powerset structure

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## Definition

Given a set $V$ with an [[extensional relation]] $\prec$, a **proto-powerset structure** is a [[function]] $\mathcal{P}:V \to V$ such that

* for all elements $b \in V$, and $c \in V$, if for all elmemets $a \in V$, $a \prec b$ implies $a \prec c$, then $b \prec \mathcal{P}(c)$.

A **powerset structure** is a function $\mathcal{P}:V \to V$ such that

* for all elements $b \in V$, and $c \in V$, $b \prec \mathcal{P}(c)$ if and only if for all elements $a \in V$, $a \prec b$ implies $a \prec c$.

Uniqueness of $\mathcal{P}(c)$ follows from $\prec$ being an [[extensional relation]].

## Foundational concerns

In any [[material set theory]], instead of postulating the mere existence of a set $\mathcal{P}$ in which for all sets $b$, if for all sets $a$, $a \in b$ implies $a \in c$, then $b \in \mathcal{P}$, one could add a primitive unary operation $\mathcal{P}(c)$ which takes material sets $c$ and returns a material set $\mathcal{P}(c)$ such that for all $b$, if for all $a$, $a \in b$ implies $a \in c$, then $b \in \mathcal{P}(c)$.

## See also

* [[axiom of power sets]]

* [[pairing structure]]

* [[union structure]]

* [[natural numbers structure]]

[[!redirects powerset structure]]
[[!redirects powerset structures]]


pp-wave spacetime

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#### Gravity
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[[!include gravity contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _pp-wave spacetime_ is a [[spacetime]], i.e. an exact solution to [[Einstein's equations]] for [[gravity]], containing nothing but [[radiation]] ([[gravitational waves]] and/or [[electromagnetic waves]]) with one fixed [[wave vector]].

The [[Ricci flat]] pp-wave spacetimes are examples of [[universal spacetimes]].

## Properties

### As Penrose limits

Every [[spacetime]] looks like a pp-wave near a [[lightlike]] [[geodesic]]. This is called the _Penrose limit_. (see [Blau 11](#Blau11))

### Relation to BMN matrix model

The _[[BMN matrix model]]_ ([BMN 02, Section 5 and Appendix B](#BerensteinMaldacenaNastase02)) is a deformation of the [[BFSS matrix model]] by [[mass]]-terms which correspond to a deformation of the [[Minkowski spacetime]] background to a [[pp-wave spacetime]].

## Examples

* [[M-wave]]

## Related concepts

* [[flat space holography]]

## References

### General

* {#Blau11} [[Matthias Blau]], _Plane waves and Penrose limits_, 2011 ([pdf](http://www.blau.itp.unibe.ch/lecturesPP.pdf), [[BlauPlaneWavesAndPenroseLimit.pdf:file]])

See also

* Wikipedia, _[pp-wave spacetime](https://en.wikipedia.org/wiki/Pp-wave_spacetime)_

[[!include pp-waves as Penrose limits of AdS spacetimes -- references]]

[[!redirects pp-wave spacetimes]]

[[!redirects pp-wave]]
[[!redirects pp-waves]]

[[!redirects PP-wave spacetime]]
[[!redirects PP-wave spacetimes]]

[[!redirects PP-wave]]
[[!redirects PP-waves]]

[[!redirects Penrose limit]]
[[!redirects Penrose limits]]


pp-wave spacetimes > history

see at _[[pp-wave spacetime]]_


pp-waves as Penrose limits of AdS spacetimes -- references

### pp-Waves as Penrose limits of $AdS_p \times S^q$ spacetimes

Discussion of [[pp-wave spacetimes]] as [[Penrose limits]] ([[Inönü-Wigner contractions]]) of [[anti de Sitter spacetime|AdSp]] [[product manifold|x]] [[n-sphere|S^q]] spacetimes and of the corresponding limit of [[AdS-CFT duality]]:

* {#BerensteinMaldacenaNastase02} [[David Berenstein]], [[Juan Maldacena]], [[Horatiu Nastase]], Section 2 of: _Strings in flat space and pp waves from $\mathcal{N} = 4$ Super Yang Mills_, JHEP 0204 (2002) 013 ([arXiv:hep-th/0202021](https://arxiv.org/abs/hep-th/0202021))

* N. Itzhaki, [[Igor Klebanov]], [[Sunil Mukhi]], _PP Wave Limit and Enhanced Supersymmetry in Gauge Theories_, JHEP 0203 (2002) 048 ([arXiv:hep-th/0202153](https://arxiv.org/abs/hep-th/0202153))

* [[Nakwoo Kim]], Ari Pankiewicz, Soo-Jong Rey, [[Stefan Theisen]], _Superstring on PP-Wave Orbifold from Large-N Quiver Gauge Theory_, Eur. Phys. J. C25:327-332, 2002 ([arXiv:hep-th/0203080](https://arxiv.org/abs/hep-th/0203080))

* E. Floratos, [[Alex Kehagias]],
_Penrose Limits of Orbifolds and Orientifolds_, JHEP 0207 (2002) 031 ([arXiv:hep-th/0203134](https://arxiv.org/abs/hep-th/0203134))

* E. M. Sahraoui, E. H. Saidi, _Metric Building of pp Wave Orbifold Geometries_, Phys.Lett. B558 (2003) 221-228 ([arXiv:hep-th/0210168](https://arxiv.org/abs//hep-th/0210168))

Review:

* Darius Sadri, [[Mohammad Sheikh-Jabbari]], _The Plane-Wave/Super Yang-Mills Duality_, Rev. Mod. Phys. 76:853, 2004 ([arXiv:hep-th/0310119](https://arxiv.org/abs/hep-th/0310119))

* {#Ydri18} [[Badis Ydri]], Section 3.1.10 of: _Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory_ ([arXiv:1708.00734](https://arxiv.org/abs/1708.00734)), published as: _Matrix Models of String Theory_, IOP 2018 ([ISBN:978-0-7503-1726-9](https://iopscience.iop.org/book/978-0-7503-1726-9))

See also:

* Michael Gutperle, Nicholas Klein, *A Penrose limit for type IIB $AdS_6$ solutions* ([arXiv:2105.10824](https://arxiv.org/abs/2105.10824))


PPS

PPS is the acronym for __Preuves, Programmes et Systèmes.__  It is officially forms part of [[IRIF]], that is the Institut de Recherche en Informatique Fondamentale, a mixed research group (UMR 7126) of the CNRS and  the Université Paris Diderot (aka Paris 7).

It is very well known for the research of its members in Theoretical Computer Science.

Its members include:

*  [[Pierre-Louis Curien]];

* [[Philippe Gaucher]];

* [[Paul-André Melliès]];

* [[Francois Métayer]];

* [[Yves Guiraud]].

(more to be added ?)

plus other members of the $\pi r^2$-[[pi-r-squared team|team]].


Pr(infinity,1)Cat

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## Idea

$Pr(\infty,1)Cat$ is the [[(∞,1)-category]] of [[locally presentable (∞,1)-categories]] and [[(∞,1)-colimit]]-preserving [[(∞,1)-functors]] between them ([[Lawvere distributions]]).

Recall that a [[presentable (∞,1)-category]] is a [[localization of an (∞,1)-category|localization]] of a [[(∞,1)-category of (∞,1)-presheaves]]. In particular it has all small [[colimits]]. An [[(∞,1)-functor]] $C \times D \to E$ from the cartesian [[product]] of two presentable $(\infty,1)$-categories is _bilinear_ if it respects colimits in both variables.

It turns out that there is a _universal_ such bilinear functor

$$
  C \times D \to C \otimes D
  \,,
$$

which thereby defines a [[tensor product of presentable (∞,1)-categories]]. This defines a [[monoidal (infinity,1)-category|monoidal structure]] on presentable $(\infty,1)$-categories, which is in fact [[symmetric monoidal (infinity,1)-category|symmetric]].

The collection $Pr(\infty,1)Cat$ of presentable $(\infty,1)$-cateories with colimit-preserving [[(∞,1)-functors]] between them (i.e. with "*linear*" functors between them!), is an $(\infty,1)$-generalization of the category $Set Mod$ of ordinary categories and [[bimodules]] or [[profunctors]], or [[distributors]] between them. See [[distributor]] and in particular the discussion there about the equivalent reformulation in terms of colimit-preserving functors.

Using $Pr(\infty,1)Cat$ with its notion of "linearity" one obtains a very general notion of $\infty$-linear algebra. This is described at [[geometric ∞-function theory]].

## Definition

### Unstable version

Write $Pr(\infty,1)Cat_1$ for the [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-categories]] whose

* objects are [[presentable (∞,1)-category|presentable (∞,1)-categories]];

* morphisms are [[(∞,1)-colimit]]-preserving [[(∞,1)-functors]].

### Stable version

The symmetric monoidal structure on presentable $(\infty,1)$-categories restricts to one on presentable [[stable (∞,1)-category|stable (∞,1)-categories]].

The tensor unit of stable presentable $(\infty,1)$-categories is the [[stable (∞,1)-category of spectra]].

## Properties

### Adjoint functor theorem

A functor between locally presentable $(\infty,1)$-categories:

* is a [[left adjoint]] if and only if it preserves small colimits (hence, in particular, is accessible), and
* is a [[right adjoint]] if and only if it preserves small limits and is accessible.

### Hom-objects

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

For $C, D \in Pr(\infty,1)Cat$, let

$$
  Func^L(C,D) \subset Func(C,D)
$$

be the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-functors]] on those that preserve all small [[(∞,1)-colimits]] (equivalently, are left adjoints).

=--

Evidently $Func^L(C,D)$ would be the hom-$(\infty,1)$-category of an enhancement of $Pr(\infty,1)Cat$ to an $(\infty,2)$-category; its maximal sub-$\infty$-groupoid is the hom-$\infty$-groupoid $Pr(\infty,1)Cat(C,D)$.

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

For all $C,D$ we have that $Func^L(C,D)$ is itself locally presentable.

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.3.8]].

### Embedding into $(\infty,1)Cat$
 {#EmbeddingIntoCat}

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

All small [[limit in a quasi-category|limits and colimits]] exists in $Pr(\infty,1)Cat$. The limits are preserved by the embedding $Pr(\infty,1)Cat \hookrightarrow $ [[(∞,1)Cat]].

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.3.13]].

### Tensor product

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

Let $C_1, \cdots, C_n$ be a finite collection of locally presentable $(\infty,1)$-categories. There exists a locally presentable $(\infty,1)$-category $C_1 \otimes \cdots \otimes C_n$ and an [[(∞,1)-functor]]

$$
  C_1 \times \cdots \times C_n \to C_1 \otimes \cdots \otimes C_n
$$

(the [[tensor product]]) such that

1. it preserves [[(∞,1)-colimit]]s in each variable;

1. for every $D \in Pr(\infty,1)Cat$, composition with $f$ produces an [[equivalence of (∞,1)-categories]]

$$
  Func_{(\infty,1)}^L(C_1 \otimes \cdots \otimes C_n,D)
  \stackrel{\simeq}{\to}
  Func^{L}_{(\infty,1)}(C_1 \times \cdots \times C_n,D)
  \hookrightarrow
  Func^{}_{(\infty,1)}(C_1 \times \cdots \times C_n,D)
$$

onto the full [[sub-(∞,1)-category]] of those functors, that preserves colimits in each argument.

=--

This is  ([Lurie, NA, theorem 4.1.4](#LurieNoncommutative)).

This tensor product makes $Pr(\infty,1)Cat$ a [[symmetric monoidal (∞,1)-category]].  Indeed it is even [[closed monoidal (infinity,1)-category|closed]], since the [[hom-objects]] $Func^L$ supply [[right adjoints]] to the tensor product ([Lurie 09, Rem. 5.5.3.9](#Lurie09HTT)).

The definition of the tensor product is easy: $C \otimes D = Cts(C^{op},D)$, where $Cts$ denotes the category of continuous (small-limit-preserving) functors.  Equivalently, this is $Cocts(C,D^{op})^{op}$, where $Cocts$ denotes the category of cocontinuous (small-colimit-preserving) functors; it is tempting to write this as $Func^L(C,D^{op})^{op}$, but note that $D^{op}$ is not itself locally presentable.  However, $D^{op}$ is nevertheless cocomplete, so any cocontinuous functor $C \to D^{op}$ has a right adjoint.

