field bundle

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## Idea

In [[Lagrangian field theory]] a _[[field history]]_ is a [[section]] of some [[fiber bundle]] $E \overset{fb}{\to} \Sigma$ over [[spacetime]]/[[worldvolume]] $\Sigma$. This is then called the _field bundle_.

The [[de Rham complex]] of the [[jet bundle]] of the field bundle, equipped with its canonical horizontal/vertical bigrading, is the _[[variational bicomplex]]_ of the [[Lagrangian field theory]].

For more see at _[[A first idea of quantum field theory]]_ the chapter _[Fields](A+first+idea+of+quantum+field+theory#Fields)_.

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

The notion of field bundle captures only some aspects of the general notion of spaces of physical fields. For instance the configurations of the _[[B-field]]_ do not form sections of a fiber bundle (but of a [[fiber infinity-bundle|fiber 2-bundle]]) and hence for instance the [magnetic charge anomaly](magnetic+charge#MagneticChargeAnomaly) has no real description by field bundles. See at _[[field (physics)]]_ in the section _[The idea of field bundles and its problems](field+%28physics%29#IdeaOfFieldBundlesAndItsProblems)_ for more.

=--

## Examples

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###### Example

For $E \to X$ the trivial bundle $X \times \mathbb{R} \to X $ or $X \times \mathbb{C} \to X$ a smooth section is simply a [[smooth function]] on $X$ with values in the [[real numbers]] or [[complex numbers]], respectively. In [[physics]] this is called a _[[scalar field]]_.

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###### Example

More generally, for $V$ a [[vector space]], the trivial field bundle $X \times V \to X$ has as secton $V$-valued functions. These are also called _linear [[sigma-model]]_ fields.

Still more generally, if $Y$ is any manifold, then the sections of the trivial field bundle $X \times Y \to X$ are called _non-linear [[sigma-model]]_ fields.

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###### Example

For $X$ a [[smooth manifold]] and $E \to X$ some [[tensor product]] of copies of the [[tangent bundle]] $T X \to X$ and the [[cotangent bundle]] $T^* X \to X$ a section is a [[tensor field]] of the corresponding rank.

Many physical fields are related to tensor fields, but few are genuinely tensor fields. See at _[[field (physics)]]_ for more on this.

=--

## Related concepts

* [[fiber bundles in physics]]

* [[field (physics)]]

* [[extended phase space]]

## References

The traditional idea of field bundle is discussed for instance around section 7.3.3 of

* Laurent Claessens, _Field theory from a bundle point of view_ (2011) ([pdf](http://student.ulb.ac.be/~lclaesse/lectures.pdf)) .

Detailed discussion of field bundles in [[gauge theory]] with a fixed [[instanton sector]]/[[principal bundle]]-class is around section 2.5 of

* {#BDS} [[Marco Benini]], [[Claudio Dappiaggi]], [[Alexander Schenkel]], _Quantized Abelian principal connections on Lorentzian manifolds_, Communications in Mathematical Physics 2013 ([arXiv:1303.2515](http://arxiv.org/abs/1303.2515))

and the issue is highlighted more explicitly in

* {#Schenkel14} [[Alexander Schenkel]], _On the problem of gauge theories
in locally covariant QFT_, talk at _[Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html)_ Trento, 2014 ([[SchenkelTrento2014.pdf:file]])

* {#Schreiber14} [[Urs Schreiber]], _Higher field bundles for gauge fields_, talk at _[Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html)_ Trento, 2014 ([[schreiber:Higher field bundles for gauge fields|web]])

The issue was then fixed in

* {#BeniniSchenkelSzabo15} [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Homotopy colimits and global observables in Abelian gauge theory_ ([arXiv:1503.08839](http://arxiv.org/abs/1503.08839))

precisely by retaining to groupoids/stacks of fields, hence using a higher stacky field bundle.

A discussion of the problems of the traditional notion and its rectification in [[higher geometry]] is at

* section _[Fields](geometry%20of%20physics#Fields)_ of _[[geometry of physics]]_

[[!redirects field bundles]]

[[!redirects field fiber]]
[[!redirects field fibers]]


field extension

> This page is about adjunctions of a set to a [[field]] in field theory. For the notion of adjunction in [[2-category theory]], see [[adjunction]].

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## Idea

If a [[subset]] $k$ of a [[field]] $K$ is a [[subfield]], then we call the larger field $K$ an **extension** of the smaller field $k$.

More generally, if $k \to K$ is *any* [[ring]] [[homomorphism]] between fields, then it must be an [[injection]], so we may treat it as a field extension.

## Adjunctions of a set to a field

Let $k$ be a [[field]], let $K$ be a field extension of $k$, and let $S \subseteq K$ be a subset of $K$. $k \subseteq K$ is also a subset of $K$. Then the **adjunction** of $S$ to $k$, or the field **generated** by $S$ over $k$, is the [[initial object|initial]] subfield $k(S) \subseteq K$ such that $k \subseteq K$ and $S \subseteq K$.

## Properties

* [[Galois theory]]

* [[class field theory]]

## Factorization system

Every field extension can be factorized as a [[purely transcendental extension]] followed by an [[algebraic extension]]. Indeed, by [[Zorn's lemma]], we may construct a transcendence basis (i.e. maximal algebraically independent set) $B$, and the purely transcendental part is the subfield generated by $B$.

Unfortunately, this does not yield an [[orthogonal factorization system]]: given a field $K$, we may form the field $K (x)$ of [[rational functions]] over $K$, which is a purely transcendental extension of $K$, and we may form the [[algebraic closure]] $\overline{K (x)}$, which is an algebraic extension of $K (x)$; but we have the following commutative diagram,
$$\array{
K & \to & K (x^2) \\
\downarrow & & \downarrow \\
K (x) & \to & \overline{K (x)}
}$$
where $K (x^2)$ is the subfield of $K (x)$ generated by $x^2$, and $K (x^2) \to \overline{K (x)}$ is algebraic, yet there is no homomorphism $K (x) \to K (x^2)$ making both evident triangles commute.

## Examples

* [[cyclotomic field extension]]

* [[anti-cyclotomic field extension]]

## Related concepts

* [[transcendence degree]]

* [[separable field extension]]

* [[regular field extension]]

* [[ramified extension]]

* [[Schur index]]

* [[group extension]]

* [[algebraic extension]]

## References

* Wikipedia, [Field extension](https://en.wikipedia.org/wiki/Field_extension)

[[!redirects field extension]]
[[!redirects field extensions]]

[[!redirects extension of fields]]
[[!redirects extensions of fields]]


field history

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## Idea

Given a [[Lagrangian field theory]] with [[field bundle]] $E \overset{fb}{\to} \Sigma$ over [[spacetime]] $\Sigma$, then a _field history_ is a smooth [[section]] of this field bundle $\Phi \in \Gamma_\Sigma(E)$.

For every [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ the [[restriction]] $\Phi\vert_{\Sigma_p}$ of $\Phi$ to $\Sigma_p$ may be thought of as the _field configuration_ "at that time" and hence the change of $\Phi\vert_{\Sigma_p}$ as the choice of Cauchy surface changes reflects the "history" of these configurations, whence the name.

For a [[sigma model]] field theory a field history may be thought of as a _trajectory_.

For more see at _[[geometry of physics -- perturbative quantum field theory]]_ the chapter _[Fields](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#Fields)_.

## Related concepts

* [[field observable]]

* [[field (physics)]]

* [[configuration space (physics)]]

[[!redirects field histories]]

[[!redirects trajectory]]
[[!redirects trajectories]]


field net

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## Idea

In the [[AQFT]] formalization of [[quantum field theory]] a [[local net of observables]] assigns to each region of [[spacetime]] the [[algebra]] of [[observables]] localized in that region. But in typical constructions of [[quantum field theories]] this algebra is obtained from an algebra of quantum fields that are _not_ all observable by quotienting out a [[gauge group]] [[action]]. A **field net** corresponding to a [[local net of observables]] is a [[net of C-star-systems]] which formalizes this idea, notably so that the quotient by the group action reproduces the given local net of observables.

## Reconstruction

Given a [[local net of observables]] $\mathcal{A}$ the corresponding field algebra, according to [Doplicher-Roberts](#DoplicherRoberts) the the following.

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

The _field algebra_ $\mathcal{F}_0$ corresponding to $\mathcal{A}$ is the collection of [[equivalence classes]] of triples $(A, \rho, \psi)$ consisting

* $A \in \mathcal{A}$ an [[observable]];

* $\rho : \mathcal{A} \to \mathcal{A}$ a [[superselection sector]], hence an object of the [[DHR category]] $\Delta_{DHR}$ of $\mathcal{A}$;

* $\psi \in E(\rho)$ a [[vector]] in the space [[Hilbert space]] assigned by the [[fiber functor]] of the [[Doplicher-Roberts reconstruction theorem]] to superselection sector $\rho$;

where the [[equivalence relation]] identifies for $T : \rho \to \rho'$ an intertwiner two such triples by the rule

$$
  (A T, \rho, \psi)
  \sim
  (A, \rho', E(T)\psi)
  \,.
$$

This becomes an [[associative algebra|algebra]] by defining the product on representatives as

$$
  (A_1, \rho_1, \psi_1) \cdot (A_2, \rho_2, \psi_2)
  =
  (A_1 \rho_1(A_2), \rho_1 \otimes \rho_2, \psi_1 \otimes \psi_2)
  \,.
$$

With a bit more work a [[star algebra]] structure is defined.

This becomes a net by assigning to an open $\mathcal{O}$ the subalgebra of such triples with $A \in \mathcal{A}(\mathcal{O})$ and $\rho$ localized in $\mathcal{O}$.

Finally, one can construct a representation of this in [[bounded operators]], and taking the [[norm]] closure there gives the field net $\mathcal{F}$.

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## References

The construction of a field net for every local net of observables in [[DHR superselection theory]] is due to

* [[Sergio Doplicher]], [[John Roberts]], _Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics_, Comm. Math. Phys. 131 (1) (1990)
 {#DoplicherRoberts}

A detailed review is in sections 9 and 10 of

* [[Hans Halvorson]], _Algebraic quantum field theory_ ([pdf](http://www.princeton.edu/~hhalvors/aqft.pdf))

A purely [[category theory|category theoretic]] construction of the field-net analog of the [[DHR category]] is given in

* [[Michael Müger]], _Galois Theory for Braided Tensor Categories
and the Modular Closure_, Adv. Math. 150, 151-201 (2000) ([arXiv:math.CT/9812040](http://arxiv.org/abs/math.CT/9812040))

[[!redirects field nets]]


field observable

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## Idea

Given [[Lagrangian field theory]] with [[field bundle]] $E \overset{fb}{\to} \Sigma$ over some [[spacetime]] $\Sigma$, so that a [[field history]] is a smooth [[section]] $\Phi \in \Gamma_\Sigma(E)$, then for every point $x \in \Sigma$ (every [[event]]) and every [[coordinate function]] $\phi^a \in C^\infty(E\vert_{U_x}$) on the [[fibers]] of $E$ over a [[neighbourhood]] of $X$ there is the [[observable]]

$$
  \mathbf{\Phi}^a(x)
  \;\colon\;
  \Gamma_\Sigma(E)
    \longrightarrow
  \mathbb{C}
$$

which reads out the value of that component of the [[field history]] $\Phi$ at that point ([[event]])

$$
  \mathbf{\Phi}^a(x) \;\colon\; \Phi \mapsto \Phi^a(x)
  \,.
$$

Hence in the case that $E$ is a [[vector bundle]], then under the identification of [[linear observables]] with [[distributions]] ([this prop.](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#LinearObservablesAreTheCompactlySupportedDistributions)), these field observable are the [[Dirac delta distributions]].

[[!include perturbative observables -- table]]

In general such $\mathbf{\Phi}^a(x)$ itself is far from being [[on-shell]] [[gauge invariance|gauge invariant]] and hence far from being an element in the [[BV-BRST cohomology]] of the theory. But since most observables that are may be expressed as algebraic combinations of these "point evaluation observables" they are of particular interest. Since they manifestly reflect the value of [[field histories]], it is common to terminologically conflate the field observables $\mathbf{\Phi}^a(x)$ with "the fields" of the field theory.

## Related entries

* [[operator product expansion]]

* [[A first idea of quantum field theory]], [this example](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#PointEvaluationObservables)

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

[[!redirects field observables]]

[[!redirects point-evaluation observable]]
[[!redirects point-evaluation observables]]

[[!redirects point-evaluation field observable]]
[[!redirects point-evaluation field observables]]


field of fractions

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## Idea

For a commutative [[integral domain]] $D$, the set of all nonzero elements $D^\times$ is a multiplicative set, the corresponding [[commutative localization]] $D\to (D^\times)^{-1} D$ is an injective homomorphism of rings and $Q(D) = (D^\times)^{-1} D$ is a field, called the field of fractions or the quotient field of $D$. Its elements are fractions $a/b$ where $a\in D$ and $b\in D^\times$ which are by the definition the equivalence classes of pairs $(a,b) \in D\times D^\sharp$ and $(a,b)\sim (c,d)$ iff $a d = b c$. The addition is given by the formula
$$
\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd}
$$
and multiplication by $(a/b)(c/d) = (ac)/(bd)$.

