P. Ghez

## Selected Writings

On [[C-star category|$C^\ast$-categories]] and introducing the special case of [[W-star category|$W^\ast$-categories]]:

* {#GhezLinaRoberts85} [[P. Ghez]], [[Ricardo Lima]], [[John E. Roberts]], *$W^\ast$-categories*,  Pacific J. Math. **120** 1 (1985) 79-109 [[euclid:pjm/1102703884](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-120/issue-1/Wast-categories/pjm/1102703884.full)]

category: people


P. K. Mitter

* [InSpire page](https://inspirehep.net/authors/2394407)

## Selected writings

On [[non-perturbative quantum field theory]]:

* [[Gerard ’t Hooft]], [[Arthur Jaffe]], [[Gerhard Mack]], [[P. K. Mitter]], [[Raymond Stora]] (eds.) *Nonperturbative Quantum Field Theory*,  NATO Science Series B **185** (1988) [[doi:10.1007/978-1-4613-0729-7](https://doi.org/10.1007/978-1-4613-0729-7)]

category: people


p1-structure

## Idea

[[c-structure]] for $c$ the 1st [[Pontryagin class]] $p_1$, o.e. trivialization of $p_1$.

In the presence of [[Spin structure]] then $p_1$-structure is also called _[[string structure]]_. In the presence of (just) [[orientation]] (i.e. [[w1-structure]]) on [[3-manifolds]] then $p_1$-structure is closely related to [[Atiyah 2-framing]] structure.

[[!redirects p1-structures]]


Pablo Dartnell

* [mathematics genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=39123)

## related $n$Lab entry

* [[jet group]]

category: people


Paige Randall North

* [website](https://paigenorth.github.io/)

## Selected writings

* [[Benedikt Ahrens]], [[Paige Randall North]], [[Niels van der Weide]], *Semantics for two-dimensional type theory*, LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, August 2022, No. 12, Pages 1–14, ([doi:10.1145/3531130.3533334](https://doi.org/10.1145/3531130.3533334))

* [[Benedikt Ahrens]], [[Paige Randall North]], [[Michael Shulman]], [[Dimitris Tsementzis]], *The Univalence Principle* ([arXiv:2102.06275](https://arxiv.org/abs/2102.06275))

On [[univalence]] in [[homotopy type theory]]:

* [[Benedikt Ahrens]], [[Paige Randall North]], *Univalent foundations and the equivalence principle*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[arXiv:2202.01892](https://arxiv.org/abs/2202.01892), [doi:10.1007/978-3-030-15655-8](https://doi.org/10.1007/978-3-030-15655-8)]

On [[2-dimensional type theory]]:

* [[Benedikt Ahrens]], [[Paige Randall North]], [[Niels van der Weide]], *Semantics for two-dimensional type theory*, LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, 12 (2022) 1–14, [[doi:10.1145/3531130.3533334](https://doi.org/10.1145/3531130.3533334)]

category: people

[[!redirects Paige North]]


Paine on a Category Theory Demonstrations program

This entry is about the inner workings of a web-based program that generates componentwise illustrations of simple [[limit]]s and [[colimit]]s in [[Set]] as developed at

* J. Paine, [Category Theory Demonstrations](http://www.j-paine.org/cgi-bin/webcats/webcats.php)

***

In this section, I'll explain coproducts by showing you a program that implements the categorical coproduct of sets. We programmers live our working lives surrounded by data structures and subroutines, entities that become as concrete to us — as "thing-like", as "manipulable" — as teacups and bricks. So if I can implement some categorical calculations as programs, I hope it will give them a sense of reality that the maths alone might lack. This way of explaining category theory isn't new, actually: it's the idea behind the following book, which implements categorical constructions in the programming language ML:

* [_Computational Category Theory_ (dead link)]( http://www.cs.manchester.ac.uk/~david/categories/book/book.pdf), by [David Rydeheard](http://www.cs.manchester.ac.uk/~david/) and [Rod Burstall](http://www.dcs.ed.ac.uk/home/rb/).

## Preliminaries: functional programming in Python 3.0 ##

I'll implement my categorical coproduct in the programming language [Python 3.0](http://www.python.org/download/releases/3.0/). This is because: its syntax makes it easy to read; many programmers already know it; and it's free, so you can download it from the above link and try the examples yourself.

But most importantly, Python supports [functional programming] (http://en.wikipedia.org/wiki/Functional_programming). That is, functions are [first-class citizens](http://en.wikipedia.org/wiki/First-class_object), meaning that one can assign them to variables and store them in datatypes, pass them into other functions as arguments and back from other functions as results, and construct them at runtime.

In the next few paragraphs, I'll introduce Python 3.0 and then functional programming in Python. I'll do so by using it as a desk calculator, where I type commands — usually, expressions to be evaluated — and get back a result. If you want to try this, you can type them either into the Python command-line window, or into Python's IDLE graphical user-interface window.

As a start, this shows Python evaluating simple arithmetic expressions. What I type is after the three greater-than signs; Python's reply is on a line by itself:

    >>> 1
    1
    >>> 1+3
    4
    >>> 1+2*5
    11

I can store a result in a variable, and then use the variable as all or part of an expression. I'm showing you this because although the example below only stores numbers in variables, I shall later want to store functions. Please note that eleven = 1+2*5 means "put the result of evaluating 1+2*5 into the variable eleven". Like many languages, Python uses the equals sign for assignment, which may be confusing if you're not a programmer:

    >>> eleven = 1+2*5
    >>> eleven
    11
    >>> eleven * 2
    22

I can call a function. The from math import * below merely gets me the built-in arithmetic functions, including exp, and isn't really very interesting:

    >>> from math import *
    >>> exp(1)
    2.7182818284590451

This is where it begins getting interesting. Below, I put the _function_ exp into the variable myfunc. I can then apply myfunc in the same way I did exp:

    >>> myfunc = exp
    >>> myfunc(1)
    2.7182818284590451

Instead of using exp as a function value, I can write a "lambda expression":

    >>> identity = lambda x: x
    >>> identity(5)
    5
    >>> double = lambda x: 2*x
    >>> double(5)
    10

A lambda expression denotes a function without naming it or hiving it off into a separate definition. Then, in the same way that 1 denotes an integer and can be used as a constant in a language of arithmetic expressions, a lambda expression can be used in a language of expressions over functions. You can see this in the interaction below, where I have applied a lambda expression to 5. The syntax is the same as for double(5), except that in place of double, we have the lambda expression:

    >>> (lambda x: 2*x)(5)
    10

By the way, lambda expressions arose as part of the lambda calculus. \[ Possibly put some of this on a new lambda calculus page \] This is a mathematical model of computation,  invented by the logician [Alonzo Church](http://en.wikipedia.org/wiki/Alonzo_Church) as a foundation for logic. (One topic he wanted to investigate was Gödel's incompleteness theorem, hoping it might not apply to logic founded on the lambda calculus.) It consists of a language of expressions over values which include functions, and a set of transformation rules for evaluating these expressions. His notation for functions used Greek lambda to signal a function's argument: in it, the identity function is written $\lambda x.x$. Python's lambda x: x, and lambda expressions in other programming languages, are keyboard approximations to this notation. I should note that the phrase "lambda expression" seems to be ambiguous: some people use it only for functions written using lambda notation, while others use it to mean any expression in the lambda calculus. I suspect, though I haven't yet looked at Church's original paper, that he used it in the first sense. You'll find some online references about the lambda calculus at the n-Category Café thread [_Classical vs Quantum Computation (Week 1)_](http://golem.ph.utexas.edu/category/2006/10/classical_vs_quantum_computati.html), and a history of it at:

* [_History of Lambda-calculus and
Combinatory Logic_](http://www-maths.swan.ac.uk/staff/jrh/papers/JRHHislamWeb.pdf), by Felice Cardone and J. Roger Hindley.

... I now have all the prerequisites needed for some functional programming....

    >>> compose = lambda f, g: lambda x: g(f(x))

    >>> double_then_exp = compose( double, exp )
    >>> double_then_exp(1)
    7.3890560989306504
    >>> exp(2)
    7.3890560989306504

## Categorical definition of coproduct ##

I'll start with the definition of coproduct, which I've lightly edited from Section 3, _Coproducts and Colimits_ in Chapter III. _Universals and Limits_, of:

* [_Categories for the Working Mathematician_], by Saunders Mac Lane.

> For any [[category]] $C$, a _coproduct diagram_ consists of [[object|objects]] $a$, $b$ and $c$ of $C$, and a pair of [[morphism|arrows]] $i:a\to c$ and $j:b\to c$. This pair has the familiar [[universal property]]: For any pair of arrows $f:a\to d$ and $g:b\to d$, there is a unique arrow $h:c\to d$, such that $f = i;h$ and $g = j;h$.

Before going on to an example, I must just explain that I am using semicolon to denote composition, with the meaning that $(i;h)(x)$ = $h(i(x))$. That is, we apply $i$ first, then $h$. As the page on [[composition]] explains, this is an alternative to the standard mathematical notation $h \circ i$: computer scientists often prefer the semicolon notation, because it reflects the order in which the functions get applied.

To create an example of a coproduct diagram, I'll start by defining the sets $a$, $b$ and $c$:

    a = { 'dimanche', 'lundi', 'mardi' };
    b = { 'Mittwoch', 'Donnerstag', 'Freitag', 'Samstag' };
    c = { 'Sunday', 'Monday', 'Tuesday', 'Wednesday', 'Thursday', 'Friday', 'Saturday' };

Python 3.0 has adopted mathematical notation for sets, enclosing them in curly brackets. Quotes enclose string constants, which I'm using to create arbitrary set elements.

Now I need the pair of arrows $i$ and $j$. I'll define these as functions:

    # Maps a to c.
    def i( x ):
      if x == 'dimanche':
        return 'Sunday';
      elif x == 'lundi':
        return 'Monday';
      elif x == 'mardi':
        return 'Tuesday';

    # Maps b to c.
    def j( x ):
      if x == 'Mittwoch':
        return 'Wednesday';
      elif x == 'Donnerstag':
        return 'Thursday';
      elif x == 'Freitag':
        return 'Friday';
      elif x == 'Samstag':
        return 'Saturday';

...\[Explain somewhere why I have made a and b not subsets of c. In examples where they are, perhaps readers don't see the purpose of the injections, so I want to make that explicit. Also, I shall later use this \[but maybe not appropriate topic for nLab? Though relevant to HDA&N\] to talk about "glueing data" (database merging, sensor fusion.)\]

One important point is that, to keep my program simple, I'm cheating a little bit. In category theory, essential parts of every arrow are its [[source]] and its [[target]]. Thus, the arrow $i$ that maps $a$ to $c$ is _different_ from an arrow that maps $a$ into — say — the set $\{'Sunday','Monday','Tuesday'\}$, or the set $\{'Sunday','Monday','Tuesday','Friday'\}$. This is so even if their mappings are the same. In computing terminology, we would say that essential parts of every arrow are its argument type and its result type.

If I represent $i$ and $j$ by functions as above, I can't represent that in my code. But I shall do so by writing the source and target in a comment above every function. \[Perhaps I'll show a version later that does represent arrows. Use a dict with source, target, and body fields.\]

Now I'll show the definition of coproduct again:

> For any category $C$, a _coproduct diagram_ consists of objects $a$, $b$ and $c$ of $C$, and a pair of arrows $i:a\to c$ and $j:b\to c$. This pair has the familiar universal property: For any pair of arrows $f:a\to d$ and $g:b\to d$, there is a unique arrow $h:c\to d$, such that $f = i;h$ and $g = j;h$.

I have created the objects $a$, $b$ and $c$ and the arrows $i$ and $j$: \[insert diagram a<--i--c--j-->b. It would be really nice to (a) be able to click on the objects and arrows and see their content; (b) drag them around and manipulate them.\]. Now look at the sentence following "universal property". I'm going to change the wording slightly:

> You give me any pair of arrows $f:a\to d$ and $g:b\to d$, and I will give you a unique arrow $h:c\to d$, such that $f = i;h$ and $g = j;h$.

But this sounds like a function talking! "You give me these arrows $f$ and $g$," it commands me, "and I will give you a new arrow $h$." \[ link in: CftWM 63, after (2), and the phrase "uniquely determined" \] So I am now going to implement a function that takes arrows $f$ and $g$ and returns our $h$. In fact, I want you to see the coproduct diagram \[a<--i--c--j-->b\] as such a function, so I am first going to implement another function that turns it into one.

