absolute convergence

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### Context
#### Analysis
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[[!include analysis - contents]]
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\tableofcontents

## Definition

### For series in the real numbers

In [[real analysis]], given a sequence $a:\mathbb{N} \to \mathbb{R}$ in the [[real numbers]], the series $\sum_{n = 0}^\infty a_n$ is **absolutely convergent** if the [[sequence]] of [[partial sums]] is a [[Cauchy sequence]], and the [[sequence]] of [[partial sums]] of the series $\sum_{n = 0}^\infty \vert a_n \vert$ is also a Cauchy sequence.

### For series in Banach spaces

In [[functional analysis]], given a sequence $a:\mathbb{N} \to B$ in a [[Banach space]] $B$, the series $\sum_{n = 0}^\infty a_n$ is **absolutely convergent** if the [[sequence]] of [[partial sums]] is a [[Cauchy sequence]] in $B$, and the [[sequence]] of [[partial sums]] of the series $\sum_{n = 0}^\infty \Vert a_n \Vert$ is a Cauchy sequence in the real numbers.

## See also

* [[conditional convergence]]
* [[Banach space]]

## External links

* Wikipedia, *[Absolute convergence](https://en.wikipedia.org/wiki/Absolute_convergence)*

[[!redirects converges absolutely]]
[[!redirects absolutely converges]]
[[!redirects absolutely convergent]]

[[!redirects absolutely convergent series]]


absolute de Rham cohomology

#Contents#
* table of contents
{:toc}

## Idea

Absolute de Rham cohomology is a [[de Rham cohomology]] theory for [[arithmetic motives]] which provides a spectral interpretation of [[L-functions]]. It is viewed as a phenomenon in hypothetical "absolute" geometry "over a field of one element". It is hoped by some that it may help to approach [[Riemann hypothesis]].

## Related concepts

* [[absolute cohomology]]

## References

* Nagoya conference [Witt vectors, foliations, and absolute de Rham cohomology](http://kskedlaya.org/conference2010/), 2010, [participants](http://kskedlaya.org/conference2010/participants.html), [schedule](http://kskedlaya.org/conference2010/schedule.html)

* Kedlaya overview slides [pdf](http://kskedlaya.org/slides/nagoya2010.pdf) from Nagoya talk

* MathOverflow question [Riemann hypothesis via absolute geometry](http://mathoverflow.net/questions/69389/riemann-hypothesis-via-absolute-geometry)


absolute differential form

# Absolute differential forms
* table of contents
{: toc}

## Idea

It\'s well known that one can integrate a [[differential form]] on an [[orientation|oriented]] [[submanifold]].  Less well known (but also true), one can integrate a [[differential pseudoform]] on an [[pseudoriented]] (transversely oriented) submanifold.  But in classical differential geometry, one also sees forms that can be integrated on *unoriented* submanifolds.

I call these _absolute_ forms.  The term 'absolute' suggests a lack of additional required structure, in this case some sort of orientation on the domain of integration.  It also suggests [[absolute value]], since many of the examples from classical differential geometry involve absolute values.  Indeed, we can define the absolute value of a form or a pseudoform to be an absolute form, although not every absolute form arises in this way.

The main theorem of absolute forms is that, if $\omega$ is a (pseudo)-$p$-form and $R$ is a (pseudo)-oriented $p$-dimensional submanifold, then
$$ {\left| \int_R \omega \right|} \leq \int_{|R|} {|\omega|} ,$$
where ${|\omega|}$ is an absolute $p$-form (the absolute value of $\omega$), $|{R}|$ is simply $R$ with its (pseudo)-orientation ignored, and the absolute value on the left is the ordinary absolute value of scalars.  This theorem also applies if we start with an absolute $p$-form $\omega$, (although in that case $R$ starts out unoriented and so is the same as ${|R|}$).  If $R$ is a [[de Rham cohomology|de Rham]] chain (a formal [[linear combination]] of appropriately oriented submanifolds), we also take absolute values of the formal coefficients in ${|R|}$.  (This operation does not respect the usual notion of equality of chains, but the theorem is true all the same.)

## Definitions

Let $X$ be a [[differentiable manifold]] (or similar sort of space), and let $p$ be a [[natural number]] (typically $0 \leq p \leq n$, where $n$ is the [[dimension]] of $X$).  Recall that an (exterior differential) __$p$-[[exterior differential form|form]]__ $\omega$ on $X$ is a [[function]] that assigns a [[real number]] (or whatever is the relevant sort of scalar) $\omega_c(v_1,\ldots,v_p)$ to a point $c$ in $X$ and a $p$-[[tuple]] $(v_1,\ldots,v_p)$ of [[tangent vectors]] at $c$, [[multilinear map|multilinearly]] and [[alternating form|alternating]] in the $v_i$.  Similarly, a __$p$-[[pseudoform]]__ $\omega$ on $X$ is a function that assigns a scalar $\omega_c^o(v_1,\ldots,v_p)$ to a point $c$ in $X$, a local [[orientation]] $o$ at $c$, and a $p$-tuple $(v_1,\ldots,v_p)$ of tangent vectors at $c$, multilinearly and alternating in the $v_i$ and reversing sign under a reversal of $o$.

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

An __absolute $p$-form__ $\omega$ on $X$ is a function that assigns a scalar $\omega_c(v_1,\ldots,v_p)$ to a point $c$ in $X$ and a $p$-[[tuple]] $(v_1,\ldots,v_p)$ of [[tangent vectors]] at $c$ and that satisfies the following conditions:

1.  Fixing $c$, $\omega_c({-})$ shall be [[uniformly continuous map|uniformly continuous]].

2.  The $p$-tuple $(v_1,\ldots,v_p)$ shall be [[linearly independent subset|linearly independent]] if $\omega_c(v_1,\ldots,v_p) \ne 0$.  Thus, although $\omega_c$ is not linear, we may still call it [[alternating form|alternating]]; however (as a consequence of 3), it is actually [[symmetric function|symmetric]].

3.  Fix a $p$-dimensional subspace $S$ of the tangent space at $c$ and an orientation $o$ of $S$.  Now given a linearly independent $p$-tuple $(v_1,\ldots,v_p)$ from $S$ (that is a [[basis]] of $S$), let $\omega_c(v_1,\ldots,v_p)_S^o$ be $\pm\omega_c(v_1,\ldots,v_p)$ according to whether the orientation of $S$ induced by the $v_i$ matches $o$, and extend this by continuity to all $p$-tuples from $S$ (which extension must be unique and exists by 1&2).  The resulting function $\omega_c({-})_S^o$ shall be [[multilinear map|multilinear]] (and so also alternating, by 2).
=--

The multilinearity condition here is rather weaker than for a (pseudo)-form, since it applies only within a $p$-dimensional subspace.  Shifting one vector even slightly outside of $S$ loses all connection provided by multilinearity, which is why we need a continuity condition; continuity holds for (pseudo)-forms automatically.

An absolute $p$-form $\omega$ is __[[continuous map|continuous]]__ if it is jointly continuous in all of its data ($c$ as well as the $v_i$).  Since the domain of the function $\omega$ is a manifold (a [[vector bundle]] over $X$, although $\omega$ is not a map of vector bundles), we can even discuss [[differentiable map|differentiability]], [[smooth map|smoothness]], and even [[analytic map|analyticity]] of $\omega$ when $X$ has the relevant structure.

An absolute $0$-form is the same thing as a $0$-form.  An absolute $n$-form on an $n$-dimensional manifold $X$ is essentially the same thing as an $n$-pseudoform; with the notation from condition 3, the only possibility for $S$ is the entire tangent space $T_c{X}$, and we have
$$ \tilde\omega_c^o(v_1,\ldots,v_n) = \omega_c(v_1,\ldots,v_n)_{T_c{X}}^o $$
to relate the $n$-pseudoform $\tilde{\omega}$ to the absolute $n$-form $\omega$.  Finally, the only absolute $p$-form for $p \gt n$ is $0$.

At a point $c$, an absolute $p$-form $\omega$ is:

* __indefinite__ if $\omega_c(v_1,\ldots,v_p) \gt 0$ for some (necessarily [[linearly independent subset|linearly independent]]) $p$-tuple of vectors and $\omega_c(v_1,\ldots,v_p) \lt 0$ for some $p$-tuple,

* __semidefinite__ if not indefinite,

* __definite__ (and hence semidefinite) if $\omega_c(v_1,\ldots,v_p) \ne 0$ for every independent $p$-tuple of vectors at $c$,

* __positive__ (and hence semidefinite) if $\omega_c(v_1,\ldots,v_p) \geq 0$ for every $p$-tuple of vectors (it is enough when they are independent),

* __negative__ (and hence semidefinite) if $\omega_c(v_1,\ldots,v_p) \leq 0$ for every (independent) $p$-tuple of vectors.

All these are at a point $c$; $\omega$ satisfies the condition tout court if it holds for all $c$.

Given an absolute $p$-form $\omega$, its __[[absolute value]]__ ${|\omega|}$ is a positive semidefinite absolute $p$-form:
$$ {|\omega|}_c(v_1,\ldots,v_p) \coloneqq {|\omega_c(v_1,\ldots,v_p)|} .$$
If we start with a $p$-form $\omega$, then the same definition defines a positive absolute $p$-form ${|\omega|}$.  If we start with a $p$-pseudoform $\omega$, then essentially the same definition still works; we use either orientation to evaluate $\omega$ with the same result.  Note that ${|\omega|}$ is continuous if $\omega$ is.  However, we may *not* conclude that ${|\omega|}$ is differentiable just because $\omega$ is differentiable (or even analytic).  On the other hand, ${|\omega|}$ inherits differentiability properties from $\omega$ wherever $\omega \ne 0$.  (Even then, however, we cannot inherit analyticity, except in $1$ dimension.)

Given two absolute $p$-forms $\omega$ and $\eta$, their __sum__ $\omega + \eta$ is an absolute $p$-form:
$$ (\omega + \eta)_c(v_1,\ldots,v_p) \coloneqq \omega_c(v_1,\ldots,v_p) + \eta_c(v_1,\ldots,v_p) .$$
Given an absolute $p$-form $\omega$ and a scalar field $f$, their __product__ $f \omega$ is an absolute $p$-form:
$$ (f \omega)_c(v_1,\ldots,v_p) \coloneqq f(c) \omega_c(v_1,\ldots,v_p) .$$
In this way, the space of absolute $p$-forms is a [[module]] over the [[associative algebra|algebra]] of scalar fields and the space of [[sections]] of a [[vector bundle]].  For now, we decline to define products of absolute forms of aribtrary rank.

Given an absolute $p$-form $\omega$ on $X$, a manifold $U$, and a [[continuously differentiable map]] $R\colon U \to X$, the __pullback__ $R^*\omega$ is an absolute $p$-form on $U$:
$$ (R^*\omega)_c(v_1,\ldots,v_p) \coloneqq \omega_{R(c)}(R_*v_1,\ldots,R_*v_p) .$$
Here, $R_*v_i$ is the [[pushforward]] of $v_i$ under $R$.  Note that $R^*\omega$ is continuous if $\omega$ is; we can also pull back differentiability and analyticity properties that $\omega$ and $R$ both have.

Given a continuous absolute $p$-form $\omega$ on $X$, a $p$-dimensional manifold $U$, and a continuously differentiable map $R\colon U \to X$, the __integral__ $\int_R \omega$ is a scalar:
$$ \int_R \omega \coloneqq \int_U R^*\omega .$$
On the right-hand side, $R^*\omega$ is a continuous absolute $p$-form on $U$, but since $U$ is $p$-dimensional, this is essentially the same as a continuous $p$-pseudoform on $U$, and we already know how to integrate this (see [[integration of differential forms]]).

