κ-ary exact category

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#### Regular and Exact categories
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#### Category Theory
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# $\kappa$-ary regular and exact categories
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## Idea

The notions of [[regular category]], [[exact category]], [[coherent category]], [[extensive category]], [[pretopos]], and [[Grothendieck topos]] can be nicely unified in a theory of "familial regularity and exactness."  This was apparently first noticed in [Street 1984](#Street), and expanded on by [Shulman 2012](#Shulman) with a generalized theory of [[exact completion]].

## Sinks and relations

### Definition

Let $C$ be a [[finitely complete category]].  By a *[[sink]]* in $C$ we mean a [[family]] $\{f_i\colon A_i \to B\}_{i\in I}$ of [[morphisms]] with common [[target]].  A sink $\{f_i\colon A_i \to B\}$ is **[[extremal epimorphism|extremal epic]]** if it doesn't factor through any proper [[subobject]] of $B$.  The _[[pullback]]_ of a sink along a morphism $B' \to B$ is defined in the evident way.

By a (many-object) **[[relation]]** in $C$ we will mean a family of objects $\{A_i\}_{i\in I}$ together with, for every $i,j\in I$, a monic span $A_i \leftarrow R_{i j} \to A_j$ (that is, a [[subobject]] $R_{i j}$ of $A_i \times A_j$.  We say such a relation is:

* *[[reflexive relation|reflexive]]* if $R_{i i}$ contains the [[diagonal morphism|diagonal]] $A_i \to A_i \times A_i$, for all $i$,

* *[[transitive relation|transitive]]* if the pullback $R_{i j} \times_{A_j} R_{j k}$ factors through $R_{i k}$, for all $i,j,k$,

* *[[symmetric relation|symmetric]]* if $R_{i j}$ contains, hence is equal to, the transpose of $R_{j i}$ for all $i,j$, and

* a *[[congruence]]* if it is reflexive, transitive, and symmetric; this is an internal notion of (many-object) [[equivalence relation]].

Abstractly, reflexive and transitive relations can be identified with categories [[enriched category|enriched]] in a suitable [[bicategory]]; see [Street 1984](#Street).  Congruences can be identified with enriched $\dagger$-[[dagger-categories|categories]].

A __quotient__ for a relation is a [[colimit]] for the diagram consisting of all the $A_i$ and all the spans $A_i \leftarrow R_{i j} \to A_j$.  And the __kernel__ of a sink $\{f_i\colon A_i\to B\}$ is the relation on $\{A_i\}$ with $R_{i j} = A_i \times_B A_j$.  It is evidently a congruence.

Finally, a sink is called **[[effective epimorphism|effective-epic]]** if it is the quotient of its kernel.  It is called **universally effective-epic** if any pullback of it is effective-epic.

### Examples

* If ${|I|} = 1$, a congruence is the same as the ordinary internal notion of [[congruence]].  In this case [[quotient object|quotients]] and [[kernel pair|kernels]] reduce to the usual notions.

* If ${|I|} = 0$, a congruence contains no data and a sink is just an object in $C$.  The empty congruence is, trivially, the kernel of the empty sink with any target $B$, and a quotient for the empty congruence is an [[initial object]].

* Given a family of objects $\{A_i\}$, define a congruence by $R_{i i}=A_i$ and $R_{i j}=0$ (an initial object) if $i \neq j$.  Call a congruence of this sort _trivial_ (empty congruences are always trivial).  Then a quotient for a trivial congruence is a [[coproduct]] of the objects $A_i$, and the kernel of a sink $\{f_i\colon A_i\to B\}$ is trivial iff the $f_i$ are disjoint monomorphisms.

## $\kappa$-ary regularity and exactness

### Definition

Let $\kappa$ be an [[arity class]].  We call a sink or relation **$\kappa$-ary** if the cardinality ${|I|}$ is $\kappa$-small.  As usual for arity classes, the cases of most interest have special names:

* When $\kappa = \{1\}$ we say **unary**.
* When $\kappa = \omega$ is the set of finite cardinals, we say **finitary**.
* When $\kappa$ is the class of all cardinal numbers, we say **infinitary**.

+-- {: .num_theorem #KRegular}
###### Theorem
For a category $C$, the following are equivalent:

1. $C$ has [[finite limits]], every $\kappa$-ary sink in $C$ factors as an extremal epic sink followed by a monomorphism, and the pullback of any extremal epic $\kappa$-ary sink is extremal epic.