...\[To be continued\]


Painleve transcendent

## Overview of Painlevé transcendents

In the study of ordinary differential equations one of the important things is the behaviour of [[monodromies]] and closely related singularities of solutions. In the linear case, the poles possibly appear just at the poles of coefficients of the solutions. In the nonlinear case the solutions can appear elsewhere and generally propagate with change of initial conditions. Very important is if the singularities do not move or monodromies don't change with change of parameters.
An [[ordinary differential equation]] (ODE) satisfies the __Painlevé property__ if all solutions are single valued around every movable singularity (this is more effective way to express that they can be made single-valued functions by uniformization). A class of such "good" nonlinear equations has been defined by __Paul Painlevé__ ([wikipedia](http://en.wikipedia.org/wiki/Paul_Painlev%C3%A9)), who discovered at the end of 19th century a truly remarkable fact that all _second order_ ODEs of the form
$$
u'' = F (z; u, u'),
$$
where $F$ is rational in $u,u'$ and analytic in $z$ and which satisfy the Painlevé property have solutions which can be expressed in terms of well known functions like elementary and hypergeometric functions and only 6 new kinds of *transcendental functions* called _Painlevé I-VI_. Furthermore he obtained a complete classification of such equations (automatically, $F$ is at most quadratic in $u,u'$) in 50 classes (44+6) up to a number of standard transformations. Painlevé transcendents are now of central importance in the study of [[integrable systems]].

There are also some noncommutative versions which are still purely understood.

Painlevé VI is 2nd order nonlinear ODE governing the isomonodromic deformations of [[Fuchsian differential equation]]s
$$
\frac{d f}{d z} = \left(\frac{A_1}{z} + \frac{A_2}{z-t}+ \frac{A_3}{z-1}\right),
$$
where $A_i\in sl_2(\mathbf{C})$ and parameter $t$ varies through $\mathbf{P}^1\backslash\{0,1,\infty\}$.

## Literature and links

#### General

* eom/[Painlevé equation](https://encyclopediaofmath.org/wiki/Painlev
%C3%A9_equation)
* wikipedia: [Painlevé transcendents](http://en.wikipedia.org/wiki/Painlev%C3%A9_transcendents), [isomonodromic deformation](https://en.wikipedia.org/wiki/Isomonodromic_deformation)
* _One hundred years of PVI, the Fuchs–Painlevé equation_, J. Phys. A, special issue, Preface,
[pdf](http://www.researchgate.net/profile/Marta_Mazzocco2/publication/243413584_One_hundred_years_of_PVI_the_FuchsPainlev_equation/links/00b495315eb2aa177d000000)

Related $n$Lab entries include [[Picard-Vessiot theory]], [[Riemann-Hilbert problem]], [[integrable system]], [[monodromy]], [[holonomic quantum field]]

#### Classical works

* P. Painlevé, _Sur les équations differentielles du second ordre et d'ordre superieur, dont l'integrale génerale est uniforme_, Acta Math. __25__ (1902), pp. 1–86.

* Richard Fuchs, Comptes Rendus de l'Académie des Sciences Paris  __141__ (1905) 555–558

* B. Gambier, _Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes_ Acta Math. __33__ (1910) 1–55

#### Collection one century later

* _The Painlevé property, one century later_, Robert Conte, ed. CRM Series in Mathematical Physics, Springer, 1999 [doi](https://doi.org/10.1007/978-1-4612-1532-5)

Among the papers in the volume:

* [[Boris Dubrovin]], _Painlevé transcendents in two-dimensional topological field theory_, 287-412

Another collection

* Théories asymptotiques et équations de Painlevé, Séminaires et Congrès __14__, Société Mathématique de France 2006

#### Contemporary works

Modern viewpoint to R. Fuchs work

* [[Yuri Manin]], _Sixth Painlevé equation, universal elliptic curve, and mirror of $\mathbf{P}^2$_, In: Geometry of Differential Equations, ed. by A. Khovanskii, A. Varchenko, V. Vassiliev. Amer. Math. Soc. Transl. (2), vol. 186 (1998) 131–151 [alg–geom/9605010](https://arxiv.org/abs/alg-geom/9605010)

Connection to [[mirror symmetry]] is in

* [[Yu. I. Manin]], _Rational curves, elliptic curves, and the Painlevé equation_, Surveys in modern mathematics, London Math. Soc. Lec. Note Ser. __321__, Cambridge Univ. Press 2005, 24-33 [doi](https://doi.org/10.1017/CBO9780511614156.004) MR2166923

An arithmetic analogue is studied in

* Alexandru Buium, [[Yuri I. Manin]], _Arithmetic differential equations of Painlevé VI type_, in: Arithmetic and Geometry, Luis Dieulefait, Gerd Faltings, D. R. Heath-Brown, B. Z. Moroz, Cambridge Univ. Press 2015, [arxiv/1307.3841](https://arxiv.org/abs/1307.3841)

* [[Emma Previato]], _Curves in isomonodromy and isospectral deformations: Painlevé VI as a case study_, In: Algebraic curves and their applications. Contemporary Math. 724 (2018) 258-276, Amer. Math. Soc.

* M. Olshanetsky, _Painlevé type equations and Hitchin systems_, Proc. "Integrability: the Seiberg-Witten and Whitham Equations", Edinburgh, Sep. 1998 [math-ph/9901019](https://arxiv.org/abs/math-ph/9901019)
* AM Levin, MA Olshanetsky, _Hierarchies of isomonodromic deformations and Hitchin systems_, [pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.263.2907&rep=rep1&type=pdf)

* Martin D. Kruskal, [[Nalini Joshi]], Rod Halburd, _Analytic and asymptotic methods for nonlinear singularity analysis: a review and extensions of tests for the Painlevé property_, 1996 [pdf](http://www.ucl.ac.uk/~ucahrha/Publications/Pond-97.pdf)

* Hiroshi Umemura, _On the irreducibility of the first differential equation of Painlevé_, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo (1988) 771–789; _Second proof of the irreducibility of the first differential equation of Painlevé_, Nagoya Math. J. __117__, 125-171 [doi](https://doi.org/10.1017/S0027763000001835)
* Robert Conte, Micheline Musette, _The Painlevé handbook_

* Henryk Żołądek, _The monodromy group_, Monografie Matematyczne __67__, 588 pp. Birkhäuser 2006

* A. A. Kapaev, _Quasi-linear Stokes phenomenon for the Painlevé first equation_, J. Phys. A: Math. Gen. __37__, 11149 (2004) [doi](http://dx.doi.org/10.1088/0305-4470/37/46/005)

* A. A. Bolibruch, A. R. Its, A. A. Kapaev, _On the Riemann–Hilbert–Birkhoff inverse monodromy problem and the Painlevé equations_, Алгебра и анализ, 16:1 (2004), 121–162
* M. V. Babich, _Isomonodromic deformations and Painlevé equations_, Constr. Approx. 41:3 (2015) 335-356 [doi](https://doi.org/10.1007/s00365-015-9286-2)
* Leonid O. Chekhov, Marta Mazzocco, Vladimir N. Rubtsov, _Painlevé monodromy manifolds, decorated character varieties, and cluster algebras_, IMRN __24__ (2017) 7639–7691 [arxiv/1511.03851](https://arxiv.org/abs/1511.03851) [doi](https://doi.org/10.1093/imrn/rnw219); _Algebras of quantum monodromy data and decorated character varieties_, n: Geometry and physics: A Festschrift in honour of Nigel Hitchin, Oxford Univ. Press 2018 [arxiv/1705.01447](https://arxiv.org/abs/1705.01447); _Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras_, Adv. Math. __376__ (2021) 107442 [arxiv/1905.02772](https://arxiv.org/abs/1905.02772) [doi](https://doi.org/10.1016/j.aim.2020.107442)
* Marta Mazzocco, [[Vladimir Rubtsov]], _Confluence on the Painlevé monodromy manifolds, their Poisson structure and quantisation_, [arXiv/1212.6723](https://arxiv.org/abs/1212.6723)

* Marco Bertola, _Fredholm determinants and (noncommutative) Painlevé II equation_, slides, [pdf](http://www.impan.pl/~fasde/presentations/Bertola.pdf)

* A. Okounkov, E. Rains, _Noncommutative geometry and Painlevé equations_, Algebra & Number Theory 9(6) 1363–1400 (2015) [doi](https://doi.org/10.2140/ant.2015.9.1363)

* Kazuo Okamoto, _Polynomial Hamiltonians associated with Painlevé equations, I_, Proc. Jap. Acad. Ser. A, Math. Sci. __56__:6 [doi](https://10.3792/pjaa.56.264); _II, Differential equations satisfied by polynomial Hamiltonians_, Proc. Jap. Acad. Ser. A, Math. Sci. __56__:8 [doi](https://doi.org/10.3792/pjaa.56.367) [euclid](https://projecteuclid.org/euclid.pja/1195516717)

* H. Sakai, _Rational surfaces associated with affine root systems and geometry of the Painlevé equations_, Commun. Math. Phys. __220__ (2001) 165--229

*  M. A. Bershtein, A. I. Shchechkin, _Bilinear equations on Painlevé $\tau$ functions from CFT_, Commun. Math. Phys. __339__, 1021--1061 (2015) [doi](https://doi.org/10.1007/s00220-015-2427-4)

Multidimensional generalizations of Painlevé VI are introduced in

* G. Aminov, S. Arthamonov, A. Levin, M. Olshanetsky, A. Zotov, _Painlevé field theory_, [arXiv/1306.3265](https://arxiv.org/abs/1306.3265)
* [[Nalini Joshi]], Pieter Roffelsen, _On the monodromy manifold of $q$-Painlevé VI and its Riemann-Hilbert problem_, [arXiv/2202.10597](https://arxiv.org/abs/2202.10597)
* [[Nikita Nekrasov]], _Blowups in BPS/CFT correspondence, and Painlevé VI_, [arXiv:2007.03646](https://arxiv.org/abs/2007.03646)

#### Model-theoretic aspects

* Joel Nagloo, _Geometric triviality of the strongly minimal second Painlevé equations_ Annals of Pure and Applied Logic, 166(3):358–368, 2015.
* J. Nagloo, A. Pillay, _On  the  algebraic  independence  of  generic  Painlevé transcendents_, Compos.Math. __150__ (2014) 668-678.
* [[James Freitag]], _Algebraic relations between solutions of Painlevé equations_, [arXiv:1710.03304](https://arXiv.org/abs/1710.03304)
* Guy Casale, [[James Freitag]], Joel Nagloo, _Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups_, Annals of Mathematics __192__ (2020), 721–765 [doi](https://doi.org/10.4007/annals.2020.192.3.2)

[[!redirects Painlev%C3%A9 transcendent]]
[[!redirects Painlev%C3%A9 transcendents]]
[[!redirects Painlevé transcendents]]
[[!redirects Painlevé transcendent]]
[[!redirects Painlevé property]]
[[!redirects Painlevé equation]]
[[!redirects Painlevé equations]]
[[!redirects Painleve equation]]


pair

A _pair_ is usually an [[n-tuple]] for $n = 2$, that is an _[[ordered pair]]_.  Such a pair is a [[term]] of [[product type]], an [[element]] of a [[Cartesian product]], a [[generalized element]] of a [[product|product object]], etc.

But a _pair_ may also be an _[[unordered pair]]_.  The existence of these is a basic axiom of [[ZFC]] and its variations, the [[axiom of pairing]].

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


pair set

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

\tableofcontents

## Idea

In [[material set theory]], [[sets]] and [[elements]] are the same thing, so unordered pairs and pair sets are the same thing. However, in other foundations of mathematics, sets and elements are not the same thing, so an unordered pair is an element, while a **pair set** is a set.

## Definition

### In material set theory

If $A$ is a [[set]] and $x$ and $y$ are [[elements]] of $A$, then the __pair set__ $\{x,y\}$ is a subset of $A$ such that for all elements for which $z \in \{x,y\}$ holds, $z = x$ or $z = y$. In material set theory, these are also called [[unordered pairs]].

In [[material set theory]], we may apply this when $x$ and $y$ are not previously given as elements of any set $A$.  In that case, the existence of the unordered pair is given by the [[axiom of pairing]].

### In structural set theory

If $A$ is a [[set]] and $x$ and $y$ are [[elements]] of $A$, then the __pair set__ $\{x,y\}$ is a subset of $A$ with injection $i:\{x,y\} \hookrightarrow A$ such that for all elements $z \in \{x,y\}$, $i(z) = x$ or $i(z) = y$, and for all other sets $B$ with injection $j:B \hookrightarrow A$ such that for all elements $w \in B$, $i(w) = x$ or $i(w) = y$, there is an injection $k:B \hookrightarrow \{x,y\}$ such that for all elements $w \in B$, $i(k(w)) = j(w)$.

## Properties

The pair set $\{x,x\} = \{x\}$ is a [[singleton]].

## See also

* [[unordered pair]]

* [[axiom of pairing]]

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


pairing

# Pairings
* table of contents
{: toc}

## Idea

When $a$ and $b$ are elements of [[sets]], the __pairing__ of $a$ and $b$ is the [[ordered pair]] $(a,b)$.

It is natural to extend this to [[generalised elements]] in any [[category]] with binary [[products]].

For products of higher arity, one can say __tripling__, __quadrupling__, etc, or just __tupling__.

## Definition

Let $X$ and $Y$ be [[objects]] of some [[category]] $C$, and suppose that [[the]] [[product]] $X \times Y$ exists in $C$.

