Lindelöf topological space

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Named after [[Ernst Leonard Lindelöf]].

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## Definition

A [[topological space]] is a _Lindelöf space_ if every [[open cover]] has a [[countable cover|countable]] sub-cover.

## Examples

* Every [[compact topological space]] is Lindelöf.

* Every [[second-countable topological space]] [[second-countable spaces are Lindelöf|is Lindelöf]].

* Every [[sigma-compact topological space]] is Lindelöf.

## Properties

[[regular Lindelöf spaces are paracompact]]

## Related properties

[[!include topology - countability axioms]]

## References

On [[product topological space|product spaces]] with Lindelöf spaces:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *On productively  Lindelöf spaces*, Scient. Math. Japon. **65** (2007) 23–36 [[pdf](https://www.math.mcgill.ca/barr/papers/alster.pdf), [[BKR-ProductivelyLindeloef.pdf:file]]]

See also:

* Wikipedia, _[Lindelöf space](https://en.wikipedia.org/wiki/Lindel%C3%B6f_space)_

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Lindenbaum number

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## Idea

The _Lindenbaum number_ of a [[set]] $X$ is the smallest [[well-ordered set]] that is not a quotient of $X$ (as a bare set). It is a 'surjective analogue' of the [[Hartogs number]].

## Definition

Given a set $X$, denote by $\aleph^*(X)$, the set of [[ordinals]] $\alpha$ that admit a [[surjection]] $X\to \alpha$. Equivalently, it is the set of order isomorphism classes of well-orderings of quotients of $X$.

## Properties

By construction, if there were a surjection function $X \to \aleph^*(X)$, then $\aleph^*(X)$ would be order isomorphic to one of its own elements, and hence to a proper initial segment of itself, which is impossible.

If $\aleph(X)$ denotes the [[Hartogs number]] of $X$, then we also have:

* there is an injection $\aleph(X) \to \aleph^*(X)$,

* for well-orderable $X$, then $\aleph(X) \simeq \aleph^*(X)$,

* $X$ and $\aleph^*(X)$ are comparable if and only if $X$ is well-orderable, and this holds for both $\lt$ and $\lt^*$, the surjective preordering on cardinals,

* it is consistent that for any well-orderable cardinal $\kappa$, there is a Dedekind-finite set $X$ (i.e. an infinite set such that $\aleph(X)=\aleph_0$) and $\aleph^*(X)\geq\kappa$ ([Monro 1975](#Monro75), so in particular the gap between $\aleph(X)$ and $\aleph^*(X)$ can be arbitrarily large,

* the statement that $\aleph(X)=\aleph^*(X)$ for all $X$ is equivalent to the axiom of choice for well-ordered families (i.e. if $A$ is a well-orderable family of non-empty sets, then $A$ admits a choice function). This was proved by Pincus, and appears in [Pelc 1978](#Pelc78),

* there is an injection $\aleph^*(X) \to \aleph(P(X))$ ([Diener 1992](#Diener92)).

## Related entries

* [[Hartogs number]]
* [[successor]]
* [[diagonal argument]]

# References

The concept appeared as exercise 7.1.6 in

* Karel Hrbacek and Thomas Jech, _Introduction to Set Theory_, 3rd Edition, Marcel Dekker (1999)

[Note: the exercise is also in chapter 8 in the 1984 2nd edition, not sure about the 1st edition]

And is mentioned in

* {#Diener92} Karl‐Heinz Diener, _On the transitive hull of a $\kappa$-narrow relation_, Math. Logic Quarterly, Volume 38, Issue1 (1992) pp 387-398, doi:[10.1002/malq.19920380137](https://doi.org/10.1002/malq.19920380137)

The name appears to be due to

* Asaf Karagila, _Embedding Posets Into Cardinals With $DC_\kappa$_, arXiv:[1212.4396v1](https://arxiv.org/abs/1212.4396v1) (published as _Embedding Orders Into The Cardinals With $DC_\kappa$_ Fund. Math. 226 (2014), 143-156. and omitting the Lindenbaum number)

note: version 1 of the arXiv paper!

* {#Monro75} G. P. Monro, _Independence results concerning Dedekind-finite sets_, J. Austral. Math. Soc. 19 (1975), 35–46.

* {#Pelc79} Andrzej Pelc, _On some weak forms of the axiom of choice in set theory_. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 7, 585–589.


Lindenbaum-Tarski algebra

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## Idea

The **Lindenbaum--Tarski algebra** of a [[theory]] in [[propositional logic]] is an algebraic (and order-theoretic) structure built out of its [[formula]]s (modulo provable equivalence) and [[connectives]].  It thus carries an algebraic structure that corresponds to the logic in question and is generally the [[free]] such structure generated by the axioms of the theory.

## Examples

* The Lindenbaum--Tarski algebra of a [[theory]] in [[intuitionistic logic|intuitionistic]] propositional logic is a [[Heyting algebra]].

* The Lindenbaum--Tarski algebra of a theory in classical propositional logic is a [[Boolean algebra]].

## The Lindenbaum--Tarski algebra of a normal modal logic

Associated to any [[normal modal logic]], $\Lambda$ in $\mathcal{L}_\omega(n)$ (and more
generally) is an algebra $\mathfrak{A}^\Lambda_\omega$,which is a [[algebraic models for modal logics|BAO]] of type $n$,
i.e. $n$ (modal) operators, $m_i$.  This is called the **Lindenbaum--Tarski algebra** of $\Lambda$, and is a quotient of the [[term algebra]] of  $\mathcal{L}_\omega(n)$, i.e. of the free universal algebra of type $n$ formed by the $\mathcal{L}_\omega(n)$-formulae using the connectives $\vee$, $\wedge$, $\neg$, $\bot$, $\top$, and the $\Diamond_i$.  The Lindenbaum--Tarski algebra is formed from this free algebra by using the congruence $\simeq_\Lambda$, where

$$\phi \simeq_\Lambda\psi  \iff  {\vdash}_\Lambda \phi\leftrightarrow \psi.$$

(Recall the usual rules of notation: $\phi \to \psi$ is $\neg\phi\vee\psi$;
$\phi\leftrightarrow \psi$ is $(\phi\to \psi)\wedge (\psi\to\phi)$; and
${\vdash}_\Lambda \phi $ means $\phi \in \Lambda$.)

 The elements of $\mathfrak{A}^\Lambda_\omega$ are the equivalence
classes:

$${\left\|\phi\right\|} = \{\psi \mid \vdash_\Lambda \phi\leftrightarrow \psi\},$$

with the operations

*  ${\left\|\phi\right\|} + {\left\|\psi\right\|} = {\left\|\phi \vee \psi\right\|}$;

*  ${\left\|\phi\right\|} \cdot {\left\|\psi\right\|}= {\left\|\phi \wedge \psi\right\|}$;

*  ${\|\phi\|^-} = {\|\neg\phi\|}$;

*  $0 = {\|\bot\|}$;

*  $1 = {\|\top\|}$;

and

*  $m_i {\|\phi\|} = {\|\Diamond_i\phi\|} $.

As we assumed that $\Lambda$ was normal, the normality schemata:

 $(K)$: $\Diamond(\psi \vee \chi)\to \diamond \psi\vee \diamond \chi$;

 $(N)$, in the form, $\neg \Diamond_i(\bot)$

and monotonicity :

if $\psi \to \chi \in \Lambda$ then $\Diamond_i \psi \to \Diamond_i \chi \in \Lambda$.

 imply

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###### Lemma

$m_i$ is a normal operator.
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###### Proof

We first note that $\vdash_\Lambda \Diamond(\psi\vee \phi) \leftrightarrow \Diamond_i \psi \vee \Diamond_i \phi$, then the result follows by simply writing down what is required and staring at it for a moment!
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### Canonical models

The Lindenbaum--Tarski algebra for a modal logic, $\Lambda$,   gives rise a canonical Kripke model. This has the set of $\Lambda$-[[maximal consistent]] formulae as its set of states.

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Lindström's theorem

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## Idea

_Lindström's theorem_, or more precisely, Lindström's theorems, are important results in [[abstract model theory]] due to the Swedish logician [[Per Lindström]] that give syntax-free characterizations of (classical untyped) [[first-order logic]] ('syntax-free' in the sense of not referring to the specific [[syntax|syntactic]] make up of the formulas).

###The first theorem of Lindström###
This  ([[semantics|semantic]]) version of Lindström's characterization is the best known and is usually meant when one speaks of ' _the_ theorem of Lindström' . It states that the validity of the downward [[Löwenheim-Skolem theorem]] (a sentence that holds in some [[model]] already holds in an at most countable model) and the [[compactness theorem]] (a set of sentences has a model if every finite subset has a model) characterizes up to expressive power standard [[first-order logic]] among all [[formal logic|logics]] satisfying certain regularity conditions[^fine]:

[^fine]: These conditions formalize some classical background assumptions on logical systems like the restriction of semantics to standard set-theoretic structures and the existence of [[negation]] in the sense that for every sentence $\varphi$ there is $\psi$ such that $\mathfrak{A}\models\varphi$ iff not $\mathfrak{A}\models\psi$.

A gain in expressive power comes at the price of the loss of the validity of at least one of these theorems e.g. in [[second-order logic]] both theorems fail to hold.

_Barwise_ (1977, p.45, cf. below) has called the first Lindström theorem

>one of the first, and still most striking, results in abstract model theory

and [VäänänenWesterståhl, p. 102](#VaananenWesterstahl2010) write that the

> 11-page paper (ed.: the authors refer to (#Lindstrom1969) ) is without a doubt Pelle Lindström's most important single contribution to mathematical logic. In it, he uses a surprisingly general notion of a logic [...]

and go on to write [VäänänenWesterståhl, p. 104](#VaananenWesterstahl2010)

> Lindström's Theorem tells us that any proper extension of [[first-order logic]] has to detect something non-trivial about the set-theoretic [[universe]]. On the other hand, Lindström's theorem reveals a deep robustness of first-order logic: it does not matter which way you define your syntax - as long as your semantics obeys certain basic principles [...] you will get the same logic.

###The second theorem of Lindström###
The _second_ Lindström theorem is of a more proof-theoretic nature and involves the concept of an _effective logical system_ which draws on [[computability theory]]. It achieves a characterization of standard first-order logic in the class of all effective regular logical systems by the two properties of having an enumerable set of tautologies and validity of the Löwenheim-Skolem theorem.

##Discussion##
These results are sometimes viewed as an explanation of the privileged role that standard first-order logic plays in mathematical logic.

In the end, what lesson one draws from them (once the classical background assumptions accepted), largely hinges on how one assesses the role of the _[[Löwenheim-Skolem theorem]]_ in mathematical logic, which enjoys a rather controversial status and has encountered skepticism already by _Thoralf Skolem_, the man himself. This scepticism is clearly present in the following comment by _Hao Wang_ (1974, quoted after Flum(1985), pp.77-78):

>..what is established (by Lindström's theorem) is not that first-order logic is the only possible logic but rather that is the only possible logic when we in a sense deny the reality to the concept of uncountability ...

Further alternative characterizations and extensions to other logics are known and constitute the field of [[abstract model theory]].

## Related concepts

* [[predicate logic]]

* [[higher order logic]]

* [[Löwenheim-Skolem theorem]]

* [[compactness theorem]]

* [[abstract model theory]]

## References

* {#Lindstrom1969} [[Per Lindström]]: _On Extensions of Elementary Logic_, Theoria 35 pp.1-11 (1969)

* Barwise (ed.): _Handbook of Mathematical Logic_ , Elsevier Amsterdam 1977.

A textbook account of the results appears in:

* Ebbinghaus, Flum, Thomas : _Einführung in die mathematische Logik_, Wissenschaftliche Buchgesellschaft Darmstadt 1986.

