, M. Müller‐Preussker, B. V. Martemyanov, A. I. Veselov, _Topology and confinement at $T \neq 0$: calorons with non‐trivial holonomy_, AIP Conference Proceedings 892, 213 (2007) ([doi:10.1063/1.2714375](https://doi.org/10.1063/1.2714375))

### Relation to skyrmions, instantons, monopoles

The construction of [[Skyrmions]] from [[instantons]] is due to

* {#AtiyahManton89} [[Michael Atiyah]], N S Manton, _Skyrmions from instantons_, Phys.  Lett.  B, 222(3):438–442, 1989 (doi:10.1016/0370-2693(89)90340-7)

The relation between [[skyrmions]], [[instantons]], [[calorons]], [[solitons]] and [[monopoles]] is usefully reviewed and further developed in

* {#Cork18a} [[Josh Cork]], _Calorons, symmetry, and the soliton trinity_, PhD thesis, University of Leeds 2018 ([web](http://etheses.whiterose.ac.uk/22097/))

* {#Cork18b} [[Josh Cork]], _Skyrmions from calorons_, J. High Energ. Phys. (2018) 2018: 137 ([arXiv:1810.04143](https://arxiv.org/abs/1810.04143))

### In confinement/mass gap

Discussion as part of a solution to the [[confinement]]/[[mass gap]]-problem:

* {#Greensite11} Jeff Greensite, _An Introduction to the Confinement Problem_, Lecture Notes in Physics, Volume 821, 2011 ([doi:10.1007/978-3-642-14382-3](https://link.springer.com/book/10.1007%2F978-3-642-14382-3))

* P. Gerhold, E.-M. Ilgenfritz, M. Müller-Preussker, _An $SU(2)$ KvBLL caloron gas model and confinement_, Nucl.Phys.B760:1-37, 2007 ([arXiv:hep-ph/0607315](https://arxiv.org/abs/hep-ph/0607315))

* {#LarsenShuryak14}  Rasmus Larsen, [[Edward Shuryak]], _Classical interactions of the instanton-dyons with antidyons_, Nucl. Phys. A **950**, 110 (2016) ([arXiv:1408.6563](https://arxiv.org/abs/1408.6563))

* {#LarsenShuryak15} Rasmus Larsen, [[Edward Shuryak]], _Interacting Ensemble of the Instanton-dyons and Deconfinement Phase Transition in the SU(2) Gauge Theory_, Phys. Rev. D 92, 094022, 2015 ([arXiv:1504.03341](https://arxiv.org/abs/1504.03341))

### Relation to NS5-brane in heterotic string theory

* {#SasakiYata16} Shin Sasaki and Masaya Yata, _Non-geometric five-branes in heterotic supergravity_,  JHEP **1611**, 064, 2016 ([arXiv:1608.01436](https://arxiv.org/abs/1608.01436))

[[!redirects calorons]]


caloron correspondence

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## Idea

For $G$ a [[topological group]] (or [[Lie group]]) and $P \to X$ a $G$-[[principal bundle]], we have that forming [[mapping spaces]] out of the [[circle]] yields a free [[loop group]]-[[principal bundle]] over the [[free loop space]] of $X$:

$$
  \array{
    \mathcal{L}G &\to& \mathcal{L} P
    \\
    && \downarrow
    \\
    && \mathcal{L} X
  }
  \,.
$$

In the special case that $X = Y \times S^1$ is a [[Cartesian product]] with the [[circle]], then one can consider the [[subspace]] of the [[free loop space]] of $X$ on those loops whose [[projection]] on the $S^1$-factor is the [[identity]]. This subspace is of course [[equivalence|equivalent]] to $Y$, giving a canonical inclusion

$$
  i \;\colon\; Y \hookrightarrow \mathcal{L} (Y \times S^1)
  \,.
$$

(Abstractly, this is the [[adjunct]] of the [[identity]] under the [[internal hom]]-[[adjunction]].)

Along this inclusion one can [[pullback|pull back]] the $\mathcal{L}G$-principal bundle over $\mathcal{L}X$. The **caloron correspondence** is the statement that if

$$
  \Omega_{Y} P \hookrightarrow \mathcal{L} P
$$

in turn is the subspace on those loops in $P$ which map $0 \in S^1$ to a chosen [[section]] of $P$ over $Y \times \{0\}$, then forming the pullback $i^\ast \Omega_Y P$ in

$$
  \array{
    i^\ast \Omega_Y P &\to& \Omega_Y P
    \\
    \downarrow && \downarrow
    \\
    Y &\stackrel{i}{\longrightarrow}& \mathcal{L}(Y \times S^1)
  }
$$

constitutes an [[equivalence of groupoids]] between that of $G$-[[principal bundles]] over $Y \times S^1$ and [[loop group]]-principal bundles over $Y$.

## Related concepts

* [[caloron]]

* [[Nahm transform]]

* [[double dimensional reduction]]

## References

The term "caloron correspondence" originates in

* H. Garland, [[Michael Murray]], _Kac-Moody monopoles and periodic instantons_. Comm. Math. Phys., 120(2):335–351, 1988.

A review and further developments are in

* [[Michael Murray]], [[Raymond Vozzo]], _The caloron correspondence and higher string classes for loop groups_, J. Geom. Phys., 60(9):1235-1250, 2010 ([arXiv:0911.3464](http://arxiv.org/abs/0911.3464))

See also

* Pedram Hekmati, [[Michael Murray]], [[Raymond Vozzo]], _The general caloron correspondence_, J. Geom. Phys., 62(2):224-241, 2012 ([arXiv:1105.0805](http://arxiv.org/abs/1105.0805))

[[!redirects caloron correspondences]]


Calvin R. Putnam

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=16918)

## Selected writings

On [[commutators]] in [[operator algebra]]:

* [[Calvin R. Putnam]], *Commutation Properties of Hilbert Space Operators and Related Topics*, Ergebnisse der Mathematik und ihrer Grenzgebiete **36**, Springer (1967) [[doi:10.1007/978-3-642-85938-0](https://doi.org/10.1007/978-3-642-85938-0)]

category: people

[[!redirects Calvin Putnam]]
[[!redirects C. R. Putnam]]


Camell Kachour

Dr Camell Kachour, Ph.D Macquarie University, Supervisors : Michael Batanin and Ross Street.

Education :

10/2013 PhD thesis, Macquarie University, Sydney. Title: Aspects of Globular Higher Category Theory

Talks :

02/2007 Université de Paris 7, Paris, SIC. Définition algébrique des cellules non-strictes

06/2008 Université du Littoral Côte d'Opale, Calais, CT2008. Operadic definition of the non-strict cells

01/09/2010 Macquarie University, Sydney. Steps toward the Operadic definition of the Weak ω-Category of Weak ω-Categories (part 1): Operadic approach of the weak higher transformations

08/09/2010 Macquarie University, Sydney. Steps toward the Operadic definition of the Weak ω-Category of Weak ω-Categories (part 2): Weak higher transformations as models for Theories

15/09/2010 Macquarie University, Sydney. Steps toward the Operadic definition of the Weak ω-Category of Weak ω-Categories (part 3): The Red Operad

14/09/2011 Macquarie University, Sydney. An algebraic approach to weak ω-groupoids

20/12/2011 Université de Paris 7, Paris, PPS seminar. Approche algébrique des ω-groupoïdes faibles

19/08/2012 Macquarie University, Sydney. The importance of ω-operad of coendomorphism in Higher Category Theory (part 1)

05/09/2012 Macquarie University, Sydney. The importance of ω-operad of coendomorphism in Higher Category Theory (part 2)

27/11/2014 USTHB, Algiers. Hommage Catégorique à Alexander Grothendieck

16/01/2015 Warsaw University, Warsaw. Algebraic Models of (∞,n)-Categories on the Globular Setting

05/06/2015 Université de Paris 7, Paris. Opérades supérieures de Batanin et Structures infini-graphiques

11/09/2015 Université de Paris 7, Paris. Vers l'infini-catégorie faible des infini-catégories faibles dans le contexte globulaire

International congress :

06/2008 CT2008, Calais, France.

10/2008 International Conference, Brussels, Belgium, Royal Flemish Academy.

06/2010 CT2010, Genova, Italia. 06/2014 CT2014, Cambridge, UK.

