real numbers type

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## Idea

There are many possible definitions of the (two-sided, Dedekind) real numbers type in [[dependent type theory]]. If the dependent type theory has a [[type universe]] or a [[type of all propositions]], then one can translate the impredicative definition of the [[Dedekind real numbers]] over to dependent type theory.

However, in [[dependent type theory]] without a [[type universe]] or a [[type of all propositions]], the impredicative definition of the [[Dedekind real numbers]] no longer works. Nonetheless, it is possible to define the type of [[real numbers]] $\mathbb{R}$ via [[inference rules]] as a type with a type family of [[domains]] of the [[structural Dedekind cut]] associated with each [[real number]].

## Definitions

We shall assume the minimum amount of type formers in [[dependent type theory]] to define [[Dedekind cuts]] of the rational numbers:

* [[dependent product types]], [[dependent sum types]], [[identity types]], [[empty type]], [[unit type]], [[sum types]], [[bracket types]], and a [[natural numbers type]]

The types in the dependent type theory form the [[(infinity,1)-category theory|(infinity,1)-categorical]] version of a [[Heyting category]] with [[exponential objects]] and a [[natural numbers object]].

In addition, if one has a [[type of all propositions]] $\mathrm{Prop}$ in the [[dependent type theory]], one can define the type of real numbers as the type of all (two-sided) Dedekind cuts, which are [[subtypes]] or [[predicates]] $C:(\mathbb{Q} + \mathbb{Q}) \to \mathrm{Prop}$ of the [[sum type]] $\mathbb{Q} + \mathbb{Q}$ which satisfy the following axioms:

* Dedekind cuts are bounded:
$$\exists q:\mathbb{Q}.C(\mathrm{inl}(q)) \quad \exists r:\mathbb{Q}.C(\mathrm{inr}(r))$$

* Dedekind cuts are rounded:
$$\prod_{q:\mathbb{Q}} C(\mathrm{inl}(q)) \simeq \exists r:\mathbb{Q}.(q \lt r) \times C(\mathrm{inr}(r))$$
$$\prod_{r:\mathbb{Q}} C(\mathrm{inr}(r)) \simeq \exists q:\mathbb{Q}.(q \lt r) \times C(\mathrm{inl}(q))$$

* Dedekind cuts are disjoint:
$$\prod_{q:\mathbb{Q}} \neg (C(\mathrm{inl}(q)) \times C(\mathrm{inl}(r)))$$

* Dedekind cuts are located:
$$\prod_{q:\mathbb{Q}} \prod_{r:\mathbb{Q}} (q \lt r) \to (C(\mathrm{inl}(q)) \vee C(\mathrm{inr}(r))$$

Usually, Dedekind cuts are presented as pairs of predicates $(L, U):(\mathbb{Q} \to \mathrm{Prop}) \times (\mathbb{Q} \to \mathrm{Prop})$, but this is equivalent to the above definition, since for any types $A$, $B$, and $C$, one has that
$$((A + B) \to C) \simeq (A \to C) \times (B \to C)$$

The predicate $\mathrm{isCut}(C)$ is defined as the product of all the axioms above, and the type of real numbers is defined as
$$\mathbb{R} \coloneqq \sum_{C:(\mathbb{Q} + \mathbb{Q}) \to \mathrm{Prop}} \mathrm{isCut}(C)$$

In dependent type theory with a [[type universe]] $U$, one can take the type of all $U$-small propositions by the dependent sum type

$$\mathrm{Prop}_U \coloneqq \sum_{A:U} \mathrm{isProp}(A)$$

and define locally $U$-small Dedekind cuts $C:(\mathbb{Q} + \mathbb{Q}) \to \mathrm{Prop}_U$ and the (locally $U$-small) real numbers $\mathbb{R}_U$ using $\mathrm{Prop}_U$.

So, what happens if the dependent type theory has neither a [[type of all propositions]] nor a [[type universe]], so that we are working in fully [[predicative mathematics]]? It is no longer possible to quantify over all subtypes of $\mathbb{Q} + \mathbb{Q}$, so one cannot define the type of real numbers impredicatively. Also, without impredicativity, a [[subtype]] of the sum type $\mathbb{Q} + \mathbb{Q}$ is a type $C$ with an [[embedding of types]] $e:C \hookrightarrow (\mathbb{Q} + \mathbb{Q})$. One can show from the definition of embedding that given an element $x:\mathbb{Q} + \mathbb{Q}$, the [[fiber type]] of $e$ at $x$ is a [[mere proposition]]
$$\mathrm{isProp}\left(\sum_{z:C} e(z) =_{\mathbb{Q} + \mathbb{Q}} x\right)$$
so everywhere in the definition above where one would have used the type family $C(x)$ one can instead use the [[dependent sum type]] $\mathrm{fiber}(e, x) \coloneqq \sum_{z:C} e(z) =_{\mathbb{Q} + \mathbb{Q}} x$.

Then a Dedekind cut consists of a type $C$ with an embedding of types $e:C \hookrightarrow (\mathbb{Q} + \mathbb{Q})$, which satisfies the following axioms:

* Dedekind cuts are bounded:
$$\exists q:\mathbb{Q}.\mathrm{fiber}(e, \mathrm{inl}(q)) \quad \exists r:\mathbb{Q}.\mathrm{fiber}(e, \mathrm{inr}(r))$$

* Dedekind cuts are rounded:
$$\prod_{q:\mathbb{Q}} \mathrm{fiber}(e, \mathrm{inl}(q)) \simeq \exists r:\mathbb{Q}.(q \lt r) \times \mathrm{fiber}(e, \mathrm{inr}(r))$$
$$\prod_{r:\mathbb{Q}} \mathrm{fiber}(e, \mathrm{inr}(r)) \simeq \exists q:\mathbb{Q}.(q \lt r) \times \mathrm{fiber}(e, \mathrm{inl}(q))$$

* Dedekind cuts are disjoint:
$$\prod_{q:\mathbb{Q}} \neg (\mathrm{fiber}(e, \mathrm{inl}(q)) \times \mathrm{fiber}(e, \mathrm{inl}(r)))$$

* Dedekind cuts are located:
$$\prod_{q:\mathbb{Q}} \prod_{r:\mathbb{Q}} (q \lt r) \to (\mathrm{fiber}(e, \mathrm{inl}(q)) \vee \mathrm{fiber}(e, \mathrm{inr}(r))$$

The predicate $\mathrm{isCut}(C, e)$ is defined as the product of all the axioms above, and the type of all Dedekind cuts with [[domain]] $C$ is given by the dependent sum type

$$\mathrm{Cuts}(C) \coloneqq \sum_{e:C \hookrightarrow \mathbb{Q} + \mathbb{Q}} \mathrm{isCut}(C, e)$$

If there existed a type universe, one can simply define the real numbers as the type of Dedekind cuts in $U$;

$$\mathbb{R}_U \coloneqq \sum_{C:U} \mathrm{Cuts}(C)$$

However, while there is no type universe $U$, one can add inference rules for the type of real numbers $\mathbb{R}$ such that $\mathbb{R}$ behaves as the [[dependent sum type]] $\sum_{C:U} \mathrm{Cuts}(C)$ if the hypothetical $U$ contained every single type in the type theory.

Formation rules for the real numbers type:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{R} \; \mathrm{type}}$$

Introduction rules for real numbers type:

$$\frac{\Gamma \vdash C \; \mathrm{type}}{\Gamma \vdash \mathrm{inR}_C:\mathrm{Cuts}(C) \to \mathbb{R}}$$

Elimination rules for real numbers type:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{R} \vdash \mathrm{CutDomain}(x) \; \mathrm{type}} \quad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{R} \vdash \mathrm{cut}(x):\mathrm{Cuts}(\mathrm{CutDomain}(x))}$$

Computation rules for real numbers type:

$$\frac{\Gamma \vdash C \; \mathrm{type} \quad c:\mathrm{Cuts}(C)}{\Gamma \vdash \mathrm{CutDomain}(\mathrm{inR}_C(c)) \equiv C \; \mathrm{type}} \quad \frac{\Gamma \vdash C \; \mathrm{type} \quad c:\mathrm{Cuts}(C)}{\Gamma \vdash \mathrm{cut}(\mathrm{inR}_C(c)) \equiv c:\mathrm{Cuts}(C)}$$

Uniqueness rules for real numbers type:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{R} \vdash \mathrm{inR}_{\mathrm{CutDomain}(x)}(\mathrm{cut}(x)) \equiv x:\mathrm{Cuts}(\mathrm{CutDomain}(x))}$$

The inference rules for the real numbers type imply that the real numbers type are a [[Tarski universe]] with universal type family $\mathrm{CutDomain}(x)$ indexed by $x:\mathbb{R}$, where $\mathrm{CutDomain}(x)$ is the [[domain]] of the [[Dedekind cut]]
$$\mathrm{cut}(x):\mathrm{CutDomain}(x) \hookrightarrow (\mathbb{Q} + \mathbb{Q})$$
associated with the real number $x:\mathbb{R}$. This makes the real numbers type similar to other [[type universes]] such as the [[type of all propositions]] or the [[type of finite types]].

### As a terminal object

An [[Archimedean ordered field]] can be defined in any dependent type theory with the following types:

* [[dependent product types]], [[dependent sum types]], [[identity types]], [[empty type]], [[unit type]], [[sum types]], [[bracket types]], and a [[natural numbers type]]

Then the real numbers type $\mathbf{R}$ is the homotopy terminal Archimedean ordered field.

If the [[dependent type theory]] has [[quotient sets]], then it is provable that $\mathbf{R}$ is Cauchy complete. From the definition of homotopy terminal Archimedean ordered field, $\mathbf{R}$ is an algebra of the endofunctor $X \mapsto \mathcal{C}(X)$ 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 type of Archimedean ordered field 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 homotopy terminal Archimedean field, $\mathbf{R}$ is an algebra of the endofunctor $X \mapsto \mathcal{D}(X)$ 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 type of Archimedean ordered field is a Dedekind complete Archimedean ordered field.

If a real numbers type $R$ as defined in the previous section also exists, then it is a [[subtype]] of this real numbers type $\mathbb{R}$ defined as the terminal Archimedean ordered field, since $R$ is provably Archimedean ordered, and any [[ring homomorphisms]] between two field objects is an [[embedding of types]], and by definition of terminal object, there is always a ring homomorphism $R \to \mathbb{R}$.

## Properties

### Algebraic structure on real numbers

One can add two Dedekind cuts together; this involves adding inference rules for adding the two domains of the Dedekind cuts as well as for adding the embeddings of the Dedekind cuts. The rules for adding the witnesses that the embeddings are Dedekind cuts are provable from the fact that the type $\mathrm{isCut}(A, e_A)$ is always a [[mere proposition]], and so if inhabited, then it is [[contractible]], so are not included.

The formation rules for addition of two Dedekind cuts are as follows:

$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \\
\Gamma \vdash e_A:A \hookrightarrow (\mathbb{Q} + \mathbb{Q}) \quad \Gamma \vdash e_B:B \hookrightarrow (\mathbb{Q} + \mathbb{Q}) \\
\Gamma \vdash c_A:\mathrm{isCut}(A, e_A) \quad \Gamma \vdash c_B:\mathrm{isCut}(B, e_B)
\end{array}}{\Gamma \vdash A +_{(e_A, c_A), (e_B, c_B)} B \; \mathrm{type}}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \\
\Gamma \vdash e_A:A \hookrightarrow (\mathbb{Q} + \mathbb{Q}) \quad \Gamma \vdash e_B:B \hookrightarrow (\mathbb{Q} + \mathbb{Q}) \\
\Gamma \vdash c_A:\mathrm{isCut}(A, e_A) \quad \Gamma \vdash c_B:\mathrm{isCut}(B, e_B)
\end{array}}{\Gamma \vdash e_A + e_B:(A +_{(e_A, c_A), (e_B, c_B)} B) \hookrightarrow (\mathbb{Q} + \mathbb{Q})}$$

In addition, we have rules defining the addition of two embeddings of Dedekind cuts:

$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \\
\Gamma \vdash e_A:A \hookrightarrow (\mathbb{Q} + \mathbb{Q}) \quad \Gamma \vdash e_B:B \hookrightarrow (\mathbb{Q} + \mathbb{Q}) \\
\Gamma \vdash c_A:\mathrm{isCut}(A, e_A) \quad \Gamma \vdash c_B:\mathrm{isCut}(B, e_B)
\end{array}}{\Gamma, q:\mathbb{Q} \vdash \mathrm{fib}(e_A + e_B, \mathrm{inl}(q)) \equiv \exists r, s:\mathbb{Q}.\mathrm{fib}(e_A, \mathrm{inl}(r)) \wedge \mathrm{fib}(e_B, \mathrm{inl}(s)) \wedge q = r + s \; \mathrm{type}}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \\
\Gamma \vdash e_A:A \hookrightarrow (\mathbb{Q} + \mathbb{Q}) \quad \Gamma \vdash e_B:B \hookrightarrow (\mathbb{Q} + \mathbb{Q}) \\
\Gamma \vdash c_A:\mathrm{isCut}(A, e_A) \quad \Gamma \vdash c_B:\mathrm{isCut}(B, e_B)
\end{array}}{\Gamma, q:\mathbb{Q} \vdash \mathrm{fib}(e_A + e_B, \mathrm{inr}(q)) \equiv \exists r, s:\mathbb{Q}.\mathrm{fib}(e_A, \mathrm{inr}(r)) \wedge \mathrm{fib}(e_B, \mathrm{inr}(s)) \wedge q = r + s \; \mathrm{type}}$$

as well as rules defining the addition of two domains of Dedekind cuts:

