rational number

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###Context###
#### Arithmetic
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[[!include arithmetic geometry - contents]]
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#### Algebra
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[[!include higher algebra - contents]]
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# Rational numbers
* table of contents
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## Definition

A _rational number_ is a [[fraction]] of two [[integer]] [[numbers]].

### As a field

The [[field]] of **rational numbers**, $\mathbb{Q}$, is the [[field of fractions]] of the [[commutative ring]] of [[integers]], $\mathbb{Z}$, hence the field consisting of formal [[fractions]] ("[[ratios]]") of [[integers]].

### As a commutative ring

Let $(\mathbb{N}^+,1:\mathbb{N}^+,s:\mathbb{N}^+\to \mathbb{N}^+)$ be the set of positive integers. The positive integers are embedded into every [[commutative ring]] $R$: there is an injection $inj:\mathbb{N}^+\to\R$ such that $inj(1) = 1$ and $inj(s(n)) = inj(n) + 1$ for all $n:\mathbb{N}^+$.

Suppose $R$ has an injection $inv:\mathbb{N}^+\to\R$ such that $inj(n) \cdot inv(n) = 1$ and $\inv(n) \cdot \inj(n) = 1$ for all $n:\mathbb{N}^+$. Then $R$ is called a __[[commutative algebra|$\mathbb{Q}$-algebra]]__, and the commutative ring of __rational numbers__ $\mathbb{Q}$ is the [[initial object|initial]] commutative $\mathbb{Q}$-algebra.

It can then be proven from the ring axioms and the properties of the integers that every rational number apart from zero and has a multiplicative inverse, making $\mathbb{Q}$ a field.

### As an sequential colimit in CRing

Let $\mathbb{Z}[1/n!]$ be the [[localization of a commutative ring|localization]] of the [[integers]] $\mathbb{Z}$ [[localisation of a commutative ring away from an element|away from]] the [[factorial]] $n!$, and for each $i \in \mathbb{N}$, there is a unique commutative [[ring homomorphism]] $h_{i}:\mathbb{Z}[1/i!]\to\mathbb{Z}[1/(i + 1)!]$ defined by the [[universal property]] of localisation of $\mathbb{Z}[1/i!]$ away from $i + 1$. Then the commutative ring of __rational numbers__ $\mathbb{Q}$ is the [[sequential colimit]] of the [[diagram]]

$$\mathbb{Z}[1/0!] \overset{h_0}\to \mathbb{Z}[1/1!] \overset{h_1}\to \mathbb{Z}[1/2!] \overset{h_2}\to \ldots$$

### As an abelian group

Let $(\mathbb{N}^+,1:\mathbb{N}^+,s:\mathbb{N}^+\to \mathbb{N}^+)$ be the set of positive integers and let $(\mathbb{Z},0,+,-,1)$ be the [[free abelian group]] on the set ${1}$.

Let $A$ be an [[abelian group]] containing $\mathbb{Z}$ as an abelian subgroup. The positive integers are embedded into the [[function algebra|function abelian group]] $A \to A$, with $id_A:A \to A$ being the [[identity function]] on $A$; i.e. there is an injection $inj:\mathbb{N}^+\to (A \to A)$ such that $inj(1) = id_A$ and $inj(s(n)) = inj(n) + id_A$ for all $n:\mathbb{N}^+$.

Suppose $A$ has an injection $inv:\mathbb{N}^+\to (A \to A)$ such that for all $n:\mathbb{N}^+$, $inj(n) \circ inv(n) = id_A$ and $inv(n) \circ inj(n) = id_A$. Then the abelian group of __rational numbers__ $\mathbb{Q}$ is the [[initial object|initial]] such abelian group.

### As an initial object in a category

Let $\mathbb{Z}$ be the [[integers]] and let
$$\mathbb{Z}_{#0} \coloneqq \{a \in \mathbb{Z} \vert a \lt 0 \vee 0 \lt a \}$$
be the set of integers apart from zero.

For lack of a better name, let us define a *set with rational numbers* to be a set $A$ with a function $\iota \in \mathbb{Z} \times \mathbb{Z}_{#0} \to A$ with domain the Cartesian product $\mathbb{Z} \times \mathbb{Z}_{#0}$ and codomain $A$, such that

$$\forall a \in \mathbb{Z}. \forall b \in \mathbb{Z}_{#0}. \forall c \in \mathbb{Z}. \forall d \in \mathbb{Z}_{#0}. (a \cdot d = c \cdot b) \implies (\iota(a, b) = \iota(c, d))$$

A *homomorphism of sets with rational numbers* between two sets with rational numbers $A$ and $B$ is a function $f \in B^A$ such that

$$\forall a \in \mathbb{Z}. \forall b \in \mathbb{Z}_{#0}. f(\iota_A(a, b)) = \iota_B(a, b)$$

The *category of sets with rational numbers* is the category $SwRN$ whose objects $Ob(SwRN)$ are sets with rational numbers and whose morphisms $Mor(A,B)$ for sets with rational numbers $A \in Ob(SwDF)$, $B \in Ob(SwRN)$ are homomorphisms of sets with rational numbers. The set of **rational numbers**, denoted $\mathbb{Q}$, is defined as the initial object in the category of sets with rational numbers.

### In dependent type theory

In [[dependent type theory]], there are multiple different definitions of the type of rational numbers.

#### As a dependent sum type

Let $\mathbb{Z}$ be the [[integers]], let

  $$\mathbb{Z}_{\neq 0} \coloneqq \sum_{a:\mathbb{Z}} \max(a,-a) \gt 0$$

be the non-zero integers, and let $\gcd:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ be the [[greatest common divisor]] on the integers. Then the rational numbers is defined as the type of pairs of integers and non-zero integers such that the greatest common divisor is equal to one:

  $$\mathbb{Q} \coloneqq \sum_{x:\mathbb{Z}} \sum_{y:\mathbb{Z}_{\neq 0}} \gcd(x, \pi_1(y)) =_\mathbb{Z} 1$$

We define the function $(-)/(-) : \mathbb{Z} \times \mathbb{Z}_{\neq 0} \rightarrow \mathbb{Q}$ for all integers $a:\mathbb{Z}$ and non-zero integers $b:\mathbb{Z}_{\neq 0}$ by

$$a/b \coloneqq (a \div \gcd(a, \pi_1(b)), \pi_1(b) \div \gcd(a, \pi_1(b)), p(a, b))$$

where $p(a, b)$ is the witness that the greatest common divisor of $a \div \gcd(a, \pi_1(b))$ and $\pi_1(b) \div \gcd(a, \pi_1(b))$ is equal to 1, and $a \div b$ is [[integer]] [[division]], since integers are an [[Euclidean domain]] and thus have a division and remainder operation.

#### As a higher inductive type

In [[homotopy type theory]], the type of **rational numbers**, denoted $\mathbb{Q}$, is defined ([UBP13, §11.1](#UBP13)) as the [[higher inductive type]] generated by:

* A function $(-)/(-) : \mathbb{Z} \times \mathbb{Z}_{\neq 0} \rightarrow \mathbb{Q}$, where

  $$\mathbb{Z}_{\neq 0} \coloneqq \sum_{a:\mathbb{Z}} \max(a,-a) \gt 0$$

* A dependent product of functions between identities representing that equivalent fractions are equal:

  $$equivfrac : \prod_{a:\mathbb{Z}} \prod_{b:\mathbb{Z}_{\neq 0}} \prod_{c:\mathbb{Z}} \prod_{d:\mathbb{Z}_{\neq 0}} (a \cdot \pi_1(d) = c \cdot \pi_1(b)) \to (a/b = c/d)\,,$$

* A set-truncator

  $$\tau_0: \prod_{a:\mathbb{Q}} \prod_{b:\mathbb{Q}} isProp(a=b)$$

## Properties

Let $i$ be the inclusion of the non-zero integers into the integers.

### Commutative ring structure on the rational numbers ###

+--{: .num_defn}
###### Definition
The rational number **zero** $0 \in \mathbb{Q}$ is defined as

$$0 \coloneqq 0/1$$
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###### Definition
The binary operation **addition** $(-)+(-):\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$ is defined as

$$a/b + c/d \coloneqq (a \cdot i(d) + c \cdot i(b))/(b \cdot d)$$

for $a \ in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$, $c \in \mathbb{Z}$, $d \in \mathbb{Z}_{#0}$.
=--

+--{: .num_prop}
###### Proposition
For any $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q + r = r + q$.
=--

+--{: .proof}
###### Proof
TODO
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###### Definition
The unary operation **negation** $-(-):\mathbb{Q} \to \mathbb{Q}$ is defined as

$$-(a/b) \coloneqq (-a)/b$$

for $a \in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$
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###### Definition
The binary operation **subtraction** $(-)-(-):\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$ is defined as

$$a/b - c/d \coloneqq (a \cdot i(d) - c \cdot i(b))/(b \cdot d)$$

for $a \in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$, $c \in \mathbb{Z}$, $d \in \mathbb{Z}_{#0}$
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###### Definition
The rational number **one** $1 \in \mathbb{Q}$ is defined as

$$1 \coloneqq 1/1$$
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###### Definition
The binary operation **multiplication** $(-)\cdot(-):\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$ is defined as

$$a/b \cdot c/d \coloneqq (a \cdot c)/(b \cdot d)$$

for $a \in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$, $c \in \mathbb{Z}$, $d \in \mathbb{Z}_{#0}$
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###### Definition
The right $\mathbb{N}$-action **exponentiation** $(-)^{(-)}:\mathbb{Q} \times \mathbb{N} \to \mathbb{Q}$ is defined as

$$(a/b)^n \coloneqq (a^n)/(b^n)$$

for $a \in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$, $n \in \mathbb{N}$.
=--

This makes the rational numbers into a commutative ring.

### Order structure on the rational numbers ###

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###### Definition
The [[predicate]] *is positive*, denoted as $isPositive(a/b)$, is defined as

$$isPositive(a/b) \coloneqq (a \gt 0) \wedge (i(b) \gt 0) \vee (a \lt 0) \wedge (i(b) \lt 0)$$

for $a \in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$.
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###### Definition
The [[predicate]] *is negative*, denoted as $isNegative(a/b)$, is defined as

$$isNegative(a/b) \coloneqq (a \gt 0) \wedge (i(b) \lt 0) \vee (a \lt 0) \wedge (i(b) \gt 0)$$

for $a \in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$.
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###### Definition
The [[predicate]] *is zero*, denoted as $isZero(a/b)$, is defined as

$$isZero(a/b) \coloneqq a = 0$$

for $a \in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$.
=--

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###### Definition
The [[predicate]] *is non-positive*, denoted as $isNonPositive(a/b)$ is defined as

$$isNonPositive(a/b) \coloneqq \neg isPositive(a/b)$$

for $a \in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$.
=--

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###### Definition
The [[predicate]] *is non-negative*, denoted as $isNonNegative(a/b)$, is defined as

$$isNonNegative(a/b) \coloneqq \neg isNegative(a/b) $$

for $a \in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$.
=--

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###### Definition
The [[predicate]] *is non-zero*, denoted as $isNonZero(a/b)$, is defined as

$$isNonZero(a/b) \coloneqq isPositive(a/b) \vee isNegative(a/b)$$

for $a \in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$.
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###### Definition
The [[relation]] *is less than*, denoted as $p \lt q$, is defined as

$p \lt q \coloneqq isPositive(q - p)$

for $p:\mathbb{Q}$, $q:\mathbb{Q}$.
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###### Definition
The dependent type *is greater than*, denoted as $p \gt q$, is defined as

$p \lt q \coloneqq isNegative(q - p)$

for $p:\mathbb{Q}$, $q:\mathbb{Q}$.
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###### Definition
The dependent type *is apart from*, denoted as $p \# q$, is defined as

$p \lt q \coloneqq isNonZero(q - p)$

for $p:\mathbb{Q}$, $q:\mathbb{Q}$.

=--

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###### Definition
The dependent type *is less than or equal to*, denoted as $p \leq q$, is defined as

$p \lt q \coloneqq isNonNegative(q - p)$

for $p:\mathbb{Q}$, $q:\mathbb{Q}$.

=--

+--{: .num_defn}
###### Definition
The dependent type *is greater than or equal to*, denoted as $p \leq q$, is defined as

$p \lt q \coloneqq isNonPositive(q - p)$

for $p:\mathbb{Q}$, $q:\mathbb{Q}$.
=--

### Pseudolattice structure on the rational numbers ###

+--{: .num_defn}
###### Definition
The **ramp function** $ramp:\mathbb{Q} \to \mathbb{Q}$ is defined as

$$ramp(a/b) \coloneqq (ramp(a) \cdot ramp(i(b)) + ramp(-a) \cdot ramp(i(-b)))/(b \cdot b)$$

for $a \in \mathbb{Z}$, $b \in \mathbb{Z}_{#0}$, $ramp:\mathbb{Z} \to \mathbb{Z}$.
=--

+--{: .num_defn}
###### Definition
The **minimum** $min:\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$ is defined as

$$\min(p,q) \coloneqq p - ramp(p - q)$$

for $p:\mathbb{Q}$, $q:\mathbb{Q}$.
=--

+--{: .num_defn}
###### Definition
The **maximum** $max:\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$ is defined as

$$\max(p, q) \coloneqq p + ramp(q - p)$$

for $p:\mathbb{Q}$, $q:\mathbb{Q}$.
=--

+--{: .num_defn}
###### Definition
The **absolute value** $\vert(-)\vert:\mathbb{Q} \to \mathbb{Q}$ is defined as

$$\vert p \vert \coloneqq \max(p, -p)$$

for $p:\mathbb{Q}$.
=--

+--{: .num_defn}
###### Definition
The **metric** $\rho:\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$ is defined as

$$\rho(p, q) \coloneqq \max(p, q) - \min(p, q)$$

for $p:\mathbb{Q}$, $q:\mathbb{Q}$.
=--

### Uniform structure on the rational numbers ###

We define the ternary relation $x \sim_\epsilon y \coloneqq \rho(x, y) \lt \epsilon$ for $x \in \mathbb{Q}$, $y \in \mathbb{Q}$, and $\epsilon \in \mathbb{Q}_+$, called "$x$ and $y$ are within a distance of $\epsilon$". One could show that the rational numbers are a [[uniform space]] with respect to the [[uniformity]] [[ternary relation]] $x \sim_\epsilon y$.

