flexible limit

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### Context
#### 2-Category theory
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[[!include 2-category theory - contents]]
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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# Contents
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## Idea

A **flexible limit** is a [[strict 2-limit]] whose [[weighted limit|weight]] is [[cofibrant object|cofibrant]].  This implies that flexible limits are also [[2-limits]] (in the non-strict sense, which for us is the default -- recall that these are traditionally called [[bilimits]]).

Furthermore, all [[PIE-limits]] and therefore all [[strict pseudo-limits]] are flexible; thus any [[strict 2-category]] which admits all flexible limits also admits all $2$-limits.  A number of [[strict 2-categories]] admit all flexible limits, but not all strict $2$-limits, and this is a convenient way to show that they admit all $2$-limits.

## Definition

Let $D$ be a small [[strict 2-category]].  Write $[D,Cat]$ for the strict 2-category of [[strict 2-functors]], [[strict 2-natural transformation|strict 2-natural transformations]], and [[modifications]], and $Ps(D,Cat)$ for the strict 2-category of strict 2-functors, [[pseudonatural transformations]], and modifications.  The inclusion
$$ [D,Cat] \to Ps(D,Cat) $$
(as a [[wide subcategory]]) has a strict [[left adjoint]] $Q$, which is the [[pseudo morphism classifier]] for an appropriate [[strict 2-monad]].  Therefore, for any functor $\Phi \colon D\to Cat$, we have $Q\Phi \colon D\to Cat$ such that pseudonatural transformations $\Phi \to \Psi$ are in natural bijection with strict 2-natural transformations $Q\Phi \to \Psi$.

The [[counit of an adjunction|counit]] of this adjunction is a canonical strict 2-natural transformation $q\colon Q\Phi \to \Phi$.  We say that $\Phi$ is **flexible** if this transformation has a [[section]] in $[D,Cat]$.

## Examples

All [[PIE-limits]] are flexible.  This includes [[products]], [[inserters]], [[equifiers]] by definition, and also [[descent object|descent objects]], [[iso-inserters]], [[comma objects]], [[Eilenberg–Moore objects]], and so on.  In fact, PIE-limits have a characterization similar to the definition above of flexible limits: they are the coalgebras for $Q$ regarded as a 2-[[comonad]].

The [[split idempotent|splitting of idempotents]] is flexible, but not PIE.  Moreover, in a certain sense it is the "only" such.  Precisely, flexible limits are the [[saturated class of limits|saturation]] of each of the following classes of limits:

* PIE-limits together with splitting of idempotents
* PIE-limits together with splitting of idempotent equivalences
* strict pseudo-limits together with splitting of idempotents
* strict pseudo-limits together with splitting of idempotent equivalences
* [[products]], [[powers]], splitting of idempotents, [[pullbacks]] of [[normal isofibrations]]
* [[products]], [[powers]], splitting of idempotents, [[pullbacks]] of [[amnestic isofibrations]]

## Related pages

* Flexible limits may be characterised as the **persistent** [[double limits]].

* [[coherence theorem for bicategories with finite limits]]
* [[PIE-limit]]

## References

* G. J. Bird, [[Max Kelly]], [[John Power]], [[Ross Street]], _Flexible limits for 2-categories_, Journal of Pure and Applied Algebra **61** Issue 1 (1989) pp 1-27. doi:[10.1016/0022-4049(89)90065-0](http://dx.doi.org/10.1016/0022-4049(89%2990065-0)

* [[John Bourke]]. _Accessible aspects of 2-category theory_. Journal of Pure and Applied Algebra 225.3 (2021): 106519.

[[!redirects flexible limit]]
[[!redirects flexible limits]]

[[!redirects flexible 2-limit]]
[[!redirects flexible 2-limits]]


flexible magmoid

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### Context
#### Category theory
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[[!include category theory - contents]]
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#### Algebra
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[[!include higher algebra - contents]]
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#### Categorification
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[[!include categorification - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Just as a [[groupoid]] is the [[oidification]] of a [[group]] and a [[ringoid]] is the oidification of a [[ring]], an flexible magmoid should be the oidification of a [[flexible magma]].

## Definition

A __flexible magmoid__ is a [[magmoid]] $Q$ such that for every two objects $a,b \in Ob(Q)$ and for every two morphisms $f:a \to b$ and $g:b \to a$,

$$
  f \circ (g \circ f) = (f \circ g) \circ f
  \,.
$$

## Examples

* Every [[category]] is a flexible magmoid.

* A one-object flexible magmoid is a [[flexible magma]].

* A [[Mod]]-enriched flexible magmoid is called a flexible [[linear magmoid]].

## Related concepts

[[!include oidification - table]]


Floer homology

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

_Floer homology_ is a common name for several similar frameworks of infinite-dimensional analogues of [[Morse homology]], related to certain [[Fredholm operator]]s which arise as [[elliptic operator]]s on certain manifolds. Some of these are related to [[symplectic topology]]/[[symplectic geometry|geometry]] or [[contact geometry]] and another group is related to aspects of the geometry of 3-[[manifolds]]. One of these uses the [[Chern-Simons theory]] [[action functional]] as a [[Morse function]] on the space of [[connection on a bundle|connection]]s on $G$-[[principal bundle]]s over the given 3-manifold.

The most important flavours are the **Floer-Oh homology for [[action functional]]s** which is important in the study of [[symplectic manifold]]s, its version for Lagrangean intersection theory and the instanton Floer homology studied by [[Andreas Floer]] and important in the study of 3-manifolds and the mathematics of [[gauge theories]].

Floer homology for the [[action functional]] is a more systematic approach to the phenomena which were discovered several years before Floer in a historical article of [[Mikhail Gromov]] on [[J-holomorphic curves]].

## Examples and Variants

### Instanton Floer homology

For $\Sigma$ a 3-[[dimension]]al [[compact space|compact]] [[smooth manifold]] and $G$ a [[simply connected]] [[compact space|compact]] [[Lie group]] let $[\Sigma,\mathbf{B}G_{conn}]$ be the [[space]] of $G$-[[connection on a bundle|connections]] on $\Sigma$, which is equivalently the [[groupoid of Lie algebra valued forms]] on $\Sigma$ in this case.

The _instanton Floer homology_ groups of $\Sigma$ are something like the "mid-dimensional" [[singular homology]] groups of the [[configuration space]] $[\Sigma,\mathbf{B}G_{conn}]$.

More precisely, there is canonically the [[Chern-Simons action functional]]

$$
  S_{CS} : [\Sigma,\mathbf{B}G_{conn}] \to U(1)
$$

on this space of connections, and one can form the corresponding [[Morse homology]].

The [[critical locus]] of $S_{CS}$ is the space of flat $G$-connections (vanishing [[curvature]]), whereas the [[flow line]]s of $S_{CS}$ correspond to the [[Yang-Mills instanton]]s on $\Sigma \times [0,1]$.

For more see

* [[instanton Floer homology]].

## References

The original articles are

* [[Andreas Floer]],

  * _The unregularized gradient flow of the symplectic action_ , Comm. Pure Appl. Math. 41 (1988), 775–813.

  * _An instanton-invariant for 3-manifolds_ , Comm. Math. Phys. 118 (1988), no. 2, 215–240.

  * _Morse theory for Lagrangian intersections- , J. Differential Geom. __28__ (1988), no. 3, 513–547.

  * _Cuplength estimates on Lagrangian intersections_ , Comm. Pure Appl. Math. __42__ (1989), no. 4, 335–356.

  * _Symplectic fixed points and holomorphic spheres_ , Comm. Math. Phys. __120__, no. 4 (1989), 575–611.

  * _Witten's complex and infinite dimensional Morse Theory_ , J. Diff. Geom. __30__ (1989), p. 202–221.

Surveys are in

* [[Yong-Geun Oh]], _Symplectic topology and Floer homology_, [pdf](https://web.archive.org/web/20130603153154/https://math.wisc.edu/~oh/all.pdf)
 {#Oh}

* Darko Milinković, _Floer homology in classical mechanics and quantum field theory_, [ps](http://www.mphys5.phy.bg.ac.rs/proceedings3/Milinkovic.ps)

* [[Simon Donaldson]], _Floer homology groups in Yang-Mills theory_ Cambridge University Press (2002) ([pdf](http://catdir.loc.gov/catdir/samples/cam031/2001035888.pdf))

* Qingtao Chen, _Introduction to Floer Homology and its
relation with TQFT_ (2005) ([pdf](http://math.berkeley.edu/~alanw/242papers05/chen.pdf))

On [[holomorphic function|Holomorphic]] Floer Theory:

* [[Maxim Kontsevich]], [[Yan Soibelman]]. *Holomorphic Floer theory I: exponential integrals in finite and infinite dimensions*. (2024) ([arXiv:2402.07343](https://arxiv.org/abs/2402.07343))

See also

* [wikipedia](http://en.wikipedia.org/wiki/Floer_homology)


floor

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### Context
#### Arithmetic
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[[!include arithmetic geometry - contents]]
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# Floors and ceilings
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{: toc}

## Idea

By the *floor* and *ceiling* of a [[real number]] one means the [[integers]], obtained by rounding the real number down or up, respectively.