## Examples

Examples of absolute forms from classical differential geometry include:

*  Absolute $0$-forms are the same as ordinary $0$-forms.

*  Absolute $n$-forms on an $n$-[[dimensional]] manifold are the same as $n$-pseudoforms (and hence the same as [[absolutely continuous measure|absolutely continuous]] [[Radon measures]]).

*  In [[complex analysis]], ${|\mathrm{d}z|}$ is an absolute $1$-form sometimes used in [[contour integration]].  This literally is the absolute value of the differential of the identity map $z$.

*  More generally, the [[arclength]] element $\mathrm{d}s = {\|\mathrm{d}\mathbf{x}\|}$ on a [[Riemannian manifold]] is an absolute $1$-form.  Neither $\mathrm{d}s$ nor (in general) $\mathrm{d}\mathbf{x}$ is actually the differential of anything, but $\mathrm{d}\mathbf{x}$ is the canonical [[tangent vector|vector]]-valued $1$-form (which, on an [[affine space]], really is the differential of the [[identity map]] $\mathbf{x}$), and we really can use the metric to take the norm of such a form to get an absolute $1$-form.

*  Similarly, the [[surface area]] element $\mathrm{d}S$ on a Riemannian manifold is an absolute $2$-form, and we can continue into higher dimensions (although the classical [[volume element]] $\mathrm{d}V$ in $\mathbb{R}^3$ is already covered as a $3$-pseudoform).  In principle, we ought to be able to write down expression for $\mathrm{d}S$ etc in terms of $\mathrm{d}s$, although so far the only thing that I know how to do is $\mathrm{d}S = {\|{\mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}\|} / 2$, where $\hat\times$ indicates a wedge product of vector-valued forms whose vectors are multiplied by the [[cross product]]. (This can be generalized to any finite-dimensional area in any finite-dimensional Riemannian manifold; in particular, $\mathrm{d}V = {|{\mathrm{d}\mathbf{x} \hat\cdot \mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}|} / 6$.)

+-- {: .num_remark}
The reason that $\mathrm{d}s$ determines $\mathrm{d}S$ is that lengths determine areas, as through Heron\'s Formula.  However, our usual ways of deriving one form from another (thinking of them, at a given point, as maps from lists of vectors to numbers) involve only rerranging (permuting, duplicating, and contracting) the inputs and applying operations to the outputs.  But this cannot be sufficient; knowing only the lengths of two vectors $v$ and $w$ (which vectors are the inputs to $\mathrm{d}S$, and which lengths are the outputs of $\mathrm{d}s$ at those inputs) does not tell you the area of the parallelogram that they span (which is the output of $\mathrm{d}S$).  You also need the length of $v - w$ (or of $v + w$).  So while we can express $\mathrm{d}S_c$ ($\mathrm{d}S$ at a point $c$) as $(v, w \mapsto 2 H(\mathrm{d}s(v), \mathrm{d}s(w), \mathrm{d}s(v - w))$ (where $H$ is the Heron function, the real-valued function of three real variables that takes the lengths of the three sides of a triangle and returns its area), or equivalently as $(v, w \mapsto 2 H({\|v\|}, {\|w\|}, {\|{v - w}\|}))$, we cannot express $\mathrm{d}S$ with a formula anything along the lines of ${\|{\mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}\|} / 2$ using $\mathrm{d}s$ directly instead of $\mathrm{d}\mathbf{x}$.  (And so you may as well describe $\mathrm{d}S_c$ as $(v, w \mapsto {\|{v \times w}\|})$ instead.)
=--

## Related concepts

* exterior [[differential forms]] and [[pseudoforms]] (the more well-known variations);

* [[cojet differential forms]] (and [[cogerm differential forms]] more generally) generalize both exterior and absolute $1$-forms and pseudo-$1$-forms;

* [[coflare differential form]]s are a common generalization of exterior forms, absolute forms, and cojet forms (although not all cogerm forms).

## References

Near the end of a Usenet post from 2002, we see a definition of $\int_R {|\omega|}$ for $\omega$ a (pseudo)-$p$-form and $R$ a $p$-dimensional submanifold, but without a broader context for ${|\omega|}$ itself:

* [[Toby Bartels]] and Ralph Hartley; [Densitized Pseudo Twisted Forms](https://groups.google.com/group/sci.physics.research/msg/424da828e75b6b90?dmode=source)
  {#Usenet}

Apparently absolute $p$-forms (at least if continuous) are the same as even $p$-[[densities]] as defined by Gelfand; see this MathOverflow answer:

* Juan Carlos Álvarez Paiva; answer to [Why do I need densities in order to integrate on a non-orientable manifold?](http://mathoverflow.net/questions/90455/why-do-i-need-densities-in-order-to-integrate-on-a-non-orientable-manifold/90714#90714).
  {#MathOverflow}

[[!redirects absolute differential form]]
[[!redirects absolute differential forms]]
[[!redirects absolute form]]
[[!redirects absolute forms]]
[[!redirects differential absolute form]]
[[!redirects differential absolute forms]]

[[!redirects absolute value of a form]]
[[!redirects absolute value of a differential form]]
[[!redirects absolute value of a pseudoform]]
[[!redirects absolute value of a pseudo-form]]


absolute extensor

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#### Topology
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# Contents
* table of contents
{: toc}

## Idea

A [[topological space]] $K$ is called an _absolute extensor_ if for

1. $X$ any [[nice topological space|nice]] [[topological space]]

1. $A \overset{i}{\hookrightarrow} X$ any [[closed subspace]]

1. $f\colon A \longrightarrow K$ any [[continuous function]]

there is an [[extension]] to a [[continuous function]] $\tilde{f}:X\to K$, i.e., such that $\tilde{f}=f\circ i$:
$$
  \array{
    A &\overset{f}{\longrightarrow}& K
    \\
    {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\exists \tilde f}}
    \\
    X
  }
$$

Here "[[nice topological space]]" is variously taken to mean [[metrizable topological space]] or at least [[normal topological space]].

A variation of this concept, **absolute neighborhood extension**, only requires the extension to exist over a neighborhood of $A$ in $X$.

## Examples

The [[Tietze extension theorem]] implies that the [[real line]] $\mathbb{R}$ equipped with its [[Euclidean space]] [[metric topology]] is an absolute extensor. It follows that so are the [[closed interval]] [[subspace]] $[0,1] \subset \mathbb{R}$ and the [[circle]] $S^1$.

Products of absolute extensors are absolute extensors, including the [[Hilbert cube]].

The two point discrete space $S^0$ as well as any sphere $S^n$ is an absolute neighborhood extensor, but not an absolute extensor.

## Related concepts

* [[Tietze extension theorem]]

* [[quasi-finite CW-complex]]

* [[absolute retract]]

## References

* [[eom]], _[Absolute neighbourhood extensor](https://www.encyclopediaofmath.org/index.php/Absolute_neighbourhood_extensor)_

[[!redirects absolute extensors]]

[[!redirects absolute neighbourhood extensor]]
[[!redirects absolute neighbourhood extensors]]

[[!redirects absolute neighborhood extensor]]
[[!redirects absolute neighborhood extensors]]


absolute Galois group

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#### Group Theory
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#Contents#
* table of contents
{:toc}

## Idea

The _absolute Galois group_ of a [[field]] $k$ is that of the [[field extension]] $k \hookrightarrow k_s$ which is the [[separable closure]] of $k$. When $k$ is a [[perfect field]] this is equivalently the Galois group of the [[algebraic closure]] $k \hookrightarrow \overline{k}$.

## Definition

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

Let $k$ be a [[field]]. Let $k_s$ denote the [[separable closure]] of $k$. Then the [[Galois group]] $Gal(k\hookrightarrow k_s)$ of the [[field extension]] $k\hookrightarrow k_s$ is called *absolute Galois group of $k$*.

=--

## Properties

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

By general [[Galois theory]]
we have $Gal(K\hookrightarrow K_s)\simeq \pi_1(Spec\; K)$ is equivalent to the [[fundamental group]] of the [[spectrum of a commutative ring|spectrum]] [[scheme]] $Spec K$

=--

An instance of [[Grothendieck's Galois theory]] is the following:

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

$$\begin{cases}
Sch_{et}\to Gal(k\hookrightarrow k_s)-Set
\\
X\mapsto X(k_s)
\end{cases}$$

from the category of [[étale scheme|étale schemes]] to the category of sets equipped with an [[action]] of the absolute Galois group is an equivalence of categories.
=--

+-- {: .num_prop}
###### Proposition
Recall the every [[profinite group]] appears as the [[Galois group]] of some [[Galois extension]]. Moreover we have:

Every [[projective object|projective]] [[profinite group]] appears as an absolute [[Galois group]] of a [[pseudo algebraically closed field]].
=--

## Examples

### Of the rational numbers
 {#OfTheRationalNumbers}

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

There is no direct description (for example in terms of [[generators and relations]]) known for the absolute Galois group $G_\mathbb{Q} \coloneqq Gal(\mathbb{Q}\hookrightarrow \overline{\mathbb{Q}})$ of the [[rational numbers]] (with $\overline{\mathbb{Q}}$ being the [[algebraic numbers]]).

However [[Belyi's theorem]] implies that there is a faithful [[action]] of $G_\mathbb{Q}$ on the [[children's drawing|children's drawings]].
=--

+-- {: .num_theorem}
###### Theorem
**(Drinfeld, Ihara, Deligne)**

There is an inclusion of the absolute Galois group of the rational numbers into the [[Grothendieck-Teichmüller group]] (recalled e.g. as [Stix 04, theorem 6](#Stix04)).

=--

## References

### General

* {#Stix04} [[Jakob Stix]], _The Grothendieck-Teichmüller group and Galois theory of the rational numbers_, 2004 ([[StiXGaloisAndGT.pdf:file]])

Discussion of the [[p-adic numbers|p-adic]] absolute Galois group as the [[etale fundamental group]] of a [[quotient]] of some [[perfectoid space]] is in

* {#Weinstein04} [[Jared Weinstein]], _$Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ as a geometric fundamental group_ ([arXiv:1404.7192](https://arxiv.org/abs/1404.7192), [workshop](https://www.mathi.uni-heidelberg.de/~G.QpAsPi1geom/))

See also

* Wikipedia, _[Absolute Galois group](https://en.wikipedia.org/wiki/Absolute_Galois_group)_

### In string theory

Discussion in the context of [[string theory]] includes

* [[Gregory Moore]], _Arithmetic and Attractors_ ([arXiv:hep-th/9807087](https://arxiv.org/abs/hep-th/9807087), abridged version in [arXiv:hep-th/9807056](https://arxiv.org/abs/hep-th/9807056))

category: Galois theory

[[!redirects absolute Galois groups]]

[[!redirects absolute Galois theory]]


absolute idealism

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### Context
#### Philosophy
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## Related concepts

* [[doctrine of ideas]]

* [[idealism]]

  [[objective idealism]], [[subjective idealism]]

* [[exceptional naturalism]]

* [[empiricism]]

## References

* Wikipedia, _[Absolute idealism](http://en.wikipedia.org/wiki/Absolute_idealism)_

* [[Hegel]], _[[Science of Logic]]_

* [[Roger Penrose]]'s _[Mind-Maths-Matter](Roger+Penrose#MindMathsMatter)_


absolute pushout

[[!redirects absolute pushouts]]

#Contents#
* table of contents
{:toc}

## Idea

An **absolute pushout** is a [[pushout]] which is [[preserved limit|preserved]] by any [[functor]] whatsoever.  In general this happens because the pushout is a pushout for purely "diagrammatic" reasons.  See [[absolute colimit]] for more.

