true proposition

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# Contents
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## Idea

In [[logic]], the __true__ proposition, or __truth__, is the [[proposition]] which is always true.

The truth is commonly denoted $true$, $T$, $\top$, or $1$.  These may be pronounced 'true' even where it would be ungrammatical for an adjective to appear in ordinary English.

## Definitions

In [[natural deduction]] the [[inference rules]] for true is given by

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \top \; \mathrm{prop}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Xi \vdash \top \; \mathrm{true}}$$

### In classical logic

In [[classical logic]], there are two [[truth values]]: true and [[false]].  Classical logic is perfectly symmetric between truth and falsehood; see [[de Morgan duality]].

### In constructive logic

In [[constructive logic]], $true$ is the [[top element]] in the [[poset]] of [[truth values]].

Constructive logic is still [[two-valued logic|two-valued]] in the sense that any truth value which is not true is [[false]].

### In linear logic

In [[linear logic]], there is both _additive_ truth, denoted $\top$, and _multiplicative_ truth, denoted $1$.  As the notation suggests, it is $\top$ that is the [[top element]] of the lattice of linear truth values.  (In particular, $1 \vdash \top$ but $\top \nvdash 1$.)

### In a topos

In terms of the [[internal logic]] of a [[topos]] (or other [[category]]), $true$ is the [[top element]] in the [[poset of subobjects]] of any given [[object]] (where each object corresponds to a [[context]] in the internal language).

However, not every topos is [[two-valued topos|two-valued]] (even if it is [[boolean topos|boolean]], so there may be other truth values besides $true$ and $false$.

### In type theory

In [[type theory]] with [[propositions as types]], truth is represented by the [[unit type]].

### In homotopy type theory

In [[homotopy type theory]], truth is represented by any [[contractible type]].

## Examples

### In the topos $Set$

In the archetypical [[topos]] [[Set]], the terminal object is the [[singleton]] [[set]] $\{*\}$ (the [[point]]) and the poset of subobjects of that is classically $\{\emptyset \hookrightarrow *\}$.  Then truth is the singleton set $\{*\}$, seen as the [[improper subset]] of itself.  (See [Internal logic of Set](http://ncatlab.org/nlab/show/internal+logic#LogicOfSet) for more details).

The same is true in the archetypical [[(∞,1)-topos]] [[∞Grpd]]. From that perspective it makes good sense to think of

* a set as a 0-[[truncated]] $\infty$-groupoid: a [[0-groupoid]];

* a [[subsingleton]] set as a $(-1)$-[[truncated]] $\infty$-groupoid: a [[(?1)-groupoid]];

* the singleton set as the $(-2)$-[[truncated]] $\infty$-groupoid: the unique (up to equivalence) [[(?2)-groupoid]].

In this sense, the object $true$ in [[Set]] or [[∞Grpd]] may canonically be thought of as being [[generalized the|the]] unique [[(?2)-groupoid]].

## Related concepts

* **true**, [[unit type]]

* [[false]], [[empty type]]

* [[contradiction]], [[inconsistency]], [[consistency]]

* [[correspondence theory of truth]]

[[!include homotopy n-types - table]]

[[!redirects true]]
[[!redirects True]]
[[!redirects true proposition]]
[[!redirects true propositions]]
[[!redirects true statement]]
[[!redirects true statements]]
[[!redirects true sentence]]
[[!redirects true sentences]]

[[!redirects truth]]
[[!redirects Truth]]
[[!redirects the truth]]


truncated object

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#### Higher category theory
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# Truncated objects
* table of contents
{: toc}

## Idea

A **$k$-truncated object** in an [[n-category]] is an object which "behaves [[internally]] like a $k$-category".  More precisely, since an object of an $n$-category can behave at most like an $(n-1)$-category, a $k$-truncated object behaves like a $min(k,n-1)$-category.  More generally, a **$(k,m)$-truncated object** in an [[(n,r)-category]] is an object which behaves internally like a $min((k,m),(n-1,r-1))$-category.

## Definition

Let $C$ be an $(n,r)$-category, where $n$ and $r$ can range from $-2$ to $\infty$ inclusive.  An object $x\in C$ is **$(k,m)$-truncated** if for all objects $a\in C$, the $(n-1,r-1)$-category $C(a,x)$ is in fact a $(k,m)$-category.

## Examples

* In a [[1-category]]:
  * every object is $0$-truncated,
  * the $(-1)$-truncated objects are the [[subterminal objects]], and
  * the $(-2)$-truncated objects are the [[terminal objects]].

* In a [[2-category]]:
  * every object is $1$-truncated,
  * the $0$-truncated objects are the [[discrete object in a 2-category|discrete objects]],
  * the $(1,0)$-truncated objects are the __groupoidal objects__, and
  * the $(0,1)$-truncated objects are the __posetal objects__.

* In an $(\infty,1)$-category, the $k$-truncated objects (which are automatically $(k,0)$-truncated) are also called **$k$-types**.  See [[n-truncated object of an (∞,1)-category]].

## Properties

### Reflectivity

If the $(n,r)$-category has sufficient [[familial regularity and exactness|exactness properties]], then the $(k,m)$-truncated objects form a [[reflective subcategory]].  More generally, in such a case there is a [[orthogonal factorization system|factorization system]] $(E,M)$ such that $M/1$ is the category of $(k,m)$-truncated objects.  (Note that this is *not* a [[reflective factorization system]], but it is often a [[stable factorization system]].)  For example:

* In any category with a [[terminal object]], the subcategory of terminal objects is reflective, and corresponds to the factorization system $E = $ all morphisms, $M=$ [[isomorphisms]].

* In a [[regular category]], the category of [[subterminal objects]] is reflective, and corresponds to the factorization system $E=$ [[regular epimorphisms]], $M=$ [[monomorphisms]].

* In a [[regular 2-category]], the same holds true, where $E$ is the class of regular $1$-epimorphisms ([[eso morphisms]]) and $M$ the class of $1$-monomorphisms ([[ff morphisms]]).  With additional exactness conditions, the categories of $0$-truncated, $(1,0)$-truncated, and $(0,1)$-truncated objects are also reflective; see [[michaelshulman:truncation in an exact 2-category|here]].

* In an [[(∞,1)-topos]], the $n$-truncated objects are reflective and we have the [[(n-connected, n-truncated) factorization system]].

## Related concepts

* [[(n+1,1)-category of n-truncated objects]]

* [[n-truncated object of an (∞,1)-category]]

* [[n-truncation modality]]

* [[decategorification]]

[[!redirects truncated object]]
[[!redirects truncated objects]]
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[[!redirects truncated objects in an n-category]]
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[[!redirects k-truncated objects in an n-category]]
[[!redirects (k,m)-truncated object in an (n,r)-category]]
[[!redirects (k,m)-truncated objects in an (n,r)-category]]

[[!redirects groupoidal object]]
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[[!redirects posetal objects]]
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[[!redirects posetal objects in a 2-category]]


truncation

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* table of contents
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## Idea

In [[homotopy theory]] and [[homotopy type theory]], by *truncation* one means the [[reflective subcategory|reflection]] onto [[truncated objects]].

## Related entries

* For discussion in  [[homotopy type theory]] (mostly) see:

  * [[h-level]]

  * [[contractible type]], [[mere proposition]], [[h-set]], [[h-groupoid]]

  * [[cone type]] or (-2)-truncation

  * [[propositional truncation]] or (-1)-truncation

  * [[set truncation]] or 0-truncation

  * [[n-truncation modality]]

* For discussion in [[higher category theory]], see:

  * [[truncated object]]

  * [[homotopy n-type]]

  * [[Postnikov tower]]

  * [[truncation of a chain complex]]

  * [[n-truncated object of an (infinity,1)-category]]

  * [[(n+1,1)-category of n-truncated objects]]

  * [[(n-connected, n-truncated) factorization system]]

[[!redirects truncated]]
[[!redirects truncation]]
[[!redirects truncations]]

[[!redirects (-2)-truncated]]
[[!redirects (-2)-truncation]]
[[!redirects (-2)-truncations]]

[[!redirects (-1)-truncated]]
[[!redirects (-1)-truncation]]
[[!redirects (-1)-truncations]]

[[!redirects 0-truncated]]
[[!redirects 0-truncation]]
[[!redirects 0-truncations]]

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[[!redirects 1-truncation]]
[[!redirects 1-truncations]]

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[[!redirects 2-truncation]]
[[!redirects 2-truncations]]

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[[!redirects 3-truncation]]
[[!redirects 3-truncations]]

[[!redirects 4-truncated]]
[[!redirects 4-truncation]]
[[!redirects 4-truncations]]

[[!redirects n-truncated]]
[[!redirects n-truncation]]
[[!redirects n-truncations]]

[[!redirects k-truncated]]
[[!redirects k-truncation]]
[[!redirects k-truncations]]


truncation of a chain complex

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## Definition

For $C_\bullet$ a [[chain complex]], the _truncation_ $(\tau_{\leq} C)_\bullet$ at some $n \in \mathbb{N}$ is the chain complex defined by

$$
  (\tau_n C)_i =
  \left\{
    \array{
      0 & | i \gt  n \\
      C_n/B_n & | i = n \\
      C_n & | i \lt n
    }
  \right.
 \,,
$$

where $B_n = im(d_n)$.

For connective chain complexes this is the notion of [[truncated object in an (infinity,1)-category]] realized in the [[(infinity,1)-category of chain complexes]].

## References

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_


truss

## Idea

While a [[ring]] is a set with two operations, (Abelian) group with respect to one and semigroup with respect to another with a standard two-sided distributivity law, there are also interesting examples of a "brace-like" distributivity law instead. Truss is a formalization/generalization of the usage of the alternative distributivity which uses a [[heap]] as a weakening of a concept of a group. Trusses appear in a number of subjects including the study of solutions to [[quantum Yang-Baxter equation]], ideal ring extensions, [[Hopf-Galois extension]]s, [[inverse semigroup]]s etc.

If we drop Abelianess of the "additive" group then we talk about a more general notion of a skew truss (which is therefore to [[near-ring]] roughly what a truss is to a ring).

## Definition

A __left truss__ is a [[heap]] $(A,[.,.,.])$ together with another "multiplicative" operation which distributes with the ternary operation of the heap from the left
$$
a\cdot [b,c,d] = [a\cdot b,a\cdot c,a\cdot d]
$$
A [[truss]] is a left and right truss.

A left truss is a __left [[brace]]__ if it is Abelian group under the multiplicative operation.

Alternatively, we can use the binary operations only, together with some sort of a cocycle (measuring nonstandardness of distributive law). So let $(A,+)$ be a group (though additive, not necessarily commutative!) and $(A,\cdot)$ a [[semigroup]].
As usually, consider the associated heap operation of $(A,+)$,
$$
[b,c,d] = b - c + d
$$
(where $-$ means adding, in this order, the additive inverse).
Then the following are equivalent:

1. $(A,[.,.,.])$ is a truss

2. $\exists \sigma: A\to A$ such that $a\cdot(b+c) = (a\cdot b)-\sigma(a)+(a\cdot c)$

3. $\exists\lambda : A\times A\to SA$ such that
$a\cdot(b+c) = (a\cdot b) + \lambda(a,c)$

4. $\exists\mu : A\times A\to SA$ such that
$a\cdot(b+c) = \mu(a,b)+(a\cdot c)$

5. $\exists\kappa,\tilde\kappa$ such that $a\cdot(b+c) = \kappa(a,b)+\tilde\kappa(a,c)$

## Examples

* All even numbers form a (nonunital) subring of the ring of integers. Another coset, the class $2\mathbb{Z}+1$ of all odd numbers is closed under multiplication, but not under addition. However, if we introduce the new addition $a\oplus b = a - 1 + b$ then
$2\mathbb{Z}+1$ is closed under both operation and the brace-like distributive law holds $a\cdot(b \oplus c) = a\cdot b \ominus a \oplus a\cdot c$.

## Properties

## Literature

* [[Tomasz Brzeziński]], _Trusses: Between braces and rings_, Trans. Amer. Math. Soc. __372__ (2019), 4149-4176 [doi](https://doi.org/10.1090/tran/7705); preprint  [pdf](https://core.ac.uk/download/pdf/187088098.pdf)

* T. Brzeziński, _Enter truss_, Northatlantic NCG seminar 2021 (video 2hr [yt](https://www.youtube.com/watch?v=F0MiZMxjQlk))

* Tomasz Brzeziński, Bernard Rybołowicz, _Modules over trusses vs modules over rings: direct sums and free modules_, Algebra and Representation Theory 25, 1–23 (2022) [doi](https://doi.org/10.1007/s10468-020-10008-8)

* Ryszard R. Adruszkiewicz, Tomasz Brzeziński, Bernard Rybołowicz, _Ideal ring extensions and trusses_, [arXiv:2101.09484](https://arxiv.org/abs/2101.09484)

category: algebra


truth value

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# Contents
* table of contents
{: toc}

## Idea

[[classical mathematics|Classically]], a __truth value__ is either $\top$ ([[true]]) or $\bot$ ([[false]]), hence an [[element]] of the [[boolean domain]].