1. For every $x \in \mathbb{Q}$ and $\epsilon \in \mathbb{Q}_+$, $x \sim_\epsilon x$

2. For every $x \in \mathbb{Q}$, $y \in \mathbb{Q}$, $z \in \mathbb{Q}$ and $\delta \in \mathbb{Q}_+$, $\epsilon \in \mathbb{Q}_+$, $x \sim_\delta y$ and $y \sim_\epsilon z$ implies that $x \sim_{\delta + \epsilon} z$.

3. For every $x \in \mathbb{Q}$, $y \in \mathbb{Q}$ and $\epsilon \in \mathbb{Q}_+$, $x \sim_\epsilon y$ implies that $y \sim_\epsilon x$.

4. For every $x \in \mathbb{Q}$, $y \in \mathbb{Q}$, $x \sim_1 y$ if and only if $\rho(x, y) \lt 1$.

5. For every $x \in \mathbb{Q}$, $y \in \mathbb{Q}$ and $\delta \in \mathbb{Q}_+$, $\epsilon \in \mathbb{Q}_+$, $x \sim_{\min(\delta, \epsilon)} y$ implies that $x \sim_{\delta} y$ and $x \sim_{\epsilon} y$.

6. For every $x \in \mathbb{Q}$, $y \in \mathbb{Q}$ and $\delta \in \mathbb{Q}_+$, $\epsilon \in \mathbb{Q}_+$, $\delta \leq \epsilon$ and $x \sim_{\delta} y$ implies that $x \sim_{\epsilon} y$.

### Algebraic closure

The [[algebraic closure]] $\overline{\mathbb{Q}}$ of the rational numbers is called the [[field]] of _[[algebraic numbers]]_. The [[absolute Galois group]] $Gal(\overline{\mathbb{Q}}\vert \mathbb{Q})$ has some curious properties, see [there](absolute+Galois+group#OfTheRationalNumbers).

### Topologies

There are several interesting [[topological space|topologies]] on $\mathbb{Q}$ that make $\mathbb{Q}$ into a [[topological group]] under addition, allowing us to define interesting fields by taking the [[complete space|completion]] with respect to this topology:

1.  The _[[discrete topology]]_ is the most obvious, which is already complete.

2.  The _absolute-value topology_ is defined by the [[metric space|metric]] $d(x,y) \coloneqq {|x - y|}$; the completion is the field of [[real numbers]]. The rational numbers are thus a [[Hausdorff space]].

    (This topology is [[totally disconnected space|totally disconnected]] ([this exmpl.](totally+disconnected+space#RationalNumbersAreTotallyDisconnected)))

3.  Fixing a [[prime number]] $p$, the _$p$-adic topology_ is defined by the [[ultrametric]] $d(x,y) \coloneqq 1/n$ where $n$ is the highest exponent on $p$ in the [[prime factorization]] of ${|x - y|}$; the completion is the field of $p$-[[adic number|adic numbers]].

According to [[Ostrowski's theorem]] this are the only possibilities.

Interestingly, (2) cannot be interpreted as a [[locale|localic]] group, although the completion $\mathbb{R}$ can.  (Probably the same holds for (3); I need to check.)

### Analysis

One could analytically define the concepts of [[limit of a function]] and [[continuous function]] with respect to the absolute-value topology, and prove that the limit of a function satisfy the [[algebraic limit theorem]]. Since the [[reciprocal function]] on the rational numbers is well defined for non-zero rational numbers $\mathbb{Q}_{\neq 0}$, given a continuous function $f:I \to \mathbb{Q}$ for open interval $I \subseteq \mathbb{Q}$ the [[difference quotient]] function exists and thus the [[derivative]] is well-defined for continuous functions. One could thus define [[smooth functions]] on the rational numbers, and because the rational numbers are a Hausdorff space, [[analytic functions]] on the rational numbers, despite the fact that the rational numbers are a totally disconnected space.

For example,

* A rational metric space is a type $X$ with a function $d_X:X \times X \to \mathbb{Q}_{\geq 0}$ such that

  * for all $x \in X$ and $y \in X$, $d_X(x, y) = d_X(y, x)$
  * for all $x \in X$, $y \in X$, and $z \in X$, $d_X(x, z) \leq d_X(x, y) + d_X(y, z)$
  * for all $x \in X$, $x = y \iff d_X(x, y) = 0$

One could show that any [[open interval]], [[half-open interval]], and [[closed interval]] in the rational numbers is a rational metric space, with its metric inherited from the Euclidean metric on the rational numbers $d_\mathbb{Q}(x, y) = {|a - b|}$. Thus one could do analysis with [[partial functions]] in the [[rational numbers]], such as the [[reciprocal function]] $x \mapsto \frac{1}{x}$ and the [[rational functions]] on the rational numbers.

Then, given a rational metric space $(X, d_X)$:

* if it exists, the (Weierstrass) [[limit of a function]] $f:X \to \mathbb{Q}$ from $X$ to the rational numbers approaching an element $a \in X$ is a rational number $L \in \mathbb{Q}$ with a function on the positive rational numbers $M:\mathbb{Q}_+ \to \mathbb{Q}_+$ such that for all positive rational numbers $\epsilon \in \mathbb{Q}_+$ and for all elements $b \in X$, $0 \lt d_X(a, b) \lt M(\epsilon)$ implies ${|f(b) - L|} \lt \epsilon$

* similarly, if it exists, the (Bourbaki) [[limit of a function]] $f:X \to \mathbb{Q}$ from $X$ to the rational numbers approaching an element $a \in X$ is a rational number $L \in \mathbb{Q}$ with a function on the positive rational numbers $M:\mathbb{Q}_+ \to \mathbb{Q}_+$ such that for all positive rational numbers $\epsilon \in \mathbb{Q}_+$ and for all elements $b \in X$, $d_X(a, b) \lt M(\epsilon)$ implies ${|f(b) - L|} \lt \epsilon$

* a [[pointwise continuous function]] from $X$ to the rational numbers is a function $f:X \to \mathbb{Q}$ with a function from $X$ to the set of endofunctions on the positive rational numbers $M:X \to \mathbb{Q}_+ \to \mathbb{Q}_+$ such that for all positive rational numbers $\epsilon \in \mathbb{Q}_+$ and for all rational numbers $a \in X$ and $b \in X$, $\delta_X(a, b) \lt M(a, \epsilon)$ implies that ${|f(a) - f(b)|} \lt \epsilon$.

* a [[uniformly continuous function]] from $X$ to the rational numbers is a function $f:X \to \mathbb{Q}$ with a function on the positive rational numbers $M:\mathbb{Q}_+ \to \mathbb{Q}_+$ such that for all positive rational numbers $\epsilon \in \mathbb{Q}_+$ and for all elements numbers $a \in X$ and $b \in X$, $\delta_X(a, b) \lt M(\epsilon)$ implies that ${|f(a) - f(b)|} \lt \epsilon$.

Now, given an [[injection]] $i:X \hookrightarrow \mathbb{Q}$, so that $X$ is a [[subset]] of the rational numbers,

* a [[pointwise differentiable function]] from $X$ to the rational numbers is a function $f:X \to \mathbb{Q}$ with a [[derivative]] function $f':X \to \mathbb{Q}$ on the rational numbers and a function from $X$ to the set of endofunctions on the positive rational numbers $M:X \to \mathbb{Q}_+ \to \mathbb{Q}_+$ such that for all positive rational numbers $\epsilon \in \mathrm{Q}_+$ and for all elements $a \in X$ and $b \in X$, ${|f(a) - f(b)|} \lt M(a, \epsilon)$ implies that
$${|f(b) - f(a) - f'(a)(i(b) - i(a)))|} \lt \epsilon {|f(a) - f(b)|}$$

* a [[uniformly differentiable function]] from $X$ to the rational numbers is a function $f:X \to \mathbb{Q}$ with a [[derivative]] function $f':X \to \mathbb{Q}$ on the rational numbers and a function on the positive rational numbers $M:\mathbb{Q}_+ \to \mathbb{Q}_+$ such that for all positive rational numbers $\epsilon \in \mathrm{Q}_+$ and for all elements $a \in X$ and $b \in X$, ${|f(a) - f(b)|} \lt M(\epsilon)$ implies that
$${|f(b) - f(a) - f'(a)(i(b) - i(a)))|} \lt \epsilon {|f(a) - f(b)|}$$

* a [[smooth function]] in rational numbers is a function $f:\mathbb{Q} \to \mathbb{Q}$ with a [[sequence]] of functions $D^{(-)}f:\mathbb{N} \to (\mathbb{Q} \to \mathbb{Q})$ and a sequence of functions $M^{(-)}f:\mathbb{N} \to (\mathbb{Q}_+ \to \mathbb{Q}_+)$ in the positive rational numbers, such that

  * for every element $x \in X$, $(D^{0}f)(x) = f(x)$

  * for every natural number $n \in \mathbb{N}$, for every positive rational number $\epsilon \in \mathbb{Q}_+$, for every rational number $h \in \mathbb{Q}$ such that $0 \lt | h | \lt M^{n}f(\epsilon)$, and for every rational number $x \in \mathbb{Q}$,
$$\left|f(x + h) - \sum_{i=0}^n \frac{h^i (D^{i}f)(x)}{i!}\right| \lt \epsilon |h^n|$$

Alternatively, we could use [[infinitesimals]]. Let $\mathbb{Q}[\epsilon]/(\epsilon^2 = 0)$ be the dual rational numbers, and let $\mathbb{Q}[[\epsilon]]$ be the commutative ring of [[formal power series]] on the rational numbers. The former is equivalent to the product $\mathbb{Q} \times \mathbb{Q}$ and the latter is equivalent to the [[function set]] $\mathbb{Q}^\mathbb{N}$.

Both of these are commutative [[rational algebras]] with ring homomorphism $h:\mathbb{Q} \to \mathbb{Q}[\epsilon]/(\epsilon^2 = 0)$ and $k:\mathbb{Q} \to \mathbb{Q}[[\epsilon]]$.

* A function $f:X \to \mathbb{Q}$ is **differentiable** if it has a function $\mathrm{lift}_f:\mathbb{Q}[\epsilon]/(\epsilon^2 = 0) \to \mathbb{Q}[\epsilon]/(\epsilon^2 = 0)$ such that $\mathrm{lift}_f(h(i(a))) = h(i(a))$ and a function $f':X \to \mathbb{Q}$ such that for all elements $a \in X$,

$$\mathrm{lift}_f(h(i(a)) + \epsilon) = h(i(a)) + h(f'(a)) \epsilon$$

* A function $f:X \to \mathbb{Q}$ is **smooth** if it has a function $\mathrm{lift}_f:\mathbb{Q}[[\epsilon]] \to \mathbb{Q}[[\epsilon]]$ such that $\mathrm{lift}_f(h(i(a))) = h(i(a))$ and a sequence of functions $\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times X \to \mathbb{Q}$ with $\frac{d^0 f}{d x^0}\left(a\right) = a$ for all $a \in X$, such that for all natural numbers $n \in \mathbb{N}$
$$\mathrm{lift}_f(h(j(a)) + \epsilon) = \sum_{i = 0}^{\infty} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i$$

## Related concepts

* [[natural number]]

* [[integer]]

* **rational number**

  * a finite [[field extension]] of $\mathbb{Q}$ is called a _[[number field]]_

  * [[dyadic rational number]]

  * [[decimal rational number]]

* [[Q/Z]]

* [[algebraic number]]

* [[Gaussian number]]

* [[irrational number]]

* [[quadratic irrational number]]

* [[real number]]

* [[p-adic number]]

* [[rational numbers object]]

* [[rational interval coalgebra]]

* [[rational root theorem]]

## References

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

See also:

* Wikipedia, *[Rational number](https://en.wikipedia.org/wiki/Rational_number)*

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

* {#UBP13} [[Univalent Foundations Project]], §11.1 of: *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

and specifically in [[Agda]]:

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

[[!redirects rational number]]
[[!redirects rational numbers]]
[[!redirects field of rational numbers]]
[[!redirects rationals]]


rational numbers object

[[natural numbers object]] $\hookrightarrow$ rational numbers object $\hookrightarrow$ [[real numbers object]]


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###Context###
#### Rational homotopy theory
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[[!include differential graded objects - contents]]
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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#### Homological algebra
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[[!include homological algebra - contents]]
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> under construction

#Contents#
* table of contents
{:toc}

## Idea

It is a classical fact that the [[rationalization]] of classical [[homotopy theory]] (of [[topological spaces]] or [[simplicial sets]]) -- called _[[rational homotopy theory]]_ -- is considerably more tractable than general [[homotopy theory]], as exhibited by the existence of small concrete [[dg-algebra|dg-algebraic]] models for [[rational homotopy types]]: [[minimal Sullivan algebras]] or equivalently their [[dual]] [[dg-coalgebras]]. A similar statement holds for the rationalization of [[stable homotopy theory]] i.e. the [[homotopy theory]] of [[spectra]] (of [[topological spaces]] or [[simplicial sets]]): rational spectra are equivalent to rational [[chain complexes]], i.e. to [[dg-modules]] over $\mathbb{Q}$. This is a dg-model for _[[rational stable homotopy theory]]_ compatible with that of classical rational homotopy theory in tat the [[stabilization]] adjunction that connects classical [[homotopy theory]] to [[stable homotopy theory]] is, under these identifications, modeled by the [[forgetful functor]] from dg-(co-)algebras to [[chain complexes]]

$$
  \array{
    & \text{classical homotopy theory}
     &&
    \text{stable homotopy theory}
    \\
    & spaces & \underoverset{\underset{\Sigma^\infty}{\longrightarrow}}{\overset{\Omega^\infty}{\longleftarrow}}{} & spectra
    \\
    & & \text{stabilization}
    \\
    \text{rationally}
    &
    \text{dg(co)algebras}
    &
     \underoverset{\underset{underlying}{\longrightarrow}}{\overset{free}{\longleftarrow}}{}
    &
    \text{chain complexes}
  }
$$

