local Langlands conjecture

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## Idea

The _local Langlands conjectures_ are certain [[conjectures]] in the context of the [[Langlands program]]. Where the genuine [[Langlands correspondence]] concerns [[global fields]], the local Langlands correspondence concerns [[local fields]].

In the case of the [[algebraic group]] $GL_n$ (the [[general linear group]]), the local Langlands conjectures assert a correspondence between:

1. F-semisimple Weil-Deligne [[representations]] (see Def. \ref{WeilDeligneRepresentation} below) of the [[Weil group]] of a  [[local field]] $F$;

1. [[irrep|irreducible]] admissible [[representations]] of $GL_n(F)$ (see Def. \ref{IrreducibleAndAdmissibleSubgroups} below),

generalizing  local [[class field theory]] from [[abelian group|abelian]] [[Galois groups]] to non-abelian Galois groups.

More generally, it states that for a [[local field]] $F$ and a [[reductive group]] $G$, the [[isomorphism classes]] of irreducible admissible representations of $G(F)$ are partitioned into _L-packets_ by the equivalence classes of [[L-parameters]].

## Definitions:

We discuss some of the basic definitions, see also [Tate 77](#Tate77).

Throughout:

* $F$ is a [[p-adic field]] with $\kappa$ denoting its [[residue field]];

* $G$ is a [[reductive group]].

\begin{definition}\label{IrreducibleAndAdmissibleSubgroups}
**([[irreducible representations|irreducible]] and admissible representations)**
A [[linear representation]] of $G(F)$ on a [[vector space]] $V$ is called:

*  *[[irrep|irreducible]]* if the only $G(F)$-[[invariant subspaces]] are $0$ and $V$.

* *admissible* if for any [[open subspace|open]] [[subgroup]] $U$ of $G(F)$ the [[fixed subspace]] $V^{U}$ is [[finite-dimensional vector space|finite-dimensional]].

\end{definition}

\begin{definition}\label{WeilGroup}
**([[Weil group]])**\linebreak
 The *[[Weil group]]* $W_F$ is the subgroup of the [[Galois group]] $\mathrm{Gal}(\overline{F}/F)$ defined as the [[inverse image]] of [[Frobenius automorphisms]] $\mathrm{Frob}^{\mathbb{Z}}\subset \mathrm{Gal}(\overline{\kappa}/\kappa)$ under the [[surjective]] map $\mathrm{Gal}(\overline{F}/F)\to\mathrm{Gal}(\overline{\kappa}/\kappa)$.
\end{definition}

\begin{definition}\label{WeilDeligneRepresentation}
**([[Weil-Deligne representation]])**\linebreak
A *[[Weil-Deligne representation]]* is a pair $(\rho_{0},N)$ where $\rho_{0}:W_{F}\to GL_{n}(\mathbb{C})$ is a representation of the [[Weil group ]] (Def. \ref{WeilGroup}) and $N$ is a nilpotent _monodromy operator_ satisfying $\rho_{0}(\sigma)N\rho_{0}(\sigma)^{-1}=\left\Vert\sigma\right\Vert N$ for all $\sigma\in W_{F}$, where $\Vert\sigma\Vert$ is the [[valuation]] of the corresponding element of $F^{\times}$ under the isomorphism of local [[class field theory]].
\end{definition}

## Examples

### Local Langlands correspondence for $GL_1$

Let $F$ be a $p$-adic field. The only irreducible admissible representations of $GL_1(F)$ ($=F^\times$) are continuous group homomorphisms $\chi:F^\times\to\mathbb{C}^{\times}$. The $1$-dimensional Weil-Deligne representations $(\rho_{0},N)$ of $W_F$ must have monodromy operator $N=0$ and must factor through the abelianization $W_F^{\mathrm{ab}}$. The local Langlands correspondence in this case is the same as local class field theory, and sends $\chi$ to the Weil-Deligne representation $(\rho_{0},0)$, where $\rho_{0}$ is the composition $W_{F}\to W_{F}^{\mathrm{ab}}\xrightarrow{\mathrm{rec}^{-1}}F^{\times}\xrightarrow{\chi}\mathbb{C}^{\times}$, with $\mathrm{rec}:F^{\times}\xrightarrow{\sim}W_{F}^{\mathrm{ab}}$ being the Artin reciprocity map of local [[class field theory]].

### Local Langlands correspondence for $GL_2$

\begin{definition}

Let $F$ be a $p$-adic field, $p\neq 2$. Given continuous admissible characters $\chi_{1}$ and $\chi_{2}$, we define $I(\chi_{1},\chi_{2})$ to be the vector space of function $\phi:GL_{2}(F)\to \mathbb{C}$ which are [[locally constant function|locally constant]] with respect to the $p$-adic topology on $GL_{2}(F)$ and satisfying

$$\phi\left(\begin{pmatrix}a & b \\0 & d\end{pmatrix}g\right)=\chi_{1}(a)\chi_{2}(d)\Vert a/d\Vert^{1/2}\phi(g).$$

Letting $GL_2(F)$ act on this by translation gives us a representation of $GL_2(F)$. This representation is said to belong to the *principal series*.

\end{definition}

Whether a principal series is irreducible or not is governed by the following theorem:

\begin{theorem}[Bernstein-Zelevinsky]

If $\chi_{1}/\chi_{2}\neq\Vert\cdot\Vert^{\pm 1}$ then $I(\chi_{1},\chi_{2})$ is irreducible.

If $\chi_{1}/\chi_{2}=\Vert\cdot\Vert^{1}$ then we have an exact sequence
$$0\to\rho\to I(\chi_{1},\chi_{2})\to S(\chi_{1},\chi_{2})\to 0$$
where $\rho$ is the $1$-dimensional representation of $GL_2(F)$ given by $\chi_{1}\Vert\cdot\Vert^{1/2}\det$, and $S(\chi_{1},\chi_{2})$ is an irreducible representation.

If $\chi_{1}/\chi_{2}=\Vert\cdot\Vert^{-1}$ then we have an exact sequence
$$0\to S(\chi_{1},\chi_{2})\to I(\chi_{1},\chi_{2})\to \rho\to 0$$
with $\rho$ and $S(\chi_{1},\chi_{2})$ as above.

\end{theorem}

Therefore the irreducible admissible representations of $GL_2(F)$ are the following:

* Principal series representations $I(\chi_{1},\chi_{2})$ for $\chi_{1}/\chi_{2}\neq \Vert\cdot\Vert^{\pm 1}$.

* Special representations $S(\chi_{1},\chi_{1}\times\Vert\cdot\Vert)$.

* $1$-dimensional representations $\chi\circ\det$.

* "Base change" representations $BC_{E}^{F}(\psi)$ for $E$ a quadratic extension of $F$ and $\psi$ an admissible character $\psi:E\to\mathbb{C}^{\times}$.

Let $\rho_{i}:W_{F}\to \mathbb{C}^{\times}$ be the representation of the Weil group associated to the character $\chi_{i}:F^{\times}\to \mathbb{C}^{\times}$ by the local Langlands correspondence for $GL_1$. Then the local Langlands correspondence associates to each irreducible admissible representation of $GL_2(F)$ a $2$-dimensional Weil-Deligne representation as follows:

* To the principal series representation $I(\chi_{1},\chi_{2})$ we associate the Weil-Deligne representation $(\rho_{1}\oplus\rho_{2},0)$.

* To the special representation $S(\chi_{1},\chi_{1}\times\Vert\cdot\Vert)$, we associate the Weil-Deligne representation $\left(\begin{pmatrix}\Vert\cdot\Vert\rho_{1} & 0\\0 & \rho_{1}\end{pmatrix},\begin{pmatrix} 0 & 1\\0 & 0\end{pmatrix}\right)$.

* To the $1$-dimensional representation $\chi_{1}\circ\det$, we associate the Weil-Deligne representation $\left(\begin{pmatrix}\rho_{1}\times\Vert\cdot\Vert^{1/2} & 0\\0 & \rho_{1}\times\Vert\cdot\Vert^{-1/2}\end{pmatrix},0\right)$.

* To the "base change" representation $BC_{E}^{F}(\psi)$ we associate the Weil-Deligne representation ($Ind_{W_{E}}^{W_{F}}\sigma,0$), where $\sigma$ is the unique nontrivial element of $\mathrm{Gal}(E/F)$.

## Geometrization

[[Laurent Fargues]] and [[Peter Scholze]] have developed a geometric approach to the local Langlands conjecture in [FarguesScholze21](#FarguesScholze21). In this formulation the local Langlands correspondence is expressed as an equivalence between the derived category of certain $\ell$-adic sheaves (this requires the theory of [[solid abelian groups]] from [[condensed mathematics]]) on the moduli stack $\mathrm{Bun}_{G}$ of $G$-torsors on the [[Fargues-Fontaine curve]] on one hand, and the derived category of certain coherent sheaves on the [[moduli stack of L-parameters]] (denoted $Z^{1}(W_{F},\widehat{G})_{\mathcal{O}_{L}}/\widehat{G}$ in [FarguesScholze21](#FarguesScholze21)) in the other (compare [[geometric Langlands correspondence]]). More precisely:

\begin{conjecture}
There is an equivalence
$$\mathcal{D}(\mathrm{Bun}_{G},\mathcal{O}_{L})^{\omega}\cong\mathcal{D}_{\mathrm{coh},\mathrm{Nilp}}^{b,\mathrm{qc}}(Z^{1}(W_{F},\widehat{G})_{\mathcal{O}_{L}}/\widehat{G})$$
of stable $\infty$-categories equipped with actions of $\Perf(Z^{1}(W_{F},\widehat{G})_{\mathcal{O}_{L}}/\widehat{G})$.
\end{conjecture}

In their work Fargues and Scholze are able to associate an L-parameter $\varphi_{A}:W_{F}\to\widehat{G}(\mathbb{Q}_{\ell})$ to a Schur object $A$ in $D(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ using the method of [[Vincent Lafforgue]] (originally developed for the global Langlands correspondence for function fields over $\mathbb{F}_{q}$) involving excursion operators. This is also known as the "automorphic to Galois" direction in the global setting.