(In [[constructive mathematics]], this is not so simple, although it still holds that any truth value that is not true is false.)

More generally, a __truth value__ in a [[topos]] $T$ is a morphism $1 \to \Omega$ (where $1$ is the [[terminal object]] and $\Omega$ is the [[subobject classifier]]) in $T$.  By definition of $\Omega$, this is equivalent to an (equivalence class of) monomorphisms $U\hookrightarrow 1$.  In a [[two-valued topos]], it is again true that every truth value is either $\top$ or $\bottom$, while in a [[Boolean topos]] this is true in the [[internal logic]].

Truth values form a [[partial order|poset]] (the __poset of truth values__) by declaring that $p$ precedes $q$ iff the [[conditional]] $p \to q$ is true.  In a topos $T$, $p$ precedes $q$ if the corresponding subobject $P\hookrightarrow 1$ is contained in $Q\hookrightarrow 1$.  Classically (or in a two-valued topos), one can write this poset as $\{\bot \to \top\}$.

The poset of truth values is a [[Heyting algebra]]. Classically (or internal to a Boolean topos), this poset is even a [[Boolean algebra]]. It is also a [[complete lattice]]; in fact, it can be characterised as the [[initial object|initial]] complete lattice. As a complete Heyting algebra, it is a [[frame]], corresponding to the one-point [[locale]].

When the set of truth values is equipped with the [[specialization topology]], the result is [[Sierpinski space]].

A truth value may be interpreted as a $0$-[[0-poset|poset]] or as a $(-1)$-[[(-1)-groupoid|groupoid]]. It is also the best interpretation of the term '$(-1)$-[[(-1)-category|category]]', although this doesn\'t fit all the patterns of the [[periodic table]].

## Related concepts

* [[boolean domain]]

* [[bit]], [[qbit]]

* [[logic gate]]

* In [[synthetic topology]] with a [[dominance]], some truth values are [[open truth value|open]].

* [[semi-decidable proposition]]

[[!include homotopy n-types - table]]

[[!redirects truth value]]
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[[!redirects truth-value]]
[[!redirects truth-values]]

[[!redirects set of truth values]]
[[!redirects proset of truth values]]
[[!redirects poset of truth values]]
[[!redirects lattice of truth values]]
[[!redirects frame of truth values]]
[[!redirects algebra of truth values]]


Tsit-Yuen Lam

[[!redirects Tsit-Tuen Lam]]

* [personal page](https://math.berkeley.edu/~lam/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Tsit_Yuen_Lam)

## Selected writings

Introducing the [[Artin-Lam induction exponent]]:

* {#Lam68} [[Tsit-Yuen Lam]], *Artin exponent of finite groups*, Journal of Algebra **9** (1968) 94-119 (doi:10.1016/0021-8693(68)90007-0, [pdf](https://core.ac.uk/download/pdf/82635891.pdf))

## Related $n$Lab entries

* [[divisible group]]

category: people

[[!redirects Tsit Yuen Lam]]


Tsuguhiko Asakawa

* [inspire page](https://inspirehep.net/authors/1018018)

## Selected writings

Towards an enhancement of the [[BFSS matrix model]] by [[topological K-theory]] to account for the [[K-theory classification of D-brane charge]] ([[K-matrix model]]):

* {#AsakawaSugimotoTerashima01} [[Tsuguhiko Asakawa]], [[Shigeki Sugimoto]], [[Seiji Terashima]], *D-branes, Matrix Theory and K-homology*, JHEP 0203 (2002) 034 [[arXiv:hep-th/0108085](https://arxiv.org/abs/hep-th/0108085), [doi:10.1088/1126-6708/2002/03/034](https://doi.org/10.1088/1126-6708/2002/03/034)]

Relating [[Dirac structures]] to [[D-branes]]:

* {#AsakawaSasaWatamura} [[Tsuguhiko Asakawa]], Shuhei Sasa, Satoshi Watamura, _D-branes in Generalized Geometry and Dirac-Born-Infeld Action_ [[arXiv:1206.6964](http://arxiv.org/abs/1206.6964)]

category: people


TT deformation

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#Contents#
* table of conents
{:toc}

## Idea

A _$T \bar T$-deformation_ of a [[conformal field theory]] is a deformation induced by the term $T \bar T$ quadratic in the [[energy-momentum tensor]]

## References

### General

The original discussion for [[2d CFT]]:

* [[Alexander Zamolodchikov]], _Expectation value of composite field $T \bar T$ in two-dimensional quantum field theory_ ([arXiv:hep-th/0401146](https://arxiv.org/abs/hep-th/0401146))

Generalization to [[CFT]] of arbitary dimension

* Marika Taylor, _TT deformations in general dimensions_ ([arXiv:1805.10287](https://arxiv.org/abs/1805.10287))

Review:

* [[John Cardy]], _The $T \bar T$ deformation of quantum field theory_, 2018 ([pdf](https://icmp2018.org/sites/icmp/files/slides/0900_cardy_symposia_thursday.pdf), [[CardyTTDeformation.pdf:file]])

* Yunfeng Jiang, _Lectures on solvable irrelevant deformations of 2d quantum field theory_ ([arXiv:1904.13376](https://arxiv.org/abs/1904.13376))

* [[Monica Guica]], _$T\bar T$ deformations and holography: review and open questions_, talk at *[[Strings 2022]]* [[indico:4940816](https://indico.cern.ch/event/1085701/contributions/4940816/), [slides](https://indico.cern.ch/event/1085701/contributions/4940816/attachments/2481655/4260285/slides.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=bfe514d2-b4c8-4f03-a2d1-33ee630b74f6)]

See also

* George Jorjadze, [[Stefan Theisen]], _Canonical maps and integrability in $T \bar T$ deformed 2d CFTs_ ([arXiv:2001.03563](https://arxiv.org/abs/2001.03563))

* [[John Cardy]], Benjamin Doyon, _$T \bar T$deformations and the width of fundamental particles_ ([arXiv:2010.15733](https://arxiv.org/abs/2010.15733))

### In $AdS_3/CFT_2$

The interpretation of the $T \bar T$-deformation in [[AdS3-CFT2 duality]] is due to

* Lauren McGough, Márk Mezei, [[Herman Verlinde]], _Moving the CFT into the bulk with $T \bar T$_, J. High Energ. Phys. (2018) 2018: 10 ([arXiv:1611.03470](https://arxiv.org/abs/1611.03470))

### Of the DBI action

Proposal of a non-abelian [[DBI-action]] via $T \bar T$-deformation:

* T. Daniel Brennan, Christian Ferko, [[Savdeep Sethi]], _A Non-Abelian Analogue of DBI from $T \bar T$ ([arXiv:1912.12389](https://arxiv.org/abs/1912.12389))

[[!redirects TT deformations]]

[[!redirects TT-deformation]]
[[!redirects TT-deformations]♧

[[!redirects T-barT deformation]]
[[!redirects T-barT deformations]]


tube lemma

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## Idea

The **tube lemma** may refer to one of several fundamental lemmas in topology ([[general topology|point-set topology]] to be more exact[^1]) that underlie many arguments involving [[compact spaces]]. That is to say: much like the [[Yoneda lemma]] of pure [[category theory]], there are various closely related statements that can be said to fall under the rubric "tube lemma"; we consider several examples here.

[^1]: The use of the phrase "point-set topology" signals that the tube lemma proper does refer to *points*, as opposed to the "point-free" or "pointless" topology as developed in the theory of [[locales]]. Point-free analogues of the main consequences of the tube lemma, such as the fact that the product of two compact topological spaces is again compact, are quite nontrivial and require a significantly different approach.

## Statements and proofs
  {#StatementAndProofs}

+-- {: .num_prop #ProjectingOutCompactSpaceIsClosedMap}
###### Proposition
**([[closed-projection characterization of compactness]])

If $X$ is any [[topological space|space]] and $Y$ is a [[compact topological space|compact]], then the [[projection map]] $p: X \times Y \to X$ out of the [[product topological space]] is a [[closed map]], i.e., under [[direct image]] (the [[left adjoint]] $\exists_p: P (X \times Y) \to P(X)$ to the [[inverse image]] map $p^\ast: P(X) \to P(X \times Y)$), [[closed sets]] in $X \times Y$ are mapped to closed sets in $X$.

=--

Assuming the [[principle of excluded middle]], the statement of prop. \ref{ProjectingOutCompactSpaceIsClosedMap} is equivalent to an alternative formulation:

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

let $\forall_p: P(X \times Y) \to P(X)$ be the [[right adjoint]] to $p^\ast: P(X) \to P(X \times Y)$. (In [[classical logic]], $\forall_p = \neg \exists_p \neg$.) Then

* If $Y$ is compact, then $\forall_p: P(X \times Y) \to P(X)$ takes [[open sets]] in $X \times Y$ to open sets in $X$. (Compare [[overt space]].)

=--

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

This alternative may be proven directly as follows. Let $W \subseteq X \times Y$ be an open set, and suppose $\{x\} \subseteq \forall_p(W)$. This means precisely that $p^\ast\{x\} = \{x\} \times Y \subseteq W$. We are required to show that there is a neighborhood $O$ of $x$ such that $O \subseteq \forall_p(W)$, which is to say $O \times Y \subseteq W$.

=--

This $O \times Y$ may be pictured as a little open "tube" around $\{x\} \times Y$ which fits inside $W$, thus explaining the name of the eponymous lemma.

+-- {: .num_lemma #tube}
###### Lemma
**(Tube lemma)**

Let $X$ be a [[topological space]] and let $Y$ be a [[compact topological space]]. For a point $x \in X$ and $W$ open in $X \times Y$, if $\{x\} \times Y \subseteq W$, then there is an open neighborhood $O$ of $x$ such that the "tube" $O \times Y$ around $\{x \} \times Y$ is still contained: $O \times Y \subseteq W$.

=--

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

Let $\mathcal{O}_x$ denote the set of open neighborhoods of $x$, and for $U \in \mathcal{O}_x$, let $U\multimap W$ be the largest open $V$ of $Y$ such that $U \times V \subseteq W$. The collection $\{U \multimap W: U \in \mathcal{O}_x\}$ is an open cover of $Y$; since $Y$ is compact, there is a finite subcover $U_1 \multimap W, U_2 \multimap W, \ldots, U_n \multimap W$. Then $O = U_1 \cap \ldots \cap U_n \in \mathcal{O}_x$ is the desired open. For, $O \subseteq U_i$ implies $(U_i \multimap W) \subseteq (O \multimap W)$, so that

$$Y = \bigcup_{i=1}^n (U_i \multimap W) \subseteq O \multimap W$$

and this implies $O \times Y \subseteq O \times (O \multimap W) \subseteq W$.

=--

+-- {: .proof #ProofOfTubeLemmaViaClosedProjectionCharacterization}
###### Proof
(of the tube lemma \ref{tube} via [[closed-projection characterization of compactness]])

Let

$$
  C \coloneqq (X \times Y) \backslash W
$$

be the [[complement]] of $W$. Since this is closed, by prop. \ref{ProjectingOutCompactSpaceIsClosedMap} also its projection $p_X(C) \subset X$ is closed.

Now

$$
  \begin{aligned}
    \{x\} \times Y \subset W
      & \;\Leftrightarrow\;
    \{x\} \times Y \; \cap \; C = \emptyset
    \\
    & \;\Rightarrow\;
    \{x\} \cap  p_X(C) = \emptyset
  \end{aligned}
$$

and hence by the closure of $p_X(C)$ there is (by [this lemma](Introduction+to+Topology+--+1#UnionOfOpensGivesClosure)) an open neighbourhood $U_x \supset \{x\}$ with

$$
  U_x \cap p_X(C) = \emptyset
  \,.
$$

This means equivalently that $U_x \times Y \cap C = \emptyset$, hence that $U_x \times Y \subset W$.

=--

## Consequences

### Tychnoff theorem

+-- {: .num_cor}
###### Corollary
**(binary [[Tychonoff theorem]])**

The [[product topological space]] $X \times Y$ of two [[compact topological spaces]] $X, Y$ is itself compact.

=--

+-- {: .proof}
###### Proof
Let $\mathcal{U}$ be any open cover of $X \times Y$, and let $\mathcal{B}$ be the collection of opens of $X \times Y$ that are unions of finitely many elements of $\mathcal{U}$. For each $x \in X$, we have that $\{x\} \times Y$ is compact since $Y$ is, so there is $B \in \mathcal{B}$ such that $\{x\} \times Y \subseteq B$, whence there is $U \in \mathcal{O}_x$ with $U \times Y \subseteq B$ by the tube lemma.

It follows that the collection

$$\{U\; open\; in\; X: U \times Y \subseteq B \; for\; some\; B \in \mathcal{B}\}$$

covers $X$, whence by compactness of $X$ there is a finite subcover $U_1, \ldots, U_n$ for which $U_i \times Y \subseteq B_i$ for some $B_i \in \mathcal{B}$, and then $B = B_1 \cup \ldots \cup B_n$ belongs to $\mathcal{B}$ and is all of $X \times Y$.
=--

### Exponentiability of the compact-open topology

The tube lemma may be used to show that the [[mapping space]] from a [[locally compact topological space]] to any topology spaces equipped with the [[compact-open topology]] is an [[exponential object]] in the [[category]] [[Top]] of topological spaces. See [there](compact-open+topology#Exponentiability).

