display of a p-divisible group

## Idea

In general an assignment of an object of linear algebra to a $p$-divisible group is called a [[Dieudonné theory]].

There is a Dieudonné theory assigning to a formal $p$-divisible group $X$ over an excellent [[p-adic ring]] $R$ an object called *a display*. On the display one can read off the structural equations for the [[Cartier module]] of $X$.

## Definition

Let $R\in CRing$ a commutative unitary ring. Let $W(R)$ denote the ring of [[Witt vectors]] of $R$. Let

$$w_n:\begin{cases}
W(R)\to R
\\
(x_0,\dots,x_i,\dots)\mapsto x_0^{p^n}+p x_1^{p^{n-1}}+ \dots + p^n x_n
\end{cases}$$

denote the morphism of rings assigning to Witt vector its correspnding [[Witt polynomial]]. Let

$$w_n:\begin{cases}
W(R)\to W(R)
\\
(x_0,\dots,x_i,\dots)\mapsto (0,x_0,\dots,x_i,\dots)
\end{cases}$$

denote the [[Verschiebung morphism]] which is a morphism of the underlying additive groups. Let $p$ be a [[prime number]] and let

$$F:W(R)\to W(R)$$

denote the [[Frobenius morphism]]. Then Frobenius, Verschiebung, and the Witt-polynomial morphism satisfy the ''$p$-adic Witt-Frobenius identities'':

$$w_n(F(x))=w_{n+1}(x),\; n\ge 0$$

$$w_n(V(x))=w_{n-1}(x),\; n\gt 0$$

$$w_0(V(x))=0$$

$$FV=p$$

$$VF(xy)=xV(y)$$

+-- {: .num_defn}
###### Definition
A *$3n$-display over R* is defined to be a quadruple $(P, Q, F, V^{-1})$ where $P$ is a finitely generated projective $W(R)$-module, $Q \subset P$ is a submodule and $F$ and $V^{-1}$ are $F$-linear maps $F : P \to P$, $V^{-1}: Q \to P$.

The following properties are satisfied:

1. $ker(w_0)P \subset  Q \subset P$ and $P /Q$ is a direct summand of the $W (R)$−module $P /ker(w_0)P$.

1. $V^{-1} : Q \to P$ is a $F$-linear epimorphism.

1.  For $x \in P$ and $w \in W (R)$, we have $V^{-1} ( V wx) = wF x$.

=--

+-- {: .num_defn}
###### Definition
Let $(\alpha_{ij})$ be a invertible matrix satisfying

a) $F e_j=\sum_{i=1}^h\alpha_{ij} e_i, \; j=1,\dots ,d$

b) $V^{-1} e_j=\sum_{i=1}^h\alpha_{ij} e_i, \; j=1,\dots ,h$

Let $(\beta_{kl})$ denote the inverse matrix of $(\alpha_{ij})$. Let $B:=(w_0(\beta_{kl})modulo\; p)_{k,l=d+1,\dots,k}$ let $B^{(p)}$ deonte the matrix obtained by raising all entries to the $p$-th power. $(\alpha_{ij})$ is said to satisfy *the $V$-nilpotence condition* if there is a natural number $N\in \mathbb{N}$ such that $B^{p^{N-1}}\cdot\dots\cdot B^{(p)}\cdot B=0$.

Then a $3n$-display satisfying the $V$-nilpotence condition locally on the [[spectrum]] $Spec R$ is called a *display*.
=--

## Properties

The following theorem compares the Dieudonné theory in terms of displays and the [[crystalline Dieudonné theory]] of Grothendieck and Messing [Messing](#Messing).

+-- {: .num_theorem}
###### Theorem
Let $\mathcal{P}:=(P,Q,F,V^{-1})$ be a display over $R$. Then there is an isomorphism of [[crystal|crystals]]

$$D_\mathcal{P}\stackrel{\sim}{\to}\mathbf{D}_{B T_\mathcal{P}}(S)$$

where on the right side is the crystal defined in [Messing](#Messing).
=--

## Examples

## References

* [[Lawrence Breen]], [[Pierre Berthelot]], [[William Messing]], Théorie de Dieudonné cristalline II {#Messing}

* T. Zink, the display of a formal p-divisible group, to appear in [[Astérisque]], [pdf](http://www.math.uni-bielefeld.de/~zink/display.pdf)

* T. Zink, Windows for displays of p-divisible groups. in:Moduli of Abelian Varieties, Progress in Mathematics 195, Birkhäuser 2001, [pdf](http://www.math.uni-bielefeld.de/~zink/Texel.pdf)


displayed category

# Displayed categories

* table of contents
{: toc}

## Idea

A **displayed category** over a [[category]] $C$ is the "classifying map" of a category over $C$.  That is, it is equivalent to the data of a category $D$ and a functor $F:D\to C$, but organized differently as a "functor" associating to each object or morphism of $C$ the [[fiber]] over it.  The operation (which is an [[equivalence]]) taking a displayed category to the corresponding functor $F:D\to C$ is a generalization of the [[Grothendieck construction]]. This Grothendieck construction is an instance of the [[T-Grothendieck construction]] where $T$ is the theory of categories.

Displayed categories are particularly useful in [[type theory]] (especially [[internal categories in homotopy type theory]]) and to preserve the [[principle of equivalence]], since they allow a more "category-theoretic" formulation of various notions (such as [[identity-on-objects]] and [[Grothendieck fibrations]] and strict [[creation of limits]]) that, if stated in terms of a functor $F:D\to C$, would involve equality of objects.

## Definitions

### By components

A **displayed category** $D$ over a category $C$ consists of

* for each object $a:C$, a type $D(a)$
* for each object $a:C$ and $b:C$, morphism $f:a \to b$, and element $x:D(a)$ and $y:D(b)$, a set of morphisms $Hom_f(x, y)$ from $x$ to $y$ over $f$
* for each object $a:C$ and element $x:D(a)$, a morphism $id_x:Hom_{id_a}(x, x)$
* for each object $a:C$, $b:C$, and $c:C$, morphism $f:a \to b$ and $g:b \to c$, and elements $x:D(a)$, $y:D(b)$, $z:D(c)$, a function:
$$(-)\circ_{a, b, c, f, g, x, y, z}(-):Hom_g(y, z) \times Hom_f(x, y) \to Hom_{g \circ f}(x, z)$$
such that
* for all morphisms $k:Hom_f(x, y)$, there is the dependent identification
$$\lambda(k):id_y \circ k =_* k$$
* for all morphisms $k:Hom_f(x, y)$, there is the dependent identification
$$\rho(k):k \circ id_x =_* k$$
* for all morphisms $k:Hom_f(x, y)$, $l:Hom_f(y, z)$, $m:Hom_f(z, w)$,
$$(k \circ l) \circ m =_* k \circ (l \circ m)$$

...

### Abstract definition

A **displayed category** over a category $C$ is a [[lax functor]] from $C$, regarded as a bicategory with only identity 2-cells, to the [[bicategory]] [[Span]].

Better, it is a lax [[double functor]] from $C$, regarded as a [[double category]] "horizontally" with only identity vertical arrows and 2-cells, to the (pseudo) double category $Span$ with sets as objects, functions as vertical arrows, and spans as horizontal arrows.  Although this produces an equivalent notion, it is better because a **displayed functor** is then a [[vertical transformation]] between such lax double functors.

A displayed category may equivalently be described as a *[[normal lax functor|normal]]* lax functor from $C$ to [[Prof]] (either the bicategory or the double category, as appropriate), meaning one that strictly preserves identities.  Formally, this is because $Prof = Mod(Span)$, where $Mod(-)$ denotes the double category of horizontal monads and modules, and $Mod$ is a [[right adjoint]] to the inclusion of [[virtual double categories]] and [[normal lax functor|normal (lax) functors]] into all (lax) functors; see [(CS, Prop. 5.14)](#CS).

Equivalently, it is a [[double profunctor]] between $C$ and the terminal double category $1$, i.e., a double presheaf on $C$.

## Correspondence to slices

The category over $C$ corresponding to a displayed category $D:C\to Span$ is the pullback
$$ \array{ D & \to & Span_* \\ \downarrow & & \downarrow \\ C & \to & Span }$$
Where $Span_*=Span(Set_*)$ is the bicategory (or double category) of [[pointed sets]] and pointed spans.  This is a strict pullback, which exists in the 2-category of bicategories (or double categories) and lax functors because the projection $Span_* \to Span$ is not just lax but  strong.  Equivalently, it is the pullback
$$ \array{ D & \to & Prof_* \\ \downarrow & & \downarrow \\ C & \to & Prof }$$
where $Prof_*$ consists of pointed categories and pointed profunctors, a pointed profunctor $H:(A,a_0)⇸(B,b_0)$ being equipped with an element of $H(a_0,b_0)$.

This construction induces an [[equivalence of categories]] $Disp(C) \to Cat/C$, which restricts to the following equivalences:

* A displayed category factors through the inclusion $Set \hookrightarrow Span$ (or equivalently $Set \hookrightarrow Prof$) if and only if $F:D\to C$ is a [[discrete opfibration]].  Similarly, it factors through $Set^{op}$ if and only if $F$ is a discrete fibration.

* A factorization of a displayed category $C\to Prof$ through the inclusion $Cat^{op} \hookrightarrow Prof$ (as co[[representable profunctors]]) is equivalent to giving $F:D\to C$ the structure of a [[cloven fibration|cloven]] [[prefibred category]], i.e. equipped with a choice of weakly cartesian liftings.  The factorization is a [[pseudofunctor]] precisely when $F:D\to C$ is a [[Grothendieck fibration]]; in this case we see the usual [[Grothendieck construction]] of a pseudofunctor.

* Similarly, factorizations through $Cat\hookrightarrow Prof$ corresponds to cloven Grothendieck (pre)opfibrations.

* An arbitrary displayed category $C\to Prof$ is a pseudofunctor if and only if $F:D\to C$ is a [[Conduché functor]], i.e. an exponentiable morphism in $Cat$.

### The construction, explicitly
As mentioned above, there is a construction $\int$ turning a normal lax functor $F:C \to Prof$ into a functor $\pi_F : \int F \to C$. Here we spell it out.

But first, let us explictly remark how the opposite construction works. Using notation from [Benaboù's lectures](#Benabou), the normal lax functor $dP : C \to Prof$ associated to a functor $P:E \to C$ is obtained by 'taking fibers'. On objects, it maps $x:C$ to the category $P^{-1}x$ of objects and maps of $E$ mapping to $x$ and its identity arrow, respectively. On morphisms, it maps an arrow $f:x \to y$ to the profunctor $P^{-1}y^{op} \times P^{-1}x \to Set$ mapping a choice of objects $y'$ and $x'$ 'lifting' $y$ and $x$ respectively to the set of maps $f^\sharp : x'\to y'$ that project down to $f$, i.e.~the 'fiber over $f$' of the arrow part of $P$ (which is defined given two objects).