In the other direction (analogous to the "Galois to automorphic direction") Fargues and Scholze approach this via the construction of the _spectral action_, which involves the use of [[infinity groupoids]] (which they refer to as _anima_). Subsequent work of Johannes  Anschütz and Arthur-Cesar Le Bras in [AnschutzLeBras21](#AnschutzLeBras21) use this spectral action to show the existence of the [[Hecke eigensheaf]].

## $p$-adic and mod $p$ local Langlands

The [[p-adic local Langlands correspondence]] concerns $p$-adic representations of the [[absolute Galois group]] of a $p$-adic field $F$, as opposed to complex Weil-Deligne or $\ell$-adic representations. It is related to [[p-adic Hodge theory]]. It should also be compatible with a [[mod p local Langlands correspondence]], which has coefficients in finite fields. Currently the only known case of the $p$-adic and mod $p$ local Langlands correspondences is the case $GL_2(\mathbb{Q}_{p})$, see also [Breuil2010](#Breuil2010).

## Related concepts

* [[Fargues-Fontaine curve]]

* [[global analytic geometry]]

## References

See also:

* Wikipedia, _[Local Langlands conjectures](http://en.wikipedia.org/wiki/Local_Langlands_conjectures)_.

* Colin Bushnell, Guy Henniart, _The local Langlands conjecture for $GL(2)$_

* [Topics on Automorphic Forms](http://www.math.columbia.edu/~chaoli/docs/AutomorphicForm.html), course notes by Chao Li from a course taught by Jack Thorne
* [Kevin Buzzard's MSRI Summer School on automorphic forms](http://wwwf.imperial.ac.uk/~buzzard/MSRI/)

Basic definitions are discussed in:

* {#Tate77} [[John Tate]], *Number theoretic background*, in: *Automorphic forms, representations and L-functions*, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. (1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26. Amer. Math. Soc., Providence, RI ([ISBN:978-0-8218-3371-1](https://bookstore.ams.org/pspum-33-2), [pdf](http://www.math.ucsd.edu/~csorense/teaching/math205/Tate_Weil.pdf), [[TateNumberTheory.pdf:file]])

An approach via the [[Fargues-Fontaine curve]]:

* {#FarguesScholze21} [[Laurent Fargues]], [[Peter Scholze]], _Geometrization of the local Langlands correspondence_ ([arXiv:2102.13459](https://arxiv.org/abs/2102.13459))

The above work is partially inspired by the following work on the _global_ Langlands correspondence for function fields over $\mathbb{F}_{q}$:

* [[Vincent Lafforgue]], _Chtoucas pour les groupes réductifs et paramétrisation de Langlands
globale_ [arXiv:1209.5352](http://arxiv.org/abs/1209.5352)

The spectral action of Fargues and Scholze is used to show the existence of the [[Hecke eigensheaf]] in

* {#AnschutzLeBras21} Johannes Anschütz and Arthur-Cesar Le Bras, _Averaging functors in Fargues' program for GL_n_ [arXiv:2104.04701](https://arxiv.org/abs/2104.04701)

The $p$-adic and mod $p$ local Langlands correspondence is discussed in:

* {#Breuil2010} [[Christophe Breuil]], *The emerging p-adic Langlands program*, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010 ([pdf](https://www.imo.universite-paris-saclay.fr/~breuil/PUBLICATIONS/ICM2010.pdf))

[[!redirects local Langlands conjectures]]

[[!redirects local Langlands program]]
[[!redirects local langlands program]]

[[!redirects local Langlands duality]]
[[!redirects local Langlands dualities]]

[[!redirects local Langlands correspondence]]
[[!redirects local Langlands correspondences]]


local Lie group

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## Idea

A __local Lie group__ (rarely also called _Lie groupuscule_)
is a  local / [[stalk]]wise version of a [[Lie group]], containing information about the product operation in arbitrary small [[open neighborhoods]] of the unit element; the multiplication is defined only when the factors are sufficiently close to the unit element, and if the consecutive products of triples in both ways are defined they are associative. To every local Lie group one functorially associates its Lie algebra.

Every real [[Lie algebra]] is a Lie algebra of some local Lie group. Or in more modern and precise phrasing, the category of real local Lie groups is equivalent to the category of finite-dimensional real Lie algebras. This has been proved by [[Sophus Lie]] as his famous [[Lie's three theorems|third theorem]]. The extension to the global Lie theory has been possible only after works of [[Élie Cartan]], who extended the equivalence to the equivalence between the category of finite-dimensional real Lie algebras and connected simply connected Lie groups.

## Related concepts

[[!include infinitesimal and local - table]]

[[!redirects local Lie groups]]
[[!redirects Lie groupuscule]]

[[!redirects local group]]
[[!redirects local groups]]


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#Contents#
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## Idea ##

See [[model structure on simplicial presheaves]].

## Definition ##

There are many [[model category]] structures on the category of [[simplicial presheaf|simplicial presheaves]] derived from the [[model structure on simplicial sets]].

The _local_ such model structures are of interest in that they model [[infinity-stack homotopically|infinity-stacks]] so that they are a [[presentable (infinity,1)-category|presentation]] of the [[(infinity,1)-category of (infinity,1)-sheaves]] on the given [[site]].

They can be thought of as being obtained from global model structures, of which there are two:

* the  **[[global model structure on simplicial sheaves|global projective]]** model structure has weak equivalences and fibrations being _objectwise_ those of [[simplicial set]]s;

* the  **[[global model structure on simplicial sheaves|global injective]]** model structure has weak equivalences and cofibrations being _objectwise_ those of [[simplicial set]]s;

These two model structures are Quillen equivalent
(*DHI04* [p. 5](http://www.math.uiuc.edu/K-theory/0563/spre.pdf#page=5) with the Quillen equivalence given by the identity functor). They can be defined on any domain category $S$, not necessarily a [[site]]. If we do have a structure of a [[site]] on $S$ then there is a  notion of _local weak equivalences_ of simplicial presheaves on $S$, defined below.
One gets _local_ projective and _local_ injective model structures by applying left [[Bousfield localization]] of the above model structures at local weak equivalences (see [p. 6](http://arxiv.org/PS_cache/math/pdf/0205/0205027v2.pdf#page=6) of *DHI04*)

* the  **local projective** model structure (weak equivalences are locally (usually stalkwise) and cofibrations are those that have the left [[lifting property]] against objectwise acyclic fibrations);

* the **local injective** (weak equivalences are locally (usually stalkwise) and fibrations are those that have the right [[lifting property]] against the objectwise acyclic cofibrations).

**Warning** Since the (homotopy classes) of weak equivalences do not form a [[small set]], the general existence theorem recalled at [[Bousfield localization of model categories]] does not apply. The existence of the Bousfield localization has to be shown by hand. For the injective structure this is what Joyal and Jardine accomplished.

Again, the injective and projective local model structures are Quillen equivalent by the identity functors between the underlying categories and hence provide projective and injective versions of the corresponding homotopy theory of [[infinity-stack homotopically|infinity-stacks]].

In the local injective structure all objects are cofibrant, so that the [[opposite category]] of simplicial presheaves with the local injective model structure is a [[category of fibrant objects]].

Both local model structures are proper [[enriched category|simplicially enriched]] categories (*DHI04* [p. 5](http://www.math.uiuc.edu/K-theory/0563/spre.pdf#page=6)).

The _local injective_ model structure on simplicial presheaves is originally due to Jardine, following the construction of the Quillen equivalent local [[model structure on simplicial sheaves]] by Joyal. It was only later realized in _DHI04_ as a left [[Bousfield localization]] of the global injective model structure.

In between the injective and the projective model structures there are many other model structures obtained by varying the class of generating global cofibrations.

In the following let $S$ be a small [[site]] and denote by $SimpPr(S)$ be the category of [[simplicial presheaf|simplicial presheaves]] on $S$.

## Local model structures ##

One usually says that a **local model structure** on a category of [[presheaf|presheaves]] is one whose weak equivalences are not defined objectwise but on [[cover]]s and/or on [[stalk]]s.

### Local weak equivalences ###

There are different equivalent ways to define local weak equivalences of simplicial presheaves on a site $S$.

### In terms of sheaves of homotopy groups

(see section 2 of [Jardine 2007](#Jardine07))

We want to say that a local weak equivalence of simplicial presheaves is one which is "over each point" an isomorphism of homotopy groups.

we need the following terminology about sheaves of simplicial homtopy groups:

* for $X$ a [[simplicial set]], write $\pi_0(X)$ for its set of connected components and $\pi_n(X,x)$, $n \geq 1$, $x \in X_0$, for its $n$th simplicial homotopy group at $x$ (the homotopy group of its [[geometric realization]]), $\pi_n(X,x) = \pi_n(|X|, x)$. This yields functors
$\pi_0 : SimpSet \to Set$ and $\pi_n : SimpSet \to Grps$.

* By postcomposition these functors induce functors
$
  \pi_0 : SimpSet^{S^{op}} \to Set^{S^{op}}
$
and
$
  \pi_n : SimpSet^{S^{op}} \to Grps^{S^{op}}
$.

* By postcomposition with the sheafification functor this yields functors
$
  \tilde \pi_0 : SimpSet^{S^{op}} \to Sh(S)
$
and
$
  \tilde \pi_n : SimpSet^{S^{op}} \to Sh(S,Grps)
$.

