convergence

> This entry is about the notion of limit in [[analysis]] and [[topology]]. For the notion of the same name in [[category theory]] see at _[[limit]]_.

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# Contents
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## Idea

A _limit_ of a [[sequence]] (or [[net]]) of points $(x_i)$ in a [[topological space]] (or other [[convergence space]]) $X$ is a point $x$ such that the sequence eventually gets arbitrarily close to $x$.  We can also speak of a limit of a [[filter]] on $X$.

The notion is of particular and historical importance in [[analysis]], where it serves to define for instance the notion of [[derivative]].

## Definitions

The precise definition depends on what sort of space $X$ is.

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

If $X$ a [[topological space]] and $I$ the set of [[natural numbers]] (or more generally any [[directed set]]) and $\nu = (x_i)_{i\in I}\colon I \to X$ is a [[sequence]] (or a [[net]]) of points in $X$, one says that a point $x \in X$ is a **limit** of $\nu$ or that $\nu$ **converges** to $x$ if for each [[neighbourhood]] $U$ in $X$ of $x$ there exists an $n \in I$ such that $x_i \in U$ for each $i \geq n$.
=--

An important special case (the original) is:

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

If $X$ the [[real line]] and $I$ the set of [[natural numbers]] (or more generally any [[directed set]]) and $\nu = (x_i)_{i\in I}\colon I \to X$ is a [[sequence]] (or a [[net]]) of [[real numbers]], one says that a point $x \in X$ is a **limit** of $\nu$ or that $\nu$ **converges** to $x$ if for each [[positive number]] $\epsilon$ there exists an $n \in I$ such that ${|x_i - x|} \lt \epsilon$ for each $i \geq n$.
=--

An important generalization (possibly the most general) is:

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

If $X$ a [[convergence space]] and $I$ the set of [[natural numbers]] (or more generally any [[directed set]]) and $\nu = (x_i)_{i\in I}\colon I \to X$ is a [[sequence]] (or a [[net]]) of points in $X$, one says that a point $x \in X$ is a **limit** of $\nu$ or that $\nu$ **converges** to $x$ if the [[eventuality filter]] of $\nu$ converges to $x$ (which is a primitive concept in convergence spaces).
=--

Other types of [[space]] for which we might put in definitions (or which might have definitions on their own pages) are ([[extended metric space|extended]]) ([[quasimetric space|quasi]])-([[pseudometric space|pseudo]])-[[metric spaces]], [[premetric spaces]], ([[quasiuniform space|quasi]])-[[uniform spaces]], [[pretopological spaces]], and (quasi)-[[uniform convergence spaces]].

### Notation

When one of the conditions above holds, we may write any of the following, where '$\to$' is read as 'converges to':

* $\nu \to x$,
* $x_i \to_i x$,
* $x_i \to x$ (suppressing the index $i$ by abuse of notation).

Or we may write any of the following, were $\lim$ is read as 'the set of limits of':

* $x \in \lim \nu$,
* $x \in \lim_i x_i$,
* $x \in \lim x_i$ (suppressing the index $i$ by abuse of notation).

Of course, the right-hand side has a meaning by itself, as the set of limits itself (a [[subset]] of the [[underlying set]] of $X$, or a [[subspace]] of $X$ itself).

If $X$ is a [[Hausdorff space]], then there is at most one point $x$ with the property that the sequence (or net) $\nu$ converges to $x$.  Then we may write any of the following, were now $\lim$ is read as 'the limit of':

* $x = \lim \nu$,
* $x = \lim_i x_i$,
* $x = \lim x_i$ (suppressing the index $i$ by abuse of notation).

Now the right-hand side by itself is the possibly undefined [[term]] for the limit itself (if it exists).

### Limits of filters

More generally than [[sequences]], and equivalently to [[nets]], we may speak of limits of [[filters]] on $X$.  This concept is axiomatized directly in the concept of [[convergence space]]. In the case of a [[topological space]] $X$, a filter of subsets of $X$ converges to a point $x$ if every neighbourhood of $x$ is contained in the filter.

In the definitions above, equivalent nets (those with equal [[eventuality filters]]) always converge to the same point.  As every [[proper filter]] is the eventuality filter of some net, a proper filter converges to $x$ if any of these nets converges to $x$; the [[improper filter]] converges to every point.  (In [[constructive mathematics]], we may cover all filters by saying: $F$ converges to $x$ if, on the assumption that $F$ is proper, any of its nets converges to $x$.)

## Properties

### Relation to limits in the sense of category theory
 {#RelationToLimitsInCategoryTheory}

The [[limits]] of [[category theory]] are a great generalization of an analogy with the limits discussed here.  It turns out, however, that limits in [[topological spaces]] (at least) can be viewed as category-theoretic limits. For now, see [this math.sx answer](http://math.stackexchange.com/a/62800).

## Examples

* [[geometric series]]

## Related concepts

* [[filter]]

* [[convergence space]]

* [[series]]

* [[limit point]]

* [[radius of convergence]]

* [[uniform convergence]], [[pointwise convergence]]

* [[divergent series]]

* [[limit of a function]]

## References

* Wikipedia, _[Limit (mathematics)](http://en.wikipedia.org/wiki/Limit_%28mathematics%29)_

Discussion of this history of the concept, with emphasis on its roots all the way back in [[Zeno's paradoxes of motion]] is in

* {#Boyer49} Carl Benjamin Boyer, _The history of the Calculus and its conceptual development_, Dover 1949

category: analysis

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convergence radius

# Radius of convergence
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## Idea

The radius of convergence of a [[power series]] tells how far out the series will [[convergence|converge]].

## Definition

Let $a = (a_0, a_1, \ldots)$ be an [[infinite sequence]] of [[complex numbers]], let $\zeta$ be a particular complex number, and consider the [[power series]]
\[ \sum_n a_n (z - \zeta)^n .\label{series}\]
The __radius of convergence__ is the [[supremum]] of the [[positive numbers]] $\epsilon$ such that (eq:series) converges for ${|z - \zeta|} \lt \epsilon$.  (This supremum is a nonnegative [[lower real]] in $[0,\infty]$.)

The phrasing above is ambiguous.  If $\epsilon$ is less than the radius of convergence, does this mean that (eq:series) converges for *every* $z$ with ${|z - \zeta|} \leq \epsilon$ or for *some* such $z$?  It doesn\'t matter:

+-- {: .num_theorem #radii}
###### Theorem
If (eq:series) converges for some $z$ with ${|z - \zeta|} = \epsilon$, then (eq:series) converges for every $z$ with ${|z - \zeta|} \lt \epsilon$.
=--
(We do *not* say that (eq:series) converges for $z$ with ${|z = \zeta|} = \epsilon$, but this has no effect on the supremum.)

## Properties

The radius of convergence is clearly independent of $\zeta$.  It can be calculated quite easily from the coefficients $a_i$:

+-- {: .num_theorem #Hadamard}
###### Theorem (Cauchy–Hadamard)
The radius of converge of (eq:series) is
$$ R = \liminf_n {|a_n|}^{-1/n} .$$
=--

For ${|z - \zeta|} \lt R$, (eq:series) is (pretty much by definition) an [[analytic function]] of $z$.  There is a partial converse:

+-- {: .num_theorem #analytic}
###### Theorem
If a function $f$ is analytic at all $z$ with ${|z - \zeta|} \lt R$, then there is a power series (eq:series) that converges to $f(z)$ for all $z$ with ${|z - \zeta|} \lt R$.
=--
(Specifically, $a_n = f^{(n)}(\zeta)/n!$.)

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

Over the [[real numbers]], Theorem \ref{analytic} fails.  I\'m sure (says one of this pages authors) that there are interesting things to say about series in [[adic numbers]], [[matrices]], and things like that, but I don\'t know them.

=--

## Related concepts

* [[asymptotic series]]

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convergence space

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# Convergence spaces
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## Idea

A _convergence space_ is a generalisation of a [[topological space]] based on the concept of [[convergence]] of [[filters]] (or [[nets]]) as fundamental.  The basic concepts of [[point-set topology]] ([[continuous functions]], [[compact space|compact]] and [[Hausdorff space|Hausdorff]] [[topological spaces]], etc) make sense also for convergence spaces, although not all theorems hold.  The [[category]] $Conv$ of convergence spaces is a [[quasitopos]] and may be thought of as a [[nice category of spaces]] that includes [[Top]] as a [[full subcategory]].

## Definitions

A __convergence space__ is a [[set]] $S$ together with a [[relation]] $\to$ from $\mathcal{F}S$ to $S$, where $\mathcal{F}S$ is the set of [[filters]] on $S$; if $F \to x$, we say that $F$ __converges__ to $x$ or that $x$ is a __limit__ of $F$.  This must satisfy some axioms:

1. Centred: The [[principal ultrafilter]] $F_x = \{ A \;|\; x \in A \}$ at $x$ converges to $x$;
2. Isotone: If $F \subseteq G$ and $F \to x$, then $G \to x$;
3. Directed: If $F \to x$ and $G \to x$, then some filter contained in the [[intersection]] $F \cap G$ converges to $x$.  In light of (2), it follows that $F \cap G \to x$ itself.  (Strictly speaking, the relation should not be called [[directed set|directed]] unless also every point is a limit of some filter, but this follows from 1.)

It follows that $F \to x$ if and only if $F \cap F_x$ does.  Given that, the convergence relation is defined precisely by specifying, for each point $x$, a filter of [[subfilter|subfilters]] of the principal ultrafilter at $x$.  (But that is sort of a tongue twister.)

A filter $F$ __clusters__ at a point $x$, written $F \rightsquigarrow x$, if there exists a [[proper filter]] $G$ such that $F \subseteq G$ and $G \to x$.

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

The morphisms of convergence spaces are the [[continuous functions]]; a function $f$ between convergence spaces is __continuous__ if $F \to x$ implies that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the [[filterbase]] $\{f(A) \;|\; A \in F\}$.  In this way, convergence spaces form a [[concrete category]] $Conv$.

Note that the definition of 'convergence' varies in the literature; at the extreme end, one could define it as any relation whatsoever from $\mathcal{F}S$ (or even from the [[proper class|class]] of all nets on $S$, see [[preconvergence space]]) to $S$, but that is so little structure as to be not very useful.  An intermediate notion is that of [[filter space]], in which (3) is not required.  Here we follow the terminology of [Lowen-Colebunders](#LC).

## Examples

In [[measure theory]], given a [[measure space]] $X$ and a [[measurable space]] $Y$, the space of [[almost-everywhere defined]] [[measurable functions]] from $X$ to $Y$ becomes a convergence space under [[convergence almost everywhere]].  In general, this convergence space does not fit into any of the examples below.

A [[pseudotopological space]] is a convergence space satisfying the _star property_:

*  If $F$ is a filter such that every proper filter $G \supseteq F$ clusters at $x$, then $F$ converges to $x$.

Assuming the [[ultrafilter theorem]] (a weak version of the [[axiom of choice]]), it\'s enough to require that $F$ converges to $x$ whenever every [[ultrafilter]] that refines $F$ converges to $x$ (or clusters there, since these are equivalent for ultrafilters).

A [[subsequential space]] is a pseudotopological space that may be defined using only [[sequences]] instead of arbitrary nets/filters.  (More precisely, a filter converges to $x$ only if it refines (the eventuality filter of) a [[infinite sequence|sequence]] that converges to $x$.)

A [[pretopological space]] is a convergence space that is _infinitely filtered_:

*  If $(F_\alpha)_\alpha$ is any family of filters each of which converges to $x$, then $\bigcap_\alpha F_\alpha$ converges to $x$.

In particular, the [[intersection]] of all of the filters converging to $x$ (the [[neighbourhood filter]] of $x$) also converges to $x$.  Note that every pretopological space is pseudotopological.

