inner product space

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### Context
#### Linear algebra
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[[!include higher linear algebra - contents]]
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# Inner product spaces
* table of contents
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## Idea

An _inner product_ on a [[vector space]] $V$ (also "scalar product" in the sense of: with values in "[[scalars]]", namely in the [[ground field]] $\mathbb{K}$) is a pairing $\langle\text{-},\text{-}\rangle$ of [[vectors]] to [[scalars]]
$$
  v_1,\, v_2
  \,\in\,
  V
  \;\;\;\;\;\;\;\;\;
    \vdash
  \;\;\;\;\;\;\;\;\;
  \langle v_1,\, v_2 \rangle
  \,\in\,
  \mathbb{K}
$$
which is  [[bilinear form|bilinear]] *or rather* -- namely if $\mathbb{K}$ is understood with a [[star-algebra|star]]-[[involution]] (such as the [[complex numbers]] under [[complex conjugation]]) -- [[sesquilinear form|sesquilinear]], in which case one also speaks of a *[[Hermitian inner product]]*, for definiteness.

Often one requires such a pairing to be non-degenerate or even [[positive-definite]] in order to qualify as an inner product, standard conventions depend on context.

For example, the ([[Hermitian inner product|Hermitian]]) inner product on a [[Hilbert space]] is required to be positive definite, as is that on [[tangent spaces]] in [[Riemannian geometry]], but the inner product on [[tangent spaces]] in [[pseudo-Riemannian geometry]] is only required to be non-degenerate.

The group of [[automorphisms]] of an inner product space is the [[orthogonal group of an inner product space]].

## Definitions

Let $V$ be a [[vector space]] over the [[field]] (or more generally a [[ring]]) $\mathbb{K}$.  Suppose that $\mathbb{K}$ is equipped with an [[involution]] $r \mapsto \overline{r}$, called _conjugation_; in many examples, this will simply be the [[identity function]], but not always (for the [[complex numbers]] one typically consider the involution by [[complex conjugation]]).

Then a ([[Hermitian inner product|Hermitian]]) _inner product_ on $V$ is a function
$$
  \langle\text{-},\, \text{-}\rangle
  \;\colon\;
  V \times V \to k
$$
that is (1--3) _sesquilinear_ (or _bilinear_ when the involution is the identity) and (4) _conjugate-symmetric_ (or _symmetric_ when the involution is the identity).  That is:

1.  $ \langle 0, x \rangle = 0 $ and $ \langle x, 0 \rangle = 0 $;
2.  $ \langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle $ and $ \langle x, y + z \rangle = \langle x, y \rangle + \langle x, z \rangle $;
3.  $ \langle c x, y \rangle = \bar{c} \langle x, y \rangle $ and
$\langle x, c y \rangle = \langle x, y \rangle c$;
4.  $ \langle x, y \rangle = \overline{\langle y, x \rangle} $.

Here we use the _physicist\'s convention_ that the inner product is antilinear (= conjugate-linear) in the first variable rather than in the second, rather than the _mathematician\'s convention_, which is the reverse.

Note that we use the same ring as values of the inner product as for [[scalars]], and that $\langle x, c y \rangle = \langle x, y \rangle c$ is written with $c$ on the right for the case that we deal with noncommutative division ring.

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Are the two conventions really equivalent when $k$ is noncommutative?  —Toby
=--

(The axiom list above is rather redundant.  First of all, (1) follows from (3) by setting $c = 0$; besides that, (1--3) come in pairs, only one of which is needed, since each half follows from the other using (4).  It is even possible to derive (3) from (2) under some circumstances.)

An __inner product space__ is simply a vector space equipped with an inner product.

We define a function ${\|{-}\|^2}\colon V \to k$ by ${\|x\|^2} = \langle x, x \rangle$; this is called the __norm__ of $x$.  As the notation suggests, it is common to take the norm of $x$ to be the square root of this expression in contexts where that makes sense, but for us ${\|{-}\|^2}$ is an atomic symbol.  The norm of $x$ is __real__ in that it equals its own conjugate, by (4).

## Definiteness
   {#definite}

Notice that, by (1), $\langle 0, y \rangle = 0$ for all $y$.  In fact, the [[subset]] $\{ x \;|\; \forall y,\; \langle x, y \rangle = 0 \}$ is a [[linear subspace]] of $V$.  Of course, we also have ${\|0\|^2} = 0$, but $\{ x \;|\; {\|x\|^2} = 0 \}$ may not be a subspace.  These observations motivate some possible conditions on the inner product:

*  The inner product is __semidefinite__ if $\{ x \;|\; {\|x\|^2} = 0 \}$ *is* closed under addition (and hence is a subspace); it\'s __indefinite__ if there are $x$ and $y$ with ${\|x\|^2} = 0$ and ${\|y\|^2} = 0$ but ${\|x + y\|^2} \ne 0$.
*  The inner product is __nondegenerate__ if $x = 0$ whenever $\langle x, y \rangle = 0$ for all $y$; it\'s __degenerate__ if there is $x$ with $x \ne 0$ but $\langle x, y \rangle = 0$ for all $y$.
*  The inner product is __definite__ if $x = 0$ whenever ${\|x\|^2} = 0$; there ought to be a term for the condition that there is some $x \ne 0$ with ${\|x\|^2} = 0$, so let\'s call it __nondefinite__.

(In [[constructive mathematics]], we usually want an [[inequality relation]] relative to which the vector-space operations and the inner product are [[strongly extensional function|strongly extensional]], to make sense of the conditions with $\ne$ in them.  We can also use [[contrapositives]] to put $\ne$ in the other conditions, which makes them stronger if the inequality relation is [[tight relation|tight]].)

An inner product is definite iff it\'s both semidefinite and nondegenerate.  Semidefinite inner products behave very much like definite ones; you can mod out by the elements with norm $0$ to get a [[quotient vector space|quotient space]] with a definite inner product.  In a similar way, every inner product space has a nondegenerate quotient.

Now suppose that $k$ is equipped with a [[partial order]].  (Note that the [[complex numbers]] are standardly so equipped, with $a \leq b$ iff $b - a$ is a nonnegative real.)  Then we can consider other conditions on the inner product:

*  The inner product is __positive semidefinite__, or simply __positive__, if ${\|x\|^2} \geq 0$ always.
*  The inner product is __positive definite__ if it is both positive and definite, in other words if ${\|x\|^2} \gt 0$ whenever $x \ne 0$.
*  The inner product is __negative semidefinite__, or simply __negative__, if ${\|x\|^2} \leq 0$ always.
*  The inner product is __negative definite__ if it is both positive and definite, in other words if ${\|x\|^2} \lt 0$ whenever $x \ne 0$.

In this case, we have these theorems:

*  A positive or negative inner product really is semidefinite (as the terminology implies).
*  Conversely, a semidefinite inner product is either positive or negative, at least if we use [[classical logic]].  (In [[constructive mathematics]], we can say that a semidefinite inner product is either positive or negative, indeed is positive [[xor]] negative, as long as it is nonzero in the sense that at least some $\lang{x,y}\rang$ is [[apart]] from zero.  And of course, the zero inner product is the only one that is both positive and negative.)
*  Hence, a definite inner product is either positive or negative definite.  (We can strengthen this to 'xor' and state it constructively if we assume that $V$ is nonzero in that there exists $x$ with $x \ne 0$.)
*  Conversely, an inner product is indefinite if and only if some norms are positive and some are negative.  (No constructive caveats here!)

Negative (semi)definite inner products behave very much like positive (semi)definite ones; you can turn one into the other by multiplying all inner products by $-1$.

The study of positive definite inner product spaces (hence essentially of all semidefinite inner product spaces over partially ordered fields) is essentially the study of [[Hilbert spaces]].  (For Hilbert spaces, one usually uses a [[topological field]], typically $\mathbb{C}$, and requires a completeness condition, but this does not effect the algebraic properties much.)  The study of indefinite inner product spaces is very different; see the [English Wikipedia article](http://secure.wikimedia.org/wikipedia/en/wiki/Krein_space) on [[Krein space]]s for some of it.

