fragment

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[formal logic]] one speaks of a "fragment" of a [[theory]] when referring to just part of it. Specifically, when rules of [[type formation]] or some of the [[natural deduction|inference rules]] are omitted, one says that one retains a "fragment" of the original theory.

This terminology is jargon and has not been formalized. What exactly counts as a fragment of a theory and what not differs a bit from author to author.

## Examples

### First-order logic in type theory
 {#FirstOrderFragment}

In [[type theory]] [[first order logic]] of [[propositions]] is contained via "[[propositions as types]]" essentially as the sub-theory dealing with types of type [[Prop]], hence with [[bracket types]] or what in [[homotopy type theory]] are called [[(-1)-types]].

Some authors speak of the "first-order fragment" of type theory when referring to this restriction (e.g. [Bezem-Hendriks-Nivelle 02](#BezemHendriksNivelle2002), [Hendriks 13](#Hendriks13)).

### Multiplicative conjunction in linear logic

By default [[linear logic]] has a wealth of [[conjunctions]]. Discarding or ignoring all of them except the [[multiplicative conjunction]] yield the fragment of _[[multiplicative linear logic]]_.

## References

* {#BezemHendriksNivelle2002} Marc Bezem, Dimitri Hendriks, Hans de Nivelle, _Automated Proof Construction in Type Theory using Resolution_, 2002 ([pdf](http://www.phil.uu.nl/ozsl/articles/Hendriks01.pdf))

* {#Hendriks13} Dimitri Hendriks, _Metamathematics in Coq_, 2013 ([pdf](http://www.phil.uu.nl/ozsl/articles/Hendriks03.pdf))

[[!redirects fragments]]


frame

> This entry is about the notion of _frame_ in [[topos theory]]. For other notions, see [[frame (disambiguation)]].

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The notion of _frame_ is a generalization of the notion of [[category of open subsets]] of a [[topological space]]. A _frame_ is like a category of open subsets in a space possibly more general than a topological space: a [[locale]]. This in turn is effectively defined to be anything that has a collection of open subsets that behaves essentially like the open subsets of a topological space do.

## Definition

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

A **frame** $\mathcal{O}$ is

* a [[partial order|poset]]

* that has

  * all [[small limit|small]] [[coproduct | coproducts]], called [[join|joins]] $\bigvee$

  * all [[finite limits]], called [[meet|meets]] $\wedge$

* and which satisfies the _infinite distributive law_:

  $$
    x \wedge (\bigvee_i y_i) \leq \bigvee_i (x\wedge y_i)
  $$

  for all $x, \{y_i\}_i$ in $A$

  (Note that the converse holds in any case, so we have equality.)

A _frame homomorphism_ is a homomorphism of posets that preserves finite meets and arbitrary joins. Frames and frame homomorphisms form the category [[Frm]].

The formal duals of frames, hence the objects in the [[opposite category]] [[Locale]] $:=$ [[Frm]]${}^{op}$ are called [[locale | locales]].

=--

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

The notation $\mathcal{O}$ to supposed to remind us that every frame is like a [[frame of opens]] of the corresponding [[locale]].

=--

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

So in particular a frame is a _[[lattice]]_ which has not only finite [[joins]] but all small joins. Being small itself, it is a [[suplattice]], and hence a [[complete lattice]]: it also has all (small) meets.

Furthermore, as the distributive law certainly holds when the joins in question are finite, it is a _[[distributive lattice]]_.

=--

## Properties

A useful way to understand frames and [[locales]] is as the simplest nontrivial special cases of [[(n,1)-topos | (n,1)-toposes]]. So we start in

* [As (0,1)-toposes](#As0Topos)

with some remarks on this, and only then turn to

* [General properties](#GeneralProperties)

of frames, which should make more sense this way.

### As a $(0,1)$-topos
 {#As0Topos}

The notion of _frame_ -- or rather its formal dual, the notion of _[[locale]]_ -- is the special case of the notion of [[(n,1)-topos | (n,1)-toposes]] for $n = 0$: [[(0,1)-topos | (0,1)-toposes]].

The axioms on a frame are nothing but [[Giraud's axioms]] on [[sheaf topos | sheaf toposes]], restricted to [[(0,1)-categories]]:

given the existence of finite limits and arbitrary colimits, the _infinite distributive law_ expresses that a frame has [[universal colimits]]: they are stable under [[pullback]]. (For notice that in a poset [[pullback | pullbacks]] and [[product | products]] coincide.)

Then a morphism of frames is precisely (the [[inverse image]] of) a [[geometric morphism]]: a morphism preserving finite limits and arbitrary colimits.

### General
 {#GeneralProperties}

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

A frame is automatically a [[Heyting algebra]].

=--

In [[category theory|category theoretic]] terms this means that it ia a [[cartesian closed category]], hence that

for every object $x \in \mathcal{O}$ the functor

$$
  x \wedge (-) : \mathcal{O} \to \mathcal{O}
$$

that forms the [[product]] with $x$ has a [[right adjoint]]

$$
  x \Rightarrow (-) : \mathcal{O} \to \mathcal{O}
$$

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

This exists by the [[adjoint functor theorem]], using that there is only a finite number of morphisms between any two objects (one or none) and that finite limits exist in $\mathcal{O}$.

=--

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

Therefore one may think of a of a frame as a [[cartesian closed category|cartesian closed]] [[suplattice]], or a cartesian [[quantale]].

But notice that the frame [[homomorphism | homomorphisms]] are not required to preserve the Heyting implication.

=--

### As sites

A frame is naturally equipped with the structure of a [[site]]:

a family of morphism $\{U_i \to U\}$ is [[covering]] precisely if $U$ is the [[union]] of the $U_i$

$$
  \bigvee_i U_i = U
  \,.
$$

For more on this see [[locale]].

## Formal duals: locales

The [[opposite category]] to the category [[Frm]] is the category [[Loc]] of [[locale | locales]] (possibly slightly generalized [[topological space | topological spaces]])

$$
  Loc := Frm^{op}
$$

Conversely, any [[topological space]] has a [[frame of opens|frame of open subsets]].  (In fact, one definition of a topological space is a set equipped with a subframe of its [[power set]].)

## Examples

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

For $X$ a [[topological space]], the [[category of open subsets]] of $X$ is a frame: the [[frame of opens]].

=--

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

For $X$ a [[set]], the [[power set]] $\mathcal{P}(X)$ of $X$ is the [[frame of opens]] corresponding to the [[discrete topology]] on $X$.

=--

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

The [[poset of truth values]] $\Omega$ is a [[frame]] equivalent to the [[frame of opens]] corresponding to the [[discrete topology]] on a [[singleton]].

=--

+-- {: .num_prop}
###### Proposition
A [[sup-lattice|complete]] [[decidable equality|decidable]] [[linear order]] is a frame.
=--

+-- {: .proof}
###### Proof
For any $x$ and collection $y_i$, the inequality $\bigvee_i x \wedge y_i \leq x \wedge \bigvee_i y_i$ holds automatically. For the reverse inequality, we note this follows trivially in case $\bigvee_i y_i \leq x$, since in that case we have $y_i \leq x$ for all $i$, whence

$$x \wedge \bigvee_i y_i = \bigvee_i y_i = \bigvee_i x \wedge y_i.$$

Otherwise we are in the case $x \lt \bigvee_i y_i$, where we must show the inequality

$$x \wedge \bigvee_i y_i = x \leq \bigvee_i x \wedge y_i$$

But this inequality must hold, else $\bigvee_i x \wedge y_i \lt x$ which would imply $y_i \lt x$ for all $i$, whence $\bigvee_i y_i \leq x$, contradiction.
=--

## Related concepts

* [[Heyting algebra]]
* [[Grothendieck topos]]
* [[coframe]]

## References

Frames in [[univalent foundations]]:

* Ayberk Tosun, _Formal Topology in Univalent Foundations_, ([pdf](https://odr.chalmers.se/handle/20.500.12380/301098), [slides](https://www.cs.bham.ac.uk/~axt978/talks/lab-lunch-formal-topology.pdf))

[[!redirects frames]]
[[!redirects morphism of frames]]
[[!redirects morphisms of frames]]
[[!redirects homomorphism of frames]]
[[!redirects homomorphisms of frames]]
[[!redirects frame homomorphism]]
[[!redirects frame homomorphisms]]
[[!redirects frame morphism]]
[[!redirects frame morphisms]]


frame (disambiguation)

Some unrelated concepts that go by the term _frame_:

* [[frame]] (in [[order theory]], [[locale theory]], [[topos theory]], etc)

* [[frame of a vector space]] / [[frame bundle]]

* [[Kripke frame]] in [[modal logic]]

category: disambiguation


frame bundle

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

# Frame and coframe bundles
* table of contents
{: toc}

## Definition

### Traditional

Given a $k$-[[vector bundle]] $p\colon E \to M$ of finite [[rank]] $n$, its __frame bundle__ (or bundle of [[frame of a vector space|frames]] in $E \to M$) is the bundle $F E \to M$ over the same base whose [[fiber]] over $x \in M$ is the set of all vector space [[basis|bases]] of $E_x = p^{-1}(x)$. The frame bundle has a natural action of $GL_n(k)$ given by an ordered change of basis which is free and transitive, i. e., the frame bundle is a [[principal bundle|principal]] $GL_n(k)$-bundle.

The __frame bundle of a [[manifold]]__ $M$ is the [[principal bundle]] $F T M \to M$ (also denoted $F M \to M$) of frames in the [[tangent bundle]] $T M$.

In the finite-dimensional case, the dual $GL_n$-principal bundle $(F T)^* M$ is the __coframe bundle__ of the manifold. This means that $F^* M = (F T)^* M$ is the associated bundle to $F T M \times_{GL_n(k)}GL_n(k)$ where the left action of $GL_n(k)$ on $GL_n(k)$ is given by right multiplication by inverses $g. h = h\cdot g^{-1}$. Also $F T M\cong (F T)^* M\times_{GL_n(k)} GL_n(k)$ using the same formula. Furthermore, the right action of $GL_n(k)$ on this associated bundle is given by left multiplication by inverses on $GL_n(k)$ factor.

Coframe bundle $F^* M$ has the following independent description. One looks at the set $\mathcal{U}(M)$ of tuples of the form $(p,(U,h))$ where $p\in U$ and $(U,h)$ is chart of the smooth structure on $M$, $U\subset M$, $h : U\to \mathbf{R}^n$ (an atlas where $U$-s make a basis of topology suffices). $GL_n(k)$ acts on the right on $\mathcal{U}(M)$ by

$$
(p, (U, h)) A := (p, (U, A^{-1} h)).
$$
Then $((p,(U,h))A)A' = (p,(U,h)) (AA')$ holds.
The total space $F^* M$ of the coframe bundle by the definition, as a set, consists of classes of equivalence of tuples in $\mathcal{U}(M)$ where $(p,(U,h)) \sim (p',(U',h'))$ iff $p = p'$ and the [[Jacobian matrix]] of the transition between charts at $h'(p)$ is the unit matrix: $J_{h'(p)}(h\circ (h')^{-1}) = I$. The left action of $GL_n(k)$ is induced on the quotient. There is an obvious projection $\pi: [(p,(U,h)]\mapsto p$. To define the differential and principal bundle structure one charts $F^* M\to M$ with local trivializations from the neighborhoods of the form $U\times GL_n(k)$, transfers the structure and checks that the transition functions are of the appropriate smoothness class and right $GL_n(k)$-equivariant. The basic prescription is that to every chart $(U,h)$ one defines a map

$$
\phi_{h} = \pi^{-1}(U)\to U \times GL_n(k),\,\,\,\,\,\,z\mapsto (\pi(z), J_{h(\pi(z))}(h'\circ h^{-1})),
$$
where $z = [(\pi(z), (U',h'))]$ with $\pi(z)\in U'\cap U$. This does not depend on the choice of the chart $(U',h')$ around $\pi(z)$. There is an equivariance
$$
J_{h(\pi(z A))}(h'\circ h^{-1})) = A^{-1} J_{h(\pi(z))}(h'\circ h^{-1}))
$$
and on intersection of $(U,h)$ and $(V,g)$
$$
J_{h(\pi(z))}(h'\circ h^{-1})) = J_{g(\pi(z))}(h'\circ g^{-1})J_{h(\pi(z))}(g\circ h^{-1})
$$
Then $\phi_h$ is onto and

$$
(\phi_h \circ (\phi_g)^{-1})(p,A) = (p, A J_{h(p)}(g\circ h^{-1})
$$

what shows that the transition functions are smooth (where $GL_n(k)$ has the standard differential structure).

### In differential cohesion
 {#InDifferentialCohesion}

In a context of [[differential cohesion]], then the frame bundle (or [[higher order frame bundle]]) of a $V$-[[manifold]] is the [[principal bundle]] ([[principal infinity-bundle]]) to which the [[infinitesimal disk bundle]] is the canonically [[associated bundle]] ([[associated infinity-bundle]])

See at _[differential cohesion -- Frame bundles](differential+cohesion#GLnTangentBundles)_.