When viewed as [[functions]] from $\mathbb{R}$ to itself (or to $\mathbb{Z}$), these are standard examples of functions exhibiting partial notions of [[continuity]]: the floor function is both [[right-continuous map|right-continuous]] and [[upper semicontinuous map|upper semicontinuous]], while the ceiling function is both [[left-continuous map|left-continuous]] and [[lower semicontinuous map|lower semicontinuous]].  They are [[step functions]] used to approximate definite [[integrals]] of [[continuous maps]] and otherwise to relate integrals and [[series]].  They provide convenient notation to express various notions of [[rounding]].  From the perspective of [[order theory]], the maps may be seen as the [[right adjoint]] and [[left adjoint]] (respectively) of the [[inclusion map]] from $\mathbb{Z}$ to $\mathbb{R}$.

## Definitions

Given a [[real number]] $x$, the __floor__ of $x$, denoted $\lfloor{x}\rfloor$ or $[x]$, is the largest integer $n$ such that $n \leq x$, and the __ceiling__ of $x$, denoted $\lceil{x}\rceil$, is the smallest integer $n$ such that $n \geq x$.

Typically the notation $\lfloor{x}\rfloor$ is used when both floor and ceiling appear, but $[x]$ is often easier when only the floor is considered.  Since
$$ \lceil{x}\rceil = -\lfloor{-x}\rfloor ,$$
this is not actually a restriction.

The floor of $x$ is also called the __integer part__ of $x$, and then one refers as well to the __fractional part__ of $x$, denoted $\{x\}$, defined by
$$ \{x\} = x - [x] .$$

One must of course prove that the floor of $x$ exists; this fails in [[constructive mathematics]], although the floor of $x$ exists for [[full set|almost all]] $x$, and all of these functions can still be defined as continuous maps between appropriate [[locales]].

## Properties

### As adjoint functors
  {#AsAdjointFunctors}

We discuss how the floor and celing functions are left and right [[adjoint functors]] to the inclusion of the [[integers]] into the [[real numbers]], if both are regarded as [[posets]], and hence as [[categories]].

+-- {: .num_example #PartiallyOrderedSetsAsSmallCategories}
###### Example
**([[preordered sets]] as [[thin categories]])**

Let $(S, \leq)$ be a [[preordered set]]. Then this induces a [[small category]] whose [[set]] of [[objects]] is $S$, and which has precisely one morphism $x \to y$ whenever $x \leq y$, and no such morphism otherwise:

\[
  \label{RelationsAsMorphismInPartiallyOrderedSet}
  x \overset{\exists !}{\to} y
  \phantom{AAA}
  \text{precisely if}
  \phantom{AAA}
  x \leq y
\]

Conversely, every [[small category]] with at most one morphism from any object to any other, called a _[[thin category]]_, induces on its set of objects the [[structure]] of a  [[partially ordered set]] via (eq:RelationsAsMorphismInPartiallyOrderedSet).

Here the [[axioms]] for [[preordered sets]] and for [[categories]] match as follows:

| |  $\phantom{A}$[[reflexive relation|reflexivity]]$\phantom{A}$ | $\phantom{A}$[[transitive relation|transitivity]]$\phantom{A}$ |
|----|-----|------|
| $\phantom{A}$[[partially ordered sets]]$\phantom{A}$ | $\phantom{A}$ $x \leq x$ $\phantom{A}$ | $\phantom{A}$ $(x \leq y \leq z) \Rightarrow (x \leq z)$ $\phantom{A}$ |
| $\phantom{A}$[[thin categories]]$\phantom{A}$ | $\phantom{A}$[[identity morphisms]]$\phantom{A}$ | $\phantom{A}$[[composition]]$\phantom{A}$ |
{: style='margin:auto'}

=--

+-- {: .num_prop #FloorAndCeilingAsAdjointFunctors}
###### Proposition
**([[floor]] and [[ceiling]] as [[adjoint functors]])**

Consider the canonical inclusion

$$
  \mathbb{Z}_{\leq}
    \overset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow}
  \mathbb{R}_{\leq}
$$

of the [[integers]] into the [[real numbers]], both regarded as [[preorders]] in the standard way ("lower or equal"). Regarded as [[full subcategory]]-inclusion of the corresponding [[thin categories]], via Example \ref{PartiallyOrderedSetsAsSmallCategories}, this inclusion functor has both a left and right [[adjoint functor]]:

* the [[left adjoint]] to $\iota$ is the [[ceiling function]]

* the [[right adjoint]] to $\iota$ is the [[floor function]]

\[
  \label{AdjointTriple}
  \lceil(-)\rceil \;\;\dashv\;\; \iota \;\;\dashv\;\; \lfloor (-) \rfloor
  \,.
\]

Hence this induces an [[adjoint modality]], as discussed there.

The [[adjunction unit]] and [[adjunction counit]] express that each real number is in between its floor and ceiling

$$
  \iota \lfloor x \rfloor \;\leq\; x \;\leq\; \iota \lceil x \rceil
$$

Hence the [[modal objects]] in both cases are the [[integers]] among the [[real numbers]], while every real number is ceiling-submodal and floor-supmodal.

=--

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

First of all, observe that we indeed have [[functors]]

$$
  \lfloor(-)\rfloor \;,\;
  \lceil(-)\rceil
  \;\;\colon\;
  \mathbb{R}
    \longrightarrow
  \mathbb{Z}
$$

since floor and ceiling preserve the ordering relation.

Now in view of the identification of [[preorders]] with [[thin categories]] in Example \ref{PartiallyOrderedSetsAsSmallCategories}, the hom-isomorphism defining [[adjoint functors]] of the form $\iota \dashv \lfloor(-)\rfloor$ says for all $n \in \mathbb{Z}$ and $x \in \mathbb{R}$, that we have

$$
  \underset{ \in \mathbb{Z}}{\underbrace{n \leq \lfloor x \rfloor}}
  \;\Leftrightarrow\;
  \underset{ \in \mathbb{R}}{\underbrace{n \leq x }}
  \,.
$$

This is clearly already the defining condition on the [[floor]] function $\lfloor x \rfloor$.

Similarly, the  hom-isomorphism defining [[adjoint functors]] of the form $\lceil(-)\rceil \dashv \iota$ says that for all $n \in \mathbb{Z}$ and $x \in \mathbb{R}$, we have

$$
  \underset{ \in \mathbb{Z}}{\underbrace{\lceil x \rceil \leq n}}
  \;\Leftrightarrow\;
  \underset{ \in \mathbb{R}}{\underbrace{x \leq n }}
  \,.
$$

This is evidently already the defining condition on the [[floor]] function $\lfloor x \rfloor$.

Notice that in both cases the condition of a _[[natural isomorphism]]_ in both variables, as required for an [[adjunction]], is automatically satisfied: For let $x \leq x'$ and $n' \leq n$, then naturality means, again in view of the identifications in Example \ref{PartiallyOrderedSetsAsSmallCategories}, that

$$
  \array{
    (n \leq \lfloor x \rfloor) &\Leftrightarrow& (n \leq x)
    \\
    \Downarrow && \Downarrow
    \\
    (n' \leq \lfloor x' \rfloor) &\Leftrightarrow& (n' \leq x')
    \\
    \\
    \in \mathbb{Z} && \in \mathbb{R}
  }
$$

where the logical implications are equivalently functions between [[sets]] that are either [[empty set|empty]] or [[singletons]]. But Functions between such sets are unique, when they exist.

=--

## References

Wikipedia summarizes the basic properties:

* English Wikipedia. Floor and ceiling functions. [Web](https://en.wikipedia.org/wiki/Floor_and_ceiling_functions).

Chapter 6 of these notes uses the floor and ceiling functions throughout to relate [[integrals]] and [[series]] when teaching [[infinitesimal calculus]]:

* Toby Bartels. One-variable Calculus. [Web](http://tobybartels.name/MATH-1600/2017FA/calcbook/).

[[!redirects floor]]
[[!redirects floors]]
[[!redirects floor operation]]
[[!redirects floor operator]]
[[!redirects floor function]]
[[!redirects floor functor]]
[[!redirects floor map]]
[[!redirects floor mapping]]

[[!redirects ceiling]]
[[!redirects ceilings]]
[[!redirects ceiling operation]]
[[!redirects ceiling operator]]
[[!redirects ceiling function]]
[[!redirects ceiling functor]]
[[!redirects ceiling map]]
[[!redirects ceiling mapping]]


flop transition

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### Context
#### Geometry
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[[!include higher geometry - contents]]
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#### String theory
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[[!include string theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _flop transition_ is a continuous path in a space of _noncommutative 2-geometries_ that starts at an ordinary complex 3-dimensional [[Calabi-Yau space]], then passes through a point that does not correspond to an ordinary geometry (the _[[Gepner model|Gepner point]]_) and emerges afterwards again as an ordinary CY-geometry -- but now with different [[topology]].

