posite

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

# Posites
* table of contents
{: toc}

## Idea

The notion of  _posite_ (or _$(0,1)$-site_, less properly _$0$-site_) is a [[decategorification]] of the notion of _[[site]]_ (which may also be called an $1$-site or $(1,1)$-site). For [[categorifications]] thereof, see [[(∞,1)-site]] and [[2-site]].

Just as a site is a [[category]] with a [[coverage]], whose ([[set]]-valued) [[sheaves]] form a [[Grothendieck topos]], so a posite is a [[poset]] with a coverage, whose $(0,1)$-sheaves ([[truth value]]-valued sheaves) form a [[locale]].  On the other hand, we can also consider a posite as a site in its own right, in which case the $1$-sheaves on it form a [[localic topos]].

As locale theory serves as an approach to [[topology]] in which locales take the role traditionally held by [[topological spaces]], so posites take the role traditionally held by [[topological bases]].

A [[formal topology]] may be generated by a posite equipped with a [[positivity predicate]].

## Definitions

+-- {: .un_defn}
###### Definition (posite)

A __posite__ is a [[site]] whose underlying category is [[thin category|thin]].
=--

In other words, a posite is a [[poset]] (or more generally a [[preordered set]]) equipped with a [[Grothendieck topology]] or a [[coverage]].  (The name is a portmanteau/blend of 'poset' and 'site'.)

The definition of coverage may be simplified a little in this case.

+-- {: .un_defn}
###### Definition (coverage on a poset)

Let $S$ be a poset (or proset).  A __coverage__ on $S$ is a [[binary relation]] $\lhd$ between $S$ and its [[power set]] that satisfies these conditions:

*  If $u \lhd V$ and $v \in V$, then $v \leq u$;
*  If $u \lhd V$ and $u' \leq u$, then $u' \lhd V'$ for some $V'$ such that, for every $v' \in V'$, there is a $v \in V$ such that $v' \leq v$.

Then a __posite__ is precisely a poset (or proset) equipped with a coverage.
=--

If $S$ is a [[meet-semilattice]], then there is another alternative defintion.  (Actually, it is not necessary that the [[poset]] $S$ have *all* finitary [[meets]] but only *[[bounded meet|bounded]]* ones.  That is, if $v, w \leq u$ for some fixed $u$, then $v \wedge w$ must exist, but not otherwise.  Such a poset $S$ is a semilattice if and only if it has a [[top element]].  Compare the notion of [[locally cartesian category]].)

+-- {: .un_defn}
###### Definition (cartesian coverage on a locally cartesian posite)

Let $S$ be a poset (or proset) with all bounded binary meets.  A __cartesian coverage__ on $S$ is a [[binary relation]] $\lhd$ between $S$ and its [[power set]] that satisfies these conditions:

*  If $u \lhd V$ and $v \in V$, then $v \leq u$;
*  If $u \lhd V$ and $u' \leq u$, then $u' \lhd \{ v \wedge u' \;|\; v \in V \}$.

Then a __locally cartesian posite__ is such a poset (or proset) equipped with such a coverage.
=--

It is a theorem that every posite whose underlying poset has all bounded binary meets is equivalent (in the sense that there is an isomorphism of sites as defined below) to a locally cartesian posite.

In any case, given an element $u$ of $S$ and a [[subset]] $V$ of $S$, if
$$ u \lhd V ,$$
then we say that $V$ is a __basic [[cover]]__ of $u$.

+-- {: .un_defn}
###### Definition (morphism of posites)

Given two posites $S$ and $T$, a __morphism of posites__ from $S$ to $T$ is a [[function]] $f\colon S \to T$ such that:

*  If $u \leq v$ in $S$, then $f(u) \leq f(v)$ in $T$;
*  Given any $u$ in $T$, for some $v$ in $S$, we have $u \leq f(v)$;
*  If $w \leq f(u), f(v)$ in $T$, then for some $x \leq u, v$ in $S$, we have $w \leq f(x)$; and
*  If $u \lhd V$, then $f(u) \lhd \{ f(v) \;|\; v \in V \}$.
=--

Clauses (1--3) simply state that $f$ is a [[flat functor]]; when $S$ and $T$ are semilattices, we can replace this with the requirement that $f$ preserves meets (including the top element).  Clause (4) requires $f$ to respect covers.

## Ideals and sheaves

One often looks at [[sheaves]] on [[sites]].  On posites, one can either look at sheaves or at ideals.

+-- {: .un_defn}
###### Definition (ideal)

Given a posite $S$, an __ideal__ on $S$ (also called a __$(0,1)$-sheaf__; compare $(\infty,1)$-[[(infinity,1)-sheaf|sheaf]]) is a [[subset]] $I$ of $S$ satisfying these conditions:

1.  If $u \leq v$ and $v \in I$, then $u \in I$; and
2.  If $u \lhd V$ and $V \subseteq I$, then $u \in I$.
=--

In other words, it is a [[sheaf]] valued in [[truth values]] (rather than in [[sets]] as is the default).

A subset $I$ that satisfies condition (1) alone states that $I$ is a __[[lower subset]]__ (a [[(0,1)-presheaf]]).

The ideals on a posite form a [[frame]] $Id(S)$ (or $Sh_{0,1}(S)$) under inclusion, which may alternately be interpreted as a [[locale]].

Every element of $S$ may be interpreted as an ideal on $S$; given two elements $u,v$ of $S$, $u$ belongs to the __ideal [[represented functor|represented]]__ by $v$ if and only if $u \leq v$; that is, $v$ represents its [[down set]].

The frame of ideals is given by a [[universal property]].  If $L$ is a frame, we make $L$ into a posite using its canonical coverage (see the examples below).  Then the operation from elements of $S$ to ideals on $S$ is a morphism of posites; furthermore, given any morphism of posites from $S$ to a frame $L$ (with its canonical coverage), there exists a unique frame [[homomorphism]] from $Id(S)$ to $L$ that makes the obvious triangle commute.

If instead we treat a posite as a special kind of site and look at its $1$-sheaves, we have this result:

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

The [[topos of sheaves]] on any posite $P$ is a [[localic topos]].
=--

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

This is obvious if we use the definition of "localic topos" which says that it is generated by [[subterminal objects]], since the (sheafifications of) representable functors always generate a topos of sheaves, and for a posite these are subterminal.
=--

The relevant locale is precisely the locale of ideals.

## Interpretations

Given a posite $S$, we may think of the elements of $S$ as __basic opens__ of a space which is generated from $S$ using its coverage, to produce a [[locale]].  Thus a posite is precisely a __base__ for a locale.  See the example coming from a topological space below to see how this can work precisely for a [[topological locale]].

A coverage can be seen as a [[sequent]] calculus; the interpretation of
$$ u \lhd V $$
in topology is analogous to the interpretation of
$$ p \lhd Q $$
in [[logic]].  Note that we have a single proposition on the left but a set of propositions (interpreted [[disjunction|disjunctively]]) on the right, the reverse of an [[intuitionistic logic|intuitionistic]] sequent.

## Examples

### Using ideals

Any poset (or proset) $S$ has a __[[canonical coverage]]__, in which

*  $u \lhd V$ if and only if, for any $u' \leq u$, $u'$ is a [[join]] of some $V'$ such that, for every $v' \in V'$, there is a $v \in V$ such that $v' \leq v$.

If $S$ is a [[frame]], then we can simplify this condition:

*  $u \lhd V$ if and only if $u$ is a join of $V$;

and then the canonical coverage is cartesian.  In this case, $Id(S)$ is [[natural isomorphism|naturally isomorphic]] to $S$ itself.  (Analogously, if $E$ is a [[Grothendieck topos]] with its canonical coverage, then $Sh(E)$, the [[topos of sheaves]] on $E$, is naturally [[equivalence of categories|equivalent]] to $E$.)

Let $X$ be a [[topological space]], and let $S$ be a [[topological base]] of $X$.  Then $S$ is a semilattice under [[intersection]]; we make $S$ into a posite by defining

*  $u \lhd V$ if and only if $u$ is the [[union]] of $V$ in $X$ (which is not necessarily the same as a join in $S$).

The frame generated by this posite is [[natural isomorphism|naturally isomorphic]] to the [[frame of opens]] of $X$.  If $X$ is a [[sober space]], then this frame, interpreted as a [[locale]], may be identified with $X$.

The [[locale of real numbers]] is generated from the locally cartesian poset $S \coloneqq \mathbb{Q}^op \times \mathbb{Q}$, where $\mathbb{Q}$ is the usual poset of [[rational numbers]], equipped with the cartesian coverage given by

*  $(x,y) \lhd \empty$ whenever $x \geq y$;
*  $(w,z) \lhd \{(w,x),(y,z)\}$ whenever $w \leq y \lt x \leq z$; and
*  $(w,z) \lhd \{ (x,y) \;|\; w \lt x \lt y \lt z \}$.

Often you will see $\pm\infty$ added to $\mathbb{Q}$ in a description of this locale, which adds a [[top element]] to $S$ and so makes it into a semilattice, but that makes no difference to the generated locale in this case.

### Using sheaves

The [[double negation topology]] used in the sheaf-theoretic approach to [[forcing]] is a posite.  More generally, [[classical mathematics|Classical]] [[set theory|set-theoretic]] [[forcing]] is done exclusively on posites.  One "reason" for this (in the sense of a reason why set-theorists have not been forced to look beyond such models) is explained in [the paper by Freyd](#Freyd) below.

The [[semilattice of commutative subalgebras]] of a $C^*$-[[C-star algebra|algebra]] $A$ is a posite (with trivial topology) internal to whose sheaf topos we have a [[constructive Gelfand duality theorem]] for $A$, even if $A$ was externally non-commutative.

The [[classifying topos]] of any [[propositional logic|propositional]] [[geometric theory]] is localic, and can be presented as sheaves on a posite of [[propositions]].

## Use in predicative mathematics

We call a frame (or locale) $L$ __accessible__ if it is [[isomorphism|isomorphic]] to $Id(S)$ for some [[small category|small]] posite $S$.  (Compare the concept of [[accessible category]], in the case of a [[cocomplete category|cocomplete]] [[pretopos]].)

In [[classical mathematics]], an accessible frame must be small, but this fails in [[predicative mathematics]].  (Conversely, any small frame is trivially accessible; take $S$ to be $L$ with its canonical coverage.)  Since many predicativists have philosophical objections to working with large objects at all, they may prefer to work with small posites directly.  Whatever the philosophy, we may use small posites in place of accessible locales, or at any rate use this to prove that the category of the latter is [[essentially moderate category|essentially moderate]].

[[formal topology|Formal topology]] is a programme for [[topology]] which is based on using small posites.  However, formal topologists also require a positivity predicate on their posites; the intended interpretation is that $u$ is positive iff its corresponding [[open subspace]] is [[inhabited space|inhabited]].  This seems to be needed for a good theory of [[compact spaces]] (and related notions) in a predicative constructive framework.

## References

*  Most of this article is based on _[[Stone Spaces]]_, but with a different presentation.