## Properties

### The canonical differential 1-form
 {#TheCanonical1Form}

The frame bundle $Fr(X)$ carries a canonical [[differential 1-form]] with values in $\mathbb{R}^n$.

$$
  \alpha \in \Omega^1(Fr(X), \mathbb{R}^n)
$$

This is defined as follows. Let $p \in Fr(X)$ be a point in the frame bundle $\pi \colon Fr(X)\to X$ over some point $x \in X$, hence a linear isomorphism $p \colon T_x \simeq \mathbb{R}^n$. For $v \in T_p Fr(X)$ a [[tangent vector]] to the frame bundle, its projection $\pi_\ast v \in T_x X$ is a [[tangent vector]] to $X$. Then the value of $\alpha$ on $v$ is the image of this $\pi_\ast(v)$ under the isomorphism $p$

$$
  \alpha(v) \coloneqq p(\pi_\ast(v))
  \,.
$$

([Sternberg 64, section VII, (2.2)](#Sternberg64))

### Relation to $G$-structures

A choice sub-bundle of a frame bundle which is a $G$-[[principal bundle]] for $G\hookrightarrow GL(n)$ defines a _[[G-structure]]_. See there for more.

## Related concepts

* a [[section]] of a frame bundle is also called a _[[frame field]]_.

* [[higher order frame bundle]] ([[jet bundle]]-version)

## References

* Wikipedia (English) [frame bundle](http://en.wikipedia.org/wiki/Frame_bundle)

* {#Sternberg64} [[Shlomo Sternberg]], _Lectures on differential geometry_, Prentice Hall 1964; Russian transl. Mir 1970

[[!redirects coframe bundle]]
[[!redirects frame bundles]]


frame field

#Contents#
* table of contents
{:toc}

## Definition

A _frame field_ is a [[section]] of a [[frame bundle]] (of a [[manifold]]), hence a collection of [[vector fields]] that form a [[basis of a vector space|basis]] of the [[tangent space]] at each point.

## Related concepts

* [[G-structure]]

* [[vector field]]

[[!redirects frame fields]]


frame of a vector space

A [[basis]] for a [[vector space]].

One speaks of the [[frame bundle]] of the [[tangent bundle]] of a [[smooth manifold]].

[[!redirects local frame]]


frame of opens

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

Given a [[topological space]] $X$, the [[open subspaces]] of $X$ form a [[poset]] which is in fact a [[frame]].  This is the __frame of open subspaces__ of $X$.  When thought of as a [[locale]], this is the __[[topological locale]]__ $\Omega(X)$.  When thought of as a [[category]], this is the __[[category of open subsets]]__ of $X$.

Similarly, given a [[locale]] $X$, the open subspaces of $X$ form a poset which is in fact a frame.  This is the __frame of open subspaces__ of $X$.  When thought of as a locale, this is simply $X$ all over again.  When thought of as a category, this is a [[site]] whose [[topos of sheaves]] is a __[[localic topos]]__.

The frame of open subsets of the [[point]] is given by the power set of a singleton, or more generally by the [[subobject classifier|object of truth values]] of the ambient topos.

[[!redirects frame of opens]]
[[!redirects frames of opens]]
[[!redirects frame of open sets]]
[[!redirects frames of open sets]]
[[!redirects frame of open subsets]]
[[!redirects frames of open subsets]]
[[!redirects frame of open subspaces]]
[[!redirects frames of open subspaces]]


framed bicategory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For large classes of examples of [[bicategory|bicategories]] the [[k-morphism|1-morphisms]] naturally are of one of two different types:

1. special morphisms that may be taken to compose strictly among themselves;

1. more general morphisms that behave like [[bimodule]]s.

The archetypical example is indeed the [[bicategory]] whose objects are [[algebra]]s, and whose 1-morphisms are [[bimodule]]s between these: every ordinary algebra homomorphism $f : A \to B$ induces an $A$-$B$-bimodule ${}_f B$ and this operation induces a 2-functor from the [[category]] of algebras and algebra homomorphisms into the [[bicategory]] of algebras and bimodules, which is the identity on objects.

For usefully working with bicategories of this kind, it is typically of crucial importance to remember this extra information. A _framing_ on a bicategory is a way to encode this.

This is essentially the same as a _[[proarrow equipment]]_ on a [[bicategory]]. See there for more.

## See also
* [[companion pair]]
* [[conjunction]]
* [[proarrow equipment]]

## References

* {#Shulman08} [[Mike Shulman]], *Framed bicategories and monoidal fibrations*, Theory and Applications of Categories **20** 18 (2008) 650–738 [[tac:2018](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html), [arXiv:0706.1286](https://arxiv.org/abs/0706.1286)]

* [[Nicola Gambino]], [[Joachim Kock]], _Polynomial functors and polynomial monads_, ([arXiv:0906.4931](http://arxiv.org/abs/0906.4931))

* [[Thomas Fiore]], [[Nicola Gambino]], [[Joachim Kock]], _Monads in double categories_, ([arXiv:1006.0797](http://arxiv.org/abs/1006.0797))

* [[Patrick Schultz]], _Regular and exact (virtual) double categories_, ([arXiv:1505.00712](http://arxiv.org/abs/1505.00712))

* [[Patrick Schultz]], [[David Spivak]], [[Christina Vasilakopoulou]], Ryan Wisnesky, _Algebraic Databases_, ([arXiv:1602.03501](http://arxiv.org/abs/1602.03501))

* Pierre-Evariste Dagand, [[Conor McBride]], _A Categorical Treatment of Ornaments_, ([arXiv:1212.3806](http://arxiv.org/abs/1212.3806))

[[!redirects framed bicategories]]
[[!redirects fibrant double category]]
[[!redirects fibrant double categories]]


framed combinatorial topology

# Contents
* automatic toc
{: toc}

## Idea

Framed combinatorial topology studies combinatorial-topological concepts, such as regular cells, in the context of [[directed topological space#framed_spaces|framed-directed topological spaces]].

## References

* {#DornDouglas21} [[Christoph Dorn]] and [[Christopher Douglas]], _Framed combinatorial topology_, 2021 ([arXiv](https://arxiv.org/abs/2112.14700), [latest](https://cxdorn.github.io/fct-book/))
* {#Heidemann2023} [[Lukas Heidemann]], _Framed Combinatorial Topology with Labels in $\infty$-Categories_, 2023 ([arXiv](https://arxiv.org/abs/2305.06288))


framed elliptic curve

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Over the [[complex numbers]], a _framed elliptic curve_ is an [[elliptic curve]] equipped with an [[order|ordered]] [[basis]] $\{a,b\}$ of the [[homology]] of the curve such that the [[intersection number]] $a\cdot b$ is 1 (e.g. [Hain 08, def. 1.13](#Hain08)).

The standard construction of the [[moduli stack of elliptic curves]] is in fact via the [[orbifold quotient]] of the space of framed elliptic curves [[action|acted]] on by the [[modular group]].

A framing on an elliptic curve is in a way the extreme case of a _[[level structure on an elliptic curve]]_.

A framed elliptic curve is in particular naturally a [[framed manifold]] (a framed surface).

## Related concepts

* [[moduli stack of elliptic curves]]

* [[moduli space of framed manifolds]]

## References

* {#Hain08} [[Richard Hain]], _Lectures on Moduli Spaces of Elliptic Curves_ ([arXiv:0812.1803](http://arxiv.org/abs/0812.1803))

[[!redirects framed elliptic curves]]


framed link

### Idea

A **framed link** is a link where the circles forming the components are viewed as having a thickness, albeit an arbitrarily small thickness.
That is to say, instead of the components of the links being embedded circles, they are embedded solid tori.

The thickening can be considered in one direction only which gives embedded ribbons.  This is an equivalent definition which can be useful as it makes the distinction clearer between an embedding and the same embedding composed with, say, a [[Dehn twist]].

A [[link diagram]] can be made into a diagram of a framed links by giving it the **blackboard framing**: this views each segment of the link diagram as a ribbon lying on the "blackboard".

Two diagrams represent the same link diagram if and only if they can be related by a modified version of the [[Reidemeister moves]] in which the first move is replaced by the following move.

[[!include modified Reidemeister move 1 - SVG]]

## Definition

The precise notion of a framing on a link is perhaps most naturally given in terms of smooth structures on manifolds. We'll start from there and later give other versions.

Let $M$ be an oriented 3-manifold, for example $S^3$. The tangent bundle $p: T M \to M$ is a bundle with structure group $GL(3) = GL(\mathbb{R}^3)$ which in fact admits a reduction to $GL_+(3)$ (invertible linear transformations with positive determinant) since $M$ is oriented. A further reduction to the structure group $SO(3)$ can be effected by a number of means, e.g., by equipping $M$ with a Riemannian metric.

Now let

$$L: S^1 \sqcup \ldots \sqcup S^1 \hookrightarrow M$$

be a smoothly embedded link (of oriented circles). The normal bundle of $L$ is the cokernel of the bundle inclusion

$$T(S^1 \sqcup \ldots \sqcup S^1) \to L^\ast T M$$

(as bundles over $S^1 \sqcup \ldots \sqcup S^1$); it is naturally associated with a principal $SO(2)$-bundle. A *framing* of $L$ is simply a choice of section $\sigma$ of this principal $SO(2)$-bundle. It can be thought of as a smooth choice of unit normal vectors along points of the embedded link, and can be visualized as "ribbons", one for each component $S^1$, with one edge of the ribbon being that component.

The $SO(2)$-bundle or circle bundle over each component $S^1$ can be seen as an embedded [[torus]] in $M$. The framing itself induces a diffeomorphism on this torus which takes an element $(\xi, p)$ over $p \in S^1$ to $(\sigma(p)\xi, p)$.

However, it is often convenient to consider such framings, or rather their associated torus diffeomorphisms, only up to isotopy. For each component $S^1$, the isotopy class is specified by an integer which gives the number of 360 degree clockwise rotations of the unit normal or clockwise twists of the ribbon as one traverses the component in the direction of its orientation. Thus a shortcut is to say a framing of an oriented link is given by assigning an integer to each link component.

[[!redirects framed links]]
[[!redirects Framed link]]
[[!redirects Framed links]]
[[!redirects framed knot]]
[[!redirects framed knots]]
[[!redirects Framed knot]]
[[!redirects Framed knots]]


framed little 2-disk operad

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

#Contents#
* table of contents
{:toc}

## Definition

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

The **framed little 2-disk operad** is the [[operad]] $fD_2$ in [[Top]] whose [[topological space]] $fD_2(n)$ of $n$-ary operations is the space of maps

$$
  \coprod_n D \to D
$$

from $n$-copies of the 2-[[ball]] to itself, which restrict on each component to a map that is a combination of

* a translation

* a dilatation

* a rotation

of the disk (regarded via its standard [[embedding]] $D \hookrightarrow \mathbb{R}^2$ into the 2-simensional [[Cartesian space]]) such that the images of all disks are disjoint.
=--

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

This differs from the [[little 2-disk operad]] by the fact that rotations of the disks are admitted. Under passing to chains and then to [[homology]], this operation gives rise to the BV-operator in a [[BV-algebra]]. See [Properties](#Properties) below.

=--

## Properties {#Properties}

+-- {: .num_theorem}
###### Theorem

The [[homology]] of the framed little 2-disk operad in chain complexes is the [[BV-operad]] $BV$ the operad for [[BV-algebra]]s:

$$
  BV \simeq H_\bullet(fD_2)
  \,.
$$

=--

This is due to ([Getzler](#Getzler)).

+-- {: .num_theorem}
###### Theorem

The framed little disk operad is [[formal dg-algebra|formal]] in characteristic zero.

This means that there is a zig-zag of [[quasi-isomorphism]]s

$$
  C_\bullet(fD_2)
   \stackrel{\simeq}{\leftarrow}
   \stackrel{\simeq}{\to}
   \cdots
   \stackrel{\simeq}{\leftarrow}
   \stackrel{\simeq}{\to}
   H_\bullet(fD_2)
  \,.
$$

=--

This is due to [Severa 09](#Severa09),
[Giansiracusa-Salvatore 09)](#GiansiracusaSalvatore09). See also ([Valette, slide 35](#Valette)).

Accordingly one makes the following definition:

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

The operad for [[homotopy BV-algebra]]s is any cofibrant [[resolution]] of $BV \simeq  H_\bullet(fD_2)$, or equivalently of $C_\bullet(fD_2)$.

=--

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

Write $R \beta_j$ for the [[ribbon braid group]] on $j$ elements and $P R \beta_j$ for the [[kernel]] of the surjection $R \beta_j \to \Sigma_j$ onto the [[symmetric group]].

Say that a [[ribbon operad]] $P$ is an **$R_\infty$-operad** if the [[ribbon braid group]]s  act freely and properly on $P$ and if each [[topological space]] $P(k)$ is [[contractible]].

=--

+-- {: .num_theorem}
###### Theorem

If $P$ is an $R_\infty$-operad, then the sequence of quotient spaces $\{P(n)/P R \beta_n\}$  forms a symmetric operad equivalent to the frame little disks operad.

=--

This is ([Wahl, lemma 1.5.17](#Wahl)).

## Related concepts

* **framed little 2-disk operad**

* [[framed little n-disk operad]]

* [[BD operad]]

[[!include deformation quantization - table]]

## References

The framed little 2-disk operad was introduced in

* {#Getzler} [[Ezra Getzler]], _Batalin-Vilkovisky algebras and two-dimensional topological field theories_ , Comm. Math. Phys. 159 (1994), no. 2, 265–285. ([arXiv](http://arxiv.org/abs/hep-th/9212043))

For the relation to ribbons see

* {#Wahl} [[Nathalie Wahl]], _Ribbon braids and related operads_ PhD thesis, Oxford (2001) ([pdf](http://eprints.maths.ox.ac.uk/43/1/wahl.pdf)).

