Raymond F. Streater

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

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=77928)

## Selected writings

On [[extremal quantum channels]]:

* [[L. J. Landau]], [[Raymond F. Streater]],  *On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras*, Linear Algebra and its Applications **193** (1993) 107-127 [doi:10.1016/0024-3795(93)90274-R]

On [[non-perturbative quantum field theory|non-perturbative]]/[[AQFT|algebraic]] [[quantum field theory]] and its key [[theorems]] ([[spin-statistics theorem]], [[PCT theorem]], ...):

* [[Raymond F. Streater]], [[Arthur S. Wightman]], *PCT, Spin and Statistics, and All That*, Princeton University Press (1989, 2000) [[ISBN:9780691070629](https://press.princeton.edu/books/paperback/9780691070629/pct-spin-and-statistics-and-all-that), [jstor:j.ctt1cx3vcq](https://www.jstor.org/stable/j.ctt1cx3vcq)]

category: people

[[!redirects Raymond Streater]]

[[!redirects Ray Streater]]
[[!redirects Ray F. Streater]]


Raymond Laflamme

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

## Selected writings

On [[quantum error correction]]:

* [[Emanuel Knill]], [[Raymond Laflamme]], *A Theory of Quantum Error-Correcting Codes*, Phys. Rev. Lett. 84:2525-2528, 2000 ([arXiv:quant-ph/9604034](https://arxiv.org/abs/quant-ph/9604034))

Mathematical argument that [[quantum computation]] with [[quantum error correction]] is practically feasible:

* [[Emanuel Knill]], [[Raymond Laflamme]], [[Wojciech H. Zurek]], *Resilient Quantum Computation: Error Models and Thresholds*, Proceedings of the Royal Society A **454** 1969 (1998) [[arXiv:quant-ph/9702058](https://arxiv.org/abs/quant-ph/9702058)]

Realization of [[quantum error correction]] in [[experiment]]:

* D. G. Cory, M. D. Price, W. Maas, [[Emanuel Knill]], [[Raymond Laflamme]], [[Wojchiek H. Zurek]], T. F. Havel, and S. S. Somaroo, *Experimental Quantum Error Correction*, Phys. Rev. Lett. 81, 2152 (1998) ([doi:10.1103/PhysRevLett.81.2152](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.2152))

Introducing [[measurement-based quantum computation]]:

* [[Emanuel Knill]], [[Raymond Laflamme]], [[Gerard J. Milburn]], *A scheme for efficient quantum computation with linear optics*, Nature **409** (2001) 46–52 [[doi:10.1038/35051009](https://doi.org/10.1038/35051009)]

On [[NMR]] [[spin resonance qbits]]:

* [[Raymond Laflamme]], [[Emanuel Knill]], et al., *Introduction to NMR Quantum Information Processing*, Proceedings of the International School of Physics "Enrico Fermi" **148** Experimental Quantum Computation and Information [[arXiv;quant-ph/0207172](https://arxiv.org/abs/quant-ph/0207172)]

Introducing the *[[DQC1]]* [[quantum complexity theory|quantum complexity]] [[complexity class|class]] ("one clean qbit"):

* {#KnillLaflamme98} [[Emanuel Knill]], [[Raymond Laflamme]], *On the Power of One Bit of Quantum Information*, Phys. Rev. Lett. **81** (1998) 5672-5675 [[arXiv:quant-ph/9802037](https://arxiv.org/abs/quant-ph/9802037), [doi:10.1103/PhysRevLett.81.5672](https://doi.org/10.1103/PhysRevLett.81.5672)]

* [[David Poulin]], [[Raymond Laflamme]], [[Gerard J. Milburn]], [[Juan Pablo Paz]], *Testing integrability with a single bit of quantum information*, Phys. Rev. A **68** 22302 (2003) [[arXiv:quant-ph/0303042](https://arxiv.org/abs/quant-ph/0303042), [doi:10.1103/PhysRevA.68.022302](https://doi.org/10.1103/PhysRevA.68.022302)]

On [[quantum computation]]:

* [[Phillip Kaye]], [[Raymond Laflamme]], [[Michele Mosca]], *An Introduction to Quantum Computing*, Oxford University Press (2007) [[pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149125645.pdf), [ISBN:9780198570493](https://global.oup.com/academic/product/an-introduction-to-quantum-computing-9780198570493?cc=at&lang=en&)]

category: people


Raymond Puzio

* [webpage](https://plus.google.com/117975038274580909142/posts)

## Related entries

* [[Planet Math]]

category: people


Raymond Stora

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

* Ph. Blanchard, [[Jose Gracia-Bondia]], S. Lazzarini and I. Todorov, _Stora’s legacy: perturbative renormalization_, to appear.

## Selected writings

On [[non-perturbative quantum field theory]]:

* [[Gerard ’t Hooft]], [[Arthur Jaffe]], [[Gerhard Mack]], [[P. K. Mitter]], [[Raymond Stora]] (eds.) *Nonperturbative Quantum Field Theory*,  NATO Science Series B **185** (1988) [[doi:10.1007/978-1-4613-0729-7](https://doi.org/10.1007/978-1-4613-0729-7)]

## Related entries

* [[BRST complex]], [[BV-BRST formalism]]

* [[main theorem of perturbative renormalization]]

category: people


Raymond Vozzo

* [website](http://maths.adelaide.edu.au/raymond.vozzo/)

## Selected writings

Discussion of [[twisted cohomology|twists]] of [[KR-theory]] by [[HZR-theory]] in degree 3 via [[bundle gerbes]] ([[Jandl gerbes]]) suitable for classifying [[D-brane charge]] on [[orientifolds]]:

* [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Real bundle gerbes, orientifolds and twisted KR-homology_, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number  ([arXiv:1608.06466](http://arxiv.org/abs/1608.06466), [doi:10.4310/ATMP.2019.v23.n8.a5](https://dx.doi.org/10.4310/ATMP.2019.v23.n8.a5))

* [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Sign choices for orientifolds_ ([arXiv:1905.06041](https://arxiv.org/abs/1905.06041))

## Related entries

* [[bundle gerbes]]
* [[caloron correspondence]]
* [[KR cohomology theory]]

category:people


Razvan Gelca

* [webpage](http://www.math.ttu.edu/~rgelca/)

## related $n$Lab entries

* [[theta function]], [[Chern-Simons theory]]

category: people


reader-writer (co)monads -- table

| ([[comonad|co]])[[monad]] name | [[underlying]] [[endofunctor]] | ([[comonad|co]])[[monad]] [[structure]] induced by |
|--|--|--|
| [[reader monad]] |  $W \to (\text{-})$ on [[cartesian closed category|cartesian]] types | [unique comonoid structure](comonoid#InACartesianMonoidalCategory) on $W$ |
| [[coreader comonad]] |  $W \times (\text{-})$ on [[cartesian closed category|cartesian]] types | [unique comonoid structure](comonoid#InACartesianMonoidalCategory) on $W$ |
| [[writer monad]] | $A \otimes (\text{-})$ on [[closed monoidal category|monoidal]] types | chosen [[monoid object|monoid structure]] on $A$ |
| [[cowriter comonad]] | $\array{A \to (\text{-}) \\ \\ A \otimes (\text{-})}$ on [[closed monoidal category|monoidal]] types | chosen [[monoid object|monoid structure]] on $A$

 chosen [[comonoid object|comonoid structure]] on $A$ |
| [[Frobenius monad|Frobenius]] (co)writer |  $\array{A \to (\text{-}) \\ A \otimes (\text{-})}$ on [[closed monoidal category|monoidal]] types | chosen [[Frobenius monoid|Frobenius monoid structure]] |


real absolute value > history

< [[absolute value]]


real algebraic K-theory

#Contents#
* table of contents
{:toc}

## Idea

_Real algebraic K-theory_ is the the $\mathbb{Z}_2$-[[equivariant cohomology]] refinement of  [[algebraic K-theory]] in analogy to how [[KR-theory]] ("real topological K-theory") is the $\mathbb{Z}_2$-equivariant refinement of [[topological K-theory]].

## References

The main text is an unpublished book draft

* [[Lars Hesselholt]], [[Ib Madsen]], _Real algebraic K-theory_ ([manuscript](http://www.math.ku.dk/~larsh/papers/s05/book.pdf))

Some details are in

* Dotto, _Stable real K-theory and real topological Hochschild homology_ ([arXiv:1212.4310](http://arxiv.org/abs/1212.4310))


real analytic space

#Contents#
* table of contents
{:toc}

## Idea

The analog of [[complex analytic space]] with the [[complex numbers]] replaced by the [[real numbers]].

## Properties

Whitney: every [[paracompact topological space|paracompact]] [[smooth manifold]] admits a real analytic structure.

Use the [[Whitney embedding theorem]] to realize the smooth manifold as the [[zero locus]] of non-degenerate [[smooth functions]], then use the [[Weierstrass approximation theorem]] to replace these by [[polynomials]]. The resulting zero locus gives the desired real-analytic version of the manifold.

([from here](http://www.math.niu.edu/~rusin/known-math/95/real.anal)):

One can also use Whitney's methods to show that two diffeomorphic real analytic manifolds are real-analytic equivalent provided that they both admit embeddings in Euclidean space, equivalently provided that global, real-valued, real analytic functions distinguish points.  This preliminary result also holds true for any other category of functions and manifolds, or Nash category, with an implicit and inverse function theorem and a few other features.  For example, it's true for Nash manifolds.  The hard part is then showing that every real analytic manifold does indeed embed in Euclidean space.  This is the Morrey-Grauert theorem.  Grauert's proof, the more sophisticated one, takes as its starting point a complex analytic manifold which is a tubular neighborhood of the real analytic manifold.  It then applies some hard stuff in several complex variables to prove the existence of many global, complex analytic functions on the complex manifold.

## Related concepts

* [[GAGA]]

## References

### General

* {#Huisman02} [[Johannes Huisman]], _The exponential sequence in real algebraic geometry and Harnack's Inequality for proper reduced real schemes_, Communications in Algebra, Volume 30, Issue 10, 2002 ([pdf](http://pageperso.univ-brest.fr/~huisman/rech/publications/exphi.pdf))

### In physics

Generalization of the [[time tube theorem]] to [[AQFT on curved spacetimes|AQFT on curved]] but [[real analytic space|real analytic]] [[spacetimes]]:

* [[Alexander Strohmaier]], *On the Local Structure of the Klein–Gordon Field on Curved Spacetimes*, Letters in Mathematical Physics **54** (2000) 249–261 [[doi:10.1023/A:1010927625112](https://doi.org/10.1023/A:1010927625112), [arXiv:math-ph/0008043](https://arxiv.org/abs/math-ph/0008043)]

* [[Alexander Strohmaier]], [[Edward Witten]], *Analytic states in quantum field theory on curved spacetimes* [[arXiv:2302.02709](https://arxiv.org/abs/2302.02709)]

* [[Alexander Strohmaier]], [[Edward Witten]], *The Timelike Tube Theorem in Curved Spacetime* [[arXiv:2303.16380](https://arxiv.org/abs/2303.16380)]

[[!redirects real analytic spaces]]

[[!redirects real analytic geometry]]

[[!redirects real-analytic manifold]]
[[!redirects real-analytic manifolds]]

[[!redirects real analytic manifold]]
[[!redirects real analytic manifolds]]


real closed field

# Contents
* automatic table of contents goes here
{: toc}

## Definition

An [[ordered field]] $F$ is **real closed** if it satisfies the following two properties:

* Any non-negative element $x \geq 0$ in $F$ has a [[square root]] in $F$;

* Any odd-degree [[polynomial]] with coefficients in $F$ has a root in $F$.

Notice that the order on a real closed field is definable from the algebraic structure: $x \leq y$ if and only if $\exists_z x + z^2 = y$. (In particular, there is a _unique_ ordering on a real closed field, defined by taking the positive elements to be precisely the nonzero squares.) In fact, the [[category]] of real closed fields and order-preserving field homomorphisms is a [[full subcategory]] of the category [[Field]] of [[fields]] and field homomorphisms.

## Properties

Real closed fields can be equivalently characterized by any of the following properties:

1. $F$ is not [[algebraically closed field|algebraically closed]], but some finite [[field extension|extension]] is. This extension is necessarily $F[\sqrt{-1}]$. See also [[fundamental theorem of algebra]].

2. As a field, $F$ is [[elementary equivalence|elementarily equivalent]] to the field of real numbers.

3. The [[intermediate value theorem]] holds for all [[polynomial functions]] with coefficients in $F$.

4. $F$ is an ordered field that has no ordered algebraic extension.

In fact, there is a [[completion]] of any [[ordered field]] to a real closed field, in the following sense:

+-- {: .un_theorem}
###### Theorem

The full inclusion of the category of real closed fields and field homomorphisms to the category of ordered fields and ordered field homomorphisms has a left adjoint.
=--

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

We give a brief sketch of proof, referring to Lang's _Algebra_ ($3^{rd}$ edition), section IX.2, for more details.