On [[forcing]], see:

*  [[Peter Freyd]], _All topoi are localic or why permutation models prevail_, Journal of Pure and Applied Algebra 46 (1987) pp 49--58.
{#Freyd}

*  [This MO question](http://mathoverflow.net/questions/13480/set-theoretic-forcing-over-sites) and its answer.

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posite > history

< [[posite]]


position

Position is perhaps the most easily-measured [[observable]].

It is one of the basic [[coordinates]] in both [[Lagrangean mechanics]] (based on [[state space]], given by position and [[velocity]]) and in [[Hamiltonian mechanics]] (based on [[phase space]], given by position and [[momentum]]).

See also [[Position, Velocity, and Acceleration]].

## Related concepts

* [[speed]]

[[!redirects positions]]


position space

In physical applications, a differential manifold $M$ is often used to parametrize the possible positions of particles in a system. In such [[models]], $M$ is often called the "position space".

Manifolds and bundles which are derived from $M$ also receive convenient names. The tangent bundle $TM$ is called the "configuration space", since the configuration of the system is specifying positions and velocity vectors. The cotangent bundle $T^* M$ is called the "phase space" (why?).

A partial list follows.

| Manifold  | Manifold Name |
| ------------- | ------------- |
| $M$ | Position Space |
| $TM$ | Configuration Space |
| $T^* M$ | Phase Space |

Reference: Betancourt and Stein. The Geometry of Hamiltonian Monte Carlo


Position, Velocity, and Acceleration

#Idea#

Consider three paths that begin at $x_0$ and end at $x_{2\tau}$ enclosing the spacetime region depicted below

$$\array{
{} & {} & {} & x_{2\tau} & {} & {} & {} \\
{} & {} & {} &  \bullet     & {} & {} & {} \\
{} & {} & {}^{v^-_{\tau}}\nearrow & {} & \nwarrow^{v_{\tau}^+} & {} & {} \\
x^-_{\tau} & \bullet & \stackrel{F^-}{\Uparrow} & \stackrel{v_0}{\uparr} & \stackrel{F^+}{\Uparrow} & \bullet & x^+_{\tau} \\
{} & {} & {}_{v_0^-}\nwarrow & {} & \nearrow_{v_0^+} & {} & {} \\
{} & {} & {} &  \bullet & {} & {} & {} \\
{} & {} & {} &    x_0   & {} & {} & {}
}$$

Staring at the above picture, we want to understand the analogy:

$$\array{
category & functor & natural transformation \\
fiber & connection & curvature \\
position & velocity & acceleration
}$$

For those who better understand the mathematics of the first line, this analogy is likely to help motivate the physics as we go down the ladder. For those who better understand the physics of the bottom line, it is hoped that this analogy will help to motivate the mathematics as we move up the ladder.

#Formulation#

First, due to the symmetry of the diagram above, we can consider the simplified region

$$\array{
{} & {} & {} & x_{\tau} & {} & {} & {} \\
{} & {} & {} &  \bullet     & {} & {} & {} \\
{} & {} & {}^{v_0}\nearrow & {}_F\Rightarrow & \searrow^{v_{\tau}} & {} & {} \\
x_0 & \bullet & {} & \stackrel{v}{\longrightarrow} & {} & \bullet & x_{2\tau}
}$$

This diagram represents a very simple strict 2-category we will denote by $P^2(\mathbb{D}^2)$ with

* Objects $\{x_0,x_\tau,x_{2\tau}\}$
* Morphisms $\{x_0\stackrel{v_0}{\to}x_{\tau},x_\tau\stackrel{v_\tau}{\to}x_{2\tau},x_0\stackrel{v}{\to}x_{2\tau}\}$
* 2-Morphism $\{v_0\stackrel{F}{\Rightarrow}v_\tau\}$

#Discussion#

_John said (from [n-Cafe](http://golem.ph.utexas.edu/category/2009/01/the_third_time_is_the_charm.html#c021749)):_

Todd wrote:

Position, velocity, acceleration; Newton?

YES!!!

Each one is clearly obtained from the previous one in a systematic way, and Newton invented this way — differentiation!

But, he needed differentiation not for first derivatives, but for second derivatives.  His grand discovery was this: the laws of motion become clear not when we try to explain an object's position, nor when we try to explain its velocity, but only when we try to explain its acceleration!

So, just as Eilenberg and Mac Lane needed categories to define functors, and needed functors to define natural transformations, so they could make precise sense of cohomology theories being 'naturally isomorphic' ...

... Newton needed position to define velocity, and velocity to define acceleration, so he could make precise sense of

$$F=ma.$$

There's even a nice analogy between these two trios.  An category is a place for an object to sit.  A functor is a way of changing an object.  A natural transformation changes a way of changing an object.

I haven't quite figured out how to exploit this analogy.  It's interesting to note that an n-category internal to Vect is automatically a chain complex and is thus related to second derivatives by the formula

$$d^2=0.$$

But somehow I should get natural transformations (chain homotopies) into the game.

Anyway, it might be a sign of genius for somebody to study a trio of concepts, each systematically derived from the previous one, where the third one is the charm.

***

_Urs said (from [n-Cafe](http://golem.ph.utexas.edu/category/2009/01/the_third_time_is_the_charm.html#c021768)):_

Let $G$ be a Lie group – and let's concentrate here on $G=\mathbb{R}$ to get closest to the Newtonian example regarded as a 1-object groupoid $\mathbf{B}\mathbb{R}$.

Functors from $\mathbf{B}\mathbb{R}$ to itself and natural transformations between these arrange themselves into the smooth 2-groupoid $\mathbf{BE}(\mathbb{R})$.

To find the infinitesimal version of this triple gadget (object = position, morphism = functor, 2-morphism = natural transformation) we proceed the way one differentiates any Lie group: we map smooth 1- and 2-parameter paths into this:

So let $P^2(\mathbb{R}^2)$ be the strict 2-groupoid of smooth paths in the plane and consider smooth 2-functors

$$F:P^2(\mathbb{R}^2)\to\mathbf{BE}(\mathbb{R}).$$

These 2-functors are in bijection with 1-forms $A$ on $\mathbb{R}^2$.

The way the equivalence works is precisely the Newtonian example:

Regarding the value of the functor $F$ on a path $\gamma$ in $\mathbb{R}^2$ as a trajectory in $\mathbb{R}$, $A$ gives the velocity of this trajectory with respect to "parameter time". For instance if you think of the standard speed parameterized path $\gamma:t\mapsto (t,0)$ in $\mathbb{R}^2$ and write $A=A_t dt$, then $A_t=v$ is precisely the velocity of the trajectory.

And the value of the 2-functor $F$ on 2-morphisms then, i.e. its value in natural transformations of functors on $\mathbf{B}\mathbb{R}$, is indeed given by the ("exterior") acceleration $dA$.

This example, incidentally, when we pass from $\mathbb{R}$ to more general groups and include linear representations, then also subsumes the first Trimblean triple as a "special case" of "category-functor-natural transformation".

***

_Urs then said (from [n-Cafe](http://golem.ph.utexas.edu/category/2009/01/the_third_time_is_the_charm.html#c021795)):_

Okay, let me try to find a setup in which this becomes very obvious.

First of all, the more obvious part of the analogy is

$$\array{
category & functor  & natural transformation \\
fiber & connection & curvature
}$$

so let me start with that.

Here a simple situation is that where

* each fiber looks like a group $G$, regarded as a one-object groupoid (hence a category), which I write as $\mathbf{B}G$;
* all functors go from $\mathbf{B}G$ to itself;
* and the natural transformations in question are the natural transformations between these functors.

And we want to assume that $G$ is a Lie group, so that we can proceed with the analogy which clearly presupposes a smooth setup.

In this case, the main subtlety in making sense of the above analogy comes from the existence of outer automorphisms on $G$. So, in the spirit of making everything as clear as possible, let me assume a situation where these vanish. For instance assume $G=E_8$ (i.e. the compact real form of $E_8$).

Then (I am saying this to be self contained for the record and for other readers) the relation between the two lines in the above analogy is established by the notioin of smooth 2-functors

$$tra:P^2(X)\to Cat$$

from the 2-groupoid $P^2(X)$ whose

* objects are points in a manifold $X$;
* morphisms are classes of smooth paths in $X$;
* 2-morphisms are classes of smooth surfaces in $X$.

Assume this 2-functor $tra$ assigns

1. to each point the fiber of a $G$-principal bundle $P$ over $X$, which we can locally identify with $G$ and hence with the category $\mathbf{B}G$
$$\array{
BG \\
category \\
fiber \\
P x
}$$
1. to each path $\gamma:x\to y$ a functor from the fiber over $x$ to the fiber over $y$.
+-- {: .un_theorem #Connection}
###### Theorem
Such smooth assignments of functors to paths are in bijection with connections $\nabla$ on the $G$-principal bundle: upon identifying the fibers with $\mathbf{B}G$ the functor is given by an automorphism of $G$ hence, by assumption, by an element of $G$, and this element is the parallel transport $P\exp(\int_\gamma \nabla)$.
=--
$$\array{
\mathbf{B}G & BG\stackrel{tra(\gamma)}{\to} \mathbf{B}G \\
category & functor \\
fiber & connection \\
P_x & \nabla
}$$
1. to each surface $\Sigma:\gamma\to\gamma':x\to y$ a _natural transformation_ of functors $tra(\gamma)\to tra(\gamma')$. Since $\mathbf{B}G$ has a single object, such natural transformations are given by elements of $G$ once we identify the fibers with $\mathbf{B}G$.
+-- {: .un_theorem #Curvature}
###### Theorem
This element of $G$ is the group element
$P_\nabla\exp(\int_\Sigma F_\nabla)$
where $F_\nabla$ is the curvature 2-form of $\nabla$ and $P_\nabla\exp(\int \cdots)$ denotes the surface ordered exponential integral of a 2-form with respect to $\nabla$ (as appearing in the non-abelian Stokes theorem).
=--

$$\array{
\mathbf{B}G & \mathbf{B}G\stackrel{tra(\gamma)}{\to}\mathbf{B}G & tra(\gamma)\stackrel{\exists !}{\Rightarrow}tra(\gamma') \\
category & functor & natural transformation \\
fiber & connection & curvature \\
P_x & \nabla & F_\nabla
}$$

So that's how this works. Before going into the further analogy

$$\array{
category & functor & natural transformation \\
fiber & connection & curvature \\
position & velocity & acceleration
}$$

let's massage the above kind of observation into a form that we are all happy with.

***

_Urs then said (from [n-Cafe](http://golem.ph.utexas.edu/category/2009/01/the_third_time_is_the_charm.html#c021833)):_

Hi John,

So maybe we can focus on the next stage […]

Okay. In my comment which started this discussion I essentially pointed out that:

the bijection/equivalence

(smooth 1-functors from paths in $X\to$ endomorphisms of $G$) $\simeq$ (1-forms)

is – manifestly the way one proves it – controlled by the concept of velocity:

if you think of $X$ as 1-dimensional, then the 1-form on $X$ with values in $Lie(G)$ is nothing but the gadget which reads in a bit of "coordinate time" for a trajectory on $G$ and spits out the velocity tangent vector of that trajectory with respect that coordinate time.