#### The generalized Grothendieck construction
The category $\int F$ is built as follows:

* objects are pairs $(x:C, x' : Fx)$

* a morphism $(x,x') \to (y,y')$ is given by a pair of morphisms $(f,f^\sharp)$ where $f:x \to y$ is an arrow in $C$ and $f^\sharp \in Ff(y',x')$

* identities are given by picking $1_x : x \to x$ and $1_x^\sharp = 1_{x'} \in F(1_x)(x',x') = \mathrm{Hom}_{Fx}(x',x')$

* composition is defined componentwise: given $(f,f^\sharp) : (x,x') \to (y,y')$ and $(g,g^\sharp):(y,y') \to (z,z')$, the composite is the pair $(gf, \ell_{f,g}(y', f^\sharp, g^\sharp))$, where $(y', f^\sharp, g^\sharp)$ is the equivalence class of the coend defining the composition $Fg \circ Ff$ (see [[profunctor]]) and $\ell_{f,g}$ is the laxator of $F$.

Finally, the functor $\pi_F$ simply discards the second component on objects and morphisms.

One can readily observe how this construction reduces to the usual [[Grothendieck construction]] when $F$ factors through $Cat$.

##### On morphisms
The above construction is functorial and turns 'morphisms of normal lax functors into $Prof$' into morphisms in $Cat/C$.
To describe this, is useful to notice the category of normal lax functors into Prof is given by $Dbl_{normal,lax}^{vertical}(hC, Cat)$, where $hC$ is the casting of $C$, a category, as a double category whose vertical and 2-cells are all trivial.

This category has as objects normal lax [[double functors]] $C \to Cat$, where $Cat$ is the [[proarrow equipment]] of categories, functors (in the vertical/tight direction), profunctors (in the horizontal/loose direction) and suitable natural transformations as 2-cells. The morphisms are then [[vertical natural transformations]] between them.

Let's spell out the definition of one of these vertical natural transformations $\phi : F \Rightarrow G$. Its components are given by functors (i.e. vertical cells in $Cat$) $\phi_x : Fx \to Gx$, indexed by objects $x:C$, and squares

\begin{centre}
\begin{tikzcd}[ampersand replacement=\&,sep=scriptsize]
	Fx \&\& Fy \\
	\\
	Gx \&\& Gy
	\arrow["Ff", "\shortmid"{marking}, from=1-1, to=1-3]
	\arrow["Gf"', "\shortmid"{marking}, from=3-1, to=3-3]
	\arrow["{\phi_x}"', from=1-1, to=3-1]
	\arrow["{\phi_y}", from=1-3, to=3-3]
	\arrow["{\mathrm{st}^\phi}"{description}, shorten <=11pt, shorten >=11pt, Rightarrow, from=1-3, to=3-1]
\end{tikzcd}
\end{centre}

indexed by morphisms $f:x\to y$ in $C$. These have, moreover, to satisfy various 'natural' conditions of compatibility with horizontal composition of squares in $Cat$ and so on.

The equivalence $\int$ converts this data back to a commutative triangle
\begin{centre}
\begin{tikzcd}[ampersand replacement=\&,row sep=scriptsize]
	{\int F} \&\& {\int G} \\
	\& C
	\arrow["{\pi_F}"', from=1-1, to=2-2]
	\arrow["{\pi_G}", from=1-3, to=2-2]
	\arrow["{\int \phi}", from=1-1, to=1-3]
\end{tikzcd}
\end{centre}

The crucial thing to notice here is that, fundamentally, such a functor over $C$ is defined 'fiberwise', in the sense that, choosen an object $x:C$, $\int \phi$ has to restrict to a functor $\int\phi_x : \pi_F^{-1}x \to \pi_G^{-1}x$. But these categories are, by definition, $Fx$ and $Gx$, so that $\int \phi_x$ can be naturally defined to be $\phi_x$ itself.

Now what's left to define is the action of $\int \phi$ on those arrows that cross fibers (in a sense, all the arrows except those that map to identities in $C$ -- or, put differently again, the above discussion pinned down the object part of $\int \phi$).

Thus let $(f,f^\sharp):(x,x') \to (y,y')$ be a morphism in $\int F$. We know how to map $(x,x')$ and $(y,y')$ using $\phi_x$ and $\phi_y$ respectively, so that $\int \phi(f,f^\sharp)$ has to be an arrow $(x,\phi_x(x')) \to (y,\phi_y(y'))$.
The exact arrow is picked by the last piece of data from the vertical transformation $\phi$, the 'strength' $\mathrm{st}^{\phi}$. In fact this is a natural transformation whose components are $\mathrm{st}^{\phi}_{x',y'} : Ff(x',y') \to Gf(\phi_x(x'),\phi_y(y'))$, which is exactly the data needed to define the arrow part of $\int \phi$.

Functoriality of $\int \phi$ is a direct consequence of the properties required to $\phi$, which are spelled out at [[vertical natural transformation]].

## Properties of functors through properties of the reindexing

Let $F:E \to B$ be a functor, call $dF : B^{op} \to Prof$ the associated lax normal functor.
By requiring a different degree of laxity to $dF$ or which subcategory of $Prof$ it hits, we can express different properties of $F$:

1. If $dF$ is a pseudofunctor, then $F$ is a [[Conduché fibration]], i.e. an exponential morphism in $Cat$
2. If $dF$ factors through the inclusion $Cat \to Prof$, then $F$ is a [[prefibration]]
3. If $dF$ is both Conduché and a prefibration, then $F$ is a [[Grothendieck fibration]]
4. Call *partial functor* a profunctor $\phi : X \to Psh(Y)$ that factors through $(y,0) : Y + 1 \to Psh(Y)$, which is the Yoneda embedding on $Y$ and picks the empty presheaf otherwise. When $dF$ factors through the subcategory of partial functors it is called a [[foliated category|foliation]].

These and more examples are discussed in [Benabou](#Benabou).

## References

The correspondence between categories over $C$ and [[normal lax functors]] $C\to Prof$ was observed by [[Jean Benabou|Bénabou]] already in 1972:

* [[Jean Benabou|Jean Bénabou]], *2-dimensional limits and colimits of distributors (or how to glue together categories)*, [[Oberwolfach]] 'Tagungsbericht', 1972, [(web)](https://oda.mfo.de/handle/mfo/1575)

A more detailed account of Benabou's work on this topic can be found at:

* Jean Bénabou, *Distributors at work*, Notes by Thomas Streicher from lectures given at TU Darmstadt, 2000, [pdf](http://www.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf) {#Benabou}

The term "displayed category", and the applications to type theory, are due to:

* [[Benedikt Ahrens]], [[Peter LeFanu Lumsdaine]], *Displayed Categories*, [arxiv](https://arxiv.org/abs/1705.04296)

An unpacking of the definition as lax functors into [[Span]] is in 2.2 of

* [[Duško Pavlović]], [[Samson Abramsky]], *Specifying Interaction Categories*, [LNCS](https://link.springer.com/chapter/10.1007/BFb0026986)

Also cited above:

* [[Geoff Cruttwell]], [[Mike Shulman]], *A unified framework for generalized multicategories* [TAC](http://tac.mta.ca/tac/volumes/24/21/24-21abs.html)
 {#CS}

[[!redirects displayed categories]]
[[!redirects displayed functor]]
[[!redirects displayed functors]]


displayed type theory

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#### Modalities, Closure and Reflection
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=--
=--
=--

\tableofcontents

## Idea

**Displayed type theory** [[KS23](#KS23)] is a [[modal type theory]] which [[synthetic mathematics|synthetically]] models an [[(infinity,1)-topos|$(\infty,1)$-topos]] and the corresponding [[(infinity,1)-category|$(\infty,1)$-category]] of [[Reedy model category|Reedy fibrant]] [[augmented simplicial set|augmented]] [[semi-simplicial object|semi-simplicial]] [[diagrams]] of objects in the $(\infty,1)$-topos. Displayed type theory has an indexed endo-[[modality]] called *display* given by [[inference rules]] which can be used to construct [[semi-simplicial types]] as [[coinductive types]] in the type theory. It has only partial internal unary parametricity, rather than full internal unary parametricity, so that it has [[semantics]] in any $(\infty,1)$-topos rather than merely the $(\infty,1)$-topos of [[cubical objects]].

## Related concepts

* [[semi-simplicial types]]

* [[modal type theory]]

## References

* {#KS23} [[Astra Kolomatskaia]], [[Michael Shulman]], *Displayed Type Theory and Semi-Simplicial Types* [[arXiv:2311.18781](https://arxiv.org/abs/2311.18781)]

[[!redirects displayed type theories]]


Disquisitiones Arithmeticae

This page collects material related to

* {#DA} [[Carl Gauss]],

  _Disquisitiones Arithmeticae_,

  1801

  ([pdf scan](http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?action=pdf&metsFile=PPN235993352&divID=LOG_0001&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN235993352%7C&targetFileName=PPN235993352_LOG_0001.pdf&))

on [[number theory]], also [[quadratic forms]], ...

## References

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

category: reference


dissipation

__Dissipation__ in a thermodynamic system is an irreversible loss of ability to do work in a system; in other words, the energy gets converted into its forms which are inaccessible for work eventually due to the second law of thermodynamics. For example, dissipation of a mechanical energy to the heat absorbed by the environment via friction.

Often dissipation can be accounted for by working with non-Hermitean effective Hamiltonians.

## Literature

Related entries: [[fluctuation-dissipation theorem]], [[Langevin equation]], [[statistical physics]]

* wikipedia: [dissipation](https://en.wikipedia.org/wiki/Dissipation)
* R. P. Feynman, F. L. Vernon, Jr. _The theory of a general quantum system interacting with
a linear dissipative system_, Annals of Physics __24__ (1963) 118--173, reprinted in Annals of Physics 281(1-2) 547--607 (2000) [doi:10.1006/aphy.2000.6017](https://doi.org/10.1006/aphy.2000.6017)
* Leggett, A. J., Chakravarty, S. D. A. F. M. G. A., Dorsey, A. T., Fisher, M. P., Garg, A., & Zwerger, W. _Dynamics of the dissipative two-state system_, Reviews of Modern Physics, 59(1) (1987) 1

category: physics


dissipative system

## References

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

* [[Erich Jantsch]], _[[The Self-Organizing Universe]]_

[[!redirects dissipative systems]]


distance (graph theory)

[[!redirects graph distance]]

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#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
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* table of contents
{:toc}

## Idea

In [[graph theory]], given an (undirected) [[graph]] the *graph distance* is, if it exists, the [[function]] that assigns to any [[subset]] of 2 [[vertices]] the [[length]] of the minimum path of edges between them, hence the [[minimum]] [[number]] of consecutive [[edges]] needed to connect the two vertices.

## Applications

### In geometric group theory

In [[geometric group theory]] the graph distance of the [[Cayley graph]] of a [[finitely generated group]] serves to equip with [[group]] with a [[metric]].

## Related concepts

* [[distance]]

## References

See also

* Wikipedia, *Distance (graph theory)*

[[!redirects graph distances]]


DistLat

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#### $(0,1)$-Category theory
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#Contents#
* tableofcontents
{:toc}

## Definition

__$Dist Lat$__ is the [[category]] whose [[objects]] are [[distributive lattices]] and whose [[morphisms]] are lattice [[homomorphisms]], that is [[functions]] which preserve finitary [[meets]] and [[joins]] (equivalently, binary meets and joins and the [[top]] and [[bottom]] elements).  $Dist Lat$ is a [[subcategory]] of [[Pos]] and a [[replete subcategory]] of [[Lat]].