$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \\
\Gamma \vdash e_A:A \hookrightarrow (\mathbb{Q} + \mathbb{Q}) \quad \Gamma \vdash e_B:B \hookrightarrow (\mathbb{Q} + \mathbb{Q}) \\
\Gamma \vdash c_A:\mathrm{isCut}(A, e_A) \quad \Gamma \vdash c_B:\mathrm{isCut}(B, e_B)
\end{array}}{\Gamma \vdash A +_{(e_A, c_A), (e_B, c_B)} B \equiv \sum_{q:\mathbb{Q}} \mathrm{fib}(e_A + e_B, \mathrm{rec}_{\mathbb{Q} + \mathbb{Q}}(\mathrm{inl}(q), \mathrm{inr}(q))) \; \mathrm{type}}$$

Then given two real numbers $x:\mathbb{R}$ and $y:\mathbb{R}$, addition on the real numbers is defined as

$$x + y \coloneqq \mathrm{inR}_{\mathrm{CutDomain}(x) +_{\mathrm{cut}(x), \mathrm{cut}(y)} \mathrm{CutDomain}(y)}(\mathrm{cut}(x) + \mathrm{cut}(y)):\mathbb{R}$$

## Related concepts

* [[real numbers object]]

* [[Dedekind real numbers]]

## References

For the impredicative definition of the real numbers using a type of propositions, see section 11.2 of:

* [[Univalent Foundations Project]], section 11.2 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_


real oriented cohomology theory

[[!redirects real-oriented cohomology theory]]

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## Idea

The concept of _real oriented cohomology theory_ ([Araki 78](#Araki78), [Araki 79](#Araki79)) is the analog of that of _[[complex oriented cohomology theory]]_ in the context of $\mathbb{Z}_2$-[[equivariant cohomology theory]].

Where [[complex oriented cohomology theory]] singles out the [[classifying space]] $B S^1 = B U(1)$ of the [[circle group]], in real-oriented cohomology theory its role is played by this same space but equipped with the $\mathbb{Z}_2$-[[∞-action]] induced by the $\mathbb{Z}_2$-[[action]] $S^1 \to S^1$ on the circle that identifies the two semi-circles.

If one thinks of the canonical embedding $S^1 \hookrightarrow \mathbb{C}$ of the circle as the unit circle in the [[complex plane]], then this involution is induced by the [[complex conjugation]] [[involution]] on the complex plane. In this sense real oriented cohomology theory is in a sense "complex oriented cohomology theory with complex conjugation divided out", and that motivates the term _real_ cohomology theory.

The original and motivating example is the real version of [[complex K-theory]] called [[KR-theory]]. The real-oriented version of [[complex cobordism]] is [[MR-theory]] and that of [[Morava E-theory]] is [[BPR-theory]].

A modern review is in ([Kriz 01, section 2](#Kriz01)).

Beware the terminological issue expanded on also at _[[KR-theory]]_: real-oriented versions of "complex" cohomology theory, being genuinely $\mathbb{Z}_2$-equivariant, are richer than just the "orthogonal" [[homotopy fixed point]] theory. In particular on $\mathbb{Z}_2$-spaces of the form $X\cup X$ on which $\mathbb{Z}_2$ acts by swapping the two copies, the real version reduces to the original version on $X$. Hence real-oriented cohomology is a genuine generalization of complex-oriented cohomology. For instance [[KR-theory]] subsumes both [[complex K-theory]] as well as [[KO-theory]] as well as [[KSC-theory]].

## Applications

In [[physics]] ([[string theory]]) where ([[gauge field|gauge]]) [[field (physics)|fields]] are [[cocycles]] in some ([[differential cohomology theory|differential]]) [[cohomology theory]], real-oriented cohomology theory is that of [[orientifold]] [[spacetimes]].

## Examples

* [[real space]]

* [[KR-theory]]

* [[MR-theory]]

* [[BPR-theory]]

* [[ER-theory]]

## Related concepts

* [[orientifold]]

* [[complex oriented cohomology theory]], [[quaternionic oriented cohomology theory]]

* [[equivariant complex oriented cohomology theory]]

\linebreak

[[!include chromatic tower examples - table]]

## References

The definition is originally due to:

* {#Araki78} [[Shôrô Araki]], *$\tau$-Cohomology Theories*, Japanese Journal of Mathematics **4** 2 (1978) 363-416 [[doi:10.4099/math1924.4.363](https://doi.org/10.4099/math1924.4.363)]

* {#Araki79} [[Shôrô Araki]], *Forgetful spectral sequences*, Osaka Journal of Mathematics **16** 1 (1979) 173-199 [[Euclid](https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-16/issue-1/Forgetful-spectral-sequences/ojm/1200771837.full)]

  > (on the relevant [[spectral sequences]])

* [[Shôrô Araki]], *Orientations in $\tau$-cohomology theories*, Japan Journal of Mathematics (N.S.) **5** 2 (1979) 403-430 [[doi:10.4099/math1924.5.403](https://doi.org/10.4099/math1924.5.403)]

Discussion of the [[Adams-Novikov spectral sequence]] in this context is in

* {#Kriz01} [[Igor Kriz]], _Real-oriented homotopy theory and an analogue of the
Adams-Novikov spectral sequence_, Topology 40 (2001) 317-399 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf))

[[!redirects real-oriented cohomology theories]]

[[!redirects real oriented cohomology theory]]
[[!redirects real oriented cohomology theories]]

[[!redirects real-oriented cohomology]]
[[!redirects real oriented cohomology]]

[[!redirects real-oriented generalized cohomology theory]]
[[!redirects real oriented generalized cohomology theory]]


real part

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## Definition

Given a [[star-algebra]] $(A ,\ast)$ in [[characteristic zero]], then for $a\in A$ any element

* its _real part_ is $Re(a) \coloneqq \frac{1}{2}(a + a^\ast)$;

* its _imaginary part_ is $Im(a) \coloneqq \frac{1}{2}(a - a^\ast)$,

where $(-)^\ast$ is the given star-[[anti-involution]] (e.g. [[complex conjugation]], [[quaternionic conjugation]], ...).

## Examples

Standard situations where real and imaginary parts are considered include the real [[normed division algebra]]

* $A$ the [[complex numbers]];

* $A$ the [[quaternions]];

* $A$ the [[octonions]]

## Related concepts

* [[imaginary number]]

[[!redirects real parts]]

[[!redirects imaginary part]]
[[!redirects imaginary parts]]


real polynomial function

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## Definition ##

### Without scalar coefficients

A **real polynomial function** is a [[polynomial function]] in the [[real numbers]], a [[function]] $f:\mathbb{R} \to \mathbb{R}$ such that

* $f$ is in the image of the [[function]] $j:\mathbb{R}^* \to (\mathbb{R} \to \mathbb{R})$ from the [[free monoid]] $\mathbb{R}^*$ on $\mathbb{R}$, i.e. the set of lists of real numbers, to the [[function algebra]] $\mathbb{R} \to \mathbb{R}$, such that

  * $j(\epsilon) = 0$, where $0$ is the zero function.
  * for all $a \in \mathbb{R}^*$ and $b \in \mathbb{R}^*$, $j(a b) = j(a) + j(b) \cdot (-)^{\mathrm{len}(a)}$, where $(-)^n$ is the $n$-th power function for $n \in \mathbb{N}$
  * for all $r \in \mathbb{R}$, $j(r) = c_r$, where $c_r$ is the constant function whose value is always $r$.

* $f$ is in the image of the canonical [[ring homomorphism]] $i:\mathbb{R}[x] \to (\mathbb{R} \to \mathbb{R})$ from the real polynomial ring in one indeterminant $\mathbb{R}[x]$ to the [[function algebra]] $\mathbb{R} \to \mathbb{R}$, which takes constant polynomials in $\mathbb{R}[x]$ to constant functions in $\mathbb{R} \to \mathbb{R}$ and the indeterminant $x$ in $\mathbb{R}[x]$ to the identity function $\mathrm{id}_\mathbb{R}$ in $\mathbb{R} \to \mathbb{R}$

* There exists a natural number $n$ such that the $n$-th order derivative of $f$ is equal to the zero function:
$$\frac{d^n f}{d x^n} = 0$$

### With scalar coefficients

A **real polynomial function** is a [[function]] $f:\mathbb{R} \to \mathbb{R}$ with a [[natural number]] $n \in \mathbb{N}$ and a [[list]] of length $n$ of real numbers $a:[0, n)_\mathbb{N} \to \mathbb{R}$ which satisfy one of these conditions:

* for all $x \in R$,
$$f(x) = \sum_{i:[0, n)_\mathbb{N}} a(i) \cdot x^i$$
where $x^i$ is the $i$-th [[power function]] for multiplication.

* $f$ is a [[solution]] to the $n$-th order linear homogeneous [[ordinary differential equation]]
$$\frac{d^n f}{d x^n} = 0$$
with initial conditions
$$\frac{d^i f}{d x^i}(0) = i! \cdot a(i)$$
for each natural number $i:[0, n)_\mathbb{N}$

## Properties ##

### Degree ###

#### Without scalar coefficients

Given a non-zero real polynomial function $f:\mathbb{R} \to \mathbb{R}$, the **degree** of $f$ is the maximum [[natural number]] $n$ where the $n$-th [[derivative]] of $f$ is not the zero-function:
$$\mathrm{deg}(f) \coloneqq \max_{n \in \mathbb{N}, \frac{d^{n} f}{d x^{n}} \neq 0}(n)$$

#### With scalar coefficients

The **degree** of a real polynomial function, with additional data of a [[natural number]] $n \in \mathbb{N}$ and a [[list]] of length $n$ of real numbers $a:[0, n)_\mathbb{N} \to \mathbb{R}$, is defined as the maximum natural number $i \lt n$ such that $\vert a_i \vert \gt 0$.

#### In constructive mathematics

In [[constructive mathematics]], there exist real polynomial functions for which one cannot prove that a particular natural number $i \lt n$ is the degree of the real polynomial function: i.e. the [[function algebra|function sub-$\mathbb{R}$-algebra]] of real polynomial functions is not a [[Euclidean domain]].

### Pointwise continuity ###

Given a [[real polynomial function]] $f:\mathbb{R} \to \mathbb{R}$, $f$ is a [[pointwise continuous function]] with respect to its [[metric topology]] defined through the absolute value function, subtraction, and its strict total order.

### Pointwise differentiability ###

Given a [[real polynomial function]] $f:\mathbb{R} \to \mathbb{R}$, $f$ is a [[pointwise differentiable function]] with respect to its [[metric topology]] defined through the absolute value function, subtraction, and its strict total order.

### Smoothness ###

Every real polynomial function is a [[smooth function]]. This could be shown [[coinductively]]: A smooth function is a [[pointwise differentiable function]] whose derivative is also smooth, and thus the $n$-th derivative of a smooth function is a smooth function. The zero function $f(x) = 0$ is a smooth function, every real polynomial function is a pointwise differentiable function and the $n$-th derivative of a polynomial function, with a [[natural number]] $n \in \mathbb{R}$ and a [[list]] of length $n$ of real numbers $a:[0, n)_\mathbb{N} \to \mathbb{R}$ such that one of the two conditions above is satisfied, is the zero function. Thus, every polynomial function is a smooth function.

## See also ##

* [[real number]]

* [[polynomial function]]

* [[real quadratic function]]

* [[real cubic function]]

* [[rational functions are continuous]]

## References ##

The proof of pointwise continuity using [[epsilontic analysis]] is spelled out for instance in

* Kyle Miller, _Polynomials are continuous functions_, 2014 ([pdf](https://math.berkeley.edu/~kmill/math1afa2014/poly.pdf))

See also:

* Wikipedia, _[Polynomial function](https://en.wikipedia.org/wiki/Polynomial_function)_

[[!redirects real polynomial functions]]

[[!redirects real polynomial functions are pointwise continuous]]
[[!redirects polynomials are continuous functions]]
[[!redirects polynomials are continuous]]


real polynomial functions are continuous > history

< [[real polynomial function]]


real polynomial functions are pointwise continuous > history

< [[real polynomial function]]


real projective plane

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## Idea

The [[projective plane]] $\mathbb{R}P^2$ over the [[real numbers]].

## Related concepts

* [[real projective space]], [[real projective line]]

* [[complex projective plane]]

* [[quaternionic projective plane]]

* [[octonionic projective plane]]

[[!redirects real projective planes]]


real projective space

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## Idea

The _real projective space_ $\mathbb{R}P^n$ is the [[projective space]] of the real [[vector space]] $\mathbb{R}^{n+1}$.

Equivalently this is the [[Grassmannian]] $Gr_1(\mathbb{R}^{n+1})$.

