order-type

#Contents#
* Contents
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## Idea

The order type of a [[partially ordered set]] (poset) $P$ is an invariant $ot(P)$ of the poset which is finer than its [[cardinal number]] $|P|$. The cardinal numbers classify sets according to how many elements they contain, however they don't pay attention to any potential order-theoretic properties on the sets in question -- for example

\begin{centre}
$\omega+1\cong\omega$
\end{centre}

in **Set** but not in **Pos** since the bijections in **Set** all send $\omega$ to some finite ordinal and thusly don't preserve ordering. This is expressed in terms of cardinalities and order types by observing that $|\omega+1|=|\omega|=\omega$ but $ot(\omega+1)=\omega+1\neq\omega=ot(\omega)$ since each ordinal is its own order type but generally not its own cardinal number.

##Definition

Let $P$ be a poset. We say that another poset $Q$ has the same **order type** as $P$ iff there exists an order preserving bijection $P\cong Q$. We define an equivalence relation $\sim$ on posets by

\begin{centre}
$P\sim Q\iff \exists f:P\cong Q(\text{f is an order-preserving bijection})$,
\end{centre}

and we define the **order type** of $P$, denoted $ot(P)$, to be either the equivalence class $\sim/P$ or an arbitrary representative of the equivalence class $\sim/P$ depending on context.

### Examples

We have that $ot(\alpha)\cong\alpha$ in **Pos** for each ordinal $\alpha$ by definition, and further the order type of any well-ordered set 'is' the ordinal one would expect (up to iso):

$$ot(\{0,1,4,5\})\cong4,$$

$$ot(\{0,2,4,6,\dots\omega,\omega+2,\omega+4,\omega+6\})\cong\omega+4,$$

$$\vdots$$

For well-ordered posets it seems natural to take the order type to be the ordinal representative of the equivalence class, yielding the literal equalities in the idea section above, however this is not strictly necessary.

The order type of the negative integers is denoted by $^*\omega$ and the order type of the rationals is denoted by $\eta$, and neither of these order types have representatives in the ordinals since they aren't well-ordered.

#### Discussion

The notion of cardinality for a set $X$ can be defined using the bijection equivalence relation $\sim$ in **Set**
\begin{centre}
$X\sim Y\iff \exists f:X\cong Y\in Hom_{Set}$,
\end{centre}
so $|X|$ is the [[cardinal number]] member of the equivalence class of all sets in bijection with $X$. Since all sets are in bijection with some cardinal number (assuming the [[axiom of foundation]] via Scott’s trick) and cardinal numbers are just particular ordinals (assuming the [[axiom of choice]], otherwise more care must be taken) we do not encounter the phenomenon above where some equivalence classes don't have ordinal representatives in a universe with choice.

The order type equivalence relation $\sim$ in **Pos** above is a finer parsing of the universe of partially ordered sets which doesn't admit an ordinal in each of its equivalence classes, and each ordinal defines its own unique equivalence class under this relation instead of only the cardinal numbers as with the bijection equivalence relation.

The increased 'fineness' of this parsing can be attributed to the fact that we effectively restricted the iso equivalence relation in **Set** to the subcategory **Pos** where there are fewer isos between objects and thusly more equivalence classes.

## References

* Cantor, Georg [ "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"](https://gdz.sub.uni-goettingen.de/download/pdf/PPN243919689_0077/LOG_0014.pdf)


ordered Artinian local ring

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### Context
#### Algebra
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[[!include higher algebra - contents]]
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\tableofcontents

## Definition

An ordered Artinian local ring is a [[local ring]] which is both an [[ordered local ring]] and an [[Artinian local ring]]: a [[commutative ring]] $R$ with a [[strict weak order]] $\lt$ such that the positive elements form a [[multiplicative subset]] of $R$, the sum of two positive elements is positive, every element $a \in R$ is invertible if and only if it is positive or negative, and the set of non-invertible elements forms a [[nilradical]].

## See also

* [[ordered local ring]]

* [[Artinian local ring]]

* [[ordered field]]

* [[Archimedean ordered Artinian local ring]]

[[!redirects ordered Artinian local ring]]
[[!redirects ordered Artinian local rings]]

[[!redirects ordered local Artinian ring]]
[[!redirects ordered local Artinian rings]]

[[!redirects ordered Artinian local algebra]]
[[!redirects ordered Artinian local algebras]]

[[!redirects ordered local Artinian algebra]]
[[!redirects ordered local Artinian algebras]]

[[!redirects ordered Artin local ring]]
[[!redirects ordered Artin local rings]]

[[!redirects ordered local Artin ring]]
[[!redirects ordered local Artin rings]]

[[!redirects ordered Artin local algebra]]
[[!redirects ordered Artin local algebras]]

[[!redirects ordered local Artin algebra]]
[[!redirects ordered local Artin algebras]]


ordered field

# Ordered fields
* table of contents
{: toc}

## Idea

An ordered field is a [[field]] equipped with a compatible [[strict total order]].

Note that while the adjective 'ordered' usually refers to a [[partial order]], it is traditionally used more strictly when placed before 'field'.

## Definition

Classically:

An __ordered field__ is a [[field]] $K$ equipped with a [[strict total order]] $\lt$ such that

* $0 \lt 1$

* for all $a \in R$ and $b \in R$, $0 \lt a$ and $0 \lt b$ implies that $0 \lt a + b$; alternatively, $0 \lt a + b$ implies that $0 \lt a$ or $0 \lt b$.

* for all $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a \cdot b$

One often sees the definition using a weak [[total order]] $\leq$ instead of the strict total order $\lt$. This makes no difference in [[classical mathematics]], but the definition of the strict total order is the one that generalizes to [[constructive mathematics]].

An ordered field could also be defined as a [[field]] with a [[predicate]] $\mathrm{isPositive}$ such that

* zero is not positive:

$$\not \mathrm{isPositive}(0)$$

* one is positive:

$$\mathrm{isPositive}(1)$$

* for every element $a:A$, if $a$ is not positive and $-a$ is not positive, then $a = 0$

$$\forall a:A.\neg\mathrm{isPositive}(a) \wedge \neg\mathrm{isPositive}(-a) \Rightarrow (a = 0)$$

* for every term $a:A$, if $a$ is positive, then $-a$ is not positive.

$$\forall a:A. \forall b:A. \mathrm{isPositive}(a) \Rightarrow \neg\mathrm{isPositive}(-a)$$

* for every term $a:A$, $b:A$, if $a$ is positive, then either $b$ is positive or $a - b$ is positive.

$$\forall a:A. \forall b:A. \mathrm{isPositive}(a) \Rightarrow \left(\mathrm{isPositive}(b) \vee \mathrm{isPositive}(a - b)\right)$$

* for every term $a:A$, $b:A$, if $a$ is positive and $b$ is positive, then $a + b$ is positive

$$\forall a:A. \forall b:A. \mathrm{isPositive}(a) \wedge \mathrm{isPositive}(b) \Rightarrow \mathrm{isPositive}(a + b)$$

* for every term $a:A$, $b:A$, if $a$ is positive and $b$ is positive, then $a \cdot b$ is positive

$$\forall a:A. \forall b:A. \mathrm{isPositive}(a) \wedge \mathrm{isPositive}(b) \implies \mathrm{isPositive}(a \cdot b)$$

The strict total order is defined as
$$a \lt b \coloneqq \mathrm{isPositive}(b - a)$$

### In constructive mathematics

In constructive mathematics, [[denial inequality]] is no longer a [[tight apartness relation]], and neither relation, is a [[decidable relation]]. Thus, the concept of a [[field]] splits into multiple separate notions, *[[discrete fields]]*, *[[Heyting fields]]*, and *[[Kock fields]]*, depending on whether one uses [[denial inequality]] or a separate [[tight apartness relation]] to characterize invertibility of elements. This split in the definition of field carries over to the definition of an ordered field.

* An **ordered discrete field** is an [[ordered local ring]] such that $x$ is invertible or $x = 0$.

* An **ordered Heyting field** is an [[ordered local ring]] such that $x$ is non-invertible if and only if $x = 0$.

* An **[[ordered Kock field]]** is an [[ordered local ring]] such that $x$ is invertible if and only if $\neg (x = 0)$.

Ordered Heyting fields are commonly used in [[real analysis]] with the [[Dedekind real numbers]], since the Dedekind real numbers $\mathbb{R}$ and any [[subfield]] of $\mathbb{R}$ form an ordered Heyting field. On the other hand, ordered Kock fields are commonly used in [[synthetic differential geometry]].

Furthermore, a set having a strict total order no longer implies that the set has a weak total order, and a set having a weak total order no longer implies that the set has a strict total order. In particular, the traditional definition of an ordered field as defined above no longer implies that the ordered field is a [[lattice]], that it has binary [[joins]] and [[meets]].

Thus, some authors in constructive mathematics, such as [Booij 2020](#Booij20) and [Univalent Foundations Project 2013](#UFP13), have defined an ordered field to additionally have a lattice structure on $\leq$. However, it is consistent in constructive mathematics that there are ordered Heyting fields without a lattice structure (see [here](https://mathoverflow.net/questions/463680/archimedean-ordered-fields-without-maxima-and-minima-in-constructive-mathematics) for example), and ordered Kock fields which satisfy the [[Kock-Lawvere axiom]] are not lattices with respect to $\leq$ either.

## Examples

The field $\mathbb{R}$ of [[real numbers]] is [[the]] [[Dedekind completion|Dedekind-complete]] ordered field.

The field $\mathbb{Q}$ of [[rational numbers]] is a [[subfield]] of $\mathbb{R}$ that is too small to be complete.

The field of [[surreal number]]s is a [[field extension]] of $\mathbb{R}$ that is too large to be complete.

## Properties

Every ordered field must have [[characteristic]] $0$, since we can prove by [[induction]] that $n \gt 0$ for every positive [[natural number]] $n$.

As a result, the [[rational numbers]] are the [[initial]] ordered field, and every ordered field is a [[Q-algebra|$\mathbb{Q}$-algebra]].

The [[archimedean field|archimedean]] ordered fields are precisely the [[subfield]]s of [[real number|the field of real numbers]].

+-- {: .un_prop}
######Proposition
Every Dedekind complete ordered field is archimedean.
=--

+-- {: .proof}
######Proof
Suppose otherwise: let $a, b \gt 0$ be given, and suppose $b$ is an upper bound of $a, 2a, 3a, \ldots$. Then $b - a$ is an upper bound of $0, a, 2a, \ldots$ and consequently there can be no least upper bound of the sequence, contradicting Dedekind completeness.
=--

The following is a result in classical mathematics.

+-- {: .un_prop}
###### Proposition
A field admits an order ("is orderable") if and only if it is a **real field**, i.e., if the element $-1$ is not a sum of squares.
=--

+-- {: .proof}
###### Proof
Given an ordered field, any non-zero square is positive since either $-\alpha$ or $\alpha$ is positive, and so $(-\alpha)^2 = \alpha^2$ is positive. Hence a sum of non-zero squares cannot be negative, and in particular cannot be equal to $-1$.

In the other direction, every real field $F$ may be embedded in a [[real closed field]] (this requires [[Zorn's lemma]]), and a real closed field admits a unique ordering. The restriction of this ordering to the embedded field $F$ gives an ordering on $F$.
=--

## See also

* [[interval arithmetic]]
* [[strict linearly ordered ring]]
* [[ordered local ring]]
* [[ordered Artinian local ring]]
* [[ordered reduced local ring]]

## References

* {#Booij20} Auke B. Booij, Analysis in univalent type theory ([pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf))
* {#UFP13} Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

[[!redirects ordered field]]
[[!redirects ordered fields]]
[[!redirects linearly ordered field]]
[[!redirects linearly ordered fields]]
[[!redirects totally ordered field]]
[[!redirects totally ordered fields]]
[[!redirects strictly ordered field]]
[[!redirects strictly ordered fields]]

[[!redirects ordered discrete field]]
[[!redirects ordered discrete fields]]
[[!redirects ordered Heyting field]]
[[!redirects ordered Heyting fields]]


ordered group

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# Ordered groups
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## Idea

An ordered group is both a [[poset]] and a [[group]] in a compatible way.  The concept applies directly to other constructs with group structure, such as ordered [[abelian groups]], ordered [[vector spaces]], etc.  However, for [[ordered ring]]s, [[ordered fields]], and so on, additional compatibility conditions are required.