All of this definiteness terminology may now be applied to an *[[linear operator|operator]]* $T$ on $V$, since $(x, y) \mapsto \langle{x, T y}\rangle$ is another inner product (on $\dom T$, if necessary).  See [[positive operator]].

## Examples

* [[Euclidean space]];

* [[Hilbert spaces]] (over $\mathbb{C}$), for example $\mathbb{C}^n$;

* Finite-dimensional modules over $\mathbb{H}$, the [[quaternions]].

* [[semisimple Lie algebras]] with the negative of the [[Killing form]] are positive-definite inner product spaces.

* [[inner product abelian groups]]

## Related concepts

* [[orthogonal structure]]

* [[self-dual object]]

* [[inner product of vector bundles]]

* [[bra-ket]]

## References

Original discussion of [[Hermitian inner products]] in the context of defining [[Hilbert spaces]] (as mathematical foundations of [[quantum mechanics]]):

* {#vonNeumann30} [[John von Neumann]], p. 64 in: *Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren*, Math. Ann. **102** (1930) 49–131 [[doi:10.1007/BF01782338](https://doi.org/10.1007/BF01782338)]

* [[John von Neumann]]:

  p. 21 in: *Mathematische Grundlagen der Quantenmechanik*, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  pp. 38 in: *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

Textbook accounts in the context of [[operator algebra]]:

* [[Richard V. Kadison]], [[John R. Ringrose]], §2.1 in: *Fundamentals of the theory of operator algebras* Vol I *Elementary Theory*, Graduate Studies in Mathematics **15**, AMS (1997) [[ISBN:978-0-8218-0819-1](https://bookstore.ams.org/gsm-15)]

* [[Bruce Blackadar]], §I.1.1 in: *Operator Algebras -- Theory of $C^\ast$-Algebras and von Neumann Algebras*, Encyclopaedia of Mathematical Sciences **122**, Springer (2006) [[doi:10.1007/3-540-28517-2](https://doi.org/10.1007/3-540-28517-2)]

Discussion of plain inner products (without star-involution) in terms of [[self-dual objects]] in suitable [[symmetric monoidal category|symmetric]] [[monoidal categories]]:

* {#Selinger12} [[Peter Selinger]], _Autonomous categories in which $A \simeq A^\ast$_, talk at QPL 2012 ([[SelingerSelfDual.pdf:file]])

[[!redirects inner product]]
[[!redirects inner products]]
[[!redirects inner product space]]
[[!redirects inner product spaces]]

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


inquiry

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Inquiry).

[[!redirects enquiry]]


inquiry driven system

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview).


inquiry driven system > history

< [[inquiry driven system]]


inserter

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###Context###
#### 2-Category theory
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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#Contents#
* table of contents
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## Idea

An _inserter_ is a particular kind of [[2-limit]] in a [[2-category]], which universally inserts a [[2-morphism]] between a pair of parallel 1-morphisms.

## Definition

Let $f,g\colon A\rightrightarrows B$ be a pair of parallel 1-morphisms in a 2-category.  The **inserter** of $f$ and $g$ is a universal object $V$ equipped with a morphism $v\colon V\to A$ and a 2-morphism $\alpha\colon f v \to g v$.

More precisely, universality means that for any object $X$, the induced functor
$$Hom(X,V) \to Ins(Hom(X,f),Hom(X,g)) $$
is an equivalence, where $Ins(Hom(X,f),Hom(X,g))$ denotes the category whose objects are pairs $(u,\beta)$ where $u\colon X\to A$ is a morphism and $\beta\colon f u \to g u$ is a 2-morphism.  If this functor is an *isomorphism* of categories, then we say that $V\xrightarrow{v} A$ is a **strict inserter**.

Inserters and strict inserters can be described as a certain sort of [[weighted limit|weighted]] 2-limit, where the diagram shape is the [[walking structure|walking]] parallel pair $P = (\cdot \rightrightarrows\cdot)$ and the weight $P\to Cat$ is the diagram
$$1 \;\rightrightarrows\; I$$
where $1$ is the [[terminal category]] and $I$ is the [[interval category]].  Note that inserters are not equivalent to any sort of [[conical limit|conical]] 2-limit.

An inserter in $K^{op}$ (see [[opposite 2-category]]) is called a **coinserter** in $K$.

## Example: the strict 2-category of categories

In the strict 2-category [[Cat]] of categories, inserters can be concretely described as follows.

The input data is two categories, $A$ and $B$,
and two functors, $F,G\colon A\to B$.
The objects of the inserter are pairs $(X,b)$, where $X\in A$
and $b\colon F(X)\to G(X)$.
Morphisms $(X,b)\to(X',b')$ are morphisms $f\colon X\to X'$ such that
$b'\circ F(f)=G(f)\circ b$.
The functor from the inserter to $A$ discards the data of $b$.

The inserter in $Cat$ is also called the category of [[dialgebras]].

## Properties

* Any inserter is a [[faithful morphism]], and also a [[conservative morphism]], but not in general a [[fully faithful morphism]].

* Any strict inserter is, in particular, an inserter.  (This is not true for all strict 2-limits.)

* Strict inserters are (by definition) a particular class of [[PIE-limits]].

## Examples

* Let $C$ denote a category and $F : C \rightarrow C$ denote a functor. Then the notion of an [[algebra for an endofunctor]] of $F$ corresponds to the inserter of $F$ and $\mathrm{id}_C$, and the notion of a [[coalgebra for an endofunctor]] of $F$ corresponds to the inserter of $\mathrm{id}_C$ and $F$.

## References

* G.J. Bird, G.M. Kelly, A.J. Power, R.H. Street, _Flexible limits for 2-categories_, J. Pure and Applied Algebra __61__:1, 1-27, 1989, doi:10.1016/0022-4049(89)90065-0

* [[Fernando Lucatelli Nunes]], _Freely generated $n$-categories, coinserters and presentations of low dimensional categories_, [arxiv/1704.04474](https://arxiv.org/abs/1704.04474)

[[!redirects inserters]]
[[!redirects strict inserter]]
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[[!redirects coinserter]]
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[[!redirects strict coinserter]]
[[!redirects strict coinserters]]


instanton

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#### Fields and quanta
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#### Topological physics
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# Contents
* table of contents
{: toc}

## Idea

### In quantum field theory

In [[quantum field theory]] an _instanton_ is a [[configuration space|field configuration]] with a "topological twist": not in the connected component of the trivial field configurations. Specifically for [[gauge fields]] which mathematically are represented by [[principal connections]], an instanton is a nontrivial underlying [[principal bundle]] (or similarly non-trivial [[associated bundle|associated]] [[vector bundle]]).

The term derives from the special case of instantons on a [[sphere]] but modeled as field configurations on a [[Euclidean space]] constrained to vanish asymptotically. These look like solutions localized in [[spacetime]]: "at an instant".

Instantons enter the [[axial anomaly]]/[[chiral anomaly]] in the [[standard model of particle physics]] which is thought to be a source of [[baryogenesis]] in the early universe. Generally the [[QCD]] [[vacuum]] state is argued to consist of a [[superposition]] of all possible instanton sectors, see at _[[QCD instanton]]_.

### In string theory

More generally, in [[string theory]] a [[brane]] which wraps a completely [[spacelike]] cycle in [[target space]] is called an _instanton_, since the [[worldvolume]] of such a brane is localized in the [[time]]-direction of target space. Under passing to the [[effective quantum field theory]] of the string theory, this reproduces many instantons in the sense of quantum field theory above.