The field of fractions $Q(D)$ is unique minimal field in which the integral domain $D$ is embedded in the sense that every field $K\supset D$ contains the subfield isomorphic to $Q(D)$, namely consisting of all the fractions $a/d$ with $a\in D$, $d\in D^\sharp$ taken in the sense of division in $K$.

In the noncommutative case, the notion of a quotient skewfield is not always, and not uniquely defined. One class where this notion works well is the case or (left or right) [[Ore domain]]s where the set of all nonzero elements form a (left or right) Ore set and the Ore localization at that set defines a [[skewfield of fractions]].

Not every noncommutative integral domain can be embedded at all into a division ring. Suppose it does, say $D\hookrightarrow F$. Then all the say left fractions $d^{-1} b$ with $d,b\in D, d\neq 0$ do not form a subfield, namely we can not in general add a fractions to a fraction with common denominator, so this candidate for a subfield may have noninvertible elements.

## Related concepts

* [[sheaf of rational functions]]

* [[localization]]

[[!redirects fields of fractions]]

[[!redirects quotient field]]

[[!redirects fraction field]]
[[!redirects fraction fields]]


field of moduli

\tableofcontents

\section{Idea}

Given a [[curve]] $X$ in [[algebraic geometry]] over some field $F$, especially some kind of [[number field]], the field of moduli of $X$ with respect to $F$ is, roughly speaking, the largest field invariant under those automorphisms of $\overline{F}$, the [[algebraic closure]] of $F$, which do not change $X_{\overline{F}}$, the base change of $X$ to a [[scheme]] over $\overline{F}$.

Automorphisms of $\overline{F}$ which fix $F$ are fundamental in [[Galois theory]], forming the [[absolute Galois group]] of $F$, and indeed in mathematics generally: the absolute Galois group of the [[rational number|rational numbers]] is a very deep object, for example. Considering the field of moduli of a curve is a kind of geometric analogue of this.

\section{Definition}

There are slightly different ways to give a precise definition. We choose the following formulation (see [Mochizuki2015](#Mochizuki2015) for example, although the notion goes back at least to the era of [[André Weil]]).

\begin{defn} \label{DefinitionFieldOfModuli} Let $X$ be a [[curve]] in the sense of algebraic geometry, defined over a [[field]] $F / \mathbb{Q}$. Given an automorphism $\sigma$ of $\overline{F}$, one can ask that the [[scheme]] $X_{\overline{F}} = X \times_{F} \overline{F}$ (where the projection morphisms are the obvious ones) is isomorphic to the [[base change]] of $X_{\overline{F}}$ along $\sigma$. The _field of moduli_ of $X$ with respect to $F$ is the sub-field of $\overline{F}$ fixed by all such automorphisms  $\sigma$ which belong to the [[Galois group]] $Gal\left(\overline{F} / \mathbb{Q} \right)$, namely which fix $\mathbb{Q}$. \end{defn}

\begin{rmk} The field of moduli of $X$ with respect to $F$ is a finite [[field extension|extension]] of $F$. \end{rmk}

\begin{rmk} A curve $X$ over $F$ need not be definable over $F_{mod}$, that is, there may not exist any scheme over $F_{mod}$ which is isomorphic to $X_{\overline{F}}$ after base change along the canonical morphism $\overline{F} \rightarrow F_{mod}$. \end{rmk}

\section{References}

Definition \ref{DefinitionFieldOfModuli} is 5.1 (ii) in the following.

* {#Mochizuki2015} [[Shinichi Mochizuki]], _Topics in absolute anabelian geometry. III: Global reconstruction algorithms_, J. Math. Sci., Tokyo 22, No. 4, 939-1156 (2015)


field strength

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## Idea

In [[gauge theory]] cocycles in [[differential cohomology]] model _gauge fields_.

By definition, every [[differential cohomology]] theory $\Gamma^\bullet(-)$ comes with a characteristic curvature form morphism

$$
  F : \bar \Gamma^\bullet(X) \to \Omega^\bullet(X)\otimes \pi_*\Gamma \otimes \mathbb{R}
  \,,
$$

the (generalized) [[Chern character]].

For $c \in \Gamma^\bullet(X)$ a cocycle representing a gauge field in [[gauge theory]], its image $F(c) \in \Omega^\bullet(X)$ is the **field strength** of the gauge field. If we think of this cocycle as being (a generalization of) a [[connection on a bundle]], this is essentially the [[curvature]] of that connection.

Often gauge fields are _named_ after their field strength. For instance the field strength of the [[electromagnetic field]] is the $2$-form $F \in \Omega^2(X)$ whose components are the electric and the magnetic fields.

## Examples

* [[electromagnetic field strength]]

## Related concepts

[[!include gauge field - table]]

[[!redirects field strengths]]


field theory

In [[physics]]:

* [[field (physics)]] [[theory (physics)]]

* [[classical field theory]]

* [[prequantum field theory]]

* [[quantum field theory]]

  * [[algebraic quantum field theory]]

  * [[functorial quantum field theory]]

  * [[local quantum field theory]]

  * [[relativistic quantum field theory]]

  * [[Lagrangian quantum field theory]]

  * [[perturbative quantum field theory]]

  * [[non-perturbative quantum field theory]]

In [[mathematics]]:

* [[class field theory]]

[[!redirects field theories]]

category: disambiguation


field theory with boundaries and defects - table

| singularity | [[field theory with singularities]] |
|--|--|
| [[boundary condition]]/[[brane]] | [[boundary field theory]] |
| [[domain wall]]/[[bi-brane]] | [[QFT with defects]] |


field with one element

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#F-un#
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{:toc}

## Overview

Various phenomena in the context of  [[algebraic geometry]]/[[arithmetic geometry]] (and particularly in the context of [[algebraic groups]]) over  [[finite fields]] $\mathbb{F}_q$ turn out to make perfect sense _as expressions in $q$_ when extrapolated to the case $q=1$, and to reflect interesting (combinatorial, representation theoretical...) facts, even though, of course, there is no actual [[field]] with a single element (since in a field by definition the elements 1 and 0 are distinct).

Motivated by such observations, [[Jacques Tits]] envisioned in ([Tits 57](#Tits57)) a new kind of geometry adapted to the explanation of these identities. [[Christophe Soulé]] then expanded on Tits' ideas by introducing the notion of _field with one element_ and studying its fine arithmetic invariants. While there is no field with a single element in the standard sense of _[[field]]_, the idea is that there is some other object, denoted $\mathbb{F}_1$, such that it does make sense to speak of "geometry over $\mathbb{F}_1$". Following the French pronunciation one also writes $F_{un}$ (and is thus led to the inevitable pun).

In the [[relative point of view]] the $S$-schemes are schemes with a morphism of schemes over a base scheme $S$; but every $S$-scheme is a scheme over [[Spec(Z)]]. In __absolute algebraic geometry__ all "generalized schemes" should live over $Spec(F_1)$ and $Spec(F_1)$ should live **below** $Spec(\mathbb{Z})$; this is similar to the fact that the quotient stacks like $[*/G]$ live below the single point $*$ (there is a direct image functor from sheaves on a point to sheaves over $[*/G]$). One of the principal and very bold hopes is that the study of $F_{un}$ should lead to a natural proof of [[Riemann conjecture]] (see also MathOverflow [here](http://mathoverflow.net/questions/69389/riemann-hypothesis-via-absolute-geometry)). It was originally suggested by ([Manin 95](#Manin95)) that the [[Riemann hypothesis]] might be solved by finding an $\mathbb{F}_1$-[[analogy|analogue]] of [[André Weil]]'s proof for the case of [[arithmetic curves]] over the [[finite fields]] $\mathbb{F}_q$.

A first proposal for what a [[variety]] "over $\mathbb{F}_1$" ought to be is due to ([Soulé 04](#Soule)).  After that a plethora of further proposals appeared, including ([Connes-Consani 08](#ConnesConsani08)).

## Borger's absolute geometry
 {#BorgerAbsoluteGeometry}

Maybe an emerging consensus is that the preferred approach is [[Borger's absolute geometry]] ([Borger 09](#Borger09)). Here the structure of a [[Lambda-ring]] on a ring $R$, hence on $Spec(R) \to Spec(\mathbb{Z})$,  is interpreted as a collection of lifts of all [[Frobenius morphisms]] and hence as [[descent]] data for descent to $Spec(\mathbb{F}_1)$ (which is defined thereby).  This definition yields an [[essential geometric morphism]] of [[gros topos|gros]] [[etale toposes]]

$$
  Et(Spec(\mathbb{Z}))
   \stackrel{\overset{}{\longrightarrow}}{\stackrel{\overset{}{\longleftarrow}}{\underset{}{\longrightarrow}}}
  Et(Spec(\mathbb{F}_1))
  \,,
$$

where on the right the notation is just suggestive, the [[topos]] is a suitable one over [[Lambda-rings]]. Here the middle [[inverse image]] is the [[forgetful functor]] which forgets the Lambda structure, and its [[right adjoint]] [[direct image]] is given by the [[arithmetic jet space]] construction (via the [[ring of Witt vectors]] construction).

This proposal seems to subsume many aspects of other existing proposals (see e.g. [Le Bruyn 13](#LeBruyn13)) and stands out as yielding an "absolute [[base topos]]" $Et(Spec(\mathbb{F}_1))$ which is rich and genuinely interesting in its own right.

## Function field analogy

[[!include function field analogy -- table]]

## Algebra over $\mathbb{F}_1$
 {#AlgebraOverF1}

### Modules/Vector spaces over $\mathbb{F}_1$
 {#Modules}

It makes good sense to identify the concept of finite rank [[modules]]/[[finite-dimensional vector spaces]] over the field with one element with that of ([[pointed set|pointed]]) [[finite sets]]

$$
  \mathbb{F}_1 Mod_{fin}
  \;\simeq\;
  Set^{\ast/}_{fin}
$$

and hence the [[symmetric group]] $\Sigma_n$ on $n$ [[elements]] with the [[general linear group]] over $\mathbb{F}_1$:

$$
  GL(n,\mathbb{F}_1)
  \;\simeq\;
  \Sigma_n
$$

(e.g. [Cohn 04, "puzzle 1"](#Cohn04), [Durov 07, 2.5.6](#Durov07), [Snyder 07](#Snyder07))

### Algebraic K-theory
 {#AlgebraicKTheory}

With the identification $\mathbb{F}_1 Mod \simeq FinSet^{\ast/}$ from [above](#Modules) it follows that the [[algebraic K-theory]] over $\mathbb{F}_1$ is _[[stable cohomotopy]]_:

$$
  \begin{aligned}
    K \mathbb{F}_1
    & \coloneqq\;
    K(\mathbb{F}_1 Mod)
    \\
    & \simeq\;
    K(FinSet)
    \\
    & \simeq
    \mathbb{S}
  \end{aligned}
  \,.
$$

Here in the second step we used the definition of algebraic K-theory for ordinary [[commutative rings]] as the [[K-theory of a permutative category|K-theory of the permutative category]] of modules ([this example](K-theory+of+a+permutative+category#examples#OrdinaryAlgebraicKTheoryFromPermutativeCategoryOfProjectiveModules)), in the second step we used the identification of modules over $\mathbb{F}_1$ with [[pointed set|pointed]] [[finite sets]] from [above](#Modules), and finally we used the identification of the [[K-theory of a permutative category|K-theory of the permutative category]] of [[finite set]] with the [[sphere spectrum]] ([this example](K-theory+of+a+permutative+category#StableCohomotopyIsKTheoryOfFinSet)), which is the spectrum representing [[stable cohomotopy]], by definition.

The perspective that the [[K-theory]] $K \mathbb{F}_1$ over $\mathbb{F}_1$ should be [[stable Cohomotopy]] has been highlighted in ([Deitmar 06, p. 2](#Deitmar06), [Guillot 06](#Guillot06), [Mahanta 17](#Mahanta17), [Dundas-Goodwillie-McCarthy 13, II 1.2](#DundasGoodwillieMcCarthy13), [Morava](#MoravaSomeBackground), [Connes-Consani 16](#ConnesConsani16)).). Generalized to [[equivariant stable homotopy theory]], the statement that [[equivariant K-theory]] $ K_G \mathbb{F}_1$ over $\mathbb{F}_1$ should be [[equivariant stable Cohomotopy]] is discussed in [Chu-Lorscheid-Santhanam 10, 5.3](#ChuLorscheidSanthanam10).

[[!include Segal completion -- table]]

## Related concepts

* [[Lambda-ring]],

* [[blue scheme]]

* [[tropical geometry]].