First, for each ordered field $F$, there is a real closed algebraic extension $F \to R$ that is order-preserving (theorem 2.11). This is called a **real closure** of the ordered field $F$.

Second, any two real closures of $F$ are uniquely isomorphic (theorem 2.9); in fact, the proof shows there is at most one order-preserving homomorphism over $F$ between any two real closures. Therefore we may speak of _the_ real closure of $F$, which we denote as $\widebar{F}$.

Finally, let $F \to R$ be any order-preserving field homomorphism to a real closed field $R$. We must show that $F \to R$ extends uniquely to a homomorphism $i: \widebar{F} \to R$. Any such homomorphism $i$ must factor through the subfield $R' \hookrightarrow R$ consisting of elements $\alpha \in R$ that are algebraic over $F$, since $\widebar{F}$ is algebraic over $F$. But this subfield is also real closed. Therefore, by the preceding paragraph, there is at most one homomorphism $\widebar{F} \to R'$ extending $F \to R'$, and the proof is complete.
=--

## Examples

1. The [[real number]]s form a real closed field.

2. Real [[algebraic number]]s form a real closed field, which is the real closure of the ordered field of [[rational numbers]].

3. A field of nonstandard real numbers (as in Robinson [[nonstandard analysis]]) is real closed.

4. [[surreal number|Surreal numbers]] form a (large) real closed field.

5. If $F$ is real closed, then the field of [[Puiseux series]] over $F$ is also real closed.

6. More generally, given a real closed field $F$, the field of [[Hahn series]] over $F$ with [[valuation ring|value group]] $G$ (a linearly ordered group) is real closed provided that $G$ is [[divisible group|divisible]].

7. Any [[o-minimal structure|o-minimal]] ordered ring structure $R$ is a real closed field.

8. Given an o-minimal ordered ring $R$, the field of [[germ]]s at infinity of definable functions $R \to R$ in any o-minimal expansion of $(R, 0, 1, +, -, \cdot, \lt)$ is real closed. (By "germ at infinity", we mean an equivalence class of functions for which $f \equiv g$ if and only if $f(x) = g(x)$ for all sufficiently large $x$.)

## Infinites and infinitesimals

Each real closed field $R$ contains a [[valuation ring|valuation]] subring $B \hookrightarrow R$ consisting of the "bounded" or archimedean elements, i.e., elements $x \in R$ such that $-n \leq x \leq n$ for some integer multiple $n$ of the identity. An element in the complement of $B$ is an **infinite** element of $R$, and the reciprocal of an infinite element is an **infinitesimal** element. The field of fractions of $B$ is clearly $R$.

We remark that any real closed field contains a copy of the field of real [[algebraic numbers]], which as before we denote by $\widebar{\mathbb{Q}}$ (not to be confused with the algebraic closure of $\mathbb{Q}$). Each of the elements of $\widebar{\mathbb{Q}}$ is archimedean.

Let $B^\ast$ be the group of units of $B$. The quotient $R^\ast/B^\ast$ is the **value group** of $R$. It can be viewed as the "group of orders of infinities and infinitesimals" of $R$. If $R$ is real closed, then the value group is a linearly ordered [[divisible group]] (divisible because we can take $n^{th}$ roots of positive elements in $R$). The structure of the value group as ordered group is an important invariant of the real closed field.

In the other direction, to each ordered divisible abelian group $G$, there exists a real closed field having $G$ as its value group. For example, one may form the [[Hahn series]] over $\widebar{\mathbb{Q}}$ with value group $G$.

## In constructive mathematics

In constructive mathematics, the various definitions of real closed field bifurcate, because of different definitions of an odd-degree polynomial. One could define an odd-degree polynomial as a polynomial whose coefficient for an odd number $n$ is [[denial inequality|not equal to]] zero and whose coefficient for all $i \gt n$ for is equal to zero. On the other hand, one could also define an odd-degree polynomial as a polynomial whose coefficient for an odd number $n$ is [[tight apartness relation|apart from]] zero and whose coefficient for all $i \gt n$ for is equal to zero. These two definitions are different from each other, the real numbers satisfy the second, while they do not satisfy the first.

## Related concepts

* [Euclidean geometry -- Tarski's axioms](Euclidean+geometry#TarskiAxioms)

## References

* [[Serge Lang]], _Algebra_ (3rd edition), Addison-Wesley, 1993.

* David Marker, _Notes on Real Algebra_ [(link)](http://www.math.uic.edu/~marker/orsay/real_algebra.pdf)

[[!redirects real closed field]]
[[!redirects real closed fields]]
[[!redirects real-closed field]]
[[!redirects real-closed fields]]


real cohomology

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

#Contents#
* table of contents
{:toc}

## Idea

By _real cohomology_ one usually means [[ordinary cohomology]] with [[real number]] [[coefficients]], denoted $H^\bullet\big(-, \mathbb{R}\big)$.

Hence, with the pertinent conditions on the [[domain]] [[space]] $X$ satisfied, its real cohomology $H^\bullet\big(-, \mathbb{R}\big)$ is what is computed by the [[Cech cohomology]] or [[singular cohomology]] or [[sheaf cohomology]] of $X$ with coefficients in $\mathbb{R}$.

In particuar, for $X$ a [[smooth manifold]], the [[de Rham theorem]] says that real cohomology of $X$ is also computed by the [[de Rham cohomology]] of $X$

$$
  H^\bullet\big( X, \mathbb{R}\big)
  \;\simeq\;
  H^\bullet_{dR}\big( X \big)
  \,.
$$

More generally,  for $X$ a smooth manifold with smooth [[action]] of a [[connected topological space|connected]] [[compact Lie group]], the [[equivariant de Rham theorem]] says that the real cohomology of the [[homotopy quotient]] (e.g. [[Borel construction]]) of $X$ is computed by the [[Cartan model]] for [[equivariant de Rham cohomology]] on $X$.

| [[cohomology]] | [[Borel equivariant cohomology|Borel]]-[[equivariant cohomology]] |
|---|---|
| [[real cohomology|real]] [[ordinary cohomology]] | real [[equivariant ordinary cohomology]] |
| [[de Rham cohomology]] | [[equivariant de Rham cohomology]]

## Properties

* [[de Rham theorem]]

* [[PL de Rham theorem]]

## Related concepts

* [[real vector space]]

* [[integral cohomology]], [[rational cohomology]], [[complex cohomology]]

* [[de Rham cohomology]], [[de Rham theorem]]

* [[equivariant de Rham cohomology]], [[equivariant de Rham theorem]]

* [[rational homotopy theory]]

* [[rational Cohomotopy]]


real cubic function

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

# Contents
* table of contents
{: toc}

## Definition ##

A **real cubic function** is a [[cubic function]] in the [[real numbers]]. Equivalently, it is a [[solution]] to the fourth-order linear homogeneous [[ordinary differential equation]]
$$\frac{d^4 f}{d x^4} = 0$$
with initial conditions
$$f(0) = d$$
$$\frac{d f}{d x}(0) = c$$
$$\frac{d^2 f}{d x^2}(0) = 2 b$$
$$\frac{d^3 f}{d x^3}(0) = 6 a$$

## Properties ##

### Extrema ###

The [[derivative]] of a real cubic function $f:\mathbb{R} \to \mathbb{R}$, $f(x) \coloneqq a x^3 + b x^2 + c x + d$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, $d \in \mathbb{R}$ is the function $\delta_x f:\mathbb{R} \to \mathbb{R}$, $\delta_x f(x) \coloneqq 3a x^2 + 2b x + c$. The discriminant of the derivative is given by $\Delta_\partial = (2b)^2 - 4 (3a) c = 4b^2 - 12a c$. If the discriminant of the derivative is positive $\Delta_\partial \gt 0$, then there are two local extrema in the cubic function. Otherwise, $f$ is [[monotonic]] on the entire domain of $a \gt 0$, or [[antitonic]] on the entire domain if $a \lt 0$.

Using the [[smooth quadratic formula]], we find that for positive discriminant the two extrema are located at

$$x_1 = \frac{-2b + \mathrm{sqrt}_\mathrm{sm}(4b^2 - 12a c)}{6a} = \frac{-2b + 2 \mathrm{sqrt}_\mathrm{sm}(b^2 - 3a c)}{6a} = \frac{-b + \mathrm{sqrt}_\mathrm{sm}(b^2 - 3a c)}{3a}$$

$$x_2 = \frac{-2b - \mathrm{sqrt}_\mathrm{sm}(4b^2 - 12a c)}{6a} = \frac{-2b - 2 \mathrm{sqrt}_\mathrm{sm}(b^2 - 3a c)}{6a} = \frac{-b - \mathrm{sqrt}_\mathrm{sm}(b^2 - 3a c)}{3a}$$

For negative discriminant, there is no extrema. For zero discriminant, there is a [[saddle point]] at

$$x = -\frac{b}{3a}$$

If $a = 0$, the real cubic function is [[degenerate]]; it becomes a [[real quadratic function]].

### Inflection point ###

THe inflection point of the real cubic function $f:\mathbb{R} \to \mathbb{R}$, $f(x) \coloneqq a x^3 + b x^2 + c x + d$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, $d \in \mathbb{R}$ occurs at the zero of the second derivative of $f$:

$$\partial_x f(x) = 6 a x + 2b = 0$$

which occurs at

$$x = -\frac{b}{3a}$$

The inflection point is to real cubic functions what the extremum was to real quadratic functions.

### In constructive mathematics ###

In [[classical mathematics]], the [[law of trichotomy]] holds in the real numbers, so the three cases above cover every real number. However, in [[constructive mathematics]], trichotomy does not hold in the real numbers, and as a result, there exists real cubic functions $f:\mathbb{R} \to \mathbb{R}$ such that one cannot decide whether $f$ has two local extrema, a saddle point, or zero local extrema. Similarly, there exists real cubic functions $f:\mathbb{R} \to \mathbb{R}$ such that one cannot decide whether the inflection point occurs to the left, the right, or on the $y$-intercept line.

## Depressed cubic functions ##

An unnormalized depressed cubic function is a real cubic function whose inflection point occurs at $x = 0$, and has the canonical form of $g(x) \coloneqq a x^3 + p x + q$, with $\vert a \vert \gt 0$. $g$ is called a depressed cubic function when $a$ is normalized to $1$.

Every real cubic function $f:\mathbb{R} \to \mathbb{R}$, $f(x) \coloneqq a x^3 + b x^2 + c x + d$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, $d \in \mathbb{R}$, is a translation of an unnormalized depressed cubic function $g:\mathbb{R} \to \mathbb{R}$ by the equation

$$g(x) = f\left(x - \frac{b}{3a}\right)$$

$$a x^3 + p x + q = a \left(x - \frac{b}{3a}\right)^3 + b \left(x - \frac{b}{3a}\right)^2 + c \left(x - \frac{b}{3a}\right) + d$$

$$a x^3 + p x + q = a x^3 - 3 a \frac{b}{3a} x^2 + 3 a \left(\frac{b}{3a}\right)^2 x - a \left(\frac{b}{3a}\right)^3 + b x^2 - 2 b \frac{b}{3a} x + b \left(\frac{b}{3a}\right)^2 + c x - c \left(\frac{b}{3a}\right) + d$$

$$a x^3 + p x + q = a x^3 + \frac{b^2}{3a} x - \frac{b^3}{27a^2} - \frac{2 b^2}{3a} x + \frac{b^3}{9 a^2} + c x - \frac{b c}{3a} + d$$

$$a x^3 + p x + q = a x^3 + \frac{3a c - b^2}{3a} x + \frac{2 b^3 - 9 a b c + 27 a^2 d}{27a^2}$$

thus,

$$p = \frac{3a c - b^2}{3a}$$

$$q = \frac{2 b^3 - 9 a b c + 27 a^2 d}{27a^2}$$

### Discriminant ###

The discriminant of an unnormalized depressed cubic function $g(x) \coloneqq a x^3 + p x + q$ is given by

$$\Delta = -4 p^3 - 27 a q^2$$

Substituting in the values above for $p$ and $q$, the discriminant of a general real cubic function $f:\mathbb{R} \to \mathbb{R}$, $f(x) \coloneqq a x^3 + b x^2 + c x + d$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, $d \in \mathbb{R}$ is

$$\Delta = -4 \left(\frac{3a c - b^2}{3a}\right)^3 - 27 a \left(\frac{2 b^3 - 9 a b c + 27 a^2 d}{27a^2}\right)^2 = \frac{4(b^2 - 3a c)^3 - (2 b^3 - 9 a b c + 27 a^2 d)^2}{27 a^3}$$

## Exact zeroes ##

### Partial inverse functions ###

Given a real cubic function $f(x) \coloneqq a x^3 + b x^2 + c x + d$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, $d \in \mathbb{R}$ such that $\vert a \vert \gt 0$, the following cases are possible:

* If the discriminant of the derivative of $f$ is positive, $\Delta_\partial \gt 0$, then there are three branches of the partial inverse function of $f$.