## Properties

$Dist Lat$ is given by a finitary [[variety of algebras]], or equivalently by a [[Lawvere theory]], so has all the usual properties of such categories.  The __[[free object|free]] distributive lattice__ on a [[set]] $X$ is the set of [[finite]] antichains of the [[finite power set]] $\mathcal{P}_{fin}$ of $X$; a collection $\mathcal{C}$ of  [[subsets]] of $X$ is an __antichain__ if, whenever $A \subseteq B$ for $A, B \in \mathcal{C}$, we have $A = B$ (that is, no two distinct elements of $\mathcal{C}$ may be comparable).  Note that $\mathcal{P}_{fin}X$ is the free [[semilattice]] on $X$; we may interpret an element of it as a finitary join of elements of $X$.  Then we interpret an element of $\mathcal{P}_{fin}\mathcal{P}_{fin}X$ as a finitary meet of elements of $\mathcal{P}_{fin}X$.  However, if two of the elements of $\mathcal{P}_{fin}X$ that appear in this meet are comparable, then we need only the smaller of them, so we take only the irredundant elements of $\mathcal{P}_{fin}\mathcal{P}_{fin}X$.  (We could equally well take joins of meets instead of meets of joins; then we keep only the larger of two meets that appear in a given join.)

category: category

[[!redirects DistLat]]
[[!redirects Dist Lat]]

[[!redirects free distributive lattice]]


distributed computing

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#### Computing
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#Contents#
* table of contents
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## Idea

(...)

## Related pages

* [[computing]]

* [[parallel computing]]

* [[arithmetic cryptography]],

* [[multiagent system]]

* [[blockchain]]

## Literature

* Wikipedia:

  *[distributed computing](https://en.wikipedia.org/wiki/Distributed_computing)*,

  *[communication protocol](https://en.wikipedia.org/wiki/Communication_protocol)*,

  *[gossip protocol](https://en.wikipedia.org/wiki/Gossip_protocol)*,

  *[message-oriented middleware](https://en.wikipedia.org/wiki/Message-oriented_middleware)*

* K. Birman, _Reliable distributed systems: technologies, web services
and applications_, Springer 2005

* M. van Steen, A. Tanenbaum, _Distributed systems_, 3rd ed., 2017.

* M. J. Fischer, N. Lynch, M. S. Paterson, _Impossibility of distributed consensus with one faulty process_ [JACM](https://projects.csail.mit.edu/jacm/jacm85.html) 32:2, 374-382 (1985) [pdf](https://groups.csail.mit.edu/tds/papers/Lynch/jacm85.pdf)

A small treatise on distributed computing concepts used in building [[blockchain]] architectures

* Roger Wattenhofer, _The science of the blockchain_, 115 pp. Inverted Forest Publishing 2016

category: computer science

[[!redirects distributed system]]
[[!redirects distributed systems]]
[[!redirects distributed computing system]]


distribution

> This entry is about the concept of distributional densities in functional analysis. For a [[distribution on an affine group scheme]] see there. For the concept of distribution in differential geometry and Lie theory see at _[[distribution of subspaces]]_.

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## Idea
 {#Idea}

In [[functional analysis]], the concept of _distributional density_, usually just called _distribution_ for short, is a generalization of the concept of [[density]], hence of something that may be [[integration|integrated]] against a [[bump function]] to produce a [[number]]. If a non-degenerate background [[density]]/[[volume form]] $dvol$ is fixed, then each other density is a [[function]] relative to $dvol$, and hence with such an identification understood distributional densities are _[[generalized functions]]_, namely objects that may arise as potentially singular [[limit of a sequence|limits]] of [[sequences]] of [[smooth functions]] (i.e. of [[non-singular distributions]]).  Famous examples of such are the [[delta distributions]] and the [[Heaviside distribution]] which behave like constant functions with an infinitely sharp spike or kink, respectively.

Distributional densities appear notably as [[fundamental solutions]] to linear [[partial differential equations]] (such as for the [[wave equation]]/[[Klein-Gordon equation]], whose fundamental solutions are the [[propagators]] of [[free fields|free quantum fields]]), which is the context in which the concept was originally introduced. The study of their singularity structure (encoded by their [[singular support]] and their [[wave front set]]) is a fundamental tool in [[PDE theory]] (for instance in the [[propagation of singularities theorem]]), known as _[[microlocal analysis]]_. Distributions are also fundamental in the rigorous construction of [[perturbative quantum field theory]], where they appear in the variant as [[operator-valued distributions]].

Often distributions are considered by default just on [[open subsets]] of [[Euclidean space]] with its canonical [[volume form]] tacitly understood. But the concept of distributions makes sense more generally on general [[smooth manifolds]] (at least). If these are equipped with the structure of a ([[pseudo-Riemannian manifold|pseudo-]]) [[Riemannian manifold]] then the induced [[volume form]] again identifies distributions with generalized functions.

More in detail, given an actual [[density]]/[[volume form]] $dvol$ on some [[smooth manifold]] $X$, then the operation of [[integration]] of [[bump functions]] (elements in the [[topological vector space]] $C^\infty_c(X)$) against $dvol$ yields the [[continuous linear functional]]

$$
  \array{
    C^\infty_c(X)
      &\longrightarrow&
    \mathbb{R}
    \\
    b &\mapsto& \int_{x \in X} b(x) dvol(x)
  }
  \,.
$$

However, not every [[continuous linear functional]] on $C^\infty_c(X)$ arises this way. For example for $x_0 \in X$ any point, the simple [[evaluation]] map

$$
  \delta_{x_0}
  \;\colon\;
  b
  \mapsto
  b(x_0)
$$

is also a continuous linear functional on $C^\infty_c(X)$ (the "[[delta distribution]]"). While this is not the integral against any [[bump function]] times a fixed density, it is the [[limit of a sequence|limit]] of integrations against any [[sequence]] of bump functions (times the fixed density) whose [[support]] narrows in on $x_0$. Therefore one defines a distributional density simply to be _any_ [[continuous linear functional]] on $C^\infty_c(X)$.

Various immediate variants of this definition may be considered. For instance if the space of "test functions" $C^\infty_c(X)$ is generalized to that of all [[smooth functions]], then one speaks of _[[compactly supported distributions]]_, or if it is enlarged just to the [[Schwartz space]] of functions with all [[derivatives]] rapidly decreasing, then one speaks of [[tempered distributions]]. These are important as on them there is a a good concept of [[Fourier transform of distributions]].

Most of the usual constructions of [[differential calculus]] generalize from [[smooth functions]] to distributions, notably there is a concept of [[derivative of distributions]] (defined by generalizing the formula for [[integration by parts]]). A key subtlety is that, however, some standard operations on functions become only [[partial function|partially defined]] on distributions, namely only when their singularity structure is compatible. In particular there is a concept of [[pullback of distributions]] and the [[product of distributions]] compatible with that of smooth functions, but defined only whenever the [[wave front sets]] of the distributions involved satisfy suitable compatibility conditions. Taking this subtlety into account for the [[operator-valued distributions]] appearing in [[perturbative quantum field theory]] is what leads to the concept of [[Wick algebras]] ("normal ordering"), see there for more.

Due to their potentially singular nature, there is more freedom in the [[extension of distributions]] than there is for smooth functions. Notably for extensions from the complement of a single point to that point the freedom is in choosing a [[point-supported distribution]], and these are precisely the [[derivative of a distribution|derivatives]] of [[delta-distributions]]. In the construction of [[time-ordered products]] of [[operator-valued distributions]] it is precisely this freedom in choosing point-[[extensions of distributions]] which in [[perturbative quantum field theory]] is known as "[[renormalization]]".

## Definition
 {#Definitions}

We first recall the

* [Traditional definition](#TraditionalDefinition).

Then we consider the axiomatic reformulation in terms of [[monads]] following [Kock 11](#Kock11).

* [Characterization by monads](#CharacterizationByMonads)

### As continuous linear functionals
 {#TraditionalDefinition}

Distributions come in various flavors, depending on what spaces of functions they act on. The functions they act on are called **test functions**; typically they are smooth functions on domains in Euclidean space satisfying some boundedness property.

The widest (and generally the default) notion is as follows.

+-- {: .num_defn #CompactlySupportedTestFunctions}
###### Definition
**(compactly supported test functions)**

For $X \subset \mathbb{R}^n$ a [[smooth manifold]] given as an [[open subset]] of a [[Euclidean space]], the [[topological vector space]] $C^\infty_c(X)$ of _compactly supported test functions_ is the following

1. the underlying set is the set of [[bump functions]], hence of [[smooth functions]] $X \to \mathbb{R}$ to the [[real numbers]] with [[compact support]];

1. equipped with evident [[real vector space]] structure given by pointwise addition and pointwise multipication of functions.

1. equipped with the [[topological space|topology]] which is the [[metric topology]] induced from the family of [[seminorms]]

   $$\rho_{K, \alpha}(f) = \sup_{x \in K} |\partial^{\alpha} f|$$

   where $K \subseteq U$ is [[compact topological space|compact]] and $\alpha = (\alpha_1, \ldots, \alpha_n)$ is a multi-index and

   $$\partial^{\alpha} = \frac{\partial^{\alpha_1}}{\partial x^{\alpha_1}} \ldots \frac{\partial^{\alpha_n}}{\partial x^{\alpha_n}}$$

   is the corresponding [[differential operator]].

=--

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

The [[topological vector space]] $C^\infty_c(X)$ of compactly supported test functions (def. \ref{CompactlySupportedTestFunctions}) is [[locally convex topological vector space|locally convex]] and [[complete space|complete]] with respect to its [[uniformity]]; it is in fact an LF-space: an [[inductive limit]] of [[Fréchet spaces]] $C_c^{\infty}(K)$ (each of which has [[empty set|empty]] [[interior]] as a [[subspace]] of $C_c^{\infty}(U)$, so by the [[Baire category theorem]], $C_c^{\infty}(U)$ is not itself a [[Fréchet space]]).

=--

+-- {: .num_defn #DistributionOnOpenSubsetOfEuclideanSpace}
###### Definition
**(distribution on Euclidean space)**

Let $X \subset \mathbb{R}^n$ be a [[smooth manifold]] given as an [[open subset]] of [[Euclidean space]].

A **distribution on** $X$ is a [[continuous linear functional]] of the form

$$C^\infty_c(X) \longrightarrow \mathbb{R}$$

from the [[locally convex space|locally convex]] [[topological vector space]] of compactly supported test functions (def. \ref{CompactlySupportedTestFunctions}) to the [[real numbers]].

The space of distributions on $X$ is denoted $\mathcal{D}'(X)$ (see also remark \ref{NotationForSpaceOfDistributions}). There is an obvious bilinear pairing

$$
  \array{
    \mathcal{D}'(X) \times C_c^{\infty}(X) &\longrightarrow& \mathbb{R}
    \\
    (S, \phi) &\mapsto& S(\phi)
  }
$$

given by [[evaluation]].

Often one writes $\langle S, \phi\rangle$ instead of $S(\phi)$. The space of distributions can be given the weak $*$-topology, meaning the smallest topology rendering the maps

$$\langle -, \phi\rangle \;\colon\; \mathcal{D}'(U) \to \mathbb{R}$$

continuous for all test functions $\phi$.  As $C_c^\infty(U)$ is reflexive, this agrees with the weak topology.