To show that $Cts(C^{op},D)$ is locally presentable, recall that $C \simeq Ind_\kappa A$ for some small $\kappa$-cocomplete $A$.  Thus $Cts(C^{op},D) \simeq \kappa Cts (A^{op},D)$, where $\kappa Cts$ denotes the category of $\kappa$-limit-preserving functors; this is an accessible localization of $Fun(A^{op},D)$ and hence locally presentable.

To show that $Cts(C^{op},D)$ has the right universal property, we compute

$$
\begin{aligned}
Func^L(C,Func^L(D,E))
&\simeq \kappa CoCts (A,Func^L(D,E)) & \quad \text{(by the universal property of }\; C = Ind_\kappa(A))\\
&\simeq \kappa Cts (A^{op},Func^L(D,E)^{op})^{op} &\quad \text{(by taking opposites)}\\\\
&\simeq \kappa Cts (A^{op},CtsAcc(E,D))^{op} &\quad \text{(by the adjoint functor theorem)}\\
&\simeq CtsAcc(E,\kappa Cts (A^{op},D))^{op} &\quad (*)\\
&\simeq Func^L(\kappa Cts (A^{op},D),E) &\quad \text{(by the adjoint functor theorem)}
\end{aligned}
$$

To justify the step $(*)$, note that both sides are equivalent to some subcategory of functors $A^{op} \times E \to D$, and in both cases part of the requirement is that $\kappa$-limits be preserved in the $A^{op}$-variable and all small limits be preserved in the $E$-variable.  The remaining condition on the left is that for each $a\in A$ the corresponding functor $E\to D$ is accessible, i.e. preserves $\lambda_a$-filtered colimits for some cardinal $\lambda_a$.  On the right, the remaining condition is that the [[currying|curried]] functor $E\to \kappa Cts (A^{op},D)$ is accessible, i.e. preserves $\mu$-filtered colimits for some cardinal $\mu$.  Since $\mu$-filtered colimits commute with $\kappa$-limits as long as $\mu\ge \kappa$, such colimits in $\kappa Cts (A^{op},D)$ are pointwise, so we are at least talking about the same colimits, so the right-hand condition implies the left (take each $\lambda_a = \mu$).  On the other hand, because $A$ is small, if the left-hand condition holds we can take $\mu = \sup_{a\in A} \lambda_a$ to make the right-hand condition hold.

### Colimits

Above we remarked that the forgetful functor $Pr(\infty,1)Cat \to (\infty,1)Cat$ preserves limits.  It does not preserve colimits, but $Pr(\infty,1)Cat$ does have colimits; they are constructed by passing to right adjoints and taking the limit there.

In particular, this implies there are some cases in which colimits coincide with limits.  For instance:

+--{: .num_prop}
###### Proposition
Small coproducts in $Pr(\infty,1)Cat$ coincide with small products.
=--
+--{: .proof}
###### Proof
Let $\prod_i C_i$ be a product in $Pr(\infty,1)Cat$.  Then we have
$$
\begin{aligned}
Func^L(\prod_i C_i, D)
&\simeq Cts(D, \prod_i C_i)\\
&\simeq \prod_i Cts(D,C_i)\\
&\simeq \prod_i Func^L(C_i,D)
\end{aligned}
$$
=--

To be precise, the next result requires an enhancement of $Pr(\infty,1)Cat$ to an $(\infty,2)$-category so that we can talk about [[powers]] and [[copowers]] by small $(\infty,1)$-categories.  However, the proof gives an explicit universal property that makes sense without having to define that whole $(\infty,2)$-category.

\begin{proposition}
The [[copower]] of $C\in Pr(\infty,1)Cat$ by a small $(\infty,1)$-category $A$ is the [[(infinity,1)-category of (infinity,1)-presheaves|$\infty$-category of $\infty$-presheaves]] $C^{A^{op}}$.
\end{proposition}

\begin{proof}
Let $P A$ denote the [[(∞,1)-presheaf (∞,1)-category]] on $A$.  This is its [[free cocompletion]], so we have $Func^L(P A,C) \simeq C^A$.  Dually, $(P A)^{op}$ is the [[free completion]] of $A^{op}$, so we have $Cts((P A)^{op},C) \simeq C^{A^op}$.  Now we have:
$$
\begin{aligned}
Func^L(C,D)^A
&\simeq Func^L(C,D^A)\\
&\simeq Func^L(C,Func^L(P A,D))\\
&\simeq Func^L(C \otimes P A, D)\\
&\simeq Func^L(Cts((P A)^{op},C),D)\\
&\simeq Func^L(C^{A^op},D)
\end{aligned}
$$
\end{proof}

In general, this sort of argument should work for all lax colimits of lax functors; for instance, [[Kleisli objects]] should also coincide with [[Eilenberg-Moore objects]] (though this also requires enhancing $Pr(\infty,1)Cat$ to an $(\infty,2)$-category).  The corresponding 1-categorical fact is that in any [[bicategory]] with [[local colimits]] (colimits in each hom-category distributed over by composition), lax colimits of lax functors are also lax limits.

### Relation to $\infty$-profunctors
 {#RelationToInfinityProfunctors}

In analogy to the situation for [[profunctors]] between [[1-categories]] (see [there](profunctor#FuncsOnPresheaves)), [[(infinity,1)-profunctors|$\infty$-profunctors]]

$$
  \mathcal{C} ⇸ \mathcal{D}
$$

are equivalently plain but [[(infinity,1)-colimit|$\infty$-colimit]]-[[preserved limit|preserving]] [[(infinity,1)-functors|$\infty$-functors]]

$$
  PSh_\infty(\mathcal{C})
  \xrightarrow{\;}
  PSh_\infty(\mathcal{D})
$$

between the corresponding [[(infinity,1)-categories of (infinity,1)-presheaves|$\infty$-categories of $\infty$-presheaves]].

In this way [[small (infinity,1)-categories|small $\infty$-categories]] with [[(infinity,1)-profunctors|$\infty$-profunctors]] between them form a [[full sub-(infinity,1)-category|full sub-$\infty$-category]] of of the $\infty$-category [[Pr(∞,1)Cat]] that of [[presentable (infinity,1)-categories|presentable $\infty$-categories]] with [[cocontinuous functor|cocontinuous]] [[(infinity,1)-functor|$\infty$-functors]] between them.

### As $\infty$-vector spaces

In some context it makes good sense to think of $Pr(\infty,1)Cat$ as a model for an $(\infty,1)$-category of "$\infty$-vector spaces", or at least "$\infty$-abelian groups".  For instance, the fact that the $(\infty,2)$-category $Pr(\infty,1)Cat$ has local colimits, making certain limits and colimits coincide, is a sort of categorification of the fact that of [[Vect]] and [[Ab]] are [[additive categories]] in which finite products and coproducts coincide.

More on this analogy is at [[integral transforms on sheaves]].  Here a small $(\infty,1)$-category $S$ is to be thought of as a _basis_ and the locally presentable $(\infty,1)$-category $C \hookrightarrow PSh_{(\infty,1)}(C)$ as the $\infty$-vector space spanned by this basis. The colimits in $C$ play the role of addition of vectors and the fact that morphisms in $Pr(\infty,1)Cat$ are colimit-presserving means that they play  the role of _linear_ maps between vector spaces. This is described also at [[Lawvere distribution]].

The monoidal product $\otimes : Pr(\infty,1)Cat \times Pr(\infty,1)Cat \to Pr(\infty,1)Cat$ plays the role of the [[tensor product]] of vector spaces, with a morphism out of $C \otimes D$ being a bilinear morphism out of $C \times D$, and the fact that $Pr(\infty,1)Cat$ is closed monoidal reflects the fact that [[Vect]] is closed monoidal.  The construction of the tensor product as $Cts(C^{op},D)$ corresponds to the fact that, at least for finite-dimensional $V$, we have $V\otimes W \cong Hom(V^\ast,W)$.

(A related decategorification of $Pr(\infty,1)Cat$ is the category [[Sup]] of [[suplattices]], which can also be thought of as analogous to abelian groups or vector spaces.)

\begin{remark}
**(Application)**
\linebreak
Combined with the fact that the embedding $Pr(\infty,1)Cat \hookrightarrow (\infty,1)Cat$ preserves limits (prop. \ref{LimitsAndColimits}), this yields some useful statements.

For instance with $Pr(\infty,1)Cat$ regarded as $\infty Vect$, for any [[∞-group]] $G$ with [[delooping]] [[∞-groupoid]] $\mathbf{B}G$, we may think of an [[(∞,1)-functor]] $\rho : \mathbf{B}G \to Pr(\infty,1)Cat$ as a linear [[representation]] of $G$: the single object of $\mathbf{B}G$ is sent to a presentable $(\infty,1)$-category $V$ and the morphisms in $\mathbf{B}G$ then define an [[action]] of $G$ on that.
\end{remark}

## Related concepts

* [[PrCat]]

* [[Ho(CombModCat)]]

* [[2Ab]], [[2Ring]]

## References

The $(\infty,1)$-category $Pr(\infty,1)Cat$ is introduced in section 5.5.3 of

* {#Lurie09HTT} [[Jacob Lurie]], _[[Higher Topos Theory]]_

The monoidal structure on $Pr(\infty,1)Cat$ is described in section 4.1 of

* [[Jacob Lurie]], _[[Noncommutative Algebra]]_
{#LurieNoncommutative}

That this is in fact a symmetric monoidal structure is discussed in section 6 of

* [[Jacob Lurie]], _[[Commutative Algebra]]_

(see proposition 6.14 and remark 6.18).  This monoidal structure is exhibited as a restriction of a monoidal structure on all cocomplete $(\infty,1)$-categories in section 4.8 of

* [[Jacob Lurie]], _[[Higher Algebra]]_.

[[!redirects symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories]]

[[!redirects symmetric monoidal (∞,1)-category of presentable (∞,1)-categories]]
[[!redirects symmetric monoidal (infinity,1)-category of presentable (inf]]
[[!redirects Pr(∞,1)Cat]]

category: category


practical foundation of mathematics

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[foundations of mathematics]] the term **practical foundations** (following a term introduced in ([Taylor](#Taylor))) refers to emphasis on conceptually natural formalizations.

> In my own education I was fortunate to have two teachers who used the term "foundations" in a common-sense way (rather than in the speculative way of the Bolzano-Frege-Peano-Russell tradition). This way is exemplified by their work in Foundations of Algebraic Topology, published in 1952 by [Eilenberg](#EilenbergSteenrod) (with Steenrod), and The Mechanical Foundations of Elasticity and Fluid Mechanics, published in the same year by Truesdell. The orientation of these works seemed to be "concentrate the essence of practice and in turn use the result to guide practice". ([Lawvere 2003: 213](#Lawvere))

   Formal systems of interest here are [[natural deduction]] in [[type theories]], which allow natural expressions for central concepts in mathematics, notably via their [[categorical semantics]] and the conceptual strength of [[category theory]] (see [Harper](#Harper)).

## References

* [[Paul Taylor]], _[[Practical Foundations of Mathematics]]_, Cambridge University Press ([web](http://www.cs.man.ac.uk/~pt/Practical-Foundations/html/index.html))
{#Taylor}

* {#Lawvere} [[William Lawvere]], _Foundations and applications: axiomatization and education_, Bulletin of Symbolic Logic 9 (2003), 213-224 ([ps](http://www.math.ucla.edu/~asl/bsl/0902/0902-006.ps))

Under [[computational trinitarianism]] this corresponds to a practical foundation in [[program|programming]], laid out in

* {#Harper} [[Robert Harper]], _[[Practical Foundations for Programming Languages]]_ ([web](http://www.cs.cmu.edu/~rwh/plbook/book.pdf))

A foundation for [[algebraic topology]] in this practical spirit is laid out in

* {#EilenbergSteenrod} [[Samuel Eilenberg]], [[Norman Steenrod]], _[[Foundations of Algebraic Topology]]_

[[!redirects practical foundation]]
[[!redirects practical foundations]]
[[!redirects practical foundations of mathematics]]


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

The book

* [[Robert Harper]]

  **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)

lays the foundations of the theory of [[programming languages]] in terms of [[type theory]].
In over 500 pages, the author formally specifies and step-by-step extends established [[type systems]] and reasons about type safety and [[operational semantics]] (not so much about [[dependent types]] and [[categorical semantics]]). The book also contains a discussion of [[formal logic]] in type theory and features of variants of the `Algol` [[programming language]], in particular.

