linear type theory

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### Context
#### Type theory
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[[!include type theory - contents]]
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#### Monoidal categories
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[[!include monoidal categories - contents]]
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#### Linear algebra
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

_Linear type theory_ is the [[linear logic]]-version of [[type theory]]. In the definition of ([Seely 89, prop. 1.5](#Seely89)), following ([Girard 87](#Girard87)), this is the [[internal language]] of/has [[categorical semantics]] in [[star-autonomous categories]].

More generally, the term "linear" (as in *linear type theory*) has over time come to connote [[internal languages|internal]] [[type theories]], and related logical frameworks ([[sequent calculus|sequent calculi]], [[natural deduction]]) that correspond to (usually symmetric) monoidal categories, where particularly the monoidal product is not assumed to be the cartesian product (reviews include [Blute-Panangaden-Seely 94](#BlutePanangadenSeely94)). Indeed, this general notion still faithfully follows the original motivation for the term "linear" as introduced in ([Girard 87](#Girard87)), since the non-cartesianness of the [[tensor product]] means the absence of a [[diagonal]] map and hence the impossibility of [[functions]] to depend on more than a single (linear) copy of their [[variables]].

In the corresponding logical frameworks, the non-cartesianness means one drops the contraction and weakening rules of inference associated with cartesian structure, but retains the exchange rule corresponding to the symmetry constraint. This general type of "linear logic" takes on many flavors: in addition to the Girard-style language that is naturally interpreted in [[star-autonomous categories]], one has languages for [[closed monoidal category|monoidal biclosed categories]], [[closed monoidal category|symmetric monoidal closed categories]], [[compact closed categories]], and others, collectively representing the "multiplicative" core of linear logic as understood in this general sense.

In addition to this "multiplicative" core, one also often studies modalities (comonads) which relate these symmetric monoidal products to cartesian products and coproducts (so-called "additive" operations in linear logic).

Extending these considerations, the [[dependent type theory]]-version of linear type theory should be _[[dependent linear type theory]]_.

Linear (dependent) type theory can be argued to be the proper [[type theory]] for [[quantum logic]]. In this context the linearity of the type theory, hence the absence of [[diagonal maps]] in its [[categorical semantics]], corresponds to the [[no-cloning theorem]] in [[quantum physics]].

## Properties

### Categorical semantics

(...)

In [[star-autonomous categories]].

| Conjunctive operator | $\leftarrow$[[de Morgan duality]] $\rightarrow$ | Disjunctive operator |
| -------------------- | - | -------------------- |
| $\top$               |   | $0$                  |
| $1$                  |   | $\bot$               |
| $\&$                 |   | $\oplus$             |
| $\otimes$            |   | $\parr$              |
| $^\bot$              |   | $^\bot$              |
| $\bigwedge$          |   | $\bigvee$            |
| $!$                  |   | $?$                  |

(...)

### The canonical co-modality

The original axiomatics for [[linear type theory]] in ([Girard 87](#Girard87)) contain in addition to the structures corresponding to a ([[star-autonomous category|star-autonomous]]) [[symmetric monoidal category|symmetric]] [[closed monoidal category]] a certain (co-)[[modality]] traditionally denoted "$!$", the [[exponential modality]].
In ([Benton 95](#Benton95), [Bierman 95](#Bierman95))
it is noticed (reviewed in ([Barber 97, p. 21 (26)](#Barber97))) that a natural [[categorical semantics]] for this modality identifies it with the [[comonad]] that is induced from a [[strong monoidal adjunction]]

$$
  (\Sigma \dashv R)
  \;\colon\;
  \mathcal{C}
  \stackrel{\overset{\Sigma}{\leftarrow}}{\underset{R}{\longrightarrow}}
  S
$$

between the [[closed monoidal category|closed]] [[symmetric monoidal category]] $\mathcal{C}$ which interprets the given [[linear type theory]] and a [[cartesian monoidal category]] $S$.

If there is only the [[strong monoidal functor]] $\Sigma \;\colon\; S \longrightarrow \mathcal{C}$ without possibly a [[right adjoint]], then ([Barber 97, p. 21 (27)](#Barber97)) speaks of the _structural [[fragment]]_ of [[linear type theory]].

In ([Ponto-Shulman 12](#PontoShulman12)) it is observed that this in turn is canonically induced if $\mathcal{C} \simeq \mathcal{C}_\ast$ is the linear type theory over the trivial context of a [[dependent linear type theory]]/[[indexed closed monoidal category]] with category of contexts being $S$. See at _[indexed monoidal (infinity,1)-category -- Exponential modality and Fock spaces](https://ncatlab.org/nlab/show/indexed+monoidal+%28infinity%2C1%29-category#TheCanonicalComodality)_ for more on this.

## History of linear categorical semantics
 {#HistoryCategoricalSemantics}

There is a long history of logical frameworks of "linear type" and their categorical semantics in various [[doctrines]] of monoidal and closed monoidal categories, even though much of the early history predates Girard's introduction of linear logic (and which therefore do not bear the term "linear"). We give a thumbnail sketch of some of this below.

The founding father could be said to be [[Joachim Lambek]], who was the first to apply deductive systems and particularly the method of Gentzen cut-elimination to what we would now recognize as [[categorification]]s of various [[fragments]] of propositional logic. It is noteworthy that one of the various first applications of this categorified cut-elimination was to the (*linear*) doctrine of [[monoidal biclosed categories]], in which Lambek was interested in connection with his linguistic research ([Lambek 58](#Lambek58)); see ([Lambek 68](#Lambek68), [Lambek 69](#Lambek69)). A principal goal in these papers was to study (and partially solve) the coherence problem for such structures.

+-- {: .num_remark}
###### Remark
Kelly and Mac Lane then adapted the essential cut-elimination techniques of Lambek to study the (difficult) coherence problem for closed symmetric monoidal categories, in their landmark study [KM](#KM71). Their work cleverly avoided the explicit introduction of logical equipment, presumably in the service of greater self-containment or greater categorical readership, although the debt to Lambek's pioneering work is clear enough (and _is_ made explicit).
=--

By the late 1970's, the connections between [[proof theory]], coherence theory, and the categorical semantics of deductive systems had been well elaborated, and again it is noteworthy that there was especial focus on categorical doctrines of general linear type. A largish batch of various [[fragments]] of classical logic, particularly in sequent calculus form, together with their categorical semantics, was collected in a book-length study ([Szabo 78](#Szabo78)) by Manfred Szabo, who had been Lambek's student. (While the categorical semantics of various deductive systems were carefully spelled out in Szabo's book, this work was unfortunately tainted by overly ambitious claims by Szabo regarding his purported solutions to the relevant coherence problems, in terms of normalization algorithms which turned out not to be Church-Rosser; see [Jay 90a](#Jay90a).)

Nor were the logical frameworks restricted to those of sequent calculus type. In particular, the Soviet logician Minc and his students had also developed a type theory whose semantics is naturally valued in closed monoidal categories; see [Minc 77](#Minc77). This too was in view of coherence problems.

Interest in these problems was of course reawakened and reinvigorated in the light of Girard's extraordinarily insightful 1987 paper [Linear Logic](#Girard87). The connection with closed symmetric monoidal categories and $\ast$-autonomous categories was picked up in very short order by the categorical community; an early account of this connection was given by [Seely 89](#Seely89). It was not long before Girard's very fruitful exposition of linear proof theory and [[cut-elimination]] in terms of [[proof net|proof nets]], specifically for the [[multiplicative conjunction|multiplicative]] [[fragment]] MLL which is the basic technical core of Girard's work, was seen as closely related to "extranaturality" graphs (aka KM-graphs) used by Kelly-Mac Lane in their work on the coherence theory for closed symmetric monoidal categories -- as well as by other "Australian category theorists" who had developed Kelly's theory of clubs (which again tended to be most successful for doctrines of linear type).

These aspects are covered in ([Blute 91](#Blute91)), who treated a number of coherence problems and particularly the method of Lambek/Kelly-Mac Lane as efficiently streamlined through the method of proof nets with connection given to KM-graphs and clubs. Note that all of those applications were towards various [[fragments]] of MLL or doctrines of linear type.

From approximately the same time period we also have some work of Barry Jay, who had developed a calculus of terms and their normalization to study coherence for closed monoidal categories. See for instance ([Jay 89](#Jay89), [Jay 90b](#Jay90b)). (This does not seem to owe allegiance to Girard's methods particularly, but it does indicate some contemporaneous thinking on linear types, terms, and their categorical semantics.)

+-- {: .num_remark}
###### Remark
It should be noted that the "classical" techniques developed by Lambek, Kelly-Mac Lane, and Blute merely yield _partial_ coherence results, in effect evading the notorious "problem of the unit" which had made the full coherence problem for closed monoidal categories seem fearsomely difficult. However, Trimble in his (unpublished) 1994 thesis showed how to re-interpret Girard's proof nets so that they naturally took the unit into account, and was then able to give a _full_ coherence result for MILL by arranging cut-free nets into a strongly normalizing and confluent rewrite system. Note that while full coherence results for this case had also been proposed by Voreadou and Dole (students of Mac Lane), by Minc and Jay, and also incorrectly by Szabo, none of these earlier approaches were in a position to take advantage of the proof net formalism. For another account of this sort of formalism, specifically for the case of $\ast$-autonomous categories, [[weakly distributive category|weakly distributive categories]], and full coherence thereof, see [Blute-Cockett-Seely-Trimble96](#BluteCockettSeelyTrimble96).
=--

Meanwhile, the type theory and its categorical semantics for _full_ linear logic, going beyond MLL and taking into account the modalities and the "additive" operations of LL, was also advanced during this time period. A comprehensive description is given in [Benton-Bierman-de Paiva-Hyland 92](#BentonBiermanPaivaHyland92), with an important antecedent in de Paiva's work on the Dialectica interpretation [de Paiva 89](#dePaiva89). See also [Hyland-de Paiva 93](#HylandPaiva93), [Bierman 95](#Bierman95), [Barber 97](#Barber97)

See also [Schalk 04](#Schalk04), [Melliès 09](#Mellies09). Schalk and Melliès have also worked on game-theoretic semantics of theories of linear type -- which itself has a long and intricate history.

Finally, in addition to these usual "denotational" categorical semantics, linear logic also has an "operational" categorical semantics, called the Geometry of Interaction, in traced monoidal categories.

## Related concepts

* [[quantum logic]], [[quantum programming]]

* [[quantum lambda-calculus]]

* [[dependent linear type theory]]

* [[propositions as projections]]

* [[modal type theory]]

* [[geometric homotopy type theory]]

* [[stable homotopy type]]

## References

### General

The original article on [[linear logic]] is

* {#Girard87} [[Jean-Yves Girard]], _Linear logic_,   Theoretical Computer Science 50:1, 1987.  ([pdf](http://girard.perso.math.cnrs.fr/linear.pdf))

reviewed in:

* {#Girard11} [[Jean-Yves Girard]], part III of _[[Lectures on Logic]]_, European Mathematical Society 2011

Linear type theory as such is made explicit in:

* [[Simon John Ambler]], *First Order Linear Logic in Symmetric Monoidal Closed Categories*, PhD thesis, Edinburgh (1991) [[ECS-LFCS-92-194](https://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-194), [pdf](http://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-194/ECS-LFCS-92-194.pdf), [[Ambler-FOLL.pdf:file]]]

  > (with [[categorical semantics]] in [[symmetric monoidal category|symmetric]] [[closed monoidal categories]])

* Masahito Hasegawa. _Logical predicates for intuitionistic linear type theories_, Typed Lambda Calculi and Applications:
4th International Conference, TLCA '99, ed. J.-Y. Girard, Lecture Notes in Computer Science 1581, Springer, Berlin, 1999 ([pdf](http://www.kurims.kyoto-u.ac.jp/~hassei/papers/tlca99.pdf))

* [pdf](http://www.cs.cmu.edu/~fp/courses/15816-f01/handouts/lintt.pdf)

* {#MihályiNovitzká13} Daniel Mihályi, Valerie Novitzká, *What about Linear Logic in Computer Science?*, Acta Polytechnica Hungarica **10** 4 (2013) 147-160 [[pdf](http://acta.uni-obuda.hu/Mihalyi_Novitzka_42.pdf), [[MihalyiNovitzka-LinearLogic.pdf:file]]]

### Lambek's syntactic calculus

Several decades before Girard's article on linear logic, Lambek introduced a sequent calculus without weakening, contraction, or exchange, and described its applications to modeling aspects of natural language (generalizing previous work by Ajdukiewicz and Bar-Hillel on [[categorial grammar]]):

* {#Lambek58} [[Joachim Lambek]]. _The mathematics of sentence structure_ . American mathematical monthly, 154-170, 1958. [link](http://www.jstor.org/stable/2310058)

Lambek called his system the "syntactic calculus", while nowadays it is often called _Lambek calculus_ in linguistics.