## Converse of the tube lemma

As explained [above](#StatementAndProofs), the three statements

1. Tube lemma, Lemma \ref{tube}

1. $\forall_p: P(X \times Y) \to P(X)$ preserves openness if $X$ is compact

1. $\exists_p: P(X \times Y) \to P(X)$ preserves closedness if $X$ is compact (prop. \ref{ProjectingOutCompactSpaceIsClosedMap})

are virtually tautologically equivalent, and in this sense any one of these forms may be referred to as the tube lemma. In this section, we indicate that the converse of the tube lemma holds, stated as follows.

+-- {: .num_theorem #converse}
###### Theorem
**([[closed-projection characterization of compactness]])**
If for all spaces $X$ the projection $p: X \times Y \to X$ is closed (form 3), then $Y$ is compact.
=--

Various proofs may be given. If one likes the characterization of compactness that says every [[ultrafilter]] converges, then a neat conceptual proof runs as follows: given a space $Y$ satisfying the hypothesis, take $X$ to be the [[Stone-Cech compactification]] $\beta Y$ of the discrete space $Y_d$ on the underlying set of $Y$, and let $C \subseteq \beta Y \times Y$ be the convergence relation (i.e., $(U, y)$ belongs to $C$ iff the ultrafilter $U$ converges to $y$). One checks that $C$ is closed. Then the direct image $p(C)$ is closed by hypothesis. Notice also that $p(C)$ contains the set of principal ultrafilters $prin(y)$ ($prin(y)$ converges to $y$ after all), and this set may be identified with the dense set $Y_d \subset \beta Y$. Being closed and dense, $p(C)$ is all of $\beta Y$, but this simply says that every ultrafilter on $Y$ converges, so that $Y$ is compact.

The characterization of compactness via ultrafilter convergence has a slight drawback of relying on the [[ultrafilter theorem]], a kind of choice principle. If one doesn't like this, there are various workarounds that avoid it, but let it be said that all proofs (some of which are collected [here](#Trim)) have basically the same intuitive character. Given $Y$, one forms a space $X$ by adjoining one or more ideal points to the discrete space $Y_d$, where the open sets around an ideal point correspond to elements of some filter $F$ that we want to show clusters around, or converges to, some point $y$ (in order to show compactness of $Y$). Just as for the convergence relation in the proof sketched above, one passes to the closure $C$ of the diagonal $Y \to Y_d \times Y \hookrightarrow X \times Y$. The image $p(C)$ is closed and contains the dense subset $Y_d$, and so contains any ideal point $p$, and thus $(p, y) \in C$ for some $y$. This turns out to mean $F$ converges or clusters to $y$, as desired.

Here is a more precise enactment of one such proof.

+-- {: .proof}
###### Proof
Let $Y$ satisfy the hypothesis of Theorem \ref{converse}; to prove $Y$ is compact, suppose $\mathcal{C}$ is some collection of closed sets such that every finite intersection of elements of $\mathcal{C}$ is [[inhabited set|inhabited]]. Let $X = Y_d \sqcup \{\infty\}$, formed by adjoining an ideal point $\infty$ to the discrete space $Y_d$ on the underlying set of $Y$, and stipulating that whenever $K \in \mathcal{C}$, the set $K \sqcup \{\infty\}$ is an open neighborhood of $\infty$. The topology thus generated consists of arbitrary subsets $U \subseteq Y$ together with sets $F \sqcup \{\infty\}$ where $F$ belongs to the filter generated by $\mathcal{C}$. Note that the finite intersection property of $\mathcal{C}$ guarantees that the filter is *proper*, meaning in particular $\infty$ is not an open point, equivalently that the closure of $Y_d$ in $X$ is all of $X$.

We have a diagonal embedding $Y \stackrel{\Delta}{\to} Y_d \times Y \hookrightarrow X \times Y$; let $C$ be the closure of $Y$ in $X \times Y$. Of course $p(C) \subseteq X$ contains $Y_d$, and is closed by hypothesis, so as we just observed, $p(C) = X$. So $\infty \in p(C)$; this means that $(\infty, y) \in C$ for some $y \in Y$. By closure of $C$, for each $K \in \mathcal{C}$ the neighborhood $(K \sqcup \{\infty\}) \times U$ of $(\infty, y)$ meets $\Delta(Y) \subseteq X \times Y$. This just says every open $U \in \mathcal{O}_y$ intersects $K$; since $K$ is closed in $Y$, this means $y \in K$. Thus

$$y \in \bigcap_{K \in \mathcal{C}} K$$

and the non-emptiness of this intersection proves that $Y$ is compact.
=--

## References

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

* [[Todd Trimble]], *Characterizations of compactness*, [nLab link](https://ncatlab.org/toddtrimble/published/Characterizations+of+compactness).


tubular neighborhood

[[!redirects tubular neighbourhood]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

For $i : X \hookrightarrow Y$ an [[embedding]] of [[manifold]]s, a **tubular neighbourhood** of $X$ in $Y$ is

* a real [[vector bundle]] $E \to X$;

* an extension of $i$ to an [[isomorphism]]

  $$
    \hat i : E \to U_{i(X)}
  $$

  with an [[open neighbourhood]] of $X$ in $Y$.

=--

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

The [[derivative]] of $\hat i$ provides an [[isomorphism]] of $E$ with the [[normal bundle]] $\nu_{X/Y}$ of $X$ in $Y$.

=--

## Properties

### General

+-- {: .num_prop}
###### Proposition
**(tubular neighbourhood theorem)**

Every [[embedding]] does admit a tubular neighbourhood.

=--

For instance ([DaSilva, theorem 3.1](#DaSilva)).

Moreover, tubular neighbourhoods are unique _up to [[homotopy]]_ in a suitable sense:

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

For an embedding $i : X \to Y$, write $Tub(i)$ for the [[topological space]] whose underlying set is the set of tubular neighbourhoods of $i$ and whose [[topology]] is the [[subspace topology]] of $Hom(N_i X, Y)$ equipped with the [[C-infinity topology]].

=--

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

If $X$ and $Y$ are [[compact space|compact]] [[manifold]]s, then $Tub(i)$ is [[contractible]] for all [[embedding]]s $i : X \to Y$.

=--

This appears as ([Godin, prop. 31](#Godin)).

### Pullbacks of tubular neighbourhoods

(...) [[propagating flow]] (...) ([Godin](#Godin)).

## Related concepts

* [[tubular neighbourhood of a mapping space]]

* key application: [[Pontrjagin-Thom collapse map]]

## References

* {#DaSilva} [[Ana Cannas da Silva]], §3 of: _Prerequisites from differential geometry_ [[pdf](http://www.math.princeton.edu/~acannas/appendix_dg.pdf)]

* [[Ana Cannas da Silva]], §6.2 in: *Lectures on Symplectic Geometry*, Lecture Notes in Mathematics **1764**, Springer (2008) [[doi:10.1007/978-3-540-45330-7](https://doi.org/10.1007/978-3-540-45330-7), [pdf](https://www.math.tecnico.ulisboa.pt/~acannas/Books/lsg.pdf)]

The homotopical uniqueness of tubular neighbourhoods is discussed in

* {#Godin} [[Veronique Godin]], _Higher string topology operations_ (2007)([arXiv:0711.4859](http://arxiv.org/abs/0711.4859))

For an analogue in homotopical algebraic geometry see

* [[Marc Levine]], _Motivic tubular neighborhoods_, [pdf](http://www.uni-due.de/~bm0032/publ/TubNbdDocumenta.pdf)

See also

* Wikipedia, _[Tubular neighborhood](http://en.wikipedia.org/wiki/Tubular_neighborhood)_

[[!redirects tubular neighborhoods]]

[[!redirects tubular neighbourhood]]
[[!redirects tubular neighbourhoods]]

[[!redirects tubular neighbourhood theorem]]

[[!redirects tubular neighborhood theorem]]


tubular neighborhood theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

Let $M$ be a [[smooth manifold]]. By an ([[embedding|embedded]]) **[[submanifold]]** we mean a [[immersion of smooth manifolds|smooth immersion]] of [[smooth manifolds]] $i \colon X \to M$ that is a topological embedding of $X$ as a [[closed subspace]] of $M$.

In that case, we have that for each $x \in X$, the [[tangent space]] $T_x X$ is included in the subspace $T_{i(x)} M$. We define the normal fiber $N_x$ to be the quotient $T_{i(x)} M/T_x X$ and the **[[normal bundle]]** (with respect to the embedding $i$) to be the space $N X$ consisting of pairs $\{(x, v): x \in X, v \in N_x\}$, forming a vector bundle over $X$ in an evident way. We let $i_0 \colon X \to N X$ denote the zero section. An [[open neighborhood]] $V$ of the zero section is **convex** if its intersection with $N_x$ is a [[convex subset]] of the vector space $N_x$.

+-- {: .num_theorem}
###### Theorem
**(Tubular Neighborhood theorem)**

For any [[submanifold]] $i \colon X \hookrightarrow M$, there is an [[open neighborhood]] $U$ of $i(X)$ in $M$ and a convex open neighborhood $V$ of $i_0(X)$ in $N X$ and a [[diffeomorphism]] $\phi: U \to V$ such that the diagram

$$
  \array{
    X & &
    \\
    \mathllap{i} \downarrow & \searrow{}^{\mathrlap{i_0}} &
    \\
    U & \underset{\phi}{\longrightarrow} & V
}
$$

[[commuting diagram|commutes]]. Such $U$ is called a **[[tubular neighbourhood]]** of $i(X)$.

=--

See for instance ([Silva 06, theorem 6.5](#Silva06))

## References

* {#Silva06} [[Ana Cannas da Silva]], §6.2 in: *Lectures on Symplectic Geometry*, Lecture Notes in Mathematics **1764**, Springer (2008) [[doi:10.1007/978-3-540-45330-7](https://doi.org/10.1007/978-3-540-45330-7), [pdf](https://www.math.tecnico.ulisboa.pt/~acannas/Books/lsg.pdf)]

[[!redirects tubular neighbourhood theorem]]


tubular neighbourhood

[[!redirects tubular neighbourhood]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

For $i : X \hookrightarrow Y$ an [[embedding]] of [[manifold]]s, a **tubular neighbourhood** of $X$ in $Y$ is

* a real [[vector bundle]] $E \to X$;

* an extension of $i$ to an [[isomorphism]]

  $$
    \hat i \;\colon\; E \to U_{i(X)}
  $$

  with an [[open neighbourhood]] of $X$ in $Y$.

=--

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

The [[derivative]] of $\hat i$ provides an [[isomorphism]] of $E$ with the [[normal bundle]] $\nu_{X/Y}$ of $X$ in $Y$.

=--

## Properties

### General

+-- {: .num_prop}
###### Proposition
**([[tubular neighbourhood theorem]])**

Every [[embedding]] of [[smooth manifolds]] does admit a tubular neighbourhood, def. \ref{TubularNeighbourhood}.

=--

For instance ([DaSilva, theorem 3.1](#DaSilva)).

Moreover, tubular neighbourhoods are unique _up to [[homotopy]]_ in a suitable sense:

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

For an embedding $i : X \to Y$, write $Tub(i)$ for the [[topological space]] whose underlying set is the set of tubular neighbourhoods of $i$ and whose [[topology]] is the [[subspace topology]] of $Hom(N_i X, Y)$ equipped with the [[C-infinity topology]].

=--

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

If $X$ and $Y$ are [[compact space|compact]] [[manifold]]s, then $Tub(i)$ is [[contractible]] for all [[embedding]]s $i : X \to Y$.

=--

This appears as ([Godin, prop. 31](#Godin)).

### Equivariant version
 {#PropertiesEquivariant}

These statements generalize to [[equivariant differential topology]]:

+-- {: .num_prop #FixedLociOfSmoothProperActionsAreSubmanifolds}
###### Proposition
**([[fixed loci]] of [[smooth function|smooth]] [[proper actions]] are [[submanifolds]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]].

Then the $G$-[[fixed locus]] $X^G \hookrightarrow X$ is a [[smooth manifold|smooth]] [[submanifold]].

If in addition $X$ is equipped with a [[Riemannian metric]] and $G$ acts by [[isometries]] then the [[submanifold]] $X^G$ is a [[totally geodesic submanifold]].

=--

(e.g. [Ziller 13, theorem 3.5.2](#Ziller13), see also [this MO discussion](https://math.stackexchange.com/a/1739784/58526))

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

Let $x \in X^G \subset X$ be any [[fixed point]]. Since this is in particular a closed invariant [[submanifold]], Prop. \ref{ExistenceOfGInvariantTubularNeighbourhoods} applies and shows that an [[open neighbourhood]] of $x$ in $X$ is $G$-equivariantly [[diffeomorphism|diffeomorphic]] to a [[linear representation]] $V \in RO(G)$. The [[fixed locus]] $V^G \subset V$ of that is hence [[diffeomorphism|diffeomorphic]] to an [[open neighbourhood]] of $x$ in $\Sigma$.