## Definitions

+-- {: .num_defn #DefBasic}
###### Definition
A particular [[pushout]] [[diagram]] in a particular [[category]] $C$ is an **absolute pushout** if it is [[preserved limit|preserved]] by every [[functor]] with [[domain]] $C$.
=--

Equivalently, since the [[Yoneda embedding]] is the [[free cocompletion]] of $C$:

+-- {: .num_defn #DefYoneda}
###### Definition
A particular [[pushout]] diagram in a particular [[category]] $C$ is an **absolute pushout** if it is [[preserved limit|preserved]] by the [[Yoneda embedding]] $C \hookrightarrow [C^{op},Set]$.
=--

## Split pushouts

We propose the following notion of **split pushout**. See [Isaacson, Def. 3.22](#Isaacson).

+-- {: .num_defn #DefSplitPushout}
###### Definition
A [[commutative diagram|commutative]] square

\begin{center}\begin{tikzcd}
A \ar[r, "p"] \ar[d, "q"'] & B \ar[d, "m"] \\ C \ar[r, "n"'] & P
\end{tikzcd}\end{center}

defines a **split pushout** if there exist [[sections]] $p s = 1$, $q t = 1$, $m u = 1$
\begin{center}\begin{tikzcd}
A \ar[r, "p"'] \ar[d, "q"] & B \ar[d, "m"'] \ar[l, bend right, "s"'] \\ C \ar[r, "n"] \ar[u, bend left, "t"] & P \ar[u, bend right, "u"']
\end{tikzcd}\end{center}
so that $p t = u n$.
=--

## Split pushouts are absolute pushouts

+-- {: .num_prop }
###### Proposition
Split pushouts are absolute pushouts.
=--

+-- {: .proof}
###### Proof
Note that split pushouts are preserved by arbitrary functors, so it suffices to show that a split pushout is a pushout in the category in which it lives. To that end consider, a [[cone]] under the [[span]] $(p,q)$:
\begin{center}\begin{tikzcd}
A \ar[r, "p"] \ar[d, "q"'] & B \ar[d, "b"] \\ C \ar[r, "c"'] & X
\end{tikzcd}\end{center}
Upon composing with the sections to $q$ and $m$
\begin{center}\begin{tikzcd}
C \ar[d, "t"'] \ar[r, "n"]\ar[dd, bend right, equals]& P \ar[d, "u"]  \\
 A \ar[r, "p"] \ar[d, "q"'] & B \ar[d, "b"] \\ C \ar[r, "c"'] & X
\end{tikzcd}\end{center}
we see that $c$ factors through the claimed pushout $P$ as $c = (b  u) n$. We must verify that $b$ also factors as $b = (b  u)  m$. Since $p$ is an epimorphism, it suffices to prove that $b p = b u m p$, which follows easily:
$$ b p = c q = c q t q = b p t q = b u n q = b u m p.$$
This produces the desired factorization. Finally, since $m$ is an epimorphism, such factorizations are unique.
=--

Note the proof that a split pushout defines a pushout square in the category in which it lives did not require $p$ to be a _split_ epimorphism. However, arbitrary functors do not preserve epimorphisms. They do however preserve split epimorphisms, and thus the section guarantees that the image of $p$ will define an epimorphism in any category.

## General characterization

\begin{theorem}
A commutative square
\begin{center}\begin{tikzcd}
A \ar[r, "p"] \ar[d, "q"'] & B \ar[d, "m"] \\ C \ar[r, "n"'] & P
\end{tikzcd}\end{center}
is an absolute pushout if and only if either there exist

1. A section $u:P\to B$, such that $m u = 1_P$.
2. Morphisms $r_1,\dots,r_k : B \to A$ and $s_1,\dots,s_k : B\to A$, for some $k\ge 1$, such that $p s_1 = 1_B$, $q s_i = q r_i$ for all $i$, $p r_i = p s_{i+1}$ for all $i\lt k$, and $p r_k = u m$.
3. Morphisms $t_1,\dots,t_{\ell+1} : C \to A$ and $v_1,\dots,v_{\ell} : C\to A$, for some $\ell \ge 0$, such that $q t_1 = 1_C$, $p t_i = p v_i$ for all $i\lt \ell$, $q v_i = q t_{i+1}$ for all $i\le \ell$, and $p t_{\ell+1} = u n$.

or the transpose thereof (i.e. interchanging $B$ with $C$ and so on).
\end{theorem}
\begin{proof}
For "if", suppose given a commutative square
\begin{center}\begin{tikzcd}
A \ar[r, "p"] \ar[d, "q"'] & B \ar[d, "b"] \\ C \ar[r, "c"'] & X.
\end{tikzcd}\end{center}
Since $m$ is a (split) epimorphism (by $u$), any factorization of this square through the given one will be unique, so it suffices to show that such a factorization exists.  Define $x = b u:P\to X$.  Then we have
$$ x m = b u m = b p r_k = c q r_k = c q s_k = b p s_k = b t r_{k-1} = \dots = b p s_1 = b $$
and
$$ x n = b u n = b p t_\ell = c q t_\ell = c q v_{\ell-1} = b p v_{\ell-1} = b p t_{\ell-1} = \dots = c q t_1 = c. $$
The transposed case is of course dual.

Conversely, suppose the given square is an absolute pushout.  Thus, in particular the induced square
\begin{center}\begin{tikzcd}
\hom(P,A) \ar[r, "p"] \ar[d, "q"'] & \hom(P,B) \ar[d, "m"] \\ \hom(P,C) \ar[r, "n"'] & \hom(P,P)
\end{tikzcd}\end{center}
is a pushout in $Set$.  Thus, in particular, the function $\hom(P,B)+\hom(P,C) \to \hom(P,P)$ is surjective, and thus for $1_P\in \hom(P,P)$ there must be either a $u:P\to B$ such that $m u = 1_P$ or a $u':P\to C$ such that $n u' = 1_P$.  WLOG assume the former.

Now the induced square
\begin{center}\begin{tikzcd}
\hom(B,A) \ar[r, "p"] \ar[d, "q"'] & \hom(B,B) \ar[d, "m"] \\ \hom(B,C) \ar[r, "n"'] & \hom(B,P)
\end{tikzcd}\end{center}
is also a pushout in $Set$.  We have two elements $1_B, u m \in \hom(B,B)$ that become equal in $\hom(B,P)$ (since $m 1_B = m = (1_P) m = m u m$), and in a pushout in $Set$ this means they must be related by a zigzag of elements of the vertex $\hom(B,A)$.  Unraveling this explicitly produces the morphisms $r_i,s_i$.

Similarly, from the induced square
\begin{center}\begin{tikzcd}
\hom(C,A) \ar[r, "p"] \ar[d, "q"'] & \hom(C,B) \ar[d, "m"] \\ \hom(C,C) \ar[r, "n"'] & \hom(C,P)
\end{tikzcd}\end{center}
and the elements $1_C\in\hom(C,C)$ and $s n \in \hom(C,B)$, we obtain the morphisms $t_i,v_i$.
\end{proof}

In particular, when $k=1$ and $\ell=0$, the above data reduces to

1. A section $u:P\to B$, such that $m u = 1_P$.
2. Morphisms $r,s : B \to A$ such that $p s = 1_B$, $q s = q r$, and $p r = u m$.
3. A morphism $t : C \to A$ such that $q t = 1_C$ and $p t = u n$.

This is precisely the data of the above-defined notion of split pushout, together with the additional morphism $r:B\to A$ such that $q r = q s$ and $p r = u m$.  However, given a split pushout as above we can define $r = t q s$ and check $q r = q t q s = q s$ and $p r = p t q s = u n q s = u m p s = u m$.  This gives another proof that any split pushout is an absolute pushout.

## Examples

In their study of [[generalized Reedy categories]], Berger and Moerdijk introduce the notion of an [[Eilenberg-Zilber category]], one of the axioms of which demands that spans of split epimorphisms admit absolute pushouts. In practice, this seems to be the case because the pushout of these split epimorphisms is a split epimorphism as above, often with an additional section $v$ of $n$ satisfying the additional equation that $v m = q s$.

## References

The general characterization of absolute pushouts appears as Proposition 5.5 in:

* Robert Paré, Robert On absolute colimits. J. Algebra 19 (1971), 80–95.

The above notion of split pushout appears in Definition 3.22 of:

* {#Isaacson} Isaacson, Samuel B. "Symmetric cubical sets." Journal of Pure and Applied Algebra 215.6 (2011): 1146-1173. ([doi](https://doi.org/10.1016/j.jpaa.2010.08.001))

The Berger-Moerdijk definition of an Eilenberg-Zilber category appears in:

* {#BergerMoerdijk} [[Clemens Berger]] and [[Ieke Moerdijk]], _On an extension of the notion of Reedy category_ (2008) ([arXiv:0809.3341](http://arxiv.org/abs/0809.3341))


absolute retract

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## Definition

A [[metrisable space|metrisable]] [[topological space]] $Y$ is an _absolute neighborhood retract_  (ANR) if ([Borsuk 32, p. 222](#Borsuk32)) for _any_ [[embedding of topological spaces|embedding]] $Y \subset Z$ as a [[closed subspace]] in a [[metrisable topological space]] $Z$, $Y$ is a [[neighborhood retract]] of $Z$.

A [[metrisable space|metrisable]] topological space $Y$ is an _absolute retract_ if for _any_ embedding $Y\subset Z$ as a [[closed subspace]] in a [[metrisable topological space]] $Z$, $Y$ is a [[retract]] of $Z$.

## Properties
 {#Properties}

\begin{prop}\label{ANRIsLocalProperty}
  **(ANR is a local property for metrizable spaces)** \linebreak
  A [[metrizable topological space]] which admits an [[open cover]] by [[absolute neighbourhood retracts]] is itself an [[absolute neighbourhood retract]].
\end{prop}

(review in [Hu 65, III Thm. 8.1](#Hu65))

\begin{prop}
A [[metrisable topological space]] is an [[absolute retract]] precisely if it is

1. a [[contractible topological space]],

1. an [[absolute neighborhood retract]].

\end{prop}

([Hu 65, Prop. II.7.2](#Hu65))

## Examples
 {#Examples}

\begin{example}
  Every ([[finite number|finite]]-[[dimension of a manifold|dimensional]])
  [[metrizable topological space|metrizable]] [[locally Euclidean topological space]] -- in particular every [[topological manifold]] -- is an [[absolute neighbourhood retract]].

\end{example}

By Prop. \ref{ANRIsLocalProperty} (see also [Hu 65, III Cor. 8.3](#Hu65)).