Any [[topological space]] is a convergence space, and in fact a pretopological one: we define $F \to x$ if every [[neighbourhood]] of $x$ belongs to $F$.  A convergence space is __topological__ if it comes from a topology on $S$.  The [[full subcategory]] of $Conv$ consisting of the topological convergence spaces is [[equivalence of categories|equivalent]] to the category [[Top]] of topological spaces.  In this way, the definitions below are all suggested by theorems about topological spaces.

Every [[Cauchy space]] is a convergence space.

## Properties

The [[improper filter]] (the [[power set]] of $S$) converges to every point.  On the other hand, a convergence space $S$ is __[[Hausdorff space|Hausdorff]]__ if every [[proper filter]] converges to at most one point; then we have a [[partial function]] $\lim$ from the proper filters on $S$ to $S$.  A topological space is Hausdorff in the usual sense if and only if it is Hausdorff as a convergence space.

A convergence space $S$ is __[[compact space|compact]]__ if every proper filter clusters at some point; that is, every proper filter is contained in a convergent proper filter.  Equivalently (assuming the [[ultrafilter theorem]]), $S$ is compact iff every ultrafilter converges.  A topological space is compact in the usual sense if and only if it is compact as a convergence space.

The topological convergence spaces can be characterized as the pseudotopological ones in which the convergence satisfies a certain "associativity" condition.  In this way one can (assuming the ultrafilter theorem) think of a topological space as a "[[generalized multicategory]]" parametrized by ultrafilters.  In particular, note that a compact Hausdorff pseudotopological space is defined by a single function $\mathcal{U}S \to S$, where $\mathcal{U}S$ is the set of ultrafilters on $S$, such that the composite $S \to \mathcal{U}S \to S$ is the identity.  That is, it is an [[algebra for an endofunctor|algebra]] for the [[pointed object|pointed]] [[endofunctor]] $\mathcal{U}$.  The compact Hausdorff _topological_ spaces (the [[compacta]]) are precisely the [[algebra for a monad|algebras]] for $\mathcal{U}$ considered as a [[monad]].  If we treat $\mathcal{U}$ as a monad on [[Rel]], then the [[lax algebra|lax algebras]] are the topological spaces in their guise as [[relational beta-modules]].

## Topological structure

Given a convergence space, a filter $F$ __star-converges__ to a point $x$, written $F \to^* x$, if every proper filter that refines $F$ clusters at $x$.  (Assuming the [[ultrafilter theorem]], $F$ star-converges to $x$ iff every ultrafilter that refines $F$ converges to $x$.)  The relation of star convergence makes any convergence space into a [[pseudotopological space]] with a weaker convergence.  In this way, $Ps Top$ becomes a [[reflective subcategory]] of $Conv$ over $Set$.

>Note: the term 'star convergence' and its symbol '$\to^*$' are my own, formed from 'star property' above, which I got from _[[HAF]]_.  Other possibilities that I can think of: 'ultraconvergence', 'universal convergence', 'subconvergence'.  ---[[Toby Bartels|Toby]]

Given a convergence space, a set $U$ is a __[[neighbourhood]]__ of a point $x$, written $x \in^\circ U$, if $U$ belongs to every filter that converges to $x$; it follows that $U$ belongs to every filter that star-converges to $x$.  The relation of being a neighbourhood makes any convergence space into a [[pretopological space]], although the pretopological convergence is weaker in general.  In this way, $Pre Top$ is a [[reflective subcategory]] of $Conv$ (and in fact of $Ps Top$) over $Set$.

Other pretopological notions:  The __[[preinterior]]__ of a set $A$ is the set of all points $x$ such that $x \in^\circ A$.  The __[[preclosure]]__ of $A$ is the set of all points $x$ such that every neighbourhood $U$ of $x$ meets (has [[inhabited set|inhabited]] [[intersection]] with) $A$.  For more on these, see [[pretopological space]].

Given a convergence space, a set $G$ is __[[open subset|open]]__ if $G$ belongs to every filter that converges to any point in $G$, or equivalently if $G$ belongs to every filter that star-converges to any point in $G$, or equivalently if $G$ equals its preinterior.  The class of open sets makes any convergence space into a [[topological space]], although the topological convergence is weaker in general.  In this way, $Top$ is a [[reflective subcategory]] of $Conv$ (and in fact of $Ps Top$ and $Pre Top$) over $Set$.

Other topological notions:  A set $F$ is __[[closed subset|closed]]__ if $F$ meets every neighbourhood of every point that belongs to $F$, equivalently if $F$ equals its preclosure.  The __[[interior]]__ of $A$ is the [[union]] of all of the open sets contained in $A$; it is the largest open set contained in $A$.  The __[[closure]]__ of $A$ is the [[intersection]] of all of the closed sets that contain $A$; it is the smallest closed set that contains $A$.  (For a topological convergence space, the interior and closure match the preinterior and preclosure.)

+-- {: .num_theorem #subcats}
###### Theorem
The inclusions $Top \to Pre Top \to Ps Top \to Conv$ are all inclusions of [[full subcategories]] over $Set$.  That is, they all agree on what a [[continuous function]] is.
=--

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

The only hard part is proving that, if $f(\mathcal{F}) \to^* f(x)$ whenever $\mathcal{F} \to x$ in a [[pretopological space]], then $x \in^\circ f^*(V)$ whenever $f(x) \in^\circ V$.  This is usually proved [[proof by contradiction|by contradiction]] and flagrant use of [[axiom of choice|choice]]: supposing that $f(x) \in^\circ V$ but $x \notin^\circ f^*(V)$, then every neighbourhood $U$ of $x$ must satisfy $U \nsubseteq f^*(V)$, so choose for each such $U$ a point $y_U$ such that $y_U \in U$ but $f(y_U) \notin V$, defining a net $y$ (indexed by neighbourhoods of $x$ ordered by reverse inclusion), such that $y \to x$, but $\neg\big(f(y) \rightsquigarrow f(x)\big)$, so $y \nrightarrow^* f(x)$, getting a contradiction.

But the theorem is in fact perfectly [[constructive mathematics|constructive]]: the filter $\mathcal{N}_x$ of neighbourhoods of $x$ converges to $x$, so $f(\mathcal{N}_x) \to^* f(x)$; all that really matters is that $f(\mathcal{N}_x) \rightsquigarrow f(x)$, so that for each $V \ni^\circ f(x)$ and $U \ni^\circ x$, for some $W$ with $x \in^\circ W \subseteq U$, $f(W) \subseteq V$, so $W \subseteq f^*(V)$, making $f^*(V)$ a neighbourhood of $x$.
=--

## Cluster spaces

The notion of clustering generalizes convergence.

A __cluster space__ is a [[set]] $S$ together with a [[relation]] $\rightsquigarrow$ from $\mathcal{F}S$ to $S$; if $F \rightsquigarrow x$, we say that $F$ __clusters__ at $x$ or that $x$ is a __cluster point__ of $F$.  The axioms are as follows:

1. Centred: The [[principal ultrafilter]] $F_x \rightsquigarrow x$;
2. Antitone: If $F \supseteq G$ and $F \rightsquigarrow x$, then $G \rightsquigarrow x$;
3. Codirected: If $F \cap G \rightsquigarrow x$ then $F \rightsquigarrow x$ or $G \rightsquigarrow x$.
4. Nontrivial: If $F \rightsquigarrow x$, then $F$ is proper.

Note that the direction of isotony and directedness for clustering is the reverse of that for convergence (hence 'antitone' and 'codirected').  Nontriviality is the nullary version of directedness (equivalent to the statement that the improper filter never clusters at any point), which we explicitly need this time.  Alternatively, we can take $\rightsquigarrow$ as a relation only on the *proper* filters; then nontriviality may be omitted from the axioms (as was done in the original reference, [Muscat 2015](#Muscat2015)).

Every convergence space is a cluster space (using the usual definition of $\rightsquigarrow$ from $\to$), and many of the notions of convergence generalize to cluster spaces, including continuous functions, open/closed sets, neighborhood filters, pre-closure, compactness, etc.

This definition of a cluster space does not seem to work in [[constructive mathematics]].  In particular, Muscat\'s proof that every convergence space satisfies the directedness axiom relies on [[excluded middle]], and the [[lesser limited principle of omniscience]] follows if it holds in the [[real line]], for example.  It\'s not clear yet what if any alternative will work better.

## References

* Eva Lowen-Colebunders (1989). Function Classes of Cauchy Continuous Maps. Dekker, New York, 1989.
  {#LC}

* Joseph Muscat (2015). An axiomatization of filter clustering. Conference: 2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD). [DOI:10.1109/FSKD.2015.7381908](https://doi.org/10.1109/FSKD.2015.7381908).
  {#Muscat2015}

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convergence vector space

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# Convergence vector space
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## Idea

A **convergence vector space** is a generalisation of a [[topological vector space]] where the topological structure is replaced by a [[convergence space|convergence structure]].

## Definition

+-- {: .num_defn #cvsp}
###### Definition
A **convergence vector space** is a [[vector space]], say $V$, together with the structure of a [[convergence space]], say $\Lambda$, which is compatible with the vector space structure in that both addition and scalar multiplication are jointly continuous.
=--

Some authors may impose the condition that the convergence structure be separated.  Any [[topological vector space]] defines a convergence vector space.

## Multilinear Mappings

Convergence structures become particularly useful when considering evaluation mappings in functional analysis.  For a [[locally convex topological vector space]] $E$ with continuous dual $E^*$, the evaluation mapping $E \times E^* \to \mathbb{F}$ is continuous with respect to the product topology if and only if $E$ is normable.  This proves to be somewhat of a limitation when one works outside the category of normable [[LCTVS]].  However, there is a convergence structure on $E \times E^*$ which makes the evaluation mapping continuous.

+-- {: .num_defn #cmulti}
###### Definition
Let $E$ and $F$ be convergence vector spaces and $n \in \mathbb{N}$.  Let $\mathcal{L}^n(E,F)$ be the space of continuous mappings $E^n \to F$ which are linear in each argument (see [[continuous multilinear operator]]).  The **continuous convergence structure** on $\mathcal{L}^n(E,F)$ is the coarsest convergence structure which makes the evaluation map $\mathcal{L}^n(E,F) \times E^n \to F$ continuous.

We write $\mathcal{L}_c^n(E,F)$ for the resulting convergence space.
=--

+-- {: .num_lemma #cmultlem}
###### Lemma
The convergence space $\mathcal{L}_c^n(E,F)$ is a convergence vector space.  It satisfies the following universal property for a [[convergence space]] $X$ then a mapping $X \to \mathcal{L}_c^n(E,F)$ is continuous if and only if the associated map $X \times E^n \to F$ is continuous.
=--

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Converging Realities -- Toward a common philosophy of physics and mathematics

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This page collect hyperlinks related to the book

* [[Roland Omnès]],

  _Converging Realities: Toward a common philosophy of physics and
mathematics_

  Princeton University Press, 2005

on the [[philosophy of mathematics]] and the [[philosophy of physics]].

A review by [[David Corfield]] is here: _[pdf](http://www.math.ucr.edu/home/baez/corfield/omnes.pdf)_

The thesis of the book -- called _physism_ -- is the following (from p. 215)

> There are basic [[axioms]] for [[logic]] and [[mathematics]]. These axioms are laws of [[physics]]. They are recognized through two inseparable criteria: their fecundity in the construction of mathematics and their necessity for a statement of the law of physics. Their fecundity can be explained in view of the universality, subtlety, and richness of the laws: the basic axioms must be fecund enough to allow a statement of the laws in the language of mathematics. Conversely, they generate every possible field of mathematics. New laws, new axioms, new fields, are possible and they may be discovered by further research. [[consistency|Consistency]] is equally necessary in mathematics and in the laws of physics, which are inseparable. Consistency cannot be explained, but it stands as one of the two criteria of [[truth]]. The other one is [[experiment|experimental]] falsification of a mathematical proposition purporting to express a law of nature.

category: reference


conversion rule

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## Idea

In type theory, a conversion rule is a rule which constrains the result of combining [[term introduction]] with [[term elimination]]. Conversion rules come in two types: **[[beta conversion]] rules** or **computation rules**, and **[[eta conversion]] rules** or **uniqueness rules**. Computation rules in particular are important, because they are used in [[inductive definitions]]. For example, the rules for [[addition]] of [[natural numbers]] has two computation rules, which derive from the two computation rules of the [[natural numbers type]].