The formality of $fD_2$ was shown in

* {#GiansiracusaSalvatore09} [[Jeffrey Giansiracusa]], [[Paolo Salvatore]], _Formality of the framed little 2-discs operad and semidirect products_ , in: _Homotopy theory of function spaces and related topics_, Cont. Math. 519, AMS, pp. 115-121  ([arxiv 0911.4428](http://arxiv.org/PS_cache/arxiv/abs/0911/0911.4428))

and

* {#Severa09} [[Pavol Severa]], _Formality of the chain operad of framed little disks_,  ([arxiv AT 0902.3576](http://arxiv.org/PS_cache/arxiv/abs/0902/0902.3576))

Discussion of [[homotopy BV-algebra]]s is in

* [[Imma Galvez-Carrillo]], [[Andy Tonks]], [[Bruno Valette]], _Homotopy Batalin-Vilkovisky algebras_ ([arXiv:0907.2246](http://arxiv.org/abs/0907.2246))

see also

* [[Imma Galvez-Carrillo]], [[Vassily Gorbounov]], [[Andy Tonks]], _Homotopy Gerstenhaber structures and vertex algebras_, [math/0611231.QA](https://arxiv.org/abs/math/0611231)

Slides of a talk summarizing this are at

* {#Valette} [[Bruno Valette]], _Homotopy Batalin-Vilkovisky algebras_ ([pdf](http://math.unice.fr/~brunov/download/Homotopy%20BV.pdf))

[[!redirects E2-operad]]
[[!redirects E2-operads]]

[[!redirects E2 operad]]
[[!redirects E2 operads]]

[[!redirects framed little disk operad]]
[[!redirects framed little disks operad]]

[[!redirects framed little disk operads]]
[[!redirects framed little disks operads]]

[[!redirects framed little disc operad]]
[[!redirects framed little discs operad]]

[[!redirects framed little disc operads]]
[[!redirects framed little discs operads]]

[[!redirects BV-operad]]


framed little n-disk operad

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

#Contents#
* table of contents
{:toc}

## Idea

The _framed little $n$-disk operad_ is an [[operad]] in [[chain complex]]es that whose $n$-ary operations come from embedding $n$-dimensional disks in a larger one _and rotating_ them.  It is joint generalization of the [[little k-cubes operad]] and the [[framed little 2-disk operad]].

## Properties

+-- {: .un_theorem}
###### Theorem

The shifted rational [[singular homology]] $H_{\bullet + d}(S X, \mathbb{Q})$ of the $n$-[[sphere]] space $S X = X^{S^n}$ of an [[oriented]] [[manifold]] $X$ of [[dimension]] $d$ naturally has the structure of a [[BV-algebra|n-BV-algebra]].

=--

This appears as ([CohenVoronov, theorem 5.3.3](#CohenVoronov)).

## References

Section 5.3 of

* [[Ralph Cohen]], [[Alexander Voronov]], _Notes on string topology_ ([arXiv:math/0503625](http://arxiv.org/abs/math/0503625))
{#CohenVoronov}


framed manifold

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

#Contents#
* table of contents
{:toc}

## Idea

In one sense of the term, a _framing_ of a [[manifold]] is a choice of trivialization of its [[tangent bundle]], hence a choice of [[section]] of the corresponding [[frame bundle]].

A manifold that admits a framing is also called a **parallelizable manifold**. A manifold equipped with a framing is also called a **parallelized manifold**.

More generally, one means by a _framing_ not a trivialization of the tangent bundle itself, but

* of the [[normal bundle]] if the manifold is understood  [[embedding|embedded]] in some [[Cartesian space]] $\mathbb{R}^d$

* of the _[[stable tangent bundle]]_.

Accordingly, a _framed cobordism_ is a [[cobordism]] equipped with a framing on the underlying manifold (see also at [[MFr]]).

For $dim(X)$ the [[dimension]] of the manifold and $n \geq dim(X)$, then one also speaks of an _$n$-framing_ to mean a trivialization of the "$n$-stabilized tangent bundle"  $T X \oplus \mathbb{R}^{n-dim(X)}$ (where the right [[direct sum|direct summand]] denotes the trivial real [[vector bundle]] of [[rank]] $n - dim(X)$). These $n$-framed manifolds appear in particular in the construction of the  [[cobordism category]] of framed $n$-dimensional cobordisms. The [[cobordism hypothesis]] asserts essentially that the [[(∞,n)-category of cobordisms]] with $n$-framing is the [[free construction|free]] [[symmetric monoidal (∞,n)-category with duals]].

Beware that there is also the term "[[2-framing]]" due to ([Atiyah](#Atiyah)), which is related but different.

## Examples

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

Every [[Lie group]] is a parallelizable manifold.

=--

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

Every non-zero [[invariant vector field]] on the Lie group provides an everywhere non-vanishing section of the tangent bundle.

=--

The following is obvious:

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

Every 3-[[dimension|dimensional]] [[manifold]] with [[spin structure]] admits a framing.

=--

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

That a 3-manifold $X$  has [[spin structure]] means that we have a [[reduction of the structure group]] of the tangent bundle to the [[spin group]], and hence the tangent bundle is classified by a map $X \to B Spin(3)$. But $Spin(3)$ has vanishing [[homotopy groups]] in degree $0 \leq k \leq 2$. Therefore its [[delooping]] [[classifying space]] $B Spin(3)$ has vanishing homotopy groups below degree 4, and hence every morphism out of a 3-dimensional manifold into it is homotopically constant.

=--

But in fact, the following stronger statement is also true.

+-- {: .num_prop #EveryOrientable3ManifoldsIsParallelizable}
###### Proposition

Every [[orientation|orientable]] 3-[[dimension|dimensional]] [[manifold]] admits a framing.

=--

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

By the argument in the proof of
prop. \ref{SpinManifoldAdmitsFraming}, the only possible
obstruction is the [[second Stiefel-Whitney class]] $w_2$.  By the discussion at _[[Wu class]]_, this vanishes on an oriented manifold precisely if the second [[Wu class]] vanishes. This, in turn, is by definition defined to represent the [[Steenrod square]] under [[cup product]], and this vanishes on a 3-manifold by degree reasons.

=--

+-- {: .num_remark #In4dFirstOrderGravity}
###### Remark

Prop. \ref{EveryOrientable3ManifoldsIsParallelizable} has some impact in the context of the [[first order formulation of gravity]], where one is interested in [[vielbein]] fields on 4-dimensional [[spacetime manifolds]], and in particular in [[globally hyperbolic spacetimes]], which, as [[topological spaces]], are the [[Cartesian product]] of the [[real line]] (time) with a [[3-manifold]] (space). Prop. \ref{EveryOrientable3ManifoldsIsParallelizable} implies that already when that spatial 3-manifold is orientable, then the whole globally hyperbolic spacetime admits a framing. (See also at _[[teleparallel gravity]]_.)

=--

+-- {: .num_theorem}
###### Theorem

The $n$-[[spheres]] that admit a framing are precisely only

* the 0-sphere $S^0 = \ast \coprod \ast$, the unit [[real numbers]]

* the 1-sphere $S^1$, the [[circle]] underlying the [[circle group]] (the unit [[complex numbers]]);

* the 3-sphere $S^3$, underlying the [[special unitary group]] $SU(2)$, is isomorphic to the unit [[quaternions]];

* the 7-sphere $S^7$, which underlies a [[Moufang loop]] internal to [[Diff]], namely the unit [[octonions]],

where the algebras appearing are precisely the four [[normed division algebras]].

=--

This is due to ([Adams 58](#Adams58)), proven with the [[Adams spectral sequence]].

### Stable homotopy elements

Left invariant framings $\mathcal{L}$ on compact connected Lie groups $G$ with $dim(G)=k$ give rise to elements $[G,\mathcal{L}]$ in the stable homotopy group $\pi_k^s$ of spheres. One can restrict attention to semisimple Lie groups $G$ since this construction behaves well with respect to products, $G \to T\times G/T$ gives a framed diffeomorphism and  $[S^1,\mathcal{L}] \in \pi_1^s$ is the generator. The following facts are assembled from ([Ossa 1982](#Ossa82)) and ([Minami 2016](#Minami16)).

* For any *semisimple* compact connected Lie group $G$, Ossa proved that $72[G,\mathcal{L}]=0$. In particular, the only possible torsion is at the primes 2 and 3.

* The left invariant framings on $SO(n),Spin(n)$ and $SU(n)$ give the zero element for $n\geq 7$, those on $Sp(n)$ for $n\geq 4$, as do those on $F_4,E_6,E_7,E_8, SO(4),SO(6),SU(5),SU(6)$. (this means that the undetermined cases of rank 4 in Ossa's Table 1 are, in fact, all 0)

* The left invariant framings on the remaining groups ($SU(2)=Spin(3)=Sp(1),SU(3),SU(4)=Spin(6),Sp(2)=Spin(5),Sp(3),SO(3),SO(5)$ and $G_2$) represent known nonzero classes in $\pi_\ast^s$.

## Properties

### Relation to intersection pairing and Kervaire invariant
 {#IntersectionPairing}

On a framed manifold, there is a canonical [[quadratic refinement]] of the [[intersection pairing]]. The associated invariant is the _[[Kervaire invariant]]_.

## Properties

### Moduli of framings

The [[homotopy type]] of the [[moduli space of framed manifolds|moduli space of framings]] on a fixed manifold is a disjoint union of subgroups of the oriented [[mapping class group]] which fix a given isotopy type of framings.

## Related concepts

* [[normal framing]], [[normal twisted framing]]

* [[2-framing]]

* [[framed elliptic curve]]

* [[moduli space of framed manifolds]]

* [[teleparallel gravity]]

Formalization in [[differential cohesion]] is discussed [there](http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#GLnTangentBundles).

## References

### General

Review in the context of the [[Kervaire invariant]] includes

* [[Akhil Mathew]], _The Kervaire invariant I_ ([web](https://amathew.wordpress.com/2012/10/01/the-kervaire-invariant-i/#more-3888))

The theorem about the parallizablitiy of spheres is due to

* {#Adams58} [[John Adams]], _On the Non-Existence of Elements of Hopf Invariant One_ Bull. Amer. Math. Soc. 64, 279-282, 1958, Ann. Math. 72, 20-104, 1960.

Relation to existence of [[flat connections]] on the [[tangent bundle]] is discussed in

* {#Thorpe65} [[John Thorpe]], _Parallelizablility and flat manifolds_, 1965 ([[ThorpeParallelizable.pdf:file]])

### For 2-manifolds

The [[moduli space of framed surfaces]] is discussed in

*  [[Oscar Randal-Williams]], _Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces_ ([arXiv:1001.5366](http://arxiv.org/abs/1001.5366))

### For 3-manifolds

* [[Rob Kirby]], [[Paul Melvin]], _Canonical framings for 3-manifolds_, Turkish Journal of Mathematics, volume 23, number 1,1999 ([[KirbyMelvon3Framings.pdf:file]])

The relation to "[[2-framings]]" is discussed in

* {#Atiyah} [[Michael Atiyah]], _On framings of 3-manifolds_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/atiyahfr.pdf))

### Left invariant framings on compact connected Lie groups

* {#Ossa82} Erich Ossa, Lie groups as framed manifolds, Topology, 21 (1982), 315–323, ([doi](https://doi.org/10.1016/0040-9383%2882%2990013-1))

* {#Minami16} Haruo Minami, _On framed simple Lie groups_, ([pdf](http://projecteuclid.org/euclid.jmsj/1468956166))

For an extension to [[p-compact groups]] see

* [[Tilman Bauer]], _p-compact groups as framed manifolds_, ([paper](https://people.kth.se/~tilmanb/publication/p-compact-groups-as-framed-manifolds/))

[[!redirects framed manifolds]]

[[!redirects framed cobordism]]
[[!redirects framed cobordisms]]

[[!redirects parallelizable manifold]]
[[!redirects parallelized manifold]]
[[!redirects parallelizable manifolds]]
[[!redirects parallelized manifolds]]

[[!redirects framing]]
[[!redirects framings]]

[[!redirects parallelizable sphere]]
[[!redirects parallelizable spheres]]

[[!redirects parallelizable]]

[[!redirects n-framing]]
[[!redirects n-framings]]
[[!redirects 1-framing]]
[[!redirects 1-framings]]
[[!redirects 3-framing]]
[[!redirects 3-framings]]
[[!redirects 4-framing]]
[[!redirects 4-framings]]
[[!redirects 5-framing]]
[[!redirects 5-framings]]


Frames and Locales

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

This entry provides a hyperlinked index for the book

* [[Jorge Picado]], [[Aleš Pultr]],

  _Frames and Locales.  Topology without points_,

  Frontiers in Mathematics, Birkhäuser (2012)

on basics of [[locales]] and [[pointfree topology]].

# Contents
* table of contents
{:toc}

The following lists chapterwise linked lists of keywords to relevant and related existing entries, as far as they already exist.

