Frobenius pseudomonoid

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# Frobenius pseudomonoids

* table of contents
{: toc}

## Idea

The concept of **Frobenius pseudomonoid** is the [[categorification]] of that of [[Frobenius algebra]].  It can be defined in any [[monoidal bicategory]].  Since Frobenius pseudomonoids are closely related to [[star-autonomous categories]], they are sometimes called **$\ast$-autonomous pseudomonoids**.

## Definitions

Like a Frobenius algebra, a Frobenius pseudomonoid can be defined in many essentially equivalent ways.  Let $(K,\otimes,I)$ be a [[monoidal bicategory]], and let $(A,\mu,\eta)$ be a [[pseudomonoid]] in $K$.

1. $A$ is Frobenius if it is equipped with a morphism $\epsilon : A \to I$ such that the composite $A\otimes A \xrightarrow{\mu} A \xrightarrow{\epsilon} I$ is the counit of a specified [[2-adjunction]] $A\dashv A$.  ([Lauda06](#Lauda06)), ([Street04](#Street04))

1. $A$ is Frobenius if it is equipped with a specified [[2-adjunction]] $A\dashv A$, with counit $p:A\otimes A \to I$, and an isomorphism $p\circ (\mu\otimes 1) \cong p\circ (1\otimes \mu)$.  ([Day-Street 03](#DayStreet03))

There should also be a definition in terms of an interacting pseudomonoid and pseudocomonoid structure, but I have not been able to find this in the literature.

Note that if $K$ is a [[compact closed bicategory]], then the 2-adjunction $A\dashv A$ can equivalently be expressed as an equivalence $A\simeq A^\circ$ from $A$ to its specified dual object.

## Relation to $\ast$-autonomy

+-- {: .un_theorem}
###### Theorem
A [[star-autonomous category]] is equivalent to a Frobenius pseudomonoid in the monoidal bicategory [[Prof]] whose multiplication $A\otimes A \to A$, unit $I\to A$, and duality equivalence $A\simeq A^\circ$ are [[representable functor|representable]] [[profunctors]] (i.e. [[functors]]).
=--
+-- {: .proof}
###### Proof
See [Day-Street 03](#DayStreet03) and [Street04](#Street04).
=--

Note that the morphisms $\epsilon : A\to I$, $p:A\otimes A\to I$, and the induced comultiplication $A\to A\otimes A$ are *not* representable.  A general Frobenius pseudomonoid in $Prof$, without any representability conditions, may be called a [[promonoidal category|pro]]-$\ast$-autonomous category.

+-- {: .un_remark}
###### Remark
There is another relation between Frobenius algebras and $\ast$-autonomous categories: ([Egger10](#Egger10)) shows that cocomplete $\ast$-autonomous [[posets]] are equivalently Frobenius algebras in the $\ast$-autonomous category [[Sup]].
=--

+-- {: .un_remark}
###### Remark
This characterisation of $\ast$-autonomous algebras in terms of pseudo-Frobenius algebras can further be refined to characterise [[autonomous categories]]. Namely, an autonomous category is a representable pseudo-Frobenius algebra in [[Prof]] whose pseudomonoid and pseudocomonoid structure not only are adjoint, but which also satsify "locality" coherence equations [Bartlett-Vicary 10](#BartlettVicary10).
=--

## References

* [[Aaron Lauda]], *Frobenius algebras and ambidextrous adjunctions*, 2006; [TAC](http://tac.mta.ca/tac/volumes/16/4/16-04abs.html)
 {#Lauda06}

* [[Aaron Lauda]], *Frobenius algebras and planar open string topological field theories*, 2005; [arxiv](https://arxiv.org/abs/math/0508349)
 {#Lauda05}

* [[Brian Day]] and [[Ross Street]], *Quantum categories, star autonomy, and quantum groupoids*, 2003, [arxiv](https://arxiv.org/abs/math/0301209)
 {#DayStreet03}

* [[Bruce Bartlett]] and [[Jamie Vicary]], *Compact categories as dagger-Frobenius pseudoalgebras*, 2010, [QPL2010](https://www.youtube.com/watch?v=rZp1dPJxG3I)
 {#BartlettVicary10}

* [[Ross Street]], *Frobenius monads and pseudomonoids*, 2004, [DOI](http://dx.doi.org/10.1063/1.1788852), [pdf](http://maths.mq.edu.au/~street/Frob.pdf)
 {#Street04}

* [[Jeff Egger]], *The Frobenius relations meet linear distributivity*, 2010 [TAC](http://tac.mta.ca/tac/volumes/24/2/24-02abs.html)
 {#Egger10}

[[!redirects Frobenius pseudomonoids]]
[[!redirects star-autonomous pseudomonoid]]
[[!redirects star-autonomous pseudomonoids]]
[[!redirects *-autonomous pseudomonoid]]
[[!redirects *-autonomous pseudomonoids]]


Frobenius reciprocity

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#Contents#
* table of contents
{:toc}

## Definition

The term _Frobenius reciprocity_ has a meaning

* [In representation theory](#InRepresentationTheory)

* [In category theory](#InCategoryTheory).

(For different statements of a similar name see the disambiguation at _[[Frobenius theorem]]_.)

### In representation theory
 {#InRepresentationTheory}

In [[representation theory]], **Frobenius reciprocity** is the statement that the [[induction functor]] for [[group representation|representations of groups]] (or in some other [[algebraic categories]]) is [[left adjoint]] to the [[restriction]] functor.  Sometimes it is used for a [[decategorification|decategorified]] version of this statement as well, on [[characters]].

Specifically for $H \hookrightarrow G$ an [[subgroup]] inclusion, there is an [[adjunction]]

$$
  (Ind \dashv Res) \;\colon\;
  Rep_G
    \stackrel{\overset{Ind}{\leftarrow}}{\underset{Red}{\longrightarrow}}
  Rep_H
$$

between the [[categories]] of $G$-[[representations]] and $H$-[[representations]], where for $\rho$ an $H$-representation, $Ind(\rho) \in Rep(G)$ is the [[induced representation]].

Sometimes also the _[[projection formula]]_

$$
  Ind(Res(W) \otimes V) \cong W \otimes Ind(V)
$$

is referred to as _Frobenius reciprocity_ in representation theory (e.g. [here on PlanetMath](http://planetmath.org/frobeniusreciprocity)).

### In cartesian categories
 {#InCategoryTheory}

In [[category theory]], Frobenius reciprocity is a condition on a pair of [[adjoint functors]] $f_! \dashv f^*$.  If both categories are [[cartesian closed]], then the adjunction is said to satisfy **Frobenius reciprocity** if the [[right adjoint]] $f^* \colon \mathcal{Y} \to \mathcal{X}$ is a [[cartesian closed functor]]; that is, if the canonical map $f^*(b^a) \to f^*(b)^{f^*(a)}$ is an [[isomorphism]] for all objects $a,b$ of $\mathcal{Y}$.

Each of the functors $-^a$, $-^{f^*(a)}$ and $f^*$ has a [[left adjoint]], so by the calculus of [[mates]], this condition is equivalent to asking that the canonical "[[projection formula|projection]]" morphism

\[
  \label{ProjectionMorphism}
  \pi
  \,\colon\,
  f_! (f^*a \times c)
  \longrightarrow
  a \times f_!(c)
\]

is an isomorphism for each $a$ in $\mathcal{Y}$ and $c$ in $\mathcal{X}$.

This holds for instance for the [[base change]] between [[slice categories]] $\mathcal{C}_{/b}$, $\mathcal{C}_{/b'}$ of a [[finitely complete category]] $\mathcal{C}$ along a morphism $f \colon b' \to b $ -- by the [[pasting law]] in $\mathcal{C}$:

\[
  \label{PastingLawForLexCategories}
  \array{
    f^\ast a \times_b c
    &\longrightarrow&
    f^\ast a
    &\longrightarrow&
    a
    \\
    \big\downarrow
    &&
    \big\downarrow
    &&
    \big\downarrow
    \\
    c
    &\underset{p_C}{\longrightarrow}&
    b'
    &\underset{f}{\longrightarrow}&
    b
  }
  \;\;\;\;\;\;\;\;
  \simeq
  \;\;\;\;\;\;\;\;
  \array{
    a
      \times_b
    f_! c
    &\longrightarrow&
    a
    \\
    \big\downarrow && \big\downarrow
    \\
    c &\underset{ f \circ p_c }{\longrightarrow}& b
    \mathrlap{\,.}
  }
\]

The condition (eq:ProjectionMorphism) clearly makes sense also if the categories are [[cartesian monoidal category|cartesian]] but not necessarily [[closed category|closed]], and is the usual formulation found in the literature.  It is equivalent to saying that the adjunction is a [[Hopf adjunction]] relative to the cartesian monoidal structures.

This terminology is most commonly used in the following situations:

* When $f^*$ and $f_!$ are the [[inverse image|inverse]] and [[direct image]] functors along a map $f$ in a [[hyperdoctrine]].  Here $S$ is a category and $P \colon S^{op} \to Cat$ is an $S$-[[indexed category]] such that each category $P X$ is [[cartesian closed]] and each functor $f^* = P f$ has a [[left adjoint]] $\exists_f$ ([[existential quantifier]], also written $f_!$).  Then $P$ is said to satisfy Frobenius reciprocity, or the **Frobenius condition**, if each of the [[adjunctions]] $\exists_f\dashv f^*$ does. If the categories $P X$ are cartesian but not closed then it still makes sense to ask for Frobenius reciprocity in the second form above, and in that case its logical interpretation is that $\exists x . (\phi \wedge \psi)$ is equivalent to $(\exists x.\phi) \wedge \psi$ if $x$ is not free in $\psi$.

* When $f^*$ is the [[inverse image]] part of a [[geometric morphism]] between [[(n,1)-topoi]] and $f_!$ is a [[left adjoint]] of it, if the [[adjunction]] $f_!\dashv f^*$ satisfies  Frobenius reciprocity, then the geometric morphism is called [[locally n-connected (n+1,1)-topos|locally (n-1)-connected]].  In particular, if $n=0$ so that we have a [[continuous map]] of [[locales]], then a left adjoint $f_!$ satisfying Frobenius reciprocity makes it an [[open map]], and if $n=1$ so that we have 1-[[topoi]], then it is [[locally connected geometric morphism|locally connected]] (see also _[[open geometric morphism]]_).  This usage of "Frobenius reciprocity" is sometimes also extended to the dual situation of [[proper map]]s of locales and topoi.

### In six operations yoga
 {#InWirthmuellerContexts}

The projection formula plays a notable role in Grothendieck's yoga of [[six operations]].  For example if an [[adjoint triple]] $(f_! \dashv f^\ast \dashv f_\ast)$ between [[symmetric monoidal category|symmetric]] [[closed monoidal categories]] is a _[[Wirthmüller context]]_ ([May 05](#May05)), $f^\ast$ is a strong [[closed monoidal functor]].  This implies the projection formula, i.e. the existence of a [[natural isomorphism]] of the form

$$
  \pi
    \;\colon\;
  f_!
  \big(
     f^\ast a \otimes c
  \big)
  \overset{\sim}{\longrightarrow}
  a \otimes (f_! c)
  \,.
$$

The [[projection formula]] also holds in a [[Grothendieck context]] or a [[Verdier-Grothendieck context]] ([May 05](#May05)).

## Properties

### Closed monoidal functors and the projection formula

The following result isolates the connection between [[closed functor|closed functors]] and the projection formula.  We begin with some context.

Recall that a monoidal category $\mathcal{Y}$ is **[left closed](https://ncatlab.org/nlab/show/closed+monoidal+category#left_right_and_biclosed_monoidal_category)** if each functor $a \otimes - \colon \mathcal{Y} \to \mathcal{Y}$ has a right adjoint $[a, -] \colon  \mathcal{Y} \to \mathcal{Y}$, called the internal hom.   We can similarly define right closed monoidal categories.   A symmetric or even braided monoidal category is left closed if and only if it is right closed, and one then simply calls it [[closed monoidal category|closed]], but for maximum generality we consider the merely monoidal case.

A functor $F$ between left closed monoidal categories is **lax closed** it if preserves the internal hom and the unit object up to a specified map

$$
  \hat{F}
  \;\colon\;
  F[a,b] \to [F a,F b],
  \qquad
  F_0 \colon I \to F(I)
  \,,
$$

[[natural transformation|natural]] in both variables and obeying some [[coherence laws]] listed at *[[closed functor]]*.  If these are [[natural isomorphisms]] we call the functor **[[strong closed functor|strong closed]]**.