Let $G$ be some object of $C$, and let $a\colon G \to X$ and $b\colon G \to Y$ be [[morphisms]] of $C$.  Then, by definition of [[product]], there exists a unique morphism $(a,b)\colon G \to X \times Y$ such that the obvious diagrams commute.

If we think of $a$ and $b$ as $G$-indexed elements of $X$ and $Y$, then $(a,b)$ is a $G$-indexed element of $X \times Y$.

## Examples

If $C$ is the [[Set|category of sets]] and $G$ is the [[point]], then $a$ and $b$ are simply elements, in the usual sense, of $X$ and $Y$; then $(a,b)$ is an element of $X \times Y$, the usual ordered pair $(a,b)$.

If $Y$ and $G$ are each $X$, with $a$ and $b$ each the [[identity morphism]] on $X$, then the pairing $(id_X,id_X)$ is the [[diagonal morphism]] $\Delta_X\colon X \to X^2$.

## Pairings versus products

Since taking [[products]] (when these always exist) is a [[functor]], we can apply this operation to any two morphisms.  That is, if $a\colon G \to X$ and $b\colon H \to Y$ are morphisms in a category $C$, and if the products $G \times H$ and $X \times Y$ exist, then we have a morphism $a \times b\colon G \times H \to X \times Y$.  This is *not* the pairing $(a,b)$, for which the source is always $G$.

A pairing is the composite of a product and a [[diagonal morphism]]:
$$ G \overset{\Delta_G}\to G \times G \overset{a \times b}\to X \times Y ;$$
conversely, a product is a pairing of two composites:
$$ \array {
   G \times H \to G \overset{a}\to X
,\\
   G \times H \to H \overset{b}\to Y
.} $$

If $G$ and $H$ are each [[terminal object|terminal]], however, then $(a,b)$ and $a \times b$ are the same [[global element]] of $X \times Y$.  Thus, both product morphisms and pairings are generalisations of [[ordered pairs]] in [[Set]].

[[!redirects pairing]]
[[!redirects pairings]]

[[!redirects tripling]]
[[!redirects triplings]]

[[!redirects quadrupling]]
[[!redirects quadruplings]]

[[!redirects tupling]]
[[!redirects tuplings]]


pairing structure

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

\tableofcontents

## Definition

### Pairing structure

Given a [[set]] $V$ with an [[extensional relation]] $\prec$ on $V$, a **pairing structure** is a [[binary function]] $P:V \times V \to V$ such that

* for all $a \in V$ and $b \in V$, $a \prec P(a, b)$ and $b \prec P(a, b)$

### Unordered pairing structure

An **unordered pairing structure** on $V$ is pairing structure $\{-,-\}:V \times V \to V$ where additionally

* for all sets $z$, $z \in \{x, y\}$ implies that $z = x$ or $z = y$.

Uniqueness of $\{x, y\}$ follows from $\prec$ being an [[extensional relation]].

### Ordered pairing structure

An **ordered pairing structure** on $V$ is a pairing structure $(-,-):V \times V \to V$ which satisfies [[product extensionality]]:

* for all $a \in V$, $a' \in V$, $b \in V$, $b' \in V$, $P(a, b) = P(a', b')$ if and only if $a = a'$ and $b = b'$

## Foundational concerns

In any [[material set theory]], instead of postulating the mere existence of a set $P$ in which $a \in P$ and $b \in P$ one could add a primitive binary operation $P(a, b)$ which takes material sets $a$ and $b$ and returns a material set $P(a, b)$ such that for all $a$ and $b$, $a \in P(a, b)$ and $b \in P(a, b)$

## See also

* [[axiom of pairing]]

* [[union structure]]

* [[powerset structure]]

* [[natural numbers structure]]

* [[product extensionality]]

## References

* [[Håkon Robbestad Gylterud]], [[Elisabeth Bonnevier]], *Non-wellfounded sets in HoTT* ([arXiv:2001.06696](https://arxiv.org/abs/2001.06696))

[[!redirects pairing structure]]
[[!redirects pairing structures]]

[[!redirects unordered pairing structure]]
[[!redirects unordered pairing structures]]

[[!redirects ordered pairing structure]]
[[!redirects ordered pairing structures]]


Paley-Wiener-Schwartz theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The Paley-Wiener-Schwartz theorem characterizes [[compact support|compactly supported]] [[smooth functions]] ([[bump functions]]) and more generally [[compactly supported distributions]] in terms of the decay property of their [[Fourier-Laplace transform]] ([[Fourier transform of distributions|of distributions]]).

Conversely this means that for a general distribution those [[covectors]] along which its [[Fourier transform of distributions|Fourier transform]] does not suitably decay detect the singular nature of the distribution (its failure to be represented by a smooth function) in a refined form which records not only the [[singular support of a distribution|singular support]] (the point at which that covector is based), but also the "direction of [[propagation of singularities theorem|propagation of singularities]]". The collection of these covectors is called the _[[wave front set]]_ of the distribution. The study of [[functional analysis]] with attention to the [[wave front sets]] is called _[[microlocal analysis]]_. This plays a central role in the definition of various operations on distributions, such as the [[pullback of distributions]] and the [[product of distributions]].

## Statement

+-- {: .num_theorem #PWSTheorem}
###### Theorem
**(Paley-Wiener-Schwartz theorem)**

For $n \in \mathbb{N}$ the vector space $C^\infty_c(\mathbb{R}^n)$ of [[compact support|compactly supported]] [[smooth functions]] ([[bump functions]]) on [[Euclidean space]] $\mathbb{R}^n$ is (algebraically and topologically) [[isomorphism|isomorphic]], via the [[Fourier transform]], to the space of [[entire functions]] $F$ on $\mathbb{C}^n$ which satisfy the following estimate: there is a [[positive number|positive]] [[real number]] $B$ such that for every [[integer]] $N \gt 0$ there is a real number $C_N$ such that:

$$
  \underset{\xi \in \mathbb{C}^n}{\forall}
  \left(
    {\vert  F(\xi) \vert}
    \le C_N (1 + {\vert \xi\vert })^{-N} \exp{ (B \; |Im(\xi)|)}
  \right)
  \,.
$$

More generally, the space of [[compactly supported distributions]] on $\mathbb{R}^n$ of [[order of a distribution|order]] $N \in \mathbb{N}$ is isomorphic via [[Fourier transform of distributions]] to those [[entire functions]] on $\mathbb{C}^n$ for which there exist [[positive number|positive]] [[real numbers]] $B, C \in \mathbb{R}_{\gt 0}$

$$
  \underset{\xi \in \mathbb{C}^n}{\forall}
  \left(
    {\vert F(\xi)  \vert}
      \le
    C_N (1 + {\vert \xi\vert })^{N} \exp{ (B \; |Im(\xi)|)}
  \right)
  \,.
$$

(Notice that the [[Fourier-Laplace transform]] [[Fourier transform of a distribution|of a]] [[compactly supported distribution]] is guaranteed to be an [[entire holomorphic function]], by [this prop.](Fourier+transform+of+distributions#FourierTransformOfCompactlySupportedDistributions).)

=--

(e.g. [Hoermander 90, theorem 7.3.1](#Hoermander90))

In fact

+-- {: .num_prop #DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}
###### Proposition
**(decay of [[Fourier transform of distributions|Fourier transform]] of [[compactly supported functions]])**

A [[compactly supported distribution]] $u \in \mathcal{E}'(\mathbb{R}^n)$ is non-[[singular support of a distribution|singular]], hence given by a [[compactly supported function]] $b \in C^\infty_{cp}(\mathbb{R}^n)$ via $u(f) = \int b(x) f(x) dvol(x)$, precisely if its [[Fourier transform of a distribution|Fourier transform]] $\hat u$ ([this def.](compactly+supported+distribution#FourierTransformOfCompactlySupportedDistribution)) satisfies the following decay property:

For all $N \in \mathbb{N}$ there exists $C_N \in \mathbb{R}_+$ such that for all $\xi \in \mathbb{R}^n$ we have that the [[absolute value]] ${\vert \hat v(\xi)\vert}$ of the Fourier transform at that point is bounded by

$$
  {\vert \hat v(\xi)\vert}
  \;\leq\;
  C_N \left( 1 + {\vert \xi\vert} \right)^{-N}
  \,.
$$

=--

(e.g. [Hoermander 90, around (8.1.1)](#Hoermander90))

## Related entries

* [[Fourier inversion theorem]]

* [[Parseval's theorem]]

## References

* {#Hoermander90} [[Lars Hörmander]], section 7.3 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

See also

* Wikipedia, _[Paley-Wiener theorem](https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem)_

[[!redirects Paley-Wiener theorem]]


Panagiotis Konstantis

* [webpage](http://www.mi.uni-koeln.de/~pako/)

## Selected writings

On [[Cohomotopy]] sets  in relation to [[quaternionic line bundles]] over [[5-manifolds]]:

* [[Panagiotis Konstantis]], _Vector bundles and cohomotopies of spin 5-manifolds_, Homology, Homotopy and Applications Volume 23 (2021) Number 1 ([arXiv:1812.06547](https://arxiv.org/abs/1812.06547), [doi:10.4310/HHA.2021.v23.n1.a9](https://dx.doi.org/10.4310/HHA.2021.v23.n1.a9))

and of [[spin structure|spin]]-manifolds in co-degree 1:

* [[Panagiotis Konstantis]], _A counting invariant for maps into spheres and for zero loci of sections of vector bundles_,  Abh. Math. Semin. Univ. Hambg. (2020) ([arXiv:1911.03214](https://arxiv.org/abs/1911.03214), [doi:10.1007/s12188-020-00228-6](https://doi.org/10.1007/s12188-020-00228-6))

category: people


Panagis Karazeris

* [webpage](http://www.math.upatras.gr/~pkarazer/)

category: people


Pante Stanica

Pantelimon (Pante) Stanica is a Romanian-American mathematician, whose interests are in cryptographic Boolean functions, Number Theory, Combinatorics and Discrete Mathematics. He is currently a professor at Naval Postgraduate School in Monterey, California.

*****************************************************************

Check out his [homepage](http://faculty.nps.edu/pstanica/) for more info.

category: people

[[!redirects Pante Stanica]]
[[!redirects mathguy]]
[[!redirects mathwiz]]


Paolo Aluffi

* [homepage](https://www.math.fsu.edu/~aluffi/)

## Selected writings

On [[algebra]]:

* {#Aluffi09} [[Paolo Aluffi]], *Algebra: Chapter 0*, Graduate Studies in Mathematics **104**,  AMS (2009) [[ISBN:978-1-4704-1168-8](https://bookstore.ams.org/gsm-104/)]

category: people


Paolo Aschieri

__Paolo Aschieri__ is a mathematical physicist currently at Alessandria, Italy. His interests include noncommutative geometry, field theories (classical and quantum, gauge and gravity, magnetic-electric duality, noncommutative field theories), quantum groups, gerbes and deformation quantization. His coauthors include Castellani, Wess, Jurčo, Schupp, Dimitrijević...

* [webpage](http://fisicateorica.mfn.unipmn.it/people/paoloaschieri.html)

## Selected writings

A model for the [[supergravity C-field]] in terms of [[nonabelian bundle 2-gerbe|nonabelian]] [[bundle 2-gerbes]]:

* [[Paolo Aschieri]], [[Branislav Jurčo]], *Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory*, JHEP 0410:068 (2004) [[arXiv:hep-th/0409200](https://arxiv.org/abs/hep-th/0409200), [doi:10.1088/1126-6708/2004/10/068](https://doi.org/10.1088/1126-6708/2004/10/068)]

[[!redirects P. Aschieri]]


Paolo Bertozzini

* [webpage](http://www.paolo-th.site90.net)

category: people


Paolo Capriotti

* [webpage](https://www.paolocapriotti.com/about/index.html)

## Selected writings

On a version of the [[Homotopy Type System]] with application to [[semi-simplicial types in homotopy type theory]]:

* [[Danil Annenkov]], [[Paolo Capriotti]], [[Nicolai Kraus]], [[Christian Sattler]]: *Two-Level Type Theory and Applications* [[arXiv:1705.03307](https://arxiv.org/abs/1705.03307)]

On [[quotient types]] and [[higher inductive types]]:

* [[Thorsten Altenkirch]], [[Paolo Capriotti]], [[Gabe Dijkstra]], [[Nicolai Kraus]], [[Fredrik Forsberg]], *Quotient inductive-inductive types*, Foundations of Software Science and Computation Structures (FoSSaCS 2018), Lecture Notes in Computer Science **10803** (2018) [[doi:10.1007/978-3-319-89366-2_16](https://doi.org/10.1007/978-3-319-89366-2_16), [arXiv:1612.02346](https://arxiv.org/abs/1612.02346)]

## Related entries

* [[internal category in homotopy type theory]]

* [[category object in an (infinity,1)-category]]

category: people


Paolo Di Vecchia

* [webpage](http://www.nordita.org/people/staff/index.php?u=divecchi)

## Selected writings

Making manifest the local [[worldsheet]] [[supersymmetry]] of the [[spinning string]] (via the supersymmetric [[Polyakov action]]):

* [[Lars Brink]], [[Paolo Di Vecchia]], [[Paul Howe]], _A locally supersymmetric and reparametrization invariant action for the spinning string_ , Physics Letters B **65** Issue 5, 20 (1976) 471-474 [doi:10.1016/0370-2693(76)90445-7]

On [[spinning particles]] and their [[worldline]] [[supersymmetry]]:

* [[Lars Brink]], [[Paolo Di Vecchia]], [[Paul Howe]], *A Lagrangian formulation of the classical and quantum dynamics of spinning particles*, Nucl. Phys. B **118** (1977) 76 [doi:10.1016/0550-3213(77)90364-9]

On the [[theta angle]] in [[QCD]]:

* {#CrewtherDiVecchiaVenezianoWitten79} [[Rodney Crewther]], [[Paolo Di Vecchia]], [[Gabriele Veneziano]], [[Edward Witten]], _Chiral estimate of the electric dipole moment of the neutron in quantum chromodynamics_, Phys. Lett. B 88 (1979) 123-127 ([CERN](http://cds.cern.ch/record/133382)).