A more recent overview of Lindström's results can be found here:

* [[Jouko Väänänen]], _Lindström's Theorem_ . Ms. ([pdf](http://www.math.helsinki.fi/logic/opetus/lt/lindstrom_theorem1.pdf))

* {#VaananenWesterstahl2010} [[Jouko Väänänen]], Dag Westerståhl: In Memoriam: Per Lindström. Theoria **76** (2010), 100-107

Further material (including a discussion of the broader implications of Lindström's results for logic) can be found in

* Jörg Flum, _Characterizing Logics_ , ([pdf](http://projecteuclid.org/download/pdf_1/euclid.pl/1235417268)), chap. III of

* Barwise, Feferman (eds.), _Model-theoretic Logics_ , Springer Heidelberg 1985 ([toc](http://projecteuclid.org/euclid.pl/1235417263#toc))

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line 2-bundle

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## Idea

Given a [[2-ring]] $R$, a _line 2-bundle_ or _2-line bundle_ is a _[[2-module bundle]]_ whose typical fiber is a [[2-line]] over $R$.

## Definition

### Recalling general 2-vector bundles

Let $R$ be a [[commutative ring]], or more generally an [[E-∞ ring]]. By the discussion at [[2-vector space]] consider the [[2-category]]

$$
  2 Vect_R \simeq Alg_R
$$

[[equivalence of 2-categories|equivalent]] to that whose objects are [[associative algebras]] (or generally [[A-∞ algebra|algebras]]) $A$ over $R$, (being placeholders for the 2-vector space $A Mod$ which is the category of [[modules]] over $A$) whose [[1-morphisms]] are [[bimodules]] between these algebras (inducing linear functors between the corresponding 2-vector spaces = categories of modules) and whose [[2-morphisms]] are [[homomorphisms]] between those.

Under [[Isbell duality]] and by the discussion at _[Modules -- as generalized vector bundles](module#RelationToVectorBundlesInIntroduction)_  we may think of this 2-category as being that of (generalized) 2-vector bundles over a [[space]] called $Spec R$.

### Line 2-bundles

The [[2-category]] $2 Vect_R \simeq Alg_R$ is canonically a [[monoidal 2-category]]. An object in $2 Vect_R$ is a _[[line]]_ if it is an _invertible object_ with respect to this tensor product, hence if it is an _[[Azumaya algebra]]_. In terms of the above this means that it represents a 2-vector bundle over $Spec R$ which is a **line 2-bundle**.

The full inclusion

$$
  \mathbf{Br}(R) \hookrightarrow 2 Vect_R \simeq Alg_R
$$

of the maximal [[2-groupoid]] on the line 2-bundles over $Spec R$ is a [[braided 3-group]], the _[[Picard group|Picard 3-group]]_ of $Spec R$.  See _[Relation to Brauer group](#RelationToBrauerGroup)_ below for more.

## Properties

### Relation to Brauer group and Picard group
 {#RelationToBrauerGroup}

The [[braided 3-group]] $\mathbf{Br}(R)$ of line 2-bundles over $Spec R$ has as [[homotopy groups]]

* $\pi_0(\mathbf{Br}(R))$ the [[Brauer group]] of $R$;

* $\pi_1(\mathbf{Br}(R))$ the [[Picard group]] of $R$, hence the group of ordinary [[line bundles]] over $R$;

* $\pi_2(\mathbf{Br}(R))$ the [[group of units]] of $R$.

## Examples

### Super line 2-bundles and twisted K-theory
 {#SuperLine2BundlesAndTwistedK}

See at _[[super line 2-bundle]]_.

## Related concepts

* [[line bundle]]

* [[holomorphic line 2-bundle]], [[algebraic line 2-bundle]]

* [[line n-bundle]]

* [[line n-bundle with connection]]

[[!include moduli of higher lines -- table]]

## References

Line 2-bundles in [[supergeometry]] as a model for the [[B-field]] and [[orientifolds]] are discussed (even if not quite explicitly in the language of higher bundles) in

* [[Daniel Freed]], _Lectures on twisted K-theory and orientifolds_ ([pdf](http://www.ma.utexas.edu/users/dafr/ESI.pdf))
 {#Freed}

based on

* [[Peter Donovan]], [[Max Karoubi]], _Graded Brauer groups and $K$-theory with local coefficients_, Publications Mathématiques de l'IHÉS, 38 (1970), p. 5-25  ([numdam](http://www.numdam.org/item?id=PMIHES_1970__38__5_0))
 {#DonovanKaroubi}

and

* [[C. T. C. Wall]], _Graded Brauer groups_, J. Reine Angew. Math. 213 (1963/1964), 187-199.
 {#Wall}

and developing constructions in

* [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))
 {#Precis}

More on [[super line 2-bundles]] is secretly in

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _Loop groups and twisted K-theory I_ ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906))
 {#FreedHopkinsTeleman}

The above higher supergeometric story is made explicit in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_
 {#FSS}

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line 3-bundle

## Idea

[[line n-bundle]] for $n = 3$

## Examples

* [[supergravity C-field]]

## Related entries

[[!include moduli of higher lines -- table]]

[[!redirects supergravity C-field]]


line bundle

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## Definition

A _line bundle_ is a [[vector bundle]] of rank (or [[dimension]]) $1$, i.e. a vector bundle whose typical fiber is a $1$-dimensional [[vector space]] (a [[line]]).

For complex vector bundles, [[complex line]] bundles are canonically [[associated bundles]] of [[circle group]]-[[principal bundles]].

## Properties

The class of line bundles has a nicer behaviour (in some ways) than the class of vector bundles in general.  In particular, the [[dual vector bundle]] of a line bundle $L$ is a [[weak inverse]] of $L$ under the tensor product of line bundles.  Thus the isomorphism classes of line bundles form a [[group]].

## Examples

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###### Example

The [[Möbius strip]] is the unique, up to [[isomorphism]], non-trivial [[real vector bundle|real]] line bundle over the [[circle]].

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###### Example

Over any [[manifold]] there is canonically the _[[density line bundle]]_ which is the [[associated bundle]] to the [[principal bundle]] underlying the [[tangent bundle]] by the [[determinant]] homomorphism.

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Similarly:

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###### Example

Every [[orientable]] [[complex manifold]] carries a [[complex vector bundle|complex]] line bundle of top-degree [[holomorphic differential forms]]. This is called its _[[canonical line bundle]]_.

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###### Example

The line bundle on the [[2-sphere]] whose [[first Chern class]] is a generator of $H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z}$ is the [[pullback bundle]] of the [[universal complex line bundle]] (example \ref{UniversalComplexLineBundle}) along the map

$$
  S^2 \longrightarrow B U(1) \simeq B^2 \mathbb{Z}
$$

which itself represents the generator in the second [[homotopy group]] $\pi_2(S^2) \simeq \mathbb{Z}$.

Beware that this is _not_ the "[[canonical line bundle]]" from example \ref{CanonicalLineBundle}, but "half" of it, its [[theta characteristic]]. See at _[[geometric quantization of the 2-sphere]]_ for more on this.

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###### Example

The [[classifying space]]/[[Eilenberg-MacLane space]] $B U(1) \simeq B^2 \mathbb{Z}$ carries the [[circle group]]-[[universal principal bundle]]. The corresponding [[associated bundle]] via the canonical [[action]] of $U(1)$ on $\mathbb{C}$ is the [[universal complex line bundle]].

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###### Example

The [[product]] of any [[space]] $X$ with the [[moduli stack]] $Pic_X$ of line bundles over it (its [[Picard stack]]) carries a tautological line bundle. This is called the _[[Poincaré line bundle]]_ of $X$.

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## Related concepts

* [[complex vector space]]

  * [[complex line]]

* [[complex vector bundle]]

  * [[complex line bundle]]

* [[holomorphic line bundle]], [[algebraic line bundle]]

* [[theta characteristic]]

* [[cubical structure on a line bundle]]

* [[fusion bundle]]

* [[bundle gerbe]]

* [[circle n-bundle with connection]]

* [[line 2-bundle]]

* [[line n-bundle]]

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line integral

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# Line integrals
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## Idea

A _line integral_ is an [[integral]] along a [[curve]].  These are sometimes called _path integrals_ (not to be confused with the [[path integral]] in [[quantum physics]], which is integration over a space of curves rather than along a curve in some space) and _[[contour integrals]]_ (especially in [[complex analysis]]).

By the modern understanding of the [[integration of differential forms]], one integrates [[differential 1-forms]] ([[cotangent vector fields]]) along [[orientation|oriented]] [[curves]], and so this would be the natural way to understand a line integral.  However, there are several slightly different line integrals, and not all of them are reducible to integration of $1$-forms along oriented curves.  Sometimes we have to fall back on more basic notions, ultimately the integration of [[pseudoform|pseudo]]-$1$-forms on a $1$-dimensional space.

## Definitions

Here we assume familiarity with integration of differential forms and pseudoforms, defining in terms of them the various classical notions of line integral.

In each case, the classical notation (developed before the rigorous treatment of analysis in the 19th century) is not taken literally by the classical definition (developed after the rigorous treatment and still found in calculus texts), and this definition is accompanied by a [[parametrization|reparametrisation]] theorem.  We would like to make sense of the classical notation and eliminate the reparametrisation theorems (or at least make them all special cases of [a general theorem to be proved once for all](differential+form#IntegrationTheorem)) using differential forms.  Often we find that we can relax some of the restrictions in the classical definition as well.

### Line integral of a vector field

Classically, we have a [[Cartesian space]] $X$, a [[continuous map]] $\vec{F}\colon X \to X$, and a [[continuously differentiable map]] $C\colon [a,b] \to X$; the __line integral__ of $\vec{F}$ along $C$ is defined as
$$ \int_C \vec{F} \cdot \mathrm{d}\vec{r} \coloneqq \int_a^b \vec{F}(C(t)) \cdot C'(t) \,\mathrm{d}t ,$$
where the integral on the right is a [[Riemann integral]] and $C'$ is the [[derivative]] of $C$ componentwise.  If $\phi\colon [e,f] \to [a,b]$ is a continuously differentiable [[monotone function|increasing]] [[bijection]], then
$$ \int_{C \circ \phi} \vec{F} \cdot \mathrm{d}\vec{r} = \int_C \vec{F} \cdot \mathrm{d}\vec{r} ,$$
the reparametrisation theorem.

We can start to justify the classical notation by interpreting $\vec{r}$ as the same map $[a,b] \to X$ as $C$; then we have
$$ \int_C \vec{F} \cdot \mathrm{d}\vec{r} \coloneqq \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \,\mathrm{d}t ,$$
in which case the classical notation simply seems to be suppressing some of the notation in the formal definition on the right.  In particular, we interpret $\mathrm{d}\vec{r}$ as meaning $\vec{r}'(t) \,\mathrm{d}t$.

But this suggests that we should really be looking at differential forms.  Since $X$ is a Cartesian space, we may identify it with any of its [[tangent spaces]] and so identify $\vec{F}$ with a [[tangent vector field]] on $X$, or equivalenty a vector-valued $0$-form.  Since $\vec{r}$ is only serving to parametrise the curve, it should be interpreted as something trivial, in this case the [[identity map]] on $X$, also viewed as a vector-valued $0$-form.  Then $\mathrm{d}\vec{r}$ is a vector-valued $1$-form (which we can do since a Cartesian space has a trivial [[connection]]), and $\vec{F} \cdot \mathrm{d}\vec{r}$ is an ordinary $1$-form.  Finally, since the curve $C$ is given up to an increasing reparametrisation, it is an [[orientation|oriented]] $1$-submanifold of $X$.  Now we may interpret the notation
$$ \int_C \vec{F} \cdot \mathrm{d}\vec{r} $$
literally as the integral of a $1$-form.  It is now no longer necessary that $C$ be given by a continuously differentiable parametrisation; the vector-valued $0$-form $\vec{r}$ is continuously differentiable regardless, and so we only need $C$ to be a [[rectifiable curve]] (although it is a theorem that such a curve must have a parametrisation ---by [[arclength]] if nothing else--- that is continuously differentiable [[almost everywhere]], so that the classical definition still covers this using a [[Riemann integral]]).

