formally smooth scheme

#Contents#
* table of contents
{:toc}

## Idea

A _formally smooth scheme_ is a [[scheme]] $X$ with the property that every map into it admits an _[[infinitesimal space|infinitesimal]] lift_ :

for any "test space", i.e. an [[affine scheme]] $U=Spec(A)$ with underlying [[reduced scheme]] $Red(U)$ (obtained from $U$ by removing all [[infinitesimal space|infinitesimal]] directions, compare [[formal scheme]]) every map $Red(U) \to X$ lifts to a map $U \to X$.

Similarly, one can talk about formally smooth [[algebraic spaces]], [[algebraic stacks]] and so on.

## Definition

A [[scheme]] $X$ over some ground [[field]] $k$, given by a morphism $X \to Spec(k)$, is **formally smooth** if that morphism is a [[formally smooth morphism]], as described there.

If the [[scheme]] $X$ is regarded as an absolute (as opposed to relative) scheme, then the morphism $X \to Spec(\mathbb{Z})$ is required to be a [[formally smooth morphism]] for $X$ to be formally smooth.

>(Nowadays people will also start to consider schemes over the [[field with one element]]...).

## General definition

There is an interpretation of formal smoothness of (set-valued) functors as an [[epipresheaf]] condition with respect to Sasha Rosenberg's generalization of [[Grothendieck topologies]], so-called [[Q-category]] formalism, namely the Q-category of nilpotent (infinitesimal) thickenings. For more on this see _[[formally smooth morphism]]_.

## References

Discussion of formally smooth schemes in the general context of [[formally smooth morphisms]] is at

* [[Maxim Kontsevich]], [[Alexander Rosenberg]], _Noncommutative spaces_, preprint MPI-2004-35 ([ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=2331))

* T. Brzeziński, _Notes on formal smoothness_, *in*: Modules and Comodules (series _Trends in Mathematics_). T Brzeziński, JL Gomez Pardo, I Shestakov, PF Smith (eds), Birkhäuser, Basel, 2008, pp. 113-124 ([doi](http://dx.doi.org/10.1007/978-3-7643-8742-6), [arXiv:0710.5527](http://arxiv.org/abs/0710.5527))


formally unramified morphism

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[space]] $X$ is called **formally unramified** if every [[morphism]] $Y \to X$ into it has for every [[infinitesimal object|infinitesimal thickening]] of $Y$ at most one [[infinitesimal object|infinitesimal]] extension.

(If all thickenings exist it is called a [[formally smooth morphism]]. If the thickening exist uniquely, it is called a [[formally etale morphism]].)

Traditionally this has been considered in the context of [[geometry]] over formal duals of [[ring]]s and [[associative algebra]]s. This we discuss in the section ([Concrete notion](#ConcreteNotion)). But generally the notion makes sense in any context of infinitesimal cohesion. This we discuss in the section
[General abstract notion](#GeneralAbstractNotion).

The concept of formally unramified morphisms is the infinitesimal version of that of [[unramified morphisms]].

## General abstract notion
 {#GeneralAbstractNotion}

### Definition

Let

$$
  \mathbf{H}
   \stackrel{\overset{u^*}{\hookrightarrow}}{\stackrel{\overset{u_*}{\leftarrow}}{\underset{u^!}{\to}}}
  \mathbf{H}_{th}
$$

be a triple of [[adjoint functor]]s with $u^*$ a [[full and faithful functor]] that preserves the [[terminal object]].

We may think of this as exhibiting [[cohesive (∞,1)-topos#InfinitesimalCohesion|infinitesimal cohesion]] (see there for details, but notice that in the notation used there we have $u^* = i_!$, $u_* = i^*$ and $u^! = i_*$).

We think of the objects of $\mathbf{H}$ as [[cohesive topos|cohesive space]]s and of the objects of $\mathbf{H}_{th}$ as such cohesive spaces possibly equipped with [[infinitesimal object|infinitesimal extension]].

As a class of examples that is useful to keep in mind consider a [[Q-category]] $(cod \dashv \epsilon \dashv dom) : \bar A \to A$  of infinitesimal thickening of rings and let

$$
  ((u^* \dashv u_* \dashv u^!)
   :
   \mathbf{H}_{th} \to \mathbf{H})
   :=
  ([dom,Set] \dashv [\epsilon, Set] \dashv [codom,Set] :
   [\bar A, Set] \to [A,Set])
$$

be the corresponding Q-category of copresheaves.

For any such setup there is a canonical [[natural transformation]]

$$
  u^* \to u^!
  \,.
$$

Details of this are in the section Adjoint quadruples at [[cohesive topos]].

From this we get for every morphism $f : X \to Y$ in $\mathbf{H}$ a canonical morphism

\[
  \label{MorphismIntoPullback}
  u^* X \to u^* Y \prod_{u^! Y} u^! X
  \,.
\]

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

A morphism $f : X \to Y$ in $\mathbf{H}$ is called **formally unramified** if (eq:MorphismIntoPullback) is a [[monomorphism]].

=--

This appears as ([KontsevichRosenberg, def. 5.1, prop. 5.3.1.1](#KontsevichRosenbergSpaces)).

The dual notion, where the morphism is required to be an [[epimorphism]] is that of [[formally smooth morphism]]s. If both conditions hold, hence if the morphism is in fact an [[isomorphism]], one speaks of [[formally etale morphism]]s.

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

An object $X \in \mathbf{H}$ is called **formally unramified** if the morphism $X \to *$ to the [[terminal object]] is formally unramified.

=--

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

The object $X$ is formally unramified precisely if

$$
  u^* X \to u^! X
$$

is a [[monomorphism]].

=--

This appears as ([KontsevichRosenberg, def. 5.3.2](#KontsevichRosenbergSpaces)).

### Properties

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

Formally unramified morphisms are closed under composition.

=--

This appears as ([KontsevichRosenberg, prop. 5.4](#KontsevichRosenbergSpaces)).

## Concrete notion
 {#ConcreteNotion}

For the moment see the discussion at [[unramified morphism]].

## Related concepts

* [[unramified morphism]]

* [[ramification of ideals]]

* [[formally smooth morphism]] and **formally unramified morphism** $\Rightarrow$ [[formally etale morphism]].

[[!redirects formally unramified morphisms]]
[[!redirects formally unramified]]


formally étale morphism

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###Context###
#### Étale morphisms
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[[!include etale morphisms - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

A [[morphisms]] of [[spaces]] $X \overset{p}{\longrightarrow} Y$ is called _formally étale_ if it has a [[lifting property]] as [[étalé spaces]] do locally, but for _[[infinitesimal space|infinitesmal]]_ extensions: If for $Z \overset{f}{\to} Y$ any morphism and $\Re(Z) \to X$ a lift of its restriction along its [[reduction]] $\Re(Z) \to Z$, there is a unique extension to a complete lift.

(If there exists at least one such infinitesimal extension, it is called a [[formally smooth morphism]]. If there exists at most one such extension, it is called a [[formally unramified morphism]]. The formally étale morphisms are precisely those that are both formally smooth and formally unramified.)

Traditionally this has been considered in the context of [[geometry]] over formal duals of [[ring]]s and [[associative algebra]]s. This we discuss in the section ([Concrete notion](#ConcreteNotion)). But generally the notion makes sense in any context of differential cohesion. This we discuss in the section
[General abstract notion](#GeneralAbstractNotion).

## General abstract notion
 {#GeneralAbstractNotion}

### Definition

Let

$$
  \mathbf{H}
   \stackrel{\overset{u^*}{\hookrightarrow}}{\stackrel{\overset{u_*}{\leftarrow}}{\underset{u^!}{\to}}}
  \mathbf{H}_{th}
$$

be an [[adjoint triple]] of [[functor]] with $u^*$ a [[full and faithful functor]] that preserves the [[terminal object]].

We may think of this as exhibiting differential cohesion (see there for details, but notice that in the notation used there we have $u^* = i_!$, $u_* = i^*$ and $u^! = i_*$).

We think of the objects of $\mathbf{H}$ as [[cohesive topos|cohesive space]]s and of the objects of $\mathbf{H}_{th}$ as such cohesive spaces possibly equipped with [[infinitesimal object|infinitesimal extension]].

As a class of examples that is useful to keep in mind consider a [[Q-category]]

$$
  (cod \dashv \epsilon \dashv dom) : \bar A \to A
$$

of infinitesimal thickening of rings and let

$$
  ((u^* \dashv u_* \dashv u^!) : \mathbf{H}_{th} \to \mathbf{H}) :=
  ([dom,Set] \dashv [\epsilon, Set] \dashv [cod,Set] : [\bar A, Set] \to [A,Set])
$$

be the corresponding Q-category of copresheaves.

For any such setup there is a canonical [[natural transformation]]

$$
  \phi : u^* \to u^!
  \,.
$$

Details of this are in the section Adjoint quadruples at [[cohesive topos]].

From this we get for every morphism $f : X \to Y$ in $\mathbf{H}$ a canonical morphism

\[
  \label{MorphismIntoPullback}
  u^* X \to u^* Y \prod_{u^! Y} u^! X
  \,.
\]

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

A morphism $f : X \to Y$ in $\mathbf{H}$ is called **formally étale** if (eq:MorphismIntoPullback) is an [[isomorphism]].

=--

This appears as ([KontsevichRosenberg, def. 5.1, prop. 5.3.1.1](#KontsevichRosenbergSpaces)).

In other words, $f$ is formally étale if the $f$-component naturality square

$$
  \array{
    u^* X &\stackrel{u^* f}{\to}& u^* Y
    \\
    {}^{\mathllap{\phi_X}}\downarrow && \downarrow^{\mathrlap{\phi_Y}}
    \\
    u^! X &\stackrel{u^! f}{\to}& u^! Y
  }
$$

of the [[natural transformation]] $\phi$ is a [[pullback]] [[diagram]].

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

The partial notions of this condition are: if the above morphism is a [[monomorphism]] then $f$ is a [[formally unramified morphism]], if it is an [[epimorphism]] then $f$ is a [[formally smooth morphism]].

=--

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

An object $X \in \mathbf{H}$ is called **formally étale** if the morphism $X \to *$ to the [[terminal object]] is formally étale.

=--

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

The object $X$ is formally étale precisely if

$$
  u^* X \to u^! X
$$

is an [[isomorphism]].

=--

This appears as ([KontsevichRosenberg, def. 5.3.2](#KontsevichRosenbergSpaces)).

### Properties

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

Formally étale morphisms are closed under composition.

=--

This appears as ([KontsevichRosenberg, prop. 5.4](#KontsevichRosenbergSpaces)).

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

This follows by the [[pasting law]] for pullbacks: let $f : X \to Y$ and $g : Y \to Z$ be two formally étale morphisms.  Then by definition both of the small squares in

$$
  \array{
    u^* X &\stackrel{u^* f }{\to}& u^* Y &\stackrel{u^* g}{\to}& u^* Z
    \\
    \downarrow && \downarrow && \downarrow
    \\
    u^! X &\stackrel{u^! f }{\to}& u^! Y &\stackrel{u^! g}{\to}& u^! Z
  }
$$

are pullback squares. Hence so is the total outer square.

=--

Using also the other case of the [[pasting law]],  the above proof shows more:

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

If

$$
  \array{
    && Y
    \\
    & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
    \\
    X &&\stackrel{h}{\to}&& Z
  }
$$

is a [[commuting diagram]] such that $g$ and $h$ are formally étale, then also $f$ is formally étale.

=--

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

Formally étale morphisms are closed under [[retract]]s.

=--

This means that if $f : X \to Y$ is formally étale and

$$
  \array{
    A &\to & X &\to& A
    \\
    \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{p}}
    \\
    B &\to& Y &\to& B
  }
$$

is a [[commuting diagram]] such that the two horizontal composites are [[identities]], then also $p$ is formally étale.