* If the discriminant of the derivative of $f$ is zero, $\Delta_\partial = 0$, then there are two branches of the partial inverse function of $f$.

* If the discriminant of the derivative of $f$ is negative, $\Delta_\partial \lt 0$, then there is only one branch of the partial inverse function of $f$.

In all cases, the partial inverse functions are solutions of the first-order nonlinear [[ordinary differential equation]]

$$(3 a F^2 + 2 b F + c) \frac{d F}{d x} = 1$$

with specific initial conditions.

## See also ##

* [[real number]]

* [[cubic function]]

* [[real quadratic function]]

* [[real polynomial function]]

## References ##

See also:

* Wikipedia, _[Cubic function](https://en.wikipedia.org/wiki/Cubic_function)_
* Wikipedia, _[Cubic equation](https://en.wikipedia.org/wiki/Cubic_equation)_

[[!redirects real cubic functions]]


real form

## Related concepts

* [[form of an algebraic group]]

## References

### In Lie theory

* wikipedia, _[Real form (Lie theory)](http://en.wikipedia.org/wiki/Real_form_%28Lie_theory%29)_

[[!redirects real forms]]

[[!redirects split real form]]
[[!redirects split real forms]]


real homotopy theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Homological algebra
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In analogy to [[rational homotopy theory]], the idea of *real homotopy theory* is to study those aspects of [[homotopy types]] that are visible when the [[ground ring]] is the [[real numbers]], such as their [[real cohomology]]-[[cohomology groups|groups]] and the [[tensor product of abelian groups|tensor product]] of their [[homotopy groups]] with the additive [[abelian group]] of [[real numbers]]. This is of central relevance in relation to [[differential cohomology theory]], which needs real instead of rational [[coefficients]] in order to connect to [[de Rham complexes]] of [[differential forms]] on [[smooth manifolds]].

But a technical issue with generalizing the [[fundamental theorem of dg-algebraic rational homotopy theory]] to the case of real homotopy theory is that the
[[PL de Rham complex|PL de Rham]]-[[Quillen adjunction between simplicial sets and connective dgc-algebras]] -- which does exist over any [[ground field]] $k$ of [[characteristic zero]] and relates to the one over the [[rational numbers]] by [[derived functor|derived]] [[extension of scalars]] (all reviewed in [FSS 2020, Sec. 3.2](#FSS20)) -- models $k$-[[localization]] only for $k = \mathbb{Q}$ the [[rational numbers]] -- then: [[rationalization]].

From the point of [[algebra]], the issue is (see [this Remark](core+of+a+ring#NoOtherCharZeroFieldIsSolid)) that other $k$ are not [[solid rings]], so that the iterations of the [[tensor product of abelian groups|tensor products]] $(-) \otimes_{\mathbb{Z}} k$ on [[homotopy groups]] are not, in general, [[isomorphism|isomorphic]] to each other, for $k \neq \mathbb{Q}$.

On the other hand, from the point of [[topology]] the issue is that the [[real numbers]] $k = \mathbb{R}$ are naturally a [[topological ring]], whose topology, however, is ignored in the standard [[Quillen adjunction between simplicial sets and connective dgc-algebras]] over the real numbers.

Therefore, the *real homotopy theory* of [Brown & Szczarba 1995](#BrownSzczarba95) replaces, in the [[fundamental theorem of dg-algebraic rational homotopy theory]], the plain [[real cohomology]]-groups by real [[continuous cohomology]] and plain [[dgc-algebras]] with [[topological ring|topological algebras]].

> The authors of ([Brown-Szczarba 95](#BrownSzczarba95)) show how the Bousfield-Gugenheim approach to [[rational homotopy theory]] may be extended to the framework of [[continuous cohomology]]. In the usual theory, the extension from the [[rational numbers|rationals]] to the [[real numbers|reals]] has always been effected by a formal tensoring with the reals. While this allows a connection to [[de Rham complex|de Rham theory]], it has always been preferable to have a direct construction of real homotopy theory akin to the [[Postnikov tower]]-like construction of [[rational homotopy theory]]. Unfortunately, in the usual situation, such a construction is impossible due to the incompatibility of the algebra of $\mathbb{R}$ (as an [[abelian group]]) with its natural [[topology]]. The authors of ([Brown-Szczarba 95](#BrownSzczarba95)) remedy this problem by creating a theory of [[minimal models]] which incorporates the usual topology of the reals and a generalization of [[van Est's theorem]] saying that $H^*(K(\mathbb{R},n); \mathbb{R})$ is a freely generated algebra on one generator (a result which is not true in the discrete topology case). By using the natural topology of $\mathbb{R}$, there is an immediate connection to [[continuous cohomology]] and applications such as [[characteristic classes]] of [[foliations]].

> (from the [Zentralblatt review](http://mirrors.library.cornell.edu/ZMATH/zmath/en/zmath/search/?an=0865.55009) of ([Brown-Szczarba 95](#BrownSzczarba95)))

{#RelationToSchematicHomotopyType} If one thinks of [[simplicial topological spaces]] as [[simplicial presheaves]] on (some [[small category|small]] version of) [[Top]], and [[local model structure on simplicial presheaves|hence]] as presentations for [[topological infinity-stack|topological $\infty$-stacks]], then localization of these at the topological real line coefficient is the topological analog of passing to *[[schematic homotopy types]]*, in the sense of [Toën 2006](schematic+homotopy+type#Toen06), which generalizes to [[D-topological infinity-groupoid|D-topological $\infty$-groupoids]] (and further to [[smooth infinity-groupoids|smooth $\infty$-groupoids]] etc.) along the lines of [Stel 2010](function+algebras+on+infinity-stacks#Stel).

## Related concepts

* [[rational homotopy theory]]

* [[p-adic homotopy theory]]

* [[schematic homotopy type]]

## References
 {#References}

### As continuous-real cohomology
 {#ReferencesAsContinuousRealCohomology}

On [[real homotopy theory]] ([[rational homotopy theory]] over the [[real numbers]]) using [[continuous cohomology|continuous]] [[real cohomology]] and [[topological algebra|topological]] [[dgc-algebras]]:

* {#BrownSzczarba95} [[Edgar Brown]], [[Robert H. Szczarba]], _Real and Rational Homotopy Theory_, Chapter 17 in: *[[Handbook of Algebraic Topology]]*,  Elsevier, 1995 869-915 ([doi:10.1016/B978-044481779-2/50018-3](https://doi.org/10.1016/B978-044481779-2/50018-3), [ZB](http://mirrors.library.cornell.edu/ZMATH/zmath/en/zmath/search/?an=0865.55009))

  > (also introducing [[Borel-equivariant rational homotopy theory]])

with emphasis on globally [[Kan complex|Kan fibrant]] [[simplicial topological spaces]] (such as [[simplicial topological groups]]):

* {#BrownSzczarba89} [[Edgar H. Brown]], [[Robert H. Szczarba]], *Continuous cohomology and real homotopy type*, Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 ([doi:10.1090/S0002-9947-1989-0929667-6](https://doi.org/10.1090/S0002-9947-1989-0929667-6), [jstor:2001017](https://www.jstor.org/stable/2001017))

with arbitrary [[fundamental group]]:

* {#BrownSzczarba90} [[Edgar Brown]], [[Robert H. Szczarba]], _Continuous cohomology and Real homotopy type II_ Asterisque 191, Societe Mathematique De France (1990) ([numdam:AST_1990__191__45_0](http://www.numdam.org/book-part/AST_1990__191__45_0))

* [[Edgar Brown]], [[Robert H. Szczarba]], *Real and rational homotopy theory for spaces with arbitrary fundamental group*, Duke Mathematical Journal 71. (1993): 229-316 ([doi:10.1215/S0012-7094-93-07111-6](https://projecteuclid.org/journals/duke-mathematical-journal/volume-71/issue-1/Rational-and-real-homotopy-theory-with-arbitrary-fundamental-groups/10.1215/S0012-7094-93-07111-6.short))

### As rationalization followed by extension of scalars

Discussion of the plain [[Quillen adjunction between simplicial sets and connective dgc-algebras]] over the real numbers, and its relation to [[rationalization]] followed by [[derived functor|derived]] [[extension of scalars]]:

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalizaed]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

following

* [[Pierre Deligne]], [[Phillip Griffiths]], [[John Morgan]], [[Dennis Sullivan]], *Real homotopy theory of Kähler manifolds*, Invent Math **29** (1975) 245-274 [[doi:10.1007/BF01389853](https://doi.org/10.1007/BF01389853)]

For discussion with focus on [[Hopf invariants]] and the [[Whitehead integral formula]] see the references

* [there](Hopf+invariant#ReferencesWhiteheadIntegralFormula).


real Hopf fibration

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#### Homotopy theory
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#Contents#
* table of contents
{:toc}

## Idea

The _real Hopf fibration_ is the [[fibration]]

$$
  \array{
     S^0 &\hookrightarrow& S^1
     \\
     && \downarrow^{\mathrlap{p_{\mathbb{R}}}}
     \\
     && S^1
  }
$$

of the [[1-sphere]] over itself with [[fiber]] the [[0-sphere]], which is induced via the [[Hopf construction]] from the product operation

$$
  \mathbb{R} \times \mathbb{R} \stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}
  \mathbb{R}
$$

on the [[real numbers]].

This may also be understood as the [[Spin(2)]]-[[double cover]] of [[SO(2)]].

## Realizations

Here are different but equivalent ways of realizing this explicitly:

### Via projective space

If the domain $S^1$ is regarded as the [[unit sphere]] $\{(x,y)  | {\vert x\vert}^2 + {\vert y\vert}^2 = 1\}$ in $\mathbb{R}\times \mathbb{R}$ and the codomain $S^1$ is regarded as the real [[projective space]], then $p$ is given simply by

$$
  p \colon (x,y) \mapsto [x;y] = [x/y; 1]
  \,,
$$
One can view the real Hopf fibration as the boundary of a [[Möbius strip]], which is the non-trivial double cover of the circle.

As an element in the first ([[stable homotopy group|stable]]) [[homotopy group]] [[homotopy groups of spheres|of spheres]] $\pi_1(S^1) \simeq \mathbb{Z}$, the real Hopf fibration represents $p_{\mathbb{R}} = 2$.

### Via join construction
  {#ViaJoinConstruction}

We spell out the real Hopf fibration realized as the [[Hopf construction]]

$$
  \array{
     X \ast X
       &\overset{H_{\mathbb{R}}}{\longrightarrow}&
     \Sigma X
     \\
     =
     &&
     =
     \\
     (X \times [0,1] \times X)_{/\sim}
     &&
     (X \times [0,1])_{/\sim}
     \\
     (x,t,y) &\mapsto& \big( f(x,y), t\big)
  }
$$

(as defined [there](Hopf+construction#Definition)) on

$$
  X
    \coloneqq
  \mathbb{Z}/2
$$

regarded with its [[cyclic group]]-[[structure]]

$$
  \array{
    (\mathbb{Z}/2) \times (\mathbb{Z}/2)
      &\overset{f}{\longrightarrow}&
    (\mathbb{Z}/2)
    \\
    (x,y) &\mapsto& x + y
  }
$$

Here the [[equivalence relation]] for the [[suspension]] $\Sigma X = (X \times [0,1])_{/\sim}$ on the right is

$$
  (x_1,0) \sim (x_2,0) \;\,,\;\;  (x_1, 1) \sim (x_2, 1) \;\;\; \forall x_1,x_2 \in X
$$

while the [[equivalence relation]] for the [[join of topological spaces|join]] $X\ast X = (X \times [0,1] \times X)_{/\sim}$ on the left is

$$
  (x, 0, y_1) \simeq (x,0,y_2) \;\;,\;\; (x_1,1,y) \sim (x_2, 1, y)
  \,.
$$

So on the left we have a [[circle]] $S^1$ realized by gluing four copies of the [[interval]] $[0,1]$ labeled in $\mathbb{Z}/2 \times \mathbb{Z}/2$ by identifying them pairwise at $0 \in [0,1]$ and pairwise _the other way_ at $1 \in [0,1]$. while on the right we have a [[circle]] $S^1$ realized by  two copies, labeled by $\mathbb{Z}/2$.