The [[lattice field theory|lattice]]-formulation of the [[CP^N sigma-model|$\mathbb{C}P^N$-model]]:

* [[Paolo Di Vecchia]], A. Holtkamp, R. Musto, F. Nicodemi, R. Pettorini, *Lattice $CP^{N-1}$ models and their large-$N$ behaviour*, Nuclear Physics B **190** 4 (1981) 719-733 [doi:10.1016/0550-3213(81)90047-X]

On the early history of [[string theory]]:

* Andrea Cappelli, Elena Castellani, Filippo Colomo, [[Paolo Di Vecchia]] (eds.), *The Birth of String Theory*,  Cambridge University Press (2012) [[doi:10.1017/CBO9780511977725](https://doi.org/10.1017/CBO9780511977725)]

* [[Paolo Di Vecchia]], *The birth of string theory* [[arXiv:0704.0101](https://arxiv.org/abs/0704.0101)]

  published as: *From the S-matrix to string theory*, Section 11 in: [[Paolo Di Vecchia]] et al. (eds.), *The Birth of String Theory*, Campridge University Press (2011) 156-178 [[doi:10.1017/CBO9780511977725](https://doi.org/10.1017/CBO9780511977725)]

category: people


Paolo Gregori

* [Google Scholar page](https://scholar.google.be/citations?user=hbYa3noAAAAJ)

* [Spire page](https://inspirehep.net/authors/1633516)

## Selected writings

On [[resurgence]] and [[non-perturbative effects]] in the [[SYK model]] and [[AdS2-CFT1 duality]]:

* [[Paolo Gregori]], [[Ricardo Schiappa]], *From Minimal Strings towards Jackiw-Teitelboim Gravity: On their Resurgence, Resonance, and Black Holes* ([arXiv:2108.11409](https://arxiv.org/abs/2108.11409))

category: people


Paolo Mancosu

* [Institute page](https://philosophy.berkeley.edu/mancosu/)

## Selected writings

On the history of [[intuitionistic mathematics]] concerning contributors such as [[L.E.J. Brouwer]], [[Hermann Weyl]] and [[David Hilbert]]:

* [[Paolo Mancosu]] (ed.), *From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s*, Oxford University Press 1998.

category: people


Paolo Pasti

* [webpage](http://en.didattica.unipd.it/offerta/docente/FC75C5CDA5B9043EC22C8D3710567C83)

## Selected writings

On the [[superembedding approach]] for [[super p-branes|super $p$-branes]]/[[Green-Schwarz sigma-models]]:

* {#BPSTV95} [[Igor Bandos]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], [[Dmitry Volkov]], _Superstrings and supermembranes in the doubly supersymmetric geometrical approach_, Nucl. Phys. B **446** (1995) 79-118 [[arXiv:hep-th/9501113](https://arxiv.org/abs/hep-th/9501113)]

On [[self-dual higher gauge fields]]:

* [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On Lorentz Invariant Actions for Chiral P-Forms_, Phys.Rev. D55 (1997) 6292-6298 ([arXiv:hep-th/9611100](https://arxiv.org/abs/hep-th/9611100))

On the [[Green-Schwarz sigma-model]] of the [[M5-brane]]:

* {#BLNPST97} [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _Covariant Action for the Super-Five-Brane of M-Theory_, Phys. Rev. Lett. 78 (1997) 4332-4334 ([arXiv:hep-th/9701149](http://arxiv.org/abs/hep-th/9701149))

* {#PST97} [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _Covariant Action for a D=11 Five-Brane with the Chiral Field_, Phys. Lett. B398 (1997) 41 ([arXiv:hep-th/9701037](https://arxiv.org/abs/hep-th/9701037))

* [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On the equivalence of different formulations of the M Theory five--brane_, Phys. Lett. B408 (1997) 135-141 ([arXiv:hep-th/9703127](http://arxiv.org/abs/hep-th/9703127))

## Related $n$Lab entries

* [[M5-brane]]

* [[AdS-CFT]]

category: people


Paolo Perrone

* [home page](http://www.paoloperrone.org)

Related nLab pages:

* [[categorical probability]]
* [[monads of probability, measures, and valuations]]
* [[Markov category]]
* [[category of couplings]]
* [[partial evaluation]]

### Selected writings

* [[Tobias Fritz]] and [[Paolo Perrone]], _A Probability Monad as the Colimit of Spaces of Finite Samples_, Theory and applications of Categories 34, 2019. ([arXiv:1712.05363](http://arxiv.org/abs/1712.05363))

* [[Tobias Fritz]] and [[Paolo Perrone]], _Stochastic order on metric spaces and the ordered Kantorovich monad_, Advances in Mathematics 366, 2020. ([arXiv:1808.09898](http://arxiv.org/abs/1808.09898))

* [[Tobias Fritz]], Tomáš Gonda and [[Paolo Perrone]], _De Finetti's theorem in categorical probability_. Journal of Stochastic Analysis, 2021. ([arXiv:2105.02639](https://arxiv.org/abs/2105.02639))

* [[Paolo Perrone]] and [[Walter Tholen]], _Kan extensions are partial colimits_, Applied Categorical Structures 30, 2022. ([arXiv:2101.04531](https://arxiv.org/abs/2101.04531))

* Sean Moss and [[Paolo Perrone]], _Probability monads with submonads of deterministic states_, LICS 2022. ([arXiv:2204.07003](https://arxiv.org/abs/2204.07003))

* Carmen Constantin, [[Tobias Fritz]], [[Paolo Perrone]] and Brandon Shapiro, _Partial evaluations and the compositional structure of the bar construction_, Theory and Applications of Categories 39, 2023. ([arXiv:2009.07302](https://arxiv.org/abs/2009.07302))

* Carmen Constantin, [[Tobias Fritz]], [[Paolo Perrone]] and Brandon Shapiro, _Weak cartesian properties of simplicial sets_, Journal of Homotopy and Related Structures, 2023. ([arXiv:2105.04775](https://arxiv.org/abs/2105.04775))

* Sean Moss and [[Paolo Perrone]], _A category-theoretic proof of the ergodic decomposition theorem_, Ergodic Theory and Dynamical Systems, 2023. ([arXiv:2207.07353](https://arxiv.org/abs/2207.07353))

* [[Paolo Perrone]], _Markov Categories and Entropy_, IEEE Transactions on Information Theory 70(3), 2024. ([arXiv:2212.11719](https://arxiv.org/abs/2212.11719))

category:people

[[!redirects P. Perrone]]
[[!redirects Perrone]]

[[!redirects paoloperrone]]


Paolo Piccinni

* [Institute page](https://gomppublic.uniroma1.it/Docenti/Render.aspx?UID=0fa687f9-d804-4349-9f37-9a8918acdd11)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=169553)

## Selected writings

On [[Pontryagin classes]] for [[Sp(n).Sp(1)]]:

* [[Paolo Piccinni]], Giuliano Romani, _A generalization of symplectic Pontrjagin classes to vector bundles with structure group $Sp(n)\cdot Sp(1)$_,  Annali di Matematica pura ed applicata (1983) 133: 1 ([doi:10.1007/BF01766008](https://doi.org/10.1007/BF01766008))

* [[Paolo Piccinni]], _Vector fields and characteristic numbers on hyperkàhler and quaternion Kâhler manifolds_, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1992) Volume: 3, Issue: 4, page 295-298 ([dml:244204](https://eudml.org/doc/244204))

On the [[octonionic Hopf fibration]]:

* {#OrneaPartonPiccinniVuletescu12} Liviu Ornea, Maurizio Parton, [[Paolo Piccinni]], Victor Vuletescu, _Spin(9) geometry of the octonionic Hopf fibration_, Transformation Groups (2013) 18: 845 ([arXiv:1208.0899](http://arxiv.org/abs/1208.0899), [doi:10.1007/s00031-013-9233-x]( https://doi.org/10.1007/s00031-013-9233-x))

* {#PartonPiccinni18} Maurizio Parton, [[Paolo Piccinni]], _The Role of Spin(9) in Octonionic Geometry_, ([arXiv:1810.06288](https://arxiv.org/abs/1810.06288)).

On [[Cayley 4-planes]]:

* {#OrneaPiccini00} Liviu Ornea, [[Paolo Piccinni]], _Cayley 4-frames and a quaternion-Kähler reduction related to Spin(7)_, Proceedings of the International Congress of Differential Geometry in the memory of A. Gray, held in Bilbao, Sept. 2000 ([arXiv:math/0106116](https://arxiv.org/abs/math/0106116))

## Related $n$Lab entries

* [[Sp(n).Sp(1)]]

* [[Cayley 4-form]]

* [[octonionic Hopf fibration]]

category: people


Paolo Piccione

* [Wikipedia entry](https://en.wikipedia.org/wiki/Paolo_Piccione)

## Selected writings

On [[Lorentzian structure]] as [[G-structure]]:

* Leandro A. Lichtenfelz, [[Paolo Piccione]], Abdelghani Zeghib, _On the Isometry Group of Lorentz Manifolds_, in: Miguel Sanchez, Miguel Ortega, Alfonso Romero (eds.) _Recent Trends in Lorentzian
Geometry_, Springer 2013 ([doi:10.1007/978-1-4614-4897-6_12](https://doi.org/10.1007/978-1-4614-4897-6_12))

On [[flat orbifolds]]:

* {#BettiolDerdzinskiPiccione18} Renato G. Bettiol, Andrzej Derdzinski, [[Paolo Piccione]], _Teichmüller theory and collapse of flat manifolds_, Annali di Matematica (2018) 197: 1247 ([arXiv:1705.08431](https://arxiv.org/abs/1705.08431), [doi:10.1007/s10231-017-0723-7](https://doi.org/10.1007/s10231-017-0723-7))

category: people


Paolo Salvatore

Paolo Salvatore is an Italian mathematician who works in Dipartimento di Matematica
Università di Roma "Tor Vergata". His research interests include [[configuration spaces of points]], [[string topology]], [[operads]], [[spaces of knots]], and [[cohomology operations]].

* [homepage](http://www.mat.uniroma2.it/~salvator/)

## Selected writings

On [[group completion]] of [[configuration spaces of points]]:

* [[Paolo Salvatore]], _Configuration spaces with summable labels_, In: Aguadé J., Broto C., [[Carles Casacuberta]]  (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel, 2001 ([arXiv:math/9907073](https://arxiv.org/abs/math/9907073))

On the [[framed little 2-disk operad]]:

* {#GiansiracusaSalvatore09} [[Jeffrey Giansiracusa]], [[Paolo Salvatore]], _Formality of the framed little 2-discs operad and semidirect products_ , in: _Homotopy theory of function spaces and related topics_, Cont. Math. 519, AMS, pp. 115-121  ([arxiv 0911.4428](http://arxiv.org/PS_cache/arxiv/abs/0911/0911.4428))

and relation to [[BV-algebra]]:

* {#SalvatoreWahl} [[Paolo Salvatore]], [[Nathalie Wahl]], _Framed discs operads and Batalin-Vilkovisky algebras_ ([pdf](http://www.math.ku.dk/~wahl/SalvatoreWahl.pdf))

category: people


Paolo Zanardi

* [personal page](https://www.isi.it/en/people/paolo-zanardi)

* [institute page](https://dornsife.usc.edu/cf/faculty-and-staff/faculty.cfm?pid=1016223)

* [GoogleScholar page](https://scholar.google.com/citations?user=Tsn9P-YAAAAJ)

## Selected writings

The idea of [[adiabatic quantum computation]] ([[topological quantum computation]]) formulated in the generality of [[Berry phases]] for any [[loops]] in any external parameter space ([[holonomic quantum computation]]):

* [[Paolo Zanardi]], [[Mario Rasetti]], *Holonomic Quantum Computation*, Phys. Lett. A **264** (1999) 94-99 [doi:10.1016/S0375-9601(99)00803-8, [arXiv:quant-ph/9904011](https://arxiv.org/abs/quant-ph/9904011)]

* [[Jiannis Pachos]], [[Paolo Zanardi]], [[Mario Rasetti]], *Non-Abelian Berry connections for quantum computation*, Phys. Rev. A **61** (2000) 010305 [[arXiv:quant-ph/9907103](https://arxiv.org/abs/quant-ph/9907103), [doi:10.1103/PhysRevA.61.010305](https://doi.org/10.1103/PhysRevA.61.010305)]

* [[Jiannis Pachos]], [[Paolo Zanardi]], *Quantum Holonomies for Quantum Computing*, Int. J. Mod. Phys. B **15** (2001) 1257-1286 [[arXiv:quant-ph/0007110](https://arxiv.org/abs/quant-ph/0007110), [doi:10.1142/S0217979201004836](https://doi.org/10.1142/S0217979201004836)]

category: people


para construction

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

#Contents#
* table of contents
{: toc}

## Idea
The **Para construction** is a variety of strictly related functorial constructions that produce a category of parametric morphisms.