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

By applying the [[natural transformation]] $ \phi : u^* \to u^!$ to this diagram we obtain a retract diagram in the category of squares, given by the naturality squares of $\phi$ on $f$ and $p$, where the middle square is a pullback square. By this proposition at _[[retract]]_ this implies that also the retracting square is a pullback, which means that $p$ is formally étale.

=--

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

If $u^*$ preserves [[pullbacks]], then formally étale morphisms are stable under pullback.

=--

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

Consider a [[pullback]] diagram

$$
  \array{
    A \times_Y X &\to& X
    \\
    {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{f}}
    \\
    A &\to& Y
  }
$$

where $f$ is formally étale.

Applying the [[natural transformation]] $\phi : u^* \to u^!$ to this yields a square of squares. Two sides of this are the [[pasting]] composite

$$
  \array{
    u^* A \times_Y X &\to& u^* X &\stackrel{\phi_X}{\to}&  u^! X
    \\
    \downarrow && \downarrow^{\mathrlap{u^* f}}
                      && \downarrow^{\mathrlap{u^! f}}
    \\
    u^* A &\to& u^* Y &\stackrel{\phi_Y}{\to}& u^! Y
  }
$$

and the other two sides are the pasting composite

$$
  \array{
     u^* A \times_Y X &\stackrel{\phi_{A \times_Y X}}{\to}& u^! A \times_Y A
     &\stackrel{}{\to}& u^! X
     \\
     \downarrow^{} && \downarrow && \downarrow^{\mathrlap{u^! f}}
     \\
     u^* A &\stackrel{\phi_A}{\to}& u^! A &\to& u^! Y
  }
  \,.
$$

Counting left to right and top to bottom, we have that

* the first square is a pullback by assumption on $u^*$;

* the second square is a pullback, since $f$ is formally étale.

* the fourth square is a pullback since $u^!$ is [[right adjoint]] and so also preserves pullbacks;

* also the total bottom rectangle is a pullback, since it is equal to the bottom  total rectangle;

* therefore finally the third square is a pullback, by the [[pasting law]], hence also $p$ is formally étale.

=--

## Concrete notions
 {#ConcreteNotion}

We discuss realizations of the above general abstract definition in concrete models of the axioms.

See also the concrete notions of [[formally smooth morphism]] and [[formally unramified morphism]].

### In differential geometry

The [[category]] [[SmoothMfd]] of [[smooth manifold]]s may naturally be thought of as sitting inside the more general context of the [[cohesive (∞,1)-topos]] [[Smooth∞Grpd]] of [[smooth ∞-groupoid]]s. This is canonically equipped with a notion of [[differential cohesion|infinitesimal cohesion]] exhibited by its inclusion into [[SynthDiff∞Grpd]]. This implies that there is an intrinsic notion of [[formally étale morphism]]s of smooth $\infty$-groupoids in general and of smooth manifolds in particular

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

A [[smooth function]] is a formally étale morphism in this sense precisely if it is a [[local diffeomorphism]] in the traditional sense.

=--

See this section for more details.

### In algebraic geometry

* [[formally étale morphism of schemes]]

### In noncommutative geometry

See ([RosenbergKontsevich, section 5.8](#RosenbergKontsevich))

## Related concepts

[[formally smooth morphism]] and [[formally unramified morphism]] $\Rightarrow$ **formally étale morphism**

[[!include cohesion - table]]

## References
 {#References}

The idea of defining étale morphisms $f$ as those that get send to a pullback square by a natural transformation goes back to lectures by [[André Joyal]] in the 1970s.

See the introduction and see section 4 of

* [[Eduardo Dubuc]], _Axiomatic etal maps and a theory of spectrum_, Journal of pure and applied algebra,  149 (2000)
 {#Dubuc}

The identification of the natural transformation in question with that induced by an [[adjoint triple]] ("[[Q-categories]]") and the relation to _formal_ étaleness is observed (apparently independently?) in

* [[Maxim Kontsevich]], [[Alexander Rosenberg]], _Noncommutative spaces_, preprint MPI-2004-35 ([[KontsevichRosenbergNCSpaces.pdf:file]], [ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=2331), [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=2303))
 {#KontsevichRosenbergSpaces}

Formalization and discussion in the context of [[cohesive (∞,1)-toposes]] is in section 2.5.3 (and defn 5.3.19) of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

[[!redirects formally etale morphism]]

[[!redirects formally etale]]

[[!redirects formally étale morphism]]
[[!redirects formally étale morphisms]]

[[!redirects formally etale morphisms]]

[[!redirects formally étale map]]
[[!redirects formally étale maps]]

[[!redirects formally etale map]]
[[!redirects formally etale maps]]


formally étale morphism of schemes

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of [[formally étale morphism]] between [[schemes]].

## Definition

We first state the traditional

* _[Explicit definition in components](#ExplicitlyInComponents)_

Then we give the abstract

* _[Characterization by reduction/infinitesimal shape/de Rham Space](#CharacterizationByReductionModality)_

### Explicitly in components
 {#ExplicitlyInComponents}

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

A [[morphism]] of [[commutative rings]] $R \hookrightarrow A$
is called _formally étale_ if for every ring $B$ and for every [[nilpotent ideal]] $I \subset B$ and for every [[commuting diagram]] of the form

$$
  \array{
    B/I &\leftarrow& A
    \\
    \uparrow && \uparrow
    \\
    B &\leftarrow& R
  }
$$

there is a unique diagonal morphism

$$
  \array{
    B/I &\leftarrow& A
    \\
    \uparrow &\swarrow& \uparrow
    \\
    B &\leftarrow& R
  }
$$

that makes both triangles commute.

=--

(e.g. [Stacks Project 57.9, 57.12](#StackProject))

By [[Isbell duality|formal duality]] and [[sheafification|locality]] this yields a notion of formally étale morphisms of [[affine varieties]] and of [[schemes]].

### Characterization by reduction/infinitesimal shape
 {#CharacterizationByReductionModality}

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

Write $CRing_{fin}$ for the [[category]] of [[finitely generated ring|finitely generated]] [[commutative rings]] and write $CRing_{fin}^{ext}$ for the category of [[infinitesimal ring extensions]].
Write

$$
  Red \;\colon\; CRing_{fin}^{ext} \longrightarrow CRing_{fin}
$$

for the [[functor]] which sends an [[infinitesimal ring extension]] to the underlying [[commutative ring]] (in the maximal case this sends a commutative ring to its [[reduced ring]], whence the name of the functor), and write

$$
  i \;\colon\; CRing_{fin} \hookrightarrow CRing_{fin}^{ext}
$$

for the [[full subcategory]] inclusion that regards a ring as the trivial infinitesimal extension over itself.

=--

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

There is an [[adjoint triple]] of [[idempotent monad|idempotent]] ([[comonad|co]]-)[[monads]]

$$
  (Red \dashv \int_{inf} \dashv \flat_{inf}) \;\colon\; PSh((CRing_{fin}^{ext})^{op}) \longrightarrow PSh((CRing_{fin}^{ext})^{op})
$$

where the [[left adjoint]] [[comonad]] $Red$ is given on [[representable functor|representables]] by the [[reduced scheme|reduction]] functor of def. \ref{ReductionOnRings} (followed by the inclusion).

=--

This statement and the following prop. \ref{FormalEtalenessBydeRhamSpace} is a slight paraphrase of an observation due to ([Kontsevich-Rosenberg 04](Q-category#KontsevichRosenbergSpaces)).

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

The  functors from def. \ref{ReductionOnRings} form an
[[adjoint pair]] $(Red \dashv i)$ because an extension element
can only map to an extension element; so for $\widehat R \to R$
an [[infinitesimal ring extension]] of $R = Red(\widehat R)$,
and for $S$ a commutative ring with
$i(S) = (S \to S)$ its trivial extension, there is a [[natural isomorphism]]

$$
  Hom_{CRing_{fin}^{ext}}(\widehat R, i(S))
  \simeq
  Hom_{CRing_{fin}}(R,S)
  \,.
$$

This exhibits $CRing_{fin}$ as a [[reflective subcategory]]
of $CRing_{fin}^{ext}$.

$$
  (Red \dashv i)
   \;\colon\;
  CRing_{fin}
   \stackrel{\overset{Red}{\leftarrow}}{\underset{i}{\hookrightarrow}}
  CRing_{fin}^{ext}
  \,.
$$

Via [[Kan extension]] this [[adjoint pair]] induces an [[adjoint quadruple]] of [[functors]] on [[categories of presheaves]]

$$
  PSh(CRing_{fin}^{op})
    \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^\ast = Red_!}{\leftarrow}}{\stackrel{\overset{Red^\ast}{\hookrightarrow}}{\underset{Red_\ast}{\leftarrow}}}}
  PSh((CRing_{fin}^{ext})^{op})
  \,.
$$

The [[adjoint triple]] to be shown is obtained from composing these adjoints pairwise.

That $Red$ coincides with the reduction functor on representables is a standard property of [[left Kan extension]] (see [here](Kan+extension#LeftKanOnRepresentables) for details).

=--

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

These considerations make sense in the general abstract context of
"[[differential cohesion]]" where the [[adjoint triple]] of prop. \ref{DifferentialCohesionModality} would be called:

([[reduction modality]] $\dashv$ [[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]]).

=--

Due to the [[full subcategory]] inclusion $i_!$ in the proof of prop. \ref{DifferentialCohesionModality} we may equivalently regard presheaves on $(CRing_{fin})^{op}$ (e.g. [[schemes]]) as presheaves on $(CRing_{fin}^{ext})^{op}$ (e.g. [[formal schemes]]). This is what we do implicitly in the following.

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

A morphism $f \;\colon\; Spec A \to Spec R$ in $CRing_{fin}^{op} \hookrightarrow PSh(CRing_{fin}^{op})$ is
formally étale, def. \ref{ExplicitDefinition}, precisely if it is $\int_{inf}$-[[modal type|modal]] relative $Spec R$, hence if the [[natural transformation|naturality square]] of the [[infinitesimal shape modality]]-[[unit of a monad|unit]]

$$
  \array{
    Spec A &\longrightarrow& \int_{inf} Spec A
    \\
    \downarrow && \downarrow
    \\
    Spec R &\longrightarrow& \int_{inf} Spec R
  }
$$

is a [[pullback]] square.

=--

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

Evaluated on $I \hookrightarrow R  \to R/I \in CRing_{fin}^{ext}$ any object, by the [[Yoneda lemma]] and the $(Red \dashv \int_{inf})$-[[adjunction]] the naturality square becomes

$$
  \array{
    CRing(A,B) &\longrightarrow& CRing(A,B/I)
    \\
    \downarrow && \downarrow
    \\
    CRing(R,B) &\longrightarrow& CRing(R,B/I)
  }
  \,.
$$

in [[Set]]. Chasing elements through this shows that this is a [[pullback]] precisely if the condition in def. \ref{ExplicitDefinition} holds.

=--

The basic stability property of [[étale morphisms]], which we need in the following, immediately follows from this characterization:

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

For $\stackrel{f}{\to} \stackrel{g}{\to}$ two composable morphisms, then

1. if $f$ and $g$ are both (formally) étale, then so is their composite $g \circ f$;

1. if $g$ and $ g\circ f$ are (formally) étale, then so is $f$;

1. the [[pullback]] of a (formally) étale morphism along any morphism is again (formally) étale.

=--

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

With prop. \ref{FormalEtalenessBydeRhamSpace} this is equivalently the
statement of the [[pasting law]] for [[pullback]] diagrams.

=--

## Properties

### Relation to étale morphism

Formally étale morphisms of schemes which are in addition [[locally of finite presentation]] are equivalently [[étale morphisms of schemes]].

Relaxing this finiteness condition yields the notion of [[weakly étale morphisms]].

