proton

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## Idea

A _proton_ is a [[bound state]] of two [[up quarks]] and a [[down quark]], hence a [[baryon]].

[[!include flavors of fundamental particles -- table]]

The [[bound states]] of [[protons]] and [[neutrons]] (jointly: [[nucleons]]) in turn are [[nucleus (physics)|nuclei]], the cores of [[atoms]]. The proton alone is the nucleus of the [[hydrogen atom]].

In the [[standard model of particle physics]] the proton is stable, and [[experiment]] highly constrains possible rare [[proton decay]] (see there). But some [[model (physics)|models]] beyond the standard model (notably many, but not all, [[GUT]] [[model (physics)|models]]) do imply rare [[proton decay]].

## Properties

### Mass

A [[proton]] has [[rest mass]] about a [[GeV]]:

$$
  m_{proton} \simeq 0.938 GeV
  \,.
$$

### Spin

The contribution of the constituents ([[quarks]], [[gluons]]) to the total [[spin]] of the proton turns out to be subtle ("[[proton spin crisis]]") [Thomas 08](#Thomas08).

## Related entries

* [[neutron]]

* [[Lambda baryon]]

* [[proton spin crisis]]

* [[proton radius problem]]

## References

See also

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

On the [[spin]] of the proton (cf. *[[proton spin crisis]]*):

* {#Thomas08} Anthony Thomas, _The spin of the proton_, Progress in Particle and Nuclear Physics Volume 61, Issue 1, July 2008, Pages 219–228 Quarks in Hadrons and Nuclei — 29th Course International Workshop on Nuclear Physics ([arXiv:0805.4437](https://arxiv.org/abs/0805.4437))

* Anthony Thomas, _The resolution of the proton spin crisis_, 2008 ([pdf](http://www4.rcf.bnl.gov/~nouicer/BNL_seminars/AWT_2.pdf))

On the [[mass]] of the proton and the [[mass gap problem]] ($\sim$[[confinement]]-problem) of [[QCD]]:

* {#Roberts21} [[Craig Roberts]], *On Mass and Matter*, AAPPS Bulletin **31** 6 (2021) [[arXiv:2101.08340](https://arxiv.org/abs/2101.08340), [doi:10.1007/s43673-021-00005-4](https://doi.org/10.1007/s43673-021-00005-4)]

On the [[pressure]] of the proton, via [[lattice QCD]]:

* [[Phiala Shanahan]], [[William Detmold]], *Pressure Distribution and Shear Forces inside the Proton*, Physical Review Letters **122** 7 (2019) [[doi:10.1103/PhysRevLett.122.072003](https://doi.org/10.1103/PhysRevLett.122.072003), [arXiv:1810.07589](https://arxiv.org/abs/1810.07589)]

On proton stability:

* [[Juven Wang]], Zheyan Wan, Yi-Zhuang You, *Proton Stability: From the Standard Model to Beyond Grand Unification*, Phys. Rev. D **106** (2022) 025016 [[arXiv:2204.08393](https://arxiv.org/abs/2204.08393), [doi:10.1103/PhysRevD.106.025016](https://doi.org/10.1103/PhysRevD.106.025016)]

[[!redirects protons]]


proton decay

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## Idea

In the present [[standard model of particle physics]] the [[proton]] is a stable [[bound state]] (of [[quarks]]). But in some hypothetical extensions of the standard model, notably in many [[GUT]] [[model (physics)|models]], the proton would be unstable, albeit with an extremely long decay time, and hence could decay (e.g. [Kolešová-Malinský 14](#KolesovaMalinski14)).

[[experiment|Experimental]] searches for rare proton decay show that the minimum mean lifetime of the proton is at least on the order of $10^{31}$ [[years]] (e.g. [Super-Kamiokande 15](#SuperKamiokande15)), probably higher.

Since the age of the [[observable universe]] is about $1.3 \cdot 10^{10}$ [[years]], this translates to saying that on average at most one proton in 10 kilotons of protons (essentially 10 kilotons of [[hydrogen atoms]]) decays per [[year]].

More recently it has been claimed that [[GUT]] [[model (physics)|models]] may entirely avoid proton decay after all ([Mütter-Ratz-Vaudrevange 16](#MuetterRatzVaudrevange16), [Fornal-Grinstein 17](#FornalGrinstein17)), in particular in [[gauge-Higgs grand unification]] such as [[Spin(11)]]- ("[[SO(11)]]"-) and [[Spin(12)]]- ("[[SO(12)]]"-) models: ([Hosotani-Yamatsu 15](#HosotaniYamatsu15), [Furui-Hosotani-Yamatsu 16, Sec. 2.6](#FuruiHosotaniYamatsu16) [Hosotani 17, Section 6](#Hosotani17))

## Related concepts

* [[doublet-triplet splitting problem]]

* [[GUT]]

## References

### General

* [[Jogesh Pati]], _Discovery of Proton Decay: A Must for Theory, a Challenge for Experiment_ ([arXiv:hep-ph/0005095](http://arxiv.org/abs/hep-ph/0005095))

* {#KolesovaMalinski14} Helena Kolešová, [[Michal Malinský]], _Proton lifetime in the minimal $SO(10)$ GUT and its implications for the LHC_, Phys. Rev. D 90, 115001 (2014) ([arXiv:1409.4961](http://arxiv.org/abs/1409.4961))

See also

* Wikipedia, _[Proton decay](https://en.wikipedia.org/wiki/Proton_decay)_

### Experimental constraints

Experimental results include

* {#SuperKamiokande15} [[Super-Kamiokande]], _Search for Nucleon and Dinucleon Decays with an Invisible Particle and a Charged Lepton in the Final State at the Super-Kamiokande Experiment_ ([arXiv:1508.05530](https://arxiv.org/abs/1508.05530))

### Proton stability in GUTs

Claims that proton decay may be entirely avoided in [[GUTs]] after all are discussed in:

* {#MuetterRatzVaudrevange16} Andreas Mütter, Michael Ratz, Patrick K.S. Vaudrevange, _Grand Unification without Proton Decay_ ([arXiv:1606.02303](https://arxiv.org/abs/1606.02303))

  (claims that many [[string theory]] and [[supergravity]] models have this property)

* {#FornalGrinstein17} Bartosz Fornal, Benjamin Grinstein, _$SU(5)$ Unification without Proton Decay_, Phys. Rev. Lett. 119, 241801 (2017) ([arXiv:1706.08535](https://arxiv.org/abs/1706.08535))

and in particular in [[gauge-Higgs grand unification]] (such as [[Spin(11)]]- ("[[SO(11)]]"-) and [[Spin(12)]]- ("[[SO(12)]]"-) models):

* {#HosotaniYamatsu15} Yutaka Hosotani, Naoki Yamatsu, _Gauge–Higgs grand unification_, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 11, November 2015 ([arXiv:1504.03817](https://arxiv.org/abs/1504.03817), [doi:10.1093/ptep/ptw116](https://doi.org/10.1093/ptep/ptw116))

* {#FuruiHosotaniYamatsu16} Atsushi Furui, Yutaka Hosotani, Naoki Yamatsu, _Toward Realistic Gauge-Higgs Grand Unification_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 9, September 2016, 093B01 ([arXiv:1606.07222](https://arxiv.org/abs/1606.07222))

* {#Hosotoni16} Yutaka Hosotani, _Gauge-Higgs EW and Grand Unification_, International Journal of Modern Physics AVol. 31, No. 20n21, 1630031 (2016)  ([arXiv:1606.08108](https://arxiv.org/abs/1606.08108))

Discussion in  [[SemiSpin(32)]]-[[heterotic string|heterotic]] [[string phenomenology]]:

* Saul Ramos-Sanchez, Section 5.4 of _Towards Low Energy Physics from the Heterotic String_,   Fortsch. Phys. 10:907-1036, 2009 ([arXiv:0812.3560](https://arxiv.org/abs/0812.3560))

See also

* Yuta Hamada, Masahiro Ibe, Yu Muramatsu, Kin-ya Oda, Norimi Yokozaki, _Proton Decay and Axion Dark Matter in $SO(10)$ Grand Unification via Minimal Left-Right Symmetry_ ([arXiv:2001.05235](https://arxiv.org/abs/2001.05235))


proton radius problem

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## Idea

What is called the *ptoton radius problem* or *proton radius puzzle* is an apparent disagreement of computations in [[QCD]] and its [[effective field theories]] with [[experiment]], akin to the [[proton spin crisis]] (see there for more), but concerning the [[proton]]'s [[radius]].

## References

* Gil Paz, *The Proton Radius Puzzle* ([arXiv:1909.08108](https://arxiv.org/abs/1909.08108))

* Clara Peset, Antonio Pineda, Oleksandr Tomalak, *The proton radius (puzzle?) and its relatives* ([arXiv:2106.00695](https://arxiv.org/abs/2106.00695))

See also:

* Wikipedia, *[Proton radius puzzle](https://en.m.wikipedia.org/wiki/Proton_radius_puzzle)*

[[!redirects proton radius puzzle]]


proton spin crisis

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## Idea

In the [[standard model of particle physics]], the [[hadrons]], such as the [[proton]], are supposedly [[bound states]] of three [[constituent quark|constituent]] [[quarks]], the latter being the fundamental objects in [[quantum chromodynamics]]. In fact, quarks are thought to be _[[confinement|confined]]_ to such colour-charge-neutral [[bound states]]. However, at the relevant [[energy]]-scale [[QCD]] is "strongly coupled" (the [[coupling constant]] is large) so that established methods of [[perturbative quantum field theory]] do not apply (possibly standard pQFT has to be supplied with an [[interacting vacuum state]], or more general methods of [[non-perturbative quantum field theory]] are needed, all of which remains largely open). As a result, the expected _[[confinement]]_ of [[quarks]] to [[hadrons]] is theoretically an open problem, known in [[mathematical physics]] essentially as the _[[mass gap problem]]_.

Therefore, together with the very existence of [[hadrons]], also their basic properties, such as [[mass]] and [[spin]] do not currently have a derivation from first principles in [[QCD]]. There exists qualitative understanding and there exists computer simulation in [[lattice QCD]]. When [[experiment]] showed that also these heuristics drastically failed to provide an understanding of the [[spin]] of the [[proton]] in terms of the spin of its constitutents [[quarks]] and of the [[gluons]] binding these quarks together, the situation was called the _proton spin crisis_ or _proton spin puzzle_.

## Related issues

* [[proton radius problem]]

## References

* {#Thomas08} Anthony Thomas, _The spin of the proton_, Progress in Particle and Nuclear Physics Volume 61, Issue 1, July 2008, Pages 219–228 Quarks in Hadrons and Nuclei — 29th Course International Workshop on Nuclear Physics ([arXiv:0805.4437](https://arxiv.org/abs/0805.4437))

* Anthony Thomas, _The resolution of the proton spin crisis_, 2008 ([pdf](http://www4.rcf.bnl.gov/~nouicer/BNL_seminars/AWT_2.pdf))

* Xiangdong Ji, Feng Yuan, Yong Zhao, _Proton spin after 30 years: what we know and what we don't?_ ([arXiv:2009.01291](https://arxiv.org/abs/2009.01291))

Apparent resolution by [[lattice QCD]]-computations:

* C. Alexandrou et al., *Complete flavor decomposition of the spin and momentum fraction of the proton using lattice QCD simulations at physical pion mass*, Phys. Rev. D 101, 094513 ([arXiv:2003.08486](https://arxiv.org/abs/2003.08486), [doi:10.1103/PhysRevD.101.094513](https://doi.org/10.1103/PhysRevD.101.094513))

See also

* Wikipedia, _[Proton spin crisis](https://en.wikipedia.org/wiki/Proton_spin_crisis)_


protoring

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## Definition ##

A **protoring** is a [[commutative monoid]] $(M, 0, +)$ with a [[strict total order]] $\lt$ such that

* $a \lt b$ if and only if $a + c \lt b + c$.