## Properties

### General

\begin{proposition}
Every continuous map $\mathbb{R}P^n\rightarrow\mathbb{R}P^n$ for $n$ even has a [[fixed point]]. This does not hold for $n$ odd as in this case the continuous map $\mathbb{R}P^n\rightarrow\mathbb{R}P^n, [x_0:x_1:\ldots:x_{n-1}:x_n]\mapsto[x_1:-x_0:\ldots:x_n:-x_{n-1}]$ does not have a fixed point.
\end{proposition}

([Hatcher 02, page 155 exercise 2](#Hatcher02))

([Hatcher 02, page 180](#Hatcher02))

\begin{proposition}
For $n\neq 1,3,7$, the smallest number $k$, so that there exists an [[immersion]] of real projective space $\mathbb{R}P^n$ into [[cartesian space]] $\mathbb{R}^{k-1}$, is exactly the [[topological complexity]] $\operatorname{TC}(\mathbb{R}P^n)$ (with convention $\operatorname{TC}(*)=1$).
\end{proposition}

([Farber & Tabachnikov & Yuzvinsky 02, Theorem 12](#FarberTabachnikovYuzvinsky02))

\begin{proposition}
For $n=1,3,7$ one has
$$
\operatorname{TC}(\mathbb{R}P^n)
=n+1
$$
for the [[topological complexity]] (with convention $\operatorname{TC}(*)=1$).
\end{proposition}

([Farber & Tabachnikov & Yuzvinsky 02, Proposition 18](#FarberTabachnikovYuzvinsky02))

### Cell structure
 {#CellStructure}

+-- {: .num_prop #RealProjectiveSpaceCWComplex}
###### Proposition
**([[CW-complex]] structure)**

For $n \in \mathbb{N}$, the real projective space $\mathbb{R}P^n$ admits the structure of a [[CW-complex]].

=--

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

Use that $\mathbb{R}P^n \simeq S^n/(\mathbb{Z}/2)$ is the [[quotient space]] of the [[Euclidean space|Euclidean]] [[n-sphere]] by the $\mathbb{Z}/2$-action which identifies [[antipode|antipodal points]].

The standard CW-complex structure of $S^n$ realizes it via two $k$-cells for all $k \in \{0, \cdots, n\}$, such that this $\mathbb{Z}/2$-action restricts to a homeomorphism between the two $k$-cells for each $k$. Thus $\mathbb{R}P^n$ has a CW-complex structure with a single $k$-cell for all $k \in \{0,\cdots, n\}$.

=--

### Homotopy groups

+-- {: .num_prop #HomotopyGroupsOfRealProjectiveSpace}
###### Proposition
**([[homotopy groups]] of real projective space)**

The [[homotopy groups]] of real projective plane can be calculated with the long exact sequence of homotopy groups ([Hatcher 02, Theorem 4.41.](#Hatcher02)) of the [[fiber bundle]] $S^0\rightarrow S^n\rightarrow\mathbb{R}P^n$ and are given by
\[
  \label{ListOfHomotopyGroupsOfComplexProjectiveSpace}
    \pi_k
    \big(
      \mathbb{R}P^n
    \big)
    \;=\;
    \left\{
    \begin{array}{ll}
      \ast & k = 0
      \\
      \mathbb{Z} & k = 1, n = 1
      \\
      \mathbb{Z}_2 & k = 1, n > 1
      \\
      \pi_k
      \big(
        S^n
      \big)
      &
      k \geq 1, n > 0
    \end{array}
    \right.
\]

=--

### Homology and cohomology

+-- {: .num_prop #HomologyAndCohomologyOfRealProjectiveSpace}
###### Proposition
**([[homology]] and [[cohomology]] of real projective space)**

The [[ordinary homology]] [[homology groups|groups]] of real projective space $\mathbb{R}P^n$ can be calculated using its CW structure and are given by
\[
  \label{HomologyOfComplexProjectiveSpace}
    H_k
    \big(
      \mathbb{R}P^n
    \big)
    \;=\;
    \left\{
    \begin{array}{ll}
       \mathbb{Z} & k = 0 \quad or \quad k = n \quad if \quad odd
       \\
       \mathbb{Z}_2 & k \quad odd \quad and \quad 1 \leq k \leq n
       \\
       1 & otherwise
    \end{array}
    \right.
\]

([Hatcher 02, Example 2.42](#Hatcher02))

Similarly the [[ordinary cohomology]] [[cohomology groups|groups]] of $\mathbb{R}P^n$ are
\[
  \label{CohomologyOfRealProjectiveSpace}
    H^k
    \big(
      \mathbb{R}P^n
    \big)
    \;=\;
    \left\{
    \begin{array}{ll}
       \mathbb{Z} & k = 0 \quad or \quad k = n \quad if \quad odd
       \\
       \mathbb{Z}_2 & k \quad odd \quad and \quad 1 \leq k \leq n-1
       \\
       1 & otherwise
    \end{array}
    \right.
\]

=--

One has $H_{n-1}(\mathbb{R}P^n)\cong\mathbb{Z}_2$ für $n$ odd and $H_{n-1}(\mathbb{R}P^n)\cong 1$ for $n$ even, hence $\mathbb{R}P^n$ is [[orientable]] iff $n$ is odd.

([Hatcher 02, Corollary 3.28.](#Hatcher02))

### Relation to the $\mathbb{Z}/2$-classifying space
 {#RelationToClassifyingSpace}

The infinite real projective space $\mathbb{R}P^\infty \coloneqq \underset{\longrightarrow}{\lim}_n \mathbb{R}P^n$ is the [[classifying space]] for [[real line bundles]]. It has the [[homotopy type]] of the [[Eilenberg-MacLane space]] $K(\mathbb{Z}/2,1) = B \mathbb{Z}/2$.

### Kahn-Priddy theorem

* [[Kahn-Priddy theorem]]

## Related concepts

* [[complex projective space]]

* [[quaternionic projective space]]

* [[octonionic projective space]]

* [[classifying space]]

## References

See also:

* [[Topospaces]], _[Real projective space](https://topospaces.subwiki.org/wiki/real_projective_space)_
* Wikipedia, _[Real projective space](https://en.wikipedia.org/wiki/Real_projective_space)_

[[Topological complexity]] of real projective space and connection with their [[immersion]]:

* {#FarberTabachnikovYuzvinsky02} [[Michael Farber]], [[Serge Tabachnikov]], [[Sergey Yuzvinsky]], _Topological robotics: motion planning in projective spaces_ (2002), [arXiv:math/0210018](https://arxiv.org/abs/math/0210018)

Homotopy groups, homology and cohomology of real projective space:

* {#Hatcher02} [[Allen Hatcher]], _Algebraic Topology_, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)]

Computation of [[cohomotopy]]-sets of real projective space:

* {#West71} Robert West, _Some Cohomotopy of Projective Space_, Indiana University Mathematics Journal Indiana University Mathematics Journal Vol. 20, No. 9 (March, 1971), pp. 807-827 ([jstor:24890146](https://www.jstor.org/stable/24890146))

[[!redirects real projective spaces]]


real quadratic function

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# Contents
* table of contents
{: toc}

## Definition ##

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

## Properties ##
 {#Properties}

### Extrema ###

Given a real quadratic function $f(x) \coloneqq a x^2 + b x + c$, if $\vert a \vert \gt 0$, after [[completing the square]], the quadratic function may be written as

$$
  f(x) = a \left(x + \frac{b}{2a}\right)^2 - \frac{b^2 - 4 a c}{4a}
$$

The [[extremum]] occurs at

$$x = -\frac{b}{2a}$$

and the extremum is a [[minimum]] if $a \gt 0$ and a [[maximum]] if $a \lt 0$. The value of $f$ at the extremum is

$$f(x) = \frac{4 a c - b^2}{4a}$$

When $a \gt 0$, the real quadratic function is said to be concave up, while when $a \lt 0$, the real quadratic function is said to be concave down.

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

### 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 quadratic functions $f:\mathbb{R} \to \mathbb{R}$ such that one cannot decide whether $f$ is concave up, concave down, or degenerate. Similarly, there exists real quadratic functions $f:\mathbb{R} \to \mathbb{R}$ such that one cannot decide whether the extremum is a minimum, is a maximum, or does not exist due to the degeneracy of $f$.

## Exact zeroes ##

### Partial inverse functions ###

Given a real quadratic function $f:\mathbb{R} \to \mathbb{R}$, $f(x) \coloneqq a x^2 + b x + c$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$ such that $\vert a \vert \gt 0$, since $f$ is [[continuously differentiable]] if $\vert a \vert \gt 0$, due to the [[inverse function theorem]], there are two branches of the partial inverse function of $f$, $g$ and $h$, such that if $a \gt 0$, then $g$ and $h$ have domain

$$\left(c - \frac{b^2}{4a}, \infty\right)$$

and if $a \lt 0$, $g$ and $h$ have domain

$$\left(-\infty,c - \frac{b^2}{4a}\right)$$

In both cases, $g$ and $h$ are defined as solutions to the first-order nonlinear [[ordinary differential equation]]

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

with the initial conditions for the principal branch $g$ are

$$g\left((a + b + c) + \left(c - \frac{b^2}{4a}\right)\right) = 1 - \frac{b}{2a}$$

and the initial conditions for the reflected branch $h$ are

$$h\left((a - b + c) + \left(c - \frac{b^2}{4a}\right)\right) = -1 - \frac{b}{2a}$$

As a result, $g$ and $h$ are automatically pointwise [[continuously differentiable]], and for every $x$ in the domain of $g$ and $h$,

$$g(x) \gt -\frac{b}{2a}$$

$$h(x) \lt -\frac{b}{2a}$$

Since this differential equation is separable, one could use separation of variables to see that $g$ and $h$ satisfy the [[functional equations]] $a g(x)^2 + b g(x) + c = x$ and $a h(x)^2 + b h(x) + c = x$.

If the discriminant $\Delta = b^2 - 4 a c$ is greater than zero $\Delta \gt 0$, then the quadratic function $f$ has two zeroes apart from each other. The larger of the two zero of $f(x)$ is at $g(0)$ and the smaller zero of $f(x)$ is at $h(0)$. If the discriminant is less than or equal to zero $\Delta \leq 0$, then zero isn't in the domain of either $g$ and $h$.

### Square roots ###

If $a = 1$, $b = 0$, and $c = 0$, then the real quadratic function above is the square function $f(x) \coloneqq x^2$. The principal branch of the continuous partial inverse function of the square function is called the [[continuous principal square root]] $\mathrm{sqrt}_\mathrm{cont}:(0, \infty) \to \mathbb{R}$, and is the solution to the differential equation

$$(2 \mathrm{sqrt}_\mathrm{cont}) \frac{d \mathrm{sqrt}_\mathrm{cont}}{d x} = 1$$

with initial condition $\mathrm{sqrt}_\mathrm{cont}(1) = 1$. The reflected branch of the continuous partial inverse function of the square function is simply the negation of the principal square root $-\mathrm{sqrt}_\mathrm{cont}:(0, \infty) \to \mathbb{R}$.

### Quadratic formula ###

For any quadratic function $f$, the principal branch of its inverse has a canonical definition in terms of the continuous principal square root:

$$g(x) = \frac{-b + \mathrm{sqrt}_\mathrm{cont}(4ax + b^2 - 4ac)}{2a}$$

and the reflected branch has a canonical definition in terms of the continuous principal square root as well:

$$h(x) = \frac{-b - \mathrm{sqrt}_\mathrm{cont}(4ax + b^2 - 4ac)}{2a}$$

The **continuous quadratic formula** is the evaluation of the two branches at zero, if zero is in the domain of $g$ and $h$:

$$g(0) = \frac{-b + \mathrm{sqrt}_\mathrm{cont}(b^2 - 4ac)}{2a}$$

$$h(0) = \frac{-b - \mathrm{sqrt}_\mathrm{cont}(b^2 - 4ac)}{2a}$$

The continuous quadratic formula is only valid for real quadratic functions with positive discriminant, unlike the [[quadratic formula]] in the [[discrete field]] of [[quadratic irrational numbers]], which is valid for [[quadratic functions]] with [[rational number|rational]] [[coefficients]] and a non-negative discriminant. But there's an analytic reason for why the quadratic formula should be restricted to real quadratic functions with positive discriminant: for any real quadratic function $f$ with a non-positive discriminant $\Delta \leq 0$, there is no interval on the real numbers such that the conditions required for the [[intermediate value theorem]] are fulfilled: if $a \gt 0$, then $f(x) \geq 0$ for all real numbers $x$, and if $a \lt 0$, then $f(x) \leq 0$ for all real numbers $x$.

### Negative and zero discriminant ###

As stated above, when a real quadratic function $f$ has a non-positive discriminant, $\Delta \leq 0$, if $a \gt 0$, then $f(x) \geq 0$ for all real numbers $x$, and if $a \lt 0$, then $f(x) \leq 0$ for all real numbers $x$. When $f$ has a negative discriminant, $\Delta \lt 0$, then $f$ is provably always [[tight apartness relation|apart]] from zero: if $a \gt 0$, then $f(x) \gt 0$ for all real numbers $x$, and if $a \lt 0$, then $f(x) \lt 0$ for all real numbers $x$. As a result, $f$ has no zeroes in the real numbers if the determinant of $f$ is negative.