In "Towards a geometry of interaction" (1989), Girard references Lambek's 1958 article, and writes that "this work must be acknowledged as a true ancestor to linear logic; its connection to linguistics can be seen as the first serious evidence against the exclusive focus on mathematics" (p.81).  On the other hand, Lambek later wrote that his original motivations were actually in [[homological algebra]]:

> I would now call this system "bilinear logic", meaning "non-commutative linear logic" or "logic without Gentzen's three structural rules". My original name had been "syntactic calculus", because of its intended application to linguistics; but actually it had been developed in collaboration with George Findlay [1955] in an attempt to understand some basic homological algebra. Alas, our paper was rejected on the grounds that most of our results were to appear in a book by Cartan and Eilenberg [1956], the publication of which had been delayed because of a paper shortage. (Lambek, "Bilinear logic in algebra and linguistics", _Advances in Linear Logic_, CUP, 1995)

### Categorical semantics

The following articles are about "logics of linear type" and their categorical semantics, although they predate the use of the word "linear" in Girard's sense. Most have a view motivated in part by categorical coherence problems.

* {#Lambek68} [[Joachim Lambek]], _Deductive systems and categories_, Mathematical Systems Theory 2 (1968), 287-318.

* {#Lambek69} [[Joachim Lambek]], _Deductive systems and categories II_, Lecture Notes in Math. 86, Springer-Verlag (1969), 76-122.

* {#Minc77} G.E. Minc, _Closed categories and the theory of proofs_, translated from an article in Russian in Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Mat. Instituta im. V.A. Steklova AN SSSR 68 (1977), 83-114. Republished in Grigorii E. Mints, _Selected Papers in Proof Theory_, Bibliopolis (1992).

* {#Szabo78} M.E. Szabo, _Algebra of Proofs_, Studies in Logic and the Foundations of Mathematics, vol. 88 (1978), North-Holland.

* {#Jay89} C.B. Jay, _Languages for monoidal categories_, JPAA 59 (1989), 61-85.

* {#Jay90a} C.B. Jay, _Coherence in category theory and the Church-Rosser property_. Unpublished preprint (1990), cited in the following reference.

* {#Jay90b} C.B. Jay, _The structure of free closed categories_, JPAA 66 (1990), 271-285.

The [[categorical semantics]] of linear type theory in [[star-autonomous categories]] was first described in

* {#Seely89} [[R. A. G. Seely]],  _Linear logic, $\ast$-autonomous categories and cofree coalgebras_, _Contemporary Mathematics_ 92, 1989.  ([[SeelyLinearLogic.pdf:file]], [ps.gz](http://www.math.mcgill.ca/rags/nets/llsac.ps.gz))

with the first non-syntactic, mathematical categorical model in the same volume:

* {#dePaiva89} [[Valeria de Paiva]], _The Dialectica Categories_, _Contemporary Mathematics_ 92, 1989. ([web] (http://www.cs.bham.ac.uk/~vdp/publications/dial87.pdf))

As mentioned above, generalization of this semantics, restricted to the "multiplicative fragment" but covering doctrines of more general symmetric monoidal closed type (with specific reference to Girard's work) is given in

* {#Blute91} [[Richard Blute]], _Linear logic, coherence, and dinaturality_, Dissertation, University of Pennsylvania 1991, published in Theoretical Computer Science archive Volume 115 Issue 1, (July 5, 1993) 3-41.

Blute's thesis derives coherence theorems for various doctrines of "linear type", including a coherence theorem for symmetric monoidal [[closed categories]] due to Kelly and Mac Lane:

* {#KM71} [[Max Kelly]], [[Saunders MacLane]], _Coherence in closed categories_, JPAA 1 (1971), 97-140 ([web](http://www.sciencedirect.com/science/article/pii/0022404971900132))

An decent review of some of this is in

* {#BlutePanangadenSeely94} [[Richard Blute]], [[Prakash Panangaden]], [[R. A. G. Seely]], _Fock Space: A Model of Linear Exponential Types_ (1994) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.6825))

Extensions of the coherence-theoretic applications of proof nets in Blute's thesis, to give full solutions to coherence problems, were given in

* {#Trimble94} [[Todd Trimble]], _Linear Logic, Bimodules, and Full Coherence for Autonomous Categories, Rutgers 1994

* {#BluteCockettSeelyTrimble96} [[Richard Blute]], Cockett, [[R. A. G. Seely]], [[Todd Trimble]], _Natural deduction and coherence for weakly distributive categories_, JPAA 113 (1996), 229-296. ([web](http://www.sciencedirect.com/science/article/pii/002240499500159X))

Type theory for full linear logic, together with its categorical interpretation, was developed in

* {#BentonBiermanPaivaHyland92} [[Nick Benton]], Gavin Bierman, [[Valeria de Paiva]], [[Martin Hyland]], _Term assignments for intuitionistic linear logic_. Technical report 262, Cambridge 1992 ([citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.8666))

* {#HylandPaiva93} [[Martin Hyland]], [[Valeria de Paiva]], _Full Intuitionistic Linear Logic_ (extended abstract). Annals of Pure and Applied Logic, 64(3), pp.273-291, 1993. ([pdf](http://www.cs.bham.ac.uk/~vdp/publications/fill.pdf))

* {#Barber97} Andrew Graham Barber, _Linear Type Theories, Semantics and Action Calculi_, PhD Thesis, Edinburgh 1997 ([web](http://www.lfcs.inf.ed.ac.uk/reports/97/ECS-LFCS-97-371/), [pdf](http://www.lfcs.inf.ed.ac.uk/reports/97/ECS-LFCS-97-371/ECS-LFCS-97-371.pdf))

Further review and discussion is in

* {#Bierman95} Gavin M. Bierman, _What is a categorical model of intuitionistic linear logic? ([web] (https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.31.8687&rep=rep1&type=pdf))

The more general case of general [[symmetric monoidal categories]] is reviewed and further discussed in

* {#Schalk04} Andrea Schalk, _What is a categorical model for linear logic?_ ([pdf](http://www.cs.man.ac.uk/∼schalk/notes/llmodel.pdf))

* [[Richard Blute]], [[Philip Scott]], *Category theory for linear logicians*, in: *Linear Logic in Computer Science*, Cambridge University Press (2004) 3-64 [[doi:10.1017/CBO9780511550850.002](https://doi.org/10.1017/CBO9780511550850.002), [pdf](https://www.site.uottawa.ca/~phil/papers/catsurv.web.pdf)]

* {#Mellies09} [[Paul-André Melliès]] , _Categorial Semantics of Linear Logic_, in _Interactive models of computation and program behaviour_, Panoramas et synthèses 27, 2009 ([pdf](http://www.pps.univ-paris-diderot.fr/~mellies/papers/panorama.pdf))

The interpretation of Girard's $!$-modality as the [[comonad]] induced from a [[monoidal adjunction]] between the closed [[symmetric monoidal category]] and a [[cartesian closed category]] is due to

* {#Bierman95} G. Bierman, _On Intuitionistic Linear Logic_ PhD thesis, Computing
Laboratory, University of Cambridge, 1995 ([pdf](http://research.microsoft.com/~gmb/papers/thesis.pdf))

* {#Benton95} [[Nick Benton]], *A mixed linear and non-linear logic: Proofs, terms and models*, in *Computer Science Logic. CSL 1994*, Lecture Notes in Computer Science **933** [[doi:10.1007/BFb0022251](https://doi.org/10.1007/BFb0022251), [[BentonLinearLogic.pdf:file]]]

Interpretation of this in [[dependent linear type theory]] is in

* {#PontoShulman12} [[Kate Ponto]], [[Mike Shulman]], _Duality and traces for indexed monoidal categories_, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 ([arXiv:1211.1555](http://arxiv.org/abs/1211.1555))

Further discussion of linear type theory is for instance in

* _Chapter 7, Linear type theory_ [pdf](http://www.cs.cmu.edu/~fp/courses/linear/handouts/lintt.pdf)

* Anders Schack-Nielsen, Carsten Schürmann, _Linear contextual modal type theory_ [pdf](http://www.itu.dk/~carsten/papers/lcmtt.pdf)

* Bernardo Toninho, Luís Caires, Frank Pfenning, _Dependent Session Types via Intuitionistic Linear Type Theory_ [pdf](http://ctp.di.fct.unl.pt/~lcaires/papers/dstypes.pdf)

* Iliano Cervesato, [[Frank Pfenning]], _A Linear Logical Framework_, 1996, ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.21.1152))

Discussion of application of linear logic to [[quantum logic]], [[quantum computing]] and generally to [[quantum physics]] includes

* {#Pratt92} [[Vaughan Pratt]], *Linear logic for generalized quantum mechanics*, in Proceedings of *[Workshop on Physics and Computation (PhysComp'92)](https://www.computer.org/csdl/proceedings/phycmp/1992/12OmNx19jVS)* (1992) 166-180 [[doi:10.1109/PHYCMP.1992.615518](https://doi.ieeecomputersociety.org/10.1109/PHYCMP.1992.615518), [pdf](http://boole.stanford.edu/pub/ql.pdf), [[Pratt-LinearLogicForQuantum.pdf:file]]]

* [[Ross Duncan]], *Types for quantum mechanics* (2006) [[pdf](http://personal.strath.ac.uk/ross.duncan/papers/rduncan-thesis.pdf), [pdf](https://www.cs.ox.ac.uk/people/ross.duncan/talks/2005/pps-22-05-2005.pdf)]

* {#CCGP09} Gianpiero Cattaneo, Maria Luisa Dalla Chiara, Roberto Giuntini and Francesco Paoli, section 9 of _Quantum Logic and Nonclassical Logics_, p. 127 in  Kurt Engesser, Dov M. Gabbay, Daniel Lehmann (eds.) _Handbook of Quantum Logic and Quantum Structures: Quantum Logic_, 2009 North Holland

* [[Ugo Dal Lago]], Claudia Faggian, _On Multiplicative Linear Logic, Modality and Quantum Circuits_ ([arXiv:1210.0613](http://arxiv.org/abs/1210.0613))

Discussion of linear type theory without [[units]] is in

* [[Robin Houston]], _Linear Logic without Units_ ([arXiv:1305.2231](http://arxiv.org/abs/1305.2231))

Discussion of [[inductive types]] in the context of linear type theory is in

* Stéphane Gimenez, _Towards Generic Inductive Constructions in
Systems of Nets_ ([pdf](http://www.imn.htwk-leipzig.de/WST2013/papers/paper_16.pdf))

[[!redirects linear type theories]]

[[!redirects linear type]]
[[!redirects linear types]]

[[!redirects linear homotopy type]]
[[!redirects linear homotopy types]]

[[!redirects linear type]]
[[!redirects linear types]]

[[!redirects linear type system]]
[[!redirects linear type systems]]


linear-non-linear logic

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### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

Linear-non-linear logic is a presentation of intuitionistic [[linear logic]] that decomposes the $!$ modality into an [[adjunction]] of between a syntax for a [[cartesian product|cartesian]] logic and a linear logic with cartesian variables.