Moreover, conversion rules use [[equality]]. The usage of equality in this manner is called **conversional equality**, and in the context of beta conversion rules is also called **computational equality**. In [[type theory]], there are three notions of equality, [[judgmental equality]], [[propositional equality]], and [[typal equality]], all of which could be used for conversional equality.

For example, the [[beta conversion]] rule for the [[dependent product type]] could be expressed in judgmental, propositional, and typal equality:

* Judgmental beta conversion rules for dependent product types:

$$\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \lambda(x:A).b(x)(a) = b[a/x]:B[a/x]}$$

* Propositional beta conversion rules for dependent product types:

$$\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \lambda(x:A).b(x)(a) =_{B[a/x]} b[a/x] \; \mathrm{true}}$$

* Typal beta conversion rules for dependent product types:

$$\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \beta_\Pi:\lambda(x:A).b(x)(a) =_{B[a/x]} b[a/x]}$$

Similarly, the [[eta-conversion]] rule for the [[sum type]] could be expressed in judgmental, propositional, and typal equality:

* Judgmental eta conversion rules for sum types:

$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B \quad \Gamma, z:A + B \vdash u:C \quad \Gamma, a:A \vdash u[\mathrm{inl}(a)/z] \equiv c[a/x]:C[\mathrm{inl}(a)/z] \quad \Gamma, b:B \vdash u[\mathrm{inr}(b)/z] \equiv d[b/y]:C[\mathrm{inr}(b)/z]}{\Gamma \vdash u[e/z] \equiv \mathrm{ind}_{A + B}^C(c[\mathrm{inl}(e)/x], d[\mathrm{inr}(e)/y], e):C[e/z]}$$

* Propositional eta conversion rules for sum types:

$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B \quad \Gamma, z:A + B \vdash u:C \quad \Gamma, a:A \vdash u[\mathrm{inl}(a)/z] \equiv_{C[\mathrm{inl}(a)/z]} c[a/x] \; \mathrm{true} \quad \Gamma, b:B \vdash u[\mathrm{inr}(b)/z] \equiv_{C[\mathrm{inr}(b)/z]} d[b/y] \; \mathrm{true}}{\Gamma \vdash u[e/z] \equiv_{C[e/z]} \mathrm{ind}_{A + B}^C(c[\mathrm{inl}(e)/x], d[\mathrm{inr}(e)/y], e) \; \mathrm{true}}$$

* Typal eta conversion rules for sum types:

$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B \quad \Gamma, z:A + B \vdash u:C \quad \Gamma, a:A \vdash i_\mathrm{inl}(u):u[\mathrm{inl}(a)/z] =_{C[\mathrm{inl}(a)/z]} c[a/x] \quad \Gamma, b:B \vdash i_\mathrm{inr}(u):u[\mathrm{inr}(b)/z] =_{C[\mathrm{inr}(b)/z]} d[b/y]}{\Gamma \vdash \eta_{A + B}(u, e):u[e/z] =_{C[e/z]} \mathrm{ind}_{A + B}^C(c[\mathrm{inl}(e)/x], d[\mathrm{inr}(e)/y], e)}$$

The paradigmatic example of conversional equality is a pair of terms like "$(\lambda x. x+x)(2)$" and "$2+2$", where the second is obtained by $\beta$-reduction from the first.  In a type theory that includes definitions by [[recursion]], expansion of a recursor is also computational equality. For instance, if addition is defined by recursion, then "$2+2$" (that is, $s(s(0))+s(s(0))$) reduces via this rule to "$4$" (that is, $s(s(s(s(0))))$).

## Contextual conversion rules

There are also **contextual conversion rules**. These differ from the usual beta-conversion rules in that in that there is an additional context member $\Delta$ attached to the end of the context $\Gamma, x:A$ so that the full context becomes $\Gamma, x:A, \Delta$. By definition, $\Delta$ is dependent upon $x:A$, and the conclusion usually involves substituting $x:A$ by some given term $a:A$ in the context, becoming $\Gamma, \Delta[a/x]$.

For example, using the example of the [[beta-conversion]] rule for the [[dependent product type]], the contextual beta-conversion rules are given by the following:

* Judgmental contextual beta-conversion rules for dependent product types:

$$\frac{\Gamma, x:A, \Delta \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma, \Delta[a/x] \vdash \lambda(x:A).b(x)(a) \equiv b[a/x]:B[a/x]}$$

* Propositional contextual beta-conversion rules for dependent product types:

$$\frac{\Gamma, x:A, \Delta \vert \Phi \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma, \Delta[a/x] \vert \Phi[a/x] \vdash \lambda(x:A).b(x)(a) \equiv_{B[a/x]} b[a/x] \; \mathrm{true}}$$

* Typal contextual beta-conversion rules for dependent product types:

$$\frac{\Gamma, x:A, \Delta \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma, \Delta[a/x] \vdash \beta_\Pi:\lambda(x:A).b(x)(a) =_{B[a/x]} b[a/x]}$$

Similarly, one could define contextual eta-conversion rules for the [[sum type]]:

* Judgmental contextual eta-conversion rules for sum types:

$$\frac{\Gamma, z:A + B, \Delta \vdash C \; \mathrm{type} \quad \Gamma, x:A, \Delta[\mathrm{inl}(x)/z] \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B, \Delta[\mathrm{inr}(y)/z] \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B \quad \Gamma, z:A + B, \Delta \vdash u:C \quad \Gamma, a:A, \Delta[\mathrm{inl}(a)/z] \vdash u[\mathrm{inl}(a)/z] \equiv c[a/x]:C[\mathrm{inl}(a)/z] \quad \Gamma, b:B, \Delta[\mathrm{inr}(b)/z] \vdash u[\mathrm{inr}(b)/z] \equiv d[b/y]:C[\mathrm{inr}(b)/z]}{\Gamma, \Delta[e/z] \vdash u[e/z] \equiv \mathrm{ind}_{A + B}^C(c[\mathrm{inl}(e)/x], d[\mathrm{inr}(e)/y], e):C[e/z]}$$

* Propositional contextual eta-conversion rules for sum types:

$$\frac{\Gamma, z:A + B, \Delta \vert \Phi \vdash C \; \mathrm{type} \quad \Gamma, x:A, \Delta[\mathrm{inl}(x)/z] \vert \Phi[\mathrm{inl}(x)/z] \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B, \Delta[\mathrm{inr}(y)/z] \vert \Phi[\mathrm{inr}(y)/z] \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B \quad \Gamma, z:A + B, \Delta \vert \Phi \vdash u:C \quad \Gamma, a:A, \Delta[\mathrm{inl}(a)/z] \vert \Phi[\mathrm{inl}(a)/z] \vdash u[\mathrm{inl}(a)/z] \equiv_{C[\mathrm{inl}(a)/z]} c[a/x] \; \mathrm{true} \quad \Gamma, b:B, \Delta[\mathrm{inr}(b)/z] \vert \Phi[\mathrm{inr}(b)/z] \vdash u[\mathrm{inr}(b)/z] \equiv_{C[\mathrm{inr}(b)/z]} d[b/y] \; \mathrm{true}}{\Gamma, \Delta[e/z] \vert \Phi[e/z] \vdash u[e/z] \equiv_{C[e/z]} \mathrm{ind}_{A + B}^C(c[\mathrm{inl}(e)/x], d[\mathrm{inr}(e)/y], e) \; \mathrm{true}}$$

* Typal contextual eta-conversion rules for sum types:

$$\frac{\Gamma, z:A + B, \Delta \vdash C \; \mathrm{type} \quad \Gamma, x:A, \Delta[\mathrm{inl}(x)/z] \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B, \Delta[\mathrm{inr}(y)/z] \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B \quad \Gamma, z:A + B, \Delta \vdash u:C \quad \Gamma, a:A, \Delta[\mathrm{inl}(a)/z] \vdash i_\mathrm{inl}(u):u[\mathrm{inl}(a)/z] =_{C[\mathrm{inl}(a)/z]} c[a/x] \quad \Gamma, b:B, \Delta[\mathrm{inr}(b)/z] \vdash i_\mathrm{inr}(u):u[\mathrm{inr}(b)/z] =_{C[\mathrm{inr}(b)/z]} d[b/y]}{\Gamma, \Delta[e/z] \vdash \eta_{A + B}(u, e):u[e/z] =_{C[e/z]} \mathrm{ind}_{A + B}^C(c[\mathrm{inl}(e)/x], d[\mathrm{inr}(e)/y], e)}$$

## See also

* [[formation rule]], [[introduction rule]], [[elimination rule]]
* [[beta-conversion]], [[eta-conversion]]
* [[inductive definition]]
* [[equality]]

## References

Contextual dependent product types and contextual identity types are defined in the appendix of:

* [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))

where the computation rules are contextual computation rules.

[[!redirects conversion rule]]
[[!redirects conversion rules]]
[[!redirects computation rule]]
[[!redirects computation rules]]
[[!redirects uniqueness rule]]
[[!redirects uniqueness rules]]

[[!redirects contextual conversion rule]]
[[!redirects contextual conversion rules]]
[[!redirects contextual computation rule]]
[[!redirects contextual computation rules]]
[[!redirects contextual uniqueness rule]]
[[!redirects contextual uniqueness rules]]

[[!redirects conversional identity]]
[[!redirects conversional identities]]
[[!redirects conversional equality]]
[[!redirects conversional equalities]]

[[!redirects computational identity]]
[[!redirects computational identities]]
[[!redirects computational equality]]
[[!redirects computational equalities]]


convex analysis

## Related concepts

* [[analysis]]

* [[convex function]]

* [[Legendre transform]]

## References

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


convex cone

A subset $C$ of a [[vector space]] $V$ is a **convex cone** if $a x + b y$ belongs to $C$, for any positive scalars $a, b$, and any $x, y$ in $C$.

A convex cone is therefore a [[cone]] which is a [[convex space]].


convex function

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

# Contents
* table of contents
{: toc}

## Idea

A _convex function_ is a [[real numbers|real]]-valued [[function]] defined on a [[convex set]] whose [[graph]] is the boundary of a [[convex set]].

There is another context where people say a function is convex if it is a [[Lipschitz function]] between [[metric spaces]] with Lipschitz constant (or Lipschitz modulus) 1. These are different concepts of convexity, although there are relations between convexity and Lipschitz continuity, as we shall see below.

## Definition

Let $D$ be a [[convex space]]. A [[function]] $f \colon D \to \mathbb{R}$ is **convex** if the [[set]] $\{(x, z) \in D \times \mathbb{R}: z \geq f(x)\}$ is a convex subspace of $D \times \mathbb{R}$. Equivalently, $f$ is **convex** if for all $x, y \in D$,

$$f(t x + (1-t) y) \leq t f(x) + (1-t) f(y)$$

whenever $0 \leq t \leq 1$.

This definition obviously extends to functions $f \colon D \to I$ where $I$ is an [[interval]] of $\mathbb{R}$ (whether open or closed or half-open, it doesn't matter).

The function $f$ is called **concave** if it satisfy the reverse inequality to the one given above, or equivalently if $-f$ is convex.

A function $f$ is __strictly convex__ if the inequality holds strictly whenever $0 \lt t \lt 1$.  In high-school mathematics, one often says _concave upward_ for strictly convex and _concave downward_ for strictly concave.

## Examples

* A [[homomorphism]] of [[convex sets]], i.e. a  convex-linear map, $D \to \mathbb{R}$, is of course convex.