> [p. xvii:] Types are the central organizing principle of the theory of programming languages. Language features are manifestations of type structure. The syntax of a language is governed by the constructs that define its types, and its semantics is determined by the interactions among those constructs. The soundness of a language design—the absence of ill-defined programs—follows naturally. The purpose of this book is to explain this remark.

The preview of the second edition (2016) is available as a [pdf](https://www.cs.cmu.edu/~rwh/pfpl/2nded.pdf). A description of the changes is [here](https://existentialtype.wordpress.com/2016/04/11/practical-foundations-of-programming-languages-second-edition/).

## Related entries

* [[Carl A. Gunter]]: *The semantics of types in programming languages*, in: *[[Handbook of Logic in Computer Science]]* Vol 3: *Semantic Structures*, Oxford University Press (1995) [[pdf](http://seclab.illinois.edu/wp-content/uploads/2011/04/Gunter94.pdf)]

* [[Paul Taylor]]

  _[[Practical Foundations of Mathematics]]_

  ([web](http://www.paultaylor.eu/~pt/prafm/index.html))

* [[William Lawvere]], [[Robert Rosebrugh]],

  _[[Sets for Mathematics]]_,

  Cambridge UP  2003 ([book homepage](http://www.mta.ca/~rrosebru/setsformath/), [GoogleBooks](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics), [pdf](http://patryshev.com/books/Sets%20for%20Mathematics.pdf))

* [[computational trilogy]]

category: reference, applications

[[!redirects Practical Foundations for Programming Languages]]
[[!redirects PFPL]]


Practical Foundations of Mathematics

The book

* [[Paul Taylor]]:

  \linebreak

  **Practical Foundations of Mathematics**

  \linebreak

  Cambridge Studies in Advanced Mathematics **59**

  Cambridge University Press (1999)

  ISBN 0-521-63107-6

  [webpage](http://www.paultaylor.eu/~pt/prafm/index.html)

is a description of how (in Taylor\'s opinion) the [[foundations]] of mathematics should really be done, with an eye towards matching how mathematics is done in practice (with the consequence that the system is no stronger than necessary).

The result is actually a series of foundations, most [[constructive mathematics|constructive]], suitable for different sorts of mathematics.  Ultimately, these are described as [[internal logic|logic in]] categories defined by [[sketch]]es and equipped with distinguished [[pullback]]-stable classes of _display morphisms_.

The book includes a self-contained, though dense, introduction to [[category theory]]. Before the three chapters on category theory comes a chapter "Posets and Lattices", which "does for posets everything that is later done for categories" (per Taylor's [summary](http://www.paultaylor.eu/~pt/prafm/summary.html)); compare [[category theory vs order theory]].

## Resources

The text is [available online](http://www.PaulTaylor.EU/Practical-Foundations/index.html) in a somewhat unreadable format.

There is also a [summary](http://www.paultaylor.eu/~pt/prafm/summary.html) in a readable format. This is basically an expanded table of contents together with an abbreviated introduction, with a link into the above-mentioned online text for each section.

A useful survey of some of the topics discussed there is also in

* [[Paul Taylor]], _Foundations for computable topology_ ([web](http://www.paultaylor.eu/ASD/foufct/abstract.html))
 {#topology}

which is an exposition of Taylor's [[Abstract Stone Duality]].

## Related entries

* [[Robert Harper]],

  _[[Practical Foundations for Programming Languages]]_,

  Cambridge University Press (2016), ([webpage](http://www.cambridge.org/us/academic/subjects/computer-science/programming-languages-and-applied-logic/practical-foundations-programming-languages-2nd-edition?format=HB))

* [[William Lawvere]], [[Robert Rosebrugh]],

  _[[Sets for Mathematics]]_,

  Cambridge UP  2003 ([book homepage](http://www.mta.ca/~rrosebru/setsformath/), [GoogleBooks](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics), [pdf](http://patryshev.com/books/Sets%20for%20Mathematics.pdf))

category: reference

[[!redirects Practical Foundations]]
[[!redirects Practical Foundations of Mathematics]]
[[!redirects PFM]]


praeclarum theorema

The material previously on this page may be found at [Inquiry Into Inquiry](http://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/).


Prakash Panangaden

Prakash is a researcher in the Computer Science department of McGill University. His research interests include (a) probabilistic systems (b) concurrency and (c) quantum information theory.

* [website](http://www.cs.mcgill.ca/~prakash/)

## Selected writings

On [[measurement-based quantum computation]]:

* [[Vincent Danos]], [[Elham Kashefi]], [[Prakash Panangaden]], *The Measurement Calculus*, Journal of the ACM, **54** 2 (2007) [[arXiv:0704.1263](https://arxiv.org/abs/0704.1263), [doi:10.1145/1219092.1219096](https://doi.org/10.1145/1219092.1219096)]

On [[minimal objects]]:

* Nick Bezhanishvili, Clemens Kupke, [[Prakash Panangaden]], *Minimization via Duality*, in: *Logic, Language, Information and Computation. WoLLIC 2012*, Lecture Notes in Computer Science **7456**, Springer (2012) [[doi:10.1007/978-3-642-32621-9_14](https://doi.org/10.1007/978-3-642-32621-9_14), [pdf](https://staff.science.uva.nl/n.bezhanishvili/Papers/Wollic.pdf)]

category: people


Praphulla Koushik

I am Praphulla Koushik. I am a research scholar in School of Mathematics, IISER Thiruvananthapuram, India. I am interested in Differential geometry, Algebraic geometry, Category theory.


Pratik Rath

* [GoogleScholar page](https://scholar.google.com/citations?user=imnkPa4AAAAJ&hl=en)

## Selected writings

On [[holography]], [[holographic entanglement entropy]], [[holographic Renyi entropy]] and [[quantum error correcting codes]]:

* [[Pratik Rath]], *Aspects of Holography And Quantum Error Correction*, 2020 ([pdf](https://digitalassets.lib.berkeley.edu/etd/ucb/text/Rath_berkeley_0028E_19810.pdf), [[RathHolographyAndQEC.pdf:file]])

* [[Chris Akers]], [[Pratik Rath]], *Holographic Renyi entropy from quantum error correction*,  J. High Energ. Phys. 2019, 52 (2019) ([arXiv:1811.05171](https://arxiv.org/abs/1811.05171))

category: people


PrCat

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Categories of categories
+--{: .hide}
[[!include categories of categories - contents]]
=--
#### Compact objects
+-- {: .hide}
[[!include compact object - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

When working with [[locally presentable categories]], one is typically interested only in the [[colimit]]-[[preserved limit|preserving]] [[functors]] between them, hence (by the [[adjoint functor theorem]]) equivalently the [[left adjoint]] functors.

One hence considers the [[very large category]] $PrCat$ whose objects are [[locally presentable categories]], and whose morphisms are [[left adjoint functors]].

The analog of this

* in [[(∞,1)-category theory]] is [[Pr(∞,1)Cat]];

* in [[model category]]-theory is [[Ho(CombModCat)]].

## Related concepts

* [[CatAdj|$Cat_{Adj}$]]

[[!include locally presentable categories - table]]


pre-abelian category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

A **pre-abelian category** is an [[additive category]] (an [[Ab-enriched category]] with finite [[biproduct|biproducts]]) such that every [[morphism]] has a [[kernel]] and a [[cokernel]].

=--

Equivalently:

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

A **pre-abelian category** is an [[Ab-enriched category]] category with all [[finite limits]] and finite [[colimits]].

=--

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

These two definitions are indeed equivalent.

=--

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

By the discussion [here](finite+limit#Properties) the existence of finite limits is equivalent to that of finite [[products]] and [[equalizers]]. But an [[equalizer]] of two morphisms $f$ and $g$ in an [[Ab]]-[[enriched category]] is the same as a [[kernel]] of $f-g$. Dually for finite colimits, coequalizers and cokernels.

=--

## Properties

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

For every object $c\in C$ in a pre-abelian category, the operations of [[kernel]] and [[cokernel]] form a [[Galois connection]] between the [[preorders]] $Sub(c)$ of [[monomorphisms]] ([[subobjects]]) into $c$ and $Quot(c)$ of [[epimorphisms ]]out of $c$.

In particular, $f:b\to c$ is a [[kernel]] iff $f = ker(coker(f))$ and dually.

=--

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

Every [[morphism]] $f:A\to B$ in a pre-abelian category has a canonical decomposition

$$
  A\stackrel{p}\to \coker(\ker f)\stackrel{\bar{f}}\to\ker(\coker f)\stackrel{i}\to B
$$

where $p$ is a [[cokernel]], hence  an [[epimorphism|epi]], and $i$ is a [[kernel]], and hence [[monomorphism|monic]].

=--

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

If $\bar f$ in the above decomposition is always an [[isomorphism]], then the pre-abelian category is called an _[[abelian category]]_.

=--

## Examples

* Of course every [[abelian category]] is pre-abelian.

* The category $TF$ of [[torsion subgroup|torsion-free]] [[abelian groups]] is [[reflective subcategory|reflective]] in all of [[Ab]].  Therefore, it is a complete and cocomplete $Ab$-enriched category, and therefore in particular pre-abelian.  However, it is not abelian; the monomorphism $2:\mathbb{Z}\to \mathbb{Z}$ is not a kernel.

## Related concepts

The concept "pre-abelian category" is part of a sequence of concepts of [[additive and abelian categories]].

[[!redirects pre-abelian categories]]
[[!redirects preabelian categories]]
[[!redirects preabelian category]]


pre-additive category

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

# Preadditive categories
* table of contents
{: toc}

## Definition

Some authors take *pre-additive category* to mean *[[Ab-enriched category|$Ab$-enriched category]]*, for example:

* [[Nicolae Popescu]], _[[Abelian categories with applications to rings and modules]]_, London Math. Soc. Monographs __3__, Academic Press (1973) [[MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)]

* [[Charles Weibel]], *[[An Introduction to Homological Algebra]]*, Cambridge University Press (1994) [[doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136), [pdf](https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf)]

* [[Masaki Kashiwara]], [[Pierre Schapira]]: Section 8 in: *[[Categories and Sheaves]]*, Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006) [[doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf)]

* [[The Stacks Project]], *12.3 Preadditive and additive categories* [[09SE](https://stacks.math.columbia.edu/tag/09SE)]

Other authors require in addition the existence of a [[zero object]] (which, this being the [[empty diagram|empty]] [[coproduct]], is the first step towards requiring more general [[finite coproducts]] as for an [[additive category]], whence the terminology):

* Sandro M. Roch, Def. 1.0.4 in: *A brief introduction to abelian categories* (2020) [[pdf](https://page.math.tu-berlin.de/~roch/files/abelian_categories.pdf)]

## Related concepts

* [[pseudo-abelian category]]

[[!redirects preadditive category]]
[[!redirects preadditive categories]]
[[!redirects pre-additive category]]
[[!redirects pre-additive categories]]


pre-Calabi-Yau algebra

Related notions in $n$Lab: [[Calabi-Yau algebra]], [[Calabi-Yau category]], [[Calabi-Yau object]]

* [[Maxim Kontsevich]], Alex Takeda, [[Yiannis Vlassopoulos]], _Pre-Calabi-Yau algebras and topological quantum field theories_, [arXiv:2112.14667](https://arxiv.org/abs/2112.14667)
* [[Maxim Kontsevich]], [[Yannis Vlassopoulos]], Natalia Iyudu, _Pre-Calabi-Yau algebras and [[double Poisson bracket]]s_, [arXiv:1906.07134](https://arxiv.org/abs/1906.07134)
* Maxim Kontsevich, Yannis Vlassopoulos, Natalia Iyudu, _Pre-Calabi-Yau algebras as noncommutative Poisson structures_, J. Algebra _567_ (2021) 63--90 [doi](https://doi.org/10.1016/j.jalgebra.2020.08.029)
* Natalia Iyudu, [[Maxim Kontsevich]], _Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology_, [arXiv:2011.11888](https://arxiv.org/abs/2011.11888); _Pre-Calabi-Yau algebras and ξ∂-calculus on higher cyclic Hochschild cohomology_, preprint IHES M-19-14 (2019) [pdf](https://preprints.ihes.fr/2019/M/M-19-14.pdf)
* Maxim Kontsevich, Alex Takeda, Yiannis Vlassopoulos, _Smooth Calabi-Yau structures and the noncommutative Legendre transform_, [arXiv:2301.01567](https://arxiv.org/abs/2301.01567)

> We elucidate the relation between smooth Calabi-Yau structures and pre-Calabi-Yau structures. We show that, from a smooth Calabi-Yau structure on an A∞-category A, one can produce a pre-Calabi-Yau structure on A; as defined in our previous work, this is a shifted noncommutative version of an integrable polyvector field. We explain how this relation is an analogue of the Legendre transform, and how it defines a one-to-one mapping, in a certain homological sense. For concreteness, we apply this formalism to chains on based loop spaces of (possibly non-simply connected) Poincaré duality spaces, and fully calculate the case of the circle.