## Examples

* [[Yang-Mills instanton]]

  * [[instanton in QCD]]

  * [[BPST-instanton]]

* [[Coleman-De Luccia instanton]]

* [[S-brane]]

* [[wrapped brane]]

  * [[membrane instanton]]

  * [[D3-brane instanton]]

  * [[M5-brane instanton]]

## Related concepts

* [[Yang-Mills instanton]]

* [[instanton sector]]

* [[small instanton]]

* [[non-perturbative effect]], [[non-perturbative quantum field theory]], [[non-perturbative string theory]]

* [[renormalon]]

* [[quantum tunneling]]

* at [[thermal field theory|finite temperature]]: [[caloron]]

  * [[caloron correspondence]]

* [[renormalon]]

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

* [[brane]]

* [[Spin(7)-instanton]]

* [[worldsheet instanton]], [[membrane instanton]]

* [string theory FAQ -- Isn't it fatal that the string perturbation series does not converge?](string+theory+FAQ#NonConvergenceOfPerturbationSeries)

[[!include gauge field - table]]

## References
 {#References}

### In quantum field theory and specifically Yang-Mills theory

* [[Daniel Freed]], [[Karen Uhlenbeck]], _Instantons and Four-Manifolds_, Mathematical Sciences Research Institute Publications, Springer 1991 ([doi:10.1007/978-1-4613-9703-8](https://link.springer.com/book/10.1007/978-1-4613-9703-8))

* [[Nicholas Manton]], [[Paul Sutcliffe]], _Topological solitons_, Cambridge Monographs on Math. Physics, [gBooks](http://books.google.com/books?id=e2tPhFdSUf8C)

* [[Werner Nahm]], _Self-dual monopoles and calorons_, in _Group theoretical methods in physics_ (Trieste, 1983), pages 189-200. Springer, Berlin (1984) ([journal](http://www.springerlink.com/index/y005405w74862327.pdf))

* {#Tong05} [[David Tong]], _TASI Lectures on Solitons_ ([arXiv:hep-th/0509216](https://arxiv.org/abs/hep-th/0509216)), _Lecture 1: Instantons_ ([pdf](http://www.damtp.cam.ac.uk/user/tong/tasi/instanton.pdf))

* [[Mikio Nakahara]], Section 10.5.5 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

See also literature at [[Yang-Mills instanton]].

Yang-Mills instantons on spaces other than just spheres are explicitly discussed in

* [[Gabor Kunstatter]], _Yang-mills theory in a multiply connected three space_, Mathematical Problems in Theoretical Physics: Proceedings of the VIth International Conference on Mathematical Physics Berlin (West), August 11-20,1981. Editor: R. Schrader, R. Seiler, D. A. Uhlenbrock, Lecture Notes in Physics, vol. 153, p.118-122 ([web](http://adsabs.harvard.edu/abs/1982LNP...153..118K))

based on

* [[Chris Isham]] [[Gabor Kunstatter]], Phys. Letts. v.102B, p.417, 1981. ([doi](https://doi.org/10.1016/0370-2693%2881%2991244-2))

* [[Chris Isham]] [[Gabor Kunstatter]], J. Math. Phys. v.23, p.1668, 1982. ([doi](https://doi.org/10.1063/1.525552))

A generalization is discussed in

* [[Edward Frenkel]], [[A. Losev]], [[Nikita Nekrasov]], _Instantons beyond topological theory I_ ([arXiv:hep-th/0610149](https://arxiv.org/abs/hep-th/0610149))

* [[Edward Frenkel]], [[A. Losev]], [[Nikita Nekrasov]], _Instantons beyond topological theory II_ ([arXiv:hep-th/0610149](https://arxiv.org/abs/0803.3302))

Expositions and summaries of this are in

* [[Edward Frenkel]], [[A. Losev]], [[Nikita Nekrasov]], _Notes on instantons in topological field theory and beyond_ ([arXiv:hep-th/0702137](https://arxiv.org/abs/hep-th/0702137))

* [[Jacques Distler]], _Localized_ (2006) ([blog post](https://golem.ph.utexas.edu/~distler/blog/archives/001030.html))

### Relation to Skyrmions, calorons, monopoles

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

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

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

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

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

### In string theory

In [[string theory]] (for [[D-branes]]).

* {#Witten00} [[Edward Witten]], _World-Sheet Corrections Via D-Instantons_, JHEP 0002:030, 2000 ([arXiv:hep-th/9907041](http://arxiv.org/abs/hep-th/9907041))

* Albion Lawrence, [[Nikita Nekrasov]], _Instanton sums and five-dimensional gauge theories_, Nucl.Phys. B513 (1998) 239-265 ([arXiv:hep-th/9706025](http://arxiv.org/abs/hep-th/9706025))

The study of M-brane instantons originates around

* {#BeckerBeckerStrominger95} [[Katrin Becker]], [[Melanie Becker]], [[Andrew Strominger]], _Fivebranes, Membranes and Non-Perturbative String Theory_, Nucl.Phys.B456:130-152,1995 ([arXiv:hep-th/9507158](http://arxiv.org/abs/hep-th/9507158))

Specifically [[membrane instantons]] are further discussed in

* [[Jeffrey Harvey]], [[Gregory Moore]], _Superpotentials and Membrane Instantons_ ([arXiv:hep-th/9907026](http://arxiv.org/abs/hep-th/9907026))

and 5-brane instantons in

* [[Edward Witten]], _Non-Perturbative Superpotentials In String Theory_, Nucl.Phys.B474:343-360,1996 ([arXiv:hep-th/9604030](http://arxiv.org/abs/hep-th/9604030))

In the context of [[F-theory]] and [[M5-brane]] instantons:

* Max Kerstan, [[Timo Weigand]], _Fluxed M5-instantons in F-theory_ ([arXiv:1205.4720](http://arxiv.org/abs/1205.4720))

[[!redirects instanton]]
[[!redirects instantons]]


instanton in QCD

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#Contents#
* table of contents
{:toc}

## Idea

In [[QCD]] [[instanton]] configurations of the [[gauge field]] control the experimentally observed [[vacuum]] structure of the theory.

One considers QCD on a [[Minkowski spacetime]] [[Wick rotation|Wick rotated]] to a Euclidean $\mathbb{R}^4$ and assumes the [[field strength]] to [[vanish at infinity]]. This makes the gauge field configurations be equivalently $SU(3)$-[[principal connections]] on the 4-[[sphere]] $S^4$. The [[second Chern class]] of the underlying $SU(3)$-[[principal bundle]] is called the [[Yang-Mills instanton]]-number of the gauge field configuration. The theoretical/experimentally observed [[vacuum]] of [[QCD]] is some [[superposition]] of gauge fields in various instanton number sectors.

In particular the **instanton liquid model** in QCD assumes that the [[vacuum]] (ground state of QCD) is populated by instanton field configurations of average radius $1/3 fm$ ([[femtometres]]) with a density of 1 such instanton per $fm^4$. Some details of this model remain subtle, see notably ([Schaefer-Shuryak](#SchaeferShuryak)) for a survey.

This instanton sea model suffers however from infrared divergencies that either require a finite-volume cutoff or appeal to other effects (e.g. [Witten 98](#Witten98)).

But [[lattice gauge field theory]] simulations demonstrate numerically that the QCD vacuum is filled with instantons (e.g. [Gruber 13, esp. sections 5 and 7](#Gruber13)). Whether or not their density and distribution may be or has been theoretically derived from first principles is still being discussed (e.g. [Schaefer 02](#Schaefer02))

The _Witten-Veneziano formula_ ([Witten 79](#Witten79), [Veneziano 79](#Veneziano79)) relates topological instanton charge to the mass of the $\eta'$-[[particle]] in the limiting case that the gauge group is $SU(N)$ for $N \to \infty$. Discussion of this relation for the experimentally observed finite case of $N = 3$ includes ([Creutz 11](#Creutz11)).

## Experimental evidence

An instanton vacuum is argued to account for

* formation of the gluon condensate ([[Mikhail Shifman]], [[Arkady Vainshtein]] and [[Vladimir Zakharov]], Nucl. Phys. B 147 (1979) 385).

* spontaneous [[chiral symmetry breaking]] ([Diakonov 96](#Diakonov96))

* [[baryogenesis]] in the early universe by inducing baryon number non-conservation

## Critical discussion of the literature
 {#CriticalDiscussionOfTheLiterature}

In the physics literature a style of discussion of instantons is wide-spread where the motivation of the formulas considered is dubious. In these references (for instance [Schaefer-Shuryak 96](#SchaeferShuryak) and [Forkel 00](#Forkel) and many more) it is said that

1. a YM field of minimal energy, hence of vanishing [[field strength]] is one that is pure gauge;

1. that therefore on a [[Minkowski spacetime]] $\mathbb{R}^4$ in [[temporal gauge]] and assuming fields are [[vanishing at infinity]], these are given by [[connection on a bundle|connection]] 1-forms on $S^3$ of the form $U^{-1} d U$ for $U \colon S^3 \to G$ a smooth function, and

1. that therefore equivalence classes of such fields are given by [[homotopy]] classes of maps $S^3 \to G$, hence by the third [[homotopy group]] $\pi_3(G)$.