In the simplest case, one starts with a monoidal category $\mathcal{C}$ and builds a category $\mathbf{Para}(\mathcal{C})$ whose morphisms $A \to B$ are pairs $(P,f)$ where $P:\mathcal{C}$ is a _parameter_ and $f: A \otimes P \to B$ is a morphism in $\mathcal{C}$.

Such a morphism might be visualised using the [[string diagram|graphical]] language of monoidal categories (below, left). However, this notation does not emphasise the special role played by $P$, which is part of the data of the morphism itself. Parameters and data in [[machine learning]] have different semantics; by separating them on two different axes, we obtain a graphical language which is more closely tied to these semantics (below, right).

This gives us an intuitive way to compose parameterised maps:

The Para construction is of relevance in categorical cybernetics, since controlled processes such as machine learning models, economic agents and Bayesian reasoners are straightforwardly modeled as parametric processes of some kind.

Indeed, the notation $\mathbf{Para}(\mathcal{C})$ was originally introduced in ([Fong, Spivak and Tuyeras 2019](#FongSpivakTuyeras2019)), and then successively refined in ([Gavranovic 2019](#Gavranovic19)) and ([Cruttwell et al. 2021](#Cruttwell2021)) for applications to machine learning.  ([Capucci et al. 2020](#Capucci2020)) generalized the construction from monoidal categories to [[actegories]], in order to capture examples from other areas of categorical cybernetics.  A still further generalization and systematization of the Para construction is being developed by [[David Jaz Myers]] and [[Matteo Capucci]], see ([Myers 2022](#Myers22)) and ([Capucci 2023](#Capucci23)).

Nonetheless, the 1-categorical version of Para already appears e.g. in ([Hermida & Tennent 2012](#HermidaTennent2012)), with precursors in ([Pavlovic 1997](#Pavlovic1997)).
Also, [Baković  2008, Thm. 13.2](#Bakovic2008) introduces Copara (under a different name) in the very general context of actions of [[bicategories]].

## Definition

### For monoidal categories

\begin{definition}
For $(\mathcal{C}, I, \otimes)$ a [[monoidal category]], $\mathbf{Para}(\mathcal{C})$ is the [[bicategory]] given by the following data:

* Its [[objects]] are the objects of $\mathcal{C}$.

* A [[1-morphism]] $A \to B$ is a choice of a parameter object $P \colon \mathcal{C}$ and a morphism in $\mathcal{C}$ of the form

  $$
    f \colon A \otimes P \to B
    \,.
  $$

* A [[2-morphism]] $(P, f) \Rightarrow (Q, g)$ is a morphism $r \colon P \to Q$ in $\mathcal{C}$ such that the following [[commuting diagram|diagram commutes]]:

\begin{tikzcd}
  {A \otimes P}
  \\
  {A \otimes Q} & B
  \arrow["{A \otimes r}", from=1-1, to=2-1]
  \arrow["f", from=1-1, to=2-2]
  \arrow["g"', from=2-1, to=2-2]
\end{tikzcd}

* Identity morphisms are given by the right [[unitors]]:

  $$
    (I, \rho) \colon X \otimes I
      \xrightarrow{\rho_X}
    X
    \,.
  $$

* The composite of a map $f \colon A \otimes P \to B$ and $g \colon B \otimes Q \to C$ is given by the animation above. In symbols, this is the $P \otimes Q$-parameterised map defined as

  $$
    A \otimes P \otimes Q
      \xrightarrow{f \otimes Q}
    B \otimes Q \xrightarrow{g} C
  $$

* The data of [[associators]] and [[unitors]] for the bicategory, as well as their coherence diagrams, are defined using those of $\mathcal{C}$.

\end{definition}

If $\mathbf{C}$ is strict monoidal, then $\mathbf{Para}(\mathcal{C})$ is a [[2-category]].
One can show that if $\mathbf{C}$ is [[commutative monoidal category|commutative]], then $\mathbf{Para}(\mathcal{C})$ is symmetric monoidal, since commutativity of $\otimes$ allows one to exhibit an interchange.
It is believe this still holds if $\mathbf{C}$ is just symmetric, making $\mathbf{Para}(\mathcal{C})$ a [[symmetric monoidal bicategory]].

### For actegories

\begin{definition}
Let $(\mathcal{M}, I, \otimes)$ be a [[monoidal category]] and let $(\mathcal{C}, \odot)$ be a right $\mathcal{M}$-[[actegory]]. Then $\mathbf{Para}(\odot)$ is a [[bicategory]] with the following data:

* Its 0-cells are the objects of $\mathcal{C}$.
* A 1-cell $A \to B$ is a choice of a parameter object $P : \mathcal{M}$ and a morphism
$$
f : A \odot P \to B
$$
in $\mathcal{C}$.
* A 2-cell $(P, f) \Rightarrow (Q, g)$ is a morphism $r : P \to Q$ in $\mathcal{M}$ such that the following commutes:

\begin{tikzcd}
	{A \odot P} \\
	{A \odot Q} & B
	\arrow["{A \odot r}"', from=1-1, to=2-1]
	\arrow["f", from=1-1, to=2-2]
	\arrow["g"', from=2-1, to=2-2]
\end{tikzcd}

* Identity morphisms are given by the unitors:
$$
(I, \eta) : X \odot I \xrightarrow{\eta_X} X
$$
* The composition of a map $f : A \odot P \to B$ and $g : B \odot Q \to C$ is defined as
$$
A \odot (P \otimes Q) \cong (A \odot P) \odot Q \xrightarrow{f \odot Q} B \odot Q \xrightarrow{g} C
$$
* The data of associators and unitors for the bicategory, as well as their coherence diagrams, are defined using those of $\mathcal{M}$ and $\odot$.

\end{definition}

Again, it is folklore that this bicategory is symmetric monoidal when $\odot$ is a *[[monoidal actegory|symmetric monoidal action]]*, meaning $\mathcal{C}$ is monoidal, $\mathcal{M}$ is symmetric and $\odot$ is a symmetric monoidal functor.

Also, this construction is 2-functorial from $\mathcal{M}\mathbf{Act}$ to $\mathbf{Bicat}$.

### Dual construction: copara and bipara
Given a *left* $\mathcal{M}$-actegory, one can produce a bicategory of *coparametric morphisms* $f: A \to P \odot B$ by dualizing the above construction in the obvious way.
This is known as $\mathbf{Copara}(\odot)$.

Likewise, given an $\mathcal{M}$-[[biactegory]] there is a bicategory $\mathbf{Bipara}(\odot, \odot')$ whose 1-cells are *biparametric morphisms* $f: A \odot P \to Q \odot B$, whose composition rule uses the bimodulator of the biactegory.

### Generalizations

The Para construction naturally generalizes in five different ways (with the first four described in ([Myers 2022](#Myers22))):

{#AsADoubleCategory} **(1.)** Move from bicategories to [[double categories]] (in the weak sense): $\mathbf{Para}(\mathcal{C})$ is usefully thought of as a [[double category]] whose [[tight map|tight category]] is still $\mathcal{C}$, whose [[loose map|loose maps]] are parametric morphisms, and whose squares are reparametrizations that commute suitably:

\begin{tikzcd}[sep=20pt]
  A
  \ar[
    rr,
    "{ (P,f) }",
    "{\ }"{name=s, pos=.9, swap}
  ]
  \ar[
    dd, "{ h }"{swap}
  ]
  &&
  B
  \ar[
    dd, "{ k }"
  ]
  &&
  A \odot P
  \ar[
    rr,
    "{ f }"
  ]
  \ar[
    dd,
    "{ h \odot r }"{swap}
  ]
  &&
  B
  \ar[
    dd,
    "{ k }"
  ]
  \\
  &&&
  :=
  \\
  A'
  \ar[
    rr,
    "{ (P', f') }"{swap},
    "{\ }"{name=t, pos=.1}
  ]
  &&
  B'
  \ar[
    from=s,
    to=t,
    Rightarrow,
    shorten=10pt,
    "{ r }"
  ]
  &&
  A' \odot P'
  \ar[
    rr,
    "{ f' }"{swap}
  ]
  &&
  B'
\end{tikzcd}

**(2.)** Allow for *colax actions*. This includes [[comonads]] (and [[graded monad|graded comonads]] more generally), since they are colax actions of the terminal monoidal category; and applying Para to a comonad yields a double categorical version of the [[coKleisli category]] of the comonad.

**(3.)** Allow for "dependent parameters", i.e. for situation in which the parameter $P$ of a parametric morphism $f: A \odot P \to B$ actually depends on $A$, thus allowing for $f$ of the form $f: (a:A) \times P(a) \to B$ (notation is suggestive). This makes double categories such as [[span|$\mathbf{Span}(\mathcal{C})$]] instances of the Para construction (where the left leg encodes the parameter dependency and the right leg is the parametric morphism itself).

**(4.)** Have the construction take place in [[complete category|complete]] 2-categories, as opposed to $\mathbf{Cat}$. This allows to describe Para for structured categories and to apply it to different 2-categorical structures (e.g. indexed categories).

**(5.)** Internal vs. external parameterisation: given a $\mathcal{M}$-actegory $(\mathcal{C}, \odot)$, the Para construction describes parameterisation which is _internal_ to the category $\mathcal{C}$. For instance, given an action on the category [[Vect]], a morphism $f : A \odot P \to B$ is a linear map. But often we are interested in morphisms linear only in one variable, for instance, a function $f : \mathbf{Set}(P, \mathbf{Vect}(A, B))$. In such cases we're interested in _external_ parameterisation, one which is captured by $\mathcal{C}$ being an object of $\mathcal{M}Cat$, instead of $\mathcal{M}Act$. A unified perspective on these two approaches is that of a [[locally graded category]].

## In categorical cybernetics

Para has been developed to model effectively the compositional structure of controlled processes, such as those involving agents. The examples better developed are in [[deep learning]] and [[game theory]], see ([Capucci et al. 2020](#Capucci2020)).

The idea that agents are parametric functions is quite old. In fact, agents take in an input $A$ and produce an output $B$. However, they have additional inputs/outputs not available to the "outside world", i.e. not part of their composition interface.
These can be the _weights_ (or _parameters_) of a neural network (or a more general ML model), the _strategies_ of a player in an economic game, the control signal of a controlled ODE or a Markov decision process.

Thus agents are naturally modelled as parametric morphisms.

## Examples

* The [[Cayley graph]] of a monoid action (together with its category structure) is a low-dimensional example of a Para construction for actegories.

* When the base category is set to be the category of [[optics (in computer science)|optics]], then $\mathbf{Para(\mathbf{Optic(\mathcal{C})})}$ recovers the category of [[neural networks]] defined in ([Capucci et al. 2020](#Capucci2020)).

* The "monoidal indeterminates" construction of [Hermida and Tennent](#HermidaTennent2012) is a Para construction for actegories. Hermida and Tennent consider a symmetric monoidal functor $j:\mathbf{\Sigma}\to \mathcal{C}$ (hence: a symmetric monoidal action of $\mathbf{\Sigma}$ on $\mathcal{C}$) and provide a symmetric monoidal bicategory $\mathcal{C}(\mathbf{x}:j)$ and ordinary monoidal category $\mathcal{C}[\mathbf{x}:j]$.
The morphisms $X\to Y$ in $\mathcal{C}(\mathbf{x}:j)$ are pairs $(w,f)$ where $w$ is in $\mathbf{\Sigma}$ and $f:X\otimes j(w)\to Y$ in $\mathcal{C}$.

* If we quotient $\mathbf{Para}(\mathcal{C})$ to get a 1-category by equating connecting components of 1-cells, this is a [[semicartesian monoidal category|semicocartesian]] monoidal category. In fact it is the semicocartesian [[reflection]] of $\mathcal{C}$. (See ([Hermida and Tennent](#HermidaTennent2012), Corollary 2.13).) Similarly, quotienting $\mathbf{CoPara}(\mathcal{C})$
gives the semicartesian reflection of $\mathcal{C}$.