Published and Accepted Articles :

[1] Kamel Kachour, Définition algébrique des cellules non-strictes, Cahiers de Topologie et de Géométrie Différentielle Catégorique (2008), volume 1, pages 1–68.

[2] Camell Kachour, Operadic Definition of the Non-strict Cells, Cahiers de Topologie et de Géométrie Différentielle Catégorique (2011), volume 4, pages 1–48.

[3] Camell Kachour, Algebraic Definition of weak (∞,n)-Categories, Published in Theory and Applications of Categories (2015), Volume 30, No. 22, pages 775-807

[4] CamellKachour, ω-OperadsofCoendomorphismsandFractal ω-OperadsforHigherStructures., Published in Categories and General Algebraic Structures with Applications (2015).

[5] Camell Kachour, Operads of higher transformations for globular sets and for higher magmas, Published in Categories and General Algebraic Structures with Applications (2015).

[6] Camell Kachour, Steps toward the Weak ω-category of the Weak ω-categories in the globular setting, Published Categories and General Algebraic Structures with Applications (2015).

Works in Progress :

[7] Camell Kachour and Jacques Penon, Higher Operads for the Weak Higher Transformations and Stable Pseudo-Algebras, Work in progress (2015).

[8] Camell Kachour, Model of Higher Stacks on the Globular Setting, Work in Progress (2015).

Experience Teaching :

1996–2008 Examiner, Ministère de l'éducation nationale, Paris. Examiner for undergraduate level students (level for class elites "Special Mathematics" and "Superior Mathematics" in France)

Languages :

French Mother tongue; English Fluently
Interests : Music Mountain Hikking Japanese Judo

Camell Kachour,
67 rue Brancion, 75015 Paris, France.
camell.kachour@gmail.com


Cameron Smith

[website](http://cameronraysmith.net)

category:people


Camille Laurent-Gengoux

* [webpage](http://www-math.univ-poitiers.fr/~laurent/)

## Selected writings

On [[twisted equivariant K-theory]] via [[bundle gerbes]] over [[differentiable stacks]]:

* [[Jean-Louis Tu]], [[Ping Xu]], [[Camille Laurent-Gengoux]], _Twisted K-theory of differentiable stacks_, Annales Scientifiques de l’École Normale Supérieure Volume 37, Issue 6, November–December 2004, Pages 841-910 ([arXiv:math/0306138](https://arxiv.org/abs/math/0306138), [doi:10.1016/j.ansens.2004.10.002](https://doi.org/10.1016/j.ansens.2004.10.002))

On [[fiber bundles]] and [[principal bundles]] over [[Lie groupoids]] (e.g. [[orbifolds]]):

* [[Camille Laurent-Gengoux]], [[Jean-Louis Tu]], [[Ping Xu]], _Chern-Weil map for principal bundles over groupoids_, Math. Z. 255, 451–491 (2007) ([arXiv:math/0401420](https://arxiv.org/abs/math/0401420), [doi:10.1007/s00209-006-0004-4](https://doi.org/10.1007/s00209-006-0004-4))

## Related $n$Lab entries

* [[Lie-Poisson structure]]

category: people


Camilo Arias Abad

* [website](https://sites.google.com/a/unal.edu.co/camilo-arias-abad/)

## Selected writings

On [[infinity-local systems|$\infty$-local systems]] in the sense of [[(infinity,1)-vector bundles|$(\infty,1)$-vector bundles]] with [[flat infinity-connections|flat $\infty$-connections]]:

* [[Camilo Arias Abad]], [[Florian Schätz]]: *The $A_\infty$ de Rham theorem and integration of representations up to homotopy*, International Mathematics Research Notices, **2013** 16 (2013) 3790–3855 [[arXiv:1011.4693](https://arxiv.org/abs/1011.4693), [doi:10.1093/imrn/rns166](https://doi.org/10.1093/imrn/rns166)]

On [[braid group representations]]:

* [[Camilo Arias Abad]], *Introduction to representations of braid groups*, Rev. colomb. mat. vol.49 no.1 (2015) ([arXiv:1404.0724](https://arxiv.org/abs/1404.0724), [doi:10.15446/recolma.v49n1.54160](https://doi.org/10.15446/recolma.v49n1.54160))

On enhancing the [[Chern-Weil homomorphism]] from [[ordinary cohomology]]-groups to [[dg-categories]] of [[infinity-local systems]]:

* [[Camilo Arias Abad]], Santiago Pineda Montoya, Alexander Quintero Velez, *Chern-Weil theory for $\infty$-local systems* $[$[arXiv:2105.00461](https://arxiv.org/abs/2105.00461)$]$

category: people

[[!redirects Calimo Abad]]


Can Kozcaz

## Selected writings

On [[M-strings]]:

* [[Babak Haghighat]], [[Amer Iqbal]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _M-Strings_, Commun. Math. Phys. 334, 779–842 (2015) ([arXiv:1305.6322](https://arxiv.org/abs/1305.6322), [doi:10.1007/s00220-014-2139-1](https://doi.org/10.1007/s00220-014-2139-1))

* [[Babak Haghighat]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _On orbifolds of M-Strings_, Physical Review D 89.4 (2014): 046003 ([arXiv:1310.1185](https://arxiv.org/abs/1310.1185))

category: people


cancellative binary function

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## Idea

A version of [[cancellative monoid|cancellative]] [[magmas]] where the binary function's [[codomain]] need not  coincide with its [[domain]] set. This allows the concept to be generalised from [[magmas]] and [[monoids]] to other binary functions such as [[actions]] and [[modules]].

## Definition

Given sets $A$, $B$, and $C$, a binary function $f:A \times B \to C$ is **left cancellative** if for all $c \in B$ and $d \in B$, if $a \cdot c = a \cdot d$ for all $a \in A$, then $c = d$, and it is **right cancellative** if for all $a \in A$ and $b \in A$, if $a \cdot c = b \cdot c$ for all $c \in A$, then $a = b$. It is **cancellative** if it is both left cancellative and right cancellative.

## See also

* [[cancellative monoid]]
* [[torsion-free module]]
* [[integral domain]]

[[!redirects cancellative binary functions]]


cancellative category

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## Idea

The concept of a _(left/right) cancellative category_ is the generalization of the concept of [[cancellative monoid]] from [[monoids]] to [[categories]].

## Definition

In [[category theory]], a category $\mathcal{C}$ is **left cancellative** if all [[morphisms]] in $\mathcal{C}$ are [[monomorphisms]] (for arbitrary morphisms $f,h_0,h_1$ of $\mathcal{C}$, if $f\circ h_0=f\circ h_1$, then $h_0=h_1$). $\mathcal{C}$ is **right cancellative** if all [[morphisms]] in $\mathcal{C}$ are [[epimorphisms]] (for arbitrary morphisms $f,h_0,h_1$ of $\mathcal{C}$, if $h_0 \circ f=h_1\circ f$, then $h_0=h_1$). $\mathcal{C}$ is **cancellative** if it is both left cancellative and right cancellative.

## Examples

### Cancellative categories

 * any [[cancellative monoid]], regarded as a category with a single object

 * any [[groupoid]]

 * any [[poset]], or indeed any [[preorder]]

### Left cancellative categories

 * the category [[Field]] of [[fields]] (with [[ring]] [[homomorphisms]] as the morphisms)

 * the category of nontrivial [[vector spaces]] (over the field of [[real numbers]] or [[complex numbers]]) equipped with positive definite [[inner products]]

## Related concepts

* [[cancellative monoid]]

## References

* M. V. Lawson and A. R. Wallis, A categorical description of Bass-Serre theory ([arXiv:1304.6854v5](https://arxiv.org/abs/1304.6854))

* M. V. Lawson, "Ordered Groupoids and Left Cancellative Categories"
Semigroup Forum, Volume **68**, Issue 3, (2004),  458–-476

[[!include oidification - table]]

[[!redirects cancellative categories]]

[[!redirects left cancellative category]]
[[!redirects left cancellative categories]]
[[!redirects left-cancellative category]]
[[!redirects left-cancellative categories]]

[[!redirects right cancellative category]]
[[!redirects right cancellative categories]]
[[!redirects right-cancellative category]]
[[!redirects right-cancellative categories]]

[[!redirects all morphisms monic]]
[[!redirects all arrows monic]]
[[!redirects all morphisms epic]]
[[!redirects all arrows epic]]

[[!redirects all morphisms monic and epic]]
[[!redirects all arrows monic and epic]]
[[!redirects all morphisms epic and monic]]
[[!redirects all arrows epic and monic]]


> This entry is about regular elements in [[ring theory]] and [[commutative algebra]]. For regular elements in [[formal logic]] and [[topology]], see [[regular element]]. For regular elements in [[physics]]/[[quantum field theory]] see at _[[regularization (physics)]]_.