Classical [[homotopy theory]] and [[stable homotopy theory]] are unified and jointly generalized in [[parameterized stable homotopy theory]], whose [[objects]] are [[parameterized spectra]], parameterized over a classical [[homotopy type]]. The _rational parameterized stable homotopy theory_ to be discussed here is supposed to be the rationalization of this joint generalization, unifiying and jointly generalizing the algebraic model of [[rational topological spaces]] by [[Sullivan algebras]] and of $H \mathbb{Q}$-[[module spectra]] by [[chain complexes]].

$$
  \array{
    & \text{stable homotopy theory}
    &&
    \text{parameterized stable homotopy theory}
    &&
    \text{classical homotopy theory}
    \\
    &
    \text{spectra}
     &\overset{\phantom{\text{include}}}{\longrightarrow}&
    \text{parameterized spectra}
     &\overset{\phantom{\text{project}}}{\longrightarrow}&
    \text{spaces}
    \\
    &
    &\text{include}&
    &\text{project}&
    \\
    \text{rationally}
    &
    \text{chain complexes}
    &\overset{\phantom{\text{include}}}{\longrightarrow}&
    \text{dg-modules}
    &\overset{\phantom{\text{project}}}{\longrightarrow}&
    \text{dg(co)algebras}
  }
$$

Here we (intend to) show that, accordingly, rational parameterized homotopy theory is presented by the the [[opposite category|opposite]] of the [[homotopical category]] of [[dg-modules]] over cochain [[differential graded-commutative algebras]] in non-negative degrees.

> under construction

## Preliminaries

### Chain complexes

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

Write

* $Ch_{\bullet,\mathbb{Q}}$ for the [[category of chain complexes]] of [[modules]]/[[vector spaces]] over $\mathbb{Q}$ (i.e. [[differential]] of degree -1)

* $Ch^{\bullet}_{\mathbb{Q}}$ for the category of [[cochain complexes]] (i.e. differential of degree +1).

For $n \in \mathbb{N}$ write

* $Ch_{\geq n, \mathbb{Q}} \hookrightarrow Ch_{\bullet,\mathbb{Q}}$ for the [[full subcategory]] of the [[chain complexes]] concentrated in degree $\geq n$;

* $Ch^{\geq n}_{\mathbb{Q}} \hookrightarrow Ch^\bullet_{\mathbb{Q}}$ for the [[full subcategory]] of the [[cochain complexes]] concentrated in degree $\geq n$.

For $V \in \mathbb{Q} Mod$ a rational [[vector space]], and for $n \in \mathbb{N}$, we write $V[n]$ both for the [[chain complex]] as well as for the [[cochain complex]] concentrated on $V$ in degree $n$.

=--

### dg-Algebras

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

Write $dgcAlg^{\geq 0}_{\mathbb{q}}$ for the [[category]] of cochain [[dgc-algebras]] over the [[rational numbers]] concentrated in non-negative degrees.

Say that a [[morphism]] in this category is

1. a _weak equivalence_ if it is a  [[quasi-isomorphisms]] on the [[forgetful functor|underlying]] [[chain complexes]];

1. a _fibration_ if it is degreewise [[surjection]];

1. a _cofibration_ it it is a [[relative Sullivan algebra]] inclusion,

We write

$$
  (dgcAlg^{\geq 0}_{\mathbb{Q}})_{proj}
$$

for the category $dgcAlg^{\geq 0}_{\mathbb{Q}}$ equipped with these three [[classes]] of morphisms.

=--

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

The [[homotopical category]] $(dgcAlg^{\geq 0}_{\mathbb{Q}})_{proj}$ from def. \ref{dgcAlgebras} is a [[model category]], to be called the _[[projective model structure on dgc-algebras]] in non-negative degrees_.

=--

([Bousfield-Gugenheim 76, theorem 4.3](#BousfieldGugenheim76))

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

For $S \in sSet$ a [[simplicial set]], write

$$
  \Omega^\bullet_{poly}(S)
   \in
  dgcAlg^{\geq 0}_{\mathbb{Q}}
$$

for the [[polynomial differential forms]] with [[rational number|rational]] [[coefficients]] on $S$.

=--

([Bousfield-Gugenheim 76, def. 2.1](#BousfieldGugenheim76))

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

Write $\mathbb{Q}[0] \coloneqq (\mathbb{Q}[0], d = 0)$ for the [[dgc-algebra]] concentrated on the [[ground field]] in degree 0, necessarily with vanishing [[differential]]. This is the [[initial object]] in $dgcAlg^{\geq 0}_{\mathbb{Q}}$.

Write

$$
  (dgcAlg^{\geq 0}_{\mathbb{Q}})_{/\mathbb{Q}[0]}
$$

for the [[slice category]] of that of all [[dgc-algebras]] (def. \ref{dgcAlgebras}) over $\mathbb{Q}[0]$. Hence an [[object]] in this category is a pair consisting of a [[dgc-algebra]] $A$ and a dg-algebra [[homomorphism]] of the form

$$
  \epsilon_A \;\colon\; A \longrightarrow \mathbb{Q}[0]
  \,.
$$

This is equivalently called a _$\mathbb{Q}[0]$-[[augmented algebra|augmented]] dgc-algebra_. The [[kernel]] of the augmentation map $\epsilon$

$$
  ker_{\epsilon_{A}} \in Ch^{\geq 0}_{\mathbb{Q}}
$$

is the _[[augmentation ideal]]_ of $(A,\epsilon)$.

Since $\mathbb{Q}[0] \in dgcAlg^{\geq 0}_{\mathbb{Q}}$ carries a unique augmentation $\epsilon = id$, we still write $\mathbb{Q}[0]$ for the [[ground field]] regarded as an augmented dgc-algebra. As such this is now a [[zero object]].

Furthermore write

$$
  \left(
   (dgcAlg^{\geq 0}_{\mathbb{Q}})_{/\mathbb{Q}[0]}
  \right)_{proj}
$$

for the [[slice model structure]] induced on this by the [[projective model structure on dgc-algebras]] according to prop. \ref{dgcAlgProjectiveModelStructure}.

=--

See also [Bousfield-Gugenheim 76, 4.11](#BousfieldGugenheim76)

### Simplicial Lie algebras

$$
  (LieAlg_k)^{\Delta^{op}}_{proj}
    \underoverset
      {\underset{N}{\longrightarrow}}
      {\overset{N^\ast}{\longleftarrow}}
      {\bot}
  (dgLieAlg_k)_{proj}
    \underoverset
      {\underset{CE}{\longrightarrow}}
      {\overset{\mathcal{L}}{\longleftarrow}}
      {\bot}
  dgcoAlg_k
$$

(...)

## Statement

We want to claim the following:

For every $\mathfrak{g} \in (LieAlg_{\mathbb{Q}})^{\Delta^{op}}$
there is a [[Quillen equivalence]]

$$
  SeqSpec\left(
    \mathfrak{g}Mod
  \right)_{stable}
    \underoverset
      {\underset{SeqSpec((Sym)^{op})}{\longrightarrow}}
      {
        \overset{SeqSpec( ker_{\epsilon(-)}^{op} )}{\longleftarrow}
      }
      {\simeq_{Qu}}
  SeqSpec\left(
    \mathfrak{g} / (LieAlg_{\mathbb{Q}})^{\Delta^{op}} / \mathfrak{g}
  \right)_{stable}
$$

Idea of proof: the analogous statement for simplicial Lie algebras replaced by rational [[simplicial algebras]] $cAlg_{\mathbb{A}}^{\Delta^{op}}$ is [Schwede 97, theorem 3.2.3](#Schwede97). Apart from the connectivity of the $Sym$-construction, all that this proof uses is that simplicial commutative algebras form a [[proper model category|right proper]] [[simplicial model category]]. But also the [[model structure on simplicial Lie algebras]] is right proper and simplicial.

## Related concepts

* [[rational model for a suspension]]

* [[rational homotopy theory]]

* [[rational stable homotopy theory]]

* [[rational parametrized homotopy theory]]

* [[rational equivariant homotopy theory]]

* [[rational equivariant stable homotopy theory]]

## References

A classical reference on plain [[rational homotopy theory]] is:

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], *[[On PL deRham theory and rational homotopy type]]*, Memoirs of the AMS **179** (1976) [[ams:memo-8-179](https://bookstore.ams.org/memo-8-179)]

The equivalence between $H R$-[[module spectra]] (unparametrized) and $R$-[[chain complexes]] is due to

* {#Schwede97} [[Stefan Schwede]], section 3 of _Spectra in model categories and applications to the algebraic cotangent complex_, Journal of Pure and Applied Algebra 120 (1997) 104 ([pdf](http://www.math.uni-bonn.de/people/schwede/modelspec.pdf))

* {#Shipley02} [[Brooke Shipley]], _$H \mathbb{Z}$-algebra spectra are differential graded algebras_ , Amer. Jour. of Math. 129 (2007) 351-379. ([arXiv:math/0209215](http://arxiv.org/abs/math/0209215))

Discussion of rational fiberwise [[suspension spectra]]:

* [[Michael C. Crabb]], [[Ioan Mackenzie James]], around Prop. 15.8 of: *Fiberwise homotopy theory*, Springer Monographs in Mathematics, Springer (1998) $[$[doi:10.1007/978-1-4471-1265-5](https://doi.org/10.1007/978-1-4471-1265-5), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/crabbjames.pdf) ,[pdf](https://www.gbv.de/dms/ilmenau/toc/243540361.PDF)$]$

* [[Yves Félix]], Aniceto Murillo [[Daniel Tanré]], _Fibrewise stable rational homotopy_, Journal of Topology, Volume 3, Issue 4, 2010, Pages 743–758 ([doi:10.1112/jtopol/jtq023]( https://doi.org/10.1112/jtopol/jtq023))

A discussion of full-blown rational parametrized stable homotopy theory is due to

* [[Vincent Braunack-Mayer]], _[[schreiber:thesis Braunack-Mayer|Rational parameterized stable homotopy theory]]_, Zurich, 2018

* [[Vincent Braunack-Mayer]], _Strict algebraic models for rational parametrised spectra I_,  Algebraic & Geometric Topology 21 (2021) 917–1019 [[arXiv:1910.14608](https://arxiv.org/abs/1910.14608), [doi:10.2140/agt.2021.21.917](https://doi.org/10.2140/agt.2021.21.917)]

Application to mathematical analysis of [[duality between M-theory and type IIA string theory]]:

* [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of Super M-Branes rational parameterized stable homotopy theory]]_, Communications in Mathematical Physics 371: 197 (2019) ([arXiv:1806.01115](https://arxiv.org/abs/1806.01115), [doi:10.1007/s00220-019-03441-4](https://doi.org/10.1007/s00220-019-03441-4))

* [[Vincent Braunack-Mayer]], _[[schreiber:Parametrised homotopy theory and gauge enhancement]]_, talk at _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_ Durham Symposium 2018 , Fortschritte der Physik (2019) ([doi:10.1002/prop.201910003](https://onlinelibrary.wiley.com/doi/abs/10.1002/prop.201910003)m [arXiv:1903.02862](https://arxiv.org/abs/1903.02862))

[[!redirects rational parametrized stable homotopy theory]]

[[!redirects parametrized rational stable homotopy theory]]


rational parametrized homotopy theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Rational parametrized homotopy theory is _[[parametrized homotopy theory]]_ in the approximation of [[rational homotopy theory]].

## Properties

+-- {: .num_prop}
###### Proposition
**([[rational fibration lemma]])**

Let

$$
  \array{
    F &\overset{fib(p)}{\longrightarrow}& A
    \\
    && \big\downarrow{}^{\mathrlap{p}}
    \\
    && B
  }
$$

be a [[Serre fibration]] of [[connected topological space|connected]] [[topological spaces]], with [[fiber]] $F$ (over any base point) also connected.

If in addition

1. the [[fundamental group]] $\pi_1(B)$ [[nilpotent module|acts nilpotently]] on the [[homology groups]] $H_\bullet(F,k)$

   (e.g. if $B$ is [[simply connected topological space|simply connected]], or if the fibration is a [[principal bundle]]);

1. at least one of $A$, $F$ has rational [[finite type]]

then the [[cofiber]] of any [[relative Sullivan model]] for $p$ is a [[Sullivan model]] for $F$.
=--

([Félix-Halperin-Thomas 00, Theorem 15.3](#FelixHalperinThomas00), following [Halperin 83, Section 16](#Halperin83))

Moreover, if $CE(\mathfrak{l}B)$ is a [[minimal Sullivan model]] for $B$, then the cofiber of the corresponding minimal [[relative Sullivan model]] for $p$ is the [[minimal Sullivan model]] $CE(\mathfrak{l}F)$ for $F$:

\[
  \label{SullivanModelForFiber}
  \array{
    CE(\mathfrak{l}F)
    &\overset{
      cofib
      \big(
        CE(\mathfrak{l}p)
      \big)
    }{\longleftarrow}&
    CE(\mathfrak{l}_{{}_B}A)
    \\
    && \big\uparrow{}^{\mathrlap{ CE(\mathfrak{l}p) }}
    \\
    && CE(\mathfrak{l}B)
  }
\]

([Félix-Halperin-Thomas 00, Corollary on p. 199](#FelixHalperinThomas00), [Felix-Halperin-Thomas 15, Theorem 5.1](#FelixHalperinThomas15))

But this cofiber, being the cofiber of a [[relative Sullivan model]] and hence of a [[cofibration]] in the [[projective model structure on dgc-algebras]], is in fact the [[homotopy cofiber]], and hence is a model for the [[homotopy fiber]] of the [[rationalization|rationalized]] fibration.