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

A **local weak equivalence** of simplicial presheaves is a morphism $f : X \to Y$ such that

1. the morphism $\tilde \pi_0(f) : \tilde \pi_0 X \to \tilde \pi_0 Y$ is an isomorphism in $Sh(S)$;

2. the diagrams
$$
  \array{
     \tilde \pi_n X &\to& \tilde \pi_n Y
     \\
     \downarrow && \downarrow
     \\
     \tilde X_0
     &\to&
     \tilde Y_0
  }
$$
are pullback diagrams in $Sh(S,SimpSet)$, for all $n \geq 1$, where $\tilde X_0$ denotes the sheaf associated to $X_0$.
=--

Equivalently a morphism $f : X \to Y$ of simplicial presheaves is, equivalently, a local weak equivalence if all induced morphisms of sheaves
$$
  \tilde \pi_n (X|_U, x) \to \tilde \pi_n Y|_U, f(x)
$$
are isomorphisms for all $U \in S$, for $X|_U, Y|_U$ the pullbacks to the over-category site $S/U$, for all $x \in X_0(U)$ and all $n \geq 0$.

If the site $S$ _has enough points_ then this condition is equivalent to saying that $f$ is a weak equivalence in the [[model structure on simplicial sets]] over every [[stalk]] (see [Jardine 2001 p. 363](#Jardine01), [Jardine 2015 pp. 63](#Jardine15)).

###In terms of local liftings###

(see _DI02_, i.e. Dugger and Isaksen, Weak equivalences of simplicial presheaves )

If $X$ and $Y$ are local fibrations there is a characterisation in terms of local homotopy liftings. Write $P$ for the pushout of the diagram $\partial \Delta^n \leftarrow \partial \Delta^n\times \Delta^1 \rightarrow \Delta^n\times \Delta^1$. Then there are two maps $\Delta^n\rightarrow P$ by restriction of $\Delta^n\times \Delta^1\rightarrow P$ along the cofaces.

Then a local weak equivalence is a morphism $f : X \to Y$ such that for all commuting diagrams

$$
  \array{
    \partial \Delta^n \otimes U &\to& X
    \\
    \downarrow^{i_U} && \downarrow^f
    \\
    \Delta^n \otimes U &\to& Y
  }
$$

with $U$ simplicially constantly representable there exists a covering sieve $R$ of $U$ such that for every $V\in R$ there are morphisms $g:\Delta^n \otimes V\rightarrow X$ and $h:P\otimes V\rightarrow Y$ for which $g\circ i_V=\partial\Delta^n \otimes V\rightarrow \partial\Delta^n \otimes U \rightarrow X$ and $\Delta^n\otimes V \rightarrow \Delta^n \otimes U\rightarrow Y= \Delta^n\otimes V\rightarrow P\otimes V\rightarrow Y$ and in addition the square

$$
\array{
    \Delta^n\otimes V &\to& X
    \\
    \downarrow && \downarrow^f
    \\
    P \otimes U &\to& Y
  }
$$

commutes.

## Examples ###

* Every object-wise weak equivalence is in particular a local weak equivalence.

## Local injective model structure ##

The **local injective model structure** on simplicial presheaves on a [[site]] $C$ is the **left** [[Bousfield localization]]
$SPr(C)_{loc inj}$
of the [[global model structure on simplicial presheaves|injective global model structure]] $SPr(C)_{inj}$ at the class of local weak equivalences described above.

So

* cofibrations are precisely the objectwise cofibrations of simplicial sets, i.e. the monomorphisms in $SPr(S)$;

* weak equivalences are the _local weak equivalences_ from above.

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

The inclusion of sheaves into simplicial presheaves
$SimpSh(S) \hookrightarrow SimpPr(S)$ and the [[sheafification]] functor  $SimpPr(S) \to SimpSh(S)$ constitute a  [[Quillen equivalence]] with respect to the above _local injective model structure_ on $SimpPr(S)$ and the local [[model structure on simplicial sheaves]].

=--

([Jardine 2007, Thm. 5](#Jardine07))

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

The fibrant objects in the local injective model
structure $SPr(C)_{loc inj}$ are those simplicial presheaves that

1. are fibrant in the global injective model structure;

2. satisfy [[descent]] for all [[hypercover]]s.

=--

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

_DHI04_, theorem 1.1

=--

## Local projective model structure ##

The **local projective model structure** on simplicial presheaves on a [[site]] $C$ is the **left** [[Bousfield localization]]
$SPr(C)_{loc proj}$
of the [[global model structure on simplicial presheaves|projective global model structure]] $SPr(C)_{proj}$ at the class of local weak equivalences described above.

So

* cofibrations are precisely the cofibrations in the global projective structure (defined by left lifting property with respect to global Kan fibrations)

* weak equivalences are the _local weak equivalences_ from above.

**Remark**. Notice that this is still using **left** Bousfield localization. If we used right Bousfield localization the local projective fibrations would simply be the global Kan fibrations. Instead we have the following.

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

The local injective model structure $SPr(C)_{loc inj}$ is [[Quillen equivalence|Quillen equivalent]] to the
"universal homotopy theory" $U C/S$ constructed by

1. formally adding [[homotopy limit|homotopy colimits]] to the category $C$ to create the category $U C$.

1. imposing relations requiring that for every [[hypercover]] $U \to X$, the morphism $hocolim_n U_n \to X$ is a weak equivalence.

=--

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

_DHI04_, theorem 1.2

=--

In $U C/S$ the fibrant objects have a simpler description
than in $SPr(C)_{loc inj}$: they still need to satisfy [[descent]] but the implicit fibrancy condition with respect to the global injective structure is replaced by the fibrancy condition with respect to the global projective structure

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

The fibrant objects in $U C/S$ are those simplicial presheaves $A$ that

1. are objectwise fibrant (i.e. take values in [[Kan complex]]es)

1. satisfy [[descent]] for all [[hypercover]]s.

=--

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

_DHI04_, theorem 1.3

=--

## Intermediate model structures ##

One can regard the projective and the injective model structure as two extrema of a poset of model structures on simplicial presheaves; see [[intermediate model structure]].

## References

Eg.

* {#JardineLecture} [[John F. Jardine]], *Simplicial presheaves*, Journal of Pure and Applied Algebra **47** (1987) 35-87 [doi:10.1016/0022-4049(87)90100-9, [pdf](http://math.uchicago.edu/~amathew/simplicialpresheaves.pdf)]

* {#Jardine01} [[J. F. Jardine]], *Stacks and the homotopy theory of simplicial sheaves*, Homology, Homotopy and Applications **3** 2 (2001)  361–384 [[doi:10.4310/HHA.2001.v3.n2.a5](https://dx.doi.org/10.4310/HHA.2001.v3.n2.a5), [euclid](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-3/issue-2/Stacks-and-the-homotopy-theory-of-simplicial-sheaves/hha/1139840259.full)]

* {#Jardine07} [[John F. Jardine]], *Simplicial Presheaves*, lecture notes, Fields Institute (2007) [[pdf](https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf), [webpage](https://www.uwo.ca/math/faculty/jardine/courses/fields/fields_lectures.html)]

* {#Jardine15} [[John F. Jardine]], *[[Local homotopy theory]]*, Springer Monographs in Mathematics (2015) [[doi:10.1007/978-1-4939-2300-7](https://doi.org/10.1007/978-1-4939-2300-7)]

For more see at *[[model structure on simplicial presheaves]]*.

The local projective model structure on simplicial presheaves appears as theorem 1.6 in

* Benjamin Blander, _Local projective model structure on simplicial presheaves_ ([pdf](http://www.math.uiuc.edu/K-theory/0462/combination2.pdf))

Its analog for sheaves, theorem 2.1 there, is due to

* K. Brown, Gersten, ...

That the local projective model structure (directly defined) is indeed the left Bousfield localization of the global projective model structure is lemma 4.3 there.

[[!redirects local model structures on simplicial presheaves]]

[[!redirects injective local model structure on simplicial presheaves]]
[[!redirects injective local model structures on simplicial presheaves]]

[[!redirects projective local model structure on simplicial presheaves]]
[[!redirects projective local model structures on simplicial presheaves]]


local model structure on simplicial sheaves

#Contents#
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{:toc}

## Idea

See [[model structure on simplicial presheaves]] for background, idea and introduction.

The [[local model structure on simplicial presheaves]] has a slight variant with [[presheaf|presheaves]] replaced by [[sheaf|sheaves]].

This is originally due to Joyal. Later Jardine introduced the [[local model structure on simplicial presheaves]] and proved that both are [[Quillen equivalence|Quillen equivalent]].

## Definition

For $C$ a [[site]] and $SSh(C)$ the category of [[sheaf|sheaves]] on $X$ with values in [[simplicial set]]s, Joyal's **local model structure** on $SSh(X)$ is (from a revisionist point of view) the restriction of the _injective_ [[local model structure on simplicial presheaves]] from all presheaves to sheaves.

## Related concepts

* [[model structure on simplicial presheaves]]

## References

* [[J. F. Jardine]], page 9 of: _Fields lectures: simplicial presheaves_ ([pdf](http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf))


local object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[reflective localization|Reflective localizations]] of categories and higher categories in the sense of [[left adjoint]] functors $L : C \to C'$ to inclusions $C' \hookrightarrow C$ of full subcategories (as in particular for [[geometric embeddings]]) are characterized by the collection $S \subset Mor(C)$ of morphisms of $C$ which are sent by $L$ to isomorphisms, or more generally to equivalences, as well as by the collection of objects which are _local_ with respect to these morphisms, in that these morphisms behave as equivalences with respect to homming into objects.

## Definition for ordinary categories

### Local objects

Let $C$ be a [[category]] and $S$ a collection of [[morphisms]] in $C$. Then an [[object]] $c \in C$ is **$S$-local** if the [[hom-functor]]
$$
  C(-,c) : C^{op} \to Set
$$
sends morphisms in $S$ to [[isomorphisms]] in [[Set]], i.e. if
for every $s : a \to b$ in $S$, the function
$$
  C(s,c) : C(b,c) \to C(a,c)
$$
is a [[bijection]].