However, this operation from $\vec{F}$ to $\vec{F} \cdot \mathrm{d}\vec{r}$ is something more fundamental in differential geometry; in fact, $\vec{F} \cdot \mathrm{d}\vec{r}$ is simply $\vec{F}^\flat$, the [[cotangent vector field]] that corresponds to the tangent vector field $\vec{F}$.  Understanding this, we wish to generalise $X$ to any (pseudo)-[[Riemannian manifold]].  In this case, we cannot interpret $\vec{r}$ literally, but we may still interpret $\mathrm{d}\vec{r}$ as a vector-valued $1$-form, indeed as the tautological one that (viewing a $1$-form as an operation on vector fields) is the [[identity map]] on vector fields.  With this understanding, the classical notation still makes sense, although it is probably easier to write
$$ \int_C \vec{F} \cdot \mathrm{d}\vec{r} = \int_C \vec{F}^\flat $$
as the most general definition of the line integral of a vector field along an oriented curve in a (pseudo)-Riemannian manifold.

### Line integral of a scalar field

Classically, we again have a [[Cartesian space]] $X$, now a [[continuous map]] $f$ from $X$ to $\mathbb{R}$ or $\mathbb{C}$, and again a [[continuously differentiable map]] $C\colon [a,b] \to X$; the __line integral__ of $f$ along $C$ is defined as
$$ \int_C f \mathrm{d}s \coloneqq \int_a^b f(C(t)) {\|C'(t)\|} \,\mathrm{d}t ,$$
where again the integral on the right is a [[Riemann integral]] and $C'$ is the [[derivative]] of $C$ componentwise.  If $\phi\colon [e,f] \to [a,b]$ is a continuously differentiable [[bijection]] (whether increasing *or* decreasing), then
$$ \int_{C \circ \phi} f \mathrm{d}s = \int_C f \mathrm{d}s ,$$
the reparametrisation theorem.

We cannot interpret $\mathrm{d}s$ as the differential of anything; rather, $\mathrm{d}s$ is the magnitude of the line integral element $\mathrm{d}\vec{r}$ from the previous section. In particular, identifying $\vec{r}$ with $C$ again, we have
$$ \int_C f \mathrm{d}s \coloneqq \int_a^b f(\vec{r}(t)) {\|\vec{r}'(t)\|} \,\mathrm{d}t ,$$
so $\mathrm{d}s$ seems to mean ${\|\vec{r}'(t)\|} \,\mathrm{d}t$.  Using the standard orientation on $[a,b]$ to change the $1$-form $\mathrm{d}t$ to the pseudo-$1$-form ${|\mathrm{d}t|}$, this is consistent with $\mathrm{d}s = {\|\mathrm{d}\vec{r}\|}$.

But for this to really make sense, we have to see what kind of object $\mathrm{d}s$ is on $X$.  It is neither a $1$-form (which can be integrated along an oriented curve) nor a pseudo-$1$-form (which can be integrated along a pseudo-oriented curve, that is a transversely oriented curve); it is instead an *[[absolute differential form|absolute]]* $1$-form, which can be integrated along an *unoriented* curve.  If we interpret $\mathrm{d}\vec{r}$ as the canonical vector-valued $1$-form again, then we can take its magnitude to get a positive semidefinite (nonnegative-scalar-valued, and in this case actually definite) absolute $1$-form, so $\mathrm{d}s = {\|\mathrm{d}\vec{r}\|}$ is literally true.

It would be nice to have notation without fake differentials and with the one piece of structure that actually plays a role: the metric.  If we multiply two $1$-forms using the [[symmetric product]] (instead of the [[exterior product]] as usual for differential forms), then we get a symmetric [[bilinear form]], and in this way the (symmetric) square of the arclength element $\mathrm{d}s$ is the metric $g$.  Since $\mathrm{d}s$ is positive, we can reasonably call it the *principal* [[square root]] of $g$.  Thus,
$$ \int_C f \mathrm{d}s = \int_C f \sqrt{g} $$
defines the line integral of a scalar field along an unoriented curve in a Riemannian manifold.

On a *pseudo*-Riemannian manifold, $g$ itself may not be positive and so may not have a square root.  In that case, we can take the absolute value of $g$ first and use
$$ \int_C f \mathrm{d}s = \int_C f \sqrt{|g|} ,$$
although this is most intuitive for curves that are consistently [[timelike curve|timelike]] or [[spacelike curve|spacelike]].  It\'s also possible to keep the previous formula and allow one of these two types of curve to have [[purely imaginary number|imaginary]] arclength; which one depends on conventions.  In any case, a line integral along a [[lightlike curve]] is zero.

### Contour integral of a complex function

Classically, we have the [[complex plane]] $\mathbb{C}$, a [[continuous map]] $f\colon \mathbb{C} \to \mathbb{C}$, and a [[continuously differentiable map]] $C\colon [a,b] \to \mathbb{C}$; the __[[contour integral]]__ (or __line integral__ again) of $f$ along $C$ is defined as
$$ \int_C f \mathrm{d}z \coloneqq \int_a^b f(C(t)) C'(t) \,\mathrm{d}t ,$$
where the integral on the right is a [[Riemann integral]] and $C'$ is the [[derivative]] of $C$.  If $\phi\colon [e,f] \to [a,b]$ is a continuously differentiable [[monotone function|increasing]] [[bijection]], then
$$ \int_{C \circ \phi} f \mathrm{d}z = \int_C f \mathrm{d}z ,$$
the reparametrisation theorem.

We start to justify the classical notation by interpreting $z$ as the same map $[a,b] \to \mathbb{C}$ as $C$; then we have
$$ \int_C f \mathrm{d}z \coloneqq \int_a^b f(z(t)) z'(t) \,\mathrm{d}t ,$$
in which case the classical notation seems again to be suppressing some of the notation in the formal definition.  In particular, $\mathrm{d}z$ is interpreted as $z'(t) \,\mathrm{d}t$.

But again, we should really be looking at differential forms.  We may identify the space $\mathbb{C}$ with the [[scalar field]] and so identify $f$ with a [[scalar field]] on $\mathbb{C}$, or equivalenty a $0$-form.  Since $z$ is only serving to parametrise the curve, it should be interpreted as the [[identity map]] on $\mathbb{C}$, also viewed as a $0$-form.  Then $\mathrm{d}z$ is a $1$-form, and $f \mathrm{d}z$ is a $1$-form.  Finally, since the curve $C$ is given up to an increasing reparametrisation, it is an [[orientation|oriented]] $1$-submanifold of $X$.  Now we may interpet the notation
$$ \int_C f \mathrm{d}z $$
literally as the integral of a $1$-form.  It is now no longer necessary that $C$ be given by a continuously differentiable parametrisation; the vector-valued $0$-form $z$ is continuously differentiable regardless, and so we only need $C$ to be a [[rectifiable curve]] (although it is a theorem that such a curve must have a continuously differentiable parametrisation after all).

### Absolute contour integral of a complex function

Classically, we have the [[complex plane]] $\mathbb{C}$, a [[continuous map]] $f\colon \mathbb{C} \to \mathbb{C}$, and a [[continuously differentiable map]] $C\colon [a,b] \to \mathbb{C}$; the __absolute contour integral__ (or whatever one calls it) of $f$ along $C$ is defined as
$$ \int_C f {|\mathrm{d}z|} \coloneqq \int_a^b f(C(t)) {|C'(t)|} \,\mathrm{d}t ,$$
where the integral on the right is a [[Riemann integral]] and $C'$ is the [[derivative]] of $C$.  If $\phi\colon [e,f] \to [a,b]$ is a continuously differentiable [[bijection]] (whether increasing or decreasing), then
$$ \int_{C \circ \phi} f {|\mathrm{d}z|} = \int_C f {|\mathrm{d}z|} ,$$
the reparametrisation theorem.

As before, if $z$ is interpreted as the same map $[a,b] \to \mathbb{C}$ as $C$; then we have
$$ \int_C f {|\mathrm{d}z|} \coloneqq \int_a^b f(z(t)) {|z'(t)|} \,\mathrm{d}t ,$$
so ${|\mathrm{d}z|}$ seems to mean ${|z'(t)|} \,\mathrm{d}t$.  As before, if we use the standard orientation on $[a,b]$ to change the $1$-form $\mathrm{d}t$ to the pseudo-$1$-form ${|\mathrm{d}t|}$, then ${|\mathrm{d}z|}$ is the absolute value of $\mathrm{d}z$ from before.

Of course, we should really be looking at differential forms.  Again, $f$ is a [[scalar field]] on $\mathbb{C}$, or equivalenty a $0$-form.  Again $z$ is the [[identity map]] on $\mathbb{C}$, viewed as a $0$-form.  Then $\mathrm{d}z$ is a $1$-form, its absolute value ${|\mathrm{d}z|}$ is a (positive definite) [[absolute form|absolute]] $1$-form, and $f {|\mathrm{d}z|}$ is a (more general) absolute $1$-form.  Since $C$ is given up to an arbitrary reparametrisation, it is an unoriented $1$-submanifold of $X$, which is just what we need for an absolute $1$-form.  Now we may interpet the notation
$$ \int_C f {|\mathrm{d}z|} $$
literally as the integral of an absolute $1$-form.

This is actually a special case of the line integral of a scalar field, since
$$ {|\mathrm{d}z|} = {|\mathrm{d}(x + \mathrm{i}y)|} = {|\mathrm{d}x + \mathrm{i} \,\mathrm{d}y|} = \sqrt{(\mathrm{d}x)^2 + (\mathrm{d}y)^2} = \sqrt{g} = \mathrm{d}s ,$$
since $\mathrm{d}x^2 + \mathrm{d}y^2$ is the standard metric on $\mathbb{C}$.

## Properties

### Cauchy's theorems

* [[Cauchy integral formula]]

* [[Cauchy integral theorem]]

## References

Here are a couple of old Usenet posts that explain how line integrals of scalar fields should be viewed in terms of forms and pseudoforms.

* [first](https://groups.google.com/group/sci.physics.research/msg/2774cbbc982e200e?dmode=source)
* [second](https://groups.google.com/group/sci.physics.research/msg/424da828e75b6b90?dmode=source)

These are obsolete with the concept of [[absolute forms]], but they contain more explicit calculations.

See also

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

[[!redirects line integral]]
[[!redirects line integrals]]
[[!redirects contour integral]]
[[!redirects contour integrals]]
[[!redirects contour integration]]

[[!redirects contour integration]]
[[!redirects contour integrations]]


line Lie-n algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _line Lie $n$-algebra_ over a ground [[field]] $k$ is the [[Lie n-algebra]] analog of the abelian (trivial) 1-[[dimensional]] [[Lie algebra]] on $k$.

## Definition

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

For $n \in \mathbb{N}, n \geq 1$ the **line Lie $n$-algebra**

$$
  b^{n-1} k \in L_\infty \stackrel{CE}{\hookrightarrow} dgAlg^{op}
$$

is the [[L-∞ algebra]] whose [[Chevalley-Eilenberg algebra]]

$$
  CE(b^{n-1} \mathbb{R}) =
    (\wedge^\bullet \langle c\rangle , d = 0)
$$

is the free graded-commutative algebra on a single generator $C$ in degree $k$ equipped with the trivial differential

$$
  d c = 0
  \,.
$$

=--

+-- {: .num_example}
###### Example

For $n = 2$ then the _line Lie 2-algebra_ is the [[Lie 2-algebra]] that comes from the [[differential crossed module]] $(\mathbb{R} \to 0)$.