* The commutative subsemigroup of strictly positive elements $M_+ \coloneqq \{x \in M \vert 0 \lt x\}$ form a [[semiring]] $(M, +, 1, \cdot)$.

* $a \lt b$ if and only if $a \cdot c \lt b \cdot c$ and $c \cdot a \lt c \cdot b$.

## Examples ##

* Every [[ordered integral domain]] is a protoring.

## See also ##

* [[ordered group]]

* [[ordered ring]]

* [[ordered integral domain]]

* [[archimedean protoring]]

* [[difference protoring]]

## References

* Davorin Lešnik, Synthetic Topology and Constructive Metric Spaces, ([arxiv:2104.10399](https://arxiv.org/abs/2104.10399))

[[!redirects protorings]]


provability logic

[[!redirects Provability Logic]]
[[!redirects Gödel-Löb logic]]
[[!redirects Goedel-Loeb logic]]
[[!redirects provability logics]]
[[!redirects Provability logics]]

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##Idea

**Provability logic** studies the properties of a formal provability predicate  $\Box F$ interpreted as 'F is provable'. This gives rise to [[modal logics]] kindred to **K4** and its variations.

For the best known of these systems, the **Gödel-Löb logic** of provability $GL$, topological interpretations have been pioneered by H. Simmons and L. Esakia in the 1970s.

## GL in Hilbert-style

The language of $GL$ has propositional variables $p_0,p_1,\dots$ and logical connectives $\to$, $\bot$, $\top$ and one unary modality $\Box$. Besides all tautologies of classical propositional logic, GL has the following axiom schemes:

* $\Box (\varphi\to\psi) \to (\Box\varphi\to\Box\psi)$ . (Normality)

* $\Box (\Box \varphi\to \varphi)\to \Box\varphi $ . ([[Löb's axiom]])

with inference rules

* $\array{
\arrayopts{\rowlines{solid}}
\varphi ,\quad\varphi\to\psi\\
\psi
}$  (Modus ponens)

*  $\array{
\arrayopts{\rowlines{solid}}
\varphi\\
\Box\varphi
}$  (Necessitation)

$GL$ proves the _transitivity rule_ $\Box\varphi\to\Box\Box\varphi$ but not the _reflexivity rule_ $\Box\varphi\to\varphi$ whence it is an extension of $K4$ but not of $S4$.

##Links

* [Stanford Encyclopedia of Philosophy entry](http://plato.stanford.edu/entries/logic-provability/)

##Related entries

* [[scattered topos]]

* [[Magari algebra]]

* [[epistemic logic]]

* [[Brouwer-Heyting-Kolmogorov interpretation]]

* [[Gödel's incompleteness theorem]]

* [[arithmetic pretopos]]

##References

* S. Artemov, L. Beklemishev, _Provability logic_ , pp.229-403 in Gabbay, Guenthner (eds.), _Handbook of Philosophical Logic vol. 13_ , Kluwer Dordrecht 2004$^2$.  ([draft](http://sartemov.ws.gc.cuny.edu/files/2012/10/Artemov-Beklemishev.-Provability-logic.pdf))

* L. Beklemishev, D. Gabelaia, _Topological interpretations of provability logic_ , arXiv:1210.7317 (2012).  ([abstract](http://arxiv.org/abs/1210.7317))

* R. M. Solovay, _Provability interpretations of modal logic_ , Israel J. Math. **28** (1976) pp.33–71.


proxi-Banach ring

[[!redirects pseudo-Banach ring]]
[[!redirects Pseudo-Banach ring]]
[[!redirects pseudo-Banach rings]]
[[!redirects proxi-Banach ring]]
[[!redirects proxi-Banach rings]]
[[!redirects proxi-normed ring]]
[[!redirects proxi-normed rings]]

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## Idea and motivations

The notion of proxi-Banach ring is a generalization of the notion of [[Banach ring]], essentially obtained by replacing the usual triangular inequality $|a+b|\leq |a|+|b|$ that appears in the theory of [[metric spaces]] by the more natural (in a categorical sense) weak triangular inequality
$|a+b|\leq C\cdot\max(|a|,|b|)$
that appears in the theory of [[proxi-metric spaces]].

Their use allows the definition of an $\mathbb{R}_+^*$-action on various categories of [[global analytic spaces]].

They are also the natural objects that appear as $\pi_0$ of spectral rings in
[[spectral global analytic geometry]].

## Definition

A pseudo-Banach ring is a ring object in the rig category $(\mathbb{R}_{+\leq}^\mathrm{Sets},\oplus_\infty,\otimes_m)$
of $\mathbb{R}_+$-graded sets (with bounded maps between them).
This means a set $R$ with a norm
$|\cdot|_R:R\to \mathbb{R}_+$ such that there exists $C$, $D$ and $E$, $F$ with

* $|a+b|\leq C\cdot \max(|a|,|b|)$,

* $|ab|\leq D\cdot|a|\cdot|b|$,

* $|-a|\leq E|a|$,

* $|0|=0$,

* $|1|\leq F\cdot 1$ (one often supposes additionally that $|1|=1$).

## The free module

If $X$ is an $\mathbb{R}_+$-graded set, and $R$ is a pseudo-Banach ring, one may define an associated pseudo-normed free module $R^{(X)}$ by putting on
the usual free module the following convenient grading (not given by the usual $\ell^1$ grading): if $\sum a_x\{x\}$ is an element with support $supp(a)$, one may write it $\sum a_i\{x_i\}$, and parenthesize it in binary terms. In the case of a support of cardinal three, we may write for example
$$P(a_1\{x_1\}+a_2\{x_2\}+a_3\{x_3\})=((a_1\{x_1\}+a_2\{x_2\})+a_3\{x_3\}).$$
Each parenthesis contains only two terms, and there is a finite set of choices for the position of parenthesis, denoted $\mathcal{P}a(supp(a))$. We then define by induction ($C$ denotes the norm, i.e., infimum of the $C$ constants for the addition map), for $P$ an element of this set,
$$|(u)+(v)|_{C,P}:=C\cdot \max(|u|_{C,Pu},|v|_{C,Pv}).$$
One then takes the maximum of all these expressions to define a natural pseudo-norm on $R^{(X)}$ by
$$
\left|\sum a_x\{x\}\right|_{\infty,C}:=\max_{P\in \mathcal{P}a(supp(a))}\left|P(\sum a_x\{x\})\right|_{C,P}.
$$
If $C=1$, i.e., in the non-archimedean case, this gives back the usual non-archimedean (maximum) seminorm on the free module.

## Modules and their operations

Using the free module, and the coproduct and product of $\mathbb{R}_+$-graded sets, one defines direct sums and tensor products of modules, and throught the symmetric algebra construction, the free $\mathbb{R}_+$-graded algebra (algebra of convergent "power series" with coefficients in $R$ and polyradii in $X$).

One may also work with the category of ind-pseudo-Banach ring, and of ind-pseudo-Banach modules over them, to develop an overconvergent version of the theory.

## A natural flow

The action of $\mathbb{R}_+^*$ on all these objects is simply given by acting on the grading through
$$|\cdot|\mapsto |\cdot|^t:=e^{t\mathrm{ln}|\cdot|}.$$

## Related subjects

[[global analytic geometry]]

[[overconvergent global analytic geometry]]

[[spectral global analytic geometry]]

## Reference

For the category of $\mathbb{R}_+$-graded sets:

* [[Frédéric Paugam]] _Overconvergent global analytic geometry_ ([arXiv:1410.7971](http://arxiv.org/abs/1410.7971))


proxi-metric space

[[!redirects pseudo-metric space]]
[[!redirects pseudo-metric]]
[[!redirects pseudo-metric spaces]]
[[!redirects pseudo-metrics]]
[[!redirects proxi-metric]]
[[!redirects proxi-metric spaces]]

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## Idea

The category of [[metric spaces]] with Lipschitz maps between them is not good enough from a categorical viewpoint: it has finite limits (the max metric on the product) but not all finite colimits: the coproduct doesn't always exist. To overcome this difficulty, one needs to enlarge this category in various ways.

The first enlargement is obtained by replacing the usual triangular inequality
$d(x,y)\leq d(x,z)+d(z,y)$ by the proxi-metric inequality: there exists $C\geq 1$ such that
$$d(x,y)\leq C\cdot \max(d(x,z),d(z,y)).$$
If $C=1$, we get back the ultrametric inequality, and the usual triangular inequality corresponds to the case $C=2$.

This gives a category of so-called proxi-metric spaces stable by the $\mathbb{R}_+^*$-flow given by
$$d\mapsto d^t:=e^{t\log(d)}.$$

To get enough finite colimits, one needs to restrict to bounded proximetric spaces, and then, one may take an ind-pro or an ind-completion to get a good category of generalized metric spaces, i.e., a convenient setting for the development of a metric stable homotopy theory, based on the use of the interval $[0,1]$
or a normed version of it.

The aim of this normed/metrized stable homotopy theory is to develop topological cohomological invariants for [[proxi-normed rings]].

## Related subjects

[[global analytic geometry]]

[[spectral global analytic geometry]]


proximity space

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## Idea

In addition to the well-known [[topological spaces]], many other structures can be used to found topological reasoning on sets, including [[uniform spaces]] and _proximity spaces_.  Proximity spaces provide a level of structure in between topologies and uniformities; in fact a proximity is equivalent to an equivalence class of uniformities with the same [[totally bounded space|totally bounded]] [[reflection]].

Proximity spaces are often called _nearness spaces_, but this term has other meanings in the literature. (See for example [this article](http://ijmsa.yolasite.com/resources/43--newnew.pdf).) One can clarify with the term _set--set nearness space_.  The same goes for the term _apartness space_, which is another way to look at the same basic idea.  (See [[apartness space]] for a point--set version and [[apartness relation]] for a point--point version.)

## Notation and terminology

A __proximity space__ is a kind of [[structured set]]: it consists of a [[set]] $X$ (the set of __[[points]]__ of the space) and a __proximity structure__ on $X$.  This proximity structure is given by any of various [[binary relations]] between the [[subsets]] of $X$:

*  The __proximity relation__ or __nearness relation__ $\delta$; subsets $A$ and $B$ are __proximal__  or __near__ if $A \;\delta\; B$;
*  The __apartness relation__ $\bowtie$; $A$ and $B$ are __apart__ if $A \bowtie B$.
*  The __neighbourhood relation__ $\ll$; $A$ is a __proximal neighbourhood__ of $B$ if $B \ll A$.

The conditions required of these relations are given below in the Definitions.  (We say 'proximal neighbourhood' instead of simply 'neighbourhood' to avoid misapplying intuition from general topology.  That $A \ll B$ doesn\'t necessarily mean that $A \subseteq Int(B)$; sometimes it means that $Cl(A) \subseteq Int(B)$, which is stronger, or something else.)

In [[classical mathematics]], these relations are all interdefinable:

*  $A \;\delta\; B \;\iff\; \neg(A \bowtie B) \;\iff\; \neg(A \ll B^{\mathsf{c}})$;
*  $A \bowtie B \;\iff\; \neg(A \;\delta\; B) \;\iff\; A \ll B^{\mathsf{c}}$;
*  $A \ll B \;\iff\; \neg(A \;\delta\; B^{\mathsf{c}}) \;\iff\; A \bowtie B^{\mathsf{c}}$.

(Here, $\neg$ indicates [[negation]] of truth values, while $B^{\mathsf{c}}$ is the [[complement]] of $B$ in $X$, i.e. $X\setminus B$.)

In [[constructive mathematics]], any one of these relations may be taken as primary and the others defined using it; thus we distinguish, constructively, between a __set--set nearness space__, a __set--set apartness space__, and a __set--set neighbourhood space__.