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## Definition

Given a [[commutative ring]] $R$, an [[element]] $e \in R$ is *left cancellative* or *left regular* if $e \cdot a = e \cdot b$ for all $a \in R$ and $b \in R$, then $a = b$.

$$\mathrm{isLeftCancellative}(e) \coloneqq \forall a,b \in R.(e \cdot a = e \cdot b) \implies (a = b)$$

An element $e \in R$ is *right cancellative* or *right regular* if $a \cdot e = b \cdot e$ for all $a \in R$ and $b \in R$, then $a = b$.

$$\mathrm{isRightCancellative}(e) \coloneqq \forall a,b \in M.(a \cdot e = b \cdot e) \implies (a = b)$$

An element $e \in R$ is *cancellative* or *regular* if it is both left cancellative and right cancellative.

$$\mathrm{isCancellative}(e) \coloneqq \mathrm{isLeftCancellative}(e) \wedge \mathrm{isRightCancellative}(e)$$

## Multiplicative subset of cancellative elements ##

The **multiplicative subset of cancellative elements** in $R$ is the [[multiplicative subset]] of all cancellative elements in $R$

$$\mathrm{Can}(R) \coloneqq \{e \in M \vert \mathrm{isCancellative}(e)\}$$

The term 'cancellative element' could be replaced with the synonym 'regular element'. If the synonym 'regular element' is used in place of 'cancellative element', such as in [Lombardi & Quitté 2010](#LombardiQuitté2010), then this multiplicative subset is typically written as $\mathrm{Reg}(R)$.

This is also called the **[[filter of a ring|filter]] of regular elements** in $R$, as in [Lombardi & Quitté 2010](#LombardiQuitté2010). Similarly as above, 'regular element' could be replaced with 'cancellative element'.

Since the multiplicative identity element $1$ is always cancellative, the multiplicative subset of all cancellative elements forms a [[cancellative monoid]].

## Properties ##

\begin{proposition}
For a commutative ring $R$, given elements $a, b \in R$, if the product $a \cdot b$ is cancellative, then both $a$ and $b$ are cancellative.
\end{proposition}

By this proposition, a commutative ring $R$ is a [[approximate integral domain|strict approximate integral domain]] if the following condition holds:

* For all elements $a, b \in R$, if the $a + b$ is cancellative, then either $a$ is cancellative or $b$ is cancellative.

\begin{theorem}
For a [[approximate integral domain|strict approximate integral domain]] $R$, the addition and multiplication operations on $R$ are strongly extensional with respect to the canonical apartness relation $\#$ defined by $x \# y$ iff $x - y$ is cancellative.

In this way $R$ becomes an internal commutative [[ring object]] in the category $\mathrm{Apart}$, consisting of sets with apartness relations and maps (strongly extensional functions) between them.
\end{theorem}

\begin{proof}
Recall that products $X \times Y$ in the category of sets with apartness relations is the cartesian product of the underlying sets equipped with the apartness relation defined by $(x, y) \# (x', y')$ iff $x \# x'$ in $X$ *or* $y \# y'$ in $Y$. Recall also that a function $f: X \to Y$ between sets with apartness relations is *strongly extensional* if $f(x) \# f(y)$ implies $x \# y$.

For addition, for all elements $a \in R$, $b \in R$, $a' \in R$, $b' \in R$, if $(a + b) \# (a' + b')$, then $a + b - (a' + b') = (a - b') + (a - b')$ is cancellative, so $a - a'$ or $b - b'$ is cancellative since for all elements $a \in R$ and $b \in R$, if the sum $a + b$ is cancellative, then either $a$ is cancellative or $b$ is cancellative, whence $(a, b) # (a', b')$. Thus addition is strongly extensional.

For multiplication, for all elements $a \in R$, $b \in R$, $a' \in R$, $b' \in R$, if $a \cdot b # a' \cdot b'$, then $a \cdot b - a' \cdot b'$ is cancellative. Write $a \cdot b - a' \cdot b' = (a - a') \cdot b + a' \cdot (b - b')$. Since for all elements $a \in R$ and $b \in R$, if the sum $a + b$ is cancellative, then either $a$ is cancellative or $b$ is cancellative, either $(a - a') \cdot b$ is cancellative or $a' \cdot (b - b')$ is cancellative. Since for all elements $a \in R$ and $b \in R$, if $a \cdot b$ is cancellative, then $a$ is cancellative and $b$ is cancellative, either $a - a'$ is cancellative or $b - b'$ is, whence $(a, b) # (a', b')$. So multiplication is also strongly extensional.
\end{proof}

### Relation with zero-divisors ###

\begin{theorem}
Given a commutative ring $R$, if an element $e \in R$ is a cancellative element, then it is a non-[[zero-divisor]], where non-zero-divisor is defined as an element $a \in R$ such that for every element $b:R$, $b \cdot a = 0$ or $a \cdot b = 0$ implies that $b = 0$.
\end{theorem}

\begin{proof}
Suppose that $e \in R$ is cancellative. This means that for all elements $a \in R$ and $b \in R$, $a \cdot e = b \cdot e$ implies that $a = b$ and $e \cdot a = e \cdot b$ implies that $a = b$. For the first equation, subtracting $b \cdot e$ from both sides of the equation leads to $(a - b) \cdot e = 0$, and for the second equation, subtraction $e \cdot b$ from both sides of the equation leads to $e \cdot (a - b) = 0$. Subtracting $b$ from both sides of $a = b$ leads to $a - b = 0$. Defining the element $c = a - b$ results in the condition that for every element $c \in R$, $c \cdot e = 0$ implies that $c = 0$ and $e \cdot c = 0$ implies that $c = 0$, which implies that $e$ is a non-[[zero-divisor]].
\end{proof}

\begin{theorem}
Given a commutative ring $R$, if an element $e \in R$ is a non-zero-divisor, then it is cancellative.
\end{theorem}

\begin{proof}
Suppose that $e \in R$ is a non-zero-divisor. This means that for every element $c \in R$, $c \cdot e = 0$ or $e \cdot c = 0$ implies that $c = 0$. However, since $R$ is a commutative ring, if $c \cdot e = 0$, then $e \cdot c = 0$, so the statement $c \cdot e = 0$ or $e \cdot c = 0$ implies that $c = 0$ implies the statement that $c \cdot e = 0$ implies that $c = 0$ and $e \cdot c = 0$ implies that $c = 0$. Since $R$ is an [[abelian group]], by definition of an abelian group, the [[image]] of the binary subtraction function $a - b$ is $R$ itself, and thus, one could replace $c$ with $a - b$ for elements $a \in R$ and $b \in R$, resulting in the statement that $(a - b) \cdot e = 0$ implies that $(a - b) = 0$ and $e \cdot (a - b) = 0$ implies that $(a - b) = 0$. Adding $b \cdot e$ to each side of the first equation, $e \cdot b$ to each side of the second equation, and $b$ to each side fo the third equation leads to the statement that $a \cdot e = b \cdot e$ implies that $a = b$ and $e \cdot a = e \cdot b$ implies that $a = b$, which is precisely the definition of cancellative element. Thus, every non-zero-divisor is a cancellative element.
\end{proof}

Thus, we have established that cancellative elements and non-zero-divisors are the same thing in a commutative ring. However, the proof relies on the [[abelian group]] structure of commutative rings, and this property does not necessarily hold in other algebraic structures where the concepts of cancellative element and non-zero-divisor make sense, such as in [[rigs]] or [[absorption monoids]].