Any [[lax monoidal functor]] betweeen left [[closed monoidal categories]] is lax closed (for a sketch of the argument see *[[closed functor]]*), but a [[strong monoidal functor]] may not be [[strong closed functor|strong closed]].

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

Suppose $f_! \dashv f^\ast$ is an [[adjoint functor|adjunction]] between left [[closed monoidal categories]]. Then [[natural transformations]]

$$
  \phi
   \;\colon\;
  f^*[a,b] \to [f^*a, f^*b]
$$

correspond [[bijection|bijectively]] to [[natural maps]]

$$
  \pi
    \;\colon\;
  f_!
  \big(
    (f^\ast a) \otimes c
  \big)
    \longrightarrow
  a \otimes (f_! c)
  \,.
$$

Furthermore, $\phi$ is an isomorphism if and only if $\pi$ is, in which case we say the **[[projection formula]]** holds.

=--

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

Suppose $\mathcal{X}$ and $\mathcal{Y}$ are left closed monoidal categories and $f_! \colon \mathcal{X} \to \mathcal{Y}$ is left adjoint to $f^* \colon \mathcal{Y} \to \mathcal{X}$.    Suppose we have a natural map

$$ \phi \colon f^*[a,b] \to [f^*a, f^*b]  $$

for $a,b \in \mathcal{Y}$.  Thus we obtain a natural map

$$ \mathcal{X}(c, f^*[a,b]) \to \mathcal{X}(c, [f^*a, f^*b]), $$

for arbitrary $c \in \mathcal{X}$ (now natural in all three variables).   By hom-tensor adjointness and the fact that $f_!$ is the left adjoint of $f^*$ we can rewrite this as

$$ \mathcal{Y}(f_! c, [a,b]) \to \mathcal{X}(f^*a \otimes c,f^*b). $$

Using both these facts again we obtain

$$ \mathcal{Y}(a \otimes f_! c, b) \to \mathcal{X}(f_!(f^\ast a \otimes c), b)\,. $$

By the [[Yoneda lemma]] this gives the desired natural map

$$
  \pi \colon\;
  f_!(f^\ast a \otimes c) \longrightarrow a \otimes f_! c\,.
$$

By running through this calculation one can see that if $\phi$ is a invertible then all the other natural maps listed above are too, including $\pi$.  Conversely, starting with $\pi$ we can run the argument backwards and get $\phi$, and if $\pi$ is invertible then so is $\phi$.

=--

It follows that if $f^*$ is strong closed, the projection formula holds.  Also if $f$ is strong monoidal and the projection formula holds, $f^*$ is strong closed.

### Relation to Frobenius laws (in Frobenius algebras)

The name "Frobenius" is sometimes used to refer to other conditions on adjunctions, known as "Frobenius laws". The formal structure of the Frobenius law appears in the notion of [[Frobenius algebra]], in the axiom which relates multiplication to comultiplication, and recurs in another form isolated by Carboni and Walters in their studies of cartesian bicategories and bicategories of relations. Namely, if $\delta \colon 1 \to \otimes \Delta$ denotes the diagonal transformation on a cartesian bicategory (e.g., $Rel$), with right adjoint $\delta^\dagger$, then there is a canonical map

$$\delta \delta^\dagger \stackrel{\phi}{\to} (1 \otimes \delta^\dagger)(\delta \otimes 1)$$

mated to the coassociativity isomorphism

$$(1 \otimes \delta)\delta \to (\delta \otimes 1)\delta$$

and the **Frobenius law** here is the assumption that _the 2-cell $\phi$ is an isomorphism_. (There are two Frobenius laws actually; the other is that a similar canonical map

$$\delta \delta^\dagger \stackrel{\phi'}{\to} (\delta^\dagger \otimes 1)(1 \otimes \delta),$$

mated to the inverse coassociativity, is also an isomorphism. However, it may be shown that if one of the Frobenius laws holds, then so does the other; see the article [[bicategory of relations]].)

It is very easy to make a slip and call the Frobenius law "Frobenius reciprocity", perhaps all the more because there are close connections between the two. One example occurs in the context of bicategories of relations, as follows.

Given a locally posetal [[cartesian bicategory]] $B$ and any object $c$ of $B$, one may construct a hyperdoctrine of the form

$$\hom_B(i-, c)\colon Map(B)^{op} \to Semilat$$

where $i: Map(B) \to B$ is the inclusion, and $Semilat$ is the 2-category of meet-semilattices. Here $r \in \hom(i b, c)$ is thought of as a relation from $b$ to $c$, and for a map $f: a \to b$, the relation $f^\ast r$ is the pulling back

$$f^\ast r \coloneqq (a \stackrel{f}{\to} b \stackrel{r}{\to} 1)$$

along $f$, and one may show that $f^\ast-$ preserves finite local meets. Indeed, the pushforward or quantification along $f$ takes $q: a \to 1$ to

$$\exists_f q \coloneqq (b \stackrel{f^\dagger}{\to} a \stackrel{q}{\to} 1)$$

and $\exists_f \dashv f^\ast$ because $f^\dagger$ is _right_ adjoint to the map $f$. Because $f^\ast-$ is a right adjoint, it preserves local meets.

Frobenius reciprocity in this context, ordinarily written as

$$r \wedge \exists_f q = \exists_f (f^\ast r \wedge q),$$

can then be restated for the hyperdoctrine $\hom_B(i-, c)$; it takes the form

$$r \wedge q f^\dagger = (r f \wedge q)f^\dagger$$

for any map $f: a \to b$ and predicates $q \in \hom(a, c)$, $r \in \hom(b, c)$.

Meanwhile, recall that a **bicategory of relations** is a (locally posetal) cartesian bicategory in which the Frobenius laws hold.

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

Frobenius reciprocity holds in each hyperdoctrine $\hom_B(i-, c)$ associated with a bicategory of relations.
=--

+-- {: .proof}
###### Proof (sketch)
One first proves that a bicategory of relations is a compact closed bicategory in which each object $b$ is self-dual. The unit here is given by

$$\eta_b = (1 \stackrel{\varepsilon^\dagger}{\to} b \stackrel{\delta}{\to} b \otimes b)$$

and the counit by

$$\theta_b = (b \otimes b \stackrel{\delta^\dagger}{\to} b \stackrel{\varepsilon}{\to} 1).$$

Using this duality, each relation $r: b \to c$ has an opposite relation $r^{op} \colon c \to b$ given by

$$c \stackrel{c \otimes \eta_b}{\to} c \otimes b \otimes b \stackrel{1 \otimes r \otimes 1}{\to} c \otimes c \otimes b \stackrel{\theta_c \otimes b}{\to} b.$$

It may further be shown that in a bicategory of relations, if $f: a \to b$ is a map, then its right adjoint $f^\dagger$ equals the opposite $f^{op}$. Therefore Frobenius reciprocity becomes the equation
$$r \wedge q f^{op} = (r f \wedge q)f^{op}$$
but in fact this is just a special case of the more general modular law, which holds in a bicategory of relations as shown [here](http://rfcwalters.blogspot.com/2009/10/categorical-algebras-of-relations.html) in a blog post by Walters. The modular law in turn depends crucially upon the Frobenius laws.
=--

Thus, in this instance, _Frobenius reciprocity follows from the Frobenius laws_.

+-- {: .num_prop #FRtoFL}
###### Proposition
In a locally posetal cartesian bicategory, the Frobenius laws follow from Frobenius reciprocity.
=--

+-- {: .proof}
###### Proof
Again, Frobenius reciprocity in a (locally posetal) cartesian bicategory $B$ means that for any map $f: a \to b$ and any two relations $q \in B(a, c)$, $r \in B(b, c)$, the canonical inclusion
$$(q \wedge r f)f^\dagger \leq q f^\dagger \wedge r$$
is an equality. One (and therefore both) of the Frobenius laws will follow by taking the following choices for $f$, $q$, and $r$:

$$f = \delta_x, \qquad q = \varepsilon_{x}^{\dagger} \otimes 1_x, \qquad r = \varepsilon_x \otimes 1_x \otimes \varepsilon_{x}^{\dagger}$$

where $\delta_x: x \to x \otimes x$ is the diagonal map and $\varepsilon_x: x \to 1$ is the projection. The remainder of the proof is best exhibited by a string diagram calculation, which is given here: [[Frobenius-reciprocity.pdf:file]].
=--

## Examples

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

Generally, for $\mathbf{H}$ a [[topos]] and $f \;\colon\; X \longrightarrow Y$ any [[morphism]], then the induced [[base change]] [[etale geometric morphism]]

$$
  (f_! \dashv f^\ast \dashv f_\ast)
  \;\colon\;
  \mathbf{H}_{/X}
  \to
  \mathbf{H}_{/Y}
$$

has [[inverse image]] $f^\ast$ a [[cartesian closed functor]] and hence (see there) exhibits Frobenius reciprocity.

=--

## Related concepts

* [[Beck-Chevalley condition]]

* [[Wirthmüller context]]

## References
 {#References}

The term 'Frobenius reciprocity', in the context of [[hyperdoctrines]], was introduced in

* [[Bill Lawvere]], _[[Equality in hyperdoctrines and comprehension schema as an adjoint functor]]_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970) 1-14 [[pdf](https://ncatlab.org/nlab/files/LawvereComprehension.pdf)]

Lawvere defines Frobenius reciprocity by either of the two equivalent conditions (see "Definition-Theorem" on p.6), and notes that "one of these kinds of identities is formally similar to, and reduces in particular to, the Frobenius reciprocity formula for permutation representations of groups" (p.1).

Related discussion is in:

* {#Seely83} [[Robert A. G. Seely]], p. 511 of: *Hyperdoctrines, Natural Deduction and the Beck Condition*, Zeitschr. f. math. Logik und Grundlagen d. Math. **29** (1983) 505-542 [[doi:10.1002/malq.19830291005]( https://doi.org/10.1002/malq.19830291005), [pdf](https://www.math.mcgill.ca/seely/ZML/ZML.PDF)]

* {#Pavlović96} [[Duško Pavlović]], p. 164 in: *Maps II: Chasing Diagrams in Categorical Proof Theory*, Logic Journal of the IGPL, **4** 2 (1996) 159–194 [[doi:10.1093/jigpal/4.2.159](https://doi.org/10.1093/jigpal/4.2.159), [pdf](http://www.isg.rhul.ac.uk/dusko/papers/1996-mapsII-IGPL.pdf)]

A textbook source is around lemma 1.5.8 in

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

General discussion in the context of projection formulas in [[monoidal categories]] (not necessarily cartesian) is in

* {#May05} [[Halvard Fausk]], [[Po Hu]], [[Peter May]],  *Isomorphisms between left and right adjoints*, Theory and Applications of Categories, **11** 4 (2003) 107-131 [[tac:11-04](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), [pdf](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf)]

Manifestations of the Frobenius reciprocity formula, [in the sense of category theory](#InCartegoryTheory), recur throughout mathematics in various forms (push-pull formula, projection formula); see for example this Math Overflow post:

* Andrea Ferretti, Ubiquity of the push-pull formula, MO Question 18799, March 20, 2010. [(link)](http://mathoverflow.net/questions/18799/ubiquity-of-the-push-pull-formula)

Further MO discussion includes

* [Wrong-way Frobenius reciprocity for finite groups representations](http://mathoverflow.net/questions/132272/wrong-way-frobenius-reciprocity-for-finite-groups-representations)

[[!redirects Frobenius reciprocity]]
[[!redirects Frobenious reciprocity]]
[[!redirects Frobenius law]]
[[!redirects Frobenius condition]]
[[!redirects Frobenius axiom]]


Frobenius theorem

* [[Frobenius reciprocity]]

* [[Frobenius theorem (differential topology)]]

[[!redirects Frobenius theorems]]


#Contents#
* table of contents
{:toc}

## Idea

A statement of necessary and sufficient conditions for the existence of [[foliations]] of [[manifolds]] by maximal solution manifolds to a [[differential equation]].

(For different statements of similar name see the disambiguation at _[[Frobenius theorem]]_.)

## References

* Wikipedia, _[Frobenius theorem (differential topology)](http://en.wikipedia.org/wiki/Frobenius_theorem_%28differential_topology%29)_


Frobenius-Perron dimension

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

#Contents#
* table of contents
{:toc}

## Idea

In the theory of [[fusion rings]]/[[fusion categories]], it is often very useful to have a good theory of dimensions of objects. Dimensions are useful linearization tool, as well as for defining invariants.