These points are not really correct, even though the conclusion about the classification of instanton sectors turns out to be right.

The correct statement is that winding-number sectors/instanton sectors are equivalence classes of $G$-[[principal bundles]] on the [[one-point compactification]] $S^4$ of [[Minkowski spacetime]] $\mathbb{R}^4$, and these happen to indeed be given by homotopy classes of maps $S^3 \to G$, with $S^3$ regarded as the "equator" of $S^4$. This is for instance stated explicitly in ([Nakahara](#Nakahara)) and used in many references, such as ([Witten 85](#Witten85)).

Before we come to this, first notice the problems with the above items from the standard physics reviews:

1. [[gauge field|Gauge fields]] with vanishing [[field strength]] are [[flat connections]]. These are classified by [[group]] [[homomorphisms]] from the [[fundamental group]] of the given [[space]] or [[spacetime]] to the [[gauge group]]. Hence not every such field configuration is "pure gauge", this is only the case if the [[fundamental group]] is trivial. Of course this is the case on $\mathbb{R}^n$ and $S^n$, so the first item above is correct in these cases. But beware that these authors usually justify the third item above by referring to gauge transformations between gauge fields, which is then again wrong, see the next point.

1. Flat connection on $S^3$ are not classified by homotopy classes of maps $S^3 \to G$. Instead, since $\pi_1(S^3)$ is the trivial group, there is only the trivial such flat connection. This is actually manifest in the formula for such connections which is preferred in these references, which is "$U^{-1} d U$" with $U \colon S^3 \to G$ a gauge transformation. That formula exhibits precisely the trivial gauge connection $d$ as gauge equivalent to $U^{-1} d U$.

What would give the intended classification is if one considered connections of the form $U^{-1} d U$ only modulo those gauge transformations $h \colon S^3 \to G$ which can be smoothly extended to maps $D^4 \to G$. This is what these references seem to be implicitly assuming. But this is not the gauge transformation law for connections... this is instead the gauge transformation law for transition functions of $G$-principal bundles.

And this brings us to the correct mathematical interpretation of instantons, as in ([Nakahara](#Nakahara)) and similar:

We are looking for gauge field configurations on $\mathbb{R}^4$ that have [[field strength]] [[vanishing at infinity]]. This means we are looking for $G$-[[principal connections]] on the [[one-point compactification]] $S^4$ of $\mathbb{R}^4$ such that the [[curvature]] 2-form vanishes at one "pole".

Now by the discussion at [[principal bundle]], $G$-principal bundles can be constructed for instance by [[cocycles]] in [[Cech cohomology]] of $S^4$ with [[coefficients]] in $G$. In general this means to specify $G$-valued transitions functions on overlaps of a [[good open cover]] of $S^4$. But a standard argument shows that for spheres the following [[covering]] is already sufficient: we cover the $n$-sphere $S^n$ by two hemi-$n$-spheres that overlap a bit at the equator, which is hence of the form $S^{n-1} \times [-\epsilon, \epsilon]$. A Cech cocycle on such a cover is then simply given by one single $G$-valed transition function $S^{n-1} \to G$. This is known as the _[[clutching construction]]_.

Moreover, a [[gauge transformation]] of such a Cech cocycle is given by multiplying with the restriction of $G$-valued functions on the two hemi-$n$-spheres $\simeq D^n$ to the equator. This just means that those transition functions $S^3 \to G$ are gauge-trivial which may be extended smoothly to funtions $D^4 \to G$.

In conclusion, the desired classification of instanton solutions by homotopy classes of maps $U \colon S^3 \to G$ is precisely the [[clutching construction]] of $G$-principal bundles.

Finally, in order to add the [[principal connections]] to the picture, we think of one of the two semi-$n$-spheres to be the neighbourhood of infinity and hence, since the [[field strength]] is demanded to be [[vanishing at infinity]], take the local gauge field to be the vanishing [[groupoid of Lie algebra valued 1-forms|Lie algebra valued differential 1-form]] there. But then the rules for [[Cech cohomology]] with coefficients in the [[groupoid of Lie algebra valued 1-forms]], and using the above [[clutching construction]], it follows that on the equator overlap the gauge field has to be $U^{-1} d U \in \Omega^1( S^3 \times (-\epsilon, \epsilon) )$. This we may then extend in any desired way to a (non-flat) gauge connection over the remaining semi-$n$-sphere $D^4$, hence over the actual spacetime.

Notice that this is precisely the argument which for $G = U(1)$ the [[circle group]] and for $n = 2$ is known as the argument of _[[Dirac charge quantization]]_ (which is also often misrepresented in the physics literature...)

For more see at _[Yang-Mills instanton -- SU(2)-instantons from the correct maths to the traditional physics story](Yang-Mills+instanton#FromTheMathsToThePhysicsStory)_.

## Related concepts

* [[interacting vacuum]]

* [[fiber bundles in physics]]

* [[non-perturbative effect]]

* [[Yang-Mills instanton]]

  * [[theta vacuum]]

  * [[BPST-instanton]]

* [[quantum tunneling]]

* at [[thermal field theory|finite temperature]]: [[caloron]]

  * [[caloron correspondence]]

* [[CP violation]]

* [[baryogenesis]]

* [vortex atoms -- Similarity to concepts in modern particle physics](http://ncatlab.org/nlab/show/On+Vortex+Atoms#SimilarityToParticlePhysics)

## References
 {#References}

### General

(See also the general references at _[[instanton]]_.)

Textbook account:

* [[Robert E. Marshak]], Section 10.3 of: *Conceptual Foundations of Modern Particle Physics*, World Scientific 1993 ([doi:10.1142/1767](https://doi.org/10.1142/1767))

Physics-style  surveys include the introductory lecture notes

* [[Marcos Marino]], _Instantons and Large $N$ -- An introduction to non-perturbative methods in QFT_ ([pdf](http://laces.web.cern.ch/laces/LACES10/notes/instlargen.pdf))

* {#Forkel} Hilmar Forkel, _A Primer on Instantons in QCD_ ([arXiv:hep-ph/0009136](http://arxiv.org/abs/hep-ph/0009136))

* [[Edward Shuryak]], _Physics of QCD Instantons_, Proceedings of the International School of Subnuclear Physics Erice, Sicily, Italy, August – September 2002 ([web](http://www.worldscientific.com/doi/abs/10.1142/9789812796653_0003))

and the fairly detailed account (with lots of pointers to the literature)

* {#SchaeferShuryak98} [[Thomas Schaefer]], [[Edward Shuryak]], _Instantons in QCD_, Rev. Mod. Phys.70:323-426,1998 ([arXiv:hep-ph/9610451](http://arxiv.org/abs/hep-ph/9610451))

  {#SchaeferShuryak} section III D: relation to [[confinement]]

* Dmitri Diakonov, _Instantons at work_ ([arXiv:hep-ph/0212026](http://arxiv.org/abs/hep-ph/0212026))

See also the survey in

* Marcus Hutter, _Instantons in QCD: Theory and Application of the Instanton Liquid Model_ ([arXiv:hep-ph/0107098](http://arxiv.org/abs/hep-ph/0107098))

On possible experimental signatures at the [[LHC]]:

* Simone Amoroso, Deepak Kar, Matthias Schott, _How to discover QCD Instantons at the LHC_ ([arXiv:2012.09120](https://arxiv.org/abs/2012.09120))

Detailed argument for the [[theta-vacuum]] structure from [[chiral symmetry breaking]] is offered in

* [[Curtis Callan]], R.F. Dashen, [[David Gross]], _The Structure of the Gauge Theory Vacuum_, Phys.Lett. 63B (1976) 334-340 ([spire](http://inspirehep.net/record/3673?ln=en))

* G. Morchio, [[Franco Strocchi]], _Chiral symmetry breaking and theta vacuum structure in QCD_, Annals Phys.324:2236-2254, 2009 ([arXiv:0907.2522](https://arxiv.org/abs/0907.2522))