This is really what "parallel transport" of 1-forms means: find the trajectory such that its velocity is at every point prescribed by the given 1-form.

Think in the case $X=\mathbb{R}$ of the functor as describing a particle which propagates on a group manifold. The fact that the functor is smooth says that we have a smooth trajectory. The fact that it is functorial means that this trajectory is obtained by integrating a velocity 1-form.

Now, in my comment which started this discussion I next said that on 2-cells our parallel transport computes the "exterior acceleration": given a 1-parameter family of trajectories, the 2-form that controls the value of a smooth 2-functor on it is the exterior differential of the "velocity 1-form".

At first it is admittedly a bit of a stretch as far as the Newtonmian concept of acceleration goes to call this curvature 2-form the "acceleration" here, though it does consist of second derivatives of the original trajectories.

But then notice the Newton-force law for charged relativitic particles: the Lorentz-force law. it says that the acceleration bivector  $\dot{v} v$ of the relativistic particle equals the (electric component of) the curvature 2-form. So we have

$$\array{
fiber & inner degrees of freedom at given position \\ connection & action functional in terms of velocity \\ curvature & force law in terms of acceleration.
}$$

There must be better ways to say this. But maybe this helps to indicate what I am thinking of.

#References#

* [Connections on non-abelian Gerbes and their Holonomy](http://arxiv.org/abs/0808.1923)
* [Orbifold String Topology: Paths in Smooth Categories](http://golem.ph.utexas.edu/string/archives/000735.html)
* [A connection whose curvature is the Lie bracket](http://arxiv.org/abs/0803.3321)
* [Spacetime Calculus](http://modelingnts.la.asu.edu/html/STC.html)

category: drafts


positive characteristic

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

#Contents#
* table of contents
{:toc}

## Idea

To work "in positive characteristic" means to work over a [[ground ring]]/[[field]] of [[positive number|positive]] [[characteristic]].

Whereas much of "classical" mathematics took place/takes place over [[fields]] of [[characteristic]] 0, such as the [[real numbers]] and the [[complex numbers]], much of modern mathematics and in particular [[number theory]] and [[arithmetic geometry]] crucially incorporates also the "exotic" case of positive characteristic, such as [[finite fields]].

## Related concepts

* [[characteristic zero]]

* [[p-adic numbers]]

* [[supersingular elliptic curve]]

* [[number theory]], [[arithmetic geometry]]

## References

* Anatoly N. Kochubei, _Analysis in Positive Characteristic_, Cambridge Tracts in Mathematics, 2009


positive cone

# Positive cones
* table of contents
{: toc}

## Idea

Geometrically, the positive elements of a [[vector space]] with a [[partial order]] form a [[cone]], called the positive cone.  The concept makes sense even more generally but is particularly important in [[operator algebras]].

## Definition

Let $V$ be a [[partially ordered set]] and let $0$ be an [[element]] of $V$.  Then the __positive cone__ $V^+$ of the pointed poset $(V,0)$ is the [[up set]] of $0$:
$$ V^+ \coloneqq \{x\colon V \;|\; 0 \leq x\} .$$

If $V$ is an [[operator algebra]], or more generally an [[ordered group]], then we use the usual [[identity element]] $0$ here.

## Properties

Let $V$ be an [[ordered group]]; that is, $V$ is a poset (as above) and also a [[group]] (written additively but not necessarily [[abelian group|commutative]]) with this compatibility property:

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

Then the positive cone $V^+$ satisfies these properties:

*  If $a, -a \in V^+$ (where $-a$ is the [[inverse element|inverse]] of $a$), then $a = 0$;
*  Conversely, $0 \in V^+$;
*  If $a, b \in V^+$, then $a + b \in V^+$;
*  If $a \in V^+$, then $b + a - b \in V^+$ (which is trivial if $V$ is commutative).

Conversely, if $V^+$ is any subset of the group $V$ with these properties, then $V$ becomes an ordered group with either of these equivalent definitions:

*  $a \leq b$ iff $b - a \in V^+$,
*  $a \leq b$ iff $-a + b \in V^+$.

In this way, we have a [[bijection]] between ordered group structures and positive cones in a group.

## The extended positive cone

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

The [[finite measures]] on a given [[measurable space]] form an ordered vector space $V$, and the positive cone $V^+$ consists precisely of the finite [[positive measures]].  But we often want to allow positive measures to take infinite values.  The space of (possibly infinite) positive measures is the _extended_ positive cone of $V$.
=--

I only know the general definition in some rather limited cases:

+-- {: .num_defn}
###### Definition (positive cone of a von Neumann algebra)

Let $V$ be a $W^*$-[[W-star-algebra|algebra]], and let $V_*$ be its [[predual]].  Recall that $V$ is the space of [[continuous map|continuous]] [[linear maps]] from $V_*$ to the [[base field]].  The __extended positive cone__ $\bar{V}^+$ of $V$ is the space of lower [[semicontinuous map|semicontinuous]] linear maps from the positive cone $V_*^+$ of $V_*$ to the space $\bar{\mathbb{R}}^+ = [0,\infty]$ of [[extended real number|extended]] positive [[real numbers]].
=--

The extended positive real numbers are really showing up in their guise as the nonnegative [[lower real numbers]] (the appropriate targets for a lower semicontinuous map), and the extended positive cone is really a generalisation of the nonnegative upper reals.  In particular, the extended positive cone of $\mathbb{R}$ itself is $[0,\infty]$.

This doesn\'t include the motivating example, but the following generalisation does:

+-- {: .num_defn}
###### Definition (positive cone of an ordered module over a von Neumann algebra)

Let $V$ be a [[module]] over the $W^*$-algebra $A$, and let $V$ have the structure of an ordered group such that the action of $A$ preserves order (in that a positive element acting on a positive element gives a positive element).  Then the __extended positive cone__ $\bar{V}^+$ of $V$ consists of formal infinitary $\bar{A}^+$-[[linear combination]]s of positive elements of $V$ modulo the (hopefully) obvious [[equivalence relation]].
=--

When applied to the space of finite measures on a [[localisable measurable space]] $X$ (acted on by the $W^*$-algebra $L^\infty(X)$), this should give the positive measures on $X$ (but I need to check the details).

## References

The extended positive cone of a $W^*$-algebra is Definition 4.4 of:

*  [[Masamichi Takesaki]], Theory of Operator Algebras II, Springer; [Google books](http://books.google.com/books?id=-4GyR1VlQz4C&pg=PA215&dq="extended+positive+part").

[[!redirects positive cone]]
[[!redirects positive cones]]

[[!redirects extended positive cone]]
[[!redirects extended positive cones]]
[[!redirects extended positive part]]
[[!redirects extended positive parts]]


positive element

# Positive elements
* table of contents
{: toc}

## Idea

In [[classical mathematics]], an element of a [[poset]] is positive if and only if it not a [[bottom element]].

A more subtle definition is needed in [[constructive mathematics]]; analogously to how "nonempty sets" may need to be replaced by [[inhabited sets]].  In mathematics that is both constructive and [[predicative mathematics|predicative]], positivity can be axiomatised but not defined.

## Definitions

Let $L$ be a [[preordered set]], and let $x$ be an element of $L$.

+-- {: .num_defn #preddef}
###### Definition
**(in [[classical mathematics]], and in [[predicative mathematics]] with [[classical logic]])**

An element $x \in L$ is __positive__ if it is not a [[bottom element]], equivalently if there exists an element $y \in L$ such that $x \leq y$ is false: $\neg(x \leq \bigvee \empty)$.
=--

+-- {: .num_defn #constdef}
###### Definition
**(in [[classical mathematics]], and in impredicative [[constructive mathematics]])**

An element $x$ of $L$ is __positive__ if whenever $x$ is the [[join]] of some [[subset]] $A$ of $\{x\}$ in $L$, $A$ is [[inhabited subset|inhabited]] (and so all of $\{x\}$): $\forall P\colon Prop,\; x = \bigvee \{x \;|\; P\} \;\Rightarrow\; P$.
=--

It follows that any arbitrary subset $A$ of $L$ is inhabited if $x$ is the join of $A$.  (This is because if $x = \bigvee A$, then $x = \bigvee \{x \;|\; A\ is\ inhabited\}$.)  But it\'s important that the definition only quantifies over [[truth values]], which is why this definition is classically predicative, and may even be acceptable to some constructive predicativists who don\'t accept [[function sets]].

+-- {: .num_defn #predconstdef}
###### Definition
**(in [[predicative mathematics|predicative]] [[constructive mathematics|constructive]] mathematics)**

A **positivity predicate** on $L$ is a [[predicate]] $\lozenge{x}$, pronounced "$x$ is positive", such that

*  If $x$ is the join of a subset and $x$ is positive, then some element of that subset is positive: $\lozenge{x} \;\wedge\; x = \bigvee A \;\vdash\; \exists\, u \in A,\; \lozenge{u}$.

*  If $x$ equals a join on the assumption that $x$ is positive, then $x$ really is that join: $\lozenge{x} \;\Rightarrow\; x = \bigvee A \;\vdash\; x = \bigvee A$.
=--

These definitions are equivalent under certain circumstances.  Specifically:

* In predicative mathematics with classical logic, one can prove that every preorder has a unique positivity predicate, which must match Definition \ref{preddef}.

* In impredicative constructive mathematics, one can prove that if a preorder admits a positivity predicate, then it is given by Definition \ref{constdef}.  However, in general, the predicate defined in Definition \ref{constdef} is not a positivity predicate in the sense of Definition \ref{predconstdef}.

* In [[classical mathematics]] (with [[classical logic]] and impredicativity), all of the definitions are equivalent.

## Examples

Every [[power set]] has a positivity predicate: $x$ is positive iff $x$ is [[inhabited subset|inhabited]].  (Of course, there are few power sets in predicative mathematics, but often it is enough to think of the power set as a [[proper class]].)

The positive predicate on a [[locale]] plays a role in the definition of an [[overt space]].  Locale theory is often considered constructively but impredicatively; in predicative constructive mathematics, a positivity predicate is used in the corresponding theory of [[formal topology]].

In impredicative constructive mathematics, a sufficient condition for a complete poset to possess a positivity predicate is that any element is a join of positive elements. To show this, it suffices to prove that for any element $a$, it holds that $a = \bigvee \{ a \;|\; \lozenge{a} \}$ (that is, $a = \bigvee \{ b \;|\; b = a \;\wedge\; \lozenge{b} \}$). The remaining properties can then be easily verified. By assumption, $a = \sup B$ for some set $B$ containing only positive elements. For any $b \in B$, it holds that $b \leq a$, thus that $a$ is positive and thus that $b \leq \bigvee \{ a \;|\; \lozenge{a} \}$. Therefore $\sup B \leq \bigvee \{ a \;|\; \lozenge{a} \}$ holds as well. The other inequality is trivial.

Every [[atom]] of $L$ is positive, and indeed an atom is precisely a [[minimal element|minimal]] positive element.

## Properties

Although classically trivial, a key property of positivity in the constructive context is this:
$$ \bigvee A = \bigvee A^+ ,$$
where $A^+$ is the set of positive elements of $A$.