## Definitions

From the previous section, we have a [[set]] $X$ and we are discussing [[binary relations]] $\delta, \bowtie, \ll$ on $X$.  These are required to satisfy the following conditions; in each row, the conditions for the various relations are all equivalent (classically).  In these conditions, $x, y$ are points, while $A, B, C$ are subsets, and we require them for all points or subsets.  We list the conditions roughly in order of increasing optional-ness, then define terminology for relations satisfying them.

| Name | Condition for nearness | Condition for apartness | Condition for proximal neighbourhoods |
| - | - | - | - |
| Isotony (left) | If $A \supseteq B \;\delta\; C$, then $A \;\delta\; C$ | If $A \subseteq B \bowtie C$, then $A \bowtie C$ | If $A \subseteq B \ll C$, then $A \ll C$ |
| Isotony (right) | If $B \;\delta\; C \subseteq D$, then $B \;\delta\; D$ | If $B \bowtie C \supseteq D$, then $B \bowtie D$ | If $B \ll C \subseteq D$, then $B \ll D$ |
| Additivity (left, nullary) | It is false that $\emptyset \;\delta\; A$ | $\emptyset \bowtie A$ | $\emptyset \ll A$ |
| Additivity (right, nullary) | It is false that $A \;\delta\; \emptyset$ | $A \bowtie \emptyset$ | $A \ll X$ |
| Additivity (left, binary) | If $A \cup B \;\delta\; C$, then $A \;\delta\; C$ or $B \;\delta\; C$ | If $A \bowtie C$ and $B \bowtie C$, then $A \cup B \bowtie C$ | If $A \ll C$ and $B \ll C$, then $A \cup B \ll C$ |
| Additivity (right, binary) | If $A \;\delta\; B \cup C$, then $A \;\delta\; B$ or $A \;\delta\; C$ | If $A \bowtie B$ and $A \bowtie C$, then $A \bowtie B \cup C$ | If $A \ll B$ and $A \ll C$, then $A \ll B \cap C$ |
| Reflexivity (general) | If $A$ meets $B$ (their [[intersection]] is [[inhabited subset|inhabited]]), then $A \;\delta\; B$ | If $A \bowtie B$, then $A$ and $B$ are [[disjoint set|disjoint]] | If $A \ll B$, then $A \subseteq B$ |
| Reflexivity (for singletons) | $\{x\} \;\delta\; \{x\}$ | It is false that $\{x\} \bowtie \{x\}$ | If $\{x\} \ll A$, then $x \in A$ |
| Transitivity | If for every $D \subseteq X$, either $A \;\delta\; D^{\mathsf{c}}$ or $D \;\delta\; B$, then $A \;\delta\; B$ | If $A \bowtie B$, then for some $D \subseteq X$, both $A \bowtie D^{\mathsf{c}}$ and $D \bowtie B$ | If $A \ll B$, then for some $D \subseteq X$, both $A \ll D$ and $D \ll B$ |
| Symmetry (weak) | if $A \;\delta\; B$ then $B \;\delta\; A$ | if $A \bowtie B$ then $B \bowtie A$ | if $A \ll B$ then $B^{\mathsf{c}} \ll A^{\mathsf{c}}$ |
| Symmetry (strong) | $A \;\delta\; B$ iff $B \;\delta\; A$ | $A \bowtie B$ iff $B \bowtie A$ | $A \ll B$ iff $B^{\mathsf{c}} \ll A^{\mathsf{c}}$ |
| Local Decomposability | | If $\{x\}\bowtie B$, then for some $C$ and $D$, $\{x\}\bowtie D$, $C\bowtie B$, and $C\cup D = X$. | If $\{x\}\ll B$, then for some $C$ and $D$, $\{x\}\ll C$, $D\ll B$, and $C^{\mathsf{c}}\cup D = X$. |
| Decomposability | | If $A\bowtie B$, then for some $C$ and $D$, $A\bowtie D$, $C\bowtie B$, and $C\cup D = X$. | If $A\ll B$, then for some $C$ and $D$, $A\ll C$, $D\ll B$, and $C^{\mathsf{c}}\cup D = X$. |
| Separation | If $\{x\} \delta \{y\}$, then $x = y$ | Unless $\{x\} \bowtie \{y\}$, then $x = y$ | $x = y$ if, for all $A$, $y \in A$ whenever $\{x\} \ll A$ |
| Perfection (left) | If $A \;\delta\; B$, then $\{x\} \;\delta\; B$ for some $x \in A$ | $A \bowtie B$ if $\{x\} \bowtie B$ for all $x \in A$ | $A \ll B$ if $\{x\} \ll B$ for all $x \in A$ |
| Perfection (right) | If $A \;\delta\; B$, then $A \;\delta\; \{y\}$ for some $y \in B$ | If $A \bowtie \{y\}$ for all $y \in B$, then $A \bowtie B$ | If $A \ll C_i$ for all $i$, then $A \ll \bigcap_i C_i$ |

When both left and right rules are shown, we only need one of them if we have Symmetry, but we need both if we lack Symmetry (or if we are using proximal neighbourhoods in constructive mathematics; see below).  Even so, Isotony is usually given on both sides, since it is convenient to combine both directions into a single statement.  On the other hand, Isotony is equivalent to the converse of binary Additivity, so sometimes these are combined instead (so Isotony does not explicitly appear), usually on only one side when Symmetry is used.

Whether made explicit or not, Isotony is very fundamental.  In particular, it allows us to assert Reflexivity only for singletons, although this is often not done (to avoid mentioning points).

In [[classical mathematics]], the weak Symmetry axioms are equivalent to the strong versions, while Decomposability is equivalent to Transitivity (and Local Decomposability is of course a special case of Decomposability).  In [[constructive mathematics]], strong Symmetry for $\ll$ is unreasonably strong (as is binary Additivity for $\delta$), while Local Decomposability and Decomposability appear genuinely stronger than Transitivity (but not unreasonably so); see remarks below.  (We don't bother to state a version of (Local) Decomposability for $\delta$ because the other axioms for $\delta$ seem unreasonably strong constructively, while Decomposability is automatic in classical mathematics.)

A __topogeny__ is a relation that satisfies both forms of Isotony and all four forms of Additivity.  A __topogenous relation__ or __quasipreproximity__ is a topogeny that also satisfies Reflexivity; a __quasiproximity__ is a quasipreproximity that also satisfies Reflexivity and Transitivity.  A topogeny is __symmetric__ if it satisfies Symmetry; a __preproximity__ is a symmetric quasipreproximity, and a __proximity__ is a symmetric quasiproximity.  A topogeny is __separated__ if it satisfies Separation.  A topogeny is __perfect__ if it satisfies left Perfection, __coperfect__ if it satisfies right Perfection, and __biperfect__ if it satisfies both; a symmetric topogeny is usually called simply __perfect__ if it satisfies any form of Perfection, because then it must satisfy both (except in constructive mathematics using proximal neighbourhoods).  Constructively, a proximity space satisfying Regularity may be called __proximally regular__.

A __(quasi)-(pre)-proximity space__ is a set equipped with a (quasi)-(pre)-proximity.  All of these terms may be used with _nearness_, _apartness_, or _proximal neighbourhoods_, as explained in the previous section; nearness is usually the default.

+-- {: .un_remark}
###### Warnings

Some authors require a (quasi)-proximity to be separated; conversely, some authors do not require a (quasi)-proximity to satisfy Transitivity (so 'pre' is the default for them).  The term 'topogeny' is also not found in the literature (except [[toddtrimble:topogeny|here]], in a generalization of nearness spaces); it is derived from '[[topogenous relation]]', a term used in the theory of [[syntopogenous spaces]] for a quasipreproximity.  The terminology for Perfection also comes from syntopogenous spaces.

Transitivity is sometimes called _normality_, due to its superficial similarity with the notion of a [[normal space]].  However, the underlying topology of a proximity space need not be normal in the topological sense.  If regarded as a [[separation axiom]], the Transitivity axiom is more analogous to [[completely regular space|complete regularity]] than to normality, and at least in classical mathematics every proximity space is completely regular --- but this is not true in constructive mathematics, or for quasi-proximity spaces, so it is debatable whether it should be attributed to the Transitivity axiom.  When comparing proximities to topologies via [[syntopogenies]], Transitivity is instead analogous to the fact that the closure of a set is closed (or the interior of a set is open), which we require of all topologies; thus arguably we should require it of all (quasi-)proximities.  If this is removed from the definition of topological space, then we have a [[pretopological space]], which is the origin of the 'pre' terminology above.
=--

If $X$ and $Y$ are (quasi)-(pre)-proximity spaces, then a [[function]] $f: X \to Y$ is said to be **proximally continuous** if $A \;\delta\; B$ implies $f_*(A) \;\delta\; f_*(B)$, equivalently if $A \bowtie B$ whenever $f_*(A) \bowtie f_*(B)$, equivalently if $f^*(C) \ll f^*(D)$ whenever $C \ll D$.  In this way we obtain [[categories]] $QProx$ and $Prox$; the [[forgetful functors]] $QProx \to Set$ and $Prox \to Set$ (taking a space to its set of points) make them into [[topological concrete categories]].

### In constructive mathematics

As remarked above, the strong Symmetry axiom for $\ll$ is too strong in constructive mathematics.  In fact, if it is satisfied by a nontrivial space $X$, then [[excluded middle]] follows: for any [[truth value]] $P$ such that $\neg\neg P$ is true, let $A=X$ and $B = \{ x\in X \mid P \}$; then $B^{\mathsf{c}} = \emptyset \ll \emptyset = A^{\mathsf{c}}$, so strong Symmetry gives $A\ll B$, hence $A\subseteq B$ and so $P$ holds.

On the other hand, while the Symmetry axioms for $\bowtie$ and $\delta$ may seem stronger than the allowable weak symmetry axiom for $\ll$, this is arguably an illusion.  For given a proximal neighborhood space with $\ll$ (and weak Symmetry), if we define $A\bowtie B$ to mean $A\ll B^{\mathsf{c}}$, we obtain a proximal apartness space (with Symmetry).  In fact, the Transitivity axiom for $\bowtie$ implies that if $A\bowtie B$ then $A\bowtie (B^{\mathsf{c}})^{\mathsf{c}}$, so that (with Symmetry) $\bowtie$ is completely determined by its restriction to subsets that are complements.

In this sense, a proximal neighborhood space actually contains more information than a proximal apartness space; it is unclear whether the latter gives rise to the former constructively.  On the other side, we can of course define $A\;\delta \; B$ to mean $\neg(A\bowtie B)$, but with this definition we cannot even prove binary Additivity for $\delta$.  In fact, binary Additivity for $\delta$ seems too strong: it is apparently not satisfied even by a metric space.

We can, however, show the following constructively.

+--{: .un_theorem}
###### Theorem
(Quasi-)proximal apartness spaces satisfying Decomposability are equivalent to (quasi-)proximal neighborhood spaces satisfying Decomposability.
=--
+--{: .proof}
###### Proof
As above, in one direction we define $A\bowtie B$ to mean $A\ll B^{\mathsf{c}}$.  In the other, we define $A\ll B$ to mean that there exists a $C$ with $A\bowtie C$ and $B\cup C = X$.  Most of the axioms are easy.  To prove right binary Additivity for $\ll$, given $A\ll B$ and $A\ll C$, we have $A\bowtie D$ and $A\bowtie E$ with $D\cup B = X$ and $E\cup C = X$, whence $A\bowtie D\cup E$ with $(D\cup E) \cup (B\cap C) = X$ by distributing $\cup$ over $\cap$.

To prove Decomposability (and hence Transitivity) for $\ll$, given $A\ll B$, we have $A\bowtie C$ with $B\cup C = X$, whence by Decomposability for $\bowtie$ we have $D,E$ with $A\bowtie D$, $E\bowtie C$, and $D\cup E = X$.  By Decomposability again, we have $D_1,D_2,E_1,E_2$ with $A\bowtie D_1$, $D_2\bowtie D$, and $D_1\cup D_2 = X$, while $E\bowtie E_1$, $E_2\bowtie C$, and $E_1\cup E_2=X$.  Then $A\ll D_2$ (because of $D_1$) and $E_2 \ll B$ (because of $C$), while $D\subseteq D_2^{\mathsf{c}}$ (since $D_2\bowtie D$) and $E\subseteq E_2$ (because $E\bowtie E_1$ and so $E\subseteq E_1^{\mathsf{c}} \subseteq E_2$) and so $X = D\cup E = D_2^{\mathsf{c}} \cup E_2$.