## References

After the very first observations by Tits, pioneers were [[Christophe Soulé]] and Kapranov and Smirnov. More recently there are extensive works by [[Alain Connes]] and Katia Consani, [[Nikolai Durov]], [[James Borger]] and [[Oliver Lorscheid]].

### Expositions

* {#Cohn04} Henry Cohn, _Projective geometry over $\mathbb{F}_1$ and the Gaussian binomial coefficients_, American Mathematical Monthly 111 (2004), 487-495 ([arXiv:math/0407093](https://arxiv.org/abs/math/0407093))

* [[Lieven Le Bruyn]], _Looking for $F_{un}$_, [blog](http://www.neverendingbooks.org/looking-for-f_un)

* {#Snyder07} [[Noah Snyder]], _[The field with one element](https://sbseminar.wordpress.com/2007/08/14/the-field-with-one-element/)_, 2007

* Javier López Peña, [[Oliver Lorscheid]], _Mapping $F_1$-land:An overview of geometries over the field with one element_, [arXiv/0909.0069](http://arxiv.org/abs/0909.0069)

* [[John Baez]], _This Week's Finds 259_ ([html](http://math.ucr.edu/home/baez/week259.html) [blog](http://golem.ph.utexas.edu/category/2007/12/this_weeks_finds_in_mathematic_19.html))

* [[Alain Connes]], _Fun with $\mathbf{F}_1$_, 5 min. [video](http://www.dailymotion.com/video/x6xe0g_article-alain-connes_tech)

* [[Lieven Le Bruyn]], _The field with one element_, seminar notes 2008 ([web](http://win.ua.ac.be/~lebruyn/b2hd-LeBruyn2008d.html))

* {#Lorscheid14} [[Oliver Lorscheid]], _Lectures on $\mathbb{F}_1$_, 2014 ([pdf](http://www.impa.br/~lorschei/Lectures_on_F1.pdf))

* [[Oliver Lorscheid]], _$\mathbb{F}_1$ for everyone_, 2018 ([arXiv:1801.05337](https://arxiv.org/abs/1801.05337))

* [[Tim Hosgood]], _Under $\mathop{Spec}\mathbb{Z}$: a reader's companion_, master thesis, 2016 ([pdf](https://thosgood.com/assets/files/under-spec-z.pdf)).

### Original articles ###

* {#Tits57} [[Jacques Tits]], _Sur les analogues algebriques des groupes semi-simples complexes_. In Colloque d'algebre superieure, tenu a Bruxelles du 19 au 22 decembre 1956, Centre Belge de Recherches Mathematiques, pages 261-289. Etablissements Ceuterick, Louvain, 1957.

* {#Soule04} [[Christophe Soulé]], _Les varietes sur le corps a un element_ Mosc. Math. J., 4(1):217-244, 312, 2004 ([pdf](http://www.ams.org/distribution/mmj/vol4-1-2004/soule.pdf))

* {#Manin95} [[Yuri Manin]], _Lectures on zeta functions and motives (according to Deninger and Kurokawa)_ Asterisque, (228):4, 121-163, 1995. Columbia University Number Theory Seminar ([pdf](http://streams1.nts.jhu.edu/mathematics/ncgdocs/manin-zeta-absmotives.pdf))

* {#ToenVaquie05} [[Bertrand Toen]], [[Michel Vaquie]], _Under Spec Z_ ([arXiv:math/0509684](http://arxiv.org/abs/math/0509684))

Around (0.4.24.2) in

* {#Durov07} [[Nikolai Durov]], _New Approach to Arakelov Geometry_ ([arXiv:0704.2030](http://arxiv.org/abs/0704.2030))

the algebraic structure of $\mathbb{F}_1$ is regarded as being the [[maybe monad]], hence [[modules]] over $\mathbb{F}_1$ are defined to be [[algebra over a monad|monad-algebras]] over the [[maybe monad]], hence [[pointed sets]].

Other approaches include

* [[Alain Connes]], [[Caterina Consani]], [[Matilde Marcolli]], _Fun with $\mathbf{F}_1$_, [arxiv/0806.2401](http://arxiv.org/abs/0806.2401)

* [[Yuri Manin]], _Cyclotomy and analytic geometry over $F_1$_, [arxiv/0809.1564](http://arxiv.org/abs/0809.1564)

* {#ConnesConsani08} [[Alain Connes]], [[Caterina Consani]], _On the notion of geometry over $\F_1$_, [arxiv/0809.2926](http://arxiv.org/abs/0809.2926);

  _Schemes over $\F_1$ and zeta functions_, [arxiv/0903.2024](http://arxiv.org/abs/0903.2024); _Characteristic one, entropy and the absolute point _, in: Noncommutative Geometry, Arithmetic, and Related Topics, 21st Meeting of the Japan-U.S. Math. Inst., Baltimore 2009, JHUP (2012), pp. 75–139, [arxiv/0911.3537](http://arxiv.org/abs/0911.3537);

  _From monoids to hyperstructures: in search of an absolute arithmetic_, [arxiv/1006.4810](http://arxiv.org/abs/1006.4810);

  _On the arithmetic of the BC-system_, [arxiv/1103.4672](http://arxiv.org/abs/1103.4672); _Projective geometry in characteristic one and the epicyclic category_, [arxiv/1309.0406](http://arxiv.org/abs/1309.0406)

* [[M. Marcolli]], [[Ryan Thorngren]], _Thermodynamical semirings_, [arXiv/1108.2874](https://arxiv.org/abs/1108.2874)

* Bora Yalkinoglu, _On Endomotives, Lambda-rings and Bost-Connes systems_,  With an appendix by Sergey Neshveyev, [arxiv/1105.5022](http://arxiv.org/abs/1105.5022)

The approach in terms of [[Lambda-rings]] due to

* {#Borger09} [[James Borger]], _Lambda-rings and the field with one element_ ([arXiv/0906.3146](http://arxiv.org/abs/0906.3146))

with details in

* {#Borger08} [[James Borger]], _The basic geometry of Witt vectors, I: The affine case_ ([arXiv:0801.1691](http://arxiv.org/abs/0801.1691))

* {#Borger10} [[James Borger]], _The basic geometry of Witt vectors, II: Spaces_ ([arXiv:1006.0092](http://arxiv.org/abs/1006.0092))

More discussion relating to this includes

* {#LeBruyn13} [[Lieven Le Bruyn]], _Absolute geometry and the Habiro topology_ ([arXiv:1304.6532](http://arxiv.org/abs/1304.6532))

### Relation to stable homotopy theory

The interpretation of [[stable cohomotopy]] as [[algebraic K-theory]] over $\mathbb{F}_1$ is amplified in the following articles:

* {#Deitmar06} [[Anton Deitmar]], _Remarks on zeta functions and K-theory over $\mathbb{F}_1$_ ([arXiv:math/0605429](https://arxiv.org/abs/math/0605429))

* {#Guillot06} [[Pierre Guillot]], _Adams operations in cohomotopy_ ([arXiv:0612327](https://arxiv.org/abs/math/0612327))

* {#MoravaSanthanam} [[Jack Morava]], Rekha Santhanam, _Power operations and absolute geometry_, 2012 ([pdf](http://www.lemiller.net/media/slidesconf/AbsolutePower.pdf))

* {#Mahanta17} [[Snigdhayan Mahanta]], _G-theory of $\mathbb{F}_1$-algebras I: the equivariant Nishida problem_, J. Homotopy Relat. Struct. 12 (4), 901-930, 2017 ([arXiv:1110.6001](https://arxiv.org/abs/1110.6001))

* {#Berman18} John D. Berman, p. 92 of _Categorified algebra and equivariant homotopy theory_ ([arXiv:1805.08745](https://arxiv.org/abs/1805.08745))

* {#ChuLorscheidSanthanam10} Chenghao Chu, [[Oliver Lorscheid]], Rekha Santhanam, _Sheaves and K-theory for $\mathbb{F}_1$-schemes_, Advances in Mathematics, Volume 229, Issue 4, 1 March 2012, Pages 2239-2286 ([arxiv:1010.2896](https://arxiv.org/abs/1010.2896))

see also

* {#MoravaSomeBackground} [[Jack Morava]], _Some background on Manin's theorem $K(\mathbb{F}_1) \sim \mathbb{S}$_ ([pdf](http://www.alainconnes.org/docs/Morava.pdf), [[MoravaSomeBackground.pdf:file]])

* {#ConnesConsani16} [[Alain Connes]], [[Caterina Consani]], _Absolute algebra and Segal's Gamma sets_, Journal of Number Theory 162 (2016): 518-551 ([arXiv:1502.05585](https://arxiv.org/abs/1502.05585))

### In quantum field theory

Speculation on [[quantum field theory]] over $\mathbb{F}_1$ (via [[wonderful compactifications]] of [[configuration spaces of points]])

* Dori Bejleri, [[Matilde Marcolli]], *Quantum field theory over $\mathbb{F}_1$*, Journal of Geometry and Physics
**69** (2013) 40-59 [[arXiv:1209.4837](https://arxiv.org/abs/1209.4837), [doi:10.1016/j.geomphys.2013.03.002](https://doi.org/10.1016/j.geomphys.2013.03.002)]

* Seyed Khaki, *Original $\mathbb{F}_1$ in emergent spacetime* [[arXiv:2401.07822](https://arxiv.org/abs/2401.07822)]

[[!redirects field of one element]]
[[!redirects Fun]]
[[!redirects absolute geometry]]
[[!redirects absolute algebraic geometry]]
[[!redirects absolute ground field]]
[[!redirects F1]]


field-strength renormalization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[renormalization]]

## References

General introduction:

* [[Michael Peskin]], [[Daniel Schroeder]], Section 7.1 of: _An Introduction to Quantum Field Theory_, ([spire:407703](https://inspirehep.net/literature/407703), [ISBN 9780201503975](https://www.routledge.com/An-Introduction-To-Quantum-Field-Theory/Peskin-Schroeder/p/book/9780201503975))

Discussion in the context of [[holographic QCD]]/[[holographic light front QCD]]:

* [[Stanley Brodsky]], [[Guy de Teramond]], Alexandre Deur, _AdS/QCD, Light-Front Holography, and the Nonperturbative Running Coupling_,  International Journal of Modern Physics A Vol. 25, No. 27n28, pp. 5009-5024 (2010)  ([arXiv:1002.4660](https://arxiv.org/abs/1002.4660), [doi:10.1142/S0217751X10050822](https://doi.org/10.1142/S0217751X10050822))

[[!redirects field strength renormalization]]


fields (physics)


fields and quanta - table

**[[field (physics)|fields]] and [[fundamental particle|particles]] in [[particle physics]]**

**and in the [[standard model of particle physics]]**:

**[[force]] [[field (physics)|field]] [[gauge field|gauge]] [[bosons]]**

* [[photon]] - [[electromagnetic field]] ([[abelian group|abelian]] [[Yang-Mills field]])

* [[W-boson|W]], [[Z-boson|Z]], [[B-boson]] - [[electroweak field]] ([[Yang-Mills field]])

* [[gluon]] - [[strong nuclear force]] ([[Yang-Mills field]])

* [[graviton]] - [[field (physics)|field]] of [[gravity]]

* [[infraparticle]]

**[[scalar field|scalar]] [[bosons]]**

* [[Higgs boson]], [[inflaton]] ([[scalar field]])

**[[matter]] [[field (physics)|field]] [[fermions]]** ([[spinors]], [[Dirac fields]])

[[!include flavors of fundamental particles -- table]]

(also: [[antiparticles]])

**[[effective field theory|effective particles]]**

* [[Goldstone bosons]]

**[[hadrons]]** ([[bound states]] of the above [[quarks]])

* [[meson]]

  * [[scalar meson]]

    * [[σ-meson]]

    * [[pion]]

    * [[kaon]]

    * [[D-meson]]

    * [[B-meson]]

  * [[vector meson]]

    * [[ω-meson]]

    * [[ρ-meson]]

    * [[f1-meson]]

    * [[a1-meson]]

    * [[b1-meson]]

    * [[h1-meson]]

    * [[K*-meson]]

  * [[tensor meson]]

  * [[quarkonium]]

    * [[charmonium]]

    * [[ϒ-meson]]

* [[exotic meson]]

  * [[XYZ meson]]

  * [[tetraquark]]

* [[baryon]]

  * [[nucleon]]

    * [[proton]]

    * [[neutron]]

    * [[chemical element]]

      * [[carbon]]

      * [[nitrogen]]

  * [[Lambda baryon]]

  * [[pentaquark]]

**[[solitons]]**

* [[Skyrmion]]

* [[caloron]]

* [[instanton]]

**in [[grand unified theory]]**

* [[leptoquark]]

* [[Z'-boson]]

**[[MSSM|minimally extended supersymmetric standard model]]**

**[[superpartners]]**

bosinos:

* [[gravitino]] - [[field (physics)|field]] of [[supergravity]] ([[Rarita-Schwinger field]])

* [[gaugino]] - [[super Yang-Mills theory|super Yang-Mills field]]

* [[gluino]]

[[sfermions]]:

* [[squark]]

**[[dark matter]] candidates**

* [[WIMP]]

* [[axion]]

**Exotica**

* [[preon]]

* [[graviphoton]]

* [[dilaton]]

* [[monopole]]

* [[dual graviton]]

* [[giant graviton]]

**[[auxiliary fields]]**

* [[ghost field]]

* [[antifield]]

* [[antighost field]], [[Nakanishi-Lautrup field]]


Fields medal

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

A prize for work in [[mathematics]].