I. Spaces and [[lattice|Lattices]] of Open Sets

 1. [[Sober spaces]]
 2. The axiom $T_D$: another case of spaces easy to reconstruct
 3. Summing up
 4. Aside: several technical properties of $T_D$-spaces

II. [[Frames]] and [[Locales]]. Spectra

 1. [[Frames]]
 2. [[Locales]] and [[localic maps]]
 3. Points
 4. Spectra
 5. The unit σ and [[spatiality]]
 6. The unit λ and [[sobriety]]

III. [[Sublocales]]

 1. [[Extremal monomorphisms]] in [[Loc]]
 2. [[Sublocales]]
 3. The co-frame of [[sublocales]]
 4. Images and preimages
 5. Alternative representations of [[sublocales]]
 6. Open and [[closed sublocales]]
 7. Open and [[closed localic maps]]
 8. [[Closure]]
 9. Preimage as a homomorphism
 10. Other special sublocales: one-point sublocales, and [[Boolean locale|Boolean]] ones
 11. [[Sublocales]] as quotients. Factorizing frames is surprisingly easy

IV. Structure of [[Localic Morphisms]]. The Categories [[Loc]] and [[Frm]]

 1. Special morphisms. Factorizing in Loc and Frm
 2. The down-set functor and free constructions
 3. [[Limits]] and a [[colimit]] in [[Frm]]
 4. [[Coproducts]] of [[frames]]
 5. More on the structure of [[coproduct]]
 6. [[Epimorphisms]] in [[Frm]]

V. [[Separation Axioms]]

 1. Instead of $T_1$: [[subfit]] and [[fit]]
 2. Mimicking the [[Hausdorff axiom]]
 3. I-Hausdorff frames and [[regular monomorphisms]]
 4. Aside: Raney identity
 5. Quite like the classical case: [[Regular]], [[completely regular]] and [[normal]]
 6. The categories [[RegLoc]], [[CRegLoc]], [[HausLoc]] and [[FitLoc]]

VI. More on [[Sublocales]]

 1. Subspaces and [[sublocales]] of spaces
 2. Spatial and induced [[sublocales]]
 3. [[Complemented sublocales]] of spaces are spatial
 4. The [[zero-dimensionality]] of $Sl(L)^op$ and a few consequences
 5. Diﬀerence and pseudodifference, residua
 6. [[Isbell’s Development Theorem]]
 7. Locales with no non-spatial sublocales
 8. Spaces with no non-induced sublocales

VII. [[Compactness]] and [[Local Compactness]]

 1. Basics, and a technical lemma
 2. [[Compactness]] and [[separation]]
 3. Kuratowski-Mrówka characterization
 4. Compactification
 5. [[Well below]] and [[rather below]]. Continuous completely regular frames
 6. Continuous is the same as [[locally compact]].  [[Hofmann-Lawson duality]]
 7. One more spatiality theorem
 8. [[Supercompactness]]. Algebraic, superalgebraic and [[supercontinuous frames]]

VIII. (Symmetric) Uniformity and Nearness

 1. Background
 2. [[Uniformity]] and [[nearness]] in the point-free context
 3. Uniform homomorphisms. Modelling embeddings. Products
 4. Aside: admitting nearness in a weaker sense
 5. Compact uniform and nearness frames. Finite covers
 6. [[Completeness]] and [[completion]]
 7. Functoriality. [[CUniFrm]] is coreﬂective in [[UniFrm]]
 8. An easy completeness criterion

IX. [[Paracompactness]]

 1. [[Full normality]]
 2. [[Paracompactness]], and its various guises
 3. An elegant, specifically point-free, characterization of [[paracompactness]]
 4. A pleasant surprise: paracompact (co)reflection

X. More about Completion

 1. A variant of the [[completion of uniform frames]]
 2. Two applications
 3. Cauchy points and the resulting space
 4. [[Cauchy spectrum]]
 5. [[Cauchy completion]]. The case of countably generated uniformities
 6. Generalized Cauchy points

XI. [[Metric Frames]]

 1. Diameters and metric diameters
 2. Metric spectrum
 3. [[Uniform Metrization Theorem]]
 4. [[Metrization theorems]] for plain frames
 5. Categories of [[metric frames]]

XII. [[Entourages]]. [[Asymmetric Uniformity]]

 1. [[Entourages]]
 2. [[Uniformities]] via [[entourages]]
 3. [[Entourages]] versus [[covers]]
 4. [[Asymmetric uniformity]]: the classical case
 5. [[Biframes]]
 6. [[Quasi-uniformity]] in the point-free context via [[paircovers]]
 7. The adjunction $QUnif \leftrightarrows QUniFrm$
 8. [[Quasi-uniformity]] in the point-free context via [[entourages]]

XIII. [[Connectedness]]

 1. A few observations about sublocales
 2. [[Connected]] and disconnected locales
 3. [[Locally connected]] locales
 4. A weird example
 5. A few notes

XIV. Frame of Reals and Real Functions

 1. The frame $L(R)$ of [[reals]]
 2. Properties of $L(R)$
 3. $L(R)$ versus the usual space of reals
 4. The metric uniformity of $L(R)$
 5. Continuous real functions
 6. [[Cozero]] elements
 7. More general real functions
 8. Notes

XV. [[Localic Groups]]

 1. Basics
 2. The category of [[localic groups]]
 3. [[Closed Subgroup Theorem]]
 4. The multiplication μ is open. The semigroup of open parts
 5. Uniformities
 6. Notes

Appendix I. [[Posets]]

 1. Basics
 2. [[Zorn’s Lemma]]
 3. [[Suprema]] and [[infima]]
 4. [[Semilattices]], [[lattices]] and [[complete lattices]]. Completion
 5. [[Galois connections]] (adjunctions)
 6. (Semi)lattices as algebras. [[Distributive lattices]]
 7. [[Pseudocomplements]] and [[complements]].  [[Heyting]] and [[Boolean algebras]]

Appendix II. Categories

 1. [[Categories]]
 2. [[Functors]] and [[natural transformations]]
 3. Some basic constructions
 4. More special morphisms. Factorization
 5. [[Limits]] and [[colimits]]
 6. [[Adjunction]]
 7. [[Adjointness]] and (co)limits
 8. Reflective and [[coreflective subcategories]]
 9. [[Monads]]
 10. Algebras in a category

category: reference


Franc Forstneric

__Franc Forstnerič__ is a Professor of Mathematics at the University of Ljubljana, Slovenia. His research mainly focuses on several complex variables.

* [homepage](http://www.fmf.uni-lj.si/~forstneric)

## Selected writings

On [[Oka manifolds]] and the [[Oka principle]]:

* [[Franc Forstnerič]], _The Oka principle for sections of stratified fiber bundles_, Pure Appl. Math. Quarterly (Special Issue in honor of Joseph J. Kohn), 6 (2010), no. 3, 843--874 ([arxiv/0705.0591](http://arxiv.org/abs/0705.0591), [doi:10.4310/PAMQ.2010.v6.n3.a11](https://dx.doi.org/10.4310/PAMQ.2010.v6.n3.a11))

* {#Forstneric09a} [[Franc Forstnerič]], _Oka manifolds_, Comptes Rendus Mathematique, Acad. Sci. Paris 347 (2009), 1017–20 ([arXiv:0906.2421](https://arxiv.org/abs/0906.2421), [doi:10.1016/j.crma.2009.07.005](https://doi.org/10.1016/j.crma.2009.07.005))

* {#Forstneric09b} [[Franc Forstnerič]], _Oka maps_, Comptes Rendus Mathematique, Acad. Sci. Paris, Ser. I 348 (2010) 145-148 ([arxiv/0911.3439](http://arxiv.org/abs/0911.3439), [doi:10.1016/j.crma.2009.12.004](https://doi.org/10.1016/j.crma.2009.12.004))

* {#ForstnericLarusson11} [[Franc Forstnerič]], [[Finnur Lárusson]], *Survey of Oka theory*, New York J. Math., 17a (2011), 1-28 ([arXiv:1009.1934](https://arxiv.org/abs/1009.1934), [eudml:232963](https://eudml.org/doc/232963))

* {#Forstneric11} [[Franc Forstnerič]], _Stein manifolds and holomorphic mappings -- The homotopy principle in complex analysis_, Springer 2011 ([doi:10.1007/978-3-642-22250-4](https://link.springer.com/book/10.1007/978-3-642-22250-4))

* [[Franc Forstnerič]], *Gromov's contribution to the Oka principle*, 2012 ([pdf](https://www.fmf.uni-lj.si/~forstneric/papers/2012Gromov%27scontribution.pdf), [[Forstneric_GromovContributionToOka.pdf:file]])

* {#Forstneric13} [[Franc Forstnerič]] (appendix by [[Finnur Lárusson]]), *Oka manifolds: From Oka to Stein and back*, Annales de la Faculté des sciences de Toulouse, Mathématiques, Série 6, Tome 22 (2013) no. 4, pp. 747-809 ([numdam:AFST_2013_6_22_4_747_0](https://afst.centre-mersenne.org/item/?id=AFST_2013_6_22_4_747_0))

* [[Franc Forstnerič]], *Developments in Oka theory since 2017* ([arXiv:2006.07888](https://arxiv.org/abs/2006.07888))

and in relation to [[minimal surfaces]]:

* [[Antonio Alarcon]], [[Franc Forstnerič]], *New complex analytic methods in the theory of minimal surfaces: a survey*,  Journal of the Australian Mathematical Society, 106(3), 287-341 ([arXiv:1711.08024](https://arxiv.org/abs/1711.08024), [doi:10.1017/S1446788718000125](https://doi.org/10.1017/S1446788718000125))

## Related $n$Lab entries

* [[Stein manifold]]

* [[Oka manifold]]

* [[Oka principle]]

category: people

[[!redirects Franc Forstnerič]]
[[!redirects Franc Forstneric]]
[[!redirects Franc Forstnerich]]


Frances Kirwan

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

## Selected writings

On [[geometric invariant theory]]:

*  {#MumfordFogartyKirwan65} [[David Mumford]], [[John Fogarty]], [[Frances Clare Kirwan]], _Geometric invariant theory_, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) __34__, Springer-Verlag (1965/1982) ([ISBN:978-3-540-56963-3](https://www.springer.com/gp/book/9783540569633), [pdf](https://altexploit.files.wordpress.com/2017/08/mumford-d-fogarty-j-geometric-invariant-theory-springer-1982.pdf))

On [[homotopy type of spaces of rational maps]]:

* [[Frances Kirwan]], *On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles*, Ark. Mat. 24(1-2): 221-275 (1985) ([doi:10.1007/BF02384399](https://www.projecteuclid.org/journals/arkiv-for-matematik/volume-24/issue-1-2/On-spaces-of-maps-from-Riemann-surfaces-to-Grassmannians-and/10.1007/BF02384399.full))

category: people

[[!redirects Frances Clare Kirwan]]


Francesco Benini

* [Institute page](https://www.sissa.it/tpp/members.php?ID=209)

* [Google Scholar page](https://scholar.google.it/citations?user=xY858rUAAAAJ&hl=en)

## Selected writings

On [[5-brane webs]] and [[D=4 N=2 super Yang-Mills theory]]:

* [[Francesco Benini]], [[Sergio Benvenuti]], [[Yuji Tachikawa]], _Webs of five-branes and $\mathcal{N}=2$ superconformal field theories_, JHEP 0909:052 (2009) [[arXiv:0906.0359](https://arxiv.org/abs/0906.0359)]

On [[anomaly cancellation]] via the [[Green-Schwarz mechanism]] from a [[higher gauge theory]]-perspective

* [[Francesco Benini]], [[Clay Cordova]], [[Po-Shen Hsin]], *On 2-Group Global Symmetries and Their Anomalies*,  J. High Energ. Phys. **2019** 118 (2019) ([arXiv:1803.09336](https://arxiv.org/abs/1803.09336), doi:10.1007/JHEP03(2019)118)

category: people


Francesco D'Andrea

* [webpage](http://wpage.unina.it/francesco.dandrea/)

## Selected writings

On the [[spectral triple|spectral]] [[Riemannian geometry]] of the [[fuzzy 2-sphere]]:

* [[Francesco D'Andrea]], [[Fedele Lizzi]], [[Joseph Várilly]], _Metric Properties of the Fuzzy Sphere_, Lett. Math. Phys.103 (2013), 183-205 ([arXiv:1209.0108](https://arxiv.org/abs/1209.0108))

category: people


Francesco Petruccione

* [institute page](https://quantum.ukzn.ac.za/prof-f-petruccione/)

* [GoogleScholar page](https://scholar.google.com/citations?user=chM4fT4AAAAJ&hl=en)

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

## Selected writings

On [[open quantum system|open]] [[quantum mechanics|quantum mechanical]] [[quantum systems|systems]]:

* [[Heinz-Peter Breuer]], [[Francesco Petruccione]], *The Theory of Open Quantum Systems*, Oxford University Press (2007) [[doi:10.1093/acprof:oso/9780199213900.001.0001](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001)]

category: people


Francesco Toppan

* [website](http://www.cbpf.br/~toppan/)

category: people


Francesco Vissani

* [Institute page](https://www.gssi.it/comitato-scientifico/item/834-francesco-vissani)

* [GoogleScholar page](https://scholar.google.it/citations?user=W1geGQIAAAAJ&hl=en)

## Selected writings

On [[neutrinos]] in [[particle physics]]:

* Guido Fantini, Andrea Gallo Rosso, [[Francesco Vissani]], Vanessa Zema, _The formalism of neutrino oscillations: an introduction_, in: *The State of the Art of Neutrino Physics* World Scientific (2018)
[[arXiv:1802.05781](https://arxiv.org/abs/1802.05781), [doi:10.1142/10600](https://doi.org/10.1142/10600)]

* [[Francesco Vissani]], _What is matter according to particle physics and why try to observe its creation in lab?_, Universe **7** 3 (2021) 61 [[arXiv:2103.02642](https://arxiv.org/abs/2103.02642), [doi:10.3390/universe7030061](https://doi.org/10.3390/universe7030061)]

* [[Francesco Vissani]], *First steps towards understanding neutrinos* [[arXiv:2310.07834](https://arxiv.org/abs/2310.07834)]


Francis Borceux

_Francis Borceux_ is a [[category theory|category theorist]] at Louvain (Belgium).