Since a [[zero-divisor]] is defined in the nLab as not being a non-zero-divisor,

\begin{theorem}
In a commutative ring $R$, an element $e \in R$ is a [[zero-divisor]] if and only if it is non-cancellative
$$\mathrm{isZeroDivisor}(e) \iff \neg \mathrm{isCancellative}(e)$$
\end{theorem}

### Integral domains ###

The theorems relating cancellative elements to zero-divisors provide alternative definition of the various (commutative) [[integral domains]] in constructive mathematics in terms of cancellative elements, in analogy with the definition of [[fields]] in terms of invertible elements:

\begin{definition}
A commutative ring $R$ is a **integral domain** if an element is non-cancellative (or equivalently, a zero-divisor) iff it is zero. In addition to $0\neq 1$, this condition means that every non-cancellative element (or equivalenty, zero-divisor) is zero.
\end{definition}

\begin{definition}
A commutative ring $R$ is a **Heyting integral domain** if it is an integral domain and additionally, for all elements $a \in R$ and $b \in R$, if the sum $a + b$ is cancellative, then either $a$ is cancellative or $b$ is cancellative.
\end{definition}

\begin{theorem}
In addition to $0 \# 1$, the above condition in a Heyting integral domain then means that every element apart from $0$ is cancellative.
\end{theorem}

\begin{definition}
A commutative ring $R$ is a **discrete integral domain** if all elements $e \in R$ are cancellative [[xor]] equal to zero. This condition means that every element is either $0$ or cancellative, and it also implies that $0 \neq 1$.
\end{definition}

### Fields ###

Given the above definitions of an integral domain, a [[field]] could be defined as an integral domain where every cancellative element is a [[unit]], or equivalently, an integral domain whose multiplicative subset of cancellative elements is the group of units.

### Ring of fractions ###

For every commutative ring $R$, the [[ring of fractions]] is defined in [Quinn2009](#Quinn2009) to be the [[localization of a commutative ring|localization]] of $R$ at the multiplicative subset of cancellative elements, $R[\mathrm{Can}(R)^{-1}]$. This is similar to the [[Grothendieck group]] construction of a general [[cancellative monoid]]: the multiplicative subset of cancellative elements in $R[\mathrm{Can}(R)^{-1}]$ is the [[group of units]] $R[\mathrm{Can}(R)^{-1}]^\times$ in $R[\mathrm{Can}(R)^{-1}]$.

If $R$ is an [[integral domain]], then $R[\mathrm{Can}(R)^{-1}]$ is the [[field of fractions]] of $R$.

### Unique factorization domains ###

In [Lombardi & Quitté 2010](#LombardiQuitté2010), a [[unique factorization domain]] is defined as a [[GCD domain]] $R$ for which the [[quotient object|quotient]] monoid of the multiplicative subset of cancellative elements by the group of units $\mathrm{Can}(R)/R^\times$ admits a complete factorization.

## See also ##

* [[multiplicative subset]]

* [[filter of a ring]]

* [[cancellative monoid]]

* [[integral domain]]

* [[ring of fractions]]

* [[field of fractions]]

* [[group of units]]

* [[field]]

* [[torsion-free module]]

* [[GCD ring]]

* [[prefield ring]]

## References

* {#LombardiQuitté2010} [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

* {#Quinn2009} [[Frank Quinn]], *Proof Projects for Teachers* (2009) [[pdf](https://personal.math.vt.edu/fquinn/education/pfs4teachers0.pdf), [[Quinn-ProofProjects.pdf:file]]]

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cancellative midpoint algebra

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## Idea

The idea of a cancellative midpoint algebra comes from [[Peter Freyd]].

## Definition

A __cancellative midpoint algebra__ is a [[midpoint algebra]] $(M,\vert)$ that satisfies the cancellative property:

* for all $a$, $b$, and $c$ in $M$, if $a \vert b = a \vert c$, then $b = c$

## Examples

The [[rational numbers]], [[real numbers]], and the [[complex numbers]] with $a \vert b \coloneqq \frac{a + b}{2}$ are examples of cancellative midpoint algebras.

The [[trivial group]] with $a \vert b = a \cdot b$ is a cancellative midpoint algebra.

## Related concepts

* [[midpoint algebra]]

* [[symmetric cancellative midpoint algebra]]

* [[closed midpoint algebra]]

## References

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

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


cancellative monoid

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## Definition

### In set theory

A [[monoid]] $(A, \cdot, 1)$ is called _left cancellative_ if

$$
  \underset{a,b,z \in A}{\forall}
  \left(
    \left(
      z \cdot a = z \cdot b
   \right)
     \Rightarrow
   \left(
     a = b
   \right)
  \right)
$$

and called _right cancellative_ if

$$
  \underset{a,b,z \in A}{\forall}
  \left(
    \left(
      a \cdot z = b \cdot z
   \right)
     \Rightarrow
   \left(
     a = b
   \right)
  \right)
$$

It is called _cancellative_ if it is both _left cancellative_ and _right cancellative_.

### In infinity-groupoid theory

A [[monoid]] ([[n-truncated|$0$-truncated]] [[An-space|$A_3$-space]]) $(A, \cdot, 1)$ is called _left cancellative_ if for all objects $a \in A$ and $b \in A$ the [[homotopy fiber]] of the functor $L:A \to A$, defined as $L_a(z) \coloneqq a \cdot z$, at $b$ is $(-1)$-truncated, and is called _right cancellative_ if for all elements $a \in A$ and $b \in B$ the [[homotopy fiber]] of the functor $R:A \to A$, defined as $R_a(z) \coloneqq z \cdot a$ at $b$ is $(-1)$-truncated. It is called _cancellative_ if it is both _left cancellative_ and _right cancellative_.

## Related concepts

* [[Grothendieck group]]
* [[cancellative category]]
* [[integral monoid]]
* [[GCD ring]]
* [[multiplicative submonoid of cancellative elements]]
* [[cancellative binary function]]

[[!include oidification - table]]

## References

See also

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

[[!redirects cancellative monoids]]

[[!redirects cancellative semi-group]]
[[!redirects cancellative semi-groups]]

[[!redirects left cancellative]]
[[!redirects left-cancellative]]

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[[!redirects right-cancellative]]


canonical

See:

* [[canonical bundle]],
* [[canonical commutation relation]],
* [[canonical coordinate]],
* [[canonical extension]],
* [[canonical form]],
* [[canonical Hilbert space of half-densities]],
* [[canonical hypergroup]],
* [[canonical model structure]],
* [[canonical model structure on categories]],
* [[canonical model structure on operads]],
* [[canonical model structure on groupoids]],
* [[canonical momentum]],
* [[canonical morphism]],
* [[canonical presentation]],
* [[canonical relation]],
* [[canonical topology]],
* [[canonical transformation]].

category: adjective

[[!redirects canonical]]


canonical bundle

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## Idea

For $X$ a [[space]] with a notion of _[[dimension]]_ $n = dim(X) \in \mathbb{N}$ and a notion of ([[Kähler differential form|Kähler]]) [[differential forms]] on it, the _canonical bundle_ or [[canonical sheaf]] over $X$ is the [[line bundle]] (or its  [[sheaf]] of [[sections]]) of $n$-forms on $X$, the $dim(X)$-fold [[exterior product]]

$$
  L_{can} \coloneqq \Omega^n_X
$$

of the bundle $\Omega^1_X$ of 1-forms.

The [[first Chern class]] of this bundle is also called the **canonical [[characteristic class]]** or just the **canonical class** of $X$.

The inverse of the canonical line bundle (i.e. that with minus its [[first Chern class]]) is called the _anticanonical line bundle_.

Over an [[algebraic variety]], the [[divisor (algebraic geometry)|divisor]] corresponding to the canonical line bundle is called the _[[canonical divisor]]_.

A [[square root]] of the canonical class, hence another characteristic class $\Theta$ such that the [[cup product]] $2 \Theta = \Theta \cup \Theta$ equals the canonical class is called a _[[Theta characteristic]]_ (see also _[[metalinear structure]]_).

## Examples

### In complex analytic geometry
 {#InComplexAnalysis}

For $X$ [[complex manifold]] regarded over the complex numbers, then [[Kähler differential forms]] are _holomorphic_ forms. Hence the canonical bundle for $dim_{\mathbb{C}}(X) = n$ is $\Omega^{n,0}$ (see also at _[[Dolbeault complex]]_), a [[complex line bundle]].

For $X$ a [[Riemann surface]] of [[genus of a surface|genus]] $g$, the [[degree of a coherent sheaf|degree]] of the canonical bundle is $2 g - 2$. This means it is divisible by 2 and hence there are "[[Theta characteristic]]" square roots.

In particular the [[first Chern class]] of the canonical bundle on the [[2-sphere]] is _twice_ that of the [[basic line bundle on the 2-sphere]], the generator in $H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z}$. See also at _[[geometric quantization of the 2-sphere]]_.