* In particular, if we start from the category $\mathbf{FdIsometry}$ of [[finite-dimensional vector space|finite dimensional]] [[Hilbert spaces]] and [[isometry|isometries]], the 1-cells of $\mathbf{CoPara}(\mathbf{FdIsometry})$ are [[Stinespring factorization theorem|Stinespring dilations]] of [[quantum channels]], and the quotiented 1-category is equivalent to the category of [[quantum channels]] (see ([Huot and Staton 2018](#HuotStaton2018))).

## Related concepts

* [[action of a monoidal category]]
* [[lens (in computer science)]]

## References

The Para construction first appears (though not under that name) in:

* {#HermidaTennent2012} [[Claudio Hermida]], [[Robert Tennent]]. _Monoidal indeterminates and categories of possible worlds_. Theoretical Computer Science vol 430. 2012. Preliminary version in MFPS 2009. [doi:j.tcs.2012.01.001](https://doi.org/10.1016/j.tcs.2012.01.001)

* {#Pavlovic1997} [[Duško Pavlović]], _Categorical logic of names and abstraction in action calculi_. Math. Structures Comput. Sci. 7 (6) (1997) 619–637. [pdf](https://www.kestrel.edu/people/pavlovic/papers/clna-MSCS97.pdf)

* {#Bakovic2008} [[Igor Baković ]], _Bigroupoid 2-torsors_. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics, ([pdf](https://edoc.ub.uni-muenchen.de/9209/1/Bakovic_Igor.pdf))

The terminology "Para construction" first appears in:

* {#FongSpivakTuyeras2019} [[Brendan Fong]], [[David Spivak]], Rémy Tuyéras, _Backprop as Functor: A compositional perspective on supervised learning_, 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 2019, pp. 1-13, 2019. ([arXiv:1711.10455](https://arxiv.org/abs/1711.10455), [LICS'19](https://ieeexplore.ieee.org/abstract/document/8785665))

It has then be developed for the purposes of machine learning in:

* {#Gavranovic2019} [[Bruno Gavranović]], _Compositional Deep Learning_ [[arXiv:1907.08292](https://arxiv.org/abs/1907.08292)]

* {#Cruttwell2021} [[G.S.H. Cruttwell]], [[Bruno Gavranović]], [[Neil Ghani]], Paul Wilson, Fabio Zanasi, _Categorical Foundations of Gradient-Based Learning_, ([arXiv:2103.01931](https://arxiv.org/abs/2103.01931))

and generalized further to [[actegories]] for the purposes of categorical cybernetics in:

* {#Capucci2020} [[Matteo Capucci]], [[Bruno Gavranović]], [[Jules Hedges]], Eigil Fjeldgren Rischel, _Towards Foundations of Categorical Cybernetics_, ([arXiv:2015.06332](https://arxiv.org/pdf/2105.06332.pdf))

The generalization of the Para construction being developed by [[David Jaz Myers]] and [[Matteo Capucci]] is expounded in the following talks:

* {#Myers22} [[David Jaz Myers]], _The Para construction as a distributive law_, talk at the Virtual Double Categories Workshop (2022) [[slides](https://bryceclarke.github.io/virtual-double-categories-workshop/slides/david-jaz-myers.pdf), [video](https://www.youtube.com/watch?v=zB_ifewP8Yk)]

* {#Capucci23} [[Matteo Capucci]], _Constructing triple categories of cybernetic processes_ [[slides](https://zenodo.org/record/8221550/files/main.pdf), [video](https://youtu.be/jYxnUy1vH7Q)]

* [[Matteo Capucci]], *Para Construction as a Wreath Product*, [talk at](CQTS#CapucciJan2024) *[[CQTS]]* (Jan 2024) [slides: [[Capucci-ParaAsWreathProduct.pdf:file]]]

A view on [[quantum channels]] via CoPara:

* {#HuotStaton2018} Mathieu Huot, Sam Staton, _Universal properties in quantum theory_ (QPL 2018) ([pdf](https://www.mathstat.dal.ca/qpl2018/papers/QPL_2018_paper_68.pdf)).

[[!redirects copara construction]]
[[!redirects bipara construction]]


para-complex structure

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## Idea
Intuitively, a **para-complex structure** is to a [[complex structure]] what the [[para-complex numbers]] (see [here](https://ncatlab.org/nlab/show/composition+algebra#para_and_nonunital_composition_algebras) for the moment) are to the [[complex numbers]].

## Definition

### Para-complex structure on vector spaces
 {#ComplexStructureOnAVectorSpace}

\begin{definition}
\label{LinearParaComplexStructure}
Let $V$ be a [[dimension|finite dimensional]] [[real vector space]]. A **para-complex structure** on $V$ is a
nontrivial [[involution]] $I\in \text{End}(V)$ , i.e., $I^2=\text{Id}$ and $I\neq Id$, such that the two eigenspaces $V^{\pm}:= \text{ker}(Id\mp I)$ of $I$ are of the same dimension. A vector space $V$ endowed with a para-complex structure is known as a **para-complex vector space**.

\end{definition}

\begin{definition}
\label{AlmostParaComplexMfld}
An **almost para-complex manifold** is a [[smooth manifold]] $M$ with an endomorphism field $I\in \Gamma(\text{End}TM)$ such that for all $p\in M$, $I_p$ is a para-complex structure on $T_p M$. A splitting
$$
TM = L_{+} \oplus L_{-}
$$
on eigenspaces associated with eigenvalues $\pm$ of $J$ is an **almost para-complex structure** on $M$.
\end{definition}

## Related concepts
* [[para-Hermitian manifold]]

## References

General:

* [[Vicente Cortés]], Christoph Mayer, [[Thomas Mohaupt]], and Frank Saueressig. *Special geometry of Euclidean supersymmetry 1. Vector multiplets.* Journal of High Energy Physics 2004, no. 03 (2004): 028. \[[doi:10.1088/1126-6708/2004/03/028](https://doi.org/10.1088/1126-6708/2004/03/028)\]


para-Hermitian manifold

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#Idea#

The analogue of [[Hermitian manifold]] for [[para-complex structure]].

## Definition

\begin{definition}
An *almost para-Hermitian manifold* $(M,J,\eta)$ is an [[para-complex structure|almost para-complex manifold]] $(M,J)$ endowed with a [[pseudo-Riemannian metric]] $\eta$ of signature $(d,d)$, for $\text{dim}(M)=2d$, such that
$$
\eta(J(X),J(Y))=-\eta(X,Y)
$$

\end{definition}

## Related concepts

* [[Hermitian manifold]]
* [[para-complex structure]]

## References

General:

* Stefan Ivanov, Simeon Zamkovy, *ParaHermitian and paraquaternionic manifolds*, Differential Geometry and its Applications **23** 2 (2005) 205-234 [[doi:10.1016/j.difgeo.2005.06.002](https://doi.org/10.1016/j.difgeo.2005.06.002)]

As [[target spacetimes]] in a context of [[type II geometry]]:

* Vincenzo Emilio Marotta, [[Richard J. Szabo]], *Born Sigma-Models for Para-Hermitian Manifolds and Generalized T-Duality*,  Reviews in Mathematical Physics **33** 09 (2021)  2150031 [[arXiv:1910.09997](https://arxiv.org/abs/1910.09997), [doi:10.1142/S0129055X21500318](https://doi.org/10.1142/S0129055X21500318)]

[[!redirects para-Hermitian manifolds]]

[[!redirects paraHermitian manifold]]
[[!redirects paraHermitian manifolds]]


para-quaternionic structure

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## Idea

The analogue of [[quaternionic structure]] for [[para-quaternions]].

## Definition

### Para-quaternionic structure on vector spaces
 {#ComplexStructureOnAVectorSpace}
\begin{definition}
\label{DefParaQuaternionicStructure}
A **para-quaternionic structure** on a [[vector space]] $V$ is a [[Lie algebra|Lie subalgebra]] $Q\subset \text{End}(V)$ of the [[endomorphism Lie algebra]] which admits a [[linear basis]] $(J_1, J_2, J_3)$ such that $J_3 = J_1 J_2$ and $J^2_{\alpha}= \epsilon_{\alpha} Id$, where $(\epsilon_1,\epsilon_2,\epsilon_3)=(-1,1,1)$.
\end{definition}

### Para-quaternionic Kähler manifold

\begin{definition}
A [[pseudo-Riemannian manifold]] $(M, g)$ of dimension $\geq 5$ endowed with a parallel [[distribution of subspaces|distribution]] $Q_p\subset \text{End}(T_p M)$ of $g$-[[skew-symmetric matrix|skew-symmetric]] para-quaternionic structures is called a **para-quaternionic Kähler manifold**.
\end{definition}

The metric $g$ of a para-quaternionic Kähler manifold has signature $(2n, 2n)$ and is [[Einstein manifold|Einstein]].

## References

General:

* [[Dmitry Vladimirovich Alekseevsky]], and [[Vicente Cortés]]. *The twistor spaces of a para-quaternionic Kähler manifold.* Osaka J. Math. 45(1): 215-251 (March 2008).

* David E. Blair, J. Davidov and O. Muskarov: *Hyperbolic twistor spaces*, Rocky Mountain J. Math. 35 (2005), 1437–1465.

* David E. Blair. *A product twistor space*, Serdica Math. J. 28 (2002), 163–174.


parabola

## Related concepts

* [[conic section]]

  * [[ellipse]], [[hyperbola]]

* [[circle]]

[[!redirects parabolas]]


parabolic geometry

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## Idea

The [[Cartan geometry]] induced by [[parabolic subgroup]] inclusions is called _parabolic geometry_.

>Parabolic geometries are Cartan geometries of type $(G, P)$, where $G$ is a semisimple Lie group and $P \subset G$ is a parabolic subgroup. The corresponding homogeneous spaces $G/P$ are the so–called generalized flag manifolds which are among the most important examples of homogeneous spaces. Under the conditions of regularity and normality, parabolic geometries always are equivalent to underlying structures. ([Cap 05](#Cap05))

Parabolic geometries include [[conformal geometry]], [[projective geometry]], almost quaternionic structures, almost Grassmannian structures, hypersurface type [[CR structures]], systems of 2nd order ODEs, and various bracket-generating distributions. An example of a Cartan geometry that is not parabolic is [[Riemannian geometry]].

A key application of parabolic Cartan geometry is to the construction of curved generalizations of the [[BGG resolution]].

All parabolic geometries admit a fundamental curvature quantity called harmonic curvature $\kappa_H$, which is a complete obstruction
to flatness. The [[Weyl tensor]] is the specific instance of $\kappa_H$ in conformal geometry.

## Examples

* [[conformal geometry]] via [[conformal connections]]

* [[CR geometry]]

## Related concepts

* [[tractor bundle]]

* [[Schubert calculus]]

* [[twistor space]]

## References

General discussion includes

* [[Andreas Čap]], H. Schichl, _Parabolic Geometries and Canonical Cartan Connections_ Hokkaido Math. J. 29 no.3 (2000), 453-505

* {#Cap05} [[Andreas Čap]], _Two constructions with parabolic geometries_, [arXiv:0504389](http://arxiv.org/abs/math/0504389)

* [[Felipe Leitner]], part 1, section 5 of _Applications of Cartan and Tractor Calculus to Conformal and CR-Geometry_, 2007 ([pdf](http://elib.uni-stuttgart.de/opus/volltexte/2009/3922/pdf/HabilKopf.pdf))

* {#CapSlovak09} [[Andreas Čap]], [[Jan Slovák]],  _Parabolic Geometries I -- Background and General Theory_, AMS 2009

Relation to [[twistor spaces]] is discussed in

* {#Cap01} [[Andreas Čap]], _Correspondence spaces and twistor spaces for parabolic geometries_, J. Reine Angew. Math. 582 (2005) 143-172 ([arXiv:math/0102097](http://arxiv.org/abs/math/0102097))

Discussion of [[BGG sequences]] in parabolic geometry includes

* [[David Calderbank]], Tammo Diemer, _Differential invariants and curved Bernstein-Gelfand-Gelfand sequences_, J.Reine Angew.Math. 537 (2001) 67-103 ([arXiv:math/0001158](http://arxiv.org/abs/math/0001158))

for more see the references at _[BGG resolution -- References -- Curved generalization to parabolic Cartan geometries](BGG+resolution#ReferencesCurved)_.