In fact:

\begin{example}\label{ParacompactBanachManifoldsAreANRs}
Every [[paracompact topological space|paracompact]] [[Banach manifold]] is an [[absolute neighbourhood retract]].
\end{example}
By [Palais 1966, Cor. to Thm. 5 on p. 3](#Palais66).

\begin{proposition}\label{ClosedInclusionIntoANRIsANRIffhCofibration}
  Let $X$ be an [[absolute neighbourhood retract]] (ANR) and $A \xhookrightarrow{i} X$ a [[closed subspace]]-inclusion. Then $A$ is an ANR precisely iff the inclusion $i$ is a [[Hurewicz cofibration]].
\end{proposition}
([Aguilar, Gitler & Prieto 2002, Thm. 4.2.15](#AGP02))

\begin{example}
  Every [[finite number|finite]]-[[dimension of a CW complex|dimensional]] [[locally finite CW complex|locally finite]] [[CW-complex]] is an [[absolute neighbourhood retract]].
\end{example}
([Dugundji 52](#Dugundji52), [Kodama 56](#Kodama56), review in [Hu 65, III Cor. 8.4](#Hu65))

## Related concepts

* [[retract]],

* [[neighborhood retract]]

* [[deformation retract]]

## References

The notion of _absolute neighbourhood retract_ is due to

* {#Borsuk32} [[Karol Borsuk]], _Über eine Klasse von lokal zusammenhängenden Räumen_, Fund. Math **19** (1932) 220-242 ([dml:212574](https://eudml.org/doc/212574), [pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm19/fm19119.pdf))

Further development:

* [[Olof Hanner]], _Some theorems on absolute neighbourhood retracts_, Arkiv För Matematik Band 1 nr 30 (1950) ([doi:10.1007/BF02591376](https://doi.org/10.1007/BF02591376))

* {#Dugundji52} [[James Dugundji]], _Note on CW polytopes_, Portugaliae mathematica (1952) **11** 1 (1952) 7-10-b ([dml:114693](https://eudml.org/doc/114693))

* {#Kodama56} Yukihiro Kodama, _Note on an absolute neighborhood extensor for metric spaces_, Journal of the Mathematical Society of Japan **8** 3 (1956) 206-215 ([doi:10.2969/jmsj/00830206](https://doi.org/10.2969/jmsj/00830206))

* [[Karol Borsuk]], _Concerning the classification of topological spaces from the stand point of the theory of retracts_, Fundamenta Mathematicae **46** (3) (1959) 321-330 ([dml:213516](https://eudml.org/doc/213516))

Discussion for [[infinite-dimensional manifolds]]:

* {#Palais66} [[Richard S. Palais]], *Homotopy theory of infinite dimensional manifolds*, Topology **5** 1 (1966) 1-16 (doi:10.1016/0040-9383(66)90002-4)

Textbook accounts and review:

* [[Karol Borsuk]], _Theory of retracts_, Vol. 44. Państwowe Wydawn. Naukowe, 1967

* {#Hu65} [[Sze-Tsen Hu]], _Theory of Retracts_, Wayne State University Press (1965) ([google-books](https://books.google.ae/books/about/Theory_of_Retracts.html?id=GVTvAAAAMAAJ&redir_esc=y))

* [[Sibe Mardešić]], _Absolute Neighborhood  Retracts and Shape  Theory_ ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/mardesic.pdf))

* {#AGP02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, Def. 4.2.10 in: _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586), [toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

See also:

* [[eom]], _[Absolute retract for normal spaces](https://www.encyclopediaofmath.org/index.php/Absolute_retract_for_normal_spaces)_

* [[eom]], _[Retract of a topological space](https://encyclopediaofmath.org/wiki/Retract_of_a_topological_space)_

* Wikipedia, _Absolute neighbourhood retract_

[[!redirects absolute retracts]]

[[!redirects absolute neighborhood retract]]
[[!redirects absolute neighborhood retracts]]

[[!redirects absolute neighbourhood retract]]
[[!redirects absolute neighbourhood retracts]]

[[!redirects ANR]]
[[!redirects ANRs]]


absolute value

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# Contents
* table of contents
{: toc}

## Terminology

In [[field]] theory, what we call an 'absolute value' here is often called a 'valuation'.  However, there is also a more general notion of [[valuation]] used in field theory, which is what we call 'valuation'.  The notion of absolute value is also used in [[functional analysis]], where it may be called a 'multiplicative norm' (rather than merely submultiplicative, as norms on [[Banach algebras]] are required to be).

## Definition

For $k$ a [[rig]] (typically either a [[field]] or at least an [[integral domain]], or else an [[associative algebra]] over such), an **absolute value** on $k$ is a (non-trivial) multiplicative [[seminorm]], or equivalently a finite real-valued [[valuation]].

This means it is a [[function]]

$$
  {\vert {-} \vert}\colon k \to \mathbb{R}
$$

to the [[real numbers]] such that for all $x, y \in k$

1. ${\vert x \vert} \geq 0$;

2. ${\vert x \vert} = 0$ precisely if $x = 0$;

3. ${\vert x \cdot y \vert} = {\vert x \vert} {\vert y \vert}$;

4. ${\vert x + y \vert} \leq {\vert x \vert} + {\vert y \vert}$ (the [[triangle inequality]]).

If the last triangle inequality is strengthened to

* ${\vert x + y \vert} \leq max({\vert x \vert}, {\vert y \vert})$

then ${\vert {-} \vert}$ is called an [[ultrametric]] or **non-archimedean** absolute value, since then for any $x, y \in k$ with $\vert x \vert \lt \vert y \vert$ then for all natural numbers $n$, $\vert n x \vert \leq \vert x \vert \lt \vert y \vert$.  If the opposite holds, that whenever $\vert x \vert \lt \vert y \vert$ (and $x\neq 0$) there exists a natural number $n$ with $\vert n x \vert \gt \vert y \vert$, then it is called **archimedean**.

Two absolute values ${\vert {-} \vert}_1$ and ${\vert {-} \vert}_2$ are called _equivalent_ if for all $x \in k$

$$
  ({\vert x \vert}_1 \lt 1)
   \Leftrightarrow
  ({\vert x \vert}_2 \lt 1)
  \,.
$$

An [[equivalence class]] of absolute values is also called a **[[place]]**.

A [[field]] equipped with an absolute value which is a [[complete metric space]] with respect to the corresponding [[metric]] is called a [[complete field]].

## Examples

### Trivial absolute value

Every field admits the trivial absolute value ${\vert {-} \vert}_0$ defined by

$$
  {\vert x \vert}_0 =
  \left\{
    \array{
       0 & if\; x = 0
       \\
       1 & otherwise
    }
  \right.
  \,.
$$

This is non-archimedean.

### On the real and complex numbers
 {#OnTheRealAndComplexNumbers}

Since the [[real numbers]] are a [[sequentially Cauchy complete]] [[Archimedean field]], the standard absolute value ${\vert {-} \vert_\infty}$ on the [[real numbers]] is

$$
  {\vert x \vert_\infty} = \lim_{n \to \infty} x \tanh(n x)
$$

With the standard absolute value function defined, the maximum and principal square root functions could be defined on the real numbers as well.

The standard absolute value on the [[complex numbers]] is

$$
  {\vert x + i y \vert_\infty}
  =
  \sqrt{x^2 + y^2}
  \,.
$$

These standard absolute values are archimedean, and with respect to these standard absolute values, both $\mathbb{R}$ and $\mathbb{C}$ are [[complete field|complete]] and hence are complete [[archimedean valued fields]]. Notice that $\mathbb{R}$ is in addition an [[ordered field]] and as such also an [[archimedean field]].

Similar norms exist on the [[quaternions]] and [[octonions]], showing that absolute values can be of interest on noncommutative and even nonassociative [[division rings]].

### On the rational numbers

The standard absolute value [above](#OnTheRealAndComplexNumbers) restricts to the standard absolute value on the [[rational numbers]]

$$
  {\vert {-} \vert_\infty}\colon \mathbb{Q} \to \mathbb{R}
  \,.
$$

Moreover, for any [[prime number]] $p$ and [[positive number]] $\epsilon \lt 1$, there is an absolute value ${\vert {-} \vert_{p,\epsilon}}$ on $\mathbb{Q}$ defined by

$$
  \left\vert \frac{k}{l} p^n\right\vert_{p,\epsilon}
  =
  \epsilon^n
$$

whenever $n$ is an [[integer]] and $k$ and $l$ are nonzero [[integers]] not divisible by $p$ (and ${\vert 0 \vert_{p,\epsilon}} = 0$).

These are called the **$p$-adic absolute values**.  Given $p$, they are all equivalent (the open unit ball consists of all rational numbers whose denominator in lowest terms is not divisible by $p$), so there is a unique **$p$-adic [[place]]**.  For most purposes, only the place matters, and one may write simply $|q|_p$; however, if one wants a specific absolute value, then the usual choice is to use $\epsilon = 1/p$ (so that ${|p^n|_p} = p^{-n}$ whenever $n$ is an integer).

The $p$-adic absolute value is non-archimedean. The [[complete field|completion]] $\mathbb{Q}_p$ of $\mathbb{Q}$ under this absolute value is called the field of [[p-adic numbers]], which is therefore a [[non-archimedean field]].

[[Ostrowski's theorem]] says that these examples exhaust the non-trivial absolute values on the [[rational numbers]]. Therefore the [[real numbers]] and the [[p-adic numbers]] are the only possible field completions of $\mathbb{Q}$.

### On Laurent power series

The field of [[Laurent series]] $k[ [ T] ]$ over a [[field]] $k$ is a [[complete field]] with respect to the absolute value that sends a series to $\epsilon^n$ for a fixed $0 \lt \epsilon \lt 1$ and with $n$ the lowest integer such that the $n$th coefficient of the series is not $0$.

## See also

* [[real square root]]

* [[sign function]]

## References

* {#BoschGuntzerRemmert84} [[Siegfried Bosch]], [[Ulrich Güntzer]], [[Reinhold Remmert]], §1.5, §1.6 _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

Discussion in [[point-free topology]]:

* [[Ming Ng]], [[Steven Vickers]], *A Point-Free Look at Ostrowski's Theorem and Absolute Values* [[arXiv:2308.14758](https://arxiv.org/abs/2308.14758)]

[[!redirects absolute value]]
[[!redirects absolute values]]

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[[!redirects Archimedean absolute value]]
[[!redirects archimedean absolute values]]
[[!redirects Archimedean absolute values]]

[[!redirects archimedean]]
[[!redirects non-archimedean]]

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[[!redirects non-archimedean valuations]]

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[[!redirects real absolute values]]


absolute zero

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#Contents#
* table of contents
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## Idea

In [[thermodynamics]] the term "absolute zero", short for "absolute zero temperature", refers to absolute vanishing of the [[temperature]].

## Related concepts

* [[quantum phase transition]]

* [[temperature]], [[inverse temperature]]

* [[topological order]], [[topological entropy]]

## References

See also:

* Wikipedia, *[Absolute zero](https://en.wikipedia.org/wiki/Absolute_zero)*

[[!redirects absolute zero temperature]]


absolutely continuous function

# Absolutely continuous functions
* table of contents
{: toc}

## Idea

The basic idea behind a [[continuous function]] is that the output of the function can be made to change by only a small amount so long as the input is allowed to change by only a small amount.  There are, of course, different ways to make this precise, including [[uniformly continuous functions]] and [[Lipschitz continuous functions]].  With an absolutely continuous function, you allow multiple changes to multiple inputs to be combined into a single total change (and you consider the [[absolute values]] of the changes, so that they won\'t cancel).

The result is a notion of function that gets along well with the [[fundamental theorem of calculus]] in the context of the [[Lebesgue integral]] on the [[real line]].  Absolute continuity is weaker than Lipschitz continuity but stronger than mere (pointwise) continuity.

## Definitions

The first definition below is the most elementary; that the others are equivalent are important theorems.

Let $a$ and $b$ be [[real numbers]], and let $f$ be a real-valued [[function]] on the [[interval]] $[a,b]$.  Then $f$ is __absolutely continuous__ on $[a,b]$ iff:

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

Given any [[positive number]] $\epsilon$, for some positive number $\delta$, given any [[natural number]] $n$ and any $2n$-tuple of elements of $[a,b]$, interpreted as an increasing $n$-tuple of nonoverlapping subintervals of $[a,b]$, if the total length of the intervals is less than $\delta$, then the total variation of $f$ on the intervals is less than $\epsilon$.  That is (after $\epsilon$ and $\delta$), given $a \leq a_1 \leq b_1 \leq a_2 \leq b_2 \leq \cdots \leq a_n \leq b_n \leq b$, if
$$ \sum_{i = 1}^n (b_i - a_i) \lt \delta ,$$
then
$$ \sum_{i = 1}^n {|{f(b_i) - f(a_i)}|} \lt \epsilon .$$
=--

Various trivial variations of this may be met with: the comparison with $\delta$ and/or $\epsilon$ may be weak instead of strict; the number of subintervals may be infinite (so long as they are still nonoverlapping), since an infinite sum (of nonnegative numbers, as we have here) is simply a [[supremum]] of finite sums; and of course we may start by specifying that the $2n$ numbers come in order as the endpoints of the $n$ subintervals, rather than starting with any $2n$ numbers and then putting them in order and forming the subintervals from those.  (Note that putting them in order is fine even in [[constructive analysis]], since choosing the $i$th element in order from a list of [[rational numbers]] is continuous, so may be extended constructively to real numbers, although we can\'t assume that the final list is a permutation of the original list.)