=--

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

Without the assumption of [[proper action]] in Prop. \ref{FixedLociOfSmoothProperActionsAreSubmanifolds} the conclusion generally fails. See [this MO comment](https://math.stackexchange.com/a/1739768/58526) for a counter-example.

=--

+-- {: .num_prop #GActionOnNormalBundleToFixedLocus}
###### Proposition/Definition
**($G$-action on [[normal bundle]] to [[fixed locus]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]].

Then linearization of the $G$-action aroujnd the [[fixed locus]] $X^G \subset X$ equips the [[normal bundle]] $N_X\left( X^G\right)$ with [[smooth function|smooth]] and [[fiber]]-wise [[linear map|linear]]  $G$-[[action]].

=--

(e.g. [Crainic-Struchiner 13, Example 1.7](#CrainicStruchiner13))

+-- {: .num_prop #ExistenceOfGInvariantRiemannianMetrics}
###### Proposition
**(existence of $G$-[[invariant Riemannian metric]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[compact Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]].

Then there exists a [[Riemannian metric]] on $X$ with its [[invariant]] with respect to the $G$-[[action]], hence such that all elements of $G$ act by [[isometries]].

=--

([Bredon 72,VI Theorem 2.1](#Bredon72), see also [Ziller 13, Theorem 3.0.2](#Ziller13))

+-- {: .num_defn #GInvariantTubularNeighbourhood}
###### Definition
**($G$-[[equivariant tubular neighbourhood]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]].

For $\Sigma \subset X^G \subset X$ a [[closed manifold|closed]] [[smooth manifold|smooth]] [[submanifold]] inside the [[fixed locus]], a _$G$-invariant [[tubular neighbourhood]]_ $\mathcal{N}(\Sigma \subset X)$ of $\Sigma$ in $X$ is

1. a [[smooth vector bundle]] $E \to \Sigma$ equipped with a [[fiber]]-wise [[linear map|linear]] $G$-[[action]];

1. an equivariant [[diffeomorphism]] $E \overset{}{\longrightarrow} X$ onto an [[open neighbourhood]] of $\Sigma$ in $X$ which takes the [[zero section]] identically to $\Sigma$.

=--

+-- {: .num_prop #ExistenceOfGInvariantTubularNeighbourhoods}
###### Proposition
**(existence of $G$-[[equivariant tubular neighbourhoods]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]].

If $\Sigma \overset{\iota}{\hookrightarrow} X$ is a [[closed manifold|closed]] [[smooth manifold|smooth]] [[submanifold]] inside the $G$-[[fixed locus]]

\begin{center}
  \begin{xymatrix}
    \Sigma
    \ar@{^{(}->}[rr]^-{\iota^G}
    \ar@{^{(}->}[dr]_{\iota}
    &&
    X^G
    \ar@{^{(}->}[dl]
    \\
    &
    X
  \end{xymatrix}
\end{center}

then

1. $\Sigma$ admits a $G$-invariant [[tubular neighbourhood]] $\Sigma \subset U \subset X$ (Def. \ref{GInvariantTubularNeighbourhood});

1. any two choices of such $G$-invariant tubular neighbourhoods are $G$-equivariantly [[isotopy|isotopic]];

1. there always exists an $G$-invariant tubular neighbourhood parametrized specifically by the [[normal bundle]] $N(\Sigma \subset X)$ of $Sigma$ in $X$, equipped with its induced $G$-[[action]] from Def. \ref{GActionOnNormalBundleToFixedLocus}, and such that the $G$-equivariant [[diffeomorphism]] is given by the [[exponential map]]

   $$
     \exp_\epsilon
       \;\colon\;
     N(\Sigma \subset X)
       \overset{\simeq}{\longrightarrow}
     \mathcal{N}(\Sigma \subset X)
   $$

with respect to a $G$-invariant [[Riemannian metric]] (which exists according to Prop. \ref{ExistenceOfGInvariantRiemannianMetrics}):

=--

The existence of the $G$-invariant tubular neighbourhoods is for instance in [Bredon 72 VI Theorem 2.2](#Bredon72), [Kankaanrinta 07, theorem 4.4](#Kankaanrinta07). The uniqueness up to equivariant isotopy is in [Kankaanrinta 07, theorem 4.4, theorem 4.6](#Kankaanrinta07). The fact that one may always use the [[normal bundle]] appears at the end of the proof of [Bredon 72 VI Theorem 2.2](#Bredon72), and as a special case of a more general statement about invariant tubular neighbourhoods in [[Lie groupoids]] it follows from [Pflaum-Posthuma-Tang 11, Theorem 4.1](#PflaumPosthumaTang11) by applying the construction there to each point in $\Sigma$ for one and the same choice of background metric. See also for instance [Pflaum-Wilkin 17, Example 2.5](#PflaumWilkin17).

### Pullbacks of tubular neighbourhoods

(...) [[propagating flow]] (...) ([Godin](#Godin)).

## Related concepts

* [[tubular neighbourhood of a mapping space]]

* key application: [[Pontrjagin-Thom collapse map]]

## References

* {#Thom54} [[René Thom]], p. 21 of: _[[Quelques propriétés globales des variétés différentiables]]_, Comment. Math. Helv. 28, (1954). 17-86 ([doi:10.1007/BF02566923](https://doi.org/10.1007/BF02566923), [dml:139072](https://eudml.org/doc/139072), [digiz:GDZPPN002056259](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf))

Basics on tubular neighbourhoods are reviewed for instance in

* {#DaSilva} [[Ana Cannas da Silva]], section 3 of _Prerequisites from differential geometry_ ([pdf](http://www.math.princeton.edu/~acannas/appendix_dg.pdf))

* {#Kochman96} [[Stanley Kochman]], section 1.2 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

Discussion in the generality of [[equivariant differential topology]] includes

* {#Bredon72} [[Glen Bredon]], Chapter VI.2 of _Introduction to compact transformation groups_, Academic Press 1972 ([pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

* {#Bredon72} [[Glen Bredon]], _Introduction to compact transformation groups_, Academic Press 1972 ([pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

* {#Kankaanrinta07} [[Marja Kankaanrinta]], _Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions_,     Algebr. Geom. Topol. Volume 7, Number 1 (2007), 1-27 ([euclid:agt/1513796653](https://projecteuclid.org/euclid.agt/1513796653))

* {#PflaumPosthumaTang11} [[Markus Pflaum]], Hessel Posthuma, X. Tang, _Geometry of orbit spaces of proper Lie groupoids_, Journal für die reine und angewandte Mathematik (Crelles Journal) 2014.694 ([arXiv:1101.0180](https://arxiv.org/abs/1101.0180), [doi:10.1515/crelle-2012-0092](https://doi.org/10.1515/crelle-2012-0092))

* {#Ziller13} Wolfgang Ziller, _Group actions_, 2013 ([pdf](https://www.math.upenn.edu/~wziller/math661/LectureNotesLee.pdf))

* {#CrainicStruchiner13} Marius Crainic, Ivan Struchiner, _On the linearization theorem for proper Lie groupoids_,  Annales scientifiques de l'École Normale Supérieure, Série 4, Volume 46 (2013) no. 5, p. 723-746 ([numdam:ASENS_2013_4_46_5_723_0](http://www.numdam.org/item/ASENS_2013_4_46_5_723_0/) [doi:10.24033/asens.2200](https://doi.org/10.24033/asens.2200))

* {#PflaumWilkin17} [[Markus Pflaum]], Graeme Wilkin, _Equivariant control data and neighborhood deformation retractions_, Methods and Applications of Analysis, 2019 ([arXiv:1706.09539](https://arxiv.org/abs/1706.09539))

The homotopical uniqueness of tubular neighbourhoods is discussed in

* {#Godin} [[Veronique Godin]], _Higher string topology operations_ (2007)([arXiv:0711.4859](http://arxiv.org/abs/0711.4859))

For an analogue in homotopical algebraic geometry see

* [[Marc Levine]], _Motivic tubular neighborhoods_, [pdf](http://www.uni-due.de/~bm0032/publ/TubNbdDocumenta.pdf)

see also

* wikipedia [tubular neighborhood](http://en.wikipedia.org/wiki/Tubular_neighborhood)

[[!redirects tubular neighbourhoods]]

[[!redirects tubular neighborhood]]
[[!redirects tubular neighborhoods]]


tubular neighbourhood of a mapping space

[[!redirects tubular neighbourhood of mapping spaces]]
[[!redirects tubular neighbourhoods of mapping spaces]]

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

#Contents#
* table of contents
{:toc}

## Idea ##

The result of [[evaluation fibration of mapping spaces]] extends to more general evaluation maps between [[mapping spaces]].  One way to interpret that result is that the inclusion $C^\infty(S,p;M,q) \to C^\infty(S,M)$ has a [[tubular neighbourhood]].  Providing $M$ has enough [[diffeomorphism]]s, this is true of more general inclusions where they are defined by "coincidences".  That is to say, if $P$ is a condition on maps $S \to M$ that prescribes where certain points "coincide", then the submanifold of $C^\infty(S,M)$ of smooth maps satisfying this condition will have a tubular neighbourhood in the manifold of all smooth maps.


Tudor D. Stanescu

* [Institute page](https://physics.wvu.edu/faculty-and-staff/faculty/tudor-stanescu)

## Selected writings

On [[quantum materials]] and [[topological quantum computation]]:

* [[Tudor D. Stanescu]], *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))

category: people

[[!redirects Tudor Stanescu]]


Tudor Dimofte

* [webpage](http://theory.caltech.edu/~tdd/)

## Selected writings

On [[perturbative quantum field theory|perturbative]] [[Chern-Simons theory with complex gauge group]]:

* {#DGLZ09} [[Tudor Dimofte]], [[Sergei Gukov]], [[Jonatan Lenells]], [[Don Zagier]], _Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group_, Commun. Num. Theor. Phys. **3** (2009) 363-443 [[arXiv:0903.2472](http://arxiv.org/abs/0903.2472), [doi:10.4310/CNTP.2009.v3.n2.a4](https://dx.doi.org/10.4310/CNTP.2009.v3.n2.a4)]

* {#Dimofte16} [[Tudor Dimofte]],  _Perturbative and nonperturbative aspects of complex Chern-Simons Theory_, Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 44 ([arXiv:1608.02961](https://arxiv.org/abs/1608.02961))

On [[holomorphic blocks]]:

* {#BeemDimoftePasquetti12} [[Christopher Beem]], [[Tudor Dimofte]], [[Sara Pasquetti]], _Holomorphic Blocks in Three Dimensions_ ([arXiv:1211.1986](http://arxiv.org/abs/1211.1986))

On [[Coulomb branch]] [[singularities]] in [[D=3 N=4 super Yang-Mills theory]]:

* [[Mathew Bullimore]], [[Tudor Dimofte]], [[Davide Gaiotto]], _The Coulomb Branch of 3d $\mathcal{N}=4$ Theories_, Commun. Math. Phys. (2017) 354: 671 ([arXiv:1503.04817](https://arxiv.org/abs/1503.04817))

  (relation to [[Nahm's equations]] and the [[moduli space of monopoles]])

Discussion of [[topological quantum field theory]] with an eye towards [[supersymmetry|supersymmetric]] field theory, in terms of the [[ordinary homology]] of [[configuration spaces of points]] (see also at _[[correlators as differential forms on configuration spaces of points]]_):

* [[Christopher Beem]], [[David Ben-Zvi]], [[Mathew Bullimore]], [[Tudor Dimofte]], [[Andrew Neitzke]], _Secondary products in supersymmetric field theory_ ([arXiv:1809.00009](https://arxiv.org/abs/1809.00009))

category: people


Tudor Ratiu

__Tudor S. Ratiu__ is a Romanian-American mathematician. His main works are in mechanics and dynamical systems from the point of view of differential geometry.