* Wikipedia, _[Fields medal](http://en.wikipedia.org/wiki/Fields_Medal)_

[[!redirects Fields medals]]


Fierz identity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What are called _Fierz identities_ in [[physics]] are the relations that hold between [[multilinear map|multilinear]]  expression in [[spinors]]. For example for all [[Majorana spinors]] $\psi$ in Lorentzian spacetime dimension 4,5,7,11, then the following identity holds (example \ref{TheM2andM5CocyclesAsFierzIdentities} below):

$$
  \left(\overline{\psi} \wedge \Gamma_{a b} \psi\right)
  \wedge
  \left(\overline{\psi} \wedge \Gamma^b \psi\right)
  \;=\;
  0
  \,.
$$

(Here $\overline{(-)}$ denotes the Majorana conjugate, $\Gamma_a$ are a Clifford representations, the "$\wedge$"-signs denotes symmetrization in the spinor components and summation over repeated indices is understood. The details of this are discussed [below](#BilinearFierzIdentities).)

In [D'Auria-Fré-Maina-Regge 82](#DAuriaFreMainaRegge82) it was pointed out that all Fierz identities may be understood as expressing the product operation in the [[representation ring]] of the [[spin group]] (in some given dimension): for $\{S_i\}_{i \in I}$ denoting [[isomorphism classes]] of [[irreducible representations|irreducible]] [[spin representations]], then, by definition of [[irreps]], their [[tensor product of representations]] decomposes again as a [[direct sum]] of [[irreducible representations]]

$$
  S_i \otimes S_j  = \underset{k}{\oplus} C_{i j}{}^k S_k
$$

with "[[Clebsch-Gordan coefficients]]" $C_{i j}{}^k$. These coefficients are effectively the Fierz identities.

For example for Lorentzian dimension 11 with $(\tfrac{1}{2})^5$ denoting the unique irreducible [[Majorana spinor]] representation, then one finds ([D'Auria-Fré 82b, section 3](#DAuriaFre82b)) that the symmetric part in the quadruple tensor product of this representation with itself decomposes as a direct sum of irreps as follows

$$
  \left\{
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
  \right\}_{sym}
   \;\simeq\;
  (0)^5
    \;\oplus\;
  (1)^3 (0)^2
    \;\oplus\;
  (1)^4 (0)
    \;\oplus\;
  (1)^5
    \;\oplus\;
  (2) (0)^4
    \;\oplus\;
  (2)(1)(0)^3
    \;\oplus\;
  (2)^2 (0)^3
    \;\oplus\;
  (2)^2 (1)^3
    \;\oplus\;
  (2)^5
$$

where the symbols refer to [[Young diagrams]] canonically labeling representations (details are in example \ref{11dQuadrilinearCGCoefficients} below).

The point is that the expression $ \left(\overline{\psi} \wedge \Gamma_{a b} \psi\right) \wedge \left(\overline{\psi} \wedge \Gamma^b \psi\right)$ from above is a spinor quadrilinear which transforms in the vector representation $(1)(0)^4$ (due to its one free spacetime index). But that vector representation $(1)(0)^4$ is missing from the [[direct sum]] above, meaning that the spinor quadrilinear has vanishing components in this vector representation, hence that this expression vanishes identically.

## In terms of cochains on super-Minkowski spacetimes

We discuss Fierz identities as identities among multispinorial elements of the [[Chevalley-Eilenberg algebra]] $CE(\mathbb{R}^{d-1,1\vert N})$ of [[super-Minkowski spacetime]] $\mathbb{R}^{d-1,1\vert N}$, regarded as the super-translation [[supersymmetry]] [[super Lie algebra]]. In this form Fierz identities encode [[cocycles]] in the [[supersymmetry]] super-[[Lie algebra cohomology]], such as those which serve as [[higher WZW terms]] characterizing [[super p-branes]]. We follow [Castellani-D'Auria-Fré 82, section II.8](#CDF).

### Bilinear Fierz identities
  {#BilinearFierzIdentities}

Given a fixed [[real spin representation]] $N$, then the
odd [[coordinates]] $\{\theta^\alpha\}_{\alpha = 1}^{dim_{\mathbb{R}}(N) }$ of the [[super Minkowski spacetime]] [[supermanifold]]
$\mathbb{R}^{d-1,1\vert N}$ span, by construction, precisely that representation space, and hence so do the
spinorial components of the [[super vielbein]] form

$$
  \psi^\alpha = \mathbf{d}\theta^\alpha
   \;\;\;
   \in \Omega^{\bullet}_{li}(\mathbb{R}^{d-1,1\vert N})
   \simeq
    CE(\mathbb{R}^{d-1,1\vert N})
  \,,
$$

since in the construction of [[super differential forms]] on $\mathbb{R}^{d-1,1\vert N}$, the de Rham operator
$\mathbf{d}$ acts on the odd coordinates just formally, by sending the generator $\theta^\alpha$ to the new generator
named $\mathbf{d} \theta^\alpha$.

Therefore we may identify the [[spin representation]] $N$ with the [[linear span]] (over $\mathbb{R}$) of these elements

$$
  N  \simeq \langle \mathbf{d}\theta^\alpha \rangle_{\alpha = 1}^{dim_{\mathbb{R}}(N) }
  \,,
$$

were the [[spin group]] acts on the elements on the right in the defining way (see at _[[geometry of physics -- supersymmetry]]_): a spinorial rotation in a plane $\omega = \{\omega^{a b}\}$ by an angle $\alpha$ acts by

$$
  R_\omega(\psi) \coloneqq \exp(\tfrac{\alpha}{4} \omega^{a b} \Gamma_{a b} ) \psi
  \,.
$$

We may build new [[spin representations]] from this one by forming multilinear expressions in the [[super vielbein]]. For example the elements in $CE(\mathbb{R}^{d-1,1\vert N})$ of the form

$$
  \begin{aligned}
    \overline{\psi} \wedge \Gamma_a \psi
    &=
    \left(C_{\alpha \alpha'} \Gamma_a{}^{\alpha'}_{\beta}\right) \, \psi^\alpha \wedge \psi^{\beta}
    \\
    & =
    \left(C_{\alpha \alpha'} \Gamma_a{}^{\alpha'}_{\beta}\right) \, \mathbf{d}\theta^\alpha \wedge \mathbf{d}\theta^\beta
  \end{aligned}
$$

span, as the spacetime index $a$ ranges in $\{0, 1, \cdots, d-1\}$, a $d$-dimensional [[real vector space]]

$$
  \left\langle
     \,\overline{\psi} \wedge \Gamma_a \psi\,
  \right\rangle_{a = 0}^{d-1}
$$

which still carries a linear [[action]] of the [[spin group]], induced from the spin action on the $\psi$-s:

$$
  \begin{aligned}
    R_\omega(\overline{\psi} \wedge \Gamma_a \psi)
    & =
    \overline{\left( \exp(\tfrac{\alpha}{4}\omega^{a b}\Gamma_{a b} ) \psi \right)}
      \wedge \Gamma_a
    \left( \exp(\tfrac{\alpha}{4}\omega^{a b}\Gamma_{a b} \psi ) \right)
    \\
    & =
    \overline{\psi} \wedge \exp(-\tfrac{\alpha}{4} \omega^{a b} \Gamma_{a b}) \Gamma_a \exp(\tfrac{\alpha}{2}\omega^{a b} \Gamma_{a b})
     \psi
    \\
    & =
    \overline{\psi}
      \wedge
      (R_\omega(\Gamma_a))
    \psi
    \end{aligned}
  \,.
$$

Of course similarly we obtain elements

$$
  \overline{\psi} \Gamma_{a_1 \cdots a_p} \psi
$$

which, if they are non-vanishing at all, span the representation

$$
  \wedge^p \mathbb{R}^d
$$

Now observe that we may say all this more abstractly as follows:

1. the elements $(\psi \wedge \overline{\psi})^{\alpha \beta}$ span the symmetrized [[tensor product of representations]]

   $$
     \{N \otimes N\}_{sym}
       \;\simeq\;
     \langle
       \,  (\psi \wedge \overline{\psi})^\alpha{}_\beta \,
     \rangle_{\alpha,\beta = 1}^{dim_{\mathbb{R}}(N)}
   $$

1. for given $p \in \mathbb{N}$, then the elements of the form $\overline{\psi} \wedge \Gamma_{a_1 \cdots a_p} \psi$ form a [[subrepresentation]] thereof, equivalent to the vector representation $\wedge^p\mathbb{R}^{d}$

1. hence there is a [[direct sum]] decomposition

   $$
     \left\{N \otimes N\right\}_{sym}
       \;\simeq\;
      \underset{p \in \mathbb{N}}{\bigoplus}
      c_p \left(\wedge^p \mathbb{R}^d\right)
   $$

   in the [[category of representations]] of the [[spin group]], which expresses the (symmetrized) [[tensor product of representations]] of the [[Majorana spinor]] representation as a [[direct sum]] of skew-symmetrized tensor products of the vector representation.

Indeed this direct sum decomposition is exhaustive:

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

For $d \in \mathbb{N}$ and $N$ a [[Majorana spinor]] representation of $Spin(d-1,1)$, then the following identity holds:

$$
  (\psi \wedge \overline{\psi})^\alpha{}_\beta
   \;=\;
  \tfrac{1}{dim_{\mathbb{R}}(N)}
  \left(
     \left(
       \overline{\psi}\psi
     \right)
     +
     \left(
       \overline{\psi} \Gamma_a \psi
     \right)
     (\Gamma^a)^\alpha{}_\beta
     +
     \tfrac{1}{2!}
     \left(
       \overline{\psi} \Gamma_{a_1 a_2} \psi
     \right)
     (\Gamma^{a_1 a_2})^\alpha{}_\beta
      +
     \cdots
  \right)
  \,.
$$

=--

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

By the discussion there, the [[Majorana spinor]] representation is a real sub-representation of a complex _Dirac representation_ $\mathbb{C}^{(2^\nu)}$. The latter has the special property that

1. the [[Clifford algebra]] contains the full [[matrix algebra]];

1. for $p \geq 1$ the Clifford elements $\Gamma_{a_1 \cdots a_p}$ have vanishing [[trace]].

The first point implies that there exists coefficients $X^{a_1 \cdots a_p} \in \mathbb{C}$ for $p \in \mathbb{N}$ such that

$$
  \psi \wedge \overline{\psi}
    =
  \tfrac{1}{dim_{\mathbb{R}}(N)}
  \left(
    X
    +
    X^a \Gamma_a
    +
    X^{a b} \Gamma_{a b}
    +
    \cdots
  \right)
  \,.
$$

The second condition then implies that multiplying this expression with $\Gamma^{a_1 \cdots a_p}$ and taking the trace projects out the coefficient $X^{a_1 \cdots a_p}$:

$$
  \begin{aligned}
    X^{a_1 \cdots a_p}
      & =
    \frac{1}{p! dim_{\mathbb{R}}(N)}
    tr_N
    \left(
      \left(
        X
        +
        X^a \Gamma_a
        +
        X^{a b} \Gamma_{a b}
        +
        \cdots
      \right)
      \Gamma^{a_1 \cdots a_p}
   \right)
   \\
   & =
   \tfrac{1}{p!}
   tr_N
   \left(
     \psi \wedge \overline{\psi} \, \Gamma^{a_1 \cdots a_p}
   \right)
   \\
   & =
   \tfrac{1}{p!}
   \left(
     \overline \psi \wedge \Gamma^{a_1 \cdots a_p} \psi
   \right)
  \end{aligned}
  \,.
$$

Notice that it is the last step, identifying the trace over $\psi \wedge \overline{\psi} \Gamma^{a_1 \cdots a_p}$ with the $\psi$-$\psi$ component of the matrix $\Gamma^{a_1 \cdots a_p}$, where we use the _symmetrization_ of the spinor tensor product, namely the identity $\psi^\alpha \wedge \overline{\psi}_\beta = \overline{\psi}_\beta \wedge \psi^\alpha$.