* The [[norm]] function $\mathbb{C} \to \mathbb{R}$ is convex. This follows readily from multiplicativity ${|x y|} = {|x|} \cdot {|y|}$ and the [[triangle inequality]] ${|x + y|} \leq {|x|} + {|y|}$.

* More generally, for a [[normed vector space]] $V$, the norm function ${\|(-)\|}$ is convex, again by the triangle inequality and the scaling axiom ${\|\alpha v\|} = {|\alpha|} \cdot {\|v\|}$ for [[scalars]] $\alpha$.

* For any twice-[[differentiable function]] $f \colon (a, b) \to \mathbb{R}$, the second derivative $f''$ is nonnegative iff $f$ is convex; this may be proven using the [[mean value theorem]]. Examples include the [[exponential function]] $\exp: (-\infty, \infty) \to \mathbb{R}$ and the $p$-power function $[0, \infty) \to \mathbb{R}: t \mapsto t^p$ if $p \geq 1$.

* More generally, for an open convex region $D \subseteq \mathbb{R}^n$ in a [[Euclidean space]], a twice-differentiable function $f: D \to \mathbb{R}$ is convex iff its [[Hessian]] is a [[semidefinite element|positive semidefinite]] [[bilinear form]].

* Any positive $\mathbb{R}$-[[linear combination]] of convex functions on $D$ is again convex.

* The pointwise [[maximum]] of two convex functions on $D$ is again convex.

* If $g: V \to W$ is a homomorphism or convex-linear map between [[convex spaces]], and if $f: W \to \mathbb{R}$ is convex, then $f \circ g: V \to \mathbb{R}$ is convex.

In the next two examples, $I \subseteq \mathbb{R}$ is an [[interval]].

* It is not generally true that a composition $D \stackrel{g}{\to} I \stackrel{f}{\to} \mathbb{R}$ of convex functions is convex. For example, this fails for the case $D = \mathbb{R}$ and $g(x) = x^2 + 1$ and $f: [1, \infty) \to \mathbb{R}$ given by $f(x) = x^{-1}$.

* However, if $f: I \to \mathbb{R}$ is both [[monotone function|monotone increasing]] and convex, then for any convex function $g: D \to I$, the composite $f \circ g: D \to \mathbb{R}$ is convex (as is easily shown).

A special case of the last class of examples are functions of the form $e^f = \exp \circ f$ where $f$ is convex. We say a function $f: D \to (0, \infty)$ is __log-convex__ if $\log f$ is convex. We see then that log-convex functions are also convex.  The [[identity function]] on $\mathbb{R}$ is an example of a convex function that is *not* log-convex (or use $x \mapsto x^2$ for a strict example).

## Properties

+-- {: .num_lemma #squeeze}
###### Lemma
If $f: (a, b) \to \mathbb{R}$ is convex and $u \lt v \lt w \in (a, b)$, then

$$\frac{f(u) - f(v)}{u - v} \leq \frac{f(u) - f(w)}{u - w} \leq \frac{f(v) - f(w)}{v - w}.$$

Consequently, the function $x \mapsto \frac{f(x) - f(v)}{x - v}$ on the domain $u \lt x \lt v$ is increasing and is bounded above by $\frac{f(v)-f(w)}{v-w}$; similarly, this function on the domain $v \lt x \lt w$ is increasing and bounded below by $\frac{f(u)-f(v)}{u-v}$.
=--

The proof is virtually trivial; just write $v$ as a convex combination $t u + (1-t)w$ and use the definition of convexity.

+-- {: .num_prop}
###### Proposition
A convex function $f: (a, b) \to \mathbb{R}$ is continuous.
=--

+-- {: .proof}
###### Proof
For any point $x_0 \in (a, b)$ there are $s, t, u, v \in (a, b)$ with $s \lt t \lt x_0 \lt u \lt v$. Then for any $x \in (t, x_0) \cup (x_0, u)$ we have by Lemma \ref{squeeze}

$$\frac{f(s)-f(t)}{s-t} \leq \frac{f(x) - f(x_0)}{x-x_0} \leq \frac{f(u)-f(v)}{u-v}$$

so that $K = \max\{\left\vert \frac{f(s)-f(t)}{s-t}\right\vert, \left\vert \frac{f(u)-f(v)}{u-v}\right\vert\}$ serves as a Lipschitz constant, i.e., we have

$$|f(x) - f(x_0)| \leq K|x - x_0|$$

for all $x \in (t, u)$. This is enough to force continuity at the point $x_0$.
=--

In the converse direction, we have the following result which frequently arises in practice.

+-- {: .num_prop}
###### Proposition
If $D \subseteq \mathbb{R}^n$ is a convex set and $f: D \to \mathbb{R}$ is a continuous function such that

$$f\left(\frac{x+y}{2}\right) \leq \frac{f(x) + f(y)}{2},$$

then $f$ is convex.
=--

+-- {: .proof}
###### Proof
For any fixed $x, y \in D$, the function $h: [0, 1] \to \mathbb{R}$ defined by

$$h(t) = t f(x) + (1-t)f(y) - f(t x + (1-t)y)$$

is continuous. The [[inverse image]] $h^{-1}([0, \infty))$ is closed in $[0, 1]$ and, by an easy induction argument, contains the set of [[dyadic rational numbers]] $t \in [0, 1]$, which is dense in $[0, 1]$. Being closed and dense, $h^{-1}([0, \infty))$ is all of $[0, 1]$, i.e., $h(t) \geq 0$ for all $t \in [0, 1]$, but this is precisely the condition that $f$ is convex.
=--

Another easy consequence of Lemma \ref{squeeze} is

+-- {: .num_prop}
###### Proposition
For a convex function $f: (a, b) \to \mathbb{R}$, the right-hand and left-hand derivatives

$$\underset{x \searrow x_0}{\lim} \frac{f(x)-f(x_0)}{x-x_0}, \qquad \underset{x \nearrow x_0}{\lim} \frac{f(x)-f(x_0)}{x-x_0}$$

of $f$ exist at every point $x_0 \in (a, b)$, and the left-hand derivative at $x_0$ is less than or equal to the right-hand derivative at $x_0$.
=--

It further follows from Lemma \ref{squeeze} that if $f$ is convex and we define $(D f)(x_0)$ to be the average of the right-hand and left-hand derivatives, then $D f$ is monotone increasing and hence is discontinuous at at most countable many points of jump discontinuity (whence $D f$ is [[Riemann integration|Riemann-integrable]], for instance).

## Related concepts

* [[convex analysis]]

* [[Legendre transform]]

* [[Lipschitz function]]

## References

See also

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

[[!redirects convex map]]
[[!redirects convex maps]]
[[!redirects convex function]]
[[!redirects convex functions]]
[[!redirects log-convex function]]
[[!redirects log-convex functions]]

[[!redirects concave map]]
[[!redirects concave maps]]
[[!redirects concave function]]
[[!redirects concave functions]]
[[!redirects log-concave function]]
[[!redirects log-concave functions]]


convex powerset of distributions monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
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#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
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# Contents
* table of contents
{: toc}

## Idea

The _convex powerset of distributions monad_ is a [[monad]] (or family of monads) that encompasses both [[nondeterministic computation|non-determinism]] (via [[power sets]]) and [[probability]] (via [[probability distributions]]).

## Definition

The finitary version of the monad is defined as follows. First recall that for any [[finite set]] $n$, the set of [[probability distributions]] $\Delta(n)\subseteq [0,1]^n$,

$$\Delta(n)=\{(r_1,\dots,r_n)\in[0,1]^n | \textstyle \sum_{i=1}^n r_i=1\}$$

is the [[topological simplex|topological $(n-1)$-simplex]]. A subset $S\subseteq \Delta(n)$ is _[[convex set|convex]]_ if it is closed under [[convex combinations]], i.e. if $\vec r\in S$ and $\vec s\in S$ then $(a\vec r+(1-a) \vec s)\in S$ for any $a\in[0,1]$.

Any finite set of points $\{\vec r_1\dots \vec r_k\}\subseteq \Delta(n)$ _generate_ a convex set (the [[convex hull]]):

$$[\vec r_1\dots \vec r_k] = \{
a_1\vec r_1 + \dots +a_k\vec r_k |
(a_1,\dots, a_k)\in [0,1]^k \& \textstyle \sum_{j=1}^k a_j=1
\}$$

(Recall that the convex hull is not uniquely determined by such a set of points, since some points in the generating set maybe convex combinations of others.)

Let $C(n)$ be the set of finitely generated convex subsets of $\Delta(n)$.

* $C(n)$ has the structure of a [[semilattice]], with $S\oplus T$ the [[union]] of $S$ and $T$ closed under convex combinations.

* It also has the structure of a [[convex space]], with
$[\vec r_1\dots \vec r_k]+_p [\vec s_1\dots \vec s_l]$ defined to be

$$
\array{[p\vec r_1+(1-p)\vec s_1,&\dots,&p\vec r_k+(1-p)\vec s_1,\\
\dots,&\dots,&\dots,\\
p\vec r_1+(1-p)\vec s_l,&\dots,&p\vec r_k+(1-p)\vec s_l].}$$

We thus build a monad structure for $C(n)$:

* The [[unit of a monad|unit]], $\eta_n:n\to C(n)$, picks out the singleton sets containing the basis vectors: $\eta_n(1) = \{(1,0,\dots,0)\}$, $\eta_n(2) = \{(0,1,\dots,0)\}$, and so on.

* The bind, or [[Kleisli extension]], takes a function
$f:m\to C(n)$ to a function $f^*:C(m)\to C(n)$, given by using the convex and semilattice structures:

$$
f^*([\vec r_1\dots \vec r_k]) =
(r_{1,1}f(1)+\dots+r_{1,m}f(m))
\oplus \dots\oplus
(r_{k,1}f(1)+\dots+r_{k,m}f(m)).
$$

## Presentation

The finitary convex powerset of distributions monad is presented by an [[equational theory]]. This has the theory of [[convex spaces]] ($+_r$) together with the theory of [[semilattices]] ($\oplus$) and the axiom

$$ x +_r (y \oplus z) = (x +_r y) \oplus (x +_r z) $$

for all $r\in (0,1)$.

(It may be tempting to think that this is a [[distributive law]] of monads, but it is not, because two sets of distributions are equated if they have the same convex hull. For instance, $x\oplus y = x \oplus y \oplus (x+_0.5 y)$, for:

$$ x \oplus y =
(x \oplus y) +_0.5 (x \oplus y)
=
(x +_0.5 x)\oplus (x +_0.5 y) \oplus (y +_0.5 x) \oplus (y +_0.5 y)
=
x \oplus y \oplus (x +_0.5 y) \ .)$$

The theory is not [[commutative algebraic theory|commutative]], and so the monad is not a [[commutative monad]].
Indeed

$$  (x +_0.5 z) \oplus (y +_0.5 x) \neq
(x \oplus y) +_0.5 (z \oplus x)$$

since the right hand side is equal to
$(x +_0.5 z) \oplus (y+_0.5 z) \oplus x \oplus (y+_0.5 x)$,
and we can compare the convex hulls of the two points of the left-hand side $[(0.5,0,0.5),(0.5,0.5,0)]$  and the four points of the right-hand side [(0.5,0,0.5),(0,0.5,0.5),(1,0,0),(0.5,0.5,0)], which are different convex subsets of the 2-simplex (triangle).

## References

Convex powersets of distributions have been considered by various authors over the last few decades.

A recent extension to metric spaces, including a literature survey is:

* Matteo Mio, Valeria Vignudelli, *Monads and Quantitative Equational Theories for Nondeterminism and Probability*,in:  *CONCUR 2020*, LIPIcs **171** (2020) 28:1-28:18, [[doi:10.4230/LIPIcs.CONCUR.2020.28](https://doi.org/10.4230/LIPIcs.CONCUR.2020.28), [hal:03028173](https://hal.science/hal-03028173), [arXiv:2005.07509](https://arxiv.org/abs/2005.07509)]

The theory in the finitely generated case is shown to be a "distributive tensor" of probability and non-determinism, and axioms and geometric nature is spelt out in Chapter 6 of:

* Kwok-Ho Cheung, *Distributive Interaction of Algebraic
Effects*, PhD thesis, Oxford 2017 [[university archive](https://ora.ox.ac.uk/objects/uuid:66106628-0a71-4564-bc34-c398db766818)].