[[étale morphism of schemes|étale morphism]] $\Rightarrow$ [[pro-étale morphism of schemes|pro-étale morphism]] $\Rightarrow$ [[weakly étale morphism of schemes|weakly étale morphism]] $\Rightarrow$ [[formally étale morphism of schemes|formally étale morphism]]

## Related concepts

* [[formally unramified morphism]], [[formally smooth morphisms]]

* [[basics of étale cohomology]]

## References

The traditional formulation is for instance in

* [[The Stacks Project]], _[57.9. Formally smooth, étale, unramified transformations](http://stacks.math.columbia.edu/tag/04G3)_ _[57.12. Formally étale morphisms](http://stacks.math.columbia.edu/tag/04GB)_
 {#StackProject}

The  characterization via a ([[reduction modality]] $\dashv$ [[infinitesimal shape modality]]) is more or less explicit in

* [[Carlos Simpson]], [[Constantin Teleman]], _de Rham Theorem for $\infty$-stacks_ 1997 ([pdf](http://math.berkeley.edu/~teleman/math/simpson.pdf))
 {#SimpsonTeleman97}

* [[Maxim Kontsevich]], [[Alexander Rosenberg]], section 4 of _Noncommutative spaces_, preprint MPI-2004-35 ([[KontsevichRosenbergNCSpaces.pdf:file]], [ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=2331), [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=2303))
 {#KontsevichRosenberg04}

The formalization as [[differential cohesion]] is discussed in

* [[Urs Schreiber]], sections 3.5, 3.10 of _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))

[[!redirects formally étale morphisms of schemes]]

[[!redirects formally etale morphism of schemes]]
[[!redirects formally etale morphisms of schemes]]


ForMath

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#### Constructivism, Realizability, Computability
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## Idea

A project for formalization of parts of [[mathematics]] ([[formal proof]]) in formal languages supported by [[proof assistants]] (particularly in [[Coq]]).

## Related projects

* [[UniMath project]]

* [[Xena project]]

* [[The QED Project]]

* [[Archive of Formal Proofs]]

## Related entries

* [[MathScheme]]

* [[formal proof]]

## References

* [ForMath Website](http://wiki.portal.chalmers.se/cse/pmwiki.php/ForMath/ForMath)

* {#ForMath} [[Thierry Coquand]], _ForMath: Formalisation of Mathematics_ research project ([web](http://wiki.portal.chalmers.se/cse/pmwiki.php/ForMath/ForMath))

* {#MathClasses} Eelis van der Weegen, [[Bas Spitters]], Robbert Krebbers, Matthieu Sozeau, Tom Prince, [[Jelle Herold]],

  _Math Classes_, Coq Library for basic mathematical structures ([web](http://math-classes.org/))

[[!redirects ForMath project]]


formes tordues > history

< [[formes tordues]]


formula

## Related concepts

* [[theory]], [[signature (in logic)]]

## References

* Wikipedia, _[Well-formed formula](https://en.wikipedia.org/wiki/Well-formed_formula)_

[[!redirects formulas]]
[[!redirects formulae]]

[[!redirects well-formed formula]]
[[!redirects well-formed formulas]]


Fosco Loregian

I am an Italian postdoc.

I'm a former student of [[Domenico Fiorenza]]. I owe him my mathematical everything.

I've been a postdoc in all these places:

* University of Western Ontario, (CA)
* Masaryk University (CZ)
* Max Planck Institute for Mathematics (D)
* Center of Mathematics, University of Coimbra (PT)

Currently, I am in Tallinn in the [compositionality](https://compose.ioc.ee) group of [Pawel Sobocinski](https://www.ioc.ee/~pawel/).

Here's a list of the papers I've written: [link](https://arxiv.org/search/math?searchtype=author&query=Loregian%2C+F)

[This](http://tetrapharmakon.github.io) is my personal website. It's way more up to date than the nLab...

Write me an email any time!

## Selected writings

On [[formal category theory]]:

* [[Ivan Di Liberti]], [[Simon Henry]], [[Mike Lieberman]], [[Fosco Loregian]], _Formal category theory_, course notes (2017) [[pdf](https://tetrapharmakon.github.io/stuff/course_muni-formal.pdf), [[DLHLL-FormalCategoryTheory.pdf:file]]]

* [[Ivan Di Liberti]], [[Fosco Loregian]], _On the unicity of formal category theories_ [[arXiv:1901.01594](https://arxiv.org/abs/1901.01594)]

Review of [[cohesion]] ([[cohesive toposes]]) and [[differential cohesion]]:

* {#Loregian19} [[Fosco Loregian]], *Cohesion in Rome*, talk notes, Dec. 2019  ([pdf](https://tetrapharmakon.github.io/stuff/cohesive_rome.pdf), [[LoregianCohesion.pdf:file]])

On [[optics (in computer science)]]:

* [[Bryce Clarke]], Derek Elkins, [[Jeremy Gibbons]], [[Fosco Loregian]], [[Bartosz Milewski]], Emily Pillmore, [[Mario Román]], _Profunctor optics, a categorical update_, 2020. ([arXiv:2001.07488](https://arxiv.org/abs/2001.07488))

{#RiehlLoregian2020} On ([[discrete fibration|discrete]]) [[Grothendieck fibrations]] and [[profunctors]]:

*  [[Emily Riehl]] and [[Fosco Loregian]], _Categorical notions of fibration_, Expositiones Mathematicae **38** 4 (2020) 496-514 [[arXiv:1806.06129](https://arxiv.org/abs/1806.06129), [doi:10.1016/j.exmath.2019.02.004](https://doi.org/10.1016/j.exmath.2019.02.004)]

On [[ends]] and [[coends]]:

* [[Fosco Loregian]], _Coend calculus_, Cambridge University Press 2021 ([arXiv:1501.02503](http://arxiv.org/abs/1501.02503), [doi:10.1017/9781108778657]( https://doi.org/10.1017/9781108778657), ISBN:9781108778657).

On [[t-structures]] and [[reflective factorization systems]]:

* [[Domenico Fiorenza]], [[Fosco Loregian]], *$t$-Structures are normal torsion theories*, Appl Categor Struct **24** (2016) 181–208 [[arxiv:1408.7003](http://arxiv.org/abs/1408.7003), [doi:10.1007/s10485-015-9393-z](https://doi.org/10.1007/s10485-015-9393-z)]

* [[Domenico Fiorenza]], [[Fosco Loregian]], [[Giovanni Marchetti]], *Hearts and towers in stable $\infty$-categories*,  J. Homotopy Relat. Struct. **14**  (2019) 993–1042 [[arXiv:1501.04658](https://arxiv.org/abs/1501.04658), [doi:10.1007/s40062-019-00237-0](https://doi.org/10.1007/s40062-019-00237-0)]

* [[Fosco Loregian]], [[Simone Vrili]] *Triangulated factorization systems and t-structures*, Journal of Algebra **550** (2020) 219-241 [[doi:10.1016/j.jalgebra.2019.12.021](https://doi.org/10.1016/j.jalgebra.2019.12.021)]

On the [[adjoint functor theorem]] in the context of [[lax-idempotent 2-monads]]:

* {#ALL23} [[Nathanael Arkor]], [[Ivan Di Liberti]], [[Fosco Loregian]], _Adjoint functor theorems for lax-idempotent pseudomonads_, [[arXiv:2306.10389](https://arxiv.org/abs/2306.10389)]

## Related entries

* [[orthogonal factorization system in a derivator]]

* [t-structure](https://ncatlab.org/nlab/show/t-structure#in_stable_categories)

* [[end]]

category: people

[[!redirects Fosco]]
[[!redirects Fosco Loregian]]


Fosco Loregian > history

< [[Fosco Loregian]]


Fotis Farakos

* [webpage](http://www.muni.cz/sci/people/233932)

## Selected writings

On [[higher curvature corrections]] to [[D=4 supergravity]] and relation to [[quintessence]]:

* [[Fotis Farakos]], _Quintessence from higher curvature supergravity_ ([arXiv:2003.09366](https://arxiv.org/abs/2003.09366))

On [[membranes]] in [[D=4 supergravity]] ([[super 2-brane in 4d]]):

* [[Igor Bandos]], [[Fotis Farakos]] Stefano Lanza, L. Martucci,  [[Dmitri Sorokin]], _Higher Forms and Membranes in 4D Supergravities_, in: Proceedings of _[[Higher Structures in M-Theory 2018]]_, Fortschritte der Physik, Special Issue Volume 67, Issue 8-9 ([arXiv:1903.02841](https://arxiv.org/abs/1903.02841), [doi:10.1002/prop.201910020](https://doi.org/10.1002/prop.201910020))

## Related $n$Lab entries

* [[supergravity]]

* [[Starobinsky model of cosmic inflation]]

category: people


foundation of mathematics

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# Contents
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## Idea
 {#Idea}

In the context of _foundations of mathematics_ or _[[mathematical logic]]_ one studies formal systems -- [[theories]] -- that allow us to formalize much if not all of [[mathematics]] (and hence, by extension, at least aspects of mathematical fields such as fundamental [[physics]]).

There are two different attitudes to what a desirable or interesting foundation should achieve:

1. In **[[proof theory|proof-theoretic]] foundations** the emphasis is on seeing which formal systems, however convoluted they may be conceptually, allow us to formalize and prove which [[theorems]].

   The archetypical such system is [[ZFC]] [[set theory]]. Other formal systems of interest here are [[elementary function arithmetic]] and [[second order arithmetic]], because they are [[proof theory|proof-theoretically]] weak, and still can derive "almost all of undergraduate mathematics" ([Harrington](#Harrington)).

1. In **[[practical foundations]]** (following a term introduced in ([Taylor](#Taylor))) the emphasis is on conceptually natural formalizations that _concentrate the essence of practice and in turn use the result to guide practice_ ([Lawvere](#Lawvere)), as in ([Eilenberg-Steenrod](#EilenbergSteenrod), [Harper](#Harper)).

   Formal systems of interest here are [[ETCS]] or flavors of [[type theory]], which allow natural expressions for central concepts in mathematics (notably via their [[categorical semantics]] and the conceptual strength of [[category theory]]).

For a philosophical treatment of foundations see [[foundations and philosophy]].

## Deductive systems

Almost all [[foundations of mathematics]] are expressed in some foundational [[deductive system]].

One versatile deductive system [[natural deduction]], which could be used to define certain foundation of mathematics using [[logic over type theory]], such as all [[set theories]], including [[categorical set theories]] like [[ETCS]], as well as the foundations of mathematics using only [[type theory]], including [[set-level type theories]] as well as [[higher foundations]] such as [[homotopy type theory]], as well as typed [[higher-order logic]].

Alternatives include [[sequent calculus]] for logic over untyped theories, such as [[unsorted set theory]] and untyped higher-order logic, as well as [[lambda-calculus]] for [[type theories]].

### Dependent type theory

Dependent type theory itself supports various [[foundations of mathematics]] via the [[propositions as some types]] interpretation of [[dependent type theory]]. Suppose that a dependent type theory has a separate type judgment as well as [[dependent product types]], [[dependent sum types]], [[identity types]], [[weak function extensionality]], [[propositional truncations]], [[empty type]], [[unit type]], [[sum types]]. All the operations in [[predicate logic]] are derivable from said type formers. Then:

* One can add a [[cumulative hierarchy]] to the dependent type theory and work entirely in the cumulative hierarchy for [[material set theory]]

* One can add a [[category of sets]] to the dependent type theory and work entirely in the category of sets for [[structural set theory]]

* One can add a [[type universe]] satisfying certain [[axioms]] and [[axiom schemata]], such as [[universe extensionality]], closure under identity types, closure under dependent sum types, closure under dependent product types, [[propositional resizing]], [[axiom of replacement|replacement]], [[axiom of infinity|infinity]], and [[axiom of choice|choice]], to the dependent type theory and work entirely in the universe for [[univalent type theory]] or [[univalent foundations]]. Adding internal universe types as [[small]] [[object classifiers]] as well as all [[higher inductive types|higher inductive]] and [[coinductive types]] to the universe results in [[homotopy type theory]].