Now, suppose that $f$ has zero discriminant. This means that $b^2 - 4ac = 0$, and after [[completing the square]], the resulting function is equal to

$$f(x) = a \left(x + \frac{b}{2a}\right)^2$$

By definition of an [[ordered field]], there are no multiplicative [[nilpotent elements]] in the real numbers whose absolute value is greater than zero, and thus every nilpotent element is equal to zero. As a result,

$$f(x) = a \left(x + \frac{b}{2a}\right)^2 = 0$$

$$\left(x + \frac{b}{2a}\right)^2 = 0$$

$$x + \frac{b}{2a} = 0$$

$$x = -\frac{b}{2a}$$

There is only one zero, which occurs at the [[extremum]] of the real quadratic function.

### 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 quadratic functions $f:\mathbb{R} \to \mathbb{R}$ such that one cannot decide whether the discriminant of $f$ is positive, negative, or zero. As a result, there exist real quadratic functions where one cannot decide the number of zeroes the function has. Furthermore, one cannot prove that a real quadratic function with positive determinant has exactly two zeroes.

## Newton's method ##

Given a real quadratic function $f:\mathbb{R} \to \mathbb{R}$, $f(x) \coloneqq a x^2 + b x + c$ for real numbers $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, where $\vert a \vert \gt 0$, suppose we want to approximate a zero of $f$. There is an algorithm called **[[Newton's method]]** which would allow us to do so. The real quadratic function has [[discriminant]] $\Delta = b^2 - 4 a c$ and has 2 zeroes if $\Delta \gt 0$, 1 zero at the extremum if $\Delta = 0$, or no zeroes of $\Delta \lt 0$. (In [[constructive mathematics]], there are real quadratic functions where it isn't possible to determine the number of zeroes the real quadratic function has.)

Assume that $\Delta \gt 0$. Select a real number $x_0 \in \mathbb{R}$ as the initial guess, where $2 a x_0 + b \gt 0$ or $2 a x_0 + b \lt 0$, and the next guess would be defined as

$$x_{i+1} = x_i - \frac{a x_i^2 + b x_i + c}{2 a x_i + b}$$

Newton's method isn't valid when $2 a x_0 + b = 0$ because of division by zero. When $2 a x_0 + b \gt 0$, Newton's method converges towards the zero at $x \gt -\frac{b}{2a}$, and when $2 a x_0 + b \lt 0$, Newton's method converges towards the zero $x \lt -\frac{b}{2a}$.

## See also ##

* [[real number]]

* [[quadratic function]]

* [[quadratic formula]]

* [[real square root function]]

* [[real cubic function]]

* [[real polynomial function]]

## References ##

See also:

* Wikipedia, _[Quadratic function](https://en.wikipedia.org/wiki/Quadratic_function)_

[[!redirects real quadratic functions]]

[[!redirects real quadratic formula]]
[[!redirects real quadratic formulae]]
[[!redirects real quadratic formulas]]
[[!redirects continuous quadratic formula]]
[[!redirects continuous quadratic formulae]]
[[!redirects continuous quadratic formulas]]


real rational functions are pointwise continuous

[[!redirects rational functions are continuous]]

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## Statetement

Every [[rational function]] on the [[real numbers]] is a [[pointwise continuous function]] on its [[domain]] of definition.

This follows directly from the fact that [[real polynomial functions are pointwise continuous]]

## References

A proof using [[epsilontic analysis]] is spelled out for instance around corollary 3.16 here:

* [pdf](https://www.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch3.pdf)


real reciprocal function

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## Definition

### Implicit definition

In [[real analysis]], the [[reciprocal]] $\frac{1}{x}$ is a [[partial function]] implicitly defined over the [[apartness|non]]-[[zero]] [[real numbers]] by the [[equation]] $x \left(\frac{1}{x}\right) = 1$. This is the definition commonly used when defining the real numbers as a [[field]].

### By the exponential and natural logarithm functions

The __reciprocal__ $\frac{1}{(-)}:(-\infty,0)\union(0,\infty)\to\mathbb{R}$ is piecewise defined as

$$
\frac{1}{x} \coloneqq
\begin{cases}
-e^{-\ln(-x)} & x \in (-\infty,0) \\
e^{-\ln(x)} & x \in (0,\infty)
\end{cases}
$$

This definition implies that the reciprocal is [[analytic function|analytic]] in each of the two [[connected components]] of the [[domain]].

### By power series

Let us define the functions $f \colon (-1,1)\to\mathbb{R}$ and $g:(-1,1)\to\mathbb{R}$ from the [[open interval|open subinterval]] of the real numbers $(-1,1) \subset \mathbb{R}$ to the real numbers $\mathbb{R}$ as the locally [[convergence|convergent]] [[power series]]

$$
  f(x)\coloneqq \sum_{n=0}^{\infty} x^n
$$

$$
  g(x)\coloneqq \sum_{n=0}^{\infty} (-1)^n x^n
$$

The __reciprocal__ $\frac{1}{(-)}:(-\infty,0)\union(0,\infty)\to\mathbb{R}$ is then piecewise defined as

$$
\frac{1}{x} \coloneqq
\begin{cases}
\lim_{a\to 0^-} (-a) \sum_{n=0}^{\infty} a^n (x-f(a+1))^n & x \in (-\infty,0) \\
\lim_{a\to 0^+} a \sum_{n=0}^{\infty} (-a)^n (x+g(a-1))^n & x \in (0,\infty)
\end{cases}
$$

Equivalently, given an element $p \in (0, 1)$, the __reciprocal__ $\frac{1}{(-)}:(-\infty,0)\union(0,\infty)\to\mathbb{R}$ is then piecewise defined as

$$
\frac{1}{x} \coloneqq
\begin{cases}
\lim_{i \to \infty} p^i \sum_{n=0}^{\infty} (-p^i)^n (x-f(-p^i+1))^n & x \in (-\infty,0) \\
\lim_{i \to \infty} p^i \sum_{n=0}^{\infty} (-p^i)^n (x+g(p^i-1))^n & x \in (0,\infty)
\end{cases}
$$

This definition implies that the reciprocal is [[analytic function|analytic]] in each of the two [[connected components]] of the [[domain]].

## Related concepts

* [[logarithm]]

* [[analytic function]]

* [[limit of a function]]

* [[power series]]

* [[reciprocal function]]


real space

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## Idea

In the context of [[real-oriented cohomology theory]], notably in _[[KR-theory]]_, by "real space" ("real manifold") one means a [[topological space]] (a [[manifold]]) equipped with an [[group action|action]] of the [[cyclic group|cyclic]] [[group of order 2]] $\mathbb{Z}/2$ (a "non-linear [[real structure]]").

In the context of [[string theory]] real spaces appear as [[orientifold]] [[target space|target]] [[spacetimes]]. The involution [[fixed points]] here are known as _[[O-planes]]_.

## Examples

### Real circles

There are three non-equivalent real structures on the [[circle]] $S^1$, usually denoted

* $S^{2,0}$ for the trivial involution;

* $S^{1,1}$ for the reflection involution (identifying the two semi-circles);

* $S^{0,2}$ for the antipodal involution (rotation by $\pi$).

Accordingly [[real-oriented cohomology theory]] is bigraded in a way modeled on this bigrading.

(This is standard notation, but maybe $S^{1,0}$, $S^{\tfrac{1}{2}, \tfrac{1}{2}}, S^{0,1}$ would be more suggestive. Indeed the quotients in the first and the last case are actually circles, while in the second case it is the semi-circle.)

### Complexified cartesian spaces

The complex $n$-dimensional complexified cartesian space ${\mathbb{C}}^n$ equipped with its conjugative involution is a real space. Explicitly, this involution sends $(z^1,\ldots, z^n)$ to $(\bar{z^1},\ldots, \bar{z^n})$.

### Complex projective spaces.

The complex $n$-dimensional complex projective space ${\mathbb{P}}^n_{\mathbb{C}}$ equipped with a conjugation involution is a real space. For each choice of affine chart, the conjugation involution of this chart (which is biholomorphic to ${\mathbb{C}}^n$) extends to a conjugation involution on ${\mathbb{P}}^n_{\mathbb{C}}$. Any two conjugation involutions are ${\mathbb{Z}}/2$-equivariantly diffeomorphic.

For example, the Riemann surface ${\mathbb{P}}^1_{\mathbb{C}}$ is diffeomorphic to the 2-sphere $S^2$ and its conjugation involution is the antipodal action.

### Complex general and special linear groups.

For $A$ a square matrix, the determinant of its conjugation transpose equals the conjugate of its determinant. In symbols, ${\mathrm{det}} A^* = \overline{{\mathrm{det}} A}$. Hence sending a square matrix to its conjugate transpose is an involution on the complex general $GL(n,{\mathbb{C}})$ and special $SL(n,{\mathbb{C}})$.

In particular, the real space $SL(1,{\mathbb{C}})$ equipped with this conjugate transpose involution is equivariantly diffeomorphic to $S^{0,2}$, the circle equipped with its antipodal action (dicussed above).

## Properties

### Relation to real cobordism

There is a "real" analog of [[complex cobordism cohomology theory]] $MU$, the [[MR cohomology theory]] $M \mathbb{R}$.

While $\pi_\bullet(M \mathbb{R})$ is _not_ the [[cobordism ring]] of [[real manifolds]], still every [[real manifold]] does give a class in $M \mathbb{R}$ ([Kriz 01, p. 13](#Kriz01)). For details see here: [pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm161/fm16115.pdf).

## Related concepts

* [[equivariant cohomology]]

* [[real structure]]

* [[real oriented cohomology theory]]

  * [[KR-theory]]

## References

* {#Atiyah66} [[Michael Atiyah]], *K-theory and reality*, The Quarterly Journal of Mathematics. Oxford. Second Series **17** 1 (1966)  367-386 [[doi:10.1093/qmath/17.1.367](https://doi.org/10.1093/qmath/17.1.367), [[AtiyahKReal.pdf:file]], ISSN:0033-5606]

* {#Kriz01} [[Igor Kriz]], _Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence_, Topology 40 (2001) 317-399 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf))

[[!redirects real spaces]]

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

[[!redirects Real space]]
[[!redirects Real spaces]]


real square root function

[[!redirects real square root]]

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## Idea ##

The different types of [[square root]] [[partial functions]] on the [[real numbers]] that satisfy the [[functional equation]] $f(x)^2 = x$ on some [[subset]] of the real numbers.

## Definition ##

This following definition of a principal square root function comes from a joint proof of the existence of a principal square root function in the non-negative real numbers in constructive mathematics by Madeleine Birchfield [here](https://mathoverflow.net/questions/426103/proof-in-constructive-mathematics-that-the-principal-square-root-function-exists) and François G. Dorais  [here](https://mathoverflow.net/a/426110). Most of the following text has been copied from the two sources.

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

There exists a square root function $\mathrm{sqrt}:(0, \infty) \to \mathbb{R}$ defined by
$$\mathrm{sqrt}(x) \coloneqq e^{\frac{1}{2} \ln(x)}$$

Let us define continuous functions $f_n:[0, \infty) \to [0, \infty)$:
$$f_n(x) = \begin{cases}
1/2^n & \mathrm{when}\; x \leq 1/4^n \\
e^{\frac{1}{2} \ln(x)} & \mathrm{when}\; x \geq 1/4^n \\
\end{cases}$$
As stated, that requires knowing whether $x \leq 1/4^n$ or $x \geq 1/4^n$, but it is possible to work around this by patching three functions together:

 * $f^{0}_k:[0,1/4^k) \to \mathbb{R}$ is the linear function $f(x) = 2^k x$,
 * $f^{+}_k:(0,\infty) \to \mathbb{R}$ is defined as $\min(2^k x,\exp(\frac{1}{2} \ln(x)))$.

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

Now the sequence of functions $(f_n)_{n=0}^\infty$ so defined converges uniformly on any bounded interval to a continuous function $\sqrt{(-)}:[0, \infty) \to [0, \infty)$ called the **principal square root** function. It is easily seen that $(\sqrt{x})^2 = x$ and $\sqrt{x^2} = x$.

The principal square root function is used to define the [[Euclidean metric]] in [[Euclidean spaces]].

## Other square root functions

According to ([Richman 2012](#Richman12)), given the existence of a principal square root function, there are an uncountable number of functions that satisfy the [[functional equation]] $f(x)^2 = x$ on some [[subset]] of the [[real numbers]]. Each of these could be called a real "square 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 $f(x) \coloneqq (-1)^{1_{\mathbb{Q}}(x)} \sqrt{x}$ is a real square root function, even though it is nowhere continuous, and not defined on the entire half-open interval $[0, \infty)$.

## See also

* [[square root]], for square roots in more general mathematical structures

* [[real quadratic function]]

* [[real absolute value]]

* [[sign function]]

* [[Euclidean field]]

* [[real nth 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 square root function]]
[[!redirects square root functions]]
[[!redirects principal square root function]]
[[!redirects principal square root functions]]

[[!redirects metric square root]]
[[!redirects metric square roots]]
[[!redirects metric principal square root]]

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

[[!redirects continuous square root]]
[[!redirects continuous square roots]]
[[!redirects continuous principal square root]]

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


real structure

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

#Contents#
* table of contents
{:toc}

## Definition

A _real structure_ on a [[complex vector space]] $V$ is an [[antilinear map]] $\sigma \colon V \to V$ which is an [[involution]].