The theory is further spelt out, amongst other things, in:

* Filippo Bonchi, Ana Sokolova, Valeria Vignudelli: *The Theory of Traces for Systems with Nondeterminism, Probability, and Termination*, Logical Methods in Computer Science **18** 2 (2022) \[doi:10.46298/lmcs-18(2:21)2022\]

The case with unbounded, infinite joins is earlier considered here:

* [[Bart Jacobs]], *Coalgebraic trace semantics for combined possibilitistic and probabilistic systems*, Proceedings of CMCS 2008, ENTCS **203** 5 (2008) 131-152 [[doi:10.1016/j.entcs.2008.05.023](https://doi.org/10.1016/j.entcs.2008.05.023)]

In [[domain theory]], convex powersets of distributions are explored and axiomatized amongst other approaches by

* Daniela Varacca: *Probability, Nondeterminism and Concurrency: Two Denotational Models for
Probabilistic Computation*. PhD thesis, Univ. Aarhus (2003) [[brics:DS/03/14](https://www.brics.dk/DS/03/14)]

and analyzed as "convex powercones" by

* [[Michael W. Mislove]], *Nondeterminism and probabilistic choice: Obeying the laws*, in: Proc. CONCUR
2000, Lecture Notes in Computer Science **1877**, Springer (2000) 350–364 [[doi:10.1007/3-540-44618-4\_26](https://doi.org/10.1007/3-540-44618-4_26)].

* [[Regina Tix]], [[Klaus Keimel]], [[Gordon D. Plotkin]]: *Semantic domains for combining probability and non-determinism*, Electronic Notes in Theoretical Computer Science, **222** (2009) 3–99 [[doi:10.1016/j.entcs.2009.01.002](https://doi.org/10.1016/j.entcs.2009.01.002)]

* Jean Goubault-Larrecq: *Prevision domains and convex powercones*, in: Proc. FOSSACS 2008, Lecture Notes in Computer Science **4962**, Springer (2008) 318–333 [[doi:10.1007/978-3-540-78499-9\_23](https://doi.org/10.1007/978-3-540-78499-9_23)]

[[!redirects convex powerset of distributions monads]]


convex set

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## Definition

A [[subset]] $S$ of a [[real number|real]] [[affine space]] $X$ is **convex** if for any two points $x,y \in S$, the straight line segment connecting $x$ with $y$ in $X$ is also contained in $S$.  In other words, for any $x,y\in S$, and any $t\in [0,1]$, we have also $t x + (1-t) y \in S$.

## Properties

Every convex set is [[star-shaped]] about each of its points, and hence [[contractible space|contractible]] provided it is [[inhabited set|inhabited]].

## Related notions

* One generalization of convexity to [[Riemannian manifold | Riemannian manifolds]] and [[metric space | metric spaces]] is [[geodesic convexity]].

* An abstract generalization of the notion of a convex set is that of a [[convex space]]. There is a nice characterization of those convex spaces which are isomorphic to convex subsets of real affine space.

* The **convex hull** of a subset is the smallest convex subset containing it.

[[!redirects convex hull]]
[[!redirects convex hulls]]
[[!redirects convex set]]
[[!redirects convex sets]]

[[!redirects convex subset]]
[[!redirects convex subsets]]
[[!redirects convex sub-set]]
[[!redirects convex sub-sets]]

[[!redirects convext set]]

[[!redirects convexity]]


convex space

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## Idea

A **convex space** (also called **barycentric algebra** and other terms, invented independently many times) is a [[set]] [[extra structure|equipped]] with a notion of taking weighted averages, or convex-[[linear combinations]], of its elements.  Do not confuse this with an (abstract) _[[convex set]]_, which a special kind of convex space, also defined below.

The [[category]] of convex spaces is the category of algebras of a [[Lawvere theory]], being the affine part of the theory of $K$-(semi)modules with only the idempotent operations.  This definition is used by [Meng (1989)](#Meng), and many basic properties of the category are detailed therein.   Equivalently, the category of convex spaces is the [[Eilenberg-Moore category|category of algebras]] of a [[finitary monad]], as explained by [Jacobs (2010)](#Jacobs10) and [Swirszcz](#Swirszcz).

The category of convex spaces is is complete, cocomplete, symmetric monoidal closed under the (usual) tensor product construction, and has a cogenerator [Börger and Kemper (1994)](#Borger).   The subcategory consisting of the single object, the unit interval, is dense (left-adequate) in the category.  This follows from Isbell's theorem on left adequate subcategories for algebraic theories, using the fact that the free convex space on 2 elements is the unit interval.

Axiomatically, a convex space can be characterized as a [[set]] $X$ equipped with a family of [[functions]] $c_p : X \times X \to X$ satisfying some natural [[axioms]] (described below).

For examples all [[nonunital ring|nonunital]] [[commutative rings]] are convex spaces, with the map $c_p(x,y) = x + p(y-x)$.

The [[monad]] assigning to any set the free convex space on that set is a [[finitary monad|finitary]] [[commutative monad]] (see [Jacobs (2010)](#Jacobs10)). We can thus follow Durov in thinking of it as a [[generalized ring]].  This allows us to think of convex spaces as 'modules' of a generalized ring, very much as [[vector spaces]] are modules of a field.  This is also true of the relatives of convex spaces: [[affine space|affine spaces]] and [[conical space|conical spaces]].  For example, all **affine spaces** are convex spaces as defined below.

Of particular importance are convex spaces parametrized by the interval $P = [0,1]$ or the Boolean algebra $P = \{0,1\}$. These two algebras are dual, in a certain sense described by [Jacobs (2009)](#Jacobs09). This duality is functorial, and therefore is present for convex spaces for general $P$. This leads to the notion of a [[dual convex space]].

## Definition

A __convex space__ is a [[set]] $X$ equipped with:

* a multiplicatively closed [[subset]] $Q$ of a [[semiring|(semi)]][[ring]] $P$, so that for each element $p\in Q$ there exists an element $q\in Q$ such that $p+q=1$, and

* an operation $c_p: X \times X \to X$ defined for all $p\in Q$,

such that the following identities always hold:

*  $c_0(x,y) = x$,
*  $c_p(x,x) = x$ for all $p \in P$,
*  $c_p(x,y) = c_{1-p}(y,x)$ for all $p \in P$ (in semiring case replace $1-p$ by $q$ and require that $p+q = 1$),
* $c_p(x, c_q(y,z)) = c_{p q}(c_r(x,y),z)$ for all $p,q,r \in P$ satisfying $p(1 - q) = (1 - p q)r$.

As a consequence of the first and third axioms, $c_1(x,y) = c_0(y,x) = y$.

This defines convex spaces as a [[variety of algebras]], with one binary operation for each $p$.

The intended interpretation is that $c_p(x,y) = x + p(y-x) = (1-p)x + p y$. i.e., $c_p(x,y)$ is the $p$-weighted average of $x$ and $y$, where $x$ gets weight $1-p$ and $y$ gets weight $p$. By thinking of $p$ as a continuous parameter, this interpretation has the advantage of "starting" at $x$, then moving toward $y$ at "rate" $p$.

This interpretation is '[[bias|biased]]', in the sense that the centered choice $p=0$ favors $x$. It is also possible to give an '[[bias|unbiased]]' definition, which characterizes to convex-linear combinations of many points. This is an $n$-ary operation parametrised by a list $p := (p_1,\ldots,p_n)$ satisfying $\sum_{i = 1}^n p_i = 1$. If $x := (x^1,\ldots, x^n)$, then $c_p(x) := \sum_i p_i x^i$.

A [[homomorphism]] of convex spaces may be called a __convex-linear map__ or an __affine linear map__ (since an [[affine space]] is a convex space with extra properties, as in the examples below).  It should probably *not* be called a 'convex map', which (between affine spaces) is something more general.

## Examples

Any real [[vector space]] is a convex space, with $c_p(x,y) = x + p(y-x)$.  In the unbiased version, any convex-linear combination is a [[linear combination]].  Note that a convex-linear map between vector spaces may not be a linear map, since it may not preserve the identity; thus, a vector space is a convex space with [[extra structure]].

More generally, any real [[affine space]] is a convex space; since $p + (1 - p) = 1$, the expression for $c_p$ in a vector space is valid in an affine space.  In the unbiased version, any convex-linear combination is an [[affine linear combination]].  Now any convex-linear map between affine spaces is an affine linear map (and conversely); an affine space is a convex space with [[extra properties]].

Still more generally, any [[convex subset]] (that is, one containing the entire line segment between two given points) of a real [[affine space]] is a convex space (again with extra properties, which are described algebraically below).

The [[Boolean field]] $\{0,1\}$ is a convex space with $c_p(x,y) = x \vee y = x + y - x y$ whenever $0 \lt p \lt 1$ (with $c_0(x,y) = x$ and $c_1(x,y) = y$ as always); this cannot be realised as a subset of a vector space.  This can be generalised to any (possibly unbounded) [[semilattice]].  (It would be nice to find an example like this that can be defined constructively; this one relies on [[excluded middle]].)

## Abstract convex sets

There is a nice abstract converse to the example of a [[convex subset]] of an affine space.  A convex space is __cancellative__ if $y = z$ whenever $c_p(x,y) = c_p(x,z)$ for some $x$ and $p \ne 0$.  We may call a cancellative convex space an __abstract convex set__.  The justification for this terminology is this

+-- {: .un_theorem}
**Theorem** (Thm. 2 in the paper by Stone)

A convex space is cancellative if and only if it is isomorphic (as a convex space) to a convex subset of some real affine space.
=--

Compare this with the theorem that a [[monoid]] is [[cancellative monoid|cancellative]] if and only if it is isomorphic to a submonoid of some [[group]].

Of course, most of the examples given above are cancellative, being manifestly given as convex subsets of real affine space.  However, the last example --- a semilattice with $c_p(x,y) = x \vee y$ whenever $0 \lt p \lt 1$ --- is non-cancellative.

## Related concepts

* [[super convex space]]
* [[convex function]]
* [[distribution monad]]
* [[monads of probability, measures, and valuations]]
* [[vector space]], [[affine space]], [[conical space]]

## References

Convex spaces have been rediscovered many times under many different names.  References tend to define $c_p$ only for $0 \lt p \lt 1$, but it seems obvious that it\'s best to include the edge cases as well.  Classically, it makes no difference, but the definition above is probably better in [[constructive mathematics]].

*  _[[Handbook of Analysis and its Foundations]]_, Section 12.7 (short and to the point).

* {#Borger} Börger & Kemper, Cogenerators for convex spaces, Applied Categorical Structures, Vol. 2 (1994), 1-11.

*  Romanowska, Smith, Orłowska; Abstract barycentric algebras; [pdf](http://staff.science.uva.nl/~gfontain/tacl09-abstracts/tacl2009_submission_48.pdf).  This generalises from $[0,1]$ to an arbitrary $L \Pi$-algebra ($L$ for 'Łukasiewicz', $\Pi$ for 'product', so think of $[0,1]$ as a space of fuzzy truth values).

*  Romanowska & Smith (1985); Modal Theory: An Algebraic Approach to Order, Geometry, and Convexity; Res. Exp. Math. 9; Heldermann-Verlag, Berlin, 1985.

* Marshall Harvey Stone, _Postulates for the barycentric calculus_, Ann. Mat. Pura. Appl. (4), 29:25–30, 1949.