Lastly, if $\bowtie$ satisfies Symmetry and $A\ll B$, we have $A\bowtie C$ with $B\cup C = X$, whence by Decomposability for $\bowtie$ we have $D,E$ with $A\bowtie D$, $E\bowtie C$, and $D\cup E = X$.  Then by Symmetry, we have $B^{\mathsf{c}} \subseteq C\bowtie E$ while $X = E\cup D \subseteq E\cup A^{\mathsf{c}}$, so $B^{\mathsf{c}} \ll A^{\mathsf{c}}$.

To show that these constructions are inverses, if we start with $\ll$ then by a round-trip we obtain $A\ll' B$ defined to mean that there exists $C$ with $A\ll C^{\mathsf{c}}$ and $B\cup C = X$.  Now on one hand, $B\cup C = X$ implies $C^{\mathsf{c}}\subseteq B$, so $A\ll C^{\mathsf{c}} \subseteq B$.  But on the other, if $A\ll B$ then Decomposability gives $C$ and $D$ with $A\ll C$, $D\ll B$, and $C^{\mathsf{c}}\cup D = X$, whence $A \ll (C^{\mathsf{c}})^{\mathsf{c}}$ and $X = D \cup C^{\mathsf{c}} \subseteq B \cup C^{\mathsf{c}}$.

On the other side, if we start with $\bowtie$ then by a round-trip we obtain $A\bowtie' B$ defined to mean that there exists $C$ with $A\bowtie C$ and $C \cup B^{\mathsf{c}}= X$.  Now on one hand, if $C \cup B^{\mathsf{c}}= X$ then $B \subseteq C$, so $A\bowtie C$ implies $A\bowtie B$.  But on the other, if $A\bowtie B$ then Decomposability gives $C$ and $D$ with $A\bowtie C$ and $D\bowtie B$ and $C\cup D = X$, whence $D\subseteq B^{\mathsf{c}}$ and so $X = C\cup D \subseteq C \cup B^{\mathsf{c}}$.
=--

In other words, the two constructively reasonable ways to define a (quasi-)proximity space give the same result in the Decomposable case.

On the other hand, the Decomposability axiom is very strong, constructively; as noted [below](#Uniform), the standard way to make a proximity space from a uniform space does not satisfy it.  Thus constructive treatments of proximity such as [Bridges-Vita](#BridgesVita) have usually not included Decomposability, instead using the weaker Transitivity axiom (called the *Efremovic property* in [Bridges-Vita](#BridgesVita)).

## Relations to other topological structures

### Preorders

Given points $x, y$ of a (quasi)-proximity space, let $x \leq y$ mean that $x$ belongs to every proximal neighbourhood of $\{y\}$, or equivalently (via Isotony) that $\{y\} \;\delta\; \{x\}$.  By Reflexivity, $\leq$ is [[reflexive relation|reflexive]]; by Transitivity, $\leq$ is [[transitive relation|transitive]].  (In fact, we can use these to deduce that $x \leq y$ iff every proximal neighbourhood of $\{y\}$ is a proximal neighbourhood of $\{x\}$, which is manifestly reflexive and transitive.)  Therefore, $\leq$ is a [[preorder]].

If the quasiproximity satisfies Symmetry, then this preorder is [[symmetric relation|symmetric]] and hence an [[equivalence relation]].

Regardless of Symmetry, a (quasi)-proximity space is separated iff this preorder is the [[equality relation]].  That is, $x = y$ if $x$ belongs to every proximal neighbourhood of $\{y\}$, or equivalently if every proximal neighbourhood of $\{y\}$ is a proximal neighbourhood of $\{x\}$, or equivalently if $\{x\}$ is near $\{y\}$, or equivalently if $\{x\}$ is not apart from $\{y\}$.  This may be viewed as a converse of simplified Reflexivity, which states that $\{x\} \;\delta\; \{y\}$ whenever $x = y$.

Conversely, given a set equipped with a preorder $\leq$, let $A \;\delta\; B$ if $x \leq y$ for some $x \in A$ and some $y \in B$, or equivalently let $A \bowtie B$ if $x \leq y$ for no $x \in A$ and no $y \in B$, or equivalently let $A \ll B$ if $x \leq y$ for $x \in A$ implies $y \in B$.  Then we have a quasiproximity space which is symmetric iff $\leq$ is.

In this way, we get the category $Proset$ of preordered sets as a [[reflexive subcategory]] of $QProx$, with the category $SymPreOrd$ of [[symmetric prosets]] (sets equipped with equivalence relations) as a reflexive subcatgory of $Prox$.

In [[constructive mathematics]], a quasi-proximal nearness space (despite its general unreasonableness) or neighborhood space gives rise to a preordered set as above, while a proximal apartness space yields an [[inequality relation]] that is an [[apartness relation]] if we have Local Decomposability.

### Topological spaces

Every proximity space is a [[topological space]]; let $x$ belong to the closure of $A \subseteq X$ iff $\{x\} \;\delta\; A$, or equivalently let $x$ belong to the interior of $A$ iff $\{x\} \ll A$.  This topology is always [[completely regular space|completely regular]], and is [[Hausdorff space|Hausdorff]] (hence [[Tychonoff space|Tychonoff]]) iff the proximity space is separated; see [[separation axiom]].  Proximally continuous functions are [[continuous map|continuous]] for the induced topologies, so we have a functor $Prox \to Top$ over $Set$.

Conversely, if $(X,\tau)$ is a completely regular topological space, then for any $A, B \subseteq X$ let $A \bowtie B$ iff there is a continuous function $f: X\to [0,1]$ such that $f(x) = 0$ for $x \in A$ and $f(x) = 1$ for $x \in B$.  This defines a proximity structure on $X$, which induces the topology $\tau$ on $X$, and which is separated iff $\tau$ is a Hausdorff (hence Tychonoff) topology.

In general, a completely regular topology may be induced by more than one proximity.  However, if it is moreover [[compact topological space|compact]], then it has a unique compatible proximity, given above.  In the case of a [[compact Hausdorff space]] (or more generally any [[normal regular space]]), we then have $A \ll B$ iff $Cl(A) \subseteq Int(B)$.

The construction of a topology works just as well for a quasi-proximity, but the result is no longer necessarily completely regular.  In fact, every topology can be induced by some quasi-proximity: define $A \ll B$ if $A \subseteq int(B)$.  Moreover, in this way topological spaces can be identified with (left) perfect quasi-proximity spaces.

In [[constructive mathematics]], the three now-inequivalent kinds of (quasi-)proximity space (nearness, apartness, and neighborhood) yield three inequivalent notions of topological space: [[closure operators]], [[point-set apartness spaces]], and [[topological spaces]] (equivalently [[neighborhood]] spaces).  Note in particular that the Transitivity axiom ensures that (in the three cases) the closure of a set is closed, that the final axiom of a point-set apartness space holds (if $x\bowtie A$ then $x\bowtie \{ y \mid \neg(y\bowtie A) \}$), and that the interior of a set is open.  Hence, we can identify left perfect quasi-proximal apartness or neighborhood spaces with point-set apartness spaces and topological spaces respectively, by defining $A \ll B$ if $A \subseteq int(B)$ as above in the second case, and $A\bowtie B$ if $x\in A \Rightarrow x\bowtie B$ in the first.

If a quasi-proximal apartness or neighborhood space satisfies Local Decomposability, then its underlying point-set apartness space or topological space is [[locally decomposable space|locally decomposable]] (hence the name for the former axiom).  Conversely, the canonical quasi-proximial apartness or neighborhood space obtained from a point-set apartness space or topological space satisfies Local Decomposability if the original space was locally decomposable.

Unlike in the classical case, the topology underlying a proximity space (which, recall, includes Symmetry) apparently need not be [[completely regular space|completely regular]] or even [[regular space|regular]].  However, we can say:

+--{: .un_theorem}
###### Theorem
Let $X$ be a proximal apartness or neighborhood space (with Symmetry).  Then:

* If $X$ satisfies Local Decomposability, its underlying topological space is regular.
* If $X$ satisfies Decomposability and [[countable choice]] holds, then its underlying topological space is completely regular.
=--
+--{: .proof}
###### Proof
Suppose first that $X$ is a Locally Decomposable proximal neighborhood space and $U$ is an open set containing $x$, which means $\{x\} \ll U$.  By Local Decomposability, we have sets $C$ and $D$ such that $\{x\} \ll C$ and $D\ll U$ and $X = D \cup C^{\mathsf{c}}$.  Transitivity then gives us a set $B$ with $\{x\} \ll B \ll C$, and Symmetry then tells us that $C^{\mathsf{c}} \ll B^{\mathsf{c}}$.  Let $V = int(B^{\mathsf{c}})$ and $W = \int(B)$; then $V\cap W = \emptyset$, $W$ is an open neighborhood of $x$, and $X = D \cup C^{\mathsf{c}} \subseteq U \cup V$.  Thus, $X$ is regular.

Now suppose $X$ is a Decomposable proximal neighborhood space and we have countable choice.  Then given any opens $A,B$ with $A\ll B$, essentially the same proof as above shows that $A$ is "well-inside" $B$.  Thus, applying Transitivity repeatedly (making countably many choices) we can obtain a "scale" of interpolating well-inside open subsets indexed by (dyadic) rational numbers, and thereby a function $f:X\to \mathbb{R}$ separating $A$ and $B$, as in the classical proof of Urysohn's lemma.  (With only Transitivity instead of Decomposability, we would obtain only a function valued in the [[lower real numbers|lower]] or upper real numbers, rather than the [[located real numbers]].)

The arguments for proximal apartness spaces are essentially the same.
=--

Conversely, if $X$ is a completely regular topological space, then the above definition that $A \bowtie B$ iff they are separated by a continuous function $f: X\to [0,1]$ is also constructively valid, and produces a proximity space satisfying Symmetry and Decomposability.

### Uniform spaces {#Uniform}

If $Y$ is a [[uniform space]], then for all $A, B \subseteq Y$ let $A \;\delta\; B$ iff $V \cap (A \times B)$ is [[inhabited set|inhabited]] for every [[entourage]] (aka vicinity) $V$. This defines a biperfect proximity structure on $Y$.  In terms of the other relations, this means $A\bowtie B$ iff $V \cap (A \times B) = \emptyset$ for some entourage $V$, and $A\ll B$ iff $V[A] \subseteq B$ for some entourage $V$, where $V[A] = \{ y \mid \exists x\in A, (x,y)\in V \}$.

[[uniformly continuous map|Uniformly continuous functions]] are proximally continuous for the induced proximities, so we have a functor $Unif \to Prox$ over $Set$.  Moreover, the composite $Unif \to Prox \to Top$ is the usual "underlying topology" functor of a uniform space, i.e. the topology induced by the uniformity and the topology induced by the proximity structure are the same.

Conversely, if $X$ is a proximity space, consider the family of sets of the form
$$ \bigcup_{k=1}^n (A_k \times A_k) $$
where $(A_k)_k$ is a [[list]] (a finite family) of sets such that there exists a same-length list of sets $(B_k)_k$ with $B_k \ll A_k$ and $X = \bigcup_{k=1}^n B_k$.  These sets form a base for a [[totally bounded uniformity]] on $X$, which induces the given proximity.

In fact, this is the *unique* totally bounded uniformity which induces the given proximity: every proximity-class of uniformities contains a unique totally bounded member.  Moreover, a proximally continuous function between uniform spaces with totally bounded [[codomain]] is automatically uniformly continuous.  Therefore, the forgetful functor $Unif \to Prox$ is a left adjoint, whose right adjoint also lives over $Set$, is [[fully faithful functor|fully faithful]], and has its [[essential image]] given by the totally bounded uniform spaces.