Equivalently this is a [[real vector space]] $W$ and an [[isomorphism]] $V \simeq W \otimes_{\mathbb{R}} \mathbb{C}$ of $V$ with its [[complexification]].

Here $W = Eig(\sigma,1) \hookrightarrow V$ is the [[eigenspace]] of $\sigma$ for [[eigenvalue]] 1 and $W \otimes \{i\} = Eig(\sigma,-1) \hookrightarrow V$ is the [[eigenspace]] for [[eigenvalue]] -1.

## Examples
 {#Examples}

\begin{example}\label{RealStructuresOnComplexLine}
**(real structures on the complex line)**
\linebreak
The standard real structure on the [[complex line]] $\mathbb{C}$ is [[complex conjugation]] $\overline{x + \mathrm{i} y} \,\equiv\, x - \mathrm{i}y$:
$$
  \array{
    \mathbb{C}
    &\longrightarrow&
    \mathbb{C}
    \\
    z
      &\mapsto&
    \overline{z}
    \mathrlap{\,.}
  }
$$
The [[fixed locus]] of this real structure are the [[real numbers]] under their defining embedding into the complex numbers:
$$
  \mathbb{C}^{\overline{(\text{-})}}
  \;\simeq\;
  \mathbb{R}
    \hookrightarrow
  \mathbb{R} \oplus \mathrm{i} \mathbb{R}
    \,\equiv\,
  \mathbb{C}
$$
with
$$
  \array{
    \mathbb{R} \otimes_{{}_{\mathbb{R}}} \mathbb{C}
    &\xrightarrow{ \;\sim\; }&
    \mathbb{C}
    \\
    (x', x + \mathrm{i} y)
    &\mapsto&
    x' x + \mathrm{i} x' y
    \mathrlap{\,.}
  }
$$

Restricting the canonical complex-[[bilinear form]]
\[
  \label{CanonicalComplexBilinearFormOnComplexLine}
  \array{
    \mathbb{C} \otimes_{\mathbb{C}} \mathbb{C}
    &\longrightarrow&
    \mathbb{C}
    \\
    (z_1, z_2)
    &\mapsto&
    z_1 z_2
  }
\]
along this inclusion yields the canonical $\mathbb{R}$-bilinear form on $\mathbb{R}$:
$$
  \array{
    \mathbb{R} \otimes_{\mathbb{R}} \mathbb{R}
    &\hookrightarrow&
    \mathbb{C} \otimes_{\mathbb{C}} \mathbb{C}
    &\longrightarrow&
    \mathbb{C}
    \\
    (x_1, x_2)
    &\mapsto&
    (x_1, x_2)
    &\mapsto&
    x_1 x_2
    \mathrlap{\,.}
  }
$$

Another real structure on $\mathbb{C}$ is:
$$
  \array{
    \mathbb{C}
    &\longrightarrow&
    \mathbb{C}
    \\
    z
      &\mapsto&
    -\overline{z}
    \mathrlap{\,.}
  }
$$
Now the fixed locus is the [[imaginary numbers]] $\mathrm{i} \mathbb{R}$
$$
  \mathbb{C}^{-\overline{(\text{-})}}
  \;\simeq\;
  \mathrm{i}\mathbb{R}
    \hookrightarrow
  \mathbb{R} \oplus \mathrm{i} \mathbb{R}
    \,\equiv\,
  \mathbb{C}
$$
with
$$
  \array{
    \mathrm{i}\mathbb{R} \otimes_{{}_{\mathbb{R}}} \mathbb{C}
    &\xrightarrow{ \;\sim\; }&
    \mathbb{C}
    \\
    (\mathrm{i} x', x + \mathrm{i} y)
    &\mapsto&
    - x' y + \mathrm{i} x' x
    \mathrlap{\,.}
  }
$$
Restricting the canonical complex bilinear form (eq:CanonicalComplexBilinearFormOnComplexLine) along this inclusion now yields minus the canonical real bilinear form
$$
  \array{
    \mathbb{R} \otimes_{\mathbb{R}} \mathbb{R}
    &\hookrightarrow&
    \mathbb{C} \otimes_{\mathbb{C}} \mathbb{C}
    &\longrightarrow&
    \mathbb{C}
    \\
    (x_1 ,\, x_2)
    &\mapsto&
    (\mathrm{i} x_1 ,\, \mathrm{i}x_2)
    &\mapsto&
    - x_1 x_2
    \mathrlap{\,.}
  }
$$
\end{example}

\begin{example}
**(real structures on the complex plane)**
\linebreak
On the [[complex plane]] $\mathbb{C}^2 \,\equiv\, \mathbb{C} \oplus \mathbb{C}$ we have first of all the real structures inherited from $\mathbb{C}$ (Ex. \ref{RealStructuresOnComplexLine}).
$$
  \array{
    \mathbb{C}^2
    &\longrightarrow&
    \mathbb{C}^2
    \\
    (z_1, z_2) &\mapsto& (\overline{z_1}, \overline{z_2})
  }
$$
and
$$
  \array{
    \mathbb{C}^2
    &\longrightarrow&
    \mathbb{C}^2
    \\
    (z_1, z_2) &\mapsto& -(\overline{z_1}, \overline{z_2})
    \,.
  }
$$
In addition, there is for instance
$$
  \array{
    \mathbb{C}^2
    &\longrightarrow&
    \mathbb{C}^2
    \\
    \left[
    \array{
      z_1
      \\
      z_2
    }
    \right]
    &\mapsto&
    \left[
    \array{
      +\overline{z_1}
      \\
      -\overline{z_2}
    }
    \right]
    \,.
  }
$$
whose fixed locus is
$$
  \array{
    \mathbb{R}^2
    &\hookrightarrow&
    \mathbb{C}^2
    \\
    \left[
    \array{
      x_1
      \\
      x_2
    }
    \right]
    &\mapsto&
    \left[
    \array{
      x_1
      \\
      \mathrm{i} x_2
    }
    \right]
    \mathrlap{\,,}
  }
$$
the restriction to which of the canonical complex bilinear form on the complex plane
\[
  \label{CanonicalComplexBilinearFormOnComplexPlane}
  \array{
     \mathbb{C}^2
     \otimes_{\mathbb{C}}
     \mathbb{C}^2
     &\longrightarrow&
     \mathbb{C}
     \\
     \left(
       \left[
       \array{
         z_1
         \\
         z_2
       }
       \right]
       ,\,
       \left[
       \array{
         z'_1
         \\
         z'_2
       }
       \right]
     \right)
     &\mapsto&
     z_1 z'_1
       +
     z_2 z'_2
  }
\]
is the [[hyperbolic form]] on the real [[plane]]:
$$
  \array{
     \mathbb{R}^2 \otimes_{\mathbb{R}} \mathbb{R}^2
     &\hookrightarrow&
     \mathbb{C}^2
     \otimes_{\mathbb{C}}
     \mathbb{C}^2
     &\longrightarrow&
     \mathbb{C}
     \\
     \left(
       \left[
       \array{
         x_1
         \\
         x_2
       }
       \right]
       ,\,
       \left[
       \array{
         x'_1
         \\
         x'_2
       }
       \right]
     \right)
     &\mapsto&
     \left(
       \left[
       \array{
         x_1
         \\
         \mathrm{i}x_2
       }
       \right]
       ,\,
       \left[
       \array{
         x'_1
         \\
         \mathrm{i} x'_2
       }
       \right]
     \right)
     &\mapsto&
     x_1 x'_1 - x_2 x'_2
     \mathrlap{\,.}
  }
$$

Yet another real structure is
$$
  \array{
    \mathbb{C}^2
    &\longrightarrow&
    \mathbb{C}^2
    \\
    \left[
    \array{
      z_1
      \\
      z_2
    }
    \right]
    &\mapsto&
    \left[
    \array{
      \overline{z_2}
      \\
      \overline{z_1}
    }
    \right]
    \,.
  }
$$
whose [[fixed locus]] is
$$
  \array{
    \mathbb{R}^2
    &\hookrightarrow&
    \mathbb{C}^2
    \\
    \left[
    \array{
      x_1
      \\
      x_2
    }
    \right]
    &\mapsto&
    \left[
    \array{
      x_1 + \mathrm{i} x_2
      \\
      x_1 - \mathrm{i} x_2
    }
    \right]
    \,,
  }
$$
the restriction to which of the canonical complex bilinear form (eq:CanonicalComplexBilinearFormOnComplexPlane)
is twice the [[hyperbolic form]] on $\mathbb{R}^2$
$$
  \array{
     \mathbb{R}^2 \otimes_{\mathbb{R}} \mathbb{R}^2
     &\hookrightarrow&
     \mathbb{C}^2
     \otimes_{\mathbb{C}}
     \mathbb{C}^2
     &\longrightarrow&
     \mathbb{C}
     \\
     \left(
       \left[
       \array{
         x_1
         \\
         x_2
       }
       \right]
       ,\,
       \left[
       \array{
         x'_1
         \\
         x'_2
       }
       \right]
     \right)
     &\mapsto&
     \left(
       \left[
       \array{
         x_1 + \mathrm{i}x_2
         \\
         x_1 - \mathrm{i}x_2
       }
       \right]
       ,\,
       \left[
       \array{
         x'_1 + \mathrm{i}x'_2
         \\
         x'_1 - \mathrm{i}x'_2
       }
       \right]
     \right)
     &\mapsto&
     2 x_1 x'_1 - 2 x_2 x'_2
     \mathrlap{\,.}
  }
$$
\end{example}

\begin{example}\label{RealStrucAsDaggerSelfDualHermitian}
**(real structure as dagger-self-dual Hermitian structure)**
\linebreak
  Let $\mathscr{V}$ be a [[finite-dimensional vector space|finite-dimensional]] [[complex vector space]] equipped with *both*

1. a non-degenerate [[sesquilinear form|sesquilinear]] ([[Hermitian form|Hermitian]]) [[inner product]] ${\langle - \vert - \rangle}$,

1. a non-degenerate symmetric [[bilinear form|bilinear]] [[inner product]] ${(-\vert-)}$

such that

* the Hermitian-[[adjoint operator|adjoint]] ${(-\vert-)}^\dagger$ of the bilinear pairing is the [[coevaluation map]] that exhibits $\mathscr{V}$ as a [[self-dual object]],

then this induces a [[real structure]] on $\mathscr{V}$.

Moreover, for $\mathscr{W}$ another [[finite-dimensional vector space|finite-dimensional]] [[complex vector space]] equipped with a compatible pair of such structures, then the linear maps
$$
  \mathscr{V}
  \longrightarrow
  \mathscr{W}
$$
which preserve both structures also preserve that real structure, hence come from $\mathbb{R}$-[[linear maps]] of underlying [[real vector spaces]].
\end{example}
\begin{proof}
To start with, we note some generalities:

1. the non-degenerate sesquilinear form ${\langle -\vert -\rangle}$ is equivalently given by an [[antilinear map]]
$a \colon \mathscr{V} \to \mathscr{V}^\ast$ to the [[dual linear space]], via

   $$
     {\langle - \vert -\rangle}
       \,=\,
     a(-)(-)
   $$

1. the non-degenerate bilinear form ${(- \vert -)}$ is equivalently given by a complex [[linear map]] $l \,\colon\, \mathscr{V} \to \mathscr{V}^\ast$, via

   $$
     {(- \vert -)}
       \,=\,
     l(-)(-)
   $$

(where the second argument means [[evaluation]]).

In [[bra-ket notation]] this is suggestively written as:

$$
  {( v \vert} \;\;\coloneqq\;\; l {\vert v )}
  \,\;\;\;\;\;
  {\langle w \vert} \;\;\coloneqq\;\; a {\vert w \rangle}
  \,.
$$

For example, the [[Hermitian adjoint]] $f^\dagger \,\colon\, W \to \mathscr{V}$ of any linear map $f \,\colon\, \mathscr{V} \to \mathscr{W}$

is

$$
  f^\dagger {\vert w \rangle}
  \;\coloneqq\;
  a^{-1}\big( {\langle w \vert} f\big)
  \;=\;
  a^{-1} \circ f^\ast \circ a {\vert w \rangle}
  \,.
$$

Moreover, if $\big\{ {\vert w \rangle} \big\}_{w \in W}$ is any [[orthonormal basis]] of $\mathscr{V}$ with respect to $\langle - \vert - \rangle$, then the [[coevaluation map]] which exhibits $\mathscr{V}^\ast \,\coloneqq\, Map_{\mathbb{C}}(\mathscr{V}, \mathbb{C})$ as the [[linear dual space]] to $\mathscr{V}$ may be written as

\begin{tikzcd}[row sep=-1pt]
  &
  \mathcal{V}^\ast
  \\
  \mathbb{C}
  \ar[r, "{ \mathrm{coev} }"]
  &
  \otimes
  \\
  & \mathcal{V}
  \\[3pt]
  1
   \ar[r, phantom, "{\mapsto}"]
  &
  \underset{w \in W}{\sum}
  \vert w \rangle \langle w \vert
\end{tikzcd}
and hence equivalently as
\begin{tikzcd}[row sep=-1pt]
  &
  \mathcal{V}
  \ar[r, "{ a }"]
  &
  \mathcal{V}^\ast
  \\
  \mathbb{C}
  \ar[r, "{ \mathrm{coev} }"]
  &
  \otimes
  \\
  &
  \mathcal{V}^\ast
  \ar[r, "{ a^{-1} }"]
  &
  \mathcal{V}
  \mathrlap{\,,}
\end{tikzcd}
which we use at the very end below.