* [Wikipedia entru](https://en.wikipedia.org/wiki/Tudor_Ratiu)

## Selected writings

* Jerrold E Marsden, Tudor S Ratiu, _Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems_, Springer 2013
* Juan-Pablo Ortega, Tudor S Ratiu, _Momentum maps and Hamiltonian reduction_, Progress in Mathematics 222, Birkhäuser 2003.
* Daniel Beltiţă, [[Tudor S. Ratiu]], _Geometric representation theory for unitary groups of operator algebras_, Advances in Mathematics __208__:1 (2007) 299--317 [doi](https://doi.org/10.1016/j.aim.2006.02.009) arXiv:[math.RT/0501057](https://arxiv.org/abs/math/0501057)
* François Demoures, François Gay-Balmaz, Tudor S. Ratiu, _Multisymplectic variational integrators and space/time symplecticity_,  Analysis and Applications __14__:03 (2016) 341--391 [arXiv:1310.4772](https://arxiv.org/abs/1310.4772) [doi](https://doi.org/10.1142/S0219530515500025)

category: people

[[!redirects Tudor S. Ratiu]]


Tullio Regge

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

* [[Leonardo Castellani]],  [[Anna Ceresole]], [[Riccardo D'Auria]], [[Pietro Fré]] (eds.): _Tullio Regge: An Eclectic Genius_, World Scientific 2019 ([doi:10.1142/11643](https://doi.org/10.1142/11643))

## Selected writings

Precursor discussion to the [[D'Auria-Fré formulation of supergravity]]:

* [[Yuval Ne'eman]], [[Tullio Regge]], *Gravity and supergravity as gauge theories on a group manifold*, Physics Letters B **74** 1–2 (1978) 54-56 [doi:10.1016/0370-2693(78)90058-8, [spire:6328](https://inspirehep.net/literature/6328)]

  also: Rivista del Nuovo Cimento **1** 5 (1978) 1–43

* {#DAuriaFre80} [[Riccardo D'Auria]], [[Pietro Fré]],  [[Tullio Regge]], _Graded Lie algebra, cohomology and supergravity_, Riv. Nuov. Cim. **3** fasc. 12 (1980) [[spire:156191](http://inspirehep.net/record/156191)]

## Related entries

* [[Regge calculus]]

category: people


Tully-Fisher relation

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#Contents#
* table of contents
{:toc}

## Idea

In [[astronomy]], the _Tully-Fisher relation_ is a relation between the asymptotic behaviour of [[galaxy rotation curves]], witnessed by the width of their [[21cm hydrogen line]], and the total visible [[baryon|baryonic]] [[mass]] of the [[galaxies]].

For most observed [[galaxies]] their rotation curves do not follow the predictions of standard [[gravity]] applied to the visible [[baryon|baryonic]] [[mass]] distribution in the galaxy. This has led to the speculation that either there is further invisible but gravitating [[dark matter]] making up most of the mass of galaxies, or the laws of [[gravity]] are modified on galactic scales ("[[MOND]]").

Recent computer simulation ([FIRE](https://fire.northwestern.edu)) of [[galaxy|galactic]] [[structure formation]] using the [[standard model of cosmology|standard]] [[cold dark matter]] [[model (physics)|model]] qualitatively reproduces the peculiar [[galactic rotation curves]] that motivated dark matter (or [[MOND]], for that matter) in the first place ([Hopkins et al. 17, Figure 4, Figure 5](#Hopkins17)):

\begin{center}

\end{center}

> graphics grabbed from ([Hopkins et al. 17](#Hopkins17))

and also reproduces well the baryonic [[Tully-Fisher relation]] ([El-Badry et al. 18, Figure 4](#ElBadry18)) which used to be an issue in the [[standard model of cosmology|standard]] [[cold dark matter]] [[model (physics)|model]]:

\begin{center}

\end{center}

> graphics grabbed from ([El-Badry et al. 18](#ElBadry18))

More in detail, [[galaxy rotation curves]] exhibit a close correlation between [[angular velocity]] and total _visible_ enclosed math at any given [[radius]], called the _[[radial acceleration relation]]_ (RAR) or _[[mass-discrepancy acceleration relation]]_. This, too, is reproduced in [[cold dark matter]]-[[model (in theoretical physics)]]  computer simulation ([Santos-Santos et al. 15](#SantosSantosEtAl15), [Cintio-Lelli 15](#CintioLelli15), [Keller-Wadsley 16](#KellerWadsley16), [Ludlow et al 16](#LudlowEtAl16)). These early simulations were not found conclusive in [Lelli et al 16, section 8.2](#LelliEtAl16). But more detailed analysis ([PSF 18](#PSF18)) and more refined simulation [Dutton-Maccio-Obreja-Buck 19](#DuttonMaccioObrejaBuck19) has then been claimed to confirm the statement.

A conceptual explanation via stellar feedback is discussed in [GBFH19](#GBFH19).

## References

See also

* Wikipedia, _[Tully-Fisher relation ](https://en.wikipedia.org/wiki/Tully%E2%80%93Fisher_relation)_

Computer simulation of [[dark matter]] [[structure formation]] on [[galaxy]]-scales

* {#Hopkins17} Hopkins et al. _FIRE-2 Simulations: Physics versus Numerics in Galaxy Formation_. Monthly Notices of the Royal Astronomical Society, Volume 480, Issue 1, 11 October 2018, Pages 800–863 ([arXiv:1702.06148](https://arxiv.org/abs/1702.06148), [doi:10.1093/mnras/sty1690](https://doi.org/10.1093/mnras/sty1690))

* {#ElBadry18} El-Badry et al. _Gas Kinematics in FIRE Simulated Galaxies Compared to Spatially Unresolved HI Observations_, Monthly Notices of the Royal Astronomical Society, Volume 477, Issue 2, 21 June 2018, Pages 1536–1548 ([arXiv:1801.03933](https://arxiv.org/abs/1801.03933), [doi:10.1093/mnras/sty730](https://doi.org/10.1093/mnras/sty730))

On the [[mass-discrepancy acceleration relation]]:

* {#CintioLelli15} Arianna Di Cintio, Federico Lelli, _The mass discrepancy acceleration relation in a $\Lambda CDM$ context_, Monthly Notices of the Royal Astronomical Society: Letters, Volume 456, Issue 1, 11 February 2016, Pages L127–L131 ([arXiv:1511.06616](https://arxiv.org/abs/1511.06616), [doi:10.1093/mnrasl/slv185](https://doi.org/10.1093/mnrasl/slv185))

* {#SantosSantosEtAl15} Isabel M. Santos-Santos et al. _The distribution of mass components in simulated disc galaxies_ ([arXiv:1510.02474](https://arxiv.org/abs/1510.02474))

* {#KellerWadsley16} B.W. Keller, J.W. Wadsley, _$\Lambda CDM$ is Consistent with SPARC Radial Acceleration Relation_ ([arXiv:1610.06183](https://arxiv.org/abs/1610.06183))

* {#LudlowEtAl16} Aaron D. Ludlow et. al. _The Mass-Discrepancy Acceleration Relation: a Natural Outcome of Galaxy Formation in Cold Dark Matter halos_, Phys. Rev. Lett. 118, 161103 (2017) ([arXiv:1610.07663](https://arxiv.org/abs/1610.07663))

* {#PSF18} Chiara Di Paolo, Paolo Salucci, Jean Philippe Fontaine, _The Radial Acceleration Relation (RAR): the crucial cases of Dwarf Discs and of Low Surface Brightness galaxies_, ApJ 2019 ([arXiv:1810.08472](https://arxiv.org/abs/1810.08472))

* {#DuttonMaccioObrejaBuck19} Aaron A. Dutton, Andrea V. Macciò, Aura Obreja, Tobias Buck, _NIHAO XVIII: Origin of the MOND phenomenology of galactic rotation curves in a LCDM universe_ ([arXiv:1902.06751](https://arxiv.org/abs/1902.06751))

Critical comments on the early simulations in

* {#LelliEtAl16} Federico Lelli, Stacy S. McGaugh, James M. Schombert, Marcel S. Pawlowski, section 8.2 of _One Law To Rule Them All: The Radial Acceleration Relation of Galaxies_ ([arXiv:1610.08981](https://arxiv.org/abs/1610.08981))

Conceptual explanation via stellar feedback:

* {#GBFH19} Michael Y. Grudić, Michael Boylan-Kolchin, Claude-André Faucher-Giguère, [[Philip Hopkins]], _Stellar feedback sets the universal acceleration scale in galaxies_ ([arxiv:1910.06345](https://arxiv.org/abs/1910.06345))

[[!redirects Tully-Fisher relations]]

[[!redirects baryonic Tully-Fisher relation]]
[[!redirects baryonic Tully-Fisher relations]]

[[!redirects radial acceleration relation]]
[[!redirects radial acceleration relations]]

[[!redirects RAR]]

[[!redirects mass-discrepancy acceleration relation]]
[[!redirects mass-discrepancy acceleration relation]]
[[!redirects mass discrepancy acceleration relation]]


Tung Tran

* [InSpire page](https://inspirehep.net/authors/1510638)

## Selected writings

On the [[IKKT matrix model]] in relation to [[higher spin gauge theory]]:

* [[Harold Steinacker]], [[Tung Tran]], *Spinorial higher-spin gauge theory from IKKT in Euclidean and Minkowski signatures* [[arXiv:2305.19351](https://arxiv.org/abs/2305.19351)]

* [[Harold C. Steinacker]], [[Tung Tran]], *Soft limit of higher-spin interactions in the IKKT model* [[arXiv:2311.14163](https://arxiv.org/abs/2311.14163)]

* [[Harold C. Steinacker]], [[Tung Tran]], *Spinorial description for Lorentzian $\mathfrak{hs}$-IKKT* [[arXiv:2312.16110](https://arxiv.org/abs/2312.16110)]

category: people


tuple

# Tuples
* table of contents
{: toc}

## Idea

The generalisation of [[ordered pair]] to something having more positions is usually called a *tuple* (or _ordered tuple_). More particularly one gets the term _$n$-tuple_, which refers to a list, $(x_1, \ldots, x_n)$, with $n$ entries from some set, $X$; here $n$ is a [[natural number]].  It thus corresponds to an element in the $n$-fold [[product set]], $X^n$. The various elements $x_i$ of the $n$-tuple are usually called its **components** and sometimes it is useful to call the set of components the **support** or **range** of the tuple.

## Variations

*  An [[ordered pair]] is a $2$-tuple.  A $3$-tuple is a _[[triple]]_, a $4$-tuple is a _[[quadruple]]_, a $5$-tuple is a _[[quintuple]]_ etc.

*  The notion of $1$-tuple is trivial; a $1$-tuple from $X$ is equivalent to an element of $X$.  But if you wish to emphasize that you have a $1$-tuple, then it may be called a _[[singleton list|singleton]]_.

*  The notion of $0$-tuple is also trivial, but in a different way; there is (for each set $X$) a single $0$-tuple from $X$.  See [[empty list]] for notations, but in the end it hardly matters what you call it.

*  The term 'tuple' is usually used for an $n$-tuple for a *specific* number $n$.  If we wish to speak of an $n$-tuple for an *arbitrary* $n$ (particularly without specifying that $n$), then we may speak of a [[list]] (which has other terminology, described on that page).  Then the set of lists is the [[disjoint union]] over $n$ of the sets of $n$-tuples.

*  The term 'tuple' is usually used for an $n$-tuple for a *finite* number $n$.  If we wish to speak of an $n$-tuple for an *infinite* (or possibly infinite) $n$, then we may speak of a [[sequence]].

## Formalisation

See [[ordered pair]] for methods of formalising ordered pairs (which are $2$-tuples) in various [[foundations of mathematics]].  Some of these generalise immediately to $n$-tuples for arbitrary $n$; otherwise, we may define $n$-tuples [[recursion|recursively]]: a triple is an ordered pair whose (say) first component is an ordered pair; a quadruple is an ordered pair whose first component is a triple, etc.

## Related concepts

* A 2-tuple is a [[pair]].

* A 3-tuple is [[triple]].

* **tuple**, [[sequence]], [[function]]

* [[dependent tuple]], [[dependent sequence]], [[dependent function]]

* [[list]], [[indexed set]]

[[!redirects tuple]]
[[!redirects tuples]]
[[!redirects n-tuple]]
[[!redirects n-tuples]]
[[!redirects ordered tuple]]
[[!redirects ordered tuples]]
[[!redirects ordered n-tuple]]
[[!redirects ordered n-tuples]]


tuple extensionality

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#### Type theory
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#### Category theory
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\tableofcontents

## Definition

### In set theory

Let $\mathbb{N}$ be the set of natural numbers, let $n:\mathbb{N}$ be an arbitrary natural number, let $\mathrm{Fin}(n)$ be the [[finite set]] with $n$ elements, and let $A$ be an arbitrary set. Then **$n$-tuple extensionality** states that for every [[tuple]] $a:\mathrm{Fin}(n) \to A$ and $b:\mathrm{Fin}(n) \to A$, $a =_{\mathrm{Fin}(n) \to A} b$ if and only if $a(i) =_A b(i)$ for all elements $i \in \mathrm{Fin}(n)$.