* Alex Takeda, _The noncommutative Legendre transform and Calabi-Yau structures_, Purdue Topology Seminar [youtube](https://www.youtube.com/watch?v=na-wj6aOw2E)
* Johan Leray, [[Bruno Vallette]], _Pre-Calabi--Yau algebras and homotopy double Poisson gebras_, [arXiv:2203.05062](https://arxiv.org/abs/2203.05062)
* Wai-Kit Yeung, _Pre-Calabi-Yau structures and moduli of representations_, [arXiv:1802.05398](https://arxiv.org/abs/1802.05398); _Ribbon dioperads and modular ribbon properads_, [arXiv:2202.13269](https://arxiv.org/abs/2202.13269)
* [[David Fernández]], Estanislao Herscovich, _Double quasi-Poisson algebras are pre-Calabi-Yau_, [arXiv:2002.10495](https://arxiv.org/abs/2002.10495)
* Marion Boucrot, _Morphisms of pre-Calabi-Yau categories and morphisms of cyclic $A_\inf$-categories_, [arXiv:2304.13661](https://arxiv.org/abs/2304.13661)

category: algebra


pre-images preserve unions and intersections

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

#Contents#
* table of contents
{:toc}

## Statement

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

Let $f \colon X \longrightarrow Y$ be a [[function]] between [[sets]]. Let $\{S_i \subset Y\}_{i \in I}$ be a set of [[subsets]] of $Y$. Then

1. $f^{-1}\left( \underset{i \in I}{\cup}  S_i\right) = \left(\underset{i \in I}{\cup} f^{-1}(S_i)\right)$ (the [[pre-image]] under $f$ of a [[union]] of subsets is the union of the pre-images)

1. $f^{-1}\left( \underset{i \in I}{\cap}  S_i\right) \subset \left(\underset{i \in I}{\cap} f^{-1}(S_i)\right)$ (the [[pre-image]] under $f$ of the [[intersection]] of the subsets is the intersection of the pre-images).

=--

For details see at _[[interactions of images and pre-images with unions and intersections]]_.

## Related statements

* [[images preserve unions]]

* [[countable unions of countable sets are countable]]

[[!redirects pre-images preserve intersections]]


pre-Lie algebra

# Pre-Lie algebras
* table of contents
{: toc}

## Definition

A **pre-Lie algebra** is a [[vector space]] $A$ equipped with a bilinear operation $\cdot: A \times A \to A$ such that
\[      [L_a, L_b] = L_{[a,b]}  \label{basic_identity} \]
for all $a, b\in A$.  Here $L_a$ is the operation of left multiplication by $a$:
$$ L_a b = a \cdot b $$
and $[L_a, L_b]$ is the usual commutator of operators using [[composition]]:
$$ [L_a, L_b] = L_a L_b - L_b L_a $$
while $[a,b]$ is the commutator defined using the $\cdot$ operation:
\[ [a,b] = a \cdot b - b \cdot a  \label{torsion_free_identity} \]

Unravelling this, we see a pre-Lie algebra is vector space $A$ equipped with a bilinear operation $\cdot: A \times A \to A$ such that
\[ a \cdot (b \cdot c) - b \cdot (a \cdot c) =
(a \cdot b) \cdot c - (b \cdot a) \cdot c  \label{identity} \]
More precisely, this is a **left** pre-Lie algebra.  We can also define right pre-Lie algebras.

Every [[associative algebra]] is a pre-Lie algebra, but not conversely.  The reason pre-Lie algebras have the name they do is that this weakening of the concept of associative algebra is still enough to give a [[Lie algebra]]!  In other words: it is well-known that if $A$ is an associative algebra, the operation
$$  [a,b] = a \cdot b - b \cdot a $$
makes $A$ into a Lie algebra.  But this is also true for pre-Lie algebras!  It is a fun exercise to derive the [[Jacobi identity]] from equation \eqref{identity}.

## Examples

First, given a [[manifold]] with a flat torsion-free [[connection]] $\nabla$ on its [[tangent bundle]], we can make the space of [[tangent vector fields]] into a pre-Lie algebra by defining
$$  v \cdot w = \nabla_v w $$
The definition of 'flat' is precisely \eqref{basic_identity}, whereas that of 'torsion-free' is precisely \eqref{torsion_free_identity}.  The Lie algebra arising from this pre-Lie algebra is just the usual Lie algebra of vector fields.

Second, suppose $O$ is a [[linear operad]], and let $A$ be the free $O$-[[algebra of an operad|algebra]] on one generator.  As a vector space we have
$$  A = \bigoplus_{n} O_n/S_n \, .$$
Here $S_n$ is the [[symmetric group]], which acts on the space $O_n$ of $n$-ary operations of $O$.  Moreover, $A$ becomes a pre-Lie algebra in a manner described here:

* Dominique Manchon, [A short survey on pre-Lie algebras](http://math.univ-bpclermont.fr/~manchon/biblio/ESI-prelie2009.pdf)

Third, the [[Hochschild cohomology|Hochschild]] [[chain complex]] of any associative algebra, with grading shifted down by one, can be given the structure of a '[[graded object|graded]] pre-Lie algebra', as discovered by Gerstenhaber and described here:

* Justin Thomas, [Graduate student seminar: Hochschild cohomology](http://www.math.northwestern.edu/~jdthomas/Talk%20Notes/Hoch%20Cohomology.pdf)

In fact it was Gerstenhaber who coined the term 'pre-Lie algebra', for this reason.

## Relation to the work of Connes and Kreimer

Connes and Kreimer formalized the process of [renormalization](http://ncatlab.org/nlab/show/renormalization#hopfalgebraic_renormalization_3) using a certain [[Hopf algebra]] built from [[Feynman diagram]]s.  More abstractly we can understand the essence of their construction using a Hopf algebra built from rooted trees, as explained here:

* John Baez, This Week's Finds in Mathematical Physics, Week 299.  ([web](http://math.ucr.edu/home/baez/week299.html))  ([blog](http://golem.ph.utexas.edu/category/2010/06/this_weeks_finds_in_mathematic_60.html))

The key is to form the free pre-Lie algebra on one generator, then turn this into a Lie algebra as described above, and then form the universal enveloping of that, which is a cocommutative Hopf algebra.  Finally, the restricted dual of this cocommutative Hopf algebra is the commutative Hopf algebra considered by Connes and Kreimer here:

* Alain Connes and Dirk Kreimer, Hopf algebras, renormalization and noncommutative geometry, _Commun. Math. Phys._ **199** (1998), 203--242 ([arXiv](http://arxiv.org/abs/hep-th/9808042))

Pre-Lie algebras are algebras of a [[linear operad]] called $PL$.  The space $PL_n$ has a basis given by labelled rooted [[trees]] with $n$ vertices, and the $i$th partial composite $s \circ_i t$ is given by summing all the possible ways of inserting the tree $t$ inside the tree $s$ at the vertex labelled $i$.  For details see:

* Frédéric Chapoton, Muriel Livernet, Pre-Lie algebras and the rooted trees operad, _Int. Math. Res. Not._ 2001 (2001), 395-408.

The free pre-Lie algebra on one generator is thus
$$\bigoplus_{n} PL_n /S_n  \, $$
so the description of $PL_n$ in terms of rooted trees gives a kind of 'explanation' of the relation between the Connes--Kreimer Hopf algebra and rooted trees.

## Self-referentiality ##

Pre-Lie algebras have a strange self-referential feature.  Every operad of a large class gives a pre-Lie algebra, but the operad for pre-Lie algebras is one of this class!  This raises the following interesting puzzle.

As we have seen above, for any linear operad $O$, the free $O$-algebra with one generator becomes a pre-Lie algebra.  But the operad for pre-Lie algebra is an operad of this type.  So, the free pre-Lie algebra on one generator becomes a pre-Lie algebra in this way.   But of course it already *is* a pre-Lie algebra!  Do these pre-Lie structures agree?

The answer is _no._  For an explanation, see page 7 here:

* Dominique Manchon, A short survey on pre-Lie algebras.  ([pdf](http://math.univ-bpclermont.fr/~manchon/biblio/ESI-prelie2009.pdf))

## References ##

The best overall introduction to pre-Lie algebras seems to be that by Dominique Manchon, cited above.  For two more introductions, try the following:

* John Baez, This Week's Finds in Mathematical Physics, Week 299.  ([web](http://math.ucr.edu/home/baez/week299.html))  ([blog](http://golem.ph.utexas.edu/category/2010/06/this_weeks_finds_in_mathematic_60.html))

* Frédéric Chapoton, Operadic point of view on the Hopf algebra of rooted trees.  ([pdf](http://www-math.unice.fr/~patras/CargeseConference/ACQFT09_FredericCHAPOTON.pdf))


pre-metric electromagnetism

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

#Contents#
* table of contents
{:toc}

## Idea

Inspection of the [[equations of motion]] for [[classical field theory|classical]] [[electromagnetism]] ([[Maxwell's equations]]) readily reveals that --- either in [[vacuum]], in a [[dielectric medium]] or in a [[background field]] of [[gravity]] --- they may equivalently be (re)written (in fact this is close to their original historical form) as

1. a system of purely "topological", "cohomological" or "exterior" [[partial differential equations]]

   on *twice* the number of physical fields,

1. a linear "[[constitutive equation]]" imposing a "duality" relation on the fields

   reducing them to the physical degrees of freedom,

where it is *only the second but not the first* part that involves the "[[background field|background]]" structure, namely the [[dielectric tensor]] characterizing the ambient [[dielectric medium]] and/or the [[pseudo-Riemannian metric]] characterizing the [[field (physics)|field]] of [[gravity]].

This fact itself is fairly evident (certainly in modern formulations of Maxwell's equations, see [below](#Details)), but some authors have highlighted it as possibly being of deeper relevance, starting with [Kottler (1922a)](#Kottler22a), [(1922b)](#Kottler22b), [Cartan (1924) §80](#Cartan24), [Dantzig (1934)](#Dantzig34)  and more recently [Hehl & Obukhov (2003)](#HehlObukhov03), [Delphenich (2005a)](#Delphenich05a), [(2005b)](#Delphenich05a) (see the comprehensive account by [Delphenich (202x)](#DelphenichBook)), the broad idea being that the second step above should be regarded as, indeed, secondary and possibly subject to conceptual re-evaluation (e.g. [Delphenich (2015)](#Delphenich15)).

We point out [below](#RRFieldInGravitationalBackground) (see also [Freed (2002)](#Freed02)) that, while the explicit perspective of "pre-metric electromagnetism" is maybe not widely appreciated under this name, in fact exactly the same idea -- just with the [[electromagnetic field]] replaced by the (hypothetical) "[[RR-field]]" and then often called the "democratic" instead of "premetric" formulation --  is secretly at the heart of the the widely recognized [[conjecture]] of [[K-theory classification of D-brane charge]] as well as of related conjectures, such as *[[schreiber:Hypothesis H]]*.

## Premetric electromagnetism
 {#Details}

### Electromagnetism in gravitational backgrounds

The [[equations of motion]] of [[classical field theory|classical]] [[electromagnetism]] ([[Maxwell's equations]]) on any [[spacetime manifold]] $(X,g)$ read, in modern [[differential form]]-formulation:

\[
  \label{MaxwellEquationsInTraditionalDifferentialForm}
  \begin{array}{rcl}
    \mathrm{d}\,  F &=& 0 \mathrlap{\,,}
    \\
    \mathrm{d} \, \star_g F &=& j
  \end{array}
\]

where

* the [[differential 2-form]] $F \,\in\, \Omega^2(X)$ is the [[Faraday tensor]],

* the [[differential 3-form]] $j \,\in\, \Omega^3(X)$ is the electromagnetic [[current]] density

* $\mathrm{d} \,\colon\,\Omega^p(X) \longrightarrow \Omega^{p+1}(X)$ denotes the [[de Rham differential]],

and last not least

* $\star_g \,\colon\, \Omega^p(X) \longrightarrow \Omega^{4-p}(X)$ denotes the [[Hodge star operator]] induced by the [[pseudo-Riemannian metric]] of the given [[Lorentzian manifold]] $(X,g)$.