The semantics of linear-non-linear logic is more direct than that of intuitionistic linear logic: it is based on a symmetric [[monoidal adjunction]] where the source of the left adjoint is a [[cartesian monoidal category]].

## Related

* If the linear context is restricted to having at most one variable, the judgmental structure of linear-non-linear logic coincides with [[call-by-push-value|call-by-push-value with stacks]].

## References

* [[Nick Benton]], [_A mixed linear and non-linear logic: Proofs, terms and models_, CSL 1994](https://link.springer.com/chapter/10.1007/BFb0022251)


linearly compact module

# Linearly compact vector spaces, modules, rings, objects#
* table of contents
{:toc}

## Motivation

Linearly compact vector spaces were introduced in the development of the idea of duality. The (algebraic) linear dual of the discrete infinite-dimensional vector space is of larger cardinality so the original space is not isomorphic to the dual of its dual. But if a natural formal topology (which comes say from the filtration of the original space by its finite-dimensional subspaces; the formal topology on the dual is equivalent to consider the dual cofiltration) is given to the algebraic dual, then it makes sense to take the space of _continuous_ linear functional and we recover the original vector space. More precisely this amounts to an embedding of the category of (discrete) vector spaces into the category of linearly compact vector spaces, the latter category has a duality which extends the duality for finite-dimensional vector spaces.

A standard reference for the basics is the Dieudonné's book on formal groups.

The next definition is copied from Tom Leinster's note that's listed below.

## Definition

A **linearly compact vector space** over a field $k$ is a topological vector space $V$ over $k$ such that:

* The topology is linear: the open affine subspaces form a basis for the topology.
* Any family of closed affine subspaces with the finite intersection property has nonempty intersection.
* The topology is Hausdorff.

More generally, a __linearly compact module__ $M$
over a [[topological ring]] $R$
is a Hausdorff linearly topologized
(meaning there is a basis of neighborhoods
of 0 consisting of open submodules)
such that every family of closed cosets
with the finite intersection property
(meaning finite subfamilies have nonempty intersections)
has nonempty intersection.

## Examples

Every [[pseudocompact module]] is linearly compact.

## Related concepts

* [[pseudocompact module]]

## Literature

* [[L. S. Pontrjagin]], _Über stetige algebraische Körper_, Ann. of Math. 33 (1932) 163-174
* [[N. Jacobson]], _Totally disconnected locally compact rings_,  Amer. J. Math. 58 (1936) 433-449; _A note on topological fields_, Amer. J. Math. 59 (1937) 889-894
* N. Jacobson , O. Taussky , Locally compact rings, Proc. Nat. Acad. Sci. U.S.A. 21 (1935) 106-108
* Solomon Lefschetz, _Algebraic topology_, Amer. Math. Soc. Colloq. Pub. 27, 1942, reprinted 1963
* I. Kaplansky, _Topological rings_, Amer. J. Math. 69 (1947) 153-183; _Topological methods in valuation theory_, Duke Math. J. 14 (1947) 527-541; _Locally compact rings I_, Amer. J. Math. 10 (1948) 447-459; _II_, Amer. J. Math. 13 (1951) 20-24; _III_, Amer. J. Math. 14 (1952) 929-935; _Topological representations of algebras II_, Trans. Amer. Math. Soc. 68
(1950) 62-75
* Daniel Zelinsky, _Linearly compact modules and rings_, American Journal of Mathematics __75__, No. 1 (Jan., 1953), pp. 79-90 [jstor](http://www.jstor.org/stable/2372616)
* O. Goldman, C. H. Sah, _On a special class of locally compact rings_, J. Algebra 4 (1966) 71-95; _Locally compact rings of special type, J. Algebra 11 (1969) 363-454
* [[Tom Leinster]], [Where Do Linearly Compact Vector Spaces Come From?](https://golem.ph.utexas.edu/category/2012/09/where_do_linearly_compact_vect.html), $n$-Cat Café 2012
* Tom Leinster, _Codensity and the ultrafilter monad_, [arxiv/1209.3606](http://arxiv.org/abs/1209.3606)
* [[Jean Dieudonné]], _Introduction to the theory of formal groups_, Dekker, New York 1973.
* J. Dieudonné, _Linearly compact spaces and double vector spaces over sfields_, Amer. J. Math. 73, No. 1 (Jan., 1951), pp. 13-19 [jstor](http://www.jstor.org/stable/2372154)

Linearly compact rings and modules are treated in chapter VII, linear compactness and semisimplicity, in

* S. Warner, _Topological rings_, North-Holland Math. Studies 178, 1993

A similar concept: *profinite $k$-modules* - is treated in

* A. Yekutieli, _On the Structure of Behaviors_, Linear Algebra and its Applications  392 (2004), 159-181.

The definitions of linearly compact subcategories and linearly compact objects in (co)Grothendieck categories can be found in the chapter on duality in

* N. Popescu, [[Abelian categories with applications to rings and modules]], London Mathematical Society Monographs 3, Academic Press 1973, xii + 470 pp
*  U. Oberst, _Duality theory for Grothendieck categories and linearly compact rings_, J. Algebra __15__ (1970), p. 473 --542, [pdf](http://www.sciencedirect.com/science/article
* Mihail Ursul, _Topological rings satisfying compactness conditions_, Math. and its applications 549, Kluwer 2002, [gBooks](https://books.google.hr/books?id=OGDzOJjcjOQC)

[[!redirects linearly compact ring]]
[[!redirects linearly compact object]]
[[!redirects linearly compact modules]]
[[!redirects linearly compact vector spaces]]
[[!redirects linearly compact vector space]]


linearly distributive category

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###Context###
#### Monoidal categories
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[[!include monoidal categories - contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

A **linearly distributive category** is a [[category]] with two [[monoidal structures]] $\otimes$ ("times") and $\invamp$ ("par")  that "distribute" in the sense that there is a map (not necessarily invertible)

$$
   \delta \colon A \otimes (B \parr C) \longrightarrow (A \otimes B) \parr C
$$

Linearly distributive categories are the "representable" form of [[polycategories]], and they are also what "remains" of a [[star-autonomous category]] when the involution is forgotten but the dual tensor $A\invamp B \coloneqq (A^* \otimes B^*)^*$ is remembered.

Another way to say essentially the same thing is that a $\ast$-autonomous category is like a [[compact closed category]] in which the [[unit]] and [[counit]] of the [[dual objects]] refer to two different tensor products: we have $\top \to A \parr A^*$ but $A^* \otimes A \to \bot$, where $(\otimes,\top)$ and $(\parr,\bot)$ are two different monoidal structures.  A linearly distributive category encodes the necessary relationship between two such monoidal structures such that this makes sense, i.e. such that the [[triangle identities]] can be stated.

## Definition

A **linearly distributive category** (also called a **weakly distributive category**) is a [[monoidal category]] in two ways, with monoidal structures $(\otimes, \top)$ and $(\parr, \bot)$ equipped with extra associator natural transformations

\[ \delta^L:A\otimes (B\parr C) \to (A \otimes B) \parr C \]

\[ \delta^R:(A\parr B) \otimes C \to A \parr (B \otimes C) \]

and the obvious six pentagon equations and four triangle equations that make the monoidal structures work together nicely.  (Some of these equations can be interpreted as saying that the $\delta$s are [[tensorial strengths]] for the functors $(A\parr -)$ and $(-\parr C)$ with respect to the $\otimes$ monoidal structure.)  A **symmetric linearly distributive category** is one in which both monoidal structures are symmetric and three squares commute relating the symmetries to the $\delta$s.

These extra associators are sometimes called "distributors", and should not be confused with [[profunctors]].  The term is a pun on the distributivity of multiplication over addition, but "linearized" so that each variable only appears once in the result.

**Warning:** a [[distributive category]] *cannot* be made into a linearly distributive category with $\otimes = \times$ and $\parr = +$ unless it is a [[partial order]] (but every [[distributive lattice]] is indeed a linearly distributive category) (for a proof, see [Cockett-Seely 1997](#CockettSeely97)).  This mistake is easy to make and even appears in print in one or two places.

## Linear functors and transformations

A **linear functor** $F:C\to D$ between linearly distributive categories consists of a pair of functors $F_\otimes, F_\parr$ such that, $F_\otimes$ is a [[lax monoidal functor]] with respect to $\otimes$, $F_\parr$ is a [[colax monoidal functor]] with respect to $\parr$, and there are "linear strengths"

$$ F_\otimes (A\parr B) \to F_\parr(A) \parr F_\otimes (B) $$

$$ F_\otimes (A\parr B) \to F_\otimes(A) \parr F_\parr(B) $$

$$ F_\otimes (A) \otimes F_\parr(B) \to F_\parr(A\otimes B)$$

$$ F_\parr (A) \otimes F_\otimes(B) \to F_\parr(A\otimes B)$$

satisfying appropriate coherence conditions ([Cockett-Seely 1999](#CockettSeely99)).  A **linear natural transformation** $F\to G$ consists of a monoidal transformations $F_\otimes \to G_\otimes$  and $F_\parr \to G_\parr$ satisfying further coherence conditions.  This defines a 2-category of linearly distributive categories, which has the property of embedding the 2-category of $\ast$-autonomous categories and arbitrary lax monoidal functors (see below).

On the other hand, a linear functor in this sense is more general than a functor between underlying [[polycategories]] (see below).  The latter probably coincide with linear functors having the property that $F_\otimes = F_\parr$ as functors (so that the same functor $F$ is $\otimes$-lax monoidal and $\parr$-colax monoidal) and the linear strengths are identities.  We might call these **Frobenius linear functors**, since such a $1\to C$ is the same as a [[Frobenius algebra]] in $C$, and when specialized to the monoidal case $\otimes=\parr$ they coincide with [[Frobenius monoidal functors]].

Yet stronger is the notion of *strong* Frobenius linear functor, in which the lax and colax monoidal structures are [[strong monoidal functor|strong]].

## Related structures

### Star-autonomous category

Any [[star-autonomous category]] has an underlying symmetric linearly distributive category where we define $A\parr B \coloneqq (A^* \otimes B^*)^*$, or equivalently $A\parr B \coloneqq A^* \multimap B$, where $\multimap$ is the internal-hom.  The left distributor, for instance, is obtained as follows:
$$\array{\arrayopts{\rowlines{solid}}
\array{\arrayopts{\rowlines{solid}}
(A\otimes B)^* \longrightarrow (A\otimes B)^* \\
A \otimes (A\otimes B)^* \longrightarrow B^*} \qquad
\array{\arrayopts{\rowlines{solid}}
B^* \multimap C \longrightarrow B^* \multimap C\\
B^* \otimes (B^* \multimap C) \longrightarrow C\\
B^* \longrightarrow (B^* \multimap C) \multimap C} \\
A \otimes (A\otimes B)^* \longrightarrow (B^* \multimap C) \multimap C \\
A\otimes (B^*\multimap C) \longrightarrow (A \otimes B)^* \multimap C
}$$

Conversely, we say that a symmetric linearly distributive category with monoidal structures $(\otimes,\top)$ and $(\parr,\bot)$ has a **negation** if every object $A$ has a [[dualizable object|dual]] $\neg A$ and arrows

$$\top \to A \parr \neg A, \qquad \neg A \otimes A \to \bot$$

subject to triangular equations

$$1_A = \left( A \cong \top \otimes A \to (A \parr \neg A) \otimes A \stackrel{dist}{\to} A \parr (\neg A \otimes A) \to A \parr \bot \cong A \right)$$

$$\,$$

$$1_{\neg A} = \left( \neg A \cong \neg A \otimes \top \to \neg A \otimes (A \parr \neg A) \stackrel{dist}{\to} (\neg A \otimes A) \parr \neg A \to \bot \parr \neg A \cong \neg A \right).$$

Then symmetric linearly distributive categories with negations are the same as $\ast$-autonomous categories; see ([Cockett-Seely 1997](#CockettSeely97)).  Moreover:

* the [[forgetful functor]] from $\ast$-autonomous categories to symmetric linearly distributive ones has a [[left adjoint]] that "freely adjoins negations".  I don't know which kind of functors this statement refers to, but it's probably strong Frobenius ones.