* [[Tobias Fritz]], Convex spaces I: definition and examples.
[arXiv/0903.5522](http://arxiv.org/abs/0903.5522)

* [[John Baez]], [[Tobias Fritz]], [[Tom Leinster]], [[johnbaez:Convex spaces and an operadic approach to entropy]], $n$Lab draft

* {#Jacobs09} [[Bart Jacobs]], _Duality for convexity_ [arXiv/0911.3834](http://arxiv.org/abs/0911.3834)

* {#Jacobs10} [[Bart Jacobs]], Convexity, duality and effects. In: IFIP International Conference on Theoretical Computer Science. Springer. 2010, pp. 1–19.  ([pdf](https://link.springer.com/content/pdf/10.1007/978-3-642-15240-5_1.pdf))

* J.R. Isbell, Adequate subcategories, Illinois Journal of Math, 4, 541-552  (1960).

* Shiri Artstein-Avidan, Vitali Milman, _The concept of duality in convex analysis, and the characterization of the Legendre transform_, Annals of Math. __169__, n.2, 661-674 (2009)

* Joe Flood, _Semiconvex geometry_, J. Austral. Math. Soc. Ser. A **30** (1980/81), 496-–510.

* {#Swirszcz} T. Swirszcz, _Monadic functors and categories of convex sets_ , Preprint No. **70**, _Proc. Inst. Math. Pol. Acad. Sci._, Warsaw; _Monadic functors and convexity_, _Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys._ **22** (1974), 39--42.

* Stanley P. Gudder, _Convexity and mixtures_, SIAM Review **19** (1977), 221--240; _A general theory of convexity_, Milan Journal of Mathematics, **49** (1979), 89--96.

* {#Meng} Xiao-qing Meng, _Categories of convex sets and of metric spaces with applications to stochastic programming and related areas_, PhD thesis ([[Meng.djvu|djvu:file]])

Many other references, and a discussion of how convex spaces have been repeatedly rediscovered, can be found at the $n$-Category Café post [Convex Spaces](http://golem.ph.utexas.edu/category/2009/04/convex_spaces.html).

[[!redirects convex space]]
[[!redirects convex spaces]]
[[!redirects barycentric algebra]]
[[!redirects barycentric algebras]]

[[!redirects convex combination]]
[[!redirects convex combinations]]
[[!redirects convex linear combination]]
[[!redirects convex linear combinations]]
[[!redirects convex-linear combination]]
[[!redirects convex-linear combinations]]

[[!redirects convex linear function]]
[[!redirects convex linear functions]]
[[!redirects convex-linear function]]
[[!redirects convex-linear functions]]
[[!redirects convex linear map]]
[[!redirects convex linear maps]]
[[!redirects convex-linear map]]
[[!redirects convex-linear maps]]


convolution algebra

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## Idea

There are many related constructions of algebras, topological algebras and so on which bear the name of a _convolution algebra_.

The basic mechanism is usually the

* [Convolution of maps from a coalgebra to an algebra](#ConvolutionOfMapsFromACoalgebraToAnAlgebra) .

The probably most widespread example of this is the

* [Convolution algebra of a group](#OfAGroup)

This is a special case of the

* [Convolution algebra of a groupoid/category](#OfAGroupoid)

### Of maps from a coalgebra to an algebra
 {#ConvolutionOfMapsFromACoalgebraToAnAlgebra}

Let $k$ be a commutative unital [[ring]], $(C,\Delta)$ a (counital) $k$-[[coalgebra]] and $(A,m)$ an associative (unital) $k$-[[associative unital algebra|algebra]]. Then the set of linear maps
$$
\mathrm{Hom}_k(C,A)
$$
has a structure of an associative (unital) algebra, called __convolution algebra__, in which the product of two linear maps $f,g$ is given by
$$f\star g = m\circ(f\otimes g)\circ\Delta. $$

### Of maps from a comonoid to a monoid in a closed monoidal category

\begin{proposition}
Let $(\mathcal{C}, \otimes, I, \multimap)$ be a [[closed monoidal category]],  $(A,\Delta,\eta)$ a [[comonoid]] in $\mathcal{C}$ and $(B,\nabla,\epsilon)$ a [[monoid]]. Then  $A \multimap B$ is a monoid.
\end{proposition}

\begin{proposition}
Let $(\mathcal{C}, \otimes, I, \multimap)$ be a [[closed monoidal category]], $Mon(\mathcal{C})$ the [[category of monoids]] of $\mathcal{C}$ and $Comon(\mathcal{C})$ the [[category of comonoids]] of $\mathcal{C}$. We then have a functor:
$$
Comon(\mathcal{C})^{op} \times Mon(\mathcal{C}) \rightarrow Mon(\mathcal{C})
$$
which associate $A \multimap B$ to $(A,B)$
\end{proposition}

### Of a group
 {#OfAGroup}

Given a [[finite group]] $G$ and a [[ring]] $R$, the space of [[functions]] $C(G,R)$ inherits the _convolution product_ defined by

$$
  f_1 \star f_2 \colon g \mapsto \sum_{g_1 \cdot g_2 = g} f_1(g_1) \cdot f_2(g_2)
  \,.
$$

This is the non-commutative product operation that appears in the [[Hopf algebra]] structure on $C(G,R)$.

### Of a groupoid/category
 {#OfAGroupoid}

More generally, there is convolution of functions on morphisms of a [[groupoid]]. See at _[[groupoid convolution algebra]]_ for details.

[[!redirects convolution algebras]]


convolution product

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## Idea

A _convolution product_ is the binary operation on [[ring]]-valued (or more generally [[magma]]-valued) [[functions]] $f$ on a [[group]] $G$ or more generally on the set of [[morphisms]] $\mathcal{G}$ of a [[groupoid]], which is given by [[sum|summing]] or more generally [[integration|integrating]] products of values on complementary elements of the schematic form

$$
  (f_1 \star f_2)(g) \;=\; \sum_{h \in G} f_1(h)\cdot f_2( g \cdot h^{-1} )
$$

where $h^{-1}$ denotes the [[inverse element]] or more generally the [[inverse morphism]] of $h$.

For example if $G$ is a [[finite group]], then this defines the [[group algebra]], or more generally if $G$ is a suitable [[topological group]] or [[Lie group]] or more generally a suitably [[topological groupoid]] or [[Lie groupoid]], then an [[integral]]-version of this formula defines the _[[groupoid convolution algebra]]_.

Often the convolution product is considered in [[analysis]] for suitably [[integrable functions]] $f \colon \mathbb{R} \to \mathbb{R}$ on the [[real line]] $\mathbb{R}^1$ regarded as a [[group]] with respect to [[addition]] of [[real numbers]]:

$$
  (f_1 \star f_2)(x)
  \;\coloneqq\;
  \int_{t \in\mathbb{R}} f_1(t) f_2(x-t) \, d t
  \,.
$$

This generalizes to a [[convolution product of distributions]].

These convolution products play a central role in [[Fourier analysis]].

A [[categorification]] of the concept of convolution is the concept of _[[Day convolution]]_.

## Related concepts

* [[group algebra]]

* [[groupoid algebra]], [[category algebra]]

[[!redirects convolution products]]

[[!redirects convolution]]
[[!redirects convolutions]]

category: disambiguation


convolution product of distributions

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## Idea

The generalization of the [[convolution product]] on [[smooth function]] to [[distributions]], viewing distributions as [[generalized functions]] (via [this prop.](non-singular+distribution#NonSingularDistributionsAreDenseInAllDistributions)).

## Definition

+-- {: .num_defn #ConvolutionOfADistributionWithSmoothFunction}
###### Definition
**(convolution of a [[distribution]] with a [[smooth function]])**

Let $u \in \mathcal{D}'(\mathbb{R}^n)$ be a [[distribution]], and $f \in C^\infty_0(\mathbb{R}^n)$ a [[compactly supported smooth function]]. Then the convolution of the two is the smooth function

$$
  u \star f \in C^\infty(\mathbb{R}^n)
$$

defined by

$$
  (u \star f)(x) \;\coloneqq\; u\left( f(x-(-))\right)
  \,.
$$

=--

([Hörmander 90, top of section 4.1](#Hoermander90))

+-- {: .num_defn #ConvolutionOfADistributionWithACompactlySupportedDistribution}
###### Definition
**(convolution of a [[distribution]] with a [[compactly supported distribution]])**

Let $u_1, u_2 \in \mathcal{D}'(\mathbb{R}^n)$ be two [[distributions]], such that at least one of them is a [[compactly supported distribution]] in $\mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{D}'(\mathbb{R}^n)$, then their convolution product

$$
  u_1 \star u_2 \;\in \; \mathcal{D}'(\mathbb{R}^n)
$$

is the unique distribution such that for $f \in C^\infty(\mathbb{R}^n)$ a [[smooth function]], it satisfies

$$
  (u_1 \star u_2) \star f = u_1 \star (u_2 \star f)
  \,,
$$

where on the right we have twice a convolution of a distribution with a smooth function according to def. \ref{ConvolutionOfADistributionWithSmoothFunction}.

=--

([Hörmander 90, def. 4.2.2](#Hoermander90))

## Properties

### Relation to pointwise product of distributions

+-- {: .num_prop #FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct}
###### Proposition
**([[Fourier transform of distributions]] interchanges [[convolution of distributions]] with pointwise product)**

Let

$$
  u_1 \in \mathcal{S}'(\mathbb{R}^n)
$$

be a [[tempered distribution]] and

$$
  u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)
$$

be a [[compactly supported distribution]].

Observe here that the [[Paley-Wiener-Schwartz theorem]] implies that the [[Fourier transform of distributions]] of $u_1$ is a [[non-singular distribution]]
$\widehat{u_1} \in C^\infty(\mathbb{R}^n)$ so that the product $\widehat{u_1} \cdot \widehat{u_2}$ is always defined.

Then the [[Fourier transform of distributions]] of the convolution product of distributions (def. \ref{ConvolutionOfADistributionWithACompactlySupportedDistribution}) is the product of the [[Fourier transform of distributions]]:

$$
  \widehat{u_1 \star u_2}
  \;=\;
  \widehat{u_1} \cdot \widehat{u_2}
  \,.
$$

=--

(e.g. [Hörmander 90, theorem 7.1.15](#Hoermander90))

For the converse: [Friedlander-Joshi 98, p. 102](#FriedlanderJoshi98)

+-- {: .num_remark #ProductOfDistributionsViaFourierTransformOfConvolution}
###### Remark
**([[product of distributions]] via [[Fourier transform of distributions]])**

Prop. \ref{FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct} together with the [[Fourier inversion theorem]] suggests to _define_ the [[product of distributions]] $u_1 \cdot u_2$ for [[compactly supported distributions]] $u_1, u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$ by the formula

$$
  \widehat{ u_1 \cdot u_2 }
  \;\coloneqq\;
  (2\pi)^{-n} \widehat{u_1} \star \widehat{u_2}
  \,.
$$

For this to make sense, the [[convolution product]] of the [[smooth functions]] on the right needs to exist, which is not guaranteed. The condition that this exists is the [[Lars Hörmander|Hörmander]]-condition on the _[[wave front set]]_ of $u_1$ and $u_2$. See at _[[product of distributions]]_ for more.