$$
  S^1 \longrightarrow S^1
  \,.
$$

A full path around the circle on the left is given, in terms of the above [[coordinates]] in $[0,1] \times (\mathbb{Z}_2 \times \mathbb{Z}_2)$, by

$$
  \array{
   (0,(0,0))
    &\rightsquigarrow&
   (1,(0,0))
   \\
   &&
    \sim
   \\
   &&
   (1,(1,0))
    &\rightsquigarrow&
   (0,(1,0))
   \\
   &&&&
     \sim
   \\
    &&&&
    (0,(1,1))
    &\rightsquigarrow&
    (1,(1,1))
    \\
    &&&&&&
      \sim
    \\
    &&&&&&
     (1,(0,1))
    & \rightsquigarrow&
     (0,(0,1))
    \\
    &&&&&&&&
       \sim
    \\
    &&&&&&&&
     (0,(0,0))
  }
$$

As we map this path over to the other $S^1$ by adding up the two coordinate labels in $\mathbb{Z}/2$, we trace out the following path on the right, with coordinates in $[0,1] \times \mathbb{Z}/2$:

$$
  \array{
   (0,(0+0 = 0))
    &\rightsquigarrow&
   (1,(0 + 0 = 0))
   \\
   &&
    \sim
   \\
   &&
   (1,(1 + 0 = 1))
    &\rightsquigarrow&
   (0,(1 + 0 = 1))
   \\
   && &&
     \sim
   \\
    && &&
    (0,(1+1 = 0))
    &\rightsquigarrow&
    (1,(1+1 = 0))
    \\
    && && &&
      \sim
    \\
    && && &&
     (1,(0 + 1 = 1))
      &\rightsquigarrow&
     (0,(0 + 1 = 1))
    \\
    && && && &&
       \sim
    \\
    && && && &&
     (0,(0 + 0 = 0))
  }
$$

That's twice around the circle on the right, for once on the left, manifestly showing that the real Hopf fibration is the non-trivial [[double cover]] of the [[circle]] by itself:

$$
  S^1 \overset{\cdot 2}{\longrightarrow} S^1
  \,.
$$

## Related concepts

* [[complex Hopf fibration]]

* [[quaternionic Hopf fibration]]

* [[octonionic Hopf fibration]]


real irreducible spin representations -- table

| $d$ | $Spin(d-1,1)$ | minimal real spin representation $S$ | $dim_{\mathbb{R}} S\;\;$ |  $V$ in terms of $S^\ast$ |  [[supergravity]] |
|--|--|--|--|--|--|
| 1 | $\mathbb{Z}_2$ | $S$ real | 1 | $V \simeq (S^\ast)^{\otimes}^2$ |  |
| 2 | $\mathbb{R}^{\gt 0} \times \mathbb{Z}_2$ | $S^+, S^-$ real | 1 | $V \simeq ({S^+}^\ast)^{\otimes^2} \oplus ({S^-}^\ast)^{\otimes 2}$ |  |
| 3 | $SL(2,\mathbb{R})$ | $S$ real | 2 |  $V \simeq Sym^2 S^\ast$ |  |
| 4 | $SL(2,\mathbb{C})$ | $S_{\mathbb{C}} \simeq S' \oplus S''$ | 4 | $V_{\mathbb{C}} \simeq {S'}^\ast \oplus {S''}^\ast$ | [[d=4 N=1 supergravity]] |
| 5 | $Sp(1,1)$ | $S_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W$ | 8 |  $\wedge^2 S_0^\ast \simeq \mathbb{C} \oplus V_{\mathbb{C}}$ |  |
| 6 | [[SL(2,H)]] | $S^\pm_{\mathbb{C}} \simeq S_0^\pm  \otimes_{\mathbb{C}} W$ | 8 | $V_{\mathbb{C}} \simeq \wedge^2 {S_0^+}^\ast \simeq (\wedge^2 {S_0^-}^\ast)^\ast$ |  |
| 7 |  |  $S_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W$ | 16 |  $\wedge^2 S_0^\ast \simeq V_{\mathbb{C}} \oplus \wedge^2 V_{\mathbb{C}}$ |  |
| 8 |  | $S_{\mathbb{C}} \simeq S^\prime \oplus S^{\prime\prime}$ | 16 | ${S'}^\ast {S''}^\ast \simeq V_{\mathbb{C}} \oplus \wedge^3 V_{\mathbb{C}}$ |  |
| 9 |  | $S$ real | 16 |  $Sym^2 S^\ast \simeq \mathbb{R} \oplus V \wedge^4 V$ |  |
| 10 |   | $S^+ , S^-$ real | 16 | $Sym^2(S^\pm)^\ast \simeq V  \oplus \wedge_\pm^5 V$ | [[type II supergravity]] |
| 11 |   | $S$ real | 32 | $Sym^2 S^\ast \simeq V \oplus \wedge^2 V \oplus \wedge^5 V$ | [[11-dimensional supergravity]] |


real nth root function

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\tableofcontents

## Idea ##

Given an integer $n$, The different types of $n$-th [[root]] [[partial functions]] on the [[real numbers]] that satisfy the [[functional equation]] $f(x)^n = x$ on some [[subset]] of the real numbers.

## Definition ##

Let us define the real numbers to be a [[sequentially Cauchy complete]] Archimedean ordered field, since that is the minimum requirement for which the inverse function theorem is true.

There exists an $n$-th root function for every integer $n \in \mathbb{Z}$ apart from zero, defined on the positive real numbers by the equation:

$$0 \lt \vert n \vert \vdash \sqrt[n]{x} \coloneqq \exp(\frac{1}{n} \ln(x))$$

Now, assume that $n \gt 1$. Then one could extend the $n$-th root function to a function defined on the half-open interval $[0, \infty)$. Let us define continuous functions $f_k:[0, \infty) \to [0, \infty)$:
$$f_k(x) = \begin{cases}
1/n^k & \mathrm{when}\; x \leq 1/n^{2 k} \\
\exp(\frac{1}{n} \ln(x)) & \mathrm{when}\; x \geq 1/n^{2 k} \\
\end{cases}$$
As stated, that requires knowing whether $x \leq 1/n^{2 k}$ or $x \geq 1/n^{2 k}$, but it is possible to work around this by patching three functions together:

 * $f^{0}_k:[0,1/n^{2 k}) \to \mathbb{R}$ is the linear function $f(x) = n^k x$,
 * $f^{+}_k:(0,\infty) \to \mathbb{R}$ is defined as $\min(n^k x,\exp(\frac{1}{n} \ln(x)))$.

Since these functions agree on their overlap, and their domains comprise all of $[0, \infty)$ we do get a total function $f_k:[0, \infty) \to [0, \infty)$ as a result.

Now the sequence of functions $(f_k)_{k=0}^\infty$ so defined converges uniformly on any bounded interval to a continuous function $\sqrt[n]{(-)}:[0, \infty) \to [0, \infty)$ called the **principal nth root** function. It is easily seen that $(\sqrt[n]{x})^n = x$ and $\sqrt[n]{x^n} = x$.

Additionally, if $n$ is an odd integer greater than 1, then there exist $n$-th root functions defined on the entirety of the real numbers.

Let us patch this sequence of triples of functions together:

 * $f^{-}_k:(-\infty,0) \to \mathbb{R}$ is defined as $\max(-n^k x,-\exp(\frac{1}{n} \ln(-x)))$
 * $f^{0}_k:(-1/n^{2 k},1/n^{2 k}) \to \mathbb{R}$ is the linear function $f(x) = n^k x$,
 * $f^{+}_k:(0,\infty) \to \mathbb{R}$ is defined as $\min(n^k x,\exp(\frac{1}{n} \ln(x)))$.

Since these functions agree on their overlap, and their domains comprise all of $\mathbb{R}$ we do get a total function $f_k:\mathbb{R} \to \mathbb{R}$ as a result.

Now the sequence of functions $(f_k)_{k=0}^\infty$ so defined converges uniformly on any bounded interval to a continuous function $\sqrt[n]{(-)}:\mathbb{R} \to \mathbb{R}$ called the **nth root** function for odd integer $n$. It is easily seen that $(\sqrt[n]{x})^n = x$ and $\sqrt[n]{x^n} = x$.

## Other nth root functions

According to ([Richman 2012](#Richman12)), given the existence of an nth root function, there are an uncountable number of functions that satisfy the [[functional equation]] $f(x)^n = x$ on some [[subset]] of the [[real numbers]]. Each of these could be called a real "nth root function".

For example, let $1_{\mathbb{Q}}:\mathbb{R} \to \mathbb{R}$ be the constructive Dirichlet indicator function, defined as $1_{\mathbb{Q}}(x) \coloneqq 1$ for every rational number $q \in \mathbb{Q}$, and $1_{\mathbb{Q}}(x) \coloneqq 0$ for every real number $x$ apart from every rational number $q \in \mathbb{Q}$

$$\forall q \in \mathbb{Q}. \vert x - q \vert \gt 0$$

Then the function $f(x) \coloneqq (-1)^{1_{\mathbb{Q}}(x)} \sqrt[n]{x}$ is a real nth root function, even though it is nowhere continuous, and not defined on the entire open interval $(n, \infty)$.

## See also

* [[root]]
* [[real square root function]]

## References

* {#Richman12} [[Fred Richman]], *Algebraic functions, calculus style*. Communications in Algebra, Volume 40, Issue 7, July 2012, Pages 2671-2683 [[doi:10.1080/00927872.2011.584337](https://doi.org/10.1080/00927872.2011.584337)]

[[!redirects real nth root function]]
[[!redirects real nth root functions]]

[[!redirects real root function]]
[[!redirects real root functions]]

[[!redirects principal nth root function]]
[[!redirects principal nth root functions]]

[[!redirects principal root function]]
[[!redirects principal root functions]]


real number

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# Contents
* table of contents
{: toc}

## Idea

A _real number_ is a [[number]] that may be [[Dedekind completion|approximated]] by [[rational numbers]].  Equipped with the operations of [[addition]] and [[multiplication]] induced from the rational numbers, real numbers form a _[[field]]_, commonly denoted _$\mathbb{R}$_. The underlying set is the _[[Dedekind completion|completion]]_ of the [[ordered field]] $\mathbb{Q}$ of rational numbers: the result of adjoining to $\mathbb{Q}$ [[suprema]] for every [[inhabited set|inhabited]] [[bounded set|bounded subset]] with respect to the natural [[order|ordering]] of rational numbers.

The [[set]] of real numbers also carries naturally the [[mathematical structure|structure]] of a [[topological space]] and as such $\mathbb{R}$ is called the  _[[real line]]_ also known as _[[continuum|the continuum]]_. Equipped with both the topology and the field structure, $\mathbb{R}$ is a [[topological field]] and as such is the [[complete space|uniform completion]] of $\mathbb{Q}$ equipped with the [[absolute value]] [[metric space|metric]].

Together with its [[cartesian products]] -- the [[Cartesian spaces]] $\mathbb{R}^n$ for [[natural numbers]] $n \in \mathbb{N}$ -- the real line $\mathbb{R}$ is a standard formalization of the idea of _continuous space_.  The more general concept of ([[smooth manifold|smooth]]) _[[manifold]]_ is modeled on these Cartesian spaces. These, in turn are standard models for the notion of [[space]] in particular in [[physics]] (see _[[spacetime]]_), or at least in [[classical physics]]. See at _[[geometry of physics]]_ for more on this.

## History

The original idea of a real number came from [[geometry]]; one thinks of a real number as specifying a _point on a line_, with _[[line]]_ understood as the abstract idea of the object that a pencil and a ruler draw on a piece of paper.  (More precisely, given two distinct points on the line, called $0$ and $1$, you get a [[bijection]] between the points and the real numbers.)

[[Euclid]] (citing [[Eudoxus]]) dealt with ratios of geometric magnitudes, which give positive real numbers; an arbitrary real number is then a difference of ratios of magnitudes.  However, the Greeks did not think of such ratios as [[number]]s; that appears to have been an insight of the Arabs.  See more at [[Eudoxus real number]].

A big project of the 19th century (at least in hindsight) was the 'arithmetisation of analysis': showing how real numbers could be defined completely in terms of rational numbers (and the desired classes of functions on them could be defined in terms of the general point-set notion of [[function]]).  Two successful approaches were developed in 1872, [[Richard Dedekind]]\'s definition of real numbers as certain sets of rational numbers (called _[[Dedekind cuts]]_) and [[Georg Cantor]]\'s definition as certain sequences of rational numbers (called _[[Cauchy sequences]]_).

A more modern approach is instead to characterise the properties that the set of real numbers must have and to prove that this is [[generalized the|categorical]] (unique up to a unique bijection preserving those properties).  Then the important result of the 19th-century programme is simply that this is consistent (that there exists at least one such set).  One can even use [[David Hilbert|Hilbert]]\'s or [[Alfred Tarski|Tarski]]\'s axioms for geometry to do this characterisation, coming full circle back to geometry.

Exactly how to define or characterise real numbers is still important in [[constructive mathematics]] and [[topos theory]] with its [[internal logic]].  For more on this, see [[real numbers object]] and the examples below.