For the next definition, fix a model of [[nonstandard analysis]].

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

Given any [[hypernatural number]] $n$ in the model and any $2n$-tuple of elements of the nonstandard extension of $[a,b]$, interpreted as an increasing $n$-tuple of nonoverlapping subintervals of $[a,b]$, if the total length of the intervals is [[infinitesimal number|infinitesimal]], then the total variation of $f$ on the intervals is infinitesimal.  That is, given [[hyperreal numbers]] $a \leq a_1 \leq b_1 \leq a_2 \leq b_2 \leq \cdots \leq a_n \leq b_n \leq b$, if
$$ \sum_{i = 1}^n (b_i - a_i) \approx 0 ,$$
then
$$ \sum_{i = 1}^n {|{f^*(b_i) - f^*(a_i)}|} \approx 0 .$$
=--

See [Tuckey 1993](#Tuckey1993), pages 34--36.

That the next definition is equivalent is the [[fundamental theorem of calculus]] for the [[Lebesgue integral]] on the [[real line]].

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

There exists a Lebesgue-integrable function $g$ on $[a,b]$ such that $f(x) = f(a) + \int_{x=a} g(x) \,\mathrm{d}x$ for $x \in [a,b]$.  (This is a [[semidefinite integral]].)
=--

In this case, $g$ must equal the [[derivative]] $f'$ [[almost everywhere]] on $[a,b]$.  (So in particular, $f$ is differentiable almost everywhere with a Lebesgue-integrable derivative, although this is not enough without requiring that $f$ be an indefinite integral of its derivative.)

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

The function $f$ is [[uniformly continuous]] on $[a,b]$, $f$ is of [[bounded variation]] on $[a,b]$, and the [[direct image]] under $f$ of any [[null subset]] of $[a,b]$ is null.
=--

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

The [[Stieltjes measure]] $\mathrm{d}f$ is [[absolutely continuous measure|absolutely continuous]] with respect to [[Lebesgue measure]] on $[a,b]$.
=--

The last of these is the source of the term 'absolutely continuous' as applied to measures.

## Generalizations

One may easily generalize the [[codomain]] of the elementary definition of absolutely continuous functions to any [[metric space]].

## Examples

The cube-root function is absolutely continuous (on any bounded interval) but not Lipschitz continuous (on any interval containing $0$).

The [[Cantor function]] is *not* absolutely continuous, even though it is continuous, and differentiable almost everywhere, with a Lebesgue-integrable derivative.

## References

* Curtis Tuckey. 1993. Nonstandard Methods in the Calculus of Variations. CRC Press. [Google books snippet](https://books.google.com/books?id=RvrC1mw9tIEC&q=%22absolutely+continuous%22#v=snippet&q=%22absolutely%20continuous%22&f=false).
  {#Tuckey1993}

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[[!redirects absolutely continuous functions]]
[[!redirects absolutely continuous mapping]]
[[!redirects absolutely continuous mappings]]
[[!redirects absolutely continuous map]]
[[!redirects absolutely continuous maps]]


absolutely continuous measure

# Absolutely continuous measures
* table of contents
{: toc}

## Idea

A [[measure]] $\mu$ is absolutely continuous with respect to $\nu$ if we can think of $\mu$ as a weighted variation on $\nu$.

## Definition

Fix a [[measurable space]] $X$ and let $\mu$ and $\nu$ be two [[measures]] on $X$.

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

The measure $\mu$ is __absolutely continuous__ with respect to $\nu$ if every $\nu$-[[full set]] is also $\mu$-full.
=--

Because [[null sets]] are more familiar than [[full sets]], we may equivalently express things as follows (but this is not correct in [[constructive mathematics]]):

+-- {: .num_defn}
###### Definition (classical)

The measure $\mu$ is __absolutely continuous__ with respect to $\nu$ if every $\nu$-[[null set]] is also $\mu$-null.
=--

If $\mu$ and $\nu$ are [[positive measures]], then we may also express this as follows:

+-- {: .num_defn}
###### Definition (positive, classical)

The positive measure $\mu$ is __absolutely continuous__ with respect to $\nu$ if, for every [[measurable set]] $A$, $\mu(A) = 0$ if $\nu(A) = 0$.
=--

Since the [[absolute value]] of a measure is a positive measure, we can also express the general definition as follows:

+-- {: .num_defn}
###### Definition (classical)

The measure $\mu$ is __absolutely continuous__ with respect to $\nu$ if, for every [[measurable set]] $A$, ${|\mu|}(A) = 0$ if ${|\nu|}(A) = 0$.
=--

## Generalisation

Since only the [[full sets]] (or, classically, the [[null sets]]) matter, we do not need to have the full structure of a [[measure]].  Sometimes we equip a [[measurable space]] with a $\delta$-[[delta-filter|filter]] $\mathcal{F}$ of full sets (or a $\sigma$-[[sigma-ideal|ideal]] $\mathcal{N}$ of null sets) without specifying a measure that produces these.  (For example, a [[smooth manifold]] is so equipped, effectively the full/null sets under [[Lebesgue measure]], even though there is no canonical such measure, since these sets are the same regardless of [[coordinate chart]].)  Then we say:

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

The measure $\mu$ is __absolutely continuous__ with respect to $\mathcal{F}$ (or $\mathcal{N}$) if every element of $\mathcal{F}$ is $\mu$-[[full set|full]] (or every element of $\mathcal{N}$ is $\mu$-[[null set|null]]).
=--

In fact, only the full/null sets of $\mu$ matter either, but until somebody has use for the notion of one $\delta$-filter (or $\sigma$-ideal) being absolutely continuous with respect to another, I will refrain from writing it down.  (See [[centipede mathematics]].)

## Defaults

If one calls a measure on the [[real line]] 'absolutely continuous', this means with respect to [[Lebesgue measure]].

This generalises to any [[cartesian space]] (with Lebesgue measure) or indeed to any [[smooth manifold]] of [[finite dimension]] (where there is no canonical Lebesgue measure but a family of local ones and so still a notion of Lebesgue-full and Lebesgue-null sets).

Or, this generalises to any [[compact group]] (with [[Haar measure]]) or indeed to any [[locally compact group]] (where there is no canonical Haar measure but a proportional family of them and so still a canonical notion of Haar-full and Haar-null sets).

## Properties

Let $\nu$ be a measure, and let $f \in L^1(\nu)$ be an [[absolutely integrable function]] with respect to $\nu$; then [[integration]] defines a measure $f \nu$:
$$ (f \nu)(E) = \int_E f \nu = \int_E f(x) \nu(\mathrm{d}x) .$$
This measure $f \nu$ is absolutely continuous with respect to $\nu$.  Conversely, given any absolutely continuous measure $\mu$, there is (at most) a unique (up to [[almost equality]]) absolutely integrable function $f$ such that $\mu = f \nu$; and this function must exist if $\mu$ and $\nu$ are [[localizable measure|localizable]].  This converse is the subject of the [[Radon–Nikodym theorem]].

A function $f\colon \mathbb{R} \to \mathbb{R}$ is [[absolutely continuous function|absolutely continuous]] iff the [[Stieltjes integral|Lebesgue–Stieltjes measure]] $\mathrm{d}f$ is absolutely continuous with respect to [[Lebesgue measure]].  (All such functions are [[continuous function|continuous]], and this example is actually the origin of the term.)

[[!redirects absolutely continuous measure]]
[[!redirects absolutely continuous measures]]


absolutely convex subset

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## Definition

A subset $B$ of a [[vector space]] $V$ over $\mathbb{R}$ or $\mathbb{C}$ is **absolutely convex** if $\lambda x + \mu y \in B$ whenever $x, y \in B$ and ${|\lambda|} + {|\mu|} \le 1$.

## Properties

Absolutely convex subsets are closely related to [[semi-norms]].
Given a vector space $V$ and an absolutely convex subset $B \subseteq V$, we define $V_B$ to be the linear span of $B$ in $V$.  Then let $\mu_B \colon V_B \to \mathbb{R}$ be the [[Minkowski functional]] of $B$.  That is, $\mu_B$ is given by:

$$
\mu_B(v) = \inf\{ t \gt 0 : t v \in B\}
$$

Then $\mu_B$ is a semi-norm on $V_B$ and $E_B \coloneqq V_B/\ker \mu_B$ is a [[normed space]].

# Related concepts

* [[convex set]]
* [[balanced set]]

[[!redirects absolutely convex]]
[[!redirects absolutely convex set]]
[[!redirects absolutely convex sets]]
[[!redirects absolutely convex subset]]
[[!redirects absolutely convex subsets]]


absolutely dense functor

[[!redirects absolute dense functor]]
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## Definition

An **absolutely dense functor** $F \colon A \to B$ is a [[functor]] which is equivalently characterised by the following conditions:

1. $F$ is [[dense]] and $Lan_F F$ is an [[absolute Kan extension]].

1. $F^* = [F^{op}, Set] \colon [B^{op}, Set] \to [A^{op}, Set]$ is [[fully faithful]].

1. The [[counit]] of the adjunction $F_* \dashv F^* : [B^{op}, Set] \to [A^{op}, Set]$ is invertible.

1. $F$ is [[corepresentably fully faithful]] (also called a [[lax epimorphism]]) in [[Cat]].

1. For every functor $G : B^{op} \times B \to C$, there is a canonical isomorphism ([MathOverflow answer](https://mathoverflow.net/a/354097))
$$ \int_{b \in B} G(b, b) \cong \int_{a \in A} G(F a, F a) $$
(This is a notion of [[final functor|initiality]] for [[ends]].)

More generally, an **absolutely dense morphism** in a [[proarrow equipment]] $K \to M$ is a 1-cell $f : a \to b$ in $K$ for which the counit $f_* \circ f^* \cong 1_b$.

## Properties

Every simultaneously reflective and coreflective subcategory of a presheaf category is itself a presheaf category and is induced by precomposition along an absolutely dense functor: this is the main result of the paper of El Bashir--Velebil below.

## References

(Absolutely dense functors are called **Cauchy dense** in the following paper of [[Brian Day]].)

- [[Brian Day]]. _Density presentations of functors_. Bulletin of the Australian Mathematical Society 16.3 (1977): 427-448.

- [[Jiri Adamek]], [[Robert El Bashir]], [[Manuela Sobral]], [[Jiri Velebil]]. _On functors which are lax epimorphisms_. TAC. (2001)

- Robert El Bashir and [[Jiri Velebil]]. _Simultaneously reflective and coreflective subcategories of presheaves_. Theory and Applications of Categories 10.16 (2002): 410-423.

- [[Fernando Lucatelli Nunes]] and [[Sousa Lurdes]]. _On lax epimorphisms and the associated factorization_. Journal of Pure and Applied Algebra 226.12 (2022)

[[!redirects absolutely dense]]
[[!redirects absolutely dense functors]]
[[!redirects absolutely dense morphism]]
[[!redirects absolutely dense morphisms]]


absorbing subset

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## Definition

Let $V$ be a [[vector space]] over $\mathbb{R}$ or $\mathbb{C}$.  A subset $B$ of $V$ is said to be **absorbing** if $\bigcup_{r \gt 0} r B = V$.