=--

Some of the coefficients in prop. \ref{BilinearFierzDecomposition} may vanish identically. These are the _bilinear Fierz identities_, of the form

$$
  \overline{\psi} \Gamma_{a_1 \cdots a_p} \psi = 0
  \,.
$$

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

Let $d = 11$. Write $\mathbf{32}$ or $(\tfrac{1}{2})^5$ for the [[Majorana spinor]] representation of $Spin(d-1,1)$. Then

$$
  \left\{
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
  \right\}_{sym}
     \;\simeq\;
  \underset{\simeq \mathbb{R}^d}{\underbrace{(1)^1 (0)^4}}
    \;\oplus\;
  \underset{\simeq \wedge^2 \mathbb{R}^d}{\underbrace{(1)^2 (0)^3}}
    \;\oplus\;
  \underset{\wedge^5 \mathbb{R}^d}{\underbrace{(1)^5}}
  \,.
$$

=--

([D'Auria-Fré 82b (3.1)](#DAuriaFre82b))

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

Since we know from prop. \ref{BilinearFierzDecomposition} that the right hand side has to be some direct sum of representations of the form $\wedge^p \mathbb{R}^d$, it is sufficient to check that there is only one choice of sum such that [[dimensions]] match on both sides of the equation.

Now the dimension of $\{N \otimes N\}_{sym}$ is that of the space of symmetric $32 \times 32$ matrices:

$$
  dim_{\mathbb{R}}
  \left(
     \{\mathbf{32} \otimes \mathbf{32}\}_{sym}
  \right)
   \;=\;
  \frac{1}{2}
  \left(
    32 \times 33
  \right)
   =
  528
$$

while the dimension of $\wedge^p \mathbb{R}^d$ is the [[binomial coefficient]]

$$
  dim_{\mathbb{R}}(\wedge^p \mathbb{R}^d)
   \;=\;
  \left(
    11 \atop p
  \right)
  \,.
$$

Hence the claim follows from the fact that

$$
  \begin{aligned}
    528
     & = 11 + 55 + 462
    \\
    & =
  \left(11 \atop 1\right)
   +
  \left(11 \atop 2\right)
   +
  \left(11 \atop 5\right)
  \end{aligned}
  \,.
$$

=--

### Quadrilinear Fierz identities
  {#QuadraticFierzIdentities}

Now we consider the [[direct sum]] decomposition of the [[tensor product of representations]] of _four_ copies of a [[spin representation]]. This yields the quadrilinear Fierz identities.

+-- {: .num_example #11dQuadrilinearCGCoefficients}
###### Example

The group $Spin(10,1)$ has [[rank of a Lie group|rank]] 5, and hence its [[irreducible representation|irreducible]] vector representations are labeled by [[Young diagrams]] consisting of five rows. For instance

$$
  (2)^2 (1)^2 (0)
$$

denotes the representation whose elements may be identified with tensors of the form

$$
  X_{\array{ a_1 & a_2 \\ a_3 & a_4 \\ a_5 }}
$$

which are

1. skew-symmetric in indices in the same column;

1. symmetric and trace-less in indices in the same row.

Write again $(\tfrac{1}{2})^5$ for the [[Majorana spinor]] representation. Then the following identity holds in the [[representation ring]]:

$$
  \left\{
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
  \right\}_{sym}
   \;\simeq\;
  \left.
    \array{
    (0)^5
    \\
    \oplus
    \\
    (2) (0)^4
    \\
    \oplus
    \\
    (1)^3 (0)^2 \oplus  (2)(1)(0)^3
    \\
    \oplus
    \\
    (1)^4 (0) \oplus (2)^2 (0)^3
    \\
    \oplus
     \\
    (1)^5
     \\
     \oplus
     \\
    (2)^2 (1)^3
    \\
    \oplus
    \\
    (2)^5
   }
  \right.
$$

=--

([D'Auria-Fré 82b (3.3) ](#DAuriaFre82b))

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

As before, this is supposed to follow already by matching total dimensions on both sides

$$
  \frac{32 \times 33 \times 34 \times 35}{4 \times 3 \times 2}
   \;=\;
  \left.
    \array{
    1
    \\
    +
    \\
    65
    \\
    +
    \\
    165 +  429
    \\
    +
    \\
    330 + 1144
    \\
    +
     \\
    462
     \\
     +
     \\
    17160
    \\
    +
    \\
    32604
   }
  \right.
$$

=--

More in detail we have the following decompositions, in the notation from [above](#QuadraticFierzIdentities).

\[
  \label{Fierz11dA}
  \left(\overline{\psi} \wedge \Gamma_{a_1} \psi\right)
   \wedge
  \left( \overline{\psi} \wedge \Gamma_{a_2} \psi \right)
   \;=\;
  X^{(\mathbf{65})}_{\array{a_1 \\ a_2}}
   +
  \frac{1}{11} \delta_{\array{a_1 a_2}}X^{(\mathbf{1})}
\]

Here for instance the symbol $X^{(\mathbf{65})}_{\array{a_1 \\ a_2}}$ denotes the projection of the term on the left into the direct summand given by the [[representation]] $(2)(0)^4$ of dimension $65$. Similarly:

\[
  \label{Fierz11dB}
  \left(\overline{\psi} \wedge \Gamma_{a_1 a_2} \psi\right)
    \wedge
  \left(\overline{\psi} \wedge \Gamma_{a_3}\right)
    \;=\;
  X^{(\mathbf{429})}_{\array{ a_1 & a_2 \\ a_3}}
   +
  X^{(\mathbf{165})}_{\array{a_1 a_2 a_3}}
\]

\[
  \label{Fierz11dC}
  \left(
    \overline{\psi}\Gamma_{a_1 a_2} \psi
  \right)
  \left(
    \overline{\psi} \Gamma_{a_3 a_4}
  \right)
   \;=\;
  X^{(\mathbf{1144})}_{\array{a_1 a_2 \\ a_3 a_4}}
   +
  X^{(\mathbf{330})}_{\array{a_1 a_2 a_3 a_4}}
   +
  \tfrac{4}{9}\delta_{\array{ [a_1 \\ [a_3} } X^{(\mathbf{65})}_{\array{a_2] \\ a_4] } }
  -
  \tfrac{2}{11} \delta_{\array{a_1 & a_2 \\ a_3 & a_4}} X^{(\mathbf{1})}
\]

\[
  \label{Fierz11dD}
  \left(
    \overline{\psi}
      \wedge
      \Gamma_{a_1 \cdots a_5}
    \psi
  \right)
  \wedge
  \left(
    \overline{\psi}
      \wedge
    \Gamma_{a_6}
   \psi
  \right)
  \;=\;
  \epsilon_{a_1 \cdots a_6}{}^{b_1 \cdots b_5}
  X^{(\mathbf{462})}_{b_1 \cdots b_5}
  +
  X^{(\mathbf{4290})}_{\array{a_1 & \cdots & a_5 \\ a_6}}
  +
  \frac{15}{7}
  \delta_{a_6 [ a_1} X^{(\mathbf{330})}_{\array{a_2 & \cdots & a_5}}
\]

and some more.

([D'Auria-Fré 82b table 2 ](#DAuriaFre82b))

As a corollary:

+-- {: .num_example #TheM2andM5CocyclesAsFierzIdentities}
###### Example

For $d = 11$ then

1. the following Fierz identity holds:

   $$
     \left(
       \overline{\psi} \wedge \Gamma_{a b} \psi
     \right)
     \wedge
     \left(
        \overline{\psi} \wedge \Gamma^b \psi
     \right)
     \;= \; 0
     \,.
   $$

   (this is the cocycle condition for the [[higher WZW term]] of the [[M2-brane]] ([Bergshoeff-Sezgin-Townsend 87](#BergshoeffSezginTownsend87)), [AETW 87](#AETW87))

1. the following Fierz identity holds:

   $$
     \left(
       \overline{\psi}
         \wedge
         \Gamma_{a_1 \cdots a_4 b}
       \psi
     \right)
     \wedge
     \left(
       \overline{\psi}
        \wedge
        \Gamma^{b}
        \psi
     \right)
     \;=\;
     3
     \left(
     \overline{\psi}
        \Gamma_{[a_1 a_2}
     \psi
     \right)
     \wedge
     \left(
       \overline{\psi}
          \Gamma_{a_3 a_4]}
       \psi
     \right)
   $$

   (this is the cocycle condition for the [[higher WZW term]] of the [[M5-brane]] ([BLNPST 97](#BLNPST97), [FSS 15](#FSS15))).

=--

([D'Auria-Fré 82b (3.13) and (3.28) ](#DAuriaFre82b))

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

The first identity is the result of equation (eq:Fierz11dB) after tracing over the indices $a_2$ and $a_3$. Under this trace both summands on the right of (eq:Fierz11dB) vanish: $X^{(\mathbf{429})}_{\array{ a_1 & a_2 \\ a_3}}$ because it is trace-free in indices in a column, and $X^{(\mathbf{165})}_{\array{a_1 a_2 a_3}}$ because it is skew-symmetric in all indices.

The second identity follows from taking the trace over the indices $a_5 and a_6$ in (eq:Fierz11dD) and of skew-symmetrizing over all indices in (eq:Fierz11dC). By the symmetry properties of the tensors on the right of both equations, in both cases all tensors vanish except, in both cases, the contribution proportional to $X^{(\mathbf{330})}_{[a_1 \cdots a_3]}$, which both identities share. So it only remains to check that the proportionality factor is 3, as claimed. By writing out the skew-symmetrization in the last term in (eq:Fierz11dD) one finds:

{#ComputationOfRelativeCoefficientInM5Cocycle}
$$
  \begin{aligned}
    \frac{15}{7}
    \delta^{a_1 a_6}
    \delta_{a_6 [a_1}
    X^{(\mathbf{330})}_{a_2 \cdots a_5]}
    & =
    \frac{15}{7}
    \delta^{a_1}{}_{[a_1} X^{(\mathbf{330})}_{a_2 \cdots a_5]}
    \\
    & =
    \frac{15}{7}
    \frac{1}{5!}
    \sum_{ \left\{\sigma \atop { {\text{permutation of}} \atop {\{1,\cdots , 5\}} } \right\}}
    (-1)^{\vert \sigma\vert }
    \delta^{a_1}{}_{a_{\sigma(1)}}
    X_{a_{\sigma(2)} \cdots a_{(\sigma(5))}}
    \\
    & =
    \frac{15}{7}
    \frac{1}{5!}
    \sum_{\left\{\sigma \atop { {\text{permutation of}} \atop {\{1,\cdots , 4\}} } \right\} }
    (-1)^{\vert \sigma\vert }
    \left(
      \underset{= 11}{\underbrace{\delta^{a_1}_{a_1}}}
      X^{(\mathbf{330})}_{a_{\sigma(1)}\cdots a_{\sigma(4)}}
      -
      4
      \delta^{a_1}{}_{a_{\sigma(1)}}
      X_{a_1 a_{\sigma(2)} \cdots a_{\sigma(4)}}
    \right)
   \\
   & =
    \frac{15}{7}
    (11-4)
    \frac{1}{5}
    \;
    \underset{X^{(\mathbf{330})}_{a_1\cdots a_4}}{
    \underbrace{
    \frac{1}{4!}
    \sum_{ \left\{ \sigma \atop { {\text{permutation of}} \atop {\{1,\cdots , 4\}} } \right\} }
    (-1)^{\vert \sigma\vert}
    X^{(\mathbf{330})}_{a_{\sigma(1)}\cdots a_{\sigma(4)}}
    }
    }
    \\
    & =
    3 \; X^{(\mathbf{330})}_{a_{\sigma(1)} \cdots a_{\sigma(4)}}
  \end{aligned}
$$

where we used that $X^{(\mathbf{330})}_{a_1 \cdots a_4}$ is already skew-symmetric in all indices.

=--

+-- {: .num_example #QuadraticFierzIdentitiesIn5d}
###### Example

On D=5 [[number of supersymmetries|N = 2]] [[super Minkowski spacetime]] ([[5d supergravity]]) there are quadrilinear Fierz identities of this form:

<img src="https://ncatlab.org/nlab/files/FierzIdentitiesForBraneCocyclesIn5d.png" width="600">

([Castellani-D'Auria-Fré 91 (III.5.50)](#CDF)).

=--

## Related concepts

* [[Clebsch-Gordan coefficient]]

## References

Named after [[Markus Fierz]].

The interpretation of Fierz identities as relations satisfied by [[Clebsch-Gordan coefficients]] in the [[representation ring]] of the [[spin group]] originates in

* {#DAuriaFreMainaRegge82} [[Riccardo D'Auria]], [[Pietro Fré]],  E. Maina, [[Tullio Regge]]  _A New Group Theoretical Technique for the Analysis of Bianchi Identities and Its Application to the Auxiliary Field Problem of $D=5$ Supergravity_, Annals Phys. 139 (1982) 93 ([doi:10.1016/0003-4916(82)90007-0](http://dx.doi.org/10.1016/0003-4916(82)90007-0), [spire](http://inspirehep.net/record/167640/))

where it was applied to $Spin(4,1)$ (relevant in [[5-dimensional supergravity]]).