## Definition

Let $G$ be a [[group]] (written additively but not necessarily [[abelian group|commutative]]), and let $\leq$ be a [[partial order]] on the [[underlying set]] of $G$.  Then $(G,\leq)$ is an __ordered group__ if this compatibility condition (*translation invariance*) holds:

*  If $a \leq b$, then $a + c \leq b + c$ and $c + a \leq c + b$.

+-- {: .num_remark}
###### Remark
An ordered group is *not* the same thing as a [[group object]] in $Pos$. The trouble is that requiring the inversion map $x \mapsto x^{-1}$ to preserve order (i.e., to be monotone, not antitone) is much too restrictive. Rather, an ordered group is a [[monoid object]] in the [[cartesian monoidal category]] $Pos$ which has the [[stuff, structure, property|property]] that its underlying monoid in $Set$ is a group.
=--

If $G$ is an [[abelian group]], then we have an __ordered abelian group__; in this case, only one direction of translation invariance needs to be checked.

It works just as well to talk of partially ordered [[monoids]], using the same condition of translation invariance.  Equivalently, an __ordered monoid__ is a thin [[monoidal category]], or a monoidal $(0,1)$-category.

## Properties

The order $\leq$ is determined entirely by the group $G$ and the [[positive cone]] $G^+$:
$$ G^+ \coloneqq \{x\colon G \;|\; 0 \leq x\} .$$
It\'s possible to *define* an ordered group in terms of the positive cone (by specifying precisely the conditions that the positive cone must satisfy); see [[positive cone]] for this.

However, this characterisation probably can\'t be made to work for ordered monoids (although I haven\'t checked for certain).

## Examples

The underlying additive group of any [[ordered field]] is an ordered group.

In particular, the underlying additive group of the field $\mathbb{R}$ of [[real numbers]] is an ordered group.

Although the field $\mathbb{C}$ of [[complex numbers]] is not an ordered field (since it is not *[[strict total  order|strictly totally]]* ordered), its underlying additive group is still an ordered group (where $a \leq b$ means that $b - a$ is a nonnegative real number).

Given a [[topological vector space]] $V$, we often consider its [[dual vector space]] $V^*$, consisting of the [[continuous map|continuous]] [[linear maps]] from $V$ to its [[base field]], which is usually either $\mathbb{R}$ or $\mathbb{C}$.  This inherits a partial order from the [[target]] field, and then the underlying additive group is an ordered group; in fact, we have an [[ordered algebra]].  (This is the main sort of example that I know of, but that probably just reflects my own limited knowledge.)

More generally, if $V$ is any [[set]], $G$ is any ordered group, and $F$ is any collection of [[functions]] from $V$ to $G$, as long as $F$ is a [[subgroup]] of the group of all functions from $V$ to $G$, then $F$ is an ordered group.

Non-abelian examples include free groups and torsion-free nilpotent groups.  The basics of the theory for both abelian and nonabelian ordered groups can be found in Birkhoff's Lattice Theory.

## Related concepts

* [[pseudolattice ordered abelian group]]

* [[totally ordered abelian group]]

* [[Tarski group]]

* [[protoring]]

## References

*  English Wikipedia: [Partially ordered group](https://en.wikipedia.org/wiki/Partially_ordered_group).
   {#Wikipedia}

The [[stable category]] of preordered groups is studied in

* [[Aline Michel]], _The stable category of preordered groups_, 2023. [[arXiv:2309.11953](https://arxiv.org/abs/2309.11953)]

This categorical properties of preordered groups was also studied in

* [[Maria Manuel Clementino]], [[Nelson Martins-Ferreira]], and [[Andrea Montoli]], _On the categorical behaviour of preordered groups_, Journal of Pure and Applied Algebra, Vol. 223, No. 10, 2019. [[doi:doi.org/10.1016/j.jpaa.2019.01.006](https://doi.org/10.1016/j.jpaa.2019.01.006)]

[[!redirects ordered group]]
[[!redirects ordered groups]]
[[!redirects partially ordered group]]
[[!redirects partially ordered groups]]
[[!redirects partially-ordered group]]
[[!redirects partially-ordered groups]]

[[!redirects ordered abelian group]]
[[!redirects ordered abelian groups]]
[[!redirects partially ordered abelian group]]
[[!redirects partially ordered abelian groups]]
[[!redirects partially-ordered abelian group]]
[[!redirects partially-ordered abelian groups]]

[[!redirects ordered monoid]]
[[!redirects ordered monoids]]
[[!redirects partially ordered monoid]]
[[!redirects partially ordered monoids]]
[[!redirects partially-ordered monoid]]
[[!redirects partially-ordered monoids]]


ordered integral domain

# Ordered integral domains
* table of contents
{: toc}

## Idea

An ordered integral domain is an [[integral domain]] equipped with a compatible [[strict total order]].

Note that while the adjective 'ordered' usually refers to a [[total order]], which by default is non-strict, it is traditionally used more strictly when placed before 'integral domain'.

## Definition

An __ordered integral domain__ is an [[integral domain]] $R$ equipped with a [[strict total order]] $\lt$ such that:

* $0 \lt 1$

* for all $a \in R$ and $b \in R$, $0 \lt a$ and $0 \lt b$ implies that $0 \lt a + b$; alternatively, $0 \lt a + b$ implies that $0 \lt a$ or $0 \lt b$.

* for all $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a \cdot b$

## Examples

The integral domain $\mathbb{R}$ of [[real numbers]], the integral domain $\mathbb{Z}$ of [[integers]], and the integral domain $\mathbb{Q}$ of rational numbers are all ordered integral domains.

## Properties

Every ordered integral domain must have [[characteristic]] $0$, since we can prove by [[induction]] that $n \gt 0$ for every positive [[natural number]] $n$.

The [[archimedean integral domain|archimedean]] ordered integral domain are precisely the [[subobject|integral subdomains]] of [[real number|the integral domain of real numbers]].

Every [[localization of a commutative ring|localization]] of the integral domain of integers away from a subset not containing zero is a [[dense linear order]], and the [[Dedekind completion]] of the resulting integral domain is the integral domain of real numbers. In particular, the Dedekind completions of the [[dyadic rational numbers]] and the [[decimal rational numbers]] are both the real numbers.

## Related concepts

* [[pseudo-ordered ring]]

* [[ordered field]]

* [[protoring]]

## References

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

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

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

[[!redirects ordered integral domain]]
[[!redirects ordered integral domains]]


ordered Kock field

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\tableofcontents

## Idea

A definition of [[ordered field]] in [[constructive mathematics]] which uses [[denial inequality]]. It is primarily used in [[synthetic differential geometry]].

## Definition

An **ordered Kock field** is a [[Kock field]] $R$ with a [[strict weak order]] $\lt$ which is not necessarily [[connected relation|connected]]:

1. for all $a \in R$, $\neg(a \lt a)$

2. for all $a \in R$ and $b \in R$, if $a \lt b$, then $\neg(b \lt a)$

3. for all $a \in R$, $b \in R$, and $c \in R$, if $a \lt c$, then $a \lt b$ or $b \lt c$

4. $0 \lt 1$

5. for all $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a + b$

6. for all $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a \cdot b$

7. for all $a \in R$, $a \neq 0$ if and only if $a \lt 0$ or $0 \lt a$

## See also

* [[Kock field]]

* [[ordered local ring]]

## References

* [[David Jaz Myers]], *Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory* ([arXiv:2205.15887](https://arxiv.org/abs/2205.15887))


ordered local integral domain

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### Context
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## Definition

An *ordered local integral domain* is a [[local ring]] which is both an [[ordered local ring]] and a [[local integral domain]]: a [[commutative ring]] $R$ with a [[strict weak order]] $\lt$ such that the positive elements form a [[multiplicative subset]] of $R$, the sum of two positive elements is positive, every element $a \in R$ is invertible if and only if it is positive or negative, and if the product of two elements is equal to zero, than one of the two elements is equal to zero.

## Properties

Unlike the [[theory]] of [[ordered fields]], the theory of ordered local integral domains is a [[coherent theory]].

## See also

* [[ordered local ring]]

* [[local integral domain]]

* [[ordered field]]

* [[Archimedean ordered local integral domain]]

[[!redirects ordered local integral domain]]
[[!redirects ordered local integral domains]]


ordered local ring

\tableofcontents

## Idea

In the same way that a [[local ring]] is a [[Heyting field]] whose [[apartness relation]] is not [[tight apartness relation|tight]], an ordered local ring is an [[ordered field]] whose [[strict weak order]] is not necessarily [[connected relation|connected]].

## Definition

Let $R$ be a [[commutative ring]]. $R$ is an **ordered local ring** if there is a [[strict weak order]] $\lt$ such that

* $0 \lt 1$

* for all $a \in R$ and $b \in R$, $0 \lt a$ and $0 \lt b$ implies that $0 \lt a + b$; alternatively, $0 \lt a + b$ implies that $0 \lt a$ or $0 \lt b$.

* for all $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a \cdot b$

* for all $a \in R$, $a$ is invertible if and only if $a \lt 0$ or $0 \lt a$

## Properties

Every ordered local ring is a local ring with the apartness relation given by
$$a \# b \coloneqq a \lt b \vee b \lt a$$

Every ordered local ring has a [[preorder]] given by $a \leq b \coloneqq \neg (b \lt a)$.

### Quotient ordered field

Let $D$ be the [[ideal]] of all non-invertible elements in $R$. Then the quotient ring $R/D$ is an [[ordered field]].

## Examples

* Every [[ordered discrete field]] is an ordered local ring where the order relation satisfies [[trichotomy]].

* Every [[ordered Heyting field]] is an ordered local ring where every non-positive non-negative element is equal to zero.

* Every [[ordered Kock field]] is an ordered local ring in which every non-zero element is positive or negative.

* The [[dual numbers]] $\mathbb{R}[\epsilon]/\epsilon^2$ are an ordered local ring where the [[nilpotent]] [[infinitesimal]] $\epsilon \in \mathbb{R}[\epsilon]/\epsilon^2$ is a non-zero non-positive non-negative element.

## In analysis

Ordered local rings are important for modeling notions of [[infinitesimals]] and [[infinite elements]], including both the non-invertible infinitesimals common in [[synthetic differential geometry]] and the [[invertible]] infinitesimals whose reciprocals are the infinite elements common in [[nonstandard analysis]].

## See also

* [[local ring]]

* [[ordered field]]

* [[Archimedean ordered local ring]]

* [[ordered Artinian local ring]]

* [[ordered reduced local ring]]

* [[ordered local integral domain]]

[[!redirects ordered local ring]]
[[!redirects ordered local rings]]


ordered pair

# Ordered pairs
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## Idea

Given any things $a$ and $b$, the __ordered pair__ of $a$ and $b$ is a thing, usually written $(a,b)$, sometimes $[a,b]$ or $\langle{a,b}\rangle$.  The important property is
\[ \label{basic} (a,b) = (c,d) \;\Leftrightarrow\; a = c,\; b = d .\]
The things $a$ and $b$ are called the __components__ of the ordered pair $(a,b)$.  Given any two sets $X$ and $Y$, their [[Cartesian product]] is a set $X \times Y$ whose [[elements]] are precisely the ordered pairs whose components are respectively elements of $X$ and elements of $Y$.

Note that nothing is __ordered__ in an ordered pair other than how it is written out, so sometimes just the word __pair__ is used for this concept.

In terms of [[category theory]] an ordered pair is an element in a [[Cartesian product]].

The concept of  "pair" meaning a set containing just one or two members as in the [[axiom of pairing]] of [[ZFC|Zermelo–Frankel Set Theory]] is now usually distinguished as an __[[unordered pair]]__. A pair in which the components are ordered is basically an arrow between the components, which is sometimes called or analyzed as an [[interval object|interval]] within a larger context.

## Formalisations

One may wish to declare ordered pairs to exist by fiat, which was done, for example, by both [[Bourbaki]] and [[Bill Lawvere]].  In Bourbaki\'s foundational [[set theory]], $\langle{-,-}\rangle$ is a fundamental binary operation on mathematical objects satisfying two axioms: \eqref{basic} and the existence (as a set) of the Cartesian product of any two sets.  In Lawvere\'s foundational set theory, [[ETCS]], one axiom is the existence of [[products]] in the category of sets; when applied to [[global elements]], this gives their ordered pair (with the product itself being the Cartesian product), and \eqref{basic} can be proved.  Other [[structural set theories]] should contain an axiom similar to Lawvere\'s axiom of products.