A special case is
$$ a = \bigvee \{a\}^+ = \bigvee \{ a | \lozenge{a} \} $$
for any element $a$.

[[!redirects positive element]]
[[!redirects positive elements]]
[[!redirects positivity predicate]]
[[!redirects positivity predicates]]


positive energy representation

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

# Contents
* table of contents
{: toc}

## Definition

Let $G$ be a [[compact Lie group]] and write $L G$ for its _[[loop group]]_. See there for details and notation.

Write

$$
  t_\theta \colon L G \to L G
$$

for the [[automorphism]] which rotates loops by an [[angle]] $\theta$.

The corresponding [[semidirect product group]] we write $L G \rtimes S^1$

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

Let $V$ be a [[topological vector space]]. A linear representation

$$
  S^1 \to Aut(V)
$$

of the [[circle group]] is called **positive** if $\exp(i \theta)$ acts by $\exp(i A \theta)$ where $A \in End(V)$ is a [[linear operator]] with positive [[spectrum of an operator|spectrum]].

A linear [[representation]]

$$
  \rho : L G \to Aut(V)
$$

is said to have **positive energy** or to be a **positive energy representation** if it extends to a representation of the [[semidirect product group]] $L G \rtimes S^1$ such that the restriction to $S^1$ is positive.

=--

## Properties

### Relation to quantum group representations

See [this MO discussion](http://mathoverflow.net/q/178113/381).

## Related concepts

* [[quantization of loop groups]]

* [[conformal block]]

* [[equivariant elliptic cohomology]]

* [[twisted ad-equivariant K-theory]]

* [[twisted ad-equivariant Tate K-theory]]

* [[Loop Groups and Twisted K-Theory]]

## References

### General

The standard textbook on loop groups is

* Andrew Pressley, [[Graeme Segal]], _Loop groups_ Oxford University Press (1988)
  {#PressleySegal}

A review talk is

* [[Graeme Segal]], _Loop groups_ ([pdf](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0084581/chapter08.pdf))
 {#Segal}

Discussion in the context of [[string theory]] (the [[Witten genus]]) is in

* [[Kefeng Liu]], section 2.2 of _On modular invariance and rigidity theorems_, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 ([EUCLID](http://projecteuclid.org/euclid.jdg/1214456221), [pdf](http://www.math.ucla.edu/~liu/Research/loja.pdf))

### Relation to ad-equivariant K-theory

On [[twisted ad-equivariant K-theory]] of [[compact Lie groups]] and the identification with the [[Verlinde ring]] of [[positive energy representations]] of their [[loop group]]:

* [[Daniel S. Freed]], _The Verlinde algebra is twisted equivariant K-theory_, Turk. J. Math 25 (2001) 159-167 ([arXiv:math/0101038](https://arxiv.org/abs/math/0101038))

{#FHT}
* [[Daniel S. Freed]], [[Michael Hopkins]], [[Constantin Teleman]],

  1. _[[Loop Groups and Twisted K-Theory]] I_,

     J. Topology, 4 (2011), 737-789

     ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906), [doi:10.1112/jtopol/jtr019](https://doi.org/10.1112/jtopol/jtr019))

  1. _[[Loop Groups and Twisted K-Theory]] II_,

     J. Amer. Math. Soc. 26 (2013), 595-644

     ([arXiv:0511232](http://arxiv.org/abs/math/0511232), [doi:10.1090/S0894-0347-2013-00761-4](https://doi.org/10.1090/S0894-0347-2013-00761-4))

  1. _[[Loop Groups and Twisted K-Theory]] III_,

     Annals of Mathematics, Volume 174 (2011) 947-1007

     ([arXiv:math/0312155](http://arxiv.org/abs/math/0312155), [doi:10.4007/annals.2011.174.2.5](http://dx.doi.org/10.4007/annals.2011.174.2.5))

* [[Daniel S. Freed]], [[Constantin Teleman]],

  _Dirac families for loop groups as matrix factorizations_,

  Comptes Rendus Mathematique, Volume 353, Issue 5, May 2015, Pages 415-419

  ([arxiv/1409.6051](http://arxiv.org/abs/1409.6051), [doi:10.1016/j.crma.2015.02.011](https://doi.org/10.1016/j.crma.2015.02.011))

[[!redirects positive energy representation]]
[[!redirects positive energy representations]]


positive energy theorem

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

#Contents#
* table of contents
{:toc}

## Idea

In classical [[gravity]] ([[general relativity]]) the _positive energy theorem_ (formerly the _positive mass conjecture_) states that under reasonable assumptions ([[dominant energy condition]]) on the nature of the [[matter]] content (i.e. on the non-gravitational [[stress-energy tensor]]) then the [[ADM mass]] of any [[asymptotically flat spacetime]] solving the [[Einstein equations]] is non-negative, and zero precisely only for [[Minkowski spacetime]] (the [[vacuum]]).

## Related concepts

* [[BPS state]]

## References

An influential argument for the positive energy theorem for [[supergravity]] is due to

* [[Stanley Deser]], [[Claudio Teitelboim]], _Supergravity has positive energy,_  Phys.  Rev. Lett. 39, 249-252 (1977

The first proof of the positive energy theorem for bosonic Einstein gravity was given in

* [[Richard Schoen]] and [[Shing-Tung Yau]], _On the proof of the positive mass conjecture in general relativity_, Commun. Math. Phys. 65, 45 (1979).

* [[Richard Schoen]] and [[Shing-Tung Yau]], _Proof of the positive mass theorem. II_, Commun. Math. Phys. 79, 231 (1981).

An alternative proof for Einstein gravity, making crucial use of [[spinor]] algebra,  was found in

* [[Edward Witten]], _A new proof of the positive energy theorem_, Commun. Math. Phys. 80, 381 (1981).

Comparison of the relation between the proof for [[supergravity]] and the spinorial proof for Einstein gravity is made in

* {#Hull83} [[Chris Hull]], _The positivity of gravitational energy and global supersymmetry_, Comm. Math. Phys. Volume 90, Number 4 (1983), 545-561 ([Euclid](https://projecteuclid.org/euclid.cmp/1103940417))

See also

* Maung Min-Oo, _Scalar curvature rigidity of certain symmetric spaces_, 1995 ([pdf](http://ms.mcmaster.ca/minoo/mypapers/crm.pdf))

* Wikipedia, _[Positive energy theorem](http://en.wikipedia.org/wiki/Positive_energy_theorem)_

[[!redirects positive mass conjecture]]

[[!redirects positive energy conjecture]]

[[!redirects positive mass theorem]]


positive Grassmannian

[[!redirects positive Grassmanian]]
Positive Grassmanian $G_+(n,k)$ is a certain part of the Grassmannian $G(n,k)$...

Related entries: [[Grassmannian]], [[total positivity]], [[amplituhedron]]

* Alexander Postnikov, _Lectures on positive Grassmannian_, writeup by A. Morales, [pdf](http://www-math.mit.edu/~ahmorales/18.318lecs/lectures.pdf)

[[!redirects positive grassmannian]]


positive line bundle

#Contents#
* table of contents
{:toc}

## Idea

A [[holomorphic line bundle]] on a [[complex manifold]] is called _positive_ if its [[curvature]] [[differential 2-form]] is, after multiplication with $i = \sqrt{-1}$, a positive definite $(1,1)$-form.

## Properties

### Relation to ample line bundles

The [[Kodaira embedding theorem]] implies that a positive line bundle is an [[ample line bundle]] and conversely that any ample line bundle admits a Hermitean metric that makes it a positive line bundle.

### Sheaf cohomology of a positive line bundle

The _[[Kodaira vanishing theorem]]_ for [[complex geometry]] says that if $X$ is a [[Kähler manifold]] and $L$ a [[holomorphic line bundle]] on $X$ which is positive, then the [[abelian sheaf cohomology]] of $X$ with [[coefficients]] in the sheaf of [[sections]] of the [[tensor product]]

$$
  \Omega^{n,0}_X(L) \simeq L \otimes \Omega^{n,0}_X
$$

with the [[canonical line bundle]] $\Omega^{n,0}_X$ is concentrated in degree 0:

$$
  (L \; positive)
  \;\;\Rightarrow\;\;
  H^{\bullet \geq 1}(X, \Omega^{n,0}_X(L))
  = 0
  \,.
$$

## Related concepts

* When a [[prequantum line bundle]] is positive, then traditional [[geometric quantization]] with [[Kähler polarization]] coincides with [[geometric quantization by push-forward]], see [this remark](geometric+quantization#SectionsAndEulerQuantizationCoincidesOnPositiveBundles).

* [[stable vector bundle]],

## References

* Wikipedia, _[Positive line bundle](http://en.wikipedia.org/wiki/Positive_form#Positive_line_bundles)_

* [[Kefeng Liu]], Xiaofeng Sun, Xiaokui Yang, _Positivity and vanishing theorems for ample vector bundles_ ([arXiv:1006.1465](http://arxiv.org/abs/1006.1465))

[[!redirects positive line bundles]]


positive locale

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

# Contents
* table of contents
{: toc}

## Definition

An [[open sublocale]] $O \in \mathcal{O}(L)$ is **positive** if, whenever $O$ is the indexed join of open sublocales $U_i \in \mathcal{O}(L)$

$$O = \bigvee_{i:I} U_i$$

the index set $I$ is [[inhabited]].

A [[locale]] $L$ is **positive** if the top open sublocale $L \in \mathcal{O}(L)$ is positive.

## See also

* [[locale]]

* [[locally positive locale]]

* [[premetric locale]]

## References

* [[Simon Henry]], Localic Metric spaces and the localic Gelfand duality ([arXiv:1411.0898v1](https://arxiv.org/abs/1411.0898v1))

* Graham Manuell, *Uniform locales and their constructive aspects*, ([arXiv:2106.00678](https://arxiv.org/abs/2106.00678))

[[!redirects positive locales]]
[[!redirects positive open sublocale]]
[[!redirects positive open sublocales]]


positive n-form

# Positive $n$-forms
* table of contents
{: toc}

## Idea

Let $X$ be an [[orientation|oriented]] $n$-[[dimension|dimensional]] [[differentiable manifold]].  The $n$-[[differential form|forms]] on $X$ have a [[partial order]] $\leq$, which we define (via subtraction) through the notion of positive $n$-form.  This makes the [[line bundle]] of $n$-forms on $X$ into an [[oriented line bundle]].

## Definitions

An $n$-form $\omega$ is __positive semidefinite__ or just __positive__ (denoted $\omega \geq 0$) if the following equivalent conditions hold:

*  the [[integral]] of $\omega$ on any [[open submanifold]] of $X$ is nonnegative;
*  the coordinate of $\omega$ in any oriented local [[coordinate system]] on $X$ is nonnegative;
*  the value of $\omega$ at any point $p$, when applied to any positively oriented collection of $n$ [[tangent vectors]] at $p$, is nonnegative.