This was found and discussed in the context of [[string theory]] but the phenomenon is a general abstract one in the theory of [[2d SCFTs]] regarded as generalized geometries -- as described at [[2-spectral triple]].

\begin{center}
<img src="https://ncatlab.org/nlab/files/FlopTransition.jpg" width="660">
\end{center}

>  graphics grabbed from [Greene 00](#Greene00)

## Related concepts

* [[conifold tansition]]

* [[moduli space of Calabi-Yau spaces]]

## References

A survey is in

* [[Brian Greene]], _String Theory on Calabi-Yau Manifolds_ ([arXiv:hep-th/9702155](http://arxiv.org/abs/hep-th/9702155))

A rough and brief survey of the flop transition and related phenomena with many pointers to original literature is also given in

* {#Greene00} [[Brian Greene]], _Aspects of collapsing cocycles_, in Proceedings of [Strings 2000](http://www.theory.caltech.edu/~preskill/millennium.html), WorldScientific 2001  ([pdf](http://cdsweb.cern.ch/record/475955/files/0011059), [doi:10.1142/4599](https://doi.org/10.1142/4599))

See also

* [[Shing-Tung Yau]], _[Moduli of high dimensional Calabi-Yau manifolds](http://www.scholarpedia.org/article/Calabi-Yau_manifold#Moduli_of_high_dimensional_Calabi-Yau_manifolds)_

On the [[quantum volume]] of vanishing cycles in the flop transition:

* [[Brian Greene]], Yakov Kanter, _Small Volumes in Compactified String Theory_, Nucl.Phys.B497:127-145,1997 ([arXiv:hep-th/9612181](https://arxiv.org/abs/hep-th/9612181))

* [[Brian Greene]], [[Calin Lazaroiu]], _Collapsing D-Branes in Calabi-Yau Moduli Space: I_, Nucl.Phys. B604 (2001) 181-255 ([arXiv:hep-th/0001025](https://arxiv.org/abs/hep-th/0001025))

* [[Calin Lazaroiu]], _Collapsing D-branes in one-parameter models and small/large radius duality_, Nucl.Phys. B605 (2001) 159-191 ([arXiv:hep-th/0002004](https://arxiv.org/abs/hep-th/0002004))

* Mark Raugas, _D-Branes and Vanishing Cycles in Higher Dimensions_ ([arXiv:hep-th/0102133](https://arxiv.org/abs/hep-th/0102133))

[[!redirects flop transitions]]


Florian Girelli

* [institute page](https://uwaterloo.ca/applied-mathematics/people-profiles/florian-girelli)

* [InSpire page](https://inspirehep.net/authors/1040425)

* [GoogleScholar page](https://scholar.google.com/citations?user=IWOs2aoAAAAJ&hl=en)

## Selected writings

On [[higher gauge theory]] for [[principal 2-bundles]] [[connection on a 2-bundle|with connection]]:

* [[Florian Girelli]], [[Hendryk Pfeiffer]], *Higher gauge theory -- differential versus integral formulation*, J. Math. Phys. **45** (2004) 3949-3971 [[hep-th/0309173](http://arxiv.org/abs/hep-th/0309173), [doi:10.1063/1.1790048](https://doi.org/10.1063/1.1790048)]

See also:

* Aristide Baratin, Florian Girelli, Daniele Oriti,  _Diffeomorphisms in group field theories_, Physical Review D, vol. 83, Issue 10, id. 104051, [doi](http://dx.doi.org/10.1103/PhysRevD.83.104051), [arxiv/1101.0590](http://arxiv.org/abs/1101.0590)

* Florian Girelli, Etera R. Livine, _A deformed Poincaré invariance for group field theories_, Classical Quantum Gravity __27__ (2010), no. 24, 245018 [MR2011j:83018](http://www.ams.org/mathscinet-getitem?mr=2739974), [doi](http://dx.doi.org/10.1088/0264-9381/27/24/245018)

* Florian Girelli, Etera R. Livine, Daniele Oriti, _Four-dimensional deformed special relativity from group field theories_, Phys. Rev. D 81 (2010), no. 2, 024015, 14 pp. [MR2610868](http://www.ams.org/mathscinet-getitem?mr=2610868)

* Florian Girelli, _[[Snyder space]]-time: K-loop and Lie triple system_, SIGMA Symmetry Integrability Geom. Methods Appl. __6__ (2010), Paper 074, 19 pp. [journal](http://www.emis.de/journals/SIGMA/2010/074) [pdf](http://www.emis.de/journals/SIGMA/2010/074/sigma10-074.pdf) [arxiv/1009.4762](http://arxiv.org/abs/1009.4762)

* F. Girelli, E. R. Livine, _Field theories with homogenous momentum space_, in Proceedings of 25th Max
Born Symposium: The Planck Scale (Wroclaw, Poland, 2009), AIP Conf. Proc. 1196, (2009), 115–123, [arXiv:0910.3107](http://arxiv.org/abs/0910.3107)

* F. Girelli, E. R. Livine, _Scalar field theory in Snyder space-time: alternatives_, [arXiv:1004.0621](ttp://arxiv.org/abs/1004.0621)

In a broader context of [[higher gauge theory]] and relating to the [[Dirac monopole]]:

* [[Hank Chen]], [[Florian Girelli]], *Gauging the Gauge and Anomaly Resolution* [[arXiv:2211.08549](https://arxiv.org/abs/2211.08549)]

[[!redirects F. Girelli]]


Florian Gmeiner

* [webpage](http://fgmeiner.de/)

category: people


Florian Herzig

* [webpage](http://www.math.toronto.edu/~herzig/)

category: people


Florian Ivorra

* [webpage](http://perso.univ-rennes1.fr/florian.ivorra/Bienvenue_.html)

category:people


Florian Marty

* [Academia page](https://independent.academia.edu/FlorianMarty)

## Selected writings

On [[commutative monoids in symmetric monoidal categories]] regarded as [[opposite category|duals]] of generalized [[affine schemes]]:

* {#Marty09} [[Florian Marty]], *Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative*, Ph.D. Toulouse (2009) [[theses:2009TOU30071](https://www.theses.fr/2009TOU30071), [[Marty-DesOuverts.pdf:file]]]

* [[Florian Marty]], *Relative Zariski Open Objects*, Journal of K-Theory **10** 1 (2012) 9-39  [[arXiv:0712.3676](https://arxiv.org/abs/0712.3676), [doi:10.1017/is011012004jkt176 ](https://doi.org/10.1017/is011012004jkt176 )]

category: people


Florian Schätz

* [website](http://www.math.ist.utl.pt/~fschaetz/)

## Selected writings

On [[infinity-local systems|$\infty$-local systems]] in the sense of [[(infinity,1)-vector bundles|$(\infty,1)$-vector bundles]] with [[flat infinity-connections|flat $\infty$-connections]]:

* [[Camilo Arias Abad]], [[Florian Schätz]]: *The $A_\infty$ de Rham theorem and integration of representations up to homotopy*, International Mathematics Research Notices, **2013** 16 (2013) 3790–3855 [[arXiv:1011.4693](https://arxiv.org/abs/1011.4693), [doi:10.1093/imrn/rns166](https://doi.org/10.1093/imrn/rns166)]

category: people

[[!redirects Florian Schaetz]]


Florin Dumitrescu

* [mathematics genealogy](http://genealogy.math.ndsu.nodak.edu/id.php?id=138756)

## Selected writings

On [[parallel transport]]:

* [[Florin Dumitrescu]], _Connections and Parallel Transport_, Journal of Homotopy and Related Structures, vol. 5(1), 2010, pp.171–175 ([arXiv:0903.0121](http://arxiv.org/abs/0903.0121)).

## Related entries

* [[superconnection]]

* [[super parallel transport]]

category: people


Floris Takens

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

## related $n$Lab entries

* [[variational bicomplex]]

category: people


Floris van Doorn

* [webpage](http://florisvandoorn.com)

## Selected writings

On [[higher inductive types]] and ([[Atiyah-Hirzebruch spectral sequence|Atiyah-Hirzebruch]]-/[[Serre spectral sequence|Serre]]-)[[spectral sequences]] in [[homotopy type theory]] formalized in [[Lean]] (cf. *[[cohomology in homotopy type theory]]* and *[[spectral sequences in homotopy type theory]]*):

* [[Floris van Doorn]], _Spectral_ ([github.com/cmu-phil/Spectral](https://github.com/cmu-phil/Spectral))

* [[Floris van Doorn]], *On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory* (2018) [[arXiv:1808.10690](https://arxiv.org/abs/1808.10690)]

On [[infinity-groups|$\infty$-groups]] formalized in [[homotopy type theory]]:

* {#BuchholtzDoornRijke18} [[Ulrik Buchholtz]], [[Floris van Doorn]], [[Egbert Rijke]], _[[Higher Groups in Homotopy Type Theory]]_, LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (2018) 205-214 [[arXiv:1802.04315](https://arxiv.org/abs/1802.04315), [doi:10.1145/3209108.3209150](https://doi.org/10.1145/3209108.3209150)]

## Related entries

* [[Lean]]

* [[spectral sequence]], [[Serre spectral sequence]]

category: people


flow of a vector field

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###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[tangent vector field]] on a [[differentiable manifold]] $X$ then its _flow_ is the [[group]] of [[diffeomorphisms]] of $X$ that lets the points of the manifold "flow along the vector field" hence which sends them along _flow lines_ (integral curvs) that are tangent to the vector field.