The most natural way to define the dimension of an object in a fusion category is via the [[quantum dimension]]. Namely, one defines the dimension of an object to be the [[trace]] of the identity map on the object. In unitary fusion categories this works perfectly, but in higher generality there are some issues which can arise. The main issue is that trace is not defined. Even in [[spherical categories]] where there is a good notion of trace, there is no guarantee that the quantum dimension will be positive.

The key observation is that in the unitary case the quantum dimension of an object will always be the [[Frobenius-Perron theorem | Frobenius-Perron eigenvector]] of its corresponding fusion matrix. In the spherical non-unitary case the quantum dimension will still be an eigenvector of the fusion matrix, just not necessarily the Frobenius-Perron one.

The Frobenius-Perron dimension of an object is thus defined to be the Frobenius-Perron eigenvector of its corresponding fusion matrix. This is defined in higher generality than the quantum dimension, and serves the purpose of linearization equally well. It thus serves as a great tool in the study of fusion rings and fusion categories.

## Definition

### For elements of $\mathbb{N}$-rings

The most general setting for Frobenius-Perron dimensions is that of [[fusion ring | $\mathbb{N}$-rings]].

###### Definition

A _unital $\mathbb{N}$-ring_ is a [[ring]] such that

1. the underlying [[abelian group]] is [[free abelian group]];

1. there exists a finite $\mathbb{N}$-[[basis]]: a [[finite set]] $I$ of elements $X_i \in R$, $i \in I$, such that

   $$
     X_i X_j = \sum_{k \in I} c_{i j}^k X_k
   $$

   for $c_{i j}^k \in \mathbb{N}$

1. the ring unit 1 is among these basis elements.

=--

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

Let $R$ be a unital $\mathbb{N}$-ring (def. \ref{NRing}) with finite $\mathbb{N}$-[[basis]] $I$, $\vert I\vert = n \in \mathbb{N}$.

For every $X \in R$, let $(N_X)_{i j}\in \mathbb{N}$ be its [[matrix]] of left multiplication, defined by

$$
  X X_j  =  \sum_{i \in I} (N_X)_{i j} X_i
  \,.
$$

By the [[Frobenius-Perron theorem]], these matrices $N_X$ have non-negative [[eigenvalues]]. This implies that for each of them there is a [[maximum|maximal]] [[eigenvalue]]. This maximal eigenvalue is called the **Frobenius-Perron dimension** of $X$, $FPdim(X)$.

=--

### For objects of fusion categories

Let $\mathcal{C}$ be a [[fusion category]], i.e. a [[tensor category]] which is [[finite abelian category|finite]] and [[semisimple category]] (i.e. it has a [[finite number]] of [[isomorphism classes]] $[X_i]$ of [[simple objects]], all finite [[direct sums]] of these exist, and every object is isomorphic to such).

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

Then the [[isomorphism classes]] $[X]$ of objects of $\mathcal{C}$ form an $\mathbb{N}$-ring (def. \ref{NRing}) under [[tensor product]], the _[[fusion ring]]_.

The **Frobenius-Perron dimension of $X \in \mathcal{C}$ is that of its isomorphism class $[X]$ as an element of the [[fusion ring]], according to def. \ref{FPDimension}:

$$
  FPdim(X) \coloneqq FPDim([X])
  \,.
$$

=--

## References

* [[Damien Calaque]], [[Pavel Etingof]], section 4 of  _Lectures on tensor categories_ ([arXiv:0401246](https://arxiv.org/abs/math/0401246))

* {#EGNO15} [[Pavel Etingof]], Shlomo Gelaki, Dmitri Nikshych, [[Victor Ostrik]], section 3.3 in _Tensor categories_, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 ([pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf
))

* [[Pavel Etingof]], [[Dmitri Nikshych]], [[Viktor Ostrik]], "On Fusion Categories", Annals of mathematics, 2005 ([pdf](https://arxiv.org/pdf/math/0203060.pdf))


Frobenius-Perron theorem

## Idea

The Frobenius-Perron theorem is a foundational result in the theory of non-negative matrices. In its simplest form it says that the largest eigenvalue of any square matrix with positive entries is real, is of multiplicity one, and that it has an eigenvector with strictly positive entries. The largest real eigenvalue of a matrix with non-negative entries is aptly known as the Frobenius-Perron eigenvalue, and its corresponding positive eigenvector is the Frobenius-Perron eigenvector.

Seeing as matrices with positive/non-negative entries are ubiquitous across mathematics, this theorem has a plethora of applications. For instance,

* [[stochastic processes]]/[[Markov chains]], where the Frobenius-Perron eigenvector corresponds to a stable state/limiting behavior of the process.

* Algebraic graph theory - the study of linear-algebraic properties of graphs. That is, situations where one studies the linear algebra of adjacency matrices of graphs. Adjacency matrices are necessarily non-negative, and the Frobenius-Perron eigenvector can be of great utility. This theory is used strongly, for instance, in the proofs of many [[ADE classifications]].

* [[fusion rings]]/[[fusion categories]]. Here, the fusion rules of the category will naturally induce non-negative matrices. The Frobenius-Perron eigenvalue of fusion matrix corresponding to an element is known as its [[Frobenius-Perron dimension]] and serves a key role in theory.

## Statements

There are a collection of related results all known as the Frobenius-Perron theorem, which require more or less strong strong conditions and give more or less strong results. The most canonical "Frobenius-Perron theorem" is as follows:

\begin{theorem} The eigenvalue of largest absolute value for any matrix with positive entries will be a positive real number. It's corresponding eigenspace will be one-dimensional, and it will have an eigenvector with all positive entries.
\end{theorem}

Conversely, we have the following result:

\begin{theorem} If an eigenvector to a positive matrix has positive entries, then its eigenvalue must be the Frobenius-Perron eigenvalue.
\end{theorem}

This gives a very simple way of computing the Frobenius-Perron eigenvalue of a positive matrix. Namely, all one has to do is construct a positive eigenvector and measure its eigenvalue.

Often, one will be applying the Frobenius-Perron theorem to matrices with non-negative entries. Here, we can say the following:

\begin{theorem} The eigenvalue of largest absolute value for any matrix with non-negative entries will be a non-negative real number. It will have an eigenvector with all non-negative entries.
\end{theorem}

There are many non-negative matrices for which the stronger positive-matrix formulation of the theorem still holds. Namely, we have the following:

\begin{theorem} Let $A$ be a matrix such that $A^m$ is positive for some $m\geq 1$. The eigenvalue of largest absolute value for $A$ will be a positive real number. It's corresponding eigenspace will be one-dimensional, and it will have an eigenvector with all positive entries.
\end{theorem}

## Related concepts

* [[Frobenius-Perron dimension]]

* [[fusion category]]

## References

* [[Damien Calaque]], [[Pavel Etingof]], section 4 of  _Lectures on tensor categories_ ([arXiv:math/0401246](https://arxiv.org/abs/math/0401246))

* {#EGNO15} [[Pavel Etingof]], Shlomo Gelaki, Dmitri Nikshych, [[Victor Ostrik]], section 3.2 in _Tensor categories_, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 ([pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf
))

* Wikipedia, _[Perron-Frobenius theorem](https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem)_

[[!redirects Perron-Frobenius theorem]]


+-- {: .rightHandSide}
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###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

This page collects material related to the textbook

* [[Michael Dütsch]],

  _From classical field theory to perturbative quantum field theory_

  Progress in Mathematical Physics,

  Birkhaeuser 2019

  ([publisher page](https://www.springer.com/de/book/9783030047375))

which is an introductory textbook on [[perturbative quantum field theory]] in its mathematically precise formulation known as _[[perturbative algebraic quantum field theory]]_, based on [[causal perturbation theory]].

The book amplifies the axiomatization of [[retarded products]] over the (equivalent) [[time-ordered products]] and includes a special focus on [[master Ward identities]].

## Related textbooks

* [[Günter Scharf]], _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Springer 1995

* [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

* {#Rejzner16} [[Katarzyna Rejzner]], _[[Perturbative Algebraic Quantum Field Theory]]_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

* [[Urs Schreiber]], _[[geometry of physics -- perturbative quantum field theory|Perturbative Quantum Field Theory]]_, 2017

category: reference


This page collects material related to the book

* [[Mikhail Shifman]], [[Arkady Vainshtein]], [[John Wheater]] (eds.),

  _From Fields to Strings -- Circumnavigating Theoretical Physics_

  Ian Kogan Memorial Collection (In 3 Volumes)

  World Scientific 2005

  [doi:10.1142/5621](https://doi.org/10.1142/5621)

#Contents#
* table of contents
{:toc}

## Volume 1

**Topics:**

* from [[perturbative quantum field theory]] to [[string theory]]

* [[quantum chromodynamics]] and [[quantum field theory]]

* [[condensed matter physics]]

\linebreak

On the [[Schwinger effect]] of [[vacuum polarization]]:

* {#Dunne04} [[Gerald Dunne]], _Heisenberg-Euler Effective Lagrangians: Basics and Extensions_, in:  [[Misha Shifman]], [[Arkady Vainshtein]], [[John Wheater]] (eds.), _[[From Fields to Strings -- Circumnavigating Theoretical Physics]]_, pp. 445-522, World Scientific 2005 ([arXiv:hep-th/0406216](https://arxiv.org/abs/hep-th/0406216), [doi:10.1142/9789812775344_0014](https://doi.org/10.1142/9789812775344_0014))

## Volume 2

**Topics:**

* [[grand unified theories]], [[Kaluza-Klein theory]], [[cosmology]]

* [[conformal field theory]]

* [[AdS/CFT correspondence]]

## Volume 3

**Topics:**

* [[string theory]]

category: reference


This page concerns:

* Alessandra Capotosti:

  \linebreak

  **From String structures to Spin structures on loop spaces**

  \linebreak

  Ph.D. thesis,

  Università degli Studi Roma Tre, Rome, April 2016

  \linebreak

  [thesis pdf](http://www.matfis.uniroma3.it/Allegati/Dottorato/TESI/capotosti/PhD%20Thesis%202016%20A%20Capotosti.pdf), [[Capotosti-FromStringStructures.pdf:file]]

   talk slides [[Capotosti-FromStringStruc-Slides.pdf:file]]

on [[differential string structures]].

**Summary**

This thesis derives the [[transgression]] map of [[String structures]] on an $n$-dimensional [[smooth manifold]] $X$ ($n \geq 3$) to [[Spin structures]] on its [[loop space]] from the existence of a natural morphism of [[smooth infinity-stacks|smooth $\infty$-stacks]]
$$
  \mathbf{B}Spin
    \longrightarrow
  \mathbf{B}^2\big(\mathbf{B}U(1)_{conn}\big)
$$
refining the [[first fractional Pontryagin class]] (following [[schreiber:Cech Cocycles for Differential characteristic Classes|Fiorenza, Schreiber and Stasheff 2012]]). This transgression map has been known for a while in the Physics literature, and a completely rigorous proof has
been then given by [[Konrad Waldorf]] in a series of papers.

Recall that a [[Spin manifold]] $X$ is endowed with a [[String
structure]] if the map $X \overset{f_{T X}}\rightarrow B Spin(n) \overset{\frac{1}{2}p_1}\rightarrow K(\mathbb{Z}, 4)$ is homotopically trivial, where $f_{T X}$ denotes the classifying map for the [[tangent bundle]] of $X$. In this case, the
map $f_{TX}$ can be lifted to a map $X \to B String(n)$, where $B String(n)$ is defined as the [[homotopy
fiber]] of $\frac{1}{2}p_1$. By taking the [[based loop space]] of $B String(n)$ one obtains the topological [[String group]]:
$String(n) = \Omega BString(n)$.

Let now $\mathcal{L}X$ be
the [[free loop space]] of the Spin manifold $X$ and let $\mathcal{L}Spin(n)$ be the [[loop group]] of the [[Spin group]]. Since the
Spin group $Spin(n)$ is [[connected topological space|connected]], the loop space of the Spin manifold $X$ is naturally an
$\mathcal{L}Spin(n)$-manifold (an [[infinite-dimensional manifold|infinite dimensional one]]), i.e., the tangent bundle $T (\mathcal{L}X)$ is naturally associated with an $\mathcal{L}Spin(n)$-bundle over $\mathcal{L}X$ (see [Waldorf 2011, Lemma 5.1](#smooth+loop+space#Waldorf11)  and [Spera & Wurzbacher 2007](smooth+loop+space#SperaWurzbacher07)).