Supposing $X$ is unoriented, an $n$-[[pseudoform]] $\omega$ is __positive semidefinite__ if the following equivalent conditions hold:

*  the [[integral]] of $\omega$ on any [[open submanifold]] of $X$ is nonnegative;
*  the coordinate of $\omega$ in any local [[coordinate system]] on $X$ is nonnegative;
*  the value of $\omega$ at any point $p$, when applied relative to either local orientation $o$ at $p$ to any $o$-oriented collection of $n$ [[tangent vectors]] at $p$, is nonnegative.

Arguably, positivity of pseudoforms is fundamental; an orientation allows one to interpret forms as pseudoforms.

A (pseudo)-form is __positive definite__ (denoted $\omega \gt 0$) if it is also nondegenerate (nowhere zero), or equivalently if the conditions above hold with "strictly positive" in place of "nonzero" (taking integrals over only [[inhabited subset|inhabited]] submanifolds).  A positive definite form can be interpreted as a [[volume form|volume]] (pseudo)-form on $X$.

## Properties

If $X$ is oriented, then every $n$-form $\omega$ has an [[absolute value]] ${|\omega|}$, which is a positive $n$-form.  However, even if $\omega$ is [[smooth map|smooth]], still ${|\omega|}$ may only be [[continuous map|continuous]].  However, if $\omega$ is nondegenerate, then not only will ${|\omega|}$ be also nondegenerate, it will be smooth (if $\omega$ is).  Even if $X$ is unoriented, still any $n$-form or $n$-pseudoform $\omega$ will have a positive $n$-pseudoform ${|\omega|}$ as absolute value.

[[!redirects positive n-form]]
[[!redirects positive n-forms]]
[[!redirects positive n-pseudoform]]
[[!redirects positive n-pseudoforms]]

[[!redirects positive definite n-form]]
[[!redirects positive definite n-forms]]
[[!redirects positive definite n-pseudoform]]
[[!redirects positive definite n-pseudoforms]]


positive number

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

# Positive numbers
* table of contents
{: toc}

## Definition

Typically, a __positive number__ is a [[real number]] $p$ such that, using the usual [[strict total order]] on the [[real line]],
\[ p \gt 0 \label{strict}\]
For emphasis, such a number may be called __strictly positive__ or __[[positive definiteness|positive definite]]__.

A real number $p$ is __nonnegative__ if
\[ p \geq 0 ,\label{weak}\]
using the usual [[total order]] on the real line.  The only nonnegative number that is not positive is [[zero]].  Such a number may also be called __weakly positive__ or __[[positive definiteness|positive semidefinite]]__; sometimes (especially in French), it is called simply 'positive'.

The formulas (eq:strict) and (eq:weak) may be used as well for $p$ taken from various extensions and restrictions of the real line.  For example:

*  If $p$ is a [[rational number]], then we may use the usual orders on the set of rational numbers, or equivalently interpret $p$ as a special real number and use the definitions for a real number.  Thus positive rational numbers specialise positive real numbers.  (The same thing goes for other [[subalgebra|subalgebras]] of the real line.)
*  If $p$ is a [[complex number]], then we may use the usual [[strict preorder]] and [[partial order]] on the [[complex plane]], or equivalently require first that $p$ is real and then use the definitions for a real number.  Thus positive complex numbers are the same as positive real numbers. However, the term 'nonnegative' should not be used here. (The same thing goes for other [[hypercomplex number|hypercomplex]] extensions of the real line.)
*  If $p$ is a [[surreal number]], then we use the usual orders on the surreal line; we may *not* assume that $p$ is real this time.  Thus positive surreal numbers generalise positive real numbers.  (The same thing goes for other [[hypperreal number|hyperreal]] extensions of the real line.)

The set of (say real) positive numbers may be denoted $\dot{\mathbb{R}}^+$, $]0,\infty[$, $\mathbb{R}_{\gt0}$, or in many other ways; the corresponding notations for the set of nonnegative numbers are $\bar{\mathbb{R}}^+$, $[0,\infty[$, and $\mathbb{R}_{\geq0}$, while unadorned $\mathbb{R}^+$ is ambiguous.

## Applications

[[Karl Weierstrass|Karl Weierstraß]]\'s [[epsilontic analysis]] relies on repeatedly [[quantification|quantifying]] over the (strictly) positive real numbers.  These positive numbers are typically denoted $\epsilon$ and $\delta$, hence the term 'epsilontics'.  This quantification replaces the use of (individual) positive [[infinitesimal numbers]] in intuitive pre-epsilontic analysis (or in rigorous post-epsilontic [[nonstandard analysis]]).  In [[predicative mathematics]], one may prefer to quantify over positive [[rational numbers]], or even over the numbers $1/n$ for $n$ a positive [[integer]], instead.  (The important thing is to have a [[small set]] of positive numbers whose [[topological closure]] includes [[zero]].)

## Related concepts

* [[even number]], [[odd number]]

* [[sign function]]

[[!redirects positive number]]
[[!redirects positive numbers]]
[[!redirects positive real number]]
[[!redirects positive real numbers]]
[[!redirects positive real]]
[[!redirects positive reals]]

[[!redirects strictly positive number]]
[[!redirects strictly positive numbers]]
[[!redirects strictly positive real number]]
[[!redirects strictly positive real numbers]]
[[!redirects strictly positive real]]
[[!redirects strictly positive reals]]

[[!redirects nonnegative number]]
[[!redirects nonnegative numbers]]
[[!redirects non-negative number]]
[[!redirects non-negative numbers]]
[[!redirects nonnegative real number]]
[[!redirects nonnegative real numbers]]
[[!redirects non-negative real number]]
[[!redirects non-negative real numbers]]
[[!redirects nonnegative real]]
[[!redirects nonnegative reals]]
[[!redirects non-negative real]]
[[!redirects non-negative reals]]

[[!redirects weakly positive number]]
[[!redirects weakly positive numbers]]
[[!redirects weakly positive real number]]
[[!redirects weakly positive real numbers]]
[[!redirects weakly positive real]]
[[!redirects weakly positive reals]]

[[!redirects negative number]]
[[!redirects negative numbers]]
[[!redirects negative real number]]
[[!redirects negative real numbers]]
[[!redirects negative real]]
[[!redirects negative reals]]


positive operator

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

# Contents
* table of contents
{: toc}

## Definition

### On Hilbert spaces

A _positive operator_ is a [[linear operator]] $A$ on a [[Hilbert space]] $(H, \langle -,-\rangle)$ such that the [[quadratic form]]

$$
  v \mapsto \langle v, A v\rangle
$$

for $v \in H$ is [[quadratic form|positive semi-definite]].

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

Every positive operator $A$ on a Hilbert space is [[self-adjoint operator|self-adjoint]].

=--

+-- {: .proof}
###### Proof
Let $B = \frac{1}{2}(A + A^\dagger)$ and $C = \frac{1}{2}i(A^\dagger - A)$.
Then $B$ and $C$ are self-adjoint, and $A = B + iC$.  Now, $\langle v, A v \rangle = \langle v, B v \rangle + i \langle v, C v \rangle$ is real for any $v$, so $\langle v, C v \rangle = 0$ for all $v$.  Hence $C = 0$ and $A = B$.
=--

### In $C^\ast$-algebras

More generally:

+-- {: .un_defn}
###### Definition
An element $A$ of an (abstract) [[C*-algebra]] is called **positive** if it is [[self-adjoint operator|self-adjoint]] and its [[spectrum of an operator|spectrum]] is contained in $[0, \infinity)$.
=--

Here, 'positive' means positive semidefinite; see at [[inner product]] for the family of variations of this notion.  (The relevant inner product here is that associated with the quadratic form above: $v, w \mapsto \langle v, A w\rangle$.)

### In dagger-categories

Yet more generally, in a [[dagger category]] a [[morphism]] $f$ is positive if it is of the form $f = g^\dagger \circ g$ for some morphism $g$.

(eg. [Selinger 2005 Def. 4.1](#Selinger05))

## Related concepts

* [[positive operator valued measure]]

* [[normal operator]]

* [[self-adjoint operator]]

* [[state on an operator algebra]]

* [[density matrix]]

* [[quantum effect]]

* [[quantum operation]]

## References

Discussion in [[dagger-compact categories]] with an eye twoards [[completely positive maps]] on spaces of [[density matrices]]:

* {#Selinger05} [[Peter Selinger]], *Dagger-compact closed categories and completely positive maps*, Electronic Notes in Theoretical Computer Science, **170** (2007) 139-163 [[doi:10.1016/j.entcs.2006.12.018](https://doi.org/10.1016/j.entcs.2006.12.018), [[SelingerPositiveMaps.pdf:file]]]

[[!redirects positive operator]]
[[!redirects positive operators]]

[[!redirects positive linear operator]]
[[!redirects positive linear operators]]


positive type

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### Context
#### Type theory
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[[!include type theory - contents]]
=--
=--
=--

# Positive types
* table of contents
{: toc}

## Idea

In [[type theory]], a **positive type** is one whose *constructors* are regarded as primary.  Its eliminators are derived from these by the rule that "to use an element of a positive type, it is necessary and sufficient to specify what should be done for all possible ways that element could have been constructed".

The opposite notion is a [[negative type]], forming two [[polarity in type theory|polarities]].

In [[categorical semantics]] of type theory, positive types correspond to "from the left" [[universal properties]] (i.e. for mapping out).

In [[denotational semantics]], positive types behave well with respect to "[[call-by-value]]" and other [[eager evaluation]] strategies.

## Examples

* [[inductive type]]
  * [[sum type]]
  * [[product type]] (but see below)
  * [[dependent sum type]] (but see below)
  * [[empty type]]
  * [[unit type]] (but see below)
  * [[identity type]]

* [[unit types]], [[dependent sum types]] and cartesian [[product types]] can be presented both positively (as a particular inductive type) and negatively.  The two definitions are equivalent in ordinary type theory, but distinct in [[linear logic]].  The same is true for binary [[sum types]] if we allow [[sequents]] with multiple conclusions.

[[!redirects positive types]]


positive-operator-valued measure

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#### Measure and probability theory
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#### Physics
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# Contents
* table of contents
{: toc}

## Idea

Given a [[measurable space]] $X$ and a [[Hilbert space]] $H$, then a _positive-operator--valued measure_ (POVM) is essentially a [[probability measure]] on $X$ with values in the [[positive operators]] on $H$.

If all the positive operators appearing are [[projection operators]] then one speaks of a *projection valued measure* (PVM).

Any PVM induces a [[quantum channel]] and these are precisely the quantum operations which encode (projective) [[quantum measurement]], see at *[[quantum measurement channel]]*.