Often this is considered for $X \simeq \mathbb{R}^{3,1}$ being [[Minkowski spacetime]], hence for $g = \eta$ the [[Minkowski metric]], in which case this describes pure "Maxwell theory", but the exact same formulas apply in the generality that $(X,g)$ is any [[Lorentzian manifold]], in which case they form the sector of the [[equations of motion]] of [[Einstein-Maxwell theory]] which involve the [[electromagnetic field]] and its coupling to [[background field|background]] [[gravity]]. (The remaining sector are the [[Einstein equations]] for the metric field $g$ sourced by the [[stress-energy tensor]] of the Maxwell field.)

In order to bring out more manifestly that [[gravity]] (the [[pseudo-Riemannian metric]]) enters *only* through the [[Hodge star operator]], we may evidently re-write the above pair of equations (eq:MaxwellEquationsInTraditionalDifferentialForm) equivalently as follows:

\[
  \label{MaxwellEquationsInPreMetricDifferentialForm}
  \begin{array}{rcl}
    \mathrm{d}\,  F &=& 0 \mathrlap{\,,}
    \\
    \mathrm{d} \, G &=& j
  \end{array}
  \;\;\;\;
  \text{and}
  \;\;\;\;
  G \,=\, \star_g F
  \,.
\]

### Electromagnetism in dielectric media

In fact, this formulation (eq:MaxwellEquationsInPreMetricDifferentialForm) is closer to the (original) formulation of Maxwell's equations used in the case that spacetime $X$ is thought of as possibly filled with a *[[dielectric medium]]* where, *a priori*, one considers two groups of fields:

1. the [[electric field]] $\vec E$ and [[magnetic field|magnetic]] [[flux density]] $\vec B$

   which jointly constitute the [[Faraday tensor]], given (on a local [[coordinate chart]] $(t,x^1, x^2, x^3)$) by (see [there](Maxwell's+equations#InTermsOfFaradayTensor)):

  $$
    F
      \,=\,
    E_i \mathrm{d}x^i \wedge \mathrm{d} t
      \,+\,
    \epsilon_{i j k} B^i \mathrm{d} x^j \mathrm{d} x^k
  $$

as well as

1. the actual [[magnetic field]] $\vec H$ and the *dielectric displacement* (or similar) $\vec D$, with

   $$
     G
       \,=\,
     H_i \mathrm{d}x^i \wedge \mathrm{d} t
       \,-\,
     \epsilon_{i j k} D^i \mathrm{d} x^j \mathrm{d} x^k
     \,,
   $$

and then imposes *[[constitutive equations]]*
relating $(\vec E, \vec B)$ to $(\vec H, \vec D)$ and thereby expressing the [[dielectric]]-property of any electromagneric *medium* that one imagines filling the spacetime $X$.

In general, [[constitutive equations]] can be nonlinear (and even [[multi-valued function|multi-valued]], reflecting hysteresis effects), but for sufficiently small [[field strengths]] they are of the form

$$
  G \,=\, \star_\epsilon F
$$

for a [[linear map]]

$$
  \star_\epsilon
  \,\colon\,
  \Omega^2(X)
  \longrightarrow
  \Omega^2(X)
$$

much like the [[Hodge star operator]] (whence here we use similar notation for both, which is non-standard).

## Duality-symmetric higher gauge theory

While not traditionally discussed under this terminology in the literature, we may recognize ([SS23](#SS23)) [[higher gauge field|higher]] versions of "premetric" electromagnetism occuring in the [[supergravity]]- and [[string theory]]-literature as *duality-symmetric* or "democratic" formulations.

For more on the general picture see at *[[Gauss law]] -- [In higher gauge theory](Gauss+law#InHigherGaugeTheory)*.

### RR-fields in gravitational backgrounds
 {#RRFieldInGravitationalBackground}

The [[theory (physics)|theory]] of [[type II supergravity|type II]] [[D=10 supergravity]] famously contains [[higher gauge fields]] called the *[[Ramond-Ramond fields]]* or *[[RR-fields]]* for short, which may be understood as a certain higher-degree generalization of the [[electromagnetic field]]:

Where the [[electromagnetism|electromagnetic]] [[field strength]] is a single [[differential 2-form]] $F_2 \,\in\, \Omega^2(X)$ (the [[Faraday tensor]]), the [[RR-fields]] have their [[field strengths]] encoded in a [[tuple]] of [[differential forms]] in every second degree up to *half* the [[spacetime]] [[dimension of a manifold|dimension]]:

$$
  \Big\{
    F_{2p + \sigma}
    \,\in\,
    \Omega^{2p+\sigma}(X)
    \;{\Big\vert}\;
    0 \leq 2p+\kappa \leq 5
  \Big\}
  \,,
$$

where $\sigma = 0$ for [[type IIA supergravity]] and $\sigma = 1$ for [[type IIB supergravity]].

Moreover, the [[equations of motion]] of these [[RR-fields]] are of the general form of [[Maxwell's equations]] (eq:MaxwellEquationsInTraditionalDifferentialForm), in fact when the [[Kalb-Ramond field]] and the [[spinor fields]] vanish, then the equations of motion are as before, up to form degree

\[
 \label{RRFieldEquationsInTraditionalDifferentialForm}
  \begin{array}{rcl}
    \mathrm{d} F_{2p + \sigma} &=& 0
    \\
    \mathrm{d} \star_g F_{2p + \sigma} &=& 0
    \mathrlap{\,.}
  \end{array}
\]

(In more generality the right-hand sides are non-vanishing combinations with the [[Kalb-Ramond field]] -- already for the first equation! -- and the [[spinor fields]].)

But one may famously understand [[type II supergravity]] as the low-energy limit of [[type II string theory]], in which case there are subtle [[flux quantization laws]] of the total [[flux]] of these [[RR-fields]] (in variation of the [[Dirac charge quantization]]-law imposed on the Maxwell field when regarding [[electromagnetism]] as the background field for [[quantum]] [[electrons]]).

A widely-considered [[conjecture]] ([[K-theory classification of D-brane charge|here]]) says that this [[flux quantization]] is realized by

1. regarding the would-be [[Hodge duality|Hodge duals]] $\star_g F_{2p + \sigma}$ of the RR-fields as independent fields $F_{10-2p-\sigma}$, so that the "pre-metric RR-field" now has about twice as many components:

   $$
     \Big\{
       F_{2p + \sigma}
       \,\in\,
       \Omega^{2p+\sigma}(X)
       \;{\Big\vert}\;
       0 \leq 2p+\kappa \leq 5
     \Big\}
     \,,
   $$

   and then regarding/constraining such a [[tuple]] as the image under the [[Chern character]] of a class in ([[differential K-theory|differentia]]) complex [[topological K-theory]] in addition to imposing the pre-metric [[equations of motion]].

1. afterwards imposing the Hodge-duality constraint, now thought of as a "self-duality" on K-theory and remaining subject to discussion in the literature: see at *[[self-dual higher gauge field]]* the section *[Examples -- RR-Fields](self-dual+higher+gauge+theory#RRFieldsin10d).

\[
  \label{RRFieldsEquationsInPreMetricDifferentialForm}
  \array{
    \mathrm{d} \, F_{2\bullet + \sigma}
    \;=\;
    0
    \\
    F_{D-2\bullet-\sigma}
    \;=\;
    \star_g
    F_{2\bullet+\sigma}
  }
\]

(Again, this is stated so far for vanishing [[Kalb-Ramond field]], for ease of exposition here. More generally the KR-field does not vanish and the above discussion applies for [[twisted K-theory]].)

Clearly, this just the idea of "pre-metric electromagnetism" but now enacted in a variant situation. The [[string theory]]-literature refers to this mostly as the "democratic" formulation of the [[RR-fields]], see [Dall'Agata, Lechner & Tonin (1998)](#Dall'AgataLechnerTonin98) or [Mkrtchyan & Valach (2023)](#MkrtchyanValach23).

### $C$-Field in gravitational background

Somewhat similarly, the [[equations of motion]] of the [[supergravity C-field]] in [[D=11 supergravity]] are of the form

$$
  \begin{array}{rcl}
    \mathrm{d} G_4 &=& 0
    \\
    \mathrm{d} \star_g G_4 &=& \tfrac{1}{2} G_4 \wedge G_4
  \end{array}
$$

where now $G_4 \,\in\, \Omega^4(X)$ is a [[differential 4-form]]. If we regard these equations in "pre-metric" form as in the above discussion of the [[RR-fields]], by introducing an *a priori* independent 7-form $G_7 \,\in\, \Omega^7(X)$, then they read

\[
  \label{PreMetricEOMsForCField}
  \begin{array}{rcl}
    \mathrm{d} G_4 &=& 0
    \\
    \mathrm{d} G_7 &=& -\tfrac{1}{2} G_4 \wedge G_4
  \end{array}
  \;\;\;\;\;\;
  \text{and}
  \;\;\;\;\;\;
  G_7 \,=\, \star_g G_4
  \,.
\]

As for the [[RR-field]] and its conjectured [[K-theory classification of D-brane charge|charge quantization in K-theory]], in this "pre-metric" formulation it makes sense to ask which [[generalized cohomology theory]] yields the differential equations on the left, as those characterizing the image under its generalized [[Chern-Dold character|Chern-Dold-like charatcer]].

There is no [[Whitehead-generalized cohomology theory]] with this property, due to the non-linear [[Quadratic Functions in Geometry, Topology, and M-Theory|quadratic function]] $G_4 \mapsto \tfrac{1}{2} G_4 \wedge G_4$ appearing in (eq:PreMetricEOMsForCField), but there is a [[non-abelian cohomology]] theory with the correct [[schreiber:The Character Map in Non-Abelian Cohomology|nonabelian character map]]: namely [[Cohomotopy|4-Cohomotopy theory]]. This observation (due to [Sati (2013,2018), §2.5](https://ncatlab.org/schreiber/show/Hypothesis+H#ReferencesInRational)) suggests (this is "[[schreiber:Hypothesis H]]") that the [[charge quantization]] of the [[supergravity C-field]] in [[M-theory]] may be in some form of [[Cohomotopy]] theory.

## References
 {#References}

### Premetric electromagnetic fields
 {#ReferencesPremetricElectromagnetism}

The idea (if not the terminology) of pre-metric electromagnetism is attributed to:

* {#Kottler22a} [[Friedrich Kottler]], *Newtonsches Gesetz und Metrik*, Sitzungsber. Akad. Wiss. Wien, Math.-Naturw. Klasse, Abt. IIa, **131** (1922) 1-14 [Engl. transl. by [[David H. Delphenich|Delphenich]]: [pdf](http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/kottler_-_newtons_law_and_metrics.pdf), [[Kottler-NewtonsLawAndMetric.pdf:file]]]

* {#Kottler22b} [[Friedrich Kottler]], *Maxwell'sche Gleichungen und Metrik*, Sitz. Akad. Wien IIa **131** (1922) 119-146 [Engl. transl. by [[David H. Delphenich|Delphenich]]: [pdf](http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/kottler_-_maxwells_equation.pdf), [[Kottler-MaxwellEquationsAndMetric.pdf:file]]]

* {#Cartan24} [[Élie Cartan]], §80 in: *Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite)*, Annales scientifiques de l’É.N.S. 3e série, tome **41** (1924) 1-25 [[numdam:ASENS_1924_3_41__1_0](http://www.numdam.org/item?id=ASENS_1924_3_41__1_0)]

* {#Dantzig34} [[David van Dantzig]], *The fundamental equations of electromagnetism, independent of metrical geometry*, Mathematical Proceedings of the Cambridge Philosophical Society **30** 4  (1934) 421-427 [[doi:10.1017/S0305004100012664](https://doi.org/10.1017/S0305004100012664)]

Early review:

* [[Edmund T. Whittaker]]: pp. 192-196 of: *A History of the Theories of Aether and Electricity* -- Vol. 2: *The Modern Theories 1900-1926* (1953), reprinted by Humanities Press (1973) [pdf, Wikipedia entry]

  > "[...] This chapter has been concerned, for the most part, with General Relativity, which is essentially a geometrisation of physics. It may be closed with some account of a movement in the opposite direction, seeking to abolish the priviledged position of geometry in physics, and indeed inquiring how far it may be possible to construct a physics independent of geometry. Since the notion of metric is a complicated one, which requires measurements with clocks and scales, generally with *rigid* bodies, which themselves are systems of great complexity, it seems undesirable to take metric as fundamental, particularly for phenomena which are simpler and actually independent of it."