### Local morphisms

Conversely, a **morphism** $f : x \to y$ is **$S$-local if for every $S$-local object $c$ the induced morphism

$$
  C(f,c) : C(y,c) \to C(x,c)
$$

is an isomorphism.

## Definition for $(\infty,1)$-categories

### Local objects

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

Let $C$ be an [[(∞,1)-category]] and $S$ a collection of morphisms in $C$. Then an [[object]] $c \in C$ is **$S$-local** if the [[hom-functor]]
$$
  C(-,c) : C^{op} \to \infty Top
$$
evaluated on $s \in S$ induces isomorphism in the [[homotopy category]] of [[Top]].
=--

This is [5.5.4.1](http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf#page=383) in [[Higher Topos Theory|HTT]]

### Local morphisms

Conversely, a **morphism** $f : x \to y$ is **$S$-local** if for every $S$-local object $c$ the induced morphism

$$
  C(f,c) : C(y,c) \to C(x,c)
$$

induces an isomorphism in the [[homotopy category]] of [[Top]].

## Definition in model categories

Let $C$ be a [[model category]] (usefully but not necessarily a [[simplicial model category]]). And let $S \subset Mor(C)$ be a collection of [[morphism]]s in $C$.

Write $\mathbf{R}Hom_C(-,-) : C^{op}\times C \to SSet$ for the [[(infinity,1)-categorical hom-space|derived hom space functor]].

For instance if $C$ is a [[simplicial model category]] then this may be realized in terms of a cofibrant replacement functor $Q : C \to C$ and a fibrant replacement functor $P$ as

$$
  \mathbf{R}Hom_C(X,Y) = C(Q X, P Y)
  \,.
$$

+-- {: .num_defn }
###### Definition
**(local object, local weak equivalence)**

An object $c \in C$ is a **$S$-local object** if for all $s : a \to b$ in $S$ the induced morphism

  $$
    \mathbf{R}Hom_C(s,c) : \mathbf{R}Hom_C(b,c) \to \mathbf{R}Hom_C(a,c)
  $$

  is a weak equivalence (in the standard [[model structure on simplicial sets]]);

A morphism $f : x \to y$ in $C$ is an **$S$-local morphism** or **$S$-equivalence** if for every $S$-local object $c$ the induced morphism

  $$
    \mathbf{R}Hom_C(f,c) : SSet \to SSet
  $$

  is a weak equivalence.

An **$S$-localization of an object** $c$ is an $S$-local object $\hat c$ and an $S$-local equivalence $c \to \hat c$.

An **$S$-localization of a morphism** $f : c \to d$ is a pair of $S$-localizations $c \to \hat c$ and $d \to \hat d$ of objects, and a commuting square

$$
  \array{
    c &\stackrel{f}{\to}& d
    \\
    \downarrow && \downarrow
    \\
    \hat c &\to & \hat d
  }
  \,.
$$

=--

### Properties {#PropInModCat}

In [[proper model category|left proper model categories]] there is an equivalent stronger characterization of $S$-locality of cofibrations $i : A \hookrightarrow B$.

+-- {: .num_prop }
###### Proposition
**(characterization of $S$-local cofibrations)**

Let $C$ be a [[proper model category|left proper]] [[simplicial model category]] and $S \subset Mor(C)$, a collection of morphisms.

Then a cofibration $i : A \hookrightarrow B$ is an $S$-local weak equivalence precisely if for all fibrant $S$-local objects $X$ the morphism

$$
  C(B,X) \to C(A,X)
$$

is an acyclic fibration in the standard [[model structure on simplicial sets]].

=--

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

Notice that this is stronger than the statement that $\mathbf{R}Hom(B,X) \to \mathbf{R}Hom(A,X)$ is a weak equivalence not only in that it asserts in addition a fibration, but also in that it deduces this without first passing to a cofibrant replacement of $A$ and $B$.

=--

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

This is [[Higher Topos Theory|HTT, lemma A.3.7.1]].

The proof makes use of the following general construction: for $f : A \to B$ any morphism let $\emptyset \hookrightarrow A' \stackrel{\simeq}{\to} A$ be a cofibrant replacement, factor $A' \to B$ as
$A' \stackrel{i'}{\hookrightarrow} B' \stackrel{\simeq}{\to} B$ and consider the [[pushout]] diagram

$$
  \array{
    A' &\stackrel{i'}{\hookrightarrow}& B'
    \\
    \downarrow^{\mathrlap{f \in W}}
    && \downarrow_{\mathrlap{g\in W}}
    &
    \searrow^{\mathrlap{f' \in W}}
    \\
    A &\stackrel{}{\hookrightarrow}& A \coprod_{A'} B
    &\stackrel{j \in W}{\to}&
    B
  }
  \,.
$$

By [[proper model category|left properness]] the pushout $g$ of the weak equivalence $f$ along the cofibration $i'$ is again a weak equivalence and by [[category with weak equivalences|2-out-of-3]] the morphism $j$ is a weak equivalence.

Now assume that $i$ is an $S$-local equivalence. We need to show that $i^* : C(B,X) \to C(A,X)$ is an acyclic Kan fibration for all fibrant $S$-local $X$. By the very definition of [[enriched model category]] it follows from $i$ being a cofibration and $X$ being fibrant that this is a Kan fibration. So it remains to show that it is a [[weak homotopy equivalence]] of [[simplicial set]]s. We know that the corresponding induced morphism

$$
  ({i'}^* : C(B',X) \to C(A',X))
  \simeq
  (\mathbf{R}Hom(B,X) \to \mathbf{R}Hom(A,X))
$$

on the cofibrant replacement is a weak equivalence, by the assumption that $X$ is $S$-local, and also, as before, a fibration, since $i'$ is still a cofibration.

By homming the entire diagram above into $X$, and using that the [[hom-functor]] $C(-,X)$ sends [[colimit]]s to [[limit]]s, we find the [[pullback]] diagram

$$
  \array{
    C(A \coprod_{A'} B', X)
    &\to&
    C(B',X)
    \\
    {}^{q}\downarrow^{\mathrlap{\in (W\cap fib)_{SSet}}}
    &&
    {}^{{i'}^*}\downarrow^{\mathrlap{\in (W\cap fib)_{SSet}}}
    \\
    C(A,X)
    &\to&
    C(A',X)
  }
$$

in [[SSet]], which shows that $q$ is an acyclic fibration, being the pullback of an acyclic fibration.

To show that $i^*: C(B,X) \to C(A,X)$ is a weak equivalence it suffices to show that all its fibers $(i^*)^{-1})(t)$ over elements $t : A \to X$ are [[contractible]] [[Kan complex]]es. These fibers map to the corresponding fibers $q^{-1}(t)$ by precomposition with $j$. By the fact that $j$, regarded as a morphism

$$
  \array{
   && A
   \\
   & {}\swarrow
    && \searrow
   \\
   A \coprod_{A'} B' &&\stackrel{j}{\to}&& B
  }
$$

in the [[model structure on an over category|model structure on the undercategory]] $A/C$ is a weak equivalence between cofibrant objects (because $A \hookrightarrow B$ is a cofibration by assumption and $A \to A \coprod_{A'} B'$ as being the pushout of the cofibration $i'$) we have that precomposition $C(j,X)$ with $j$ is the image under the [[SSet]]-[[enriched functor|enriched]] [[hom-functor]] of a weak equivalence between cofibrant objects mapping into a fibrant object

$$
  \array{
    && A
    \\
    & \swarrow & \downarrow & \searrow^{t}
    \\
    A \coprod_{A'} B'
    &\stackrel{j}{\to}&
    B
    &\to&
    X
  }
$$

and hence, by the general properties of [enriched homs between cofibrant/fibrant objects](http://ncatlab.org/nlab/show/%28infinity,1%29-categorical+hom-space#enriched_homs_between_cofibrantfibrant_objects_6) a weak equivalence. $j^* : (i^*)^{-1}(t) \stackrel{\simeq}{\to} q^{-1}(t)$, so that indeed $(i^*)^{-1}(t)$ is contractible.

This proves the first part of the statement. For the converse statement, assume now that...

=--

### References

A classical textbook reference is section 3.2 of

* Hirschhorn, _Model categories and their localization_

A useful reference with direct ties to the [[(∞,1)-category]] story in the background is section A.3.7 of

* [[Jacob Lurie]], [[Higher Topos Theory]]

## Saturated class of morphisms

Every morphism in $S$ is $S$-local.

The collection $S$ of morphisms is called **saturated** if the collection of $S$-local morphisms coincides with $S$.

## Remarks

* a [[reflective subcategory]] as well as a [[reflective (∞,1)-subcategory]] can be realized as the full ($(\infty,1)$-)subcategory on $S$-local objects, where $S$ is the collection of morphisms sent by the corresponding [[localization of an (∞,1)-category]] to equivalences. For details on this see the discussion at [[geometric embedding]].

[[!redirects local objects]]
[[!redirects local morphism]]
[[!redirects local morphisms]]


local observable

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

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] ([[field theory]]) a _local observable_ is an [[observable]] which is an average of a function of the values of the [[field (physics)|fields]] and their [[derivatives]] at each fixed spacetime point.

If $\Phi$ is a [[field (physics)|field configuration]] over some [[spacetime]] $\Sigma$ then a local observable is a function of $\Phi$ which, assigns values of the form

$$
  \Phi
    \;\mapsto\;
  \int_\Sigma f(\Phi(x), \nabla \Phi(x), \nabla^2 \Phi(x), \cdots) b(x) dvol_\Sigma(x)
$$

where $f$ is some [[smooth function]] of its arguments and $b$ is some [[bump function]] on [[spacetime]].

This is in contrast to an observable which combines the values of fields at different spacetime points other than by forming spacetime averages. For instance if $x \neq y \in \Sigma$ are two distinct spacetime points, then

$$
  \Phi \mapsto \Phi(x) \Phi(y)
$$

is a smooth function on the space of fields, hence an [[observable]], but it is not a local observables.