## Related concepts

* [[density bundle]]

* [[volume form]]

* [[Kodaira vanishing theorem]]

[[!include square roots of line bundles - table]]

## References

In the context of [[algebraic geometry]]:

* Vladimir Lazić _Lecture 7. Canonical bundle, I and II_ (2011) ([pdf I](http://www.staff.uni-bayreuth.de/~btm113/AGlect7.pdf), [pdf II](http://www.staff.uni-bayreuth.de/~btm113/AGlect8.pdf))

See also

* Wikipedia, _[Canonical bundle](http://en.wikipedia.org/wiki/Canonical_bundle)_

[[!redirects canonical bundles]]

[[!redirects canonical class]]
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canonical commutation relation

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## Idea

In contexts related to [[quantum mechanics]] and [[quantum field theory]], by the "canonical commutation relations" (CCR) one refers to the [[commutator]] relations in [[Weyl algebras]], i.e. [[associative algebras]] generated from elements $\{a_k, a^\ast_k\}_{k \in K}$ subject to the "canonical" expressions for the [[commutators]] $[a,b] \coloneqq a \cdot b - b \cdot a$

$$
  \underset{i,j \in K}{\forall} \left(  [a_i, a_j] = 0 = [a^\ast_i, a^\ast_j] \right)
$$

$$
  \underset{i,j \in K}{\forall}\left( [a_i, a^\ast_j] = diag((a_k))_{i, j} \right)
  \,,
$$

where $diag((a_k))$ is some [[diagonal matrix]] with entries $(a_k)_{k \in K}$.

The archetypical example is the [[deformation quantization]] of the simple [[phase space]] which is the [[symplectic vector space]] $\mathbb{R}^2$ equipped with the [[symplectic form]] $\omega = \left( \array{ 0 & -1 \\ 1 & 0 } \right)$.

The resulting algebra is equivalently the [[quotient]] of the [[universal enveloping algebra]] of the [[Heisenberg Lie algebra]] $h_2$ which identifies the [[center|central element]] with a multiple of the 1 (the multiplicative neutral element, see at _[[polynomial Poisson algebra]]_ for more).

More concretely, in the [[quantization]] of a single particle propagating on the [[real line]] the [[Hilbert space]] of [[quantum states]] is identified with the the space of [[square integrable functions]] $L^2(\mathbb{R})$. On this the operators

$$
  a \coloneqq \tfrac{1}{\sqrt{2}}\left(x + i \hbar \frac{\partial}{\partial x} \right)
  \phantom{AAAAA}
  a^\ast \coloneqq \tfrac{1}{\sqrt{2}}\left(x - i \hbar \frac{\partial}{\partial x}\right)
$$

act (where "$x$" denotes the operator that multiplies a function with the canonical coordinate function, and $\frac{\partial}{\partial x}$ is the operator that forms the [[derivative]] with respect to this coordinate).

These operators satisfy the canonical commutation relations with

$$
  [a, a^\ast] = i \hbar
$$

If the particle being quantized here is equipped with [[Hamiltonian]] that represents the [[energy]] of a [[harmonic oscillator]], then one may show that the operator $a$ has the interpretation of removing one quantum of energy from the oscillator, while $a^\ast$ has the interpretation of adding one quantum.

(Accordingly the CCR relations in this case have been argued to be related to the [[combinatorics]] of placing a ball into a box and removing a ball from a box.)

More generally, in the [[quantum field theory]] of the [[free fields|free]] [[scalar field]] on [[Minkowski spacetime]] of [[dimension]] $d+1 \in \mathbb{N}$, each [[Fourier transform|Fourier mode]] [[amplitude]] $a_k$ of the field behaves independently like a [[harmonic oscillator]] and hence the [[Wick algebra]] of [[quantum observables]] of this free field is a [[Weyl algebra]] with a [[countable set]] $\{a_k, a^\ast_k\}_{k \in \mathbb{Z}^d}$ of generator, subject to the "canonical commutation relations"

$$
  [a_k, a^\ast_{k'}] = i \hbar \delta_{k, k'}
$$

(where on the right we have the [[Kronecker delta]]). Now $a_k$ is interpreted as having the effect of "annihilating" a paticle/quantum in mode $k$, while $a_k^\ast$ has the effect of "creating" one.

Therefore operators satisfying the "canonical commutation relations" are often referred to as (particle) _creation and annihilation operators_.

One a curved [[spacetime]] these relations become more complicated, see at _[[Wick algebra]]_ for more.

If the field in question is not a [[bosonic field]] but a [[fermionic field]] then all of the above has to be understood in [[superalgebra]] with the fermionic variabled in off super-degree. This yields [[anti-commutator]] relations as above, hence often called "canonical anti-commutation relations".

Under passing to [[exponentials]] the canomical commutation relations are also called the _[[Weyl relations]]_.

## Properties

* The [[Stone-von Neumann theorem]] says that for finitely many generators the canonical commutation relations (in the form of the [[Weyl relations]]) have, up to [[isomorphism]], a unique [[irreducible representation|irreducible]] [[unitary representation]]: the _[[Schrödinger representation]]_.

  [[Haag's theorem]] says that this uniqueness fails for infinitely many generators.

## Related concepts

* [[Weyl relation]], [[Weyl algebra]]

* [[Heisenberg algebra]]

* [[Stone-von Neumann theorem]]

* [[Wick algebra]]

* [[singleton representation]]

## References

Original discussion:

* {#Weyl27} [[Hermann Weyl]], (36) in: *Quantenmechanik und Gruppentheorie*, Zeitschrift für Physik **46** (1927) 1–46 [[doi:10.1007/BF02055756](https://doi.org/10.1007/BF02055756)]

More on the history of the notion:

* [[Severino C. Coutinho]], Introduction to: *A primer of algebraic D-modules*, London Math. Soc. Stud. Texts **33**, Cambridge University Press (1995) [[doi:10.1017/CBO9780511623653](https://doi.org/10.1017/CBO9780511623653)]

* [[Severino C. Coutinho]], *The Many Avatars of a Simple Algebra*, The American Mathematical Monthly **104** 7 (1997) 593-604 [[doi:10.2307/2975052](https://doi.org/10.2307/2975052), [jstor:2975052](https://www.jstor.org/stable/2975052)]

Monograph on [[commutators]] in [[operator algebra]]:

* [[Calvin R. Putnam]], *Commutation Properties of Hilbert Space Operators and Related Topics*, Ergebnisse der Mathematik und ihrer Grenzgebiete **36**, Springer (1967) [[doi:10.1007/978-3-642-85938-0](https://doi.org/10.1007/978-3-642-85938-0)]

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canonical divisor

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## Idea

Given an [[algebraic variety]], its _canonical divisor_ is the [[divisor (algebraic geometry)|divisor]] corresponding to  its [[canonical line bundle]].

Similarly, its _anticanonical divisor_ is the additive inverse of the canonical divisor in the group of divisors, hence the divisor corresponding to the [[anticanonical line bundle]].

## References

See also:

* Wikipedia, _[Canonical bundle](https://en.m.wikipedia.org/wiki/Canonical_bundle)_

[[!redirects canonjcal divisors]]

[[!redirects anticanonical divisor]]
[[!redirects anticanonical divisors]]


canonical extension

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## Idea

Canonical extension provides an algebraic formulation of [[duality]] theory and a tool to derive representation theorems. It may be regarded as an algebraic formulation of [[Stone duality]] (see [Gehrke-Vosmaer, p. 5](#GehrkeVosmaer)).

Originally ([Jonsson-Tarski](#JonssonTarski)) is was formulated for [[Boolean algebras]] with operators, but the notion was later generalized to [[distributive lattices]] and even to arbitrary [[posets]].

## Definition

### For distributive lattices

Given a [[distributive lattice]] $L$, its **canonical extension** $L^\delta$ is the [[downset]] [[lattice]] of the [[poset]] of [[prime filters]] of $L$, ordered by inclusion.