## Definitions and characterizations

There are two basic approaches possible: to define what a __real number__ is as a mathematical object, or to define the __real line__ as a specific object in some previously known [[category]].

### The terminal Archimedean ordered field

There is a well-known algebraic (more or less) characterisation of the real line as the 'terminal Archimedean ordered field' (cf. [Richman08](#Richman08)) This can be interpreted as follows:

*  A __field__ is well known in algebra; if it matters, we mean a [[Heyting field]].
*  An __ordered field__ means a *[[strictly totally ordered]]* field.
*  An __[[Archimedean ordered field]]__ is an ordered field satisfying the [[Archimedean property]].
*  An Archimedean ordered field is __terminal__ if it is a [[terminal object]] in the category of Archimedean ordered fields.

We speak of [[the]] such field because it is unique up to unique [[isomorphism]] by definition of [[terminal object]].

In [[predicative mathematics]], one can define $\mathbf{R}$ by fiat via the [[universal property]] of the real numbers as the [[terminal object|terminal]] [[Archimedean ordered field]]. From this characterization, the real numbers are a [[terminal coalgebra]] of the [[identity functor|identity endofunctor]] $X \mapsto X$ on the [[category]] of [[Archimedean ordered fields]].

If [[quotient sets]] exist and [[sequence algebras]] of Archimedean ordered fields exist, then it is provable that $\mathbf{R}$ is Cauchy complete. From the definition of terminal object, $\mathbf{R}$ is an algebra of the endofunctor $X \mapsto \mathcal{C}(X)$ in the category of Archimedean ordered fields which takes Archimedean ordered fields $X$ to the Archimedean ordered field $\mathcal{C}(X)$ of equivalence classes of Cauchy sequences in $X$. Every algebra of the endofunctor $X \mapsto \mathcal{C}(X)$ in the category of Archimedean ordered fields is a Cauchy complete Archimedean ordered field.

Similarly, if [[power sets]] of Archimedean ordered fields exist, then it is provable that $\mathbf{R}$ is Dedekind complete. From the definition of terminal object, $\mathbf{R}$ is an algebra of the endofunctor $X \mapsto \mathcal{D}(X)$ in the category of Archimedean ordered fields which takes Archimedean ordered fields $X$ to the Archimedean ordered field $\mathcal{C}(X)$ of two-sided Dedekind cuts in $X$. Every algebra of the endofunctor $X \mapsto \mathcal{D}(X)$ in the category of Archimedean ordered fields is a Dedekind complete Archimedean ordered field.

### Dedekind cuts {#Dedekind}

Consider two [[inhabited set|inhabited]] subsets, $L$ and $U$, of a [[countable set|countable]] unbounded [[dense linear order]], such as $\mathbb{Q}$ (the set of [[rational numbers]]) or  $\mathbb{Z}[1/10]$ (the set of [[decimal fractions]]), such that:

*  If $a \in L$, then $b \in L$ for some $b \gt a$.
*  If $b \in U$, then $a \in U$ for some $a \lt b$.
*  If $a \lt b$ are rational numbers, then $a \in L$ or $b \in U$.  (\*)
*  If $a \in L$ and $b \in U$, then $a \lt b$.

We may define a __Dedekind real number__ to be such a pair, which is also called a __[[Dedekind cut]]__.

If $x \coloneqq (L,U)$ is a Dedekind cut, then we write $a \lt x$ to mean that $a \in L$ and $x \lt b$ to mean that $b \in U$.

We may approximate a Dedekind cut $x$ as closely as we like by applying (\*) as often as necessary.  This will be only finitely often, for any fixed positive level of approximation, given initial upper and lower bounds (which exist since $L$ and $U$ are inhabited).

See [[Dedekind completion]] for more.

### Cauchy sequences

Classically, a real number can be given by an [[infinite sequence|infinite]] [[Cauchy sequence]] of [[decimal fractions]] $\mathbb{Z}[1/10]$, each of which is a decimal fraction that approximates the real number to a given number of decimal places. However, many real numbers have several representations, i.e.

  $$1/10 = 0.099999... = 0.10000...$$

so we need to specify an [[equivalence relation]] on the Cauchy sequences.  Thus, $\mathbb{R}$ is constructed as a [[subquotient]] of the [[function set]] $\mathbb{Z}[1/10]^{\mathbb{N}}$.

We can generalise this to any [[Cauchy sequence]] of rational numbers, and $\mathbb{R}$ is constructed as a [[subquotient]] of the [[function set]] $\mathbb{Q}^{\mathbb{N}}$.

This construction is equivalent to the construction by Dedekind cuts, at least assuming [[weak countable choice]] (which also follows from [[excluded middle]]).  Thus it is popular in both [[classical mathematics]] and traditional [[constructive mathematics]] (which accepts [[countable choice]]).  However, in stricter forms of constructive mathematics, including those used as [[internal languages]] in [[topos theory]], the Cauchy reals and Dedekind reals are not equivalent.  (On the other hand, by generalising to Cauchy [[nets]], we recover the Dedekind reals again.)

See [[Cauchy real number]] and [[generalized Cauchy real number]] for more.

### The initial sequentially modulated Cauchy complete archimedean field

There is an algebraic (more or less) characterisation of the real line as the 'initial sequentially modulated Cauchy complete archimedean field'.  This can be interpreted as follows:

*  A __field__ is well known in algebra; if it matters, we mean a [[Heyting field]].
*  An __ordered field__ means a *[[strict total order|strictly totally]]* ordered field.
*  An __archimedean field__ is an ordered field satisfying the [[archimedean property]].
*  An archimedean field is __[[sequentially modulated Cauchy complete]]__ if all [[Cauchy sequences]] in the field with a $\mathbb{N}$-[[modulus of convergence]] [[converge]].
*  A sequentially modulated Cauchy complete archimedean field is __initial__ if it is an [[initial object]] in the category of sequentially modulated Cauchy complete archimedean fields.

The initial sequentially modulated Cauchy complete ordered field results in the [[HoTT book real numbers]].

### The Dedekind complete ordered field

There is a well-known algebraic (more or less) characterisation of the real line as the 'Dedekind complete ordered field', or sometimes the 'Dedekind complete archimedean field'.  This can be interpreted as follows:

*  A __field__ is well known in algebra; if it matters, we mean a [[Heyting field]].
*  An __ordered field__ means a *[[linear order|linearly]]* ordered field.
*  An __archimedean field__ is an ordered field satisfying the [[archimedean property]].
*  An ordered field is __Dedekind complete__ if it is [[Dedekind completion|Dedekind-complete]]:

1. For all elements $a \in F$, the upwards unbounded [[open interval]] $(a,\infty)$ is inhabited.
2. For all elements $a \in F$, the downwards unbounded [[open interval]] $(-\infty,a)$ is inhabited.
3. For all elements $a \in F$ and $b \in F$, $a \lt b$ if and only if $(b,\infty)$ is a subinterval of $(a,\infty)$
4. For all elements $a \in F$ and $b \in F$, $b \lt a$ if and only if $(-\infty,b)$ is a subinterval of $(-\infty,a)$
5. For all elements $a \in F$ and $b \in F$, if $a \lt b$, then $F$ is a subinterval of the union of $(a, \infty)$ and $(-\infty, b)$
6. For all elements $a \in F$ and $b \in F$, the intersection of $(a,\infty)$ and $(-\infty,b)$ is a subinterval of $(a,b)$

In impredicative mathematics, we speak of [[the]] such field because it is unique up to unique [[isomorphism]].

+-- {: .un_theorem}
###### Theorem

Assuming impredicative foundations, there is an archimedean field $\mathbb{R}$ which is Dedekind-complete, and into which every archimedean field embeds.  Furthermore, every Dedekind-complete ordered field is isomorphic to $\mathbb{R}$, and uniquely so.
=--

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

Construct $\mathbb{R}$ using, say, Dedekind cuts of rational numbers.  Then it is well known how to prove these facts about $\mathbb{R}$, so we omit the proof for now.
=--

However, we note that the proof is valid in weak [[foundations]], in particular internal to any [[topos]] with a [[natural numbers object]].  One can actually work in even weaker foundations than that; see the constructions at [[real numbers object]].

Even weaker foundations are possible if one allows the [[underlying set]] of $\mathbb{R}$ to be [[proper class|large]].

However, if we are working in [[predicative mathematics]] with multiple [[universes]] in the foundation, the Dedekind real numbers would no longer be unique up to unique isomorphism, but rather there would be a set of Dedekind real numbers for each universe.

### The Cauchy complete ordered field

There is a well-known algebraic (more or less) characterisation of the real line as the 'Cauchy complete ordered field', or sometimes the 'Cauchy complete archimedean field'.  This can be interpreted as follows:

*  A __field__ is well known in algebra; if it matters, we mean a [[Heyting field]].
*  An __ordered field__ means a *[[linear order|linearly]]* ordered field.
*  An __archimedean field__ is an ordered field satisfying the [[archimedean property]].
*  An ordered field is __[[complete space|Cauchy complete]]__ if every [[Cauchy net]] in the field [[converges]].

In impredicative mathematics, we speak of [[the]] such field because it is unique up to unique [[isomorphism]].

+-- {: .un_theorem}
###### Theorem

Assuming impredicative foundations, there is an archimedean field $\mathbb{R}$ which is Cauchy-complete, and into which every archimedean field embeds.  Furthermore, every Cauchy-complete ordered field is isomorphic to $\mathbb{R}$, and uniquely so.
=--

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

Construct $\mathbb{R}$ using, say, Cauchy nets of rational numbers.  Then it is well known how to prove these facts about $\mathbb{R}$, so we omit the proof for now.
=--

However, we note that the proof is valid in weak [[foundations]], in particular internal to any [[topos]] with a [[natural numbers object]]. One can actually work in even weaker foundations than that; see the constructions at [[real numbers object]].

Even weaker foundations are possible if one allows the [[underlying set]] of $\mathbb{R}$ to be [[proper class|large]].

However, if we are working in [[predicative mathematics]] with multiple [[universes]] in the foundation, the Cauchy real numbers would no longer be unique up to unique isomorphism, but rather there would be a set of Cauchy real numbers for each universe.

### As an archimedean Tarski group

There is a characterisation of the real line as the 'complete archimedean Tarski group' due to Alfred Tarski. This can be interpreted as follows:

*  An __abelian group__ is well known in algebra.
*  An __linearly ordered abelian group__ is an [[abelian group]] with a [[linear order]].
*  A __densely linearly ordered abelian group__ is an linearly ordered abelian group that is also a [[dense linear order]].
*  A __Tarski group__ is a densely linearly ordered abelian group with a point $1$ such that $1 \lt 1 + 1$.
*  An __archimedean Tarski group__ is a [[Tarski group]] satisfying the [[archimedean property]].
*  An archimedean Tarski group is __complete__ if it is [[Dedekind completion|Dedekind-complete]].

See [[Tarski's axiomatization of the real numbers]] for more information.

### The locale of real numbers

Consider [[binary relations]] $\sim$ on a [[countable]] [[inhabited]] [[dense linear order]] without endpoints, such as the [[rational numbers]], satisfying these four properties:

*  If $a \geq b$, then $a \sim b$.
*  If $a \geq b \sim c \geq d$, then $a \sim d$.
*  If $a \sim b \gt c \sim d$, then $a \sim d$.
*  If $b \sim c$ whenever $a \lt b$ and $c \lt d$, then $a \sim d$.

The collection of all such relations form a [[frame]], which we may interpret (by definition) as the __[[locale of real numbers]]__.  It can also be defined as the [[localic completion]] of the rational numbers.

We may then define a __localic real number__ to be a [[point of a locale|point]] of this locale.

This agrees with the notion of Dedekind real number, even in very weak ([[predicative mathematics|predicative]] and [[constructive mathematics|constructive]]) [[foundations]].

See [[locale of real numbers]] for more.

### Unit intervals

The [[coalgebra of the real interval|unit interval]] of the real numbers $[0,1]$ could be constructed as a [[terminal coalgebra of an endofunctor]] in the [[category]] of [[intervals]]. Let $(\mathbb{R},0,+,-,1,\cdot,\lt)$ be an [[ordered field]] where $0 \lt 1$, with a [[monotone]] $f:[0,1]\to \mathbb{R}$ such that $f(0) = 0$ and $f(1) = 1$. The set $\mathbb{R}$ of real numbers is the initial such ordered field.

See also: [[dyadic interval coalgebra]], [[decimal interval coalgebra]], [[rational interval coalgebra]].

### $\mathbb{R}^+$ as a terminal coalgebra

The positive real line $\mathbb{R}^+$ may be characterized as the [[terminal object|terminal]] [[coalgebra for an endofunctor]]

Let [[Pos]] be the [[category]] of [[poset]]s with a [[forgetful functor]]

$$
  U\colon Pos \to Set
$$

Consider the endofunctor

$$
  F_1\colon Pos \to Pos
$$

defined as the [[ordinal product]]

$$
  F_1\colon X \mapsto \omega \cdot X,
$$
for $\omega \in Pos$, where $\omega \cdot X$ is the [[cartesian product]] $U(\omega) \times U(X)$ with the [[lexicographic order]].