## Examples

* The unit ball in a [[seminormed vector space]] is absorbing.
* The linear span of absolutely convex set $C$ in a vector space has $C$ as an absorbing set.

## Related concepts

* [[convex set]]
* [[balanced set]]
* [[absolutely convex set]]

category: functional analysis

[[!redirects absorbing]]
[[!redirects absorbing set]]
[[!redirects absorbing sets]]
[[!redirects absorbing subset]]
[[!redirects absorbing subsets]]


absorption category

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## Idea

The concept of an absorption category should be the [[oidification]] of an [[absorption monoid]].

## Definition

An __absorption category__ or __annihilation category__ $C$ is a [[category]] where for every two objects $a, b \in Ob(C)$ there is a morphism $0_{a\to b}: a \to b$, such that for any objects $a, b, c, d \in Ob(C)$, for any morphism $f:b \to c$, $f \circ 0_{a\to b} = 0_{a\to c}$, and for any morphism $g:d \to a$, $0_{a\to b} \circ g = 0_{d\to b}$.

Such a structure is the same thing as a [[enriched category theory|category enriched]] in the category of [[pointed sets]], taking the [[monoidal category|monoidal product]] to be the [[smash product]].

## Examples

* Every [[ringoid]] and [[algebroid]] is an absorption category.

* An [[absorption monoid]] is an absorption category with only one object.

## Related concepts

[[!include oidification - table]]

[[!redirects absorption categories]]
[[!redirects annihilation category]]
[[!redirects annihilation category]]


absorption magma

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## Definition

An __absorption magma__ or __annihilation magma__ is a [[magma]] $(M,\cdot)$ with an element $0\in M$ satisfying the absorption/annihilation axioms: for all $a \in M$, $0 \cdot a = 0$ and $a \cdot 0 = 0$.

Equivalently, this is a [[magma object]] in the [[category]] of [[pointed sets]].

## Properties

### Zero divisors

A non-zero element $a \in M$ is a __[[zero divisor]]__ if thete exists a non-zero element $b \in M$ such that $a \cdot b = 0$ or $b \cdot a = 0$.

## Examples

* [[absorption monoids]]

* Every [[magma object]] in the category of [[commutative magma|commutative]] [[unital magmas]] is an absorption magma. Particular examples of this include the multiplicative monoid in a rig (monoid objects in [[CMon]]) or a ring (monoid objects in [[Ab]].

* The multiplicative magmas of the [[octonions]] and the [[sedenions]] are absorption magmas.

## See also

* [[magma]]

* [[zero divisor]]

* [[rig]], [[ring]], [[lattice]], [[integral domain]] [[field]]

[[!redirects absorption magmas]]
[[!redirects annihilation magma]]
[[!redirects annihilation magmas]]
[[!redirects integral magma]]
[[!redirects integral magmas]]
[[!redirects division magma]]
[[!redirects division magmas]]


absorption monoid

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## Idea

Absorption monoids are the [[monoid objects]] in [[pointed sets]], in the same way that [[rings]] are the [[monoid objects]] in [[abelian groups]]. Thus, the theory of absorption monoids and the theory of rings are very similar to each other, except that rings have additive structure whereas absorption monoids do not have additive structure.

## Definition

An __absorption monoid__ or __annihilation monoid__ is a [[monoid]] $(M,1,\cdot)$ that is also an [[absorption magma]] $(M,0)$.

Equivalently, it is a [[monoid object]] in the [[category]] of [[pointed sets]], since left and right multiplication $\cdot$ by any element $x$ preserves the point $0$.

## Properties

### Initial and terminal absorption monoids

The [[initial object|initial]] absorption monoid is the [[boolean domain]] $\mathbb{2}$ with elements $0 \in \mathbb{2}$ representing [[false]], $1 \in \mathbb{2}$ representing [[true]], and $(-)\cdot(-):\mathbb{2} \times \mathbb{2} \to \mathbb{2}$ representing [[conjunction]].

The [[terminal object|terminal]] absorption monoid is the [[trivial monoid]] $\mathbb{1}$, the monoid whose underlying [[set]] is a [[singleton]]. The trivial monoid is also [[strictly terminal]].

### Absorption monoid homomorphisms

Given absorption monoids $M$ and $N$, an absorption monoid homomorphism is a function $h:M \to N$ such that

* $h(0) = 0$
* $h(1) = 1$
* for all $a \in M$ and $b \in M$, $h(a \cdot b) = h(a) \cdot h(b)$.

### Ideals and anti-ideals

A two-sided [[ideal]] of an absorption monoid $M$ is a subset $I$ of $M$ such that

* $0 \in I$
* for all elements $a \in M$ and $b \in M$, if $a \cdot b \in I$, then either $a \in I$ or $b \in I$.

A two-sided anti-ideal of an absorption monoid $M$ is a subset $A$ of $M$ such that

* $0 \notin I$
* for all elements $a \in M$ and $b \in M$, if $a \in I$ and $b \in I$, then $a \cdot b \in I$.

### Quotient absorption monoids

Given an absorption monoid $M$ and a two-sided ideal $I$, the quotient of $M$ by $I$ is the [[initial object|initial]] absorption monoid $M/I$ with absorption monoid homomorphism $i:M \to M/I$ such that for all elements $a \in I$, $i(a) = 0$.

### Invertible elements

An element $a \in M$ is an **invertible element** or a **unit** if there exists an element $b \in M$ such that $a \cdot b = 1$ and $b \cdot a = 1$.

The set of invertible elements $M^\times$ in an absorption monoid $M$ is always closed under multiplication; i.e. $M^\times$ is a [[submonoid]] of $M$. In fact, since every element is invertible, $M^\times$ forms a [[subgroup]] of $M$, called the [[group of units]].

### Division monoids

An absorption monoid $M$ is a [[division monoid]] if every non-invertible element in $M$ is equal to zero. $M$ is Heyting if there is a [[tight apartness relation]] on $M$ such that every invertible element is apart from zero, and $M$ is discrete if every element in $M$ is either zero or invertible.

### Regular elements

An element $a \in M$ is a **regular element**, **cancellative element**, or **cancellable element** if for all elements $b \in M$ and $c \in M$, $b = c$ if and only if $a \cdot b = a \cdot c$ and $c \cdot a = c \cdot b$.

The set of regular elements $\mathrm{Reg}(M)$ in an absorption monoid $M$ is always closed under multiplication; i.e. $\mathrm{Reg}(M)$ is a [[submonoid]] of $M$.

### Integral monoids

An absorption monoid $M$ is an [[integral monoid]] if every non-regular element in $M$ is equal to zero. $M$ is Heyting if there is a [[tight apartness relation]] on $M$ such that every regular element is apart from zero, and $M$ is discrete if every element in $M$ is either zero or regular.

### Ore sets and Ore absorption monoids

Given an absorption monoid $M$, an [[Ore set]] is a [[submonoid]] $S$ of $\mathrm{Reg}(M)$ such that every element of $S$ satisfies the left and right Ore conditions:

* for all $a \in S$ and $b \in M$, there exists $c \in S$ and $d \in M$ such that $a \cdot d = b \cdot c$
* for all $a \in S$ and $b \in M$, there exists $c \in S$ and $d \in M$ such that $d \cdot a = c \cdot b$

A absorption monoid is an Ore absorption monoid if $\mathrm{Reg}(M)$ is an Ore set.

### Localization and group completion

The localization of an Ore integral monoid $M$ at $\mathrm{Reg}(M)$ is a [[division monoid]]. The [[localization of a monoid|localization]] of an absorption monoid at $0$ is the [[trivial group]]; thus, the [[group completion]] of any absorption monoid is the [[trivial group]].

### Actions and modules

Given an absorption monoid $M$, an left $M$-action on a pointed set $(P, 0)$ is an ternary function $\alpha_L:M \times P \to P$ such that:

* for all elements $p \in P$, $\alpha_L(1, p) = p$
* for all elements $a \in M$, $b \in M$, and $c \in P$, $\alpha_L(a, \alpha_L(b, c)) = \alpha_L(a \cdot b, c)$
* for all elements $p \in P$, $\alpha_L(0, p) = 0$
* for all elements $a \in M$, $\alpha_L(a, 0) = 0$

A right $M$-action on a pointed set $(P, 0)$ is a binary function $\alpha_R:P \times M \to P$ such that:

* for all elements $p \in P$, $\alpha_R(p, 1) = p$
* for all elements $a \in M$, $b \in M$, and $c \in P$, $\alpha_R(\alpha_R(c, a), b) = \alpha_R(c, a \cdot b)$
* for all elements $p \in P$, $\alpha_R(p, 0) = 0$
* for all elements $a \in M$, $\alpha_R(0, a) = 0$

Given absorption monoids $M$ and $N$, an $M$-$N$-[[biaction]] on a pointed set $(P, 0)$ is a ternary function $\alpha:M \times P \times N \to P$ such that:

* for all elements $p \in P$, $\alpha(1, p, 1) = p$
* for all elements $a \in M$, $b \in M$, $c \in P$, $d \in N$, $e \in N$, $\alpha(a, \alpha(b, c, d), e) = \alpha_L(a \cdot b, c, d \cdot e)$
* for all elements $p \in P$ and $d \in N$, $\alpha_L(0, p, d) = 0$
* for all elements $a \in M$ and $d \in N$, $\alpha_L(a, 0, d) = 0$
* for all elements $a \in M$ and $p \in P$, $\alpha_L(a, p, 0) = 0$

Pointed sets equipped with left or right $M$-actions are called left or right $M$-[[module object|modules]], and pointed sets equipped with $M$-$N$-biactions are called $M$-$N$-[[bimodule object|bimodules]].

## Examples

### The multiplicative monoid of the natural numbers

The multiplicative monoid of the natural numbers $\mathbb{N}^\times$ is the [[free object|free]] commutative absorption monoid on the natural numbers, the [[initial object|initial]] commutative absorption monoid $\mathbb{N}^\times$ with a function $\mathrm{prime}:\mathbb{N} \to \mathbb{N}^\times$. $\mathbb{N}^\times$ has [[decidable equality]]. The localization of $\mathbb{N}^\times$ at the image of $\mathrm{prime}$, or equivalently at the non-zero elements of $\mathbb{N}^\times$, is the multiplicative monoid of the non-negative rational numbers, $\mathbb{Q}_{\geq 0}^\times$.

### Other examples

* The [[extended natural numbers]] $(\bar{\mathbb{N}}, 0, +, \infty)$ are an absorption monoid.

* Every [[integral monoid]] is an absorption monoid.

* The multiplicative monoid of every [[rig]] is an absorption monoid.

* Every join-[[semilattice]] with a [[top element]] and evety meet-semilattice with a [[bottom element]] is an absorption monoid.

## Related concepts

* [[monoid]]

* [[integral monoid]]

* [[zero divisor]]

* [[rig]], [[ring]], [[lattice]], [[field]]

[[!include oidification - table]]

[[!redirects absorption monoids]]
[[!redirects annihilation monoid]]
[[!redirects annihilation monoids]]

[[!redirects absorbing element]]
[[!redirects absorbing elements]]
[[!redirects annihilating element]]
[[!redirects annihilating elements]]


Abstract and Concrete Categories

This page collects links related to the book

* [[Jiri Adamek|Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]]:

  **Abstract and Concrete Categories -- The Joy of Cats**

  John Wiley and Sons, New York (1990)

  reprinted as:

  Reprints in Theory and Applications of Categories **17** (2006) 1-507

  [tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html)

  [book webpage](http://katmat.math.uni-bremen.de/acc/), [pdf](http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf)

on [[category theory]]

Originally published in the series
Pure and Applied Mathematics. A Wiley-Interscience Publication.
John Wiley & Sons, Inc., New York, 1990. xiv+482 pp. ISBN: 0-471-60922-6

Online edition appeared in 2004:

* Reprints in Theory and Applications of Categories, No. 17 (2006), 1–507.
[abstract](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html),
[pdf](http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf).