* One can add a [[Russell type of all propositions]] and a [[natural numbers type]] and work in the dependent type theory itself for [[higher-order logic]].

## Basic notions

### Objects and collections

In most foundations of mathematics, there are basic notions of objects and collections of objects.

Note that in certain foundations, the objects, collections, and membership are derived notions from some other notion. Collections are referred to a number of names, such as [[set]], [[class]], [[type]], et cetera. Objects could be referred to as [[element]] or [[term]].

For example, in [[pure set]] theories like [[ZFC]] and [[Mostowski set theory]], elements and sets are the same notion, and thus elements could be considered to be a derived notion from set, or sets could be considered to be a derived notion from element. In [[material set theory]] with [[urelements]], such as [[ZFA]], elements are the basic notion and sets are derived from elements. In [[ETCS]], while sets are basic, elements are derived from a basic notion of [[function]], and in [[fully formal ETCS]], both sets and elements are derived notions from a basic notion of [[function]]. In [[SEAR]] and [[structural ZFC]], sets and elements are both basic distinct notions.

Something similar occurs with [[class theories]], such as [[Morse-Kelley class theory]] and the framework of [[algebraic set theory]], where classes play the role of sets.

In type theory, there are also a distinction to be made. In most type theories, both terms and types are basic foundational notions. However, in [[book HoTT]], terms are the basic notion, and [[types]] are derived from terms.

In [[higher-order logic]], the [[domain of discourse]] and higher-order [[predicates]] play the role of collections, while the objects in the [[domain of discourse]] play the role of the elements.

### Membership and typing

In most foundations, there is a basic notion of membership, of an "object being in a collection".

Foundations could be contrasted between those foundations where membership is a [[membership relation]], and those foundations where membership is a [[typing judgment]]. This distinction is largely the same distinction as the distinction between [[unsorted set theories]] and [[simply sorted set theories]] on one hand, and [[dependently sorted set theories]] and [[dependent type theories]] on the other hand.

For theories with a membership relation, one could further distinguish between [[homogeneous membership relations]] (for relations on the same sort) and [[heterogeneous membership relations]] (for relations between two different sorts), as well as membership relations which are primitive to the theory, and membership relations which are derived from some other existing structure in the theory.

For theories with a typing judgment, one simply judges the object, element, or term $a$ to be in the collection, set, or type $A$, () or $a:A$. This is in contrast to membership relations, where () is a proposition with some truth value, and thus one could say whether the proposition that the object, element, or term $a$ is in collection, set, or type $A$ is true or not.

Formally, there are also a difference in the contexts. When () or $a:A$ is a judgment, then it lies in the type layer, but when () is a true proposition, then it lies in the proposition layer $\Phi$ while the typing judgment $a:$Set lies in the type layer.

### Equality and equivalence

In most foundations such as [[dependent type theory]] and [[logic over type theory]], one distinguishes various different notions of _[[equality]]_ or _[[equivalence]]_ of the terms of the language.

There are broadly three different notions of equality which could be distinguished: **[[judgmental equality]]**, **[[propositional equality]]**, and **[[typal equality]]**, each of which applies for both objects (elements, terms) and collections (sets, classes, types). Judgmental equality is defined as a basic judgment in the foundations. Propositional equality is defined as a proposition in any [[typed predicate logic]]. Typal equality is defined as a type in type theory, and is also called the [[identity type]] or [[path type]] for elements, and the [[type of equivalences]] for types.

The equality of elements used in most [[set theories]] and [[class theories]] are [[propositional equalities]]. The equality of elements used in [[dependent type theories]] are [[typal equality]]. If the dependent type theory is not an [[objective type theory]], most likely it also either has one of [[judgmental equality]] or [[propositional equality]] to use for [[definitions]].

The equality of collections differs from theory to theory. There are [[set theories]] and [[class theories]] where [[propositional equality]] is used as the notion of [[equality]] between sets and classes, while for other set theories and class theories, the notion of equality between sets and classes is rather [[bijection]] of sets and classes, rather than equality of sets and classes. Something similar happens in dependent type theory, depending on whether one uses a separate type judgment, or [[Russell universes]] without a separate type judgment: for Russell universes, the [[identity type]] is used for the equality of types in the universe, since types are terms of Russell universes. On the other hand, the dependent type theories with a separate type judgment, there is no identity type between two types, and one has to use the [[type of equivalences]] instead.

A mathematical foundation is said to respect the [[principle of equivalence]] if there is no strict [[propositional equality]] between sets. Thus, any [[unsorted set theory]] like [[fully formal ETCS]] violates the [[principle of equivalence]].

### Size issues, universes, and classes

In most foundations of mathematics, there is no way to talk about the collection of all collections; this is known as [[size issues]] in [[mathematics]]. However, there is a way around it, by the usage of [[universes]]. Essentially, instead of trying to talk about the collection of all sets, one instead talks about the collection $U$, whose elements are called **$U$-[[small sets]]** or **$U$-small types**, and whose [[subsets]] or [[subtypes]] are called **$U$-[[classes]]**.
The objects not in $U$ are called **$U$-[[large sets]]** or **$U$-large types**, and the [[subsets]] or [[subtypes]] which are not [[singleton subsets]] or subtypes are called **$U$-[[proper classes]]**. Each foundations of mathematics has their own approach to [[universes]]:

* In [[set theory]], one usually either uses [[Grothendieck universes]], or some internal model of [[set theory]] or [[class theory]], whether [[well-pointed category|well-pointed]] [[Heyting categories]], [[well-pointed category|well-pointed]] [[division allegories]], [[sets]] with [[extensional relation|extensional]] [[well-founded relation|well-founded]] [[transitive relation|transitive]] relations, [[categories with class structure]], et cetera.

* In [[class theory]], the notion of universe is already in the theory via the [[universal class]], which is by definition a [[class]] [[universe]]. The basic primitive in class theory are [[classes]], and the $U$-small classes are called [[sets]].

* In [[dependent type theory]], one uses [[univalent universe|univalent]] [[Russell universes]] or [[Tarski universes]], which are basically internal models of [[dependent type theory]].

## Effect of foundations on concrete problems
{#Concrete}

It may seem on first sight that foundational questions in mathematics are remote from "real mathematics". This is not quite so. For a list of "real world" problems that do depend on foundations see

* [[effects of foundations on "real" mathematics]]

## Related concepts

* [[New Foundations]]

* [[univalent foundations for mathematics]]

* [[synthetic mathematics]]

* [[category theory and foundations]]

\section{Philosophical musings}

* [[Gravity and grace]], Simone Weil.

## References

* [[Bertrand Russell]], _[[Principia Mathematica]]_, 1910

* {#Lawvere} [[William Lawvere]], _Foundations and applications: axiomatization and education_, Bulletin of Symbolic Logic 9 (2003), 213-224 ([ps](http://www.math.ucla.edu/~asl/bsl/0902/0902-006.ps), [Euclid](http://projecteuclid.org/euclid.bsl/1052669290))

* {#Harrington} L. A. Harrington (ed.), _Harvey Friedman's Research on the Foundations of Mathematics_, Studies in Logic and the Foundations of Mathematics (2012)

* [[Stefania Centrone]], [[Deborah Kant]], [[Deniz Sarikaya]] (editors), *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[doi:10.1007/978-3-030-15655-8](https://doi.org/10.1007/978-3-030-15655-8)]

[[practical foundations|Practical foundations]] in terms of [[type theory]] language are laid out in

* [[Paul Taylor]], _[[Practical Foundations of Mathematics]]_, Cambridge University Press ([web](http://www.paultaylor.eu/~pt/prafm/index.html))
{#Taylor}

Under [[computational trinitarianism]] this corresponds to a practical foundation in [[program|programming]], laid out in

*  [[Robert Harper]], _[[Practical Foundations for Programming Languages]]_ ([web](https://www.cs.cmu.edu/~rwh/pfpl/))
{#Harper}

A classification of [[material set theories]] and [[structural set theories]] is in

* [[Michael Shulman]] (2018). Comparing material and structural set theories. [arXiv:1808.05204](https://arxiv.org/abs/1808.05204).
{#Shulman2018}

A foundation for [[algebraic topology]] in this practical spirit is laid out in

* [[Samuel Eilenberg]], [[Norman Steenrod]], _[[Foundations of Algebraic Topology]]_
  {#EilenbergSteenrod}

A big picture intro to the comparison between set theory, type theory and topos/category theory as approaches to foundations is in

* [[Steve Awodey]], _From Sets to Types to Categories to Sets_ [pdf](http://www.andrew.cmu.edu/user/awodey/preprints/stcsFinal.pdf)

A comparative discussion of complexities of different foundations is in

* [[Freek Wiedijk]], _Is ZF a hack? Comparing the complexity of some (formalist interpretations of) foundational systems for mathematics_ ([pdf](http://www.cs.ru.nl/~freek/zfc-etc/zfc-etc.pdf))

  **Abstract** This paper presents [[Automath]] encodings (which also are
valid in LF/P) of various kinds of foundations of mathematics. Then it compares these encodings according to their size, to find out which foundation is the simplest.

   The systems analyzed in this way are two kinds of [[set theory]] (ZFC and NF), two systems based on Church's [[higher order logic]] ([[Isabelle]]/Pure and [[HOL]]), three kinds of [[type theory]] (the [[calculus of constructions]], Luo's extended calculus of constructions, and Martin-Löf predicative type theory) and one [[ETCS|foundation based on category theory]]. The conclusions of this paper are that the simplest system is type theory (the calculus of constructions) but that type theories that know about serious mathematics are not simple at all. Set theory is one of the simpler systems too. Higher order logic is the simplest if one looks at the number of concepts (twenty-five) needed to explain the system. On the other side of the scale, category theory is relatively complex, as is Martin-Löf's type theory.

* [[Colin McLarty]], _Set theory for Grothendieck's number theory_, [pdf](http://www.cwru.edu/artsci/phil/Groth%20found.pdf)

[[!redirects mathematical foundation]]
[[!redirects mathematical foundations]]

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foundational axiom

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## Idea

A  _foundational axiom_ is an [[axiom]] that would be used in [[foundations]].