This construction extends to a [[functor]]

$$
  (-)^\delta : DistLattice \to DistLattice^+
$$

from the [[category]] of [[distributive lattices]] to that of [[completely distributive lattice|completely distributive]] [[algebraic lattices]]. This is [[left adjoint]] to the corresponding [[forgetful functor]], exhibiting [[completely distributive algebraic lattices]] as a [[reflective subcategory]] of the distributive lattices

$$
  DistLattice^+ \stackrel{\overset{(-)^\delta}{\leftarrow}}{\hookrightarrow}
  DistLattice
  \,.
$$

### For Heyting algebras
 {#ForHeytingAlgebras}

The canonical extension $L^\delta$ of a [[distributive lattice]] $L$ is a [[complete lattice|complete]] and [[completely distributive lattice]].

In particular the canonical extension is a [[Heyting algebra]]. If $L$ is itself already a Heyting algebra, then $e_L : L \to L^\delta$ preserves the Heyting [[implication]]. Also, canonical extension preserves homomorphisms of Heyting algebras. Hence it restricts to a functor

$$
  (-)^\delta : HeytingAlgebra \to HeytingAlgebra
  \,.
$$

### For coherent categories
 {#ForCoherentCategories}

A [[distributive lattice]] is a [[coherent category|cogerent]] [[(0,1)-category]]. One may therefore ask if there is a generalization of canonical extension to general [[coherent categories]].

The following is considered in ([Coumans](#Coumans)).

By the discussion at [[coherent category]] and [[coherent hyperdoctrine]], we have a [[reflective sub-2-category]] embedding

$$
 (\mathcal{A} \dashv \mathcal{S}) : CoherentCat \hookrightarrow CoherentHyperdoctrine
$$

given by sending a coherent category $C$ to its [[indexed category|self-indexing]] $c \mapsto C_{/c}$.

Since a [[coherent hyperdoctrine]] takes values in [[distributive lattices]], we can apply canonical extension of distributive lattices termwise to get a functor

$$
  (-)^\delta : CoherentHyperdoctrine \to CoherentHyperdoctrine
  \,.
$$

Define then canonical extension of coherent categories to be the [[2-functor]] induced from this under the above reflection:

$$
  (-)^\delta : CoherentCat
   \stackrel{\mathcal{S}}{\to}
  CoherentHyperdoctrine
    \stackrel{(-)^\delta}{\to}
  CoherentHyperdoctrine
    \stackrel{\mathcal{A}}{\to}
  CoherentCat
  \,.
$$

Under the restriction along the inclusion $DistLat \hookrightarrow CoherentCat$ this reproduces the canonical extension of distributive lattices: for $L$ a dsitributive lattice there is an equivalence

$$
  \mathcal{A}(\mathcal{S}_L^\delta)
  \simeq
  L^\delta
  \,.
$$

(This is ([Coumans, prop 12](#Coumans))).

### For Heyting categories

There is also a joint generalization of canonical extension for Heyting algebras ([here](#ForHeytingAlgebras)) and for coherent categories ([here](#ForCoherentCategories)).

Write $HeytingCat$ for the [[2-category]] of [[Heyting categories]], and $FirstOrderHyperdoctrine$ for the 2-category of [[first-order hyperdoctrines]] (which are, in particular, [[hyperdoctrines]] with values in [[Heyting algebras]]).

Then the above restricts to a [[reflective sub-2-category]] inclusion

$$
  (\mathcal{A} \dashv \mathcal{S})
  :
  HeyteingCat
   \hookrightarrow
  FirstOrderHyperdoctrine
$$

This is ([Coumans, prop. 19](#Coumans)).

And the canonical extension of coherent categories accordingly restricts to a functor on Heyting categories

$$
  (-)^\delta : HeytingCat \to HeytingCat
  \,.
$$

Such that for $C$ a Heyting category, also the [[unit of an adjunction|unit]] $C \to C^\delta$ is a morpjism of Heyting categories.

This is ([Coumans, cor. 21](#Coumans)).

## Related concepts

* [[topos of types]]

## References

The study of canonical extensions originates in the articles

* B. Jónsson, [[Alfred Tarski]], _Boolean algebras with operators, I_, Amer. J. Math. 73 (1951), 891–939.
 {#JonssonTarski}

Reviews include

* Mai Gehrke, Jacob Vosmaer, _A view of canonical extension_ ([arXiv:1009.2803](http://arxiv.org/abs/1009.2803))
 {#GehrkeVosmaer}

* Dion Coumans, _Generalizing canonical extensions to the categorical setting_, [pdf](http://www.math.ru.nl/~coumans/Generalisingcanonicalextensiontothecategoricalsetting-DionCoumans-vs2.pdf), to appear in Annals of Pure and Applied Logic
 {#Coumans}

[[!redirects canonical extensions]]


canonical form

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## Idea

Systems of _formal_ [[logic]], such as [[type theory]], try to transform expressions into a _canonical form_ which then serves as the end result of the given [[computation]] or [[deduction]]. A formal system is said to enjoy _canonicity_ if every expression reduces to canonical form.

More precisely, in [[type theory]], a [[term]] belonging to some [[type]] is said to be of **canonical form** if it is explicitly built up using the constructors of that type.  A canonical form is in particular a [[normal form]] (one not admitting any reductions), but the converse need not hold.

For example, the terms of type $\mathbb{N}$ (a [[natural numbers type]]) of canonical form are the [[numerals]]
$$ S(S(S(\cdots (0)\cdots ))). $$

A type theory is said to enjoy **canonicity** if every term computes to a canonical form.  This is held to be an important meta-theoretic property of type theory, especially considered as a programming language or as a computational foundation for mathematics.

## Canonicity vs axioms

Adding [[axioms]] to type theory, such as the principle of [[excluded middle]] or the usual version of the [[univalence axiom]], can destroy canonicity.  The axioms result in "stuck terms" which are not of canonical form, yet neither can they be "computed" any further.

For instance, if we assume the law of excluded middle, then we can build a term $case(LEM(Goldbach),0,1) \colon \mathbb{N}$ which is $0$ or $1$ according as the Goldbach conjecture is true or false.  Clearly this term doesn't "compute", but neither is it of canonical form (a numeral).

Similarly, using the univalence axiom, we can obtain a term $p : (\mathbf{2}=\mathbf{2})$ corresponding to the automorphism of the type $\mathbf{2}$ which switches $0_\mathbf{2}$ and $1_\mathbf{2}$.  Then the term $transport(p,0_\mathbf{2})$ also has type $\mathbf{2}$, but doesn't "compute" because the computer gets "stuck" on the univalence term.

It is conjectured that univalence, unlike excluded middle, can be given a "computational" interpretation while preserving canonicity.  Some partial progress towards this can be found [here](http://homotopytypetheory.org/2011/07/27/canonicity-for-2-dimensional-type-theory/).

## See also

* [[homotopy canonicity]]

## References

Discussion of canonicity in [[homotopy type theory]] with [[univalence]] is discussed in

* Simon Huber (with [[Thierry Coquand]]), _Towards a computational justification of the Axiom of Univalence_ , talk at _TYPES 2011_ ([pdf](http://www.cse.chalmers.se/~simonhu/slides/types11.pdf))

* [[Dan Licata]], [[Robert Harper]], _Computing with Univalence_  (2012) ([pdf](http://4wft.fmf.uni-lj.si/wp-content/uploads/2012/04/Licata.pdf))

* [[Dan Licata]], _Canonicity for 2-Dimensional Type theory_  (2011) ([web](http://homotopytypetheory.org/2011/07/27/canonicity-for-2-dimensional-type-theory/))

[[!redirects canonical form]]
[[!redirects canonical forms]]
[[!redirects canonicity]]
[[!redirects canonicity for type theory]]
[[!redirects type-theoretic canonicity]]


canonical formula of myth

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>_La première expression de la réalité serait de dire que la langue (c'est à dire le sujet parlant) n'aperçoit ni l'idée a, ni la forme A, mais seulement le rapport $\frac{a}{A}$; cette expression serait encore tout à fait grossière. Il n'aperçoit pas vraiment que le rapport entre les deux rapports $\frac{a}{AHZ}$ et $\frac{abc}{A}$, ou $\frac{b}{ARS}$ et $\frac{blr}{B}$, etc. C'est là ce que nous appelons le QUATERNION FINAL.  de Saussure ([2002](#Saussure02), p.39)

## Idea

The **canonical formula of mythical transformation** is an expression proposed in 1955 by the anthropologist [[Claude Lévi-Strauss|Lévi-Strauss]] in order to account for the abstract relations occuring between characters and their attributes in a myth understood as the collection of its variants. The vagueness of the initial proposal has been lifted by several mathematical interpretations in the aftermath.