Beware that the book uses the term "quasicategory" for what  _[[Categories for the Working Mathematician]]_ calls a [[metacategory]]: a model (in the informal sense) of the first-order theory of a category, with no regard for size issues. These days a [[quasicategory]] is mostly understood to be a particular type of [[simplicial set]] that models an [[(∞,1)-category]].

category: reference

[[!redirects Abstract and Concrete Categories: The Joy of Cats]]
[[!redirects Abstract and Concrete Categories -- The Joy of Cats]]

[[!redirects Joy of Cats]]
[[!redirects The Joy of Cats]]


abstract circle

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## Idea

An _abstract circle_ in the sense of ([Moerdijk 96](#Moerdijk96)) is essentially a cellular structure of the form of a [[circle]] $S^1$, in direct analogy to how a [[linear interval]] in the abstract sense is a cellular model of the actual topological interval $[0,1]$.

Indeed, just as linear intervals have as [[classifying topos]] the category of [[simplicial sets]], so abstract circles have as classifying topos the category of [[cyclic sets]] ([Moerdijk 96](#Moerdijk96)).

## References

* {#Moerdijk96} [[Ieke Moerdijk]], _Cyclic sets as a classifying topos_, 1996 ([[MoerdijkCyclic.pdf:file]])

[[!redirects abstract circles]]


abstract elementary class

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## Idea

In [[model theory]] the notion of *abstract elementary classes* is a vast generalizations of that of [[elementary classes]] of [[structure in model theory|structures]] beyond [[first-order theory|first-order theories]] (e.g. for the infinitary logic $L_{\omega_1,\omega}$) as introduced by [[Saharon Shelah]]. Their theory is also more general than the homogeneous model theory.

An __abstract elementary class__ is a __nonempty__ class $K$ of structures for a given signature with language $L(K)$, that is closed under [[isomorphisms]] and equipped with a strong substructure relation $\prec_K$ (strong substructure relation means that if $M\prec_K N$ and $M_0\subset M$ is a substructure, then $M_0 \prec_K N$) that is a [[partial order]] satisfying the axioms on union of chains (Tarski-Vaught), coherence and downward Loewenheim-Skolem properties. More precisely, $\prec_K$ is a partial order such that

(A0) if $M,N\in K$, $M\prec_K N$ then $M\subset N$

(A1) (closure under isomorphisms)

* (a) $M\in K$ and $N$ an $L(K)$ structure with $N\cong M$, then $N\in K$

* (b) if $N_1,N_2,M_1,M_2\in K$, $f_i : N_i\cong M_i$, $i = 1,2$, $f_1\subset f_2$, with $M_1\prec_K M_2$ then $N_1\prec_K N_2$

(A2) for $M,N,P\in K$, if $M\prec_K P$, $N\prec_K P$, and $M\subset N$, then $M\prec_K N$

(A3) downward Loewenheim-Skolem. There exist a cardinal $LS(K) = LS(K,\prec_K)\geq |L(K)|+\aleph_0$ such that $\forall M\in K$, $\forall A\subset |M|$, $\exists N\in K$ with $A\subset |N|$, $N\prec_K M$, $\|N\|\leq |A|+LS(K)$.

(A4) (Tarski-Vaught chain condition) for every regular cardinal $\mu$

....

The usual elementary classes, i.e. the classes of the form $K = Mod(T)$ for a first-order theory $T$, are abstract elementary with respect to the relation $\prec_K$ of being an elementary submodel, with ${|LS(K)|} = {|L(T)|}+\aleph_0$ ($L(T)$ is the underlying language of the theory $T$).

## Related entries

* [[amalgamation]]

## References

* wikipedia [abstract elementary class](http://en.wikipedia.org/wiki/Abstract_elementary_class)

* [[Saharon Shelah]], _Classification theory for elementary abstract classes I, II, Studies in Logic (London), __18__, __20__, College Publications, London 2009

* John Baldwin, _Categoricity_, Amer. Math. Soc. 2011, [pdf](http://www.math.uic.edu/~jbaldwin/pub/AEClec.pdf)

* D. W. Kueker, _Abstract elementary classes and infinitary logic_, Ann. Pure Appl. Logic __156__ (2008), 274-286.

AECs can also be essentially identified with [[accessible categories]] in which all morphisms are [[monomorphisms]].  Some recent papers which study them from this viewpoint include:

* [[Tibor Beke]], [[Jiří Rosický]] _Abstract elementary classes and accessible categories_, Annals of Pure and Applied Logic __163__ (2012) 2008-2017, [arxiv/1005.2910](http://arxiv.org/abs/1005.2910)

* Michael Lieberman, [[Jiří Rosický]], Sebastien Vasey, *Internal sizes in μ-abstract elementary classes*, [arxiv/1708.06782](https://arxiv.org/abs/1708.06782); *Set-theoretic aspects of accessible categories*, [arxiv/1902.06777](https://arxiv.org/abs/1902.06777)

* M. J. Lieberman, _Topological and category-theoretic aspects of abstract elementary classes_, Thesis, The University of Michigan 2009, [pdf](http://deepblue.lib.umich.edu/bitstream/2027.42/63854/1/liebermm_1.pdf); defense slides [pdf](http://www.math.upenn.edu/~mlieb/defense.pdf); _Category theoretic aspects of abstract elementary classes_, Annals Pure Appl. Logic __162__ (2011), 903-915; _A topology for Galois types in AECs_, [arxiv/0906.3573](http://arxiv.org/abs/0906.3573)

On [[topos theory|topos theoretic]] methods in model theory of AECs:

* Christian Espíndola, _A topos-theoretic proof of Shelah's eventual categoricity conjecture for abstract elementary classes_, [arxiv/1906.09169](https://arxiv.org/abs/1906.09169); _A short proof of Shelah's eventual categoricity conjecture for AEC's with amalgamation, under GCH_, [arxiv/1909.13713](https://arxiv.org/abs/1909.13713)

On the [[homotopy types]] of [[classifying spaces]] of abstract elementary classes:

* [[Tim Campion]], Jinhe Ye, *Homotopy Types of Abstract Elementary Classes*, Journal of Pure and Applied Algebra
**225** 5 (2021) 106461 [[arXiv:1909.07965](https://arxiv.org/abs/1909.07965), [doi:10.1016/j.jpaa.2020.106461](https://doi.org/10.1016/j.jpaa.2020.106461)]

[[!redirects abstract elementary classes]]


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## Idea

The [[category theory|category theoretic]] notions of

* _[[category]]_ and _[[object]]_

on the one hand and of

* _[[theory]]_ and _[[model]]_

on the other have been suggested ([Lawvere](#Lawvere)) to usefully formalize, respectively, the heuristic notions

* "general" and "particular"

as well as

* "abstract" and "concrete", respectively.

We have:

* a ([[syntactic category]] of a) [[Lawvere theory]] $T$ (or the equivalent in any [[doctrine]]) $T$ is an _abstract general_ or _abstract universal_ (*abstraktes Allgemeines*)

* the [[category]] $T Mod(E)$ of $T$-[[models]]/[[algebra over an algebraic theory|algebras]] in any context $E$ is a _concrete general_ or _concrete universal_

* an [[object]] of any $T Mod(E)$ is a _particular_.

That seems to be roughly what is suggested in [Lawvere](#Lawvere). Of course one could play with this further and consider further refinement such as

* a (generating) [[object]] in $T$ is an _abstract particular_ ;

* an [[object]] of any $T Mod(E)$ is a _concrete particular_.

## Examples

### Groups

The [[syntactic category]] $T_{Grp}$ of the [[theory]] of [[group]]s is the "general abstract" of groups. Its essentially unique generating object is  _the abstract particular_ group.

The category $T_{Grp} Mod(Set) = $ [[Grp]] of all groups is the _concrete general_ of groups.

An object in there is some [[group]]: a concrete particular.

## References

The [[category theory|category-theoretic]] formalization of these notions as proposed by Bill Lawvere is disussed in print for instance in

* {#Lawvere} [[Bill Lawvere]], _Categorical refinement of a Hegelian principle_, section 1 of [[Bill Lawvere]], _[[Tools for the advancement of objective logic]] -- Closed categories and toposes_, in John Macnamara, [[Gonzalo Reyes]], _the logical foundations of cognition_, Oxford University Press (1994)

See also an email comment recorded [here](http://conceptualmathematics.wordpress.com/2012/06/09/general-concepts-and-reality-prof-f-william-lawvere/).

For discussion of "particular" and related in [[philosophy]] see also

The terminology is inspired by

* [[Georg Hegel]], _[[Science of Logic]]_,

{#HegelQuotes} for instance

> [EL§61](Science+of+Logic#EL61) If we are to believe the Critical philosophy, thought is subjective, and its ultimate and invincible mode is abstract universality or formal identity. Thought is thus set in opposition to Truth, which is no abstraction, but concrete universality. In this highest mode of thought, which is entitled Reason, the Categories are left out of account. The extreme theory on the opposite side holds thought to be an act of the particular only, and on that ground declares it incapable of apprehending the Truth. This is the Intuitional theory.

> [§71](Science+of+Logic#71) It is only after profounder acquaintance with the other sciences that logic ceases to be for subjective spirit a merely abstract universal and reveals itself as the universal which embraces within itself the wealth of the particular

> [§1323](Science+of+Logic#1323) This universal Notion, which we have now to consider here, contains the three moments: universality, particularity and individuality.

> [§1337b](Science+of+Logic#1337b) When people talk of the determinate Notion, what is usually meant is merely such an abstract universal.

> [§1599](Science+of+Logic#1599) Such a universal which merely subsumes, is an abstraction which only becomes concrete in something else, in the particular. End, on the contrary, is the concrete universal, which possesses in its own self the moment of particularity

> [PS§456b](Science+of+Logic#PS456b) This common element is either any one particular side of the object raised to the form of universality, such as, for example, in the rose, the red colour; or the concrete universal, the genus, for example, in the rose, the plant;

See also in and around the section [The genus](Science+of+Logic#TheGenus)

Survey of these Hegelian ideas includes

* {#HibbenLuft} John Grier Hibben, Eric v.d. Luft, _Hegel's Shorter Logic: An Introduction and Commentary_

where on p. 143 it says about the _Shorter Logic_:

> Particularity and individuality are related as "abstract" and "concrete", respectively. The particular is the "abstract individual". The individual is the "concrete particular". The universal is their union, and may be either "abstract" or "concrete". The so-called "concrete universal" is Hegel's gold standard for conceptual thought $[$...$]$.