In general, proximally continuous functions between given uniform spaces need not be uniformly continuous.  But in addition to total boundedness of the codomain, a different sufficient condition is that the domain be a [[metric space]].

The relation between quasi-uniformities and quasi-proximities is similar.

In [[constructive mathematics]], all three of the relations $\delta,\bowtie,\ll$ may be defined as above from a uniformity.  All the axioms of all three kinds of proximity are then provable, except Binary Additivity for $\delta$, Strong Symmetry for $\ll$, and (Local) Decomposability.  Local Decomposability follows from [[uniform regularity]] for the uniform space, but Decomposability seems unobtainable in this way; it seems to assert [[located subspace|locatedness]] of all subsets.

Moreover, constructively not every $\bowtie$- or $\ll$-proximity is induced by a uniformity in this way.  Consider, on an inhabited set $X$, the relation $A\bowtie B$ defined to mean $A = \emptyset \vee B=\emptyset$.  This is a $\bowtie$-proximity, but if it is induced by a uniformity, then [[weak excluded middle]] holds.  To see this, given a proposition $P$, fix $\xi\in X$ and let $A = \{ x \mid x=\xi \wedge P\}$ and $B = \{x \mid x=\xi \wedge \neg P\}$.  Then $A\times B = \emptyset$, hence $A\bowtie B$ in any proximity defined from a uniformity; but to say $A = \emptyset \vee B=\emptyset$ is weak excluded middle for $P$.  (This example is from [Bridges and Vita](#BridgesVita).)

However, there is a different way to obtain a proximity underlying a uniformity: define $A\bowtie B$ if there is a uniformly continuous function $f: X\to [0,1]$ such that $f(x) = 0$ for $x \in A$ and $f(x) = 1$ for $x \in B$.  Classically, this is equivalent to the usual definition, by the usual proof that any uniform space is completely regular.  Constructively, it (and a similar definition for $\ll$) yields a proximity space satisfying Decomposability, which has the same underlying topology *if* our original uniform space was already [[completely regular space|completely regular]].  It seems possible that at least with [[countable choice]], we could show that every Decomposable proximity space arises from a uniform space in this way.

### Syntopogenous spaces

A proximity space can be identified with a [[syntopogenous space]] which is both *simple* and *symmetric*, and similarly a quasi-proximity space can be identified with one that is just simple.  The relations with topological spaces and uniform spaces can then be seen as coreflections inside the category of syntopogenous spaces; see [[syntopogenous space]].

### Compactifications

The (separated) proximities inducing a given (Hausdorff) completely regular topology can be identified with (Hausdorff) [[compactifications]] of that topology.  The compactification corresponding to a proximity on $X$ is called its __Smirnov compactification__.  The points of this compactification can be taken to be __clusters__ in $X$, which are defined to be collections $\sigma$ of subsets of $X$ such that

1. If $A \in \sigma$ and $B \in \sigma$, then $A \;\delta\; B$.
2. If $A \;\delta\; C$ for all $C \in \sigma$, then $A \in \sigma$.
3. If $(A \cup B) \in \sigma$, then $A \in \sigma$ or $B \in \sigma$.
4. $\sigma$ is nonempty.

Note that conditions (1) and (2), together with Isotony for $\delta$, imply that any cluster $\sigma$ is upwards-closed.  Together with nullary Additivity for $\delta$, they also imply the nullary version of (3): $\emptyset \notin\sigma$.  However, a cluster is not generally a [[filter]]: it is not closed under binary intersections.  This is most obvious when we consider how $X$ embeds in its compactification, taking each point $x$ to the "principal cluster" $\sigma_x = \{ A \mid \{x\}\;\delta\; A \}$, i.e. the collection of sets whose closure contains $x$, which is certainly not generally closed under intersections.  (If it were a filter, of course, condition (3) would say that it is prime.)

## Proximities as profunctors {#profunctors}

As a [[poset]], the [[power set]] $\mathcal{P}X$ of $X$ may be regarded as a [[category]] [[enriched category|enriched]] over [[truth values]].  There is a notion of a [[bimodule]] over a category, also called (more specifically) a distributor or [[profunctor]].

Then a profunctor from $\mathcal{P}X$ to itself is precisely a binary relation $\ll$ on subsets of $X$ that satisfies Isotony.  Adding Reflexivity makes it a co-[[pointed object|pointed]] profunctor, and Transitivity morally makes it a [[coassociative coalgebra]] with Reflexivity as counit.  (Actually, coassociativity is trivial when enriched over truth values, as is the claim that Reflexivity, once it exists, is a counit, but we say 'coassociative' to clarify which sense of 'coalgebra' we mean.)

The sense in which Transitivity makes this a coalgebra is actually a bit involved, and it only quite works because of Additivity.  A coalgebra with a given profunctor $\ll$ as its underlying bimodule has the *structure* of an operation that, given $x \ll z$, takes this to an equivalence class of $y$ such that $x \ll y \ll z$, where $y$ is equivalent to $y'$ if $y \subseteq y'$ (or by any equivalence that follows).  By Isotony and left binary Additivity, $x \ll y \cup y' \ll z$ (or use right Additivity and $y \cap y'$); since $y, y' \subseteq y \cup y'$, we have the desired equivalence.

This suggests that if we want a notion of proximity *without* Additivity, then Transitivity must become more complicated, being a [[extra structure|structure]] rather than just a property (and a structure that should be preserved by proximal maps).

As for Additivity itself, this presumably corresponds to something more general in the world of profunctors related to [[limits]] and [[colimits]], but I haven\'t figured it out yet.

Symmetry probably doesn\'t fit into this picture very well, but who knows?

## Related concepts

[[!include generalized uniform structures - table]]

## References

*  R. Engelking, _General topology_, chapter 8.

*  [[Douglas Bridges]], Peter Schuster, and Luminita Vita, _Apartness, topology, and uniformity: a constructive view_, [pdf](http://www.math.canterbury.ac.nz/mathlsv/dagstuhl01.pdf)

* {#BridgesVita} [[Douglas Bridges]] and Luminita Vita, *Apartness and Uniformity: a constructive development*

*  S. A. Naimpally and B. D. Warrack, _Proximity spaces_, Cambridge University Press 1970

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Prym variety

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## Idea

In good cases a [[homomorphism]] $f\colon C_1 \to C_2$ of [[algebraic curves]] allows to push-forward [[algebraic line bundles]] such as to induce a homomorphism of the corresponding [[Jacobian varieties]]/[[Picard varieties]]  $J_1 \to J_2$. The _Prym variety_ of $f$ then is the [[kernel]] of this map, hence the space of algebraic line bundles on $C_1$ which pushes to the trivial line under $f$.

## Related concepts

* [[Prym-Tyurin variety]]

## References

* Wikipedia, _[Prym variety](http://en.wikipedia.org/wiki/Prym_variety)_

[[!redirects Prym varieties]]


Prym-Tyurin variety

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## Idea

The concept of _Prym-Tyurin variety_ or _generalized [[Prym variety]]_ is a slight generalization of that of [[Jacobian variety]] of an [[algebraic curve]].

It is an [[abelian variety]] $Z$ equipped with  [[principally polarized variety|principal polarization]] $\Xi$ such that there exists an [[algebraic curve]] $C$ and an embedding $\iota \colon Z \hookrightarrow Jac(C)$ of $Z$ into its [[Jacobian variety]] such that the pullback of the principal polarization $\Theta$ of $Jac(C)$ along this embedding is an integral fraction of $\Xi$:

$$
  \iota^\ast \Theta = n \Xi
  \,.
$$

Here $n$ is called the _exponent_ of $Z$ in $Jac(C)$.

(e.g. [Birkenhage-Lange](#BirkenhageLange))

## Properties

Any principally polarized abelian variety is a Prym-Tyurin variety for a curve $C$ of sufficiently high [[genus of a curve|genus]].

## Related concepts

* [[Prym variety]]

## References

* {#BirkenhageLange} Christina Birkenhake Herbert Lange, section 12.2 of _Complex abelian varieties_

[[!redirects Prym-Tyurin varieties]]

[[!redirects generalized Prym variety]]
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Prüfer group

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## Definition

For a [[prime number]] $p$, the **Prüfer $p$-group** is defined uniquely up to [[isomorphism]] as the [[group]] where every element has exactly $p$ $p^{th}$ [[roots]]. It is a [[divisible group|divisible]] [[abelian group]] which can be described in several ways, for example:

* It is the [[discrete group]] that is [[Pontrjagin duality|Pontryagin dual]] to the [[compact topological space|compact]] [[topological group]] of [[p-adic integer|p-adic integers]] (for more on this see as _[[Pontryagin duality for torsion abelian groups]]_).

* It is $\mathbb{Z}[1/p]/\mathbb{Z}$, the [[colimit]] of the sequence of inclusions

  $$0 \hookrightarrow \mathbb{Z}/(p) \hookrightarrow \ldots \hookrightarrow \mathbb{Z}/(p^n) \hookrightarrow \mathbb{Z}/(p^{n+1}) \hookrightarrow \ldots$$

As such, it is the [[initial algebra]] of the functor $F: \mathbb{Z} \darr LCHAb \to \mathbb{Z} \darr LCHAb$ that pushes out along multiplication by $p: \mathbb{Z} \to \mathbb{Z}$.

## Properties

The Prüfer $p$-groups are the only infinite groups whose [[subgroups]] are [[total order|totally ordered]] by inclusion. They are often useful as [[counterexamples in algebra]]; for example, a Prüfer group is an Artinian but not a Noetherian $\mathbb{Z}$-module.

## Related concepts

* [[Q/Z]]

* [[localization of abelian groups]]

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pseudo-abelian category > history

< [[pseudo-abelian category]]
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pseudo-connection

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## Idea

A [[connection on a bundle]] may be understood as a construction that serves to interpolate between a [[principal bundle]] and its [[characteristic class]] in [[real cohomology]], represented by [[curvature characteristic form]]s of the connection.

If one allows (and it makes good sense to allow this) curvature characteristic forms to be represented not necessarily by globally defined forms, but by [[cocycle]]s in the [[abelian sheaf cohomology]] of the truncated [[de Rham complex]] $\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \to \cdots \to \Omega^n_{cl}(X)$ then a weaker notion than that of a connection serves the same purpose: that of a _pseudo-connection_ .

Effectively a pseudo-connection on a principal bundle is locally a collection of [[Lie-algebra valued 1-form]]s, that do _not_ have to satisfy the cocycle condition familiar from connections.

That makes pseudo-connections rather empty structures. And in a precise sense this is actually the point of them: there is a (non-concrete) [[Lie groupoid]] $\mathbf{B}G_{diff}$ such that [[Cech cohomology|Cech cocycles]] with values in it are those for $G$-principal bundles with pseudo-connections. This groupoid is in fact weakly equivalent to the [[delooping]] groupoid $\mathbf{B}G$ that serves to classify just bare smooth $G$-principal bundles

$$
  \mathbf{B}G_{diff} \stackrel{\simeq}{\to} \mathbf{B}G
  \,.
$$

The purpose of pseudo-connections is precisely to form this [[resolution]] of $\mathbf{B}G$, which allows to represent an intrinsic curvature characteristic class to be modeled by an [[anafunctor]] with tip $\mathbf{B}G_{diff}$.

While for ordinary $G$-principal bundles the notion of pseudo-connection can be (and traditionally is) circumvented, it does become crucial for [[gerbe]]s and [[principal 2-bundle]]s and more generally in [[infinity-Chern-Weil theory]]. For instance an [[equivariant cohomology|equivariant]] [[gerbe]] _with_ connection is _not_ an equivariant object in the 2-category of gerbes with ordinary connections, but a certain constrained equivariant gerbe with pseudo-connection.

## Details

For the moment see [[infinity-Chern-Weil theory]].

(...)