## References

Monographs:

* [[Paul Busch]], Marian Grabowski, [[Pekka J. Lahti]], *Operational Quantum Physics*, Lecture Notes in Physics Monographs **31**, Springer (1995) [[doi:10.1007/978-3-540-49239-9](https://doi.org/10.1007/978-3-540-49239-9)]

* [[Paul Busch]], [[Pekka J. Lahti]], Juha-Pekka Pellonpää, Kari Ylinen, *Quantum Measurement*, Springer (2016) [[doi:10.1007/978-3-319-43389-9](https://doi.org/10.1007/978-3-319-43389-9)]

Review with extensive references to the literature and historical quotes:

* [[Arnold Neumaier]], *Born's rule and measurement* [[arXiv:1912.09906](https://arxiv.org/abs/1912.09906)]

See also:

* {#PreskillGenMeas04} [[John Preskill]], *Generalized measurements*, Section 3.1.2 [[pdf](http://theory.caltech.edu/~preskill/ph219/chap3_15.pdf#page=8)] in: *Quantum Computation*, lecture notes, since 2004 [[web](http://theory.caltech.edu/~preskill/ph229/)]

* {#Barnett09} [[Stephen Barnett]], Chapter 4 of: *Quantum Information*, Oxford University Press (2009) [[ISBN:9780198527633](https://global.oup.com/ukhe/product/quantum-information-9780198527633)]

* {#Flory11} [[Mario Flory]], *POVMs and superoperators* (2011) [[pdf](https://wiki.physik.uni-muenchen.de/TMP/images/8/87/POVMs.pdf), [[Flory-POVM.pdf:file]]]

* {#Wheeler12} [[Nicholas Wheeler]], *Generalized Quantum Measurement* (2012) [[pdf](https://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Generalized%20Quantum%20Measurement/Generalized%20Quantum%20Measurements.pdf), [[Wheeler-GeneralizedQuantumMeasurement.pdf:file]]]

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

Discussion in [[quantum field theory]]:

* Adam Bednorz, *General quantum measurements in relativistic quantum field theory*, Phys. Rev. D **108** 056020 (2023) [[arXiv:2305.12765](https://arxiv.org/abs/2305.12765), [doi:10.1103/PhysRevD.108.056020](https://doi.org/10.1103/PhysRevD.108.056020)]

Critical comments on the notion of POVMs as "generalized measurements":

* {#Griffiths12} [[Robert B. Griffiths]], Section 5 of: *Quantum Channels, Kraus Operators, POVMs* (2012) [[pdf](https://quantum.phys.cmu.edu/QCQI/qitd412.pdf), [[Griffiths-QuantumChannels.pdf:file]]]

  > [[p. 17:](https://ncatlab.org/nlab/files/Griffiths-QuantumChannels.pdf#page=17)] When $G_k$ has rank greater than one but is not a projector it is not so clear what one learns from the POVM when the outcome is $k$. One can assert that in this instance the system a at the time of interest had the property given by the support of $G_k$: the smallest projector P such that $P G_k = G_k$, see CQT, p. 44. This, however, need not be very informative. In particular there are cases in which the support of $G_k$ is $I_a$, in which case the fact that a had this (always true) property tells us nothing.

  > Given the generality allowed in the definition of a POVM, this lack of clarity is not too surprising. Think of a situation in which two vehicles collide and a wheel spins off of one of them. What does this tell one about the state of affairs before the collision? Probably it says something, but it might be rather difficult to say just what it is.

  > For this reason it is a bit odd to refer to a general POVM as a "measurement."

Projection valued measures are central objects in the theory of (general) [[coherent state]]s.

* Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau, _Coherent states, wavelets and their generalizations_, Graduate Texts in Contemporary Physics. Springer 2000

[[!redirects POVM]]
[[!redirects POVMs]]

[[!redirects positive-operator--valued measure]]
[[!redirects positive-operator--valued measures]]
[[!redirects positive-operator–valued measure]]
[[!redirects positive-operator–valued measures]]
[[!redirects positive-operator-valued measure]]
[[!redirects positive-operator-valued measures]]
[[!redirects positive-operator valued measure]]
[[!redirects positive-operator valued measures]]
[[!redirects positive operator valued measure]]
[[!redirects positive operator valued measures]]

[[!redirects positive-operator--valued probability measure]]
[[!redirects positive-operator--valued probability measures]]
[[!redirects positive-operator–valued probability measure]]
[[!redirects positive-operator–valued probability measures]]
[[!redirects positive-operator-valued probability measure]]
[[!redirects positive-operator-valued probability measures]]
[[!redirects positive-operator valued probability measure]]
[[!redirects positive-operator valued probability measures]]
[[!redirects positive operator valued probability measure]]
[[!redirects positive operator valued probability measures]]

[[!redirects PVM]]
[[!redirects PVMs]]

[[!redirects projection valued measure]]
[[!redirects projection valued measures]]

[[!redirects projection-valued measure]]
[[!redirects projection-valued measures]]

[[!redirects generalized quantum measurement]]
[[!redirects generalized quantum measurements]]


positivism

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#### Philosophy
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# References

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


positron

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#### Fields and quanta
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#Contents#
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{:toc}

## Idea

The [[antiparticle]] of the [[electron]].

## Related concepts

* The [[bound state]] of a positron with an [[electron]] is [[positronium]].

## References

See also

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

[[!redirects positrons]]


positronium

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* table of contents
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## Idea

In [[quantum electrodynamics]], _positronium_ is a [[bound state]] of an [[electron]] with a [[positron]] (in contrast to that of an [[electron]] with a [[proton]], which is a [[hydrogen atom]]).

## References

* [[Paul Hoyer]], Chapter II of: _Bound states -- from QED to QCD_, Lectures at "Mini-school on theoretical methods in particle physics" at the Higgs Centre for Theoretical Physics, University of Edinburgh on 30 September to 4 October 2013 ([arXiv:1402.5005](https://arxiv.org/abs/1402.5005))

* [[Paul Hoyer]], Chapter V of: _Journey to the Bound States_ ([arXiv:2101.06721](https://arxiv.org/abs/2101.06721))

* Xingbo Zhao, Kaiyu Fu, Hengfei Zhao, James P. Vary, _Positronium: an illustration of nonperturbative renormalization in a basis light-front approach_ ([arXiv:2103.06719](https://arxiv.org/abs/2103.06719))

See also:

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


pospace

# Idea and origins #
Although partially ordered spaces (pospaces) and [[directed space]]s originally arose in Analysis and General Topology, the recent interest in them has resulted from attempts to model state spaces of evolving processes, particularly those that arise in modelling concurrent programming. They also occur in models of space-time and in modelling certain modal logics.

The basic idea is that of a topological space in which points can be compared using a partial order corresponding to 'precedes'.

# Definitions #

* A **partially ordered space** or **pospace**, $X$, is a topological space with a (globally defined) closed partial order, $\leq$, so considering $\leq$ as a subset of $X \times X$, it is a closed subset.  (It follows that $X$ must be a [[Hausdorff space]].)

* A **dimap** $f: X \to Y$ between two pospaces, $X$ and $Y$, is a continuous map that respects the partial order:
$$x \leq x' \Rightarrow f(x) \leq f(x').$$

# Examples #

* The _standard directed interval_, $I_d = ([0,1], \leq)$, is a pospace that gives rise to the [[directed space]] of the same name.

* Any pospace $X$ gives rise to a [[directed space]] by taking the directed paths to be, well, directed paths: dimaps from $I_d$ to $X$. (It can be helpful to use directed paths of variable length based on $\stackrel{\to}{[0,r]}$ the directed version of the interval of length $r$.)

# Homotopy #

There are several useful notions of 'homotopy' between dimaps. Choosing between them depends on the  intended use and/or the situation being modelled.

* Suppose $a$ and $b$ are dimaps between $X$ and $Y$. A **directed homotopy** between $a$ and $b$ is a dimap
$$h: X \times I_d \to Y,$$
where $X \times I_d$ is given the product partial order. (The pospace $I_d$ is defined below amongst the examples.)

We say $a$ and $b$ are **dihomotopic**, the relation being called **directed homotopy**.

# Discussion #

_[[Eric Forgy|Eric]]_: I can imagine that "directed" stuff might begin to garner more attention here, so could we settle on a notation? For example, I've used $\uparr$ to indicate that a space is directed so that the directed interval is denoted $\uparr I$. Above, it is given as $I_d$. I am slightly partial (but not firmly set) to using $\uparr$ to denote that a space is directed or partially ordered (or even preordered).

Second, I have a "continuum" allergy and never like to see a definition that explicitly evokes the continuum to define important objects. Is there a more combinatoric or abstract way to define pospaces, [[directed space|directed spaces]], and [[directed homotopy theory|directed homotopy]] that does not evoke the continuum immediately?

How/if is the standard directed interval related to [[interval object]]?

_[[Tim Porter|Tim]]_:  I am also alergic to too much 'continuum' use.  The problem is to get a substitute that allows  and controls subdivision.  Ideas anyone?  Gabriel Minian (paper in _K-theory_ used a variant of Baues's cofibration category notion to handle various categories including that of simplicial complexes.  He used an Ind-object instead of an interval object. His theory is not a directed one, of course.  Marco Grandis also looked at the problem of subdivision in some papers and going right back there were notes by George Hoff and Marcel Evrard trying to do the homotopy theory of categories using concatentated intervals.  I can try to find references if it should proved useful.

As to combinatorial definitions, pospace is not infected with 'continuumitis', but the directed interval is. Perhaps what you want is that the directed interval is both a directed object and an interval object, but that may lead to a clash of definitions, so better an amalgam of the two definitions should be sought.

One can try to use a domain theoretic definition of things a bit like the papers such as _A Domain of spacetime intervals for General relativity_ by Keye Martin and Prakash Panangaden. (see on Prakash's page
http://www.cs.mcgill.ca/~prakash/accepted.html).

_Toby_: As Tim says, there\'s nothing about the continuum in the basic notions of pospace; that comes in only when you introduce dihomotopy, and already ordinary homotopy involves the continuum.  Really we should talk about what [[interval object]]s there may be in the category of pospaces and what homotopies with respect to them are like; the continuum is simple our first example of that.

As for notation, notice that we\'ve actually got three: $I_d$, $\uparrow I$, and $\vec{I}$ (applied to $[0,r]$ above).  Personally, I\'d just use $I$; let the context tell us that it\'s directed.  (Nobody needs special notation to distinguish the set $R$ of real numbers from the poset $R$, after all; why should we need it when we topologise?)


possible worlds semantics

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#### Modalities, Closure and Reflection
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#Contents#
* table of contents
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## Idea

In the context of [[modal logic]], by _possible world semantics_ one means the interpretation of the [[possibility]] and [[necessity]] [[modalities]] as [[existential quantification]] and [[universal quantification]] over a [[context]] of "possible worlds". Such a model is sometimes referred to as a [[Kripke frame]].

For formalization of this in terms of [[hyperdoctrines]]/[[dependent type theory]] see at _[necessity and possibility -- In first-order logic and dependent type theory](http://ncatlab.org/nlab/show/necessity+and+possibility#InFirstOrderLogicAndTypeTheory)_.

Under [[categorical semantics]] such a context interprets as an [[object]] of some [[category]] whose interpretation is similar to that of a [[probability space]] in [[measure theory]] -- which is one sensible formalization of possible worlds (e.g. [Toronto-McCarthy 10, slide 23](#TorontoMcCarthy10)).