1. $C$ has finite limits, and the kernel of any $\kappa$-ary sink in $C$ is also the kernel of some universally effective-epic sink.

1. $C$ is a [[regular category]] and has [[pullback-stable colimit|pullback-stable]] [[joins]] of $\kappa$-small families of [[subobjects]].
=--

When these conditions hold, we say $C$ is **$\kappa$-ary regular**, or alternatively **$\kappa$-ary coherent**.  There are also some other more technical characterizations; see [Shulman 2012](#Shulman).

+-- {: .num_theorem #KExact}
###### Theorem
For a category $C$, the following are equivalent:

1. $C$ has [[finite limits]], and every $\kappa$-ary congruence is the [[kernel]] of some universally effective-epic sink.

1. $C$ is $\kappa$-ary regular, and every $\kappa$-ary congruence is the kernel of some sink.

1. $C$ is both [[exact category|exact]] and $\kappa$-ary [[extensive category|extensive]].
=--

\begin{definition}\label{KappaAryExactCategory}
When these conditions hold, we say that $C$ is called a **$\kappa$-ary exact category**, or alternatively a **$\kappa$-ary pretopos**.
\end{definition}
\begin{definition}\label{KappaAryExactFunctor}
A [[functor]] $F \,\colon\, C\to D$ between $\kappa$-ary exact categories (Def. \ref{KappaAryExactCategory}) is called a **$\kappa$-ary exact functor** if it [[preserved limit|preserves]] [[finite limits]] and $\kappa$-small effective-epic (or equivalently extremal-epic) families.
\end{definition}

\begin{definition}\label{2CategoryOfKappaAryExactCategories}
**(The 2-category of $\kappa$-ary exact categories)**
\linebreak
The [[2-category]]
$EX_\kappa$
of $\kappa$-ary exact categories (Def. \ref{KappaAryExactCategory}),
$\kappa$-ary exact functors (Def. \ref{KappaAryExactFunctor})
and [[natural transformations]]
is a [[reflective subcategory|reflective]] [[full sub-2-category]] of the 2-category $SITE_\kappa$ of [[∞-ary sites]].  The reflector is called *[[exact completion]]*.
\end{definition}

### Examples

1. $C$ is [[regular category|regular]] iff it is unary regular.
2. $C$ is [[coherent category|coherent]] iff it is finitary regular.
3. $C$ is [[coherent category|infinitary-coherent]] iff it is infinitary regular.
4. $C$ is [[exact category|exact]] iff it is unary exact.
5. $C$ is a [[pretopos]] iff it is finitary exact.
6. $C$ is an [[pretopos|infinitary pretopos]] iff it is infinitary exact.

Some other sorts of [[exactness properties]] (especially [[lex-colimits]]) can also be characterized in terms of congruences, kernels, and quotients.  For instance:

7. $C$ is $\kappa$-ary [[extensive category|lextensive]] iff every $\kappa$-ary _trivial_ congruence has a pullback-stable quotient of which it is the kernel.

In [Street](#Street), there is also a version of regularity and exactness that applies even to some _large_ sinks and congruences, and implies some small-generation properties of the category as well.

### Properties

In a $\kappa$-ary regular category,

* Every extremal-epic $\kappa$-ary sink is the quotient of its kernel.
* Any $\kappa$-ary congruence that is a kernel has a quotient.

Thus, in a $\kappa$-ary exact category,

* Every $\kappa$-ary congruence has a quotient.

In a $\kappa$-ary regular category, the class of all $\kappa$-small and effective-epic families generates a [[topology]], called its $\kappa$-canonical topology.  This topology makes it a [[∞-ary site]].

## References

* {#Street} [[Ross Street]], *The family approach to total cocompleteness and toposes*,  Transactions of the AMS **284** no. 1, 1984. ([doi:10.1090/S0002-9947-1984-0742429-3](https://doi.org/10.1090/S0002-9947-1984-0742429-3))

* {#Shulman} [[Michael Shulman]], *Exact completions and small sheaves*,  *Theory and Applications of Categories*, Vol. 27, 2012, No. 7, pp 97-173.  ([tac:27-07](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html))

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κ-ary site

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# $\kappa$-ary sites

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## Idea

A *$\kappa$-ary site* is a [[site]] whose covering sieves are determined by $\kappa$-small covering families, and which has a very weak sort of finite limits.  These conditions get weaker as $\kappa$ gets larger,  until when $\kappa$ is the size of the universe, every [[small category|small]] site is $\kappa$-ary.