Therefore (eq:SullivanModelForFiber) implies that on fibrations of connected finite-type spaces where $\pi_1$ of the base acts nilpotently on the homology of the fiber: _[[rationalization]] preserves [[homotopy fibers]]_.

(This is the _fibration lemma_ orginally due to [Bousfield-Kan 72, Chapter II](#BousfieldKan72).)

## Related concepts

* [[parametrized homotopy theory]]

* [[rational parametrized stable homotopy theory]]

## References

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], Chapter II "Fiber Lemmas" of: _Homotopy Limits, Completions and Localizations_, Springer 1972 ([doi:10.1007/978-3-540-38117-4](https://doi.org/10.1007/978-3-540-38117-4))

* {#Halperin83} [[Steve Halperin]], Section 16-20 of: _Lectures on minimal models_, Mem. Soc. Math. Franc. no 9/10 (1983) ([doi:10.24033/msmf.294](https://doi.org/10.24033/msmf.294))

* {#Silveira84} [[Flavio da Silveira]], _Rational homotopy theory of fibrations_, Pacific Journal of Mathematics, Vol. 113, No. 1 (1984) ([pdf](http://msp.org/pjm/1984/113-1/pjm-v113-n1-p01-s.pdf))

* {#FelixHalperinThomas00} [[Yves Félix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], Sections 14 and 15 of: _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 ([doi:10.1007/978-1-4613-0105-9](https://link.springer.com/book/10.1007/978-1-4613-0105-9))

* {#FelixHalperinThomas15} [[Yves Félix]], [[Steve Halperin]], [[Jean-Claude Thomas]], Sections 4 and 5 of: _Rational Homotopy Theory II_, World Scientific 2015 ([doi:10.1142/9473](https://doi.org/10.1142/9473))

[[!redirects parametrized rational homotopy theory]]


rational point

#Contents#
* table of contents
{:toc}

## Idea

In [[algebraic geometry]] and [[arithmetic geometry]], a _rational point_ of an [[arithmetic scheme]] $S$ is a [[morphism]] $Spec(Q) \to S$ from the [[ring spectrum]] of the [[rational numbers]].

If $S$ is the [[algebraic variety]] of solutions to some [[polynomial]] [[equations]], then a rational point corresponds to a [[solution]] of these equations in the field of [[rational numbers]]

## Related entries

* [[Fermat curve]]

## References

* Wikipedia, _[Rational point](https://en.wikipedia.org/wiki/Rational_point)_

[[!redirects rational points]]


rational stable homotopy theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Rational stable homotopy theory is the simple special case of [[stable homotopy theory]] under [[rationalization]].

It is a classical fact that the [[rationalization]] of classical [[homotopy theory]] (of [[topological spaces]] or [[simplicial sets]]) -- called _[[rational homotopy theory]]_ -- is considerably more tractable than general [[homotopy theory]], as exhibited by the existence of small concrete [[dg-algebra|dg-algebraic]] models for [[rational homotopy types]]: [[minimal Sullivan algebras]] or equivalently their [[dual]] [[dg-coalgebras]]. A similar statement holds for the rationalization of [[stable homotopy theory]] i.e. the [[homotopy theory]] of [[spectra]] (of [[topological spaces]] or [[simplicial sets]]): rational spectra are equivalent to rational [[chain complexes]], i.e. to [[dg-modules]] over $\mathbb{Q}$. This is a dg-model for _rational stable homotopy theory_ compatible with that of classical rational homotopy theory in that the [[stabilization]] adjunction that connects classical [[homotopy theory]] to [[stable homotopy theory]] is, under these identifications, modeled by the [[forgetful functor]] from dg-(co-)algebras to [[chain complexes]]

$$
  \array{
    & \text{classical homotopy theory}
     &&
    \text{stable homotopy theory}
    \\
    & spaces & \underoverset{\underset{\Sigma^\infty}{\longrightarrow}}{\overset{\Omega^\infty}{\longleftarrow}}{} & spectra
    \\
    & & \text{stabilization}
    \\
    \text{rationally}
    &
    \text{dg(co)algebras}
    &
     \underoverset{\underset{underlying}{\longrightarrow}}{\overset{free}{\longleftarrow}}{}
    &
    \text{chain complexes}
  }
$$

## Rational spectra

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

By the [[stable Dold-Kan correspondence]], the ([[stable homotopy theory|stable]]) [[homotopy theory]] of _[[rationalization|rationalized]] [[spectra]]_, namely of $H \mathbb{Q}$-[[module spectra]] is [[equivalence of (infinity,1)-categories|equivalent]] to that of [[chain complexes]] of [[modules]]/[[vector space]] over the [[rational numbers]]

$$
  H \mathbb{Q} ModSpectra
   \;\simeq\;
  Ch_\bullet(\mathbb{Q})
  \,.
$$

=--

([Schwede-Shipley 03, theorem 5.1.6](#SchwedeShipley03), see also at _[[stable Dold-Kan correspondence]]_).

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

Observe that $H \mathbb{Q}$-module spectra are just the rational spectra. Since [[rationalization]] of spectra is the [[smashing localization]] $(-)\wedge H \mathbb{Q}$ ([here](rationalization#RationalizationOfSpectra)) every rational spectrum $X \simeq H \mathbb{Q} \wedge X$ carries an $H \mathbb{Q}$-module structure. Conversely, just as a $\mathbb{Q}$-[[module]] structure on an [[abelian group]] is unique if it exists, so an $H \mathbb{Q}$-[[module spectrum]] structure on a [[spectrum]] is essentially unique if it exists, due to the fact that the multiplication map $H \mathbb{Q} \wedge H \mathbb{Q} \to H \mathbb{Q}$ is an [[equivalence in an (infinity,1)-category|equivalence]].

=--

Theorem \ref{RationalStableDoldKanCorrespondence} parallels that of classical [[rational homotopy theory]]:

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

There is an [[equivalence of (infinity,1)-categories|equivalence of homotopy theories]] between the [[homotopy theory]] of [[nilpotent topological space|nilpotent]] [[rational topological spaces]] [[of finite type]] with that of cochain [[dgc-algebras]] over $\mathbb{Q}$ in non-negative degree

$$
  \infty Grpd_{\mathbb{Q}, nil,fin}
    \;\simeq\;
  dgcAlg^{\geq 0}_{\mathbb{Q},nil,fin}
$$

=--

## Rational stabilization
 {#RationalStabilization}

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

The following composite total [[derived functors]]

$$
  \array{
    \mathrm{Ho}( \mathrm{Spectra}_{\mathbb{Q}, \mathrm{fin}} )
    \\
    \downarrow \simeq \uparrow
    \\
    \mathrm{Ho}( \mathrm{Ch}_{\mathbb{Q},\bullet,\mathrm{fin}} )
     &
      \underoverset
        \underset{\mathbb{R}\mathrm{cn}_2}{\longrightarrow}
        \overset{\mathbb{L} i_2}{\longleftarrow}
        {\bot}
     &
    \mathrm{Ho}( \mathrm{Ch}_{\mathbb{Q}, \gt 1, \mathrm{fin}} )
    \\
    &&
    \downarrow \simeq \uparrow^{(-)^\ast}
    \\
    &&
    \mathrm{Ho}( \mathrm{Ch}^{\gt 1}_{\mathbb{Q}, \mathrm{fin}}   )^{\mathrm{op}}
     &
      \underoverset
       \underset{ (\mathbb{L}\mathrm{Sym}_{/\mathbb{Q}[0]})^{\mathrm{op}} }{\longrightarrow}
       \overset{ (\mathbb{R}( U \circ \mathrm{ker}(\epsilon_{(-)}) ))^{\mathrm{op}} }{\longleftarrow}
       {\bot}
     &
   \mathrm{Ho}( \mathrm{dgcAlg}^{\gt 0}_{\mathbb{Q}, \mathrm{fin}})_{/\mathbb{Q}[0]} )^{\mathrm{op}}
    \\
   && && \updownarrow \simeq
    \\
    && &&
    \mathrm{Ho}(\mathrm{Top}_{\mathbb{Q}, \gt 1 , \mathrm{fin}})
  }
$$

(where the key part in the middle right, involving $Sym$, is from prop. \ref{dgcAlgChAdjunction}, the left middle part involving connectivity is from prop. \ref{ChainComplexesConnectiveCover}, and the equivalences on the far left and far right are those from theorem \ref{RationalStableDoldKanCorrespondence} and theorem \ref{ClassicalRationalHomotopyTheory}, respectively)

agree with the restriction of the
[[stabilization]] [[adjoint (infinity,1)-functor|infinity-adjunction]]

$$
  Spectra
    \underoverset
      {\underset{\Omega^\infty}{\longrightarrow}}
      {\overset{\Sigma^\infty}{\longleftarrow}}
      {\bot}
  \infty Grpd^{\ast/}
$$

to simply connected rational homotopy types of finite type.

=--

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

By the nature of the classical model structure on topological spaces,
every simply connected object of $\mathrm{Ho}(\mathrm{Top}^{\ast/})$ is represented by one that is a [[retract]] of a [[transfinite composition]] of [[pushouts]] of the pointed connected generating cofibrations $S^n \to D^n$, for $n \geq 1$. More abstractly, the [[(∞,1)-category]] of simply connected homotopy types is generated under [[(∞,1)-colimits]] from the [[n-spheres]] of dimension $n \geq 2$. (See the analogous argument used in the [[Brown representability theorem]] [here](Brown%20representability%20theorem#TheClassicalPointedConnectedHomotopyCategoryAsDomainForTheAbstractBrownRepresentabilityTheorem)).

Since $\Sigma^\infty$ is a left adjoint functor it is sufficient to check that the two functors agree on $S^n$ for $n \geq 1$. That they agree on $D^n$ is immediate. We hence need to consider their value on the positive
dimensional $n$-spheres:

Let $n$ be odd. Then the [[minimal Sullivan model]] for $S^n$ is $\mathrm{Sym}( \mathbb{R}[n] )$.
Since every dgc-algebra is fibrant in the projective model structure, the value of
$\mathbb{R}(U \circ \mathrm{ker}(\epsilon_{(-)}))$ on this model is represented by
the value of $U \circ \mathrm{ker}(\epsilon_{(-)})$ on that model, which is the cochain complex
$\mathbb{Q}[n]$, hence equivalently the corresponding chain complex. This is indeed the
rationalization of $\Sigma^\infty S^n$.

Next let $n$ be even with $n \geq 1$. Then the minimal Sullivan model for $S^n$ is
$\mathrm{Sym}( \mathbb{R}[n] \oplus \mathbb{R}[2n-1], d c_{2n-1} = c_{n} \wedge c_n  )$.
The underlying chain complex of the augmentation ideal is spanned by the elements
$c_{n}^{\wedge^k}$  in degree $n k$ and by the elements
$c_{n}^{\wedge^k} \wedge c_{2n-1}$ in degree $nk+2n-1$. The former elements are all closed,
but except for $k = 1$ are all in the image of the latter elements, none of which is closed.
Hence this chain complex is quasi-isomorphic to $\mathbb{R}[n]$. Again, this is indeed
the rationalization of $\Sigma^\infty S^{n}$

=--

## Appendix

Here we collect some background definitions, notations and facts for ease of reference in the main text above.

### Chain complexes

Throughout, let $k$ be a [[field]] of [[characteristic zero]].

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

Write

1. $\mathrm{Ch}_{k,\bullet}$ for the [[category of chain complexes]] of $k$-[[vector spaces]], i.e. the category of $\mathbb{Z}$-graded $k$-vector spaces $V$ equipped with a linear map $\partial_V : V \to V$ of degree -1 such that $\partial_V \circ \partial_V = 0$ (the [[differential]]);

1. $\mathrm{Ch}_k^\bullet$ for the category of [[cochain complexes]] of $k$-vector spaces, i.e. as before, but with differential $d_V$ of degree +1.

For $V$ a $k$-vector space, and $n \in \mathbb{N}$ we write $V[n]$ for the (co-)chain complex concentrated on
$V$ in degree $k$.

For $n \in \mathbb{Z}$ write
$$
    \mathrm{Ch}_{k, \geq n}
      \overset{i_n}{\hookrightarrow}
    \mathrm{Ch}_{k,\bullet}
$$
$$
    \mathrm{Ch}_k^{\geq n}
      \overset{i^n}{\hookrightarrow}
    \mathrm{Ch}_k^\bullet
$$

for the [[full subcategories]] on the (co-)chain complexes that are concentrated in degrees $\geq n$.

=--

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

The inclusions from def. \ref{Complexes} have right resp. left adjoints, which we denote by

$$
    \mathrm{Ch}_{k,\bullet}
      \underoverset
        {\underset{\mathrm{cn}_n}{\longrightarrow}}
        {\overset{i_n}{\longleftarrow}}
        {\bot}
    \mathrm{Ch}_{k, \geq n}
$$

and

$$
    \mathrm{Ch}^{\geq 0}_k
      \underoverset
       {\underset {i_n}{\longrightarrow}}
       {\overset{\mathrm{cn}^n}{\longleftarrow}}
       {\bot}
    \mathrm{Ch}^\bullet_k
  \,.
$$

These are [[Quillen adjunctions]] with respect to the [[projective model structure on chain complexes]].