## Related concepts

* [[Kripke frame]]

* [[probability space]]

* [[many-worlds interpretation of quantum mechanics]]

* [[multiverse]]

## References
 {#References}

### General

The original articles:

* {#Kripke59} [[Saul A. Kripke]], *A Completeness Theorem in Modal Logic*, The Journal of Symbolic Logic **24** 1 (1959) 1-14 [[doi:10.2307/2964568](https://doi.org/10.2307/2964568), [jstor:2964568](https://www.jstor.org/stable/2964568), [pdf](https://www.filosoficas.unam.mx/~morado/Cursos/17Modal/Kripke1959.pdf)]

* [[Saul A. Kripke]], *Semantical Considerations on Modal Logic*, Acta Philosophical Fennica **16** (1963) 83-94 [[pdf](http://saulkripkecenter.org/wp-content/uploads/2019/03/Semantical-Considerations-on-Modal-Logic-PUBLIC.pdf)]

* {#Kripke63} [[Saul A. Kripke]], *Semantical Analysis of Modal Logic I. Normal Modal Propositional Calculi*, Mathematical Logic Quaterly **9** 5-6 (1963) 67-96 [[doi:10.1002/malq.19630090502](https://doi.org/10.1002/malq.19630090502)]

* {#Kripke65} [[Saul A. Kripke]], *Semantical Analysis of Modal Logic II. Non-Normal Modal Propositional Calculi*, in *The Theory of Models* (Proceedings of the 1963 International Symposium at Berkeley) Studies in Logic and the Foundations of Mathematics (1965) 206-220 [[doi:10.1016/B978-0-7204-2233-7.50026-5](https://doi.org/10.1016/B978-0-7204-2233-7.50026-5)]

Modern exposition:

* [[Patrick Blackburn]], [[Johan van Benthem]], *Modal logic: a semantic perspective*, Ch 1 in: *The Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) [doi:10.1016/S1570-2464(07)80004-8, [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/)]

In general see the [reference at](modal+logic#References) *[[modal logic]]*, such as:

* {#BradleySwartz79} [[Raymond D. Bradley]], [[Norman Swartz]], *Possible Worlds -- An Introduction to Logic and its Philosophy*, Hackett Publishing (1979) [[webpage](http://www.sfu.ca/~swartz/pw/), [pdf](https://www.sfu.ca/~swartz/pw/text/pw_all.pdf)]

* Rod Girle, *Possible Worlds*, McGill-Queen's University Press (2003) [[jstor:j.cttq48cx](https://www.jstor.org/stable/j.cttq48cx)]

* [[Johan van Benthem]], §2.2 of: *Modal Logic for Open Minds* (2010) [[ISBN:9781575865980](http://web.stanford.edu/group/cslipublications/cslipublications/site/9781575865980.shtml),[pdf](http://fenrong.net/teaching/mljvb.pdf), [webpage](https://archive.illc.uva.nl/lgc/MLoM/)]

* Daniel Nolan, *Topics in the Philosophy of Possible Worlds*, Routledge (2002) [[ISBN:9780415516303](https://www.routledge.com/Topics-in-the-Philosophy-of-Possible-Worlds/Nolan/p/book/9780415516303)]

* [[Valentin Goranko]], [[Martin Otto]], *Model Theory of Modal Logic*, Section 5 in: *Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 249-329 [[pdf](https://www2.mathematik.tu-darmstadt.de/~otto/papers/mlhb.pdf),  [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/), [publisher page](https://www.sciencedirect.com/bookseries/studies-in-logic-and-practical-reasoning/vol/3/suppl/C)]

and more philosophically ("modal realism"):

* David Lewis, *On the Plurality of Worlds*, Wiley (1986, 2001) [[ISBN:978-0-631-22426-6](https://www.wiley.com/en-us/On+the+Plurality+of+Worlds-p-9780631224266), [Wikipedia entry](https://en.wikipedia.org/wiki/On_the_Plurality_of_Worlds)]

See also:

* Wikipedia, *[Possible world](https://en.wikipedia.org/wiki/Possible_world)*

* Stanford Encyclopedia of Philosophy, _[Possible Worlds](http://plato.stanford.edu/entries/possible-worlds/)_

* Stanford Encyclopedia of Philosophy, _[Modal Logic -- Possible worlds semantics](http://plato.stanford.edu/entries/logic-modal/#PosWorSem)_

* {#TorontoMcCarthy10} Neil Toronto, Jay McCarthy, _From Bayesian Notation to Pure Racket_, 2010 ([pdf](http://jeapostrophe.github.io/home/static/toronto-2010ifl-slides.pdf))

[[!include possible-worlds and many-worlds -- references]]

[[!redirects possible worlds semantics of modal logic]]

[[!redirects possible world]]
[[!redirects possible worlds]]

[[!redirects possible-world]]
[[!redirects possible-worlds]]


possible-worlds and many-worlds -- references

### Possible-worlds vs. Many-worlds
 {#ManyPossibleWorldsReferences}

References which consider, in one way or another, the notions of

* *[[possible worlds]]* in [[epistemic logic|epistemic]] [[modal logic]]

* *[[many-worlds interpretation of quantum mechanics|many worlds]]* in [[interpretations of quantum mechanics]]:

in relation to each other:

* {#Bunge76} [[Mario Bunge]], *Possibility and Probability*,  in: *Foundations of Probability Theory, Statistical Interference, and Statistical Theories of Science*, Reidel (1976) 17-34 $[$[doi:10.1007/978-94-010-1438-0_2](https://doi.org/10.1007/978-94-010-1438-0_2)$]$

* Brian Skyrms, Part III of: *Possible Worlds, Physics and Metaphysics*, Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition **30** 5 (1976) 323-332 $[$[jstor:4319099](https://www.jstor.org/stable/4319099)$]$

* Paul Tappenden, p. 101 (4 of 17) in: *Identity and Probability in Everett's Multiverse*, The British Journal for the Philosophy of Science, **51** 1 (2000) 99-114 $[$[jstor:3541750](https://www.jstor.org/stable/3541750)$]$

* Daniel Nolan, p. 22 of: *Topics in the Philosophy of Possible Worlds*, Routledge (2002) $[$[ISBN:9780415516303](https://www.routledge.com/Topics-in-the-Philosophy-of-Possible-Worlds/Nolan/p/book/9780415516303)$]$

* Rod Girle, Ch. 8 of: *Possible Worlds*, McGill-Queen's University Press (2003) $[$[jstor:j.cttq48cx](https://www.jstor.org/stable/j.cttq48cx)$]$

* [[Simon Saunders]], p. 196 in: *Chance in the Everett Interpretation*, in: *Many Worlds?*, Oxford University Press (2010) 181–205 $[$[doi:10.1093/acprof:oso/9780199560561.003.0008](https://doi.org/10.1093/acprof:oso/9780199560561.003.0008)$]$

* Nuriya Nurgalieva, Lídia del Rio, *Inadequacy of Modal Logic in Quantum Settings*, EPTCS **287** (2019) 267-297 $[$[arXiv:1804.01106](https://arxiv.org/abs/1804.01106), [doi:10.4204/EPTCS.287.16](https://doi.org/10.4204/EPTCS.287.16)$]$

* [[Alastair Wilson]], p. 20 of: *Modal Metaphysics and the Everett Interpretation* (2006) $[$[philsci:2635](http://philsci-archive.pitt.edu/2635), [pdf](http://philsci-archive.pitt.edu/2635/1/modalmetaphysicsandeverett.pdf)$]$

* Vladislav Terekhovich, *Modal Approaches in Metaphysics and Quantum Mechanics* $[$[arXiv:1909.10046](https://arxiv.org/abs/1909.10046)$]$

* [[Alastair Wilson]], *The Nature of Contingency: Quantum Physics as Modal Realism*, Oxford University Press (2020) $[$[ISBN:9780198846215](https://global.oup.com/academic/product/the-nature-of-contingency-9780198846215)$]$

* Raoni W. Arroyo, Jonnas R. B. Arenhart, *Whence deep realism for Everettian quantum mechanics?*, Foundations of Physics **52** 121 (2022)  [[arXiv:2210.16713](https://arxiv.org/abs/2210.16713), [doi:10.1007/s10701-022-00643-0](https://doi.org/10.1007/s10701-022-00643-0)]

Beware that there is also

* Bas C. van Fraassen, *Modal interpretation of repeated measurement*, Philosophy of Science **64** 4 (1997) 669-676 $[$[doi:10.1086/392577](https://doi.org/10.1086/392577), [SEP review](https://plato.stanford.edu/entries/qm-modal/)$]$

which, even if some vocabulary is superficially alike, does not refer either to [[modal logic]] nor to the [[many-worlds interpretation of quantum mechanics|many-worlds]] interpretation.


possibly empty loop

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## Idea

There should be a version of a [[loop (algebra)|loop]] that may be empty, similar to how an [[associative quasigroup]] is a [[group]] that may be empty. That leads to the concept of a _possibly empty loop_, which is an example of [[centipede mathematics]].

## Definition

### As a quasigroup

A __possibly empty loop__ is a [[quasigroup]] $(G,(-)\cdot(-):G\times G\to G,(-)\backslash(-):G\times G\to G,(-)/(-):G\times G\to G)$ that satisfies the following laws:

* $a \cdot (b \backslash b) = a$
* $(b \backslash b) \cdot a = a$
* $a \cdot (b / b) = a$
* $(b / b) \cdot a = a$

for all $a$ and $b$ in $G$.

### With division only

A __possibly empty left loop__ is a set $G$ with a binary operation $(-)\backslash(-):G\times G\to G$, such that:

  * For all $a$ and $b$ in $G$, $a\backslash a=b\backslash b$
  * For all $a$ in $G$, $(a\backslash (a\backslash a))\backslash (a\backslash a)=a$

For any element $a$ in $G$, the element $a\backslash a$ is called a __right identity element__, and the element $a\backslash (a\backslash a)$ is called the __right inverse element__ of $a$. For all elements $a$ and $b$ in $G$, __left multiplication__ of $a$ and $b$ is defined as $(a\backslash (a\backslash a))\backslash b$.

A __right quotient__ on a set $G$ is a binary operation $(-)/(-):G\times G\to G$ such that:

  * For all $a$ and $b$ in $G$, $a/a=b/b$
  * For all $a$ in $G$, $(a/a)/((a/a)/a)=a$

For any element $a$ in $G$, the element $a/a$ is called a __left identity element__, and the element $(a/a)/a$ is called the __left inverse element__ of $a$. For all elements $a$ and $b$, __right multiplication__ of $a$ and $b$ is defined as $a/((b/b)/b)$.

A __possibly empty loop__ is a possibly empty left and right loop as defined above such that the following are true:

* left and right identity elements are equal (i.e. $a/a = a \backslash a$) for all $a$ in $G$

* left and right multiplications are equal (i.e. $a/((b/b)/b) = (a\backslash (a\backslash a))\backslash b$) for all $a$ and $b$ in $G$.

## Properties

* Every __possibly empty commutative loop__ is a [[commutative invertible quasigroup]].

* Every __possibly empty associative loop__ is a [[associative quasigroup]].

## Examples

* Every [[loop (algebra)|loop]] is a possibly empty loop.

* Every [[associative quasigroup]] is a possibly empty loop.

* The empty quasigroup is a possibly empty loop.