### In dependent type theory

In [[dependent type theory]], **$n$-tuple extensionality** states that given two $n$-tuples $a:\mathrm{Fin}(n) \to A$ and $b:\mathrm{Fin}(n) \to A$ there is an [[equivalence of types]] between the [[identity type]] $a =_{\mathrm{Fin}(n) \to A} b$ and the dependent tuple type $(i:\mathrm{Fin}(n)) \to (a(i) =_{A} b(i))$:
$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A \; \mathrm{type}}{\Gamma,a:\mathrm{Fin}(n) \to A, b:\mathrm{Fin}(n) \to A \vdash \mathrm{tupext}(a, b):(a =_{\mathrm{Fin}(n) \to A} b) \simeq (i:\mathrm{Fin}(n)) \to (a(i) =_{A} b(i))}$$

There is also a version of $n$-tuple extensionality for [[dependent tuple types]] called **dependent $n$-tuple extensionality**, which states that given two dependent $n$-tuples $a:(i:\mathrm{Fin}(n)) \to A(i)$ and $b:(i:\mathrm{Fin}(n)) \to A(i)$ there is an [[equivalence of types]] between the [[identity type]] $a =_{(i:\mathrm{Fin}(n)) \to A(i)} b$ and the dependent tuple type $(i:\mathrm{Fin}(n)) \to (a(i) =_{A(i)} b(i))$:
$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma, i:\mathrm{Fin}(n) \vdash A(i) \; \mathrm{type}}{\Gamma,a:(i:\mathrm{Fin}(n)) \to A(i), b:(i:\mathrm{Fin}(n)) \to A(i) \vdash \mathrm{dtupext}(a, b):(a =_{(i:\mathrm{Fin}(n)) \to A(i)} b) \simeq (i:\mathrm{Fin}(n)) \to (a(i) =_{A(i)} b(i))}$$

## Properties

Tuple extensionality is always true in [[set theory]] and [[dependent type theory]], for the same reason that [[product extensionality]] is true.

Indeed, one could express tuple extensionality as two separate cases representing nullary tuples and binary tuples:

* Every two elements in a [[singleton]] are equal to each other

* [[product extensionality]]: Given two [[ordered pairs]] in a [[product set]], they are equal if and only if the left product projections are equal to each other and the right product projections are equal to each other.

In dependent type theory, this becomes:

* Every [[identity type]] of two elements in a [[contractible type]] is a contractible type.

* [[product extensionality]]: Given two [[ordered pairs]] in a [[product set]], the identity type between the two ordered pairs is equivalent to the product type of the identity type of the left product projections and the identity type of the right product projections.

## See also

* [[extensionality]]
* [[product extensionality]]
* [[sequence extensionality]]
* [[function extensionality]]

[[!redirects tuple extensionality]]
[[!redirects n-tuple extensionality]]

[[!redirects dependent tuple extensionality]]
[[!redirects dependent n-tuple extensionality]]


Turaev-Viro model

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#### Physics
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#Contents#
* table of contents
{:toc}

## Idea

The _Turaev-Viro model_, or more precisely _Turaev-Viro-Barrett-Westbury model_, is a [[3d TQFT]] construction induced from a [[spherical category]]/[[fusion category]] $\mathcal{C}$.

If $\mathcal{C}$ is moreover a [[modular tensor category]] then there exists also the [[Reshetikhin-Turaev construction]] [[3d TQFT]]. In this case the
Turaev-Viro model is something like the "norm square" of the [[Reshetikhin-Turaev construction]] on $\mathcal{C}$.

For $G$ a [[finite group]] and $\mathcal{C} = Vect_G$ the [[category]] of $G$-[[graded vector spaces]] the Turaev-Viro model describes the $G$-[[Dijkgraaf-Witten theory]], also the [[Levin-Wen model]].

See for instance the introduction of ([Kirillov-Balsam 10](#KirillovBalson10)) for a review.

## Properties

### As an extended TQFT

The Turaev-Viro model has been constructed as a 3-2-1 [[extended TQFT]] in ([Kirillov-Balsam 10](#KirillovBalsam10), [Balsam 10a](#Balsam10a), [Balsam 10b](#Balsam10b)).

Related but different is the construction of fully extended [[3d TQFT]] from [[fusion categories]] via the [[cobordism theorem]], see at _[fusion category -- Relation to extended 3d TQFT](fusion%20category#RelationtoTQFT)_ for more on this.

## References

The original articles

* [[Vladimir Turaev]] and O. Ya. Viro, _State sum invariants of 3-manifolds and quantum 6jsymbols_, Topology 31 (1992), no. 4, 865–902, DOI 10.1016/0040-9383(92)90015-A. MR1191386

* [[Vladimir Turaev]], _Quantum invariants of knots and 3-manifolds_, de Gruyter Studies in Mathematics **18** Walter de Gruyter & Co., Berlin, 1994 ([doi:10.1515/9783110435221](https://doi.org/10.1515/9783110435221))

constructed 3-manifold invariants from quantum [[6j symbols]]. See also

* [[Adrian Ocneanu]], _Chirality for operator algebras_, In _Subfactors (Kyuzeso, 1993), pp. 39-63. World Sci. Publ., River Edge, NJ, 1994

In

* [[John Barrett]], [[Bruce Westbury]], _Invariants of piecewise-linear 3-manifolds_, Trans. Amer. Math. Soc. 348 (1996), no. 10, 3997–4022, ([doi:10.1090/S0002-9947-96-01660-1](https://doi.org/10.1090/S0002-9947-96-01660-1), MR1357878)

* [[John Barrett]], [[Bruce Westbury]], _Spherical Categories_, Adv. Math. 143:357-375, 1999 ([arXiv:hep-th/9310164](http://arxiv.org/abs/hep-th/9310164), [doi:10.1006/aima.1998.1800](https://doi.org/10.1006/aima.1998.1800))

it was shown that this construction proceeds from any [[spherical category|spherical]] [[fusion category]].

Relation to the [[Crane-Yetter model]]:

* {#BarrettGarciIslasMartins04} [[John Barrett]], J. Garcia-Islas, [[João Faria Martins]], _Observables in the Turaev-Viro and Crane-Yetter models_, J. Math. Phys. 48:093508, 2007 ([arXiv:math/0411281](http://arxiv.org/abs/math/0411281))

Refinement of the construction to an [[extended TQFT]] is in

* {#KirillovBalsam10} [[Alexander Kirillov]] Jr., [[Benjamin Balsam]], _Turaev-Viro invariants as an extended TQFT_ ([arXiv:1004.1533](http://arxiv.org/abs/1004.1533))

* {#Balsam10a} [[Benjamin Balsam]], _Turaev-Viro invariants as an extended TQFT II_ ([arXiv:1010.1222](http://arxiv.org/abs/1010.1222))

* {#Balsam10b} [[Benjamin Balsam]], _Turaev-Viro invariants as an extended TQFT II_ ([arXiv:1012.0560](http://arxiv.org/abs/1012.0560))

Discussion that relates the [[quantization of 3d Chern-Simons theory]] to the [[Reshetikhin-Turaev construction]] of a 3d-[[TQFT]] from the [[modular tensor category]] induced by $G$ is in

* {#Andersen} [[Jørgen Andersen]], _A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications_ ([arXiv:1206.2785](arxiv.org/abs/1206.2785))

and references cited there.

A relation to the [[Levin-Wen model]] is discussed in

* Alexander Kirillov Jr, _String-net model of Turaev-Viro invariants_ ([arXiv:1106.6033](http://arxiv.org/abs/1106.6033))

Discussion of the [[Ising model]] [[2d CFT]] as a boundary theory to a 3d [[TQFT]] based on the [[Turaev-Viro model]], similar to the [[CS-WZW correspondence]], and the phenomenon of [[Kramers-Wannier duality]], is in:

* {#FreedTeleman18} [[Daniel Freed]], [[Constantin Teleman]], _Topological dualities in the Ising model_ ([arXiv:1806.00008](https://arxiv.org/abs/1806.00008))

Discussion of the connection with [[Dijkgraaf-Witten theory]]

* Jerome Petit, 2006. _The invariant of Turaev-Viro from Group category_ ([math/0608614](https://arxiv.org/abs/math/0608614))

[[!redirects Turaev-Viro construction]]
[[!redirects Turaev-Viro-Barrett-Westbury model]]

[[!redirects Turaev-Viro-invariant]]
[[!redirects Turaev-Viro-invariants]]

[[!redirects Turaev-Viro invariant]]
[[!redirects Turaev-Viro invariants]]

[[!redirects TV invariant]]
[[!redirects TV invariants]]

[[!redirects TV-invariant]]
[[!redirects TV-invariants]]


Turing category

[[!redirects Turing Category]]

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#Contents#
* table of contents
{:toc}

## Definition

A Turing category is a certain [[categorification]] of [[partial combinatory algebras]] (PCAs) based on [[restriction categories]].

+-- {: .num_defn #TuringCategory}
###### Definition
**(Turing category)**

A _Turing category_ is a [[cartesian  restriction category]] $(\mathcal{C}, \bar{} )$ with a fixed [[object]] A and [[morphism]] $\bullet \colon A \times A \rightarrow A$ having the following [[universal property]]:

For each $\mathcal{C}$-morphism $f \colon X \rightarrow Y$ there is a [[section]] $s \colon Y \rightarrow A$ and [[retract]] $r \colon A \rightarrow X$, along with a total map $h \colon \mathbf{1} \rightarrow A$ making the following [[commuting diagram|diagram commute]]:

\begin{centre}
    \begin{tikzcd}
    \arrow[r, "\bullet"] A \times A & A \arrow[r, "r"] & X \arrow[ddll, "f"] \\
    \arrow[u, "\mathsf{id_A} \times h"] A \times 1 \simeq A \arrow[ur, swap,  "sfr"] & & \\
    Y \arrow[u, "s"] & &
    \end{tikzcd}
\end{centre}

=--

While restriction structure captures the function partiality one finds in [[computability theory]], the universal property stated above captures the universal coding of $\mathcal{C}$-morphisms into operations on the _Turing object_ $A$, via the application $\bullet$. Indeed one recovers, as an example, the category $\mathbf{Rec}$, of natural numbers $n, m \in \mathbf{N}$ and partial recursive functions $ \mathbf{N}^n \rightarrow \mathbf{N}^m$. This has universal applicative structure:

\begin{centre}
     \begin{tikzcd}
    \arrow[d, swap, dashed, "\exists \hspace{0.1cm} \text{total} \hspace{0.1cm} h \hspace{0.1cm} : \hspace{0.1cm} 1 \times h"] \mathbf{N} \times 1 \simeq \mathbf{N} \arrow[rd, "f_i"] \\ \mathbf{N} \times \mathbf{N} \arrow[r, swap, "\langle -- \rangle"] & \mathbf{N}
    \end{tikzcd}
\end{centre}

and universal representation $\langle i, n \rangle = f_i(n)$ of the $i$th computable function. In this case the PCA is [[Kleene's first algebra]]. More generally, the computable map category of any PCA forms a Turing category, with Turing object the PCA and its Turing morphism being the applicative structure.

And conversely, a Turing object $A$ of $\mathcal{C}$ and its Turing morphism $\bullet : A \times A \rightarrow A$ form a (relative) PCA. This is effectively a PCA-object internal to an appropriate category:

+-- {: .num_defn #RelativePCA}
###### Definition
**(Relative PCA)**

A (relative) PCA $A$ is a combinatory complete partial applicative system in a cartesian [[restriction category]] $\mathcal{D}$, i.e. a morphism $\bullet : A \times A \rightarrow A$ in $\mathcal{D}$ such that finite powers $A^n$ and $A$-computable morphisms form a well-defined cartesian restriction subcategory of $\mathcal{D}$.

=--

Note, however, that not every relative PCA in a Turing category $\mathcal{C}$ is a Turing object for $\mathcal{C}$.

### Basic properties

The arithmetical representation of a (partial) function $f : \mathbf{N} \rightarrow \mathbf{N}$ as a member of the recursive enumeration of relations $\varphi_i \subset \mathbf{N} \times \mathbf{N}$ is universal, and this representation satisfies the Kleene [[recursion theorems]]:

$\bullet$ (smn) Let $f: \mathbf{N} \rightarrow \mathbf{N}$ be a partial computable function. Then there is a partial computable function $s : \mathbf{N} \rightarrow \mathbf{N}$ such that $f(\langle i, n \rangle) = f_{s(i)} (n)$.

$\bullet$ (utm) There is a partial computable function $U : \mathbf{N} \rightarrow \mathbf{N}$ such that $U(\langle i, n \rangle ) = f_i (n)$.

As can be nearly read off the definition above, these properties hold with respect to any Turing object. This is part of the overall motivation for Turing categories, since they provide a more model-independent setting both for stating the main theorems of recursion theory, while also allowing the "intrinsic meaning" of different constructions in computability theory to stand out. For example, in [[Type II computability]] (or [[function realizability]]) it turns out that _function computability_ with respect to the PCA $\mathbf{N}^\mathbf{N}$ (either as [[Kleene's second algebra]] or the [[Baire space of sequences]]) has the topological meaning of _continuity_. Hence, despite the universal availability of Gödel-encodings of all "$A$-computations" for an appropriate data type $A$ into $\mathbf{N}$-computations (a form of the [[Church-Turing thesis]]), the theory of Turing categories and their applications to categorical realizability models, for example, are intended to give the intrinsic meaning of computational mathematics without resorting to such a universal representation, much like the success story of Type II computability.

## More Examples

(...)

## Weak Limits and Exact Completions

(...)