=--

### dgc-Algebras

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

We write $\mathrm{dgcAlg}^\bullet_k$
for the category of [[differential graded-commutative algebras]] over $k$, i.e. commutative monoids in
$\mathrm{Ch}^\bullet_k$. We write $(\mathrm{dgcAlg}^\bullet_k)_{/k[0]}$
for its [[slice category]] over $k = (k[0], d = 0) \in \mathrm{dgcAlg}^{\geq 0}$, hence for the category of [[augmented algebra|augmented]] dgc-algebras,
hence for dgc-algebras $A$ equipped with a homomorphism $\epsilon_A : A \to (k[0], d = 0)$.

Finally we write
$$
  \mathrm{dgcAlg}^{\geq 0}_k
    \hookrightarrow
  \mathrm{dgcAlg}^\bullet_k
$$
for the full subcategory of the dgc-algebras in non-negative degree, hence for the commutative monoids
in $\mathrm{Ch}^{\geq 0}$, and similarly for the augmented case
$$
  (\mathrm{dgcAlg}^{\geq 0}_k)_{/k[0]}
    \hookrightarrow
  (\mathrm{dgcAlg}^{\bullet}_k)_{/k[0]}
  \,.
$$

=--

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

There are [[adjunctions]] between the categories from def. \ref{dgcalgebras} of the form

$$
    \mathrm{dgcAlg}^{\geq 0}_k
     \underoverset
      {\underset{U}{\longrightarrow}}
      {\overset{\mathrm{Sym}}{\longleftarrow}}
      {\bot}
    \mathrm{Ch}^{\geq 0}_k
$$
and
$$
    (\mathrm{dgcAlg}^{\geq 0}_k)_{/k[0]}
     \underoverset
      {\underset{U \circ \mathrm{ker}(\epsilon_{(-)})}{\longrightarrow} }
      {\overset{\mathrm{Sym}_{/k[0]}}{\longleftarrow}}
      {\bot}
    \mathrm{Ch}^{\geq 0}_k
$$

where

1. $U$ forms the underlying chain complex;

1. $\mathrm{ker}(\epsilon_{(-)})$ forms the [[augmentation ideal]];

1. $\mathrm{Sym}$ forms the free graded-commutative dg-algebra;

1. $\mathrm{Sym}_{/k[0]}$ forms the free graded-commutative dg-algebra and regards it with its canonical augmentation over $k$.

Moreover, these are [[Quillen adjunctions]]: the first with respect to the [[projective model structure on chain complexes]] and the [[projective model structure on dgc-algebras]] and the second with respect to the corresponding [[slice model structure]].

=--

## Related concepts

* [[stable Dold-Kan correspondence]]

* [[rational homotopy theory]]

* [[rational parameterized stable homotopy theory]]

* [[rational equivariant homotopy theory]]

* [[rational equivariant stable homotopy theory]]

## References

* {#SchwedeShipley03} [[Stefan Schwede]], [[Brooke Shipley]], _Stable model categories are categories of modules_, Topology 42 (2003), 103-153 ([arXiv:math/0108143](https://arxiv.org/abs/math/0108143), doi:10.1016/S0040-9383(02)00006-X [pdf](http://www.math.uic.edu/~bshipley/classTopFinal.pdf))

[[!redirects stable rational homotopy theory]]


rational thermodynamics

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#### Physics
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[[!include physicscontents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##  Idea

What is called _rational thermodynamics_ is a proposal ([Truesdell 72](#Truesdell72)) to base the [[physics]] of  [[irreversible thermodynamics]] on a system of [[axioms]] and derive the theory from these formally.

The success of the axioms of rational thermodynamics as a theory of physical phenomena has been subject of debate. But the idea as such that [[continuum physics]] can be and should be given a clear [[axiom|axiomatic]] [[foundation]] seems to have inspired [[William Lawvere]] (see there for more), once an undergraduate student of [[Clifford Truesdell]], to base [[continuum mechanics]] on constructions in [[topos theory]], such as [[synthetic differential geometry]] and [[cohesion]].

## References

Original books include

* [[Clifford Truesdell]], _Rational thermodynamics_, Springer-Verlag, Berlin and New York, 1972
 {#Truesdell72}

* [[Clifford Truesdell]], _A first course in rational continuum mechanics_. Pure and Applied Mathematics, Academic Press, New York, San Francisco, London, 1977, xxiii + 280 pp.,  ([review](http://projecteuclid.org/euclid.bams/1183548435))

The appendix of ([Truesdell 72](#Truesdell72)) also appears as

* Bernard Coleman, David Owen, _Recent research on the foundations of thermodynamics_, in [[William Lawvere]], [[Stephen Schanuel]] (eds.), _[[Categories in Continuum Physics]]_, Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174 1986

A survey of approaches to [[irreversible thermodynamics]] is in

* Ivan Vavruch, _Conceptual problems of modern irreversible thermodynamics_, Chem. Listy 96 (2002) ([pdf](http://www.chemicke-listy.cz/docs/full/2002_05_01.pdf))

with a review of rational thermodynamics in section 3.


rational Todd class is Chern character of Thom class

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#Contents#
* table of contents
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## Statement

+-- {: .num_prop #RationalToddClassIsChernCharacterOfThomClass}
###### Proposition
**([[Todd class]] in [[rational cohomology]] is [[Chern character]] of [[Thom class]])**

Let $V \to X$ be a [[complex vector bundle]] over a [[compact topological space]]. Then the [[Todd class]] $Td(V) \,\in\, H^{ev}(X; \mathbb{Q})$ of $V$ in [[rational cohomology]] equals the [[Chern character]] $ch$ of the [[Thom class]] $th(V) \,\in\, K\big( Th(V) \big)$ in the [[complex topological K-theory]] of the [[Thom space]] $Th(V)$, when both are compared via the [[Thom isomorphisms]] $\phi_E \;\colon\; E(X) \overset{\simeq}{\to} E\big( Th(V)\big)$:

$$
  \phi_{H\mathbb{Q}}
  \big(
    Td(V)
  \big)
  \;=\;
  ch\big( th(V) \big)
  \,.
$$

More generally , for $x \in K(X)$ any class, we have

$$
  \phi_{H\mathbb{Q}}
  \big(
    ch(x)
    \cup
    Td(V)
  \big)
  \;=\;
  ch\big( \phi_{K}(x) \big)
  \,,
$$

which specializes to the previous statement for $x = 1$.

=--

([Karoubi 78, Chapter V, Theorem 4.4](#Karoubi78))

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

By the discussion at _[[universal complex orientation on MU]]_ we have:

For $V$ a [[complex vector bundle]] with [[Thom space]] $Th(V)$, its [[Thom class]] in any [[complex-oriented cohomology theory]] $E$ is classified by the composite

$$
  Th(V)
    \longrightarrow
  M U
   \overset{\sigma^E}{\longrightarrow}
  E
  \,,
$$

where $\sigma$ represents the complex orientation as a map of [[homotopy-commutative ring spectra]] on the [[Thom spectrum]] [[MU]].

In this perspective via classifying morphisms of [[ring spectra]], the statement of Prop. \ref{RationalToddClassIsChernCharacterOfThomClass} becomes that the Todd character is the composite of the complex orientation $\sigma^E$ with the [[Chern character]]

$$
  Td
    \;\colon\;
  M \mathrm{U}
    \overset{ \sigma^{KU} }{\longrightarrow}
  KU
    \overset{ ch }{\longrightarrow}
  H^{ev}\mathbb{Q}
$$

In particular, on [[cohomology rings]] $E_\bullet \coloneqq \pi_\bullet(E) \coloneqq  \widetilde E(S^\bullet)$ this composite of ring spectrum maps is the _[[Todd genus]]_ on the [[complex cobordism ring]], factored as

$$
  Td_{2\bullet}
  \;\colon\;
  \Omega^\mathrm{U}_{2\bullet}
  \;=\;
  (M\mathrm{U})_{2\bullet}
    \overset{ }{\longrightarrow}
  KU_{2\bullet}
    \overset{ch_{2\bullet}}{\longrightarrow}
  \mathbb{Q}
  \,.
$$

=--

(see [Smith 73, p. 303 (3 of 10)](#Smith73), following [Conner-Floyd 66, Section 6](#ConnerFloyd66))

## Related theorems

* [[e-invariant is Todd class of cobounding (U,fr)-manifold]]

* [[Grothendieck-Riemann-Roch theorem]]

## References

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section 6 and p. 99 (first line) in:  _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__, Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

Proof is spelled out in:

* {#Karoubi78} [[Max Karoubi]], Theorem V4.4 of: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 ([pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3))

Discussion in terms of [[Brown representability theorem|representing]] [[ring spectra|ring]] [[spectra]]:

* {#Smith73} [[Larry Smith]], Section 1 of: _The Todd character and the integrality theorem for the Chern character_, Illinois J. Math. Volume 17, Issue 2 (1973), 301-310 ([euclid:ijm/1256051760](https://projecteuclid.org/euclid.ijm/1256051760))

* Matthias Jonas Spiegel, Section 2.3.2 of: _K-Theory of Intersection Spaces_, 2013 ([doi:10.11588/heidok.00015738](https://doi.org/10.11588/heidok.00015738))

also

* {#ConnerSmith69} [[Pierre Conner]], [[Larry Smith]], Section 9 of: _On the complex bordism of finite complexes_, Publications Mathématiques de l'IHÉS, Tome 37 (1969) , pp. 117-221 ([numdam:PMIHES_1969__37__117_0](http://www.numdam.org/item/?id=PMIHES_1969__37__117_0))

[[!redirects the rational Todd class is the Chern character of the Thom class]]


rational topological space

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#### Rational homotopy theory
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **rational topological space** is a [[topological space]]
all whose (reduced) [[integral cohomology|integral homology]] groups are [[vector space]]s over the [[rational numbers]] $\mathbb{Q}$.

Every [[simply connected topological space|simply connected]] [[topological space]] has a [[rationalization]]
and passing to that rationalization amounts to forgetting all
[[torsion]] information in the homology groups and the
[[homotopy group]]s of that space. So rational spaces are a
way to approximate homotopical and cohomological characteristics
of topological spaces. The idea is that comparatively little information
(though sometimes crucial information) is lost by passing to
rationalizations, while there are powerful tools to handle and compute
with rational spaces. In particular, there is a precise sense in which
rational spaces are modeled by graded commutative
[[differential graded algebra|differential graded cochain algebra]]s.
This is the topic of [[rational homotopy theory]].

## Definition

A [[topological space]] is called _rational_ if

1. it is [[simply connected topological space|simply connected]] in that the 1st [[homotopy group]] vanishes, $\pi_1 X = 0$  (more generally we may use [[nilpotent topological spaces]] here)

1. and the following equivalent conditions are satisfied

   1. the collection of [[homotopy group]]s form a $\mathbb{Q}$-[[vector space]],

   1. the [[reduced cohomology|reduced homology]] of $X$, $\tilde H_*(X,\mathbb{Z})$ is a $\mathbb{Q}$-[[vector space]],

   1. the [[reduced cohomology|reduced homology]] of the [[loop space]] $\Omega X$ of $X$, $\tilde H_*(\Omega X,\mathbb{Z})$ is a $\mathbb{Q}$-[[vector space]].

A morphism $\ell : X \to Y$ of simply connected [[topological space]] is
called a **[[rationalization]]** of $X$ if $Y$ is a rational topological space and if $\ell$ induces an [[isomorphism]] in rational homology

$$
  H_*(\ell,\mathbb{Q}) : H_*(X,\mathbb{Q}) \stackrel{\simeq}{\to} H_*(Y,\mathbb{Q})
  \,.
$$

Equivlently, $\ell$ is a rationalization of $X$ if it induces an [[isomorphism]]
on the rationalized [[homotopy group]]s, i.e. when the morphism

$$
  \pi_* \ell \otimes \mathbb{Q} : \pi_* X \otimes \mathbb{Q}
   \to \pi_* Y \otimes \mathbb{Q} \simeq \pi_* Y
$$

is an [[isomorphism]].

A continuous map $\phi : X \to Y$ between simply connected space is a
**rational homotopy equivalence** if the following equivalent conditions are satisfied:

1. it induces an isomorphism on rationalized homotopy groups in that
   $\pi_*(\phi) \otimes \mathbb{Q}$ is an isomorphism;

1. it induces an isomorphism on rationalized homology groups in that
   $H_*(\phi,\mathbb{Q})$ is an isomorohism;

1. it induces an isomorphism on rationalized [[cohomology group]]s in that
   $H^*(\phi,\mathbb{Q})$ is an isomorphism;

1. it induces a weak [[homotopy equivalence]] on rationalizations $X_0, Y_0$ in that
   $\phi_0 : X_0 \to Y_0$ is a [[weak homotopy equivalence]].

## Properties

One of the central theorems of [[rational homotopy theory]] says:

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

Rational homotopy types of simply connected spaces $X$ are in bijective corespondence with minimal [[Sullivan model]]s $(\wedge^\bullet V,d)$

$$
  (\wedge^\bullet V , d) \stackrel{\simeq}{\to}
  \Omega^\bullet_{Sullivan}(X)
  \,.
$$

And homotopy classes of morphisms on both sides are in bijection.

=--

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

This appears for instance as corollary 1.26 in

* [[Kathryn Hess]], _Rational homotopy theory: a brief introduction_ ([arXiv](http://arxiv.org/abs/math.AT/0604626))

=--

## Examples

### Rational $n$-sphere

The **[[rational n-sphere]]** $(S^n)_0$ can be written as

$$
  (S^n)_0  :=
  \left(
     \vee_{k\geq 1} S^n_k
  \right)
  \cup
  \left(
    \coprod_{k \geq 2} D^{n+1}_k
  \right)
  \,,
$$

where...