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

The terminal coalgebra of $F_1$ is order isomorphic to the non-negative real line $\mathbb{R}^+$, with its standard order.
=--

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

This is theorem 5.1 in ([Pavlovic–Pratt 1999](#PavlovicPratt)).

There are many ways of setting up this description of $\mathbb{R}^+$, depending on the coalgebra structure $\mathbb{R}^+ \to \omega \cdot \mathbb{R}^+$ chosen. Here is one: there are evident poset isomorphisms $\mathbb{R}^+ \cong [1, \infty)$ and $\omega \cong \mathbb{N}_{\geq 2} = \{n \in \mathbb{N}: n \geq 2\}$. Define a map

$$(\alpha, \beta): [1, \infty) \to \mathbb{N}_{\geq 2} \cdot [1, \infty)$$

where $\alpha(x)$ is the smallest integer strictly greater than $x$, and $\beta(x) = 1/(\alpha(x) - x)$. The stream of integers $a_n = \alpha(\beta^n(x))$ gives a [[continued fraction]] representation of $x$ in the form

$$x = a_0 - \frac1{a_1 - \frac1{a_2 - \ldots}},$$

and the resulting bijection $[1, \infty) \to \mathbb{N}_{\geq 2} \times \mathbb{N}_{\geq 2} \times \ldots$, sending $x$ to $(a_0, a_1, \ldots)$, is in fact a poset isomorphism if we endow the right-hand side with the lexicographic order.
=--

Another way, which circumvents the use of isomorphisms $\mathbb{R}^+ \cong [1, \infty)$ and $\omega \cong \mathbb{N}_{\geq 2}$, is to define

$$(\alpha, \beta): \mathbb{R}^+ \to \omega \cdot \mathbb{R}^+$$

where $\alpha(x)$ is the floor of $x$, and $\beta(x) = 1/(1 - x + \alpha(x)) - 1$. Then $a_n = \alpha(\beta^n(x))$ gives a [[continued fraction]] representation of $x$ in the form

$$x = a_0 + \frac1{1 + \frac1{a_1 + \frac1{1 + \frac1{a_2 + \ldots}}}},$$
and the resulting [[bijection]] $\mathbb{R}_+ \to \omega \times \omega \times \ldots$, sending $x$ to $(a_0, a_1, \ldots)$, is again a poset isomorphism if we endow the right-hand side with the lexicographic order.

There are more and similar characterizations along these lines. An interesting application of the coalgebraic views is that they allow defining morphisms between the reals. A coalgebaic version Dedekind cuts or Conway numbers makes $\mathbb{R}$ into a compact category where the addition arises as the biproduct. The real vector spaces are the $\mathbb{R}$-enriched bicompletions and linear operators arise as the $\mathbb{R}$-enriched Kan extensions. ([Pavlovic 2020](#Pavlovic))

### Smooth real numbers

The **smooth real numbers** (say in a [[smooth topos]]) are only an [[ordered local ring]] rather than an [[ordered field]], because there might be non-zero non-invertible [[nilpotent]] [[infinitesimals]] in the smooth real numbers. Nevertheless, the quotient of the smooth real numbers by the [[ideal]] of non-invertible elements is one of the ordered field of real numbers as defined above.

### Other definitions

* [[Eudoxus real number]]
* [[prealgebra real number]]

## Topologies
{#Topologies}

There are alternative topologies on $\mathbb{R}$ sometimes considered:

* the [[discrete topology]] (not very interesting),
* the [[order topology]] or [[absolute-value topology]] (the usual topology of a [[Euclidean space|Euclidean]] [[metric space]]),
* the [[upper-interval topology]] or the [[lower-interval topology]],
* the [[lower semicontinuous topology]] or the [[upper semicontinuous topology]],
* the [[K-topology]].

Another variant of $\mathbb{R}$ as a topological space is the

* [[long line]].

## Generalisations

The term 'real number' was originally introduced to indicate that one is *not* considering the generalistion to [[complex numbers]] or other kinds of [[hypercomplex numbers]].  Accordingly, that term 'real' may sometimes be used for another generalisation of real numbers to indicate again that one is not considering a complexification.

The [[extended real number]]s include $\pm\infty$ as well as the real numbers; one may speak of _finite numbers_ or _bounded numbers_ to indicate that one is not considering this extension.  [[lower real|Lower reals]], [[upper reals]], and [[MacNeille real number|MacNeille reals]] are related generalisations studied in [[constructive mathematics]], although with [[excluded middle]] they are (at least if bounded) the same as ordinary real numbers; one may speak of _located numbers_ to indicate that one is not considering such extensions.

[[surreal number|Surreal numbers]] and the [[hyperreal number]]s of [[nonstandard analysis]] are two ways to include [[infinite number|infinite]] and [[infinitesimal number|infinitesimal]] versions of real numbers (besides the trivial case of $\pm\infty$); one may speak of _standard numbers_ to indicate that one is not considering such extensions (although the precise meaning of 'standard' depends on the [[universe]] that one is working in).

In [[descriptive set theory]], one often says 'real number' for an element of [[Baire space of irrational numbers|Baire space]] $\mathbb{N}^{\mathbb{N}}$.  This is not really a generalisation; by the [[Schroeder-Bernstein theorem]], the underlying sets of $\mathbb{R}$ and $\mathbb{N}^{\mathbb{N}}$ are [[bijection|isomorphic]].  Constructively, $\mathbb{N}^{\mathbb{N}}$ can still be thought of as the set of [[irrational numbers]], so this use of the term may actually be a *restriction*.

[[floating-point number|Floating-point numbers]] are often used in computer programming to represent real numbers, but they do not behave very well; one may speak of _infinite-precision numbers_ to indicate that one\'s programming environment models '[*real* real numbers](http://math.fau.edu/richman/html/mm2.htm)'.

As mentioned above, the $p$-[[adic number|adic numbers]] for various [[prime numbers]] $p$ are variations on the theme of real numbers; real numbers may be thought of as _$0$-adic numbers_.  Similarly, the real numbers are _characteristic-$0$ numbers_ since they are based on the [[prime field]] $\mathbb{Q}$; one could also start the construction with a different [[characteristic]] (although it makes more sense to get analogues of complex numbers than of real numbers).

Finally, one can consider points on a [[noncommutative geometry|noncommutative]] line instead of the usual _commutative numbers_.

So in summary, this page is about the _real, finite, located, standard, analytic, infinite-precision, $0$-adic, characteristic-$0$, commutative numbers_.

## Related concepts

* [[real numbers object]]

* [[computable real number]]

* [[Cauchy real number]]

* [[Dedekind real number]]

* [[real cohomology]], [[real homotopy theory]]

* [[p-adic numbers]]

* in [[constructive analysis]] one may use the [[completion monad]] for dealing with real numbers

* [[exact real computer arithmetic]]

* [[real-cohesive homotopy type theory]]

[[!include exceptional spinors and division algebras -- table]]

## References

For more see the references at _[[analysis]]_.

* {#Bourbaki71} [[Nicolas Bourbaki]], *Real Numbers*, Chapter IV in: *General topology*, Elements of Mathematics, Springer (1971, 1990, 1995) [[doi:10.1007/978-3-642-61701-0](https://doi.org/10.1007/978-3-642-61701-0)]

Review and history:

* [[Leo Corry]], §10.5 in: *A Brief History of Numbers*, Oxford University Press (2015) [[ISBN:9780198702597](https://global.oup.com/academic/product/a-brief-history-of-numbers-9780198702597)]

As the [[coalgebra of the real interval]]:

* {#PavlovicPratt} [[Dusko Pavlovic]], [[Vaughan Pratt]], _On coalgebra of real numbers_, Electronic Notes in Theoretical Computer Science **19** (1999) 103-117, doi:[10.1016/S1571-0661(05)80272-5](https://doi.org/10.1016/S1571-0661%2805%2980272-5), [CiteSeerx](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.5204)

A definition of the [[real numbers]] as [[generalized the|the]] [[terminal object|terminal]] [[Archimedean ordered field]] and as a [[complete space|complete]] [[Archimedean ordered field]]:

* {#Richman08} [[Fred Richman]], *Real numbers and other completions*, Mathematical Logic Quarterly **54** 1 (2008) 98-108 [[doi:10.1002/malq.200710024](https://onlinelibrary.wiley.com/doi/10.1002/malq.200710024)]

A compact category $\mathbb{R}$ making the $\mathbb{R}$-linear operators into $\mathbb{R}$-enriched Kan extensions:

* {#Pavlovic} [[Dusko Pavlovic]], _From process-propositions-as-types to categorified real numbers
and  monoidal computers_, Logic and Structure in Computer Science 1000-1070, doi:[10.1007/978-3-031-24117-8](https://doi.org/10.1007/978-3-031-24117-8), [arXiv:2007.10057](https://arxiv.org/abs/2007.10057v3)

The definition of the real numbers in [[constructive analysis]] as [[Cauchy real numbers]], namely as [[regular sequences|regular]] [[Cauchy sequences]] of [[rational numbers]] is due to:

* {#Bishop67} [[Errett Bishop]]: *[[Foundations of Constructive Analysis]]*, McGraw-Hill (1967)

* [[Errett Bishop]], [[Douglas Bridges]]: *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

Direct formalization of the definition of [[Cauchy real numbers]] from [Bishop (1967)](#Bishop67) in [[Agda]]:

* [[Martin Lundfall]], *Formalizing real numbers in Agda* (2015) [<a href="https://wcl.cs.rpi.edu/pilots/library/papers/TAGGED/4211-Lundfall%20(2015)%20-%20Formalizing%20Real%20Numbers%20in%20Agda.pdf">pdf</a>, [[Lundfall-RealNumbersInAgda.pdf:file]], [github](https://github.com/MrChico/Reals-in-agda)]

* [[Zachary Murray]], *Constructive Analysis in the Agda Proof Assistant* [[arXiv:2205.08354](https://arxiv.org/abs/2205.08354), [github](https://github.com/z-murray/honours-project-constructive-analysis-in-agda)]

review:

* [[Zachary Murray]], *Constructive Real Numbers in the Agda Proof Assistant*, [talk at *CQTS*](Center+for+Quantum+and+Topological+Systems#MurrayFeb2023), (Feb 2023) [video:[YT](https://www.youtube.com/watch?v=7Q_sjfyPJqU)]

and closely related constructions in [[Coq]] have been implemented in

* {#KrebbersSpitters11} [[Robbert Krebbers]], [[Bas Spitters]], *Type classes for efficient exact real arithmetic in Coq*, Logical Methods in Computer Science, **9** 1 (2013) lmcs:958 [[arXiv:1106.3448](http://arxiv.org/abs/1106.3448/), <a href="https://doi.org/10.2168/LMCS-9(1:1)2013">doi:10.2168/LMCS-9(1:1)2013</a>]

following a [[monad (in computer science)|monadic]] re-formulation (via the [[completion monad]]) due to

* [[Russell O'Connor]], *A Monadic, Functional Implementation of Real Numbers*, Math. Struc. Comp. Sci. **17** 1 (2007) 129-159 [[arXiv:cs/0605058](https://arxiv.org/abs/cs/0605058), [doi:10.1017/S0960129506005871](https://doi.org/10.1017/S0960129506005871)]

Overview of implementation of real numbers in [[proof assistants]] (cf. [[constructive analysis]]):

* Sylvie Boldo, Catherine Lelay, Guillaume Melquiond, *Formalization of Real Analysis: A Survey of Proof
Assistants and Libraries*, Mathematical Structures in Computer Science **26** 7 (2016) 1196-1233  [[hal:00806920](https://hal.inria.fr/hal-00806920v1), [doi:10.1017/S0960129514000437](https://doi.org/10.1017/S0960129514000437)]

A novel construction principle of the type of real numbers, as a [[higher inductive-inductive type]] in [[univalence axiom|univalent]] [[homotopy type theory]], not reliant on representatives by sequences of [[rational numbers]], and with provable [[Cauchy completeness]] even without extra [[axiom of countable choice]] is laid out (cf. *[[HoTT book real numbers]]*) in

* [[Univalent Foundations Project]]: Chapter 11 of:  *[[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)]

Something like this has been implemented in [[Coq]]:

* [[Gaëtan Gilbert]], *Formalising Real Numbers in Homotopy Type Theory*, in *CPP 2017: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs*  (2017) 112–124 [[arXiv:1610.05072](https://arxiv.org/abs/1610.05072), [doi:10.1145/3018610.3018614](https://doi.org/10.1145/3018610.3018614)]

[[!redirects real number]]
[[!redirects real numbers]]

[[!redirects real line]]
[[!redirects real number line]]
[[!redirects the continuum]]

[[!redirects located real number]]
[[!redirects located real numbers]]
[[!redirects located Dedekind real number]]
[[!redirects located Dedekind real numbers]]

[[!redirects smooth real number]]
[[!redirects smooth real numbers]]


real numbers object

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

# Real numbers object
* table of contents
{: toc}

## Idea

Recall that it is possible to define an [[internalization]] of the set of [[natural numbers]], called a [[natural numbers object]] (NNO), in any [[cartesian monoidal category]] (a category with finite [[product|products]]). In particular, the notion makes sense in a [[topos]]. But a topos supports [[intuitionistic logic|intuitionistic]] [[higher-order logic]], so once we have an NNO, it is also possible to repeat the usual construction of the [[integer|integers]], the [[rational number|rationals]], and then finally the [[real number|real numbers]]; we thus obtain an internalization of $\mathbb{R}$ in any topos with an NNO.