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=111971)

## Selected writings

Introducing the notion of [[weighted limits]] in [[enriched category theory]]:

* [[Francis Borceux]], *Une notion de $V$-limite*, in *Colloque sur l'algèbre des catégories. Amiens 1973. Résumés des conférences*, Cahiers de topologie et géométrie différentielle **14** 2 (1973) 153-223 [[numdam:CTGDC_1973__14_2_153_0](http://www.numdam.org/item/CTGDC_1973__14_2_153_0)]

* [[Francis Borceux]], [[Gregory Maxwell Kelly]], *A notion of limit for enriched categories*, Bulletin of the Australian Mathematical Society **12** 1 (1975) 49-72 [[doi:10.1017/S0004972700023637](https://doi.org/10.1017/S0004972700023637)]

On [[algebra]] [[internalization|internal]] to a [[localic topos]]:

* Francis Borceux, Gilberte Van den Bossche, _Algebra in a localic topos with applications to ring theory_, Lecture Notes in Mathematics __1038__, Springer 1983, ix+240 pp. [MR85h:18005](http://www.ams.org/mathscinet-getitem?mr=724431)

On [[category theory]], and [[categorical algebra]]:

* __[[Handbook of Categorical Algebra]]__ (1994)

On [[nonabelian homological algebra]], such as its [[diagram chasing lemmas]], in [[Mal'cev categories]], [[protomodular categories]], [[semi-abelian categories]] and [[homological categories]]:

On [[accessible categories]] and [[sketches]] in the generality of [[enriched category theory]]:

* [[Francis Borceux]], [[Carmen Quinteriro]], *Enriched accessible categories*, Bull. Austral. Math. Soc. __54__ (1996) 489-501 [[doi:10.1017/S0004972700021900](https://doi.org/10.1017/S0004972700021900)]

* [[Francis Borceux]], [[Carmen Quinteriro]], [[Jiří Rosický]], *A theory of enriched sketches*, Theory and Applications of Categories, **4** 3 (1998) 47-72 [[tac:4-03](http://www.tac.mta.ca/tac/volumes/1998/n3/4-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/1998/n3/n3.pdf)]

On [[Galois theory]]:

* [[Francis Borceux]], [[George Janelidze]]: *[[Galois Theories]]*, Cambridge Studies in Advanced Mathematics **72**, Cambridge University Press (2001) [[doi:10.1017/CBO9780511619939](https://doi.org/10.1017/CBO9780511619939)]

On [[accessible categories]] via [[doctrines of limits]]:

* {#AdámekBorceuxLackRosicky2002} [[Jiří Adámek]], [[Francis Borceux]], [[Stephen Lack]], [[Jiri Rosicky]], *A classification of accessible categories*, Journal of Pure and Applied Algebra **175** 1–3 (2002)  7-30 [doi:10.1016/S0022-4049(02)00126-3]

On [[nonabelian homological algebra]], such as its [[diagram chasing lemmas]], in [[Mal'cev categories]], [[protomodular categories]], [[semi-abelian categories]] and [[homological categories]]:

* [[Francis Borceux]], [[Dominique Bourn]], *[[Mal'cev, protomodular, homological and semi-abelian categories]]*, Mathematics and Its Applications __566__,  Kluwer 2004 ([doi:10.1007/978-1-4020-1962-3](https://link.springer.com/book/10.1007/978-1-4020-1962-3))

On [[monoid objects in monoidal categories]]:

* [[Francis Borceux]], [[George Janelidze]], [[Gregory Maxwell Kelly]], _Internal object actions_, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 ([dml:249553](https://eudml.org/doc/249553))

Introduction to [[topos theory]]:

* [[Francis Borceux]], *Some glances at topos theory*, lecture notes, Como (2018) [[pdf](https://tcsc.lakecomoschool.org/files/2018/01/Borceux.pdf), [[Borceux-ToposTheory.pdf:file]], [video playlist](https://www.youtube.com/watch?v=s_fN9euuVAY&list=PLh_3Q6ZRqWs0LBptMGClJ8OArR0fBT6Pp&index=11)]

Presenting a [[pretorsion theory]] on [[Cat]] whose torsion(-free) objects are the [[groupoids]] ([[skeletal categories]], respectively), hence whose "trivial objects" are the [[skeletal groupoids]]:

* [[Francis Borceux]], [[Federico Campanini]], [[Marino Gran]], [[Walter Tholen]], *Groupoids and skeletal categories form a pretorsion theory in $Cat$* [[arXiv:2207.08487](https://arxiv.org/abs/2207.08487)]

Some other papers

* F. Borceux, I. Stubbe, _A short introduction to enriched categories_, [pdf](http://www-lmpa.univ-littoral.fr/~stubbe/PDF/EnrichedCatsKLUWER.pdf)

* Francis Borceux, Gilberte Van Den Bossche, _Recovering a frame from its sheaves of algebras_, Journal of Pure and Applied Algebra __28__, 2, May 1983, 141-154, doi

* Francis Borceux, _On algebraic localizations_, [MR84c:18007](http://www.ams.org/mathscinet-getitem?mr=682939), in Category theory (Gummersbach, 1981), pp. 13–22, Lecture Notes in Math. __962__, Springer 1982.

That paper in the LNM Gummersbach volume has only short sketches of proofs. There is a larger French preprint version with full proofs:

* Francis Borceux, _Sur les localisations algébriques_, Preprint, Rapp. Sém. Math. Pures, Univ. Louvain, Louvain, Belgium, 1981

category: people
[[!redirects F. Borceux]]


Francis Brown

* [webpage](http://www.ihes.fr/~brown/)

## Selected writings

On the [[cosmic Galois group]]:

* {#Brown15} [[Francis Brown]], _Feynman Amplitudes and Cosmic Galois group_ ([arXiv:1512.06409](https://arxiv.org/abs/1512.06409))

* [[Francis Brown]], _Periods and Feynman amplitudes_ ([arXiv:1512.09265](https://arxiv.org/abs/1512.09265))

## Related $n$Lab entries

* [[cosmic Galois group]]

category: people


Francis Halzen

* [Wikipedia entry](https://de.wikipedia.org/wiki/Francis_Halzen)

## selected writings

* {#HalzenMartin84} [[Francis Halzen]], [[Alan Martin]], _Quarks and Leptons: An Introductory Course in Modern Particle Physics_, Wiley 1984 ([pdf](http://ajbell.web.cern.ch/ajbell/Documents/eBooks/Quarks%20&%20Leptons.pdf))

## related $n$Lab entries

* [[quark]]

* [[lepton]]

* [[QCD]]

* [[standard model of particle physics]]

category: people


Francis J. Murray

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

## Selected writings

Introducing the theory of what came to be known as *[[von Neumann algebra factors]]*:

The classification of factors into types I, II, III and the construction of examples not of type I:

* [[Francis J. Murray]], [[John von Neumann]], *On Rings of Operators*, Annals of Mathematics, Second Series, **37** 1 (1936) 116-229 [[doi:10.2307/1968693](https://doi.org/10.2307/1968693), [jstor:1968693](https://www.jstor.org/stable/1968693)]

Discussion of [[traces]] on these factors:

* [[Francis J. Murray]], [[John von Neumann]], *On rings of operators. II*, Trans. Amer. Math. Soc. **41** (1937) 208-248  [[doi:10.2307/1989620](https://doi.org/10.2307/1989620), [jstor:1989620](https://www.jstor.org/stable/1989620)]

On [[isomorphism]] of factors and proof of a single isomorphism class of approximately finite type $II_1$ factors:

* [[Francis J. Murray]], [[John von Neumann]], *On rings of operators. IV*, Annals of Mathematics, Second Series, **44** 4 (1943) 716-808 [[doi:10.2307/1969107](https://doi.org/10.2307/1969107), [jstor:1969107](https://www.jstor.org/stable/1969107)]

On decomposing [[von Neumann algebras]] as a [[direct integral]] of factors:

* {#vNeumann49} [[John von Neumann]], _On rings of operators, reduction theory_, Annals of Mathematics Second Series, Vol. 50, No. 2 (1949) [[jstor:1969463]( http://www.jstor.org/stable/1969463)]

Recollection of the history which made von Neumann turn to discussion of these "factors", motivated from considerations in the foundations of [[quantum mechanics]] and [[quantum logic]]:

* {#Rédei96} [[Miklos Rédei]], *Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead)*, Studies in History and Philosophy of Modern Physics **27** 4 (1996) 493-510 [doi:10.1016/S1355-2198(96)00017-2]

category: people


Francis Sergeraert

* [webpage](http://www-fourier.ujf-grenoble.fr/~sergerar/)

## Selected writings

On [[constructive mathematics|constructive]] methods in [[algebraic topology]] ([[constructive algebraic topology]]):

* [[Julio Rubio]], [[Francis Sergeraert]], *Constructive Algebraic Topology*, Bulletin des Sciences Mathématiques
**126** 5 (2002) 389-412 [[arXiv:math/0111243](https://arxiv.org/abs/math/0111243), doi:10.1016/S0007-4497(02)01119-3]

On [[simplicial homotopy theory]]:

* [[Francis Sergeraert]], _Introduction to Combinatorial Homotopy Theory_, 2008 ([pdf](https://www-fourier.ujf-grenoble.fr/~%20sergerar/Papers/Trieste-Lecture-Notes.pdf), [[SergeraertCombinatorialHomotopyTheory.pdf:file]])

On [[effective homology]]:

* [[Francis Sergeraert]], _Effective homology, a survey_ ([pdf](http://www-fourier.ujf-grenoble.fr/~%20sergerar/Papers/Survey.pdf))

On [[Cohomotopy]]-sets in [[computational topology]]:

* [[Martin Čadek]], [[Marek Krčál]], [[Jiří Matoušek]], [[Francis Sergeraert]], [[Lukáš Vokřínek]], [[Uli Wagner]], _Computing all maps into a sphere_, Journal of the ACM, Volume 61 Issue 3, May 2014 Article No. 1 ([arxiv:1105.6257](https://arxiv.org/abs/1105.6257))

On [[homological algebra]] in [[constructive mathematics]]:

* [[Julio Rubio]], [[Francis Sergeraert]], _Constructive homological algebra and applications_ [[arXiv:1208.3816](http://arxiv.org/abs/1208.3816)]

category: people


Francisco Belchí

* [GoogleScholar page](https://scholar.google.com/citations?user=VvXCmAcAAAAJ)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=198736)

## Selected writings

On [[Whitehead L-infinity algebras|Whitehead $L_\infty$-algebras]] in [[rational homotopy theory]]:

* [[Francisco Belchí]], [[Urtzi Buijs]], [[José M. Moreno-Fernández]], [[Aniceto Murillo]], _Higher order Whitehead products and $L_\infty$ structures on the homology of a DGL_, Linear Algebra and its Applications, **520** (2017) 16-31 [[arXiv:1604.01478](https://arxiv.org/abs/1604.01478), [doi:10.1016/j.laa.2017.01.008](https://doi.org/10.1016/j.laa.2017.01.008)]

[[!redirects Francisco Belchi]]


Francisco Marmolejo

* [webpage](https://www.matem.unam.mx/fsd/quico)

## Selected writings

On a [[2-monad|2-monadic]] treatment of [[ultracategories]]:

* [[Francisco Marmolejo]], _Ultraproducts and continuous families of models_, PhD thesis (1995) [[hdl:10222/55092](https://dalspace.library.dal.ca/handle/10222/55092), [pdf](https://dalspace.library.dal.ca/bitstream/handle/10222/55092/NN08788.PDF?sequence=1&isAllowed=y)]

On [[distributive laws]] for [[pseudomonads]]:

* {#MarmolejoDL} [[Francisco Marmolejo]], *Distributive laws for pseudomonads*, Theory and Applications of Categories, **5** 5 (1999) 81-147 [[tac](http://tac.mta.ca/tac/volumes/1999/n5/5-05abs.html)]

On [[Beck's theorem]] for [[pseudomonads]] ([[higher monadic descent]]):

* I. J. Le Creurer, [[Francisco Marmolejo]], [[Enrico M. Vitale]], _Beck's theorem for pseudo-monads_, J. Pure Appl. Algebra **173** 3 (2002) 293-313 [doi:10.1016/S0022-4049(02)00038-5]

On [[monads]] as [[extension systems]] (as in [[monads in computer science]]):

* {#MarmolejoWood10} [[Francisco Marmolejo]], [[Richard J. Wood]], *Monads as extension systems -- no iteration is necessary* [[TAC]] **24** 4 (2010) 84-113 [[24-04](http://www.tac.mta.ca/tac/volumes/24/4/24-04abs.html)]

and generalizing the perspective of [[extension systems]] to [[pseudomonads]]:

* [[Francisco Marmolejo]], [[Richard J. Wood]], *Kan extensions and lax idempotent pseudomonads*, [[TAC]] **26** 1 (2012) 1-29 [[26-01](http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html)]

* [[Francisco Marmolejo]], [[Richard J. Wood]], *No-iteration pseudomonads*, [[TAC]] **28** 14 (2013) 371-402 [[tac:28-14](http://www.tac.mta.ca/tac/volumes/28/14/28-14abs.html)]

## Related pages

* [[ultracategory]]

category: people

[[!redirects F. Marmolejo]]


Francisco Rios

* [institute page](https://www.ornl.gov/staff-profile/francisco-r-rios)