More in detail, if

$$
  \array{
    E
    \\
    \downarrow^{\mathrlap{fb}}
    \\
    \Sigma
  }
$$

is a [[fiber bundle]] regarded as the [[field bundle]] that defines the given [[field theory]], then a local observables is a function on the the space of field configurations, hence the [[space of sections]] $\Gamma_\Sigma(E)$ (or else on the subspace of those which solve the [[equations of motion]], the [[shell]]) which arises as the [[transgression of variational differential forms|transgression]] of a [[horizontal differential form]] of degree $dim(\Sigma)$ on the [[jet bundle]] $J^\infty_\Sigma(E)$ of $E$.

Products of local observables are called _multilocal observables_.

[[!include perturbative observables -- table]]

## Definition

For the moment see at _[[geometry of physics -- perturbative quantum field theory]]_ [this def.](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#LocalObservables)

## Properties

+-- {: .num_example #PolynomialLocalObservablesArePolynomialObservables}
###### Example
**([[polynomial]] [[local observables]] are [[polynomial observables]])**

A [[local observable]] which comes from a [[variational differential form|horizontal differential form]] which is a [[polynomial]] in the fields and their jets times the [[volume form]] on [[spacetime]] is a [[polynomial observable]].

These happen to be also [[microcausal observables]] ([this example](microcausal+functional#CompactlySupportedPolynomialLocalDensities)).

=--

The resultiing inclusion

$$
  LocPolyObs(E) \hookrightarrow PolyObs(E)_{cm}
$$

of the local polynomial observables into the [[microcausal polynomial observables]] is a [[dense subspace]]-inclusion. ([Fredenhagen-Rejzner 12, p. 23](#FredenhagenRejzner12))

## Related concepts

* [[Wick algebra]]

* [[spacetime support]]

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

## References

* {#BrunettiFredenhagen09} [[Romeo Brunetti]], [[Klaus Fredenhagen]], section 5.4.1 of _Quantum Field Theory on Curved Backgrounds_, chapter 5 in [[Christian Bär]], [[Klaus Fredenhagen]] (eds.), _Quantum Field Theory on Curved Spacetime_, Springer 2009

* {#BrunettiDuetschFredenhagen09} [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], section 3.2 of_Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](https://arxiv.org/abs/0901.2038))

* {#FredenhagenRejzner12} [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Perturbative algebraic quantum field theory_, In _Mathematical Aspects of Quantum Field Theories_, Springer 2016 ([arXiv:1208.1428](https://arxiv.org/abs/1208.1428))

* {#Rejzner16} [[Katarzyna Rejzner]], def. 3.14 in _[[Perturbative Algebraic Quantum Field Theory]]_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

* {#Duetsch18} [[Michael Dütsch]], def. 1.2 in _[[From classical field theory to perturbative quantum field theory]]_, 2018

[[!redirects local observables]]

[[!redirects multilocal observable]]
[[!redirects multilocal observables]]


local quantum field theory

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

#Contents#
* table of contents
{:toc}

## Idea

A central aspect of [[quantum field theory]] usually desired or demanded or expected of _fundamental_ QFTs is that it is _local_ in the rough sense that

* there is no "action at a distance", but all influences propagate (on a [[Lorentzian manifold]]) at some finite speed (that of light, usually);

* phenomena of large scale [[spacetime]]/[[worldvolume]] are entirely determined by phenomena on smaller scales;

* (space-like) separated regions of [[spacetime]] behave like [independent subsystems](quantum+mechanics#Subsystems) ([[causal locality]]).

In some [[effective quantum field theories]] locality may be violated, if fundamental local processes are averaged out to a single non-local macroscopic process, but for _fundamental_ physics of the [[observable universe]] it is thought to hold.

### Formalization in quantum field theory

Locality is formalized in the two main axiomatizations of quantum field theory as follows.

* In [[AQFT]] the [[algebras of observables]] are required to form a _[[local net]]_ meaning that

  1. there is one such algebra assigned to each suitable subset of spacetime, compatible with inclusion of such subsets,

  1. that under these inclusions algebras associated to spacelike separated regions commute with each other.

  Often in the literature the term "local quantum field theory" is meant to refer specifically to these [[AQFT]] axioms (some authors use the terms synonymously, dating from a time when this was the only axiomatization of quantum field theory considered.)

* In [[FQFT]] locality is encoded in the [[functor]]-property of the functor on the [[category of cobordisms]]: being a functor means that the assignment to a cobordism $\Sigma$ is obtained by composing the assignments to any decomposition of $\Sigma$ into small cobordisms. In particular, in _[[extended quantum field theory]]_ (now also sometimes called "fully localized" QFT) this is [[(infinity,n)-functor|n-functorial]] meaning that this gluing condition holds in all dimensions and in all directions.

### Formalization in pre-quantum field theory

There are also properties of locality in _[[prequantum field theory]]_.

A _[[Lagrangian density]]_ is called a _[[local Lagrangian]]_ if it "depends only on finitely many [[derivatives]] of the [[field (physics)|fields]]" at any point of spacetime, which formally means that it is a [[variational bicomplex|horizontal differential form]] on the [[jet bundle]] of the [[field bundle]]. Local Lagrangian densities are expected to yield local quantum field theories under [[quantization]]. This is the case at least in [[perturbative quantum field theory]] formalized via [[causal perturbation theory]]/[[perturbative AQFT]], see at _[[S-matrix]]_ the section _[Quantum observables and retarded products](S-matrix#QuantumObservables)_

## Related concepts

* [[microcausality]], [[Einstein causality]]

* [[cluster decomposition]]

* [[Lagrangian quantum field theory]]

* [[algebraic quantum field theory]]

* [[perturbative algebraic quantum field theory]]

## References

* _[[schreiber:Local prequantum field theory]]_

On [[local field theory|locality]] of [[extended functorial field theory|extended]] [[functorial field theory]]:

* [[Daniel Grady]], [[Dmitri Pavlov]], _Extended field theories are local_ ([arXiv:2011.01208](https://arxiv.org/abs/2011.01208))

Steps towards a local version of [[BV-formalism]] are undertaken in

* [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], _Classical BV theories on manifolds with boundary_ ([arXiv:1201.0290](http://arxiv.org/abs/1201.0290))

* [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], _Classical and quantum Lagrangian field theories with boundary_ ([arXiv:1207.0239](http://arxiv.org/abs/1207.0239))

* [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], _Perturbative quantum gauge theories on manifolds with boundary_ ([arXiv:1507.01221](http://arxiv.org/abs/1507.01221))

[[!redirects local quantum field theories]]

[[!redirects local field theory]]
[[!redirects local field theories]]

[[!redirects local QFT]]
[[!redirects local QFTs]]

[[!redirects local Lagrangian field theory]]
[[!redirects local Lagrangian field theories]]


Local Quantum Physics -- Fields, Particles, Algebras

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

This entry is about the textbook:

* [[Rudolf Haag]]:

  \linebreak

  **Local quantum physics -- Fields, particles, algebras**

  \linebreak

  Springer (1992, 1996)

  [ISBN 978-3-642-61458-3] (https://www.springer.com/gp/book/9783540610496)

  [ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0857.46057&format=complete)

on [[algebraic quantum field theory]] via [[local nets of observables]] satisfying the [[Haag-Kastler axioms]].

While axiomatic, the book does contain a fair bit of discussion of the relation to actual [[physics]].

Here is a hyperlinked keyword list:

#Contents#
* table of contents
{:toc}

## Chapter I Background

## Chapter II General quantum field theory

* [[quantum field theory]]

## Chapter III Algebras of local observables and fields

* [[observable]]

* [[Haag-Kastler axioms]]

* [[local net of observables]]

## Chapter IV Charges, global gauge groups and exchange symmetry

* [[global gauge group]]

* [[superselection sector]]

## Chapter V Thermal states and modular automorphisms

* [[modular theory]]

## Chapter VI Particles. Completeness of the particle picture

* [[particle]]

## Chapter VII Principles and lessons of quantum field theory

## Chapter VIII Retrospective and outlook

[[!redirects Local Quantum Physics]]


local quotient stack

#Contents#
* table of contents
{:toc}

## Idea

A _local quotient stack_ is a _[[stack]]_ with a suitable [[atlas]] by [[quotient stacks]].

## Definition

For instance ([FHT, I, A.2.2](#FHT)).

## Properties

Every local quotient stack admits a [[locally universal Hilbert bundle]]. Hence local quotient groupoids have a good [[topological K-theory]]. See at _[[groupoid K-theory]]_ for more on this.

([FHT, I, lemma 3.12](#FHT))

## References

In

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], part I of _[[Loop Groups and Twisted K-Theory]]_
 {#FHT}

the [[groupoid K-theory]] of local quotient stacks is discussed.

[[!redirects local quotient stacks]]

[[!redirects local quotient groupoid]]
[[!redirects local quotient groupoids]]


local reflexive coequalizer

A [[bicategory]] **has local reflexive coequalizers** if each of its hom-categories has [[reflexive coequalizers]] and composition preserves them.

Similarly, a [[double category]] has local reflexive coequalizers if its loose bicategory has them.

This condition is crucial for the [[virtual equipment]] of monoids and modules in a [[fibrant double category]] $\mathbb{D}$ to be representable, i.e. to have an associative tensor of bimodules ([Proposition 11.10, Shulman 2008](#Shu08)).

## References
* {#Shu08} [[Mike Shulman]], *Framed bicategories and monoidal fibrations*, Theory and Applications of Categories **20** 18 (2008) 650–738 [[tac:2018](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html), [arXiv:0706.1286](https://arxiv.org/abs/0706.1286)]


local ring

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

# Local rings
* table of contents
{: toc}

## Definitions

A **local ring** is a [[ring]] (with [[unit]], usually also assumed [[commutative ring|commutative]]) such that:

* $0 \ne 1$; and

* whenever $a + b = 1$, $a$ or $b$ is [[invertible]].