+-- {: .query}
I need to check that I remembered Bourbaki correctly; it varies with the edition.  ---Toby
=--

Instead, one may construct ordered pairs out of some more basic operation.  In a [[material set theory]], one may use _[[Kuratowski pairs]]_

$$ (a,b) \coloneqq \big\{\,\{a\},\{a,b\}\,\big\} ;$$

it is straightforward (using the [[axiom of extensionality]]) to prove that \eqref{basic} holds.  Sometimes one sees the alternative
$$ (a,b) \coloneqq \big\{a, \{a,b\}\,\big\} ;$$
but now the [[axiom of foundation]] is also needed to prove \eqref{basic}, so the first form is usually preferred.  To prove that the cartesian product of two sets is a set, one may use the axiom of separation ([[bounded separation]] is enough) to construct $X \times Y$ as a [[subset]] of the [[power set]] of the power set of the [[union]] of $X$ and $Y$, or else use the axiom of replacement ([[restricted replacement]] is enough) to construct it directly, since its elements are indexed by the sets $X$ and $Y$.

+-- {: .num_remark #class}
###### Remark
Again in the context of material set theory, there are other options for defining ordered pairs that may offer technical advantages. For example, assuming we have the natural numbers $\mathbb{N}$, and given a set $x$, let $\varphi(x)$ be the set $(x \setminus \mathbb{N}) \cup \{n+1: n \in x \cap \mathbb{N}\}$. Then define

$$(A, B) \coloneqq \{\varphi(a): a \in A\} \cup \{\varphi(b) \cup \{0\}: b \in B\}$$

and observe that the first entry $A$ may be retrieved as the set of elements of $(A, B)$ that do not contain $0$, and $B$ as the set of elements that do. One advantage is that even for classes $A, B$ (defined in ZFC as formulas), we may define a class $(A, B)$ as another formula patterned on this construction. This solves a technical exercise posed by Andrej Bauer in this [MO post](https://mathoverflow.net/a/63268/2926); compare [class functions](https://ncatlab.org/nlab/show/function#for_classes). Thus, a function between classes $f: A \to B$ may be defined as an ordered triple $(A, f, B)$ where $f$ is a class of pairs $(a, b)$ defined by a formula that is [[entire relation|entire]] and [[functional relation|functional]].
=--

In a foundational [[type theory]], ordered pairs are usually also given by fiat, but \eqref{basic} may not hold, depending on the type theory used.  Now Bourbaki\'s binary operation of pairing becomes a typed operation; given $a$ of type $X$ and $b$ of type $Y$, the ordered pair $(x,y)$ has type $A \times B$.  There are also two typed operations (either basic or definable, depending on the style of type theory used) $\pi\colon X \times Y \to X$ and $\rho\colon X \times Y \to Y$, satisfying the [[beta-rule]]s $\pi(x,y) = x$ and $\rho(x,y) = y$.  Then we can either add the [[eta-rule]] $z = (\pi z,\rho z)$, which will allow \eqref{basic} to be proved; or else take \eqref{basic} as the *definition* of equality on the product type $X \times Y$, which will then allow the eta-rule to be proved.  (Or you can do neither, and then \eqref{basic} and the eta-rule will fail.)

## Generalisations

* [[tuples]]

* [[families]]

* [[pairings]]

* [[dependent sums]]

[[!redirects ordered pair]]
[[!redirects ordered pairs]]


ordered reduced local ring

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\tableofcontents

## Definition

An *ordered reduced local ring* is a [[local ring]] which is both an [[ordered local ring]] and an [[reduced local ring]]: a [[commutative ring]] $R$ with a [[strict weak order]] $\lt$ such that the positive elements form a [[multiplicative subset]] of $R$, the sum of two positive elements is positive, every element $a \in R$ is invertible if and only if it is positive or negative, and every [[nilpotent element]] is equal to zero.

## Properties

Unlike the [[theory]] of [[ordered fields]], the theory of ordered reduced local rings is a [[coherent theory]].

## See also

* [[ordered local ring]]

* [[reduced local ring]]

* [[ordered field]]

* [[Archimedean ordered reduced local ring]]

[[!redirects ordered reduced local ring]]
[[!redirects ordered reduced local rings]]


ordered ring

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#### (0,1)-Category theory
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\tableofcontents

## Disambiguation

Traditionally, in [[classical mathematics]], an ordered ring is a [[totally ordered ring]]. However, in [[constructive mathematics]], due to the failure of [[excluded middle]], there are multiple notions of order which are not necessarily equivalent to the notion of total order but which in the presence of [[excluded middle]] are equivalent to the notion of [[total order]], and thus, there are multiple notions of ordered rings. There are also more general notions of ordered rings, where the order is a [[preorder]], a [[partial order]], a [[pseudolattice]], a [[pseudo-order]], or a [[strict preorder]].

* [[preordered ring]]

* [[totally preordered ring]]

* [[pseudolattice preordered ring]]

* [[strictly weakly ordered ring]]

* [[partially ordered ring]]

* [[totally ordered ring]]

* [[pseudolattice ordered ring]]

* [[pseudo-ordered ring]]

## External links

* Wikipedia, _Ordered ring_

category: disambiguation


orderedpair

odered pair


ordinal > history

< [[ordinal]]
[[!redirects ordinal -- history]]


ordinal analysis

#Contents#
* Contents
{:toc}

## Idea

An ordinal analysis of a formal system precisely measures its proof-theoretic strength, which is its strength at justifying [[transfinite induction]]. In practice, it also measures the system's ability to prove totality of complex computable functions.

Giving a concrete "ordinal notation" that describes the proof-theoretic strength of a system is sometimes seen as providing a constructive justification for the reasoning principles of the system. However, ordinal analysis is not limited to systems that are [[constructive]] in the sense of not assuming [[excluded middle]]. But in practice, classical systems with ordinal notations are interpretable constructively.

### Relation to Predicativity

While it is classically true that any [[deductive system#FormalSystems|effective formal system]] has some limit to its proof-theoretic strength, only systems of limited strength have had their strength concretely described by a good ordinal notation. Specifically, all systems with ordinal analyses so far are significantly weaker than full [[second-order arithmetic]]. Consequently, so far, systems that have been given ordinal notations feel more like [[predicative]] systems than like familiar impredicative systems such as [[ZFC]], [[higher-order logic|higher-order arithmetic]], and [[ETCS]].

Many have argued, however, that the boundary between predicatively-justifiable and impredicative systems lies somewhere within the range of proof-theoretic ordinals already analyzed. In other words, that the stronger systems in the table below are actually slightly impredicative, even when it isn't obvious.

## Definition

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

The **proof-theoretic ordinal** of a [[theory]] $T$ is (commonly) defined as

$$ {\vert T\vert} = \sup\{ \mathrm{otyp}({\prec}) \mid {\prec}\text{is primitive recursive and}\;
 T \vdash \mathrm{TI}({\prec},X) \}$$

where $\mathrm{otyp}({\prec})$ is the [[order-type]] of the [[well-order]]
$\prec$ and $\mathrm{TI}({\prec},X)$ means that $X$ (a free parameter) satisfies transfinite induction along $\prec$; this is the constructive way to say that $\prec$ is well-ordered. For most theories, this ordinal is equal to the $\Pi^1_1$-ordinal of the theory,

$$ {\vert T\vert}_{\Pi^1_1} := \sup\{ \mathrm{tc}(F) \mid F\;\text{is a}\;\Pi^1_1\text{-sentence and }\;
 T \vdash F \}$$

where $\mathrm{tc}(F)$ is the truth-complexity of the
$\Pi^1_1$-sentence $F$ (if $T$ is a first-order theory, we interpret
$T \vdash \forall X.F(X)$ as $T \vdash F(X)$ with a free set parameter
$X$).

=--

The proof-theoretic ordinal is always a recursive [[ordinal]] ($\lt\omega_1^{\mathrm{CK}}$) if the theory is recursive.

Note that the proof-theoretic ordinal is a rather blunt invariant, because if $F$ is any true $\Sigma^1_1$-sentence, then

$$ {\vert T\vert}_{\Pi^1_1} = {\vert T+F\vert}_{\Pi^1_1} $$

So this tells us nothing about the provable arithmetical sentences of
$T$! And indeed, all true arithmetical sentences have truth-complexity
less than $\omega$.

The solution is to refine the analysis to give a $\Pi^0_2$ ordinal
analysis of the theory, using a measure of the computational
complexity of provable $\Pi^0_2$-sentences/provably recursive
functions. However, there's no natural hierarchy of all subrecursive
functions. We can define such a computational complexity depending on a
choice of starting function and a choice of ordinal notation system
(indeed, the $\Pi^0_2$ ordinal should be thought of as an ordinal
notation, rather than an actual ordinal). But for all the ordinal
analyses we have so far, these choices turn out not to matter,
basically because any _natural ordinal notation system_ will give the
same result. A major issue is, however, that we have no definition of
what a natural ordinal notation system is (but we know one when we see
it!).

All of the ordinal analyses mentioned below also give a $\Pi^0_2$
ordinal analysis, and thus control of the computational complexity of
the provably recursive functions.

## Methods

### Upper bounds

Most ordinal analyses proceed by embedding the theory in an infinitary
proof calculus deriving judgments $\vdash^\alpha_\rho F$ meaning $F$
has an infinitary derivation of height less than $\alpha$ using only
cuts of rank less than $\rho$, or some refinement of such, for
instance using operator controlled derivations. Then one analyzes how
the height of a derivation changes during cut-elimination. For
predicative theories, we'll have

$$ \vdash^\alpha_\rho \Delta \Rightarrow \vdash^{\varphi\rho\alpha}_0 \Delta $$

One sets up the calculus so that $\vdash^\alpha_0 F$ implies
$\mathrm{tc}(F)\lt\alpha$, so this yields upper bounds for the
proof-theoretic ordinal.

### Lower bounds

Lower bounds are usually done by directly deriving
$T \vdash \mathrm{TI}({\prec_n},X)$ for a sequence of primitive recursive
well-orderings $\prec_n$ whose order-types approximate the
proof-theoretic ordinal. We get these primitive recursive
well-orderings from good ordinal notation systems, see below.

## Table of Ordinal Analyses

See below for explanations.