## Examples

[[!include foundational axiom - contents]]

[[!redirects foundational axiom]]
[[!redirects foundational axioms]]


foundational axiom - contents

**[[foundational axiom|foundational]] [[axioms]]**

* basic constructions:
  * [[axiom of cartesian products]]
  * [[axiom of disjoint unions]]
  * [[axiom of the empty set]]
  * [[axiom of fullness]]
  * [[axiom of function sets]]
  * [[axiom of power sets]]
  * [[axiom of quotient sets]]

* [[material set theory|material axioms]]:
  * [[axiom of extensionality]]
  * [[axiom of foundation]]
  * [[axiom of anti-foundation]]
  * [[Mostowski's axiom]]
  * [[axiom of pairing]]
  * [[axiom of transitive closure]]
  * [[axiom of union]]

* [[structural set theory|structural axioms]]:
  * [[axiom of materialization]]

* [[type theory|type theoretic axioms]]:
  * [[axioms of set truncation]]
    * [[uniqueness of identity proofs]]
    * [[axiom K]]
    * [[boundary separation]]
    * [[equality reflection]]
    * [[axiom of circle type localization]]
  * [[homotopy type theory|homotopy type theoretic axioms]]:
    * [[univalence axiom]]
    * [[Whitehead's principle]]

* [[axioms of choice]]:
  * [[axiom of choice]]
  * [[axiom of countable choice]]
  * [[axiom of dependent choice]]
  * [[axiom of excluded middle]]
  * [[axiom of existence]]
  * [[axiom of multiple choice]]
  * [[Markov's axiom]]
  * [[presentation axiom]]
  * [[small cardinality selection axiom]]
  * [[axiom of small violations of choice]]
  * [[axiom of weakly initial sets of covers]]

* [[large cardinal axioms]]:
  * [[axiom of infinity]]
  * [[axiom of universes]]
  * [[regular extension axiom]]
  * [[inaccessible cardinal]]
  * [[measurable cardinal]]
  * [[elementary embedding]]
  * [[supercompact cardinal]]
  * [[Vopěnka's principle]]

* strong axioms
  * [[axiom of separation]]
  * [[axiom of replacement]]

* further
  * [[reflection principle]]

* [[axiom of inequality spaces]]


foundations - contents

**[[foundations]]**

## The basis of it all

* [[mathematical logic]]

* [[deduction system]], [[natural deduction]], [[sequent calculus]], [[lambda-calculus]], [[judgment]]

* [[type theory]], [[simple type theory]], [[dependent type theory]]

* [[collection]], [[object]], [[type]], [[term]], [[set]], [[element]]

* [[equality]], [[judgmental equality]], [[typal equality]]

* [[universe]], [[size issues]]

* [[higher-order logic]]

## Set theory

[[!include set theory - contents]]

## Foundational axioms

[[!include foundational axiom - contents]]

## Removing axioms

* [[constructive mathematics]]
* [[predicative mathematics]]

[Edit this sidebar](/nlab/edit/foundations+-+contents)


Foundations > history

< [[Foundations]]
[[!redirects Foundations -- history]]


foundations and logic

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# Logic and foundations
* table of contents
{: toc}

## Main articles

* [[foundations|Foundations]]
* [[logic|Logic]]
* [[philosophy|Philosophy]]
* [[epistemology|Epistemology]]

## Axiomatic set theory

### Basic axioms and notions

* [[apartness relation]]
* [[axiom of choice]], [[axiom of extensionality]], [[axiom of foundation]], [[axiom of infinity]], [[axiom of multiple choice]], [[axiom of pairing]], [[axiom of replacement]], [[axiom of union]]
* [[COSHEP]]
* [[ETCS]]
* Trimble on ETCS: [[Trimble on ETCS I|I]], [[Trimble on ETCS II|II]],
[[Trimble on ETCS III|III]]
* [[Grothendieck universe]]
* [[internal set]]
* [[material set theory]]
* [[inhabited set]]
* [[pure set]]
* [[set theory]]

### Model theory

* [[model theory]]
* [[Mostowski's collapsing lemma]]
* [[nonstandard analysis]]
* [[forcing]]
* [[Skolem's paradox]]
* [[Vopěnka's principle]]

## Other

* [[Boolean algebra]]
* [[constructive mathematics]]
* [[equality]]
* [[evil]]
* [[Heyting algebra]]
* [[infinity-topos]]
* [[internal logic]]
* [[intuitionism]]
* [[linear logic]]
* [[predicative mathematics]]
* [[theory]]
* [[truth value]]
* [[type theory]]

[[!redirects foundations and logic]]
[[!redirects logic and foundations]]


foundations and philosophy

The boundary between [[philosophy|philosophical]] and [[logic|logical]] foundational studies of mathematics is very unclear indeed -- one has only to think of Russell's part in _Principia Mathematica_. For a discussion of more obviously logical foundational aspects of category theory see [[foundations]].

To be done

## Foundations and the philosophy of language in mathematics

Different foundations of mathematics provide different languages to use for presenting concepts in mathematics, and the symbols and words used in various foundations have different [[meaning|meanings]] from each other.

We work in a [[deductive system]] with hypothetical [[judgments]], and give examples of how different foundations of mathematics written in [[natural deduction]] result in different meanings of symbols used in the judgments used in each foundation.

For example, take the symbol $\in$ denoting membership in [[set theory]] and [[set-level type theory|set-level]] [[dependent type theories]]. In [[unsorted set theory]] and [[simply sorted set theory]], $\in$ is a proposition in the context of an element variable and a set variable. The formal judgment $a \in A \; \mathrm{prop}$ means that "'the element $a$ is in the set $A$' is a proposition", and the judgment $a \in A \; \mathrm{true}$ is commonly abbreviated as $a \in A$ and means that "the proposition that the element $a$ is in the set $A$ is true". In [[dependently sorted set theory]], and in [[layered type theory|unlayered]] [[set-level type theory|set-level]] [[dependent type theories]], $\in$ is a metatheoretic [[judgment]], rather than being part of the actual structure of the theory. The formal judgment $a \in A$ simply means that "$a$ is in the set $A$", rather than "the proposition that the element $a$ is in the set $A$ is true"; there is no propositional truth value attached to $a \in A$. Thus, the foundational notion of membership in unsorted and simply sorted set theories is semantically different from the notion of membership in dependently sorted set theories and unlayered set-level dependent type theory.

Similarly, the symbol $=$ denoting [[equality]] have different meanings in different theories.

## Philosophical papers

[[Jean-Pierre Marquis]], 1995, Category Theory and the Foundations of Mathematics: Philosophical Excavations. Synthese 103 (3).

[[Jean-Pierre Marquis]], 2008, [From a Geometrical Point of View](http://books.google.co.uk/books?id=bvy0aANuhPYC)

[[Steve Awodey]], 2004, [An Answer to Hellman's Question: "Does Category Theory Provide a Foundation for Mathematical Structuralism?"](http://www.andrew.cmu.edu/user/awodey/preprints/awodeyVhellman.pdf)

[[Ralf Kromer|Ralph Krömer]], 2007, [Tool and Object: A History and Philosophy of Category Theory](http://books.google.co.uk/books?id=nVVsp3_Jw0sC)

[[Colin McLarty]], 2005, [Learning from Questions on Categorical Foundations](http://www.cwru.edu/artsci/phil/PMLearning.pdf)

[[Colin McLarty]], 2004, 'Exploring Categorical Structuralism', Philosophia Mathematica, 12, 37–53.


Foundations of Algebraic Topology

A book published in 1952 by [[Samuel Eilenberg]] and [[Norman Steenrod]], using [[category theory]] [[foundations]] extensively to describe [[homology]] and [[cohomology]] theory.

category: reference


Foundations of Constructive Analysis

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

This pages compiles material related to the book

* [[Errett Bishop]],

  **Foundations of Constructive Analysis**

  McGraw-Hill (1967)

  > [p. ix:] This book is a piece of constructivist propaganda, designed to show that there does exist a satisfactory algebrative [to [[classical mathematics]]]. To this end we develop a large portion of abstract analysis within a constructive framework.

revised as

* [[Errett Bishop]], [[Douglas Bridges]]:

  **Constructive analysis**

  Grundlehren der mathematischen Wissenschaften **279**,

  Springer (1985)

  [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

on the foundations of [[constructive mathematics]] in general (cf. *[[Bishop's constructive mathematics]]*) and on [[constructive analysis]] in particular.

category: reference

[[!redirects Constructive Analysis]]


Foundations of Mechanics

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

This entry is to record the references

* [[Ralph Abraham]], [[Jerrold Marsden]], _Foundations of Mechanics_  Addison-Wesley (1978) ([pdf](http://authors.library.caltech.edu/25029/1/FoM2.pdf))

on [[classical mechanics]].

category: reference


four lemma

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Diagram chasing lemmas
+-- {: .hide}
[[!include diagram chasing lemmas - contents]]
=--
#### Homological algebra
+-- {: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _four lemma_ is one of the basic [[diagram chasing]] lemmas in [[homological algebra]].
It follows directly from the [[salamander lemma]]. It directly implies the [[five lemma]].

## Statement

Let $\mathcal{A}$ be an [[abelian category]].

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

Consider a [[commuting diagram]] in $\mathcal{A}$ of the form

$$
  \array{
    &\to& &\stackrel{\xi}{\to}& &\to&
    \\
    \downarrow^{\mathrlap{\tau}}
    &&
    \downarrow^{\mathrlap{f}}
    &&
    \downarrow^{\mathrlap{g}}
    &&
    \downarrow^{\mathrlap{\nu}}
    \\
    &\to& &\stackrel{\eta}{\to}& &\to&
  }
$$

where

1. the rows are [[exact sequences]],

1. $\tau$ is an [[epimorphism]],

1. $\nu$ is a [[monomorphism]].

Then

1. $\xi(ker(f)) = ker(g)$ (the [[image]] under $\xi$ of the [[kernel]] of $f$ is the [[kernel]] of $g$)

1. $im(f) = \eta^{-1}(im(g))$ (the [[preimage]] under $\eta$ of the [[image]] of $g$ is the [[image]] of $f$)

(the "strong four lemma") and hence in particular

1. if $g$ is an [[epimorphism]] then so is $f$;

1. if $f$ is a [[monomorphism]] then so is $g$

(the "weak four lemma").

=--

A direct proof from the [[salamander lemma]] is spelled out at _[salamander lemma -- implications -- four lemma](salamander+lemma#FourLemma)_.

## References

The strong/weak four lemma appears as lemma 3.2, 3.3 in chapter I and then with proof in lemma 3.1 of chapter XII of

* [[Saunders MacLane]], _Homology_ (1967) reprinted as Classics in Mathematics, Springer (1995)
 {#MacLane}

[[!redirects 4 lemma]]


four-colour theorem

#Contents#
* table of contents
{:toc}

## Idea

The **four-colour theorem**, or **four colour map theorem**, states that given any separation of a [[plane]] into contiguous regions, called a _map_, the regions can be coloured using at most four colours so that no two adjacent regions have the same colour. Regions are considered adjacent if they share a boundary segment.

## Related concepts

* [[Kepler conjecture]]

## References

See also

* Wikipedia, _[Four color theorem](https://en.wikipedia.org/wiki/Four_color_theorem)_

On the [[logical equivalence]] between the [[four-colour theorem]] and a statement about the transition from the [[small N limit]] to the [[large N limit]] for [[Lie algebra weight systems]] on [[Jacobi diagrams]] via the [['t Hooft double line construction]]:

* [[Dror Bar-Natan]], _Lie Algebras and the Four Color Theorem_, Combinatorica 17-1(1997) 43–52  ([arXiv:q-alg/9606016](https://arxiv.org/abs/q-alg/9606016), [doi:10.1007/BF01196130](https://doi.org/10.1007/BF01196130))

On a reformulation of the four-colour theorem as a statement about typing in lambda calculus:

* [[Noam Zeilberger]], _Linear lambda terms as invariants of rooted trivalent maps_ ([arXiv:1512.06751](https://arxiv.org/abs/1512.06751)).

* [[Noam Zeilberger]], _A theory of linear typings as flows on 3-valent graphs_ ([arXiv:1804.10540](https://arxiv.org/abs/1804.10540)).

[[!redirects four-color theorem]]

[[!redirects four colour theorem]]
[[!redirects four color theorem]]

[[!redirects four colour map theorem]]
[[!redirects four color map theorem]]
[[!redirects four-colour map theorem]]
[[!redirects four-color map theorem]]


Fourier integral operator

## Related concepts

* [[Fourier transform]]

* [[pseudodifferential operator]]

* [[Schwartz kernel]]

* [[Lagrangian distribution]]

## References

* Lars Hoermander, Linear differential operators
* Duistermaat

* Alejandro Uribe, _An outline of a crash course on Fourier integral operators_ ([pdf](httP://www.ms.uky.edu/~perry/CBMS/Day2-Uribe.pdf))

* Steve Zelditch, _Lagrangian distributions and Fourier integral
operators, I_, Lecture notes 2011 ([pdf](http://www.ms.uky.edu/~perry/CBMS/LagFIOSI.pdf))

* Jean-Marie Lescure (LMBP), Stéphane Vassout, _Fourier integrals operators on lie groupoids_, [arxiv/1601.00932](http://arxiv.org/abs/1601.00932)

[[!redirects Fourier integral operators]]


Fourier inversion theorem

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

#Contents#
* table of contents
{:toc}

## Idea

In [[harmonic analysis]], the _Fourier inversion theorem_ states that the [[Fourier transform]] is an [[isomorphism]] on the [[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]]. Moreover, it is its own [[inverse]] up to a prefactor and reflection at the origin.