=--

See at _[[locally convex topological vector space]]_ the section _[Continuous linear functionals](locally+convex+space#ContinuousLinearFunctionals)_ for alternative characterizations of the continuity of distributions according to def. \ref{DistributionOnOpenSubsetOfEuclideanSpace}.

Notice that other natural topologies exist, such as [[uniform convergence]] on compact subsets of $C_c^\infty(U)$ (in this case, this agrees with uniform convergence on bounded subsets which usually goes by the name of the *strong* topology).

+-- {: .num_remark #NotationForSpaceOfDistributions}
###### Remark
**(notation)**

On general grounds the symbols $D(X)$ or $\mathcal{D}(X)$ or similar would seem evident notation for the space of distributions on a smooth manifold $X$. However, [[Laurent Schwartz]] in his seminal work ([Schwartz 50](#Schwartz50)) used $\mathcal{D}(X)$ to denote the space $C^\infty_c(X)$ of compactly supported continuous functions, and then $\mathcal{D}'(X)$ for its linear continuous dual, hence for the space of distributions (see also [Hör#mander 90, below def. 2.1.1](#Hoermander90)).

=--

+-- {: .num_example}
###### Example

If $f \colon X \to \mathbb{R}$ is [[locally integrable function|locally integrable]], then for all test functions $\phi$ the [[Lebesgue integral]]

$$\langle f, \phi\rangle = \int_X f(x)\phi(x) d x $$

is defined; in this way a function $f$ locally integrable over $X$ may be regarded as a distribution on $X$ (explaining both the sense in which distributions are "generalized functions" and a reason for the angle-bracket notation for the evaluation pairing). In particular, there is an obvious inclusion

$$C_c^{\infty}(X) \hookrightarrow \mathcal{D}'(X)$$

and this inclusion turns out to be [[dense subset|dense]].

=--

Other notions of spaces of distributions, each endowed with the weak $*$-topology, include

* [[compactly supported distributions]] on $U$. These are functionals on $C^{\infty}(U)$ (test functions without compact support).

* [[tempered distributions]] (usually on $U = \mathbb{R}^n$). These are functionals on what is called the _[[Schwartz space]]_ $\mathcal{S}$: the space of smooth functions each of whose [[derivatives]] (of any order) decays rapidly (goes to zero more quickly than any negative power of $|x|$ as $|x| \to \infty$). The topology on Schwartz space is induced by the family of seminorms
$$\rho_{K, \alpha, \beta}(\phi) = \sup_{x \in K} |x^\alpha \partial^\beta \phi|$$
where $\alpha$, $\beta$ are multi-indices.

* Rapidly decaying distributions (usually on $U = \mathbb{R}^n$). These are functionals on the space of smooth functions each of whose partial derivatives (of any order) has "tempered" or moderate growth (i.e., bounded by polynomial growth).

+-- {: .num_prop #PullbackOfDistributionAlongSubmersion}
###### Proposition
**([[pullback of distributions]] along [[submersions]])**

If $X_1, X_2 \subset \mathbb{R}^n$ are two [[open subsets]] of [[Euclidean space]], and if

$$
  f \;\colon\; X_1 \overset{}{\longrightarrow} X_2
$$

is a [[submersion]] (i.e. its [[differential]] is a [[surjective function]] $d f_x \;\colon\; T_x X_1 \to T_{f(x)} X_2$ for all $x \in X_1$), then there is a unique [[continuous linear functional]]

$$
  f^\ast \;\colon\; \mathcal{D}'(X_2) \longrightarrow \mathcal{D}'(X_1)
$$

between spaces of distributions (def. \ref{DistributionOnOpenSubsetOfEuclideanSpace}) which extends the [[pullback of functions]] in that on a distribution represented by a [[bump function]] $b$ it is given by precomposition

$$
  f^\ast b = b \circ f
  \,.
$$

This is hence called the _[[pullback of distributions]]_.

=--

([Hörmander 90, theorem 6.1.2](#Hoermander90))

+-- {: .num_defn #DistributionsOnSmoothManifolds}
###### Definition
**(distributions on smooth manifolds)**

Let $X$ be a [[smooth manifold]]. Then a distribution on $X$ is an [[equivalence class]] of

1. a choice of smooth [[atlas]] $\{\mathbb{R}^n \underoverset{\simeq}{\psi_i}{\longrightarrow} U_i \subset X\}_{i \in I}$;

1. for each $i \in I$ a distribution $\phi_i \;\colon\; C^\infty(\mathbb{R}^n)\to \mathbb{R}$ on the $i$th [[chart]], as above;

1. such that for all pairs $(i,j) \in I \times I$ these component distributions are related on intersections of charts by pullback of distributions (def. \ref{PullbackOfDistributionAlongSubmersion}) along the coordinate change maps:

   $\phi_j =  (\psi_i^{-1} \circ \psi_j)^\ast  \phi_i $.

=--

([Hörmander 90, def. 6.3.3](#Hoermander90))

### As smooth linear functionals
 {#CharacterizationByMonads}

Since [[smooth functions]] on [[smooth manifolds]] are the subject of [[differential geometry]], and since spaces of [[smooth functions]] are naturally themselves [[generalized smooth spaces]], it makes sense to ask whether distribution theory is actually a native topic to [[differential geometry]].

In particular we may ask how distributions in the [[functional analysis|functional analytic]] sense relate to the _smooth linear functions_ on smooth spaces of smooth functions. Indeed, with respect to the natural formulation of differential geometry via [[functorial geometry]] ([[topos theory]]) in terms of [[diffeological spaces]], [[smooth sets]] etc. it turns out that distributional densities are equivalently the smooth linear functionals on smooth spaces of smooth functions.

This is discussed at

* _[[distributions are the smooth linear functionals]]_.

## Operations on distributions

### Inducing operations by dual extension

As $\mathcal{D}'(U)$ is dual to $C_c^\infty(U)$, each continuous linear operator on $C_c^\infty(U)$ induces a corresponding linear operator on $\mathcal{D}'(U)$ in the obvious way.  Given

$$
F\colon C_c^\infty(U) \to C_c^\infty(U)
$$

we define

$$
F^*\colon \mathcal{D}'(U) \to \mathcal{D}'(U)
$$

according to the usual formula for dualities

$$
F^* S(\phi) = S(F \phi).
$$

However, since there is an obvious inclusion $C_c^\infty(U) \to \mathcal{D}'(U)$ induced by the standard inner product on $C_c^\infty(U)$, what is more usually desired is not this dual operator but an **extension** operator.  That is, instead of $F^*$ we want an operator $F^\dagger \colon \mathcal{D}'(U) \to \mathcal{D}'(U)$ with the property that for $\phi \in C_c^\infty(U)$ then $F^\dagger(\phi) = F(\phi)$ (identifying $C_c^\infty(U)$ with its image in $\mathcal{D}'(U)$).  Being slightly more careful, let us write $\iota \colon C_c^\infty(U) \to \mathcal{D}'(U)$ for the inclusion induced by the inner product.  Then we want $F^\dagger(\iota \phi) = \iota (F(\phi))$.

If the extension exists, we have

$$
F^\dagger(\iota \phi)(\psi) = \iota(F(\phi))(\psi) = \langle F(\phi), \psi \rangle
$$

Now suppose that $F$ has an adjoint, say $F^+$, with respect to the inner product.  Note that this is not automatic since $C_c^\infty(U)$ is not a Hilbert space.  Moreover, even if $F$ extends to the Hilbert completion the Hilbertian adjoint may not work since it may not define a continuous linear map on the subspace $C_c^\infty(U)$.  But if $F^+$ does exist then we have

$$
F^\dagger(\iota \phi)(\psi) = \langle F(\phi), \psi \rangle = \langle \phi, F^+(\psi) \rangle
$$

In this case, the definition of $F^\dagger$ on the whole of $\mathcal{D}'(U)$ is obvious: simply take ${F^+}^*$.  That is, the dual operator to the adjoint to $F$.  In full, $F^\dagger \colon \mathcal{D}'(U) \to \mathcal{D}'(U)$ is defined via the formula

$$
\langle F^\dagger(S),\phi\rangle = \langle S, F^+(\phi) \rangle
$$

If the ground field is $\mathbb{C}$ then this carries through essentially unchanged except for the fact that one does **not** use the inner product on $C_c^\infty(U)$ but rather the associated bilinear pairing

$$
(\phi,\psi) = \int_U \phi \psi
$$

This is to ensure that the inclusion $C_c^\infty(U) \to \mathcal{D}'(U)$ is complex linear and not conjugate linear.  Otherwise extending operators becomes complex.

Two instances are of particular importance:

* Multiplication by a smooth function $\theta$. If $\theta$ is any smooth function on $U$ (not necessarily compactly supported), then we can define $\theta \cdot S$ by observing that this multiplication is self-adjoint:
$$\langle \theta \cdot \phi, \psi \rangle = \langle \phi, \psi \cdot \theta\rangle$$
where $\phi, \psi$ are arbitrary test functions.  Thus we define $\theta \cdot S$ by
$$\langle \theta \cdot S, \psi \rangle = \langle S, \theta \cdot \psi$$

* Differentiation. If $\partial^i$ is partial differentiation with respect to the $i^{th}$ coordinate, then for test functions $\psi$, $\phi$ we have
$$\int_U \partial^i(\psi)(x) \phi(x)\; d x = -\int_U \psi(x) \partial^i(\phi)(x)\; d x$$
by simple integration by parts and the fact that $\phi$, $\psi$ are compactly supported.  Thus differentiation is _skew_-adjoint and so we define the extension to distributions by
$$\langle \partial^i(S), \phi\rangle = -\langle S, \partial^i(\phi) \rangle$$
for all test functions $\phi$. In general,
$$\langle \partial^\alpha S, \phi\rangle = (-1)^{|\alpha|}\langle S, \partial^\alpha \phi \rangle$$
where $|\alpha| = \alpha_1 + \ldots + \alpha_n$ is the total degree of the multi-index.

Thus derivatives of distributions are defined to all orders. Some examples are given in the section "examples".

### Multiplication of Distributions
 {#MultiplicationsOfDistributions}

See at _[[multiplication of distributions]]_

## Examples

As explained above, any locally integrable function on $U$ defines a distribution on $U$. Other examples may be produced fairly cheaply by restriction of functionals on various TVS which contain the test functions.

For instance: if $C_c(U)$ denotes the space of real-valued continuous functions with compact support in $U$ (topologized by uniform convergence on compacts), then a functional $\mu: C_c(U) \to \mathbb{R}$ is essentially the same as a signed measure on $U$ (Riesz-Markov theorem), i.e., there is a unique signed measure $d m$ for which

$$\mu(\phi) = \int_U \phi d m.$$

Since the inclusion $i: C_c^\infty(U) \hookrightarrow C_c(U)$ is continuous, it follows that a measure $\mu$ defines a distribution by simple restriction along $i$:

$$C_c^\infty(U) \overset{i}{\to} C_c(U) \overset{\mu}{\to} \mathbb{R}$$

Specializing further, consider any function of bounded variation on $U = \mathbb{R}$, say a bounded monotone increasing function $\alpha$. Then the Riemann-Stieltjes integral

$$\int_{\mathbb{R}} f(x) d\alpha(x)$$

is defined for all functions $f$ with compact support; this provides a measure $d\alpha$ and hence a distribution.