By this method the Fierz identities for $Spin(9,1)$ (relevant in [[heterotic supergravity]] and [[type II supergravity]]) are discussed in

* {#DAuriaFre82a} [[Riccardo D'Auria]], [[Pietro Fré]], _Geometric Structure of $N=1, D=10$ and $N=4, D=4$ Super Yang-Mills Theory_, Nucl. Phys. B196 (1982) 205 ([spire](http://inspirehep.net/record/167639))

see also appendix C of

* [[José Figueroa-O'Farrill]], Emily Hackett-Jones, George Moutsopoulos, _The Killing superalgebra of ten-dimensional supergravity backgrounds_, Class.Quant.Grav.24:3291-3308,2007 ([arXiv:hep-th/0703192](http://arxiv.org/abs/hep-th/0703192))

and the Fierz identities for $Spin(10,1)$ (relevant in [[11-dimensional supergravity]]) were tabulated in

* {#DAuriaFre82b} [[Riccardo D'Auria]], [[Pietro Fré]], pages 12, 13 of _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) ([[GeometricSupergravityErrata.pdf:file]])

see also

* S. Naito, K. Osada, T. Fukui, _Fierz Identities and Invariance of Eleven-dimensional Supergravity Action_, Phys.Rev. D34 (1986) 536-552 ([spire](http://inspirehep.net/record/236376/?ln=de))

A textbook account of the representation ring method and summary of these results is in

* {#CDF} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], chapter II.8 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

See also

* C. C. Nishi, _Simple derivation of general Fierz-type identities_,  	Am. J. Phys. 73 (2005) 1160-1163 ([arXiv:hep-ph/0412245](http://arxiv.org/abs/hep-ph/0412245))

* [[Calin Lazaroiu]], Elena-Mirela Babalic, Ioana-Alexandra Coman, _The geometric algebra of Fierz identities in arbitrary dimensions and signatures_, JHEP09(2013)156 ([arXiv:1304.4403](https://arxiv.org/abs/1304.4403))

* Elena-Mirela Babalic, Ioana-Alexandra Coman, [[Calin Lazaroiu]], _A unified approach to Fierz identities_, AIP Conf. Proc. 1564, 57 (2013) ([arxiv:1303.1575](https://arxiv.org/abs/1303.1575))

From the point of view of [[division algebras and supersymmetry]]
the Fierz identities that give the vanishing of trilinear and of quadratic terms in spinors in certain dimensions are discussed in

* [[John Huerta]], section 2.4 _Division Algebras, Supersymmetry and Higher Gauge Theory_ ([arXiv:1106.3385](http://arxiv.org/abs/1106.3385))

The recognition of some Fierz identities as cocycle conditions defining the [[higher WZW terms]] of the [[super p-branes]] is due to

* {#HenneauxMezincescu85} [[Marc Henneaux]], Luca Mezincescu, _A Sigma Model Interpretation of Green-Schwarz Covariant Superstring Action_, Phys.Lett. B152 (1985) 340 ([web](http://inspirehep.net/record/15922?ln=en))

* {#BergshoeffSezginTownsend87}  [[Eric Bergshoeff]], [[Ergin Sezgin]], [[Paul Townsend]], _Supermembranes and eleven dimensional supergravity_, Phys.Lett. B189 (1987) 75-78, In [[Mike Duff]],  (ed.), _[[The World in Eleven Dimensions]]_ 69-72 ([pdf](http://streaming.ictp.trieste.it/preprints/P/87/010.pdf), [spire](http://inspirehep.net/record/248230?ln=en))

* {#AETW87} Anna Achúcarro, [[Jonathan Evans]], [[Paul Townsend]], [[David Wiltshire]], _Super $p$-Branes_, Phys. Lett. B **198** (1987) 441 ([spire](http://inspirehep.net/record/22286?ln=en))

* {#BLNPST97} [[Igor Bandos]], [[Kurt Lechner]], Alexei Nurmagambetov, [[Paolo Pasti]], [[Dmitri Sorokin]], Mario Tonin, _Covariant Action for the Super-Five-Brane of M-Theory_, Phys. Rev. Lett. 78 (1997) 4332-4334 ([arXiv:hep-th/9701149](http://arxiv.org/abs/hep-th/9701149))

* {#FSS15} [[nLab:Domenico Fiorenza]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], _[[schreiber:The WZW term of the M5-brane|The WZW term of the M5-brane and differential cohomotopy]]_, J. Math. Phys. 56, 102301 (2015) ([arXiv:1506.07557](https://arxiv.org/abs/1506.07557))

[[!redirects Fierz identities]]


figure 8 knot - SVG

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[[!redirects figure eight knot - SVG]]

category: svg


figure eight knot

The __figure-eight knot__ is a famous [[knot]].

[[!include figure 8 knot - SVG]]

+-- {: .un_remark}
###### Note
To include this picture in a page, write
<nowiki><code>[[!include figure 8 knot - SVG]]</code></nowiki>.
=--

category: knot theory
[[!redirects figure-eight knot]]
[[!redirects figure-eight knots]]
[[!redirects Figure-eight knot]]
[[!redirects Figure-eight knots]]
[[!redirects Figure-eight Knot]]
[[!redirects Figure-eight Knots]]
[[!redirects figure eight knot]]
[[!redirects figure eight knots]]
[[!redirects Figure eight knot]]
[[!redirects Figure eight knots]]
[[!redirects Figure eight Knot]]
[[!redirects Figure eight Knots]]
[[!redirects figure-8 knot]]
[[!redirects figure-8 knots]]
[[!redirects Figure-8 knot]]
[[!redirects Figure-8 knots]]
[[!redirects Figure-8 Knot]]
[[!redirects Figure-8 Knots]]
[[!redirects figure 8 knot]]
[[!redirects figure 8 knots]]
[[!redirects Figure 8 knot]]
[[!redirects Figure 8 knots]]
[[!redirects Figure 8 Knot]]
[[!redirects Figure 8 Knots]]


Filippo Bonchi

Filippo Bonchi is an Associate Professor in the Department of Computer Science of the University of Pisa.

* [publications](https://dblp.org/pers/hd/b/Bonchi:Filippo)

## Selected writings

On [[Functorial Semantics of Algebraic Theories]] generalized to [[relational theories]]:

* [[Filippo Bonchi]], [[Dusko Pavlovic]], [[Pawel Sobocinski]], _Functorial Semantics for Relational Theories_, ([arXiv:1711.08699](https://arxiv.org/abs/1711.08699))

category: people


filler

# Contents
* table of contents
{:toc}

## Fillers of commutative squares

If

$$ \array {
O_{0,1} & \overset{f_1}{\longrightarrow} & O_{1,1} \\
\downarrow h_0 & & \downarrow h_1 \\
O_{0,0} & \overset{f_0}{\longrightarrow} & O_{1,0} \\
}
$$

is a commutative diagram in a category $\mathcal{C}$, then a _filler_ (synonyms:  *diagonal fill-in*, [[lift]]) is:

* morphism $j\colon O_{0,0}\rightarrow O_{1,1}$ in $\mathcal{C}$ making both triangles created commute.
(That is, $f_0 = h_1 j$ and $f_1 = j h_0$.)

In certain contexts, the problem of whether there exists a filler in this sense is called a *lifting problem*.

If $j$ is uniquely determined (in an appropriate sense), then $h_0$ is said to be _orthogonal_ to $h_1$: see [[orthogonality]].

The concept of filler plays an important role in homotopy theory; see for example [[model category]].

## Horn fillers for simplicial sets

The term is used in the context of [[horns]] in simplicial sets and related structures. The term makes it possible to summarize the definition of a [[Kan complex]] in one sentence: a Kan complex is a simplicial set in which every horn has a filler.

Of course this is a special case of the preceding notion of filler: a horn filler for a horn $f: \Lambda_n^j \to X$ in a simplicial set $X$ is a diagonal fill-in of a commutative square

$$\array{
\Lambda_n^j & \hookrightarrow & \Delta_n \\
\mathllap{f} \downarrow & & \downarrow \\
X & \to & 1
}$$

going from $\Delta_n$ to $X$.

## Related concepts

[[factorization system]]

[[homotopy lifting property]]

[[orthogonality]]

[[!redirects fill-in]]
[[!redirects diagonal fill-in]]


filter

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

\tableofcontents

## Idea

A _filter_ is [[duality|dual]] to an [[ideal]].

A [[proper filter]] is equivalently the [[eventuality filter]] of a [[net]].

Filters of subsets form a [[category of filters|category]]
whose [[situs|simplicial category]] provides a somewhat more
formalisation of the intuition of "nearness" than the usual topological one; in particular, it contains the categories of topological and of uniform spaces, of simplicial sets, and of filters themselves, allowing to reformulate in terms of this category notions
such as limit, equicontinuity, locally trivial, and geometric realization.

In [[predicative mathematics]], filters of subsets are [[large category|large]], but [[locally small category|locally small]].

## Definitions

A subset $F$ of a [[partial order|poset]] $L$ is called a __filter__ if it is upward-closed and downward-[[direction|directed]]; that is:

1. If $A \leq B$ in $L$ and $A \in F$, then $B \in F$;
2. for some $A$ in $L$, $A \in F$;
3. if $A \in F$ and $B \in F$, then for some $C \in F$, $C \leq A$ and $C \leq B$.

Equivalently, a [[predicate]] $F:L \to \Omega$ is a __filter__

1. If for all $A \in L$ and $B \in L$, if $A \leq B$ and $F(A)$, then $F(B)$;
2. There exists an $A \in L$, such that $F(A)$;
3. if there exist $A \in L$ and $B \in L$ such that $F(A)$ and $F(B)$, then there is a $C \in L$ such that $F(C)$, $C \leq A$ and $C \leq B$.

One could also use a $\mathbb{N}$-overt [[dominance]] $\Sigma$, a sub-[[sigma-frame|$\sigma$-frame]] $\Sigma \subseteq \Omega$. A [[predicate]] $F:L \to \Sigma$ is a __filter__

1. If for all $A \in L$ and $B \in L$, if $A \leq B$ and $F(A)$, then $F(B)$;
2. There exists an $A \in L$, such that $F(A)$;
3. if there exist $A \in L$ and $B \in L$ such that $F(A)$ and $F(B)$, then there is a $C \in L$ such that $F(C)$, $C \leq A$ and $C \leq B$.

Sometimes the term 'filter' is used for an [[upper set]], that is any set satisfying axiom (1).  (Ultimately this connects with the use of '[[ideal in a monoid|ideal]]' in [[monoid]] theory.)

In a [[lattice]], one can use these alternative axioms:

1. If $A \in F$ and $B$ in $L$, then $A \vee B \in F$;
2. $\top \in F$;
3. if $A \in F$ and $B \in F$, then $A \wedge B \in F$.

Here, (1) is equivalent to the previous version; the others, which here say that the lattice is closed under finite [[meets]], are equivalent given (1).  (These axioms look more like the axioms for an ideal of a ring.)

You can also interpret these axioms to say that, if you think of $F$ as a function from $L$ to the set $TV$ of [[truth values]], then $F$ is a homomorphism of meet-semilattices.

### Filter of subsets

A __filter of subsets__ of a given set $S$ is a filter in the [[power set]] of $S$.  One also sees filters of open subsets, filters of compact subsets, etc, especially in topology.

### In dependent type theory

In [[dependent type theory]], let $(\Omega, \mathrm{El}_\Omega)$ be the [[type of all propositions]] with its type reflector type family. Given a type $T$, the type of all subsets of $T$ is given by the function type $T \to \Omega$. We define the [[relation]] $x:T, U:T \to \Omega \vdash x \in F$ by
$$x \in F \coloneqq \mathrm{El}_{\Omega}(F(x))$$

Let $(P, \leq)$ be a [[poset]]. Then a filter on $P$ is a [[subtype]] $F:P \to \Omega$ with dependent functions

$$p:\prod_{x:P} \prod_{y:P} (x \leq y) \times (x \in F) \to (y \in F)$$

$$q:\left[\sum_{x:P} (x \in F)\right]$$

$$r:\left[\sum_{x:P} \sum_{y:P} (x \in F) \times (y \in F) \to \left[\sum_{z:P} (z \in F) \times (z \leq x) \times (z \leq y)\right]\right]$$

The filter properties could also be turned into structure:

$$p:\prod_{x:P} \prod_{y:P} (x \leq y) \times (x \in F) \to (y \in F)$$

$$q:\sum_{x:P} (x \in F)$$

$$r:\sum_{x:P} \sum_{y:P} (x \in F) \times (y \in F) \to \sum_{z:P} (z \in F) \times (z \leq x) \times (z \leq y)$$

## Kinds of filters

A filter $F$ is __proper__ if there exists an element $A$ of $L$ such that $A \notin F$.  A filter in a lattice is proper iff $\bot \notin F$; in particular, a filter of subsets of $S$ is proper iff $\empty \notin F$.  In [[constructive mathematics]], however, one usually wants a stronger (but classically equivalent) notion: a filter $F$ of subsets of $S$ is __proper__ if every element of $F$ is [[inhabited set|inhabited]].  If $A \in F$ for every $A$ (in particular if $\empty \in F$), then we have the __[[improper filter]]__.  Compare [[proper subset]] and [[improper subset]].