## References

The term _pseudo-connection_ for generalized connections on [[bundle gerbe]]s apparently first appeared in

(...)

Not under this name, but being the same concept, they are considered on bundle gerbes in some detail in

* [[Konrad Waldorf]], ...

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pseudo-distributive law

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## Idea

The notion of a **pseudo-distributive law** is a [[vertical categorification]] of that of a [[distributive law]], relating two [[pseudomonads]] on a [[bicategory]]. There are various different levels of weakness that such a thing can exist at.  As with ordinary distributive laws, a pseudo-distributive law governs the lifting of one pseudomonad to the Eilenberg-Moore and Kleisli bicategories of the other.

## Definition

See any of the references, particularly [(Walker)](#Decagons), who simplified the definition of Marmolejo (reducing from 8 conditions to 5).

## Examples

* If $T$ is a [[pseudo-commutative 2-monad|pseudo-commutative]] [[2-monad]] on [[Cat]], then there is a pseudo-distributive law between $T$ and the 2-monad whose algebras are [[symmetric monoidal categories]]; see [(Kelly)](#Kelly).

* [[Grothendieck fibrations]] and opfibrations on a [[category]] $C$ (or more generally an object of a suitable 2-category) are the algebras for a pair of pseudomonads.  If $C$ has pullbacks, there is a pseudo-distributive law between these pseudomonads, whose joint algebras are the [[bifibrations]] satisfying the [[Beck-Chevalley condition]]; see [(von Glehn)](#vonGlehn).

* [[generalized polycategories]] are naturally defined relative to a pseudo-distributive law on a [[Prof]]-like bicategory; see [(Garner)](#Garner) for the canonical example of ordinary (symmetric) [[polycategories]].

* Pseudo-distributive laws involving [[lax-idempotent 2-monads]] have an especially nice form; see [(Marmolejo)](#MarmolejoDL) and [(Walker)](#Walker).

## Related pages

* [[distributive law]]

* [[pseudomonad]], [[relative pseudomonad]]

## References

* [[Eugenia Cheng]], [[Martin Hyland]], and [[John Power]].  _Pseudo-distributive laws_, [pdf](http://www.entcs.org/files/mfps19/83010.pdf)
 {#CHP}

* [[Max Kelly]], _Coherence theorems for lax algebras and for distributive laws_, Proceedings of the Sydney Category Seminar 1972-73.
 {#Kelly}

* [[Tamara von Glehn]], *Polynomials and models of type theory*, [PhD thesis](https://www.repository.cam.ac.uk/handle/1810/254394)
 {#vonGlehn}

* [[Richard Garner]], *Polycategories via pseudo-distributive laws*, [arXiv:math/0606735](http://arxiv.org/abs/math/0606735)
 {#Garner}

* Francisco Marmolejo, *Distributive laws for pseudomonads*, [TAC](http://tac.mta.ca/tac/volumes/1999/n5/5-05abs.html)
 {#MarmolejoDL}

* Charles Walker, *Distributive laws via admissibility*, [arXiv:1706.09575](https://arxiv.org/abs/1706.09575)
 {#Walker}

* Charles Walker. _Distributive laws, pseudodistributive laws and decagons_. [arXiv:2102.12468](https://arxiv.org/abs/2102.12468) (2021). {#Decagons}

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pseudo-equivalence relation

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## Idea

A *pseudo-equivalence relation* is like an [[equivalence relation]], but where we allow more than one element in the relation; i.e. the underlying [[directed graph]] of the relation is a [[directed pseudograph]]. In the same way that [[magmoids]] are the raw structure used to build [[semicategories]] and [[categories]], sets with pseudo-equivalence relations are the raw structure used to build [[dagger categories]] and [[groupoids]] (i.e. a groupoid without associativity, unital laws, and inverse laws).

Sets equipped with pseudo-equivalence relations are sometimes called *[[setoids]]* (i.e. in [Wilander (2012), §4](#Wilander12); [Palmgren & Wilander (2014), §2, §6](#PalmgrenWilander14); [Emmenegger & Palmgren (2020), §6](#EmmeneggerPalmgren20);  [Cipriano (2020), Def. 1.1.1](#Cipriano20)), but the term "setoid" is also used in mathematics to refer only to the sets with [[equivalence relations]]. In this article, the term "setoid" is used in the former sense of "sets equipped with pseudo-equivalence relations".

## Definition

### With a family of sets

Using graph theoretic terminology, a **pseudo-equivalence relation** on a set $V$ of [[vertices]] is a family of sets $E(a, b)$ of [[edges]] for each vertex $a \in V$ and $b \in V$, which comes with the following additional structure

* for each vertex $a \in V$ an element
$$\mathrm{refl}(a) \in E(a, a)$$

* for each vertex $a \in V$ and $b \in V$ a family of functions
$$\mathrm{sym}(a, b):E(a, b) \to E(b, a)$$

* for each vertex $a \in V$, $b \in V$, and $c \inV$, a family of functions
$$\mathrm{trans}(a, b, c):(E(a, b) \times E(b, c)) \to E(a, c)$$

### With a set

A **pseuodo-equivalence relation** on a [[set]] $V$ is a set $E$ and functions $s:E \to V$, $t:E \to V$ (a loop [[directed pseudograph]]), with with functions $\mathrm{refl}:V \to E$, $\mathrm{sym}:E \to E$, and
$$\mathrm{trans}:\{(f,g) \in E \times E \vert t(f) =_V s(g)\} \to E$$
such that

* for every $a \in V$, $s(\mathrm{refl}(a)) =_E a$
* for every $a \in V$, $t(\mathrm{refl}(a)) =_E a$
* for every $f \in E$, $s(f) =_V t(\mathrm{sym}(f))$
* for every $f \in E$, $t(f) =_V s(\mathrm{sym}(f))$
* for every $f \in E$ and $g \in E$ such that $t(f) =_V s(g)$, $s(\mathrm{trans}(f,g)) =_E s(f)$
* for every $f \in E$ and $g \in E$ such that $t(f) =_V s(g)$, $t(\mathrm{trans}(f,g)) =_E t(g)$

### Extensional functions, dagger functors, and isomorphisms

We define an **[[extensional function]]** $f:A \to B$ between two sets $A$ and $B$ with pseudo-equivalence relations $(E_A, \mathrm{refl}_A, \mathrm{sym}_A, \mathrm{trans}_A)$ and $(E_B, \mathrm{refl}_B, \mathrm{sym}_B, \mathrm{trans}_B)$ to be a function $f_V:A \to B$ and a family of functions $f_E(a, b):E_A(a, b) \to E_B(f_V(a), f_V(b))$

Composition of extensional functions is defined as composition of the vertex function and of each edge function.

An extensional function between two sets with pseudo-equivalence relations is **full** if every edge function $f_E(a, b):E_A(a, b) \to E_B(f_V(a), f_V(b))$ is a [[surjection]], and an extensional function is **faithful** if every edge function is an [[injection]]. An extensional function is **[[injective-on-objects]]** if the vertex function $f_V:V_A \to V_B$ is an [[injection]], **[[surjective-on-objects]]** if the vertex function is a [[surjection]], and **[[bijective-on-objects]]** if the vertex function is a [[bijection]].

An extensional function is a **[[dagger functor]]** if additionally it preserves the functions $\mathrm{refl}$, $\mathrm{sym}$, $\mathrm{trans}$
$$f_E(a, a)(\mathrm{refl}_A(a)) = \mathrm{refl}_B(f_V(a))$$
$$f_E(a, b)(\mathrm{sym}_A(a, b)(f)) = \mathrm{sym}_B(f_V(a), f_V(b))(f_E(a, b)(f))$$
$$f_E(a, c)(\mathrm{trans}_A(f,g)) = \mathrm{trans}_B(f_E(a, b)(f), f_E(b, c)(g)$$

An **isomorphism of setoids** is a [[full and faithful]] [[bijective-on-objects]] [[dagger functor]].

### Equivalence of definitions

Given a one-set-of-edges setoid $E\rightrightarrows V$, we define a family-of-sets-of-edges setoid by taking $E(x,y)$ to be the [[preimage]] of $(x,y)$ under the [[function]] $(s,t):E \to V\times V$.  Conversely, given a family-of-sets-of-edges setoid we define a one-set-of-edges setoid by taking $E$ to be the [[disjoint union]] of the families of edges $E = \coprod_{x,y\in V} E(x,y)$.

...

## Examples

The [[morphisms]] $E(a, b)$ of a [[category]] $V$ with a [[contravariant functor|contravariant]] [[endofunctor]] that is the [[identity-on-objects]] is a pseudo-equivalence relation where

* for every vertex $a \in V$, $b \in V$, $c \in V$, and $d \in V$ and edge $f \in E(a, b)$, $g \in E(b, c)$, and $h \in E(c, d)$
$$\mathrm{trans}(a, b, d)(f, \mathrm{trans}(b, c, d)(g, h)) = \mathrm{trans}(a, c, d)(\mathrm{trans}(a, b, c)(f, g), h)$$

* for every vertex $a \in V$, $b \in V$ and edge $f \in E(a, b)$
$$\mathrm{trans}(a, b, b)(f, \mathrm{refl}(b)) = f$$

* for every vertex $a \in V$, $b \in V$ and edge $f \in E(a, b)$
$$\mathrm{trans}(a, a, b)(\mathrm{refl}(a), f) = f$$

This includes [[dagger categories]], where additionally

* for every vertex $a \in V$, $b \in V$ and edge $f \in E(a, b)$
$$\mathrm{sym}(a, b)(\mathrm{sym}(b, a)(f)) = f$$

* for every vertex $a \in V$, $b \in V$, $c \in V$ and edge $f \in E(a, b)$, $g \in E(b, c)$
$$\mathrm{sym}(a, c)(\mathrm{trans}(a, b, c)(f, g)) = \mathrm{trans}(c, b, a)(\mathrm{sym}(a, b)(f), \mathrm{sym}(b, c)(g))$$

* for every vertex $a \in V$
$$\mathrm{sym}(a, a)(\mathrm{refl}(a)) = \mathrm{refl}(a)$$

and [[groupoids]], where

* for every vertex $a \in V$, $b \in V$ and edge $f \in E(a, b)$
$$\mathrm{trans}(a, b, a)(f, \mathrm{sym}(a, b)(f)) = \mathrm{refl}(a)$$

* for every vertex $a \in V$, $b \in V$ and edge $f:E(a, b)$
$$\mathrm{trans}(b, a, b)(\mathrm{sym}(a, b)(f), f) = \mathrm{refl}(b)$$

Additionally, a pseudo-equivalence relation on a set with one vertex is equivalent to a [[pointed set|pointed]] [[magma]] with an [[endofunction]].

## Thin setoids

Recall in graph theory that a loop directed pseudograph is **simple** or a **loop digraph** if for every vertex $a \in V$ and $b \in V$, the set of edges $E(a, b)$ is a [[subsingleton]]: for every edge $f \in E(a, b)$ and $g \in E(a, b)$, $f = g$. In the other definition, a loop directed pseudograph is a loop digraph if the functions $s:E \to V$ and $t:E \to V$ are [[jointly monic]].

A setoid is **thin** or **simple** if its underlying loop directed pseudograph is a loop digraph. In both cases, the pseudo-equivalence relation becomes an **[[equivalence relation]]**. The term "thin" originates from category theory, while the term "simple" originates from graph theory.

A thin setoid is equivalently a [[thin category|thin]] [[dagger category]], or a dagger category [[enriched category|enriched]] in [[truth values]]. A thin setoid is also a thin [[groupoid]], or a groupoid enriched in truth values.

Sometimes in the mathematical literature, setoids are thin by default.

## Core of a pseudo-equivalence relation

For any pseudo-equivalence relation $A$, the [[core]] of $A$ is defined as the maximal [[subgroupoid]] $\mathrm{Core}(A)$ of $A$.