For $n = 2 k +1$ odd, a [[Sullivan model]] for the $n$-sphere is the very simple [[dg-algebra]] with a single generator $c$ in degree $n$ and vanishing differential, i.e. the morphism

$$
  (\wedge^\bullet \langle c\rangle, d = 0)
  \to
  \Omega^\bullet_{Sullivan}(S^{2k + 1})
$$

that picks any representative of the degree $n$-cohomology of $S^{n}$ is a [[quasi-isomorphism]].

For $n = 2k$ with $k \geq 1$ there is a second generator $c_{4k+1}$ with differential

$$
  d c_{2k} = 0
$$
$$
  d c_{4k-1} = c_{2k} \wedge c_{2k}
 \,.
$$

### Rational $n$-disk

The rational $n$-disk is computed as the [[cone]] of the rational sphere:

$$
  D^{n+1}_{\mathbb{Q}} = C(S^n_{\mathbb{Q}}) = (S^n_{\mathbb{Q}}\times I)/(S^n_{\mathbb{Q}}\times\{0\}).
$$

Since the cochain complex of the $n$-disk consists of two generators $b$ and $c$ of degree $n$ and $(n+1)$, respectively, such that $\mathrm{d} b = c$, the [[cdga]] corresponding to the rational $n$-disk consists of linear combinations of $b^k$ and $c b^k$ when $n$ is even, with differentials:

$$
  \mathrm{d}(b^k)
    \;=\;
  k \cdot c b^{k-1}
$$
$$
  \mathrm{d} (c b^k)
  \;=\;
  0
  \,,
$$

and of linear combinations of $c^k b$ and $c^k$ when $n$ is odd, with differentials
$$
  \mathrm{d}(c^k b)=c^{k+1}
$$
$$
  d(c^k)=0
  \,.
$$

### Rational compact Lie-groups

For $G$ a [[compact space|compact]] [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$, let $\{\mu_{k_i}\}_{i=1}^{rank G}$ be generators of its [[Lie algebra cohomology]] with $deg \mu_{k_i} = 2 k_i-1$. Accordingly there are generators $\{P_{k_i}\}_i$ of [[invariant polynomial]]s on $\mathfrak{g}$.

Such $G$ is rationally equivalent to the product

$$
  \prod_{i = 1}^{rank G} S^{2 k_i -1}
$$

of rational $n$-spheres.

Moreover, Lie groups are [[formal homotopy type]]s, whose [[Sullivan model]] has a [[quasi-isomorphism]] to its [[cochain cohomology]].

### Rational classifying spaces of compact Lie groups {#RationalClassifyingSpace}

With $G$ as above, let $\mathcal{B}G$ be the corresponding [[classifying space]].  Then

$$
  H^\bullet(\mathcal{B}G, \mathbb{Q}) \simeq \mathbb{Q}[P_{k_1},  P_{k_2}, \cdots]
  \,,
$$

where $P_{k_i}$ is an [[invariant polynomial]] generator in degre $2 k_i$.

Indeed, also these classifying spaces are [[formal homotopy type]]s and hence a [[Sullivan model]] for $\mathcal{B}G$ is given by $(H^\bullet(\mathcal{B}G,\mathbb{R}), d=0)$.

### Quotient spaces

We may think of $\mathcal{B}G$ as the [[action groupoid]] $*// G$. The above discussion generalizes to more general such quotients.

...

### Biquotient spaces

Let $H$ be a [[compact Lie group]] and $G \subset H \times H$ a [[closed subspace|closed]] [[subgroup]] of the [[direct product group]]. This $G$ acts on $H$ by left and right multiplication

$$
  (g_1, g_2) : h \mapsto g_1 h g_2^{-1}
  \,.
$$

The corresponding [[quotient space]] is also called a _[[biquotient]]_.

...

See

* Vitali Kapovitch, _A note on rational homotopy of biquotients_ ([pdf](http://www.math.utoronto.ca/vtk/biquotient.pdf))

## Related concepts

* [[rational topological G-space]]

[[!redirects rational topological spaces]]

[[!redirects rational space]]
[[!redirects rational spaces]]
[[!redirects rational topological space]]


rational variety

A $n$-dimensional algebraic $k$-[[variety]] $M$ is __rational__ if there is a [[birational map]] $M\to P^n_k$ to the [[projective space]]. A variety is irrational if it is not rational. It is often a difficult question wheather some concrete example of a variety is rational.

[[!redirects rational varieties]]


rational vector space

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#### Linear algebra
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=--
#### Group theory
+-- {: .hide}
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=--
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=--

#Contents#
* table of contents
{:toc}

## Definition

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

Equivalently, a [[torsion-free]] [[divisible group]].

## Related concepts

* [[torsion-free module]], [[divisible group]]

* [[rational algebra]]

* [[vector G-space]]

* [[rational homotopy theory]], [[rational cohomology]]

* [[real vector space]]

* [[complex vector space]]

* [[complexification]]

## References

* Phillip A. Griffith, *Infinite Abelian group theory* Chicago Lectures in Mathematics. University of Chicago Press. (1970) [ISBN 0-226-30870-7]

[[!redirects rational vector spaces]]
[[!redirects torsion-free divisible group]]
[[!redirects torsion-free divisible groups]]


rational vertex operator algebra

A [[vertex operator algebra]] for [[rational 2d CFT]].

[[!redirects rational vertex operator algebras]]


rational zero theorem

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#### Arithmetic
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=--
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition ##

### For polynomial functions ###
The **rational zero theorem** or **rational root theorem** states:

Given a [[natural number]] $n$ and a degree $n$ [[polynomial function]] $f \colon \mathbb{Q} \to \mathbb{Q}$ on the [[rational numbers]] with [[integers]] valued [[coefficients]] $a \colon [0,n] \to \mathbb{Z} \hookrightarrow \mathbb{Q}$ and $a_n \neq 0$, defined as

$$
  f(x) \coloneqq \sum_{i = 0}^{n} a_i \, x^i
  \,,
$$

then the [[fiber]] of $f$ at $0$ is [[inhabited set|inhabited]] if one of the following is true:

* $a_0 = 0$,

* $a_0 \neq 0$ and there exists integers $m$ and $p$ such that $gcd(\vert m \vert, \vert p \vert) = 1$, $m \vert a_0$, and $p \vert a_n$.

### For polynomials ###
The **rational root theorem for polynomials** states:

Given a natural number $n$ and a degree $n$ univariate [[polynomial]] on the [[rational numbers]] $a:\mathbb{Q}[x]$ where $a_{n} = 1$, there exists a degree $1$ univariate polynomial $b:\mathbb{Q}[x]$ where $b_{1} = 1$ such that $b | a$ if and only if there exists an integer $m$ such that $gcd(\vert m \vert, \vert b_0 \vert) = 1$ and $m \cdot a_0 = b_0$.

## See also ##

* [[Gauss's lemma for polynomials]]
* [[polynomial function]]
* [[rational numbers]]
* [[integers]]

## References ##

* Wikipedia, _[Rational zero theorem](https://en.wikipedia.org/wiki/Rational_zero_theorem)_

[[!redirects rational root theorem]]


rationalization

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#### Rational homotopy theory
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=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

In [[rational homotopy theory]] one considers [[topological spaces]] $X$ only up to maps that induce [[isomorphisms]] on _rationalized_ [[homotopy groups]] $\pi_\bullet(X) \otimes_{\mathbb{Z}} \mathbb{Q}$ (as opposed to genuine [[weak homotopy equivalences]], which are those maps that induce [[isomorphism]] on the genuine [[homotopy groups]].)

Every simply connected space is in this sense equivalent to a [[rational space]]: this is its **rationalization**.

Similarly one may consider "real-ification" by considering $\pi_\bullet(X) \otimes_{\mathbb{Z}} \mathbb{R}$, etc.

## Definition
 {#Definition}

### Rationalization of a homotopy type
 {#RationalizationOfAHomotopyType}

We discuss here what it means for a map of [[homotopy types]] to exhibit the rationalization of its [[domain]].

A widely appreciated construction applies in the special case that the [[domain]] is a [[simply connected homotopy type]] (in fact, more generally in the case that it is a [[nilpotent homotopy type]]).
A slight enhancement of this construction, which is generally much less widely considered, applies to all [[connected homotopy types]]: this forms the rationalization of the [[universal cover]] (which is of course simply connected) but retains on this the information of the [[infinity-action|$\infty$-actions]] of the [[fundamental groups]] by [[deck transformations]] (see also at *[[Borel-equivariant rational homotopy theory]]*):

1. [Rationalization of simply-connected homotopy types](#RationalizationOfSimplyConnectedSpaces)

1. [$\pi_1$-Rationalization of connected homotopy types](#RationalizationOfConnectedSpaces)

#### Rationalization of simply-connected spaces
 {#RationalizationOfSimplyConnectedSpaces}

A **rationalization** of a [[simply connected space|simply connected]] [[topological space]] $X$ is a [[continuous function]] $\phi \colon X \to Y$, where

* $Y$ is a [[simply connected space|simply connected]] [[rational space]];

* $\phi$ induces an [[isomorphism]] on rationalized [[homotopy groups]]:

  $$
    \pi_\bullet(\phi)\otimes \mathbb{Q}
    \;\colon\;
    \pi_\bullet(X) \otimes \mathbb{Q}
    \stackrel{\simeq}{\longrightarrow}
    \pi_\bullet(Y) \otimes \mathbb{Q}
  $$

  or equivalently if $\phi$ induces an isomorphism on [[rational cohomology]] groups

  $$
    H^\bullet(\phi,\mathbb{Q})
    \;\colon\;
    H^\bullet(X,\mathbb{Q})
    \stackrel{\simeq}{\longrightarrow}
    H^\bullet(Y,\mathbb{Q})
    \,.
  $$

  or equivalently if $\phi$ induces an isomorphism on rational [[homology]] groups

  $$
    H_\bullet(\phi,\mathbb{Q})
    \;\colon\;
    H_\bullet(X,\mathbb{Q})
    \stackrel{\simeq}{\longrightarrow}
    H_\bullet(Y,\mathbb{Q})
    \,.
  $$

([Bousfield-Kan 72, p. 133-140](#BousfieldKan72), [Bousfield-Gugenheim 76, 11.1](#BousfieldGugenheim76), [Hess 06, Def. 1.4 with Def. 1.7](#Hess06))

(notice here that $\mathbb{Z}$ is a *[[solid ring]]*, in that $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q}\,\simeq\, \mathbb{Q}$, e.g. [here](http://www-users.math.umn.edu/~garrett/m/algebra/exercises/s19.pdf#page=2))

\begin{proposition}\label{RationalizationViaCompletionOfComponentsOfLoopGroup}
**(rationalization via $\mathbb{Q}$-completion of simplicial loop group)**
\linebreak
  The rationalization of a [[simply connected homotopy type]] is represented (say via the [[classical model structure on simplicial sets]]) by a [[reduced simplicial set]] $S \,\in\, sSet_\ast$ is given by
$$
  S_{\mathbb{Q}}
  \;\simeq\;
  \overline{W}(G S)_{\widehat{\mathbb{Q}}}
  \;\;\;
  \in
  sSet
  \xrightarrow{ L^W }
  Grp_\infty
  \,,
$$
where:

1. $G \;\colon\; sSet_\ast \xrightarrow{\;} sGrp$ is the [[simplicial loop space]]-functor to [[simplicial groups]];

1. $(-)_{\widehat{\mathbb{Q}}} \;\colon\; sGrp \to sGrp$ is degreewise the $\mathbb{Q}$-[[completion of a group|completion]] ([hence](completion+of+a+group#ExampleMalcevCompletion) [[Malcev completion]]) applied to the component group of a simplicial group;

1. $\overline{W} \;\colon\; sGrp \xrightarrow{\;} sSet$ is the [[simplicial delooping]]-operation.

\end{proposition}

([Bousfield & Kan 1971, §3](#BousfieldKan71), [Bousfield & Kan 1972, IV Prop. 4.1 (p. 109)](#BousfieldKan72), see also [Rivera, Wierstra & Zeinalian 2021, p. 7](#RiveraWierstraZeinalian21))

#### $\pi_1$-Rationalization of connected spaces
 {#PiOneRationalizationOfConnectedSpaces}

For [[connected homotopy types]] $X$ which are not necessarily [[simply connected homotopy type|simply connected]], consider their [[universal cover]] $\widehat{X}$, which sits in a [[homotopy fiber sequence]]

\[
  \label{HomotopyFiberSequenceForUniversalCover}
  \array{
    \widehat{X}
    &\longrightarrow&
    X
    \\
    \big\downarrow
    &&
    \big\downarrow
    \mathrlap{ {}^{ \eta^{[-]_1}_X } }
    \\
    \ast
    &\longrightarrow&
    B \pi_1(X)
  }
\]

over the [[delooping]]/[[classifying space]] of the [[fundamental group]] (via the [[n-truncated object of an (infinity,1)-category|1-truncation]] [[adjunction unit|unit]]).

(Notice that this fiber sequence exhibits the [[infinity-action|$\infty$-action]] of $\pi_1(X)$ on $\widehat{X}$.)