| Ordinal                                      | FOA                                        | SOA                                    | Other                          |
|----------------------------------------------|--------------------------------------------|----------------------------------------|--------------------------------|
| **Finitist systems**                         |                                            |                                        |                                |
| $\omega^3$                                   | EFA                                        | RCA$_0^*$                              |                                |
| $\omega^\omega$                              | PRA,I$\Sigma_1$                            | RCA$_0$, WKL$_0$                       | CPRC                           |
| **Predicative systems**                      |                                            |                                        |                                |
| $\varepsilon_0=\varphi(1,0)$                 | PA                                         | ACA$_0$, $\Delta_1^1$-CA$_0$, $\Sigma^1_1$-AC$_0$ | EM$_0$              |
| $\varphi(1,\varepsilon_0)$                   |                                            | ACA                                    |                                |
| $\varphi(\omega,0)$                          | ID$_1^#$                                   | $\Delta_1^1$-CR                        | EM$_0$+JR                      |
| $\varphi(\varepsilon_0,0)$                   | $\widehat{\mathrm{ID}}_1$                  | $\Delta_1^1$-CA, $\Sigma^1_1$-AC       | EM$_0$+J, ML$_1$               |
| $\Gamma_0=\varphi(1,0,0)$                    | $\widehat{\mathrm{ID}}_{\lt\omega}$, U(PA) | ATR$_0$,$\Delta^1_1$-CA+BR             | ML$_{\lt\omega}$, MLU, KPi$^0$ |
| **Meta-predicative systems**                 |                                            |                                        |                                |
| $\varphi(1,0,\varepsilon_0)$                 | $\widehat{\mathrm{ID}}_{\omega}$           | ATR                                    | KPl$^0$+F-I$_\omega$           |
| $\varphi(1,\omega,0)$                        | $\widehat{\mathrm{ID}}_{\lt\omega^\omega}$ | ATR$_0$+($\Sigma^1_1$-DC)              | KPi$^0$+$\Sigma_1$-I$_\omega$  |
| $\varphi(1,\varepsilon_0,0)$                 | $\widehat{\mathrm{ID}}_{\lt\varepsilon_0}$ | ATR+($\Sigma^1_1$-DC)                  | KPi$^0$+F-I$_\omega$           |
| $\varphi(1,\Gamma_0,0)$                      | $\widehat{\mathrm{ID}}_{\lt\Gamma_0}$      |                                        | MLS                            |
| $\varphi(2,0,0)$                             | Aut($\widehat{\mathrm{ID}}$)               |                                        | KPh$^0$                        |
| $\varphi(\omega,0,0)$                        |                                            |                                        | KPM$^0$                        |
| **Inductive types**                          |                                            |                                        |                                |
| $\psi_\Omega(\varepsilon_{\Omega+1})$        | ID$_1$                                     |                                        | KP$\omega$, CZF, ML$_1$V       |
| $\psi_\Omega(\Gamma_{\Omega+1})$             | U(ID$_1$)                                  |                                        |                                |
| $\psi_\Omega(\Omega_\omega)$                 | ID$_{\lt\omega}$                           | $\Pi^1_1$-CA$_0$,$\Delta^1_2$-CA$_0$   |                                |
| $\psi_\Omega(\Omega_\omega \varepsilon_0)$   | W-ID$_\omega$                              | $\Pi^1_1$-CA                           | W-KPl                          |
| $\psi_\Omega(\varepsilon_{\Omega_\omega+1})$ | ID$_\omega$                                | $\Pi^1_1$-CA+BI                        | KPl                            |
| $\psi_\Omega(\Omega_{\omega^\omega})$        | ID$_{\lt\omega^\omega}$                    | $\Delta^1_2$-CR                        |                                |
| $\psi_\Omega(\Omega_{\varepsilon_0})$        | ID$_{\lt\varepsilon_0}$                    | $\Delta^1_2$-CA, $\Sigma^1_2$-AC       | W-KPi                          |
| $\psi_\Omega(\varepsilon_{I+1})$             |                                            | $\Delta^1_2$-CA+BI, $\Sigma^1_2$-AC+BI | KPi, T$_0$, CZF+REA            |
| $\psi_\Omega(\Omega_{I+\omega})$             |                                            |                                        | ML$_1$W                        |
| $\psi_\Omega(\Omega_L)$                      |                                            |                                        | KPh, ML$_{\lt\omega}$W         |
| **Inductive-recursive types**                |                                            |                                        |                                |
| $\psi_\Omega(\varepsilon_{M+1})$             |                                            | $\Delta^1_2$-CA+BI+(M)                 | CZFM, KPM                      |
| $\psi_\Omega(\Omega_{M+\omega})$             |                                            |                                        | KPM$^+$, MLM, "Agda"           |

The sections of the table corresponds roughly to the kinds of reasoning captured by the systems:

* Finitistic reasoning about the natural numbers (and other finitistic inductive types)
* Predicative reasoning about the natural numbers
* Meta-predicative reasoning about the natural numbers
* Reasoning about general inductive types
* Reasoning about general inductive-recursive types

Note that these are all very much weaker than full second-order arithmetic, and of course much weaker than Zermelo or Zermelo-Fraenkel set theory.

After the meta-predicative systems, all ordinals use a notation which refers to a much larger ordinal, often patterned on a large cardinal.

Rathjen (2005) analyzed systems up to $\Delta^1_2$-CA+BI+$\Pi^1_2$-CA$^-$ (the last part is parameter-free $\Pi^1_2$-comprehension), corresponding to a stable ordinal. Arai has work on systems that are a bit stronger, up to about $\Sigma^1_3$-DC+BI, in terms of ordinal diagrams defined in terms of iterated stability, and has more recently analyzed KP+$\Pi_1$-collection, which is beyond KP+"the stable ordinal are unbounded" in strength.

With current technology, we are very far from having ordinal analyses of stronger systems such as full second order arithmetic, ZFC set theory, or impredicative type theories such as the calculus of (inductive) constructions.

### Ordinal notations

* $\varphi$ is either the binary or the ternary Veblen function
* $\Omega=\Omega_1$ is the first recursively regular ordinal
* $\Omega_\xi$ is the $\xi$th ordinal that is recursively regular or a limit of smaller recursively regulars
* $I$ is the first recursively inaccessible ordinal
* $L$ is the limit of the first $\omega$ recursively inaccessible ordinals
* $M$ is the first recursively Mahlo ordinal

The $\psi$ functions are ordinal collapsing functions.

### Systems of first-order arithmetic (FOA)

These assert some hierarchy of inductive defined sets of natural numbers.

* EFA is elementary function arithmetic
* PRA is [[primitive recursive arithmetic]]
* PA is first-order Peano arithmetic
* $\widehat{\mathrm{ID}}_\nu$ extends PA by $\nu$ iterated fixed points of monotone operators (not necessarily least).
* ID$_1^#$ is $\widehat{\mathrm{ID}}_1$ with induction only for positive formulas.
* ID$_\nu$, the theory of $\nu$-times [[iterated inductive definitions]], extends PA by $\nu$ iterated least fixed points of monotone operators.

Many of the results in the table concerning these can be summarized as

$$ {\vert \mathrm{ID}_\nu \vert} = \psi_\Omega(\varepsilon_{\Omega_\nu+1}) $$

For any of these, we get the same proof-theoretic ordinal whether we
use classical or intuitionistic logic. For the ID$_\nu$-systems, this
is due to Sieg.

The unfolding systems U(PA) and U(ID$_1$) are not exactly first-order
arithmetic systems, but capture what we can get by predicative
reasoning based on the natural numbers or one generalized inductive
definition, respectively.

### Systems of second-order arithmetic (SOA)

* Comprehension or choice principles.
* RCA$_0^*$ is the second-order version of EFA: $\exp$ is a function symbol.
* WKL is weak König's lemma.
* BR is the bar induction rule.
* BI is the bar induction axiom.
* ATR is arithmetical transfinite recursion.
* (M) says every true $\Pi^1_3$ sentence with parameters holds in a $\beta$-model of $\Delta^1_2$-CA.

A zero subscript indicates that we have the induction axiom rather
than the induction scheme, except for RCA$_0$ which has the
$\Sigma^0_1$-induction scheme.

### Kripke Platek Set theories

* KP$\omega$ is Kripke-Platek set theory, whose universe is an admissible set containing $\omega$.
* KPl asserts that the universe is a limit of admissible sets
* KPi asserts that the universe is inaccessible sets
* KPh asserts that the universe is hyperinaccessible: an inaccessible set and a limit of inaccessible sets
* KPM asserts that the universe is a Mahlo set

A superscript zero indidcates that $\in$-induction is removed.

About the Mahlo level: this also corresponds to inductive-recursive definitions (Dybjer-Setzer).

### Martin-Löf type theories

* CPRC is the Herbelin-Patey Calculus of Primitive Recursive Constructions.
* ML$_n$ is type theory without W-types with $n$ universes.
* ML$_{\lt\omega}$ is type theory without W-types with an infinite hierarchy of universes.
* MLU is type theory with a next universe operator.
* MLS is type theory without W-types with a superuniverse.
* ML$_1$V is type theory with one universe and Aczel's type of iterative sets.
* ML$_1$W is type theory with W-types and one universe.
* ML$_{\lt\omega}$W is type theory with W-types and an infinite hierarchy of universes (this corresponds to the type theory in Lean's HoTT mode).
* MLM is Martin-Löf type theory with a Mahlo universe.

### Constructive Set theories

* CZF is Aczel's constructive set theory.
* REA is the regular extension axiom.
* CZFM is CZF with a Mahlo universe

### Explicit mathematics systems

* EM$_0$ is basic [[explicit mathematics]] with elementary comprehension
* JR is the join rule, J is the join axioms
* T$_0$ is EM$_0$+J+IG, where IG is inductive generation

## References

* Aczel, The Type Theoretic Interpretation of Constructive Set Theory, Logic Colloquium 1977.

* Arai, A sneak preview of proof theory of ordinals, [arXiv 1102.0596](https://arxiv.org/abs/1102.0596).

* Arai, Proof theory for theories of ordinals III: $\Pi_N$-reflection, [arXiv 1007.0844](https://arxiv.org/abs/1007.0844).

* Arai, An ordinal analysis of $\Pi_1$-Collection, [arXiv 2112.09871](https://arxiv.org/abs/2112.09871)

* Buchholz, A simplified version of local predicativity, Proc. Proof Theory Meeting Leed '90.

* Buchholz, A Note on the Ordinal Analysis of KPM (2002).

* Buchholz, Feferman, Pohlers and Sieg, Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies, Lecture Notes in Mathematics 897 (1981).

* Dybjer and Setzer, Induction-recursion and initial algebras, APAL 124(1-3), 2003, pp. 1-47. DOI: 10.1016/S0168-0072(02)00096-9

* Feferman: Iterated inductive fixed-point theories: Application to Hancock's conjecture, in: G. Metakides (ed.): Patras Logic Symposium (North-Holland, Amsterdam, (1982), 171–196.

* Jäger, Kahle, Setzer, and Strahm, The proof-theoretic analysis of transfinitely iterated fixed point theories, Journal of Symbolic Logic

* Jäger and Strahm, Fixed point theories and dependent choice. Archive for Mathematical Logic.

* Rathjen, Ordinal notations based on a weakly Mahlo cardinal, Archive for Mathematical Logic 29 (1990), 249-263.

* Rathjen, Proof-theoretic analysis of KPM, Archive for Mathematical Logic (1991).

* Rathjen, The Recursively Mahlo Property in Second Order Arithmetic, Mathematical Logic Quarterly 42 (1996), 59-66.

* Rathjen, The strength of Martin-Löf type theory with a superuniverse. Part II, 2001, AML, DOI: 10.1007/s001530000051
* Rathjen, The strength of Martin-Löf type theory with a superuniverse. Part I, 2000, AML, DOI: 10.1007/s001530050001

* Rathjen, An ordinal analysis of parameter free $\Pi^1_2$-comprehension. Arch. Math. Log. 44(3), 263–362 (2005).

* Rathjen, The constructive Hilbert program and the limits of Martin-Löf type theory (2005), Synthese, DOI: 10.1007/s11229-004-6208-4

* Rathjen, Griffor and Palmgren, Inaccessibility in Constructive Set Theory and Type Theory, APAL 94 (1998), 181-200.

* Setzer, An upper bound for the proof theoretical strength of Martin-Löf Type Theory with W-type and one universe (2006)

* Strahm, Autonomous fixed point progressions and fixed point transfinite recursion, Proc. of Logic Colloquium '98, 449-464, [web](http://www.iam.unibe.ch/ltgpub/2000/str00a.pdf)

[[!redirects proof-theoretic ordinal]]


ordinal arithmetic

#Contents#
* table of contents
{:toc}

## Idea

[[arithmetic]] with [[ordinals]]. A kind of [[transfinite arithmetic]].

## Related concepts

* [[well-ordering theorem]]

* [[cardinal arithmetic]]

## References

See also

* Wikipedia, _[Ordinal arithmetic](https://en.wikipedia.org/wiki/Ordinal_arithmetic)_


ordinal number

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#### Arithmetic
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#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
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# Ordinal numbers
* table of contents
{: toc}

## Idea

The _ordinal numbers_ (or just _ordinals_) constitute a generalisation of a [[natural numbers]] to [[numbers]] of possibly infinite magnitudes.  Specifically, ordinal numbers generalise the concept of 'the next number after ...' or 'the index of the next item after ...'.  In particular, the next number after the natural numbers is the first infinite ordinal number.

## Definition

Naïvely, an **ordinal number** should be an [[isomorphism]] class of [[well-ordered sets]], and the **ordinal rank** of a well-ordered set $S$ would be its isomorphism class.  That is:

1. every well-ordered set has a unique ordinal number as its ordinal rank;
1. every ordinal number is the ordinal rank of some well-ordered set;
1. two well-ordered sets have the same ordinal rank if and only if they are isomorphic as well-ordered sets.

Then a **finite ordinal** is the ordinal rank of a [[finite set]], while an **infinite ordinal** or **transfinite ordinal** is the ordinal rank of an [[infinite set]].  (If you interpret both terms in the strictest sense, then there may be ordinals that are neither finite nor infinite, without some form of the [[axiom of choice]]).