For a discussion of proponents of the concrete universal in British philosophy, and of their divergences from, and continuities with, Hegelian thought, see

* Robert Stern, _Hegel, British idealism, and the curious case of the concrete universal_, British Journal for the History of Philosophy 15(1) 2007: 115 – 153 ([pdf](http://www.sheffield.ac.uk/polopoly_fs/1.101730!/file/concrete-universal-published.pdf))

where he cites the philosopher Bernard Bosanquet

> A world or cosmos is a system of members, such that every member, being _ex hypothesi_ distinct, nevertheless contributes to the unity of the whole in virtue of the peculiarities which constitute its distinctness. And the important point for us at present is the difference of principle between a world and a class. It takes all sorts to make a world; a class is essentially of one sort only. In a word, the difference is that the ultimate principle of unity and community is fully exemplified in the former, but only superficially in the latter. The ultimate principle, we may say, is sameness in the other; generality is sameness in spite of the other; universality is sameness by means of the other. (Bosanquet, The Principle of Individuality and Value, 3)

and T. M. Knox

> An abstract universal has no organic connexion with its particulars. Mind, or reason, as a concrete universal, particularizes itself into differences which are interconnected by its universality in the same way in which parts of the organism are held together by the single life which all things share. The parts depend on the whole for their life, but on the other hand the persistence of life necessitates the differentiation of the part. (Translator’s notes to Hegel’s Philosophy of Right (Oxford: Oxford University Press, 1952))

For general related discussion see also

* Wikipedia, _[Particular](http://en.wikipedia.org/wiki/Particular)_, _[Abstract particular](http://en.wikipedia.org/wiki/Abstract_particulars)_

[[!redirects abstract general]]
[[!redirects abstract generals]]

[[!redirects abstract universal]]

[[!redirects general abstract]]

[[!redirects concrete general]]
[[!redirects general concrete]]

[[!redirects concrete general]]
[[!redirects concrete generals]]

[[!redirects concrete universal]]

[[!redirects concrete particular]]
[[!redirects concrete particulars]]

[[!redirects abstract particular]]
[[!redirects abstract particulars]]

[[!redirects abstract general, concrete general and concrete particular]]
[[!redirects abstract general, concrete general, and concrete particular]]
[[!redirects abstract general, concrete general, concrete particular]]


This is a book by [[K. H. Kamps]] and [[T. Porter]] that introduces and discusses several different approaches to abstract homotopy theory.

##Reference

*  K. H. Kamps and T. Porter, _Abstract Homotopy and Simple Homotopy Theory_ ([GoogleBooks](http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1))


abstract model theory

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## Idea
_Abstract model theory_ is the study of the general properties of the [[model theory]] of extensions of (classical untyped) [[first-order logic]].

Originally motivated by _[[Lindström's theorem]]_ that characterizes first-order logic, the field has subsequently been extended to provide alternative characterizations and include different logics within its range.

The basic concept of abstract model theory is that of an _abstract logic_
which is  a triple $\mathcal{L}=(S,\Phi ,\models)$ where $\models$ is a binary relation between the class of $\mathcal{L}$-'structures' $S$ and the class of $\mathcal{L}$-'sentences' $\Phi$ to be thought of as minimalistic version of the _satisfaction_ relation.

##Remark

As the logical relations studied by abstract model theory are of a functorial nature, some category theory entered the picture already in Barwise (1974). The theory of [[institution|institutions]], aka _institution-independent model theory_ (Diaconescu 2008), constitutes abstract categorical model theory proper. In a similar abstract categorical vein is the [[geometric theory#FunctorialDefinition|functorial approach to geometric theories]] described in Johnstone (2002, sec. B4.2).

## Related entries

* [[model theory]]

* [[Lindström's theorem]]

* [[compactness theorem]]

* [[Löwenheim-Skolem theorem]]

* [[institution]]

* [[Per Lindström]]

## Link

* Wikipedia, _[Abstract model theory](http://en.wikipedia.org/wiki/Abstract_model_theory)_

## References

* Jon Barwise, _Axioms for abstract model theory_ , Annals of Mathematical Logic 7 pp.221-265, 1974.

* Barwise, Feferman (eds.), _Model-theoretic Logics_ , Springer Heidelberg 1985 (freely available online: [toc](http://projecteuclid.org/euclid.pl/1235417263#toc)) .

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

* Marta García-Matos, [[Jouko Väänänen]], _Abstract Model Theory as a Framework for Universal Logic_ , Logica Universalis 2005 pp.1-33 ([draft](http://www.math.helsinki.fi/logic/opetus/lt/unilog2005_proofs.pdf)) .

* [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. I_ , Oxford UP 2002.

* [[Jouko Väänänen]], _The Craig Interpolation Theorem and abstract model  theory_ , Synthese  164:401 (2008) (freely available online)


abstract rewriting system

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## Idea

The notion of abstract rewriting system is the simplest mathematical formalization of what is a [[rewriting system]]. It doesn't presuppose anything on the nature of the [[syntax|syntactical]] objects which are rewritten, thus the word "abstract".

## Definition

An **abstract rewriting system** $(X, \rightarrow)$ is given by a [[set]] $X$ and a [[binary relation]] $\rightarrow$ on $X$, ie. a [[subset]] of the [[cartesian product]] $X \times X$.

So this is a [[concept with an attitude]]: While an abstract re-writing system is just a [[relation]], calling this relation an abstract rewriting system indicates that one is interested in studying the behaviour of chains of related elements $x \to x_1 \to x_2 \to \cdots$ (thought of as successive stages of [[rewriting]] $x$), for instance to see if they are [[confluent category|confluent]].

## Related entries

* [[rewriting system]]

## References

* Wikipedia, [Abstract rewriting system](https://en.wikipedia.org/wiki/Abstract_rewriting_system)

* Wikipedia, Confluence_(abstract_rewriting)

[[!redirects abstract rewriting systems]]

[[!redirects abstract re-writing system]]
[[!redirects abstract re-writing systems]]


abstract scattering theory

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## Idea

In [[quantum mechanics]] based on a [[Hilbert space]] $H$ wth quantum evolution semigroup of [[unitary operators]] $U(t)$, $t\in \mathbf{R}$, Lax and Phillips formulated an abstract formulation of [[scattering]] theory, based on two distinguished subspaces of
states $\mathcal{D}_-$ of ingoing states and $\mathcal{D}_+$ of outgoing states satisfying certain system of axioms.

Apart from [[quantum mechanics]], abstract scattering theory has other interesting special cases, applying to the theory of [[automorphic form|automorphic functions]] (due to an idea of [[Israel Gelfand]], later realized by B. S. Pavlov and L. D. Faddeev), zeta function analysis and so on.

## Related concepts

* [[scattering amplitude]]

* [[scattering matrix]]

* [[scattering cross section]]

## References

* Peter D. Lax, Ralph S. Phillips,  _Scattering theory_, Bull. Amer. Math. Soc. Volume 70, Number 1 (1964), 130-142 [euclid](http://projecteuclid.org/euclid.bams/1183525789) [MR0167868](http://www.ams.org/mathscinet-getitem?mr=0167868); _Scattering theory_ (book), Academic Press 1990, 2nd ed. [gBooks](http://books.google.hr/books?id=vgqMqqCnf5AC&lpg=PP1&pg=PP1#v=onepage&q&f=false); _Scattering theory for automorphic functions, Annals of Mathematics Studies __87__, Princeton Univ. Press 1976

* B. S. Pavlov, L. D. Faddeev, _Scattering theory and automorphic functions_, Zapiski Sem. LOMI __27__ (1972) 161-193

Extension of the method which Lax and Phillips used for automorphic forms ($SL(2,\mathbf{R})$-case) to the case of more general harmonic analysis on more general Riemannian [[symmetric spaces]] is studied in

* M. A. [[Semenov-Tian-Shansky]], _Harmonic analysis on Riemannian symmetric spaces of negative curvature and the scattering theory_, Izvestiya Akademii Nauk USSR __40__:3 (1976) 562-592

category: physics


abstract Stone duality

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## Idea

Due to [[Paul Taylor]], Abstract Stone Duality (ASD) is a re-[[axiom|axiomatisation]] of the notions of *[[topological space]]* and *[[continuous function]]* in [[general topology]] in terms of a [[lambda calculus|lambda-calculus]] of [[computable function|computable]] [[continuous functions]] and [[predicates]].

Abstract Stone duality is both [[constructive mathematics|constructive]] and [[computable mathematics|computable]], thus being one approach to [[synthetic topology]].

The topology on a space is treated not as a discrete [[lattice]], but as an [[exponential object]] of the same [[category]] as the original space, with an associated λ-calculus (which includes an [[internalization|internal]] lattice structure). Every expression in the λ-calculus denotes both a continuous function and a program. ASD does not use the [[Set|category of sets]] (or any [[topos]]), but the [[full subcategory]] of [[overt space|overt]] [[discrete space|discrete]] objects plays this role (an overt object is the dual to a [[compact space|compact]] object), forming an [[arithmetic universe]] (a [[pretopos]] with [[lists]]) with general [[recursion]]; an optional 'underlying set' axiom (which is not [[predicative mathematics|predicative]]) will make this a topos.

The classical (but not constructive) theory of [[locally compact space|locally compact]] [[sober space|sober]] [[topological space]]s is a model of ASD, as is the theory of locally compact [[locale]]s over any topos (even constructively).  In "Beyond Local Compactness" on the ASD website, Taylor removes the restriction of local compactness.

## Related concepts

* [[Stone duality]]

## References

*  [[Paul Taylor]], *Abstract Stone Duality* ([www.paultaylor.eu/ASD](http://www.paultaylor.eu/ASD))

* [[Paul Taylor]], *Review of Abstract Stone Duality* (2004) [[pdf](https://cdc.ioc.ee/appsem04/webproc/short/taylor.pdf)]

On [[Dedekind real numbers]] via abstract Stone duality:

*  [[Paul Taylor]], *[Dedekind cuts](http://www.paultaylor.eu/ASD/dedras/classical)* (2007-2009?)

* [[Andrej Bauer]], [[Paul Taylor]], *The Dedekind reals in abstract Stone duality*, Mathematical Structures in Computer Science **19** 4 (2009) 757-838 [[doi:10.1017/S0960129509007695](https://doi.org/10.1017/S0960129509007695)]

review in:

* [[Andrej Bauer]], *Efficient Computation with Dedekind Reals*, talk at *[Computability and complexity in analysis 2008](https://math.andrej.com/2008/08/24/efficient-computation-with-dedekind-reals/)* and at *[Mathematics, Algorithms and Proofs 2008](http://cdsagenda5.ictp.trieste.it/full_display.php?smr=0&ida=a07167)* (2008) [[web](https://math.andrej.com/2008/08/24/efficient-computation-with-dedekind-reals/), slides: [pdf](https://math.andrej.com/wp-content/uploads/2008/08/slides-map2008.pdf), extended abstract: [pdf](https://math.andrej.com/wp-content/uploads/2008/08/abstract-cca2008.pdf)]

See also:

*  [Café discussion](http://golem.ph.utexas.edu/category/2009/01/abstract_stone_duality.html)

[[!redirects Abstract Stone Duality]]
[[!redirects abstract stone duality]]
[[!redirects ASD]]


abstract syntax tree

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## Idea

In [[formal logic]] and [[computer science]], by an *abstract syntax tree* one means a labelled [[tree]] whose nodes represent *expressions* of a [[formal language]], such as as [[programming language]], and whose edges indicate sub-expressions, meant to reflect the "abstract" [[syntax]] of the language, in contrast to the "concrete syntax" given by linear [[string (computer science)|strings]] of characters.

## Related concepts

* [[variable]], [[substitution]]

* [[word]], [[string (computer science)]]

* [[higher-order abstract syntax]]

## References

Textbook account:

* [[Robert Harper]], §1.1 in: _[[Practical Foundations for Programming Languages]]_, Cambridge University Press (2016) [[ISBN:9781107150300](http://www.cambridge.org/us/academic/subjects/computer-science/programming-languages-and-applied-logic/practical-foundations-programming-languages-2nd-edition?format=HB), [pdf](https://www.cs.cmu.edu/~rwh/pfpl/2nded.pdf)]

See also:

* Wikipedia, *[Abstract syntax tree](https://en.wikipedia.org/wiki/Abstract_syntax_tree)*

[[!redirects abstract syntax trees]]

[[!redirects abstract syntax]]