Here the [[universal cover]] $\widehat{X}$ is [[simply connected homotopy type|simply connected]] (hence in particular [[nilpotent homotopy type|nilpotent]]), so that the [above](#RationalizationOfSimplyConnectedSpaces) notion of rationalization applies to this [[fiber]]:

\begin{definition}\label{PiOneRationalization}
**($\pi_1$-rationalization)**
\linebreak
  A map $\eta^{\mathbb{Q}}_X \;\colon\; X \longrightarrow X_{\mathbb{Q}}$ of [[connected homotopy types]] is called a *$\pi_1$-rationalization* if

1. the higher [[homotopy group]] $\pi_n\big(X_{\mathbb{Q}}\big)$ are [[rational vector spaces]];

1. $\eta^{\mathbb{Q}}_X$ induces [[isomorphisms]]

   1. on the [[fundamental groups]]

      $
        \pi_1\big(\eta^{\mathbb{Q}}_{X}\big)
        \;\colon\;
        \pi_1(X)
        \xrightarrow{\;\; \sim \;\;}
        \pi_1
        \big(
          X_{\mathbb{Q}}
        \big)
      $

   1. on the the rationalization of all higher [[homotopy groups]] $\pi_n$ for $n \geq 2$:

      $
       \pi_n\big(\eta^{\mathbb{Q}}_{X}\big)
       \otimes_{\mathbb{Z}} \mathbb{Q}
       \;\colon\;
       \pi_1(X)
       \otimes_{\mathbb{Z}} \mathbb{Q}
       \xrightarrow{\;\; \sim \;\;}
       \pi_1
       \big(
         X_{\mathbb{Q}}
       \big)
       \,.
      $

\end{definition}

In generalization of Prop. \ref{RationalizationViaCompletionOfComponentsOfLoopGroup}:

\begin{proposition}
  The rationalization (Def. \ref{PiOneRationalization})
of a [[connected homotopy type]] is represented (say via the [[classical model structure on simplicial sets]]) by a [[reduced simplicial set]] $S \,\in\, sSet_\ast$ is given by
$$
  S_{\mathbb{Q}}
  \;\simeq\;
  \overline{W}(G S)_{\widehat{\mathbb{Q}/\pi}}
  \;\;\;
  \in
  sSet
  \xrightarrow{ L^W }
  Grp_\infty
  \,,
$$
where:

1. $G \;\colon\; sSet_\ast \xrightarrow{\;} sGrp$ is the [[simplicial loop space]]-functor to [[simplicial groups]];

1. $(-)_{\widehat{\mathbb{Q}/\pi}}$ is in degree $n$ the [fiberwise rationalization of](completion+of+a+group#FiberwiseCompletionOfAGroup) of the [[short exact sequences]] of  [[homotopy groups]] (obtained in the present case from the [[long exact sequence of homotopy groups]] of the [[fiber sequence]] (eq:HomotopyFiberSequenceForUniversalCover))

   $$
     1
     \to
     \pi_n
     \big(
       \widehat{X}
     \big)
     \longrightarrow
     \pi_n
     \big(
       X
     \big)
     \longrightarrow
     \pi_n
     \big(
       B \pi_1(X)
     \big)
     \to
     1
   $$

   (in the present case this is trivial in degree 1 and is the plain rationalization in higher degrees, but this formulation makes manifest that both cases are functorially compatible);

1. $\overline{W} \;\colon\; sGrp \xrightarrow{\;} sSet$ is the [[simplicial delooping]]-operation.

\end{proposition}

([Bousfield & Kan 1971, §3, p. 1008](#BousfieldKan71), see also [Rivera, Wierstra & Zeinalian 2021, p. 8](#RiveraWierstraZeinalian21) and [Ivanov 2021](#Ivanov21)).

\linebreak

### Rationalization as a localization of $Top$/$\infty Grpd$

In [[rational homotopy theory]] one considers
the [[PL de Rham complex|PL de Rham]] [[Quillen adjunction]]

$$
  (\Omega^\bullet
   \dashv K)
  \;\colon\;
  dgAlg_{\mathbb{Q}}
   \stackrel{\overset{\Omega^\bullet}{\leftarrow}}{\underset{K}{\to}}
  sSet
$$

between the [[model structure on dg-algebras]] and the standard [[model structure on simplicial sets]], where $\Omega^\bullet$ is forming [[Sullivan differential forms]]:

$$
  \Omega^\bullet(X) = Hom_{sSet}(X, \Omega^\bullet_{pl}(\Delta^\bullet_{Diff}))
  \,.
$$

The [[fundamental theorem of dg-algebraic rational homotopy theory]] says that on [[nilpotent spaces]] with [[finite type]] [[rational cohomology]] this induces an [[equivalence of categories|equivalence]] of [[homotopy categories]].

Intrinsically this should model something like the (partially) left exact [[localization of an (∞,1)-category]] of [[∞Grpd]] at those morphisms that are [[rational homotopy equivalences]].

$$
  \infty Grpd_{ratio} \stackrel{\leftarrow}{\hookrightarrow}
  \infty Grpd
  \,.
$$

Below we review classical results that says that the left [[adjoint (infinity,1)-functor]] here indeed preserves at least [[homotopy pullback]]s.

More generally, a setup by [[Bertrand Toen]] serves to provide a more comprehensive description of this situtation: see [[rational homotopy theory in an (infinity,1)-topos]].

### In homotopy type theory

In [[homotopy type theory]] and [[synthetic homotopy theory]], one considers the **rationalization** of a [[pointed type|pointed]] [[n-connected type|1-connected]] [[type]].

## Properties
  {#Properties}

### Rationalization via PL de Rham theory

+-- {: .num_defn #NilpotententFiniteRationalHomotopyTypes}
###### Definition
**(nilpotent and finite rational homotopy types)**

Write

\[
  \label{NilpotentQFiniteHomotopyTypes}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)
\]

for the [[full subcategory]] of the [[classical homotopy category]]  ([[homotopy category]] of the [[classical model structure on simplicial sets]]) on those [[homotopy types]] $X$ which are

* [[connected topological space|connected]]: $\pi_0(X) = \ast$

* [[nilpotent space|nilpotent]]: $\pi_1(X)$ is a [[nilpotent group]]

* rational [[finite type]]: $dim_{\mathbb{Q}}\big( H^n(X;,\mathbb{Q}) \big) \lt \infty$ for all $n \in \mathbb{N}$.

and

\[
  \label{NilpotentFiniteRationalHomotopyTypes}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)
\]

for the futher [[full subcategory]] on those [[homotopy types]] that are already [[rational spaces|rational]].

Similarly, write

\[
  \label{NilpotentFiniteTypedgcAlgebras}
  Ho
  \big(
     DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}}
  \big)_{fin}^{\geq 1}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
     DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}}
  \big)
\]

for the [[full subcategory]] of the [[homotopy category]] of the [[projective model structure on connective dgc-algebras]] on those [[dgc-algebras]] $A$ which are

* connected: $H^0(A) \simeq \mathbb{Q}$

* [[finite type]]: $dim_{\mathbb{Q}}\big( H^n(A) \big) \lt \infty$ for all $n \in \mathbb{N}$.

=--

([Bousfield-Gugenheim 76, 9.2](#BousfieldGugenheim76))

+-- {: .num_pro #FundamentalTheoremOfdgAlgebraicRationalHomotopyTheory}
###### Proposition
**([[fundamental theorem of dg-algebraic rational homotopy theory]])**

The [[derived adjunction]]

$$
  Ho
  \left(
    \big(
      DiffGradedCommAlgebras^{\geq 0}_{k}
    \big)^{op}_{proj}
  \right)
  \underoverset
    {
      \underset
        {\;\;\; \mathbb{R} exp \;\;\;}
        {\longrightarrow}
    }
    {
      \overset
        {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;}
        {\longleftarrow}
    }
    {\bot}
  Ho
  \big(
    HoSimplicialSets_{Qu}
  \big)
$$

of the [[Quillen adjunction between simplicial sets and connective dgc-algebras]] (whose [[left adjoint]] is the [[PL de Rham complex]]-[[functor]]) has the following properties:

* on connected, nilpotent rationally finite homotopy types $X$ (eq:NilpotentQFiniteHomotopyTypes) the [[derived adjunction unit]] is [[rationalization]]

  $$
    \array{
    Ho
    \big(
       SimplicialSets_{Qu}
    \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil}
    &
      \overset{
      }{\longrightarrow}
    &
    Ho
    \big(
       SimplicialSets_{Qu}
    \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}
    \\
    X
    &\mapsto&
    \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X)
    }
  $$

  $$
    X
    \underoverset
      {\eta_X^{der}}
      {rationalization}
      {\longrightarrow}
    \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X)
  $$

* on the [[full subcategories]] of nilpotent and finite rational homotopy types from Def. \ref{NilpotententFiniteRationalHomotopyTypes} it restricts to an [[equivalence of categories]]:

  $$
    Ho
    \left(
      \big(
        DiffGradedCommAlgebras^{\geq 0}_{k}
      \big)^{op}_{proj}
    \right)^{\geq 1}_{fin}
    \underoverset
      {
        \underset
          {\;\;\; \mathbb{R} exp \;\;\;}
          {\longrightarrow}
      }
      {
        \overset
          {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;}
          {\longleftarrow}
      }
      {\simeq}
    Ho
    \big(
      HoSimplicialSets_{Qu}
    \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}
  $$

=--

([Bousfield-Gugenheim 76, Theorems 9.4 & 11.2](#BousfieldGugenheim76))

### Preservation of homotopy pullbacks

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

The left [[derived functor]] of the [[Quillen functor|Quillen left adjoint]] $\Omega^\bullet : sSet \to dgAlg_{\mathbb{Q}}$ preserves [[homotopy pullbacks]] of objects of [[finite type]] (each rational homotopy group is a [[finite dimensional vector space]] over the [[ground field]]).

In other words in the induced pair of [[adjoint (∞,1)-functors]]

$$
  (\Omega^\bullet \dashv K)
   :
  (dgAlg_\mathbb{Q}^{op})^\circ
  \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}}
  \infty Grpd
$$

the left adjoint preserves [[limit in a quasi-category|(∞,1)-categorical pullbacks]] of objects of finite type.

=--

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

This is effectively a restatement of a result that appears below proposition 15.8 in HalperinThomas and is reproduced in some repackaged form as theorem 2.2 of [He06](http://www.math.uic.edu/~bshipley/hess_ratlhtpy.pdf). We recall the [[model category|model category-theoretic]] context that allows to rephrase this result in the above form.

Let $C = \{a \to c \leftarrow b\}$ be the pullback [[diagram]] category.

The [[homotopy limit]] functor is the right [[derived functor]] $\mathbb{R} lim_C$ for the [[Quillen adjunction]] (described in detail at [[homotopy Kan extension]])

$$
  [C,sSet]_{inj}
   \underoverset
    {\underset{lim_C}{\longrightarrow}}
    {\overset{const}{\longleftarrow}}
    {\;\;\; \bot \;\;\;}
  sSet
  \,.
$$

At [[model structure on functors]] it is discussed that composition with the Quillen pair $\Omega^\bullet \dashv K$ induces a Quillen adjunction

$$
  ([C,\Omega^\bullet] \dashv [C,K])
  \;\colon\;
  [C, dgAlg^{op}]
    \underoverset
      {\underset{[C,K]}{\longrightarrow}}
      {\overset{[C,\Omega^\bullet]}{\longleftarrow}}
      {\;\; \bot \;\;}
  [C,sSet]
  \,.
$$

We need to show that for every fibrant and cofibrant pullback diagram $F \in [C,sSet]$ there exists a [[weak equivalence]]

$$
  \Omega^\bullet \circ lim_C F
  \;\;
  \simeq
  \;\;
  lim_C   \widehat{\Omega^\bullet(F)}
  \,,
$$

here $\widehat{\Omega^\bullet(F)}$ is a [[fibrant replacement]] of $\Omega^\bullet(F)$ in $dgAlg^{op}$.

Now, every object $f \in [C,sSet]_{inj}$ is cofibrant, and it is fibrant if all three objects $F(a)$, $F(b)$ and $F(c)$ are fibrant and one of the two morphisms is a fibration. We may assume without restriction of generality that it is the morphism $F(a) \to F(c)$ that is a fibration. So we assume that $F(a), F(b)$ and $F(c)$ are three [[Kan complexes]] and that $F(a) \to F(b)$ is a [[Kan fibration]]. Then $lim_C$ sends $F$ to the ordinary [[pullback]]  $lim_C F = F(a) \times_{F(c)} F(b)$ in $sSet$, and so the left hand side of the above equivalence is

$$
  \Omega^\bullet(F(a) \times_{F(c)} F(b))
  \,.
$$

Recall that the [[Sullivan algebra]]s are the cofibrant objects in $dgAlg$, hence the fibrant objects of $dgAlg^{op}$. Therefore a fibrant replacement of $\Omega^\bullet(F)$ may be obtained by

* first choosing a [[Sullivan model]] $(\wedge^\bullet V, d_V) \stackrel{\simeq}{\to} \Omega^\bullet(c)$

* then choosing factorizations in $dgAlg$ of the composites of this with $\Omega^\bullet(F(c)) \to \Omega^\bullet(F(a)) $ and $\Omega^\bullet(F(c)) \to \Omega^\bullet(F(b))$ into cofibrations follows by weak equivalences.

The result is a diagram

$$
  \array{
    (\wedge^\bullet U^*, d_U)
    &\leftarrow&
    (\wedge^\bullet V^*, d_V)
    &\hookrightarrow&
    (\wedge^\bullet W^* , d_W)
    \\
    \downarrow^{\simeq}
    &&
    \downarrow^{\simeq}
    &&
    \downarrow^{\simeq}
    \\
    \Omega^\bullet(F(a))
    &\stackrel{}{\leftarrow}&
    \Omega^\bullet(F(c))
    &\stackrel{}{\to}&
    \Omega^\bullet(F(b))
  }
$$

that in $dgAlg^{op}$ exhibits a fibrant replacement of $\Omega^\bullet(F)$. The limit over that in $dgAlg^{op}$ is the colimit

$$
  (\wedge^\bullet U^* , d_U)
  \otimes_{(\wedge^\bullet V^* , d_V)}
  (\wedge^\bullet W^* , d_W)
$$

in $dgAlg$. So the statement to be proven is that there exists a weak equivalence

$$
  (\wedge^\bullet U^* , d_U)
  \otimes_{(\wedge^\bullet V^* , d_V)}
  (\wedge^\bullet W^* , d_W)
  \simeq
  \Omega^\bullet(F(a) \times_{F(c)} F(b))
  \,.
$$

This is precisely the statement of that quoted result [He, theorem 2.2](http://www.math.uic.edu/~bshipley/hess_ratlhtpy.pdf).