Taking this definition literally in material [[set theory]], each ordinal is then a [[proper class]] (so one could not make further sets using them as elements).  For this reason, in axiomatic set theory one usually defines an ordinal number as a particular representative of this equivalence class.  One particularly slick definition is due to von Neumann:

* An __ordinal__ is a [[transitive set|transitive]] [[pure set]] $X$ which is [[well-ordered]] by the membership relation $\in$.  Then the __ordinal rank__ of a well-ordered set $S$ is the unique ordinal number that is isomorphic (as a well-ordered set) to $S$; it is a theorem that this exists, satisfying (1--3).

These pure sets are the __von Neumann ordinals__.  In the presence of the [[axiom of foundation]], $\in$ is automatically a [[well-founded relation]], so it suffices to require that $\in$ be a [[transitive relation]] on $X^+ = X \cup \{X\}$.

In the appendix of [[John Kelley]]'s topology textbook, ordinals are defined as:

* An __ordinal__ is a class $X$ such that every element of $X$ is a subset of $X$ and for all $A,B \in X$, either $A \in B$, $B \in A$ or $A=B$.

It is later proven that in the axiomatic setting of the appendix ([[Morse-Kelley set theory]]), ordinals are sets.

From the perspective of structural [[set theory]], it breaks the [[principle of equivalence]] to care about distinctions between [[isomorphism|isomorphic]] [[objects]], and unnecessary to insist on a canonical choice of representatives for isomorphism classes.  Therefore, from this point of view it is natural to simply say:

* An __ordinal__ is a [[well-ordered set]].

However, one still may need sets of ordinals, that is sets that serve as the target of an ordinal rank function satisfying (1--3) on any (small) collection of well-ordered sets.  One can construct this as a [[quotient set]] of that collection.

## Properties

The class of ordinals is itself [[well-ordered]].  There are many equivalent ways to define this ordering.  One is that $\alpha\lt\beta$ iff $\alpha$ is isomorphic to a proper _initial segment_ of $\beta$ (that is, a subset $S\subsetneq \beta$ such that $b\in S$ and $a\lt b$ imply $a\in S$).  With the von Neumann definition, this is equivalent to simply saying that $\alpha\in\beta$.

Every ordinal $\alpha$ has a **[[successor]]** $\alpha^+$, which in the von Neumann definition is simply $\alpha^+ = \alpha \cup \{\alpha\}$.  A **limit ordinal** is any ordinal which is not a successor of any other ordinal.

In the presence of the [[axiom of choice]], a [[cardinal number]] can be defined as a special ordinal number, specifically an ordinal which is not equipollent (isomorphic as a [[set]]) to any smaller ordinal.

## Related pages

* [[countable ordinal]], [[uncountable ordinal]]

* [[ordinal arithmetic]]

* [[cardinal]]

* [[plump ordinal]]

* [[∞-number]]

* [[ordinal analysis]]

One important use of ordinals is to index transfinite constructions, such as:

* [[transfinite composition]]
* [[transfinite construction of free algebras]]

## References

For ordinal numbers in [[homotopy type theory]], see section 10.3 of:

* *Homotopy Type Theory: Univalent Foundations of Mathematics*, The [[Univalent Foundations Project]], Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

On the equivalence of set-theoretic and type-theoretic ordinals, see:

* [[Tom de Jong]], [[Nicolai Kraus]], [[Fredrik Nordvall Forsberg]], and [[Chuangjie Xu]], *Ordinals.CumulativeHierarchy.html* ([web](https://www.cs.bham.ac.uk/~mhe/TypeTopology/Ordinals.CumulativeHierarchy.html))

Basic set-theory including ordinals is developped in the appendix of:

* [[John Kelley]], _General topology_, D. van Nostrand, New York (1955), reprinted as: Graduate Texts in Mathematics, Springer (1975) [[ISBN:978-0-387-90125-1](https://www.springer.com/gp/book/9780387901251)]

[[!redirects ordinal]]
[[!redirects ordinals]]
[[!redirects ordinal number]]
[[!redirects ordinal numbers]]

[[!redirects von Neumann ordinal]]
[[!redirects von Neumann ordinals]]
[[!redirects Von Neumann ordinal]]
[[!redirects Von Neumann ordinals]]
[[!redirects von Neumann ordinal number]]
[[!redirects von Neumann ordinal numbers]]
[[!redirects Von Neumann ordinal number]]
[[!redirects Von Neumann ordinal numbers]]

[[!redirects limit ordinal]]
[[!redirects limit ordinals]]

[[!redirects limiting ordinal]]
[[!redirects limiting ordinals]]


ordinal subdivision

##Idea

The most usual method of [[subdivision]] for a [[simplicial complex]] as used in
elementary algebraic and geometric topology is the barycentric [[subdivision]].
There is however another very well structured subdivision construction
encountered which can be useful.  The basic geometric construction involves
chopping up a geometric $n$-simplex by $n$-planes parallel to a  face and halfway
between that face and the opposite vertex.

##Categorical descriptions

We first define the ordinal subdivision on the simplices, $\Delta[n]$.  (Here $\oplus$ denotes the [[ordinal sum]]

The *ordinal subdivision* of $\Delta[n]$, the
standard $n$-simplex in simplicial sets, is denoted by
$Sd(\Delta[n])$, and is defined as follows:-

$ Sd(\Delta[n]) := \int^{p,q} \Delta([p]\oplus[q],[n]) \cdot (\Delta[p] \times \Delta[q]). $

For a general simplicial set we can then define:

the _ordinal subdivision_ of a simplicial set, $X$, is
denoted by $Sd(X)$ and is defined by

$$Sd(X) := \int^{n} X_n \cdot Sd(\Delta [n]). $$
This expands to

$$ Sd(X) := \int^{n} X_n \cdot \left(\int^{p,q} \Delta([p]\oplus[q],[n])
\cdot (\Delta[p] \times \Delta[q]) \right).$$

This is related to the [total décalage](http://ncatlab.org/nlab/show/bisimplicial+set#TotalSimplicialSets) of $X$ by

$$ Sd(X) =
 \cong \int^{p,q} DEC(X)_{p,q} \cdot (\Delta[p] \times
\Delta[q]).$$

##References

* [[Phil Ehlers|P. J. Ehlers]] and [[Tim Porter]], _Ordinal subdivision and special pasting in quasicategories_, Advances in Math. 217 (2007), No 2. pp 489-518.Delta


ordinal sum

# Ordinal sums
* table of contents
{: toc}

## Idea

In [[ordinal arithmetic]], the ordinal sum is a natural addition on ordered sets.  This is particularly well behaved when considering _finite_ ordinals and so provides a useful tool when manipulating [[simplicial sets]]. With infinite ordinals as well the properties are subtler and so care has to be taken.

The basic idea is that in ordinal sum you 'first put one of the two ordinals and then the other', so that the elements of the second ordinal are all bigger than those in the first one.

Restricted to $\Delta_a$ and thus to finite ordinals, the ordinal sum induces the operation of [[join of simplicial sets]] endowing the category, $sSet_+$, of augmented simplicial sets with the structure of a [[monoidal category]].

##Recall: Finite ordinals as simplicial sets

The objects of the _augmented_ [[simplex category]] $\Delta_a$ can be identified with the finite [[total order|totally ordered]] [[set]]s, _including_ the [[empty set]], which we write in this context as

$$
  [-1] \coloneqq \emptyset
$$

so that then

$$
  [0] = *
$$

is the singleton set, as usual and

$$
  [1] = \{ 0 \lt 1\}
$$

and so on, so that $[n] = \{ 0 \lt \cdots \lt n\}$

This counting is off by one compared to the [[cardinality]] of these sets.

(We will restrict for the moment to finite ordinals and thus to the category $\Delta_a$.)

The monoidal structure on $\Delta_a$  is, at the level of the sets, just the [[disjoint union]], but we  consider the order on that union.

##The definition (initially for finite ordinals)

If we have $[n]$ and $[m]$, we form the union of the two sets, where we know the order on two elements we keep it, but if we have two elements one, $i$, say, from the $[n]$ and the other, $j$, from $[m]$ we put $i \lt j$.

###Example

As an example, consider $[1] = \{ 0 \lt 1\}$ and $[2] =  \{\overline{0} \lt \overline{1} \lt \overline{2}\}$, where the overlines are just so that we can keep track of where the different elements come from. We form the union of the two sets and the rule says that anything without an overline is less than anything with one.  This gives a linear order
$$[1] \oplus [2] = \{0 \lt 1 \lt \overline{0} \lt \overline{1} \lt \overline{2}\},$$
which is [[isomorphic]] as a [[poset]] to $[4]$. Similarly $[1] \oplus [1] = [3]$, which helps explain the picture of the related [[join of simplicial sets]] given there.

We can thus think of the operation as the **addition of cardinalities**, but must remember that $[n]$ has $n + 1$ elements. In terms of the counting 'off-by-one', this reads

$$
  ([n], [m]) \mapsto [n + m + 1]
  \,,
$$

but must remember there is also the order to keep track of.

This operation extends to give the **ordinal sum** structure  on $\Delta_a$ (for details see the discussion in the entry [[simplex category]]) making it a [[monoidal category]], whose monoidal product is the $\oplus$ operation described above.  Note however that $[m]\oplus [n]$ and $[n]\oplus [m]$, although isomorphic are not _naturally_ so.

##Ordinal sum in general
If we consider ordinals that may not be finite then clearly the idea of ordinal sum generalises, so, for ordinals, $\alpha$ and $\beta$,  $\alpha\oplus \beta$ is the set  $\alpha\sqcup \beta$ with total order in which elements in either part are ordered as originally, but where any $a\in \alpha$ is less than and $b\in \beta$.

If either ordinal is infinite, this may lead to $\alpha\oplus \beta$ and $\beta\oplus \alpha$ not being isomorphic.  For instance, $[0]\oplus\omega$ is isomorphic to $\omega$ itself, but $\omega\oplus [0]$ has a greatest element, i.e. the $0$ in $[0]$-summand, so clearly there can be no isomorphism between the two ordinals.

We thus give a *warning*:
the [[monoidal structure]] given by ordinal sum is not a symmetric monoidal category structure even when restricted to  the subcategory of finite ordinals. If one extends to consider infinite ordinals, there may even be no order preserving maps at all from $[i] \oplus [j]$ to $[j] \oplus [i]$ for $i \neq j$, let alone isomorphisms.

## Ordinal sum of categories

In "Ordinal Sums and Equational Doctrines", Lawvere defines the **ordinal sum $A +_\sigma B$ of two categories** $A$ and $B$ as the [[pushout]]

$$
 \array{
 A \times |2| \times B &\to& A \times 2 \times B \\
 \downarrow & & \downarrow \\
 A + B &\to & A +_\sigma B
 }
$$

where $A+B$ is the [[coproduct]] of $A$ and $B$, 2 is the [[interval category]], $|2|$ is its underlying discrete category of objects, and the left vertical arrow is defined by

$$
\lang a,i,b\rang = \begin{cases}a & i = 0 \\ b & i = 1\end{cases}
$$

Concretely, $A +_\sigma B$ may be described as the category $A + B$ together with exactly one arrow $a \to b$ adjoined for every object $a \in A, b \in B$ (in other words, as the [[cograph of a profunctor|collage]] of $A$ and $B$ along the terminal profunctor).

Equivalently, $A +_\sigma B$ may be described as the colimit of the diagram

$$\array{
 & & A \times B & & & & A \times B & & \\
 & \mathllap{\pi_1}\; \swarrow & & \searrow \; \mathrlap{i_0} & & \mathllap{i_1}\; \swarrow & & \searrow \; \mathrlap{\pi_2} & \\
A & & & & A \times 2 \times B & & & & B.
}$$

##Related entries

* [[join of categories]]

* [[join of simplicial sets]]

## References

* [[William Lawvere]], *Ordinal sums and equational doctrines*, in *[[Seminar on Triples and Categorical Homology Theory]]* LNM **80** (1969) 141-155 [[doi:10.1007/BFb0083085](https://doi.org/10.1007/BFb0083085)]

[[!redirects ordinal sum]]
[[!redirects ordinal sums]]
[[!redirects ordinal sum of categories]]
[[!redirects ordinal sums of categories]]


ordinary cohomology

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

#Contents#
* table of contents
{:toc}

## Idea

One says _ordinary cohomology_ for the [[cohomology theory]] [[Brown representability theorem|represented]] by the [[Eilenberg-MacLane spectrum]] $K(\mathbb{A})$ for some [[abelian group]] $A$.