The loop group $\mathcal{L}Spin(n)$ has a universal [[central extension]]
$$
1 \to U (1) \to\widetilde{\mathcal{L}Spin(n)} \to LSpin(n) \to 1
$$
 and one defines a Spin structure on $\mathcal{L}X$ as a lift of the [[structure group]] of the
tangent bundle $T (\mathcal{L}X)$ of $\mathcal{L}X$ from $\mathcal{L}Spin(n)$ to $\widetilde{\mathcal{L}Spin(n)}$, i.e., as a lift of the tangent
bundle morphism $T (\mathcal{L}X) \,\colon\, \mathcal{L}X \to B\mathcal{L}Spin(n)$ to a morphism
$$
\mathcal{L}X \to B\widetilde{\mathcal{L}Spin(n)}.
$$
Notice how a Spin structure on a loop space is not, strictly speaking a Spin structure, i.e., it
is not a morphism to $BSpin(dim \mathcal{L}X)$. On the other hand, since $\mathcal{L}X$ is infinite dimensional,
such a notion would be meaningless.

It is well known in the theoretical physics [[folklore]] that a String structure on $X$ is essentially
the same thing as a Spin structure on $\mathcal{L}X$. In a series of articles
Konrad Waldorf has given a rigorous proof of this statement, proving that there is a natural
transgression map
$$
\{\text{String structures on}\;X\} \to \{\text{Spin structures on}\;\mathcal{L}X\},
$$
which induces a bijection at the level of isomorphism classes
$$
\{\text{String structures on}\;X\}/\sim \to \{\text{Spin structures on}\;\mathcal{L}X\}/\sim,
$$
as soon as $X$ is [[compact topological space|compact]] and [[simply connected topological space|simply connected]].

In this Thesis it is shown how the transgression map considered by Waldorf is naturally obtained
from general constructions in the [[(infinity,1)-category|$\infty$-category]] of [[smooth infinity-stacks|smooth infinity-stack]]. It should however be stressed that this abstract derivation of the transgression map does not come with a proof that one gets a bijection on equivalence classes when $X$ is compact and simply connected, and the latter result need to be proved by ad hoc methods.

The crucial point in the stacky construction of the transgression map is the existence
of a natural morphism of smooth stacks
$$
 \mathbf{B}Spin
    \longrightarrow
 {\mathbf{B}}^2\big(\mathbf{B}U(1)_{conn}\big)
$$
refining the [[first fractional Pontryagin class]]. This morphism of smooth stacks appears implicitly in [Waldorf 2016](smooth+loop+space#Waldorf16) in the form of a multiplicative [[bundle gerbe]] [[connection on a bundle gerbe|with connection]] over the [[Spin group]], and as such plays
an essential role in Waldorf’s proof. In particular, the central extension $\widetilde{\mathcal{L}Spin}$ considered in the Thesis is what Waldorf calls a *fusion extension* of $\mathcal{L}Spin$.

But once the canonical
multiplicative gerbe with connection over $B Spin$ is regarded as a morphism of [[smooth infinity-stack|smooth $\infty$-stacks]], everything else follows by very general reasoning. For instance, the fact that $\widetilde{\mathcal{L}Spin}$ is a fusion extension of $\mathcal{L}Spin$ is encoded into the natural [[homotopy fiber sequence]] of smooth stacks
$$
  \array{
    \mathbf{B}\widetilde{\mathcal{L}Spin}
    & \rightarrow &
    \ast
    \\
    \big\downarrow && \big\downarrow
    \\
    \mathbf{B}\mathcal{L}Spin
    &\rightarrow&
    \mathbf{B}^2 U(1)
  }
$$
induced by the morphism $\mathbf{B}Spin \to {\mathbf{B}}^2\big({\mathbf{B}}U(1)_{conn}\big)$ and by the [[holonomy]] morphism. Also, in the Thesis is provided a direct proof of the existence of the morphism $\mathbf{B}Spin \rightarrow {\mathbf{B}}^2\big({\mathbf{B}}U(1)_{conn}\big)$ which
does not rely on Waldorf’s result and is instead based on the differential refinement of the
first fractional Pontryagin class
$$
  \frac{1}{2} \hat{\mathbf{p}}_1
  \colon
  \mathbf{B}Spin_{conn} \to \mathbf{B}^3 U(1)_{conn}
$$
constructed in [[schreiber:Cech Cocycles for Differential characteristic Classes|Fiorenza, Schreiber and Stasheff 2012]].

## Related entries

* [[stringor bundle]]

category: reference

[[!redirects Alessandra Capotosti]]


FRS-theorem on 2d rational CFT > history

see _[[FRS-theorem on rational 2d CFT]]_


FRS-theorem on rational 2d CFT

+-- {: .rightHandSide}
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#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What might be called the _FRS construction/theorem_ after [[Jürgen Fuchs|Fuchs]], [[Ingo Runkel|Runkel]] and [[Christoph Schweigert|Schweigert]], also [[Jürg Fröhlich|Fröhlich]] and [[Jens Fjelstad|Fjelstad]] ([FRS 02](#FuchsRunkelSchweigert02), [FRS 04a](#FuchsRunkelSchweigert04a), [FRS 04b](#FuchsRunkelSchweigert04b), [FRS 05](#FuchsRunkelSchweigert05), [FFRS 06](#FjeldstadFuchsRunkelSchweigert06), [FFRS 08](#FjeldstadFuchsRunkelSchweigert06)) is a rigorous construction and classification of all [[rational CFT|rational]] _full_ [[2-dimensional conformal field theories]], where "full" means that these theories are defined "at all [[genus of a surface|genera]]", namely on all 2-dimensional [[cobordisms]] (instead of just on [[tori]], which is a simple special case that much of the other literature traditionally concentrates on ("modular invariance")).

It is (was earlier) well known that the construction of [[2d CFTs]] may be decomposed into

1. a "local geometric" aspect which describes the behaviour of the theory on infinitesimal conformal cobordisms and is controled by [[vertex operator algebras]] and related structures, which in particular may be used to construct the spaces of [[conformal blocks]] of the theory, which are the spaces of _potential_ [[correlator]] functions with the right dependence of the [[conformal structure]];

1. a "global topological" aspects which consists of picking in these spaces of [[conformal blocks]] over each surface one element (one correlation function depending on conformal structure) in such a way that these choices globally fit together under gluing of 2-dimensional [[cobordisms]], a condition known as the _[[sewing constraint]]_.

What the FRS theorem effectively observes/shows is that the [[sewing constraints]] may be solved by appealing to a [[holography|holographic principle]] in form of the [[AdS3-CFT2 and CS-WZW correspondence]]. Indeed the FRS formalism may be seen as providing a mathematically precise realization of this correspondence. (That is clear from the construction in the FFRS articles, but the relation to holography was made explicit in various web posts by Urs Schreiber and then later in ([Kapustin-Saulina 11](#KapustinSaulina11))).

Namely given the [[vertex operator algebra]] that controls the CFT locally, then in the case of [[rational CFT]] (hence for a [[rational vertex operator algebra]]) its [[representation category]]  is a [[modular tensor category]] (which may alternatively be obtained from suitable representations of [[Hopf algebras]] or of [[loop groups]], but for the full construction one needs in the end an equivalence to reps of a specified VOA) and by the [[Reshetikhin-Turaev construction]] or [[Turaev-Viro construction]] this defines a [[3d TQFT]], which is essentially known to be the [[quantization of 3d Chern-Simons theory]].

The way that the FFRS construction now works is that given a [[surface]] with field insertions, then the [[CFT]] [[correlator]] of that is computed by

1. first [[triangulation|triangulating]] the surface suitably, labelling the triangulation with a [[Frobenius algebra]] object internal to the [[modular tensor category]];

1. passing to the [[orientation double cover]] of the surface, filling that with a 3d [[cobordism]] with "[[Wilson loop]]" ribbons running through it and piercing the surface at the prescribed field insertions, where the ribbons are labeled by the [[objects]] of the representation category correpsonding to the field insertions (this makes one label for boundary field insertions and two for bulk field insertions, corresponding to the two "chiral" pieces of [[closed string]] CFT)

1. applying the [[3d TQFT]] [[FQFT|functor]] to this labeled cobordism with ribbons inside to produce an element in its [[hom-object|hom-space]];

1. identifying that hom space with the space of [[conformal blocks]] and hence finally producing an actual CFT correlator.

[[FFRSStbl.gif:pic]]

Previous to the FRS construction (and still) much of the literature on classification of [[2d CFTs]] concentrated on [[modular invariance|modular invariant]] [[partition functions]], hence on CFTs evaluated on a surface which is a [[torus]]. One interesting aspect of the FRS classification of _full_ CFTs that are defined not just on the torus but on all surfaces is that it showed examples of modular invariant partition functions in the literature which had all of: one, none, or several extensions to full CFTs.

Since full [[2d CFTs]] are what is relevant in [[string theory]], where these correspond to the [[vacua]] that the [[string perturbation series]] is perturbing about (while in [[statistical physics]] often just the torus is in fact sufficient) this means that the FRS classification is part of the mathematical identification of the [[landscape of string theory vacua]]. Now, [[rational CFTs]] are known to constitute but a tiny part of that landscape, indeed a [[discrete space|discrete subset]] since any [[deformation]] of a rational CFT is non-rational. Nevertheless, rational 2d CFTs (the [[WZW model]]!) play an important roles as the "internal" parts of string theory [[KK-compactifications]]. Also, even a tiny space is infinitely more than the empty space, which is presently the part of the rest of the "landscape" that is understood rigorously.

## Related entries

* [[CFT]]

* [[holographic principle]]

* [[local-global principle]]

* [[cobordism hypothesis]]

## References

The construction and the proof of the classification result is originally described in the series of articles

* {#FuchsRunkelSchweigert02} [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], _TFT construction of RCFT correlators I: Partition functions_, Nucl. Phys. B **646**  (2002) 353-497 [[arXiv:hep-th/0204148](http://arxiv.org/abs/hep-th/0204148), doi:10.1016/S0550-3213(02)00744-7]

* {#FuchsRunkelSchweigert04a} [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], _TFT construction of RCFT correlators II: Unoriented world sheets_, Nucl.Phys. B678 (2004) 511-637 ([arXiv:hep-th/0306164](http://arxiv.org/abs/hep-th/0306164))

* {#FuchsRunkelSchweigert04b} [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], _TFT construction of RCFT correlators III: Simple currents_, Nucl.Phys. B694 (2004) 277-353 ([arXiv:hep-th/0403157](http://arxiv.org/abs/hep-th/0403157))

* {#FuchsRunkelSchweigert05} [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], _TFT construction of RCFT correlators IV: Structure constants and correlation functions_, Nucl.Phys.B715:539-638 (2005) ([arXiv:hep-th/0412290](http://arxiv.org/abs/hep-th/0412290))

* {#FjeldstadFuchsRunkelSchweigert06} [[Jens Fjelstad]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], _TFT construction of RCFT correlators V: Proof of modular invariance and factorisation_, Theor.Appl.Categor.16:342-433 (2006) ([arXiv:hep-th/0503194](http://arxiv.org/abs/hep-th/0503194))

* {#FjeldstadFuchsRunkelSchweigert08} [[Jens Fjelstad]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Uniqueness of open/closed rational CFT with given algebra of open states*, Adv. Theor. Math. Phys. **12** (2008) 1283-1375 [[hep-th/0612306](http://arxiv.org/abs/hep-th/0612306), [doi:10.4310/ATMP.2008.v12.n6.a4](https://dx.doi.org/10.4310/ATMP.2008.v12.n6.a4)]

Review:

* [[Ingo Runkel]], [[Jens Fjelstad]], [[Jürgen Fuchs]], [[Christoph Schweigert]], *Topological and conformal field theory as Frobenius algebras*, in: *Categories in Algebra, Geometry and Mathematical Physics*, Contemp. Math. **431** (2007) 225-248 $[$[doi:10.1090/conm/431](http://dx.doi.org/10.1090/conm/431), [arXiv:math/0512076](https://arxiv.org/abs/math/0512076)$]$

*  [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], _Open strings and 3d topological field theory_ ([pdf](http://www.mth.kcl.ac.uk/staff/i_runkel/PDF/ost.pdf))