Filters are often assumed to be proper by default in analysis and topology, where proper filters correspond to [[nets]].  However, we will try to remember to include the adjective 'proper'.

If the complement of a filter is an ideal, then we say that the filter is __prime__ (and equivalently that the ideal is prime).  A prime filter is necessarily proper; a proper filter in a lattice is prime iff, whenever $A \vee B \in F$, either $A \in F$ or $B \in F$.  In other words, $F: L \to TV$ must be a homomorphism of lattices.  The generalisation to arbitrary joins gives a [[completely prime filter]].

A filter is an __[[ultrafilter]]__, or __maximal filter__, if it is maximal among the proper filters.  (See that article for alternative formulations and applications.)  In a distributive lattice, every ultrafilter is prime; the converse holds in a [[Boolean algebra|Boolean lattice]].  In this case, we can say that $F: L \to TV$ is a homomorphism of Boolean lattices.

Given an element $x$ of $S$, the __principal ultrafilter__ (of subsets of $S$) at $x$ consists of every subset of $S$ to which $x$ belongs.  A principal ultrafilter is also called a __fixed ultrafilter__; more generally, a filter of subsets is __fixed__ if its [[intersection]] is [[inhabited set|inhabited]].  In contrast, if $F$ is an filter whose meet (of all elements) exists and is a [[bottom]] element (the [[empty set]] for a filter of subsets), then we call $F$ __free__.

Free ultrafilters on Boolean algebras are important in [[nonstandard analysis]] and [[model theory]].

## Filterbases

A subset $F$ of a lattice $L$ is a __filter[[base]]__ if it becomes a filter when closed under axiom (1).  Equivalently, a filterbase is any downward-directed subset.  Any subset of a meet-semilattice may be used as a filter __[[subbase]]__; form a filterbase by closing under finite meets.

A filterbase $F$ of sets is proper (that is, it generates a proper filter of sets) iff each set in $F$ is inhabited.  A filter subbase of sets is proper iff it satisfies the finite intersection property (well known in topology from a criterion for [[compact spaces]]): every finite collection from the subfilter has inhabited intersection.

## The poset of filters and push-forwards

If $f:L_1\to L_2$ is a [[monotone map]] and $F\subseteq L_1$ a filter, then $\lbrace f(A) \mid A\in F\rbrace\subseteq L_2$ is a filter base. Let $f_\ast(F)$ be the filter generated by this filter base.
The set of filters $Filters(L)$ is a poset in its own right w.r.t. inclusion and $f_\ast: (Filters(L_1),\subseteq)\to(Filters(L_2),\subseteq)$ is monotone. Therefore $L\mapsto Filters(L), f\mapsto f_\ast$ is a functor from the category of posets to itself.

If $f$ satisfies stronger properties than mere monotony, then $f_\ast$ will be better behaved as well:

* If $L$ is a meet-[[semilattice]], then $Filters(L)$ is a complete join-semilattice: $\bigvee_{i\in I} F_i = \left\langle \bigwedge_{j\in J} A_j \mid J\subseteq I\text{ finite}, A_j\in F_j\right\rangle$. If $f:L_1\to L_2$ respects finite meets, then $f_\ast$ respects all joins. If $f$ is merely monotone, then $f_\ast$ respects all filtered joins.
* If $L$ has all $|I|$-fold joins for some indexing set $I$, then $Filters(L)$ has $|I|$-fold meets: $\bigwedge_{i\in I} F_i = \lbrace \bigvee_{i\in I} A_i \mid A_i\in F_i\rbrace$. If $f$ respects $|I|$-fold joins, then $f_\ast$ respects $|I|$-fold meets.
* If $L$ is a [[distributive lattice]], then $Filters(L)$ is a [[frame]], i.e. the infinite distributive law $F \wedge \bigvee_{i\in I} G_i = \bigvee_{i\in I} (F\wedge G_i)$ holds. The other distributive law also holds for finite meets: $F \vee \bigwedge_{i\in I} G_i = \bigwedge_{i\in I} (F\vee G_i)$ if $I$ is finite. It holds more generally if $L$ has all $|I|$-fold meets, the $|I|$-fold distributive law $A\wedge \bigvee_{i\in I} B_i = \bigvee_{i\in I} (A\wedge B_i)$ holds in $L$ and $F$ is closed under $|I|$-fold meets.

* If $L$ is a complete join-semilattice, then $Filters(L)$ is a complete lattice. If $f$ respects all joins, then $f_\ast$ respects all meets. In this case $f_\ast$ has a [[left adjoint]] $f^\ast: L_2\to L_1$ which is given by $f^\ast(G) = \bigwedge \lbrace F | G\subseteq f_\ast(F)\rbrace$ so that $f^\ast(G)\subseteq F \iff G\subseteq f_\ast(F)$ holds. As a left adjoint $f^\ast$ respects all joins.

* If $(f,g)$ is an adjoint pair between $L_1$ and $L_2$, that is f$: L_1 \to L_2$, $g: L_2\to L_1$ monotone and $f(x) \leq y \iff x\leq g(y)$ for all $x\in L_1$, $y\in L_2$, then $(g_\ast,f_\ast)$ is an adjoint pair between $Filters(L_2)$ and $Filters(L_1)$. Note that the push-forward turns left adjoints into right adjoint and vice versa.

### Examples

* If $L_1=(P(X),\subseteq), L_2=(P(Y),\subseteq)$ for some sets $X,Y$ and $\alpha:X\to Y$ is any map, then $f:L_1\to L_2, A\mapsto \alpha(A)$ is monotone and respects arbitrary joins. Therefore $f_\ast: Filters(L_1)\to Filters(L_2)$ respects arbitrary meets. Note that even though $L_1, L_2$ are meet-semilattices, $f$ does not respect meets in general, because $\alpha(A\cap A')\subseteq \alpha(A)\cap\alpha(A')$ holds, but equality is equivalent to $\alpha$ being injective.
* $f$ is the left adjoint of $g: L_2\to L_1, B\mapsto \alpha^{-1}(B)$. Therefore $f_\ast$ is the right adjoint of $f^\ast=g_\ast$ which is given by taking preimage filters: $g_\ast(G) = \langle \alpha^{-1}(B) | B\in G\rangle$.
* $g$ has not only the left adjoint $f$, but also a right adjoint $h:L_1\to L_2$. It can be described as follows: Call a set $A\subseteq X$ is called $\alpha$-saturated if it is the union of fibers of $\alpha$, i.e. if $\alpha^{-1}(\alpha(A)) = A$. There is a biggest $\alpha$-saturated subset $A_\alpha$ of every $A$. The morphism $h$ is given by $h(A) := \alpha(A_\alpha) \cup (Y\setminus \alpha(X))$.
* Therefore $g_\ast$ not only preserves all joins but all meets as well.

## Application to analysis and topology

Every [[net]] $\nu: I \to S$ defines an __[[eventuality filter]]__ $E_\nu$: let $A$ belong to $E_\nu$ if, for some index $k$, for every $l \geq k$, $\nu_l \in A$.  (That is, $\nu$ is eventually in $A$.)  Note that $E_\nu$ is proper; conversely, any proper filter $F$ has a net whose eventuality filter is $F$ (as described at [[net]]).  Everything below can be done for nets as well as for (proper) filters, but filters often lead to a cleaner theory.

In a [[topological space]] $S$, a filter $F$ on $S$ __[[convergence|converges]]__ to a point $x$ of $S$ if every [[neighbourhood]] of $x$ belongs to $F$.  A filter $F$ __clusters__ at a point $x$ if every neighbourhood of $x$ intersects every element of $F$.  With these definitions, the improper filter converges to every point and clusters at no point; a proper filter, however, clusters at every point that it converges to.

The concepts of continuous function and such conditions as [[compact space|compactness]] and [[Hausdorff space|Hausdorffness]] may be defined quite nicely in terms of the convergence relation.  In fact, everything about topological spaces may be defined in terms of the convergence relation, although not always nicely.  This is because topological spaces form a [[full subcategory]] of the category of [[convergence spaces]], where the convergence relation is the fundamental concept.  More details are there.

In a [[metric space]] $S$, a filter $F$ on $S$ is __Cauchy__ if it has elements of arbitrarily small diameter.  Then a [[sequence]] is a [[Cauchy sequence]] iff its eventuality filter is Cauchy.  (This can be generalised to [[uniform spaces]].)  The concept of completion of a metric space may be defined quite nicely in terms of the Cauchy filters, although not every property (not even every uniform property) of metric spaces can be defined in this way.  As for convergence, there is a general notion of [[Cauchy space]], but the forgetful functors from metric and uniform spaces are now not full.

## Related concepts

* [[lower set]]/[[(0,1)-presheaf]], [[ideal]]/[[(0,1)-sheaf]]

* [[upper set]]/[[(0,1)-copresheaf]], **filter**/[[(0,1)-cosheaf]]

* [[category of filters]]

* [[filter of a ring]]

## References

* [Wikipedia](https://secure.wikimedia.org/wikipedia/en/wiki/Filter_%28mathematics%29).
* [[Peter Johnstone]] (1982); _[[Stone Spaces]]_; Cambridge University Press. ISBN 0-521-23893-5.

[[!redirects filters]]

[[!redirects filter of subsets]]
[[!redirects filters of subsets]]

[[!redirects proper filter]]
[[!redirects proper filters]]
[[!redirects prime filter]]
[[!redirects prime filters]]
[[!redirects fixed filter]]
[[!redirects fixed filters]]
[[!redirects free filter]]
[[!redirects free filters]]

[[!redirects filterbase]]
[[!redirects filterbases]]
[[!redirects filter base]]
[[!redirects filter bases]]
[[!redirects filter subbase]]
[[!redirects filter subbases]]


filter monad

Let $SemiLat$ be the category of meet-[[semilattices]] (i.e., [[posets]] admitting finite meets); equivalently, the category of idempotent commutative monoids. Regarding [[Sierpinski space]] $\mathbf{2} = \{0 \leq 1\}$ ambimorphically either as a [[topological space]] or as a meet-semilattice, i.e., an internal idempotent commutative monoid in $Top$ seen as a [[dualizing object]], there is a pair of [[adjoint functors]] given by homming into $\mathbf{2}$,

$$Top \stackrel{Top(-, \mathbf{2})^{op}}{\to} SemiLat^{op} \;\; \dashv \;\; SemiLat^{op} \stackrel{SemiLat(-, \mathbf{2})}{\to} Top.$$

In more detail: for a space $X$, the set of continuous maps $X \to \mathbf{2}$ is naturally identified with the set of open subsets, i.e., the topology $\mathcal{O}(X)$ (where the pointwise meet-semilattice structure on $\hom(X, \mathbf{2})$ coincides with the meets defined by intersection on $\mathcal{O}(X)$). For a meet-semilattice $L$, the set $SemiLat(L, \mathbf{2})$ is topologized as a subspace of the product space $\mathbf{2}^{{|L|}}$, a product of ${|L|}$ copies of Sierpinski space. The adjunction says that there is a natural bijection

$$\frac{L \to Top(X, \mathbf{2})}{X \to SemiLat(L, \mathbf{2})}$$

between semilattice maps above and continuous maps below.

The **filter monad** is the monad $Filt$ on $Top$ induced by this adjunction. (Compare [[ultrafilter monad]].)

For a space $X$, $Filt(X)$ is the set of meet-preserving maps $\mathcal{O}(X) \to \mathbf{2}$. This is in natural bijection with subsets of $\mathcal{O}(X)$ that are upward-closed and closed under finite intersections, i.e., [[filters]] in $\mathcal{O}(X)$. The unit of the monad evaluated at $X$ is $u_X: X \to Filt(X)$, taking $x \in X$ to the filter $u_X(x)$ of open sets containing $x$.

We can read off the multiplication $m$ on the filter monad directly from this description. For $\Phi \in Filt(Filt(X))$, we have

$$\mu_X(\Phi) \coloneqq \{U:\mathcal{O}(X)\; |\; \{F: Filt(X)\; |\; U \in F\} \in \Phi\}.$$

However, there is another description often encountered in the literature:

$$\mu_X(\Phi) = \bigcup \{\bigcap \mathcal{U}: \mathcal{U} \in \Phi\}.$$

+-- {: .proof}
###### Proof
To see $\bigcup \{\bigcap \mathcal{U}: \mathcal{U} \in \Phi\} \subseteq \mu_X(\Phi)$, suppose $U \in \bigcup \{\bigcap \mathcal{U}: \mathcal{U} \in \Phi\}$, i.e., suppose there is $\mathcal{U}$ with $\mathcal{U} \in \Phi$ and $U \in \bigcap \mathcal{U}$. Then $U \in F$ for all $F \in \mathcal{U}$, and since
already $\mathcal{U} \in \Phi$, we see $\{F \in Filt(X): U \in F\} \in \Phi$ by upward closure of $\Phi$. Hence $U \in \mu_X(\Phi)$.