More generally, we can define a real numbers object (RNO) in any category with sufficient structure; RNOs can be defined for many flavors of category, from cartesian monoidal categories up to topoi. Then we can prove that an RNO exists in any topos with an NNO, as well as in some other situations.

## Definition

Let $\mathcal{E}$ be a [[Heyting category]].  (This means, in particular, that we can interpret full [[first-order logic|first-order]] [[intuitionistic logic]] using the [[stack semantics]].)

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

A (located Dedekind) __real numbers object__ in $\mathcal{E}$ is a [[ring object]] in $\mathcal{E}$ that satisfies (in the [[internal logic]]) the axioms of a [[Dedekind-complete]] [[strictly totally ordered]] [[Heyting field]].
=--

In detail:

A _commutative ring object_ in $\mathcal{E}$ is an object $R$ equipped with morphisms $0\colon \mathbf{1} \to R$, ${-}\colon R \to R$, ${+}\colon R \times R \to R$, $1\colon \mathbf{1} \to R$, and ${\cdot}\colon R \times R \to R$ (where $\mathbf{1}$ is the [[terminal object]] of $\mathcal{E}$ and $\times$ is the [[product]] operation in $\mathcal{E}$) that make the [[ring object]] diagrams commute.

Given a commutative ring object $R$ in $\mathcal{E}$, we define a [[binary relation]] $\#$ on $R$ (that is a [[subobject]] of $R \times R$) as
$$ \{ (x,y)\colon R \times R \;|\; \exists z\colon R.\, x \cdot z = y \cdot z + 1 \} ,$$
written in the [[internal language]] of $\mathcal{E}$.  Then $R$ is a (Heyting) _field object_ if $\#$ is a tight [[apartness relation]]; that is if the following axioms (in the internal language) hold:

*  [[irreflexive relation|irreflexivity]]: $\forall x\colon R.\, \neg(x \# x)$,
*  [[symmetric relation|symmetry]]: $\forall x\colon R.\, \forall y\colon R.\, (x \# y \implies y \# x)$ (which can actually be proved using $-$),
*  [[comparison relation|comparison]]: $\forall x\colon R.\, \forall y\colon R.\, \forall z\colon R.\, (x \# z \implies (x \# y \vee y \# z))$,
*  [[tight relation|tightness]]: $\forall x\colon R.\, \forall y\colon R.\, (\neg(x \# y) \implies x = y)$.

A (strictly totally) _ordered field object_ in $\mathcal{E}$ is a field object $R$ equipped with a binary relation $\lt$ such that the following axioms hold:

*  strong [[irreflexive relation|irreflexivity]]: $\forall x\colon R.\, \forall y\colon R.\, (x \lt y \implies x \# y)$,
*  strong [[connected relation|connectedness]]: $\forall x\colon R.\, \forall y\colon R.\, (x \# y \implies (x \lt y \vee y \lt x))$,
*  [[transitive relation|transitivity]]: $\forall x\colon R.\, \forall y\colon R.\, \forall z\colon R.\, ((x \lt y \wedge y \lt z) \implies x \lt z)$,
*  respecting addition: $\forall x\colon R.\, \forall y\colon R.\, \forall z\colon R.\, (x \lt y \implies x + z \lt y + z)$,
*  respecting multiplication: $\forall x\colon R.\, \forall y\colon R.\, \forall z\colon R.\, ((x \lt y \wedge 0 \lt z) \implies x \cdot z \lt y \cdot z)$.

Given an ordered field object $R$ in $\mathcal{E}$, any object $\Gamma$ in $\mathcal{E}$, and subobjects $L$ and $U$ of $\Gamma \times R$, we say that $(L,U)$ is a _Dedekind cut_ in $R$ (parametrised by $\Gamma$) if the following axioms hold:

*  lower bound: $\forall a\colon \Gamma.\, \exists x\colon R.\, (a,x) \in L$,
*  upper bound: $\forall a\colon \Gamma.\, \exists x\colon R.\, (a,x) \in U$,
*  downward roundedness: $\forall a\colon \Gamma.\, \forall x\colon R.\, \forall y\colon R.\, ((x \lt y \wedge (a,y) \in L) \implies (a,x) \in L)$,
*  upward roundedness: $\forall a\colon \Gamma.\, \forall x\colon R.\, \forall y\colon R.\, (((a,x) \in U \wedge x \lt y) \implies (a,y) \in U)$,
*  upward openness: $\forall a\colon \Gamma.\, \forall x\colon R.\, ((a,x) \in L \implies \exists b\colon \Gamma.\, \exists y\colon R.\, ((b,y) \in L \wedge x \lt y))$,
*  downward openness: $\forall a\colon \Gamma.\, \forall x\colon R.\, ((a,x) \in U \implies \exists b\colon \Gamma.\, \exists y\colon R.\, ((b,y) \in U \wedge y \lt x))$,
*  locatedness: $\forall a\colon \Gamma.\, \forall x\colon R.\, \forall y\colon R.\, (x \lt y \implies ((a,x) \in L \vee (a,y) \in U))$,
*  separation: $\forall a\colon \Gamma.\, \forall x\colon R.\, \forall y\colon R.\, (((a,x) \in L \wedge (a,y) \in U) \implies x \lt y)$.

An ordered field object $R$ in $\mathcal{E}$ is _Dedekind complete_ if, given any object $\Gamma$ of $\mathcal{E}$ and any Dedekind cut $(L,U)$ in $R$ parametrised by $\Gamma$, there exists a morphism $x\colon \Gamma \to R$ such that
$$ L = \{ (a,b)\colon \Gamma \times R \;|\; b \lt x(a) \} ,$$
$$ U = \{ (a,b)\colon \Gamma \times R \;|\; x(a) \lt b \} .$$

Finally, a _real numbers object_ in $\mathcal{E}$ is a Dedekind-complete ordered field object.

## Properties

In the last requirement, of Dedekind completeness, we postulate (under certain conditions) the existence of a morphism $x\colon \Gamma \to R$ satisfying certain properties.
+-- {: .num_thm}
###### Theorem
This morphism is in fact unique.
=--
Here is an explicit 'external' proof:
+-- {: .proof}
###### Proof
Suppose that $x, x'\colon \Gamma \to R$ both satisfy the required properties, and consider $x \# x'$, which is a subobject of $\Gamma$.  By the strong connectedness of $\lt$, $x \# x'$ is contained in (factors through) $x \lt x' \vee x' \lt x$, which is the [[union]] of $x \lt x'$ and $x' \lt x$.

Now consider $x \lt x'$, and let $a$ be a [[generalised element]] of $\Gamma$.  If $a$ belongs to (factors through) $x \lt x'$, or equivalently $(x(a), x'(a))$ belongs to $\lt$, it follows that $(a,x(a))$ belongs to $L$.  Thus, $(x(a), x(a))$ also belongs to $\lt$, or equivalently $a$ belongs to $x \lt x$.  By the strong irreflexivity of $\lt$, this is contained in $x \# x$; by the irreflexivity of $\#$, this is contained in $\bot$ (as a subobject of $\Gamma$).  Since every $a$ that belongs to $x \lt x'$ belongs to $\bot$, $x \lt x'$ is contained in (and so equals as a subobject) $\bot$.

Similarly (either by swapping $x$ with $x'$ or by using $U$ instead of $L$), $x' \lt x$ is also $\bot$.  Therefore, $x \# x'$ is $\bot$.  By the tightness of $\#$, $x = x'$.
=--
Here is an 'internal' proof, to be interpreted in the [[stack semantics]] of $\mathcal{E}$:
+-- {: .proof}
###### Proof
Suppose that $x, x'\colon R$ both satisfy the required properties, and suppose that $x \# x'$.  By the strong connectedness of $\lt$, $x \lt x'$ or $x' \lt x$.

Now suppose that $x \lt x'$.  It follows that $x$ belongs to $L$, so $x \lt x$.  By the strong irreflexivity of $\lt$, $x \# x$; by the irreflexivity of $\#$, we have a contradiction.

Similarly (either by swapping $x$ with $x'$ or by using $U$ instead of $L$), $x' \lt x$ is also false.  Therefore, $x \# x'$ is false.  By the tightness of $\#$, $x = x'$.
=--

In the definition of a Heyting field object, all of the axioms except the last are [[coherent logic|coherent]] and therefore make sense in any [[coherent category]].
+-- {: .num_thm}
###### Theorem
An object satisfying all but the last axiom of a field object is precisely a [[local ring]] object (so in particular an RNO is a local ring object).
=--
+-- {: .proof}
###### Proof
...
=--

It would be nice to say that a Heyting category with an RNO must have an NNO; after all, $\mathbb{N}$ is contained in $\mathbb{R}$.  However, my only argument is impredicative; although I don't know a specific example, there could be a [[Π-pretopos]] with an RNO but no NNO.  However, the argument works for a [[infinitary coherent category|geometric]] Heyting category or a [[topos]].  (In light of the constructions below, the existence of an RNO is therefore equivalent to the existence of an NNO in a topos.)
+-- {: .num_thm}
###### Theorem
If $R$ is an RNO in an infinitary Heyting category or topos, then there is unique subobject $N$ of $R$ that is both a sub-[[rig]] object of $R$ and an NNO under the operations $0\colon \mathbf{1} \to N$ and $({-}) + 1\colon N \to N$.
=--
+-- {: .proof}
###### Proof
...
=--

We usually speak of *[[the]]* RNO, if one exists.  This is because any two RNOs in a Heyting category with an NNO are [[isomorphic]], in an essentially unique way.  (I can't prove this without an NNO, although the previous theorem shows that we often have one.)

+-- {: .num_thm}
###### Theorem
If $R$ and $R'$ are both RNOs in a Heyting category $\mathcal{E}$ with an NNO, then there is a unique isomorphism from $R$ to $R'$ that preserves the structures on them ($0$, $-$, $+$, $1$, $\cdot$, $\lt$).
=--
+-- {: .proof}
###### Proof
...
=--

+-- {: .num_thm}
###### Theorem
If $R$ is an RNOs in a Heyting category $\mathcal{E}$ with an NNO and $R'$ is an [[Archimedean ordered field|Archimedean ordered]] [[field extension]] of $R$, then there is a unique isomorphism from $R$ to $R'$ that preserves the structures on them ($0$, $-$, $+$, $1$, $\cdot$, $\lt$).
=--
+-- {: .proof}
###### Proof
...
=--

+-- {: .num_thm}
###### Theorem
Any RNO $R$ in a Heyting category $\mathcal{E}$ with an NNO is the [[terminal]] [[Archimedean ordered field]] object in $\mathcal{E}$.
=--
+-- {: .proof}
###### Proof
...
=--

## Constructions

We can construct a real numbers object in the following cases (presumably among others):

1. in a topos with an NNO;
2. in a $\Pi$-[[Pi-pretopos|pretopos]] with an NNO and [[weak countable choice]];
3. in a $\Pi$-pretopos with an NNO and [[subset collection]].

(Actually, (1) is a special case of (3), but the usual construction in (1) is not available in (3).  Thus, we still have three different constructions to consider.)

### In a topos with an NNO

Let $\mathcal{E}$ be an [[elementary topos]] with a natural numbers object $\mathbb{N}$. The **Dedekind real numbers object** of $\mathcal{E}$ is the object of all [[Dedekind cut|Dedekind cuts]]. To be more precise, we will need to make some auxiliary definitions.

We first construct an integers object as follows. Let $a, b\colon E \to \mathbb{N} \times \mathbb{N}$ be the [[kernel pair]] of the addition map ${+}\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, and let $\pi_1, \pi_2\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ be the [[product]] projections. We define $\mathbb{Z}$ to be the [[coequalizer]] of $(\pi_1 \circ a, \pi_2 \circ b), (\pi_1 \circ b, \pi_2 \circ a)\colon E \to \mathbb{N}$. A similar construction yields a rational numbers object $\mathbb{Q}$.