=--

### Relation to  singular support and wave front set

+-- {: .num_prop #WaveFrontSetOfCompactlySupportedDistributions}
###### Proposition
**([[wave front set]] of [[convolution of distributions|convolution of]] [[compactly supported distributions]])**

Let $u,v \in \mathcal{E}'(\mathbb{R}^n)$ be two  [[compactly supported distributions]]. Then the [[wave front set]] of their convolution of distributions (def. \ref{ConvolutionOfADistributionWithACompactlySupportedDistribution}) is

$$
  WF(u \star v)
  \;\subseteq\;
  \left\{
    (x + y, k)
    \;\vert\;
    (x,k) \in WF(u) \,\text{and}\, (y,k) \in WF(u)
  \right\}
  \,.
$$

=--

([Bengel 77, prop. 3.1](#Bengel77))

## Related concepts

* [[product of a distribution with a smooth function]]

* [[product of distributions]]

* [[Fourier transform of distributions]]

## References

* {#Hoermander90} [[Lars Hörmander]], section 4 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

* {#FriedlanderJoshi98} Friedlander F G and Joshi M 1998 _Introduction  to  the  Theory  of  Distributions_ 2nd ed. (Cambridge: Cambridge University Press)

* {#Bengel77} Gunter Bengel, _Wave front sets and Singular Supports of Convolutions_, Publ. RIMS, Kyoto Univ. 12 Suppl. (1977), 1-4 and Math. Ann. 226, 247-252 (1977) ([web](https://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002313952), [pdf](https://gdz.sub.uni-goettingen.de/download/PPN235181684_0226/PPN235181684_0226___LOG_0044.pdf))

[[!redirects convolution products of distributions]]

[[!redirects convolution of distributions]]
[[!redirects convolutions of distributions]]


Conway group

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{:toc}

## Idea

The Conway groups, $Co_{1}, Co_{2}, Co_{3}$, are three of the [[sporadic finite simple groups]]. A fourth group, $Co_0$, is the group of [[automorphisms]] of the [[Leech lattice]] with respect to [[addition]] and [[inner product]]. This latter group is not [[simple group|simple]], but $Co_1$ is the [[quotient group]] of $Co_0$ by its [[center]] of [[order of a group|order]] 2. The other two simple Conway groups are [[subgroups]] of $Co_1$.

The simple Conway groups are three of the seven [[element|members]] of the 'second generation' of the _[[Happy Family]]_ of 20 simple [[subquotients]] of the [[Monster group]].

## Properties

### As automorphism group of super VOAs

The [[Conway group]] $Co_{0}$ is the [[automorphism group|group of]] [[automorphisms of a super VOA]] of the unique chiral [[number of supersymmetries|N=1]] [[super vertex operator algebra]] of [[central charge]] $c = 12$ without fields of [[conformal weight]] $1/2$

  ([Duncan 05](#Duncan05), see also [Paquette-Persson-Volpato 17, p. 9](#PaquettePerssonVolpato17))

See also at _[[moonshine]]_.

## References

### General

See also

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

### Moonshine

* {#Duncan05} John F. Duncan, _Super-moonshine for Conway's largest sporadic group_ ([arXiv:math/0502267](https://arxiv.org/abs/math/0502267))

* {#PaquettePerssonVolpato17} [[Natalie Paquette]], Daniel Persson, Roberto Volpato, _BPS Algebras, Genus Zero, and the Heterotic Monster_ ([arXiv:1701.05169](https://arxiv.org/abs/1701.05169))

[[!redirects Conway groups]]


Conway polynomial

see page for [[Alexander polynomial]]


Conway's game of life

#Contents#
* table of contents
{:toc}

## Idea

A [[cellular automaton]].

## References

The "game of life" is due to [[John H. Conway]], first discussed in print by:

* Martin Gardner, *The fantastic combinations of John Conway's new solitaire game "life"*, Scientific American **223** (October 1970) 120-123 [[pdf](https://web.stanford.edu/class/sts145/Library/life.pdf), [web](http://ddi.cs.uni-potsdam.de/HyFISCH/Produzieren/lis_projekt/proj_gamelife/ConwayScientificAmerican.htm)]

Review:

* Wikipedia, *[Conway's game of life](https://en.m.wikipedia.org/wiki/Conway's_Game_of_Life)*

Proof of omniperiodicity:

* Nico Brown, Carson Cheng, Tanner Jacobi, Maia Karpovich, Matthias Merzenich, David Raucci, [[Mitchell Riley]], *Conway's Game of Life is Omniperiodic* [[arXiv:2312.02799](https://arxiv.org/abs/2312.02799)]

* [The Physics arXiv Blog](https://www.discovermagazine.com/blog/physics-arxiv), *[Mathematicians Prove the "Omniperiodicity" of Conway's Game of Life](https://www.discovermagazine.com/the-sciences/mathematicians-prove-the-omniperiodicity-of-conways-game-of-life)* (Dec 2024)

[[!redirects game of life]]


Cooper H. Langford

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

* William Frankena, Arthur W. Burks, *Cooper Harold Langford 1895-1964*, Proceedings and Addresses of the American Philosophical Association **38** (1964-1965) 99-101 [[jstor:3129489](https://www.jstor.org/stable/3129489)]

## Selected writings

On [[formal logic]] and giving the first modern formalizations of [[modal logic]]:

* [[Clarence I. Lewis]], [[Cooper H. Langford]], *Symbolic Logic* (1932), Dover (2000) [App. II [pdf](https://www.hist-analytic.com/Lewisstrict.pdf)]

category: people

[[!redirects Cooper Langford]]


Cooper pair

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#Contents#
* table of contents
{:toc}

## Idea

The basic kind of [[quasiparticle]] in a [[superconductor]].

## Related concepts

* [[superconductivity]], [[superfluidity]]

* [[Bose-Einstein condensate]]

* [[quantum computation]]

## References

See also:

* Wikipedia, *[Cooper pair](https://en.wikipedia.org/wiki/Cooper_pair)*

[[!redirects Cooper pairs]]

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


cooperad

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###Context###
#### Higher algebra
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#Contents#
* table of contents
{:toc}

## Idea

The notion [[duality|dual]] to [[operad]].

Cooperads are to operads as [[coalgebras]] are to [[algebras]].

Note that a cooperad is not the same notion as a one-object **comulticategory**, which is a structure akin to a [[multicategory]], except in which multimorphisms have unary domain and multiary codomain.

## References

This appears for instance in

* Ching, _Bar construction for topological operads_ ([arXiv:math/0501429](https://arxiv.org/abs/math/0501429))

[[!redirects cooperads]]
[[!redirects co-operad]]
[[!redirects co-operads]]


coordinate projection

[[projection]] on a [[coordinate]]

[[!redirects coordinate projections]]


coordinate system

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### Context
#### Manifolds and cobordisms
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# Contents
* table of contents
{: toc}

## Idea

A [[topological manifold]] $X$ of [[dimension]] $n \in \mathbb{N}$ is by definition a [[topological space]] that is locally [[homeomorphic]] to a [[Cartesian space]] $\mathbb{R}^n$. A choice of such morphism

$$
  \phi : \mathbb{R}^n \stackrel{\simeq}{\to} X|_{im(\phi)} \hookrightarrow X
$$

is a **coordinate system** or **coordinate chart** or just **chart** on the [[image]] of $\phi$.

This generalises to other sorts of [[manifolds]].

An [[atlas]] is the collection of coordinate charts defining a manifold structure.

> graphics grabbed from [[The Geometry of Physics - An Introduction|Frankel]]

## Related concepts

* [[coordinate transformation]], [[diffeomorphism]]

* [[Riemann normal coordinates]], [[Walker coordinates]]

* [[polar coordinates]]

[[!redirects coordinate system]]
[[!redirects coordinate systems]]

[[!redirects chart]]
[[!redirects coordinate chart]]

[[!redirects charts]]
[[!redirects coordinate charts]]

[[!redirects local chart]]
[[!redirects local coordinate chart]]

[[!redirects local charts]]
[[!redirects local coordinate charts]]

[[!redirects coordinate]]
[[!redirects coordinates]]

[[!redirects coordinate patch]]
[[!redirects coordinate patches]]

[[!redirects coordinate function]]
[[!redirects coordinate functions]]

[[!redirects local coordinate]]
[[!redirects local coordinates]]


coordinate-free spectrum

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#Contents#
* table of contents
{:toc}

## Idea

For various constructions in [[stable homotopy theory]] -- such as notably that of the [[symmetric monoidal smash product of spectra]] -- it is useful to use a model
for objects in the [[stable (∞,1)-category of spectra]]
and the [[stable homotopy category]] more refined than that given by [[sequential spectra]]. The notion of _coordinate-free spectrum_ is such
a refinement.

Where a [[sequential spectrum]] is a collection of [[pointed topological space]] indexed by  the [[natural numbers]] $\mathbb{N}$, a coordinate free spectrum is a collection of [[topological space]]s index by all finite dimensional subspaces of a real inner product [[vector space]] $U$ isomorphic to $\mathbb{R}^\infty$.

Notice that also spectra realized as [[excisive functors]] are coordinate-free in an evident sense, as these are indexed on all [[finite homotopy types]].

In the broader context of [[equivariant stable homotopy theory]] a coordinate-free spectrum may be thought of as the special case of a  [[G-spectrum]] over a [[G-universe]] for the special case of the trivial group $G = 1$.

## Definition

Let $U$ be a real inner product [[vector space]] isomorphic to the [[direct sum]]
$\mathbb{R}^\infty$ of countably many copies of the [[real line]] $\mathbb{R}$.

For $V \subset U$ a finite-dimensional subspace, write $S^V$ for its
[[one-point compactification]] (an $n$-dimensional [[sphere]] if $V$ is $n$-dimensional)
and for $X$ any [[pointed object|based]] [[topological space]]
write $\Omega^V X := Maps(S^V,X)$ for the [[topological space]] of basepoint-preserving
continuous maps.

For $V \subset W$ an inclusion of finite dimensional subspaces $V,W \subset U$
write $W-V$ for the [[orthogonal complement]] of $V$ in $W$.

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

A **coordinate-free spectrum** $E$ modeled on the "universe" $U$ is

* for each finite-dimensional subspace $V \subset U$ a [[pointed object|pointed]]
[[topological space]] $E_V$;

* for each inclusion $V \subset W$ of finite dimensional subspaces $V,W \subset U$
a [[homeomorphism]] of pointed [[topological space]]s

  $$
    \tilde \sigma_{V,W} : E_V \stackrel{\simeq}{\to} \Omega^{W-V} E_W
    \,.
  $$

* If we drop the requirement that the maps $\tilde \sigma_{V,W}$ be homeomorphisms, we obtain the notion of a **prespectrum**.

=--

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

The definition of coordinate free spectrum directly generalizes to that of genuine [[G-spectrum]] modeled on a [[G-universe]], leading from [[stable homotopy theory]] to [[equivariant stable homotopy theory]].

=--

## Related concepts

* [[excisive functor]]

* [[functor with smash product]]

## References

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May|P. May]], section 1 of _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_ (1995)  ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

* {#Kochmann96} [[Stanley Kochmann]], section 3.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996
[[!redirects coordinate-free spectra]]


coordination

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#### Philosophy
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#### Physics
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[[!include physicscontents]]
=--
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# Contents
* table of contents
{: toc}

##General Idea

In [[philosophy of science]] and specifically in the [[philosophy of physics]],  **coordination** concerns the relationship between [[theory (physics)|scientific theory]] and [[experiment|empirical data]]. A [[physical theory]], written, say, as a set of [[equations]] or similar data, does not by itself allow the formation of predictions. In addition we need some way of mediating between this theory and what instruments can detect about the [[observable universe]].

For example an abstract [[differential equation]] on the [[sections]] of some abstract [[fiber bundle]] does not constitute a theory of physics unless one specifies which kind of [[field (physics)|physical field]] these sections are meant to represent.

The nature of the field however determines a rule for how to measure it: If we are told that the theory is meant to be about the [[electromagnetic field]], say [[electromagnetic waves]], we test it by measuring, say, [[Lorentz forces]] on [[electrons]]; while if we are told that the theory is meant to be about [[gravity]], say [[gravitational waves]], we test it by entirely different measurements (e.g. via the [[LIGO]] experiment).

But this distinction may not be reflected in the plain [[mathematics]] that constitutes the physical, theory: In the previous example both [[electromagnetic waves]] and [[gravitational waves]] may be described,  after appropriate [[gauge fixing]], by the same kind of [[wave equation]]. Hence _coordination_ is involved in specifying what aspect of observable reality this equation is meant to be referring to.

Further aspects of coordination concern the workings of instruments such as detectors. To be able to say that a physical field has a certain strength at a point requires us to give an account of how the field affects some component of the detection process. Typically then we need to rely on an understanding that other parts of the apparatus will not be significantly affected by the physical field.