* [[Liang Kong]], _Conformal field theory and a new geometry_, in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings in Symposia in Pure Mathematics, AMS, (2011) ([arXiv:1107.3649](http://arxiv.org/abs/1107.3649))

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], p. 13 of: *Algebraic structures in two-dimensional conformal field theory*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

An explicit amplification of how the FRS formalism follows from 3-dimensional [[Chern-Simons theory]] regarded as an [[extended topological quantum field theory]] [[QFT with defects|with defects]] and the [[holographic principle]] is in

* {#KapustinSaulina11} [[Anton Kapustin]], [[Natalia Saulina]], _Surface operators in 3d TFT and 2d Rational CFT_ in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings in Symposia in Pure Mathematics, AMS, (2011)

Some more online discussion is here:

For a list of references see

* Urs Schreiber, [FRS reviews](http://golem.ph.utexas.edu/string/archives/000747.html)

A survey of the central theorem that the FRS construction solves the [[sewing constraints]] is at

* Urs Schreiber, [The FRS theorem on CFT](http://golem.ph.utexas.edu/string/archives/000813.html)

and a discussion of the converse, that every rarional 2-d CT is obtained this way is at

* Urs Schreiber, [FFRS on uniqueness of conformal field theory](http://golem.ph.utexas.edu/category/2007/01/ffrs_on_uniqueness_of_conforma.html)

On potential generalization from [[rational 2d CFT]] to [[logarithmic CFT]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], *Full Logarithmic Conformal Field theory - an Attempt at a Status Report*, Proc. of *[[Higher Structures in M-Theory 2018]]*, Fortschr. Phys.  **67** 8-9 (2019)  $[$[arXiv:1903.02838](https://arxiv.org/abs/1903.02838), [doi:10.1002/prop.201910018](https://doi.org/10.1002/prop.201910018)$]$

[[!redirects FRS theorem on rational 2d CFT]]

[[!redirects FFRS formalism]]

[[!redirects Fuchs-Runkel-Schweigert construction]]

[[!redirects FRS formalism]]

[[!redirects FRS theorem on 2d rational CFT]]

[[!redirects FFRS-formalism]]
[[!redirects FFRS-formalism]]

[[!redirects FRS-formalism]]

[[!redirects FRS-theorem]]
[[!redirects FRS theorem]]


FRW model

+-- {: .rightHandSide}
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### Context
#### Cosmology
+-- {: .hide}
[[!include cosmology -- contents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Friedmann–[[Georges Lemaître|Lemaître]]–Robertson–[[Arthur Walker|Walker]] models_ (often _FRW-models_) are class of [[model (physics)|models]] in [[cosmology]]. These are [[solutions]] to [[Einstein's equations]] describing a spatially homogeneous and isotropic expanding or contracting [[spacetime]]. Hence these are solutions used as [[model (in theoretical physics)|models]] in [[cosmology]]. Indeed, an FRW-model is part of the [[standard model of cosmology]]. (In contrast to _[[inhomogeneous cosmology]]_.)

## Details
 {#Details}

The plain FRW model parameterizes a homogenous and isotropic [[spacetime]] after [[diffeomorphism]] [[gauge fixing]] with a single parameter $t \mapsto a(t)$, the _scale factor of the universe_ depending on coordinate time $t$ (fixed by some gauge condition).

The [[equations of motion]] of the FRW model are then

1. $H^2 \coloneqq (\dot a / a)^2 = 2 \rho - k/a^2$

1. $\ddot a/ a = -(\rho + 3 p)$

1. $\dot \rho + 3 H(\rho + p ) = 0$

where

* $H \coloneqq \dot a/a$ is called the [[Hubble parameter]];

* and

  * $p$ is the [[pressure]]

  * $\rho$ is the [[density]]

  of a "perfect fluid" of [[matter]] and [[radiation]] filling the universe.

Moreover, the ratio

* $w \coloneqq p/\rho$

is part of the experimental/phenomenological input into the model, which describes which _kind_ of matter/radiation is assumed to fill [[spacetime]]

| w |  source of energy-density filling spacetime |
|--|--|
| 1/3 | radiation or relativistic matter |
| 0 | dust [[matter]] |
| 1 | stiff fluid |
| -1 | [[cosmological constant]] |

The first equation may be rewritten as

$$
  \Omega \coloneqq \Omega_R + \Omega_M + \Omega_\Lambda
  =
  1 + \frac{k}{a^2 H^2}
$$

where the _density parameter_ $\Omega$ consists of the contribution

* $\Omega_R = 2 \frac{\rho_r}{a^2}$ of [[radiation]];

* $\Omega_M = 2 \frac{\rho_r}{a^2}$ of [[matter]];

* $\Omega_\Lambda =  \frac{\Lambda}{a^2}$ of [[cosmological constant]];

## Related concepts

* [[Hubble's law]]

* [[big bang]]

* [[comoving time]]

* [[cosmic inflation]]

* [[standard model of cosmology]]

* [[structure formation]]

* [[inhomogeneous cosmology]]

## References

Named after [[Alexander Friedmann]], [[Georges Lemaître]], [[Howard Robertson]], [[Arthur Walker]].

Introduction and survey:

* [[Matthias Blau]], chapter 33 and 34 of _Lecture notes on general relativity_ ([web](http://www.blau.itp.unibe.ch/GRLecturenotes.html))

* Jorge L. Cervantes-Cota, George Smoot, _Cosmology today -- A brief review_ (2011)([arXiv:1107.1789](http://arxiv.org/abs/1107.1789))

* Konstantinos Xenos, _An Introduction to FRW Cosmology and dark energy models_ ([arXiv:2101.06135](https://arxiv.org/abs/2101.06135))

  > (emphasis on [[dark energy]] and [[inflation]])

See also:

* Wikipedia, _[Friedmann–Lemaître–Robertson–Walker metric](http://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric)_

Discussion in [[Regge calculus]]:

* Ren Tsuda, Takanori Fujiwara, _Oscillating 4-Polytopal Universe in Regge Calculus_ ([arXiv:2011.04120](https://arxiv.org/abs/2011.04120))

[[!redirects FRW models]]

[[!redirects FRW spacetime]]
[[!redirects FRW spacetimes]]

[[!redirects FLRW model]]
[[!redirects FLRW models]]
[[!redirects FLRW spacetime]]
[[!redirects FLRW spacetimes]]

[[!redirects Friedman-Robertson-Walker model]]
[[!redirects Friedman-Robertson-Walker models]]

[[!redirects Friedman-Lemaitre-Robertson-Walker model]]
[[!redirects Friedman-Lemaitre-Robertson-Walker models]]

[[!redirects Robertson-Walker model]]
[[!redirects Robertson-Walker models]]

[[!redirects Friedmann–Lemaître–Robertson–Walker metric]]


Fréchet derivative

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Fréchet derivative_ is a kind of [[functional derivative]].

There are various ways to set up [[differentiation]] in finite dimensions, the most common being the *total derivative* and the *directional derivatives*.  In [[infinite-dimensional manifold|infinite dimensions]], these become the *Fréchet derivative* and the *[[Gâteaux derivative]]* respectively.

The definition of the **Fréchet derivative** of a function is a generalisation of the notion of the *total derivative* of a function in finite dimensions.  In finite dimensions, the total derivative of a function $f \colon \mathbb{R}^n \to \mathbb{R}$ at a point $x \in \mathbb{R}^n$ is defined to be (assuming that it exists) the unique linear operator $D f_x \colon \mathbb{R}^n \to \mathbb{R}$ such that:

$$
\lim_{h \to 0} \frac{f(x + h) - f(x) - D f_x h}{{\|h\|}} = 0
$$

As all norms in finite dimensions are norm-equivalent, the choice of norm does not matter.

This generalises most easily to [[normed vector spaces]].  As it involves limits, it is generally most convenient to work with [[Banach spaces]].

## Definition

+-- {: .num_defn #frderiv}
###### Definition
Let $E$ and $F$ be [[Banach spaces]] with [[norms]] ${\|\cdot\|_E}$ and ${\|\cdot\|_F}$ respectively.
Let $U \subseteq E$ be an open subset.
A (possibly non-linear) function $f \colon U \to F$ is said to be **Fréchet differentiable** at $x \in U$ if there is a continuous linear operator $A_x \colon E \to F$ such that:
$$
\lim_{h \to 0} \frac{\|f(x + h) - f(x) - A_x h\|_F}{\|h\|_F} = 0
$$

The operator $A_x$ is called the **Fréchet derivative** of $f$ at $x$ and is written $D f_x$ or $D f(x)$.
=--

## Properties

If the Fréchet derivative exists, it is unique.

## Extensions

With the notation of Definition \ref{frderiv}, a function $f U \supseteq E \to F$ is said to be of class $C^1$ on $U$ if it is Fréchet differentiable throughout $U$ and the resulting function $x \mapsto D f_x$ is continuous as a function $U \to L(E,F)$, where $L(E,F)$ is the Banach space of continuous linear operators from $E$ to $F$.

The differentiability of $x \mapsto D f_x$ can then be questioned, and for $n \in \mathbb{N}$, the class $C^n$ is defined by iteration in the obvious way.

[[!redirects Frechet derivative]]
[[!redirects Fréchet derivatives]]
[[!redirects Frechet derivatives]]
[[!redirects Fréchet differentiable]]
[[!redirects Frechet differentiable]]


Fréchet Lie group

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###Context###
#### Lie theory
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[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _Fréchet Lie group_ is a variant of that of _[[Lie group]]_ where the underlying space is an _infinite-dimensional_ [[Fréchet manifold]].

## References

For the moment see [here](http://ncatlab.org/nlab/show/Lie+group#ReferencesOnInfiniteDimensionalLieGroups)

[[!redirects Frechet Lie group]]

[[!redirects Fréchet Lie groups
[[!redirects Frechet Lie groups]]

[[!redirects Fréchet–Lie group]]
[[!redirects Fréchet–Lie groups]]

[[!redirects Frechet–Lie group]]
[[!redirects Frechet–Lie groups]]


Fréchet manifold

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

#Contents#
* table of contents
{:toc}

## Idea

The concept of _Fréchet manifold_ is a special case of that of _[[infinite-dimensional manifold]]_:
In analogy to how a finite-dimensional [[smooth manifold]] is a [[manifold]] modeled on a [[Cartesian space]] $\mathbb{R}^n$ in [[CartSp]], a _Fréchet manifold_ is a [[manifold]] modeled on a [[Fréchet space]], such as notably $\mathbb{R}^\infty$ ([exmpl.](Fréchet+space#ProjRInfinity)).

The [[category]] of Fréchet manifolds is a [[full subcategory]] of that of [[diffeological spaces]] (prop. \ref{FFEmbeddingOfFrechetInDiffeological} below) hence of [[smooth sets]] (see [here](diffeological+space#EmbeddingOfDiffeologicalSpacesIntoTheSheafTopos)).

## Definition

### Fréchet manifolds

It is possible to define, analogous to the finite dimensional case, the notion of [[smooth functions]] between [[Fréchet spaces]], see at _[Fréchet space -- Differentiable and smooth functions](Fréchet+space#DifferentiableAndSmoothFunctions)_. Therefore, the usual definition of _[[smooth manifold]]_ carries over word by word:

+-- {: .num_defn}
###### Definition
A **Fréchet manifold** is a [[Hausdorff topological space]] with an [[atlas]] of [[coordinate charts]] taking their value in [[Fréchet spaces]], such that the coordinate transition functions are all smooth functions between Fréchet spaces.
=--

It is possible to generalize some concepts of differential geometry from the finite case to the Fréchet case, one has to be careful, however:

1. The [[dual vector space|dual]] of a Fréchet space that is not a [[Banach space]] is never a Fréchet space, therefore one cannot e.g. define both the tangent and the cotangent bundle as Fréchet manifolds. More serious is however

2. The existence and uniqueness theorems for ordinary differential equations fail in infinite dimensions, so that theorems depending on that from finite dimensional differential geometry cannot be transcribed to the infinite situation in general. It is possible to do this on a case by case basis however.