We denote by $\mathcal{P}(A)$ the [[power object]] of $A$ in $\mathcal{E}$. A **Dedekind cut** is a [[generalized element]] $(L, U)$ of $\mathcal{P}(\mathbb{Q}) \times \mathcal{P}(\mathbb{Q})$, satisfying the following conditions, expressed in the [[Mitchell-Bénabou language]] of $\mathcal{E}$ and interpreted under [[Kripke-Joyal semantics]]:

1. Non-degenerate:
   $$ \exists q\colon \mathbb{Q}.\, q \in L ;$$
   $$ \exists r\colon \mathbb{Q}.\, r \in U .$$

2. Inward-closed:
   $$ \forall q\colon \mathbb{Q}.\, \forall r\colon \mathbb{Q}.\, ((q \lt r \wedge r \in L) \implies q \in L) ;$$
   $$ \forall q\colon \mathbb{Q}.\, \forall r\colon \mathbb{Q}.\, ((r \lt q \wedge r \in U) \implies q \in U) .$$

3. Outward-open:
   $$ \forall q\colon \mathbb{Q}.\, (q \in L \implies \exists r\colon \mathbb{Q}.\, (r \in L \wedge q \lt r)) $$
   $$ \forall q\colon \mathbb{Q}.\, (q \in U \implies \exists r\colon \mathbb{Q}.\, (r \in L \wedge r \lt q)) .$$

4. Located:
   $$ \forall q\colon \mathbb{Q}.\, \forall r\colon \mathbb{Q}.\, (q \lt r \implies (q \in L \vee r \in U)) .$$

5. Mutually exclusive:
   $$ \forall q\colon \mathbb{Q}.\, \neg(q \in L \wedge q \in U) .$$

The relation $\lt$ on $\mathbb{Q}$ is the order relation constructed in the usual way. We define $\mathbb{R}$ to be the subobject of $\mathcal{P}(\mathbb{Q}) \times \mathcal{P}(\mathbb{Q})$ consisting of all Dedekind cuts as defined above.

### In a $\Pi$-pretopos with $WCC$

Summary: construct a [[Cauchy real numbers]] object and use $WCC$ ([[weak countable choice]]) to prove that it is an RNO.

Note that any [[Boolean topos]] with an NNO satisfies $WCC$, so in all we have three different constructions available in that case.

### In a $\Pi$-pretopos with an NNO and subset collection

Summary: modify the construction of a [[Cauchy real numbers]] object to use [[multi-valued function|multi-valued]] [[Cauchy sequences]].

## Examples

### In $Set$

The real numbers object in [[Set]] is the [[real line]], the usual set of (located Dedekind) [[real numbers]].  Note that this is a theorem of [[constructive mathematics]], as long as we assume that $Set$ is an elementary topos with an NNO (or more generally a [[Π-pretopos]] with [[natural numbers object|NNO]] and either WCC or subset collection).

### In sheaves on a topological space

Let $X$ be a [[topological space]], and $\mathrm{Sh}(X)$ its [[category of sheaves]]. It is well-known that $\mathrm{Sh}(X)$ is a [[Grothendieck topos]] (and so, _a fortiori_, an elementary topos), and the [[constant sheaf]] [[functor]] $\Delta\colon \mathbf{Set} \to \mathrm{Sh}(X)$ preserves [[finite limits]] and has the [[global section]] functor $\Gamma\colon \mathrm{Sh}(X) \to \mathbf{Set}$ as a [[right adjoint]]. (Hence, $\Delta$ and $\Gamma$ are the components of a [[geometric morphism]] $\mathrm{Sh}(X) \to \mathbf{Set}$.) The following claims are essentially immediate:

1. If $\mathbf{N}$ is the set of natural numbers, then $\Delta (\mathbf{N})$ must be an NNO in $\mathrm{Sh}(X)$, since $\Delta$ has a right adjoint.

2. If $\mathbf{Z}$ is the set of integers, then $\Delta (\mathbf{Z})$ is an integers object in $\mathrm{Sh}(X)$ (as defined above), since $\Delta$ preserves finite limits and colimits.

3. Similarly, if $\mathbf{Q}$ is the set of rational numbers, then $\Delta (\mathbf{Q})$ is a rational numbers object in $\mathrm{Sh}(X)$.

Thus, for every topological space $X$, the topos $\mathrm{Sh}(X)$ has a Dedekind real numbers object $\mathbb{R}$. Naïvely one might expect $\mathbb{R}$ to be isomorphic to the constant sheaf $\Delta(\mathbf{R})$, where $\mathbf{R}$ is the classical set of real numbers, but this turns out not to be the case. Instead, we have a rather more remarkable result:

+-- {: .num_theorem #DedekindRealsInToposOverTopologicalSpace}
###### Theorem

A Dedekind real numbers object $\mathbb{R}$ in the topos $\mathrm{Sh}(X)$ is isomorphic to the sheaf of real-valued [[continuous functions]] on $X$.

=--

This is shown in ([MacLane-Moerdijk, Chapter VI, §8, theorem 2](#MM94)); see also below.

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

Theorem \ref{DedekindRealsInToposOverTopologicalSpace} allows us to define various further constructions on $X$ in internal terms in $\mathrm{Sh}(X)$; for example, a [[vector bundle]] over $X$ is an internal [[projective object|projective]] $\mathbb{R}$-[[module]].

=--

### In sheaves on a gros site of topological spaces
 {#InGrosToposOnTopologicalSpaces}

+-- {: .num_theorem}
###### Theorem

Let $\{\mathbb{R}\} \hookrightarrow S \hookrightarrow Top$ be a [[small category|small]] [[full subcategory]] of [[Top]] including the [[real line]].  If $S$ is closed under forming [[open subspaces]] and pullbacks of open subspaces and we equip it with the open-cover [[coverage]], then the Dedekind real number object internal to $Sh(S)$ is [[representable functor|represented]] by $\mathbb{R}^1$.
=--

This is proven as ([MacLane-Moerdijk,  chapter VI §9, theorem 2](#MM94)) under the stronger assumption that $S$ is closed under open subspaces and finite limits, by showing that over each object in the site the argument reduces essentially to that of theorem \ref{DedekindRealsInToposOverTopologicalSpace} for that object.  However, the finite limits are not necessary; see also below.  The more general version includes the cases

* $S =$ $\{$ [[locally contractible topological spaces]] $\}$

* $S =$ $\{$ [[topological manifolds]] $\}$

(for which $Sh(S)$ is [[cohesive topos]] and $Sh_\infty(S)$ is an [[cohesive (∞,1)-topos]]).

### In a general sheaf topos
 {#InASheafTopos}

We can generalize the above theorem as follows.  Let $S$ be any [[site]], and for any object $X\in S$ let $L(X)$ denote the [[locale]] whose [[frame]] of opens is the frame of [[subobjects]] of the sheafified representable $y X \in Sh(S)$.  We have an induced functor $L:S\to Loc$.  We can also regard the ordinary real numbers $\mathbb{R}$ as a locale.

+-- {: .num_theorem}
###### Theorem
The Dedekind real number object in $Sh(S)$ is the functor $Loc(L-,\mathbb{R})$.
=--
+-- {: .proof}
###### Proof
The sheaf topos $Sh(\mathbb{R})$ is the classifying topos of the geometric theory of a real number, in the sense that for any Grothendieck topos $E$, geometric morphisms $E \to Sh(\mathbb{R})$ are equivalent to global points of the real numbers object $\mathbb{R}_E$ in $E$.  Since pullback functors are logical, they preserve the real numbers object; thus for any $X\in E$, maps $X\to \mathbb{R}_E$ are equivalent to geometric morphisms $E/X \to Sh(\mathbb{R})$.  But $Sh(\mathbb{R})$ is localic, so such geometric morphisms factor through the localic reflection of $E/X$, and therefore are equivalent to continuous $\mathbb{R}$-valued functions defined on the "little locale of $X$", i.e. the locale associated to the frame of subobjects of $X$ in $E$.

Therefore, if $E = Sh(S)$ for some site $S$, then $\mathbb{R}_E$ is the sheaf on $S$ where $\mathbb{R}_E(X)=$ the set of continuous $\mathbb{R}$-valued functions on the little locale of $y X \in E$, which is what we have called $L X$.
=--

To deduce the previous theorem from this one, it suffices to observe that if $S\subset Top$ is closed under open subspaces and their pullbacks and equipped with the open-cover coverage, then every subobject of $y X\in Sh(S)$, for any $X\in S$, is uniquely representable by an open subset of $X$.

+--{: .query}
There is some dispute about this, see [here](http://nforum.ncatlab.org/discussion/6289/when-is-the-internal-real-line-the-external-real-line/?Focus=50275#Comment_50275).

Resolution seems to be [here](http://mathoverflow.net/a/186165/381)

=--

## Generalizations

### Cauchy real numbers
It is also possible to define the notion of a [[Cauchy real number]] object and construct one in any $\Pi$-pretopos with an NNO, but as the internal logic in general lacks [[weak countable choice]], these are usually inequivalent.  (There is also potentially a difference between the *classical* Cauchy RNO and the *modulated* Cauchy RNO; see definitions at [[Cauchy real number]], to be interpreted in the [[stack semantics]].)

### Classical Dedekind real numbers
In any $\Pi$-pretopos with a NNO, one could define an object that roughly behaves like the Dedekind real numbers in classical mathematics. Instead of using [[subobjects]] of $\mathbb{Q}$ in defining a cut, one instead uses the [[decidable object|decidable morphisms]] $L, U: \mathbb{Q} \rightarrow 1\coprod 1$ to define decidable cuts, and constructs the classical Dedekind real numbers as pairs of decidable cuts. The classical and constructive Dedekind real numbers are usually inequivalent, the classical Dedekind real numbers being equivalent to classical [[Cauchy real numbers]] and the constructive Dedekind real numbers being equivalent to generalised Cauchy real numbers

## Related concepts

* [[real number]]

* [[Cauchy real number]]

* [[one-sided real numbers]]

* [[natural number]]

* [[natural numbers object]]

* [[smooth structure on a topos]]

* [[real numbers type]]

## References

Discussion in [[topos theory]] is in

* [[Peter Johnstone]], section D4.7 of  _[[Sketches of an Elephant]]_

* [[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977 (Dover reprint New York 2014). (section 6.6. pp.210-23)

* {#MM94} [[Saunders Mac Lane]], [[Ieke Moerdijk]], section VI.8 of _[[Sheaves in Geometry and Logic]]_ .

* [[Eduardo Dubuc]], _Logical Opens and Real Numbers in Topoi_ , JPAA **43** (1986) pp.129-143.

* {#Fourman75} [[Michael Fourman]], _Comparaison des Réels d'un Topos - Structures Lisses sur un Topos Elémentaire_ , Cah. Top. Géom. Diff. Cat. **16** (1975) pp.233-239. ( _Colloque Amiens 1975 proceedings_ ) (p. 18-24 in [NUMDAM](http://www.numdam.org/item?id=CTGDC_1975__16_3_217_0)))

* [[Michael Fourman]], _T$_1$ Spaces over Topological Sites_ , JPAA **27** (1983) pp.223-224.

* [[Michael Fourman]], [[Martin Hyland]], _Sheaf Models for Analysis_ , pp.280-301 in Fourman, Mulvey, Scott (eds.), _Applications of Sheaves_ , LNM **753** Springer Heidelberg 1979. ([draft](https://www.dpmms.cam.ac.uk/~martin/Research/Oldpapers/analysis79.pdf), 6.64 MB)

* [[André Joyal]], [[Gonzalo E. Reyes]], _Separably Real Closed Local Rings_ , JPAA **43** (1986) pp.271–279.

* [[Ieke Moerdijk]], [[Gonzalo E. Reyes]], _Smooth Spaces versus Continuous Spaces in Models of Synthetic Differential Geometry_ , JPAA **32** (1984) pp.143-176.

* J. Z. Reichman, _Semicontinuous Real Numbers in a Topos_ , JPAA **28** (1983) pp.81-91.

* L. Stout, _Unpleasant Properties of the Reals in a Topos_ , Cah. Top. Géom. Diff. Cat. **16** (1975) pp.320-322. ([Colloque Amiens 1975 proceedings](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1975__16_3/CTGDC_1975__16_3_217_0/CTGDC_1975__16_3_217_0.pdf),6.81 MB)

* L. N. Stout, _Topological Properties of the Real Numbers Object in a Topos_ , Cah. Top. Géom. Diff. Cat. **17** no.3 (1976) pp.295-326. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1976__17_3/CTGDC_1976__17_3_295_0/CTGDC_1976__17_3_295_0.pdf))

Discussion in [[homotopy type theory]] is in

* [[Univalent Foundations Project]], section 11 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

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