## Definition

### Traditional definition

Throughout, let $X$ be a [[differentiable manifold]] and let $v \in \Gamma(T X)$ be a continuously differentiable [[vector field]] on $X$ (i.e. of class $C^1$).

+-- {: .num_defn #IntegralCurve}
###### Definition
**(integral curves/flow lines)**

An _integral curve_ or _flow line_ of the vector field $v$ is a [[differentiable function]] of the form

$$
  \gamma
    \;\colon\;
    U
    \longrightarrow
  X
$$

for $U \subset \mathbb{R}$ an [[open interval]]  with the property that its [[tangent vector]] at any $t \in U$ equals the value of the vector field $v$ at the point $\gamma(t)$:

$$
  \underset{t \in U}{\forall}
  \left(
     d \gamma_t = v_{\gamma(t)}
  \right)
  \,.
$$

=--

+-- {: .num_defn #FlowOfAVectorField}
###### Definition
**(flow of a vector field)**

A _global flow_ of $v$ is a function of the form

$$
  \Phi \;\colon\; X \times \mathbb{R} \longrightarrow X
$$

such that for each $x \in X$ the function $\phi(x,-) \colon \mathbb{R} \to X$ is an integral curve of $v$ (def. \ref{IntegralCurve}).

A _flow domain_ is an open subset $O \subset X \times \mathbb{R}$ such that for all $x \in X$ the intersection $O \cap \{x\} \times \mathbb{R}$ is an [[open interval]] containing $0$.

A _flow_ of $v$ on a flow domain $O \subset X \times \mathbb{R}$ is a differentiable function

$$
  X \times \mathbb{R} \supset O \overset{\phi}{\longrightarrow} X
$$

such that for all $x \in X$ the function $\phi(x,-)$ is an integral curve of $v$ (def. \ref{IntegralCurve}).

=--

+-- {: .num_defn #CompleteVectorField}
###### Definition
**(complete vector field)

The vector field $v$ is called a _complete vector field_ if it admits a global flow (def. \ref{FlowOfAVectorField}).

=--

### Synthetic definition
 {#SyntheticDefinition}

In [[synthetic differential geometry]] a [[tangent vector field]] is a morphism $v \colon X \to X^D$ such that

$$
  \array{
    && X^D
    \\
    & {}^{\mathllap{v}}\nearrow & \downarrow^{\mathrlap{X^{\ast \to D}}}
    \\
    X &=& X
  }
$$

The [[internal hom]]-[[adjunct]] of such a morphism is of the form

$$
  \tilde v \;\colon\; D \longrightarrow X^X
  \,.
$$

If $X$ is sufficiently nice (a [[microlinear space]] should be sufficient) then this morphism factors through the internal [[automorphism group]] $\mathbf{Aut}(X)$ inside the internal [[endomorphisms]] $X^X$

$$
  \tilde v \;\colon\; D \longrightarrow \mathbf{Aut}(X) \hookrightarrow X^X
  \,.
$$

Then a group homomorphism

$$
  \phi_v \;\colon\; (R,+) \longrightarrow \mathbf{Aut}(X)
$$

with the property that restricted along any of the affine inclusions $D \hookrightarrow \mathbb{R}$ it equals $\tilde v$

$$
  \array{
     D &\hookrightarrow& \mathbb{R}
     \\
     & {}_{\mathllap{\tilde v}}\searrow & \downarrow^{\mathrlap{\phi}}
     \\
     && \mathbf{Aut}(X) &\hookrightarrow& X^X
  }
$$

is a _flow_ for $v$.

## Properties

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

Let $\phi$ be a global flow of a vector field $v$ (def. \ref{FlowOfAVectorField}). This yields an [[action]] of the additive group $(\mathbb{R},+)$ of [[real numbers]] on the [[differentiable manifold]] $X$ by [[diffeomorphisms]], in that

* $\phi_v(-,0) = id_X$;

* $\phi_n(-,t_2) \circ \phi_v(-,t_1) = \phi_v(-, t_1 + t_2)$;

* $\phi_v(-,-t) = \phi_v(-,t)^{-1}$.

=--

+-- {: .num_prop}
###### Proposition
**(fundamental theorem of flows)**

Let $X$ be a [[smooth manifold]] and $v \in \Gamma(T X)$ a smooth [[vector field]]. Then $v$ has a unique maximal flow (def. \ref{FlowOfAVectorField}).

This unique flow is often denoted $\phi_v$ or $\exp(v)$ (see also at _[[exponential map]]_).

=--

e.g. [Lee, theorem 12.9](#Lee)

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

Let $X$ be a [[compact topological space|compact]] [[smooth manifold]]. Then every smooth [[vector field]] $v \in \Gamma(T X)$ is a complete vector field (def. \ref{CompleteVectorField}) hence has a global flow (def. \ref{FlowOfAVectorField}).

=--

e.g. [Lee, theorem 12.12](#Lee)

## References

* {#Lee} John Lee, chapter 12 "Integral curves and flows" of _Introduction to smooth manifolds_ ([pdf](http://webmath2.unito.it/paginepersonali/sergio.console/lee.pdf))

[[!redirects flows of a vector field]]

[[!redirects flow]]
[[!redirects flows]]
[[!redirects flow line]]
[[!redirects flow lines]]

[[!redirects integral curve]]
[[!redirects integral curves]]


fluctuation-dissipation theorem

The fluctuation-dissipation theorem (fluctuation-dissipation relation) is one of the central results of [[statistical mechanics]]. The original formulation is due Nyquist (1928) and proven by

* H. B. Callen, T. A. Welton, _Irreversibility and generalized noise_,  Physical Review. 83: 34 (1951) [doi](https://doi.org/10.1103/PhysRev.83.34)

The classical form is obtained from the linear response theory by Kubo.

* Ryogo Kubo, _The fluctuation-dissipation theorem_, [pdf](http://www.mrc-lmb.cam.ac.uk/genomes/madanm/balaji/kubo.pdf)

* R. Kubo, M. Toda, _Statistical physics I, II_, Springer (see vol. II)
* Robert Zwanzig, _Nonequilibrium statistical mechanics_, Oxford University Press 2001
* U.M.B. Marconi, A. Puglisi, L. Rondoni, A. Vulpiani,  _Fluctuation-dissipation: response theory in statistical physics_, Physics Reports. 461 (4–6): 111–195 (2008)[arXiv:0803.0719](https://arxiv.org/abs/0803.0719) [doi](https://doi.org/10.1016/j.physrep.2008.02.002)

There are now generalizations in nonequilibrium thermodynamics.

* G. N. Bochkov, Yu. E. Kuzovlev, _Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem_, Physica A, 106: 443B (1991) doi

See also wikipedia: [Jarzynski equality](https://en.wikipedia.org/wiki/Jarzynski_equality), [fluctuation-dissipation theorem](https://en.wikipedia.org/wiki/Fluctuation-dissipation_theorem)

See also azimuth: [Effective thermodynamics for a marginal observer](https://johncarlosbaez.wordpress.com/2018/05/08/effective-thermodynamics-for-a-marginal-observer) (2018) and articles listed therein, including M. Polettini, M. Esposito, _Effective fluctuation and response theory_, [arxiv/1803.03552](https://arxiv.org/abs/1803.03552)

Related $n$Lab entries include [[statistical physics]], [[thermodynamics]], [[dissipation]], [[stochastic process]], [[Langevin equation]], [[Brownian motion]], [[white noise]]

category: physics
[[!redirects fluctuation-dissipation relation]]


fluid/gravity correspondence

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

#Contents#
* table of contents
{:toc}

## Idea

As a variant or special sector of the [[AdS/CFT correspondence]], the _fluid/gravity_ correspondence re-identifies aspects of ([[quantum gravity|quantum]]) [[gravity]] ([[string theory]]) on asymptotically approximately [[anti de Sitter spacetimes]] with those of certain [[fluid dynamics]] on the [[asymptotic boundary]].

## Applications

### To the quark-gluon plasma

For example some properties of the [[quark gluon plasma]] in [[quantum chromodynamics]] are captured well by [[fluid dynamics]], and after realizing [[QCD]] on [[D4-D8-brane bound states]] (the [[AdS/QCD correspondence]]) [[AdS/CFT]] provides a handle on this [[fluid dynamics]] in terms of the ([[quantum gravity|quantum]]) [[gravity]] ([[string theory]]) on an ambient [[bulk]] $\sim$ [[anti de Sitter spacetime]].

### To the relativistic Navier-Stokes equation

The fluid/gravity correspondence involves the _relativistic_ [[Navier-Stokes equation]] on the fluid side. ([BKLS 12](#BKLS12) see [Thillaisundaram 17](#Thillaisundaram17))

But it seems to remain open whether the correspondence sheds much light on the non-relativistic [[Navier-Stokes equation]] (the one whose stability is a [Millennium Problem](Navier-Stokes+equation#CMI)).