Now regarding the concrete proof:

The composite

$$
  \tau
  \;\coloneqq\;
  a^{-1} \circ l
  \;\colon\;
  \mathscr{V} \to \mathscr{V}
  \,,
$$

is an [[antilinear map|antilinear]] [[endomorphism]] of $\mathscr{V}$, hence it will be sufficient to show that it is an [[involution]] (and hence the sought-after real structure). For that, in turn, it is clearly sufficient that

\[
  \label{InterchangeOfBiAndSesquilinearForms}
  a^{-1} \circ l
  \,=\,
  l^{-1} \circ a
  \,.
\]

because then

$$
  \tau \circ \tau
  \;=\;
  \big(
    a^{-1} \circ l
  \big)
  \circ
  \big(
    a^{-1} \circ l
  \big)
  \;=\;
    a^{-1} \circ l
  \circ
    l^{-1} \circ a
  \;=\;
  id_{\mathscr{V}}
  \,.
$$

We now show that this condition (eq:InterchangeOfBiAndSesquilinearForms) is equivalent to the assumption that $(-\vert-)^\dagger$ is the coevaluation map.

Namely, with

\begin{tikzcd}[
  row sep=-1pt
]
  &[-5pt]
  \mathcal{V}
  \ar[r, equals]
  &
  \mathcal{V}
  \\
  (-\vert-)
  \;:\;
  &
  \otimes
  &
  \otimes
  \ar[r, "{ \mathrm{ev} }"]
  &
  \mathbb{C}
  \\
  &
  \mathcal{V}
  \ar[
    r,
    "{ l }"{description}
  ]
  &
  \mathcal{V}^\ast
\end{tikzcd}

we have

\begin{tikzcd}[
  row sep=-1pt
]
  && \mathcal{V}^\ast
  \ar[rrr, "{ a^{-1} }"]
  &&&
  \mathcal{V}
  \\
  &
  \mathbb{C}
  \ar[
    r,
    "{ \mathrm{coev} }"
  ]
  &
  \otimes
  \\
  &
  &
  \mathcal{V}
  \ar[r, equals]
  &
  \mathcal{V}
  \\
  (-\vert-)^\dagger
  \;:\;
  &
  \otimes
  &
  \otimes
  &
  \otimes
  \ar[r, "{ \mathrm{ev} }"]
  &
  \mathbb{C}
  \\
  &
  &
  \mathcal{V}
  \ar[
    r,
    "{ l }"{description}
  ]
  &
  \mathcal{V}^\ast
  \\
  &
  \mathbb{C}
  \ar[
    r,
    "{ \mathrm{coev} }"
  ]
  &
  \otimes
  \\
  &&
  \mathcal{V}^\ast
  \ar[rrr, "{ a^{-1} }"]
  &&&
  \mathcal{V}
  \mathrlap{\,,}
\end{tikzcd}

which by the [[zig-zag identity]] for $ev$ & $coev$ equals

\begin{tikzcd}[
  row sep=-1pt
]
  &
  &
  \mathcal{V}
  \ar[r, "{ l }"]
  &
  \mathcal{V}^\ast
  \ar[r, "{ a^{-1} }"]
  &
  \mathcal{V}
  \\
  (-\vert-)^\dagger
  \;:
  &
  \mathbb{C}
  \ar[r, "{ \mathrm{coev} }"]
  &
  \otimes
  &&
  \otimes
  \\
  &&
  \mathcal{V}^\ast
  \ar[rr, "{ a^{-1} }"]
  &&
  \mathcal{V}
  \mathrlap{\,.}
\end{tikzcd}

On the other hand, the coevaluation which exhibits $(-\vert-)$ as a [[self-dual object|self-duality]] is clearly

\begin{tikzcd}[row sep=-1pt]
  & \mathcal{V}^\ast
  \ar[r, "{ l^{-1} }"]
  &
  \mathcal{V}
  \\
  \mathbb{C}
  \ar[r, "{ \mathrm{coev} }"]
  &
  \otimes
  \\
  &
  \mathcal{V}
  \ar[r, equals]
  &
  \mathcal{V}
\end{tikzcd}

which by the above discussion equals

\begin{tikzcd}[row sep=-1pt]
  &
  \mathcal{V}
  \ar[r, "{ a }"]
  &
  \mathcal{V}^\ast
  \ar[r, "{ l^{-1} }"]
  &
  \mathcal{V}
  \\
  \mathbb{C}
  \ar[r, "{ \mathrm{coev} }"]
  &
  \otimes
  \\
  &
  \mathcal{V}^\ast
  \ar[rr, "{ a^{-1} }"]
  &
  &
  \mathcal{V}
\end{tikzcd}

This being equal to $(-\vert-)^\dagger$ above is clearly equivalent to (eq:InterchangeOfBiAndSesquilinearForms).
\end{proof}

## Related concepts

* [[complex structure]]

* [[quaternionic structure]]

* [[real space]]

* [[real oriented cohomology theory]]

  * [[KR-theory]]

## References

* Wikipedia, *[Real structure](https://en.wikipedia.org/wiki/Real_structure)*

More generally in [[spectral geometry]] (via [[spectral triples]]) and [[KR-theory]]:

* {#Connes95} [[Alain Connes]], definition 3 of _Noncommutative geometry and reality_, J. Math. Phys. 36 (11), 1995 ([pdf](http://www.alainconnes.org/docs/reality.pdf))

[[!redirects real structures]]


real vector bundle

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

#Contents#
* table of contents
{:toc}

## Definition

### Plain notion
 {#PlainNotion}

By default, by a _real vector bundle_ over a plain [[topological space]] one means a [[vector bundle]] of [[real vector spaces]] [[associated bundle|associated]] to an $O(n)$-[[principal bundle]] for $O(n)$ the [[orthogonal group]].

This is in contrast notably to [[complex vector bundles]] or [[quaternionic vector bundles]].

### In KR-theory
 {#InKRTheory}

In the context of [[KR-theory]] ("Real K-theory") and following [Atiyah 1966 p. 368](#Atiyah66) one considers a more general notion (which subsumes the plain notion above, see Exp. \ref{RealVectorBundlesAsRealVectorBundles} below):

\begin{definition}
\label{AtiyahRealVectorBundle}
**(Atiyah Real vector bundles)**
\linebreak
A *Real vector bundle* (capital "R" for disambiguation) over a [[Real space]] -- namely over

* a [[cyclic group of order 2|$\mathbb{Z}_2$]]-[[G-space|equivariant space]]

  $$
    \mathbb{Z}_2 \curvearrowright X
    \;\in\;
    \mathbb{Z}_2 Top
    \;=\;
    Func\big(
      \mathbf{B}\mathbb{Z}_2
      ,\,
      Top
    \big)
  $$

  hence a [[topological space]] equipped with a [[continuous map|continuous]] [[involution]]

  $$
    \sigma
    \,\colon\,
    X \longrightarrow X
    ,\;\;\;\;\;
    \sigma \circ \sigma = id_X
  $$

is (according to [Atiyah 1966 p. 368](#Atiyah66)):

1. a [[complex vector bundle]] $p \colon E \to X$ over the [[underlying]] [[topological space]] $X$

1. whose total space $E$ is equipped with a [[continuous map|continuous]] [[involution]] $(\text{-})^\ast \,\colon\, E \to E$

such that:

1. the bundle projection $p$ is a $\mathbb{Z}_2$-[[equivariant map]], hence such that the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       E &\overset{(\text{-})^\ast}{\longrightarrow}& E
       \\
       \mathllap{{}^p}\big\downarrow
         &&
       \big\downarrow\mathrlap{{}^p}
       \\
       X &\underset{\sigma}{\longrightarrow}& X
     }
   $$

1. the involution on $E$ is [[fiber]]-wise [[anti-linear map|anti-linear]], ie. for $x \in X$ the following [[commuting diagram|diagram commutes]]:

   $$
     \array{
       \mathbb{C} \times E_x
         &\overset{(\text{-})\cdot(\text{-})}{\longrightarrow}&
       E_x
       \\
       \mathllap{
         {}^{
           \overline{(\text{-})}
           \times
           (\text{-})^\ast
         }
       }
       \big\downarrow
         &&
       \big\downarrow\mathrlap{{}^{
         (\text{-})^\ast
       }}
       \\
       \mathbb{C} \times E_{\sigma(x)}
         &\underset{(\text{-})\cdot(\text{-})}{\longrightarrow}&
       E_{\sigma(x)}
     }
   $$

A [[homomorphism]] of such Real vector bundles is a homomorphism of the underlying complex vector bundles which respects all the involutions.
\end{definition}

\begin{remark}
  The complex numbers $\mathbb{C}$ equipped with their [[involution]] by [[complex conjugation]] under their usual multiplication operation form a [[monoid object]] [[internalization|internal]] to the category of $\mathbb{R}$-vector spaces equipped with $\mathbb{R}$-linear involution, the "[Real numbers](Hermitian+form#RealComplexNumbers)"
\[
  \label{RealComplexNumbers}
  \mathbb{Z}_2 \curvearrowright \mathbb{C}
  \;\;\;
  \in
  \;\;
  Mon\big(
    Func(\mathbf{B}\mathbb{Z}_2, Mod_{\mathbb{R}})
  \big)
  \,.
\]
The Real vector bundles of Def. \ref{AtiyahRealVectorBundle} are equivalently the vector bundles [[internalization|internal]] to the [[topos]] [[G-set|$\mathbb{Z}_2 Set$]] with the "[[ground ring|ground monoid]]" taken to be the Real numbers (eq:RealComplexNumbers).
\end{remark}

\begin{example}\label{RealVectorBundlesAsRealVectorBundles}
**(real vector bundles as Real vector bundles)**
\linebreak
  In the special case that the involution on the base space $X$ is trivial, $\sigma = id_X$, a Real vector bundle in the sense of Def. \ref{AtiyahRealVectorBundle} is a complex vector bundle over $X$ equipped continuously with a complex [[anti-linear map|anti-linear]] involution on each fiber $(\text{-})^\ast \,\colon\, E_x \to E_x$.

The resulting $\pm 1$-[[eigenspace]]-decomposition realizes each fiber as the [[complexification]]

$$
  E_x
   \;\simeq\;
  V_x \oplus \mathrm{i} V_x
  \;\simeq\;
  V_x \otimes_{\mathbb{R}} \mathbb{C}
$$

of a [[real vector space|$\mathbb{R}$-vector space]] and this decomposition is preserved by homomorphisms of Real vector bundles over such $X$, which are thus given by complexifications of morphisms of real vector bundles.

Therefore the [[full subcategory]] of Real vector bundles over a [[Real space]] $X$ whose [[involution]] is trivial is [[equivalence of categories|equivalent]] to that of plain real vector bundles over $X$ as [above](#PlainNotion).
(cf. [Atiyah 1966, p. 369](#Atiyah66))

In fact, more is true: the [[full subcategory]] of Real vector bundles on all objects whose base carries the trivial $\mathbb{Z}/2$-action is equivalent to the category [[VectBund]] of real vector bundles over varying base spaces
\[
  \label{RealVectorBundlesOverVaryingBaseSpaces}
  \array{
    \underset{X \in Top}{\textstyle{\int}}
    \mathbb{R}Vect_X
    &\xrightarrow{\; \sim \;}&
    \underset{
      (X,id) \in \mathbb{Z}/2 Top
    }{\textstyle{\int}}
    RealVect_X
    \,.
  }
\]
\end{example}

\begin{remark}\label{ComplexVectorBundlesAsRealBundles}
  Similarly to (eq:RealVectorBundlesOverVaryingBaseSpaces), there is a [[subcategory]]-inclusion of [[complex vector bundles]] over varying base spaces into Real vector bundles over varying bases spaces with [[free action]]
\[
  \label{RealVectorBundlesOverVaryingBaseSpaces}
  \array{
    \underset{X \in Top}{\textstyle{\int}}
    \mathbb{C}Vect_X
    &\xrightarrow{\; \sim \;}&
    \underset{
      (\mathbb{Z}/2 \times X) \in \mathbb{Z}/2 Top
    }{\textstyle{\int}}
    RealVect_{\mathbb{Z}/2 \times X}
    \\
    E
    &&
    E & \overline{E}
    \\
    \big\downarrow
    &\mapsto&
    \big\downarrow
    &
    \big\downarrow
    \\
    X
    &&
    \{0\} \times X
    &
    \{1\} \times X
    \,
  }
\]
where $\overline{E}$ denotes the vector bundle with fiber-wise anti-linear structure. However, as is this is not a [[full functor]]: On the right, if a bundle morphism is nontrivial on the base labels $0 \leftrightarrow 1$, the over the remaining base space it corresponds to a fiberwise [[anti-linear map]].