=--

## Properties

* For $\mathfrak{g}$ a [[Lie algebra]] a [[cocycle]] $\mu$ in degree $n$-[[Lie algebra cohomology]] on $\mathfrak{k}$ is equivalently a morphism of [[L-∞ algebra]]s

  $$
    \mu : \mathfrak{g} \to b^{n-1}\mathbb{R}
    \,.
  $$

  More generally, for $\mathfrak{g}$ an [[L-∞ algebra]], a degree-$n$ cocycle in [[∞-Lie algebra cohomology]] is given by such a morphism.

* There is a unique (up to rescaling) indecomposable [[invariant polynomial]] on $b^{n-1} \mathbb{R}$, given by the shifted copy of the generator $c$ in the [[Weil algebra]] $W(b^{n-1}\mathbb{R})$.

  Equivalently, we have

  $$
    inv(b^{n-1}\mathbb{R}) = CE(b^n \mathbb{R})
    \,.
  $$

* The [[Lie integration]] (see there) of $b^{n-1}\mathbb{R}$ is the line Lie n-group $\mathbf{B}^{n-1}\mathbb{R}$.

## Related concepts

* [[translation Lie algebra]]

* [[super translation Lie algebra]]

[[!redirects line Lie-n algebras]]

[[!redirects line Lie n-algebra]]
[[!redirects line Lie n-algebras]]

[[!redirects line Lie algebra]]
[[!redirects line Lie 2-algebra]]


line n-bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $\mathbb{G}_m$ the [[multiplicative group]], a $\mathbf{B}^{n-1}\mathbb{G}_m$ [[principal infinity-bundle]], modulated by a [[cocycle]] $X\to \mathbf{B}^n \mathbb{G}_m$, classified by a [[cohomology class]] in  in $H^n(X,\mathbb{G}_m)$.

## Examples

* [[universal Chern-Simons line 3-bundle]]

## Related concepts

* [[line bundle]]

* [[line 2-bundle]]

* [[circle n-bundle with connection]]

* [[holomorphic line n-bundle]]

[[!include moduli of higher lines -- table]]

[[!redirects line n-bundles]]


line object

> This entry discusses line objects, their multiplicative groups and additive groups in generality. For the traditional notions see at _[[affine line]]_.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## In linear algebra

In [[linear algebra]] over a [[field]] $k$, the __line__ is the field $k$ regarded as a [[vector space]] over itself. More generally, _a line_ is a vector space [[isomorphic]] to this, i.e. any 1-dimensional $k$-vector space.

The [[real line]] $\mathbb{R}$ models the naive intuition of the geometric line in [[Euclidean geometry]]. See also at _[[complex line]]_.

In many contexts of modern mathematics, however, _line_ implicitly refers to the complex line $\mathbb{C}$ (which as a real vector space is the [[complex plane]]!). For instance this is the line usually meant when speaking of [[line bundle]]s.

## Over an algebraic theory

We discuss here how in the context of [[space]]s modeled on duals of [[algebra over a Lawvere theory|algebras over]] an [[algebraic theory]] $T$, there is a canonical space $\mathbb{A}_T$ which generalizes the [[real line]] $\mathbb{R}$.

### Definition

For $T$ (the [[syntactic category]] of) any [[Lawvere theory]] we have that [[Isbell conjugation]]

$$
  (\mathcal{O} \dashv Spec)\colon
   T Alg^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}}
   Sh(C)
$$

relates $T$-algebras to the [[sheaf topos]] over duals $T \hookrightarrow C \subset T Alg^{op}$ of $T$-algebras, for $C$ a [[small category|small]] [[full subcategory]] with [[subcanonical coverage]].

By the [free T-algebra adjunction](http://ncatlab.org/nlab/show/Lawvere+theory#FreeAlgebras)

$$
  (F_T \dashv U_T)\colon T Alg \stackrel{\overset{F_T}{\leftarrow}}{\underset{U_T}
{\to}}
  Set
$$

we have the free $T$-algebra $F_T(*) \in TAlg$ on a single generator.

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

The **$T$-line object** is

$$
  \mathbb{A}_T \coloneqq Spec F_T(*) \in Sh(C)
  \,.
$$

=--

### The additive group object
 {#AdditiveGroupObject}

For $\mathcal{Ab}$ the [[Lawvere theory]] of [[abelian group]]s, say that a morphism $ab\colon \mathcal{Ab} \to T$ of Lawvere theories is an **abelian Lawvere theory**. Algebras over abelian Lawvere theories have underlying abelian groups

$$
  (ab_* \dashv ab^*)\colon
  T Alg \stackrel{\overset{ab_*}{\leftarrow}}{\underset{ab^*}{\to}}
  Ab
  \,.
$$

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

For $T$ an abelian Lawvere theory, by its underlying abelian group we have that $\mathbb{A}_T$ inherits the structure of an abelian [[group object]] in $Sh(C)$. Write

$$
  \mathbb{G}_T \in Ab(Sh(C))
$$

for this group object on $\mathbb{A}_T$.

=--

### The multiplicative group object
 {#MultiplicativeGroup}

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

For $\mathbb{A}_T$ a line object, write

$$
  (\mathbb{A}_T^\times \hookrightarrow \mathbb{A}_T)
  \in Sh(C)
$$

be the maximal [[subobject]] of the line on those [[generalized element|elements]] that have [[inverses]] under the multiplication $\mathbb{A}_T \times \mathbb{A}_T \to \mathbb{A}_T$.

This is called the **multiplicative group** of the line object, often denoted $\mathbb{G}_m$.

=--

### The group of roots of unity

See at _[[roots of unity]]_.

### Examples

* For $T$ the theory of ordinary commutative [[associative algebras]] over a [[ring]] $R$, we have that

  * $\mathbb{A}_T = \mathbb{A}_R$ is what is the _[[affine line]]_ over $R$;

  * $\mathbb{G}_m$ is the standard [algebraic multiplicative group](affine+line#MultiplicativeGroup);

  * $\mathbb{G}_a$ is the standard [algebraic additive group](affine+line#AdditiveGroup).

* For $T \coloneqq Smooth \coloneqq$ [[CartSp]] the theory of [[smooth algebra]]s, we have that $\mathbb{A}_{Smooth} = \mathbb{R}$ is the [[real line]] regarded as a [[diffeological space]].

  The additive group in this case the the additive [[Lie group]] of [[real numbers]]. The multiplicative group is the Lie group $\mathbb{R}^\times = \mathbb{R} - \{0\}$ of non-zero real numbers under multiplication.

See also _[[analytic affine line]]_.

#### Properties

##### Cohomology

For $R$ a [[ring]] and $H^n_{et}(-,-)$ the [[etale cohomology]], $\mathbb{G}_m$ the [[multiplicative group]] of the [[affine line]]; then

* $H^0_{et}(R, \mathbb{G}_m) = R^\times$ ([[group of units]])

* $H^1_{et}(R, \mathbb{G}_m) = Pic(R)$ ([[Picard group]]: iso classes of invertible $R$-modules)

* $H^2_{et}(R, \mathbb{G}_m) = Br(R)$ ([[Brauer group]] Morita classes of Azumaya $R$-algebras)

### Related concepts

* [[function algebra]]

* [[formal multiplicative group]]

* [[exponential exact sequence]]

* [[affine modality]]

* [[Picard scheme]], [[Picard modality]]

### References

The notion of a line object over general abelian Lawvere theories has been considered in

* [[nLab:Herman Stel]], _[[schreiber:master thesis Stel|∞-Stacks and their Function Algebras -- with Applications to ∞-Lie Theory]]_ (2010)
{#Stel}

in the context of [[function algebras on ∞-stacks]].

## In a monoidal category

Given a [[monoidal category]] $C$, one may define a _line object in $C$_ to be an object $L$ such that the tensoring [[functor]]  $- \otimes L : C \to C$ has an [[inverse]].

## Related concepts

* [[2-line]]

[[!include moduli of higher lines -- table]]

[[!redirects line object]]
[[!redirects line objects]]

[[!redirects line]]
[[!redirects lines]]

[[!redirects multiplicative group]]
[[!redirects additive group]]


line segment

# Line segments
* table of contents
{: toc}

## Idea

A line segment is a fundamental feature of [[geometry]], going back to [[Euclid]] (and probably much earlier), and particularly relevant in [[convex spaces]].  In general, a line segment is that which lies between two [[points]].

## Definitions

In [[synthetic geometry]], a __line segment__ may be regarded as an [[unordered pair]] $\{p,q\}$ of [[points]], called the __endpoints__ of the line segment. A __directed line segment__ is given by an *[[ordered pair|ordered]]* pair of points; so we speak of the directed line segment *from* $p$ *to* $q$ instead of the (undirected) line segment *between* $p$ *and* $q$. Generally, the endpoints are required to be distinct; otherwise the line segment is __degenerate__.

In [[analytic geometry]], where geometric figures are conceived of as sets of points, the __closed line segment__ between $p$ and $q$ consists of $p$, $q$, and every point in between, while the __open line segment__ consists of the points strictly between $p$ and $q$ but not the endpoints themselves. (One can also consider half-open/half-closed versions.) Then a *directed* line segment from $p$ to $q$ (with the same open/closed variations) is a line segment between $p$ and $q$ equipped with an [[orientation]].

In axiomatic approaches to geometry, the concept of 'between' from the previous paragraph is undefined, so that line segments are a fundamental concept.  But in a [[convex space]] (including [[Cartesian spaces]] and other [[topological vector spaces]]), where we can form linear combinations of points (as long as the coefficients are positive and have unit sum) we may say that $r$ is __[[betweenness|between]]__ $p$ and $q$ if $r = t p + (1 - t) q$ for some $t \in [0,1]$ and __strictly between__ if this is so for some $t \in (0,1)$.  Now the closed and open line segments are fully defined (in terms of the operation of taking convex-linear combinations).  In other words, the closed line segment between $p$ and $q$ is the [[convex hull]] of the set $\{p,q\}$.

A line segment should be distinguished from a __line__ (vastly generalized at [[line]]), which has no endpoints but continues to [[infinity]].  (There is also a __ray__, which has one endpoint but continues infinitely in the other direction.)  In classical [[Euclidean geometry]], lines are prominent, but they are treated as being only *potentially* infinite.  So a line at any stage of construction is always a line segment, but one of the basic constructions is to extend such a line segment to a larger one, considered to still be the same line.  In this way, a line can be thought of as an [[equivalence class]] of line segments, so equivalently as an equivalence class of pairs of (distinct) points.

+-- {: .un_remark}
###### Notation

The line segment between $p$ and $q$ is traditionally denoted $\overline{p q}$.  The directed line segment from $p$ to $q$ may be denoted $\overrightarrow{p q}$, but this is also used for other things, including the ray from $p$ through $q$ (and then a double-headed arrow is used for the line through $p$ and $q$).  Another way to write a directed line segment is $[p,q]$ (borrowing from notation for [[intervals]]), and sometimes this is even used for the undirected line segment, since it allows one to distinguish the closed line segment (as above) from the open line segment $(p,q)$.
=--

[[!redirects line segment]]
[[!redirects line segments]]

[[!redirects degenerate line segment]]
[[!redirects degenerate line segments]]
[[!redirects closed line segment]]
[[!redirects closed line segments]]
[[!redirects open line segment]]
[[!redirects open line segments]]

[[!redirects directed line segment]]
[[!redirects directed line segments]]


line type

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

\tableofcontents

## Idea

Similar to how the [[interval type]] is the abstract [[homotopy type]] for the [[unit interval]] and the [[circle type]] is the abstract [[homotopy type]] for the [[unit circle]], the line type is the abstract [[homotopy type]] for the [[real line]].