## Related concepts

* [[AdS/CFT correspondence]]

* [[AdS/QCD correspondence]]

* [[quark-gluon plasma]]

## References

The concept originates with

* [[Sayantani Bhattacharyya]], [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _Nonlinear Fluid Dynamics from Gravity_, JHEP 0802:045, 2008 ([arXiv:0712.2456](https://arxiv.org/abs/0712.2456))

Relation to the [[Navier-Stokes equation]]:

* {#BKLS12} Irene Bredberg, Cynthia Keeler, Vyacheslav Lysov, [[Andrew Strominger]], _From Navier-Stokes To Einstein_, J. High Energ. Phys. (2012) 2012: 146 ([arXiv:1101.2451](https://arxiv.org/abs/1101.2451))

Review:

* Nicola Ambrosetti, James Charbonneau, Silke Weinfurtner, _The fluid/gravity correspondence: Lectures notes from the 2008 Summer School on Particles, Fields, and Strings_ ([arXiv:0810.2631](https://arxiv.org/abs/0810.2631))

* [[Mukund Rangamani]], _Gravity & Hydrodynamics: Lectures on the fluid-gravity correspondence_, Class. Quant. Grav. 26 : 224003, 2009 ([arXiv:0905.4352](https://arxiv.org/abs/0905.4352))

* [[Veronika Hubeny]], _The Fluid/Gravity Correspondence: a new perspective on the Membrane Paradigm_, Class. Quant. Grav. 28:114007, 2011 ([arXiv:1011.4948](https://arxiv.org/abs/1011.4948))

* [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _The fluid/gravity correspondence_ ([arXiv:1107.5780](https://arxiv.org/abs/1107.5780))

* {#Thillaisundaram17} Ashok Thillaisundaram, _Aspects of fluid dynamics and the fluid/gravity correspondence_, 2017 ([CambridgeRepositor:1810/267097](https://www.repository.cam.ac.uk/handle/1810/267097), [pdf](https://www.repository.cam.ac.uk/bitstream/handle/1810/267097/ThillaisundaramThesis.pdf?sequence=1&isAllowed=y))

* Sumit Dey, Shounak De, Bibhas Ranjan Majhi, _Gravity dual of Navier-Stokes equation of an uniformly rotating fluid through parallel transport_ ([arXiv:2002.06801](https://arxiv.org/abs/2002.06801))

[[!redirects fluid-gravity correspondence]]


flux

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In [[gauge theory]] of higher abelian [[Yang-Mills theory]] type, where field configurations on some manifold $X$ are [[circle n-bundles with connection]] $\nabla$, the **magnetic flux** of the field configuration through a $(n+1)$-[[dimension]]al closed manifold $\Sigma \hookrightarrow X$ is

$$
  \Phi_\sigma(\nabla)
  \;=\;
  \textstyle{\int}_\Sigma F_\nabla
  \,,
$$

where $F_\nabla$ is the [[curvature]] $(n+1)$-form.

\[\label{FluxSchematics}\]
\begin{imagefromfile}
    "file_name": "MagneticFlux-230305.jpg",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "adapted from [Hyperphysics](#Hyperphysics)"
\end{imagefromfile}

{#FaradayIronFilings} The classical example is *[[magnetic field]]*-flux, originally observed by [[Michael Faraday]] by bringing iron filings into the [[magnetic field]] of rod magnets:

> from *Faraday's diary of experimental investigation*, vol VI, entry from 11th Dec. 1851; reproduced in [Martin (2009)](#Martin09) (p. 311 [here](http://faradaysdiary.com/ws3/faraday.pdf#page=311))

\linebreak

{#Faraday1852PlateIX} Faraday published this one year later:

> [Faraday 1852, Plate IX, p. 158](#Faraday1852)

## Examples

[[!include flux quantization in superconductors -- section]]

## Related concepts

* [[conserved current]], [[charge]]

* [[electric charge]], [[magnetic charge]]

* **flux**

  * [[flux tube]]

  * [[flux quantization]]

  * [[flux compactification]]

  * [[uncertainty of fluxes]]

## References

Origin and history:

* {#Faraday1852} [[Michael Faraday]], *Delienation of Lines of Magnetic Force by iron filings*, §37 in: *Experimental Researches in Electricity.--Twenty-Ninth Series*, Philosophical Transactions of the Royal Society of London **142** (1852) 137-159 [[doi:10.1098/rstl.1852.0012](https://doi.org/10.1098/rstl.1852.0012), [jstor:108540](https://www.jstor.org/stable/108540)]

  > (the notion of "lines of force" is introduced on pp. 154)

* {#Martin09} Thomas Martin (ed.), *Faraday's diary of experimental investigation* 1820-1862, HR Direct (2009)  [[webpage](http://faradaysdiary.com/), preview:[pdf](http://faradaysdiary.com/ws3/faraday.pdf)]

* The Royal Institution, *Michael Faradays' iron filings* [[web](https://www.rigb.org/explore-science/explore/collection/michael-faradays-iron-filings)]

* [[Raymond Brock]], *[[Faraday]]'s Lines of Force -- Modern Physics Begins*, chapter 11 in: *Quarks, Spacetime, and the Big Bang*, lecture notes [chapter: [pdf](https://web.pa.msu.edu/people/brock/file_sharing/QSandBB/QS&BB_manuscript/man_faraday_up.pdf), full text:[pdf](https://web.pa.msu.edu/people/brock/file_sharing/QSandBB/QS&BB_manuscript/QS&BB.pdf)]

Online exposition:

* {#Hyperphysics} Hyperphysics, *Magnetic Flux* [[hyperphysics.phy-astr.gsu.edu/hbase/magnetic/fluxmg.html](http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/fluxmg.html)]

[[!redirects fluxes]]

[[!redirects flux density]]
[[!redirects flux densities]]

[[!redirects flux form]]
[[!redirects flux forms]]

[[!redirects magnetic flux]]
[[!redirects electric flux]]

[[!redirects flux line]]
[[!redirects flux lines]]


flux compactification

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

#Contents#
* table of contents
{:toc}

## Idea

_Flux compactifications_ are [[Kaluza-Klein compactification]]  where a [[gauge field]] or [[higher gauge field]] has non-trivial [[field strength]], hence "generalized electromagnetic [[flux]]" in the [[fiber]]-spaces, whose repulsive [[force]] counteracts the collapsing force of [[gravity]] on the compact fiber spaces.

Applied to [[supergravity]] this may in particular yield [[perturbative string theory vacua]].

One way of achieving [[moduli stabilization]] for KK-compactifications in [[Einstein-Maxwell theory]] or [[supergravity]]/[[string theory]] is to consider [[gauge fields]] and/or [[higher gauge fields]] in the compact space. Their (higher) [[field strength]]/[[curvature]] forms ("[[fluxes]]") parameterize mass terms for the compactification moduli and hence may, under suitable conditions, stabilize them.

## Related concepts

* [[flux]], [[flux tube]], [[flux quantization]]

* [[moduli stabilization]]

* [[landscape of string theory vacua]]

* [[G2-MSSM]]

## References

* [[Michael Douglas]], [[Shamit Kachru]], _Flux compactification_, Rev. Mod. Phys. __79__ (2007) 733–796 ([arXiv:hep-th/0610102](http://arxiv.org/abs/hep-th/0610102))

* [[Frederik Denef]], [[Michael Douglas]], [[Shamit Kachru]], _Physics of string flux compactifications_, Ann.Rev.Nucl.Part.Sci.57:119-144,2007, [hep-th/0701050](http://arxiv.org/abs/hep-th/0701050), [doi](http://dx.doi.org/10.1146/annurev.nucl.57.090506.123042)

* [[Frederik Denef]], _Introduction to flux compactifications_, lecture at Summer School on particle physics, cosmology and strings - Perimeter Institute 2007, [video](http://pirsa.org/07080009)

A good survey of the story of flux compactifications in [[F-theory]] is in

* {#Denef08} [[Frederik Denef]], _Les Houches Lectures on Constructing String Vacua_, in _[[String theory and the real world]]_ ([arXiv:0803.1194](http://arxiv.org/abs/0803.1194))

For [[F-theory]]-discussion see also at _[F-theory -- References -- Flux compactification](F-theory#FluxAndInstantons)_.