More specifically, a sub-pseudo-equivalence relation of a pseudo-equivalence relation $A$ is a set $G$ with a pseudo-equivalence relation and an faithful injective-on-objects dagger functor $f:G \to A$. A sub-pseudo-equivalence relation $G$ of $A$ is a subgroupoid if $G$ additionally satisfy the groupoid equational axioms:

* for every vertex $a \in V_G$, $b \in V_G$, $c \in V_G$, and $d \in V_G$ and edge $f \in E_G(a, b)$, $g \in E_G(b, c)$, and $h \in E_G(c, d)$
$$\mathrm{trans}(a, b, d)(f, \mathrm{trans}(b, c, d)(g, h)) = \mathrm{trans}(a, c, d)(\mathrm{trans}(a, b, c)(f, g), h)$$

* for every vertex $a \in V_G$, $b \in V_G$ and edge $f \in E_G(a, b)$
$$\mathrm{trans}(a, b, b)(f, \mathrm{refl}(b)) = f$$

* for every vertex $a \in V_G$, $b \in V_G$ and edge $f \in E_G(a, b)$
$$\mathrm{trans}(a, a, b)(\mathrm{refl}(a), f) = f$$

* for every vertex $a \in V_G$, $b \in V_G$ and edge $f \in E_G(a, b)$
$$\mathrm{trans}(a, b, a)(f, \mathrm{sym}(a, b)(f)) = \mathrm{refl}(a)$$

* for every vertex $a \in V_G$, $b \in V_G$ and edge $f:E_G(a, b)$
$$\mathrm{trans}(b, a, b)(\mathrm{sym}(a, b)(f), f) = \mathrm{refl}(b)$$

Every pseudo-equivalence relation has at least one subgroupoid given by only the [[identity morphisms]] $\mathrm{refl}(a):E_A(a, a)$ of $A$. A subgroupoid $G$ of a pseudo-equivalence relation $A$ is a maximal subgroupoid if the dagger functor $f:G \to A$ is bijective-on-objects, and additionally if, for every other subgroupoid $H$ of $A$ with faithful injective-on-objects dagger functor $g:H \to A$, there is a unique faithful injective-on-objects dagger functor $h:H \to G$ such that $h \circ f = g$.

## Category of pseudo-equivalence relations

Let the category [[Set|$Set$]] be defined as a [[category]] that is [[finitely complete category|finitely complete]] and [[well-pointed category|well-pointed]] (i.e. whose [[terminal object]] is a [[extremal epimorphism|extremal]] [[separator|generating object]]). This category of sets is not even a [[regular category]], let alone an [[exact category]], as can happen in certain [[foundations of mathematics]], such as bare [[set-level type theory|set-level]] [[Martin-Löf type theory]], or [[ZFC]] and [[ETCS]] without the [[powerset]] [[axiom]]. The category $\mathrm{PseudoEquivRel}$ of sets with pseudo-equivalence relations is then the [[exact completion|ex/lex completion]] of Set.

More specifically, using the definition of pseudo-equivalence relation with two sets: if $(s_R, t_R):R\rightrightarrows X$ and $(s_S, t_S):S\rightrightarrows Y$ are two pseudo-equivalence relations, a morphism between them in $\mathrm{PseudoEquivRel}$ is defined to be a function $f:X\to Y$ with functions $f_1:R \to S$ with $s_S \circ f_1 = f \circ s_R$ and $t_S \circ f_1 = f \circ t_R$. Moreover, we declare two such functions $f,g:X\to Y$ to be *equal* if there exists a function $h:X\to S$ such that $s \circ h = f$ and $t \circ h = g$. Because $(s_S, t_S):S\rightrightarrows Y$ is a pseudo-equivalence relation, this defines an actual [[equivalence relation]] on the morphisms $f:X\to Y$, which is compatible with composition; thus we have a well-defined category $\mathrm{PseudoEquivRel}$ of pseudo-equivalence relations, which is the ex/lex completion of $Set$.

We have a [[full and faithful functor]] $Set\to \mathrm{PseudoEquivRel}$ sending an object $X$ to the pseudo-equivalence relation $(s_X, t_X):X\rightrightarrows X$. One can then verify directly that $\mathrm{PseudoEquivRel}$ is exact, that this embedding preserves finite limits, and that it is universal with respect to lex functors from $Set$ into exact categories.

If the [[presentation axiom]] (a weak form of the full axiom of choice) holds in $Set$, as a subcategory of $\mathrm{PseudoEquivRel}$, $Set$ could be thought of as the category of [[completely presented sets]], the category of sets with a [[projective presentation]]. If the [[axiom of choice]] holds in $Set$, then $Set$ is equivalent to $\mathrm{PseudoEquivRel}$, as the axiom of choice implies that $Set$ is its own free exact completion, and $Set$ is equivalent to $\mathrm{PseudoEquivRel}$ because the free functor from $Set$ to $\mathrm{PseudoEquivRel}$ is an [[equivalence of categories]].

This construction could be generalized to any [[finitely complete category]] $C$, from which the category of [[internal pseudo-equivalence relations]] of $C$ is the [[ex/lex completion]] of $C$, and denoted as $C_{ex/lex}$. If every [[epimorphism]] in $C$ is additionally a [[split epimorphism]], then $C$ is its own free exact completion.

## In type theory

Many [[set-level type theories]] use [[type|types]] which are [[0-truncated]] and thus [[h-sets]] by default because of the inclusion of [[uniqueness of identity proofs]] or [[axiom K]] to the rule system of the type theory. Sometimes, these type theories do not have [[quotient set|quotients]] and [[image factorizations]], and as a result, setoids are used instead of the native h-sets.

In [[homotopy type theory]], and more generally in any [[intensional type theory]] where not all types are [[h-sets]], such as in a type theory without [[uniqueness of identity proofs]] or [[axiom K]], the notion of "setoid" bifurcates into multiple distinct notions. In analogy with [[category]], the definitions used above in this article could be called a **strict setoid**. When the vertex types are only required to be a [[type]] rather than a [[set]], then this defines a **[[presetoid]]**. There is also the notion of a **[[univalent setoid]]**, where equality of vertices is isomorphism of vertices in the core of a presetoid.

## Related concepts

* [[Bishop set]], [[setoid]]

* [[completely presented set]]

* [[equivalence relation]]

* [[quotient type]]

* [[directed graph]]

* [[pseudo-proset]]

* [[ex/lex completion]]

* [[homotopy exact completion]]

* [[extensional function]]

* [[dagger functor]]

* [[presetoid]]

* [[univalent setoid]]

* [[internal pseudo-equivalence relation]]

[[!include oidification - table]]

[[!include types and logic - table]]

## References

For setoids defined as a set with an pseudo-equivalence relation:

* {#Wilander12} [[Olov Wilander]], *Constructing a small category of setoids*, Mathematical Structures in Computer Science **22** 1 (2012) 103-121 [[doi:10.1017/S0960129511000478](https://doi.org/10.1017/S0960129511000478), [pdf](http://www.diva-portal.org/smash/get/diva2:399799/FULLTEXT01.pdf)]

* {#PalmgrenWilander14} [[Erik Palmgren]], [[Olov Wilander]], *Constructing categories and setoids of setoids in type theory*, Logical Methods in Computer Science, **10** 3 (2014) lmcs:964 [[arXiv:1408.1364](https://arxiv.org/abs/1408.1364), doi:10.2168/LMCS-10(3:25)2014]

* {#EmmeneggerPalmgren20} [[Jacopo Emmenegger]], [[Erik Palmgren]], *Exact completion and constructive theories of sets*, J. Symb. Log. **85** 2  (2020) [[arXiv:1710.10685](https://arxiv.org/abs/1710.10685), [doi:10.1017/jsl.2020.2](https://doi.org/10.1017/jsl.2020.2)]

* {#Cipriano20} Cioffo Cipriano Junior, Def. 1.1.1 in: *Homotopy setoids and generalized quotient completion* [[pdf](https://air.unimi.it/retrieve/d7ab68cc-fc12-4fe9-8a21-617deb888f36/phd_unimi_R12371.pdf), [[Cipriano-HomotopySetoids.pdf:file]]]

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[[!redirects core of pseudo-equivalence relations]]
[[!redirects cores of pseudo-equivalence relations]]


pseudo-extranatural transformation

## Idea

Like an [[extranatural transformation]], but weakened like a [[pseudonatural transformation]].

## References

* Alexander S. Corner, _A universal characterisation of codescent objects_, Theory and Applications of Categories __34__:24 (2019) 684–713, [arxiv:1709.01332](https://arxiv.org/abs/1709.01332)

[[!redirects pseudo-extranatural transformation]]
[[!redirects pseudo-extranatural transformations]]
[[!redirects extra-pseudonatural transformation]]
[[!redirects extra-pseudonatural transformations]]

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pseudo-order

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## Idea

A *pseudo-order* or *strict total order* or *strict linear order* is the irreflexive version of a [[total order]]. This is sometimes called *linear order*, but [[linear order]] is also used to refer to the non-strict [[total order]].

A **pseudo-ordered set** or **strictly totally ordered set** is a [[set]] equipped with a pseudo-order.

In [[classical mathematics]], the distinction between strict total orders and total orders is merely a terminological technicality, which is not always observed; more precisely, there is a [[natural isomorphism|natural bijection]] between the set of total orders on a given set $S$ and the set of strict total orders on $S$, and one distinguishes them by the notation $\lt$ (for the strict total orders) and $\leq$ (for the total order).  In [[constructive mathematics]], however, they are irreducibly different.

## Definitions

A **pseudo-order** or **strict total order** on a set $S$ is a (binary) [[relation]] $\lt$ with the following properties:

* [[irreflexive relation|irreflexivity]]: $x \nless x$;
* [[asymmetric relation|asymmetry]]: if $x \lt y$, then $y \nless x$;
* [[transitive relation|transitivity]]: if $x \lt y \lt z$, then $x \lt z$;
* [[weakly linear relation|weak linearity]]: if $x \lt z$, then $x \lt y$ or $y \lt z $;
* [[connected relation|connectedness]]: if $x \nless y$ and $y \nless x$, then $x = y$.

In classical mathematics, one may see these versions of asymmetry and connectedness:

* $x \nless y$ or $y \nless x$;
* $x \lt y$ or $y \lt x$ or $x = y$.

Using [[excluded middle]], these are equivalent to asymmetry and connectedness as given above, but they need not hold for all pseudo-orders in constructive mathematics.

Actually, these axioms are overkill; to begin with, irreflexivity is simply a special case of asymmetry and so can be dropped.  Additionally, one can either drop transitivity or drop asymmetry (which then requires keeping irreflexivity); they will still follow from the other axioms.  Dropping transitivity shows manifestly the duality (see below) between pseudo-orders and total orders (even in constructive mathematics), and dropping asymmetry shows that a pseudo-order is a weakly linear [[strict preorder]]. Keeping transitivity and irreflexivity (thus allowing one to drop asymmetry) shows manifestly that a pseudo-order is a special kind of [[strict preorder]].

In classical mathematics, there are even more options.  Now weak linearity can be dropped, as it follows from transitivity and connectedness.  Also, one often combines irreflexivity, asymmetry, and connectedness into a single axiom:

* [[trichotomy]]: exactly one of $x \lt y$, $y \lt x$, $x = y$ is true.

Thus the most common definition uses only trichotomy and transitivity.

One can also interpret connectedness not as an axiom but as a definition of [[equality]], getting a pseudo-order on a [[quotient set]] of $S$.  In other words, if $\lt$ is an asymmetric and weakly linear relation on $S$, and we define $x \equiv y$ to mean that neither $x \lt y$ nor $y \lt x$, then $\equiv$ is an equivalence relation and $\lt$ induces a pseudo-order on $S/{\equiv}$.

## Examples

Classically, any [[total order]] defines an example of a pseudo-order (as explained below), and this also holds constructively in [[discrete mathematics]].  Since most pseudo-orders in these cases are usually described in terms of their total orders, the focus here is on [[constructive analysis]].  (The first item, however, is an exception.)