A review in view of the [[asymptotic series]] nature of the [[Feynman perturbation series]] is in

* {#Suslov05} [[Igor Suslov]], section 4.5 of _Divergent perturbation series_, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J. Exp. Theor. Phys. 100 (2005) 1188 ([arXiv:hep-ph/0510142](https://arxiv.org/abs/hep-ph/0510142))

A survey with emphasis on the [[strong CP problem]] is in

* [[Roberto Peccei]], _The Strong CP Problem and Axions_, Lect.Notes Phys.741:3-17,2008 ([arXiv:hep-ph/0607268](http://arxiv.org/abs/hep-ph/0607268))

A more mathematically precise account which identifies instanton solutions explicitly equivalence classes of $G$-principal bundles is around Example 9.12 (p.320) and Section 10.5.5 (p.360) of

* {#Nakahara} Nakahara, _Geometry Topology and Physics_

This perspective is for instance also the one used in

* [[Edward Witten]], _Global gravitational anomalies_,  Comm. Math. Phys. Volume 100, Number 2 (1985), 197-229. ([Euclid](http://projecteuclid.org/euclid.cmp/1103943444))
 {#Witten85}

Discussion in the context of [[chiral symmetry breaking]] includes

* {#Diakonov96} Dmitri Diakonov, _Chiral Symmetry Breaking by Instantons_ ([arXiv:hep-ph/9602375](http://arxiv.org/abs/hep-ph/9602375))

Discussion in the context of [[confinement]] includes

* Y. M. Cho, _Dimensional Transmutation by Monopole Condensation in QCD_ ([arXiv:1206.6936](http://arxiv.org/abs/1206.6936))

Discussion in the context of the [[quark-gluon plasma]] includes

* {#Shuryak01} [[Edward Shuryak]], _Nonperturbative QCD and Quark-Gluon Plasma_, lectures at Trieste, 2001 ([pdf](http://users.ictp.it/~pub_off/lectures/lns010/Shuryak/Shuryak.pdf))

* Nikolai Kochelev, _Ultra-light Glueballs in Quark-Gluon Plasma_ ([arXiv:1501.07002](http://arxiv.org/abs/1501.07002))

Discussion of the infrared problem of the QCD instanton vacuum model includes

* {#Witten98} [[Edward Witten]], _Theta Dependence In The Large N Limit Of Four-Dimensional Gauge Theories_, Phys.Rev.Lett.81:2862-2865,1998 ([arXiv:hep-th/9807109](http://arxiv.org/abs/hep-th/9807109))

Discussion of instantons in [[lattice gauge theory]] includes

* {#Moore03} Guy Moore, section 7 of _Informal lectures on lattice gauge theory_, 2003 ([pdf](http://www.physics.mcgill.ca/~guymoore/latt_lectures.pdf))

* {#Creutz11} Michael Creutz, _Anomalies, gauge field topology, and the lattice_, Annals of Physics 326 (2011) 911–925 ([pdf](http://thy.phy.bnl.gov/~creutz/mypubs/pub192.pdf))

* {#Gruber13} Florian Gruber, _Topology in dynamical Lattice QCD simulations_, 2013 ([web](http://epub.uni-regensburg.de/27631/), [pdf](http://epub.uni-regensburg.de/27631/1/dissertation.pdf))

Discussion of the $\eta'$-mass from instantons includes

* {#Witten79} [[Edward Witten]], Nucl. Phys. B 156 (1979) 269.

* {#Veneziano79} [[Gabriele Veneziano]], Nucl. Phys. B 159 (1979) 213

* {#Schaefer02} [[Thomas Schaefer]], _QCD and the eta prime Mass: Instantons or Confinement?_, Phys.Rev. D67 (2003) 074502 ([arXiv:hep-lat/0211035](http://arxiv.org/abs/hep-lat/0211035))

* _A renewed look at $\eta'$ in medium, ([arXiv:1203.6740](http://arxiv.org/abs/1203.6740))

Relation to the [[scanning map]] and [[configuration spaces of points]] in:

* {#AtiyahJones78} [[Michael Atiyah]], [[John David Stuart Jones]], _Topological aspects of Yang-Mills theory_, Comm. Math. Phys. Volume 61, Number 2 (1978), 97-118 ([arXiv:1103904210](https://projecteuclid.org/euclid.cmp/1103904210))

### Relation to skyrmions, calorons, monopoles

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

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

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

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

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

[[!redirects instantons in QCD]]
[[!redirects instanton in quantum chromodynamics]]
[[!redirects instantons in quantum chromodynamics]]
[[!redirects instanton liquid]]
[[!redirects QCD instanton liquid]]
[[!redirects instanton liquid in quantum chromodynamics]]

[[!redirects QCD vacuum]]
[[!redirects QCD vacua]]

[[!redirects QCD instanton]]
[[!redirects QCD instantons]]

[[!redirects instanton sea]]


instanton sector

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#### $\infty$-Chern-Weil theory
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#Contents#
* table of contents
{:toc}

## Definition

### In general

Given a [[gauge field]] configuration modeled by a $G$-[[principal connection]], its **instanton sector** or **charge sector** is the [[equivalence class]] of the underlying [[principal bundle]].

### In $SU(n)$-Yang-Mills theory

Notably for [[Yang-Mills theory]] on a 4-dimensional [[spacetime]] and with a [[gauge group]] the [[special unitary group]] $G = SU(n)$,  $G$-[[principal bundles]] $P$ are entirely classified by their [[second Chern class]] $c_2(P)$ and hence the value $c_2(P) \in H^4(X, \mathbb{Z})$ is the instanton sector. Given the $G$-[[principal connection]] of the gauge field the image in [[de Rham cohomology]] of this class may be expressed by the [[integration of differential forms]] $[\int_{X} \langle F_\nabla , \F_\nabla \rangle] \in H_{dR}^4(X)$, where $F_{\nabla}$ is the [[curvature]] and $\langle -,-\rangle$ the [[invariant polynomial]] which corresponds to $c_2$ under the [[Chern-Weil homomorphism]].

## Related concepts

* [[non-perturbative effect]], [[non-perturbative quantum field theory]], [[non-perturbative string theory]]

* [string theory FAQ -- Isn't it fatal that the string perturbation series does not converge?](string+theory+FAQ#NonConvergenceOfPerturbationSeries)

[[!include gauge field - table]]

[[!redirects instanton sectors]]


institution

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### Context
#### Model theory
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=--
#### Category theory
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[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

The concept of an **institution** provides an abstract category-theoretic rendering of the informal concept of a _logical system_. It was introduced by Burstall and [[Joseph Goguen|Goguen]] in the 1970s in order to get a grip on the proliferation of specification logics in programming and constitutes a genuine contribution of theoretical [[computer science]] to general logic and [[abstract model theory]].

## Definition

+-- {: .num_defn #institution}
###### Definition
An _institution_ is a quadruple $\langle\Sigma , S, M, \models\rangle$ with

* $\Sigma$ a category of "signatures",

* $S:\Sigma\to Set$ a functor assigning to an object $X\in \Sigma$ the set $S(X)$ of "sentences" of signature $X$ ,

* $M:\Sigma^{op} \to Cat$ is a functor assigning to an object $X\in\Sigma^{op}$ the category of "X-structures" and

* $\models$ is a family of "satisfaction" relations specifying for each signature $X$ and $X$-structure $\mathfrak{A}$ the set of sentences $Th(\mathfrak{A})\subseteq S(X)$ "holding in $\mathfrak{A}$",

subject to the following _satisfaction condition_:

Given a morphism of signatures $\varphi:X\to X'$ and a sentence $e\in S(X)$, the sentence $S(\varphi)(e)$ i.e. the "$\varphi$-translation" of $e$ into an $X'$-sentence, holds in a X'-structure $\mathfrak{A}'$ (in signs: $\mathfrak{A}'\models S(\varphi)(e)$) precisely if $e$ holds in $M(\varphi)(\mathfrak{A}')$:

\[ \mathfrak{A}'\models S(\varphi)(e) \; iff \; M(\varphi)(\mathfrak{A}')\models e\quad .
\]

=--

## Examples

...