## Related concepts

* [[timed set]]

## References

Turing categories were isolated categorically and named in:

* [[Robin Cockett]], [[Pieter Hofstra]], _Introduction to Turing categories_, Annals of Pure and Applied Logic, 156, 2008 ([pdf](https://www.di.ens.fr/users/longo/files/Cockett-TuringCateg.pdf))

[[!redirects turing category]]
[[!redirects turing-categories]]
[[!redirects Turing category]]
[[!redirects turing categories]]


Turing machine

# Contents
* table of contents
{: toc}

## Idea

A **Turing machine** is a [[model of computation]]. It can be thought of as a machine with a set of possible internal states that uses an infinite (in both directions) piece of tape with countably-many positions available for symbols (drawn from a specified set of symbols). There is a pointer that selects the current position on the tape. Every timestep, it can, in this order:

* Remove or change the symbol that it is currently pointing to;

* Point to the square of tape to the left or to the right of the square that it is currently pointing to;

* Change its state.

It then repeats this sequence until its state is a _halting_ state, if ever, where a halting state is one special state specified in advance.

## See also

* [[lambda-calculus]]

* [[halting problem]]

## References

### Classical computing

* Wikipedia, *[Turing machine](https://en.wikipedia.org/wiki/Turing_machine)*

### Quantum computing

On Turing machines in the context of [[quantum computing]]

* [[David Deutsch]], *Quantum theory, the Church–Turing principle and the universal quantum computer*, Proceedings of the Royal Society of London A **400** 1818 (1985) 97-117 [[doi:10.1098/rspa.1985.0070](https://doi.org/10.1098/rspa.1985.0070), [pdf](https://www.daviddeutsch.org.uk/wp-content/deutsch85.pdf)]

* Wikipedia, *[Quantum Turing machine](https://en.wikipedia.org/wiki/Quantum_Turing_machine)*

[[!redirects turing machine]]
[[!redirects Turing machines]]
[[!redirects turing machines]]

[[!redirects quantum Turing machine]]
[[!redirects quantum Turing machines]]