To make the above into an [[equivalence of categories]] one can equip the Real vector bundles moreover with involutions which square to -1 -- ie. [[complex structures]] [[internalization|internal]] to Real vector bundles! -- and require the maps to respect these further involutions. Then the functor which takes a "Real vector bundle with complex structure" over a free $\mathbb{Z}/2$-space to the $+\mathrm{i}$-[[eigenvalue]] bundle (of the internal complex structure), constitutes an equivalence with complex vector bundles.

In this sense the term "Real vector bundle" is quite appropriate.
\end{remark}

## Related concepts

* [[KO]], [[MO]]

* [[KR]], [[MR cohomology theory|MR]]

* [[Pontryagin class]]

* [[complex vector bundle]], [[quaternionic vector bundle]]

* [[real oriented cohomology theory]]

## References

### Plain notion

Real vector bundles in the plain sense are the default notion of [[vector bundles]], and as such are discussed in essentially every reference on the topic, see the list [there](vector+bundle#Literature).

### In KR-theory

The notion of "Real vector bundle" over a [[Real space]] in the sense of [[KR-theory]] was introduced in:

* {#Atiyah66} [[Michael Atiyah]], p. 368 in: *K-theory and reality*, The Quarterly Journal of Mathematics. Oxford. Second Series **17** 1 (1966)  367-386 [[doi:10.1093/qmath/17.1.367](https://doi.org/10.1093/qmath/17.1.367), [[AtiyahKReal.pdf:file]], ISSN:0033-5606]

* [[Allan L. Edelson]], *Real Vector Bundles and Spaces with Free Involutions*, Transactions of the American Mathematical Society **157** (1971) 179-188 [[doi:10.2307/1995841](https://doi.org/10.2307/1995841), [jstor:1995841](https://www.jstor.org/stable/1995841)]

* [[Allan L. Edelson]], *Real Line Bundles on Spheres*, Proceedings of the American Mathematical Society **27** 3 (1971) 579-583 [[doi:10.2307/2036501](https://doi.org/10.2307/2036501), [jstor:2036501](https://www.jstor.org/stable/2036501)]

  > (on Real [[line bundles]] over [[spheres]] [[group actions on spheres|with involution action]])

[[!redirects real vector bundles]]

[[!redirects Real vector bundle]]
[[!redirects Real vector bundles]]


real vector space

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

#Contents#
* table of contents
{:toc}

## Definition

A [[vector space]]/[[module]] over the [[field]] of [[real numbers]].

## Related concepts

* [[real vector bundle]]

* [[real structure]], [[complex structure]]

* [[real cohomology]]

* [[complex vector space]]

* [[complexification]]

* [[rational vector space]]

[[!redirects real vector spaces]]


real-cohesive (infinity,1)-topos

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###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The categorical semantics of [[Dedekind real number|real]]-[[cohesive homotopy type theory]].

## Definition

A [[cohesive (infinity,1)-topos]] $H$ is a **real cohesive (infinity,1)-topos** if $H$ has a [[Dedekind real numbers object]] $\mathbb{R}$ and it satisfies **[[axiom R-flat]]**: for each [[infinity-groupoid]] $A \in H$, $A$ is discrete if and only if the morphism $c \colon Hom(A,A^\mathbb{R})$ is an [[equivalence]].

## See also

* [[Dedekind real number]]
* [[axiom R-flat]]
* [[cohesive (infinity,1)-topos]]

## References

* [[Mike Shulman]], Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

[[!redirects real-cohesive (infinity,1)-topoi]]
[[!redirects real-cohesive (infinity,1)-toposes]]

[[!redirects real cohesive (infinity,1)-topos]]
[[!redirects real cohesive (infinity,1)-topoi]]
[[!redirects real cohesive (infinity,1)-toposes]]

[[!redirects real-cohesive (∞,1)-topos]]
[[!redirects real-cohesive (∞,1)-topoi]]
[[!redirects real-cohesive (∞,1)-toposes]]

[[!redirects real cohesive (∞,1)-topos]]
[[!redirects real cohesive (∞,1)-topoi]]
[[!redirects real cohesive (∞,1)-toposes]]

[[!redirects real-cohesive ∞-groupoid]]
[[!redirects real-cohesive ∞-groupoids]]

[[!redirects real-cohesive infinity-groupoid]]
[[!redirects real-cohesive infinity-groupoids]]

[[!redirects real cohesive ∞-groupoid]]
[[!redirects real cohesive ∞-groupoids]]

[[!redirects real cohesive infinity-groupoid]]
[[!redirects real cohesive infinity-groupoids]]

[[!redirects real-cohesive homotopy theory]]
[[!redirects real-cohesive homotopy theories]]

[[!redirects real cohesive homotopy theory]]
[[!redirects real cohesive homotopy theories]]

[[!redirects cohesive model topos]]
[[!redirects cohesive model toposes]]

[[!redirects real-cohesive model topos]]
[[!redirects real-cohesive model toposes]]

[[!redirects real cohesive model topos]]
[[!redirects real cohesive model toposes]]


real-cohesive homotopy type theory

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#### Type theory
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=--
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--

\tableofcontents

## Idea

**$\mathbb{R}$-cohesive homotopy type theory** or **real-cohesive homotopy type theory** is a version of [[cohesive homotopy type theory]] which has the [[axiom of real cohesion]]. It provides a [[synthetic mathematics|synthetic]] [[foundation]] for [[topology]], [[real analysis]], [[classical homotopy theory]], and [[algebraic topology]].

## Presentation

We assume a [[spatial type theory]] presented with crisp term judgments $a::A$. In addition, we also assume the spatial type theory has a [[Dedekind real numbers]] [[type]] $\mathbb{R}$, and $\mathbb{R}$-[[localization of a type at a family of functions|localizations]] $\mathcal{L}_{\mathbb{R}}(-)$.

Given a type $A$, let us define $\mathrm{const}_{A, \mathbb{R}}:A \to (\mathbb{R} \to A)$ to be the type of all constant functions in the [[Dedekind real numbers]] $\mathbb{R}$:
$$\delta_{\mathrm{const}_{A, \mathbb{R}}}(a, r):\mathrm{const}_{A, \mathbb{R}}(a)(r) =_A a$$
There is an equivalence $\mathrm{const}_{A, 1}:A \simeq (1 \to A)$ between the type $A$ and the type of functions from the [[unit type]] $1$ to $A$.
Given types $B$ and $C$ and a function $F:(B \to A) \to (C \to A)$, type $A$ is **$F$-[[localization of a type at a family of functions|local]]** if the function $F:(B \to A) \to (C \to A)$ is an [[equivalence of types]].

A crisp type $\Xi \vert () \vdash A$ is **discrete** if the function $(-)_\flat:\flat A \to A$ is an [[equivalence of types]].

The **axiom of $\mathbb{R}$-cohesion** states that for the crisp affine line $\Xi \vert () \vdash \mathbb{R} \; \mathrm{type}$, given any crisp type $\Xi \vert () \vdash A \; \mathrm{type}$, $A$ is discrete if and only if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{R}})$-local.

This allows us to define discreteness for non-crisp types: a type $A$ is **discrete** if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{R}})$-local.

The [[shape modality]] in $\mathbb{R}$-cohesive homotopy type theory is then defined as the $\mathbb{R}$-localization $\esh(A) \coloneqq \mathcal{L}_{\mathbb{R}}(A)$, which ensures that the shape of $\mathbb{R}$ itself is a [[contractible type]].

## See also

* [[cohesive homotopy type theory]]
* [[axiom of real cohesion]]

## References

* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

[[!redirects real-cohesive homotopy type theory]]
[[!redirects real cohesive homotopy type theory]]


real-valued-measurable cardinal

\tableofcontents

## Idea: the Banach–Ulam problem

The __Banach–Ulam problem__ asks: are there any nontrivial
[[measures]] on a [[set]] $X$ equipped with the [[σ-algebra]] $2^X$ of all [[subsets]] of $X$?

All [[measures]] are by definition countably additive.

Here a measure on $(X,2^X)$ is __trivial__ if it can be obtained by the following construction.
Start with a set $X$, a map $f\colon X\to\mathbf{R}$, and a [[σ-ideal]] $I$ on $X$.
Set $\mu(A)=\sum_{x\in A}f(x)$ if $A\in I$ and $\mu(A)=\infty$ otherwise.
Then $\mu$ is a [[measure]] on $(X,2^X)$.

Once trivial measure on $(X,2^X)$ are excluded, we can further
reduce to the case of [[probability measures]] on $(X,2^X)$
that vanish on all singleton subsets of $X$.

Recall that the additivity of a [[poset]] $P$
is the smallest cardinal $\kappa$ such that
$P$ has a subset of cardinality $\kappa$ without an upper bound.

Recall that the additivity of a [[measure]] $\mu$
is the smallest cardinal $\kappa$ such that
there is a disjoint family of cardinality $\kappa$ of measurable subsets
such that the additivity property of $\mu$ fails for this family.

Recall that any [[measure]] $\mu$ on $(X,\Sigma)$ induces a [[σ-ideal]] $N_\mu$
such that $A\in N_\mu$ if $A\subset X$ and there is $B\in\Sigma$
such that $A\subset B$ and $\mu(B)=0$.

## Definitions

A [[cardinal]] $\kappa$ is __real-valued-measurable__
if there is a $\kappa$-additive [[probability measure]]
$\mu\colon 2^\kappa\to\mathbf{R}$ that vanishes on all singleton subsets of $\kappa$.

If the [[probability measure]] only takes values 0 and 1,
then $\kappa$ is known as a [[measurable cardinal]].

A cardinal $\kappa$ is __atomlessly-measurable__
if there is an atomless $\kappa$-additive [[probability measure]]
$\mu\colon 2^\kappa\to\mathbf{R}$.

Here an __atom__ of a [[measure]] $\mu$ on $(X,\Sigma)$
is $A\in\Sigma$ such that $\mu(A)\ne0$ and if $B\in\Sigma$
satisfies $B\subset A$, then either $\mu(B)=0$ or $\mu(A\setminus B)=0$.

## Ulam's theorem

\begin{theorem}
([[Ulam]].)
Suppse $(X,2^X,\mu)$ is a nontrivial [[probability space]].

* The additivity of $\mu$ equals the additivity of the [[σ-ideal]] $N_\mu$
and is a real-valued measurable cardinal.

* Any real-valued-measurable cardinal is either atomlessly measurable or measurable.

* Any atomlessly measurable cardinal is [[weakly inaccessible]] and not greater than $\mathfrak{c}$.

* Any [[measurable cardinal]] is [[strongly inaccessible]].

* The [[Lebesgue measure]] on $\mathbf{R}$ extends to $2^{\mathbf{R}}$ if and only if there is an atomlessly measurable cardinal.

\end{theorem}

## References

* [[David H. Fremlin]], _Real-valued-measurable cardinals_.
[PDF](https://www1.essex.ac.uk/maths/people/fremlin/rvmc.pdf).


reality check

We had this error message up briefly in 2009 when we changed servers.  It was [cited in _New Scientist_](http://zenmischief.com/2010/12/defining-the-normal/), so now we have to keep it forever, so you can all verify for yourselves what geeks we are (or
were in 2009).

***

**2011-08-02** This error message is being reinstated briefly while the server on which the nLab and Azimuth Project run is being upgraded.

***

The _n_-Lab is currently experiencing some difficulties due to local fluctuations in reality.
The [[lab elf|Lab Elves]] are working hard to patch reality.
In the meantime, edits on the _n_-Lab have been temporarily disabled since the fundamentals of mathematics may vary during these spasmodic variations.
Normal service will be restored once we are sure what "normal" is.

category: meta


realizability

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### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The idea of _realizability_ is a way of making the [[Brouwer-Heyting-Kolmogorov interpretation]] of [[constructivism]] and [[intuitionistic mathematics]] precise. It is related to the [[propositions as types]] paradigm. For instance, constructively a [[proof]] of an [[existential quantification]] $\underset{x\in X}{\exists} \phi(x)$ consists of constructing a specific $x \in X$ and a proof of $\phi(x)$, which "realizes" the [[truth]] of the statement, whence the name (see e.g. [Streicher 07, section 1](#Streicher07), [Vermeeren 09, section 1](#Vermeeren09) for introductions to the rough idea, or [Bauer 05, page 12 and def. 4.7](#Bauer05) for an actual definition).

## Realizability of univalence in homotopy type theory

One possible way to find a computational interpretation for [[univalence]] in [[homotopy type theory]] is to interpret it in using realizability. Stekelburg provides a univalent universe of modest Kan complexes.

Simplicial homotopy theory can be developed in an [[extensive]] [[locally cartesian closed]] category $A$. Such categories are also called [[Heyting bialgebras]]. The category $A$ has
a class of [[nlab:small maps]] which has a bundle of small
[[assemblies]]. This provides an internal
[[Heyting bialgebra]] $S$ which we can use as a target for simplicial `sets'. There is a (Kan-) model structure on these simplicial sets.

Within $S$ we can define a universe $M$ and show that it is fibrant. This universe is even univalent.

Now, the category of assemblies in number realizability provides such a Heyting bialgebra. The modest sets, a small internally complete full subcategory, provide the univalent universe. Note that modest sets are an [[impredicative]] universe. It models the [[calculus of constructions]].