## Applications

...

## Related entries

* [[abstract model theory]]

* [[Beth definability theorem]]

* [[Grothendieck fibration]]

* [[signature]]

* [[computer science]]

* [[Joseph Goguen]]

## Links

* The late Joseph Goguen's [page on institutions](http://cseweb.ucsd.edu/~goguen/projs/inst.html).

* A Bremen based community homepage: [Flirts](http://www.informatik.uni-bremen.de/flirts/).

## References

One of the original sources is

* [[Joseph Goguen|J. Goguen]], R. Burstall, _Institutions: Abstract model theory for specification and programming_ , J. ACM **39** no.1 (1992) pp.95–146. ([ps-preprint](http://cseweb.ucsd.edu/~goguen/pps/ins.ps))

A high level introduction

* Răzvan Diaconescu, [Institution Theory](https://www.iep.utm.edu/insti-th/), Internet Encyclopedia of Philosophy

A monograph on the subject is

* Răzvan Diaconescu, _Institution-independent Model Theory_ , Birkhäuser Basel 2008.

More specific research papers are

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

* Răzvan Diaconescu, _Grothendieck institutions_ , App. Cat. Struc. **10** no.4 (2002) pp.383–402.

* Lucanu, D., Li, Y. F., & Dong, J. S. (2006).
 Semantic web languages–towards an institutional perspective.
 In Algebra, Meaning, and Computation (pp. 99-123). Springer, Berlin, Heidelberg. ([pdf](
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.119.5368&rep=rep1&type=pdf))

* Bao, J., Tao, J., McGuinness, D. L., & Smart, P. (2010).
 Context representation for the semantic web. ([pdf](https://eprints.soton.ac.uk/270829/))

[[!redirects institutions]]
[[!redirects Institution]]
[[!redirects Institutions]]


insulator

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#### Solid state physics
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#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]] an *insulator* is a [[crystal|crystalline]] material for which the [[chemical potential]] of the [[electrons]] is inside a sizeable gap between the [[valence band]] and the [[conduction band]]. (If there is a gap but just a small one, one speaks of a *[[semi-conductor]]*, instead.)

This situation implies that for net [[electric charge]] to be transported through the material (hence for it to become a [[conductor]]) an energy density above the gap width has to be applied from the outside, in order to lift electron states from the [[valence bundle]] to the [[conduction bundle]], where they can then propagate. Below this energy density the material hence resists the transport of electric current.

(graphics from [[schreiber:Topological Quantum Computation in TED-K|SS 22]])

Due to the energy gap, the [[valence bundle]] of an insulator is well-defined. If this has a non-trivial class in ([[twisted equivariant K-theory|twisted equivariant]]) [[topological K-theory]] of the [[Brillouin torus]], one speaks of a *[[topological insulator]]* (see at *[[K-theory classification of topological phases of matter]]*).

## Related concepts

* [[metal]]/[[conductor]], [[semi-conductor]], [[semi-metal]]

[[!redirects insulators]]


integer

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#### Arithmetic
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[[!include arithmetic geometry - contents]]
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# Contents
* table of contents
{: toc}

## Idea

An __integer__ is a [[number]] that is a [[natural number]] or the [[negative number|negative]] of one.

The [[ring]] $\mathbb{Z}$ of all integers may defined as the [[free group]] on one generator or as the [[initial object|initial]] [[ring]].

In keeping with a historical point of view in which integers are natural numbers with a sign attached, one may write

$$
  \mathbb{Z} = \{n, -n | n \in \mathbb{N}, 0 = -0\} = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \}
  \,.
$$

From an [[nPOV]], one may consider this as follows: $\mathbb{Z}$ is a [[filtered colimit]] of sets

$$\mathbb{N} \stackrel{1 + (-)}{\to} \mathbb{N} \stackrel{1 + (-)}{\to} \mathbb{N} \stackrel{1 + (-)}{\to} \ldots$$

whereby $-n \in \mathbb{Z}$ is represented by the element $0$ in the $n^{th}$ copy of $\mathbb{N}$ appearing in this diagram (starting the count at the $0^{th}$ copy). The resulting induced map to the colimit

$$\mathbb{N} \times \mathbb{N} \cong \sum_{m \in \mathbb{N}} \mathbb{N} \to \mathbb{Z}: (m, n) \mapsto n-m$$

imparts a monoid (in fact a group) structure on $\mathbb{Z}$ descended from the monoid structure on $\mathbb{N} \times \mathbb{N}$; compare double-entry bookkeeping in medieval mathematics (*[partita doppia](http://rfcwalters.blogspot.com/2011/02/mathematical-economics-double-entry.html)*).

As a group, $\mathbb{Z}$ is [[abelian group|abelian]] and is the [[Grothendieck group]] of the [[monoid]] (or [[semigroup]]) $\mathbb{N}$ of [[natural numbers]].

The monoid of natural numbers is naturally even a [[rig]] -- in fact the [[initial object|initial]] rig -- and this multiplicative structure extends to $\mathbb{Z}$ to make it a [[ring]] -- in fact the initial ring.

## Properties ##

* The integers form a [[commutative ring]].

* The integers have [[decidable equality]] and [[decidable relation|decidable order]].

* The integers are a [[strictly weakly ordered ring]] and a [[pseudolattice ordered ring]].

* The integers are a [[metric space]] and a [[normed space]].

* The integers are a [[Euclidean domain]]

* The integers satisfy the Archimedean property, making it into an [[Archimedean integral domain]].

### Bijection of the integers with the natural numbers

The integers are in bijection with the natural numbers. Both the integers and the natural numbers are [[submonoids]] of the [[rational numbers]], with [[pointed set|pointed]] [[monoid]] [[monomorphisms]]
$i_\mathbb{Z}:\mathbb{Z} \hookrightarrow \mathbb{Q}$ and $i_\mathbb{N}:\mathbb{N} \hookrightarrow \mathbb{Q}$ preserving addition, zero, and one. Let us take the relation on the rational numbers
$$x:\mathbb{Q}, y:\mathbb{Q} \vdash \left| x + x + \frac{1}{2} \right| =_\mathbb{Q} y + \frac{1}{2}$$
The relation
$$x:\mathbb{Z}, y:\mathbb{N} \vdash \left| i_\mathbb{Z}(x + x) + \frac{1}{2} \right| =_\mathbb{Q} i_\mathbb{N}(y) + \frac{1}{2}$$
is a [[one-to-one correspondence]], meaning that the integers are in bijection with the natural numbers.

### Sequential Cauchy completeness

Let $x:\mathbb{N} \to \mathbb{Z}$ be a sequence of integers, and let $b:\mathbb{Z}$ be an integer. Then, there is a limit relation defined as

$$\mathrm{isLimit}(x, b) \coloneqq \forall \epsilon:\mathbb{Z}_+.\exists N:\mathbb{N}.\forall n:\mathbb{N}.(n \geq N) \to (\vert x(n) - b \vert \lt \epsilon)$$

This relation is a [[functional relation]], making the integers a [[sequentially Hausdorff space]].

A **modulus of Cauchy convergence** is a function $M:\mathbb{Z}_+ \to \mathbb{N}$ with a witness

$$p(M, x):\forall \epsilon:\mathbb{Z}_+.\forall m:\mathbb{N}.\forall n:\mathbb{N}.((m \geq M(\epsilon)) \wedge (n \geq M(\epsilon))) \to (\vert x(m) - x(n) \vert \lt \epsilon)$$

$\mathbb{Z}$ is [[sequentially Cauchy complete]] if every sequence with a modulus of Cauchy convergence has a unique limit. But $\mathbb{Z}$ is sequentially Cauchy complete, because the only sequences with a unique limit are those sequences for which there exists a natural number $N:\mathbb{N}$ such that for all natural numbers $m \geq N$ and $n \geq N$, $x(m) = x(n)$. The sequence $x$ has many moduli of Cauchy convergence $M$, where the natural number $N$ is given by $N = M(1)$.