## The original formulation

In Lévi-Strauss ([1955](#LS55), p.442) the canonical formula is introduced as the "algebraic" expression:

$ f_x(a): f_y(b)\cong f_x(b):f_{a^{-1}}(y) $ .

## An example

## The quaternionic interpretation

[[Jack Morava]] ([2003](#Morava03), [2004](#Morava04)) proposed to interpret the canonical formula as the existence of an anti-isomorphism of the [[quaternion group]].

## Some history

## Related entries

* [[Claude Lévi-Strauss]]

* [[kinship]]

* [[structuralism]]

* [[quaternion group]]

* [[quaternionic set]]

* [[code loop]]

## Related pages

* French wikipedia entry: ([link](https://fr.wikipedia.org/wiki/Formule_canonique_du_mythe))

## References

* Numéro spécial _La formule canonique des mythes_, L'Homme **35** no.135 (1995). ([link](http://www.persee.fr/issue/hom_0439-4216_1995_num_35_135))

* Mauro William Barbosa de Almeida, _A fórmula canônicado mito_, ms. University of São Paulo 2009. (Available online)

* {#DaWiKi14}S. Darányi, P. Wittek, K. Kitto, *The Sphinx's New Riddle: How to Relate the Canonical Formula of Myth to Quantum Interaction*, pp.47-58 in LNCS **8369** (2014).

* Maurice Godelier, *Lévi-Strauss*, Seuil Paris 2013.

* Lucien d'Huy, *Cosmogonies - La Préhistoire des mythes*, La Découverte Paris 2020. (pp.247-252)

* {#Lacan07}Jacques Lacan, *Le mythe individuel du névrosé*, Seuil Paris 2007.

* {#Landau51}H. G. Landau, *On dominance relations and the structure of animal societies: Effect of inherent characteristics*, Bull. Math, Biophysics **13** (1951) pp.1-19,245-62.

* {#LS55}[[Claude Lévi-Strauss]], *The Structural Study of Myth*, J. American Folklore **78** no. 278 (1955) pp.428-444.

* {#LS58}[[Claude Lévi-Strauss]], *Anthropologie Structurale*, Plon Paris 1958.

* {#LS85a}[[Claude Lévi-Strauss]], *D'un oiseau à l'autre*, L'Homme **25** no.93 (1985) pp.5-12. ([link](http://www.persee.fr/doc/hom_0439-4216_1985_num_25_93_368539))

* {#LS85b}[[Claude Lévi-Strauss]], *La Potière Jalouse*, Plon Paris 1985.

* {#Lucchelli06}Juan Pablo Lucchelli, *Le mythe individuel revisité*, L'information psychatrique **82** no.2 (2006) pp.155-158. ([link](http://www.cairn.info/revue-l-information-psychiatrique-2006-2-page-155.htm))

* {#Lucchelli10}Juan Pablo Lucchelli, *Lacan et la formule canonique des mythes*, Les Temps Modernes no.660 (2010) pp.116-131.

* {#Maranda01}Pierre Maranda (ed.), *The Double Twist*, University of Toronto Press 2001.

* {#Morava03}[[Jack Morava]], *On the Canonical Formula of C. Lévi-Strauss*, arXiv:2003.0306174v2 (2003). ([abstract](http://arxiv.org/abs/math/0306174))

* {#Morava04}[[Jack Morava]], *Une interprétation mathématique de la formule canonique de Claude Lévi-Strauss*, Cahiers de L'Herne **88** (2004) pp.216-218.

* {#Morava05}[[Jack Morava]], *From Lévi-Strauss to Chaos and Complexity*,  in Mosko, Damon (eds.), *On the Order of Chaos*, Berghahn Oxford 2005.

* {#Morava20}[[Jack Morava]], *On the Canonical Formula of C. Lévi-Strauss II*, arXiv:2002.12813 (2020). ([abstract](https://arxiv.org/abs/2002.12813))

* {#Oppitz92} Michael Oppitz, *Notwendige Beziehungen - Abriß der strukturalen Anthropologie*, Suhrkamp Frankfurt am Main 1992. (pp.245ff)

* {#Petitot88}[[Jean Petitot]], *Approche morphodynamique à la formule cannonique des mythes*, L'Homme **28** no.106-107 (1988) pp.24-50. ([link](http://www.persee.fr/doc/hom_0439-4216_1988_num_28_106_368968))

* {#Rapoport49} A. Rapoport, *Outline of a Probabilistic Approach to Animal Sociology*, Bull. Math. Biophys. **11** (1949) pp.183-196, pp.273-281.

* {#Roudinesco93}Elisabeth Roudinesco, *Jacques Lacan*, Fayard Paris 1993.

* {#Saussure02}Ferdinand de Saussure, *Écrits de linguistique générale*, Gallimard Paris 2002.

* {#Scubla98}Lucien Scubla, *Lire Lévi-Strauss*, Odile Jacob Paris 1998.

* {#ThuiLeQue17}M. Thuillard, J.-L. Le Quellec, *A phylogenetic interpretation of the canonical formula of myths by Lévi-Strauss*, Cultural Anthropology and Ethnosemiotics **3** no.2 (2017) pp.1-12.

[[!redirects canonical formula]]
[[!redirects Canonical formula]]
[[!redirects canonical formula of mythical transformation]]
[[!redirects canonical formula for myth]]


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## Idea

Given a [[manifold]] $X$, there is a [[Hilbert space]] $L^2(X)$ whose elements are square-[[integration of differential forms|integrable]] [[half-densities]] on $X$, and whose [[inner product]] is given by [[tensor product|tensoring]] two half-densities to an actual [[density]] and then [[integration of differential forms|integrating]] that over $X$.

This Hilbert space is often called the _canonical Hilbert space_ of $X$, in contrast with the [[L-2-spaces]] $L^2(X,\mu)$ of $\mu$-[[integrable function|integrable]] [[functions]] for every choice of [[measure]] $\mu$ on $X$.

The notion of canonical Hilbert spaces originates in the context of [[geometric quantization]] in ([Guillemin-Sternberg](#GuilleminSternberg)). It directly corresponds to the "canonical" construction of [[groupoid convolution algebras]] (see there for details) from sections of half-density bundles without choice of a [[Haar measure]].

## References

The notion originates in [[geometric quantization]] around

* [[Victor Guillemin]], [[Shlomo Sternberg]], _Geometric asymptotics_ Math. Surveys, 14, Amer. Math. Soc., Providence, R.I., 1977; MR 58 # 24404.
 {#GuilleminSternberg}

Related lecture notes include

* Sean Bates, [[Alan Weinstein]], _Lectures on the geometry of quantization_, [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)

Applications in [[operator algebra]] theory appear in

* [[Alain Connes]], _[[Noncommutative Geometry]]_

[[!redirects canonical Hilbert space of half-densities]]
[[!redirects canonical Hilbert spaces of half-densities]]
[[!redirects canonical Hilbert-space of half-densities]]
[[!redirects canonical Hilbert-spaces of half-densities]]


canonical hypergroup

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## Idea

A **hypergroup** is a [[algebra|algebraic]] structure similar to a [[group]], but where the composition operation does not just take two elements to a single product element in the group, but to a subset of elements of the group.

It is a [[hypermonoid]] with additional groupal [[stuff, structure, property|structure and property]].

## Definition

A **canonical hypergroup** is a [[set]], $H$, equipped with a commutative binary operation,

$$
  + : H \times H \to P^*(H)
$$

taking values in [[non-empty subsets]] of $H$, and a zero element $0 \in H$, such that

1. $+$ is associative (extended to allow addition of subsets of $H$);
2. $0 + x = {\{x\}} = x + 0, \forall x \in H$;
3. $\forall x \in H, \exists ! y \in H$ such that $0 \in x + y$ (we denote this $y$ as $-x$);
4. $\forall x, y, z \in H, x \in y + z$ implies $z \in x - y$ (where $x - y$ means $x + (-y)$ as usual).

## Examples

The additive structure underlying a [[hyperring]] is a canonical hypergroup. See there for more examples.

[[!redirects hypergroup]]
[[!redirects hypergroups]]

category: algebra

[[!redirects canonical hypergroups]]


canonical model structure

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## Idea

In general, **canonical model structures** are [[model category]] structures on the categories of some flavor of [[n-category|n-categories]] for $1\le n\le \infty$ (note that $n=\infty$ or $\omega$ is allowed), which are intended to capture the correct "categorical" theory of these categories.