## Definition

The **line type** $A^1$ is the [[higher inductive type]] generated by

* an element $0:A^1$

* an [[equivalence of types]] $s:A^1 \simeq A^1$

* an identification $p:0 =_{A^1} s(0)$

Equivalently, it is the higher inductive type generated by

* an element $0:A^1$

* an [[equivalence of types]] $s:A^1 \simeq A^1$

* for each $x:A^1$, an identification $p(x):x =_{A^1} s(x)$

Equivalently, it is the higher inductive type generated by

* a function $c:\mathbb{Z} \to A^1$

* for each $z:\mathbb{Z}$, an identification $p(z):c(z) =_{A^1} c(s(z))$

where $\mathbb{Z}$ is the [[integers type]].

## Properties

The line type is [[contractible]]; see section 8.1.5 of [UFP13](#UFP13).

### Relation to the integers

The line type is equivalent to the [[propositional truncation]] of the [[integers type]].

### Relation to the circle type

The [[circle type]] $S^1$ is the [[coequalizer type]] of the pair of functions on the line type

$$
A^1\underoverset{\quad s \quad}{\mathrm{id}_{A^1}}{\rightrightarrows}A^1 \to S^1
$$

This is the analogue in [[synthetic homotopy theory]] of the fact that the unit circle is the coequalizer of the pair of functions on the [[real line]] in [[classical homotopy theory]]:

$$
\mathbb{R} \underoverset{\quad (-) + 1 \quad}{\mathrm{id}_\mathbb{R}}{\rightrightarrows}\mathbb{R} \to \mathbb{S}^1
$$

### Relation to the real numbers

In [[real-cohesive homotopy type theory]], the [[shape]] of the [[real numbers]] is equivalent to the line type $\esh(\mathbb{R}) \simeq A^1$.

## Related concepts

* [[real numbers]]

* [[integers type]]

* [[interval type]]

* [[circle type]]

## References

The line type is defined as "homotopical reals" in section 8.1.5 of

* {#UFP13} *Homotopy Type Theory: Univalent Foundations of Mathematics*, The [[Univalent Foundations Project]], Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

[[!redirects line type]]
[[!redirects line types]]

[[!redirects homotopical reals]]
[[!redirects homotopical real line]]
[[!redirects homotopical real line type]]
[[!redirects homotopical real line types]]

[[!redirects homotopical real numbers type]]
[[!redirects homotopical real numbers types]]

[[!redirects homotopical real numbers]]


line with two origins

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

# Contents
* table of contents
{: toc}

## Idea

What is called _the line with two origins_ is the [[topological space]] which results by "gluing" a copy of the [[real line]] identically to itself, except at the origin. This is a basic example of a [[non-Hausdorff topological space]] (even a non-Hausdorff [[smooth manifold]]) and of the fact that [[quotient topological spaces]] of [[Hausdorff topological spaces]] need not be Hausdorff themselves.

## Definition

Consider the [[disjoint union]] $\mathbb{R} \sqcup \mathbb{R}$ of two copies of the [[real line]] $\mathbb{R}$ regarded with its [[metric topology]]. Moreover, consider the [[equivalence relation]] on the underlying set which identifies every
point $x_i$ in the $i$th copy of $\mathbb{R}$ ($i \in \{0,1\}$) with the corresponding point in the other, the $(1-i)$th copy, except when $x = 0$:

$$
  \left(
    x_i \sim y_j
  \right)
   \;\Leftrightarrow\;
  \Big(
    \left(
      x = y
    \right)
    \,\text{and}\,
    \big(
      (
        x \neq 0
      )
      \,\text{or}\,
      (
        i = j
      )
    \big)
  \Big)
  \,.
$$

The [[quotient topological space]]

$$
  \left(
    \mathbb{R} \sqcup \mathbb{R}
  \right)/\sim
$$

by this equivalence relation is called the **line with two origins**.

## Properties

The line with two origins is a [[non-Hausdorff topological space]]:

Because by definition of the [[quotient topological space|quotient space topology]], the [[open neighbourhoods]] of $0_i \in   \left( \mathbb{R} \sqcup \mathbb{R} \right)/\sim$ are precisely those that contain subsets of the form

$$
 (-\epsilon, \epsilon)_i
  \;\coloneqq\;
 (-\epsilon,0) \cup \{0_i\} \cup (0,\epsilon)
 \,.
$$

But this means that the "two origins" $0_0$ and $0_1$ may not be [separated by neighbourhoods](separation+axioms#SeparatedByNeighbourhoods), since the intersection of
$(-\epsilon, \epsilon)_0$ with $(-\epsilon, \epsilon)_1$ is always non-empty:

$$
  (-\epsilon, \epsilon)_0
   \cap
  (-\epsilon, \epsilon)_1
  \;=\;
  (-\epsilon, 0) \cup (0, \epsilon)
  \,.
$$

The [Hausdorff reflection](Hausdorff+space#HausdorffReflection)

$$
  H \;\colon\; Top \longrightarrow Top_{Haus}
$$

of the line with two origins is the [[real line]] itself:

$$
  H\left(
    \left(
      \mathbb{R} \sqcup \mathbb{R}
    \right)/\sim
  \right)
  \;\simeq\;
  \mathbb{R}
$$

However, the line with two origins is [[T1]] and [[sober space|sober]].

It is also a [[topological manifold]], even a [[smooth manifold]], except that one often requires such to be Hausdorff to rule out examples such as this.  In a similar vein, it is [[locally compact space|locally compact]] (and a [[compact space|compact]] version can be made by starting with $[{-1,1}]$ instead of $\mathbb{R}$), [[paracompact space|paracompact]], and so forth.

## Related concepts

* [[long line]]

## References

* Wikipedia, _[Line with two origins](https://topospaces.subwiki.org/wiki/Line_with_two_origins)_

[[!redirects line with two origins]]
[[!redirects lines with two origins]]
[[!redirects lines with two origins each]]
[[!redirects line with double origin]]
[[!redirects lines with double origin]]
[[!redirects lines with double origins]]
[[!redirects line with doubled origin]]
[[!redirects lines with doubled origin]]
[[!redirects lines with doubled origins]]
[[!redirects line with doubled point]]
[[!redirects lines with doubled point]]
[[!redirects lines with doubled points]]


linear (∞,1)-category

+-- {: .rightHandSide}
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#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

If $k$ is a [[commutative ring]], $k$-linear (infinity,1)-categories are the analogue in [[(∞,1)-category]] theory of the notion of $k$-[[linear category]] in [[category theory]].

## Definition

A **$k$-linear (infinity,1)-category** is an [[additive (infinity,1)-category]] $A$ whose [[homotopy category of an (infinity,1)-category|homotopy category]] $ho(A)$ is a $k$-[[linear category]].

More generally, let $R$ be a [[commutative ring spectrum]] and let $Mod(R)$ denote the [[symmetric monoidal (infinity,1)-category]] of [[module spectra|modules]] over it. An _$R$-linear (infinity,1)-category_ is an object of the [[(infinity,1)-category of modules]] over $Mod(R)$ in the [[symmetric monoidal (infinity,1)-category]] of [[(infinity,1)-categories]].

## Properties

An $R$-linear [[(infinity,1)-category]] is naturally [[enriched (infinity,1)-category|enriched]] over the [[symmetric monoidal (infinity,1)-category]] of [[module spectra|modules]] over $R$.

## Related concepts

* [[additive (∞,1)-category]]
* [[stable (∞,1)-category]]

## References

Section 6 of

* [[Jacob Lurie]], [[Spectral Schemes]].

[[!redirects linear (∞,1)-categories]]

[[!redirects linear (infinity,1)-category]]
[[!redirects linear (infinity,1)-categories]]

[[!redirects linear infinity-category]]
[[!redirects linear infinity-categories]]

[[!redirects k-linear (infinity,1)-category]]
[[!redirects k-linear (infinity,1)-categories]]
[[!redirects k-linear (∞,1)-category]]
[[!redirects k-linear (∞,1)-categories]]


linear algebra

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

# Contents
* table of contents
{: toc}

## Idea

__Linear algebra__ over a [[skewfield]] $K$ is the study of the [[category]] $K$-[[Vect]], that is the study of [[vector spaces]] over $K$.  Sometimes one uses the term '$K$-linear algebra' to mean an [[associative algebra]] (or similar) over $K$ (compare '$K$-[[linear map]]').

Classical linear algebra is done over a [[real-closed field]] or an [[algebraically closed field]] of [[characteristic]] $0$; the latter is the simplest context, which is now pretty thoroughly understood.  Traditionally, we take these fields to be the field of [[real numbers]] and the field of [[complex numbers]], although arguably we only really use [[algebraic number|algebraic numbers]].  (From a [[constructive mathematics|constructive]] point of view, some of the classical material is valid only over [[discrete fields]], so we *must* restrict attention to algebraic numbers, or to some discrete extension, for these results to hold in their classical form.)

Fancier linear algebra is done over incomplete fields and fields with positive characteristic (and constructively over nondiscrete fields). Sometimes a generalization to categories of finitely generated projectives over a ring is considered. In infinite dimensions one rarely studies purely algebraic version, which is considered as a linear algebra, but more often one equips them with topological structure, what enters the subject of [[functional analysis]].

If one is interested in tensor products as well, then one gets a generalization called [[multilinear algebra]]: tensor algebra, tensors, exterior and symmetric algebras are some of the main characters in that theory. Study of [[determinant|determinants]] is important in the usual linear algebra but it is also closely related to the study of exterior algebras.

## Related entries

* [[matrix]], [[matrix calculus]],

* [[linear map]], [[linear equation]]

* [[rank-nullity theorem]]

* [[higher linear algebra]]

* [[linear type theory]]

## References

Historical precursors:

* [[Hermann Grassmann]], _[[Ausdehnungslehre]]_, 1844

Textbook accounts:

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

* Sheldon Axler, *Linear Algebra Done Right*, Springer (2015) [[doi:10.1007/978-3-319-11080-6](https://doi.org/10.1007/978-3-319-11080-6)]

Lecture notes:

* [[Christoph Schweigert]], *Lineare Algebra*, lecture notes, Hamburg (2022) [[pdf](https://www.math.uni-hamburg.de/home/schweigert/skripten/laskript.pdf)]

Discussion of linear algebra as [[categorical semantics]] for [[linear logic]]/[[linear type theory]]:

* [[Benoît Valiron]], [[Steve Zdancewic]], *Finite Vector Spaces as Model of Simply-Typed Lambda-Calculi*, in: *Proc. of ICTAC'14*, Lecture Notes in Computer Science **8687**, Springer (2014) 442-459 [[arXiv:1406.1310](https://arxiv.org/abs/1406.1310), [doi:10.1007/978-3-319-10882-7_26](https://doi.org/10.1007/978-3-319-10882-7_26), slides:[pdf](https://www.cs.bham.ac.uk/~drg/bll/steve.pdf), [[Zdancewic-LinearLogicAlgebra.pdf:file]]]

* {#Murfet14} [[Daniel Murfet]], *Logic and linear algebra: an introduction* [[arXiv:1407.2650](http://arxiv.org/abs/1407.2650)]

See also:

* Wikipedia, *[Linear algebra](http://en.wikipedia.org/wiki/Linear_algebra)*

* [[Jean Dieudonné]], _Linear algebra and geometry_, Translated from the French Houghton Mifflin Co., Boston, Mass. 1969, 207 pp

* I. M. Gelʹfand, _Lekcii po lineĭnoĭ algebre_ [2d ed.] Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1951; Eng. transl: _Lectures on linear algebra_, Interscience Tracts in Pure and Applied Mathematics __9__, 1961 ix+185 pp

* A. I. Kostrikin, Ju. I. Manin, _Lineĭnaya algebra i geometriya_, Moskov. Gos. Univ., Moscow, 1980. 320 pp. [MR82i:00008](http://www.ams.org/mathscinet-getitem?mr=610246); _Linear algebra and geometry_, Translated from the second Russian edition by M. E. Alferieff. Algebra, Logic and Applications, 1. Gordon and Breach Science Publishers, New York, 1989. x+309 pp. [MR91h:00008](http://www.ams.org/mathscinet-getitem?mr=1057342)

* Joel W. Robbin, _Matrix algebra using MINImal MATlab_

* N.J. Higham, _Functions of matrices. Theory and computation,  Philadelphia, PA: SIAM. 425p., 2008

[[!redirects linear algebra]]
[[!redirects linear-algebra]]
[[!redirects linear algebras]]
[[!redirects linear-algebras]]
[[!redirects linear algebraic]]
[[!redirects linear-algebraic]]


linear algebraic group

## Definition

A __linear algebraic group__ $G$ is any (Zariski closed) algebraic subgroup of $GL(n,k)$ where $k$ is a field and $n$ a natural number.