See also

* [[Barton Zwiebach]], _A first course in string theory_

* [[Raphael Bousso]], [[Joseph Polchinski]], _Quantization of four-form fluxes and dynamical neutralization of the cosmological constant_, JHEP 06, 006 (2000) [hep-th/0004134](http://arxiv.org/abs/hep-th/0004134)

* T.R. Taylor, [[Cumrun Vafa]], _RR Flux on Calabi-Yau and Partial Supersymmetry Breaking_, Phys.Lett. B474 (2000) 130-137 ([arXiv:hep-th/9912152](http://arxiv.org/abs/hep-th/9912152))

* R. Blumenhagen, B. Körs, [[Dieter Lüst]], S. Stieberger, _Four-dimensional string compactifications with D-Branes, orientifolds and fluxes_, Phys. Rept. __445__ (2007) 1–193, [hepth/0610327](http://arxiv.org/abs/hep-th/0610327).
* [[Mariana Graña]], _Flux compactifications in string theory: a comprehensive review_, [hep-th/0509003](http://arxiv.org/abs/hep-th/0509003)

* [[Jock McOrist]], [[Savdeep Sethi]], _M-theory and Type IIA Flux Compactifications_ ([arXiv:1208.0261](http://arxiv.org/abs/1208.0261))

With [[RR-field tadpole cancellation]] taken into account:

* Philip Betzler, Erik Plauschinn, _Type IIB flux vacua and tadpole cancellation_ ([arXiv:1905.08823](https://arxiv.org/abs/1905.08823))

See also:

* [[David Prieto]], *Moduli Stabilization and Stability in Type II/F-theory flux compactifications* [[arXiv:2401.13339](https://arxiv.org/abs/2401.13339)]

* Minjae Cho, Manki Kim, *A Worldsheet Description of Flux Compactifications* [[arXiv:2311.04959](https://arxiv.org/abs/2311.04959)]

On [[non-geometric string vacuum|non-geometric]] flux vacua:

* [[Ralph Blumenhagen]], A. Deser, E. Plauschinn, F. Rennecke, _Bianchi identities for non-geometric fluxes: from quasi-Poisson structures to Courant algebroids_, [arXiv:1205.1522](http://arxiv.org/abs/1205.1522)

* D. Mylonas, Peter Schupp, [[Richard Szabo]], _Membrane sigma-models and quantization of non-geometric flux backgrounds_, [arxiv/1207.0926](http://arxiv.org/abs/1207.0926)

[[!redirects flux background]]
[[!redirects flux compactifications]]


flux quantization in superconductors -- section

### Magnetic flux quantization in type II superconductors
 {#MagneticFluxQuantizationInTypeIISuperconductors}

Due to the [[Meissner-Ochsenfeld effect]], a [[superconductor]] placed in a sufficiently small external [[magnetic field]] $H_{ext} \lt H_{c_1}$ (aligned along some axis) "expels" that field , making the total  magnetic field in the [[bulk]] of the superconductor vanish. However, as the ambient magnetic field exceeds a critical value $H_{c_1}$, this behaviour changes:

* for _[[type I superconductors]]_, the superconducting state simply breaks down as $H_{ext} \gt H_{c_1}$ and the ambient magnetic field fully penetrates the material as for any normal conductor;

* for _[[type II superconductors]]_ the superconducting state eventually also breaks down as $H_{ext} \gt H_{c_2} \gt H_{c_1}$, but there is an intermediate parameter region $H_{c_1} \lt H_{ext} \lt H_{c_2}$ where both regimes mix:

\begin{imagefromfile}
    "file_name": "SuperconductorsPhaseDiagram.jpg",
    "width": 570,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [Chapman 09](superconductivity#Chapman00)"
\end{imagefromfile}

{#InThisMixedRegime} In this mixed regime, a _[[finite number]]_ of elementary units of [[magnetic flux]] enter the superconductor, carried by little _[[flux tubes]]_ inside _[[vortices]]_ of electric currents: _[[vortex strings]]_ (about a [[micron]] in diameter, e.g. [Chapman 00, p. 559](superconductivity#Chapman00)). Each vortex core carries one unit of magnetic flux -- also called a _fluxon_ --  while at some small finite distance away from all vortices, the bulk magnetic flux in the superconductor still vanishes (mathematically: it [[vanishes at infinity]], cf. eg. [Chapman 2009 (2.31)](superconductivity#Chapman00), [Timm 2020 (6.101)](superconductivity#Timm20))
:

\[\label{VortexStructure}\]

\begin{imagefromfile}
    "file_name": "SuperconductorVortexStructure-230304.jpg",
    "width": 650,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "adapted from [Loudon and Midgley 2009](superconductivity#LoudonMidgley09)"
\end{imagefromfile}

At sufficiently large density these vortices arrange into hexagonal patterns, first described by  [Abrikosov 57](superconductivity#Abrikosov57), whence also known as _Abrikosov vortices_.

\[\label{HexagonalStructure}\]
\begin{imagefromfile}
    "file_name": "AbrikosovVorticesHorizontalSchematics.png",
    "width": 250,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [Yu 2012](#Yu2012)"
\end{imagefromfile}

This flux quantization in type II superconductors is traditionally explained via the [[effective field theory]] provided by the [[Landau-Ginzburg model]]; the derivation may be found reviewed in [Chapman 00 (Section 2, culminating in (2.33))](superconductivity#Chapman00).

But, as indicated a little more explicitly in [Alvarez-Gaumé 98 (Section IV.B, culminating below IV.11)](superconductivity#AlvarezGaume98), the flux quantization as such is mathematically a direct consequence of the global topological nature of the [[electromagnetic field]], the argument being the direct 2-dimensional analog of the quantization of [[instantons in QCD]] in 4d (see also at [SU(2)-Instantons -- From the correct maths to the traditional physics story](Yang-Mills+instanton#FromTheMathsToThePhysicsStory)) and in fact is but a slight variation of the argument for [[Dirac charge quantization]] of [[magnetic monopoles]]:

Namely, the [[electromagnetic field]] is a [[connection on a circle bundle]],  and hence the [[Brown representability theorem|cohomology classifying space]] for the topological class of the [[electromagnetic field]] is the [[classifying space]] $B \mathrm{U}(1)$ of the [[circle group]], which, being an [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$, has second [[homotopy group]] the [[integers]]:

$$
  \pi_2(B \mathrm{U}(1))
    \;\simeq\;
  \big\{  S^2 \to B \mathrm{U}(1)\big\}_{\big/hmpty}
    \;\simeq\;
  \mathbb{Z}
  \,.
$$

This implies that on every [[spacetime]] which looks, up to [[homotopy equivalence]], like a [[2-sphere]] to the electromagnetic field, the [[magnetic flux]] is identified with an element in the group of [[integers]], hence is _quantized_ in integral multiples of some [[physical unit|unit]] flux.

For the case of a (hypothetical) [[magnetic monopole]] (e.g. a magnetically [[charged black hole]]), it is the [[spacetime]] _around_ the monopole (the [[complement]] of its [[worldline]] $\mathbb{R}^{0,1}$ in the ambient (asymtptotically) [[Minkowski spacetime]] $\mathbb{R}^{3,1}$) which has the [[homotopy type]] of a [[2-sphere]] $\mathbb{R}^{3,1} \setminus \mathbb{R}^{0,1} \,\simeq\, \mathbb{R}^{0,1} \times \mathbb{R}_{rad} \times S^2$; this implies the [[Dirac charge quantization]] of the [[magnetic monopole]]'s [[magnetic charge]]:

\begin{imagefromfile}
    "file_name": "DiracChargeQuantizationViaClassifyingMap.jpg",
    "width": 630,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

For the case of the type II [[superconductor]] it is instead the transversal [[vanishing at infinity]] of the magnetic field (i.e. the [[Meissner-Ochsenfeld effect]] away from the vortices) which implies that the classifying map of the [[electromagnetic field]] sees not the full transversal [[Euclidean plane]] $\mathbb{R}^2$ but its [[one-point compactification]] $\big( \mathbb{R}^2 \big)^{cpt} \,\simeq\, S^2$, which introduces an effective [[2-sphere]] topology onto spacetime (same as the [[3-sphere]] in the discussion of [[Skyrmions]] and the [[4-sphere]] in the discussion of [[instantons]]): $\mathbb{R}^{1,1} \times \big( \mathbb{R}^2 \big)^{cpt} \,\simeq\, \mathbb{R}^{1,1} \times S^2$. This implies the superconductor's magnetic flux quantization:

\begin{imagefromfile}
    "file_name": "VortexFluxQuantizationViaClassifyingMap.jpg",
    "width": 630,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

The argument that is given in most references, via consideration of the [[period of a differential form|period]] of the [[vector potential]] on a large [[circle]] around the superconductor (e.g. [Timm 20, Section 5.3](superconductivity#Timm20)), is secretly just the analysis of this picture through the [[clutching construction]] (the direct 2d analog of the discussion at _[SU(2)-Instantons -- From the correct maths to the traditional physics story](Yang-Mills+instanton#FromTheMathsToThePhysicsStory)_).


flux tube

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
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#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

\tableofcontents

## Idea

Generally, in [[gauge theory]] by a *flux tube* one means a cylindrical/string-shaped region inside which a given kind [[flux]] is concentrated. For example

* [[magnetic flux]] through a long [[solenoid]] carrying electric [[current]] (this is represented by the "flux lines" originally introduced by [[Michael Faraday]]);

* similarly: [magnetic flux tubes inside a superconductor](flux#MagneticFluxQuantizationInTypeIISuperconductors)

* "[[color charge|color]]" flux tubes between [[quarks]] in [[quantum chromodynamics]] (also: "[[Wilson lines]]")

  (which become the [[strings]] of [[string theory]] under [[AdS-CFT correspondence]]/[[AdS-QCD correspondence]])

[[!include Polyakov gauge-string duality -- section]]

## Related concepts

* [[flux]], [[flux quantization]], [[flux compactification]], [[uncertainty of fluxes]]

* the [[AdS-QCD correspondence]] postulates that [[QCD]]-flux tubes, once properly [[quantization|quanitized]], behave like  [[open strings]] propagating in a higher-dimensional [[spacetime]], whose endpoints (the [[quarks]]) are constrained to a 3+1-dimensional [[intersecting brane model|brane intersection]]

* [[large N limit]]

## References

### General

* [[Raymond Brock]], *Faraday's Lines of Force -- Modern Physics Begins*, chapter 11 in: *Quarks, Spacetime, and the Big Bang*, lecture notes [chapter: [pdf](https://web.pa.msu.edu/people/brock/file_sharing/QSandBB/QS&BB_manuscript/man_faraday_up.pdf), full text:[pdf](https://web.pa.msu.edu/people/brock/file_sharing/QSandBB/QS&BB_manuscript/QS&BB.pdf)]

See also:

* Wikipedia, *[Flux tube](https://en.wikipedia.org/wiki/Flux_tube)*

[[!include Polyakov gauge-string duality -- references]]

[[!redirects flux tubes]]


Flyspeck project

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## Idea

[[proof]] of the [[Kepler conjecture]] using the [[computer proof assistant]] [[HOL Light]].