* This functor also has a [[right adjoint]] that picks out (in the linear sense) the "nucleus" of [[dualizable objects]] ([Cockett-Seely 1999](#CockettSeely99)).  This adjunction uses lax monoidal functors between $\ast$-autonomous categories and linear functors between linearly distributive categories.

* Analogous facts hold relating non-symmetric linearly distributive categories with [[non-symmetric star-autonomous categories]].

### Polycategories

Any linearly distributive category has an underlying [[polycategory]] where we define $Hom(A_1,\dots,A_n; B_1,\dots,B_m)$ to be $Hom(A_1\otimes \cdots \otimes A_n; B_1 \parr \cdots \parr B_m)$.
The polycategorical compositions are obtained using the distributors.  For instance, we should be able to compose $f:(A,B) \to (C,D,E)$ with $g:(F,D,G) \to H$ to get $g\circ f : (F,A,B,G) \to (C,H,E)$; in a linearly distributive category this is the composite
$$\begin{aligned}
  F\otimes (A\otimes B)\otimes G
  &\xrightarrow{1_F\otimes f\otimes 1_G} F\otimes (C\parr D\parr E)\otimes G\\
  &\xrightarrow{\delta} ((F\otimes (C\parr D)) \parr E)\otimes G\\
  &\xrightarrow{\delta} ((C\parr (F\otimes D)) \parr E)\otimes G\\
  &\xrightarrow{\cong} ((C\parr E) \parr (F\otimes D))\otimes G\\
  &\xrightarrow{\delta} (C\parr E) \parr ((F\otimes D)\otimes G)\\
  &\xrightarrow{1_{C\parr E}\parr g} (C\parr E) \parr H\\
  &\xrightarrow{\cong} C\parr H\parr E
\end{aligned}$$
Conversely, a polycategory that is suitably "representable" yields a linearly distributive category.  Also, the forgetful functor from linearly distributive categories and *strong Frobenius* linear functors to polycategories has a left adjoint ([Cockett-Seely 1997](#CockettSeely97)).

### Symmetric monoidal categories

Any [[symmetric monoidal category]] becomes a linearly distributive category in which both monoidal structures are the same; all the required structure and coherence conditions follow straight from coherence for symmetric monoidal categories.  Note that such examples are almost never [[star-autonomous]].

These examples satisfy the extra property that the distributors $\delta$ are [[isomorphisms]].  However, not every linearly distributive category in which the distributors are isomorphisms are of this form; the rest are "shifted monoidal categories" in which $A \parr B \coloneqq A \otimes \bot^{-1} \otimes B$, where $\bot$ is a chosen $\otimes$-invertible object.

### Linear bicategories

[[linear bicategory|Linear bicategories]] are a [[horizontal categorification]] of linearly distributive categories.

## Internal structures

Some structures normally defined in a monoidal category can more generally be defined in a linearly distributive category.  This includes [[dual objects]] (yielding the above "negations") and also [[Frobenius algebras]] ([Egger 2010](#Egger2010)).

## As lax Frobenius algebras

Linearly distributive categories that are both closed and co-closed can be identified with lax [[Frobenius algebras]] in the [[polycategory]] [[MVar]] of [[multivariable adjunctions]]; see [this blog post](https://golem.ph.utexas.edu/category/2017/11/starautonomous_categories_are.html).

## History

A key feature of linearly distributive categories is the existence of [[morphisms]] of the form

$$
   \delta \colon A \otimes (B \parr C) \longrightarrow (A \otimes B) \parr C
$$

which exhibit linear [[distributivity]].  The importance of this map had previously  been overlooked in the [[linear logic]] community due to the (continuing) tendency to write [[sequents]]  in a one-sided manner.  However, the importance of the map was eventually realized by a number of people, albeit usually in the case of [[posets]]. [Cockett-Seely 1997](#CockettSeely97) referred to [Lambek 1993](#Lambek93) for a survey of some of those developments.

This concept was called a _weakly distributive category_  when it was presented by [[Robin Cockett]] and [[Robert Seely]] at the 1991 Durham LMS symposium on applications of categories in computer science ([Cockett-Seely 1992](#CockettSeely92) and [Cockett-Seely 1997](#CockettSeely97)). Independently, [[Valeria de Paiva]], a student of [[Martin Hyland]], also presented this concept in her PhD thesis as the category $\mathbf{GC}$, and credited conversations with [[Jean-Yves Girard]] at a 1987 Boulder conference for inspiring the construction. ([de Paiva 1991](#dePaiva91))

Subsequently, in recognition of the concept's fundamental role in the [[categorical semantics]] for the [[proof theory]] of [[linear logic]], and the fact that such categories are not a "weakening" of [[distributive categories]] in any standard sense, these categories were called _linearly distributive categories_.

Linearly distributive categories allowed the modularization of the [[proof theory]] of [[linear logic]] and a more systematic approach to the then unsolved [[coherence]] problems associated with the units in linear logic. This program of work culminated in a solution which was published in ([BCST 1996](#BCST))

These techniques also solved a classical open problem in [[closed monoidal categories]] concerning the behaviour of the triple dual.  [[Todd Trimble]], who was completing his PhD. at the time, provided a vital insight to the group on how this problem could be solved and this unlocked the solution to the general problem.

Linearly distributive categories are to [[polycategories]] as [[monoidal categories]] are to [[multicategories]].  Namely, when one represents "the commas" of a polycategory one obtains a linearly distributive category.  Crucially, the key coherence issues of (the multiplicative fragment of) linear logic concerning the units is already present at the linearly distributive level -- but it is no longer cluttered by the other features of linear logic.  Thus, this provided a clean substrate from which to study and isolate this coherence problem.

Contrary to the general direction of work in coherence (and, indeed, in the linear logic community), the solution to the problem, which was given in ([BCST 1996](#BCST)), involved an expansion reduction rewriting system modulo equations. At that time, the prevailing proof theoretic strategy for solving coherence problems was simply to [[cut elimination|cut-eliminate]]: usually this suffices to leave proofs in a normal form. However, this system was notable as this strategy clearly failed: it was necessary to leave some equations in the system. These equations, of course, involved the possible "rewiring"s of the units.

At that time this was generally thought to be a rather unsatisfactory and unnecessarily technical solution especially when compared to the normal forms to which the community was accustomed. However, in 2014, Willem Heijltjes and Robin Houston ([HH2014](#HH)) showed that equality of morphism is a [[PSPACE]]-complete problem. This essentially implies that there is no practical way round this problem beyond a search of the rewiring equivalence class.

Linearly distributive categories have now been used as the substrate for a number of systems.

## Related pages

* [[distributive law]]

* [[distributivity for monoidal structures]]

  * [[distributive monoidal category]]

  * [[bipermutative category]]

  * [[linearly distributive category]]

  * [[rig category]]

* [[polycategory]], [[star-autonomous category]]

* A linearly distributive category in which the units of the two monoidal structures coincide is an (iso-)[[mix category]].

* [[linear bicategory]]

## References

* {#Szabo} M.E. Szabo, *Polycategories*, Comm. Algebra 3 (1975) 663-689.  ([doi](http://www.tandfonline.com/doi/abs/10.1080/00927877508822067))

* {#dePaiva91} [[Valeria de Paiva]], _The Dialectica categories_, University of Cambridge Technical Report UCAM-CL-TR-213, 1991. ([pdf](http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.pdf))

* {#CockettSeely92} [[Robin Cockett]], [[Robert Seely]],  _Weakly Distributive Categories_, in M.P. Fourman, P.T. Johnstone, A.M. Pitts, eds., _Applications of Categories in Computer Science_, London Mathematical Society Lecture Note Series 177 (1992), pp. 45–65. ([book website](https://www.cambridge.org/au/academic/subjects/mathematics/logic-categories-and-sets/applications-categories-computer-science-proceedings-london-mathematical-society-symposium-durham-1991))

* {#Lambek93} [[Joachim Lambek]], _From categorial grammar to bilinear logic_, in Kosta Dosen and Peter Schroder-Heister, eds., _Substructural Logics_, Clarendon Press, 1993, pp. 207–237. ([book website](https://global.oup.com/academic/product/substructural-logics-9780198537779))

* {#BCST}  [[Richard Blute|R. Blute]], [[Robin Cockett|J.R.B. Cockett]], [[Robert Seely|R.A.G. Seely]], [[Todd Trimble|T. Trimble]], _Natural deduction and coherence for weakly distributive categories_, J. Pure Appl. Algebra, 113 (1996), pp. 229–296. ([web](http://www.sciencedirect.com/science/article/pii/002240499500159X))

* {#CockettSeely97} [[Robin Cockett]], [[Robert Seely]],  _Weakly Distributive Categories_, J. Pure Appl. Algebra, 114 (1997) 2, pp 133-173. (corrected version: [ps.gz](http://www.math.mcgill.ca/rags/linear/wdc.ps.gz), [pdf](http://www.math.mcgill.ca/rags/linear/wdc-fix.pdf))

* {#CockettSeely99} [[Robin Cockett]] and [[Robert Seely]], _Linearly distributive functors_, J. Pure Appl. Algebra 143 (1999), pp. 155–203. ([doi](https://doi.org/10.1016/S0022-4049%2898%2900110-8))

* [[Paul-André Melliès]], [Categorical Semantics of Linear Logic](http://www.pps.univ-paris-diderot.fr/~mellies/papers/panorama.pdf), published in [[Pierre-Louis Curien]], Hugo Herbelin, Jean-Louis Krivine, Paul-André Melliès. _Interactive models of computation and program behaviour_, Panoramas et Synthèses 27, Société Mathématique de France, 2009.

* {#HH} Willem Heijltjes, [[Robin Houston]], _No proof nets for MLL with units: proof equivalence in MLL is PSPACE-complete_, CSL-LICS '14 Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), Article No. 50. ([web](http://dl.acm.org/citation.cfm?id=2603126))

Frobenius algebras in linearly distributive categories are discussed in

* {#Egger2010} [[Jeff Egger]], *The Frobenius relations meet linear distributivity*, Theory and Applications of Categories, Vol. 24, 2010, No. 2, pp 25-38. ([web](http://tac.mta.ca/tac/volumes/24/2/24-02abs.html))

[[!redirects weakly distributive category]]
[[!redirects weakly distributive categories]]

[[!redirects linearly distributive categories]]


linearly independent subset

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### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Linearly independent subsets
* table of contents
{: toc}

## Idea

A [[concrete set|set]] of [[vector]]s is linearly dependent if one can be written as a [[linear combination]] of the others, and linearly independent otherwise.  In the latter case, the vectors in the set form a [[basis of a vector space|basis]] of their [[linear span|span]].

## Definitions

Let $K$ be a [[rig]], and let $V$ be a (left or right) [[module]] over $K$.  (Often $K$ is a [[field]] so that $V$ is a [[vector space]], but this is unnecessary.)  Let $S$ be a [[subset]] of the [[underlying set]] ${|V|}$ of $V$.