An algebraic group over a field is linear iff it is affine
as a $k$-scheme.

An algebraic group scheme is *affine* if the underlying scheme is [[affine scheme|affine]].

The category of [[affine group scheme]]s over a field $k$ is the [[opposite category|opposite]] of the category of commutative [[Hopf algebras]] over $k$.

## Literature

Related notions include [[algebraic group]], [[affine algebraic group]], [[hyperalgebra]], [[distribution on an affine group scheme]]

* [[Armand Borel]], _Linear algebraic groups_, Springer
* G. Hochschild, _Algebraic groups and Hopf algebras_, Illinois J. Math. __14__:1 (1970), 52-65 [euclid](https://projecteuclid.org/euclid.ijm/1256053299)
* Gerhard P. Hochschild, _Basic theory of algebraic groups and Lie algebras_,  Graduate Texts in Mathematics 75, 1981 [doi](10.1007/978-1-4613-8114-3)
* wikipedia: [linear algebraic group](https://en.wikipedia.org/wiki/Linear_algebraic_group)
* Akira Masuoka, _Hopf algebraic techniques applied to super algebraic groups_, [arXiv:1311.1261](https://arxiv.org/abs/1311.1261)

category: algebra, algebraic geometry


linear bicategory

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#### 2-Category theory
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[[!include 2-category theory - contents]]
=--
=--
=--

# Linear bicategories

* table of contents
{: toc}

## Idea

The notion of *linear bicategory* (or *linearly distributive bicategory* for long) is a [[horizontal categorification]] of the notion of [[linearly distributive category]], analogous to how [[bicategories]] are a horizontal categorification of [[monoidal categories]].

## Definition

+-- {: .un_defn}
###### Definition
A **linear bicategory** consists of

1. A set of objects $x,y,z$.
1. For each $x,y$ a hom-category $B(x,y)$.
1. Two bicategory structures $(\otimes,\top)$ and $(\parr,\bot)$ on these hom-categories.  Thus we have two compositions $\otimes,\parr : B(y,z) \times B(x,y) \rightrightarrows B(x,z)$ and two units $\top_x,\bot_x \in B(x,x)$, each coherently associative and unital.
1. Linear distributivities $(X \parr Y) \otimes Z \to X \parr (Y\otimes Z)$ and $X \otimes (Y \parr Z) \to (X\otimes Y) \parr Z$, satisfying the usual coherence laws for a [[linearly distributive category]].
=--

## Examples

* Linear bicategories with one object coincide with the [[deloopings]] of (non-symmetric) [[linearly distributive categories]].
* Any [[allegory]] whose hom-sets are [[Boolean algebras]] is a linear bicategory, with $\otimes$ the usual composition and with $X \parr Z = \neg (\neg X^\circ \otimes \neg Y^\circ)^\circ$.  In particular, this includes the bicategory of [[relations]] in any [[Boolean category]], such as [[Set]] (assuming [[classical logic]]).
* Any ordinary [[bicategory]] can be regarded as a linear bicategory with $\otimes = \parr$.
* If $B$ is a linear bicategory (such as the delooping of a [[linearly distributive category]]) such that $(B,\otimes,\top)$ has [[local colimits|local coproducts]] and $(B,\parr,\bot)$ has local products, then the bicategory $Mat(B)$ (whose objects are families of objects of $B$ and whose morphisms are matrices of 1-cells in $B$) is again a linear bicategory.  For instance, if $B$ is the [[Boolean algebra]] $\mathbf{2}$ of [[truth values]], then $Mat(\mathbf{2}) \cong Rel(Set)$.

## Linear adjoints

+-- {: .un_defn}
###### Definition
A **linear adjunction** in a linear bicategory consists of 1-cells $f:X\to Y$ and $g:Y\to X$ along with a unit $\eta : \top_X \to g \parr f$ and $\epsilon : f \otimes g \to \bot_Y$ satisfying versions of the usual [[triangle identities]]  that include the linear distributivities, as for [[dual objects]] in a [[linearly distributive category]]
=--

A linear bicategory in which every 1-cell has both a linear right adjoint and a linear left adjoint is a horizontal categorification of a [[non-symmetric star-autonomous category]].  But in [Cockett-Koslowski-Seely](#CKS00) this is called a "closed linear bicategory", with the term "$\ast$-linear bicategory" reserved for something stronger analogous to a [[cyclic star-autonomous category]].

## References

* [[Robin Cockett|Cockett]] and Koslowski and [[Robert Seely|Seely]], *Introduction to linear bicategories*, Mathematical Structures in Computer Science, 10 (2), 2000 (165 - 203)
 {#CKS00}

[[!redirects linear bicategories]]
[[!redirects linearly distributive bicategory]]
[[!redirects linearly distributive bicategories]]
[[!redirects linear adjoint]]
[[!redirects linear adjoints]]
[[!redirects linear adjunction]]
[[!redirects linear adjunctions]]


linear category > history

< [[linear category]]


linear chord diagram

+-- {: .rightHandSide}
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### Context
#### Graph theory
+-- {: .hide}
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=--
#### Knot theory
+-- {: .hide}
[[!include knot theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _linear chord diagram_ or _arc diagram_ or _rooted chord diagram_  is a trivalent finite [[undirected graph]] with an embedded oriented [[line]] and all [[vertices]] on that line.

Equivalemtly this is just an [[n-tuple]] equipped with a [[partition]] into [[pairs]].

The following shows a generic example of a linear chord diagram:

The graphics on the right shows all linear chord diagrams with precisely four [[vertices]].

Closing up the line of a linear chord diagram to a [[circle]] and remembering the [[ordering]] of vertices only op to [[cyclic permutation]], it becomes a _[[round chord diagram]]_, usually just called a [[chord diagram]]. Conversely, a linear chord diagram is equivalently a round chord diagram with one of its [[vertices]] singled out.

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Applications

### Wick's theorem

The [[combinatorics]] of contractions in [[Wick's theorem]] is governed by linear chord diagrams:

Let $\{Z_i\}$ be a set of [[quantum fields]]/[[random variables]] which are [[free fields]]/[[multivariate normal distribution|multivariate normally distributed]] with

$$
  \big\langle Z_i Z_j \big\rangle = k_{i j}
  \,.
$$

Then [[Wick's theorem]] says that the [[expectation value]] of the product of $n \in \mathbb{N}$ of these fields/random variables is the [[sum]] over linear chord diagrams with $n$ vertices of the product over the edges $e_i \to e_j$ of the given chord diagram of the factors $k_{i j}$.

For example, for $n = 4$, [[Wick's theorem]] says this:

## Related concepts

[[!include chord diagrams and weight systems -- table]]

## References

* Everett Sullivan, _Linear chord diagrams with long chords_ ([arXiv:1611.02771](https://arxiv.org/abs/1611.02771))

[[!redirects linear chord diagrams]]

[[!redirects arc diagram]]
[[!redirects arc diagrams]]

[[!redirects rooted chord diagram]]
[[!redirects rooted chord diagrams]]

[[!redirects rooted chord diagram]]
[[!redirects rooted chord diagrams]]
[[!redirects arc diagram]]
[[!redirects arc diagrams]]


linear code

#Contents#
* table of contents
{:toc}

## Idea

In [[coding theory]], a _linear code_ is an [[error correcting code]] which is [[linear algebra|linear]], in that it is a [[linear subspace]] of a [[vector space]] over a [[finite field]] $\mathbb{F}_q$. Specifically a *[[binary linear code]]* is a linear code over $\mathbb{F}_2$.

A *linear code* is a [[linear subspace]] of a [[vector space]] of [[finite number|finite]] [[dimension]] $d$ over a [[prime field]] $\mathbb{F}_p$, i.e. a vector space [[isomorphism|isomorphic]] to $(\mathbb{F}_p)^d$, for some [[prime number]] $p$ (often $p = 2$). The dimension $d$ of the vector space is also called the _length_ of the linear code.

## Examples

* [[Hamming code]]

## Related concepts

* [[binary Golay code]]

* [[binary linear code]]

* [[Mathieu group]]

* [[combinatorial design]]

* [[Leech lattice]]

* [[adinkra]]

* [[quantum error correcting code]]

## References

* [[Patrick Morandi]], _Error Correcting Codes and Algebraic Curves_ , lecture notes New Mexico State University 2001. ([pdf](http://www.math.nmsu.edu/~pmorandi/math601f01/LectureNotes.pdf))

* Jay A. Wood, _Spinor groups and algebraic coding theory_ , J.Combinatorial Th. Series A **51** (1989) pp.277-313. (available online)

See also:

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

[[!redirects linear codes]]


linear combination

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

# Linear combinations
* table of contents
{: toc}

## Idea

Linear combinations are the most general operations in the [[operads]] for [[modules]] over a [[rig]] (including modules over a [[ring]] and [[vector spaces]] over a [[field]]).

## Definition

Let $R$ be a rig, and let $V$ be a (left) $R$-module.  By 'scalar', we mean an element of $R$; by 'vector', we mean an element of $V$.  Given any [[natural number]] $n$ and any $n$-[[tuple]] $(a_1,\ldots,a_n)$ of scalars (so in short, given a [[finite list]] of scalars), we have an $n$-ary operation on $V$ that maps $(x_1,\ldots,x_n)$ to
$$ a_1 x_1 + \cdots + a_n x_n .$$
The result of this operation is the __linear combination__ of the vectors $x_1$ through $x_n$ with respective __[[coefficients]]__ $a_1$ through $a_n$.

Variations:  If $R$ is a non-unital rig (or more generally if $M$ is a non-unital module), then a linear combination may also have a term $x_0$ with no coefficient.  If $R$ is non-associative (or more generally if $M$ is a non-associative module), then the term with $x_i$ takes the form
$$ a_{i,1} (a_{i,2} \cdots (a_{i,i_m} x_i){\cdots})) .$$
If $R$ is non-commutative and $V$ is a right $R$-module, the term with $x_i$ takes the form $x_i a_i$.  If $V$ is an $R$-$S$-[[bimodule]], then the term with $x_i$ takes the form $a_i x_i b_i$, with $a_i\colon R$ and $b_i\colon S$.  Of course, all of these variations may be combined.