=--

> check the following

+-- {: .num_cor}
###### Corollary

Rationalization preserves homotopy pullbacks of objects of finite type.

=--

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

The theory of [[Sullivan model]]s asserts that rationalization of a space $X$ (a simplicial set $X$) is the derived unit of the derived adjunction $(\Omega^\bullet \dashv K)$, namely that the rationalization is modeled by $K$ applied to a Sullivan model $(\wedge^\bullet V^*, d)$ for $\Omega^\bullet(X)$.

$$
  X \to K \Omega^\bullet(X) \stackrel{\simeq}{\leftarrow} K \widehat {\Omega^\bullet(X)} := K (\wedge^\bullet V^* , d_V)
  \,.
$$

Being a Quillen right adjoint, the right derived functor of $K$ of course preserves homotopy limits. Hence the composite $K \circ \widehat{\Omega^\bullet(-)}$ preserves homotopy pullbacks between objects of finite type.

=--

### Preservation of homotopy fibers

See at _[[rational fibration lemma]]_.

### Rationalization of spectra
 {#RationalizationOfSpectra}

On [[spectra]], rationalization is a [[smashing localization]], given by [[smash product]] with the [[Eilenberg-MacLane spectrum]] $H \mathbb{Q}$. (e.g. [Bauer 11, example 1.7 (4)](#Bauer11)).

For more see at _[[rational stable homotopy theory]]_.

## Related concepts

* [[p-localization]], [[p-completion]]

* [[equivariant rationalization]]

* [[fracture square]]

* [[rational homotopy theory]]

  * [[Borel-equivariant rational homotopy theory]]

* [[fundamental theorem of dg-algebraic rational homotopy theory]]

* [[rational equivariant homotopy theory]]

* [[rational equivariant stable homotopy theory]]

## References

Classical accounts:

* {#BousfieldKan71} [[Aldridge Bousfield]], [[Daniel Kan]], *Localization and completion in homotopy theory*, Bull. Amer. Math. Soc. **77** 6 (1971) 1006-1010 [[doi:10.1090/S0002-9904-1971-12837-9](https://doi.org/10.1090/S0002-9904-1971-12837-9), [pdf](https://www.ams.org/journals/bull/1971-77-06/S0002-9904-1971-12837-9/S0002-9904-1971-12837-9.pdf)]

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], p. 133-140 in: _[[Homotopy Limits, Completions and Localizations]]_, Lecture Notes in Mathematics Vol. 304, Springer 1972 ([doi:10.1007/978-3-540-38117-4](https://doi.org/10.1007/978-3-540-38117-4))

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], Def. 11.1 in: _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))

Review:

* {#Hess06} [[Kathryn Hess]], Def. 1.7 in: _Rational homotopy theory: a brief introduction_, contribution to _[Summer School on Interactions between Homotopy Theory and Algebra](https://jdc.math.uwo.ca/summerschool/)_, University of Chicago, July 26-August 6, 2004, Chicago ([arXiv:math.AT/0604626](http://arxiv.org/abs/math.AT/0604626)), chapter in Luchezar Lavramov, [[Dan Christensen]], [[William Dwyer]], [[Michael Mandell]], [[Brooke Shipley]] (eds.), _Interactions between Homotopy Theory and Algebra_, Contemporary Mathematics 436, AMS 2007 ([doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436))

* {#Bauer11} [[Tilman Bauer]], _Bousfield localization and the Hasse square_ (2011) ([pdf](https://math.mit.edu/events/talbot/2007/tmfproc/Chapter09/Bousfield.pdf), [[Bauer_BousfieldLocalization.pdf:file]]), chapter 6 in:  [[Christopher Douglas]], [[John Francis]], [[André Henriques]], [[Michael Hill]] (eds.), _Topological Modular Forms_, Mathematical Surveys and Monographs Volume 201, AMS 2014  ([ISBN:978-1-4704-1884-7](https://bookstore.ams.org/surv-201))

* [[Tyler Lawson]], Example 8.12 in: *An introduction to Bousfield localization* ([arXiv:2002.03888](https://arxiv.org/abs/2002.03888)) in: [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds.), *[[Stable categories and structured ring spectra]]*  MSRI Book Series, Cambridge University Press

Review and further developments:

* {#RiveraWierstraZeinalian21} [[Manuel Rivera]], [[Felix Wierstra]], [[Mahmoud Zeinalian]], *Rational homotopy equivalences and singular chains*, Algebr. Geom. Topol. **21** (2021) 1535-1552 [[arXiv:1906.03655](https://arxiv.org/abs/1906.03655), [doi:10.2140/agt.2021.21.1535](https://doi.org/10.2140/agt.2021.21.1535)]

* {#Ivanov21} [[Sergei O. Ivanov]], *An overview of rationalization theories of non-simply connected spaces and non-nilpotent groups* [[arXiv:2111.10694](https://arxiv.org/abs/2111.10694)]

[[!redirects rationalizations]]

[[!redirects rationalisation]]
[[!redirects rationalisations]]


Raul Gomez

I´m a H.C. Wang Assistant Professor at the mathematics department of Cornell University. My research interest is on representation theory of reductive algebraic groups and its applications to algebraic geometry, harmonic analysis and number theory.


Ravenel's conjectures

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[Ravenel's spectra]]

* [[nilpotence theorem]]

## References

Due to

* {#Ravenel84} [[Douglas Ravenel]], Section 10 of: _Localization with Respect to Certain Periodic Homology Theories_, American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 ([doi:10.2307/2374308](https://doi.org/10.2307/2374308), [jstor:2374308](https://www.jstor.org/stable/2374308))

See also:

* Wikipedia, _[Ravenel's conjectures](https://en.wikipedia.org/wiki/Ravenel%27s_conjectures)_

[[!redirects Ravenel conjectures]]


Ravenel's spectrum

> under construction

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

#Contents#
* table of contents
{:toc}

## Idea

Due to [[Bott periodicity]], the [[coprojection]] $SU(n) \hookrightarrow \mathrm{U}$ of any [[special unitary group]] $SU(n)$ into the [[stable unitary group]] induces a map $\Omega SU(n) \longrightarrow B \mathrm{U} \to B\mathrm{O}$ from the [[based loop space]] of $SU(n)$ to the [[classifying space]] of $\mathrm{U}$ and hence of $\mathrm{O}$. This may be regarded as defining a notion of $\Omega^2 SU(n)$-[[tangential structure]].

The corresponding [[Thom spectra]] $M \big(\Omega^2 SU(n)\big)$ were denoted "$X(n)$" in [Ravenel 84, Section 3](#Ravenel84), used there for analysis of the [[Adams spectral sequence]] (see also [Ravenel 86, Section 6.5](#Ravenel86)) and influential on [[Ravenel's conjectures]] (notably the [[nilpotence theorem]]); and thus have come to be known as _Ravenel's spectra_.

These spectra turn out to be finite-rank analogs of [[MU]] in [[complex oriented cohomology theory]] as one passes from full complex orientation to complex orientation up to rank ("degree") $n$ ([Hopkins 84, Section 1.2](#Hopkins84)). For instance, just as [[MU]] is [[p-local spectrum|p-locally]] a [[wedge sum]] of [[suspensions]] of [[BP]], so Ravenel's spectra are $p$-locally wedge sums of suspensions of spectra that Ravenel denoted $T(k)$.

## Definition

For $n \in \mathbb{N}$, $n \geq 1$, consider the [[composition|composite]] [[morphism]] of [[homotopy types]]

$$
  B \Omega^2 SU(n)
  \simeq
  \Omega SU(n)
  \simeq
  \Omega^2
  B SU(n)
  \overset{
    \Omega^2 B i
  }{
   \longrightarrow
  }
  \Omega^2 B U
  \underoverset
    {\simeq}
    {\beta}
    {\longrightarrow}
  B U
  \longrightarrow
  B O
  \,,
$$

where

1. On the left we have [[looping and delooping]] [[weak homotopy equivalence|equivalences]] ([this Prop.](May+recognition+theorem#RecognitionForGroupsInAnInfinityTopos)), using that the [[based loop space]] $\Omega SU(n)$ is [[connected homotopy type|connected]];

1. $SU(n) \overset{i}{\longrightarrow} SU \coloneqq \underset{\underset{k}{\longrightarrow}} SU(k)$ is the [[coprojection]] of [[SU(n)]] into the [[special unitary group|special]] [[stable unitary group]]:

1. $\beta$ is the [[Bott periodicity]] [[weak homotopy equivalence|equivalence]].

Regarded as a universal [[tangential structure]], this induces the corresponding [[Thom spectrum]] $M\big(\Omega^2 SU(n)\big)$ (introduced as "$X(n)$" in [Ravenel 84, Section 3](#Ravenel84)).

(...)

## Properties

(...)

This carries the finite-rank analog of the [[universal complex orientation of MU]]: [Hopkins 84, Prop. 1.2.1](#Hopkins84)

(...)

## Examples

For $n=1$ we have that $SU(1) = 1$ is the [[trivial group]], so that Ravenel's spectrum at this stage is the [[sphere spectrum]]

$$
  M \Omega^2 SU(1)
  \simeq
  M 1
  \simeq
  \mathbb{S}
$$

On the other hand, the [[colimit]] of Ravenel's spectra as $n \to \infty$  is [[MU]], essentially by construction:

$$
  \underset{
     \underset{n}{\longrightarrow}
   }{lim}
   M \Omega^2 SU(n)
   \simeq
   M \Omega^2 SU
   \simeq
   M U
   \,.
$$

Hence the [[tower]] of Ravenel's spectra interpolates between the [[sphere spectrum]] and [[MU]]

$$
  \mathbb{S}
  \longrightarrow
  M \Omega^2 SU(2)
  \longrightarrow
  M \Omega^2 SU(3)
  \longrightarrow
  \cdots
  \longrightarrow
  M U
  \,.
$$

Accordingly, the [[Brown representability theorem|corresponding]] [[tower]] of [[Whitehead generalized cohomology theories]] interpolated between [[stable Cohomotopy]] and complex [[cobordism cohomology]].

## Related concepts

[[!include flavours of cobordism cohomology theories -- table]]

## References

[[!include finite-dimensional complex orientation and Ravenel's spectra -- references]]

[[!redirects Ravenel's spectra]]

[[!redirects MΩΩSU(n)]]
[[!redirects MOmegaSUn]]

[[!redirects Ravenel spectrum]]
[[!redirects Ravenel spectra]]


Ravi Vakil

* [webpage](http://math.stanford.edu/~vakil/)

## writings

* _Foundations Of Algebraic Geometry_, Course notes ([web](http://math.stanford.edu/~vakil/216blog/))

## related $n$Lab entries

* [[algebraic geometry]]

category: people


Ray Otsuki

* [inspire page](https://inspirehep.net/authors/1755618)

## Selected writings

On [[exotic branes]] in [[exceptional field theory]]:

* {#BermanMusaevOtsuki18} [[David S. Berman]], [[Edvard T. Musaev]], [[Ray Otsuki]], *Exotic Branes in Exceptional Field Theory: $E_{7(7)}$ and Beyond*,  J. High Energ. Phys. **2018** 53 (2018) [[arXiv:1806.00430v4](https://arxiv.org/abs/1806.00430), doi:10.1007/JHEP12(2018)053]

* {#BMO19} [[David Berman]], [[Edvard Musaev]], [[Ray Otsuki]], _Exotic Branes in M-Theory_, contribution to: *[Dualities and Generalized Geometries, Corfu Summer Institute 2018](http://www.physics.ntua.gr/corfu2018/dg.html)* [[arXiv:1903.10247](https://arxiv.org/abs/1903.10247)]

category: people


Ray Vanstone

* [website]() ?

category: people


Raymond A. Ryan

* [institute page](https://www.nuigalway.ie/our-research/people/mathematics-statistics-and-applied-mathematics/rayryan/)

## Selected writings

On [[tensor product of Banach spaces]]:

* {#Ryan02} [[Raymond A. Ryan]],  _Introduction to Tensor Products of Banach Spaces_.  Springer Monographs in Mathematics. Springer-Verlag London. 2002.   ([doi:10.1007/978-1-4471-3903-4](https://www.springer.com/gp/book/9781852334376), [toc pdf](https://d-nb.info/962673145/04))

category: people

[[!redirects Raymond Ryan]]


Raymond Brock

* [institute page](https://msu.edu/honoredfaculty/directory/brock-l-raymond.html)

## Selected writings

On modern fundamental [[physics]] (such as on [[quarks]], [[spacetime]] and the [[big bang]]):

* [[Raymond Brock]], *Quarks, Spacetime, and the Big Bang* lecture notes [full text:[pdf](https://web.pa.msu.edu/people/brock/file_sharing/QSandBB/QS&BB_manuscript/QS&BB.pdf)]

## Related entries

* [[flux]], [[flux tube]], [[Michael Faraday]]

category: people


Raymond D. Bradley

* [institute page](http://www.sfu.ca/philosophy/people/faculty/profiles/ray-bradley.html)

* [In Memoriam: Raymond D. Bradley, Professor Emeritus (1930 - 2022)](https://www.sfu.ca/philosophy/events/news/2022/obituary--raymond-bradley.html)

## Selected writings

On [[modal logic]] and its [[possible worlds semantics]]:

* {#BradleySwartz79} [[Raymond D. Bradley]], [[Norman Swartz]], *Possible Worlds -- An Introduction to Logic and its Philosophy*, Hackett Publishing (1979) [[webpage](http://www.sfu.ca/~swartz/pw/), [pdf](https://www.sfu.ca/~swartz/pw/text/pw_all.pdf)]

category: people

[[!redirects Raymond Bradley]]