A prototypical example of this is provided by the Heaviside function: $H(x) = 1$ if $x \gt 0$, else 0. ("Heaviside": what a great pun!) Here we have, for all $f \in C_c(\mathbb{R})$,

$$\langle f, d H \rangle = \int_{\mathbb{R}} f(x) d H(x) = f(0)$$

As a distribution, the Heaviside measure is the famous **[[Dirac distribution]]**. The long-standing intuitive practice among physicists and engineers is to write

$$d H(x) = \delta_0(x) d x$$

where of course the function $H(x)$ doesn't have a derivative in the classical sense (i.e., as a function), but as a distribution, it does. Meanwhile, $H(x)$ is itself the derivative of a continuous function: $G(x) = \max\{x, 0\}$.

For an example of a distribution on $\mathbb{R}$ which does not arise from a measure, consider the derivative of the Dirac distribution. (As a functional, it maps a test function $\phi$ to $-\phi'(0)$.)

These examples are by no means curiosities. A fairly deep theorem is that _every_ distribution arises as a linear combination of derivatives of continuous functions:

**Theorem:** Let $S$ be a distribution on an open domain $U \subseteq \mathbb{R}^n$. Then, there exist a finite collection $A$ of multi-indices $\alpha$ and continuous functions $g_\alpha$ defined on $U$ for which

$$S = \sum_{\alpha \in A} \partial^\alpha g_\alpha$$

## Applications

The theory of distributions (and more generally of [[microlocal analysis]]) is central in [[perturbative quantum field theory]] in its rigorous incarnation via [[causal perturbation theory]]/[[perturbative algebraic quantum field theory]].

For example the reason for [[normal ordered products]] in [[Wick algebras]] is given by the Hörmander criterion on [[wave front sets]] for the [[product of distributions]] to be well defind, and [[renormalization]] is understood to be the freedom in choosing [[extension of distributions]] of the resulting products of [[Feynman propagators]].

See also _[[operator-valued distribution]]_ and _[[Wightman axioms]]_.

A brief survey of applications of distribution theory to perturbative quantum field theory may be found [here](http://www1.jinr.ru/Archive/Pepan/v-31-7a/I_ktp_04_t.pdf).

Within mathematics, distributions are quite commonplace; for example, [[Georges de Rham]] appropriated them for his theory of [[current (distribution theory)|currents]]. Distribution theory has also long been used in the theory of partial differential equations. Here is a sample theorem:

* **Theorem (Ehrenpreis, Malgrange):** Let $D$ be a linear differential operator on $\mathbb{R}^n$ with constant coefficients. Given a compactly supported smooth function $f$ on $\mathbb{R}^n$, there exists a smooth solution $u$ to the equation $D u = f$.

A proof is given in these [notes](http://www-math.mit.edu/~helgason/hormander.pdf) by Helgason. The basic idea is to prove there exists a **fundamental solution** of $D$, i.e., a distribution $T$ such that $D T = \delta_0$. Then $u = f * T$ is smooth. The existence of a fundamental solution involves a theorem of Paley-Wiener type.

## Variants

### In synthetic differential geometry

There is another point of view on distributions: that they _can_ be modeled by actual functions provided that one admits infinite and infinitesimal quantities of the type used in Robinson [[nonstandard analysis]]. One particular approach is to formulate axiomatically the theory of distributions so that it can be interpreted in [[smooth toposes]] that model the axioms of [[synthetic differential geometry]] and support a suitable notion of invertible [[infinitesimal object|infinitesimal objects]] and infinitely large integers.

This is discussed in ([Moerdijk-Reyes 91](#MoerdijkReyes91)).

which closely mirrors the original treatment in Robinson's book Non-standard Analysis. Examples of models that support these axioms are the toposes $\mathcal{Z}$ and $\mathcal{B}$ described there.

### Currents

In $\mathbb{R}^n$ the distributions and generalized functions boil down to the same thing, so the terminology identifies them. But on a [[smooth manifold]], the distributions/generalized densities (functionals on test functions) and generalized functions (functionals on test *densities*) do not agree. See V. Guillemin, S. Sternberg: _Geometric asymptotics_ ([free online](http://www.ams.org/online_bks/surv14)). While generalized functions pull back, distributions/generalized densities push forward (under some conditions, though).

More generally one can study generalized differential $k$-forms in local coordinates they look like $\sum f_\alpha dx^{\alpha_1}\wedge \cdot \wedge dx^{\alpha_k}$. Usually they are called **[[current (distribution theory)|currents]]**. They are useful e.g. in the study of higher dimensional residua in higher dimensional complex geometry (cf. *Principles of algebraic geometry* by Griffiths and Harris) and in geometric measure theory (cf. the monograph by Federer).

### Hyperfunctions and Coulombeau distributions

Sometimes one considers larger spaces of distributions, where worse singularities than in Schwaryz theory are allowed. Most well known are the theory of _[[hyperfunctions]]_ and the theory of **Coulombeau distributions**.

### Distributions from nonstandard analysis

Distributions can be alternatively described using [[nonstandard analysis]], see there.

## Related concepts

* [[non-singular distribution]]

* [[order of a distribution]]

* [[support of a distribution]]

  * [[point-supported distribution]]

* [[scaling degree of a distribution]]

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

* [[derivative of a distribution]]

* [[generalized solution of a partial differential equation]]

* [[extension of distributions]]

* [[tensor product of distributions]]

* [[product of distributions]]

* [[convolution product of distributions]]

* [[current (distribution theory)]]

* [[Hörmander topology]]

* [[Fourier transform of distributions]]

* [[Lawvere distribution]] ([[categorification]] of the concept of distributions)

## References

See also _[[hyperfunction]]_, [[ultradistribution]] and references therein.

### Traditional

Generalized functions were introduced by S. L. Sobolev in 1935, and independently (under the name _distributions_) by [[Laurent Schwartz]] in the 1940's, who unaware of Sobolev's work developed an extensive theory for them. For an infinite-dimensional variant used in the foundation of Feynman [[path integral]] see also [[Connes distribution]].

The original articles include

* {#Schwartz50} [[Laurent Schwartz]], _Théorie des distributions_, 1–2 , Hermann  (1950–1951)

* [[Israel M. Gelfand]], [[Georgiy E. Shilov]], _Generalized functions_, Acad. Press  (1966-1968), AMS (2016) [[ISBN:978-1-4704-2885-3](https://bookstore.ams.org/view?ProductCode=CHELGELFSET)] transl. from И. М. Гельфанд, Г. Е. Шилов Обобщенные функции, вып. 1-3, М.:Физматгиз, 1958; 1: Обобщенные функции и действия над ними, 2: Пространства основных обобщенных функций, 3: Некоторые вопросы теории дифференциальных уравнений

* E. Magenes, G. Stampacchia, _Teoria delle distribuzioni_, Lectures Given at a Summer School of the Centro Internazionale Matematico Estivo (C.i.m.e.) Held in Saltino (Firenza) Italy, September 1-9, 1961; C.I.M.E., Ed. Cremonese, Roma, 1961; reprinted as CIME __24__, Springer 2011 [doi](http://dx.doi.org/10.1007/978-3-642-10967-6)

* {#Treves67} [[François Trèves]], _Topological Vector Spaces, Distributions and Kernels_ (Academic Press, New York, 1967)

Modern accounts:

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

* Walter Rudin, chapter 6 of _Functional analysis_, McGraw-Hill, 1991

* M. Grosser, E. Farkas, M. Kunzinger, R. Steinbauer, _On the foundations of nonlinear generalized functions I, II_, Mem. Amer. Math. Soc. __153__ (2001)

* M. Kunzinger, R. Steinbauer, _Foundations of a nonlinear distributional geometry_, Acta Appl. Math. __71__, 179-206 (2002)

In [[differential geometry]] ([[current (distribution theory)|currents]]):

* [[Victor Guillemin]], [[Shlomo Sternberg]], Chapter VI of: *Geometric asymptotics*, Mathematical Surveys and Monographs **14**, AMS (1977) [[ams:surv-14](https://bookstore.ams.org/surv-14)]

Lecture notes:

* Hasse Carlsson, _Lecture notes on distributions_ ([pdf](http://www.math.chalmers.se/~hasse/distributioner_eng.pdf))

and several chapters of the course

* Erik P. van den Ban, _Analysis on manifolds_ (2009) ([web](https://www.staff.science.uu.nl/~ban00101/anman2009/anman2009.html) - especially lectures 1-3)

Applications of distributions in [[physics]] are discussed in

* V. S. Vladimirov, _Generalized functions in mathematical physics_. Moskva, Nauka 1980, Mir 1979; _Equations of mathematical physics_, Mir 1984

* [[Nikolay Bogolyubov]], A. A. Logunov, I.T. Todorov, _Introduction to axiomatic quantum field theory_, Benjamin  (1975)

Application of distributions in [[perturbative quantum field theory]] is discussed in

* {#Scharf95} [[Günter Scharf]] _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Springer 1995 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0844.53052&format=complete))

For more on this see the references at _[[perturbative AQFT]]_.

See also

* Springer online [[eom]], _[generalized function](http://eom.springer.de/g/g043810.htm)_

References on [[Colombeau algebra]] include

* J. F. Colombeau, _New generalized functions and multiplications of distributions_, North Holland, Amsterdam (1984); _Elementary introduction in new generalized functions_, North Holland (1985)
* N. Djapić, S. Pilipović, _Microlocal analysis of Colombeau's generalized functions on a manifold_, Indag. Math. N.S. 7, 293–309 (1996)
* Stevan Pilipović, Milica Žigić, _Suppleness of the sheaf of algebras of generalized functions on manifolds_, J. Math. Anal. Appl. __379__:2 (2011) 482–486, [arxiv/1101.4552](http://arxiv.org/abs/1101.4552), [MR2784335](http://www.ams.org/mathscinet-getitem?mr=2784335), [doi](http://dx.doi.org/10.1016/j.jmaa.2010.12.060)

### In terms of smooth toposes
 {#InTermsOfSmoothToposes}

Discussion of distributions in terms morphisms out of [[internal homs]] in a [[smooth topos]] ([[distributions are the smooth linear functionals]]) is in

* {#MoerdijkReyes91} [[Ieke Moerdijk]], [[Gonzalo Reyes]], around prop. 3.6 of _[[Models for Smooth Infinitesimal Analysis]]_, Springer 1991

and for the [[Cahiers topos]] in

* {#KockReyes03} [[Anders Kock]], [[Gonzalo Reyes]], _Some calculus with extensive quantities: wave equation_, Theory and Applications of Categories , Vol. 11, 2003, No. 14, pp 321-336 ([TAC](http://www.tac.mta.ca/tac/volumes/11/14/11-14abs.html))

* {#KockReyes04} [[Anders Kock]], [[Gonzalo Reyes]], _Categorical distribution theory; heat equation_ ([arXiv:math/0407242](https://arxiv.org/abs/math/0407242))

* {#Kock11} [[Anders Kock]], _Commutative monads as a theory of distributions_ ([arxiv/1108.5952](http://arxiv.org/abs/1108.5952))

using results of

* {#FroelicherKriegl88} [[Alfred Frölicher]], [[Andreas Kriegl]], section 5 of  _Linear spaces and differentiation theory_, Wiley 1988 ([pdf](http://www.fuw.edu.pl/~kostecki/scans/froelicherkriegl1988.pdf))

and following the general conception of "[[intensive and extensive]]" in

* {#Lawvere86} [[William Lawvere]], Introduction to _[[Categories in Continuum Physics]], Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174. 1986

Similar [[sheaf theory|sheaf theoretic]] discussion of distributions as morphisms of [[smooth spaces]] is in

* {#Paugam12} [[Frédéric Paugam]], section 3.2 of _Towards the mathematics of quantum field theory_, 2012 ([pdf](https://webusers.imj-prg.fr/~frederic.paugam/documents/enseignement/master-mathematical-physics.pdf))

category: analysis

[[!redirects distributions]]

[[!redirects Schwartz distribution]]
[[!redirects Schwartz distributions]]

[[!redirects linear distribution]]
[[!redirects linear distributions]]

[[!redirects distributional density]]
[[!redirects distributional densities]]


distribution monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The distribution monad is a [[monad]] on [[Set]], whose [[algebra over a monad|algebras]] are [[convex spaces]].