To see $\mu_X(\Phi) \subseteq \bigcup \{\bigcap \mathcal{U}: \mathcal{U} \in \Phi\}$, suppose $U \in \mu_X(\Phi)$, and then put $\mathcal{U}' = \{F: Filt(X)\; |\; U \in F\}$. Then $\mathcal{U}' \in \Phi$, and $U \in \bigcap \mathcal{U}'$ by definition, so $U \in \bigcup \{\bigcap \mathcal{U}: \mathcal{U} \in \Phi\}$.
=--


filter of a ring

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## Definition

Given a [[ring]] $R$, a **left filter** of $R$ is a [[multiplicative subset]] $(F, 1, \cdot)$ (here we assume that multiplicative subsets have the [[unit]] and are thus [[submonoid]]s of $R$) of $(R, 1, \cdot)$, with [[monoid]] [[monomorphism]] $i:F \hookrightarrow R$, such that for all elements $z \in F$ and $y \in R$, if there is an element $w \in R$ such that $i(z) = w \cdot y$, then there is an element $x \in F$ such that $i(z) = i(x) \cdot y$.

A **right filter** of $R$ is a [[multiplicative subset]] $(F, 1, \cdot)$ of $(R, 1, \cdot)$, with [[monoid]] [[monomorphism]] $i:F \hookrightarrow R$, such that for all elements $z \in F$ and $y \in R$, if there is an element $w \in R$ such that $i(z) = y \cdot w$, then there is an element $x \in F$ such that $i(z) = y \cdot i(x)$.

A **two-sided filter** is a [[multiplicative subset]] $(F, 1, \cdot)$ of $(R, 1, \cdot)$ with [[monoid]] [[monomorphism]] $i:F \hookrightarrow R$ that is both a left filter and a right filter.

## See also

* [[multiplicative subset]]
* [[localization of a ring]]
* [[ideal of a ring]]

## References

* [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))


filter space

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## Idea

A _filter space_ is a generalisation of a [[topological space]] based on the concept of [[convergence]] of [[filters]] (or [[nets]]) as fundamental.  The [[category]] of filter spaces is a [[quasitopos]] and may be thought of as a [[nice category of spaces]] that includes [[Top]] as a [[full subcategory]]. Filter spaces include [[convergence spaces]], [[Choquet spaces]], and [[Kuratowski limit spaces]] as [[full subcategory|full sub]]-[[quasitoposes]].

## Definitions

A __filter space__ is a [[set]] $S$ together with a [[relation]] $\to$ from $\mathcal{F}S$ to $S$, where $\mathcal{F}S$ is the set of [[filters]] on $S$; if $F \to x$, we say that $F$ __converges__ to $x$ or that $x$ is a __limit__ of $F$.  This must satisfy some axioms:

1. Centred: The [[principal ultrafilter]] $F_x = \{ A \;|\; x \in A \}$ at $x$ converges to $x$;
2. Isotone: If $F \subseteq G$ and $F \to x$, then $G \to x$;

The definition can also be phrased in terms of [[nets]]; a net $\nu$ converges to $x$ if and only if its [[eventuality filter]] converges to $x$.

The morphisms of filter spaces are the [[continuous functions]]; a function $f$ between filter spaces is __continuous__ if $F \to x$ implies that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the [[filterbase]] $\{f(A) \;|\; A \in F\}$.  In this way, filter spaces form a [[concrete category]] $Filt$, which is a [[quasitopos]].

A filter space that satisfies an additional directedness criterion is precisely a [[convergence space]]; see there for a variety of intermediate notions leading up to ordinary [[topological spaces]].

## References

* [[Martin Hyland]], _Filter spaces and continuous functionals_, 1979 [PDF](https://www.dpmms.cam.ac.uk/~martin/Research/Oldpapers/filter79.pdf)

[[!redirects filter spaces]]


filtered (infinity,1)-category

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## Idea

This is the analog of a [[filtered category]] in the context of [[(∞,1)-categories]].

The main purpose of considering filtered (∞,1)-categories is to define [[filtered (∞,1)-colimits]], which are the colimits that commute with [[finite (∞,1)-limits]].

## Definition

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

Let $\kappa$ be a [[regular cardinal]], and let $C\in \sSet$ be an (∞,1)-category, incarnated as a [[quasicategory]].

$C$ is called **$\kappa$-filtered** if for all $\kappa$-small $K\in\sSet$ and every morphism $f\colon K\to C$ there is a morphism $\hat p\colon \rcone(K)\to C$ extending $f$, where $\rcone(K)$ denotes the (right) [[cone]] of the simplicial set $K$.  $C$ is called **filtered** if it is $\omega$-filtered.

=--

## Properties

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

An [[(∞,1)-category]] $K$ is filtered precisely if [[(∞,1)-colimits]] of shape $K$ in [[∞Grpd]] commute with all [[finite (∞,1)-limits]], hence if

$$
  {\lim_\to} : Func(K, \infty Grpd) \to \infty Grpd
$$

is a left [[exact (∞,1)-functor]].

=--

This is [[Higher Topos Theory|HTT, prop. 5.3.3.3]].

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

A filtered $(\infty,1)$-category is in particular a [[sifted (∞,1)-category]].

=--

This appears as ([Lurie, prop. 5.3.1.20](#Lurie)). Since [[sifted (∞,1)-colimits]] are precisely those that commute with finite [[products]], this is a direct reflection of the fact that finite products are a special kind of [[finite (∞,1)-limits]].

+-- {: .num_cor}
###### Corollary

For $C$ a filtered $(\infty,1)$-category, the [[diagonal]] [[(∞,1)-functor]] $\Delta : C \to C \times C$ if a [[cofinal (∞,1)-functor]].

=--

+-- {: .un_prop}
###### Proposition
A filtered $(\infty,1)$-category is weakly [[contractible]], i.e. when incarnated as a [[quasicategory]], it is [[weak homotopy equivalence|weakly equivalent]] to a [[point]] in the [[Kan-Quillen model structure]] on simplicial sets.
=--
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###### Proof
This is ([Lurie, Lemma 5.3.1.18](#Lurie)).
=--

## Related concepts

* [[sifted category]], [[sifted colimit]], [[sifted (∞,1)-category]], [[sifted (∞,1)-colimit]]

* [[directed set]], [[filtered category]], **filtered (∞,1)-category**

* [[filtered homotopy colimit]]

## Reference

Section 5.3.1 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_
 {#Lurie}

[[!redirects filtered quasicategory]]
[[!redirects filtered (infinity,1)-categories]]
[[!redirects filtered (∞,1)-category]]
[[!redirects filtered (∞,1)-categories]]

[[!redirects cofiltered (∞,1)-category]]
[[!redirects cofiltered (∞,1)-categories]]

[[!redirects filtered (∞,1)-colimit]]
[[!redirects filtered (infinity,1)-colimit]]

[[!redirects filtered (∞,1)-colimits]]
[[!redirects filtered (infinity,1)-colimits]]


filtered category

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## Idea

The concept of _filtered category_ is a [[categorification]] of the concept of [[directed set]]:  In addition to having an [[upper bound]] (but not necessarily a [[coproduct]]) for every [[pair]] of [[objects]], there must also be an upper bound (but not necessarily a [[coequaliser]]) for every pair of [[parallel morphisms]].

A [[diagram]] $F \colon D\to C$ where $D$ is a filtered category is called a **filtered diagram**.  A [[colimit]] of a filtered diagram is called a **[[filtered colimit]]**.

The dual notion of *filtered category* is that of  *[[cofiltered category]]*: a category whose [[opposite category|opposite]] is filtered.

## Definitions

### Ordinary filteredness

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

A **(finitely) filtered category**  is a [[category]] $C$ in which every [[finite category|finite]] [[diagram]] has a [[cocone]].

=--

More in details, this requirement is that: For any finite category $D$ and any [[functor]] $F \colon D\to C$, there exists an object $c\in C$ and a [[natural transformation]] $F\to \Delta c$ where $\Delta c:D\to C$ is the constant diagram at $c$. If $D^+$ is the result of freely adjoining a terminal object to a category $D$, then the condition is the same as that any functor $F: D \to C$ with finite domain admits an extension $\tilde{F}: D^+ \to C$.

Equivalently, filtered categories can be characterized as those categories where, for every finite diagram $J$, the [[diagonal functor]] $\Delta \colon C \to C^J$ is [[final functor|final]]. This point of view can be generalized to other kinds of categories whose colimits are well-behaved with respect to a type of limit, such as [[sifted categories]].

All this may be rephrased in more elementary terms by saying that:

* There exists an object of $C$  (the case when $D=\emptyset$)

* For any two objects $c_1,c_2\in C$, there exists an object $c_3\in C$ and morphisms $c_1\to c_3$ and $c_2\to c_3$.

* For any two [[parallel morphisms]] $f,g:c_1\to c_2$ in $C$, there exists a morphism $h:c_2\to c_3$ such that $h f = h g$.

Just as all finite [[colimit|colimits]] can be constructed from initial objects, binary coproducts, and coequalizers, so a cocone on any finite diagram can be constructed from these three.

In [[constructive mathematics]], the elementary rephrasing above is equivalent to every [[Bishop-finite]] diagram admitting a cocone.

### Higher filteredness

More generally, if $\kappa$ is an infinite [[regular cardinal]] (or an [[arity class]]), then a **$\kappa$-filtered category** is one such that any diagram $D\to C$ has a cocone when $D$ has $\lt \kappa$ arrows, or equivalently that any functor $F: D \to C$ whose domain has fewer than $\kappa$ morphisms admits an extension $\tilde{F}: D^+ \to C$. The usual filtered categories are then the case $\kappa = \omega$, i.e., where the $D$ have fewer than $\omega$ morphisms (in other words are finite).  (We could also say in this case "$\aleph_0$-filtered", but $\omega$-filtered is more usual in the literature.)

Note that a [[preorder]] is $\kappa$-filtered as a category just when it is $\kappa$-[[direction|directed]] as a preorder.

In [ABLR](#ABLR), they use the term **$\infty$-filtered** for a category that is $\kappa$-filtered for every cardinal $\kappa$. Thus, an $\infty$-filtered category is equivalently one in which every small diagram admits a cocone.

### Generalized filteredness

Even more generally, if $\mathcal{J}$ is a class of small categories, a category $C$ is called **$\mathcal{J}$-filtered** if $C$-colimits commute with $\mathcal{J}$-limits in [[Set]].  When $\mathcal{J}$ is the class of all $\kappa$-small categories for an infinite regular cardinal $\kappa$, then $\mathcal{J}$-filteredness is the same as $\kappa$-filteredness as defined above.  See [ABLR](#ABLR).

If $\mathcal{J}$ is the class consisting of the [[terminal category]] and the [[empty category]] --- which is to say, the class of $\kappa$-small categories when $\kappa$ is the finite regular cardinal $2$ --- then being $\mathcal{J}$-filtered in this sense is equivalent to being [[connected category|connected]].  Note that this is not what the explicit definition given above for infinite regular cardinals would specialize to by simply setting $\kappa=2$ (that would be simply [[inhabited set|inhabitation]]).

## Examples

* A filtered [[preorder]] is the same as a [[direction|directed]] one: a **filtered [[(0,1)-category]]**.

* Every category with a [[terminal object]] is filtered.

* Every category which has finite [[colimit]]s is filtered.

* A [[product]] of filtered categories is filtered.

## Related concepts

* [[sifted category]], [[sifted (∞,1)-category]]

* [[directed set]], **filtered category**, [[filtered (∞,1)-category]]

* [[direct category]]

* [[filtered colimit]]

## References

* {#ABLR} [[Jiří Adámek]], [[Francis Borceux]], [[Stephen Lack]], and [[Jiří Rosický]], _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]

[[!redirects filtered categories]]
[[!redirects filtrant category]]
[[!redirects filtrant categories]]
[[!redirects filtered diagram]]
[[!redirects filtered diagrams]]


filtered chain complex

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#### Homological algebra
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#Contents#
* table of contents
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## Definition

A _filtered chain complex_ is a [[filtered object]] in a [[category of chain complexes]].

## Examples

* [[Hodge filtration]]

## Related concepts

[[!include filtered objects -- contents]]

## References

* [[Ross Street]], _Homotopy classification of filtered complexes_, J. Australian Math. Soc. 15 (1973) 298-318 ( and  in his thesis:[pdf](http://maths.mq.edu.au/~street/Thesis.pdf))

[[!redirects filtered chain complexes]]

[[!redirects filtered cochain complex]]
[[!redirects filtered cochain complexes]]

[[!redirects filtered complex]]
[[!redirects filtered complexes]]