## References

* {#Hales06} [[Thomas Hales]], _Introduction to the Flyspeck Project_, 2006 ([drops:2006/432](http://drops.dagstuhl.de/opus/volltexte/2006/432/), [pdf](https://pdfs.semanticscholar.org/0f72/54bb124913ca58f814dd0c6ab2585dc05c77.pdf))

* [Flyspeck project home page](http://www.proof-technologies.com/flyspeck/)

* [github.com/flyspeck/flyspeck](https://github.com/flyspeck/flyspeck)


focal point

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## Definition

A [[point]] of a [[topological space]] is called **focal** ([Freyd-Scedrov](#FreydScedrov)) if its only [[open neighbourhood]] is the entire space.

## Examples

* The closed point $\bot$ of [[Sierpinski space]] $\mathbf{2}$ is a focal point.

* The vertex $v$ of a Sierpinski cone (or [[scone]]) $s(X)$ on a space $X$, given by a pushout in $Top$
$$\array{
1 \times X & \to & 1 \\
\mathllap{\bot \times 1_X} \downarrow & & \downarrow \mathrlap{v} \\
\mathbf{2} \times X & \to & s(X),
}$$
is a focal point. This construction is in fact the same as generically adding a focal point to $X$.

* The [[prime spectrum]] of a ring $A$ has a focal point iff $A$ is a [[local ring]]. In this case, the focal point is given by the unique maximal ideal of $A$.

## Properties

The [[category of sheaves]] over (the [[site of open subsets]]) of a topological space with focal point is a [[local topos]].

Every [[topos]] has a [[free construction|free]] "completion" to a "focal space", given by its [[Freyd cover]].

## In locale theory

A [[locale]] $X$ is called **local** if in any covering of $X$ by opens $U_i$, at least one $U_i$ is $X$.

The locale associated to an [[sober space]] is local in this sense if and only if the space possesses a focal point (see [Johnstone, discussion preceding lemma C1.5.6](#Johnstone)). Locale theoretically, this point is then given by the [[frame|frame homomorphism]]
$$ \mathcal{O}(X) \to \Omega, \quad U \mapsto \{ \star | U = X \}, $$
where $\Omega$ is the [[frame of opens]] of the [[point]].

## References

* [[Peter Freyd]], A. Scedrov, _Geometric logic_, (North-Holland, Amsterdam)
 {#FreydScedrov}
* [[Peter Johnstone]], _[[Sketches of an Elephant]]_ {#Johnstone}

[[!redirects focal points]]


Fock space

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## Idea

In [[physics]] the symmetric [[tensor algebra]] on a [[space of quantum states]] of some [[quantum mechanical system]] is called its _bosonic Fock space_. This is regarded in turn as the space of [[quantum states]] of arbitrarily many copies of the system. In particular if the original system describes some [[boson|bosonic]] [[particle]] species, then its Fock space is the space of quantum states of arbitrarily many such particles.

Similarly the [[exterior algebra]]/[[Grassmann algebra]] is called the _[[fermion|fermionic]] [[Fock space]]_.

The process of passing from a given [[space of quantum states]] to its Fock space is also known as (or rather: is part of) what is called _[[second quantization]]_.

Fock spaces hence appear as [[spaces of quantum states]] of [[free fields]] in [[quantum field theory]]. In [[perturbative quantum field theory]] they are still used indirectly in for non-free field theories.

## Related entries

* [[tensor algebra]]

* [[second quantization]]

* [[quantum many-body physics]]

[[!include states and observables -- content]]

## References

Named after _[[Vladimir Aleksandrovich Fock]]_.

### General

A textbook account with an eye towards [[perturbative algebraic quantum field theory]] is in

* [[Michael Dütsch]], appendix E of _[[From classical field theory to perturbative quantum field theory]]_

See also

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

### In linear type theory

The Fock space construction may be axiomatized as the [[exponential modality]] in [[linear type theory]]. This is discussed in the following articles.

* {#BlutePanangadenSeely94} [[Richard Blute]], [[Prakash Panangaden]], [[R. A. G. Seely]], _Fock Space: A Model of Linear Exponential Types_ (1994) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.6825))

* {#Fiore07} [[Marcelo Fiore]], _Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic_, Lecture Notes in Computer Science Volume 4583, 2007, pp 163-177 ([pdf](http://www.cl.cam.ac.uk/~mpf23/papers/Types/diff.pdf))

* {#Vicary07} [[Jamie Vicary]], _A categorical framework for the quantum harmonic oscillator_, International Journal of Theoretical Physics December 2008, Volume 47, Issue 12, pp 3408-3447 ([arXiv:0706.0711](http://arxiv.org/abs/0706.0711))

  (in the context of [[finite quantum mechanics in terms of dagger-compact categories]])

[[!redirects Fock spaces]]


Fock-Goncharov coordinates

Fock-Goncharov coordinates are particular coordinates on the moduli space of flat connections, introduced in

* V. Fock, [[A. B. Goncharov]], _Moduli spaces of local systems and higher Teichmüller theory_, Publ. Math. Inst. Hautes Études Sci. 103, 1–211 (2006)

See also

* V. Fock, [[A. B. Goncharov]], _Cluster X-varieties, amalgamation and Poisson-Lie groups_, [arxiv](http://www.arXiv.org/abs/math/0508408)
* Davide Gaiotto, [[Gregory Moore|Gregory W. Moore]], Andrew Neitzke, _Spectral networks and snakes_, Annales Henri Poincaré, March 2013, [doi](http://dx.doi.org/10.1007/s00023-013-0238-8)
* Martin Scorsese, _Goncharov_, 1973.

site: algebraic geometry, physics
[[!redirects Fock–Goncharov coordinates]]


focusing

##Idea

Focusing is an idea originating in the [[proof theory]] of [[linear logic]].  Originally, it was introduced as a technique for making proof search in the [[sequent calculus]] tractable, by restricting the potential application of rules based on observations about the [[polarity in type theory|polarity]] of connectives ([Andreoli 1992](#Andreoli92)).  More abstractly, the _focalization_ property can be seen as a [[canonical form]] theorem for sequent calculus proofs, which holds in a wide range of systems beyond linear logic.

>Focusing, introduced by Jean-Marc Andreoli in the context of classical linear logic [Andreoli 1992], defines a normal form for sequent calculus derivations that cuts down on the number of possible derivations by eagerly applying invertible rules and grouping sequences of non-invertible rules. A focused sequent calculus is defined relative to some non-focused sequent calculus; focalization is the property that every non-focused derivation can be transformed into a focused derivation. ([Simmons](#Simmons))

##Related concept

* [[polarity in type theory]]

##References

* {#Andreoli92} [[Jean-Marc Andreoli]], Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297–347, 1992 ([pdf](https://www.cs.cmu.edu/~fp/courses/15816-s12/misc/andreoli92jlc.pdf))
* [[Daniel Licata]], [[Noam Zeilberger]], [[Robert Harper]], _Focusing on Binding and Computation_, ([pdf](http://noamz.org/papers/lzh08focbind-tr.pdf))
* {#Simmons} [[Robert Simmons]], _Structural focalization_, [arXiv:1109.6273, cs.LO](http://arxiv.org/abs/1109.6273)
* [[Robert Simmons]], _What does focusing tell us about language design?_, ([blog post](http://requestforlogic.blogspot.co.uk/2012/03/what-does-focusing-tell-us-about.html))
* Taus Brock-Nannestad and Carsten Schürmann, _Focused Natural Deduction_, ([pdf](http://www.itu.dk/people/tbro/fnd.pdf))
* Chuck Liang and Dale Miller. Focusing and polarization in linear,
intuitionistic, and classical logics. Theoretical Computer Science,
410(46):4747–4768, November 2009. ([pdf](http://www.lix.polytechnique.fr/Labo/Dale.Miller/papers/tcs09fixed.pdf))
* Stéphane Graham-Lengrand. _Polarities & Focussing: a journey from Realisability to Automated Reasoning._ Habilitation thesis, Université Paris-Sud, 2014. ([pdf](http://www.lix.polytechnique.fr/~lengrand/Work/HDR/Dissertation/Main.pdf))

[[!redirects focussing]]


Fodor Bogomolov

__Fodor Bogomolov__ is an algebraic geometer.

* [Bogomolov's laboratory](http://bogomolov-lab.ru) of algebraic geometry and its applications

About Bogomolov-Beauville-Fujiki form in $H^2(M)$ (comongh from deformation theory) for hyperkaehler variety $M$:

* Eyal Markman, _The Beauville-Bogomolov class as a characteristic class_, [arxiv/1105.3223](http://arxiv.org/abs/1105.3223)


Fokker-Planck equation

Also called Fokker-Planck-Kolmogorov (forward) equation.