## Related entries

* [[realizability topos]]

* [[realizability model]]

* [[realizability interpretation]]

* [[effective topos]], [[Kleene-Vesley topos]]

* [[propositional axiom of choice]]

* [[Lifschitz realizability]]

[[!include computable mathematics -- table]]

## References

Realizability originates with the interpretation of [[intuitionistic mathematics|intuitionistic]] [[number theory]], later developed as _[[Heyting arithmetic]]_, in

* {#Kleene45} [[Stephen Kleene]], _On the interpretation of intuitionistic number theory_ Journal of Symbolic Logic, 10:109–124, 1945. [link](http://www.jstor.org/stable/2269016)

A historical survey of realizability (including [[realizability topos|categorical realizability]]) is in

* {#vanOosten00} [[Jaap van Oosten]], _Realizability: An Historical Essay_, 2000 ([link](http://www.staff.science.uu.nl/~ooste110/realizability/history.ps.gz), [pdf](https://pdfs.semanticscholar.org/0eb4/60525d1580fb9f184b43b974499bce2a2ea7.pdf))

A quick survey is in

* {#Vermeeren09} [[Stijn Vermeeren]], _Realizability Toposes_, 2009 ([pdf](http://stijnvermeeren.be/download/mathematics/essay.pdf))

being a summary of

* [[Martin Hyland]], _The effective topos_, in The L.E.J. Brouwer Centenary
Symposium (A. S. Toelstra and D. van Dalen, eds.), North-Holland Publishing
Company, 1982, pp. 165–216

A modern textbook account is

* {#vanOosten08} [[Jaap van Oosten]], _Realizability: an introduction to its categorical side_, Studies in Logic and the Foundations of Mathematics, vol. 152, Elsevier, 2008 ([preface pdf](http://www.staff.science.uu.nl/~ooste110/boekbegin.pdf))

Lecture notes include

* {#Streicher07} [[Thomas Streicher]], _Realizability_ (2007/08) ([pdf](http://www.mathematik.tu-darmstadt.de/~streicher/REAL/REAL.pdf))

and

* {#Bauer05} [[Andrej Bauer]], _Realizability as connection between constructive and computable mathematics_, in T. Grubba, P. Hertling, H. Tsuiki, and [[Klaus Weihrauch]], (eds.) _CCA 2005 - Second International Conference on Computability and Complexity in Analysis_, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, , vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. ([pdf](http://math.andrej.com/data/c2c.pdf))

based on

* {#Bauer00} [[Andrej Bauer]], _The Realizability Approach to
Computable Analysis and Topology_, PhD thesis CMU (2000) ([pdf](http://andrej.com/thesis/thesis.pdf))

* Peter Lietz, _From Constructive Mathematics to Computable Analysis via the Realizability Interpretation_ ([pdf.gz](http://www.mathematik.tu-darmstadt.de/~streicher/THESES/lietz.pdf.gz))

For realizability of univalent universes:

* Wouter Stekelenburg, _Realizability of Univalence: Modest Kan complexes_, [arXiv](http://arxiv.org/abs/1406.6579)

See also

* [[Steven Awodey]], [[Andrej Bauer]], _Sheaf toposes for realizability_ ([pdf](http://www.andrew.cmu.edu/user/awodey/preprints/stfr.pdf))

* Wouter Pieter Stekelenburg, _Realizability Categories_, PhD thesis, Utrecht 2013 ([arXiv:1301.2134](http://arxiv.org/abs/1301.2134))

* [[Martin Hyland]], _Variations on realizability: realizing the propositional axiom of choice_,  Mathematical Structures in Computer Science, Volume 12, Issue 3, June 2002, pp. 295 - 317  ([doi:10.1017/S0960129502003651](https://doi.org/10.1017/S0960129502003651), [pdf](https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2002/vor02.pdf))

[[!redirects realizer]]
[[!redirects realizers]]


realizability model

+-- {: .rightHandSide}
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### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

## Related concepts

* [[realizability topos]]

* [[realizability]]

## References

* [[Steven Awodey]], [[Andrej Bauer]], _Sheaf toposes for realizability_ ([pdf](http://www.andrew.cmu.edu/user/awodey/preprints/stfr.pdf))

[[!redirects realizabilitxy models]]


realizability topos

+-- {: .rightHandSide}
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### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A realizability topos is a [[topos]] which embodies the [[realizability interpretation]] of [[intuitionistic mathematics|intuitionistic]] [[number theory]] (due to Kleene) as part of its [[internal logic]]. Realizability toposes form an important class of [[elementary toposes]] that are not [[Grothendieck toposes]], and don't even have a [[geometric morphism]] to [[Set]].

The input datum for forming a realizability topos is a [[partial combinatory algebra]], or PCA.

* When the PCA is [[Kleene's first algebra]] $\mathcal{K}_1$, the resulting topos is called the [[effective topos]] $RT(\mathcal{K}_1)$.

* When the PCA is [[Kleene's second algebra]] $\mathcal{K}_2$ then $RT(\mathcal{K}_2)$ is the [[function realizability topos]].

## Constructions

There are a number of approaches toward constructing realizability toposes. One is through [[tripos]] theory, and another is through assemblies (actually the latter is a family of related approaches).

Let $A$ be a [[partial combinatory algebra|PCA]] --- in [[Set]], for simplicity, but similar constructions usually work over other base toposes.

### Via tripos theory

There is a [[tripos]] whose base category is $Set$ and for which the preorder $P_A(X)$ of $X$-indexed predicates is the set $P(A)^X$ of functions from $X$ to the [[powerset]] $P(A)$ of $A$.  The order relation sets $\phi \le \psi$ if there exists $a\in A$ such that $b\in \phi(x)$ implies $a\cdot b \in \psi(x)$ for all $x$; note that $a$ must be chosen uniformly across all $x\in X$.

Applying the tripos-to-topos construction to this tripos produces the realizability topos over $A$.  See [[tripos]] for details.

### Via assemblies

+-- {: .num_defn}
###### Definition
An **assembly** $X$ consists of a set ${|X|}$ and a function $[-]_X \colon {|X|} \to P_{\ge 1}(A)$, where $P_{\ge 1}(A)$ denotes the set of [[inhabited set|inhabited]] subsets of $A$.  An assembly is **partitioned** if $[-]_X$ takes values in singletons, i.e. is a function ${|X|} \to A$.

A **morphism** $X \to Y$ between assemblies is a function $f \colon {|X|} \to {|Y|}$ for which there exists $a \in A$ such that for all $x\in X$ and $b\in [x]_X$, $a\cdot b$ is defined and belongs to $[f(x)]_Y$.
=--

The categories of assemblies and partitioned assemblies are denoted $Ass_A$ and $PAss_A$ respectively.

+-- {: .num_prop}
###### Proposition
$Ass_A$ and $PAss_A$ are finitary [[extensive category|lextensive]].  Moreover, $Ass_A$ is [[regular category|regular]] and [[locally cartesian closed category|locally cartesian closed]].
=--

+-- {: .num_defn}
###### Theorem
$Ass_A$ is the [[reg/lex completion]] of $PAss_A$.  Therefore, [[ex/lex completion]] of $PAss_A$ coincides with the [[ex/reg completion]] of $Ass_A$.  This category is a [[topos]], called the **realizability topos** of $A$.
=--

+-- {: .num_remark #pax}
###### Remark
A general result about the ex/lex completion $C_{ex/lex}$ of a left exact category $C$ is that it has enough regular [[projective object|projectives]], meaning objects $P$ such that $\hom(P, -) \colon C_{ex/lex} \to Set$ preserves regular epis. In fact, the regular projective objects coincide with the objects of $C$ (as a subcategory of $C_{ex/lex}$). Of course, when $C_{ex/lex}$ is a topos, where every epi is regular, this means $C_{ex/lex}$ has enough projectives, or satisfies (external) [[COSHEP]]. It also satisfies internal [[COSHEP]], since binary products of projectives, i.e., products of objects of $C$, are again objects of $C$ (see [this result](/nlab/show/internally+projective+object#enough)).
=--

+-- {: .num_remark #ac}
###### Remark
The fact that a realizability topos is an ex/lex completion depends on the [[axiom of choice]] for [[Set]], since it requires the partitioned assemblies to be projective objects therein.  In the absence of the axiom of choice, the projective objects in a realizability topos are the (isomorphs of) partitioned assemblies whose underlying set is [[projective object|projective]] in [[Set]].  Thus, if [[COSHEP]] holds in [[Set]], then a realizability topos is the ex/wlex completion of the category of such "projective partitioned assemblies" (wlex because this category may not have finite limits, only weak finite limits).  Without some choice principle, the realizability topos may not be an ex/wlex completion at all; but it is still an ex/reg completion of $Ass_A$.
=--

## Properties

### Axiomatic characterization
 {#Characterization}

The following is a statement characterizing realizability toposes which is analogous to the [[Giraud axioms]] characterizing [[Grothendieck toposes]].

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

A [[locally small category]] $\mathcal{E}$ is ([[equivalence of categories|equivalent]] to) a realizability topos precisely if

1. $\mathcal{E}$ is [[exact category|exact]] and [[locally cartesian closed category|locally cartesian closed]];

1. $\mathcal{E}$ has [[enough projectives]] and the [[full subcategory]] $Proj(\mathcal{E}) \hookrightarrow \mathcal{E}$ has all [[finite limits]];

1. the [[global section]] [[functor]] $\Gamma \coloneqq \mathcal{E}(\ast,-) \colon \mathcal{E}\longrightarrow $ [[Set]]

   1. has a [[right adjoint]]  $\nabla \colon Set \hookrightarrow \mathcal{E}$ (which is necessarily a [[reflective subcategory|reflective inclusion]] making $\nabla \Gamma$ a finite-limit preserving [[idempotent monad]]/[[closure operator]]);

   1. $\nabla$ factors through $Proj(\mathcal{E})$;

1. there exists an object $D \in Proj(\mathcal{E})$ such that

   1. $D$ is $\nabla\Gamma$-[[separated presheaf|separated]] (in that its $(\Gamma \dashv \nabla)$-[[unit of a monad|unit]] is a [[monomorphism]]);

   1. all $\nabla \Gamma$-[closed](closure+operator#InducedClosureOnSlices) [[regular epimorphisms]] have the [[left lifting property]] against $D\to \ast$;

   1. for every [[projective object]] $P$ there is a $\nabla \Gamma$-[closed morphism](closure+operator#InducedClosureOnSlices) $P \to D$.

=--

This is due to ([Frey 14](#Frey14))

## Related concepts

* [[realizability]]

* [[realizability model]]

## References

* {#Vermeeren09} [[Stijn Vermeeren]], _Realizability Toposes_, 2009 ([pdf](http://stijnvermeeren.be/download/mathematics/essay.pdf))

* Matías Menni, Exact completions and toposes. Ph.D. Thesis, University of Edinburgh (2000). ([web](http://www.lfcs.inf.ed.ac.uk/reports/00/ECS-LFCS-00-424/))

A characterization of realizability toposes analogous to the [[Giraud axioms]] for [[Grothendieck toposes]] is given in

* {#Frey14} [[Jonas Frey]], _Characterizing partitioned assemblies and realizability toposes_ ([arXiv:1404.6997](http://arxiv.org/abs/1404.6997))

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realizer

## Idea

A formula is [[true]] in a [[model]] if it has a *realizer* in the model, and a formula is valid if it has a realizer in every model.

## References

* Steven Cole Kleene, on the interpretation of intuitionistic number theory, JSL, 10:109-124, 1945

* Igor Privara, Branislav Rovan, Peter Ruzicka (eds.), lecture notes in computer science, 841, [google books](http://books.google.de/books?id=r91z_7XJTHoC&pg=PA94&lpg=PA94&dq=realizer+Kleene&source=bl&ots=gCWjr2zXHJ&sig=XolOkGMtSDEfbZYU_DqPcfrWrPI&hl=de&sa=X&ei=L0SuT4TgFMjHsgbX0pSoBA&ved=0CFwQ6AEwAQ#v=onepage&q=realizer%20Kleene&f=false),  chapter 5 realizability models, p.93

* [[Thomas Streicher]], *Realizability Models for $CZF + \neg Pow$*, [unpublished note](http://www.mathematik.tu-darmstadt.de/~streicher/CIZF/rmczfnp.pdf).  In this note WISC is called $TTCA_f$ ($TTCA$ stands for "type-theoretic collection axiom).


reasoning

## Related concepts

* [[logic]]

* [[natural deduction]]

* [[inductive reasoning]]

* [[deductive reasoning]]

* [[abductive reasoning]]

* [[Bayesian reasoning]]


Rebecca R. Winarski

* [personal page](https://sites.google.com/site/rebeccawinarski/)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=183546)

## Selected writings

On the relation between [[mapping class groups]] and [[braid groups]]:

* {#MargalitWinarski21} [[Dan Margalit]], [[Rebecca R. Winarski]], *Braid groups and mapping class groups: The Birman–Hilden theory*, Bull. London Math. Soc. **53** 3 (2021) 643-659 [[arXiv:1703.03448](https://arxiv.org/abs/1703.03448), [doi:10.1112/blms.12456](https://doi.org/10.1112/blms.12456)]

category: people

[[!redirects Rebecca Winarski]]