  > "The movement was initiated by Friedrich Kottler of Vienna, who in 1922 published two papers [[Kottler (1922a)](#Kottler22a), [(1922b)](#Kottler22b)]"

Further amplification of the *[[constitutive equation|constitutive]] map* that relates the pre-metrically independent fields:

* [[E. Jan Post]], *The constitutive map and some of its ramifications*, Annals of Physics **71** 2 (1972) 497-518 [doi:10.1016/0003-4916(72)90129-7]

The terminology "pre-metric electromagnetism" seems to be due to [[David H. Delphenich]], who gives much further discussion:

* {#Delphenich05a} [[David H. Delphenich]], *On the Axioms of Topological Electromagnetism*, Annalen Phys. **14** (2005) 347-377 [[arXiv:hep-th/0311256](https://arxiv.org/abs/hep-th/0311256), [doi:10.1002/andp.20055170601](https://doi.org/10.1002/andp.20055170601)]

* {#Delphenich05b} [[David H. Delphenich]], *Symmetries and pre-metric electromagnetism*, Ann. Phys. **14** (2005) 663-704 [[arXiv:gr-qc/0508035](https://arxiv.org/abs/gr-qc/0508035), [doi:10.1002/andp.200510159](https://doi.org/10.1002/andp.200510159)]

* {#Delphenich07} [[David H. Delphenich]], *On linear electromagnetic constitutive laws that define almost-complex structures*, Annalen Phys. **16** (2007) 207-217 [[doi:gr-qc/0610031](https://arxiv.org/abs/gr-qc/0610031), [doi:10.1002/andp.200610227](https://doi.org/10.1002/andp.200610227)]

* {#Delphenich08} [[David H. Delphenich]], *Spinors and Pre-Metric Electromagnetism*, Advances in Applied Clifford Algebras **18** (2008) 567–578 [[doi:10.1007/s00006-008-0115-6](https://doi.org/10.1007/s00006-008-0115-6)]

* {#Delphenich15} [[David H. Delphenich]], *Pre-metric electromagnetism as a path to unification*, in: *Unified Field Mechanics*, World Scientific (2015) 215-220 [[arXiv:1512.05183](https://arxiv.org/abs/1512.05183), [doi:10.1142/9789814719063_0023](https://doi.org/10.1142/9789814719063_0023)]

Monographs:

* {#DelphenichBook} [[David H. Delphenich]], *Pre-Metric Electromagnetism* [part I:[pdf](http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/intro-chap_vi.pdf), part II:[pdf](http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/chapter_vii-xii.pdf), full:[[Delphenich-PreMetricElectromagnetism.pdf:file]]]

* {#HehlObukhov03} [[Friedrich W. Hehl]], [[Yuri N. Obukhov]], *Foundations of Classical Electrodynamics -- Charge, Flux, and Metric*, Progress in Mathematical Physics **33**, Springer (2003) [[arXiv:10.1007/978-1-4612-0051-2](https://doi.org/10.1007/978-1-4612-0051-2)]

The general principle of the pre-metric formulation, understood as a form of [[self-dual higher gauge theory]], also appears in:

* {#Freed02} [[Daniel Freed]], Exp. 3.8 of *[[Dirac charge quantization and generalized differential cohomology]]*, Surveys in Differential Geometry **7**, Int. Press (2000) 129-194 [[arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220), [doi:10.4310/SDG.2002.v7.n1.a6](https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6), [spire:537392](https://inspirehep.net/literature/537392)]

leading over to generalization of the idea such as to [[RR-fields]].

Further discussion:

* {#HehlItinObukhov16} [[Friedrich W. Hehl]], [[Yakov Itin]], [[Yuri N. Obukhov]], *On Kottler's path: origin and evolution of the premetric program in gravity and in electrodynamics*, International Journal of Modern Physics D **25** 11 (2016) 1640016  [[arXiv:1607.06159](https://arxiv.org/abs/1607.06159), [doi:10.1142/S0218271816400162](https://doi.org/10.1142/S0218271816400162)]

* {#BBSS15} [[Christian Becker]], [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], Rem. 2.3 in: _Abelian duality on globally hyperbolic spacetimes_, Commun. Math. Phys. **349** (2017) 361-392 [[arXiv:1511.00316](https://arxiv.org/abs/1511.00316), [doi:10.1007/s00220-016-2669-9](https://doi.org/10.1007/s00220-016-2669-9)]

* [[Yakov Itin]], *Premetric representation of mechanics, electromagnetism and gravity*, International Journal of Geometric Methods in Modern Physics **15** supp01 (2018) [[doi:10.1142/S0219887818400029](https://doi.org/10.1142/S0219887818400029)]

* [[Yuri N. Obukhov]], *Premetric approach in gravity and electrodynamics*, in: *The Fifteenth Marcel Grossmann Meeting* (2022) 654-659 [[doi:10.1142/9789811258251_0087](https://doi.org/10.1142/9789811258251_0087), [arXiv:1904.00180](https://arxiv.org/abs/1904.00180)]

* {#LazaroiuShahbazi23} [[Calin Lazaroiu]], [[Carlos S. Shahbazi]], around Def. 1.16 in: *The duality covariant geometry and DSZ quantization of abelian gauge theory*, Advances in Theoretical and Mathematical Physics **26** (2022) 2213–2312 [[arXiv:2101.07236](https://arxiv.org/abs/2101.07236), [doi:10.4310/ATMP.2022.v26.n7.a5](https://dx.doi.org/10.4310/ATMP.2022.v26.n7.a5)]

* {#LazaroiuShahbazi23} [[Calin Lazaroiu]], [[Carlos S. Shahbazi]], around (3) in: *The geometry and DSZ quantization of four-dimensional supergravity*, Letters in Mathematical Physics  **113** 4 (2023) [[arXiv:2101.07778](https://arxiv.org/abs/2101.07778), [doi:10.1007/s11005-022-01626-y](https://doi.org/10.1007/s11005-022-01626-y)]

  > (in the generality allowing "[[U-duality]]-twists" among several abelian gauge fields, motivated by application to [[D=4 supergravity]])

### Duality-symmetric RR-fields
 {#PremetricRRFieldReferences}

References which make the "democratic" ("[[pregeometric RR-fields|pre-geometric]]") formulation (eq:RRFieldsEquationsInPreMetricDifferentialForm) of the RR-fields in [[type II supergravity]] manifest:

[[!include self-duality for pregeometric RR-fields -- references]]

### Duality-symmetric C-field
 {#PremetricCFieldReferences}

[[!include self-duality for pregeometric C-field -- references]]

### General

* {#SS23} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Flux Quantization on Phase Space]]* [[arXiv:2312.12517](https://arxiv.org/abs/2312.12517)]

[[!redirects premetric electromagnetism]]

[[!redirects pre-geometric electromagnetism]]
[[!redirects pregeometric electromagnetism]]

[[!redirects pre-metric electrodynamism]]
[[!redirects premetric electrodynamism]]

[[!redirects pre-geometric electrodynamism]]
[[!redirects pregeometric electrodynamism]]

[[!redirects premetric electromagnetics]]
[[!redirects premetric electromagnetics]]

[[!redirects pre-geometric electromagnetics]]
[[!redirects pregeometric electromagnetics]]

[[!redirects pre-metric electrodynamics]]
[[!redirects premetric electrodynamics]]

[[!redirects pre-geometric electrodynamics]]
[[!redirects pregeometric electrodynamics]]

[[!redirects pre-metric RR-field]]
[[!redirects pre-metric RR-fields]]

[[!redirects premetric RR-field]]
[[!redirects premetric RR-fields]]

[[!redirects pre-geometric RR-field]]
[[!redirects pre-geometric RR-fields]]

[[!redirects pregeometric RR-field]]
[[!redirects pregeometric RR-fields]]

[[!redirects pre-metric C-field]]
[[!redirects pre-metric C-fields]]

[[!redirects premetric C-field]]
[[!redirects premetric C-fields]]

[[!redirects pre-geometric C-field]]
[[!redirects pre-geometric C-fields]]

[[!redirects pregeometric C-field]]
[[!redirects pregeometric C-fields]]

[[!redirects duality-symmetric gauge theory]]
[[!redirects duality-symmetric gauge theories]]

[[!redirects duality-symmetric higher gauge theory]]
[[!redirects duality-symmetric higher gauge theories]]

[[!redirects duality-symmetric electromagnetism]]

[[!redirects duality-symmetric RR-field]]
[[!redirects duality-symmetric RR-fields]]

[[!redirects duality-symmetric C-field]]
[[!redirects duality-symmetric C-fields]]

[[!redirects duality-symmetric Maxwell theory]]
[[!redirects duality-symmetric Maxwell theories]]

[[!redirects duality-symmetric abelian Yang-Mills theory]]
[[!redirects duality-symmetric abelian Yang-Mills theories]]


pre-net

[[!redirects Pre-net]]
## Idea

A pre-net is a [[Petri net]] equipped with an ordering on the input and output of each event. These freely generate strict [[symmetric monoidal categories]] whereas [[Petri nets]] freely generate [[commutative monoidal categories]]. Pre-nets are the same as the [[tensor schemes]] defined by [Joyal and Street](#schemes).

## Definition

A pre-net is a pair of functions
\begin{centre}
\begin{tikzcd}
E \ar[r, shift left=.5ex,"s"] \ar[r,shift right=.5ex,"t",swap] & P^*
\end{tikzcd}
\end{centre}
where $E$ is the set of _events_, $P$ is the set of _places_, and $P^*$ is the [[free monoid]] on the set $P$.

A morphism of of pre-nets $(f \colon E \to E', g \colon P \to P')$ is a pair of functions between the sets of events and places making the following diagrams commute

\begin{centre}
\begin{tikzcd}
E \ar[r, "s"] \ar[d,"f"] & P^* \ar[d,"g^*"] \\
E' \ar[r,"s"] & P'^*
\end{tikzcd}
\end{centre}

\begin{centre}
\begin{tikzcd}
E \ar[r, "t"] \ar[d,"f"] & P^* \ar[d,"g^*"] \\
E' \ar[r,"t"] & P'^*
\end{tikzcd}
\end{centre}

This defines a category $PreNet$ of pre-nets and pre-net homomorphisms.

## References

Pre-nets were introduced in

*  [[Roberto Bruni]], [[José Meseguer]], [[Ugo Montanari]], and [[Vladimiro Sassone]]. _Functorial models for Petri nets_ Information and Computation 170, no. 2 (2001): 207-236. [pdf](https://eprints.soton.ac.uk/264742/1/prenetsIandCOff.pdf)

Tensor schemes were introduced in

* {#schemes} [[André Joyal]], and [[Ross Street]]. _The geometry of tensor calculus I_ Advances in mathematics 88, no. 1 (1991): 55-112. [pdf](https://core.ac.uk/download/pdf/82659437.pdf)

[[!redirects Pre-net]]
[[!redirects pre-nets]]
[[!redirects tensor scheme]]
[[!redirects tensor schemes]]


pre-spectrum

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

#Contents#
* table of contents
{:toc}

## Idea

In [[stable homotopy theory]], the [[stable homotopy category]] [[Ho(Spectra)]] or the [[stable (infinity,1)-category|stable $(\infty,1)$-category]] [[Spectra]] of [[spectra]] is often conceived of as a [[localization]] (the [[homotopy category of a model category|homotopy category]] of a [[Bousfield localization of model categories]] or the [[reflective sub-(infinity,1)-categories|reflective sub-$(\infty,1)$-category]]) of  an ambient auxiliary [[category]] or [[(infinity,1)-category|$(\infty,1)$-category]] of "pre-spectra". The localization by *[[spectrification]]* then turns a pre-spectrum into a genuine [[spectrum]].