Here are a few equivalent ways to phrase the combined condition:

* Whenever a (finite) [[sum]] equals $1$, at least one of the summands is invertible.

* Whenever a sum is invertible, at least one of the summands is invertible.

* Whenever a sum of products is invertible, for at least one of the summands, all of its multiplicands are invertible.  (This is not entirely trivial in the noncommutative case, but it\'s still correct.)

* The invertible elements form an [[anti-ideal]]:
  * $0$ is not invertible
  * if $a + b$ is invertible, then either $a$ is invertible or $b$ is invertible
  * if $a \cdot b$ is invertible, then $a$ is invertible and $b$ is invertible

The anti-ideal of invertible elements is in fact a minimal anti-ideal, so the [[quotient object|quotient ring]] of a local ring by its anti-ideal of invertible elements is a [[Heyting field]], the __[[residue field]]__ of the local ring.

In the context of [[excluded middle]], every [[weak local ring]] is a [[local ring]]. Thus, the following is equivalent to the above definitions in classical mathematics:

* The sum of two non-invertible elements is non-invertible.
* The non-invertible elements form an [[ideal]].

\begin{remark}\label{TrivialRingAsALocalRing}
In [LombardiQuitté2010](#LombardiQuitté2010), the authors' definitions of local ring do not include the non-equational axiom $1 \neq 0$, which means that the [[trivial ring]] is a local ring and constitutes the [[terminal object]] in the [[category|categories]] of local rings.
\end{remark}

## Examples

* Every [[Heyting field]] is a local ring.

* Every [[Kock field]] is a local ring.

* For $R$ a local ring, then the [[power series ring]] $R[ [x] ]$ is also a local ring.

## Properties

### Jacobson radical

The set of non-invertible elements in a local ring is the [[Jacobson radical]].

### Kaplansky's theorem
 {#KaplanskyTheorem}

+-- {: .num_theorem}
###### Theorem
**(Kaplansky)**
A [[projective module]] over a commutative local ring is a [[free module]].
=--

An exposition of the proof may be found [here](http://blog.jpolak.org/?p=363). A [[constructive mathematics|constructive]] proof of a finitary weakening of Kaplansky's theorem proceeds as follows.

+-- {: .num_lemma}
###### Lemma
Let $A$ be a local ring. Let $\mathfrak{a}$ be a finitely generated idempotent ideal in $A$. Then $\mathfrak{a} = (0)$ or $\mathfrak{a} = (1)$.
=--
+-- {: .proof}
###### Proof
Consider $\mathfrak{a}$ as a finitely generated $A$-module. Then, by [[Nakayama's lemma]], there exists an element $x \in A$ such that $x \equiv 1$
modulo $\mathfrak{a}$ and $x \mathfrak{a} = 0$. Since $A$ is a local ring, $x$ is invertible or $1-x$ is invertible. In the first case it follows that $\mathfrak{a} = (0)$, in the second that $\mathfrak{a} = (1)$.
=--

+-- {: .num_lemma}
###### Lemma
Let $A$ be a local ring. Let $P$ be an idempotent matrix over $A$. Then $P$ is equivalent to a diagonal matrix with entries $1$ and $0$.
=--
+-- {: .proof}
###### Proof
Since $P$ is idempotent, so are its ideals $(\Lambda^i P)$
of $i$-minors:
$$ (\Lambda^i P) = (\Lambda^i (P \circ P)) =
(\Lambda^i P \circ \Lambda^i P) \subseteq (\Lambda^i P) \cdot (\Lambda^i P)
\subseteq (\Lambda^i P). $$
By the previous lemma, they are therefore each equal to $(0)$ or $(1)$. Since they form a descending chain, there exists a stage $r$ such that $(\Lambda^r P) = (1)$ and $(\Lambda^{r+1} P) = (0)$. Therefore all $(r+1)$-minors of $P$ are zero, and -- since $A$ is a local ring -- there exists at least one invertible $r$-minor. Thus $P$ can be made into a diagonal matrix of the desired form by applying row and column transformations.
=--

+-- {: .num_remark}
###### Remark
We can even show that $P$ is _similar_ to a diagonal matrix with entries $1$ and $0$: By the lemma, image and kernel of $P$ are finite free. Combining bases of these subspaces, we obtain a basis of the full space; expressing $P$ with respect to this basis, we obtain a diagonal matrix of the desired form.
=--

+-- {: .num_theorem}
###### Theorem
Let $M$ be a finitely generated module over a local ring $A$. Assume that $M$ is projective. Then $M$ is finite free.
=--
+-- {: .proof}
Fix a linear surjection $p : A^n \to M$ and a section $s : M \to
A^n$. The composition $P \coloneqq s \circ p$ is idempotent and $M$ is isomorphic to $A^n/\operatorname{ker}(P)$. Since $P$ is equivalent to a diagonal matrix with entries $1$ and $0$, this module is obviously finite free.
=--

## In geometry

In [[algebraic geometry]] or [[synthetic differential geometry]] and [[commutative algebra]], the most commonly used definition of a local commutative ring is a commutative ring $R$ with a unique maximal ideal. Hence the Spec of such an $R$ has a unique closed point. Intuitively it can be thought of as some kind of "infinitesimal neighborhood" of a closed point.

The spectrum of a ring $R$ is [[focal point|local]], i.e. in any covering of $Spec R$ by open subsets one of the subsets is already the whole of $Spec R$, if and only if $R$ is a local ring. This provides some justification for the name.

The [[topos theory]] formulation of this is a [[local topos]].

An important example of a local ring in algebraic geometry is $R = k[\epsilon]/\epsilon^2$. This ring is known as the ring of dual numbers. Intuitively, we can think of its spectrum as consisting of a closed point and a tangent vector. Indeed this is justified, as morphisms from $\operatorname{Spec} R$ to a scheme $X$ correspond exactly to pairs $(x,v)$, where $x \in X$ and $v$ is a (Zariski) tangent vector at $x$.

Local rings are also important in [[deformation theory]]. One might define an infinitesimal deformation of a scheme $X_0$ to be a deformation of $X_0$ over $\operatorname{Spec} R$ where $R$ is a local ring.

## In weak foundations

Local rings are often more useful than fields when doing mathematics [[internalization|internally]]. For one thing, the definition make sense in any [[coherent category]]. But unlike the definition of [[discrete field]] (which is also coherent), it is satisfied by a [[real-numbers object]]. Rather than mod out by the ideal of non-invertible elements, you take care to use only properties that are invariant under multiplication by an invertible element.

In [[constructive mathematics]], one could do the same thing, but it\'s more common to use the notion of [[Heyting field]]. This is closely related, however; the quotients of local rings are precisely the Heyting fields (which are themselves local rings). In fact, one can define an [[apartness relation]] (like that on a Heyting field) in any local ring: $x \# y$ iff $x - y$ is invertible. Then the local ring is a Heyting field if and only if this apartness relation is [[tight relation|tight]].

+-- {: .num_prop #internal}
###### Proposition
The addition and multiplication operations on a local ring $R$ are strongly extensional with respect to the canonical apartness relation $\#$ defined by $x \# y$ iff $x - y$ is invertible. In this way a local ring becomes an internal [[ring object]] in the category $Apart$, consisting of sets with apartness relations and maps (strongly extensional functions) between them.
=--

+-- {: .proof}
###### Proof
Recall that products $X \times Y$ in the category of sets with apartness relations is the cartesian product of the underlying sets equipped with the apartness relation defined by $(x, y) \# (x', y')$ iff $x \# x'$ in $X$ *or* $y \# y'$ in $Y$. Recall also that a function $f: X \to Y$ between sets with apartness relations is *strongly extensional* if $f(x) \# f(y)$ implies $x \# y$.

For addition, if $(x + y) \# (x' + y')$, then $x + y - (x' + y') = (x - x') + (y - y')$ is invertible, so $x - x'$ or $y - y'$ is invertible since $R$ is local, whence $(x, y) # (x', y')$. Thus addition is strongly extensional.

For multiplication, if $x y # x' y'$, then $x y - x' y'$ is invertible. Write $x y - x' y' = (x - x')y + x'(y - y')$. Since $R$ is local, either $(x - x')y$ is a unit or $x'(y - y')$ is a unit. From this we easily conclude $x - x'$ is a unit or $y - y'$ is, whence $(x, y) # (x', y')$. So multiplication is also strongly extensional.
=--

Constructively there are also possible variants of the definition of local ring.  For instance, in [Johnstone77](#Johnstone77) a **[[weak local ring]]** is defined to be a ring in which the sum of any two noninvertible elements is noninvertible. The quotients of weak local rings are the [[weak Heyting fields]], or what Johnstone called  *residue [[fields]]* (nontrivial rings in which every noninvertible element is zero); this is the origin of the name "residue field" by Johnstone.

There is also the notion of a **discrete local ring**, which is a local ring where every element is either invertible or noninvertible, or equivalently, where the [[apartness relation]] of the local ring is a [[decidable relation]]. The quotient of discrete local rings are precisely the [[discrete fields]].

## Local homomorphisms

Classically, if $R$ and $S$ are local rings with maximal ideals $\mathfrak{m}$ and $\mathfrak{n}$ respectively, then a ring [[homomorphism]] $f: R \to S$ is said to be *local* if $f(\mathfrak{m}) = \exists_f (\mathfrak{m}) \subseteq \mathfrak{n}$. Equivalently, $\mathfrak{m} \subseteq f^{-1}(\mathfrak{n})$. Taking [[complements]] and using the fact that taking [[inverse images]] preserve complements, this is equivalent to $f^{-1}(S^\times) \subseteq R^\times$ where $R^\times$ is the [[group of units]]. Of course we also have $R^\times \subseteq f^{-1}(S^\times)$ (equivalently $f(R^\times) \subseteq S^\times$) just from the fact that $f$ is a ring homomorphism, so in brief $f$ is local if $R^\times = f^{-1}(S^\times)$: an element $f(r)$ is invertible (if and) only if $r$ is.