## Related concepts

* [[quasigroup]], [[associative quasigroup]]

* [[loop (algebra)|loop]]

[[!redirects possibly empty abelian loop]]

[[!redirects possibly empty commutative loop]]


possibly empty ring

[[!redirects possibly empty nonunital ring]]
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## Idea

By replacing every element in the definition with a constant function to the element, the empty set vacuously satisfies the axioms of any algebraic structure.

## Definition

A __possibly empty ring__ is a [[set]] $R$ with a binary operation $(-)+(-):R \times R \to R$, a unary operation $-:R \to R$, a unary operation $0:R \to R$, a binary operation $(-)\cdot(-):R \times R \to R$, and a unary operation $1:R \to R$ such that that:

  * For all $a$, $b$, and $c$ in $R$, $a+(b+c)=(a+b)+c$
  * For all $a$ and $b$ in $R$, $a+b=b+a$
  * For all $a$ and $b$ in $R$, $a+0(b)=a$
  * For all $a$ and $b$ in $R$, $a+(-a)=0(b)$
  * For all $a$, $b$, and $c$ in $R$, $a\cdot (b\cdot c)=(a\cdot b)\cdot c$
  * For all $a$ and $b$ in $R$, $a\cdot 1(b)=a$
  * For all $a$ and $b$ in $R$, $1(a)\cdot b=b$
  * For all $a$, $b$, and $c$ in $R$, $a\cdot (b+c)= a\cdot b + a\cdot c$
  * For all $a$, $b$, and $c$ in $R$, $(a+b)\cdot c= a\cdot c + b\cdot c$

A possibly empty ring is __commutative__ if for all $a$ and $b$ in $R$, $a\cdot b=b \cdot a$.

## Examples

* Every [[ring]] is a possibly empty ring.

* The empty possibly empty ring is an possibly empty ring that is not a ring.

## Related concepts

* [[ring]]

* [[possibly empty field]]

[[!redirects possibly empty commutative ring]]
[[!redirects possibly empty commutative rings]]


possibly trivial field

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#Contents#
* table of contents
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## Idea

While in general, [[fields]] are non-[[trivial rings]], it is sometimes useful in [[commutative algebra]] for the [[trivial ring]] to be a regarded as a field. Thus, the idea of a **possibly trivial field**.

## Definition

Let $R$ be a [[commutative ring]], let $\mathrm{Reg}(R)$ be the [[monoid of regular elements]] in $R$, let $\neg \mathrm{Reg}(R)$ be the [[complement]] of $\mathrm{Reg}(R)$, the set of [[zero divisors]] in $R$, and let $I_{\neg \mathrm{Reg}(R)}$ be the [[two-sided ideal]] generated by $\neg \mathrm{Reg}(R)$.

A **possibly trivial field** is a [[commutative ring]] $R$ such that

* every [[element]] in $\mathrm{Reg}(R)$ is a [[unit]], or that the [[localization of a commutative ring|localization]] $R[\mathrm{Reg}(R)^{-1}]$ is isomorphic to $R$.

* every [[element]] in $I_{\neg \mathrm{Reg}(R)}$ is equal to zero, or that the [[quotient ring]] $R/I_{\neg \mathrm{Reg}(R)}$ is isomorphic to $R$.

If $R$ is the [[trivial ring]], $\neg \mathrm{Reg}(R)$ is the [[empty subset]], the ideal generated by the [[empty set|empty]] subset is the trivial ideal $0R$, and $R/0R$ is always [[isomorphic]] to $R$.

## Constructing a possibly trivial field from a commutative ring

Let $R$ be a [[commutative ring]], let $\mathrm{Reg}(R)$ be the [[monoid of regular elements]] in $R$, let $\neg \mathrm{Reg}(R)$ be the [[complement]] of $\mathrm{Reg}(R)$, the set of [[zero divisors]] in $R$, and let $I_{\neg \mathrm{Reg}(R)}$ be the [[two-sided ideal]] generated by $\neg \mathrm{Reg}(R)$.

There are two operations one could do to a commutative ring $R$, taking the quotient by $I_{\neg \mathrm{Reg}(R)}$, and localizing at $\mathrm{Reg}(R)$. Taking the quotient by $I_{\neg \mathrm{Reg}(R)}$ results in a [[possibly trivial integral domain]] $R/I_{\neg \mathrm{Reg}(R)}$, while localizing at $\mathrm{Reg}(R)$ results in a [[prefield ring]], the [[ring of fractions]] $R[\mathrm{Reg}(R)^{-1}]$. The operations are commutative: first taking the quotient by $I_{\neg \mathrm{Reg}(R)}$ then localizing at $\mathrm{Reg}(R)$ and first localizing at $\mathrm{Reg}(R)$ then taking the quotient by $I_{\neg \mathrm{Reg}(R)}$ results in isomorphic possibly trivial fields, because no element is both a zero divisor and regular.

Thus, one has a [[functor]] $L:\mathrm{CRing} \to \mathrm{Prefield}$ which takes a commutative ring to its [[ring of fractions]], and a functor $Q:\mathrm{CRing} \to \mathrm{PTIntDom}$ taking a commutative ring to its quotient possibly trivial integral domain, and because the category of prefield rings and the category of possibly trivial integral domains are subcategories of the category of commutative rings, there are also [[subfunctors]] $L':\mathrm{PTIntDom} \to \mathrm{PTField}$ and $Q':\mathrm{Prefield} \to \mathrm{PTField}$ where $L' \subseteq L$ and $Q' \subseteq Q$. The functors form a [[commutative square]] with [[natural isomorphism]] $p:L' \circ Q \cong Q' \circ L$:

$$\array{& \mathrm{CRing} & \overset{L}\rightarrow & \mathrm{Prefield} & \\
          Q & \downarrow & \cong &\downarrow & Q'\\
          & \mathrm{PTIntDom} & \underset{L'}\rightarrow& \mathrm{PTField} & \\
}$$

## In constructive mathematics

In constructive mathematics, the notion of field bifurcates into multiple notions. The definition given above results in a possibly trivial [[weak Heyting field]]. However, there are also possibly trivial [[Heyting fields]] and possibly trivial [[discrete fields]].

### Possibly trivial Heyting fields

A possibly trivial field $R$ is **Heyting** if, whenever the sum of two elements $a + b$ is invertible, then either $a$ is invertible or $b$ is invertible. These are simply called *Heyting fields* in [LombardiQuitté2010](#LombardiQuitté2010).

### Possibly trivial discrete fields

A possibly trivial field is **discrete** if every element is either zero or invertible. These are simply called *discrete fields* in [LombardiQuitté2010](#LombardiQuitté2010). Possibly trivial discrete fields have [[decidable equality]].

## See also

* [[commutative ring]]
* [[field]]
* [[possibly trivial integral domain]]
* [[weak local ring]]

## References

The concept of a possibly trivial field appeared in

* {#LombardiQuitté2010} [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

where the possibly trivial Heyting fields are called "Heyting fields" and the possibly trivial discrete field are called "discrete fields".

[[!redirects possibly trivial field]]
[[!redirects possibly trivial fields]]


possibly trivial integral domain

+-- {: .rightHandSide}
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###Context###
#### Arithmetic
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[[!include arithmetic geometry - contents]]
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#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

While in general, [[integral domains]] are non-[[trivial rings]], it is sometimes useful in [[commutative algebra]] for the [[trivial ring]] to be an integral domain. Thus, the idea of a **possibly trivial integral domain**.

## Definition

Let $R$ be a [[commutative ring]] and let $D \subseteq R$ be the subset of all [[zero divisors]] or non-[[cancellative elements]] in $D$. A **possibly trivial integral domain** is a [[commutative ring]] $R$ such that every element in the [[two-sided ideal]] $I_D$ generated by $D$ is equal to zero, or that the quotient ring $R/I_D$ is isomorphic to $R$.

If $R$ is the [[trivial ring]], $D$ is the [[empty subset]], the ideal generated by the empty subset is the trivial ideal $0R$, and $R/0R$ is always isomorphic to $R$.

## Constructing a possibly trivial integral domain from a commutative ring

From every commutative ring, one could construct a possibly trivial integral domain.

As above, let $R$ be a [[commutative ring]], let $D \subseteq R$ be the subset of all [[zero divisors]] or non-[[cancellative elements]] in $D$, and let $I_D$ be the [[two-sided ideal]] generated by $D$. The quotient of $R$ by $I_D$ is the possibly trivial integral domain of $R$.

$R/I_D$ is a possibly trivial integral domain, because the quotient of $R$ by $I_D$ takes all elements of $I_D$ to zero. If $R$ is an [[approximate integral domain]], then $R/I_D$ is an [[integral domain]]; otherwise, $R/I_D$ is the [[trivial ring]].

## In constructive mathematics

In constructive mathematics, the notion of integral domain bifurcates into multiple notions. While the definition given above results in the usual notion of an integral domain as a commutative ring whose zero divisors are equal to zero, there are also possibly trivial [[Heyting integral domains]] and possibly trivial [[discrete integral domains]], which refer directly to cancellative elements.

\begin{definition}
A possibly trivial integral domain $R$ is **Heyting** if it has a [[tight apartness relation]] and where every element apart from zero is cancellative. The trivial ring is trivially Heyting because the set of elements apart from zero is the empty subset.
\end{definition}

\begin{definition}
A possibly trivial integral domain is **discrete** if every element of $R$ is either equal to zero or cancellative. The trivial ring is trivially discrete because every element is both equal to zero and cancellative. These are simply called *integral domains* in [LombardiQuitté2010](#LombardiQuitté2010). Possibly trivial discrete integral domains have [[decidable equality]].
\end{definition}

## See also

* [[commutative ring]]
* [[integral domain]]
* [[possibly trivial field]]

## References

The concept of a possibly trivial integral domain appeared in

* {#LombardiQuitté2010} [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

where the possibly trivial discrete integral domains are simply called "integral domains".

[[!redirects possibly trivial integral domain]]
[[!redirects possibly trivial integral domains]]


PostBQP

\tableofcontents

\section{Idea}

In [[quantum complexity theory]], "PostBQP" stands (in variation of *[[BQP]]*) for *[[postselected]] bounded-error quantum polynomial time*, denoting a fundamental [[complexity class]] of [[quantum computation]].

\section{Properties}

PostBQP is closed under [[unions]] and [[intersections]] of problems in PostBQP. PostBQP is equivalent to the complexity class [[PP]], and includes [[BQP]] and [[QMA]] as subclasses.

\section{Related concepts}

* [[BQP]]

* [[QMA]]

\section{References}

* [[Scott Aaronson]], *Quantum computing, postselection, and probabilistic polynomial-time*, Proceedings of the Royal Society A, Volume 461, Issue 2063, pages 3473–3482, 08 November 2005. ([doi:10.1098/rspa.2005.1546](https://doi.org/10.1098%2Frspa.2005.1546), [arXiv:quant-ph/0412187](https://arxiv.org/abs/quant-ph/0412187))

Wikipedia:

* Wikipedia, [PostBQP](https://en.wikipedia.org/wiki/PostBQP)


Postnikov section

A stage in a [[Postnikov tower]].

[[!redirects Postnikov sections]]

[[!redirects Postnikov stage]]
[[!redirects Postnikov stages]]