For Fourier transform over [[Cartesian spaces]], see e.g. [Hörmander 90 ,theorem 7.1.5, theorem 7.1.10](#Hoermander90), [this prop.](Fourier+transform#FourierInversion).

## Statement

Let $n \in \mathbb{N}$ and consider $\mathbb{R}^n$ the [[Cartesian space]] of [[dimension]] $n$.

+-- {: .num_defn #FourierInversion}
###### Proposition
**([[Fourier inversion theorem]])**

The [[Fourier transform]] $\widehat{(-)}$ (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}) on the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) is an [[isomorphism]], with [[inverse function]] the _inverse Fourier transform_

$$
  \widecheck {(-)}
     \;\colon\;
  \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathcal{S}(\mathcal{R}^n)
$$

given by

$$
  \widecheck g (x)
    \;\coloneqq\;
   \underset{k \in \mathbb{R}^n}{\int}
      g(k) e^{2 \pi i k \cdot x}
   \, \frac{d^n k}{(2\pi)^n}
   \,.
$$

Hence in the language of [[harmonic analysis]] the function $\widecheck g \colon \mathbb{R}^n \to \mathbb{C}$ is the [[superposition]] of [[plane waves]] in which the plane wave with [[wave vector]] $k\in \mathbb{R}^n$
appears with [[amplitude]] $g(k)$.

=--

(e.g. [Hörmander, theorem 7.1.5](#Hoermander90))

## Related theorems

* [[Parseval's theorem]]

* [[Paley-Wiener-Schwartz theorem]]

## References

* {#Hoermander90} [[Lars Hörmander]], theorem 7.1.5 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

[[!redirects inverse Fourier transform]]
[[!redirects inverse Fourier transforms]]


Fourier transform

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

### Basic idea

Generally, a _Fourier transform_ is an [[isomorphism]] between the algebra of [[complex numbers|complex]]-valued [[functions]] on a suitable [[topological group]] and a [[convolution product]]-algebra structure on the [[Pontrjagin dual]] group.
The study of Fourier transforms is also called _Fourier analysis_.

Typically, such as in the case over [[Cartesian space]] (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace} below) this means to decompose
any suitable [[function]] as a [[superposition]] of _[[complex number|complex]] [[plane waves]]_, which may be thought of as the
"harmonics" of the given function. Therefore one speaks of _[[harmonic analysis]]_.

### Generalizations

The concept of Fourier transforms of functions generalizes in a variety of ways. Core part of the subject of Fourier analysis
is the generalization to _[[Fourier transform of distributions]]_ (def. \ref{FourierTransformOnTemperedDistributions} below).
The asymptotic growth of the [[Fourier transform of distributions]] reflects the singularity structure of the distributions,
in dependence of the [[direction of a vector|direction]] of the [[wave vector]] (the "[[wave front set]]"). The study of this behaviour is called _[[microlocal analysis]]_.

If the role of complex [[plane waves]] in the Fourier transform are replaced by [[wavelets]], one speaks of the _[[wavelet transform]]_.

For [[noncommutative topology|noncommutative topological]] groups, instead of continuous characters one should consider [[irreducible representation|irreducible]]  [[unitary representations]], which makes the subject much more difficult. There are also generalizations in [[noncommutative geometry]], see _[[quantum group Fourier transform]]_.

## General definition

Let $G$ be a [[locally compact space|locally compact]] [[Hausdorff topological space|Hausdorff]] [[abelian group|abelian]] [[topological group]] with [[Haar measure|invariant (= Haar) measure]] $\mu$. Then for each $f\in L_1(G,\mu)$, define its __Fourier transform__ $\hat{f}$ as a function on its [[Pontrjagin dual]] group $\hat{G}$ given by

$$
\hat{f}(\chi) = \int_G f(x) \widebar{\chi(x)} d\mu(x),\,\,\,\chi\in\hat{G}.
$$

The Fourier transform of $f\in L_1(G,\mu)$ is always continuous and bounded on $\hat{G}$; the transform of the [[convolution]] of two functions is the product of the transforms of each of the functions separately.

## Over the circle and the integers

In the classical case of __Fourier series__, where $G=\mathbb{Z}$ (the additive group of [[integers]]) and $\hat{G}=S^1$ (the [[circle group]]), the Fourier transform restricts to a unitary operator between the [[Hilbert spaces]] $L_2(S^1,d t)$ and $l_2(\mathbb{Z})$ and the Fourier coefficients are the numbers

$$c_n := \hat{f}(\chi_n) = \int_0^1 f(t) e^{-2\pi i n t} d t,$$

for $n\in\mathbb{Z}$, where the functions $\chi_n(t)= e^{2\pi i n t}$ form an orthonormal basis of $L_2(S^1,d t)$. The Fourier transform $\hat{\chi_n}$ is then viewed as the  $\mathbb{Z}$-series $\delta_n$ which in the $n$-th place has $1$ and elsewhere $0$. The Fourier transform replaces the operator of differentiation $d/d t$ by the operator of multiplication by the series $\{2\pi i n\}_{n\in\mathbb{Z}}$.

## Over compact abelian groups and discrete groups

In general, if $G$ is a [[compact space|compact]] abelian group (whose [[Pontrjagin dual]] is [[discrete group|discrete]]), one can normalize the invariant measure by $\mu(G)=1$ and $\hat{\mu}(X)=card(X)$ for $X\subset\hat{G}$. Then the Fourier transform restricts to a unitary operator from $L_2(X,\mu)$ to $L_2(\hat{G},\hat{\mu})$.

## Over cyclic groups (the discretized circle)

* [[discrete Fourier transform]]

## Over Cartesian spaces
 {#FourierTransformOnCartesianSpaces}

Throughout, let $n \in \mathbb{N}$ and write $\mathbb{R}^n$ for the [[Cartesian space]] of [[dimension]] $n$ and write $(-) \cdot (-)$ for the canonical [[inner product]] on $\mathbb{R}^n$:

$$
  k \cdot x \;\coloneqq\; \underoverset{a = 1}{n}{\sum} k_n x^n
  \,.
$$

In the following by a [[smooth function]] $f \in C^\infty(\mathbb{R}^n)$ on $\mathbb{R}^n$ we mean a smooth function with values in the [[complex numbers]].

For $f \in C^\infty(\mathbb{R}^n)$, we write $f^\ast \in C^\infty(\mathbb{R}^n)$ for its pointwise [[complex conjugate]]:

$$
  f^\ast(x) \coloneqq (f(x))^\ast
  \,.
$$

#### On functions with rapidly decreasing partial derivatives

+-- {: .num_defn #SchwartzSpace}
###### Definition
**([[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]])**

A [[complex number|complex]]-valued [[smooth function]] $f \in C^\infty(\mathbb{R}^n)$ is said to have _[[rapidly decreasing function|rapidly decreasing]] [[partial derivatives]]_ if for all $\alpha,\beta \in \mathbb{N}^{n}$ we have

$$
  \underset{x \in \mathbb{R}^n}{sup} {\vert  x^\beta \partial^\alpha f(x) \vert}
  \;\lt\; \infty
  \,.
$$

Write

$$
  \mathcal{S}(\mathbb{R}^n) \hookrightarrow C^\infty(\mathbb{R}^n)
$$

for the sub-[[vector space]] on the functions with rapidly decreasing partial derivatives regarded as a [[topological vector space]] for the [[Frechet space]] struzcture induced by the [[seminorms]]

$$
  p_{\alpha, \beta}(f) \coloneqq \underset{x \in \mathbb{R}^n}{sup} {\vert  x^\beta \partial^\alpha f(x) \vert}
  \,.
$$

This is also called the _[[Schwartz space]]_.

=--

(e.g. [Hörmander 90, def. 7.1.2](#Hoermander90))

+-- {: .num_example #CompactlySupportedSmoothFunctionsAreFunctionsWithRapidlyDecreasingDerivatives}
###### Example
**([[compactly supported function|compactly supported]] [[smooth function]] are [[functions with rapidly decreasing partial derivatives]])**

Every [[compactly supported function|compactly supported]] [[smooth function]] ([[bump function]]) $b \in C^\infty_{cp}(\mathbb{R}^n)$ has rapidly decreasing partial derivatives (def. \ref{SchwartzSpace}):

$$
  C^\infty_{cp}(\mathbb{R}^n)
   \hookrightarrow
  \mathcal{S}(\mathbb{R}^n)
  \,.
$$

=--

+-- {: .num_prop #ConvolutionProductOnSchwartzSpace}
###### Proposition
**(pointwise product and [[convolution product]] on [[Schwartz space]])

The [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) is closed under the following operations on smooth functions $f,g \in \mathcal{S}(\mathbb{R}^n) \hookrightarrow C^\infty(\mathbb{R}^n)$

1. pointwise product:

   $$
     (f \cdot g)(x) \coloneqq f(x) \cdot g(x)
   $$

1. [[convolution product]]:

   $$
     (f \star g)(x) \coloneqq \underset{y \in \mathbb{R}^n}{\int} f(y)\cdot g(x-y) \, dvol(y)
     \,.
   $$

=--

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

By the [[product law]] of [[differentiation]].

=--

+-- {: .num_prop #RapidlyDecreasingFunctionsAreIntegrable}
###### Proposition
**([[rapidly decreasing functions]] are [[integrable functions|integrable]])**

Every [[rapidly decreasing function]] $f \colon \mathbb{R}^n \to \mathbb{R}$ (def. \ref{SchwartzSpace}) is an [[integrable function]] in that its [[integral]] exists:

$$
  \underset{x \in \mathbb{R}^n}{\int} f(x) \, d^n x
  \;\lt\;
  \infty
$$

In fact for each $\alpha \in \mathbb{N}^n$ the product of $f$ with the $\alpha$-power of the [[coordinate functions]] exists:

$$
  \underset{x \in \mathbb{R}^n}{\int}
    x^\alpha f(x)\, d^n x
  \;\lt\;
   \infty
  \,.
$$

=--

+-- {: .num_defn #FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}
###### Definition
**([[Fourier transform]] of [[functions with rapidly decreasing partial derivatives]])**

The _[[Fourier transform]]_ is the [[continuous linear functional]]

$$
  \widehat{(-)}
    \;\colon\;
  \mathcal{S}(\mathbb{R}^n)
    \longrightarrow
  \mathcal{S}(\mathbb{R}^n)
$$

on the [[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpace}), which is given by [[integration]] against the [[exponential function|exponential]] [[plane wave]] functions

$$
  x \mapsto e^{- i k \cdot x}
$$

times the standard [[volume form]] $d^n x$:

$$
  \label{IntegralExpressionForFourierTransform}
  \hat f(k)
    \;\coloneqq\;
  \int_{x \in \mathbb{R}^n}
    e^{- i \, k \cdot x} f(x) \, d^n x
  \,.
$$

Here the argument $k \in \mathbb{R}^n$ of the Fourier transform is also called the _[[wave vector]]_.

=--

(e.g. [Hörmander, lemma 7.1.3](#Hoermander90))

+-- {: .num_defn #FourierInversion}
###### Proposition
**([[Fourier inversion theorem]])**

The [[Fourier transform]] $\widehat{(-)}$ (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}) on the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) is an [[isomorphism]], with [[inverse function]] the _[[inverse Fourier transform]]_

$$
  \widecheck {(-)}
     \;\colon\;
  \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathcal{S}(\mathcal{R}^n)
$$

given by

$$
  \widecheck g (x)
    \;\coloneqq\;
   \underset{k \in \mathbb{R}^n}{\int}
      g(k) e^{i k \cdot x}
   \, \frac{d^n k}{(2\pi)^n}
   \,.
$$

Hence in the language of [[harmonic analysis]] the function $\widecheck g \colon \mathbb{R}^n \to \mathbb{C}$ is the [[superposition]] of [[plane waves]] in which the plane wave with [[wave vector]] $k\in \mathbb{R}^n$
appears with [[amplitude]] $g(k)$.