This is to distinguish from more general notions of [[cohomology]], such as [[generalized (Eilenberg-Steenrod) cohomology]] and [[nonabelian cohomology]].

## Properties

### Presentation by singular cohomology

Ordinary cohomology is modeled by [[singular cohomology]].

### General

* [[universal coefficient theorem]]

* [[Künneth theorem]]

* [[Hopf degree theorem]]

* [[Hurewicz theorem]], [[Boardman homomorphism]]

* [[Eilenberg-Zilber theorem]]

### Chromatic level

[[!include chromatic tower examples - table]]

## Related concepts

* [[cohomology in homotopy type theory]]

* The refinement to [[equivariant cohomology]] is [[equivariant ordinary cohomology]]

* The refinement to [[differential cohomology]] is [[ordinary differential cohomology]].

* [[ordinary homology]]

* [[twisted ordinary cohomology]]

* [[affine modality]]

* [[comparison theorem (étale cohomology)]]

## References

[[!include cohomology -- early references]]

### General

Textbook account:

* [[Jean Gallier]], [[Jocelyn Quaintance]], *Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry*, World Scientific (2022) [[doi:10.1142/12495](https://doi.org/10.1142/12495), [webpage](https://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html)]

See the references at:

* [[singular cohomology]], [[cellular cohomology]]

* [[sheaf cohomology]]

* [[Cech cohomology]]

* [[integral cohomology]]

Discussion [[cohomology in homotopy type theory|in homotopy type theory]]:

* [[Guillaume Brunerie]], [[Axel Ljungström]], [[Anders Mörtberg]], *Synthetic Integral Cohomology in Cubical Agda*, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022) **216** (2022) $[$[doi:10.4230/LIPIcs.CSL.2022.11](https://doi.org/10.4230/LIPIcs.CSL.2022.11)$]$

  > (in [[cubical type theory|cubical]] [[Agda]])

[[!redirects ordinary cohomology theory]]


ordinary differential cohomology

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

#Contents#
* table of contents
{:toc}

## Idea

Ordinary differential cohomology is the [[differential cohomology]]-refinement of [[ordinary cohomology]], for instance realized as [[singular cohomology]].

Every [[generalized (Eilenberg-Steenrod) cohomology]]-theory has a refinement to [[differential cohomology]]. By _ordinary differential cohomology_ one refers, for emphasis, to the differential refinement of _ordinary [[integral cohomology]]_ , hence of the [[cohomology theory]] represented by the [[Eilenberg-MacLane spectrum]] $K(-,\mathbb{Z})$. To the extent that [[integral cohomology]] is often just called _cohomology_ when the context is clear, _ordinary differential cohomology_ is often called just _differential cohomology_ .

Ordinary differential cohomology classifies [[circle n-bundles with connection]]. In low degree these are ordinary [[circle bundle]]s with [[connection on a bundle|connection]]. In the next degree they are [[circle 2-group]] [[principal 2-bundles]] / [[bundle gerbes]] with [[connection on a 2-bundle|2-connection]].

## Properties

Here we write $H_{diff}^\bullet(X)$ for the ordinary differential cohomology groups of a [[smooth manifold]] $X$.

### Curvature and characteristic class
 {#CurvatureAndCharClass}
 {#AbstractProperties}

There are two [[natural transformation|natural]] morphisms:

1. The underlying [[characteristic class]]

   $$
     c : H_{diff}^\bullet(X) \to H^\bullet(X,\mathbb{Z})
   $$

   produces the class in [[integral cohomology]] that underlies a differential cocycle;

   * for $H^2_{diff}(X)$ this is called the first [[Chern class]] of a [[line bundle]];

   * for $H^3_{diff}(X)$ this is called the [[Dixmier-Douady class]] of the corresponding [[bundle gerbe]].

1. The [[curvature]]

   $$
    F : H_{diff}^\bullet(X) \to \Omega^\bullet_{cl}(X)
   $$

   produces a closed [[differential form]] of degree $n$. This happens to land in closed differential forms with integral [[period]]s (see [below](#AbstractProperties)).

The following is either a definition, if regarded as an axiomatic characterization of ordinary differential cohomology, or it is a proposition, if regarded as a property of one of the [models](#Models).

Let $X$ be a [[smooth manifold]] and $n \in \mathbb{N}$ with $n \geq 1$ Write

* $H_{diff}^n(X)$ for the ordinary differential cohomology of $X$ in degree $n$;

* $\Omega^{n-1}(X)$ for the collection of [[differential form]]s of degree $n-1$;

* $\Omega^{n-1}_{int}(X)$ for the collection of [[differential form]]s $\omega$ of degree $n-1$ that are closed and whose [[period]]s are integral: for every $\gamma : S^{n-1} \to X$ we have the [[integral]] $\int_{S^{n-1}} \gamma^* \omega \in \mathbb{Z} \hookrightarrow \mathbb{R}$. Similarly for $\Omega^n_{int}(X)$.

* $H^n(X, \mathbb{Z})$ and $H^n(X, U(1))$ for the ordinary cohomology (for instance modeled as [[singular cohomology]]) of $X$ with coefficients in the [[integer]]s or the [[circle group]] (regarded as a [[discrete group]]), respectively.

All of these sets are [[abelian group]]s: the forms under addition of forms, and the differential cohomology classes are defined or proven (depending on the approach, see above) to have abelian group structure such that  the maps to curvatures and characteristic classes, from [above](#CurvatureAndCharClass) are [[homomorphism]]s of [[abelian group]]s.

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

The differential cohomology $H_{diff}^n(X)$ of $X$ fits into [[short exact sequence]]s of abelian groups

1. **curvature exact sequence**

   \[
     \label{CurvatureSequence}
     0 \to H^{n-1}(X, U(1)) \to H^n_{diff}(X) \stackrel{F}{\to} \Omega_{int}^n(X) \to 0
   \]

1. **characteristic class exact sequence**

   \[
      \label{CharacteristicClassSequence}
      0
       \to
      \Omega^{n-1}(X)/\Omega^{n-1}_{int}(X)
       \to
      H_{diff}^n(X)
       \stackrel{c}{\to}
      H^n(X, \mathbb{Z})
       \to
      0
      \,.
   \]

=--

The first sequence (eq:CurvatureSequence) says in words: two circle $(n-1)$-bundles $n$ whose [[curvature]] coincides differ by a _flat_ circle (n-1)-bundle.

The second sequence (eq:CharacteristicClassSequence) says in words: two connections on the same circle $(n-1)$-bundle differ by a globally defined connection $(n-1)$-form, well defined up to addition of a form with integral periods.

More is true: both these sequences interlock to form the hexagonal _[[differential cohomology diagram]]_ of ordinary differential cohomology. For more see at _[differential cohomology diagram -- Examples -- Deligne coefficients](differential+cohomology+diagram#DeligneCoefficients)_.

### Models
  {#Models}

There are various equivalent [[cocycle]]-models for ordinary differential cohomology. They include

* [[Cheeger-Simons differential character]]s;

* [[Deligne cohomology]];

* [[circle n-bundles with connection]].

The last of these are often known as $U(1)$-[[gerbe]]s or [[bundle gerbe]]s with connection.

Cocycles $\nabla \in H_{diff}^2(X)$ in degree 2 ordinary differential cohomology are represented by ordinary [[circle group]]-[[principal bundle]]s with [[connection on a bundle|connection]] on $X$.  The class $c(\nabla) \in H^2(X,\mathbb{Z})$ is the [[Chern class]] of the underlying circle bundle and the form $F_\nabla \in \Omega^2_{cl}(X)$ is the [[curvature]] 2-form of the connection $\nabla$.

## Fiber integration

see

* [[fiber integration in ordinary differential cohomology]]

## Applications

### In Chern-Weil theory

For $X$ a [[smooth manifold]], $G$ a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$ and [[invariant polynomial]] $\langle -\rangle$ of degree $2n$, the [[Chern-Weil homomorphism]] may be refined to a morphism

$$
  H^1(X,G) \to H_{diff}^{2n}(X)
$$

from the first [[nonabelian cohomology]] of $X$ classifying $G$-[[principal bundle]]s to degree $2n$ ordinary differential cohomology.

The projection

$$
  H^1(X,G) \to H_{diff}^{2n}(X) \stackrel{c}{\to} H^{2n}(X,\mathbb{Z})
$$

is the integral [[characteristic class]] corresponding to the invariant polynomial and the projection

$$
  H^1(X,G) \to H_{diff}^{2n}(X) \stackrel{F}{\to} \Omega^{2n}_{cl}(X)
$$

is a differential form which represents the image of this class under $H^{2n}(X,\mathbb{Z}) \to H^{2n}(X,\mathbb{R})$ in [[de Rham cohomology]] (under the [[de Rham theorem]]).

### In physics

In [[physics]]

* the [[electromagnetic field]] is a cocycle in degree 2 ordinary differential cohomology

* the [[Kalb-Ramond field]] is a cocycle in degree 3;

* the [[supergravity C-field]] is a cocycle in degree 4.

In abelian [[higher dimensional Chern-Simons theory]] in dimension $(4k+3)$ a field configuration is a cocycle in ordinary differential geometry of degree $(2k+2)$, for $k \in \mathbb{N}$.

## In a general abstract context

Ordinary differential cohomology (and indeed a [[cocycle]] model thereof) is defined generally [[internalization|internal]] to any [[cohesive (∞,1)-topos]] $\mathbf{H}$. This is discussed at

* [cohesive (∞,1)-topos -- structures -- differential cohomology](http://ncatlab.org/nlab/show/cohesive%20%28infinity,1%29-topos%20--%20structures#DifferentialCohomology).

For the case $\mathbf{H} = $ [[Smooth∞Grpd]] this intrinsic definition reproduces the [[Deligne complex]] model. This is discussed at

* [Smooth∞Grpd -- structures -- differential cohomology](http://ncatlab.org/nlab/show/smooth%20infinity-groupoid%20--%20structures#StrucDifferentialCohomology)

## Related concepts

* [[twisted ordinary differential cohomology]]

* [[equivariant ordinary differential cohomology]]

* [[arithmetic Chow group]]

## References

For more and more original references see:

* _[[Deligne cohomology]]_

* _[[Cheeger-Simons differential characters]]_

* *[[differential cohomology]]*

Lecture notes:

* [[Byungdo Park]], *Differential cohomology and gerbes: An introduction to higher differential geometry*, lecture notes (2023) [[pdf](https://byungdo.github.io/seminars/IMSRS.pdf), [[Park-DifferentialCohomology.pdf:file]]]

  > (emphasis on relation to [[bundle gerbes]] [[connection on a bundle gerbe|with connection]])

For discussion in the broader context of generalized [[differential cohomology]] see

* [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_ ([arXiv](http://arxiv.org/abs/math.AT/0211216))

  * A pedestrian introduction of ordinary differential cocycles is in section 2.3 there.

  * The systematic construction and definition via a [[homotopy pullback]] is in section 3.2.

  * The relation to Chern-Weil theory is in section 3.3.