##History

The term *coordination* is a translation of the German *Zuordnung*. The latter has a long history of use with [[mathematics]] as something like _assignment_, hence _[[function]]_. Its use in [[philosophy]] stretches back to [[Hermann von Helmholtz]]'s *Zeichentheorie*, and was taken up later by a number of philosophers, especially by those, such as [[Ernst Cassirer]], trying to understand the implications for epistemology of the profound transformations which took place in mathematics and physics at the turn of the nineteenth century ([[special relativity]], [[general relativity]], [[quantum mechanics]]).

In the early twentieth century, discussions about coordination often took [[Henri Poincaré's]] _conventionalism_ as a starting point. Poincaré had illustrated his ideas with a thought experiment about the geometric investigations of inhabitants of a spherical world in which the temperature distribution is non-uniform, cooling off to the boundary away from the center in a prescribed way, and where all materials thermally expand with the same coefficient. An inhabitant of this world measuring with a ruler the ratio of circumference to diameter of a circle about the center of the world will find an answer greater than $\pi$.

Two interpretations then present themselves: either the geometry of the space can be taken to be [[Euclidean geometry|Euclidean]], by factoring in the changing length of a ruler through the space, or the ruler may be considered to provide a stable unit of distance and the geometry is now taken to be hyperbolic (the heat distribution having been arranged to allow this). Poincaré proposed that it is only a matter of convenience which of these accounts we choose, suggesting that we will typically retain the interpretation in terms of Euclidean geometry.

A relevant real-life experiment is Michelson and Morley's attempt in 1887 to detect the presence of the aether. Measurements of the [[speed of light]] along two perpendicular directions via interference patterns indicated it to be a constant, when one might have expected a difference according to the orientation of the direction with respect to the supposed passage of the apparatus through the aether. However another interpretation given by FitzGerald and Lorentz proposed that physical objects contract according to their motion relative to the aether in such a way that the apparatus would not be able to detect a direction-dependent speed of light.

In his *Allgemeine Erkenntnislehre* ([Schlick18](#Schlick18})), Moritz Schlick  relies heavily on the notion of coordination, proposing that it should be understood as a simple [[set theory|set-theoretic]] [[mapping]] between the system of implicitly defined terms of a [[physical theory]] and some system of given objects or elements of sensation. A few years later, Hans Reichenbach, by contrast, argued ([Reichenbach 24](#Reichenbach24)) that the act of coordination was no mere mapping between existing systems, but itself played a constitutive role in defining the objects of knowledge themselves:

>The difficulty concerning coordinative definitions is similar to that concerning elementary facts: the physical thing that is coordinated is not an immediate perceptual experience but must be constructed from such experience by means of an interpretation. If I establish the coordinative definition "a light ray is a straight line", then the coordinated physical thing, the light ray, is a construction going beyond perception. ([1924](#Reichenbach24), p. 8)

A central issue between these accounts is that the more you press on the constitutive function of coordinating principles, the more it seems that the entities referred to by physics owe their characteristics to our conceptual framing rather than to how things are in themselves.

One might hope, however, that away from toy cases such as Poincaré's heated sphere world, detailed physical investigation can determine between competing accounts. Einstein explained the superiority of his account of the Michelson-Morley experiment relative to FitzGerald-Lorentz contraction:

>Although the estimated difference between these two times is exceedingly small, Michelson and Morley performed an experiment involving interference in which this difference should have been clearly detectable. But the experiment gave a negative result — a fact very perplexing to physicists. Lorentz and FitzGerald rescued the theory from this difficulty by assuming that the motion of the body relative to the æther produces a contraction of the body in the direction of motion, the amount of contraction being just sufficient to compensate for the difference in time mentioned above. Comparison with the discussion in Section 11 shows that also from the standpoint of the theory of relativity this solution of the difficulty was the right one. But on the basis of the theory of relativity the method of interpretation is incomparably more satisfactory. According to this theory there is no such thing as a "specially favoured" (unique) co-ordinate system to occasion the introduction of the æther-idea, and hence there can be no æther-drift, nor any experiment with which to demonstrate it. Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. Thus for a co-ordinate system moving with the earth the mirror system of Michelson and Morley is not shortened, but it is shortened for a co-ordinate system which is at rest relatively to the sun. ([Einstein 1916](#Einstein16))

## In quantum mechanics

### Bohrification

The idea of _[[Bohr toposes]]_ might be regarded as an attempt to formalize coordination for [[observables]] in [[quantum mechanics]]. There the idea is roughly that to a [[quantum mechanical system]] one assigns a [[topos]] (the "[[Bohr topos]]") such that the (or some of the) [[propositions]] in the [[internal logic]] of this topos match propositions about [[observables]] of the [[physical system]].

### Relation to interpretation of QM
 {#RelationToInterpretationOfQM}

Coordination is different from -- in fact somewhat orthogonal to -- [[interpretation of quantum mechanics]]. For instance a coordination in [[quantum mechanics]] involves relating [[self-adjoint operators]] to certain [[experiments]]. Given this coordination, the theory assigns [[expectation values]] for the outcomes of the experiment, but it does not necessarily offer an "interpretation why" these propbabilities are predicted.

On the other hand, it might be that a good formulation of the coordination process makes the need/desire for interpretations diminish. This was argued by ([Tanona 10](#Tanona10)):

>...the characterization of collapse as a separate physical process is misguided because the phenomenon which collapse is supposed to address concerns not an actual process within quantum mechanical theory but rather the coordination between empirical measurements and representations of quantum systems. Until we first get clear on this relationship, it is premature to propose new processes to account for features of that relationship.

## Related concepts

* [[experiment]]

* [[theory (physics)]]

* [[statistical significance]]

* [[interpretation of quantum mechanics]]

* [[computable physics]]

##References

* {#Schlick18} Moritz Schlick, _Allgemeine Erkenntnislehre_ (1918)

* {#Reichenbach24} Hans Reichenbach, _Axiomatization of the Theory of Relativity_. Berkeley: University of California Press. (1969). Original German edition published in 1924.

* {#Ryckman91} Thomas Ryckman (1991), _Conditio sine qua non? Zuordnung in the Early Epistemologies of Cassirer and Schlick_, Synthese 88(1), pp. 57-95.

* {#Tanona10}[[Scott Tanona]], _Theory, coordination, and empirical meaning in modern physics_, in Mary Domski and Michael Dickson (eds.), _Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science_. Open Court (2010), pp. 423-454.

* {#Friedman99} [[Michael Friedman]], _Poincaré's Convetionalism and the logical positivists_, chapter 4 in _Reconsidering Logical Positivism_, Cambridge University Press 1999   ([web](https://books.google.co.uk/books?id=e9TjZc9wNUAC&pg=PA71#v=onepage&q&f=false))

* {#Einstein16} [[Albert Einstein]], _Relativity: The Special and General Theory_, 1916, New York: H. Holt and Company


cop

##Idea##

A [[coalgebra]] is half of the structure of a [[bialgebra]]. A bialgebra with antipode is a [[Hopf algebra]], and morally a "Hopf algebra without counit" is essentially a [[quantum heap]].
So a cop should be "half way toward a quantum heap".

##Definitions##

Let $(A,\otimes,1)$, or $A$ for short,
be a [[strict monoidal category]] with unit object $1$.
A __cop__ $C$ in $(A,\otimes, 1)$ is a pair
$(C,\tau)$, where $C$ is an object in $A$
and $\tau : C \rightarrow C \otimes C \otimes C$
is a morphism in $A$ satisfying the law
\begin{equation} \label{law}
(\id \otimes \id \otimes \tau) \circ \tau =
  (\tau \otimes \id \otimes \id) \circ \tau.
\end{equation}
Let $(A,\otimes,1,\sigma)$ be a strict [[symmetric monoidal
category]] and $C$ a [[monoid object|monoid (= algebra) object]] in that category, i.e.
$C$ is equipped with a product $\mu : C \otimes C \rightarrow C$
and a unit morphism $\eta : 1 \rightarrow C$ satisfying the
standard axioms.
Then an _opposite monoid_ $C_{op}$ (lower index as it is on the monoid side and upper index is for the comonoid-side duality here) is the same object $C$
equipped with product $\sigma_{C,C} \circ \mu$ and
with the same unit map $\eta$.

A __symmetric cop monoid__ $C$ in
a strict symmetric monoidal category $(A,\otimes,1,\sigma)$
is a _monoid_ object $C$ with a _morphism of monoids_
$\tau : C \rightarrow C \otimes C_{op} \otimes C$
satisfying the law (eqref:law). Notice the passage to the opposite monoid in the second tensor factor (which does not make sense for usual cops in nonsymmetric monoidal categories).
Here the tensor product has the usual tensor product
structure of a monoid in a strict symmetric monoidal category (for two monoids $C$ and $D$ one takes $(\mu \otimes \mu) \circ (\id \otimes \sigma_{D,C} \otimes \id)$ as a product on $C \otimes D$).

A __counit__ of a cop $C$ in $A$ is a morphism
$\epsilon : C \rightarrow 1$ such that
\begin{equation}
 (\id \otimes \epsilon \otimes \epsilon) \circ \tau = \id = (\epsilon
\otimes \epsilon \otimes \id) \circ \tau ,
\end{equation}
where the identification morphism
$1 \otimes 1 \otimes C \equiv C
\equiv C \otimes 1 \otimes 1$ is used.

A typical interest is in the cops in the (symmetric) monoidal category of [[vector space]]s $Vec_k$ or [[supervector space]]s $SVec_k$ over some fixed field $k$.

A __coheap monoid__ in a symmetric monoidal category is a
symmetric cop monoid such that
$(\id \otimes \mu)\circ \tau = (\mu\otimes \id)\circ\tau = \id$,
where the identification $C\otimes 1 = C = 1\otimes C$
is implicitly used. A __character__ of a monoid $(C,\mu,\eta)$ in a strict monoidal category
is a morphism $\epsilon : C \rightarrow {\bf 1}$ such that
$\epsilon \circ \eta = \id_1$ and
$(\epsilon \otimes \epsilon) = \epsilon \circ\mu$.

A __character of a symmetric cop monoid__ $C$
is any character of $(C,\eta,\mu)$ in $A$.

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

A _character_ of a coheap monoid
is a automatically a counit of the underlying cop.
=--

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

This is straightforward:
$$\array{
(\id \otimes \epsilon\otimes \epsilon)\tau =
(\id\otimes (\epsilon\circ\mu))\tau =
(\id\otimes\epsilon)(\id\otimes\mu)\tau =
(\id\otimes\epsilon)(\id\otimes\eta) = \id,
\\
(\epsilon\otimes\epsilon\otimes\id)\tau =
((\epsilon\circ\mu)\circ\id)\tau = (\id\otimes\epsilon)(\mu\otimes\id)\tau =
(\epsilon\otimes\id)(\eta\otimes\id) = \id,
}$$
where obvious identifications are implicitly
used.
=--

A __copointed cop__ is a pair $(C,\epsilon)$
of a cop $C$ and a counit $\epsilon$ of $C$.
A __copointed coheap monoid__ is a coheap monoid with
a _character_ $\epsilon$ of $C$. Warning: a copointed cop which is also a coheap is not necessarily a copointed coheap, as the counit does not
need to be a character of a coheap.

##Remarks##

* Clearly all this may be generalized to nonstrict [[monoidal categories]].

* About the language: the 'co' in 'cop' mimics the 'co' in 'coalgebra';
furthermore, in the dialect of Kent, according to the OED,
'cop' means _a small heap of hay or straw_. This way it is kind of coalgebra-like entity half way toward a (quantum) heap as in the idea section above.

##Literature##

* Zoran Škoda, _Quantum heaps, cops and heapy categories_, Mathematical Communications 12, No. 1, pp. 1-9 (2007); [math.QA/0701749](http://www.arxiv.org/abs/math.QA/0701749)