=--

(e.g. [Hörmander, theorem 7.1.5](#Hoermander90))

+-- {: .num_prop #BasicPropertiesOfFourierTransformOverCartesianSpaces}
###### Proposition
**(basic properties of the [[Fourier transform]])**

The [[Fourier transform]] $\widehat{(-)}$ (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}) on the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) satisfies the following properties, for all $f,g \in \mathcal{S}(\mathbb{R}^n)$:

1. (interchanging [[coordinate]] [[multiplication]] with [[partial derivatives]])

   $$
     \label{FourierTransformInterchangesCoordinateProductWithDerivative}
     \widehat{ x^a f } =  + i \partial_a \widehat f
     \phantom{AAAAA}
     \widehat{ - i\partial_a f} = k_a \widehat f
   $$

1. (interchanging pointwise multiplication with [[convolution product]], remark \ref{ConvolutionProductOnSchwartzSpace}):

   $$
     \label{FourierTransformInterchangesPointwiseProductWithConvolution}
     \widehat {(f \star g)} = \widehat{f} \cdot \widehat{g}
      \phantom{AAAA}
      \widehat{ f \cdot g } = (2\pi)^{-n} \widehat{f} \star \widehat{g}
   $$

1. ([[unitary operator|unitarity]], [[Parseval's theorem]])

   $$
     \underset{x \in \mathbb{R}^n}{\int} f(x) g^\ast(x)\, d^n x
     \;=\;
     \underset{k \in \mathbb{R}^n}{\int} \widehat{f}(k) \widehat{g}^\ast(k) \, d^n k
   $$

1. $$
     \label{FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor}
     \underset{k \in \mathbb{R}^n}{\int} \widehat{f}(k) \cdot g(k) \, d^n k
     \;=\;
     \underset{x \in \mathbb{R}^n}{\int} f(x) \cdot \widehat{g}(x) \, d^n x
   $$

=--

(e.g [Hörmander 90, lemma 7.1.3, theorem 7.1.6](#Hoermander90))

#### On tempered distributions
 {#FourierTransformOnTemperedDistributionsOverCartesianSpace}

The [[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpace})
serves the purpose to support the [[Fourier transform]] (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace})
together with its inverse (prop. \ref{FourierInversion}), but for many applications
one needs to apply the Fourier transform to more general functions, and in fact to _[[generalized functions]]_
in the sense of [[distributions]] (via [this prop.](non-singular+distribution#NonSingularDistributionsAreDenseInAllDistributions)).
But with the [[Schwartz space]] in hand, this generalization is readily obtained by [[formal duality]]:

+-- {: .num_defn #TemperedDistribution}
###### Definition
**([[tempered distribution]])**

A _[[tempered distribution]]_ is a [[continuous linear functional]]

$$
  u \;\colon\; \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathbb{C}
$$

on the [[Schwartz space]] (def. \ref{SchwartzSpace}) of [[functions with rapidly decaying partial derivatives]]. The [[vector space]] of all tempered distributions is canonically a [[topological vector space]] as the [[dual space]] to the [[Schwartz space]], denoted

$$
  \mathcal{S}'(\mathbb{R}^n)
    \;\coloneqq\;
  \left(
     \mathcal{S}(\mathbb{R}^n)
  \right)^\ast
  \,.
$$

=--

e.g. ([Hörmander 90, def. 7.1.7](#Hoermander90))

+-- {: .num_example #SomeNonSingularTemperedDistributions}
###### Example
**(some [[non-singular distribution|non-singular]] [[tempered distributions]])**

Every [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) induces a [[tempered distribution]] $u_f \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) by [[integration|integrating]] against it:

$$
  u_f \;\colon\; g \mapsto \underset{x \in \mathbb{R}^n}{\int} g(x) f(x)\, d^n x
  \,.
$$

This construction is a linear inclusion

$$
  \mathcal{S}(\mathbb{R}^n) \overset{\text{dense}}{\hookrightarrow} \mathcal{S}'(\mathbb{R}^n)
$$

of the [[Schwartz space]] into its [[dual space]] of [[tempered distributions]]. This is a [[dense subspace]] inclusion.

In fact already the restriction of this inclusion to the [[compactly supported function|compactly supported]] [[smooth functions]] (example \ref{CompactlySupportedSmoothFunctionsAreFunctionsWithRapidlyDecreasingDerivatives}) is a [[dense subspace]] inclusion:

$$
  C^\infty_{cp}(\mathbb{R}^n)
    \overset{dense}{\hookrightarrow}
  \mathcal{S}'(\mathbb{R}^n)
  \,.
$$

This means that every [[tempered distribution]] is a [[limit of a sequence|limit]] of a [[sequence]]
of ordinary [[functions with rapidly decreasing partial derivatives]], and in fact even the [[limit of a sequence|limit]]
of a [[sequence]] of [[compactly supported function|compactly supported]] [[smooth functions]] ([[bump functions]]).

It is in this sense that [[tempered distributions]] are "generalized functions".

=--

(e.g. [Hörmander 90, lemma 7.1.8](#Hoermander90))

+-- {: .num_example #CompactlySupportedDistibutionsAreTemperedDistributions}
###### Example
**([[compactly supported distributions]] are [[tempered distributions]])**

Every [[compactly supported distribution]] is a [[tempered distribution]] (def. \ref{TemperedDistribution}),
hence there is a [[linear map|linear]] [[injection|inclusion]]

$$
  \mathcal{E}'(\mathbb{R}^n)
    \hookrightarrow
  \mathcal{S}'(\mathbb{R}^n)
  \,.
$$

=--

+-- {: .num_example #DiracDeltaDistribution}
###### Example
**([[delta distribution]])

Write

$$
 \delta_0(-) \;\in\; \mathcal{E}'(\mathbb{R}^n)
$$

for the [[distribution]] given by point evaluation of functions at the origin of $\mathbb{R}^n$:

$$
  \delta_0(-) \;\colon\; f \mapsto f(0)
  \,.
$$

This is clearly a [[compactly supported distribution]]; hence a [[tempered distribution]] by example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}.

We write just "$\delta(-)$" (without the subscript) for the corresponding [[generalized function]] (example \ref{SomeNonSingularTemperedDistributions}),  so that

$$
  \underset{x \in \mathbb{R}^n}{\int} \delta(x) f(x) \, d^n x
  \;\coloneqq\;
  f(0)
  \,.
$$

=--

+-- {: .num_example #SquareIntegrableFunctionsInduceTemperedDistributions}
###### Example
**([[square integrable functions]] induce [[tempered distributions]])**

Let $f \in L^p(\mathbb{R}^n)$ be a function in the $p$th [[Lebesgue space]], e.g. for $p = 2$ this means that $f$ is a [[square integrable function]]. Then the operation of [[integration]] against the [[measure]] $f dvol$

$$
  g \mapsto \underset{x \in \mathbb{R}^n}{\int} g(x) f(x) \, d^n x
$$

is a [[tempered distribution]] (def. \ref{TemperedDistribution}).

=--

(e.g. [Hörmander 90, below lemma 7.1.8](#Hoermander90))

Property (eq:FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor) of the ordinary [[Fourier transform]] on [[functions with rapidly decreasing partial derivatives]] motivates and justifies the fullowing generalization:

+-- {: .num_defn #FourierTransformOnTemperedDistributions}
###### Definition
**([[Fourier transform of distributions]] on [[tempered distributions]])**

The _[[Fourier transform of distributions]]_ of a [[tempered distribution]] $u \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) is the [[tempered distribution]] $\widehat u$ defined on a smooth function $f \in \mathcal{S}(\mathbb{R}^n)$ in the [[Schwartz space]] (def. \ref{SchwartzSpace}) by

$$
  \widehat{u}(f) \;\coloneqq\; u\left( \widehat f\right)
  \,,
$$

where on the right $\widehat f \in \mathcal{S}(\mathbb{R}^n)$ is the [[Fourier transform]] of functions from def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}.
=--

(e.g. [Hörmander 90, def. 1.7.9](#Hoermander90))

+-- {: .num_example #FourierTransformOfDistributionsIndeedGeneralizedOrdinaryFourierTransform}
###### Example
**([[Fourier transform of distributions]] indeed generalizes [[Fourier transform]] of [[functions with rapidly decreasing partial derivatives]])**

Let $u_f \in \mathcal{S}'(\mathbb{R}^n)$ be a [[non-singular distribution|non-singular]] [[tempered distribution]] induced, via example \ref{SomeNonSingularTemperedDistributions}, from a [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$.

Then its [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) is the [[non-singular distribution]] induced from the [[Fourier transform]] of $f$:

$$
  \widehat{u_f} \;=\; u_{\hat f}
  \,.
$$

=--

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

Let $g \in \mathcal{S}(\mathbb{R}^n)$. Then

$$
  \begin{aligned}
    \widehat{u_f}(g)
    & \coloneqq
    u_f\left( \widehat{g}\right)
    \\
    & =
    \underset{x \in \mathbb{R}^n}{\int} f(x) \hat g(x)\, d^n x
    \\
    & =
    \underset{x \in \mathbb{R}^n}{\int} \hat f(x) g(x) \, d^n x
    \\
    & =
    u_{\hat f}(g)
  \end{aligned}
$$

Here all equalities hold by definition, except for the third: this is property (eq:FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor) from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}.

=--

+-- {: .num_example #CompactlySupportedDistributionFourierTransform}
###### Example
**([[Fourier transform of distributions|Fourier transform]] of [[compactly supported distributions]])**

Under the identification of [[smooth functions]] of bounded growth with [[non-singular distributions|non-singular]] [[tempered distributions]] (example \ref{SomeNonSingularTemperedDistributions}), the [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) of a [[tempered distribution]] that happens to be [[compactly supported distribution|compactly supported]] (example \ref{CompactlySupportedDistibutionsAreTemperedDistributions})

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

is simply

$$
  \widehat{u}(k) = u\left( e^{-  i k \cdot (-)}\right)
  \,.
$$

=--

([Hörmander 90, theorem 7.1.14](#Hoermander90))

+-- {: .num_defn #FourierTransformOfDeltaDistribution}
###### Example
**([[Fourier transform of distributions|Fourier transform]] of the [[delta-distribution]])**

The [[Fourier transform of distributions|Fourier transform]] (def. \ref{FourierTransformOnTemperedDistributions}) of the [[delta distribution]] (def. \ref{DiracDeltaDistribution}), via example \ref{CompactlySupportedDistributionFourierTransform}, is the [[constant function]] on 1:

$$
  \begin{aligned}
    \widehat {\delta}(k)
    & =
    \underset{x \in \mathbb{R}^n}{\int} \delta(x) e^{- i k x} \, d x
    \\
    & =
    1
  \end{aligned}
$$

This implies by the [[Fourier inversion theorem]] (prop. \ref{FourierInversionTheoremForDistributions}) that the [[delta distribution]] itself has equivalently the following expression as a [[generalized function]]

$$
  \begin{aligned}
    \delta(x)
    & =
    \widecheck{\widehat {\delta_0}}(x)
    \\
    & =
    \underset{k \in \mathbb{R}^n}{\int} e^{i k \cdot x} \, \frac{d^n k}{ (2\pi)^n }
  \end{aligned}
$$

in the sense that for every [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) we have

$$
  \begin{aligned}
    f(x)
    & =
    \underset{y \in \mathbb{R}^n}{\int}
    f(y) \delta(y-x)  \, d^n y
    \\
    & =
    \underset{y \in \mathbb{R}^n}{\int}
    \underset{k \in \mathbb{R}^n}{\int}
    f(y) e^{i k \cdot (y-x)}
    \, \frac{d^n k}{(2\pi)^n} \, d^n y
    \\
    & =
    \underset{k \in \mathbb{R}^n}{\int}
    e^{- i k \cdot x}
    \underset{= \widehat{f}(-k) }{
    \underbrace{
      \underset{y \in \mathbb{R}^n}{\int}
      f(y) e^{i k \cdot y}
      \, d^n y
    }
    }
    \,\, \frac{d^n k}{(2\pi)^n}
    \\
    & =
    +
    \underset{k \in \mathbb{R}^n}{\int}
    e^{i k \cdot x}
    \underset{= \widehat{f}(k) }{
    \underbrace{
      \underset{y \in \mathbb{R}^n}{\int}
      f(y) e^{- i k \cdot y}
      \, d^n y
    }
    }
    \,\, \frac{d^n k}{(2\pi)^n}
    \\
    & =
    \widecheck{\widehat{f}}(x)
  \end{aligned}
$$

which is the statement of the [[Fourier inversion theorem]] for smooth functions (prop. \ref{FourierInversion}).