### Tangent Vectors

There are several definitions of tangent vectors that are equivalent in the finite dimensional setting, but may be different in infinite dimensions. Tangent vectors can be defined to be derivations on germs of functions (algebraic definition), or as equivalence classes of smooth curves (kinematic definition). For the time being we settle with the kinematic definition:

+-- {: .num_defn}
###### Definition
**kinematic tangent vector**

The kinematic tangent vector space of a Fréchet manifold $M$ at a point $p$ consists of all pairs $(p, c'(0))$ where $c$ is a smooth curve
$$
c: \mathbb{R} \to M \; \text{with} \; c(0) = p
$$
As usual, the set of pairs $(p, c'(0)), p \in M$ forms a Fréchet manifold, the tangent bundle $TM$.
=--

The last sentence makes use of the notion of vector bundle, which can be defined exactly as in the finite dimensional setting:

### Vector bundles

+-- {: .num_defn}
###### Definition
**vector bundle**

A Fréchet manifold $V$ is a Fréchet vector bundle over $M$ with projection $\pi$, if for every point $p \in M$ there are charts of $M$ and $V$ such that $V$ is mapped locally to $U \subset F \times G$ for Fréchet spaces $F, G$, the projection $\pi$ corresponds to the projection of $U \times G$ to $U$, and the vector space structure on each fibre is induced by the vector space structure on $G$.
=--

Since, as mentioned before, the dual space of a Fréchet space that is not a Banach space is itself not a Fréchet space, we cannot define the cotangent space canonically as the dual space of the tangent space. Instead we define it directly:

### Differential forms

+-- {: .num_defn}
###### Definition
**differential form**

A differential form (a one form) $\alpha$ is a smooth map
$$
\alpha: T M \to \mathbb{R}
$$
where $TM$ is the tangent bundle.
=--

## Properties

### Relation to diffeological spaces  {#RelationBetweenDeffeologicalAndFrechetStructure}

We discuss how Fréchet manifolds form a [[full subcategory]] of that of [[diffeological spaces]].

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

Define a [[functor]]

$$
  \iota \;\colon\; FrechetManifolds \longrightarrow DiffeologicalSpaces
$$

from Fréchet manifolds to [[diffeological spaces]] (and hence to [[smooth spaces]] and  [[smooth stacks]])
in the evident way by taking for $X$ a [[Fréchet manifold]] for any $U \in $ [[CartSp]] the set of $U$-plots of $\iota(X)$ to be the set of smooth functions $U \to X$.

=--

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

The functor $\iota \colon FrechetManifolds \hookrightarrow DiffeologicalSpaces$ is a [[full and faithful functor]].

=--

This appears as ([Losik, theorem 3.1.1](#Losik94)).

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

Let $X, Y \in SMoothManifold$ with $X$ a [[compact manifold]].

Then under this embedding, the diffeological [[mapping space]] structure $C^\infty(X,Y)_{diff}$ on the mapping space coincides with the Fréchet manifold structure $C^\infty(X,Y)_{Fr}$:

$$
  \iota(C^\infty(X,Y)_{Fr})
  \simeq
  C^\infty(X,Y)_{diff}
  \,.
$$

=--

This appears as ([Waldorf, lemma A.1.7](#Waldorf)).

## Examples

### Smooth mapping spaces

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

For $X,Y$ two [[smooth manifolds]], such that in addition $X$ is [[compact topological space|compact]], then the [[mapping space]], i.e. the set of [[smooth functions]] $C^\infty(X,Y)$ is naturally a Fréchet manifold. Under the [[full subcategory]] inclusion of Fréchet manifolds into [[diffeological spaces]] and [[smooth sets]] (prop. \ref{FFEmbeddingOfFrechetInDiffeological}) this coincides with the canonical [[mapping space]] formed there.

For example [[smooth loop space]] (i.e. for $X = S^1$ the [[circle]]) are Fréchet manifolds.

For details on this see at _[[manifold structure of mapping spaces]]_.

=--

### Projective limits of smooth finite-dimensional manifolds
 {#ProjectiveLimitsOfSmoothFiniteDimensionalManifolds}

(see also [Dodson-Galanis-Vassiliou 15](#DodsonGalanisVassiliou15))

Fréchet manifolds may be thought of as [[projective limits]] of [[Banach manifolds]] (see the "added remark" at the end of [this MO comment](http://math.stackexchange.com/a/53020/58526))

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

The infinite product [[Fréchet space]] $\mathbb{R}^\infty$ ([exmpl.](Fréchet+space#ProjRInfinity)) is of course a Fréchet manifold.

=--

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

As a Fréchet manifold, $\mathbb{R}^\infty$ (example \ref{RInfinity}) should be the [[projective limit]]

$$
  \mathbb{R}^\infty \simeq \underset{\longleftarrow}{\lim}_n \mathbb{R}^n
$$

formed in the category of Fréchet manifolds.

=--

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

The point that needs checking is that for $X$ any Fréchet manifold, then a [[continuous function]]

$$
  f \colon X \longrightarrow \mathbb{R}^\infty
$$

is smooth as soon as all its components

$$
  f_n \colon X \overset{f}{\longrightarrow} \mathbb{R}^\infty
  \overset{p_n}{\longrightarrow} \mathbb{R}
$$

are smooth. This is checked for instance in ([Saunders 89, lemma 7.1.8](#Saunders89)).

=--

Conversely:

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

A function

$$
  \mathbb{R}^\infty
    \longrightarrow
  \mathbb{R}
$$

_out_ of $\mathbb{R}^\infty$ (example \ref{RInfinity}) is differentiable precisely if at each point only a [[finite number]] of its partial derivative are non-vanishing.

=--

([Saunders 89, example 7.1.6](#Saunders89))

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

As a global generalization of the pro-finite dimensional Fréchet manifold $\mathbb{R}^\infty$ of example \ref{RInfinity}, every infinite [[jet bundle]] $J^\infty E = \underset{\longleftarrow}{\lim}_k J^k E$ is a Fréchet manifold, modeled on $\mathbb{R}^\infty$ ([Saunders 89, chapter 7](#Saunders89)).

=--

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

Beware, that infinite [[jet bundles]] are also naturally thought of as [[pro-manifolds]]. This differs from the Frechet manifold structure of example \ref{JetBundlesAsFrechetManifolds}:

A morphism of [[pro-manifolds]]

$$
  f \colon J^\infty E \longrightarrow \mathbb{R}
$$

is equivalently a function that is "globally of finite order", in that there exists $k \in \mathbb{N}$ and an ordinary smooth function $f_k \colon J^k E \to \mathbb{R}$ such that $f = f_k \circ p_k$.

But by prop. \ref{DifferentiableFunctionOutOfRInfinity} a morphisms of Fréchet manifolds

$$
  f \colon J^\infty E \longrightarrow \mathbb{R}
$$

is only restricted to have finite order of partial derivatives at every point.

This is a weaker condition. In fact it seems to be also weaker than the condition of being "locally of finite order" considered in [Takens 79](jet+bundle#Takens79). (The function $f$ is _locally of finite order_ if for every point in $J^\infty E$ there exists a $k \in \mathbb{N}$ and an open neighbourhood $U_k$ of its image in $J^k E$ and a smooth function $f^U_k \colon U_k \to \mathbb{R}$ such that restricted to the pre-image of $U_k$ in $J^\infty E$ the function $f$ given by $f_k \circ p_k $).

Hence it makes sense to speak of _[[locally pro-manifolds]]_.

=--

## Related concepts

* [[Banach manifold]]

* [[Hilbert manifold]], [[ILH manifold]]

* [[convenient manifold]]

* [[F-star-algebra]]

* [[geometry of physics -- smooth sets]]

## References

Accounts include:

* [[Richard S. Hamilton]], Section I.4 in: *The inverse function theorem of Nash and Moser*, Bull. Amer. Math. Soc. **7** (1982) 65-222 [[doi:1982-07-01/S0273-0979-1982-15004-2](https://www.ams.org/journals/bull/1982-07-01/S0273-0979-1982-15004-2)]

* {#Michor80} [[Peter Michor]], _Manifolds of differentiable mappings_, Shiva Publishing (1980) [pdf](http://www.mat.univie.ac.at/~michor/manifolds_of_differentiable_mappings.pdf)

* {#KM} [[Andreas Kriegl]], [[Peter Michor]]: _The convenient setting of global analysis_, AMS (1997) ([pdf](http://www.mat.univie.ac.at/~michor/apbookh.pdf))

* V. I. Arnold,  B. A. Khesin, _Topological methods in hydrodynamics._ (Springer 1998, [ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0902.76001&format=complete))

* Boris Khesin, Robert Wendt, _The geometry of infinite-dimensional groups._ (Springer 2009, [ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1153.22001&format=complete))

The embedding into [[diffeological spaces]] is due to

* {#Losik92} [[Mark Losik]], _Fréchet manifolds as diffeologic spaces_, Russian Mathematics 36:5 (1992), 36–42.  English translation: [PDF](https://dmitripavlov.org/scans/losik-frechet-manifolds-as-diffeologic-spaces.pdf).  Russian original: М. В. Лосик, _О многообразиях Фреше как диффеологических пространствах_, Изв. вузов. Матем. (Izv. Vyssh. Uchebn. Zaved. Mat.), 1992, issue 5, 36–42, ([mathnet:ivm4812](http://mi.mathnet.ru/eng/ivm4812))

and reviewed in

* {#Losik94} [[Mark Losik]], Section 3 of: _Categorical Differential Geometry_. Cah. Topol. Géom. Différ. Catég., 35(4):274–290, 1994 ([numdam:CTGDC_1994__35_4_274_0](http://www.numdam.org/item/CTGDC_1994__35_4_274_0))

The preservation of [[mapping spaces]] under this embedding is due to

* {#Waldorf} [[Konrad Waldorf]], _Transgression to Loop Spaces and its Inverse I_ ([arXiv:0911.3212](http://arxiv.org/abs/0911.3212))

Fréchet manifold structure on [[jet bundles]] is discussed in

* {#Saunders89} [[David Saunders]], chapter 7 of _The geometry of jet bundles_, London Mathematical Society Lecture Note Series __142__, Cambridge Univ. Press 1989.

* {#DodsonGalanisVassiliou15} C. T. J. Dodson, George Galanis, Efstathios Vassiliou,, p. 109 and section 6.3 of _Geometry in a Fréchet Context: A Projective Limit Approach_, Cambridge University Press (2015)

[[!redirects Fréchet manifolds]]

[[!redirects Frechet manifold]]
[[!redirects Frechet manifolds]]


Fréchet space

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

# Fréchet spaces
* table of contents
{: toc}

## Idea

Fréchet spaces are particularly well-behaved [[topological vector spaces]] (TVSes).  Every [[Cartesian space]] $\mathbb{R}^n$ is a Fréchet space, but Fréchet spaces may have non-[[finite number|finite]] [[dimension]].  There is [[analysis]] on Fréchet spaces, yet they are more general than [[Banach spaces]]; as such, they are popular as local model spaces for possibly [[infinite-dimensional manifolds]]: _[[Fréchet manifolds]]_.

A basic example of a Fréchet space is $\mathbb{R}^\infty \coloneqq \underset{\longleftarrow}{\lim} \mathbb{R}^n$, as a [[topological space]] the _[[projective limit]]_ over the finite dimensional [[Cartesian spaces]] $\mathbb{R}^n$ (example \ref{ProjRInfinity} below) . This is not a [[Banach space]] anymore, since it does not carry a compatible [[norm]] anymore (e.g. [Saunders 89](#Saunders89), p. 253). But it evidently does carry the functions $\mathbb{R}^\infty \overset{p^n}{\longrightarrow} \mathbb{R}^n \overset{\Vert -\Vert_n}{\longrightarrow} \mathbb{R}$ for all $n \in \mathbb{N}$, where $p^n$ is the defining [[projection]] and where ${\Vert -\Vert}_n$ is the standard [[norm]] on $\mathbb{R}^n$. While not norms, these composites are [[seminorms]] on $\mathbb{R}^\infty$, they only fail the non-degeneracy condition saying that only the 0-vector has vanishing norm.

Generally, a Fréchet space is equivalently a [[real vector space]] equipped with a countable familiy of [[seminorms]], with compatibility conditions modeled on this example. See def. \ref{FrechetSpaceAsCompleteTVSWithCompatibleSeminorms} below.

Beware the clash ofterminology: a 'Fréchet topology' on a 'Fréchet topological space' is something different; this just means that a [[topological space]] satisfies the $T_1$ [[separation axiom]].  (Like all [[Hausdorff space|Hausdorff]] [[topological vector spaces]], Fréchet spaces satisfy this axiom, but they have a good deal of additional [[structure]] and [[properties]].)