* A [[lexicographic order]], even in a classical or discrete context, is more easily described as a pseudo-order than as a total order.
* The big example in analysis is the field of [[real numbers]].  Both the Dedekind reals and the Cauchy reals (even if [[weak countable choice]] fails so these are not equivalent) have a pseudo-order $\lt$ that extends the (decidable) pseudo-order on the rational numbers.  Since the corresponding partial order $\leq$ cannot be proved total (and in some classically invalid versions of constructive mathematics can even be proved not total), $\lt$ is more directly useful than $\leq$ in constructive analysis.  In any case, $\lt$ is more fundamental, since $\leq$ can be defined in terms of $\lt$ but not the other way around. (The Mac Neille real numbers have both $\lt$ and $\leq$; however, in this case, neither is necessarily a pseudo-order or a total order, nor can they necessarily be defined in terms of one another.)
* [[Baire space of irrational numbers|Baire space]] and [[Cantor space]], being representable as subspaces of the real line, of course are strictly totally ordered.  It\'s also interesting to see them as coming from the (decidable) pseudo-orders on $\mathbf{N}$ and $\mathbf{2}$, which they are $\mathbf{N}$-fold products of.

## Properties

### Relation to total orders

Using excluded middle, one can move between strict linear orders and [[total order]]s using [[negation]]; that is, the negation of a strict linear order is a total order and vice versa.  Actually one usually swaps the order too, as follows:

* $x \lt y$ iff $y \nleq x$;
* $x \leq y$ iff $y \nless x$.

To prove this, it\'s enough to see that the properties of a strict linear order are [[de Morgan duality|dual]] to the properties of a total order, as follows:

| strict linear order  |   | total order    |
| -------------------- | - | -------------- |
| irreflexivity        |   | reflexivity    |
| asymmetry            |   | totality       |
| transitivity         |   | weak linearity |
| weak linearity       |   | transitivity   |
| connectedness        |   | antisymmetry   |

Constructively, these are still the default definitions to use; that is, if one is given a strict linear order or a total order and wants to interpret the other symbol, then one does so using these definitions.  However, the result will not necessarily be a total order or a strict linear order.  To be specific, if one starts with a strict linear order $\lt$ and defines $\leq$ as above, then totality does not follow; and if one starts with a total order $\leq$ and defines $\lt$ as above, then weak linearity does not follow.  Nevertheless, at least $\leq$ will be a [[partial order]], and least $\lt$ will be a [[strict preorder]].  Furthermore, the duality between the axioms is still there, even though negation no longer mediates between them; although weak linearity need not hold for a total order constructively, the duality is preserved if one defines strict linear orders without using transitivity.

One often sees $x \lt y$ defined as $x \le y$ but $x \ne y$; this is equivalent, but doesn\'t show the de Morgan duality explicitly.  Similarly, one often sees $x \leq y$ defined as $x \lt y$ or $x = y$; this is not even equivalent constructively, although it is classically.

Keep in mind, however, that the only use of [[excluded middle]] in the classical theory is the assumption that $x = y$, $x \lt y$, and $x \leq y$ are always either true or false.  There is therefore a perfect correspondence between [[decidable subset|decidable]] strict linear orders and decidable total orders on sets with [[decidable equality]].

## See also

* [[total order]]
* [[strict weak order]]
* [[strict preorder]]
* [[trichotomous relation]]

## References

* Wikipedia, [https://en.wikipedia.org/wiki/Pseudo-order](https://en.wikipedia.org/wiki/Pseudo-order)

* Heyting, Arend (1966). Intuitionism: an introduction (2nd ed.). Amsterdam: North-Holland Pub. Co. p. 106. ISBN:978-0-444-53406-4

[[!redirects pseudo-order]]
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pseudo-ordered ring

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## Idea

A notion of [[ordered ring]] for [[pseudo-orders]]/[[strict total orders]].

## Definition

A **pseudo-ordered ring** or a **strictly totally ordered ring** is an [[ring]] $R$ with a [[pseudo-order]]/[[strict total order]] $\lt$ such that

* $0 \lt 1$

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

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

## Properties

Every pseudo-ordered ring is a [[partially ordered ring]] given by the [[negation]] of the strict total order. In the presence of [[excluded middle]], every pseudo-ordered ring is a [[totally ordered ring]].

Pseudo-ordered rings may have [[zero divisors]]. The pseudo-ordered rings which do not have zero divisors are [[ordered integral domains]].

## Examples

* the integers are a pseudo-ordered ring

* the [[Dedekind real numbers]] are a pseudo-ordered ring which in constructive mathematics cannot be proved to be a [[totally ordered ring]].

## See also

* [[strictly weakly ordered ring]]
* [[partially ordered ring]]
* [[totally ordered ring]]
* [[ordered integral domain]]
* [[ordered field]]

[[!redirects pseudo-ordered ring]]
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pseudo-orthogonal structure

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## Idea

The concept of _pseudo-orthogonal structure_ is the analog of that of _[[orthogonal structure]]_ as one generalized from Euclidean [[signature of a quadratic form|signature]] to [[Lorentz group|Lorentzian]] signature.

Given a [[smooth manifold]] $X$ of [[dimension]] $n$, then a _pseudo-orthogonal structure_ of $X$ is a [[reduction of the structure group]] of its [[frame bundle]] along the inclusion $O(n-1,1) \hookrightarrow GL(n)$ of the [[Lorentz group]] into the [[general linear group]].

In [[physics]],  specifically in the [[first-order formulation of gravity]], such a [[G-structure]] is often called a _[[vielbein]]_ field.

A pseudo-orthogonal structure induces a [[pseudo-Riemannian metric]] on $X$.

## Related concepts

* [[gravity]]

* [[Cartan geometry]]

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pseudo-proset

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## Idea

Pseudo-preorders are to [[preorders]] as [[pseudo-equivalence relations]] are to [[equivalence relations]]. A set with a pseudo-preorder is then a pseudo-preordered set or a pseudo-proset.

Pseudo-prosets are important because they are the basic structure used to build many other structures like [[categories]] and [[setoids]].

## Definition

### With a family of sets

...

### With one set

A **pseudo-preorder** on a set $A$ is a set $E$ of **morphisms** and functions $s:E \to V$, $t:E \to V$ (a loop [[directed pseudograph]]), with functions $\mathrm{id}:V \to E$ and
$$\mathrm{comp}:\{(f,g) \in E \times E \vert t(f) =_V s(g)\} \to E$$
such that

* for every $a \in V$, $s(\mathrm{id}(a)) =_E a$
* for every $a \in V$, $t(\mathrm{id}(a)) =_E a$
* for every $f \in E$ and $g \in E$ such that $t(f) =_V s(g)$, $s(\mathrm{comp}(f,g)) =_E s(f)$
* for every $f \in E$ and $g \in E$ such that $t(f) =_V s(g)$, $t(\mathrm{comp}(f,g)) =_E t(g)$

A **pseudo-preordered set** or **pseudo-proset** $A$ consists of a set $Ob(A)$ with a pseudo-preorder $Mor_A, \mathrm{id}, \mathrm{comp}$.

## Examples

A [[category]] is a pseudo-proset $A$ which additionally satisfies

* for every object $a \in Ob(A)$, $b \in Ob(A)$, $c \in Ob(A)$, and $d \in Ob(A)$ and edge $f \in Mor_A(a, b)$, $g \in Mor_A(b, c)$, and $h \in Mor_A(c, d)$
$$\mathrm{comp}(a, b, d)(f, \mathrm{comp}(b, c, d)(g, h)) = \mathrm{comp}(a, c, d)(\mathrm{comp}(a, b, c)(f, g), h)$$

* for every vertex $a \in Ob(A)$, $b \in Ob(A)$ and edge $f \in Mor_A(a, b)$
$$\mathrm{comp}(a, b, b)(f, \mathrm{id}(b)) = f$$

* for every vertex $a \in Ob(A)$, $b \in Ob(A)$ and edge $f \in Mor_A(a, b)$
$$\mathrm{comp}(a, a, b)(\mathrm{id}(a), f) = f$$

## See also

* [[preorder]]
* [[setoid]]
* [[category]]

[[!redirects pseudo-prosets]]
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pseudo-Riemannian manifold

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## Idea

Like a [[Riemannian manifold]] is a smooth [[manifold]] equipped with a [[Riemannian metric]], a pseudo-Riemannian manifold is a [[manifold]] equipped with a [[pseudo-Riemannian metric]].  [[Lorentzian manifolds]], as well as Riemannian manifolds themselves, are special cases of pseudo-Riemannian manifolds.

## Related concepts

* [[Lorentzian manifold]]

* [[causal manifold]]

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pseudo-Riemannian metric

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## Idea

The notion of _pseudo-Riemannian metric_ is a slight variant of that of _[[Riemannian metric]]_.

Where a Riemannian metric is governed by a positive-definite [[bilinear form]], a Pseudo-Riemannian metric is governed by an indefinite bilinear form.

This is equivalently the [[Cartan geometry]] modeled on the inclusion of a [[Lorentz group]] into a [[Poincaré group]].

## Definition

A **pseudo-Riemannian "metric"** is a nondegenerate [[quadratic form]] on a real vector space $\mathbb{R}^n$. A [[Riemannian metric]] is a positive-definite quadratic form on a real vector space. The data of such a quadratic form may be equivalently given by a nondegenerate symmetric bilinear pairing $\langle\, , \, \rangle$ on $\mathbb{R}^n$.

A pseudo-Riemannian metric

$$Q: \mathbb{R}^n \to \mathbb{R}$$

can always be diagonalized: there exists a basis $e_1, \ldots, e_n$ such that

$$Q(\sum_{1 \leq i \leq n} x_i e_i) = x_1^2 + \ldots + x_p^2 - x_{p+1}^2 - \ldots - x_n^2$$

where the pair $(p, n-p)$ is called the **signature** of the form $Q$. Pseudo-Riemannian metrics on $\mathbb{R}^n$ are classified by their signatures; thus we have a standard metric of signature $(p, n-p)$ where $\{e_1, \ldots, e_n\}$ is the standard basis of $\mathbb{R}^n$.

More generally, there is a notion of **pseudo-Riemannian manifold** (of type $(p, n-p)$, which is an $n$-dimensional [[manifold]] $M$ equipped with a global section

$$\sigma: M \to S^2(T^* M)$$

of the bundle of symmetric bilinear forms over $M$, such that each $\sigma(x)$ is a nondegenerate form on the
tangent space $T_x(M)$.

Certain theorems of Riemannian geometry carry over to the more general pseudo-Riemannian setting; for example, pseudo-Riemannian manifolds admit [[Levi-Civita connection]]s, or in other words a unique notion of covariant differentiation
of vector fields

$$\nabla: (X, Y) \mapsto \nabla_X(Y)$$

such that

$$X \cdot \langle Y, Z\rangle = \langle \nabla_X Y, Z\rangle + \langle Y, \nabla_X Z\rangle$$

$$[X, Y] = \nabla_X Y - \nabla_Y X$$

In that case, one may define a notion of [[geodesic]] in pseudo-Riemannian manifolds $M$, and we have a notion of "distance squared" between the endpoints along any geodesic path $\alpha: [0, 1] \to M$ (which might be a negative number of course). The term "pseudo-Riemannian metric" may refer to such distances in general pseudo-Riemannian manifolds. _(I guess.)_

A typical example of pseudo-Riemannian manifold is a [[Lorentzian manifold]], where the metric is of type $(1, n-1)$. This is particularly so in the case $n = 4$, where such manifolds are the mathematical backdrop for studying [[general relativity]] and cosmological models.

* The terminology "metric" is not optimal of course: the values of the quadratic form would need to be nonnegative to avoid terminological conflict with [[metric space|metric]] as it is more commonly understood (and even in that case, the values of "metric" refer to the square of the metric rather than the metric itself). _Caveat lector._

## Related concepts

[[!include local and global geometry - table]]

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