The archetypical examples are

* the _[[canonical model structure on categories]]_

* the _[[canonical model structure on groupoids]]_.

In [[categorification]] of these, one has

* [[canonical model structures on 2-categories]].

Canonical model structures are sometimes called "folk" model structures, but the appropriateness of this term is [questionable](https://nforum.ncatlab.org/discussion/649), especially in the cases $n \gt 1$.  Other alternatives are 'endogenous', 'standard', 'natural', and 'categorical'.

While ultimately the collection of all [[n-category|$n$-categories]] should form an $(n+1)$-category, restricting that to just invertible higher morphisms will yield an [[(n,r)-category|(n+1,1)-category]], and thus in particular an [[(∞,1)-category]].  It is this (∞,1)-category which is intended to be [[presentable (infinity,1)-category|presented]] by a canonical model structure.  In particular, the [[weak equivalences]] in a canonical model structure should be the [[equivalence of categories|category-theoretic equivalences]].

This is to be contrasted with [[Thomason model structure|Thomason model structures]] in which the weak equivalences are the morphisms that induce a [[model structure on simplicial sets|weak homotopy equivalence]] of [[nerve|nerves]].  This amounts to regarding each category, or rather its [[nerve]], as a placeholder for its _groupoidification_ (Kan fibrant replacement) and then considering the standard notion of equivalence.

In a canonical model structure for some flavor of $n$-categories, usually

* a **fibration** is a functor that lifts [[equivalence|equivalences]] in all dimensions,
* an **acyclic fibration** is a functor which is [[k-surjective functor|k-surjective]] for all $0\le k\le n$,
* a **weak equivalence** is a functor which is [[k-surjective functor|essentially k-surjective]] for all $0\le k\le n$, and
* a **cofibration** is a functor which is injective on objects and "relatively free" on $k$-morphisms for $1\le k \lt n$.  These can also be described as the morphisms [[cofibrantly generated model structure|generated]] by the inclusions
$\partial G_k \hookrightarrow G_k$ of the boundary of the $k$-[[globe]] into the $k$-[[globe]] for $0\le k \lt \infty$.

## Internalization

A common problem is to transport the (a) model structure on plain $\omega$-categories, i.e. $\omega$-categories [[internal category|internal to]] $Sets$ to another [[internalization|internal context]], notably for the case that $Sets$ is replaced with some kind of category of $Spaces$. This is relevant for the discussion of the homotopy theory of topological and smooth $\omega$-categories.

Usually, such internalization of model structures has the consequence that some properties invoked in the description of the original model structure, notably some of the lifting properties, will only continue to hold "locally".  One way to deal with this is to pass to a notion slightly weaker than that of a [[homotopy theory|model category]] called a [[category of fibrant objects]] as used in [[homotopical
cohomology theory]].

But there are also full model structures for such situations. Notice that under a suitable [[nerve]] operation all [[n-category|n-categories]] usually embed into [[simplicial set]]s. The [[models for infinity-stack (infinity,1)-toposes]] given by the [[model structure on simplicial presheaves]] then serves to present the corresponding $(\infty,1)$-category of parameterized or internal $n$-categories. See for instance also [[smooth infinity-stack]].

## Cofibrant resolutions

In

* [[Francois Métayer]], _Cofibrant complexes are free_ ([arXiv](http://arxiv.org/abs/math.CT/0701746))

it is shown that _cofibrant_ $\omega$-categories with respect to the canonical model structure are precisely the "free" ones, where "free" here means "generated from a [[polygraph]]" as described in

* [[Francois Métayer]], _Resolutions by [[polygraphs]]_ ([tac](http://www.tac.mta.ca/tac/volumes/11/7/11-07.pdf))

We had some blog discussion about this at
[Freely generated omega-categories](http://golem.ph.utexas.edu/category/2008/10/freely_generated_categories.html).

## Related concepts

* [[basic localizer]]

* [[canonical model structure on Cat]]

* [[canonical model structure on Grpd]]

* [[canonical model structure on 2Cat]]

* [[canonical model structure on Operad]]

* [[model structure on strict omega-categories]]

* [[2-trivial model structure]]

## References
 {#References}

The [[canonical model structure on categories]] was known to experts for some time before being written down formally (this is the origin of the adjective "folk", as in "[[folklore]]").

The [[canonical model structure on groupoids]] is claimed (but not proven)

* {#Anderson78} [[Donald W. Anderson]], p. 783 (12 of 37) in: *Fibrations and geometric realization*,  Bull. Amer. Math. Soc. **84** 5 (1978) 765-788 [[euclid:1183541139](http://projecteuclid.org/euclid.bams/1183541139)]

A published proof of this and the [[canonical model structure on categories]], in the further generality of [[internal category|categories internal]] to a [[Grothendieck topos]] (hence of [[2-sheaves]]) is due to

* {#JoyalTierney91} [[André Joyal]], [[Myles Tierney]], Thm. 2 (p. 221) *Strong stacks and classifying spaces*, in: *Category Theory* Lecture Notes in Mathematics **1488**, Springer (1991) 213-236 [[doi:10.1007/BFb0084222](https://doi.org/10.1007/BFb0084222)]

and more elementary proofs for the plain case (internal to [[Set]]) are given in

* [[Charles Rezk]], *A model category for categories* (1996) [[dvi](http://www.math.uiuc.edu/~rezk/cat-ho.dvi), [[Rezk_ModelCategoryForCategories.pdf:file]]]

* [[André Joyal]], _[[joyalscatlab:Model structures on Cat]]_ (2010)

and (for the [[canonical model structure on groupoids|case of groupoids]]) in

* {#Strickland00} [[Neil Strickland]], §6.1 of: _$K(n)$-local duality for finite groups and groupoids_ , Topology **39** 4 (2000) [[arXiv:math/0011109](https://arxiv.org/abs/math/0011109), doi:10.1016/S0040-9383(99)00031-2]

Further generalization to [[internal category|categories internal]] to [[finitely complete categories]]:

* [[Tomas Everaert]], [[Rudger W. Kieboom]], [[Tim Van der Linden]], _Model structures for homotopy of internal categories_, Theory and Applications of Categories,  Vol. 15, CT2004, No. 3, pp 66-94. ([tac:15-03](http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html)).

though it seems that the assumptions in this article apply to ambient categories that are [[semiabelian categories]], but do not apply to ambient categories like [[Top]].

A brief summary, together with a generalization when one assumes only weaker versions of the [[axiom of choice]], can be found at _[[folk model structure on Cat]]_.

The [[canonical model structure on 2-categories|canonical model structures for 2-categories and bicategories]] are due to

* [[Steve Lack]],

  _A Quillen Model Structure for 2-Categories_, K-Theory 26: 171–205, 2002. ([website](http://web.science.mq.edu.au/~slack/papers/qmc2cat.html))

  _A Quillen Model Structure for Bicategories_, K-Theory 33: 185-197, 2004. ([website](http://web.science.mq.edu.au/~slack/papers/qmcbicat.html))

and for $n=3$ ([[Gray-categories]]) in

* [[Steve Lack]], _A Quillen model structure for Gray-categories_ ([arxiv:1001.2366](http://arxiv.org/abs/1001.2366))

For $n = \omega$ ([[strict omega-categories]])

* [[Yves Lafont]], [[Francois Métayer]], Krzysztof Worytkiewicz, _A folk model structure on omega-cat_ ([arXiv](http://arxiv.org/abs/0712.0617))

For $n = \omega$ and all morphisms invertible, there is the [[model structure on strict omega-groupoids]]:

* [[Ronnie Brown]] and M. Golasinski, _A model structure for the homotopy theory of crossed complexes_, Cah. Top. Géom. Diff. Cat. 30 (1989) 61-82 ([pdf](http://www.bangor.ac.uk/~mas010/pdffiles/RB-golskyrev.pdf))

* [[Dimitri Ara]], [[Francois Metayer]], _The Brown-Golasinski model structure on ∞-groupoids revisited_ (pdf) Homology, Homotopy Appl. 13 (2011), no. 1, 121–142.

[[!redirects canonical model structures]]

[[!redirects folk model structure]]
[[!redirects folk model structures]]

[[!redirects natural model structure]]
[[!redirects natural model structures]]