$\kappa$-ary sites are a very general (perhaps the most general) appropriate input for $\kappa$-ary [[exact completion]].

## Definition

Let $\kappa$ be an [[arity class]].

+-- {: .un_defn}
###### Definition
A [[site]] $C$ is **weakly $\kappa$-ary** if for any covering sieve $R$ of an object $V$ in $C$, there exists a $\kappa$-small family $\{p_i: U_i \to V\}_i$ in $C$ such that (1) each $p_i \in R$, and (2) the sieve generated by $\{p_i\}$ is a covering sieve.
=--

This definition can also be rephrased purely in terms of the covering families; see ([Shulman](#Shulman)).

+-- {: .un_defn}
###### Definition
Let $C$ be a site and $G:D\to C$ a [[functor]].  A **local $\kappa$-prelimit** of $G$ is a $\kappa$-small family of [[cones]] $\{q_i: \Delta L \to G \}_i$ in $C$ such that for any cone $r:\Delta u \to G$, the sieve $\{ p: v\to u | r p$ factors through some $q_i \}$ is a covering sieve of $u$.
=--

+-- {: .un_defn}
###### Definition
A **$\kappa$-ary site** is a weakly $\kappa$-ary site which has all *finite* local $\kappa$-prelimits (i.e. whenever $D$ is a [[finite category]]).
=--

## Examples

* If $C$ has a [[trivial topology]], then a local *unary* prelimit (i.e. $\kappa=\{1\}$) is precisely a [[weak limit]].  The trivial topology is always weakly $\kappa$-ary, so a trivial site is unary just when it has weak limits.

* Any limit is, in particular, a local $\kappa$-prelimit.  Thus, any weakly $\kappa$-ary site with finite limits is $\kappa$-ary.

* If the class of *all* cones over $G$ is $\kappa$-ary, then it is a local $\kappa$-prelimit.  Thus, any $\kappa$-small and weakly $\kappa$-ary site is $\kappa$-ary.  In particular, any small site is an *infinitary* site.

* The [[regular topology]] on a [[regular category]] (including an [[exact category]]) is unary.

* The [[coherent topology]] on a [[coherent category]] (including a [[pretopos]]) is finitary.

* Generalizing the previous two examples, the class of all $\kappa$-small and effective-epic families on a [[∞-ary regular category]] (including a [[∞-ary exact category]]) is a $\kappa$-ary topology.  This is called its *$\kappa$-canonical topology*.

* The [[extensive topology]] on a (finitary) [[extensive category]] is finitary.

* The [[canonical topology]] on any [[Grothendieck topos]] is infinitary.

* The [[Zariski topology]] on $CRing^{op}$ is finitary.

## The 2-category of $\kappa$-ary sites

The 2-category $SITE_\kappa$ has $\kappa$-ary sites as its objects, and [[morphisms of sites]] as its morphisms, where we use the more general [[covering-flat functor|covering-flat]] definition of a morphism of sites.

### Properties

* $SITE_\kappa$ is equivalent to a 2-category of [[framed allegory|framed allegories]]; see ([Shulman](#Shulman)).

* $SITE_\kappa$ contains, as a full [[reflective subcategory|reflective]] sub-2-category, the 2-category of [[∞-ary exact categories]] with their $\kappa$-canonical topologies.  The reflector is called [[exact completion]].  When $\kappa$ is the size of the universe, this reflector applied to a small (hence infinitary) site constructs its [[topos of sheaves]].

## References

* [[Michael Shulman]], "Exact completions and small sheaves".  *Theory and Applications of Categories*, Vol. 27, 2012, No. 7, pp 97-173.  [Free online](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html)
{#Shulman}

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π-Kan condition

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#Contents#
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## Idea

The *$\pi_\ast$-Kan condition* ([Bousfield & Friedlander 1978, B.3](#BousfieldFriedlander78)) on a [[bisimplicial set]] guarantees that its [[geometric realization]] behaves well with respect to [[homotopy pullback]] ([Bousfield & Friedlander 1978, Thm. B.4](#BousfieldFriedlander78)).

See [this Prop.](geometric+realization+of+simplicial+topological+spaces#SufficientConditionsForRealizationToPreserveHomotopyPullback) at *[[geometric realization of simplicial topological spaces]]*.