## Selected writings

On some [[dependent linear type theory]] and [[categorical semantics]] for the [[Quipper]] [[quantum programming language]]:

* [[Francisco Rios]], [[Peter Selinger]], *A Categorical Model for a Quantum Circuit Description Language*, EPTCS **266** (2018) 164-178 [[arXiv:1706.02630](https://arxiv.org/abs/1706.02630), [doi:10.4204/EPTCS.266.11](https://doi.org/10.4204/EPTCS.266.11)]

* [[Francisco Rios]], *On a Categorically Sound Quantum Programming Language for Circuit Description*, Dalhousie University (2021) [[hdl:10222/80771](http://hdl.handle.net/10222/80771)]

category: people


Franco Rota

> not to be confused with [[Gian-Carlo Rota]]

***

* [home page](https://sites.google.com/view/rotafr)

### Selected writings

* Matteo Altavilla, Marin Petkovič, Franco Rota, _Moduli spaces on the Kuznetsov component of Fano threefolds of index 2_, Épijournal de Géométrie Algébrique (2022). ([arXiv:1908.10986](https://arxiv.org/abs/1908.10986))

On [[derived categories of coherent sheaves]]:

* [[Franco Rota]], _The derived categories of coherent sheaves_, lecture notes [[pdf](https://drive.google.com/file/d/12fhshN_lFOi54pjBdjVEPzUb1XBxz-TW/view), [[Rota-DerivedCategories.pdf:file]]]

On [[Bridgeland stability conditions]]:

* [[Franco Rota]], _An introduction to stability conditions_, lecture notes [[pdf](https://drive.google.com/file/d/1sJSwN-JTOr77fpIgRWfDXkidYZmrJ0jC/view), [[Rota-StabilityConditions.pdf:file]]]

## Related pages:

* [[coherent sheaves]]

* [[derived categories]]

* [[Bridgeland stability conditions]]

category:people


Franco Selleri

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

* [inspire page](https://inspirehep.net/authors/989373)

## Selected writings

On the [[EPR paradox]] in [[quantum physics]]:

* [[Franco Selleri]] (ed.), *Quantum Mechanics Versus Local Realism -- The Einstein-Podolsky-Rosen Paradox*, Physics of Atoms and Molecules (PAMO), Springer (1988) [[doi:10.1007/978-1-4684-8774-9](https://doi.org/10.1007/978-1-4684-8774-9)]

category: people


Franco Strocchi

* [institute page](http://pisatheorygroup.pi.infn.it/staff/strocchi-franco)

## Selected writings

On the [[Gauss law]] in relation to non-[[local field theory|locality]] of charged states in [[quantum field theory]]:

* [[Giovanni Morchio]], [[Franco Strocchi]], *Localization and symmetries*, J. Phys. A **40** (2007) 3173-3188 [[arXiv:math-ph/0607015](https://arxiv.org/abs/math-ph/0607015), [doi:10.1088/1751-8113/40/12/S17](https://doi.org/10.1088/1751-8113/40/12/S17)]

On [[quantum mechanics]]:

* [[Franco Strocchi]], *An introduction to the mathematical structure of quantum mechanics*, Advanced Series in Mathematical Physics **28**, World Scientific (2008) [[doi:10.1142/7038](https://doi.org/10.1142/7038)]

On the [[theta vacuum]] ([[Yang-Mills instanton]] vacuum) from [[chiral symmetry breaking]]:

* [[Giovanni Morchio]], [[Franco Strocchi]], _Chiral symmetry breaking and theta vacuum structure in QCD_, Annals Phys. **324** (2009) 2236-2254 [[arXiv:0907.2522](https://arxiv.org/abs/0907.2522), [doi:10.1016/j.aop.2009.07.005](https://doi.org/10.1016/j.aop.2009.07.005)]

On [[non-perturbative quantum field theory]] in the school of [[algebraic quantum field theory]]:

* [[Franco Strocchi]], _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

* [[Franco Strocchi]], _Relativistic Quantum Mechanics and Field Theory_, Found. Phys. **34** (2004) 501-527 [[arXiv:hep-th/0401143](http://arxiv.org/abs/hep-th/0401143)]

## Related entries

* [[quantum field theory]]

* [[non-perturbative quantum field theory]]

* [[AQFT]], [[Wightman axioms]]

* [[instanton in QCD]]

category: people


Francois Metayer > history

< [[Francois Metayer]]


Frank Berkshire

* [institute page](https://www.imperial.ac.uk/people/f.berkshire)

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

## Selected writings

On [[classical mechanics]]:

* [[Tom W. B. Kibble]], [[Frank Berkshire]], *Classical mechanics*, McGraw-Hill (1966, 1973, 2004) [[pdf](https://physicaeducator.files.wordpress.com/2018/02/classical-mechanics-by-kibble-and-berkshire.pdf), Wikipedia entry]

category: people

[[!redirects Frank H. Berkshire]]


Frank De Luccia

## Selected writings

On [[vacuum decay]]:

* [[Sidney Coleman]], [[Frank De Luccia]], _Gravitational effects on and of vacuum decay_ , Phys. Rev. D 21, 3305–3315 (1980) ([doi:10.1103/PhysRevD.21.3305](https://doi.org/10.1103/PhysRevD.21.3305))

category: people


Frank J. Hall

* [Institute page](https://www.mathstat.gsu.edu/profile/frank-hall/)

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=18487)

## Selected writings

On the [[Gershgorin circle theorem]]:

* {#MarsliHall13} [[Rachid Marsli]], [[Frank J. Hall]], *Geometric Multiplicities and Geršgorin Discs*, The American Mathematical Monthly Vol. 120, No. 5 (May 2013), pp. 452-455 ([jstor:10.4169/amer.math.monthly.120.05.452](https://www.jstor.org/stable/10.4169/amer.math.monthly.120.05.452))

category: people


Frank J. Oles

* [MathGenealogy page](https://mathgenealogy.org/id.php?id=186557)

* [ACM page](https://dl.acm.org/profile/81100468080)

## Selected writings

On [[semantics of programming languages]] (with an early discussion of what later was rediscovered under the name *[[lenses (in computer science)]]*):

* {#Oles82} [[Frank J. Oles]], *A category-theoretic approach to the semantics of programming languages*, Syracuse University (1982) [[pdf](https://www.cs.cmu.edu/afs/cs.cmu.edu/project/fox-19/member/jcr/www/FrankOlesThesis.pdf)]

* {#Oles86} [[Frank J. Oles]], *Type algebras, functor categories and block structure*, Algebraic methods in semantics (1986) 543–573, Daimi Report No. 156 (1983): [[doi:10.7146/dpb.v12i156.7430](https://doi.org/10.7146/dpb.v12i156.7430), [pdf](https://tidsskrift.dk/daimipb/article/view/7430/6281)]

category: people

[[!redirects Frank Oles]]


Frank Morgan

* Wikipedia entry

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=17378)

## Selected writings

On [[geometric measure theory]]:

* [[Frank Morgan]], *Geometric measure theory: a beginner's guide*, Academic Press (2016) [[doi:10.1016/C2015-0-01918-9](https://doi.org/10.1016/C2015-0-01918-9)]

## Related entries

* [[current (distribution theory)]]

category: people


Frank Neumann

* [institute page](https://le.ac.uk/people/frank-neumann)

* [MathsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=27719)

## Selected writings

On [[algebraic stacks]] and [[moduli stacks]] of [[vector bundles]]:

* [[Frank Neumann]], *Algebraic Stacks and Moduli of Vector Bundles*, impa (2011) [[pdf](https://impa.br/wp-content/uploads/2017/04/PM_36.pdf), [pdf](https://www.cimat.mx/~luis/seminarios/Pilas-algebraicas/neumann-Stacks.pdf)]

category: people


Frank Pfenning

* [webpage](http://www.cs.cmu.edu/~fp/)

## Selected writings

On [[higher-order abstract syntax]]:

* [[Frank Pfenning]], Conal Elliot, ACM SIGPLAN Notices **23** 7 (1988) 199–208 [[doi;10.1145/960116.54010](https://doi.org/10.1145/960116.54010), [pdf](https://www.cs.cmu.edu/~fp/papers/pldi88.pdf)]

On [[inductive types]] and the [[calculus of constructions]]:

* [[Frank Pfenning]], [[Christine Paulin-Mohring]], *Inductively defined types in the Calculus of Constructions*, in: *Mathematical Foundations of Programming Semantics MFPS 1989*,  Lecture Notes in Computer Science **442**, Springer (1990) $[$[doi:10.1007/BFb0040259](https://doi.org/10.1007/BFb0040259)$]$

On [[Semi-axiomatic sequent calculus]]:

* [[Henry DeYoung]], [[Frank Pfenning]], and [[Klaas Pruiksma]]. Semi-axiomatic sequent calculus. In Z. Ariola, editor, *5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)*, pages 29:1--29:22, Paris, France, June 2020. LIPIcs 167. [[bib](https://www.cs.cmu.edu/~fp/pfenning_bib.html#DeYoung20fscd) | [.pdf](http://www.cs.cmu.edu/~fp/papers/fscd20a.pdf)]

On [[linear logic]] and basics of [[type theory]]:

* [[Frank Pfenning]], *Linear Logic*, lecture notes (1998) [[pdf](https://www.cs.cmu.edu/~fp/courses/98-linear/handouts/notes.pdf), [webpage](https://www.cs.cmu.edu/~fp/courses/98-linear/handouts.html), [[Pfenning-LinearLogic98.pdf:file]]]
* [[Frank Pfenning]], *Substructural Logics*, lecture notes (2023) [[pdf](https://www.cs.cmu.edu/~fp/courses/15836-f23/misc/sublogics23.pdf), [webpage](https://www.cs.cmu.edu/~fp/courses/15836-f23/)]

On [[logical frameworks]]:

* [[Frank Pfenning]], _Logical frameworks -- a brief introduction_ (2002) [[pdf](http://www.cs.cmu.edu/~fp/papers/mdorf01.pdf)]

* [[Frank Pfenning]], _Logical frameworks_, chapter 17 in: Alan Robinson and Andrei Voronkov (eds.): _Handbook of Automated Reasoning_ (1999) 1063-1147 [[ps](http://www-2.cs.cmu.edu/twelf/notes/handbook00.ps)]

* [[Frank Pfenning]], _Logical frameworks web site_ ([web](http://www.cs.cmu.edu/~fp/lfs.html)), including an extensive bibliography and a list of implementations

On [[modal logic]]:

* {#Pfenning-Davies} [[Frank Pfenning]], Rowan Davies, *A judgemental reconstruction of modal logic*, Mathematical Structures in Comp. Sci. **11** 4 (2001) 511-540 [[pdf](http://www.cs.cmu.edu/~fp/papers/mscs00.pdf)]

* [[Frank Pfenning]], _Towards modal type theory_ (2000) [[pdf](http://www.cs.cmu.edu/~fp/talks/mantova00-talk.pdf)]

* [[Frank Pfenning]], _Intensionality, Extensionality, and Proof Irrelevance in Modal Type Theory_, in: Symposium on Logic in Computer Science (2001) 221&-230 [[web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.5884)]

On [[modal type theory]]:

* {#NanevskiPfenningPientka05} [[Aleksandar Nanevski]], [[Frank Pfenning]], Brigitte Pientka, _Contextual Modal Type Theory_ (2005) [[web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.61.2356), [slides](http://wips.cs.umn.edu/slides/pientka.pdf)]

On the [[type of natural numbers]] in [[type theory]]:

* [[Frank Pfenning]], _Lecture notes on natural numbers_ (2009) [[pdf](http://www.cs.cmu.edu/~fp/courses/15317-f09/lectures/06-nat.pdf), [[Pfenning-NaturalNumbersType.pdf:file]]]

On [[adjoint logic]]:

* Klaas Pruiksma, William Chargin, [[Frank Pfenning]], and Jason Reed, *Adjoint Logic*, 2018, [[pdf](https://www.cs.cmu.edu/~fp/papers/adjoint18b.pdf), [[PCPR18-AdjointLogic.pdf:file]]]

category: people


Frank Pfäffle

* [website](http://www.math.uni-bonn.de/people/pfaeffle/)

## related $n$Lab entries

* [[path integral]]

category: people

[[!redirects Frank Pfaffle]]


Frank Pollmann

* [Research group page](https://www.ph.tum.de/research/groups/group/TUPHTFK/?language=en)

## Selected writings

On [[symmetry-protected topological order]]:

* {#PBTO09} [[Frank Pollmann]], [[Erez Berg]], [[Ari M. Turner]], [[Masaki Oshikawa]], *Symmetry protection of topological order in one-dimensional quantum spin systems*, Phys. Rev. B **85** 075125 (2012) [[arXiv:0909.4059](https://arxiv.org/abs/0909.4059), [doi:10.1103/PhysRevB.85.075125](https://doi.org/10.1103/PhysRevB.85.075125)]

category: people


Frank Quinn

* [personal page](http://www.math.vt.edu/people/quinn/)

* Wikipedia entry

## Selected writings

On [[E-infinity spaces|$E_\infty$-spaces]] and [[E-infinity ring spectra|$E_\infty$-ring sectra]]:

* [[Peter May]] with contributions by [[Frank Quinn]], [[Nigel Ray]] and [[Jørgen Tornehave]], _$E_\infty$-Ring spaces and $E_\infty$ ring spectra_, Lecture Notes in Mathematics 577, Springer 1977 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf), [cds:1690879](https://cds.cern.ch/record/1690879))