* wikipedia [Fokker-Planck equation](http://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation)
* A. D. Fokker, _Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld_, Ann. Phys. __348__ (4. Folge 43), 810–820 (1914)
* Leo P. Kadanoff, _Statistical Physics: statics, dynamics and renormalization_, World Scientific (2000)
* Jean Zinn-Justin, _Quantum field theory and critical phenomena_, Oxford: Clarendon Press.
* R. Jordan, D. Kinderlehrer, F. Otto, _The variational formulation of the Fokker-Planck equation_ , SIAM J. Math. Anal. 29 (1998), no. 1, 1-17.
* Chuang Ren, Xing-Zhong Li, _Analytical solution of the Fokker-Planck equation for ions in a magnetic well with neutral-beam pumping_, Phys. Rev. A 46, 3558–3561 (1992) [doi](http://link.aps.org/doi/10.1103/PhysRevA.46.3558), [pdf](http://pra.aps.org/pdf/PRA/v46/i6/p3558_1)

[[!redirects Fokker-Planck-Kolmogorov equation]]


FOLDS

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## Idea

What is called _FOLDS_ (for: _[[first-order logic]] with [[dependent type|dependent sorts]]_) is a form of ([[intuitionistic logic|intuitionistic]] or [[classical logic|classical]]) [[predicate logic|predicate]] [[logic]] with a weak form of [[equality]], in which [[category theory]] (and even [[higher category theory]]) can be formulated but it is impossible to violate, say, the mathematical [[principle of equivalence]].  It also induces general notions of [[equivalence]] for [[higher structures]] and for objects *in* higher structures, which reduce to familiar notions in most or all cases.

Formally speaking, FOLDS is "merely" a "first-order [[fragment]]" of [[dependent type theory]], but its special treatment of [[equality]], its notions of [[equivalence]], and its relationship to higher-categorical structures distinguish it from DTT in general.  These aspects have been developed primarily by [[Michael Makkai]].

## Signatures

FOLDS can be formulated using [[dependent types]], [[terms]], [[contexts]], and [[judgments]], as is typical in [[type theory]].  However, it is perhaps easier for a category theorist to understand when presented as the study of the [[presheaves]] which underlie higher-categorical structures.

Both [[algebraic definition of higher category|algebraic]] and [[geometric definition of higher category|geometric]] definitions of higher-categorical structures are generally specified by giving a [[presheaf]] on some [[small category]] $D$ (such as a category of [[geometric shapes for higher structures]]), which satisfies certain [[stuff, structure, property|properties]] or is equipped with certain [[stuff, structure, property|structure]].  The category $D$ in question is usually a [[Reedy category]], and as such comes equipped with a stratification of its objects by "degree," and a division of its [[morphisms]] into "coface maps," which raise degree, and "codegeneracy maps," which lower degree.  The perspective of FOLDS is that actually the codegeneracy maps should be regarded as part of the *structure* imposed (when present, they usually assign "identities"), while the coface maps are what really describe the "geometric shape" or "dependency structure."  Note that in some cases, such as [[opetopic sets]] and [[globular sets]], there are no degeneracies to start with.

This suggests that FOLDS should study [[presheaves]] on some [[direct category]], or equivalently (changing the variance) $Set$-valued diagrams on some [[inverse category]].  If $C$ is an inverse category and $x\in C$ an object, then $x$ represents one geometric shape appearing in the structures under consideration, while all of the morphisms *out* of $x$ indicate the lower-dimensional "faces" of such a shape, i.e. the "boundary" which it fills.  However, for an arbitrary inverse category, a given shape can still have infinitely many faces in its boundary; this is difficult to deal with in logic, so we impose the extra requirement that the category have **finite fan-out**: every object $x$ is the [[source]] of only finitely many arrows altogether.  A category with finite fan-out is an inverse category iff it is also both [[skeleton|skeletal]] and [[one-way category|one-way]] (every endomorphism is an identity); thus we study skeletal one-way categories with finite fan-out.  In FOLDS these are called **simple categories**.

A FOLDS signature, or vocabulary, should then consist of a simple category of *kinds*, together with information about [[function]] and [[relation]] symbols.  However, it turns out to be easier to include *only* relation symbols; the values of functions can then be recovered as the unique objects satisfying some relation, or more generally as unique-up-to-equivalence objects equipped with some higher structure (cf. [[anafunctor]], for instance).  One reason relation symbols are easier is that they can be represented simply as additional "shapes," whose boundary consists of those elements which they relate.  Such a shape should be *maximal*, in the sense that it is not the target of any nonidentity arrow.  Thus,

* A **FOLDS signature** or **vocabulary for dependent sorts (DSV)** consists of a simple category together with a distinguished set of maximal objects, called the **relation symbols**.  The objects that are not relation symbols are called **kinds**.

## Examples

...

## Equivalences

...

## References

*  [[Michael Makkai]], _First Order Logic with Dependent Sorts, with Applications to Category Theory_, (1995) [[pdf](https://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf), [further links](http://www.math.mcgill.ca/makkai/)]

* [[Erik Palmgren]], _Categories with families, FOLDS and logic enriched type theory_ (2016) [[arXiv:1605.01586](https://arxiv.org/abs/1605.01586), [diva2:926427](http://su.diva-portal.org/smash/record.jsf?pid=diva2%3A926427)]


foliated category

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## Idea

__Foliated categories__ (French: *catégories feuilletées*), or simply __foliations__ (not to be confused with the notion of [[foliations]] in [[differential geometry]]), were introduced by [[Jean Bénabou]] in unpublished work dating back to 1984.

They are a weaker [[structure]] than [[fibered categories]], but still allow one to test for various standard properties of [[functors]] [[fibre]]-wise.

## Definition

A [[functor]] $P\colon \mathbf{X} \to \mathbf{B}$ makes its [[domain]] [[category]] $\mathbf{X}$ a *foliated category* (over $\mathbf{B}$) if the following conditions hold:

1. Every [[morphism]] $f$ in $\mathbf{X}$ factors as a $P$-vertical morphism $v$ (i.e. $P(v)$ is an identity morphism in $\mathbf{B}$), followed by a $P$-[[cartesian morphism]] $k$, i.e. $f = k\circ v$,

1. The [[class]] of [weak](Cartesian+morphism+WeakCartesianMorphisms) $P$-[[cartesian morphisms]] is closed under [[composition]].

If closure under composition is not required, we obtain the notion of **prefoliated category**.

A morphism between foliated categories $\mathbf{X}'\to \mathbf{X}$ (over $\mathbf{B}$) is a functor over $\mathbf{B}$ that sends cartesian morphisms to cartesian morphisms, and such that for every object $X'$ of $\mathbf{X}'$, and morphism $f\colon Y\to F(X')$ in $\mathbf{X}$, there is a factorisation $f = F(k)\circ v$, where $v$ is vertical in $\mathbf{X}$ and $k$ is cartesian in $\mathbf{X}'$. Bénabou calls such functors _cartesian_.

## References

* [[Jean Bénabou]], _Cartesian functors and foliated categories_, talk at Oxford (1 May 2012) [[YouTube](https://www.youtube.com/watch?v=Y1CeHmjNcuk)]

* [[Jean Bénabou]], _Foncteurs cartésiens et catégories feuilletées_, talk at *Journée Guitart*, Paris (9 November 2012) [[YouTube](https://www.youtube.com/watch?v=p3vcZDyZ4Yo), [slides](https://docs.google.com/presentation/d/17E0epsZWcDTYlwAMZ4PFCLIf4dyigS-WUA0siUtU-wY/edit?hl=fr#slide=id.gce595186_059), [[Benabou-FoliatedCategories.pdf:file]]]

* [[Jean Bénabou]], _Du vieux et du neuf sur la construction de Grothendieck_, talk at Paris-Diderot (March 2019) [[YouTube](https://youtu.be/NC0I18B45kE)]

  > (the material on foliated categories -- called simply foliations --- starts at 1:01:00).

A short summary is in [this message](https://github.com/punkdit/categories/blob/master/gmane/science/mathematics/categories/8243) on the Category Theory Mailing List.

[[!redirects foliated categories]]
[[!redirects prefoliated category]]
[[!redirects prefoliated categories]]