### Abstractly

By the [[adjunction]] between the underlying-set functor and the [[free functor]], the subset inclusion
$$ i_S\colon S \to {|V|} $$
corresponds to a [[homomorphism]]
$$ \hat{i}_S\colon K[S] \to V .$$
Although $i_S$ is (by hypothesis) a [[monomorphism]] in $Set$, $\hat{i}_S$ need not be a monomorphism in $K Mod$.

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

The subset $S$ is __linearly independent__ if $\hat{i}_S$ is a monomorphism; otherwise, $S$ is __linearly dependent__.
=--

Conversely, if we start with an (abstract!) [[set]] $S$ and a monomorphism from $K[S]$ to $V$, then the corresponding [[function]] from $S$ to ${|V|}$ must be a monomorphism (because the underlying-set functor is [[faithful functor|faithful]]).  Thus, our considering only subsets of ${|V|}$ loses no generality.

### Concretely

Given a [[linear combination]]
$$ \sum_{i=1}^n a_i v_i ,$$
this may or may not equal the [[zero vector]] $0_V$.  Of course, if every $a_i$ is the zero scalar $0_K$, then the sum must be $0_V$.

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

The subset $S$ is __linearly independent__ if, conversely, for every finite subset $\{v_1, \ldots, v_n\} \subseteq S$, we have $a_i = 0_K$ for all $i$ whenever
$$ \sum_{i=1}^n a_i v_i = 0_V ;$$
otherwise, $S$ is __linearly dependent__.
=--

Observe that the empty set is linearly independent by a vacuous implication.

### Constructively

In [[constructive mathematics]], the definitions above of linear independence are all right (and still equivalent), but the definition of linear dependence as simply the [[negation]] of linear independence is unsatisfying.  Furthermore, we sometimes want something stronger than mere linear independence.

If $K$ is a [[Heyting field]], then the field structure defines a [[tight apartness relation]] $\ne$.  Even if $K$ is not a field, we may still suppose that it is equipped with a tight apartness, or at least some [[inequality relation]] $\ne$.  If we restrict attention to modules with a compatible inequality relation and homomorphisms that preserve this, then we also get an inequality relation $\ne$ on the [[hom-sets]] of the category $K Mod$.  This allows us to define stronger notions of both linear dependence (which we take to be the default notion) and linear independence (to which we give a new name).

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

The subset $S$ is __linearly dependent__ if $\hat{i}_S$ is non-monic in the strong sense that there exist [[generalised elements]] $f, g\colon A \to K[S]$ such that
$$ A \overset{f}\underset{g}\rightrightarrows K[S] \to V $$
are equal but $f \ne g$.  Concretely, $S$ is __linearly dependent__ if some linear combination
$$ \sum_{i = 1}^n a_i v_i = 0_V $$
but at least one $a_i \ne 0_K$.
=--

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

The subset $S$ is __linearly free__ if $\hat{i}_S$ is a [[regular monomorphism]], or equivalently if it is monic in the strong sense that $f ; \hat{i}_S \ne g ; \hat{i}_S$ whenever $f \ne g$.  Concretely, $S$ is __linearly free__ if
$$ \sum_{i = 1}^n a_i v_i \ne 0_V $$
whenever at least one $a_i \ne 0_K$.
=--

Then we have the following implications (assuming that $\ne$ is tight, so that $a = b$ holds iff $a \ne b$ fails) but not (in general) their unstated converses:
$$ LF \Rightarrow LI \Leftrightarrow \neg{LD} ;$$
$$ \neg{LF} \Leftarrow \neg{LI} \Leftarrow LD .$$

It may also be instructive to look at the logical structure of each condition:
*  $LI$: $\forall (a,v),\; \sum a v = 0 \;\Rightarrow\; \forall i,\; a_i = 0$;
*  $LD$: $\exists (a,v),\; \sum a v = 0 \;\wedge\; \exists i,\; a_i \ne 0$;
*  $LF$: $\forall (a,v),\; \sum a v \ne 0 \;\Leftarrow\; \exists i,\; a_i \ne 0$.

[[!redirects linearly independent subset]]
[[!redirects linearly independent set]]
[[!redirects linearly independent]]
[[!redirects linear independence]]

[[!redirects linearly dependent subset]]
[[!redirects linearly dependent set]]
[[!redirects linearly dependent]]
[[!redirects linear dependence]]

[[!redirects linearly free subset]]
[[!redirects linearly free set]]
[[!redirects linearly free]]
[[!redirects linear freedom]]
[[!redirects linear freeness]]


lined topos

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### Context
#### Synthetic differential geometry
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[[!include synthetic differential geometry - contents]]
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#Contents#
* automatic table of contents goes here
{:toc}

## Definition

+-- {: .un_defn}
###### Definition
A **lined topos** $(\mathcal{T}, R)$ is

* a [[ringed topos]] $(\mathcal{T}, k)$

  (usually with the [[internalization|internal]] [[ring]] object $(k,+,\cdot)$ assumed to be commutative)

* and equipped with a choice $(R,+,\cdot)$ of [[internalization|internal]] commutative [[algebra]] object $(R,+,\cdot)$ over $k$ -- the **line object**.

=--

## Variations

* A [[smooth topos]] is a lined topos where the line is required to be a smooth differentiable space with [[infinitesimal object|infinitesimal]] subspaces in a certain way. This is the basic type of object studied in [[synthetic differential geometry]].

* A [[super smooth topos]] is a lined topos that is a [[smooth topos]] and in which the $k$-algebra structure on $R$ is refined to that of a $k$-[[superalgebra]].

## Constructions in lined toposes

### Path objects

The line object $R$ in a lined topos $\mathcal{T}$ canonically has the structure of a [[interval object|cartesian interval object]].

As described there, this canonically induces

* a [[cosimplicial object]] $\Delta_R : \Delta \to \mathcal{T}$

* a functor $\Pi : \mathcal{T} \to S \mathcal{T}$

  that sends each object in the topos to a [[simplicial object]]

  $$
    X \mapsto X^{\Delta_R^\bullet}
  $$

  ( which may be interpreted as presenting the
    [[schreiber:path ∞-groupoid|path ∞-groupoid]] of $X$).

### Contractible objects {#ContractibleObjects}

The following terminology is sometimes useful.

+-- {: .un_defn}
###### Terminology
**(contractible object)**

Let $(\mathcal{T} = Sh(C), R)$ be a lined [[Grothendieck topos]] with respect to a [[site]] $C$.

Call an object $X \in \mathcal{T}$  **contractible** with respect to the [[interval object]] $R$, if the [[simplicial presheaf|simplicial sheaf]] $\Pi(X) = X^{\Delta_R^\bullet} : C^{op} \to$ [[SSet]] sends each object to a [[contractible space|contractible]] [[simplicial set]].

=--

**Examples**

* **sheaves on topological spaces** Let $Top'$ be a [[small category|small]] version of the category of sufficiently nice [[topological space]]s, for instance connected [[CW complex]]es. The canonical line object in $Sh(Top)$ is ${*} \stackrel{0}{\to} [0,1] \stackrel{1}{\leftarrow} {*}$ the standard topological interval. For $X \in Top$, $\Pi(X) = X^{\Delta_R^\bullet}$ is the [[singular simplicial complex]] of $X$.  This is contractible in the above sense precisely if $X$ is a [[contractible space]] in the standard sense.

* **sheaves on cartesian spaces** Let [[CartSp]] be the full subcategory of [[Diff]] on [[smooth manifold]]s of the form $\mathbb{R}^n$, for $n \in \mathbb{N}$. The canonical line object in $\mathcal{T} = Sh(CartSp)$ is the real line regarded as an [[interval object]]

  $$
    R = ({*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*})
    \,.
  $$

  +-- {: .num_lemma}
  ###### Lemma

  In the lined topos $(\mathcal{T} = Sh(CartSp), R = \mathbb{R})$
  the [[representable functor|representable]]
  objects $\mathbb{R}^n$ are contractible with respect to $R$.

  =--

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

  This is not quite as entirely trivial as it may seem on first sight,
  but follows directly from the [[Tietze extension theorem]]
  for smooth manifolds:

  we check that for all $V \in $ [[CartSp]]
  every  [[boundary of a simplex]]
  $\partial \Delta[k] \to \Pi(\mathbb{R}^n)(V)$ extends through
  $\partial \Delta[k] \hookrightarrow \Delta[k]$:

  by the [construction of the cosimplicial object](http://ncatlab.org/nlab/show/interval+object#FundGeomInftCat)
  $\Delta_R : \Delta \to Sh(CartSp)$ we have that morphisms
  $\partial \Delta[k] \to \Pi(\mathbb{R}^n)(V)$ correspond to smooth
  maps from the
  boundary of a $V$-cylinder over the
  standard $k$-simplex in $\mathbb{R}^k \times V
  \to \mathbb{R}^n$.
  Since this is a closed subset of $\mathbb{R}^k \times V$,
  by the [[Tietze extension theorem]] these maps extend (apply the theorem
  to each of their components) to all of $\mathbb{R}^k \times V$,
  hence in particular
  to the standard $k$-simplex inside $\mathbb{R}^k$ defined by the
  interval object. This constitutes the required extension to a
  $V$-family of $k$-simplices in $\mathbb{R}^n$

  $$
    \array{
      \partial \Delta[n] &\to& (\mathbb{R}^n)^{\Delta_R^\bullet}(V)
      \\
      \downarrow & \nearrow
      \\
      \Delta[n]
    }
    \,.
  $$

  =--

* **sheaves on cartesian smooth loci** A small variation of the above example leads to [[smooth topos]]es with contractible representables:

  let $CartSp_{synth} \subset \mathbb{L}$ be the full [[subcategory]] of [[smooth loci]] on those smooth loci of the form $\mathbb{R}^n \times D_k(r)$, where $D_k(r)$ is the [[infinitesimal space]] of $k$th order infinitesimal neighbours of the origin in $\mathbb{R}^r$.

  The line object is again ${*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*}$ as in the above example. Crucially, the [[infinitesimal space]]s $D_k(r)$ all have a unique point ${*} \to D_k(r)$. Accordingly, there is also a unique morphism
  $R^n \to D_k(r)$ for all $n$. It follows that simplices in $R^n \times D_k(r)$ are simplices in $R^n$ as above, and trivial as maps to the $D_k(r)$-factor. Hence the above argument carries over to this case and shows that all the $\mathbb{R}^n \times D_k(r)$ are contractible.


Ling Zhou

* [personal page](https://lingzhou0.github.io/)

* [institute page](https://math.osu.edu/people/zhou.2568)

## Selected writings

On [[persistent homotopy]]:

* [[Facundo Mémoli]], [[Ling Zhou]], *Persistent Homotopy Groups of Metric Spaces* $[$[arXiv:1912.12399](https://arxiv.org/abs/1912.12399), [talk](Center+for+Quantum+and+Topological+Systems#ZhouAtHomotopicalPerspectivesOnTDAJune2022)$]$

category: people


linguistics

+-- {: .rightHandSide}
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### Context
#### Linguistics
+-- {: .hide}
[[!include linguistics - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

**Linguistics** is the (scientific) study of natural human language. See a reasonably good page at [Wikipedia](http://en.wikipedia.org/wiki/Linguistics). It aims to develop a toolbox for description of concrete languages (applied linguistics, descriptive grammar), their classification (language families, language typology) as well as to understand how language functions (e.g. relations to cognition and social constraints) and changes (historical and comparative linguistics). The description of a concrete language includes organizational principles called grammar and distinguished constants called lexic. The grammar is described at several hierarchical levels
(phonetics, phonology, morphology, syntax).

There is an opposition between

* individual expressions (Franc. parole) and abstract system of the language (Franc. langage)
* snapshot of a system in time (synchronic) and its development (diachronic)
* knowledge of a language (ability to speak) and knowledge about a language (theoretical description)

Linguistics sign has also a dichotomy between its meaning and its expression.