On [[TQFT]] and [[HQFT]]:

* {#Quinn95} [[Frank Quinn]], *Lectures on axiomatic topological quantum field theory*, in [[Dan Freed]], [[Karen Uhlenbeck]] (eds.) *Geometry and Quantum Field Theory* **1** (1995) [[doi:10.1090/pcms/001](https://doi.org/10.1090/pcms/001)]

On the relation between the practice of [[mathematics]] and [[physics]] (for historical context see [Hitchin 20, Sec. 8](Michael+Atiyah#Hitchin20)):

* {#JaffeQuinn93} [[Arthur Jaffe]], [[Frank Quinn]], _"Theoretical Mathematics": Towards a cultural synthesis of mathematics and theoretical physics_, Bulletin of the AMS, Volume 29,Number 1, July 1993 ([arXiv:math/9307227](http://arxiv.org/abs/math/9307227))

  > we claim that the role of rigorous [[proof]] in mathematics is functionally [[analogy|analogous]] to the role of [[experiment]] in the natural sciences ([Jaffe-Quinn 93, p. 2](JaffeQuinn93))

On rigorous [[mathematics]] and [[formal proof]]:

* {#Quinn2009} [[Frank Quinn]]: *Proof Projects for Teachers* (2009) [[pdf](https://personal.math.vt.edu/fquinn/education/pfs4teachers0.pdf), [[Quinn-ProofProjects.pdf:file]]]

* {#Quinn12} [[Frank Quinn]], *A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today*, Notices of the AMS (2012) $[$[pdf](http://www.ams.org/notices/201201/rtx120100031p.pdf)$]$

## Related entries

* [[TQFT]]

category: people


Frank Reidegeld

* [Mathematics Genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=163248)

## Selected writings

On [[G2-orbifolds]]:

* {#Reidegeld15} [[Frank Reidegeld]], _$G_2$-orbifolds from K3 surfaces with ADE-singularities_ ([arXiv:1512.05114](http://arxiv.org/abs/1512.05114), [spire:1409963](https://inspirehep.net/literature/1409963), [doi:10.17877/DE290R-18940](http://dx.doi.org/10.17877/DE290R-18940))

* [[Frank Reidegeld]], _K3 surfaces with a pair of commuting non-symplectic involutions_ ([arXiv:1809.07501](https://arxiv.org/abs/1809.07501))

## Related $n$Lab entries

* [[G2-orbifold]]

* [[ADE-orbifold]]

category: people


Frank Schindler

* [institute page](https://www.imperial.ac.uk/people/f.schindler)

* [GoogleScholar page](https://scholar.google.ch/citations?user=15-4AygAAAAJ&hl=de)

## Selected writings

On higher order [[topological insulators]] (with protected corner-modes beyond the edge-modes):

* [[Frank Schindler]], Ashley M. Cook, Maia G. Vergniory, Zhijun Wang, Stuart S. P. Parkin, [[B. Andrei Bernevig]], [[Titus Neupert]], *Higher-Order Topological Insulators*, Science Advances 01  **4** 6 (2018) eaat0346 [[arXiv:1708.03636](https://arxiv.org/abs/1708.03636), [doi:10.1126/sciadv.aat0346](https://doi.org/10.1126/sciadv.aat0346)]

On [[topological crystalline insulators]] with focus on the [[Wilson lines]] of the [[Berry connection]]:

* [[Titus Neupert]], [[Frank Schindler]], *Lecture Notes on Topological Crystalline Insulators*, in: *Topological Matter* Springer Series in Solid-State Sciences **190**, Springer (2018) [[arXiv:1810.03484](https://arxiv.org/abs/1810.03484), [doi:10.1007/978-3-319-76388-0_2](https://doi.org/10.1007/978-3-319-76388-0_2)]

category: people


Frank Verstraete

* [webpage](https://verstraete.quantum.at/)

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

## Selected writings

On [[tensor network states]]:

On [[matrix product states]] in [[solid state physics]]

* [[Frank Verstraete]], J. I. Cirac, V. Murg, _Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems_, Adv. Phys. 57,143 (2008) ([arXiv:0907.2796](https://arxiv.org/abs/0907.2796))

On [[tree tensor networks]] in [[solid state physics]]:

* Valentin Murg, Örs Legeza, Reinhard M. Noack, [[Frank Verstraete]], _Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks_, Phys. Rev. B 82, 205105 (2010) ([arXiv:1006.3095](https://arxiv.org/abs/1006.3095))

On [[tree tensor network states]] in [[quantum chemistry]]:

* Klaas Gunst, [[Frank Verstraete]], Sebastian Wouters, Örs Legeza, Dimitri Van Neck, _T3NS: three-legged tree tensor network states_, Chem. Theory Comput. 2018, 14, 4, 2026-2033 ([arXiv:1801.09998](https://arxiv.org/abs/1801.09998))

On [[tensor networks]] in [[lattice gauge theory]]:

* {#BBCCCDFJLMMRRTVV19} M.C. Bañuls, R. Blatt, J. Catani, A. Celi, J.I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C.A. Muschik, B. Reznik, E. Rico, [[Luca Tagliacozzo]], K. Van Acoleyen, [[Frank Verstraete]], U.-J. Wiese, M. Wingate, J. Zakrzewski, P. Zoller:

  _Simulating Lattice Gauge Theories within Quantum Technologies_

  ([arXiv:1911.00003](https://arxiv.org/abs/1911.00003))

category: people


Frank W. Anderson

* [institute page](https://pages.uoregon.edu/anderson/index.html)

## Selected writings

On [[rings]] and [[modules]] ([[algebra]]):

* [[Frank W. Anderson]], [[Kent R. Fuller]], _Rings and Categories of Modules_, Graduate Texts in Mathematics, **13** Springer (1992) [[doi:10.1007/978-1-4612-4418-9](https://doi.org/10.1007/978-1-4612-4418-9)]

category: people

[[!redirects Frank Anderson]]


Frank Warner

* [webpage](https://www.math.upenn.edu/people/frank-warner)

## Selected writings

On [[Hopf fibrations]] and notably the [[quaternionic Hopf fibration]]:

* {#GluckWarnerZiller86} [[Herman Gluck]], [[Frank Warner]], [[Wolfgang Ziller]], _The geometry of the Hopf fibrations_, L'Enseignement Mathématique, **32** (1986), 173-198 [[ResearchGate](https://www.researchgate.net/publication/266548925_The_geometry_of_the_Hopf_fibrations), [[GluckWarnerZiller-HopfFibrations.pdf:file]]]

On [[Hopf fibrations]] and more general [[spherical fibrations]] over [[spheres]]:

* [[Herman Gluck]], [[Frank Warner]], [[Chung Tao Yang]], _Division algebras, fibrations of spheres by great spheres and the topological determination of space by the gross behavior of its geodesics_, Duke Math. J. Volume 50, Number 4 (1983), 1041-1076 ([euclid:dmj/1077303489](https://projecteuclid.org/euclid.dmj/1077303489))

category: people


Frank Wilcek

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

## related $n$lab entries

* [[axion]]

category: people


Frank Wilczek

* [Wikipedia entry](http://en.wikipedia.org/wiki/Frank_Wilczek)

* [research group page](https://www.su.se/english/research/research-groups/quantum-frontiers?open-collapse-boxes=research-group-members)

## Selected writings

On the [[baryon current]] as a [[WZW-term]]/[[Skyrmion]] current (in the [[exponentiated pion field]]):

* [[Jeffrey Goldstone]], [[Frank Wilczek]], _Fractional Quantum Numbers on Solitons_, Phys. Rev. Lett. 47, 986 (1981) ([doi:10.1103/PhysRevLett.47.986](https://doi.org/10.1103/PhysRevLett.47.986))

Discussion of [[Spin(16)]]-[[grand unified theories]] (traditionally called "SO(16)" GUT):

* {#WilczekZee82} [[Frank Wilczek]], [[Anthony Zee]], _Families from spinors_, Phys. Rev. D 25, 553 (1982) ([doi:10.1103/PhysRevD.25.55310.1103/PhysRevD.25.553](https://doi.org/10.1103/PhysRevD.25.553))

* Goran Senjanović, [[Frank Wilczek]], [[Anthony Zee]], _Reflections on mirror fermions_,  Physics Letters B Volume 141, Issues 5–6, 5 July 1984, Pages 389-394 Physics Letters B (doi:10.1016/0370-2693(84)90269-7)

On [[anyons]]:

* {#Wilczek82a} [[Frank Wilczek]], _Magnetic Flux, Angular Momentum, and Statistics_, Phys. Rev. Lett. **48** 1982 1144 (reprinted in [Wilczek 1990, p. 163-165](#Wilczek90)) $[$[doi:10.1103/PhysRevLett.48.1144](https://doi.org/10.1103/PhysRevLett.48.1144)$]$

and introducing the terminology "anyon":

* {#Wilczek82b} [[Frank Wilczek]], *Quantum Mechanics of Fractional-Spin Particles*, Phys. Rev. Lett. **49** (1982) 957 (reprinted in [Wilczek 1990, p. 166-168](#Wilczek90)) $[$[doi:10.1103/PhysRevLett.49.957](https://doi.org/10.1103/PhysRevLett.49.957)$]$

On [[anyon statistics|anyon phases]] (specifically in the [[quantum Hall effect]]) as [[Berry phases]] of a [[quantum adiabatic theorem|adiabatic]] transport of anyon positions:

* {#AvrosSchriefferWilczek84} [[Daniel P. Arovas]], [[John Robert Schrieffer]], [[Frank Wilczek]], _Fractional Statistics and the Quantum Hall Effect_, Phys. Rev. Lett. 53, 722 (1984) $[$[doi:10.1103/PhysRevLett.53.722](https://doi.org/10.1103/PhysRevLett.53.722)$]$

The model of [[anyon statistics]] as an [[Aharonov-Bohm effect]] of a "[[fictitious gauge field]]" sourced by and coupled to each anyon:

* {#ArovasScriefferWilczekZee85} [[Daniel P. Arovas]], [[John Robert Schrieffer]], [[Frank Wilczek]], [[Anthony Zee]], *Statistical mechanics of anyons*, Nuclear Physics B **251** (1985) 117-126 (reprinted in [Wilczek 1990, p. 173-182](#Wilczek90)) $[$doi:10.1016/0550-3213(85)90252-4$]$

and the resulting [[superfluidity]]/[[superconductivity]] *of* [[anyons]]:

* [[Yi-Hong Chen]], [[Frank Wilczek]], [[Edward Witten]], [[Bertrand Halperin]], *On Anyon Superconductivity*,  International Journal of Modern Physics B **03** 07 (1989) 1001-1067 (reprinted in [Wilczek 1990](#Wilczek90))  $[$[doi:10.1142/S0217979289000725](https://doi.org/10.1142/S0217979289000725), [[CWWH-AnyonSuperfluidity.pdf:file]]$]$

On [[quantum spin liquids]] as exhibiting [[topological order]]:

* [[Xiao-Gang Wen]], [[Frank Wilczek]], [[Anthony Zee]], *Chiral spin states and superconductivity*, Phys. Rev. B **39** (1989) 11413 [[doi:10.1103/PhysRevB.39.11413](https://doi.org/10.1103/PhysRevB.39.11413)]

Review of [[anyons]]:

* {#Wilczek90} [[Frank Wilczek]], *Fractional Statistics and Anyon Superconductivity*, World Scientific (1990) [[doi:10.1142/0961](https://doi.org/10.1142/0961)]

* [[Martin Greiter]], [[Frank Wilczek]], *Fractional Statistics* [[arXiv:2210.02530](https://arxiv.org/abs/2210.02530)]

On [[axions]]:

* {#Wilcek78} [[Frank Wilczek]], _Problem of strong P and T invariance in the presence of instantons_, Phys. Rev. Lett. 40:279-82 (1978)

* {#PreskillWiseWilczek83} [[John Preskill]], M. Wise, [[Frank Wilczek]], _Cosmology of the invisible axion_, Phys. Lett. B 120:127-32 (1983)

* [[Frank Wilczek]], _Birth of axions_ in _This Week's Citation Classic_ (1991) ([pdf](http://www.garfield.library.upenn.edu/classics1991/A1991FE76900001.pdf))

* [[Frank Wilczek]], _New Ideas in Axion Searches_, talk 2017 ([pdf](http://frankwilczek.com/2017/axion_searches_01.pdf))

On [[R-parity]]:

* [[Savas Dimopoulos]] [[Stuart Raby]], [[Frank Wilczek]], _Proton decay in supersymmetric models_, Physics Letters B Volume 112, Issue 2, 6 May 1982, Pages 133-136 (doi:10.1016/0370-2693(82)90313-6)

On [[Hopf-Wess-Zumino terms]]:

* {#WilczekZee84} [[Frank Wilczek]] and [[Anthony Zee]], _Linking Numbers, Spin, and Statistics of Solitons_, Phys. Rev. Lett. 51, 2250, 1983 ([doi:10.1103/PhysRevLett.51.2250](https://doi.org/10.1103/PhysRevLett.51.2250))

On the [[landscape of string theory vacua]] and the [[multiverse]]:

* [[Frank Wilczek]], _Multiversality_ ([arXiv:1307.7376](http://arxiv.org/abs/1307.7376))

## Related nLab entries

* [[Hopf-Wess-Zumino terms]]

* [[landscape of string theory vacua]]

* [[R-parity]]

category: people


Frank Witte

* [personal page](http://www.homepages.ucl.ac.uk/~uctpfwi/)