Since the integers are the [[initial object|initial]] [[Archimedean integral domain]], the integers are also the initial sequentially Cauchy complete Archimedean integral domain. Every other sequentially Cauchy complete Archimedean integral domain is provably an [[ordered field]] and has the [[HoTT book real numbers]] $\mathbb{R}$ as an integral subdomain. That means, in the context of the [[limited principle of omniscience]], the category of sequentially Cauchy complete Archimedean integral domains is equivalent to the [[walking arrow]], with objects $\mathbb{Z}$ and $\mathbb{R}$ and homomorphism $h:\mathbb{Z} \to \mathbb{R}$.

### Cauchy completeness

Given a [[Tarski universe]] $(U, T)$ and a small type $A$ whose type reflection $T(A)$ is a [[directed set]], let $x:T(A) \to \mathbb{Z}$ be a [[net]] of integers, and let $b:\mathbb{Z}$ be an integer. Then, there is a limit relation defined as

$$\mathrm{isLimit}(x, b) \coloneqq \forall \epsilon:\mathbb{Z}_+.\exists N:T(A).\forall n:T(A).(n \geq N) \to (\vert x(n) - b \vert \lt \epsilon)$$

This relation is a [[functional relation]], making the integers a [[Hausdorff space]].

A **modulus of $U$-Cauchy convergence** is a function $M:\mathbb{Z}_+ \to T(A)$ with a witness

$$p(M, x):\forall \epsilon:\mathbb{Z}_+.\forall m:T(A).\forall n:T(A).((m \geq M(\epsilon)) \wedge (n \geq M(\epsilon))) \to (\vert x(m) - x(n) \vert \lt \epsilon)$$

$\mathbb{Z}$ is $U$-[[Cauchy complete]] if every net with a modulus of $U$-Cauchy convergence has a unique limit. But $\mathbb{Z}$ is $U$-Cauchy complete, because the only $U$-nets with a unique limit are those nets for which there exists an element $N:T(A)$ such that for all elements $m \geq N$ and $n \geq N$, $x(m) = x(n)$. The net $x$ has many moduli of $U$-Cauchy convergence $M$, where the element $N$ is given by $N = M(1)$.

Since the integers are the [[initial object|initial]] [[Archimedean integral domain]], the integers are also the initial $U$-Cauchy complete Archimedean integral domain. Every other $U$-Cauchy complete Archimedean integral domain is provably an [[ordered field]] and has the $U$-[[Dedekind real numbers]] $\mathbb{R}$ as an integral subdomain.

### Dedekind completeness

Let $\Omega$ be the [[type of all propositions]], so that the foundations is [[impredicative]]. A **Dedekind cut** is an pair $(L, U)$ of [[predicates]] such that

* there exists an integer $a:\mathbb{Z}$ such that $L(a)$
* there exists an integer $b:\mathbb{Z}$ such that $U(b)$
* for all integers $a:\mathbb{Z}$, $L(a)$ if and only if there exists an integer $b:\mathbb{Z}$ such that $a \lt b$ and $L(b)$
* for all integers $b:\mathbb{Z}$, $U(b)$ if and only if there exists an integer $a:\mathbb{Z}$ such that $a \lt b$ and $U(a)$
* for all integers $a:\mathbb{Z}$, it is not true that $L(a)$ and $U(a)$
* for all integers $a:\mathbb{Z}$ and $b:\mathbb{Z}$, if $a \lt b$, then $L(a)$ and $U(b)$.

There is a Dedekind cut for every integer $a:\mathbb{Z}$, given by $L_a(b) \coloneqq b \lt a$ and $U_a(b) \coloneqq a \lt b$. There are no other Dedekind cuts on the integers. Thus, the integers are Dedekind complete.

Since the integers are the [[initial object|initial]] [[Archimedean integral domain]], the integers are also the initial Dedekind complete Archimedean integral domain. The only other Dedekind complete Archimedean integral domain is the [[Dedekind real numbers]] $\mathbb{R}$. That means, if there is a [[type of all propositions]], the category of Dedekind complete Archimedean integral domains is equivalent to the [[walking arrow]], with objects $\mathbb{Z}$ and $\mathbb{R}$ and homomorphism $h:\mathbb{Z} \to \mathbb{R}$.

## Terminology

The underlying [[sets]] $\mathbb{Z}$ and $\mathbb{N}$ are [[isomorphic]]. Some subcultures of mathematics (and not only [[set theory|set theorists]]) use the term 'integer' synonymously for a natural number.  Computer scientists distinguish between 'unsigned integers' (natural numbers) and 'signed integers' (integers as described here).  Translations can also cause confusion with the term 'whole number'.

In [[number theory]], one generalises integers to [[algebraic integers]], an instance of the [[red herring principle]].  Accordingly, some number theorists will call the integers '[[rational integers]]' to clarify; $\mathbb{Z}$ is the [[ring of integers]] in the [[number field]] $\mathbb{Q}$ of [[rational numbers]].  (Compare, for example, [[Gaussian integers]] and [[Gaussian numbers]].)

The symbol '$\mathbb{Z}$' derives from the German word 'Zahlen', which is a generic word for 'numbers'.  (Compare [[Richard Dedekind|Dedekind]]\'s use of that word in the title of his famous book on the [[foundations]] of [[real numbers]].)

## Related concepts

* [[infinite cyclic group]]

* [[even number]], [[odd number]]

* [[natural number]], [[rational number]], [[real number]]

* [[algebraic integer]]

* [[ring of integers]]

* [[cyclotomic integer]]

* [[geometry]] modeled on the [[formal duality|formal dual]], [[Spec(Z)]], of the ring of integers is [[arithmetic geometry]]

* [[integers object]] in a [[topos]]

* [[rational zero theorem]]

* [[integers type]]

## References

The first characterization of the integers as an [[ordered integral domain]] appeared in:

* [[Hermann Grassmann]], _Lehrbuch der Arithmetik für höhere Lehranstalten_, Berlin: Enslin, 1861. ([Google Books](https://books.google.com/books?id=jdQ2AAAAMAAJ))

though the name "ordered integral domain" does not appear in the text.

History:

* [[Leo Corry]], *A Brief History of Numbers*, Oxford University Press (2015) [[ISBN:9780198702597](https://global.oup.com/academic/product/a-brief-history-of-numbers-9780198702597)]

See also:

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

Formalization in [[univalent foundations of mathematics]] ([[homotopy type theory]] with the [[univalence axiom]]):

* {#UFP13} [[Univalent Foundations Project]], Rem. 6.10.7 *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

and specifically in [[Agda]]:

* [[UniMath project]], *[agda-unimath/elementary-number-theory.integers](https://unimath.github.io/agda-unimath/elementary-number-theory.integers.html#1032)*

and specifically in [[cubical Agda]]

* [[1lab]], *[Data.Int](https://1lab.dev/Data.Int.html)*

[[!redirects integer]]
[[!redirects integers]]

[[!redirects integer number]]
[[!redirects integer numbers]]

[[!redirects additive group of integers]]
[[!redirects additive groups of integers]]


integers modulo n

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###Context###
#### Arithmetic
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[[!include arithmetic geometry - contents]]
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\tableofcontents

## Definition

Given a natural number $n$, the ring of **integers modulo $n$** is the [[quotient ring]] $\mathbb{Z}/n\mathbb{Z}$.

These are also the [[quotient]] [[rig]] $\mathbb{N}/n\mathbb{N}$.

## Properties

The integers modulo $n$ are precisely the [[finite set|finite]] [[cyclic group|cyclic]] [[rings]], since the underlying [[set]] is the [[finite set]] of [[cardinality]] $n$ and the underlying [[abelian group]] is the [[cyclic group]] of order $n$.

Given any positive integer $n$, $\mathbb{Z}/n\mathbb{Z}$ is a [[prefield ring]] whose [[monoid of cancellative elements]] consists of all integers $m$ modulo $n$ which are [[coprime]] with $n$. For $n$ a [[prime number]] this is a [[prime field]], and for $n$ a [[prime power]] this is a [[prime power local ring]].