In constructive settings it makes sense to take this last formulation as our notion of local homomorphism. In view of Proposition \ref{internal}, it makes sense to say it like this:

+-- {: .num_defn}
###### Definition
A *local homomorphism* between local rings is an internal ring homomorphism between their associated internal rings in the category $Apart$.
=--

Possible to-dos: say something about $\mathfrak{m}$-adic topology, completion, Zariski topos as classifying topos...

## Related concepts

* [[localization of a commutative ring]]

* [[strict local ring]]

* [[Jacobson radical]]

* [[valuation ring]]

* [[Zariski topos]]

* [[Henselian ring]]

* [[ordered local ring]]

* [[weak local ring]]

| [[commutative ring]] | [[reduced ring]] | [[integral domain]] |
---------------------|-----------------|-----------------|
| [[local ring]] | [[reduced local ring]] | [[local integral domain]] |
| [[Artinian ring]] | [[semisimple ring]] | [[field]] |
| [[Weil ring]] | [[field]] | [[field]] |

## References

* {#Johnstone77} [[Peter Johnstone]], *Rings, Fields, and Spectra*, Journal of Algebra **49** (1977) pp 238-260. doi:[10.1016/0021-8693(77)90284-8](https://doi.org/10.1016/0021-8693%2877%2990284-8)

* {#LombardiQuitté2010} [[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))

* {#Richman20} [[Fred Richman]], *Laurent series over $\mathbb{R}$*. Communications in Algebra, Volume 48, Issue 5, 11 Jan 2020 Pages 1982-1984 [[doi:10.1080/00927872.2019.1710166](https://doi.org/10.1080/00927872.2019.1710166)]

[[!redirects local ring]]
[[!redirects local rings]]

[[!redirects discrete local ring]]
[[!redirects discrete local rings]]


local section

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

# Contents
* table of contents
{: toc}

## Idea

For $E \to X$ a [[bundle]], (often taken to be a [[fiber bundle]] or at least typically taken to be a [[regular epimorphism|regular epimorphic]] map)  a _local section_ is a [[section]] of the [[pullback]] of the bundle along some $U \to X$, typically required to be an element of a [[covering family]] from some [[coverage]].

The assignment of local sections of some $E \to X$ to all admissible $U \to X$ is the ([[presheaf|pre]]-)[[sheaf]] of local sections assigned to a bundle.

In a [[finitely complete category|finitely complete]] [[site]] $(S,J)$ the assignment $X \mapsto \{p\colon Y \to X | p $ admits local sections over a $J$-cover $\}$ is a [[singleton pretopology]] on $S$.

## Examples

* [[coset space coprojections admitting local sections]]

* [[logarithm]]

## Related concepts

* [[global section]]

* [[flat section]]

* [[partial section]]

* [[slice of presheaves is presheaves on slice]]

[[!redirects local section]]
[[!redirects local sections]]


local site

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

#Contents#
* table of contents
{:toc}

## Definition

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

A [[site]] is called a **local site** if

* it has a [[terminal object]] $*$;

* the only [[covering]] family of $*$ is the trivial cover.

=--

This appears as ([Johnstone example C.3.6.3 (d) ](#Johnstone)).

## Properties

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

The [[category of sheaves]] $Sh(C)$ on a local site $C$ is a [[local topos]].

=--

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

Since $C$ has a terminal object, the [[global section]] functor $Sh(C) \to Set$ is given by evaluation on that object, hence is precomposition of sheaves with the inclusion $* \to C$. At the level of presheaves this has a right [[Kan extension]] functor, given by sending a set $S$ to the presheaf

$$
  \nabla S : U \mapsto S^{C(*,U)}
  \,.
$$

This is indeed a sheaf if $*$ is covered only by the trivial cover.

=--

See ([Johnstone example C.3.6.3 (d) ](#Johnstone)).

## Examples

* [[CartSp]]

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]] / [[locally ∞-connected site]]

  * [[connected site]] / [[∞-connected site]]

  * [[strongly connected site]] / [[strongly ∞-connected site]]

  * [[totally connected site]] / [[totally ∞-connected site]]

* **local site** / [[∞-local site]]

* [[cohesive site]], [[(∞,1)-cohesive site]]

## References

The definition appears as example C.3.6.3 (d) in

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
{#Johnstone}

[!redirects local sites]


local system

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _local system_  -- which is short for _local system of [[coefficients]] for [[cohomology]]_ -- is a system of coefficients for [[twisted cohomology]].

Often this is presented or taken to be presented by a [[locally constant sheaf]]. Then cohomology with coefficients in a local system is the corresponding [[sheaf cohomology]].

More generally, we say a _local system_ is a [[locally constant stack]], ... and eventually a [[locally constant ∞-stack]].

Under suitable conditions (if we have [[Galois theory]]) local systems on $X$ correspond to [[functor]]s out of the [[fundamental groupoid]] of $X$, or more generally to [[(∞,1)-functor]]s out of the [[fundamental ∞-groupoid]]. These in turn are equivalently [[flat connections]] (this relation is known as the _[[Riemann-Hilbert correspondence]]_) or generally [[flat ∞-connections]].

## Definitions

A notion of [[cohomology]] exists intrinsically within any [[(∞,1)-topos]]. We discuss local systems first in this generality and then look at special cases, such as local systems as ordinary [[sheaves]].

### General

For $\mathbf{H}$ an [[(∞,1)-sheaf (∞,1)-topos]], write

$$
  (LConst \dashv \Gamma)
  :
  \mathbf{H}
   \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
$$

for the [[terminal object|terminal]] [[(∞,1)-geometric morphism]], where $\Gamma$ is the [[global section]] [[(∞,1)-functor]] and $LConst$ the [[constant ∞-stack]]-functor.

Write $\mathcal{S} := core(Fin \infty Grpd) \in $ [[∞Grpd]] for the [[core]] [[∞-groupoid]] of the [[(∞,1)-category]] of finite $\infty$-groupoids. (We can drop the finiteness condition by making use of a higher [[universe]].) This is canonically a [[pointed object]] $* \to \mathcal{S}$, with points the terminal groupoid.

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

For $X \in \mathbf{H}$ an [[object]], a **local system** or _[[locally constant ∞-stack]]_ on $X$ is a morphism

$$
  \tilde \nabla \colon X \longrightarrow LConst \mathcal{S}
$$

in $\mathbf{H}$ or equivalently the object in the [[over-(∞,1)-topos]]

$$
  (P \to X) \in \mathbf{H}/X
$$

that is classified by $\tilde \nabla$ under the [[(∞,1)-Grothendieck construction]]

$$
  \array{
    P &\to& LConst \mathcal{Z}
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{\tilde \nabla}{\to}&  LConst \mathcal{S}
  }
$$

In other words, local systems are [[locally constant ∞-stacks]] or equivalently their classifying [[cocycles]] for [[cohomology with constant coefficients]].

=--

(See [[principal ∞-bundle]] for discussion of how [[cocycle]]s $\tilde \nabla : X \to LConst \mathcal{S}$ classify morphisms $P \to X$.)

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

If $\mathbf{H}$ happens to be a [[locally ∞-connected (∞,1)-topos]] in that there is the further [[left adjoint|left]] [[adjoint (∞,1)-functor]] $\Pi$

$$
  (\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd
$$

we call $\Pi(X)$ the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]. In this case, by the adjunction hom-equivalence we have

$$
  \mathbf{H}(X, LConst \mathcal{S})
  \simeq
  Func(\Pi(X), \mathcal{S})
  \,.
$$

This means that local systems are naturally identified with [[representations]] ($\infty$-[[permutation representation]]s, as it were) of the [[fundamental ∞-groupoid]] $\Pi(X)$:

$$
  Maps(X, LConst \mathcal{S}) \simeq Maps(\Pi(X), \mathcal{S})
  \,.
$$

This is essentially the basic statement around which [[Galois theory]] revolves.

The [[(∞,1)-sheaf (∞,1)-topos]] over a [[locally contractible space]] is locally $\infty$-connected, and many authors identify local systems on such a topological space with representations of its [[fundamental groupoid]].

=--

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

Given a local system $\tilde \nabla : X \to LConst \mathcal{S}$, the cohomology of $X$ with this **local system of coefficients** is
the intrinsic [[cohomology]] of the [[over-(∞,1)-topos]] $\mathbf{H}/X$:

  $$
    H(X,\tilde \nabla)
    :=
    \mathbf{H}_{/X}(X, P_{\tilde \nabla})
    \,,
  $$

where $P_{\tilde\nabla}$ is the [[homotopy fiber]] of $\tilde \nabla$.

=--

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

Unwinding the definitions and using the universality of the [[(∞,1)-pullback]], one sees that a [[cocycle]] $c \in \mathbf{H}(X,\tilde \nabla)$ is a [[diagram]]

$$
  \array{
    X &&\stackrel{c}{\to}&& *
    \\
    & \searrow &\swArrow& \swarrow
    \\
    && LConst \mathcal{S}
  }
$$

in $\mathbf{H}$. This is precisely a [[section]] of the [[locally constant ∞-stack]] $\tilde \nabla$.

=--

### Sheaf-theoretic case

Local systems can also be considered in abelian contexts. One finds the following version of a local system

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

A **linear local system** is a [[locally constant sheaf]] on a [[topological space]] $X$ (or manifold, analytic manifold, or algebraic variety) whose stalk is a finite-dimensional [[vector space]].