See also

* Wikipedia, _[Parabolic geometry (differential geometry)](http://en.wikipedia.org/wiki/Parabolic_geometry_(differential_geometry%29)_

A relation to [[Courant Lie 2-algebroids]] is discussed in

* {#ArmstrongLu11} Stuart Armstrong, [[Rongmin Lu]], _Courant Algebroids in Parabolic Geometry_ ([arXiv:1112.6425](http://arxiv.org/abs/1112.6425))

* Xu Xiaomeng, _Twisted Courant algebroids and coisotropic Cartan geometries_, Journal of Geometry and Physics Volume 82, August 2014, Pages 124–131 ([arXiv:1206.2282](http://arxiv.org/abs/1206.2282))

See also

* Olivier Biquard, Rafe Mazzeo, _Parabolic geometries as conformal infinities of Einstein metrics_, Archivum Mathematicum, vol. 42 (2006), issue 5, pp. 85-104 ([dml](http://dml.cz/handle/10338.dmlcz/108021), [pdf](http://math.stanford.edu/~mazzeo/Web/Papers/pgciem.pdf))

[[!redirects parabolic geometries]]
[[!redirects Parabolic geometry]]


parabolic subgroup

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## Definition

Given a [[linear algebraic group]] $G$ (i.e. an [[algebraic group|algebraic]] [[subgroup]] of the [[general linear group]] $GL(n,k)$ where $k$ is a [[field]]), a subgroup $P\subset G$ is said to be __parabolic__ if it is closed in [[Zariski topology]] and the [[quotient]] variety $G/P$ is [[projective variety|projective]]. A minimal (with respect to inclusion) parabolic subgroup of a linear algebraic group is called a [[Borel subgroup]]; in fact, given a Borel subgroup $B$, any closed subgroup $P\supset B$ is parabolic.

The [[Cartan geometry]] of parabolic subgroup inclusions is _[[parabolic geometry]]_.

## Related concepts

* [[Schubert calculus]]

* [[parabolic geometry]]

## References

* eom [parabolic subgroup](http://eom.springer.de/p/p071270.htm)

[[!redirects parabolic subgroups]]

[[!redirects standard parabolic subgroup]]
[[!redirects standard parabolic subgroups]]


paracategory

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

---

## Idea

A *paracategory* is a [[category]] where [[composition]] is only partially defined.

The definition is "unbiased" in that it comes with basic (partial) $n$-ary composition operations for all $n$, and unlike in the total case these cannot always be reduced to binary and nullary operations.  A paracategory in which all the composites are generated from binary and nullary ones is sometimes called a *precategory*, but the term *[[precategory]]* is already used in the category theory literature for a category whose objects form a [[type]] or [[infinity-groupoid]] rather than a [[set]].

## Definition

A **paracategory** is a [[quiver]] $C_1 \rightrightarrows C_0$ together with the following structure.  We write $C_n$ for the iterated pullback $C_1 \times_{C_0} \dots\times_{C_0} C_1$, the set of length-$n$ strings of composable arrows.

* [[partial function|Partial functions]] $\circ_n \colon C_n ⇀ C_1$, over $C_0\times C_0$

* $\circ_0 \colon C_0 \to C_1$ is total, so all identity arrows exist.

* $\circ_1\colon C_1 \to C_1$ is the identity.

* If $\circ_n \vec{y}$ is defined, then $\circ_{m+1+k}(\vec{x},\circ_n \vec{y},\vec{z}) = \circ_{m+n+k}(\vec{x},\vec{y},\vec{z})$ (the equality being [[Kleene equality]]).

A **functor** between paracategories is a quiver morphism $f\colon C\to D$ such that if $\circ_n \vec{x}$ is defined, then so is $\circ_n \vec{f(x)}$ and it equals $f(\circ_n \vec{x})$.  A **Kleene functor** is a functor such that if $\circ_n \vec{f(x)}$ is defined, then so is $\circ_n \vec{x}$; this is equivalently a quiver morphism such that $\circ_n \vec{f(x)}=f(\circ_n \vec{x})$ is a [[Kleene equality]].

## Examples

* If $D$ is any category and $C$ is a subclass of arrows in $D$ containing all the identities, then $C$ becomes a paracategory whose objects are the objects of $D$, whose arrows are the arrows in $C$, and where the composite of a string of arrows is defined iff the composite of that string in $D$ happens to lie in $C$.

* [[extranatural transformations]] and [[dinatural transformations]] form paracategories, since they are not always composable.  This is exploited in the definition of [[extraordinary 2-multicategory]].  This example can also be regarded as a case of the previous example, where the ambient category $D$ consists of "unnatural transformations."

* In fact, if $Cat_P$ denotes the category of categories equipped with a subclass of arrows containing the identities, then the functor $Cat_P \to ParCat$ defined above is actually a [[coreflection]], i.e. it has a [[fully faithful functor|fully faithful]] [[left adjoint]].  In particular, any paracategory is isomorphic to one obtained from a class of arrows in some category, and moreover in a universal way.

## Paracategories as generalized multicategories

Paracategories, and more general "partial algebras," can be considered as a special case of [[generalized multicategories]]; see the papers of Hermida.

## References

The definition is due to [[Peter Freyd]] in apparently unpublished work.  It has been studied further in the papers:

* [[Claudio Hermida|Hermida]] and Mateus, "Paracategories I: Internal Paracategories and Saturated Partial Algebras"

* [[Claudio Hermida|Hermida]] and Mateus, "Paracategories II: Adjunctions, fibrations and examples from probabilistic automata theory"

[[!redirects paracategories]]
[[!redirects partial category]]
[[!redirects partial categories]]


paracocyclic object

A __paracocyclic object__ in category $C$ is a [[paracyclic object|paracyclic]] (synonym: $\mathbf{Z}$-cyclic) object in $C^{\mathrm{op}}$.


paracompact Hausdorff space

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#Contents#
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## Definition

A _paracompact Hausdorff topological space_ is a  [[topological space]] that is both a [[Hausdorff topological space]] and a [[paracompact topological space]].

## Properties

* [[paracompact Hausdorff spaces are normal]]

* [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]]

* [[colimits of paracompact Hausdorff spaces]]

## Examples

* [[CW-complexes are paracompact Hausdorff spaces]]

* [[topological manifolds]] are [[locally Euclidean spaces]] which are paracompact Hausdorff (by definition)

## Related entries

* [[compact  Hausdorff space]]

## References

* Topospaces, _[Paracompact Hausdorff space](https://topospaces.subwiki.org/wiki/Paracompact_Hausdorff_space)_

[[!redirects paracompact Hausdorff spaces]]

[[!redirects paracompact Hausdorff topological space]]
[[!redirects paracompact Hausdorff topological spaces]]


paracompact Hausdorff spaces are normal

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#### Topology
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#Contents#
* table of contents
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## Statement

+-- {: .num_defn #ParacompactHausdorffSpacesAreNormal}
###### Proposition
**(Dieudonné's theorem)**

Every [[paracompact Hausdorff space]] is [[normal Hausdorff space|normal]].

In particular [[compact Hausdorff spaces are normal]].

=--

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

Let $(X,\tau)$ be a paracompact Hausdorff space

We first show that it is [[regular topological space|regular]]: To that end,
let $x \in X$ be a point, and let $C \subset X$ be a [[closed subset]] not containing $x$. We need to find disjoint open neighbourhoods $U_x \supset \{x\}$ and $U_C \supset C$.

First of all, by the Hausdorff property there exists for each $c \in C$ disjoint open neighbourhods $U_{x,c} \supset \{x\}$ and $U_c \supset \{c\}$. As $c$ ranges, the latter clearly form an open cover $\{U_c \subset X\}_{c \in C}$ of $C$, and so the union

$$
  \{U_c \subset X\}_{c \in C} \,\cup\, X \backslash C
$$

is an open cover of $X$. By paracompactness of $(X,\tau)$, there exists a locally finite refinement, and by [this lemma](locally+finite+cover#LocallyFiniteRefinementImpliesLocallyFiniteCoverWithOriginalIndexSet) we may assume its elements to share the original index set and be contained in the original elements of the same index. Hence

$$
  \{V_c \subset U_c \subset  X\}_{c \in C}
$$

is a locally finite collection of subsets, such that

$$
  U_C \coloneqq \underset{c \in C}{\cup} V_c
$$

is an open neighbourhood of $C$.

Now by definition of local finiteness there exists an open neighbourhood $W_x \supset \{x\}$ and a finite subset $K \subset C$ such that

$$
  \underset{c \in C \backslash K}{\forall}( W_x \cap V_c = \emptyset )
  \,.
$$

Consider then

$$
  U_x
   \;\coloneqq\;
  W_x
    \cap
  \left(
    \underset{k \in K}{\cap}
     \left(
       U_{x,k}
     \right)
  \right)
  \,.
$$

which is an open neighbourhood of $x$, by the finiteness of $K$.

It thus only remains to see that

$$
  U_x \cap U_C = \emptyset
  \,.
$$

But this holds because the only $V_{c}$ that intersect $W_x$ are the $V_{k} \subset U_{k}$ for $k \in K$ and each of these is by construction disjoint from $U_{x,k}$ and hence from $U_x$.

This establishes that $(X,\tau)$ is regular. Now we prove that it is normal. For this we use the same approach as before:

Let $C,D \subset X$ be two disjoint closed subsets. We need to produce disjoint open neighbourhoods for these.

By the previous statement of regularity, we may find for each $c \in C$ disjoint open neighbourhoods $U_c \supset \{c\}$ and $U_{D,c} \supset D$. Hence the union

$$
  \left\{
    U_c \subset X
  \right\}_{c \in C}
  \cup
  X \backslash C
$$

is an open cover of $X$, and thus by paracompactness has a locally finite refinement, whose elements we may, again by [this lemma](locally+finite+cover#LocallyFiniteRefinementImpliesLocallyFiniteCoverWithOriginalIndexSet), assume to have the same index set as before and be contained in the previous elements with the same index. Hence we obtain a locally finite collection of subsets

$$
  \{ V_c \subset U_c \subset X \}_{c \in C}
$$

such that

$$
  U_{C}
   \coloneqq
  \underset{c \in C}{\cup} V_c
$$

is an open neighbourhood of $C$.

It is now sufficient to see that every point $d \in D$ has an open neighbourhood $U_d$ not intersecting $U_C$, for then

$$
  U_D \coloneqq \underset{d \in D}{\cup} U_d
$$

is the required open neighbourhood of $D$ not intersecting $U_C$.

Now by local finiteness of $\{V_c \subset X\}_{c \in C}$, every $d \in D$ has an open neighbourhood $W_d$ such that there is a finite set $K_d \subset C$ so that

$$
  \underset{c \in C \backslash K_d}{\forall}
  \left(
     V_c \cap W_d = \emptyset
  \right)
  \,.
$$

Accordingly the intersection

$$
  U_d
    \coloneqq
  W_d
   \cap
  \left(
    \underset{c \in K_d \subset C}{\cap} U_{D,c}
  \right)
$$

is still open and disjoint from the remaining $V_k$, hence disjoint from all of $U_C$.

=--

## Related statement

* [[metric spaces are paracompact]]

* [[locally compact and sigma-compact spaces are paracompact]]

* [[second-countable regular spaces are paracompact]]

* [[regular Lindelöf spaces are paracompact]]

* [[Hausdorff spaces are sober]]

* [[compact subspaces of Hausdorff spaces are closed]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

* [[CW-complexes are paracompact Hausdorff spaces]]

* [[Hausdorff spaces are sober]]

* [[maps from compact spaces to Hausdorff spaces are closed and proper]]

* [[continuous images of compact spaces are compact]]

* [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]]

[[!redirects Dieudonné's theorem]]


paracompact Hausdorff spaces equivalently admit subordinate partitions of unity

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#### Topology
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* table of contents
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## Statement

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

Assuming the [[axiom of choice]] then:

Let $(X,\tau)$ be a topological space that is $T_1$ (meaning [[singletons]] are [[closed subsets]]; see [[separation property]]). Then the following are equivalent:

1. $(X,\tau)$ is [[paracompact topological space|paracompact]] and [[Hausdorff space|Hausdorff]].

1. Every [[open cover]] of $(X,\tau)$ admits a subordinate [[partition of unity]] (def. \ref{PartitionOfUnity}).

=--

We give the **proof** [below](#ParacompactHausdorffEquivalentToexistenceOfParititionsOfUnityProof).

## Background

+-- {: .num_defn #LocallyFiniteCover}
###### Definition
**([[locally finite cover]])**

Let $(X,\tau)$ be a [[topological space]].

An [[open cover]] $\{U_i \subset X\}_{i \in I}$ of $X$ is called  _locally finite_ if for all point $x \in X$, there exists a  [[neighbourhood]] $U_x \supset \{x\}$ such that it [[intersection|intersects]] only finitely many elements of the cover, hence such that  $U_x \cap U_i \neq \emptyset$ for only a [[finite number]] of $i \in I$.

=--

+-- {: .num_defn #RefinementOfOpenCovers}
###### Definition
**([[refinement]] of [[open covers]])

Let $(X,\tau)$ be a [[topological space]], and let $\{U_i \subset X\}_{i \in I}$ be a [[open cover]].

Then a _[[refinement]]_ of this open cover is a set of open subsets $\{V_j \subset X\}_{j \in J}$ which is still an [[open cover]] in itself and such that for each $j \in J$ there exists an $i \in I$ with $V_j \subset U_i$.