## Definition

There are various equivalent definitions of Fréchet spaces:

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

A __Fréchet space__ is equivalently a [[complete space|complete]] [[Hausdorff space|Hausdorff]] [[locally convex vector space]] that is [[metrisable topological space|metrisable]].  The metric can be chosen to be translation-invariant.

=--

+-- {: .num_defn #FrechetSpaceAsCompleteTVSWithCompatibleSeminorms}
###### Definition
**(via systems of [[seminorms]])**

A **Fréchet space** is a [[complete space|complete]] [[Hausdorff space|Hausdorff]] [[topological vector space]] $V$ whose [[topological space|topology]] may be given (as a [[gauge space]]) by a [[countable set|countable]] family of [[seminormed vector space|seminorms]], hence for which there exists a family of [[seminorms]]

$$
  {\Vert - \Vert}_n \;\colon\; V \longrightarrow \mathbb{R}
  \,,
  \;\;\;
   n \in \mathbb{N}
$$

such that the set of all [[open balls]] of the form

$$
  B_\epsilon^{(n)}(x)
  \coloneqq
  \left\{
    y \in V \,\vert\,  {\Vert x-y \Vert}_n \lt \epsilon
  \right\}
  \;\;\;\;\;\;\;\;
  for
  \;
  x \in V\,, \epsilon \gt 0 \,, n \in \mathbb{N}
$$

is a [[base of neighborhoods]] of $x$.

=--

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

We accept as an [[automorphism]] of Fréchet spaces any [[linear map|linear]] [[homeomorphism]]; in particular, the particular translation-invariant metric or countable family of [[seminorms]] used to prove that a space is a Fréchet space is *not* required to be preserved.  More generally, the [[morphisms]] of Fréchet spaces are the [[continuous map|continuous]] linear maps, so that Fréchet spaces form a [[full subcategory]] of the category $TVS$ of [[topological vector spaces]].

=--

## Examples

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

Every [[Banach space]] is a Fréchet space.

=--

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

If $X$ is a [[compact space|compact]] [[smooth manifold]], then the space of [[smooth map|smooth maps]] on $X$ is a Fréchet space.  This can be extended to some non-compact manifolds, in particular when $X$ is the [[real line]].

=--

+-- {: .num_example #SchwartSpaceOfFunctionsIsFrechetSpace}
###### Example
**(the [[Schwartz space]] is a Fréchet space)**

The [[Schwartz space]] $\mathcal{S}$ of functions with rapidly decreasing derivatives ([this def.](Schwartz+space#SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives))
is a Fréchet space.

=--

See [this prop.](Schwartz+space#SchwartSpaceOfFunctionsIsFrechetSpace)

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

The [[Lebesgue space]] $L^p(\mathbb{R})$ for $p \lt 1$ is *not* a Fréchet space, because it is not locally convex.

=--

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

Consider the [[direct product]] (as [[topological vector spaces]]) of a countable number of copies of the [[real line]] $\mathbb{R}$

Equivalently the [[projective limit]] (as [[topological vector spaces]])

$$
  \mathbb{R}^\infty
    \coloneqq
  \underset{\longleftarrow}{\lim}_n \mathbb{R}^n
  =
  \underset{\longleftarrow}{\lim}
  \left(
    \cdots \to \mathbb{R}^2 \overset{}{\to} \mathbb{R}^1 \to \mathbb{R}^0
  \right)
$$

over all [[Cartesian spaces]] via their canonical [[projection maps]].

(Beware that the same symbol "$\mathbb{R}^\infty$" is also used for the limit of
the same sequence but with $\mathbb{R}^n$ with discrete topology, what leads to a [[linearly compact vector space]] as well as for the [[direct sum]]/[[inductive limit]] of $\mathbb{R}\to \mathbb{R}^2\hookrightarrow\mathbb{R}^3\hookrightarrow\ldots$, which is different.)

Write

$$
  \pi_n \colon \mathbb{R}^\infty \longrightarrow \mathbb{R}^n
$$

for the induced [[projection]] maps onto the first $n$ copies and let $\|\cdot\|_n$ be the canonical [[norm]] on $\mathbb{R}^n$.

Then a compatible countable family of seminorms on $\mathbb{R}^\infty$, according to def. \ref{FrechetSpaceAsCompleteTVSWithCompatibleSeminorms}, is given by $v \mapsto {\Vert\pi_n(v) \Vert_n}$. Hence equipped with these, $\mathbb{R}^\infty$ becomes a Fréchet space.

=--

On the other hand, the locally convex direct sum of a countable number of copies of $\mathbb{R}$ is not a Fréchet space.

## Properties

### General

Fréchet spaces are [[barrelled]] and [[bornological]].

### Duals

The [[dual space|dual]] [[topological vector space]] of a Fréchet space $F$ is itself again a Fréchet space precisely only if $F$ is in fact a [[Banach space]].

This follows from the statement paragraph 29.1 (7) in ([Koethe](#Koethe)), which is: The strong dual of a locally convex metrizable TVS $F$ is metrizable iff $F$ is normable.

See also ([Saunders 89, p. 255](#Saunders89)).

### As projective limits
 {#AsProjectiveLimits}

Every [[complete topological space|complete]] [[locally convex topological vector space]] $X$ is the [[cofiltered category|cofiltered]] [[projective limit]] of [[Banach spaces]] in the [[category]] of [[locally convex spaces]]. (Note that Fréchet spaces are additionally required to be _metrisable_, so this is more general.)

To see this, choose a base $\{U_{\alpha}\}_{\alpha \in A}$ of the neighborhood filter of $0$, consisting of convex, balanced and absorbing sets and let $p_{\alpha}$ be Minkowski functional associated to $U_{\alpha}$. The Hausdorffification $X_{\alpha}$ of $(X, p_{\alpha})$ is easily seen to be a Banach space and because $A$ is directed by reverse inclusion so is $X_{\alpha}$. It is straightforward to check that $X = \underset{\longleftarrow}{\lim} X_{\alpha}$ in the category of locally convex spaces. For details, see ([Schaefer-Wolff 99, Chapter II.§5](#SchaeferWolff99),  [page 51ff](http://books.google.com/books?id=9kXY742pABoC&pg=PA51).

Now given that a Fréchet space admits a *decreasing sequence* of convex balanced and absorbing neighborhoods, it follows immediately that:

Every Fréchet space is a [[sequential limit|sequential]] [[projective limit]] of [[Banach spaces]].

Conversely, any limit of a countable sequence of Banach spaces is a Fréchet space.

See ([Schaefer-Wolff 99](#SchaeferWolff99)), Chapter II.§4, page 48f (as well as Theorem I.6.1, page 28).

(from [this math.stackexchange comment](http://math.stackexchange.com/a/53020/58526))

See also example \ref{ProjRInfinity} below.

### Path smoothness

+-- {: .num_defn #PathSmoothFunction}
###### Definition
**(path smooth linear function)

Let $V$ be a Fréchet vector space (def. \ref{FrechetSpaceAsCompleteTVSWithCompatibleSeminorms}). Then a [[linear function]]

$$
  \mu \;\colon\; V \longrightarrow \mathbb{R}
$$

is called _path smooth_ if for every [[smooth function]]

$$
  g \;\colon\; \mathbb{R} \longrightarrow V
$$

the [[composition|composite]]

$$
  \mu \circ g \;\colon\; \mathbb{R} \longrightarrow \mathbb{R}
$$

is a [[smooth function]].

=--

+-- {: .num_defn #PathSmoothLinearFunctionsOnFrechetSpaceAreContinuous}
###### Proposition
**(path-smooth linear functions on a Fréchet space are continuous)**

Let $V$ be a Fréchet vector space (def. \ref{FrechetSpaceAsCompleteTVSWithCompatibleSeminorms}). Then every [[linear function]] on $V$ which is path-smooth (def. \ref{PathSmoothFunction}) is [[continuous function|continuous]].

=--

([Moerdijk-Reyes 91, chapter II, lemma 3.7](#MoerdijkReyes91))

Prop. \ref{PathSmoothLinearFunctionsOnFrechetSpaceAreContinuous} implies for instance that _[[distributions are the smooth linear functionals]]_. See there for more.

## Differentiable and smooth functions
 {#DifferentiableAndSmoothFunctions}

It is possible to generalize some aspects of [[analysis]] (differential calculus) to Fréchet spaces (e.g. [Michor 80, chapter 8](#Michor80), [Saunders 89, p. 256](#Saunders89)).

For example the definition of the [[derivative]] of a curve is simply the same as in finite dimensions:

+-- {: .num_defn}
###### Definition
For a continuous path in a Fréchet space $f(t)$ we define
$$
f'(t) = \lim_{h \to 0} \frac{1}{h} (f(t + h) - f(t))
$$
If the limit exists and is continuous, we say that $f$ is continuously differentiably or $C^1$.
=--

And just as in the finite dimensional case, we can define the partial derivative, or rather: the [[directional derivative|directional]] or Gâteaux derivative:

+-- {: .num_defn}
###### Definition
**directional derivative**

Let $F$ and $G$ be Fréchet spaces, $U \subseteq F$ open and $P: U \to G$ a nonlinear continuous map. The derivative of $P$ at the point $f \in U$ in the direction $h \in F$ is the map
$$
D P: U \times F \to G
$$
$$
D P(f) h = \lim_{t \to 0} \frac{1}{t} ( P(f + t h) - P(f))
$$
If the limit exists and is jointly continuous in both variables we say that $P$ is continuous differentiable or $C^1$.
=--

A simple, but nontrivial example is the operator
$$
P: C^{\infty}[a, b] \to C^{\infty}[a, b]
$$
$$
P(f) \coloneqq f f'
$$
with the derivative
$$
D P(f) h = f'h + f h'
$$

It is possible to generalize the Riemann integral to Fréchet spaces, too: For a continuous path $f(t)$ on an interval $[a, b]$ in a Fréchet space $F$ we look for an element $\int_a^b f(t) d t \in F$. It turns out that such an element exists and is unique, if we impose some properties of the integral known from the finite dimensional case:

+-- {: .num_theorem}
###### Theorem
There exists a unique element $\int_a^b f(t) d t \in F$ such that

(i) for every continuous functional $\phi$ we have  $\phi(\int_a^b f(t) d t) = \int_a^b \phi(f(t)) d t$,

(ii) for every continuous seminorm ${\| \cdot \|}$ we have ${\| \int_a^b f(t) d t \|} \leq \int_a^b {\| f(t) \|} d t$

(iii) integration is linear and

(iv) additive, i.e. $ \int_a^b f(t) d t + \int_b^c f(t) d t = \int_a^c f(t) d t $

=--

There is a version of the fundamental theorem of calculus:

+-- {: .num_theorem}
###### Theorem
If P is $C^1$ and $f + t h \in Domain(P)$ for $0 \leq t \leq 1$, then
$$
P(f + h) - P(f) = \int_0^1 D P(f + t h) \;h \; d t
$$

=--

The [[chain rule]] is valid:

+-- {: .num_theorem}
###### Theorem
If $P$ and $Q$ are $C^1$ then so is their composition $Q \circ P$ and
$$
D [Q \circ P](f) h = D Q(P(f)) \; D P(f) \; h
$$
=--

The first derivative $D P$ is a function of two variables, the base point $f$ and the direction $h$. Since $D P$ is already linear in $h$, we define the second derivative with respect to $f$ only:

+-- {: .num_defn}
###### Definition
**second derivative**
The second derivative of $P$ in the direction $k$ is defined to be
$$
D^2 P(f) (h, k) = \lim_{t \to 0} \frac{1}{t} (D P(f + t k) h - D P(f) h)
$$
=--

It is a theorem that the second derivative, if it exists and is jointly continuous, is bilinear in $(h ,k)$.

We can iterate this procedure to define derivatives of arbitrary order, and thus the notion of **smooth functions between Fréchet spaces**. This allows to define the concept of smooth _[[Fréchet manifolds]]_.