=--

Regarded as a sheaf $F$ with values in [[abelian group]]s, such a linear local system serves as the coefficient for [[abelian sheaf cohomology]]. As discussed there, this is in degree $n$ nothing but the intrinsic cohomology of the $\infty$-topos with coefficients in the [[Eilenberg-MacLane object]] $\mathbf{B}^n F$.

+-- {: .num_lemma}
###### Lemma

On a connected topological space this is the same as a sheaf of sections of a finite-dimensional [[vector bundle]] equipped with flat [[connection on a bundle]]; and it also corresponds to the [[representations]] of the [[fundamental group]] $\pi_1(X,x_0)$ in the typical stalk. On an analytic manifold or a variety, there is an equivalence between the category of non-singular coherent
$D_X$-[[D-module|modules]] and local systems on $X$.

=--

## Related concepts

* [[flat vector bundle]], [[Gauss-Manin connection]]

* [[infinity-local system]]

* [[inner local system]]

* [[simplicial local system]]: within Sullivan's (1977) theory of _Infinitesimal computations in topology_, he refers to 'local systems' several times. This seems to be simplicial in nature. [[simplicial local system|This]] entry explores some of the uses of that notion based on Halperin's lecture notes on minimal models

  * D. Sullivan, _Infinitesimal computations in topology_ ([pdf](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1977__47_/PMIHES_1977__47__269_0/PMIHES_1977__47__269_0.pdf))

* [[twisted cohomology]], [[local coefficient bundle]], [[twisted infinity-bundle]]

## References

An early version of the definition of local system appears in

* [[Norman Steenrod]]: _Homology with local coefficients_, Annals 44 (1943) pp. 610 - 627,

This is before the formal notion of [[sheaf]] was published by [[Jean Leray]].  (Wikipedia's entry on [Sheaf theory](http://en.wikipedia.org/wiki/Sheaf_%28mathematics%29#History) is interesting for its historical perspective on this.)

A definition  appears as an exercise in

* {#Spanier66} [[Edwin Spanier]], _Algebraic topology_, McGraw Hill (1966), Springer (1982) [[doi:10.1007/978-1-4684-9322-1](https://link.springer.com/book/10.1007/978-1-4684-9322-1)]

> _A local system on a space $X$ is a [[covariant functor]] from the [[fundamental groupoid]] of $X$ to some [[category]]._ [p. 58]

Then the first major account with discussion of the relation to [[twisted de Rham cohomology]]:

* {#Deligne70} [[Pierre Deligne]], *Equations différentielles à points singuliers réguliers*, Lecture Notes  Math. **163**, Springer (1970) $[$[publications.ias:355](https://publications.ias.edu/node/355)$]$

Textbook accounts:

* {#Voisin02} [[Claire Voisin]] (translated by [[Leila Schneps]]), Section I 9.2.1 of: _[[Hodge theory and Complex algebraic geometry]] I_, Cambridge Stud. in Adv. Math. __76, 77__, 2002/3 ([doi:10.1017/CBO9780511615344](https://doi.org/10.1017/CBO9780511615344))

* [[Alexandru Dimca]], Section 2.5 of: *Sheaves in Topology*, Universitext, Springer (2004) $[$[doi:10.1007/978-3-642-18868-8](https://doi.org/10.1007/978-3-642-18868-8)$]$

See also:

* [[Anatoly Libgober]], [[Sergey Yuzvinsky]], *Cohomology of local systems*, Advanced Studies in Pure Mathematics **27**, Mathematics Society of Japan (2000) 169-184 [[pdf](http://homepages.math.uic.edu/~libgober/otherpapers/export/2000sergeytokyo.pdf), [doi:10.2969/aspm/02710169](https://doi.org/10.2969/aspm/02710169)]

A blog exposition of some aspects of linear local system is developed here:

* [[David Speyer]], _Three ways of looking at a local system_

  * [Introduction and connection to cohomology theories](http://sbseminar.wordpress.com/2009/04/20/three-ways-of-looking-at-a-local-system-introduction-and-connection-to-cohomology-theories/)

  * [the path groupoid approach](http://sbseminar.wordpress.com/2009/04/21/local-systems-the-path-groupoid-approach/)

  * [the infinitesimal perspective](http://sbseminar.wordpress.com/2009/04/30/three-ways-of-looking-at-a-local-system-the-infinitesimal-perspective/)

  * [the infinitesimal site](http://sbseminar.wordpress.com/2009/05/06/the-infinitesimal-site/)

A clear-sighted description of locally constant $(n-1)$-stacks / $n$-local systems as sections of constant $n$-stacks is in

* [[Pietro Polesello]], [[Ingo Waschkies]], *Higher monodromy*, Homology, Homotopy and Applications **7** 1 (2005) 109-150 [[arXiv:0407507](http://arxiv.org/abs/math/0407507)]

for [[locally constant stacks]] on [[topological space]]s. The above formulation is pretty much the evident generalization of this to general [[(∞,1)-topos]]es.

Discussion of [[Galois representations]] as encoding local systems in [[arithmetic geometry]] includes

* {#Lovering} Tom Lovering, _Étale cohomology and Galois Representations_, 2012 ([pdf](http://tlovering.files.wordpress.com/2012/06/essay-body1.pdf))

See also at _[[function field analogy]]_.

[[!redirects local systems]]

[[!redirects local system of coefficients]]
[[!redirects local systems of coefficients]]

[[!redirects cohomology with a local system of coefficients]]


local trivialization

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

#Contents#
* table of contents
{:toc}

## Idea

In the theory of [[fiber bundles]] (including [[principal bundles]], [[vector bundles]], etc.) [[internalization|internal]] to [[TopologicalSpaces]] or [[SmoothManifolds]] etc., the condition that these be _locally trivial_ is typically crucial. This means that locally, i.e. over a [[neighbourhood]] of any [[point]] in their base space, these are equivalent to a [[trivial fiber bundle]], hence to the [[projection]] out of the [[Cartesian product]] of that neighbourhood with a [[fiber]]-space (the _[[typical fiber]]_).

Key implications of local triviality are:

1. the induced bundle isomorphisms between local trivializations on [[intersections]] of their open neighbourhoods give a system of [[transition functions]] which constitute the representation of the given fiber bundle as a [[cocycle]] in [[non-abelian cohomology|non-abelian]] [[Cech cohomology]].

1. the fact that [[Serre fibrations]] are detected locally (see [there](Serre+fibration#LocalRecognition)) implies that any local fiber bundle is a [[Serre fibration]] (and even a [[Hurewicz fibration]] if it is a [[numerable fiber bundle]]).

## Definition

### For topological fiber bundles

For $p \colon E \to X$ a [[bundles]] [[internalization|in]] [[TopologicalSpaces]], hence a [[continuous function]] between [[topological spaces]], then a _local trivialization_ is

1. an [[open cover]] $\mathcal{U} \coloneqq \underset{i \in I}{\sqcup} U_i \longrightarrow X$

1. a [[topological space]] $F$, to be called the [[typical fiber]];

1. an [[isomorphism]] [[over-category|over]] $\mathcal{U}$ between the restriction of $E$ to $\mathcal{U}$ and the [[projection]] out of the [[Cartesian product]] of the cover with the typical fiber:

$$
  \array{
    U \times F
    &\underoverset{t}{\simeq}{\longrightarrow}&
    E|_{U}&\longrightarrow& E
    \\
    &
    \searrow
    &
    \downarrow &{}^{{}_{(pb)}}& \downarrow^{\mathrlap{p}}
    \\
    &&
    \mathcal{U} &\longrightarrow& X
  }
$$

If this exist, then in particular the total square here is a [[pullback square]], and one says that $p$ is _locally trivializable_.

### For equivariant topological fiber bundles

For [[equivariant bundles]] one typically needs a slightly more sophisticated notion, see [there](equivariant+bundle#NotionsOfEquivariantLocalTriviality).

## References

See the references at _[[fiber bundle]]_.

[[!redirects local trivializations]]

[[!redirects locally trivial]]

[[!redirects local trivializability]]

[[!redirects local triviality]]


local-global principle

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

#Contents#
* table of contents
{:toc}

## Idea

It is a common experience in [[mathematics]] that phenomena ("problems") are usefully decomposed into a "local" and a "global" aspect of rather different nature.

In as far as the situation has an interpretation in [[geometry]] then the local aspect is typically given by the ([[formal neighbourhoods|formal]]) [[neighbourhoods]] of all [[points]], while the global aspect is given by the underlying [[topology]]/[[homotopy type]]. The following provides some examples.

### In number theory and algebraic geometry

In [[number theory]] and hence in [[algebraic geometry]] over $Spec(\mathbb{Z})$, the "local" situation is the [[formal neighbourhoods]] of all [[primes]] $p$ (points in the [[prime spectrum]] $Spec(\mathbb{Z})$ of the [[integers]]). These are the [[formal spectra]] of the [[p-adic integers]]. In terms of [[number theory]] this means that a solution to a [[polynomial]] [[equation]] over the [[p-adic integers]] is a "local" approximation to a solution over the [[integers]] (the former is a necessary but not a sufficient condition for the latter to exist). In this form the _local-global principle_ is attributed to [[Helmut Hasse]] and also called the **Hasse principle** or similar.

(...)

### In conformal field theory

The [[FRS-theorem]] shows that every [[rational 2d CFT]] is decomposed into local data given by a [[vertex operator algebra]] and global data given by a solution of the [[sewing constraints]]...

### More

(...)

## Related concepts

* [[descent]]

## References

* [[Barry Mazur]], _On the passage from local to global in number theory_, Bulletin of the AMS Vol 29, Number 1, 1993 ([pdf](http://math.stanford.edu/~lekheng/flt/mazur2.pdf))

* Wikipedia, _[Hasse principle](https://en.wikipedia.org/wiki/Hasse_principle)_

[[!redirects local-to-global principle]]

[[!redirects Hasse principle]]
[[!redirects Hasse local-global principle]]