A characterization by the two characteristic exact sequences is discussed in

* [[James Simons]], [[Dennis Sullivan]], _Axiomatic characterization of ordinary differential cohomology_,  J Topology (2008) 1 (1): 45-56 ([arXiv:math/0701077](https://arxiv.org/abs/math/0701077), [doi:10.1112/jtopol/jtm006]( https://doi.org/10.1112/jtopol/jtm006) [web pdf](http://jtopol.oxfordjournals.org/content/1/1/45.full.pdf+html))

Discussion of [[equivariant ordinary differential cohomology]]

* [[Ernesto Lupercio]], [[Bernardo Uribe]], _Deligne Cohomology for Orbifolds, discrete torsion and B-fields_, in: _Geometric and topological methods for quantum field theory_ Proceedings, Summer School, Villa de Leyva, Colombia, July 9-27, 2001 ([arXiv:hep-th/0201184](https://arxiv.org/abs/hep-th/0201184), [spire:582101](https://inspirehep.net/literature/582101))

* [[Kiyonori Gomi]], _Equivariant smooth Deligne cohomology_, Osaka J. Math. Volume 42, Number 2 (2005), 309-337 ([euclid:ojm/1153494380](https://projecteuclid.org/euclid.ojm/1153494380))

* Andreas Kübel, [[Andreas Thom]], _Equivariant Differential Cohomology_, Transactions of the American Mathematical Society (2018) ([doi:10.1090/tran/7315](https://doi.org/10.1090/tran/7315), [arXiv:1510.06392](https://arxiv.org/abs/1510.06392))

* [[Corbett Redden]], _Differential Borel equivariant cohomology via connections_, New York Journal of Mathematics, Volume 23 (2017) 441-487 ([nyjm:23-20](http://nyjm.albany.edu/j/2017/23-20.html), [arXiv:1602.06921](https://arxiv.org/abs/1602.06921))

* [[Corbett Redden]], _An alternate description of equivariant connections_, Differential Geometry and its Applications Volume 56, February 2018, Pages 81-94 ([doi:10.1016/j.difgeo.2017.11.003](https://doi.org/10.1016/j.difgeo.2017.11.003) [arXiv:1608.01297](https://arxiv.org/abs/1608.01297))

* [[Byungdo Park]], [[Corbett Redden]], _A classification of equivariant gerbe connections_, Communications in Contemporary Mathematics **21** 02 1850001 (2019) [[doi:10.1142/S0219199718500013](https://doi.org/10.1142/S0219199718500013)]

* Cheng-Yong Du Xiaojuan Zhao, _Spark and Deligne-Beilinson cohomology on orbifolds_, Journal of Geometry and Physics Volume 104, June 2016, Pages 277-290 ([doi:10.1016/j.geomphys.2016.02.011](https://doi.org/10.1016/j.geomphys.2016.02.011))

* Cheng-Yong Du, Lili Shen, Xiaojuan Zhao, _Spark complexes on good effective orbifold atlases categorically_, Theory and Applications of Categories, 33(26):784-812, 2018 ([tac:33-26](http://www.tac.mta.ca/tac/volumes/33/26/33-26abs.html), [arXiv:1708.07551](https://arxiv.org/abs/1708.07551))

Discussion of variants of [[differential concretification|differentially concretified]] [[infinity-stack|higher]] [[moduli stacks]] of [[ordinary differential cohomology]] (higher [[bundle gerbes with connection]]) with application to [[higher gauge theory]]:

* [[Severin Bunk]], [[Carlos S. Shahbazi]], *Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology* [[arXiv:2304.06633](https://arxiv.org/abs/2304.06633)]

[[!redirects differential ordinary cohomology]]


ordinary differential equation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Equality and Equivalence
+-- {: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

An **ordinary differential equation** is a [[differential equation]] involving [[derivatives]] of a [[function]] with respect to one argument only, i.e. the function is on a [[manifold]] only of [[dimension]] $d = 1$. This function can be vector valued, what is sometimes viewed as a system of possibly coupled equations; still all of them have the derivatives taken with respect to the same parameter. (Note that a higher-order differential equation can be turned into a system of first-order equations.)

## Existence and uniqueness (Picard--Lindelöf theorem)

A basic theorem concerns existence and uniqueness of local solutions to initial value problems. Let $X$ be a [[Banach space]]; given $(t_0, y_0) \in \mathbb{R} \times X$ and $a, r \gt 0$, put $Q \coloneqq [t_0 - a, t_0 + a] \times B_r(y_0)$m where $B_r(y_0)$ is the closed [[ball]] in $X$ of radius $r$ about $y_0$.

+-- {: .num_theorem}
###### Theorem
**(Picard--Lindelöf)**
Suppose $f: Q \to X$ is a [[function]] satisfying the following conditions:

* ([[continuous function|Continuity]] in $t$): Given any $x \in B_r(y_0)$, the function $f(-,x)$ (that is $t \mapsto f(t, x)$) is continuous from $[t_0 - a, t_0 + a]$ to $X$.

* ([[Lipschitz continuous function|Lipschitz continuity]] in $y$): There is a Lipschitz constant $L$ such that
$$ {\|f(t, x) - f(t, x')\|} \leq L{\|x - x'\|} $$
for all $(t, x) \in Q$;

* ([[bounded function|Boundedness]]): There is a constant $K$ such that $\sup_{(t, x) \in Q} \|f(t, x)\| \leq K$.

Then for any $c \leq \min(a, r/K)$, there exists [[uniqueness|exactly one]] solution $y: [t_0 - c, t_0 + c] \to X$ to the initial value problem

$$ y'(t) = f(t, y(t)), \qquad y(t_0) = y_0 .$$
=--

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

We will define an [[infinite sequence]] of approximate solutions to the problem, prove that its [[convergence|limit]] exists, prove that this limit is an exact solution, and prove that this solution is unique.

1. The infinite sequence is given by __Picard iteration__:
Starting with the given constant $y_0$, [[recursion|recursively]] define

   $$ y_{n+1}(t) \coloneqq y_0 + \int_{t=t_0} f(t, y_n(t)) \,\mathrm{d}t ;$$

   that is, $y_{n+1}$ is that [[indefinite integral]] of $f(-,y_n)$ that takes the correct initial value.  (To define $y_1$, use [[abuse of notation]] to interpret $y_0(t)$ as $y_0$; that is, think of $y_0$ as a [[constant function]].)  To prove that this integral exists, use the continuity conditions and an [[inductive proof]] that each $y_n$ is continuous to show that we are integrating a continuous function.

2. Thinking of Picard iteration as an operator between Banach spaces of continuous functions, use Lipschitz continuity and boundedness to show that the [[Banach fixed point theorem]] applies, so that the sequence $(y_1, y_2, \ldots)$ [[uniform convergence|uniformly converges]] to a limit $y$.

3. Since uniform convergence of continuous functions behaves well with integration, the limiting instance of Picard iteration holds:

   $$ y(t) = y_0 + \int_{t=t_0} f(t, y(t)) \,\mathrm{d}t .$$

   By differentiating with respect to $t$, and by evaluating at $t_0$, we confirm that $y$ is a solution of the initial value problem.

4. Given any putative solution $z$, apply [[Groenwall inequality|Grönwall's inequality]] to $z - y$ to prove that $z = y$.
=--

## See also

* [[inverse function theorem]]

* [[real quadratic function]]

* [[continuous square root]]

## References

* The [English Wikipedia](https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem) has a more detailed proof.  (They assume that $X$ is $\mathbb{R}$, but this is not essential.)

[[!redirects ordinary differential equation]]
[[!redirects ordinary differential equations]]
[[!redirects ODE]]
[[!redirects ODEs]]
[[!redirects ODE-s]]
[[!redirects ODE's]]
[[!redirects ODE\'s]]
[[!redirects ODE/'s]]
[[!redirects ODE's]]

[[!redirects Picard iteration]]
[[!redirects Picard iterate]]
[[!redirects Picard iterates]]

[[!redirects Picard-Lindelof theorem]]
[[!redirects Picard-Lindelöf theorem]]
[[!redirects Picard-Lindeloef theorem]]
[[!redirects Picard–Lindelof theorem]]
[[!redirects Picard–Lindelöf theorem]]
[[!redirects Picard–Lindeloef theorem]]
[[!redirects Picard--Lindelof theorem]]
[[!redirects Picard--Lindelöf theorem]]
[[!redirects Picard--Lindeloef theorem]]


ordinary homology

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

#Contents#
* table of contents
{:toc}

## Idea

_Ordinary homology_ is [[generalized homology]] with respect to an [[Eilenberg-MacLane spectrum]] $H A$, and often understood over $H \mathbb{Z}$.

Equivalently this is computed by _[[singular homology]]_ with [[coefficients]] in $A$.

## Properties

### Relation to homotopy groups

Ordinary homology with [[coefficients]] in $\mathbb{Z}$ of a [[topological space]] $X$  serves to approximate the [[homotopy groups]] of $X$.

See for instance at _[[Hurewicz theorem]]_

### Description in terms of higher linear algebra
 {#InTermsOfHigherLinearAlgebra}

We discuss ([[twisted cohomology|twisted]]) ordinary homology and [[ordinary cohomology]] in terms of sections of [[(∞,1)-module bundles]] over the [[Eilenberg-MacLane spectrum]].

Let $k$ be a [[commutative ring]]. Write

$$
  H k \in CRing_\infty
$$

for the [[Eilenberg-MacLane spectrum]] of $k$, canonically regarded as an [[E-∞ ring]]. Write

$$
  (H k) Mod \in (\infty,1)Cat
$$

for the [[(∞,1)-category of (∞,1)-modules]] over $H k$.

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

There is an [[equivalence of (∞,1)-categories]]

$$
 (H k)Mod \simeq L_{qi} Ch_\bullet(k Mod)
$$

between the [[(∞,1)-category of (∞,1)-modules]] over the [[Eilenberg-MacLane spectrum]] $H k$ and the [[simplicial localization]] of the [[category of unbounded chain complexes]] of ordinary (1-categorical) $k$-[[modules]].

=--

This is the statement of the _[[stable Dold-Kan correspondence]]_, see at _[(∞,1)-category of (∞,1)-modules -- Properties -- Stable Dold-Kan correspondence](%28infinity%2C1%29-module#StableDoldKanCorrespondence)_.

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

Let $X$ be a [[topological space]] and write $\Pi(X) \in $ [[∞Grpd]] for its underlying [[homotopy type]] (its [[fundamental ∞-groupoid]]). Then we say that an [[(∞,1)-functor]]

$$
  \Pi(X) \to (H k) Mod
$$

is (the [[higher parallel transport]]) a _[[flat (∞,1)-bundle|flat]] [[(∞,1)-module bundle]]_ over $X$, or a _[[local system]]_ of $H k$-[[(∞,1)-modules]] over $X$.

=--

+-- {: .num_prop}
###### Proposition
**([[Riemann-Hilbert correspondence]])**

If $X$ is an [[orientation|oriented]] [[closed manifold]], then there is an [[equivalence of (∞,1)-categories]]

$$
  [\Pi(X), (H k)Mod]
  \simeq
  (T X)Mod
$$

between [[flat (∞,1)-bundle|flat]] [[(∞,1)-module bundles]]/[[local systems]] and [[L-∞ algebroid representations]] of the [[tangent Lie algebroid]] of $X$. From right to left the equivalece is established by sending an [[L-∞ algebroid representation]] given (as discussed there) by a flat $\mathbb{Z}$-graded connection on bundles of chain complexes (via prop. \ref{DoldKan}), to its [[higher holonomy]] defined in terms of [[iterated integrals]].

=--

This is the main theorem in ([Block-Smith 09](#BlockSmith09)).

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

Write

$$
  \Gamma \;\coloneqq\; \underset{\to}{\lim} \;\colon\;
  [\Pi(X), (H k)Mod]
  \to
  (H k) Mod
$$

for the [[(∞,1)-colimit]] functor.

=--

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

We may think of $\Gamma$ equivalently as

* forming flat [[sections]] of a [[flat (∞,1)-bundle|flat]] [[(∞,1)-module bundle]];

* sending a [[flat (∞,1)-bundle|flat]] [[(∞,1)-module bundle]] to its _[[Thom spectrum]]_ (see at _[Thom spectrum -- For (∞,1)-module bundles ](Thom+spectrum#ForInfinityModuleBundles)_).

=--

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

Write

$$
  \mathbb{I}_X^{H k} \;\colon\; \Pi(X) \to (K k)Mod
$$

for the [[flat (∞,1)-bundle|flat]] [[(∞,1)-module bundle]] which is constant on the chain complex concentrated on $k$ in degree 0, the [[tensor unit]] in $[\Pi(X), (H k)Mod] \simeq [\Pi(X), L_{qi}Ch_\bullet(k Mod)]$.

=--

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

For $X$ a [[topological space]], we have a [[natural equivalence]] (with the identification of prop. \ref{DoldKan} understood) of the form

$$
  \Gamma(\mathbb{I}_X^{H k})
  \simeq
  C_\bullet(X,k)
  \,,
$$

between the $(H k)$-[[(∞,1)-module]] of sections of the trivial $(H k)$-[[(∞,1)-module bundle]] $\mathbb{I}_X^{H k}$ and the [[singular chain complex]] of $X$ for ordinary homology with [[coefficients]] in $k$.