In the simplest and historically first model, pre-spectra are *[[sequential spectra]]* and the genuine [[spectra]] among these are the *[[Omega-spectra]]*.

More generally, [[highly structured spectra]] such as [[symmetric spectra]], [[orthogonal spectra]] or [[S-modules]] are all, *a priori*, pre-spectra.

## Related concepts

* [[spectrification]], [[stable homotopy type]]

* [[sequential spectrum]]

* in [[homotopy type theory]]: [[sequential spectrum type]]

## References

See the references at *[[spectrum]]*.

[[!redirects pre-spectra]]
[[!redirects prespectrum]]
[[!redirects prespectra]]


pre-symplectic current

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[local Lagrangian density]] $L \in \Omega^{p+1,0}(E)$ on a [[jet bundle]] $J^\infty(E)$ of some [[field bundle]] $E \to \Sigma$ induces a differential form

$$
  \theta \in \Omega^{p,1}(E)
$$

unique up to terms in the image of the  [[horizontal derivative]], given by

$$
  d L = \delta_{EL} L - d_H \theta
  \,,
$$

where $\delta_{EL}$ is the [[Euler-Lagrange operator]]. Its image under the [[vertical derivative]]

$$
  \omega \coloneqq d_V \theta \in \Omega^{p,2}(E)
$$

is called the _pre-symplectic current_. Accordingly $\theta$ itself is also called the _presymplectic potential current_.

This is because for any choice of $p$-dimensional submanifold $\Sigma_p \hookrightarrow \Sigma$, the [[transgression of variational differential forms]] of $\omega$ to $\Sigma_p$ is a [[presymplectic form]] on the on-shell [[space of sections]] on the [[infinitesimal neighbourhood]] of $\Sigma_p$. This is the _[[covariant phase space]]_.

## References

* _[Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively](https://www.physicsforums.com/insights/higher-prequantum-geometry-iv-covariant-phase-space-transgressively/)_

* {#GiotopoulosSati23} [[Grigorios Giotopoulos]], [[Hisham Sati]], §7 in: *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

[[!redirects pre-symplectic currents]]

[[!redirects presymplectic current]]
[[!redirects presymplectic currents]]

[[!redirects symplectic current]]
[[!redirects symplectic currents]]

[[!redirects presymplectic potential]]
[[!redirects presymplectic potentials]]

[[!redirects presymplectic potential current]]
[[!redirects presymplectic potential currents]]


prealgebra real number

[[!redirects prealgebra real numbers]]
[[!redirects pre-algebra real number]]
[[!redirects pre-algebra real numbers]]

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

#Contents#
* table of contents
{:toc}

## Idea ##

The real numbers as encountered in prealgebra.

## Definition ##

Let $R$ be an [[Archimedean integral domain]] and let $[0, 1]_R$ be the [[unit interval]] in $R$. Let $a:\mathbb{N} \to [0, 9]_\mathbb{N}$ be a sequence of decimal digits, where $n \in [0, 9]_\mathbb{N}$ is the set of all natural numbers $0 \leq n \leq 9$, and let $\mathcal{I}:[0, 9]_\mathbb{N} \to F$ be the canonical embedding of $[0, 9]_\mathbb{N}$ into $R$. The __prealgebra real numbers__ $\mathbb{R}$ is the [[initial object|initial]] [[Archimedean integral domain]] such that for every such sequence $a:\mathbb{N} \to [0, 9]_\mathbb{N}$, the [[sequence]]

$$b(n) \coloneqq \sum_{i=0}^{n} \frac{\mathcal{I}(a_i)}{10^{i+1}}$$

has a [[limit of a sequence|limit]] in the [[unit interval]] $[0, 1]_\mathbb{R}$.

## Properties ##

If the [[limited principle of omniscience]] is true, then every [[Cauchy real number]] is a prealgebra real number.

## See also ##

* [[real numbers]]
* [[limited principle of omniscience]]
* [[infinite decimal representation of a unit interval]]


preallegory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Definition

In [[homotopy type theory]] a **preallegory** is a [[dagger precategory]] $C$ such that

* for each object $a:Ob(C)$ and $b:Ob(C)$ there is a function
$$(-)\wedge_{a, b}(-):Mor_C(a, b) \times Mor_C(a, b) \to Mor_C(a, b)$$
such that for each morphism $f:Mor_C(a, b)$, $g:Mor_C(a, b)$, and $h:Mor_C(a, b)$ there are [[identifications]]
$$\mathrm{assoc}(a, b, f, g, h):f \wedge_{a, b} (g \wedge_{a, b} h) = (f \wedge_{a, b} g) \wedge_{a, b} h$$
$$\mathrm{comm}(a, b, f, g):f \wedge_{a, b} g = g \wedge_{a, b} f$$
$$\mathrm{idem}(a, b, f): f \wedge_{a, b} f = f$$

* for each object $a:Ob(C)$ and $b:Ob(C)$ and morphism $f:Mor_C(a, b)$ and $g:Mor_C(a, b)$ there is a function
$$\mathrm{monotone}(a, b, f, g): (f \wedge_{a, b} g = f) \to (f^{\dagger_{a, b}} \wedge_{b, a} g^{\dagger_{a, b}} = f^{\dagger_{a, b}})$$

* for each object $a:Ob(C)$, $b:Ob(C)$, and $c:Ob(C)$ and morphism $f:Mor_C(a, b)$, $g:Mor_C(b, c)$, and $h:Mor_C(a, c)$, there is an [[identification]]
$$\mathrm{modular}(a, b, c, f, g, h):\left((g \circ_{a, b, c} f) \wedge_{a, c} h\right) \wedge_{a, c} \left(g \circ_{a, b, c} (f \wedge_{a, b} (g^{\dagger_{b, c}} \circ_{a, c, b} h))\right) = (g \circ_{a, b, c} f) \wedge_{a, c} h$$

## Examples

* For every [[type universe]] $\mathcal{U}$ the precategory $\mathrm{Rel}_\mathcal{U}$ of $\mathcal{U}$-small sets and locally $\mathcal{U}$-small propositional-valued relations is a preallegory.

## See also

* [[precategory]]

* [[dagger precategory]]

* [[allegory]]

[[!redirects preallegory]]


precategory

> This entry is about a category whose objects form a [[type]] rather than a [[set]]. For the specific type of paracategory also known as a *precategory*, in which all the composites are generated from binary and nullary ones, see [[paracategory]].

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

In [[homotopy type theory]] (or more generally [[intensional type theory]]), when defining a [[category]] (meaning a *1-category*), one has to decide whether to impose an [[h-level]] restriction on the [[type]] of [[objects]].  A **precategory** is a category in which *no* such restriction is imposed; thus the type of objects may be a set (i.e. a 0-type), or a 1-type, or an even higher type.

Surprisingly much [[category theory]] can be done purely with precategories, and some authors call them merely "categories".  However, there are notable differences between precategories and the familiar notion of "category" from [[set theory|set-theoretic]] foundations.  Moreover, the two do not coincide under the standard interpretation of homotopy type theory into set theory (e.g. the model in [[simplicial sets]]).  Indeed, the semantics of a precategory is the 1-categorical analogue of a [[Segal space]].

Alternative notions of "1-category" in homotopy type theory include [[strict categories]], whose type of objects is a set, and [[univalent categories]], whose type of objects is a 1-type whose identifications coincide with isomorphisms for the category structure.  The latter are generally better-behaved and include naturally-occurring examples, but the former are sometimes technically useful.

There are also "even more pre-" notions of category in intensional type theory, such as [[E-categories]].

## Definition

### With a family of morphisms

A **precategory** $\mathcal{C}$ consists of

* a [[type]] of [[objects]] $Ob(\mathcal{C})$,
* for each object $A:Ob(\mathcal{C})$ and $B:Ob(\mathcal{C})$, a [[set]] $Hom(A, B)$ of [[morphisms]]
* for each object $A:Ob(\mathcal{C})$, $B:Ob(\mathcal{C})$, and $C:Ob(\mathcal{C})$, a [[binary function]]
$$(-)\circ_{A, B, C}(-) :Hom(B, C) \times Hom(A, B) \to Hom(A, C)$$
* for each object $A:Ob(\mathcal{C})$, a morphism $\mathrm{id}_A:Hom(A, A)$
* such that
  * the composition of morphisms is [[associative]]: for each object $A:Ob(\mathcal{C})$, $B:Ob(\mathcal{C})$, $C:Ob(\mathcal{C})$, and $D:Ob(\mathcal{C})$, and for each morphism $f:Hom(A, B)$, $g:Hom(B, C)$, and $h:Hom(C, D)$,
$$h \circ_{A, C, D} (g \circ_{A, B, C} f) = h \circ_{B, C, D} (g \circ_{A, B, D} f)$$
  * the composition of morphisms satisfies the left and right [[unit laws]]: for each object $A:Ob(\mathcal{C})$ and $B :Ob(\mathcal{C})$ and morphism $f:Hom(A, B)$, $\mathrm{id}_B \circ_{A, B, B} f = f$ and $f \circ_{A, A, B} \mathrm{id}_A = f$.

Of course, the latter two axioms are actually inhabitants of the [[identity type]], hence are data included in the definition just like the first four.  However, since each $Hom(A,B)$ is a set, such equalities are (typally) unique whenever they exist, so in most cases their presence as data can be ignored.

### With one type of morphism

A **precategory** $\mathcal{C}$ consists of

* a [[type]] $Ob(C)$ of __[[objects]]__;

* a type $Mor(C)$ of __[[morphisms]]__ (or __arrows__);

* functions $s:Mor(C) \to Ob(C)$, called the __[[source]]__ or __domain__, and $t:Mor(C) \to Ob(C)$, called its __[[target]]__ or __codomain__, such that $s$ and $t$ are jointly 0-truncated: the type of functions $u:Mor(C) \to (Ob(C) \times Ob(C))$ such that the following conditions are met:

  * the fibers of $u$ at each pair of objects $a:Ob(C) \times Ob(C)$ is a set

  * for the universal product projection functions $\pi_1:(Ob(C) \times Ob(C)) \to Ob(C)$ and $\pi_2:(Ob(C) \times Ob(C)) \to Ob(C)$ out of the product type and for every morphism $f:Mor(C)$, there are identifications $\mathrm{leftuniversal}(f):\pi_1(u(f)) = s(f)$ and $\mathrm{rightuniversal}(f):\pi_2(u(f)) = t(f)$

is contractible:

$$\mathrm{isContr}\left(\sum_{u:Mor(C) \to (Ob(C) \times Ob(C))} \prod_{f:Mor(C)} \mathrm{isSet}\left(\sum_{a:Ob(C) \times Ob(C)} u(f) = a\right) \times (\pi_1(u(f)) = s(f)) \times \pi_2(u(f)) = t(f)))\right)$$

This recalls the definition of a [[preorder]], where the source and target functions are required to be [[jointly monic]], i.e. jointly (-1)-truncated;

* a function
$$(-) \circ (-): \left(\sum_{f:Mor(C)} \sum_{g:Mor(C)} t(f) = s(g)\right) \to Mor(C)$$
called __composition__;

* a function $id:Ob(C) \to Mor(C)$, called the __identity__;

*  such that the following properties are satisfied:

   *  source and target are respected by composition: $s(g \circ f) = s(f)$ and $t(g\circ f) = t(g)$;

   *  source and target are respected by identities: $s(1_x) = x$ and $t(1_x) = x$;

   *  composition is [[associative]]:
      $(h \circ g)\circ f = h\circ (g \circ f)$ whenever $t(f) = s(g)$ and $t(g) = s(h)$;

   *  composition satisfies the [[unit law|left and right unit laws]]:
      if $s(f) = x$ and $t(f) = y$, then $1_y \circ f = f = f \circ 1_x$.

## Remarks

* The interpretation of the notion of precategory in the model of homotopy type theory in [[simplicial sets]] yields a [[Segal space]] in which the morphism $A_1 \to A_0 \times A_0$ has homotopically discrete fibers.

* The definition of precategory is evidently a type-theoretic notion of the definition of category with [[category#AFamilyOfCollectionsOfMorphisms|a family of collections of morphisms]].  An equivalent definition analogous to the definition of category with [[category#OneCollectionOfMorphisms|one collection of morphisms]] could be given, but it would be much less convenient to use.  In particular, the "type of all morphisms" $\sum_{A,B:Ob(\mathcal{C})} Hom(A,B)$ in a precategory is *not* itself a set, since it can incorporate nontrivial identifications from $Ob(\mathcal{C})$; the "hom-types are sets" condition in a one-type-of-morphisms definition of precategory would have to be stated as "the function $\mathcal{C}_1 \to \mathcal{C}_0$ is 0-truncated" (i.e. its fibers are sets).