Linguistic systems are often also at several levels from personal idiom/idiolect, through social sociolects, regional dialects to languages and language families. There is no fundamental agreement of entirely consistent boundary of what is a boundary level of a language as opposed to dialects or language groups. Also there is no universal criterium of what a boundary of a word is, that is what a word is.

## Syntax

Syntax is a description of how a received (valid) phrase or sentence is made out of its constituent parts and discipline studying syntax in that sense.

#### Grammar of formal languages

Linguists may deploy formal methods to analyse the [[syntax (linguistics)]] of natural language (see also [[syntax]] in the sense of logics).

[[formal grammar|Formal grammars]] characterize which [[string (computer science)|strings]] of [[words]] are considered grammatical, they come in many different flavours:

* [[context-free grammar]]

* [[pregroup grammar]]

* [[categorial grammar]]

* [[mildly context-sensitive grammar]]

* [[context-sensitive grammar]]

* [[formal grammar#unrestrictedgrammar|unrestricted grammar]]

These different frameworks come in a hierarchy of expressive power, see [[Chomsky hierarchy]].

## Semantics

Formal theories of meaning are based on the principle of [[compositionality]]: the semantics of a sentence is a function of the meanings of its words, and of its grammatical structure.

* [[Montague grammar|Montague semantics]]
* [[categorical compositional distributional semantics]]
* [[dependent type theoretic methods in natural language semantics]]

## Pragmatics

Pragmatics is concerned with the meaning of language in context.

* [[monad (in linguistics)]]
* [[language game]]

## Related Concepts

* [[syntax]], [[semantics]], [[semantics of a programming language]]

* [[semiotics]]

* [[lambda calculus]], [[type theory]]

* [[context-free grammar]]

* [[monad (in computer science)]]

* [[continuation-passing style]]

* [[linear logic]], [[modal logic]]

* [[mathematical linguistics]]

* [[transcendental grammar]]

* [[DisCoCat]], [[DisCoPy]]

## Literature

### Linguistics per se

* diverse materials and corpora can be found at extensive web site
[The Linguist List](http://www.linguistlist.org)
* Wikipedia [linguistics](http://en.wikipedia.org/wiki/Linguistics), [philology](https://en.wikipedia.org/wiki/Philology), [phonetics](https://en.wikipedia.org/wiki/Phonetics), [morphology (linguistics)](https://en.wikipedia.org/wiki/Morphology_%28linguistics%29), [syntax](https://en.wikipedia.org/wiki/Syntax), [semantics](https://en.wikipedia.org/wiki/Semantics), [pragmatics](http://en.wikipedia.org/wiki/Pragmatics), [semiotics](https://en.wikipedia.org/wiki/Semiotics), [linguistic typology](https://en.wikipedia.org/wiki/Linguistic_typology), [historical linguistics](https://en.wikipedia.org/wiki/Historical_linguistics), [comparative linguistics](https://en.wikipedia.org/wiki/Comparative_linguistics)

* [LaTeX for linguists](http://www.essex.ac.uk/linguistics/external/clmt/latex4ling)

* F. de Saussure, _Cours de linguistique générale_

* André Martinet, _Éléments de linguistique générale_, (Engl. transl.: Elements of General Linguistics)

* Adrian Akmajian, Richard A. Demers, Ann K. Farmer, Robert M. Harnish, _Linguistics, an introduction to language and communication_

* John Lyons, _Semantics_; _Linguistics semantics_

* Bernard Comrie, Language universals and linguistic typology, syntax and morphology, 1981, 1989

* Peter Ladefoged, Keith Johnson, A course in phonetics

* Steven Pinker,  _The language instinct_, 1994

* Winfred P. Lehmann, _Historical linguistics: an introduction_

* George Lakoff, _Women, fire and dangerous things_, 1987

* Ronald Langacker, _Foundations of cognitive grammar_, vol. 1, 1987, vol. 2, 1991

* Pieter A. M. Seuren, The logic of language, vol. II of Language from within; (vol. I: Language in cognition) Oxford University Press 2010

* [[David Mumford]], _Grammar isn't merely part of language_, 2016 [blog](https://www.dam.brown.edu/people/mumford/blog/2016/grammar.html)

* M. Tallerman, _Understanding syntax_ (1st ed, 1998, 5th ed. 2020)

* [[Umberto Eco]], _Semiotics and the philosophy of language_, Indiana University Press 1984

### Category theory and linguistics

* [[Joachim Lambek]], _Pregroups and natural language processing_, The Mathematical Intelligencer 28 (2006), 41–48 [pdf](http://www.math.mcgill.ca/rags/JAC/124/Lambek-Pregroups-s.pdf)
* [[Bob Coecke]], [[Mehrnoosh Sadrzadeh]], Stephen Clark, _Mathematical foundations for a compositional distributional model of meaning_, in the Lambek Festschrift, special issue of Linguistic Analysis, 2010 [arxiv/1003.4394](https://arxiv.org/abs/1003.4394)

* [[Vasily Pestun]], [[Yiannis Vlassopoulos]], *Tensor network language model*, [[arXiv:1710.10248](https://arxiv.org/abs/1710.10248)]

  > (via [[tensor networks]])

category: philosophy

[[!redirects natural language]]
[[!redirects natural languages]]

[[!redirects formal linguistics]]


linguistics - contents

**[[linguistics|Linguistics]]**

## General

* [[structuralism]]

### Philosophy of Language

* [[transcendental grammar]]

* [[language game]]

### [[mathematical linguistics|Mathematical linguistics]]

* [[language]]

* [[transcendental syntax]]

* [[syntax-semantics duality]]

#### [[syntax|Syntax]]

* [[formal grammar]]

* [[regular language]]

* [[context-free grammar]]

* [[pregroup grammar]]

* [[categorial grammar]]

* [[combinatory categorial grammar]]

* [[mildly context-sensitive grammar]]

* [[context-sensitive grammar]]

* [[dependency grammar]]

#### [[semantics|Semantics]]

* [[Montague semantics]]

* [[dependent type theoretic methods in natural language semantics]]

* [[categorical compositional distributional semantics]]

#### [[pragmatics|Pragmatics]]

* [[monad (in linguistics)]]
* [[language game]]

### Concepts

* [[compositionality]]

* [[generalized quantifier]]

* [[tree]]

* [[Chomsky hierarchy]]

### People

* [[Noam Chomsky]]

* [[Günter Hotz]]

* [[Marcus Kracht]]

* [[Joachim Lambek]]

* [[Matilde Marcolli]]

* [[Richard Montague]]

* [[Aarne Ranta]]

* [[Mehrnoosh Sadrzadeh]]

* [[Stephan Wannowski]]

[Edit this sidebar](/nlab/edit/linguistics+-+contents)


link

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#### Topology
+--{: .hide}
[[!include topology - contents]]
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=--

# Contents
* the following line creates the automatic table of contents
{: toc}

## Idea

A **link** is a generalisation of a [[knot]] where one is allowed more than one component.  Many [[knot invariants]] extend to [[link invariant]]s and for many such invariants, one needs to know this extension to compute it even for a knot.  Thus the study of links and knots is inextricably intertwined.

## Basics

+-- {: .num_defn #link}
###### Definition
A **link** is an [[embedding]] of a [[finite number]] of copies of the [[circle]].  The embedding is usually taken in $\mathbb{R}^3$, or its [[one-point compactification]], $S^3$.
=--

It is possible to generalise this to more varied sources and targets.

Links can be studied in a number of ways depending on the notion of equivalence that is used.  Coming from knot theory, one considers equivalence up to [[isotopy]]; that is, two links are equivalent if there is a homotopy between them which is an embedding for all times.  A weaker notion was consider by [Milnor](#jmLinkGroups) wherein the components of the link are allowed to pass through themselves, but not through other components.  That is, when restricted to each component it must be an immersion for all times, and the images of the components must always be disjoint.

## Examples

### Trivial links

Any [[knot]] is a link, and any [[disjoint union]] of [[unknot]]s (called an __unlink__) is a link.  We may call these 'trivial' (hopefully this name isn\'t standard for something different), in the sense of what you would know about before you study links.

### The Hopf Link

The [[Hopf link]] is the simplest non-trivial (in the sense above) link, consisting of two components linked once.

[[!include Hopf link - SVG]]

### The Borromean Link

It is possible to link together $n$ circles in such a way that removing any one makes the others fall apart.  For $n = 2$, we have the Hopf link above; for $n = 3$, we have the [[Borromean link]], or Borromean Rings.

[[!include Borromean link - SVG]]

### The Whitehead Link

The [[Whitehead link]] is an example of a link that shows the difference between the two notions of equivalence.  If the links are only allowed to move by isotopies, then the two components are linked.  However, if a link is allowed to pass through itself, then they can be unlinked.

[[!include Whitehead link - SVG]]

### Brunnian links

A __Brunnian link__ is a link which is not an unlink but which has the property that the removal of any of its components results in an unlink.  Technically, this includes the Hopf link and any knot (thanks to [this MO question](http://mathoverflow.net/questions/40724/is-the-hopf-link-a-brunnian-link) for settling that issue). The Borromean rings above are an example of a Brunnian link with three components.

## Related concepts

* [[hyperbolic link]]

* [[braid]]

* [[knot]]

## References

* [[John Milnor]],  (1954). Link groups. _Ann. of Math. (2)_, _59_, 177–195. [MR](http://www.ams.org/mathscinet-getitem?mr=71020){: #jmLinkGroups}

* {#Birman75} [[Joan S. Birman]], _Braids, links, and mapping class groups_, Princeton Univ Press (1975) [[ISBN:9780691081496](https://press.princeton.edu/books/paperback/9780691081496/braids-links-and-mapping-class-groups-am-82-volume-82), [preview pdf](https://api.pageplace.de/preview/DT0400.9781400881420_A26691398/preview-9781400881420_A26691398.pdf)]

* [[Dale Rolfsen]], _Knots and links_, AMS Chelsea __346__ (2003) [[ams:chel-346-h](https://bookstore.ams.org/chel-346-h), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/rolfsen.pdf)]

[[!redirects link]]
[[!redirects links]]
[[!redirects Link]]
[[!redirects Links]]

[[!redirects Brunnian link]]
[[!redirects Brunnian links]]


link diagram

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[[!include knot theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _link diagram_ is, roughly speaking, the combinatorial object obtained by projecting a [[link]] in 'general position' to a plane. [[Reidemeister's theorem]] establishes that one does not lose any essential information by passing from a link to its diagram, and thus it is possible to study knot theory in a way which takes link diagrams as primary: this is sometimes known as [[diagrammatic knot theory]].

## Formal definition

The formal definition of a link diagram is in itself independent of the notion of a [[link]].

\begin{defn} \label{DefinitionConnectedLinkDiagram} A _connected link diagram_ is a [[connected graph|connected]] (undirected) 4-valent [[plane graph]] $G$ equipped with the following data for every vertex $v$ of $G$.

1. A choice of division of the four edges incident to $v$ into two pairs, say $(e_{0}, e_{1})$ and $(e_{2}, e_{3})$. We refer to the two edges in the first pair as _over-edges_, and to the two edges $(e_{1}, e_{2})$ in the second pair as _under-edges_.

1. A cyclic ordering of the four edges of $G$ incident to $v$ so that the over-edges and under-edges alternate.

\end{defn}

\begin{defn} \label{DefinitionLinkDiagram} A _link diagram_ is a planar graph such that each connected component satisfies either 1. or 2. below.