TVS relationships

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<title>LCTVS</title>
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<g id="node1" class="node"><title>FD</title>
<a xlink:href="http://ncatlab.org/nlab/show/finite-dimensional topological vector space" xlink:title="Finite-Dimensional">
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</a>
</g>
<g id="node2" class="node"><title>Hi</title>
<a xlink:href="http://ncatlab.org/nlab/show/Hilbert topological vector space" xlink:title="Hilbert">
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</a>
</g>
<g id="edge2" class="edge"><title>FD->Hi</title>
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</g>
<g id="node3" class="node"><title>SC</title>
<a xlink:href="http://ncatlab.org/nlab/show/second-countable topological vector space" xlink:title="Second-Countable">
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<text fill="#008B26" text-anchor="middle" x="460" y="-445.9" style="font-family:Times New Roman;font-size:14.00;">Second-Countable</text>
</a>
</g>
<g id="edge4" class="edge"><title>FD->SC</title>
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<polygon style="fill:black;stroke:black;" points="484.219,-473.221 475,-468 478.951,-477.831 484.219,-473.221"/>
</g>
<g id="node4" class="node"><title>Nu</title>
<a xlink:href="http://ncatlab.org/nlab/show/nuclear topological vector space" xlink:title="Nuclear">
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<text fill="#008B26" text-anchor="middle" x="582" y="-445.9" style="font-family:Times New Roman;font-size:14.00;">Nuclear</text>
</a>
</g>
<g id="edge6" class="edge"><title>FD->Nu</title>
<path style="fill:none;stroke:black;" d="M536,-504C543,-495 552,-485 560,-476"/>
<polygon style="fill:black;stroke:black;" points="563.049,-477.831 567,-468 557.781,-473.221 563.049,-477.831"/>
</g>
<g id="node8" class="node"><title>Mo</title>
<a xlink:href="http://ncatlab.org/nlab/show/Montel topological vector space" xlink:title="Montel">
<polygon style="fill:none;stroke:black;" points="703,-396 641,-396 641,-360 703,-360 703,-396"/>
<text fill="#008B26" text-anchor="middle" x="672" y="-373.9" style="font-family:Times New Roman;font-size:14.00;">Montel</text>
</a>
</g>
<g id="edge8" class="edge"><title>FD->Mo</title>
<path style="fill:none;stroke:black;" d="M569,-504C588,-495 609,-483 625,-468 642,-450 655,-425 662,-406"/>
<polygon style="fill:black;stroke:black;" points="665.536,-406.585 666,-396 659.036,-403.985 665.536,-406.585"/>
</g>
<g id="node5" class="node"><title>Ba</title>
<a xlink:href="http://ncatlab.org/nlab/show/Banach topological vector space" xlink:title="Banach">
<polygon style="fill:none;stroke:black;" points="217,-396 151,-396 151,-360 217,-360 217,-396"/>
<text fill="#008B26" text-anchor="middle" x="184" y="-373.9" style="font-family:Times New Roman;font-size:14.00;">Banach</text>
</a>
</g>
<g id="edge10" class="edge"><title>Hi->Ba</title>
<path style="fill:none;stroke:black;" d="M285,-433C268,-424 245,-411 226,-401"/>
<polygon style="fill:black;stroke:black;" points="227.441,-397.797 217,-396 224.042,-403.916 227.441,-397.797"/>
</g>
<g id="node6" class="node"><title>IP</title>
<a xlink:href="http://ncatlab.org/nlab/show/inner-product topological vector space" xlink:title="Inner-Product">
<polygon style="fill:none;stroke:black;" points="345,-396 235,-396 235,-360 345,-360 345,-396"/>
<text fill="#008B26" text-anchor="middle" x="290" y="-373.9" style="font-family:Times New Roman;font-size:14.00;">Inner-Product</text>
</a>
</g>
<g id="edge12" class="edge"><title>Hi->IP</title>
<path style="fill:none;stroke:black;" d="M309,-432C306,-424 303,-415 300,-406"/>
<polygon style="fill:black;stroke:black;" points="303.226,-404.573 297,-396 296.521,-406.584 303.226,-404.573"/>
</g>
<g id="node14" class="node"><title>Re</title>
<a xlink:href="http://ncatlab.org/nlab/show/reflexive topological vector space" xlink:title="Reflexive">
<polygon style="fill:none;stroke:black;" points="628,-324 550,-324 550,-288 628,-288 628,-324"/>
<text fill="#008B26" text-anchor="middle" x="589" y="-301.9" style="font-family:Times New Roman;font-size:14.00;">Reflexive</text>
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</g>
<g id="edge14" class="edge"><title>Hi->Re</title>
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<polygon style="fill:black;stroke:black;" points="540.398,-314.478 550,-310 539.701,-307.512 540.398,-314.478"/>
</g>
<g id="node7" class="node"><title>Se</title>
<a xlink:href="http://ncatlab.org/nlab/show/separable topological vector space" xlink:title="Separable">
<polygon style="fill:none;stroke:black;" points="485,-396 401,-396 401,-360 485,-360 485,-396"/>
<text fill="#008B26" text-anchor="middle" x="443" y="-373.9" style="font-family:Times New Roman;font-size:14.00;">Separable</text>
</a>
</g>
<g id="edge18" class="edge"><title>SC->Se</title>
<path style="fill:none;stroke:black;" d="M456,-432C454,-424 452,-415 450,-406"/>
<polygon style="fill:black;stroke:black;" points="453.226,-404.573 447,-396 446.521,-406.584 453.226,-404.573"/>
</g>
<g id="node18" class="node"><title>Me</title>
<a xlink:href="http://ncatlab.org/nlab/show/metrisable topological vector space" xlink:title="Metrisable">
<polygon style="fill:none;stroke:black;" points="464,-252 378,-252 378,-216 464,-216 464,-252"/>
<text fill="#008B26" text-anchor="middle" x="421" y="-229.9" style="font-family:Times New Roman;font-size:14.00;">Metrisable</text>
</a>
</g>
<g id="edge16" class="edge"><title>SC->Me</title>
<path style="fill:none;stroke:black;" d="M475,-432C482,-422 490,-409 494,-396 498,-380 498,-375 494,-360 484,-323 460,-284 442,-260"/>
<polygon style="fill:black;stroke:black;" points="444.8,-257.9 436,-252 439.2,-262.1 444.8,-257.9"/>
</g>
<g id="node9" class="node"><title>Sc</title>
<a xlink:href="http://ncatlab.org/nlab/show/Schwartz topological vector space" xlink:title="Schwartz">
<polygon style="fill:none;stroke:black;" points="623,-396 541,-396 541,-360 623,-360 623,-396"/>
<text fill="#008B26" text-anchor="middle" x="582" y="-373.9" style="font-family:Times New Roman;font-size:14.00;">Schwartz</text>
</a>
</g>
<g id="edge24" class="edge"><title>Nu->Sc</title>
<path style="fill:none;stroke:black;" d="M582,-432C582,-424 582,-415 582,-406"/>
<polygon style="fill:black;stroke:black;" points="585.5,-406 582,-396 578.5,-406 585.5,-406"/>
</g>
<g id="node10" class="node"><title>UB</title>
<a xlink:href="http://ncatlab.org/nlab/show/ultrabornological topological vector space" xlink:title="Ultrabornological">
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</a>
</g>
<g id="edge26" class="edge"><title>Ba->UB</title>
<path style="fill:none;stroke:black;" d="M154,-360C139,-351 120,-339 104,-329"/>
<polygon style="fill:black;stroke:black;" points="105.441,-325.797 95,-324 102.042,-331.916 105.441,-325.797"/>
</g>
<g id="node11" class="node"><title>Fr</title>
<a xlink:href="http://ncatlab.org/nlab/show/Frechet topological vector space" xlink:title="Fréchet">
<polygon style="fill:none;stroke:black;" points="218,-324 150,-324 150,-288 218,-288 218,-324"/>
<text fill="#008B26" text-anchor="middle" x="184" y="-301.9" style="font-family:Times New Roman;font-size:14.00;">Fréchet</text>
</a>
</g>
<g id="edge28" class="edge"><title>Ba->Fr</title>
<path style="fill:none;stroke:black;" d="M184,-360C184,-352 184,-343 184,-334"/>
<polygon style="fill:black;stroke:black;" points="187.5,-334 184,-324 180.5,-334 187.5,-334"/>
</g>
<g id="node12" class="node"><title>DF</title>
<a xlink:href="http://ncatlab.org/nlab/show/DF topological vector space" xlink:title="DF">
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<text fill="#008B26" text-anchor="middle" x="263" y="-301.9" style="font-family:Times New Roman;font-size:14.00;">DF</text>
</a>
</g>
<g id="edge30" class="edge"><title>Ba->DF</title>
<path style="fill:none;stroke:black;" d="M204,-360C214,-351 225,-341 235,-331"/>
<polygon style="fill:black;stroke:black;" points="237.779,-333.219 243,-324 233.169,-327.951 237.779,-333.219"/>
</g>
<g id="node13" class="node"><title>No</title>
<a xlink:href="http://ncatlab.org/nlab/show/normed topological vector space" xlink:title="Normed">
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<text fill="#008B26" text-anchor="middle" x="343" y="-301.9" style="font-family:Times New Roman;font-size:14.00;">Normed</text>
</a>
</g>
<g id="edge32" class="edge"><title>Ba->No</title>
<path style="fill:none;stroke:black;" d="M217,-363C241,-352 273,-338 299,-326"/>
<polygon style="fill:black;stroke:black;" points="300.283,-329.26 308,-322 297.44,-322.863 300.283,-329.26"/>
</g>
<g id="edge34" class="edge"><title>IP->No</title>
<path style="fill:none;stroke:black;" d="M303,-360C309,-352 317,-341 324,-332"/>
<polygon style="fill:black;stroke:black;" points="326.8,-334.1 330,-324 321.2,-329.9 326.8,-334.1"/>
</g>
<g id="edge20" class="edge"><title>Mo->Re</title>
<path style="fill:none;stroke:black;" d="M651,-360C641,-351 628,-340 617,-331"/>
<polygon style="fill:black;stroke:black;" points="618.831,-327.951 609,-324 614.221,-333.219 618.831,-327.951"/>
</g>
<g id="node24" class="node"><title>Pc</title>
<a xlink:href="http://ncatlab.org/nlab/show/paracompact topological vector space" xlink:title="Paracompact">
<polygon style="fill:none;stroke:black;" points="698,-180 594,-180 594,-144 698,-144 698,-180"/>
<text fill="#008B26" text-anchor="middle" x="646" y="-157.9" style="font-family:Times New Roman;font-size:14.00;">Paracompact</text>
</a>
</g>
<g id="edge22" class="edge"><title>Mo->Pc</title>
<path style="fill:none;stroke:black;" d="M670,-360C665,-322 655,-235 649,-190"/>
<polygon style="fill:black;stroke:black;" points="652.478,-189.602 648,-180 645.512,-190.299 652.478,-189.602"/>
</g>
<g id="node20" class="node"><title>Bo</title>
<a xlink:href="http://ncatlab.org/nlab/show/bornological topological vector space" xlink:title="Bornological">
<polygon style="fill:none;stroke:black;" points="164,-180 64,-180 64,-144 164,-144 164,-180"/>
<text fill="#008B26" text-anchor="middle" x="114" y="-157.9" style="font-family:Times New Roman;font-size:14.00;">Bornological</text>
</a>
</g>
<g id="edge40" class="edge"><title>UB->Bo</title>
<path style="fill:none;stroke:black;" d="M68,-288C71,-269 76,-240 85,-216 89,-207 93,-197 98,-189"/>
<polygon style="fill:black;stroke:black;" points="101.203,-190.441 103,-180 95.084,-187.042 101.203,-190.441"/>
</g>
<g id="node15" class="node"><title>Cn</title>
<a xlink:href="http://ncatlab.org/nlab/show/convenient topological vector space" xlink:title="Convenient">
<polygon style="fill:none;stroke:black;" points="190,-252 94,-252 94,-216 190,-216 190,-252"/>
<text fill="#008B26" text-anchor="middle" x="142" y="-229.9" style="font-family:Times New Roman;font-size:14.00;">Convenient</text>
</a>
</g>
<g id="edge42" class="edge"><title>Fr->Cn</title>
<path style="fill:none;stroke:black;" d="M173,-288C168,-280 163,-270 157,-261"/>
<polygon style="fill:black;stroke:black;" points="159.916,-259.042 152,-252 153.797,-262.441 159.916,-259.042"/>
</g>
<g id="node16" class="node"><title>Cp</title>
<a xlink:href="http://ncatlab.org/nlab/show/complete topological vector space" xlink:title="Complete">
<polygon style="fill:none;stroke:black;" points="288,-252 208,-252 208,-216 288,-216 288,-252"/>
<text fill="#008B26" text-anchor="middle" x="248" y="-229.9" style="font-family:Times New Roman;font-size:14.00;">Complete</text>
</a>
</g>
<g id="edge44" class="edge"><title>Fr->Cp</title>
<path style="fill:none;stroke:black;" d="M200,-288C208,-279 217,-269 225,-260"/>
<polygon style="fill:black;stroke:black;" points="228.049,-261.831 232,-252 222.781,-257.221 228.049,-261.831"/>
</g>
<g id="node17" class="node"><title>Br</title>
<a xlink:href="http://ncatlab.org/nlab/show/Baire topological vector space" xlink:title="Baire">
<polygon style="fill:none;stroke:black;" points="360,-252 306,-252 306,-216 360,-216 360,-252"/>
<text fill="#008B26" text-anchor="middle" x="333" y="-229.9" style="font-family:Times New Roman;font-size:14.00;">Baire</text>
</a>
</g>
<g id="edge46" class="edge"><title>Fr->Br</title>
<path style="fill:none;stroke:black;" d="M218,-290C239,-280 266,-267 297,-253"/>
<polygon style="fill:black;stroke:black;" points="298.958,-255.916 306,-248 295.559,-249.797 298.958,-255.916"/>
</g>
<g id="edge48" class="edge"><title>Fr->Me</title>
<path style="fill:none;stroke:black;" d="M218,-291C221,-290 224,-289 227,-288 286,-268 305,-270 368,-252"/>
<polygon style="fill:black;stroke:black;" points="369.427,-255.226 378,-249 367.416,-248.521 369.427,-255.226"/>
</g>
<g id="node19" class="node"><title>SR</title>
<a xlink:href="http://ncatlab.org/nlab/show/semi-reflexive topological vector space" xlink:title="Semi-Reflexive">
<polygon style="fill:none;stroke:black;" points="600,-252 482,-252 482,-216 600,-216 600,-252"/>
<text fill="#008B26" text-anchor="middle" x="541" y="-229.9" style="font-family:Times New Roman;font-size:14.00;">Semi-Reflexive</text>
</a>
</g>
<g id="edge38" class="edge"><title>Re->SR</title>
<path style="fill:none;stroke:black;" d="M577,-288C572,-280 565,-270 559,-260"/>
<polygon style="fill:black;stroke:black;" points="561.8,-257.9 553,-252 556.2,-262.1 561.8,-257.9"/>
</g>
<g id="node22" class="node"><title>Bl</title>
<a xlink:href="http://ncatlab.org/nlab/show/barrelled topological vector space" xlink:title="Barrelled">
<polygon style="fill:none;stroke:black;" points="554,-180 476,-180 476,-144 554,-144 554,-180"/>
<text fill="#008B26" text-anchor="middle" x="515" y="-157.9" style="font-family:Times New Roman;font-size:14.00;">Barrelled</text>
</a>
</g>
<g id="edge36" class="edge"><title>Re->Bl</title>
<path style="fill:none;stroke:black;" d="M600,-288C610,-269 622,-238 609,-216 599,-198 581,-186 564,-178"/>
<polygon style="fill:black;stroke:black;" points="564.585,-174.464 554,-174 561.985,-180.964 564.585,-174.464"/>
</g>
<g id="edge50" class="edge"><title>Cn->Bo</title>
<path style="fill:none;stroke:black;" d="M135,-216C132,-208 128,-198 125,-190"/>
<polygon style="fill:black;stroke:black;" points="127.964,-187.985 121,-180 121.464,-190.585 127.964,-187.985"/>
</g>
<g id="node21" class="node"><title>LC</title>
<a xlink:href="http://ncatlab.org/nlab/show/locally complete topological vector space" xlink:title="Locally Complete">
<polygon style="fill:none;stroke:black;" points="314,-180 182,-180 182,-144 314,-144 314,-180"/>
<text fill="#008B26" text-anchor="middle" x="248" y="-157.9" style="font-family:Times New Roman;font-size:14.00;">Locally Complete</text>
</a>
</g>
<g id="edge52" class="edge"><title>Cn->LC</title>
<path style="fill:none;stroke:black;" d="M169,-216C182,-207 199,-196 213,-186"/>
<polygon style="fill:black;stroke:black;" points="215.621,-188.459 222,-180 211.738,-182.635 215.621,-188.459"/>
</g>
<g id="edge54" class="edge"><title>Cp->LC</title>
<path style="fill:none;stroke:black;" d="M248,-216C248,-208 248,-199 248,-190"/>
<polygon style="fill:black;stroke:black;" points="251.5,-190 248,-180 244.5,-190 251.5,-190"/>
</g>
<g id="node23" class="node"><title>QC</title>
<a xlink:href="http://ncatlab.org/nlab/show/quasi-complete topological vector space" xlink:title="Quasi-Complete">
<polygon style="fill:none;stroke:black;" points="458,-180 332,-180 332,-144 458,-144 458,-180"/>
<text fill="#008B26" text-anchor="middle" x="395" y="-157.9" style="font-family:Times New Roman;font-size:14.00;">Quasi-Complete</text>
</a>
</g>
<g id="edge56" class="edge"><title>Cp->QC</title>
<path style="fill:none;stroke:black;" d="M285,-216C305,-206 329,-195 350,-184"/>
<polygon style="fill:black;stroke:black;" points="351.283,-187.26 359,-180 348.44,-180.863 351.283,-187.26"/>
</g>
<g id="edge58" class="edge"><title>Br->Bl</title>
<path style="fill:none;stroke:black;" d="M360,-220C363,-219 366,-217 369,-216 408,-198 421,-196 467,-180"/>
<polygon style="fill:black;stroke:black;" points="467.62,-183.483 476,-177 465.406,-176.842 467.62,-183.483"/>
</g>
<g id="edge60" class="edge"><title>Me->Bo</title>
<path style="fill:none;stroke:black;" d="M378,-218C375,-218 372,-217 369,-216 287,-194 262,-200 174,-180"/>
<polygon style="fill:black;stroke:black;" points="174.492,-176.529 164,-178 173.119,-183.393 174.492,-176.529"/>
</g>
<g id="edge62" class="edge"><title>Me->Pc</title>
<path style="fill:none;stroke:black;" d="M464,-219C467,-218 470,-217 473,-216 509,-203 551,-191 584,-180"/>
<polygon style="fill:black;stroke:black;" points="585.427,-183.226 594,-177 583.416,-176.521 585.427,-183.226"/>
</g>
<g id="edge64" class="edge"><title>SR->QC</title>
<path style="fill:none;stroke:black;" d="M504,-216C485,-206 461,-195 440,-184"/>
<polygon style="fill:black;stroke:black;" points="441.56,-180.863 431,-180 438.717,-187.26 441.56,-180.863"/>
</g>
<g id="node25" class="node"><title>QB</title>
<a xlink:href="http://ncatlab.org/nlab/show/quasi-barrelled topological vector space" xlink:title="Quasi-Barrelled">
<polygon style="fill:none;stroke:black;" points="234,-108 112,-108 112,-72 234,-72 234,-108"/>
<text fill="#008B26" text-anchor="middle" x="173" y="-85.9" style="font-family:Times New Roman;font-size:14.00;">Quasi-Barrelled</text>
</a>
</g>
<g id="edge66" class="edge"><title>Bo->QB</title>
<path style="fill:none;stroke:black;" d="M129,-144C136,-135 144,-125 152,-116"/>
<polygon style="fill:black;stroke:black;" points="154.8,-118.1 158,-108 149.2,-113.9 154.8,-118.1"/>
</g>
<g id="edge68" class="edge"><title>Bl->QB</title>
<path style="fill:none;stroke:black;" d="M476,-147C473,-146 470,-145 467,-144 392,-123 304,-108 244,-99"/>
<polygon style="fill:black;stroke:black;" points="244.299,-95.5125 234,-98 243.602,-102.478 244.299,-95.5125"/>
</g>
<g id="node26" class="node"><title>Sp</title>
<a xlink:href="http://ncatlab.org/nlab/show/sequentially complete topological vector space" xlink:title="Sequentially Complete">
<polygon style="fill:none;stroke:black;" points="480,-108 310,-108 310,-72 480,-72 480,-108"/>
<text fill="#008B26" text-anchor="middle" x="395" y="-85.9" style="font-family:Times New Roman;font-size:14.00;">Sequentially Complete</text>
</a>
</g>
<g id="edge70" class="edge"><title>QC->Sp</title>
<path style="fill:none;stroke:black;" d="M395,-144C395,-136 395,-127 395,-118"/>
<polygon style="fill:black;stroke:black;" points="398.5,-118 395,-108 391.5,-118 398.5,-118"/>
</g>
<g id="node27" class="node"><title>Nm</title>
<a xlink:href="http://ncatlab.org/nlab/show/normal topological vector space" xlink:title="Normal">
<polygon style="fill:none;stroke:black;" points="679,-108 613,-108 613,-72 679,-72 679,-108"/>
<text fill="#008B26" text-anchor="middle" x="646" y="-85.9" style="font-family:Times New Roman;font-size:14.00;">Normal</text>
</a>
</g>
<g id="edge72" class="edge"><title>Pc->Nm</title>
<path style="fill:none;stroke:black;" d="M646,-144C646,-136 646,-127 646,-118"/>
<polygon style="fill:black;stroke:black;" points="649.5,-118 646,-108 642.5,-118 649.5,-118"/>
</g>
<g id="node28" class="node"><title>Mk</title>
<a xlink:href="http://ncatlab.org/nlab/show/Mackey topological vector space" xlink:title="Mackey">
<polygon style="fill:none;stroke:black;" points="206,-36 140,-36 140,-1.06581e-14 206,-0 206,-36"/>
<text fill="#008B26" text-anchor="middle" x="173" y="-13.9" style="font-family:Times New Roman;font-size:14.00;">Mackey</text>
</a>
</g>
<g id="edge74" class="edge"><title>QB->Mk</title>
<path style="fill:none;stroke:black;" d="M173,-72C173,-64 173,-55 173,-46"/>
<polygon style="fill:black;stroke:black;" points="176.5,-46 173,-36 169.5,-46 176.5,-46"/>
</g>
<g id="node29" class="node"><title>CP</title>
<a xlink:href="http://ncatlab.org/nlab/show/countably paracompact topological vector space" xlink:title="Countably Paracompact">
<polygon style="fill:none;stroke:black;" points="735,-36 557,-36 557,-1.06581e-14 735,-0 735,-36"/>
<text fill="#008B26" text-anchor="middle" x="646" y="-13.9" style="font-family:Times New Roman;font-size:14.00;">Countably Paracompact</text>
</a>
</g>
<g id="edge76" class="edge"><title>Nm->CP</title>
<path style="fill:none;stroke:black;" d="M646,-72C646,-64 646,-55 646,-46"/>
<polygon style="fill:black;stroke:black;" points="649.5,-46 646,-36 642.5,-46 649.5,-46"/>
</g>
</g>
</svg>

category: svg


Twelf

A [[logical framework]].

## Related concepts

* [[Automath]]

* [[Elf]]

## References

* [Twelf homepage](http://twelf.org/wiki/Main_Page)

## Installation notes

The precompiled binaries available on the [Twelf download page](http://twelf.org/wiki/Download) are for rather old machines.  However, it appears that they can still be run on modern machines as long as the relevant 32-bit libraries are present and have the expected names.  On Ubuntu 18.04 the following sequence of commands sufficed for at least one person to be able to run the Twelf binaries:

```
sudo dpkg --add-architecture i386
sudo apt install libgmp10:i386
sudo ln -s /usr/lib/i386-linux-gnu/libgmp.so.10 /usr/lib/i386-linux-gnu/libgmp.so.3
```

category: software