## References

* {#Vogt00} [[Dietmar Vogt]], _Lectures on Fréchet spaces_, 2000 ([pdf](http://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf))

* PlanetMath, _[Fréchet space](http://planetmath.org/frechetspace)_

* Wikipedia, _[Fréchet space](https://secure.wikimedia.org/wikipedia/en/wiki/Fréchet_space)_

* {#Koethe} Gottfried Koethe: _Topological Vector Spaces I_

* {#SchaeferWolff99} H. Schaefer, Manfred Wolff *[Topological Vector Spaces](http://books.google.com/books?id=9kXY742pABoC)*, Springer (1999)

* [[Andreas Kriegl]], _Fréchet Spaces_ lecture notes, 2016 ([pdf](http://www.mat.univie.ac.at/~kriegl/Skripten/2016SS.pdf))

Discussion of analysis on Fréchet spaces includes

* {#Michor80} [[Peter Michor]], chapter 8 of _Manifolds of differentiable mappings_, Shiva Publishing (1980) [pdf](http://www.mat.univie.ac.at/~michor/manifolds_of_differentiable_mappings.pdf)

* Richard S. Hamilton: _The Inverse Function Theorem of Nash and Moser_ (Bulletin (New Series) of the American Mathematical Society Volume 7, Number 1, July 1982)

Discussion in the context of [[jet bundles]] and [[locally pro-manifolds]] includes

* {#Saunders89} [[David Saunders]], chapter 7 of _The geometry of jet bundles_, London Mathematical Society Lecture Note Series __142__, Cambridge Univ. Press 1989.

* [[Igor Khavkine]], [[Urs Schreiber]], section 2.2 of _[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

Discussion in the context of [[synthetic differential geometry]] includes

* {#MoerdijkReyes91} [[Ieke Moerdijk]], [[Gonzalo Reyes]], around lemma 3.7 in chapter II of _[[Models for Smooth Infinitesimal Analysis]]_, Springer 1991

Refinement to [[noncommutative geometry]] by suitable smoothed [[C-star-algebras]] is discussed in

* Nikolay Ivankov, _Unbounded bivariant K-theory and an Approach to Noncommutative Fréchet spaces_ [pdf](http://hss.ulb.uni-bonn.de/2011/2624/2624.pdf)

[[!redirects Frechet space]]
[[!redirects Frechet spaces]]
[[!redirects Fréchet space]]
[[!redirects Fréchet spaces]]

[[!redirects Fréchet vector space]]
[[!redirects Fréchet vector spaces]]
[[!redirects Frechet vector space]]
[[!redirects Frechet vector spaces]]


Frédéric Chazal

* [personal page](https://geometrica.saclay.inria.fr/team/Fred.Chazal/)

## Selected writings

On [[persistence modules]] and [[stability of persistence diagrams]]:

* [[Frédéric Chazal]], [[David Cohen-Steiner]], [[Marc Glisse]], [[Leonidas J. Guibas]], [[Steve Y. Oudot]], *Proximity of persistence modules and their diagrams*, SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry (2009) 237–246 $[$[doi:10.1145/1542362.1542407](https://doi.org/10.1145/1542362.1542407)$]$

* [[Frédéric Chazal]], [[Vin de Silva]], [[Marc Glisse]], [[Steve Oudot]],  *The structure and stability of persistence modules*, SpringerBriefs in Mathematics, Springer (2016) $[$[arXiv:1207.3674](https://arxiv.org/abs/1207.3674), [doi:10.1007/978-3-319-42545-0](https://doi.org/10.1007/978-3-319-42545-0)$]$

category: people

[[!redirects Frederic Chazal]]


Frédéric Déglise

Frédéric Déglise is a mathematician working on the theory of motives and related areas of algebraic geometry, and based at the E.N.S. Lyon.

* [web](http://perso.ens-lyon.fr/frederic.deglise/)

## Selected writings

On [[triangulated categories]] of [[mixed motives]] and their [[yoga of six functors]]:

* {#CisinskiDeglise09} [[Denis-Charles Cisinski]], [[Frédéric Déglise]], *Triangulated categories of mixed motives*, Springer Monographs in Mathematics, Springer (2019) [[arXiv:0912.2110](http://arxiv.org/abs/0912.2110), [doi:10.1007/978-3-030-33242-6](https://doi.org/10.1007/978-3-030-33242-6)]

and the relevant [[model structures on chain complexes]]:

* [[Denis-Charles Cisinski]], [[Frédéric Déglise]], *Local and stable homological algebra in Grothendieck abelian categories*, Homology, Homotopy and Applications **11** 1 (2009) 219–260 [[arXiv:0712.3296](http://arxiv.org/abs/0712.3296), [hha:1251832567](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-11/issue-1/Local-and-stable-homological-algebra-in-Grothendieck-abelian-categories/hha/1251832567.full)]

## Related entries

* [[motives]]

  * [[mixed motives]]
  * [[category of motives]]
  * [[motivic homotopy theory]]
  * [[motivic cohomology]]
  * [[algebraic K-theory]]

* [[model structure on chain complexes]]

category: people

[[!redirects Frederic Deglise]]
[[!redirects Déglise]]
[[!redirects Deglise]]


Frédéric Hélein

* [wikipedia entry](http://en.wikipedia.org/wiki/Fr%C3%A9d%C3%A9ric_H%C3%A9lein)

category: people

[[!redirects Frederic Helein]]


Frédéric Paugam

Welcome on my personal webpage at the nlab.

# Position {#position}

I am currently maître de conférence (assistant professor) of
[University Paris 6](http://www.upmc.fr/) at
[Jussieu's mathematical institute](http://www.institut.math.jussieu.fr/)
in the
[Algebraic analysis group](http://www.institut.math.jussieu.fr/projets/aa/).

# Research {#myresearch}

I have two main domains of interest:

Mathematics of quantum field theory: i wrote a book whose aim was to give a coordinate free (i.e., categorical) presentation of many tools (and theories) used in the classical and quantum aspects of field theory:

* [Towards the mathematics of quantum field theory](http://people.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics.pdf).

I didn't succeed in fulfilling my curiosity in this domain.

Arithmetic geometry: i did my Phd thesis on Abelian varieties, Galois representations and Shimura varieties. My recent interests are related to [[global analytic geometry]], [[mixed geometry]] and [[global Hodge theory]], with views towards applications to
the geometric study of (analytic) number theory. You can find a presentation of some motivations in the master course notes
(preliminary form):

* [Global analytic geometry and the functional equation](http://people.imj-prg.fr/~frederic.paugam/documents/enseignement/master-global-analytic-geometry.pdf)

and at [[global Analytic geometry]].

I think that the combination of the three above tools may be quite useful to better understand the geometric meaning of functional symmetries.

# Webpage {#contact}

You can find further informations on me (publications, preprints, etc...) on my
[academic webpage](http://www.imj-prg.fr/~frederic.paugam/parts-en/index.html).

[[!redirects Frederic Paugam]]

category: people


Frédéric Piéchon

* [arXiv page](https://arxiv.org/search/cond-mat?searchtype=author&query=Pi%C3%A9chon%2C+F)

## Selected writings

Models for [[Chern insulators]] akin to the [[Haldane model]]:

* Doru Sticlet, [[Frédéric Piéchon]], [[Jean-Noël Fuchs]], Pavel Kalugin, Pascal Simon, *Geometrical engineering of a two-bands Chern insulator in two dimensions with arbitrary topological index*, Phys. Rev. B **85** 165456 (2012) [[arXiv:1201.6613](https://arxiv.org/abs/1201.6613), [doi:10.1103/PhysRevB.85.165456](https://doi.org/10.1103/PhysRevB.85.165456)]

* [[Marwa Mannaï]], [[Jean-Noël Fuchs]], [[Frédéric Piéchon]], [[Sonia Haddad]], *Stacking-induced Chern insulator* [[arXiv:2208.02491](https://arxiv.org/abs/2208.02491)]

category: people

[[!redirects Frederic Piechon]]


Frölicher space

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

{:myproof: .proof style="margin-left:2em;"}
{:mynumdef: .num_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}
{:goal: .un_remark style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

* [Idea](#idea)
* [Basics](#basics)
* [[examples of Frölicher spaces|Examples]]
* [[Frölicher spaces and Isbell envelopes|Relationship]] to [[Isbell envelope]]
* [[categorical properties of Frölicher spaces|Categorical Matters]]
* [[topological notions of Frölicher spaces|Topological Matters]]
* [[tangential notions of Frölicher spaces|Tangential Structures]]
* [References](#refs)

# Idea
{#idea}

A Frölicher space is one flavour of a [[generalized smooth space]].

Frölicher smooth spaces are determined by a rule for

* how to map the real line smoothly into it,

* and how to map out of the space smoothly to the real line.

In the general context of [[space and quantity]], Frölicher spaces take an intermediate symmetric position: they are both [[presheaf|presheaves]] as well as copresheaves on their test domain (which here is the full [[subcategory]] of manifolds on the real line) and both of these structures determine each other.

The general abstract idea behind this is described at [[Isbell envelope]].

+-- {: goal}
###### Goal
The intention for these pages is to develop the basic tools of differential topology for Frölicher spaces.
This means taking the basic pieces of "ordinary" differential topology and considering what they might look like for Frölicher spaces; including what looks the same and what looks different.

This project will both record existing structure and develop new ideas.
It is intentionally in the main area of the $n$-Lab to encourage contributions.
=--

# Basics
{#basics}

+-- {: mynumdef #FroelicherSpace}
###### Definition
A _Frölicher Space_ is a triple $(X,C_X,F_X)$ where

1. $X$ is a set;
2. $C_X$ is a set of _curves_ in $X$; that is, $C_X \subseteq Set(\mathbb{R},X)$;
3. $F_X$ is a set of _functionals_ on $X$; that is, $F_X \subseteq Set(X, \mathbb{R})$;

subject to the following saturation conditions

1. if $c\in C_X$ and $f\in F_X$, then $f c \in C^\infty(\mathbb{R}, \mathbb{R})$,
1. if $c : \mathbb{R} \to X$ is a set map with the property that $f c \in C^\infty(\mathbb{R}, \mathbb{R})$ for all $f \in F_X$ then $c \in C_X$, and
2. if $f : X \to \mathbb{R}$ is a set map with the property that $f c \in C^\infty(\mathbb{R}, \mathbb{R})$ for all $c \in C_X$ then $f \in F_X$.

A _morphism_ of Frölicher spaces, say $(X,C_X,F_X) \to (Y,C_Y,F_Y)$ is a set map $g : X \to Y$ satisfying the following (equivalent) conditions:

1. $g c \in C_Y$ for all $c \in C_X$,
2. $f g \in F_X$ for all $f \in F_Y$,
3. $f g c \in C^\infty(\mathbb{R}, \mathbb{R})$ for all $f \in F_Y$ and $c \in C_X$.
=--

Frölicher spaces and their morphisms form a category with an obvious faithful functor to the category of sets.
The properties of this category are as follows.

+-- {: .num_theorem #FroelicherCategory}
###### Theorem
The [[category]] of Frölicher spaces is [[complete category|complete]], [[cocomplete category|cocomplete]], and [[cartesian closed category|cartesian closed]].  It is [[topological category|topological]] over $Set$.  It is an [[concrete category|amnestic, transportable construct]].
=--

To its eternal shame, the category of Frölicher spaces is __not__ [[locally cartesian closed category|locally cartesian closed]].

# References
 {#refs}

The notion goes back to [[Alfred Frölicher]].

A detailed discussion of the category of Frölicher spaces and their relation to other notions of [[generalized smooth space]]s is given in

* [[Andrew Stacey]], _Comparative Smootheology_ Theory and Applications of Categories,  Vol. 25, 2011, No. 4, pp 64-117. ([tac](http://www.tac.mta.ca/tac/volumes/25/4/25-04abs.html))
{#Stacey}

This also lists all the relevant further references.

A discussion of [[Lie algebra]]s on Frölicher [[group]]s ([[group object]]s [[internalization|internal to]] the category of Frölicher spaces) is in

* Martin Laubinger, _A Lie algebra for Frölicher groups_ ([arXiv](http://arxiv.org/abs/0906.4486))

See also the unpublished thesis of Andreas Cap:

* [[Andreas Cap]], _K-theory for convenient algebras_ ([univie](http://www.mat.univie.ac.at/~michor/cap_diss.pdf))

Discussion in the context of applications to [[continuum mechanics]] is in

* [[William Lawvere]], [[Stephen Schanuel]] (eds.), _[[Categories in Continuum Physics]]_, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 986

[[!redirects Frölicher spaces]]
[[!redirects Frolicher space]]
[[!redirects Frolicher spaces]]
[[!redirects Froelicher space]]
[[!redirects Froelicher spaces]]
[[!redirects Frölicher Space]]
[[!redirects Frölicher Spaces]]
[[!redirects Frolicher Space]]
[[!redirects Frolicher Spaces]]
[[!redirects Froelicher Space]]
[[!redirects Froelicher Spaces]]