1. It is (up to homeomorphism) a circle, disjoint from the rest of the link diagram.

1. It is a connected link diagram in the sense of Definition \ref{DefinitionConnectedLinkDiagram}.

\end{defn}

\begin{rmk} It is important to note that a plane graph consists, by definition, of an abstract [[graph]] together with a _chosen_ embedding into the plane. There exist non-equivalent link diagrams which have the same underlying abstract graph, but for which the embedding in the plane is different. \end{rmk}

\begin{terminology} A component of a link diagram which satisfies 1. in Definition \ref{DefinitionLinkDiagram} is typically referred to as an _unknot_, or an _unknotted component_. \end{terminology}

\begin{terminology} \label{TerminologyCrossingsOfALinkDiagram} The vertices $v$ of a link diagram $L$ which do not belong to an unknotted component, together with the data of 1. and 2. in Definition \ref{DefinitionConnectedLinkDiagram} for $v$, are referred to as the _crossings_ of $L$.

\end{terminology}

## Obtaining a link diagram from a link

A link diagram can be obtained from a [[link]] by choosing a plane in $\mathbb{R}^{3}$, and projecting the link onto this plane. A small change in the direction of projection will ensure that it is one-to-one except at the double points, which will become the crossings of the link diagram in the sense of Terminology \ref{TerminologyCrossingsOfALinkDiagram}, where the image curve of the knot crosses itself once transversely. Which strand of the two intersecting at the double point is the 'over strand' and which is the 'under strand' is recorded as the data needed for 1. and 2. in Definition \ref{DefinitionConnectedLinkDiagram}.

Consider for example the parallel projection

$$p: \mathbb{R}^3 \to \mathbb{R}^2$$

defined by $p(x,y,z) = (x,y,0)$.

(If you prefer your knots to be in $S^3$, of course, you can remove a single point from the complement of $K$ and then project down to $\mathbf{R}^3$. It does not matter which point you use.)

A point $\mathbf{x}$ in the image $pK$ is called a *multiple point* if $p^{-1}(\mathbf{x})$ contains more than one point of $K$.  A *double point* occurs when there are exactly two points of $K$ in this, and similarly for a *triple point*, etc. Multiple points of infinite order could occur.

A knot is in *regular position* or _general position_ with respect to $p$ if there are only double points and these are genuine crossings (i.e. no tangential touching occurs in the projected curve).

Any smooth or PL link $L$ is equivalent under an arbitrarily small rotation of $\mathbf{R}^3$ to one in regular position with respect to $p$.

A proof can be found in Crowell and Fox (page 7).

\begin{terminology} A _knot diagram_ is a link diagram which arises from a projection of a [[knot]].\end{terminology}

## Related concepts

* [[Reidemeister move]]

category: knot theory

[[!redirects knot diagram]]
[[!redirects knot diagrams]]
[[!redirects link diagrams]]


link invariant

# Link invariants
* table of contents
{: toc}

## Idea

Many of the [[knot invariants]] have analogues for links. Some, in fact seem to be better thought of as being restrictions of link invariants to single component links (i.e. knots).  We repeat the statement from [[knot invariant]] with obvious adjustments.

## Definition

A __link invariant__ is map from [[isotopy]] classes of [[links]] to any kind of structure you could imagine.  These are helpful because it's much easier to check that the structures one maps to (numbers, groups, etc.) are different than it is to check that links are different.

## Examples

 * [[linking number|Linking number]]

[[!redirects link invariant]]
[[!redirects link invariants]]


linking number

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#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

# Linking numbers
* table of contents
{: toc}

## Signs of crossings

In an oriented link diagram, we can see there are two types of possible crossing. They are allocated a sign, + or -.

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One method of remembering the sign convention is to imagine an approach to the crossing along the *underpass* in the direction of the orientation:

* if the overpass passes from left to right the crossing is counted as being *positive*;
* if it passes from right to left it counts as *negative*.

## Writhe

The *writhe* of an oriented *knot* or *link* diagram is the sum of the signs of all its crossings.  If $D$ is the diagram, we denote its writhe by $w(D)$.

The writhe is used in the definition of some of the [[knot invariants]].

## Linking number

This is a variant of the writhe that is more adapted for use with links.

Suppose we have an oriented link diagram $D$ with components $C_1, \ldots, C_m$, the *linking number* of $C_i$ with $C_j$ where $C_i$ and $C_j$ are distinct components of $D$, is to be one half of the sum  of the signs of the crossings of $C_i$ with $C_j$; it will be denoted $lk(C_i,C_j)$.

The linking number of the diagram $D$ us then the sum of the linking numbers of all pairs of components:

$$Lk(D) = \sum_{1\le i\lt j\le m}lk(C_i,C_j).$$

## Examples

The writhe of the standard trefoil is 3, of the Hopf link (both components clockwise oriented) is +2, but that of the Borromean rings is 0 although it is a non-trivial link.

## Invariance?

The writhe is not an isotopy invariant, as it can be changed but twisting a stand of the knot (or link).

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

The writhe is an invariant of regular isotopy.
=--

+-- {: .un_proposition}
###### Proposition
The linking number is a [[link invariant]].
=--

+-- {: .proof}
###### Proof
We use [[Reidemeister moves]] so have to check that they do not change the linking number of a diagram. Any Reidemeister move that involves at least two components of the link (i.e. which must be an R2 or R3) leaves all linking numbers between components unchanged.  An R2 move removes or introduces two crossings of opposite sign, whilst an R3 leaves the number of crossings and their signs unaltered.
=--
We can conclude that the Hopf link is not isotopic to the two component unlink, (which is reassuring) as any assignment of orientations to the Hopf link leads to a non-zero linking number.

##Linking number is an integer

Let $n_{1}$ (resp. $n_{2}$) be the sum of the signs of crossings between a pair of components $L_{1}$ and $L_{2}$ of a link in which the over arc belongs to $L_{1}$ (resp.  $L_{2}$). The linking number is then clearly equal to $\frac{1}{2}(n_{1} + n_{2})$.

Now, take a crossing between $L_{1}$ and $L_{2}$ such that the over arc belongs to $L_{1}$. Suppose that we switch the crossing type, so that now the under arc belongs to $L_{1}$. Then, no matter what the sign of the crossing, it can be verified that the sum $n_{1} - n_{2}$ remains unchanged by this switch.

We may also obviously make crossing switches to the self-crossings of a component, or to crossings of the two components we are looking at with other components, without affecting the sum $n_{1} - n_{2}$.

In addition, it is straightforward to check that the Reidemeister moves do not change the sum $n_{1} - n_{2}$.

Now, it is obvious that any pair of components of a link can be unlinked from the link after making appropriate crossing switches. In the case that $L_{1}$ and $L_{2}$ are circles disjoint from each other and the rest of the link, we have that $n_{1} - n_{2} = 0$.

We deduce that $n_{1} - n_{2} = 0$ in all cases. Hence $n_{1} = n_{2}$, and the linking number is equal to $\frac{1}{2}(2n_{1}) = n_{1} = n_{2}$.

In particular, the linking number is an integer.

An immediate consequence of the fact that the sum of the signs of the crossings between $L_{1}$ and $L_{2}$ is even is that there must in fact be an even number of such crossings.

[[!redirects linking number]]
[[!redirects linking numbers]]


Linus Kramer

* [website](http://wwwmath1.uni-muenster.de:8013/persdb/show_perspage.php?id=381)

## Selected writings

On [[homogeneous spaces]] with the same [[rational cohomology]] as a [[product space|product]] of [[n-spheres]]:

* [[Linus Kramer]], _Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurface_,  Memoirs of the American Mathematical Society number 752 ([arXiv:math/0109133] (http://arxiv.org/abs/math/0109133), [doi:10.1090/memo/0752](http://dx.doi.org/10.1090/memo/0752), [GoogleBooks](http://books.google.com/books?id=SA8O6ihrDFkC&printsec=frontcover&hl=de&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false))

On [[topological groups]], [[locally compact groups]] and the [[Peter-Weyl theorem]]:

* [[Linus Kramer]], *Locally Compact Groups*, 2017 ([pdf](https://www.uni-muenster.de/AGKramer/content/LCManuscript.pdf), [[Kramer_LocallyCompactGroups.pdf:file]])

category: people


Linuxmetel

I learn mathematics as a hobby. I'm now mainly interested in [[algebraic geometry]] and [[commutative algebra|commutative]] [[algebra]].

X (Twitter): [@linuxmetel](https://twitter.com/linuxmetel)

category: people


Lionel Mason

Lionel Mason is professor for [[mathematics]] at Oxford.

* [webpage](https://people.maths.ox.ac.uk/lmason/)

## Selected writings

On [[twistors]] and [[twistor string theory]]:

* [[Michael Atiyah]], [[Maciej Dunajski]], [[Lionel J. Mason]], _Twistor theory at fifty: from contour integrals to twistor strings_,  Proc. R. Soc. A473: 20170530 ([arXiv:1704.07464](https://arxiv.org/abs/1704.07464), [doi:10.1098/rspa.2017.0530](http://dx.doi.org/10.1098/rspa.2017.0530))

## Related $n$Lab entries

* [[twistor string theory]]

category: people

[[!redirects Lionel J. Mason]]


Lior Yanovski

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=253040)

## Selected wrirings

On the [[Eckmann-Hilton argument]] for [[(∞,1)-operads]]:

* [[Tomer Schlank]], [[Lior Yanovski]], *The ∞-Categorical Eckmann-Hilton Argument*, Algebr. Geom. Topol. 19 (2019) 3119-3170 ([arXiv:1808.06006](https://arxiv.org/abs/1808.06006), [doi:10.2140/agt.2019.19.3119](https://doi.org/10.2140/agt.2019.19.3119))

category: people


Liouville cocycle

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## Idea

The _Liouville cocycle_ is a degree 1 [[cocycle]] in the [[groupoid cohomology]] of the [[action groupoid]] of the [[action]] of conformal rescalings on the space of [[Riemannian metric]]s on a surface.

## Definition

Recall that given a space $X$ with an [[action]] of a [[group]] $G$ on it, a $U(1)$-[[cocycle]] on the [[action groupoid]] $X// G$, i.e. a [[functor]] $X//G \to \mathbf{B} U(1)$, can be explicitly described as a function
$\lambda: G\times X \to U(1)$ such that

$$
\lambda(h g,x)=\lambda(h, g x)\lambda(g,x).
$$

Now fix a [[Riemann surface]] $\Sigma$ and take as $X$ the space of [[Riemannian metric]]s on $\Sigma$, and as $G$ the additive group of  [[real line|real]]-valued [[smooth function]]s on $\Sigma$, acting on metrics by conformal rescaling:

$$
  (f,g_{i j})\mapsto e^f g_{i j}
  \,.
$$

The **Liouville cocycle** with central charge $c\in \mathbb{R}$ is the function

$$
\lambda: C^\infty(\Sigma,\mathbb{R})\times Met(\Sigma) \to U(1)
$$

defined by

$$
\lambda(f,g)=exp(\frac{i c}{2} \int_\Sigma(d f\wedge *_g d f +4 f R_g d \mu_g)),
$$

where $*_g$ is the [[Hodge star operator]] defined by the Riemannian metric $g$, $R_g$ is the [[scalar curvarure]] and $d \mu_g$ is the [[volume form]].

## In conformal field theory

In [[conformal field theory]], the Liouville cocycle appears when one moves from genuine [[representation]]s of 2-dimensional conformal [[cobordism]]s to [[projective representation|projective representations]]. The obstruction for such a projective representation to be a genuine representation is precisely given by the central charge $c$; when $c\neq 0$, one says that the conformal field theory has a _conformal [[quantum anomaly|anomaly]]_.


Liouville integrable system

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## Idea

What is called _Liouville integrability_ is one formalization of the notion of __classical [[integrable system]]__ in [[physics]].

A [[physical system]] is called **Liouville integrable** if it admits [[canonical coordinates]] and [[canonical momenta]] which are generated from the flow of a maximal set of commuting [[Hamiltonians]].