(Here in the last step we used [[change of integration variables]] $k \mapsto -k$ which introduces one sign $(-1)^{n}$ for the new volume form,
but another sign $(-1)^n$ from the re-[[orientation]] of the integration domain. )

Equivalently, the above computation shows that the [[delta distribution]] is the [[neutral element]] for the
[[convolution product of distributions]].

=--

+-- {: .num_prop #PaleyWienerSchwartzTheorem}
###### Proposition
**([[Paley-Wiener-Schwartz theorem]])**

Let $u \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$ be a [[compactly supported distribution]] regarded as a [[tempered distribution]] by example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}. Then its [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) is a [[non-singular distribution]] induced from a [[smooth function]] that grows at most exponentially.

=--

+-- {: .num_prop #FourierInversionTheoremForDistributions}
###### Proposition
**([[Fourier inversion theorem]] for [[Fourier transform of distributions]])**

The operation of forming the [[Fourier transform of distributions]] $\widehat{u}$ (def. \ref{FourierTransformOnTemperedDistributions}) [[tempered distributions]] $u \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) is an [[isomorphism]],
with [[inverse]] given by

$$
  \widecheck{ u } \;\colon\; g \mapsto u\left( \widecheck{g}\right)
  \,,
$$

where on the right $\widecheck{g}$ is the ordinary [[inverse Fourier transform]] of $g$ according to prop. \ref{FourierInversion}.

=--

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

By def. \ref{FourierTransformOnTemperedDistributions}
this follows immediately from the [[Fourier inversion theorem]] for smooth functions (prop. \ref{FourierInversion}).

=--

We have the following distributional generalization of the basic property (eq:FourierTransformInterchangesPointwiseProductWithConvolution) from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}:

+-- {: .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]] (def. \ref{TemperedDistribution}) and

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

be a [[compactly supported distribution]], regarded as a [[tempered distribution]] via example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}.

Observe here that the [[Paley-Wiener-Schwartz theorem]] (prop. \ref{PaleyWienerSchwartzTheorem}) 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]] 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))

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

Prop. \ref{FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct} together with the [[Fourier inversion theorem]] (prop. \ref{FourierInversionTheoremForDistributions}) 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

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

which would complete the generalization of of property (eq:FourierTransformInterchangesPointwiseProductWithConvolution) from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}.

For this to make sense, the [[convolution product]] of the [[smooth functions]] on the right needs to exist, which is not guaranteed (prop. \ref{ConvolutionProductOnSchwartzSpace} does not apply here!). 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.

=--

## Related concepts

* [[harmonic]], [[fundamental frequency]]

* [[superposition]]

* [[Fourier integral operator]]

* [[Fourier transform of distributions]]

* [[pseudodifferential operator]]

* [[Poisson summation formula]]

* [[Laplace transform]], [[Fourier-Laplace transforms]]

* [[Mellin transform]]

* [[wavefront set]]

* [[wavelet transform]]

* [[Bochner's theorem]]

## References

Textbook account in [[mathematical physics]]:

* [[Michael Reed]], [[Barry Simon]]: *[[Methods of Modern Mathematical Physics]]*, Academic Press (1978), Volume II: *Fourier Analysis, Self-Adjointness* $[$[ISBN:9780125850025](https://www.elsevier.com/books/ii-fourier-analysis-self-adjointness/reed/978-0-08-092537-0)$]$

Lecture notes:

* [[Howard Georgi]], *The Physics of Waves*, Prentice Hall (1993) $[$[web](https://www.people.fas.harvard.edu/~hgeorgi/new.htm), [pdf](https://www.people.fas.harvard.edu/~hgeorgi/onenew.pdf)$]$

* {#Peacock13} [[John Peacock]], _Fourier analysis_ 2013 ([part 1 pdf](http://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part1.pdf), [part 2 pdf](http://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part2.pdf), [part 3 pdf](http://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part3.pdf), [part 4 pdf](http://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part4.pdf), [part 5 pdf](http://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part5.pdf))

* Gerald B. Folland, _A course in abstract harmonic analysis_, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. x+276 pp. [gBooks](http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC)

Discussion in the broader context of [[functional analysis]] and [[distribution]] theory:

* {#Hoermander90} [[Lars Hörmander]], chapter 7 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

* [[Sergiu Klainerman]], chapter 5 of of _Lecture notes in analysis_, 2011 ([pdf](https://web.math.princeton.edu/~seri/homepage/courses/Analysis2011.pdf))

category: analysis

[[!redirects Fourier transforms]]

[[!redirects Fourier series]]

[[!redirects Fourier integral]]
[[!redirects Fourier integrals]]

[[!redirects Fourier analysis]]

[[!redirects Fourier mode]]
[[!redirects Fourier modes]]

[[!redirects Fourier expansion]]
[[!redirects Fourier expansions]]

[[!redirects Fourier transformation]]
[[!redirects Fourier transformations]]


Fourier transform of distributions

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

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the concept of [[Fourier transform]] from suitable function to [[distributions]].

## Definition

+-- {: .num_defn #FourierTransformOfTemperedDistribution}
###### Definition
**(Fourier transform of [[tempered distributions]])**

For $n \in \mathbb{N}$, let $u \in \mathcal{S}'(\mathbb{R}^n)$ be a [[tempered distribution]]. Then its _Fourier transform_ $\hat u \in \mathcal{S}'(\mathbb{R}^n)$ is defined by

$$
  \array{
    \mathcal{S}(\mathbb{R}^n)
      &\overset{\hat u}{\longrightarrow}&
    \mathbb{C}
    \\
    f &\mapsto& \langle u, \hat f\rangle
  }
  \,,
$$

where on the right $\hat f$ is the ordinary [[Fourier transform]] of the function $f$ (an element of the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$).

=--

(e.g. [Hörmander 90, def. 7.1.9](#Hoermander90))

## Properties

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

The operation of Fourier transform of [[tempered distributions]] (def. \ref{FourierTransformOfTemperedDistribution}) induces a [[linear isomorphism]] of the space of temptered distributions with itself:

$$
  \widehat{(-)}
  \;\colon\;
  \mathcal{S}'(\mathbb{R}^n)
   \overset{\simeq}{\longrightarrow}
  \mathcal{S}'(\mathbb{R}^n)
$$

=--

(e.g. [Melrose 03, corollary 1.1](#Melrose03))

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

If $u \in \mathcal{D}'(\mathbb{R}^n) \hookrightarrow \mathcal{E}'(\mathbb{R}^n)$ happens to be a [[compactly supported distribution]], regarded as a [[tempered distribution]], then its Fourier transform according to def. \ref{FourierTransformOfTemperedDistribution} is a smooth function

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

given by

$$
  \hat u(k)
    \;\coloneqq\;
   u\left( \exp(-i \langle (-), 2 \pi k \rangle) \right)
  \,,
$$

where $\langle -,-\rangle$ denotes the canonical [[inner product]] on $\mathbb{R}^n$.

This is well-defined also on [[complex numbers]], which makes it an [[entire holomorphic function]] (by the [[Paley-Wiener-Schwartz theorem]]), called the _[[Fourier-Laplace transform]]_.

=--

(e.g. [Hörmander 90, theorem 7.1.14](#Hoermander90))

(This plays a role for instance in the [[Paley-Wiener-Schwartz theorem]].)

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

The Fourier transform (def. \ref{FourierTransformOfTemperedDistribution}) of the [[convolution of distributions]] of a [[compactly supported distribution]] $u_1 \in \mathcal{E}'$ with a [[tempered distribution]] $u_2 \in \mathcal{S}'$ is the [[product of distributions]] of their separate Fourier transforms:

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

(Here, by prop. \ref{FourierTransformOfCompactlySupportedDistributions}, $\hat u_1$ is just a smooth function, so that the product on the right is just that of a distribution with a function.)

=--

([Hörmander 90, theorem 7.1.15](#Hoermander90))

## Related concepts

* [[pullback of distributions]]

* [[product of distributions]]

* [[composition of distributions]]

* [[extension of distributions]]

* [[Paley-Wiener-Schwartz theorem]]

## References

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

* {#Melrose03} [[Richard Melrose]], chapter 1 of _Introduction to microlocal analysis_, 2003 ([pdf](http://www-math.mit.edu/~rbm/iml90.pdf))

* [[Sergiu Klainerman]], chapter 5 of _Lecture notes in analysis_, 2011 ([pdf](https://web.math.princeton.edu/~seri/homepage/courses/Analysis2011.pdf))

[[!redirects Fourie transforms of distributions]]

[[!redirects Fourier transform of a distribution]]
[[!redirects Fourier transforms of a distribution]]

[[!redirects Fourier transformation of distributions]]
[[!redirects Fourier transformations of distributions]]


Fourier-Laplace transform

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

#Contents#
* table of contents
{:toc}

## Idea

For $f$ a suitable ([[generalized function|generalized]]) [[function]] on an [[affine space]], its [[Fourier transform]] is given by $\hat f (a) \propto \int f(x) e^{i x a} d x$, while its [[Laplace transform]] is $\tilde f(a) \propto \int f(x) e^{-a x} d x $, when defined. Clearly these are two special cases of a single "transform" where $a$ is allowed to be [[complex number|complex]]; this is hence called the _Fourier-Laplace transform_.

## Definition

+-- {: .num_defn #FourierTransformOfCompactlySupportedDistribution}
###### Proposition/Definition
**(Fourier-Laplace transform of [[compactly supported distributions]])**

For $n \in \mathbb{N}$, let $u \in \mathcal{E}'(\mathbb{R}^n)$ be a compactly supported distribution on [[Cartesian space]] $\mathbb{R}^n$. Then its _[[Fourier transform of distributions]]_ is the function

$$
  \array{
    \mathbb{R}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R}
    \\
    \zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right)
  }
$$

where on the right we have the application of $u$, regarded as a [[linear function]] $u \colon C^\infty(\mathbb{R}^n) \to \mathbb{R}$, to the [[exponential function]] applied to the canonical [[inner product]] $\langle -,-\rangle$ on $\mathbb{R}^n$.

This same formula makes sense more generally for [[complex numbers]] $\zeta \in \mathbb{C}^n$. This is then called the _[[Fourier-Laplace transform]]_ of $u$, still denoted by the same symbol:

$$
  \array{
    \mathbb{C}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R}
    \\
    \zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right)
  }
$$

This is an [[entire analytic function]] on $\mathbb{C}^n$.

=--

([Hörmander 90, theorem 7.1.14](#Hoermander90))

## Related concepts

* [[Paley-Wiener-Schwartz theorem]]

* [[wave front set]]

## References

* {#Hoermander90} [[Lars Hörmander]], section 2.3 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

[[!redirects Fourier-Laplace transforms]]