## Selected writings

On [[quantum computation]], [[blockchains]] and [[machine learning]]:

* [[Melanie Swan]], [[Renato P dos Santos]], [[Frank Witte]], Between Science and Economics, Volume 2: *Quantum Computing Physics, Blockchains, and Deep Learning Smart Networks*, World Scientific 2020 ([doi:10.1142/q0243](https://doi.org/10.1142/q0243))

  * On [[holographic quantum error correction]]:

    Part 5 (Ch. 13, 14): *The AdS/CFT Correspondence and Holographic Codes* ([doi:10.1142/9781786348210_0013](https://doi.org/10.1142/9781786348210_0013), [doi:10.1142/9781786348210_0014](https://www.worldscientific.com/doi/abs/10.1142/9781786348210_0014))

category: people


Frank Wolter

* [personal page](https://cgi.csc.liv.ac.uk/~frank/)

* [insitute page](https://www.liverpool.ac.uk/computer-science/staff/frank-wolter/)

## Selected writings

On [[modal logic]]:

* {#BlackburnvanBenthemWolter07} [[Patrick Blackburn]], [[Johan van Benthem]], [[Frank Wolter]] (eds.), *The Handbook of Modal Logic*, Elsevier Amsterdam (2007) [[book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/), [publisher page](https://www.sciencedirect.com/bookseries/studies-in-logic-and-practical-reasoning/vol/3/suppl/C)]

category: people


Franka Waaldijk

[[!redirects Frank'a Waaldijk]]

# Franka Waaldijk
(changed her name in 2019, it was 'Frank Waaldijk')

* [webpage](https://www.fwaaldijk.nl/mathematics.html)
* [math science philosophy blog](https://fwaaldijk.wordpress.com)

## Some writings

* _On the foundations of constructive mathematics – especially in relation to the theory of continuous functions_, 2003 ([pdf](https://www.fwaaldijk.nl/foundations%20of%20constructive%20mathematics.pdf))

* _Modern intuitionistic topology_  (PhD thesis, [web](https://www.fwaaldijk.nl/mathematics.html#modernintuitionistic))

  on [[general topology|general]] [[topology]] in [[intuitionistic mathematics]]

## Related $n$Lab entries

* [[intuitionistic mathematics]]

* [[constructive mathematics]]

* [[computable physics]], [[Church-Turing thesis]]

* [[topology]]

category: people

[[!redirects Frank Waaldijk]]
[[!redirects Frank'a Waaldijk]]
[[!redirects Frank\'a Waaldijk]]
[[!redirects Frank’a Waaldijk]]


Franklin P. Peterson

Franklin Paul Peterson was a professor at MIT.

[Wikipedia page](https://en.wikipedia.org/wiki/Franklin_P._Peterson)

## Related concepts

* [[Peterson space]]

* [[homotopy groups with coefficients]]

## Selected writings

On [[homotopy groups with coefficients]] and [[Peterson spaces]]:

* [[Franklin P. Peterson]], _Generalized Cohomotopy Groups_.  American Journal of Mathematics 78:2 (1956), 259–281.  [doi:10.2307/2372515](http://dx.doi.org/10.2307/2372515)


Franklin Peterson

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

* E. H. Brown, F. R. Cohen, F. W. Gehring, H. R. Miller, and B. A. Taylor, _Franklin P. Peterson(1930–2000)_, Notices of the AMS, [Vol 48, No 10](https://www.ams.org/notices/200110/) ([pdf](https://www.ams.org/notices/200110/fea-peterson.pdf))

## Selected writings

On [[Cohomotopy]]:

* [[Franklin Peterson]], _Some Results on Cohomotopy Groups_, American Journal of Mathematics Vol. 78, No. 2 (Apr., 1956), pp. 243-258 ([jstor:2372514](https://www.jstor.org/stable/2372514))

category: people


Frans R. Klinkhamer

* [institute page](https://www.itp.kit.edu/memberpages/klinkhamer)

* [GoogleScholar page](https://scholar.google.com/citations?user=o71L6xgAAAAJ&hl=de)

## Selected writings

On the [[IKKT matrix model]]:

* [[Frans R. Klinkhamer]], _On the emergence of an expanding universe from a Lorentzian matrix model_, Prog. Theor. Exp. Phys. 2020, 103B03 (2020) ([arXiv:1912.12229](https://arxiv.org/abs/1912.12229))

* [[Frans R. Klinkhamer]], _IIB matrix model: Emergent spacetime from the master field_ ([arXiv:2007.08485](https://arxiv.org/abs/2007.08485))

* [[Frans R. Klinkhamer]], _IIB matrix model: Extracting spacetime points_ ([arXiv:2008.01058](https://arxiv.org/abs/2008.01058))

* [[Frans R. Klinkhamer]], _Regularized big bang and IIB matrix model_ ([arXiv:2009.06525](https://arxiv.org/abs/2009.06525))

* [[Frans R. Klinkhamer]], _M-theory and the birth of the Universe_ ([arXiv:2102.11202](https://arxiv.org/abs/2102.11202))

* [[Frans R. Klinkhamer]], *A first look at the master-field equation of the IIB matrix model* ([arXiv:2105.05831](https://arxiv.org/abs/2105.05831))

category: people


Franz Brentano

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

## Selected writings

Selected writings:

* [[Franz Brentano]], *Psychologie vom empirischen Standpunkte* (1874) [[web](https://archive.org/details/psychologievome00kraugoog/page/n8/mode/2up)]

  see: SEP: *[Brentano’s Theory of Judgement](https://plato.stanford.edu/entries/brentano-judgement/)*

## Related entries

* [[judgement]]

category: people


Franz Ciceri

* [InSpire page](https://inspirehep.net/authors/1981593)

## Selected writings

On maximal [[gauged supergravity|gauged]] [[D=2 supergravity]] via [[KK-reduction]] of [[D=11 supergravity]]:

* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Maximal $D=2$ supergravities from higher dimensions* [[arXiv:2309.07232](https://arxiv.org/abs/2309.07232)]

* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Consistent truncation of eleven-dimensional supergravity on $S^8 \times S^1$* [[arXiv:2309.07233](https://arxiv.org/abs/2309.07233)]

category: people


Franz Gross

* [inspire page](https://inspirehep.net/authors/1007452)

## Selected writings

On [[quantum chromodynamics]]:

* [[Franz Gross]], [[Eberhard Klempt]] et al., *50 Years of Quantum Chromodynamics*, EJPC [[arXiv:2212.11107](https://arxiv.org/abs/2212.11107)]

category:people


Franz Wegner

* [institute page](https://www.thphys.uni-heidelberg.de/~wegner/)

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

## Selected writings

Introducing [[lattice QFT]] in the context of the [[Ising model]]:

* {#Wegner71} [[Franz Wegner]], _Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters_, Journal of Mathematical Physics 12, 2259 (1971) ([doi:10.1063/1.1665530](https://doi.org/10.1063/1.1665530))

category: people


François Bayen

* [Scopus page](https://www.scopus.com/authid/detail.uri?authorId=22970328700)

## Selected writings

Introducing [[formal deformation quantization]]:

* {#BayenFlatoFronsdalLichnerowiczSternheimer78} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], [[André Lichnerowicz]], [[Daniel Sternheimer]], _Deformation theory and quantization. I. Deformations of symplectic structures._, Annals of Physics **111** 1 (1978) 61-110 [doi:10.1016/0003-4916(78)90224-5]

* {#BayenFlatoFronsdalLichnerowiczSternheimer78b} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], [[André Lichnerowicz]], [[Daniel Sternheimer]], _Deformation theory and quantization. II. Physical applications_, Annals of Physics **111** 1 (1978) 111-151 [doi:10.1016/0003-4916(78)90225-7]

category: people

[[!redirects Francois Bayen]]


François Conduché

François Conduché was a student of [[Ehresmann]].

* [mathematics genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=98613)

####Some publications

* F. Conduché, Au sujet de l’existence d’adjoints à droite aux foncteurs ‘image reciproque’ dans la catégorie des catégories , C. R. Acad. Sci. Paris 275 Série A (1972) pp.891-894. (gallica)

* F. Conduché, Un moyen d’extension aux catégories internes dans *A* des propriétés de la catégorie *A*, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1974), Volume: 15, Issue: 1, page 47-59 [pdf on numdam](http://archive.numdam.org/article/CTGDC_1974__15_1_47_0.pdf)

####Related pages ####

* [[Conduché functor]]

category:people

[[!redirects F. Conduché]]
[[!redirects F. Conduche]]


François David

## Selected writings

On the rigorous construction of [[Liouville theory]]

* [[François David]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Liouville Quantum Gravity on the Riemann sphere_, Communications in Mathematical Physics volume 342, pages869–907 (2016) ([arxiv:1410.7318](https://arxiv.org/abs/1410.7318))

category: people


François Englert

* [Wikipedia entry](http://en.wikipedia.org/wiki/François_Englert)

## Selected writings

On [[Freund-Rubin compactifications]] of [[D=11 supergravity]]:

* [[Francois Englert]], _Spontaneous Compactification of Eleven-Dimensional Supergravity_, Phys. Lett. B119 (1982) 339 ([spire:180130](http://inspirehep.net/record/180130))

category: people

[[!redirects Francois Englert]]


François Foltz

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=98594)

## Selected writings

Proof that the [[funny tensor product]] of [[categories]] is the only other [[symmetric closed monoidal category|symmetric closed monoidal structure]] on [[Cat]] besides the [[cartesian monoidal category|cartesian monoidal structure]]:

* {#FoltzLairKelly80} [[François Foltz]], [[Christian Lair]], [[G. M. Kelly]], *Algebraic categories with few monoidal biclosed structures or none*, Journal of Pure and Applied Algebra **17** 2 (1980) 171-177 [[pdf](https://core.ac.uk/download/pdf/82322397.pdf), doi:10.1016/0022-4049(80)90082-1]

category: people

[[!redirects Francois Foltz]]


François Gieres

* [InSpire page](https://inspirehep.net/authors/1008404)

## Selected writings

On the [[phase space]] and canonical [[quantization]] of [[Yang-Mills theory]]:

* {#BlaschkeGieres21} [[Daniel N. Blaschke]], [[François Gieres]], *On the canonical formulation of gauge field theories and Poincaré transformations*, Nuclear Physics B **965** (2021) 115366 [[arXiv:2004.14406](https://arxiv.org/abs/2004.14406), [doi:10.1016/j.nuclphysb.2021.115366](https://doi.org/10.1016/j.nuclphysb.2021.115366)]

[[!redirects Francois Gieres]]


François Lamarche

* [dblp](https://dblp.org/pid/67/72.html)

* [Mathematics Genealogy Project](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=146065)

## Selected papers

On modelling [[polymorphism]] with [[categories]]:

* [[François Lamarche]], _Modelling polymorphism with categories_, PhD thesis, McGill University, 1985. [[McGill repository](https://escholarship.mcgill.ca/concern/theses/gh93h0216)]

On [[proof nets]] for [[intuitionistic linear logic]]:

* [[François Lamarche]], _Proof Nets for Intuitionistic Linear Logic: Essential Nets_, 2008. [[HAL:inria-00347336](https://inria.hal.science/inria-00347336)]

On non-commutative [[linear logic]]:

* [[Richard Blute]], [[François Lamarche]], [[Paul Ruet]], _Entropic Hopf algebras and models of non-commutative logic_, Theory and Applications of Categories **10** 17 (2002) 424–460 [[TAC](http://www.tac.mta.ca/tac/volumes/10/17/10-17abs.html)]

[[!redirects Francois Lamarche]]

category: people


François Loeser

* [Wikipedia entry](http://en.wikipedia.org/wiki/François_Loeser)

## related $n$Lab entries

* [[motivic integration]]

category: people


François Métayer

François Métayer is a French mathematician and theoretical computer scientist, attached to the Laboratoire [[IRIF]] at Paris 7.  He researches in logic, categories, and [[rewriting]] and more particularly in the development of geometric and topological methods in this areas. He is the organiser of the [Catégories supérieures, polygraphes et homotopie](https://www.irif.fr/seminaires/cat/index) working group.

* [webpage](https://www.irif.fr/~metayer/)

* [François Métayer days](https://smimram.gitlabpages.inria.fr/lhc/francois.html) - conference in honor of François Métayer around the themes of polygraphs, homotopy and rewriting.

## Selected writings

On [[strict omega-categories]] being [[monadic]] over [[polygraphs]]:

* [[François Métayer]], _Strict $\omega$-categories are monadic over polygraphs_, Theory and Applications of Categories, Vol. 31, No. 27, 2016, pp. 799-806. [[TAC](http://www.tac.mta.ca/tac/volumes/31/27/31-27abs.html)]

On the [[module of a monad|modules]] ("algebras") over the [[state monad]]:

* [[François Métayer]], *State monads and their algebras* [[arXiv:math/0407251](https://arxiv.org/abs/math/0407251)]

category: people

[[!redirects François Métayer]]
[[!redirects François Metayer]]
[[!redirects Francois Métayer]]
[[!redirects Francois Metayer]]
[[!redirects François Métayer]]


François Trèves

* [Wikipedia entry](https://en.wikipedia.org/wiki/Fran%C3%A7ois_Tr%C3%A8ves)

## related $n$Lab entries

* [[topological vector space]], [[distributions]] and [[integral kernels]]

* [[Green hyperbolic differential equation]]

category: people

[[!redirects François Treves]]
[[!redirects Francois Treves]]