Given a [[subset]] $G$ of (the [[underlying set]] of) $V$, the set of all linear combinations of the vectors in $G$ is a [[submodule]] of $V$, the __$R$-linear span__ of $G$.

More abstractly, by the [[adjunction]] between the underlying-set functor and the [[free functor]], the subset inclusion
$$ i_G\colon G \to {|V|} $$
corresponds to a [[homomorphism]]
$$ \hat{i}_G\colon R[G] \to V .$$
Then the $R$-linear span of $G$ is the image of the homomorphism $\hat{i}_G$.

This abstract definition works more generally for any set function $f : G \to {|V|}$. The $R$-linear span of the image of $f$ in $V$ is the image of its corresponding homomorphism $\hat{f} \colon R[G] \to V$.

## Examples

Every operation on the module $V$ is a linear combination:

*  The [[identity operation]] is the linear combination of arity $1$ with coefficient $1$.

*  Addition is the linear combination of arity $2$ with coefficients $(1,1)$, and the zero element is the linear combination of arity $0$ (with no coefficients).

*  Scalar multiplication by the scalar $a$ is the linear combination of arity $1$ with coefficient $a$.

*  If $R$ is a ring (so $-1$ is a scalar), then subtraction is the linear combination of arity $2$ with coefficients $(1,-1)$, and the additive inverse is the linear combination of arity $1$ with coefficient $-1$.

*  If $R$ is [[divisible group|divisible]] (so $1/n$ is a scalar for every [[positive integer]] $n$), then the [[mean]] of $n$ vectors is the linear combination of arity $n$ with every coefficient $1/n$.

## Special cases

An __[[affine linear combination]]__ is a linear combination whose coefficients sum to $1$.  These are the operations in an [[affine space]].

If $R$ is [[ordered ring|ordered]], then a __[[conical linear combination]]__ is a linear combination whose coefficients are all positive, and a __[[convex linear combination]]__ an affine conical linear combination.  These are the operations in (respectively) a [[conical space]] and a [[convex space]].

## References

* {#Lawvere92} [[William Lawvere]], pp. 33 of: *Introduction to Linear Categories and Applications*, course lecture notes (1992) [[pdf](https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf), [[Lawvere-LinearCategories.pdf:file]]]

[[!redirects linear combination]]
[[!redirects linear combinations]]

[[!redirects linear span]]
[[!redirects linear spans]]


linear differential equation

+-- {: .rightHandSide}
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### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[differential equation]] ([[ordinary differential equation|ordinary]] or [[partial differential equation|partial]]) is called _linear_ if the [[linear combination]] of any of its [[solutions]] is still a [[solution]], hence if its space of solutions is a _[[vector space]]_.

Equivalently this means that the [[differential operator]] that corresponds to the differential equation is a [[linear operator]].

Linear differential equations may be analyzed via [[harmonic analysis]] by applying [[Fourier transform]] to decompose solutions as [[superpositions]] of [[plane wave]] "harmonics" (e.g. [Hörmander 90](#Hoermander90)).

In [[physics]] a [[field theory]] whose [[equations of motion]] is a linear partial differential equation is called a _[[free field theory]]_.

Where a general (possibly non-linear) differential equation is equivalently an object in the [[slice category]] over the [[de Rham shape]] of the space of its [[free variables]], a linear differential equation is more specifically a linear object in this slice. In the context of [[algebraic geometry]] these are the _[[D-modules]]_.

## Examples

* [[wave equation]]

* [[Dirac equation]]

* [[Klein-Gordon equation]]

* [[Schrödinger equation]]

* [[Tomonaga-Schwinger equation]]

* ...

## Related concepts

* [[superposition]]

* [[D-module]]

## References

* {#Hoermander90} [[Lars Hörmander]], _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

See also

* Wikipedia, _[Linear differential equatioon](https://en.wikipedia.org/wiki/Linear_differential_equation)_

[[!redirects linear differential equations]]

[[!redirects linear partial differential equation]]
[[!redirects linear partial differential equations]]

[[!redirects linear ordinary differential equation]]
[[!redirects linear ordinary differential equations]]


linear equation

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###Context###
#### Equality and Equivalence
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Since generally an _[[equation]]_ is the statement of [[equality]] $\phi(x) = \psi(y)$ of two [[functions]] $\phi$ and $\psi$ of [[variables]] $x$ and $y$, so a _linear equation_ is an equation between [[linear functions]].

### In 1-category theory

Typically here a _[[linear function]]_ is taken to mean an _$R$-[[linear map]]_ over some given [[ring]] $R$, hence a [[homomorphism]] $\phi : X \to Z$ or $\psi : Y \to Z$ of $R$-[[modules]] $X, Y, Z \in R$[[Mod]]. If here $Z$ is an $R$-module of [[rank]] greater than 1, one also speaks of a _system of linear equations_.

Specifically if $R = k$ is a [[field]] then these are linear maps of $k$-[[vector spaces]] and hence in this case a linear equation is a statement of equality of two [[vectors]] $\phi(x) = \psi(y)$ in some [[vector space]] $Z$ that depend linearly on vectors $x$ in a vector spaces $X$ and $y \in Y$.

Frequently this is considered specifically for the case that $g$ is a [[constant function]], hence just a fixed [[vector]]. In this case the linear equation becomes $\phi(x) = g$ for $x \in X$. If moreover $\phi$ here is represented or representable by a [[matrix]] this is typically written as

$$
  \phi \cdot\vec x = \vec g
  \,,
$$

which is the form that one finds in standard textbooks on [[linear algebra]]. If $\vec g = 0$ here this is called a _homogeneous linear equation_.

But linear equations make sense and are important in the greater generality where $R$ is not necessarily a field, and in fact in contexts more general than that of $R$-modules even. For instance [[natural isomorphisms]] between [[linear functors]] are a kind of [[categorification]] of linear equations.

### In $(\infty,1)$-category theory

Indeed, as discussed at _[[equation]]_, in the [[formal logic]] of [[type theory]] a an equation as above is a [[judgement]] of the form

$$
  x : X , y : Y \vdash (\phi(x) = \psi(y)) : Type
$$

whose solution space is the [[dependent sum]]

$$
  Sol \;\; \coloneqq \vdash \sum_{{x : X} \atop {y : Y}} (\phi(x) = \psi(y)) : Type
$$

and reading this in fact in [[homotopy type theory]] says that $X, Y, Sol$ are _[[homotopy types]]_.

The generalization of a [[ring]] $R$ to a [[homotopy type]] is an _[[E-∞-ring]]_ and that of an $R$-[[module]] $X, Y$ is a _[[module spectrum]]_.

Accordingly a linear equation in [[homotopy]]([[homotopy type theory|type]]) [[homotopy theory]] is a statement of [[equivalence]] between elements of an $R$-[[module spectrum]] that depend $R$-linearly on other $R$-module spectra. More precisely, as discussed at _[[equation]]_, the solution space to such an "$\infty$-linear equation" is the [[homotopy pullback]]

$$
  \array{
    X \times_Z^\infty Y &\to& Y
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{\psi}}
    \\
    X &\underset{\phi}{\to}& Z
  }
$$

in an [[(∞,1)-category]] of $R$-[[∞-modules]].

## Properties

### Solution spaces of homogeneous $R$-linear equations

We discuss solution space of homogeneous linear equations in the general context of [[linear algebra]] over a [[ring]] $R$ (not necessarily a [[field]]).

So let $R$ be a [[ring]] and let $N \in R$[[Mod]] be an $R$-[[module]].

Let $n_0,n_1 \in \mathbb{N}$ and let $K = (K_{i j})$ be an an $n_0 \times n_1$ [[matrix]] with entries in $R$. By [[matrix multiplication]] this defines a [[linear function]]

$$
  K \cdot (-) : N^{n_0} \to N^{n_1}
  \,.
$$

which takes the element $\vec u = (u_1, \cdots, u_{n_0}) \in N^{n_0}$ to the element $K \cdot \vec u$ with

$$
  (K \cdot \vec u)_i = \sum_{j = 1}^{n_0} K_{i j}\cdot u_j
  \,.
$$

Consider dually the linear map

$$
  (-) \cdot K^T : R^{n_1} \to R^{n_0}
$$

on the [[free modules]] over $R$.

Consider the [[quotient]] [[module]] of $R^{n_1}$ by the [[image]] of this map

$$
  R^{n_1}/ (R^{n_0} \cdot K^T)
  \,,
$$

hence the [[cokernel]] of the map, fitting in the [[exact sequence]]

$$
  R^{n_1} \stackrel{(-)\cdot K}{\to} R^{n_0} \to R^{n_1}/(R^{n_1}\cdot K^T)
  \to 0
$$

Here the morphism on the left is also called the inclusion of the [[syzygies]] of the module on the right.

Applying the [[left exact functor|left exact]] [[hom functor]] $Hom_{R Mod}(-,N)$ to this yields [[exact sequence]]

$$
  0 \to Hom_{R Mod}(R^{n_1}/(R^{n_0}\cdot K^T), N)
  \to
  N^{n_0}
  \stackrel{K \cdot(-)}{\to}
  N^{n_1}
  \,.
$$

This identifies $Hom_{R Mod}(R^{n_1}/(R^{n_0}\cdot K), N)$ as the space of solutions of the _homogeneous_ linear [[equation]] $K \cdot \vec u = 0$.

(...)

### Relation to syzygies and projective resolutions of modules

For $R$ a [[ring]], there is close relation between

1. $R$-linear equations in finitely many variables;

1. [[generators and relations|finitely generated]] $R$-[[modules]];

1. [[syzygies]] in these $R$-modules

1. and [[projective resolutions]] of these $R$-modules.

These relations we discuss in the following.

(...)

## Related concepts

* [[differential equation]]

* [[matrix calculus]]

* [[determinant]]

## References

Discussion in the context of [[syzygies]] and [[projective resolutions]] of modules is for instance in section 4.5 of

* [[Pierre Schapira]], _Categories and homological algebra_, lecture notes (2011) ([pdf](http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf))

[[!redirects linear equations]]


linear extension of a partial order

> This entry is about topological orders of directed acyclic graphs in [[graph theory]]. For topological orders of materials in [[condensed matter physics]], see [[topological order]].

***

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#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #LinearExtensionOfAPartialOrder}
###### Definition
**(linear extension)**

Given a [[set]] $X$ equipped with a [[partial ordering]] $\leq$, then a _linear extension_ is a [[linear order]] $\leq_{lin}$ on the same set $S$, such that the [[identity function]] $id_S$ is order-preserving

$$
  (S,\leq) \overset{ id_S }{\longrightarrow} \left(S,\leq_{lin}\right)
  \,.
$$

=--

{#TopologicalOrder} In [[graph theory]], a set equipped with a partial order is an acyclic [[directed  graph]], with the partial order representing the reachability relation of the graph, and any linear extension of the reachability relation is called a **topological order**.

## Properties

+-- {: .num_prop #ExistenceOfLinearExtensions}
###### Proposition
**(existence of linear extensions)**

For [[finite sets]] linear extensions (Def. \ref{LinearExtensionOfAPartialOrder}) always exist. For non-finite sets this is still the case using the [[axiom of choice]].

=--

A proof under [[axiom of choice|AC]] was first published in ([Marczewski 30](#Marczewski30)). The proposition actually follows from the weaker choice principle called the [[ultrafilter theorem|ultrafilter principle]], by appeal to the [[compactness theorem]], as follows.

+-- {: .proof}
###### Proof
It is a very simple matter to show linear extensions exist in the [[finite set|finite]] case: one may proceed by [[induction]]. Any finite $n$-element poset has a [[minimal element]] $x$ (meaning $y \leq x$ implies $y = x$). By induction the restricted partial order on $X \setminus \{x\}$ admits a linear extension, and then one may simply prepend $x$ to that linear order to complete the inductive step.