## Properties

Every **precategory** is a locally univalent [[E-category]].

In [[univalent type theory]] there is a [[structure identity principle]] for precategories:

Equality of precategories in a univalent universe is the same as having a [[fully faithful]] [[equivalent-on-objects functor]] between precategories in the univalent universe.

This is because fully faithful functors establish an equivalence of hom-sets between the two precategories, and equivalent-on-objects functors establish an equivalence of object types between the two precategories, which means that fully faithful equivalent-on-objects functors is the right notion of equivalence between precategories.

Only in univalent categories are such functors the same as [[adjoint equivalence]].

## See also

* [[category]]

* [[strict category]]

* [[univalent category]]

* [[wild category]]

* [[Segal space]]

* [[complete Segal space]]

* [[pregroupoid]]

* [[preallegory]]

* [[E-category]]

## References

* {#UFP13} Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

* {#AhrensKapulkinShulman13} [[Benedikt Ahrens]], [[Chris Kapulkin]], [[Michael Shulman]], _Univalent categories and the Rezk completion_, Mathematical Structures in Computer Science **25** 5 (*From type theory and homotopy theory to Univalent Foundations of Mathematics*),  (2015) 1010-1039   $[$[arXiv:1303.0584](https://arxiv.org/abs/1303.0584), [doi:10.1007/978-3-319-21284-5_14](https://doi.org/10.1007/978-3-319-21284-5_14)$]$

[[!redirects precategories]]
[[!redirects prefunctor]]
[[!redirects prefunctors]]


precision experiment

+-- {: .rightHandSide}
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### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[particle physics]] [[experiments]], there tends to be a dichotomy between _direct detection at high energy_ and _indirect detection via high precision_.

Traditionally, direct high energy experiments have been dominating the field since the discovery of [[heavy quark|heavy]] [[quarks]] in large accelerator experiments, culminating in the discovery of the [[Higgs boson]] at the [[LHC experiment]]. Here the particle being detected is an actual decay product of the [[scattering]] process.

But heavy [[fundamental particles]], whose mass may be beyond that of direct reaction products obtained in a given accelerator experiment, may still manifest themselves indirectly, as [[virtual particles]] contributing to [[loop order|loop corrections]] of the [[scattering amplitudes]] of all lighter particles to which they couple. Therefore, sufficiently _precise_ measurement of light particle decays at low energy may still reveal the presence and properties of virtual particles at (much) higher energy.

{#IntensityFrontier} Discerning the small statistical effects expected in precision experiments requires access to a large amount of experimental data, hence to a high *intensity* of particle scattering processes, whence the community refers to precision experiments as probing "the intensity frontier" of particle physics (see the references [below](#IntensityFrontierReferences)). This is in contrast to direct detection experiment which can do with less data as long as it concerns high energy scattering processes, whence direct detection experiments are said to be probing "the energy frontier" of particle physics.

An example for a potential effect seen in precision experiments are the [[flavour anomalies]] in [[B-meson]] decays at various experiments, including the [[LHCb experiment]], [[Belle experiment]] and [[BaBar experiment]].

Since, at the same time, the [[LHC experiment]] has not made any further _direct_ detection, beyond the [[Higgs boson]], it has been argued that indirect precision experiments will or should gain in importance in the future of [[particle physics]].

## References

General:

* {#Brambilla14} Brambilla et al., Section 5 of: _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981  ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723), [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

With focus on [[Higgs field]]-interactions:

* Gudrun Heinrich, *Collider Physics at the Precision Frontier*, Physics Reports, Volume 922, 2021, Pages 1-69 ([arXiv:2009.00516](https://arxiv.org/abs/2009.00516))

* Sven Heinemeyer, Stanislaw Jadach, Jürgen Reuter, *Theory requirements for SM Higgs and EW precision physics at the FCC-ee* ([arXiv:2106.11802](https://arxiv.org/abs/2106.11802))

In view of potential [[leptoquarks]]:

* I. Doršner, S. Fajfer, A. Greljo, J. F. Kamenik, N. Košnik, _Physics of leptoquarks in precision experiments and at particle colliders_, Physics Reports Volume 641, 17 June 2016, Pages 1-68 ([arXiv:1603.04993](https://arxiv.org/abs/1603.04993))

In view of [[flavour anomalies]]:

* {#AHRS20} Rafael Aoude, Tobias Hurth, Sophie Renner, William Shepherd, _The impact of flavour data on global fits of the MFV SMEFT_ ([arXiv:2003.05432](https://arxiv.org/abs/2003.05432))

{#IntensityFrontierReferences} On the "intensity frontier"-terminology:

* Elizabeth Clements, *New tools forge new frontiers*, [Symmetry Magazine](https://www.symmetrymagazine.org/), Aug. 2008 ([pdf](https://www.symmetrymagazine.org/sites/default/files/legacy/pdfs/200808/new_frontiers.pdf))

* J. L. Hewett et al. *Planning the Future of U.S. Particle Physics (Snowmass 2013): Chapter 2: Intensity Frontier* ([arXiv:1401.6077](https://arxiv.org/abs/1401.6077))

and in regards to the [[Belle II]] [[experiment]]:

* *Snowmass White Paper: Belle II physics reach and plans for the next decade and beyond* [[arXiv:2207.06307](https://arxiv.org/abs/2207.06307)]

  > Belle II is an experiment operating at the intensity frontier.

With emphasis on precision [[QCD]]-predictions:

* Thomas Gehrmann, Bogdan Malaescu, *Precision QCD Physics at the LHC* ([arXiv:2111.02319](https://arxiv.org/abs/2111.02319))

* Francesco Giuli, *High-precision QCD physics at FCC-ee* [[arXiv:2208.09621](https://arxiv.org/abs/2208.09621)]

[[!redirects precision experiments]]

[[!redirects precision measurement]]
[[!redirects precision measurements]]

[[!redirects intensity frontier]]


precompact space

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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

# Precompact spaces
* table of contents
{: toc}

## Idea

A _precompact space_ is one whose [[complete space|completion]] is compact.  This makes sense for [[Cauchy spaces]], but not for [[topological spaces]]. In the context of [[uniform spaces]], precompact spaces are [[totally bounded spaces]].

Precompact subsets (using the [[induced uniformity]]) are one of the useful types of subset of a [[locally convex topological vector space]], particularly with regard to defining topologies on [[dual vector space|dual spaces]].

## Definitions

Let $X$ be (in the most general situation) a [[Cauchy space]]; this includes [[uniform convergence spaces]], [[uniform spaces]], [[topological groups]], [[topological vector spaces]], and [[metric spaces]] (among other things) as special cases.

The following definitions are all equivalent even in weak [[constructive mathematics|constructive]] [[foundations]]:

+-- {: .num_defn #completion}
###### Definition
The space $X$ is **precompact** if its [[complete space|completion]] is [[compact space|compact]].
=--

+-- {: .num_defn #filters}
###### Definition
The space $X$ is __precompact__ if every [[proper filter]] on $X$ has a (proper) [[Cauchy filter|Cauchy]] refinement.
=--

+-- {: .num_defn #nets}
###### Definition
The space $X$ is __precompact__ if every [[net]] in $X$ has a [[Cauchy net|Cauchy]] [[subnet]] (under any definition of 'subnet').
=--

We also have this subsidiary notion:

+-- {: .num_defn #subspace}
###### Definition
A [[subset]] of $X$ is **precompact** if it is precompact (or ...) for the inherited structure.
=--

### Sequential precompactness

\begin{definition}
The space $X$ is __sequentially precompact__ if every [[sequence]] in $X$ has a [[Cauchy sequence|Cauchy]] [[subsequence]].
\end{definition}

## Properties

\begin{theorem}
Assuming [[weak countable choice]], every sequentially precompact [[Lawvere metric space]] is a precompact Lawvere metric space.
\end{theorem}

### Relation to totally bounded spaces

\begin{theorem}
Every precompact space is a [[totally bounded space]].
\end{theorem}

\begin{theorem}
That every [[totally bounded space|totally bounded]] [[uniform space]] is precompact is equivalent to the [[ultrafilter principle]], because the generalised [[Cantor space]] $2^K$ is complete and totally bounded for any [[cardinal number]] $K$.
\end{theorem}

\begin{theorem}
Assuming [[excluded middle]] and [[dependent choice]], every totally bounded [[metric space]] is precompact.
\end{theorem}

(I don\'t know to what extent DC and EM are actually relevant to any of this; my reference _[[HAF]]_ tacitly assumes them most of the time.)

## See also ##

* [[compact space]]

* [[totally bounded space]]

[[!redirects precompact space]]
[[!redirects precompact spaces]]

[[!redirects precompact set]]
[[!redirects precompact sets]]
[[!redirects precompact subset]]
[[!redirects precompact subsets]]
[[!redirects precompact subspace]]
[[!redirects precompact subspaces]]


preconditioner

#Contents#
* table of contents
{:toc}

## Idea

A __preconditioner__ (matrix) is a [[matrix]] which multiplies (from left or from right, thus left and right preconditioners) the matrix of [[coefficients]] of a system which discretizes a [[differential equation]] in order to get faster [[convergence]] (=smaller condition number) of iterative numerical solution techniques. As a technique, preconditioning is used also for optimization [[algorithms]].

## Literature

* wikipedia [preconditioner](http://en.wikipedia.org/wiki/Preconditioner)
* Owe Axelsson, _Iterative Solution Methods_, Cambridge University Press 1996
* J. H. Bramble, J.E. Pasciak, _A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems_,  Math. Comp. __50__ (1988) 1–17
* Tyrone Rees, Martin Stoll, Andy Wathen, _All-at-once preconditioning in PDE-constrained optimization_, Kybernetika __46__:2 (2010) 341 – 360 [pdf](http://www.kybernetika.cz/content/2010/2/341/paper.pdf)
* John W. Pearson, Martin Stoll, Andy Wathen, _Regularization-robust preconditioners for time-dependent PDE constrained optimization problems_, SIAM J. Matrix Anal. and Appl., 33(4), 1126–1152, 2012 [doi](http://dx.doi.org/10.1137/110847949)
* Michele Benzi, _Preconditioning techniques for large linear
systems: a survey_, J. of Computational Physics __182__,
418–477 (2002) [doi](http://dx.doi.org/10.1006/jcph.2002.7176)
[pdf](http://www.mathcs.emory.edu/~benzi/Web_papers/survey.pdf)

category: applications
[[!redirects preconditioning]]
[[!redirects preconditioners]]


preconvergence space

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### Context
#### Analysis
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[[!include analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

By a *preconvergence space* we shall mean the most general notion of [[topological space]] for which the notion of [[convergence]] and of [[limit of a net|limit]] for [[filters]] and [[nets]] makes sense. Notice that the possibly more natural term "[[convergence space]]" for this notion is already taken to mean a set with [[filters]] that are isotone, centered, and directed, and that definition is not the most general definition for which the notions of convergence and limit makes sense for nets.

## Definition ##

Given a set $S$, let $\mathcal{F}(S)$ denote the set of [[filters]] on $S$. A set $S$ is a __preconvergence space__ if it comes with a [[binary relation]] $x \to c$ between the $\mathcal{F}(S)$ and $S$ itself, for elements $x \in \mathcal{F}(S)$ and $c \in S$.

The definition can also be phrased in terms of [[nets]]; a net $\nu$ converges to $x$ if and only if its [[eventuality filter]] converges to $x$.

The morphisms of preconvergence spaces are the [[pointwise continuous functions]]; a function $f$ between preconvergence spaces is __pointwise continuous__ if $F \to x$ implies that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the [[filterbase]] $\{F(A) \;|\; A \in F\}$.

## See also ##

* [[limit of a filter]]
* [[Hausdorff space]]
* [[pointwise continuous function]]
* [[sequential preconvergence space]]
* [[convergence space]]

[[!redirects preconvergence space]]
[[!redirects preconvergence spaces]]


precursors

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Precursors_Of_Category_Theory).

[[!redirects precursors of category theory]]
[[!redirects Precursors of Category Theory]]


precursors > history

< [[precursors]]