=--

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

This is a classical basic (maybe [[folklore]]) statement. Here is one way to see it in full detail.

First notice that the [[(∞,1)-colimit]] of functors out of [[∞-groupoids]] and constant on the [[tensor unit]] in $(H k)Mod$ is by definition the [[(∞,1)-tensoring]] operation of $(H k)Mod$ over [[∞Grpd]]. Now if we find a [[presentable (∞,1)-category|presentation]] of $(H k)Mod$ by a [[simplicial model category]] the by the dicussion at _[(∞,1)-colomit -- Tensoring and cotensoring -- Models](limit+in+a+quasi-category#ModelsForTensoring)_ this [[(∞,1)-tensoring]] is given by the left [[derived functor]] of the [[sSet]]-[[tensoring]] in that simplicial model category.

To obtain this, use prop. \ref{DoldKan} and then the discussion at _[[model structure on chain complexes]]_ in the section _[Projective model structure on unbounded chain complexes](model%20structure%20on%20chain%20complexes#ProjectiveModelStructureOnUnboundedChainComplexes)_ which says that there is a [[simplicial model category]] structure on the [[category of simplicial objects]] in the [[category of unbounded chain complexes]] which models $L_{qi} Ch_\bullet(k Mod)$, and whose weak equivalences are those morphisms that produce [[quasi-isomorphism]] under the [[total chain complex]] functor.

In summary it follows that with any [[simplicial set]] $(\Pi(X))_\bullet in L_{whe} sSet$ representing $\Pi(X)$ (under the [[homotopy hypothesis]]-theorem) we have

$$
  \underset{\to}{\lim} (\mathbb{I}_X^{H k})
  \simeq
  \int_{[n] \in \Delta}  (\Pi(X))_n \cdot \mathbb{I}
  \,,
$$

where on the right we have the [[coend]] over the [[simplex category]] of the [[tensoring]] (of [[simplicial sets]] with [[simplicial objects]] in the [[category of unbounded chain complexes]]) of the standard cosimplicial simplex with the simplicial diagram constant on the [[tensor unit]] [[chain complex]].

The result on the right is manifestly, by the very definition of [[singular homology]], under the ordinary [[Dold-Kan correspondence]] the [[chain complex of singular simplices]]:

$$
  \int_{[n] \in \Delta}  (\Pi(X))_n \cdot \mathbb{I}
  \simeq
  \Pi(X) \otimes k
  \simeq
  C_\bullet(X,k)
  \,.
$$

=--

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

If $X \in L_{whe} Top$ is a [[Poincaré duality space]] of [[dimension]] $n$, then $\Gamma(\mathbb{I}_X^{H A}) \in (H A) Mod$ is a [[dualizable object]] which is almost self-dual except for a degree [[twisted cohomology|twist]] of (itself) degree 0:

$$
  \left(
     \Gamma\left(\mathbb{I}_X^{H A}\right)
  \right)^\vee
  \simeq
  \Sigma^{-n}
  \left(
     \Gamma\left(\mathbb{I}_X^{H A}\right)
  \right)
  \,.
$$

=--

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

Under the identifications of prop. \ref{DoldKan} and prop. \ref{SectionsOfTrivialKABundleIsAHomology} this is the theorem discussed at _[[Poincaré duality]]_ in the section _[Poincaré duality -- Refinement to homotopy theory](Poincar%C3%A9+duality#RefinementInHomotopyTheory)_.

=--

See also at _[[Dold-Thom theorem]]_.

### Ordinary homology spectra split

see at _[[ordinary homology spectra split]]_

## Related concepts

* [[singular homology]], [[Čech homology]]

* [[cellular homology]]

* [[Dold-Thom theorem]]

* [[ordinary cohomology]]

* [[string topology]]

* [[Pontrjagin ring]]

* [[stable homotopy homology theory]]

* [[bordism homology theory]]

* [[Pochhammer loop]]

## References

A classical account:

* {#Switzer75} [[Robert Switzer]], chapter 10 of: _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

In terms of [[current (distribution theory)|currents]]:

* [[Georges de Rham]], Chapter IV of: *Differentiable Manifolds -- Forms, Currents, Harmonic Forms*, Grundlehren **266**, Springer (1984) [[doi:10.1007/978-3-642-61752-2](https://doi.org/10.1007/978-3-642-61752-2)]

Comprehensive monograph:

* [[Jean Gallier]], [[Jocelyn Quaintance]], *Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry*, World Scientific (2022) [[doi:10.1142/12495](https://doi.org/10.1142/12495), [webpage](https://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html)]

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Chapter 3 of _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

The [[Riemann-Hilbert correspondence]]/[[de Rham theorem]] for $H A$-modules is established in

* {#BlockSmith09} [[Jonathan Block]], Aaron Smith, _A Riemann--Hilbert correspondence for infinity local systems_ ([arXiv:0908.2843](http://arxiv.org/abs/0908.2843))

[[!redirects ordinary homology theory]]


ordinary homology spectra split

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

#Contents#
* table of contents
{:toc}

## Statement

For $S$ any [[spectrum]] and $H A$ an [[Eilenberg-MacLane spectrum]],
then the [[smash product of spectra|smash product]] $S\wedge H A$ (the $A$-[[ordinary homology]] spectrum)
is non-canonically equivalent to a product of EM-spectra (hence a [[wedge sum]] of EM-spectra in the finite case).

([Adams 74, part II, lemma 6.1](#Adams74))

A variant for [[generalized (Eilenberg-Steenrod) cohomology]]:

Let $X$ be a [[topological space]] such that each of the [[ordinary homology|ordinary]] [[homology groups]] $H_n(X,\mathbb{Z})$ is a [[free abelian group]] on genrators $\{h_{\alpha,n}\}_{\alpha \in B_n}$. Write $c_{\alpha,n} \in H^n(X,\mathbb{Z}) \simeq Hom(H_n(X,\mathbb{Z}), \mathbb{Z})$ for the corresponding [[dual basis]].

Let $E$ be a [[multiplicative cohomology theory]] and write $h'_{n,\alpha} \in (\tau_{\leq 0} E)_n(X)$ and $c'_{n,\aloha} \in (\tau_{\leq 0} E)^n(X) $for the images of these generators under the [[0-truncated]] unit map

$$
  H \mathbb{Z} \simeq \tau_{\leq 0} \mathbb{S} \stackrel{}{\longrightarrow} \tau_{\leq 0} E
  \,.
$$

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

If one of the following conditions is satisfied

* Each $h'_{n,\alpha}$ lifts through $E_n(X) \to (\tau_{\leq 0}E)_n(X)$;

* each $H_n(X,\mathbb{Z})$ is [[finitely generated module|finitely generated]] and each $c'_{n,\alpha}$ lifts through $E^n(X) \to (\tau_{\leq 0}E)^n(X)$,

then there are non-canonical [[equivalence in an (infinity,1)-category|equivalences]] as follows:

1. $E \wedge \Sigma^\infty X_+ \simeq \underset{n,\alpha}{\vee} \Sigma^n E \;\; \in E Mod$ ;

1. $[X,E] \simeq \underset{n,\alpha}{\prod} \Sigma^{-n} E$;

1. $E_\bullet(X) \simeq \pi_\bullet(E) \otimes H_\bullet(X,\mathbb{Z})$

   and

   $E^\bullet(X)\simeq Hom(H_\bullet(X), \pi_\bullet(E))$

=--

([Lurie 10, lecture 4, prop. 7](#Lurie10), [Adams 74, part II, lemma 2.5](#Adams74))

## Related concepts

* [[splitting principle]]

## References

* {#Adams74} [[Frank Adams]], _[[Stable homotopy and generalised homology]]_, 1974

* {#Lurie10} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010 ([web](http://www.math.harvard.edu/~lurie/252x.html)), Lecture 4 _[[complex oriented cohomology theory|Complex-oriented cohomology theories]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf))

[[!redirects homology spectra that split]]

[[!redirects homology spectra split]]


ordinary mathematics

In the [[foundations of mathematics]], one sometimes comes across the phrase 'ordinary mathematics'.  This seems to mean mathematics based on foundations that are no stronger than necessary to do the standard mathematics of the 20th century.

In particular, ordinary mathematics does not use high-powered axioms such as the [[axiom of replacement]] or [[large cardinals]], thus excluding [[model theory]] and those parts of [[category theory]] that use [[Grothendieck universes]].  Limiting oneself to ordinary mathematics in this sense may be seen as a weak form of [[predicative mathematics]].

One might define it (if a formal definition is desired) as mathematics that can be formalised in [[ETCS]]; [Wikipedia](http://en.wikipedia.org/wiki/Ordinary_mathematics) defines it as mathematics that takes place in the [[von Neumann universe]] of rank $\omega + \omega$ (which is a model of $ETCS$).  One can also think of this as allowing only finitely many applications of the [[power set axiom]] (since replacement is needed to iterate this transfinitely) starting with the set of [[natural numbers]].

Compare [[classical mathematics]].  If [[constructive mathematics]] is interpreted as merely doing without the [[axiom of choice]] and [[excluded middle]] (without any attempt to be predicative), then the classical/constructive divide is independent of being ordinary.  (However, both "classical" and "constructive" are also used with connotations along the lines of ordinariness.)

More generally, in the context of any "unusual" sort of mathematics (not limited to using high-powered foundational axioms), the term "ordinary mathematics" might be used informally to refer to mathematics without that particular unusual aspect.


Ore condition

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}

### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

[[!redirects right Ore condition]]

## Idea

The (right) **Ore condition** is a simple condition on the morphisms in a category $\mathcal{C}$ in order ensure that sieves generated by singletons $\{ f\}$ behave well under [[pullback]]. It can be viewed as weaker form of the existence of pullbacks in $\mathcal{C}$.

## Definition

+-- {: .num_def #ore}
###### Definition
A [[category]] $\mathcal{C}$ is said to satisfy the (right) **Ore condition** if for any diagram
$$
\array{
  & & A \\
  & & \downarrow\\
  B & \to & C
}
$$
there is an object $D$ and arrows $D \to A, B$ such that the following diagram commutes:
$$
\array{
  D & \to & A \\
  \downarrow & & \downarrow\\
  B & \to & C
}
$$
=--

## Properties

* A category $\mathcal{C}$ obviously satisfies the Ore condition when it has pullbacks.

* When $S$ is a [[sieve]] generated by a singleton $\{ f\}$ then the pullback $h^\ast (S)$ is nonempty provided $\mathcal{C}$ satisfies the Ore condition. More generally, a category $\mathcal{C}$ satisfies the Ore condition precisely when the collection of nonempty sieves forms a [[Grothendieck topology]] on $\mathcal{C}$ (cf. [[atomic site]]).

* A [[presheaf topos]] $Set^{\mathcal{C}^{op}}$ is a [[De Morgan topos]] precisely if $\mathcal{C}$ satisfies the Ore condition (cf. [[De Morgan topos]]).

## Remark

A category $\mathcal{C}$ satisfies the [[amalgamation property]] precisely if ${\mathcal{C}^{op}}$ satifies the Ore condition. Since the former is an important property in [[model theory]], the De Morgan property is via the Ore condition dually bound to play a similar role.

## Related entries

* [[atomic site]]

* [[De Morgan topos]]

* [[amalgamation property]]

## Reference

* [[Peter Johnstone]], _Sketches of an [[Elephant]] vols. I,II_, Oxford UP 2002. (pp.79f, 1001)


Ore domain

#Contents#
* table of contents
{:toc}

## Definition

In [[algebra]], a unital [[ring]] is an __Ore domain__ if it is a [[domain (ring theory) | domain]] $R$ in which the set $R^\times$ of all nonzero elements is an [[Ore set]]. Thus one can form the [[Ore localization]] $R[(R^\times)^{-1}]$ which is then a [[skew-field]] ([[division ring]]), called the _Ore quotient ring_ (Ore quotient (skew)field). As [[Ore localizations]] of domains always do, it comes with a map $R\to R[(R^\times)^{-1}]$ which is 1-1. For most purposes, one sided Ore condition is sufficient, hence one considers also the weaker notions of left and right Ore domains.

## Related concepts

* [[Dedekind domain]]

category: algebra

[[!redirects Ore domains]]
[[!redirects Ore quotient]]