The rest is a routine application of compactness for [[propositional theories]]. Let $(X, \leq)$ be a partially ordered set, and introduce a [[signature]] consisting of [[signature|constants]] $c_x$, one for each $x \in X$, and a [[binary relation]] $L$. Introduce [[axioms]] $\neg(c_x = c_y)$ whenever $x \neq y$ in $X$, and $L(c_x, c_y)$ whenever $x \leq y$ in the poset $X$, and axioms stating that $L$ is a linear order. By the previous paragraph, the resulting theory is finitely satisfiable upon interpreting each $c_x$ as $x$. Hence the theory is satisfiable. Taking any [[model]] $M$, and interpreting the constants in $M$, and restricting $L$ to them, we obtain a linear extension on $X$.
=--

## References

* {#Marczewski30} [[Edward Marczewski]], _Sur l'extension de l'ordre partiel_, Fundamenta Mathematicae **16** (1930) 386–389 $[$[dml:212499](https://eudml.org/doc/212499), [pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm16125.pdf)$]$

See also

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

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

[[!redirects linear extension of partial orders]]
[[!redirects linear extensions of partial orders]]

[[!redirects topological ordering]]
[[!redirects topological orderings]]


linear functional

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### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

# Linear functionals
* table of contents
{: toc}

## Definition

In [[linear algebra]] and [[functional analysis]], a __linear functional__ (often just _functional_ for short) is a [[function]] $V \to k$ from a [[vector space]] to the [[ground field]] $k$.  (This is a [[functional]] in the sense of [[higher-order logic]] if the elements of $V$ are themselves functions.)  Then a __linear functional__ is a [[linear map|linear]] such function, that is a [[morphism]] $V \to k$ in $k$-[[Vect]].  In the case that $V$ is a [[topological vector space]], a __continuous linear functional__ is a [[continuous map|continuous]] such map (and so a morphism in the category [[TVS]]).  When $V$ is a [[Banach space]], we speak of __[[bounded operator|bounded]] linear functionals__, which are the same as the continuous ones.  In an algebraic context, one may also use the term __linear form__, especially to distinguish from [[bilinear forms]], [[quadratic forms]], and the like.

## Remarks

In a sense, linear functionals are co-probes for vector spaces. If the vector space $V$ in question has finite [[dimension]] and is equipped with a [[basis]], then all linear functionals are [[linear combinations]] of the [[coordinate projection]]s. These projections comprise the _[[dual basis]]_.

In infinite-dimensional [[topological vector spaces]], the notion of dual basis breaks down once spaces more general than [[Hilbert spaces]] are considered. But for [[locally convex spaces]], the [[Hahn–Banach theorem]] ensures the existence of 'enough' continuous linear functionals.  Among non-LCSes, however, there are examples such that the only continuous linear functional is the constant map onto $0 \in k$.

## Examples

* [[distribution]]

## Related concepts

* [[nonlinear functional]]

[[!redirects linear functional]]
[[!redirects linear functionals]]

[[!redirects continuous linear functional]]
[[!redirects continuous linear functionals]]

[[!redirects continuous linear map]]
[[!redirects continuous linear maps]]

[[!redirects linear continuous functional]]
[[!redirects linear continuous functionals]]

[[!redirects bounded linear functional]]
[[!redirects bounded linear functionals]]


linear functor

* This page is about functors between [[linear categories]].  For other notions of "linear functor," see [[Goodwillie calculus]] and [[polynomial functor]].

* There is a notion of map between [[linearly distributive categories]] called [[linearly distributive functor]], which is often shortened to "linear functor."

***

#Contents#
* table of contents
{:toc}

## Definition

A [[linear category]] is a category [[enriched category|enriched]] over [[Vect]], and similarly a **linear functor** is a [[functor]] [[enriched functor|enriched]] over $Vect$.

Unwrapping this a bit: given objects $x, y$ in a linear category $C$, the homset $hom(x,y)$ is equipped with the structure of a [[vector space]], and a functor $F: C \to D$ between linear categories is said to be **linear** if the map

$$ F: hom(x,y) \to hom(F(x), F(y)) $$

is linear for all $x,y \in C$.

Note that a linear functor between linear [[additive categories]] is automatically [[additive functor|additive]].

## Related concepts

* [[linear function]], [[linear equation]]

* [[smooth functor]]

[[!redirects linear functors]]


linear hyperdoctrine

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Linear hyperdoctrines

* table of contents
{: toc}

## Idea

A *linear hyperdoctrine* is a [[hyperdoctrine]] that is adapted for modeling first-order [[linear logic]].

## Definitions

Since there are many variants of linear logic, there are correspondingly many variants of linear hyperdoctrines, but all of them are some kind of [[indexed monoidal category]], or more precisely indexed monoidal poset.  We mention only a few of the most important.

+-- {: .un_defn}
###### Definition
A **MILL hyperdoctrine** is a closed indexed monoidal poset with [[indexed products]] and indexed coproducts satisfying the [[Beck-Chevalley condition]] and the [[Frobenius reciprocity]] condition.
=--

A MILL hyperdoctrine models predicate intuitionistic linear logic, with $\otimes,\mathbf{1},\multimap,\exists,\forall$.

+-- {: .un_defn}
###### Definition
A **MALL hyperdoctrine** is a MILL hyperdoctrine whose fibers are [[star-autonomous categories|*-autonomous]] [[lattices]] and whose reindexing functors preserve all the $\ast$-autonomous and lattice structure.
=--

A MALL hyperdoctrine models predicate classical linear logic without exponentials, with $\otimes,\mathbf{1},\parr,\bot,\&,\top,\oplus,\mathbf{0},\exists,\forall$.

+-- {: .un_defn}
###### Definition
A **linear-nonlinear hyperdoctrine** is a MALL hyperdoctrine $L$ together with a [[first-order hyperdoctrine]] $M$ and a fiberwise [[monoidal adjunction]] $F : M \rightleftarrows L : G$.
=--

A linear-nonlinear hyperdoctrine models full predicate classical linear logic, with the [[exponential modality]] modeled as the comonad $F G$ and $?$ as its de Morgan dual.

## References

*  [[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))
 {#Seely89}

[[!redirects linear hyperdoctrines]]


linear implication

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

# Contents
* table of contents
{: toc}

## Idea

Linear implication $D \multimap E$ is the most common version of [[implication]]/[[function type]] in [[linear logic]]/[[linear type theory]].

The [[categorical semantics]] of linear implication is as the [[internal hom]] in the non-[[cartesian closed category|cartesian]] [[closed monoidal category]] of [[types]] whose [[tensor product]] $\otimes$ corresponds to [[multiplicative conjunction]].

Closely related is the *multiplicative implication* of [[bunched implication]] logic, though they behave somewhat differently.

## Related concepts

* [[exponential conjunction]]

* [[function type]]

* [[hypothetical judgement]]

[[!include logic symbols -- table]]

## References

E.g.

* {#MihályiNovitzká13} Daniel Mihályi, Valerie Novitzká, p. 2, 5 of: *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]]]

For more see the references at *[[linear logic]]*.

[[!redirects multiplicative implication]]


linear independence > history

< [[linear independence]]


linear isometries operad

## Idea

An [[operad]].

## Applications

* [[S-module]]

## References

* Pelle Salomonsson, _The linear isometries operad in Lie--Tate homology_ ([arXiv:1007.3355](http://arxiv.org/abs/1007.3355))

[[!redirects linear isometries operads]].


linear isometry

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

#Contents#
* table of contents
{:toc}

## Idea

A *linear isometry* is a [[linear map]]

$$
  \phi
  \,\colon\,
  \mathscr{H}_1
  \longrightarrow
  \mathscr{H}_2
$$

between [[normed space|normed]] [[vector spaces]] which preserves the norm:

$$
  \big\Vert \psi(-) \big\Vert_2
  \;=\;
  \big\Vert - \big\Vert_1
  \,.
$$

In particular, when the norm is induced by ([[Hermitian form|Hermitian]]) [[inner products]] ${\langle \cdot \vert \cdot \rangle}_{i}$ (as on [[Hilbert spaces]])

$$
  {\big\Vert \psi \big\Vert}_i
  \,\equiv\,
  \sqrt{
     \left\langle \psi \vert \psi \right\rangle_i
  }
$$

then a linear isometry is an [[isometry]] in that it preserves these inner products:

$$
  \big\langle \phi(-) \vert \phi(-) \big\rangle_2
  \;=\;
  \left\langle - \vert - \right\rangle_1
$$

or equivalently, in terms of [[adjoint operators]]:

\[
  \label{InTermsOfAdjointOperators}
  \phi^\dagger \cdot \phi \,=\, Id_{\mathscr{H}_1}
  \,.
\]

\begin{remark}
**(relation to unitary operators)**
\linebreak
If in addition to (eq:InTermsOfAdjointOperators) also the reverse condition $\phi \cdot \phi^\dagger \,=\, Id_{\mathscr{H}_2}$ holds, then $\phi$ is called a *[[unitary operator]]*, which is the case iff $\phi$ is [[surjective map]].
\end{remark}

## Properties

\begin{proposition}
  \label{LinearIsometriesAreInjective}
  Linear isometries are [[injective maps]].
\end{proposition}
\begin{proof}
  For
  $\left\vert \psi \right\rangle, \left\vert \psi' \right\rangle \,\colon\, \mathscr{H}$ we need to show that
$$
  \phi\left\vert \psi \right\rangle
  \;=\;
  \phi\left\vert \psi' \right\rangle
  \;\;\;\;
  \Rightarrow
  \;\;\;\;
  \left\vert \psi \right\rangle
  \;=\;
  \left\vert \psi' \right\rangle
  \mathrlap{\,.}
$$
But by non-degeneracy of the norm this is equivalent to showing
$$
  {
  \Big\Vert
  \phi
  \big(
    \left\vert \psi \right\rangle
    -
    \left\vert \psi' \right\rangle
  \big)
  \Big\Vert
  }
  \,=\,
  0
  \;\;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;\;
  {
  \Big\Vert
    \left\vert \psi \right\rangle
    -
    \left\vert \psi' \right\rangle
  \Big\Vert
  }
  \,=\,
  0
  \mathrlap{\,.}
$$
But since $\phi$ is an isometry, here the left hand side already coincides with the right hand side and hence certainly implies it.
\end{proof}

## Related concepts

* [[isometry]]

* [[linear isomorphism]]

* [[unitary operator]]

## References

See also:

* ProofWiki, *[Linear Isometry](https://proofwiki.org/wiki/Definition:Linear_Isometry)*

* Wikipedia, *[Linear isometry](https://en.wikipedia.org/wiki/Isometry#Linear_isometry)*

[[!redirects linear isometries]]


linear isomorphism

+-- {: .rightHandSide}
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### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _linear isomophism_ is a [[linear map]] which is an [[isomorphism]] (in the given ambient [[category of modules]]/[[category of vector spaces]]), hence which has an [[inverse function]] that is also a linear map.

## Related concepts

* [[linear isometry]]

* [[unitary operator]]

[[!redirects linear isomorphisms]]


linear lambda-calculus

# Contents
* table of contents
{:toc}

## Idea

Under the [[Curry-Howard correspondence]], [[proofs]] of [[intuitionistic logic]] can be seen at [[lambda-terms|$\lambda$-terms]]. In the same way, one can see proofs of [[linear logic]] as [[terms]] of a linear $\lambda$-calculus. It allows one to have a more compact syntax for proofs of linear logic which can be more practical to analyze their computational properties. Under this correspondence, the [[cut-elimination]] of proofs corresponds to the [[beta-reduction|$\beta$-reduction]] of terms.