It can be thought of as the [[finitary monad|finitary]] prototype of a [[probability monad]].

## Definition

### Finite distributions

Let $X$ be a set. Define $D X$ as the set whose elements are functions $p:X\to[0,1]$ such that

* $p(x)\ne 0$ for only finitely many $x$, and

* $\sum_{x\in X} p(x)=1$.

Note that the sum above is finite if one excludes all the zero addenda.

The elements of $D X$ are called **finite distributions** or **finitely-[[support|supported]] probability measures** over $X$.

### Pushforward

Given a function $f:X\to Y$, one defines the **pushforward** $D f:D X\to D Y$ as follows. Given $p\in D X$, then $(D f)(p)\in D Y$ is the function
$$
y \;\mapsto\; \sum_{x\in f^{-1}(y)} p(x) .
$$
(Note that, up to zero addenda, the sum above is again finite.)

Compare with the [[pushforward of measures]].

This makes $D$ into an [[endofunctor]] on [[Set]].

### Monad structure

The unit map $\delta:X\to D X$ maps the element $x\in X$ to the function $\delta_x:X\to[0,1]$ given by
$$
\delta_x(y) \;=\; \begin{cases}
1 & y=x ;\\
0 & y\ne x .
\end{cases}
$$
Compare with the [[Dirac measures]] and [[continuous valuation#dirac_valuation|valuations]].

The multiplication map $E:D D X\to D X$ maps $\xi\in D D X$ to the function $E\xi\in D X$ given by
$$
E\xi(x) \;=\; \sum_{p\in D X} p(x) \, \xi(p).
$$
(Note that, up to zero addenda, the sum above is once again finite.)

The maps $E$ and $\delta$ satisfy the usual [[monad]] laws. The resulting monad $(D,E,\delta)$ is known as **distribution monad**, or **finitary Giry monad** (in analogy with the [[Giry monad]]), or **convex combination monad**, since the elements of $D X$ can be interpreted as formal [[convex space|convex combinations]] of elements of $X$.

This can be seen as a discrete, finitary analogue of a [[probability monad]], where one replaces [[integrals]] by [[sums]].

## Properties

* The [[algebra over a monad|algebras]] of $D$ are [[convex spaces]].

* The monad $D$ is [[finitary monad|finitary]], and [[commutative monad|commutative]].

* The [[Kleisli morphisms]] of $D$ are (finitary) [[stochastic maps]].

(...)

## See also

* [[convex space]], [[conical space]]

* [[monads of probability, measures, and valuations]]

* [[Giry monad]], [[probabilistic powerdomain]], [[extended probabilistic powerdomain]]

* [[monad]], [[monad (in computer science)]]

## References

* [[Tobias Fritz]] and [[Paolo Perrone]], _Monads, partial evaluations, and rewriting_, MFPS 36, 2020 - Section 6. ([arXiv](https://arxiv.org/abs/1810.06037))

* {#jacobs} [[Bart Jacobs]], _From probability monads to commutative effectuses_, Journal of Logical and Algebraic Methods in Programming 94, 2018.
([doi:10.1016/j.jlamp.2016.11.006](https://doi.org/10.1016/j.jlamp.2016.11.006))

* {#fritzconvex} [[Tobias Fritz]], _Convex spaces I: definitions and examples_, 2009. ([arXiv:0903.5522](https://arxiv.org/abs/0903.5522))


distribution of subspaces

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###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

> This entry is about the concept in [[differential geometry]] and [[Lie theory]]. For the concept in functional analysis see at _[[distribution]]_.

#Contents#
* table of contents
{:toc}

## Definition

A __real distribution__ on a real [[smooth manifold]] $M$ is a [[real vector bundle|real vector subbundle]] of the [[tangent bundle]] $T M$.

A __complex distribution__ is a [[complex vector bundle|complex vector subbundle]] of the [[complexification|complexified]] tangent bundle $T_{\mathbb{C}}M$ of $M$.

A __distribution of [[hyperplanes]]__ is a distribution of [[codimension]] $1$ in $T M$; a __distribution of complex hyperplanes__ is a distribution of complex codimension $1$ in $T_{\mathbb{C}} M$.

## Examples

One class of examples comes from smooth [[foliations]] by [[submanifolds]] of constant dimension $m\lt n$. Then the [[tangent vectors]] at all points to the submanifolds forming the foliation form a distribution of subspaces of dimension $m$. The distributions of that form are said to be __[[integrable distribution|integrable]]__.

... _say something about the Frobenius theorem_ ...

## References

Discussion in the context of [[geometric quantization]]:

* N. M. J. Woodhouse, _Geometric quantization_

[[!redirects distributions of subspaces]]


distribution on an affine group scheme

\tableofcontents

## Idea

The notion of *[[distributions]]* -- in the sense of [[linear functionals]] on some [[algebra of functions]] -- can to some extent be adapted from [[smooth manifolds]] to [[schemes]].

The problem is however to control the singular behaviour. A useful idea is to restrict singularities to a closed subscheme given by some ideal sheaf. The simplest case are distributions on a $k$-[[affine scheme]] $X$ over a [[field]], supported at a point $p$. For a [[group scheme]], one assumes that the point is the [[neutral element]] of the group and the algebra of distributions becomes a Hopf algebra.

## Definition

Let $X$ be a $k$-affine scheme $X$ (where $k$ is a field).
For $n$ a nonnegative integer distributions on $X$ supported at $p$ of order $\leq n$ are linear functionals $k[X]\to k$ on the ring of regular functions $k[X]$ which vanish on $(n+1)$-st power of the ideal $I_p = \{ f\in k[X] | f(p) = 0\}$ of regular functions vanishing at $p$.

If $X = G$ is a $k$-affine group scheme by the Hopf algebra of distributions we mean the algebra of distributions supported at the unit of the group with its natural Hopf algebra structure.

Largely equivalent notion, though usually differently defined is of a [[hyperalgebra of an affine algebraic group]].

## Literature

* {#CartierHopf} [[Pierre Cartier]], _A primer of Hopf algebras_, Frontiers in number theory, physics, and geometry II, 537--615, Springer 2007.; preprint IH\'ES 2006/40 [pdf](http://preprints.ihes.fr/2006/M/M-06-40.pdf)

* J. C. Jantzen, Representations of algebraic groups, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)

category: algebraic geometry

[[!redirects distribution on a linear algebraic group]]
[[!redirects distribution on an affine algebraic group]]
[[!redirects distribution on an affine scheme supported at a point]]
[[!redirects distribution on an affine scheme]]


distributional semantics

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### Context
#### Linguistics
+-- {: .hide}
[[!include linguistics - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related entries

* [[semantics]]

  * [[categorical semantics]];

  * [[denotational semantics]]

  * [[operational semantics]]

* [[DisCoCat]], [[DisCoPy]]

## References

See also:

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


distributional solution of a PDE

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### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[linear differential operator]] $P$, then a _generalized solution_ or _weak solution_ of the corresponding homogenous [[differential equation]] is a [[distribution]] $u$ (hence a "[[generalized function]]" via [this prop](non-singular+distribution#NonSingularDistributionsAreDenseInAllDistributions)) satisfying

$$
  P u = 0
  \,,
$$

where the differential operator on the left now acts via [[derivatives of distributions]]. This means that for all $b \in \mathcal{D}$ we have

$$
 u(P^\ast(b)) = 0
 \,,
$$

where $P^\ast$ is the [[formally adjoint differential operator]].

This is such that in the case that $u = u_f$ happens to be a [[non-singular distribution]] given by an ordinary [[smooth function]] $f$, then $u_f$ is a generalized solution precisely if $f$ is an ordinary solution:

$$
  \begin{aligned}
    P u_f = 0 \;\;
    & \Leftrightarrow\;\;
      u_f(P^\ast b) = 0 \phantom{AAA}\text{for all}\, b \in \mathcal{D}
    \\
    & \Leftrightarrow\;\;
     \int f P^\ast b \, dvol = 0 \phantom{AAA}\text{for all}\, b \in \mathcal{D}
     \\
     & \Leftrightarrow\;\;
     \int (P f) b \, dvol = 0 \phantom{AAA}\text{for all}\, b \in \mathcal{D}
     \\
     & \Leftrightarrow\;\;
     P f = 0
  \end{aligned}
$$

Similarly there are generalized solutions for the inhomogeneous equation, and in fact now the inhomogeneity may itself be a [[distribution]]. In particular for [[delta distribution]]-inhomogeneity

$$
  P u = \delta
$$

the generalized solutions $u$ are the _[[fundamental solutions]]_ or _[[Green's functions]]_ of $P$.

## Related concepts

* [[fundamental solution]]

* [[Green function]]

* [[formal adjoint differential operator]]

* [[Green hyperbolic differential equation]]

## References

* {#Kanwal04} [[Ram Kanwal]], section 10.2 of _Generalized Functions: Theory and Applications_, Springer 2004

[[!redirects distributional solutions of a PDE]]

[[!redirects distributional solution of a partial differential equation]]
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[[!redirects distributional solutions of partial differential equation]]

[[!redirects distributional solution of a differential equation]]
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[[!redirects distributional solutions of differential equations]]

[[!redirects distributional PDE solution]]
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[[!redirects distributional solution of a PDE]]
[[!redirects distributional solutions of a PDE]]
[[!redirects distributional solutions of PDEs]]

[[!redirects generalized solution of a PDE]]
[[!redirects generalized solutions of a PDE]]

[[!redirects generalized solution of a partial differential equation]]
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[[!redirects generalized solutions of partial differential equation]]

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[[!redirects generalized solution]]
[[!redirects generalized solutions]]

[[!redirects distributional solution to a PDE]]
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[[!redirects distributional solution to a partial differential equation]]
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[[!redirects distributional solution of a partial differential equation]]
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[[!redirects generalized solution to a PDE]]
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[[!redirects generalized solution to a partial differential equation]]
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[[!redirects generalized solutions to partial differential equation]]

[[!redirects generalized solution to a differential equation]]
[[!redirects generalized solutions to a differential equation]]

[[!redirects weak solution]]
[[!redirects weak solutions]]


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

#Contents#
* table of contents
{:toc}

## Idea

Historically, the theory of [[distributional densities]] (or just _[[distributions]]_, for short), conceived in [[functional analysis]], was phrased in terms of [[continuous linear functionals]] on [[topological vector spaces]] of [[smooth functions]]. However, since [[smooth functions]] on [[smooth manifolds]] are the subject of [[differential geometry]], and since spaces of [[smooth functions]] are naturally themselves [[generalized smooth spaces]], it makes sense to ask whether [[distribution]] theory is actually a native topic to [[differential geometry]].