=--

+-- {: .num_defn #ParacompactSpace}
###### Definition
**([[paracompact topological space]])**

A [[topological space]] $(X,\tau)$ is called _[[paracompact topological space|paracompact]]_ if every [[open cover]] of $X$ has a [[refinement]] (def. \ref{RefinementOfOpenCovers}) by a [[locally finite open cover]] (def. \ref{LocallyFiniteCover}).

=--

+-- {: .num_defn #PartitionOfUnity}
###### Definition
**([[partition of unity]])**

Let $(X,\tau)$ be a [[topological space]], and let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]]. Then a _[[partition of unity]] subordinate to the cover_ is

* a [[set]] $\{f_i\}_{i \in I}$ of [[continuous functions]]

  $$
    f_i \;\colon\; X \longrightarrow [0,1]
  $$

  (where $[0,1] \subset \mathbb{R}$ is equipped with the [[subspace topology]] of the [[real numbers]] $\mathbb{R}$  regarded as the 1-dimensional [[Euclidean space]] equipped with its [[metric topology]]);

such that with

$$
  Supp(f_i) \coloneqq Cl\left( f_i^{-1}( (0,1] ) \right)
$$

denoting the [[support]] of $f_i$ (the [[topological closure]] of the subset of points on which it does not vanish) then

1. $\underset{i \in I}{\forall} \left( Supp(f_i) \subset U_i \right)$;

1. $\left\{ Supp(f_i) \subset X \right\}_{i \in I}$  is a [[locally finite cover]] (def. \ref{LocallyFiniteCover});

1. $\underset{x \in X}{\forall} \left(  \underset{i \in I}{\sum} f_i(x) = 1 \right)$.

=--

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

Due to the second clause in def. \ref{PartitionOfUnity}, the [[sum]] in the third clause involves only a [[finite number]] of elements not equal to zero, and therefore is well defined.

=--

## Proof

The non-trivial direction to be shown is:

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

If $(X,\tau)$ is a [[paracompact topological space]], then for every [[open cover]] $\{U_i \subset X\}_{i \in I}$ there is a subordinate [[partition of unity]] (def. \ref{PartitionOfUnity}).

=--

We give the proof [below](#OpenCoverOfParacompactHausdorffSpaceAdmitsPartitionOfUnityProof).

The following says that if there exists a [[locally finite cover|locally finite]] [[refinement]] of a cover, then in fact there exists one with the same index set as the original cover.

+-- {: .num_lemma #LocallyFiniteRefinementInducesLocallyFiniteWithSameIndexSet}
###### Lemma

Let $(X,\tau)$ be a [[topological space]], let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]], and let $(\phi \colon J \to I), \{V_j \subset X\}_{j \in J})$, be a [[refinement]] to a [[locally finite cover]].

Then  $\left\{ W_i \subset X \right\}_{i \in I}$ with

$$
  W_i
    \;\coloneqq\;
  \left\{
    \underset{j \in \phi^{-1}(\{i\})}{\cup} V_j
  \right\}
$$

is still a [[refinement]] of $\{U_i \subset X\}_{i \in I}$ to a [[locally finite cover]].

=--

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

It is clear by construction that $W_i \subset U_i$, hence that we have a [[refinement]]. We need to show local finiteness.

Hence consider $x \in X$. By the assumption that $\{V_j \subset X\}_{j \in J}$ is locally finite, it follows that there exists an [[open neighbourhood]] $U_x \supset \{x\}$ and a [[finite set|finite]] [[subset]] $K \subset J$ such that

$$
  \underset{j \in J\backslash K}{\forall} \left( U_x \cap V_j = \emptyset \right)
  \,.
$$

Hence by construction

$$
  \underset{i \in I\backslash \phi(K)}{\forall} \left( U_x \cap W_i = \emptyset \right)
  \,.
$$

Since the [[image]] $\phi(K) \subset I$ is still a [[finite set]], this shows that $\{W_i \subset X\}_{i \in I}$ is locally finite.

=--

+-- {: .num_lemma #PatchesOfOpenCoverOfNormalSpaceMayBeMadeSmallerSoThatTheirClosuresAreContained}
###### Lemma
**([[shrinking lemma]])**

Let $X$ be a [[topological space]] which is [[normal topological space|normal]] and let $\{U_i \subset X\}_{i \in I}$ be a [[locally finite cover|locally finite]] [[open cover]].

Assuming the [[axiom of choice]] then:

There exists another open cover $\{V_i \subset X\}_{i \in I}$ such that the [[topological closure]] $Cl(V_i)$ of its elements is contained in the original patches:

$$
  \underset{i \in I}{\forall}
  \left(
     V_i \subset Cl(V_i) \subset U_i
  \right)
  \,.
$$

=--

+-- {: .proof #OpenCoverOfParacompactHausdorffSpaceAdmitsPartitionOfUnityProof}
###### Proof
of prop. \ref{OpenCoverOfParacompactHausdorffSpaceAdmitsPartitionOfUnity}

By paracompactness of $X$, for every open cover there exists a locally finite refinement $\{U_i \subset X\}_{i \in I}$, and by lemma \ref{LocallyFiniteRefinementInducesLocallyFiniteWithSameIndexSet} we may assume
that this has same index set. It is now sufficient to show that this locally finite cover $\{U_i \subset X\}_{i \in I}$ admits a subordinate partition of unity, since this will then also be  subordinate to the original cover.

Since [[paracompact Hausdorff spaces are normal]] we may
apply lemma \ref{PatchesOfOpenCoverOfNormalSpaceMayBeMadeSmallerSoThatTheirClosuresAreContained} to the given locally finite open cover $\{U_i \subset X\}_i$, to obtain a smaller locally finite open cover $\{V_i \subset X\}_{i \in I}$, and then apply the lemma once more to that result to get a yet smaller open cover  $\{W_i \subset X\}_{i \in I}$, so that now

$$
  \underset{i \in I}{\forall}
  \left(
     W_i \subset Cl(W_i) \subset V_i \subset Cl(V_i) \subset U_i
  \right)
  \,.
$$

It follows that for each $i \in I$ we have two disjoint [[closed subsets]], namely the [[topological closure]] $Cl(W_i)$ and the [[complement]] $X \backslash V_i$

$$
  Cl(W_i) \cap X\backslash V_i = \emptyset
  \,.
$$

Now since [[paracompact Hausdorff spaces are normal]], [[Urysohn's lemma]] says that there exist [[continuous functions]]

$$
  h_i \;\colon\; X \longrightarrow [0,1]
$$

with the property that

$$
  h_i( Cl(W_i) ) = \{1\}
  \,,
  \phantom{AAA}
  h_i( X \backslash V_i ) = \{0\}
  \,.
$$

This means in particular that $h_i^{-1}((0,1]) \subset V_i$ and hence that

$$
  Supp(h_i) =  Cl(h_i^{-1}((0,1])) \subset Cl(V_i) \subset U_i
  \,.
$$

By construction, the set of function $\{h_i\}_{i \in I}$ already satisfies conditions 1) and 2) of the three conditions on a partition of unity subordinate to $\{U_i \subset X\}_{i \in I}$ from def. \ref{PartitionOfUnity}.
It just remains to normalize these functions so that they indeed sum to unity. To that end, consider the continuous function

$$
  h \;\colon\; X \longrightarrow [0,1]
$$

defined on $x \in X$ by

$$
  h(x) \coloneqq \underset{i \in I}{\sum} h_i(x)
  \,.
$$

Notice that the [[sum]] on the right has only a [[finite number]] of non-zero summands, due to the local finiteness of the cover, so that this is well-defined.

Moreover, notice that

$$
  \underset{x \in X}{\forall} \left(  h(x) \neq 0  \right)
$$

because $\{Cl(W_i) \subset X\}_{i \in I}$ is a cover so that there is $i_x \in I$ with $x \in Cl(W_{i_x})$, and since $h_i(Cl(W_{i_x})) = \{1\}$, by the above.

Hence it makes sense to define

$$
  f_i \;\coloneqq\;  h_i/h
  \,.
$$

This is now manifestly such that $\underset{i \in I}{\sum} f_i = 1$, and so

$$
  \left\{
     f_i
  \right\}_{i \in I}
$$

is a partition of unity as required.

=--

+-- {: .proof #ParacompactHausdorffEquivalentToexistenceOfParititionsOfUnityProof}
###### Proof
of theorem \ref{ParacompactHausdorffEquivalentToexistenceOfParititionsOfUnity}

In one direction, assume that every open cover $\{U_i \subset X\}_{i \in I}$ admits a subordinate partition of unity $\{f_i\}_{i \in I}$. Then by definition (def. \ref{PartitionOfUnity}) $\{ Int(Supp(f)_i)  \subset X\}_{i \in I}$ is a locally finite open cover refining the original one, so $X$ is paracompact. Moreover, since $X$ is $T_1$, for any distinct points $x, y \in X$ there is a cover consisting of the open sets $\neg \{x\}, \neg \{y\}$ (the [[complements]] of the singletons). If $f_x, f_y$ is a subordinate partition of unity (with $f_x$ supported on $\neg \{x\}$, and similarly for $f_y$), then $f_x(x) = 0$ and $f_y(y) = 0$. Then from $f_x + f_y \equiv 1$ we derive $f_x(y) = 1$, whereupon $\{z \in X: f_x(z) \lt 1/2\}$ and $\{z \in X: f_x(z) \gt 1/2\}$ are disjoint open neighborhoods containing $x$ and $y$ respectively. Hence $X$ is Hausdorff.

The other direction is the statement of prop. \ref{OpenCoverOfParacompactHausdorffSpaceAdmitsPartitionOfUnity}.

=--

## Related statements

* [[paracompact Hausdorff spaces are normal]]

* [[compact spaces equivalently have converging subnet of every net]]

* [[Hausdorff implies sober]]

* [[compact subspaces of Hausdorff spaces are closed]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

* [[a CW-complex is a Hausdorff space]]

* [[maps from compact spaces to Hausdorff spaces are closed and proper]]

* [[continuous images of compact spaces are compact]]

* [[colimits of paracompact Hausdorff spaces]]


paracompact locale

### Definition

A [[locale]] is __paracompact__ if it is [[regular]] and every open cover has a [[locally finite]] refinement.

### Properties

Paracompact locales are very closely related to [[fully normal locales]].
In fact, for [[regular locales]] these two properties are equivalent.

Any [[metrizable locale]] is paracompact.

Any [[Lindelöf locale]] is paracompact.

A [[locale]] is paracompact if and only if it admits a [[complete uniformity]].

The [[full subcategory]] of paracompact locales is a [[reflective subcategory]]
of the category of [[completely regular locales]] as well as the category of all [[locales]].

In particular, the inclusion functor from paracompact locales to locales
preserves [[small limits]], so in particular, products of paracompact locales are paracompact.

This last property clearly distinguishes paracompact locales from paracompact spaces,
since products of paracompact spaces need not be paracompact.

### Related concepts

* [[fully normal locale]]

* [[paracompact space]]

* [[fully normal space]]


paracompact manifold

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#### Manifolds and cobordisms
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[[!include manifolds and cobordisms - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Definition

A **paracompact manifold** is a [[topological space]] that is

1. a [[paracompact space]];

1. a [[manifold]].

## Properties {#Properties}

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

Every paracompact [[smooth manifold]] admits a [[geodesic flow|complete]] [[Riemannian metric]] with bounded absolute sectional curvature and positive [[injectivity radius]].

=--

This is shown in ([Greene](#Greene)).

## References

* {#Greene} R. Greene, _Complete metrics of bounded curvature on noncompact manifolds_  Archiv der Mathematik Volume 31, Number 1

[[!redirects paracompact manifolds]]


paracompact spaces satisfying the countable chain condition are Lindelöf

## Statement

Recall that a [[topological space]] $X$ is [[paracompact space|paracompact]] if every [[open cover]] has a [[refinement]] by a [[locally finite open cover]]. Further a space is called [[Lindelöf topological space|Lindelöf]] if every [[open cover]] has a [[countable cover|countable]] sub-cover.

\begin{theorem}
Assuming the [[axiom of choice]]:

Every paracompact space $X$ satisfying the countable chain condition is Lindelöf.
\end{theorem}

\begin{proof}
The proof goes by contradiction: Assume there is an open cover $\{U_i\}_{i\in I}$ with no countable subcover. Let $\{U_j\}_{j\in J}$ be a locally finite refinement, which again must not be countable. This is to say that each point possesses an open neighborhood so small that it is only an element of finitely many $U_j$'s. Inductively (using [[Zorn's lemma]]) construct a maximal system $\{V_\lambda\}_{\lambda \in \Lambda}$ of pairwise disjoint opens being contained in at most finitely many $U_j$'s. Due to maximality $\bigcup_{\lambda\in\Lambda} V_\lambda$ is dense. This fact implies by the countable chain condition that $\Lambda$ is countable. Moreover it implies that each $U_j$ intersects at least one $V_\lambda$. But this is to say that there are at most countably many $U_j$'s. This is a contradiction.
\end{proof}

## Related statements and properties

[[!include topology - global countability axioms]]