## See also

* [[prime field]]

* [[prime power local ring]]

* [[multiplicative group of integers modulo n]]

* [[modular polynomial]]

## References

e.g. example 5 in these notes: [pdf](http://www-math.mit.edu/~dav/fields.pdf)

## External links

* Wikipedia, *[Modular arithmetic#Integers modulo n](https://en.wikipedia.org/wiki/Modular_arithmetic#Integers_modulo_n)

[[!redirects ring of integers modulo n]]
[[!redirects rings of integers modulo n]]

[[!redirects integers modulo]]

[[!redirects finite cyclic ring]]
[[!redirects finite cyclic rings]]


integers object

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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#### Induction
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[[!include induction - contents]]
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# Integers object
* table of contents
{: toc}

## Idea

Recall that a [[topos]] is a [[category]] that behaves likes the category [[Set]] of [[set|sets]].

An **integers object** [[internalization|internal to]] a topos is an [[object]] that behaves in that topos like the set $\mathbb{Z}$ of [[integers]] does in [[Set]].

## Definition

### In a topos or cartesian closed category {#CCC}

An **integers object** in a [[topos]] (or any [[cartesian closed category]]) $E$ with [[terminal object]] $1$ is

* an [[object]] $\mathbb{Z}$ in $E$

* equipped with

  * a [[morphism]] $z:1 \to \mathbb{Z}$ from the [[terminal object]] $1$;

  * an [[isomorphism]] $s : \mathbb{Z} \cong \mathbb{Z}$ ([[successor]]),

such that for all objects $A$ with morphism $z_A:1 \to A$ and isomorphism $s_A:A \cong A$, there is a unique morphism $u_A:\mathbb{Z} \to A$ such that $u_A \circ z = z_A$ and $u_A \circ s = s_A \circ u_A$.

By the [[universal property]], the integers object is unique up to [[isomorphism]].

### In a closed symmetric monoidal category

One could generalize the above definition of an integers object to any [[closed monoidal category|closed]] [[symmetric monoidal category]]: [[pointed objects in a monoidal category|pointed objects in a symmetric monoidal category]] are represented by morphisms out of the [[tensor unit]]. Thus, an integers object in a closed symmetric monoidal category $C$ with [[tensor unit]] $1$ is

* an [[object]] $\mathbb{Z}$ in $C$

* equipped with

  * a [[morphism]] $z :1 \to \mathbb{Z}$ from the [[tensor unit]] $1$;

  * an [[isomorphism]] $s : \mathbb{Z} \cong \mathbb{Z}$ (successor);

such that for all objects $A$ with morphism $z_A:1 \to A$ and isomorphism $s_A:A \cong A$, there is a unique morphism $u_A:\mathbb{Z} \to A$ such that $u_A \circ z = z_A$ and $u_A \circ s = s_A \circ u_A$.

## Free construction in a topos

The existence of an integers object in a topos $\mathcal{S}$ is equivalent to the existence of [[free group|free groups]] in $\mathcal{S}$:

+-- {: .num_prop #free_groups_in_topos}
###### Proposition
Let $\mathcal{S}$ be a topos and $\mathbf{Grp}(\mathcal{S})$ its category of internal [[group objects]]. Then $\mathcal{S}$ has an integers object precisely if the forgetful functor $U:\mathbf{Grp}(\mathcal{S})\to \mathcal{S}$ has a left adjoint.
=--

## Construction from natural numbers objects
Suppose the topos $E$ has a [[natural numbers object]] $\mathbb{N}$. Then an integers object $\mathbb{Z}$ is a [[filtered colimit]] of objects

$$\mathbb{N} \stackrel{1 + (-)}{\to} \mathbb{N} \stackrel{1 + (-)}{\to} \mathbb{N} \stackrel{1 + (-)}{\to} \ldots$$

whereby $-n:1\rightarrow\mathbb{Z}$ is represented by the morphism $z:1\rightarrow\mathbb{N}$ in the $n^{th}$ copy of $\mathbb{N}$ appearing in this diagram (starting the count at the $0^{th}$ copy). The resulting induced map to the colimit

$$\mathbb{N} \times \mathbb{N} \cong \sum_{m \in \mathbb{N}} \mathbb{N} \to \mathbb{Z}: (m, n) \mapsto n-m$$

imparts a monoid structure (in fact a group structure) on $\mathbb{Z}$ descended from the monoid structure on $\mathbb{N} \times \mathbb{N}$.

## Examples

There are many examples of natural numbers objects.

### In closed symmetric monoidal categories

\begin{example}
The [[integers]] are the integers object in the [[closed monoidal category|closed]] [[symmetric monoidal category]] [[Set]].
\end{example}

\begin{example}
In [[classical mathematics]], the [[Alexandroff compactification]] of the [[integers]], $\mathbb{Z}^* = \mathbb{Z} + \{\infty\}$, is the [[integers object]] in the closed symmetric monoidal category of [[pointed sets]] $Set_*$, with $z:\mathbb{2} \to \mathbb{Z}^*$ taking the [[boolean]] [[true]] to $\infty$ and [[false]] to $0$ and $s:\mathbb{Z}^* \cong \mathbb{Z}^*$ taking integers to its successor and $\infty$ to $\infty$. In [[constructive mathematics]], the Alexandroff compactification of the integers and the [[disjoint union]] $\mathbb{Z} + \{\infty\}$ are no longer the same; it is $\mathbb{Z} + \{\infty\}$ which remains the integers object in $Set_*$.
\end{example}

\begin{example}
The underlying abelian group of the [[ring of Laurent polynomials]] $\mathbb{Z}[X, X^{-1}]$ is the integers object in the closed symmetric monoidal category [[Ab]], with $z:\mathbb{Z} \to \mathbb{Z}[X, X^{-1}]$ taking [[integers]] to constant [[Laurent polynomials]] and the [[abelian group]] [[isomorphism]] $s:\mathbb{Z}[X, X^{-1}] \cong \mathbb{Z}[X, X^{-1}]$ multiplying Laurent polynomials by the indeterminant $X$.
\end{example}

\begin{example}
More generally, given a [[commutative ring]] $R$, the underlying $R$-module of the [[ring of Laurent polynomials]] $R[X, X^{-1}]$ is the integers object in the closed symmetric monoidal category [[RMod]], with $z:R \to R[X, X^{-1}]$ taking [[scalars]] to constant [[Laurent polynomials]] and the [[linear isomorphism]] $s:R[X, X^{-1}] \cong R[X, X^{-1}]$ multiplying Laurent polynomials by the indeterminant $X$.
\end{example}

### In a sheaf topos
 {#ExamplesInASheafTopos}

In any [[Grothendieck topos]] $E = Sh(C)$ the integers object is given by the [[constant sheaf]] on the [[set]] of ordinary integers, i.e. by the [[sheafification]] of the [[presheaf]] $C^{op} \to Set$ that is constant on the set $\mathbb{Z}$.

Similar to the case for [[natural numbers objects]], there are interesting cases in which such sheaf toposes contain objects that look like they ought to be integers objects but do not satisfy the above axioms: for instance some of the models described at [[Models for Smooth Infinitesimal Analysis]] are sheaf toposes that contain besides the standard integers object a larger object of **[[smooth natural numbers|smooth integers]]** that has [[generalized element]]s which are "infinite integers" in the sense of [[nonstandard analysis]].

## Properties

### Inverse

By definition of [[isomorphism]], there is an [[inverse]] [[isomorphism]] $p:\mathbb{Z} \cong \mathbb{Z}$, where $p \circ s = \mathrm{id}_\mathbb{Z}$ and $s \circ p = \mathrm{id}_\mathbb{Z}$.

### Initial ring object

In a category with finite products, the initial [[ring object]], an object $\mathbb{Z}$ with global elements $0:1\rightarrow\mathbb{Z}$ and $1:1\rightarrow\mathbb{Z}$, a morphism $-:\mathbb{Z}\rightarrow\mathbb{Z}$, morphsims $+:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}$ and $\times:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}$, and suitable commutative diagrams expressing the ring axioms and [[initial object|initiality]], has the structure of an integers object given by $z = 0$ and $s = x + 1$.

## Related concepts

* [[integers]]

* [[integers type]]

* [[natural numbers object]]

* [[free group]]

* [[ring object]]

* [[circle object]]

[[!redirects integers object]]
[[!redirects integers objects]]
[[!redirects integer object]]
[[!redirects integer objects]]