In particular we may ask how distributions in the [[functional analysis|functional analytic]] sense relate to the _smooth linear functions_ on smooth spaces of smooth functions. Indeed, with respect to the natural formulation of differential geometry via [[functorial geometry]] ([[topos theory]]) in terms of [[diffeological spaces]], [[smooth sets]] etc. it turns out that distributional densities are equivalently the smooth linear functionals on smooth spaces of smooth functions.

This we discuss in the section

* _[Statement](#Statement)_

below. To set the scene we briefly recall some concepts in

* _[Preliminaries](#Preliminaries)_.

Finally we discuss some

* _[Applications](#Applications)_.

## Preliminaries
 {#Preliminaries}

+-- {: .num_defn #GeneralizedSmoothSpaces}
###### Definition
**([[generalized smooth spaces]])**

In the following we write $\mathbf{H}$ for the [[category]] of [[diffeological spaces]] rather for the [[toposes]] of [[smooth sets]] or [[formal smooth set]] or [[super formal smooth sets]] etc. These form a [[sequence]] of [[full subcategory]] inclusions

$$
  SmoothManifolds
    \hookrightarrow
  DiffeologicalSpaces
    \hookrightarrow
  SmoothSets
    \hookrightarrow
  FormalSmoothSets
    \hookrightarrow
  SuperFormalSmoothSets
  \,.
$$

=--

The smallest of these categories of [[generalized smooth spaces]] that is needed for accomodating the spaces of [[smooth functions]] that distributions are defined on are [[diffeological spaces]], and the reader may want to just focus on these. We mention these more general ambient categories mostly to amplify that the identificatin of distributions with smooth linear functionals remains true in these more general contexts.

+-- {: .num_example #SmoothMappingSpace}
###### Example
**(smooth [[mapping spaces]])**

For $X \in \mathbf{H}$ any object, we write $[X,\mathbb{R}]$ for the [[mapping space]] (the [[internal hom]]). The underlying set is $C^\infty(X)$. If $X$ itself has $\mathbb{R}$-linear structure, we write

$$
  [X,\mathbb{R}]_{\mathbb{R}}
  \hookrightarrow
  [X,\mathbb{R}]
$$

for the [[subobject]] of $\mathbb{R}$-linear maps.

Concretely, for $U$ a [[smooth manifold]] (or just a [[Cartesian space]]), then the [[sheaf]] $[X,\mathbb{R}]$ assigns (see at _[[closed monoidal structure on presheaves]]_ for details)

$$
  [X,\mathbb{R}](U) = C^\infty(U \times X)
$$

and $[X,\mathbb{R}](U) \subset C^\infty(U \times X)$ is the subset of those functions $\Phi_{(-)}(-)$ such that for all $u \in U$ the function $\Phi_u \colon X \to \mathbb{R}$ is $\mathbb{R}$-linear. The [[global elements]] $\Gamma(-)$ of the [[mapping space]] constitute the ordinary [[hom set]]

$$
  \Gamma [X,\mathbb{R}] \simeq \mathbf{H}(X,\mathbb{R})
  \,.
$$

If $X$ happens to be a [[compact topological space|compact]] [[smooth manifold]] then $C^\infty(X)$ carries the structure of a [[Fréchet manifold]] (see at _[[manifold structure of mapping spaces]]_). Under the [[full subcategory|full embdding]] of [[Fréchet manifolds]] into [[diffeological spaces]], this coincides with $[X,\mathbb{R}]$ (see [there](Fréchet+manifold#RelationBetweenDeffeologicalAndFrechetStructure)).

=--

## Statement
 {#Statement}

### Compactly supported distributions

+-- {: .num_prop #CompactlySupportedDistributionsAreTheSmoothLinearFunctionals}
###### Proposition
**([[compactly supported distributions are the smooth linear functionals]])**

For $n \in \mathbb{N}$, there is a [[natural bijection]] between the underlying sets of [[compactly supported distributions]] on $\mathbb{R}^n$ ([this def.](compactly+supported+distribution#CompactlySupportedDistributionsAsContinuousLinearDualsToBumpFunctions)) and the $\mathbb{R}$-linear [[mapping space]] formed in the [[category]] $\mathbf{H}$ of [[generalized smooth spaces]] from def. \ref{GeneralizedSmoothSpaces}:

$$
  \widetilde{(-)}
  \;\colon\;
  \mathcal{E}'(\mathbb{R}^n)
    \overset{\simeq}{\longrightarrow}
  \mathbf{H}([\mathbb{R}^n, \mathbb{R}], \mathbb{R})_{\mathbb{R}}
$$

given by sending $\mu \in \mathcal{E}'(\mathbb{R}^n)$ to the [[natural transformation]] which on a test space $U \in $ [[CartSp]] takes a smoothly $U$-parameterized function $\Phi_{(-)}(-) \colon U \times \mathbb{R}^n \to \mathbb{R}$ to its evaluation in $\mu$ pointwise in $U$:

$$
  \tilde \mu(\Phi_{(-)})(u)
  \;\coloneqq\;
  \langle \mu, \Phi_{u}\rangle
  \,.
$$

=--

([Moerdijk-Reyes 91, chapter II, prop. 3.6](#MoerdijkReyes91))

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

First consider this for the case that $\mathbf{H} = $ [[SmoothSet]] (which immediately subsumes the case that $\mathbf{H} =$ [[diffeological space|DiffelogicaSpace]]).

To see that $\widetilde{(-)}$ is well defined, we need to check that the function

$$
  \array{
     U
     &\overset{
     \tilde \mu(\Phi_{(-)})}{\longrightarrow}&
     \mathbb{R}
     \\
     u &\mapsto& \langle \mu, \Phi_u\rangle
  }
$$

is [[smooth function|smooth]]. But this follows immediately since $\langle \mu,-\rangle$ by definition is [[linear function|linear]] and [[continuous function|continuous]].

To see that $\widetilde{(-)}$ is indeed a [[bijection]] for each $U$ it remains that every $\mathbb{R}$-linear smooth functional (morphisms of [[smooth sets]]) of the form

$$
  A \;\colon\; [\mathbb{R}^n,\mathbb{R}] \longrightarrow \mathbb{R}
$$

when restricted on [[global elements]] to a [[function]] of sets

$$
  A(\ast) \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}
$$

is [[continuous function|continuous]] with respect to the [[topological vector space]] structure from def. \ref{TVSOfCompactlySupportedFunctions} on the left.

Now by definition of the [[internal hom]] $A$ is actually "path-smooth" ([this def.](Frechet+space#PathSmoothFunction)) and hence the statement is implied by [this prop](Frechet+space#PathSmoothLinearFunctionsOnFrechetSpaceAreContinuous).

Finally to see that this argument generalizes to $\mathbf{H} = $ [[formal smooth set|FormalSmoothSet]] observe that the [[Weil algebra]] of every [[infinitesimally thickened point]] is a [[quotient ring]] of an algebra of smooth functions on some [[Cartesian space]] (by the [[Hadamard lemma]]). The previous argument now applies to representatives under this quotient coprojection and one checks that it is independent of the representative chosen.

=--

### General distributions

For general [[distributions]] in the sense of [[continuous linear functionals]] on the space $\mathcal{D}(\mathbb{R}^n)$ of [[compact support|compactly supported]] [[smooth functions]] ([[bump functions]]) their equivalence to smooth linear functionals is discussed in ([Kock-Reyes 04, section 2, esp. paragraph above theorem 2.3](#KockReyes04)).

(...)

## Applications
 {#Applications}

### In quantum field theory

This has some interesting consequences. For instance a major application of [[distribution]] theory is [[perturbative quantum field theory]], which, together with the rest of [[quantum physics]] is traditionally regarded to be a subject that makes heavy use of [[functional analysis]]. However, the development of [[algebraic quantum field theory]] with its emphasis of the abstract [[operator algebra]] [[algebra of observables|of observables]] and de-emphasis of [[Hilbert spaces]] [[space of quantum states|of quantum states]] (which is sometimes a convenient [[representation]] of the abstract [[operator algebra]] but more often just a distraction) combined with the development of [[perturbative quantum field theory]], where even these [[operator algebras]] are replaced by [[formal power series|formal]] [[star algebras]] of smooth functionals, has diminished the fundamental role of [[functional analysis]] in field theory. The theory of distributions survives as the tool of choice for understanding the singularity structure (or rather its absence) in the construction of the [[Wick algebra]] of [[quantum observables]] of the [[free fields]] and then in the process of [[renormalization]] via [[extension of distributions]] to the locus of coincident interaction points. But the [[quantum observables]] which are represented by these distributions are most naturally thought of as _smooth_ functions on the [[space of field histories]]. Therefore the fact that distributions in fact are the smooth linear functionals shows that their appearance in [[perturbative quantum field theory]] is a tool for studying smooth functionals, not a concept fundamental to the theory as such.

### Generalization to extensive quantities

The conception of distributions simply as elements of a double [[mapping space]] ([[internal hom]]) makes immediate that this concept exists in more generality than the traditional context. In [Lawvere 86](#Lawvere86) it was suggested that distribution theory should be considered more abstractly from such a point of general "[[extensive quantities]]". This is developed further in ([Kock 11](#Kock11)).

## References

Discussion of distributions in terms morphisms out of [[internal homs]] in the [[smooth topos]] over the site of [[smooth loci]] is in

* {#MoerdijkReyes91} [[Ieke Moerdijk]], [[Gonzalo Reyes]], around prop. II.3.6 of _[[Models for Smooth Infinitesimal Analysis]]_, Springer 1991

Discussion for the [[Cahiers topos]] is in

* {#KockReyes03} [[Anders Kock]], [[Gonzalo Reyes]], _Some calculus with extensive quantities: wave equation_, Theory and Applications of Categories , Vol. 11, 2003, No. 14, pp 321-336 ([TAC](http://www.tac.mta.ca/tac/volumes/11/14/11-14abs.html))

* {#KockReyes04} [[Anders Kock]], [[Gonzalo Reyes]], _Categorical distribution theory; heat equation_ ([arXiv:math/0407242](https://arxiv.org/abs/math/0407242))

* {#Kock11} [[Anders Kock]], _Commutative monads as a theory of distributions_ ([arxiv/1108.5952](http://arxiv.org/abs/1108.5952))

using results of

* {#FroelicherKriegl88} [[Alfred Frölicher]], [[Andreas Kriegl]], section 5 of  _Linear spaces and differentiation theory_, Wiley 1988 ([pdf](http://www.fuw.edu.pl/~kostecki/scans/froelicherkriegl1988.pdf))

and following the general conception of "[[intensive and extensive]]" in

* {#Lawvere86} [[William Lawvere]], Introduction to _[[Categories in Continuum Physics]], Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174. 1986

Similar [[sheaf theory|sheaf theoretic]] discussion of distributions as morphisms of [[smooth spaces]] is in

* {#Paugam12} [[Frédéric Paugam]], section 3.2 of _Towards the mathematics of quantum field theory_, 2012 ([pdf](https://webusers.imj-prg.fr/~frederic.paugam/documents/enseignement/master-mathematical-physics.pdf))

[[!redirects compactly supported distributions are the smooth linear functionals]]


distributive category

> For the non-cartesian case see at *[[distributive monoidal category]]*.