## Definition

A [[physical system]] given by a [[phase space]] [[symplectic manifold]] $(X, \omega)$ and equipped with a [[Hamiltonian]] $H_0 \in C^\infty(C)$ (generating time evolution) is said to be **Liouville integrable** or to be an **[[integrable system]] in the sens of [[Liouville]]**  if

* there are $(dim X -1)$ other Hamiltonian functions $\{H_i\}_{i = 0}^{dim X -1}$

* such that

  1. these all commute under the [[Poisson bracket]] with each other;

  1. the [[flow]] of the corresponding [[Hamiltonian vector fields]] generates a [[polarization]] of $(X, \omega)$.

[[!redirects Liouville integrability]]


Liouville theory

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## Idea

In the context of [[quantum field theory]], _Liouville theory_ is the name of a certain type of [[2d CFT]].

## Properties

### Relation to 3d quantum gravity

An argument (via [[Chern-Simons gravity]], but see the caveats there) that [[3d quantum gravity]] with negative [[cosmological constant]] has as [[boundary field theory]] 2d [[Liouville theory]] is due to ([Coussaert-Henneaux-vanDriel 95](#CoussaertHenneauxvanDriel95)). See also at _[[AdS3-CFT2 and CS-WZW correspondence]]_.

## Related concepts

* [[dilaton]]

* [[AGT correspondence]]

* [[Jackiw-Teitelboim gravity]]

## References

On the non-critical [[bosonic string]] via Liouville theory (cf. *[[Polyakov action]]*):

* [[Alexander Polyakov]], *Quantum geometry of bosonic strings*, Phys. Lett. B **103** (1981) 207-210 [doi:10.1016/0370-2693(81)90743-7, [pdf](http://qft.itp.ac.ru/polyakov-2.pdf)]

* [[Jean-Loup Gervais]], [[André Neveu]], *The dual string spectrum in Polyakov's quantization (I)*, Nuclear Physics B **199** 1 (1982) 59-76 [doi:10.1016/0550-3213(82)90566-1]

* [[Jean-Loup Gervais]], [[André Neveu]], *Dual string spectrum in Polyakov's quantization (II). Mode separation*, Nuclear Physics B **209** 1 (1982) 125-145 [doi:10.1016/0550-3213(82)90105-5]

* [[Thomas L. Curtright]], [[Charles B. Thorn]], *Conformally Invariant Quantization of the Liouville Theory*, Phys. Rev. Lett. **48** (1982) 1309; Erratum Phys. Rev. Lett. **48**,  (1982) 1768 [[doi:10.1103/PhysRevLett.48.1309](https://doi.org/10.1103/PhysRevLett.48.1309)]

Review and survey:

* {#AGT09} [[Luis Alday]], [[Davide Gaiotto]], [[Yuji Tachikawa]], appendix A.1 of _Liouville Correlation Functions from Four-dimensional Gauge Theories_, Lett.Math.Phys.91:167-197, 2010 ([arXiv:0906.3219](http://arxiv.org/abs/0906.3219))

* [[Harold Erbin]], *Notes on 2d quantum gravity and Liouville theory* (2015) [[pdf](http://www.lpthe.jussieu.fr/~erbin/files/liouville_theory.pdf), [[Erbin-LiouvilleTheory.pdf:file]]]

See also:

* Wikipedia, _[Liouville field theory](http://en.wikipedia.org/wiki/Liouville_field_theory)

Rigorous construction of the [[path integral]], the DOZZ formula, and the [[conformal bootstrap]] for Liouville theory:

* [[François David]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Liouville Quantum Gravity on the Riemann sphere_, Communications in Mathematical Physics volume 342, pages869–907 (2016) ([arxiv:1410.7318](https://arxiv.org/abs/1410.7318))

* [[Colin Guillarmou]], [[Rémi Rhodes]], [[Vincent Vargas]], *Polyakov's formulation of 2d bosonic string theory*, Publ. Math. IHES **130** (2019) 111–185 [[arXiv:1607.08467](https://arxiv.org/abs/1607.08467), [doi:10.1007/s10240-019-00109-6](https://doi.org/10.1007/s10240-019-00109-6)]

* [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Integrability of Liouville theory: proof of the DOZZ Formula_, Annals of Mathematics,  **191** 1 (2020) 81-166 ([arxiv:1707.08785](https://arxiv.org/abs/1707.08785), [doi:10.4007/annals.2020.191.1.2](https://doi.org/10.4007/annals.2020.191.1.2))

* [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *The DOZZ formula from the path integral*, Journal of High Energy Physics volume 2018, Article number: 94 (2018) ([arXiv:1803.05418](https://arxiv.org/abs/1803.05418) doi:10.1007/JHEP05(2018)094)

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Conformal bootstrap in Liouville Theory_ [[arXiv:2005.11530](https://arxiv.org/abs/2005.11530)]

Construction as a [[functorial field theory]] following [Segal 1988](conformal+field+theory#Segal88)

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *Segal's axioms and bootstrap for Liouville Theory* [[arXiv:2112.14859](https://arxiv.org/abs/2112.14859)]

Review in:

* [[Vincent Vargas]], *Lecture notes on Liouville theory and the DOZZ formula* ([arXiv:1712.00829](https://arxiv.org/abs/1712.00829))

* [[Rémi Rhodes]], [[Vincent Vargas]], *A probabilistic approach of Liouville field theory*, Comptes Rendus. Physique, Tome 21 (2020) no. 6, pp. 561-569. ([doi:10.5802/crphys.43](https://doi.org/10.5802/crphys.43))

An argument (via [[Chern-Simons gravity]], but see the caveats there) that [[3d quantum gravity]] with negative [[cosmological constant]] has as [[boundary field theory]] 2d [[Liouville theory]] is due to

* {#CoussaertHenneauxvanDriel95} O. Coussaert, [[Marc Henneaux]], P. van Driel, _The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant_, Class. Quant. Grav. 12 (1995) 2961-2966 ([arXiv:gr-qc/9506019](http://arxiv.org/abs/gr-qc/9506019))

* Leon A. Takhtajan, Lee-Peng Teo, _Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography_, Commun.Math.Phys. 239 (2003) 183-240 ([arXiv:math/0204318](http://arxiv.org/abs/math/0204318))

  > **Abstract**: We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity. We prove that the classical action is a Kahler potential of the Weil-Petersson metric. We also prove that Liouville action functional satisfies holography principle, i.e., it is a regularized limit of the hyperbolic volume of a 3-manifold associated with a quasi-Fuchsian group. We generalize these results to a large class of Kleinian groups including finitely generated, purely loxodromic Schottky and quasi-Fuchsian groups and their free combinations.

The relation to [[quantum Teichmüller theory]] is discussed/reviewed in:

* [[Jörg Teschner]], _On the relation between quantum Liouville theory and the quantized Teichmüller spaces_, Int. J. Mod. Phys. A 19S2:459-477,2004 ([arxiv:hep-th/0303149](http://arxiv.org/abs/hep-th/0303149))

* {#Allegretti} [[Dylan Allegretti]], _Notes on Quantum Teichmüller theory_ ([pdf](http://users.math.yale.edu/~dga4/TeichmullerNotes.pdf))

* {#Teschner14} [[Jörg Teschner]], _Quantization of moduli spaces of flat connections and Liouville theory_, proceedings of the International Congress of Mathematics 2014 ([arXiv:1405.0359](http://arxiv.org/abs/1405.0359))

Relation to the $SL(2,\mathbb{R})$-[[gauged WZW-model]] and [[analytically continued Wess-Zumino-Witten theory]]:

* [[Noboyuki Ishibashi]], _Extra Observables in Gauged WZW Models_, Nucl.Phys. B379 (1992) 199-219 ([arXiv:hep-th/9110071](http://arxiv.org/abs/hep-th/9110071))

* Jian-Feng Wu, Yang Zhou, _From Liouville to Chern-Simons, Alternative Realization of Wilson Loop Operators in AGT Duality_ ([arXiv:0911.1922](http://arxiv.org/abs/0911.1922))

Further discussion:

* Pavel Haman, Alfredo Iorio, *Classical gravitational anomalies of Liouville theory* [[arXiv:2306.09825](https://arxiv.org/abs/2306.09825)]

On [[conformal blocks]] for [[Liouville theory]]:

* [[Babak Haghighat]], [[Yihua Liu]], [[Nicolai Reshetikhin]], *Flat Connections from Irregular Conformal Blocks* [[arXiv:2311.07960](https://arxiv.org/abs/2311.07960)]

* [[Xia Gu]], [[Babak Haghighat]], [[Kevin Loo]], *Irregular Fibonacci Conformal Blocks* [[arXiv:2311.13358](https://arxiv.org/abs/2311.13358)]

category: physics

[[!redirects Liouville field theory]]

[[!redirects Liouville field]]
[[!redirects Liouville fields]]

[[!redirects Liouville gravity]]


Liouville's equation

## Related concepts

* [[Hamilton's equations]]

* [[statistical mechanics]]

* [[quantum statistical mechanics]]

* [[Bayesian interpretation of quantum mechanics]]

## References

See also

* Wikipedia, _Liouville's theorem (Hamiltonian)_


Liouville-Poincaré 1-form

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## Definition

For $X$ a [[smooth manifold]] and $T^* X \to X$ its [[cotangent bundle]], there is a unique [[differential 1-form]] on $T^* X$ itself,

$$
  \theta
   \in
  \Omega^1(T^* X)
$$

with the property that under the [[isomorphism]]

$$
  j \;\colon\; \Gamma(T^* X) \stackrel{\simeq}{\to} \Omega^1(X)
$$

between [[differential 1-forms]] and smooth [[sections]] of the [[cotangent bundle]] we have for every smooth section $\sigma \in \Gamma(T^* X)$ the identification

$$
  \sigma^* \theta = j(\sigma)
$$

between the [[pullback of a differential form|pullback]] of $\theta$ along $\sigma$ and the 1-form corresponding to $\sigma$ under $j$.

This unique differential 1-form $\theta \in \Omega^1(T^* X)$ is called the **Liouville form** or **Poincaré 1-form** or **canonical form** or **tautological form** on the cotangent bundle.

The [[de Rham differential]] $\omega \coloneqq d \theta$ is a [[symplectic form]]. Hence every [[cotangent bundle]] is canonically a [[symplectic manifold]].

On a [[coordinate chart]] $\mathbb{R}^n$ of $X$ with canonical [[coordinate functions]] denoted $(x^i)$, the cotangent bundle over the chart is $T^\ast \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n$ with canonical coordinates $((x^i), (p_j))$. In these coordinates the canonical 1-form is (using [[Einstein summation convention]])

$$
  \theta = p_i d x^i
$$

and hence the [[symplectic form]] is

$$
  \omega = d p_i \wedge d q^i
  \,.
$$

## Related concepts

* [[Maurer-Cartan form]]

[[!redirects Liouville 1-form]]
[[!redirects Poincaré 1-form]]
[[!redirects Poincare 1-form]]

[[!redirects Poincaré form]]
[[!redirects Poincare form]]

[[!redirects Liouville-Poincare 1-form]]


Liping Zou

* [spire page](https://inspirehep.net/authors/1259269)

## Selected writings

On [[hadron supersymmetry]] via [[holographic light front QCD]]:

For [[heavy hadrons]]:

* [[Marina Nielsen]], [[Stanley Brodsky]], Guy F. de Téramond, [[Hans Günter Dosch]], Fernando S. Navarra, [[Liping Zou]], _Supersymmetry in the Double-Heavy Hadronic Spectrum_, Phys. Rev. D 98, 034002 (2018) ([arXiv:1805.11567](https://arxiv.org/abs/1805.11567))

In view of [[tetraquarks]]:

* [[Hans Günter Dosch]], [[Stanley Brodsky]], [[Guy de Teramond]], [[Marina Nielsen]], [[Liping Zou]], _Exotic States in a Holographic Theory_ ([arXiv:2012.02496](https://arxiv.org/abs/2012.02496))

category: people


Lipman Bers

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

category: people