+-- {: .rightHandSide}
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#### Monoidal categories
+--{: .hide}
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#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

A [[category]] $C$  with [[finite products]] $(-)\times(-)$ and [[finite coproducts]] $(-) + (-)$ is called (finitary) **distributive** if for any $X,Y,Z\in C$ the canonical [[distributivity]] morphism
$$ X\times Y + X\times Z \longrightarrow X\times (Y+Z)$$
is an [[isomorphism]].  The canonical morphism is the unique morphism such that $X\times Y \to X\times (Y+Z)$ is $X\times i$, where $i\colon Y\to Y +Z$ is the coproduct injection, and dually for $X\times Z \to X\times (Y+Z)$.

Equivalently, a category $C$ with finite products and coproducts is distributive if for any $X \in C$, the functor $X \times -$ [[preserved limit|preserves finite coproducts]].

=--

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

This notion is part of a hierarchy of [[distributivity for monoidal structures]], and generalizes to [[distributive monoidal categories]] and [[rig categories]].  A [[linearly distributive category]] is _not_ distributive in this sense.

=--

This axiom on binary coproducts easily implies the analogous $n$-ary result for $n\gt 2$.  In fact it also implies the analogous 0-ary statement that the projection
$$ X\times 0 \to 0$$
is an isomorphism for any $X$ (see Proposition \ref{nullary} below).  Moreover, for a category with finite products and coproducts to be distributive, it actually suffices for there to be *any* [[natural transformation|natural]] family of isomorphisms $X\times Y + X\times Z \cong X\times (Y+Z)$, not necessarily the canonical ones ([Lack](#Lack)).

A category $C$ with finite products and all small coproducts is **infinitary distributive** if the statement applies to all small coproducts.  One can also consider $\kappa$-distributivity for a [[cardinal number]] $\kappa$, meaning the statement applies to coproducts of cardinality $\lt\kappa$.

Any [[extensive category]] with finite products is distributive, but the converse is not true.

## Properties

+-- {: .num_prop #monic}
###### Proposition
In a category with products and coproducts, if products distribute over binary coproducts, then [[coproduct]] [[coprojections]] are [[monomorphism|monic]].
=--

+-- {: .proof}
###### Proof
Let $i_B: B \to B + C$ be a coproduct coprojection, and suppose given maps $f, g: A \to B$ such that $i_B f = i_B g$. We observe that the coprojection

$$i: A \times B \to A \times B + A \times C$$

is monic because it has a [[retraction]] $(1_{A \times B}, \phi): A \times B + A \times C \to A \times B$. (All we need here is the existence of a map $\phi: A \times C \to A \times B$, for example the composite $A \times C \stackrel{\pi_A}{\to} A \stackrel{\langle 1_A, f \rangle}{\to} A \times B$.)

The composite of the coprojection $i$ with the canonical isomorphism $A \times B + A \times C \cong A \times (B + C)$, namely $1_A \times i_B: A \times B \to A \times (B + C)$, is therefore also monic. Given that $\langle 1_A, i_B f \rangle = \langle 1_A, i_B g \rangle: A \to A \times (B + C)$, we conclude

$$(1_A \times i_B)\langle 1_A, f \rangle = \langle 1_A, i_B f \rangle = \langle 1_A, i_B g \rangle = (1_A \times i_B)\langle 1, g \rangle,$$

whence $\langle 1_A, f\rangle = \langle 1_A, g\rangle: A \to A \times B$ since $1_A \times i_B$ is monic. It follows that $f = g$, as was to be shown.
=--

+-- {: .num_prop #nullary}
###### Proposition
If products distribute over binary coproducts, then products distribute over nullary coproducts (i.e., the projection $X \times 0 \to 0$ is an isomorphism for all objects $X$).
=--

+-- {: .proof}
###### Proof
We show that $X \times 0$ is initial. Clearly $\hom(X \times 0, Y)$ is inhabited by $X \times 0 \to 0 \to Y$ for any object $Y$. On the other hand, since the two coprojections $0 \to 0 + 0$ coincide, the same holds for the two coprojections $X \times 0 \to (X \times 0) + (X \times 0)$, by applying the distributivity isomorphism $X \times (0 + 0) \cong (X \times 0) + (X \times 0)$. This is enough to show that any two maps $X \times 0 \to Y$ coincide, since given maps $f, g : X \times 0 \to Y$, we have $f = [f, g] \circ i_1 = [f, g] \circ i_2 = g$.
=--

+-- {: .num_prop}
###### Proposition
In a distributive category, the [[initial object]] is [[strict initial object|strict]].
=--

+-- {: .proof}
###### Proof
Given an arrow $f: A \to 0$, we have that $\pi_A: A \times 0 \to A$ is a retraction of $\langle 1, f \rangle: A \to A \times 0$, so that $A$ is a retract of $A \times 0 \cong 0$. But retracts of initial objects are initial.
=--

## Examples
 {#Examples}

Every [[bicartesian closed category]] is distributive since in that case the cartesian product $X \times -$ is a [[left adjoint]] and then the distributivity requirement is an instance of the [[adjoints preserve (co-)limits|preservation of colimits by left adjoints]].
For example,

* the category [[Set]] of [[sets]],

* any [[topos]].

But closedness is far from necessary, for example

* the category [[Top]] of [[topological spaces]] with respect to forming [[product topological spaces]] and [[disjoint union topological spaces]];

is a distributive category.

These categories have in common that they are [[extensive category|extensive]]. An example[^Freyd] of a distributive category that is not extensive is given by

* the category of [[rectangular bands]] which has objects [[semigroup|semigroups]] $X$ satisfying $x x = x$ for all $x\in X$ and $x y x=x$ for all $x,y\in X$.

Since for [[poset|posets]] viewed as categories, finite products and coproducts are given by [[meets]] and [[joins]] we find:

* A [[poset]] viewed as a category is distributive iff it is a [[distributive lattice]]. Accordingly, non distributive lattices provide instances of categories with finite products and coproducts that do not distribute over each other.

A further non-example:

* The category $Pfn$ of sets and [[partial functions|partial functions]] has finite coproducts and products. The former given by [[disjoint unions]] as in [[Set]] and the latter by $(X\times Y)+X+Y$ (where $\times,+$ indicate the usual product and coproduct in $Set$) with projections $\pi_1,\pi_2$ given on the domains of definition $(X\times Y)+X$ (resp. $(X\times Y)+Y$) by $\pi_1((x,y))=x$ and $\pi_1(x)=x$ (resp. $\pi_2((x,y))=y$ and $\pi_2(y)=y$). Since the [[initial object]] $\empty$ is not [[strict initial object|strict]], $Pfn$ is not distributive.

[^Freyd]: Pointed out by [[Peter Freyd]] in [this discussion](https://www.mta.ca/~cat-dist/catlist/1999/extensive).

## Free distributive categories

\begin{proposition}

* The free distributive category on a category $\mathcal{C}$ with finite products is given by the finite [[free coproduct completion]]: $FinCoprod(\mathcal{C})$. That is, $FinCoprod(\mathcal{C})$ is distributive, and induces a bijection between distributive functors $FinCoprod(\mathcal{C})\to \mathcal{D}$ and product preserving functors
$\mathcal{C}\to \mathcal{D}$, natural in distributive categories $\mathcal{D}$.

* The free distributive category on a category $\mathcal{C}$ is the finite [[free coproduct completion]] of the finite [[free product completion]]: $FinCoprod(FinProd(\mathcal{C}))$. That is, $FinCoprod(FinProd(\mathcal{C}))$ is distributive, and induces a bijection between distributive functors $FinCoprod(FinProd(\mathcal{C}))\to \mathcal{D}$ and functors
$\mathcal{C}\to \mathcal{D}$, natural in distributive categories $\mathcal{D}$.

\end{proposition}
This is noted in [Cockett 1993, p. 280, also Corollary 5.9](#Cockett93), where [[Bob Walters]] and [[Shu-Hao Sun]] are named as advocates of this method. (However, note that Cockett does not insist on an initial object.)

In particular, then, the free distributive category, which is the free distributive category on the empty category, is the [[category of finite sets]], up to equivalence.

## Incompatibilities

Distributivity, in the sense of this page, is "incompatible" with certain other "linear" structures, at least in the sense that when combined they force the category to become [[thin category|thin]].

\begin{theorem}
If a category is distributive, and also [[linearly distributive category|linearly distributive]] with the same cartesian product and coproduct, then it is [[thin category|thin]].
\end{theorem}
\begin{proof}
This can be found in [Cockett-Seely 92](#CockettSeely92)
\end{proof}

\begin{theorem}
If a category has binary products that distribute over binary coproducts, and also has a [[monoidal category|monoidal structure]] that is [[compact closed category|compact closed]], then it is [[thin category|thin]].
\end{theorem}
\begin{proof}
See at [[compact closed category]].
\end{proof}

## Related concepts

* [[distributive law]]

* [[distributivity for monoidal structures]]

  * [[distributive monoidal category]]

  * [[bipermutative category]]

  * [[linearly distributive category]]

  * [[rig category]]

* [[extensive category]]

* [[distributive lattice]]

* [[objective number theory]]

* [[completely distributive category]]

* [[totally distributive category]]

## References

About distributive categories in general:

* {#CarboniLackWalters93} [[Aurelio Carboni]], [[Stephen Lack]], [[Bob Walters|R. F. C. Walters]], _Introduction to extensive and distributive categories_, JPAA **84** (1993) pp. 145-158 [doi:10.1016/0022-4049(93)90035-R]

* {#Cockett93} [[Robin Cockett]], *Introduction to distributive categories*, Mathematical Structures in Computer Science **3** (2009) 277-307 [doi:10.101/S0960129500000232]

* {#Lack} [[Stephen Lack]], *Non-canonical isomorphisms*, Journal of Pure and Applied Algebra **216** 3 (2012) 593-597  [[arXiv:0912.2126](http://arxiv.org/abs/0912.2126), [doi:10.1016/j.jpaa.2011.07.012](https://doi.org/10.1016/j.jpaa.2011.07.012)]

See also

* {#catlistdiscussion}[Freyd, Lack, Lawvere et al: CATLIST discussion on extensive categories 1996](https://www.mta.ca/~cat-dist/catlist/1999/extensive)

* Wikipedia, *[Distributive category](https://en.wikipedia.org/wiki/Distributive_category)*

For more see at *[[distributive monoidal category]]*.

Also cited above:

* {#CockettSeely92} [[Robin Cockett]], [[Robert Seely]],  _Weakly Distributive Categories_, in M.P. Fourman, P.T. Johnstone, A.M. Pitts, eds., _Applications of Categories in Computer Science_, London Mathematical Society Lecture Note Series 177 (1992), pp. 45–65. ([book website](https://www.cambridge.org/au/academic/subjects/mathematics/logic-categories-and-sets/applications-categories-computer-science-proceedings-london-mathematical-society-symposium-durham-1991))

* {#Houston06} [[Robin Houston]], _Finite Products are Biproducts in a Compact Closed Category_, Journal of Pure and Applied Algebra, Volume 212, Issue 2, February 2008, Pages 394-400 ([arXiv:math/0604542](http://arxiv.org/abs/math/0604542))

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