## Related entries

* [[commutativity of homotopy limits and colimits]]

## References

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf), [[BousfieldFriedlanderSpectra.pdf:file]])

* {#Anderson78} [[Donald Werner Anderson]], _Fibrations and geometric realization_,  Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. ([euclid:1183541139](http://projecteuclid.org/euclid.bams/1183541139))

* {#Lurie11} [[Jacob Lurie]], *Simplicial spaces*, Lecture 7 of: *[Algebraic L-theory and Surgery](https://www.math.ias.edu/~lurie/287x.html)*, 2011 ([pdf](https://www.math.ias.edu/~lurie/287xnotes/Lecture7.pdf))

* {#Rezk14} [[Charles Rezk]], _When are homotopy colimits compatible with homotopy base change?_, 2014 ([pdf](https://faculty.math.illinois.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf), [[RezkHomotopyColimitsBaseChange.pdf:file]])

* Edoardo Lanari, _Compatibility of homotopy colimits and homotopy pullbacks of simplicial presheaves_ ([pdf](http://algant.eu/documents/theses/lanari.pdf), [[LanariHomotopyColimitsBaseChange.pdf:file]])

  > (expanded version of [Rezk 14](#Rezk14))

* {#MazelGee14} [[Aaron Mazel-Gee]], _Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces_ ([arXiv:1412.8411](https://arxiv.org/abs/1412.8411))

* {#LurieHigherAlgebra} [[Jacob Lurie]], around Lem. 5.5.6.17 in: *[[Higher Algebra]]* (~2017)

* [[Jacob Lurie]], Thms. A.5.4.1, *[[Spectral Algebraic Geometry]]* (2018) [[pdf](https://math.harvard.edu/~lurie/papers/SAG-rootfile.pdf)]

* {#BEBP19} [[Daniel Berwick-Evans]], [[Pedro Boavida]], [[Dmitri Pavlov]], §3 of: _Classifying spaces of infinity-sheaves_, Algebraic & Geometric Topology  [[arXiv:1912.10544](https://arxiv.org/abs/1912.10544)]

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Π-pretopos

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## Definition

A **$\Pi$-pretopos** is a [[pretopos]] that is also a [[locally cartesian closed category]].

## Related concepts

* A [[ΠW-pretopos]] is a $\Pi$-pretopos that also has all [[W-types]].

* A [[predicative topos]] is $\Pi W$-pretopos with [[axiom of multiple choice]].

## References

See at _[[predicative topos]]_.

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ΠW-pretopos

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## Definition

A **$\Pi W$-pretopos** is a [[Π-pretopos]] with all [[W-types]].

## Examples
 {#Examples}

* In [[homotopy type theory]] the [[h-sets]] form a ΠW-pretopos ([Rijke-Spitter 13](#RijkeSpitter13)). See also at _[[structural set theory]]_.

* The [[inequality spaces]] and [[strongly extensional functions]] form a ΠW-pretopos.

## Related concepts

* A [[predicative topos]] is a $\Pi W$-pretopos satisfying the [[axiom of multiple choice]].

## References

See also at _[[predicative topos]]_.

That the [[h-sets]] in [[homotopy type theory]] form a ΠW-pretopos is discussed in

* {#RijkeSpitters13} [[Egbert Rijke]], [[Bas Spitters]], _Sets in homotopy type theory_, Mathematical Structures in Computer Science, Volume 25, Issue 5 (From type theory and homotopy theory to Univalent Foundations of Mathematics)  ([arXiv:1305.3835](http://arxiv.org/abs/1305.3835))

which became one chapter in

* [[Univalent Foundations Project]], _[[Homotopy Type Theory – Univalent Foundations of Mathematics]]_

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σ-meson

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## Idea

The _sigma-meson_, now also called the _$f_0(500)$-resonance_, is the [[light meson|lightest]] [[Lorentz group|Lorentz]]-[[scalar]] and [[isospin]]-[[scalar]] [[meson]].

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

Together with the [[omega-meson]] the sigma is responsible for most of the long-range [[interaction]] between [[baryons]], exhibiting the residual [[strong nuclear force]] between them, at least as modeled by the [[Walecka model]] and [[quantum hadrodynamics]].

Other models of [[quantum hadrodynamics]] do not contain the sigma at all. See [Pelaez 16](#Pelaez16) for review of the controversial and inconclusive understanding of the sigma-meson.

## Related concepts

* [[Walecka model]]

## References

### General

Original articles:

* [[Murray Gell-Mann]], M. Lévy, _The axial vector current in beta decay_, Il Nuovo Cimento (1955-1965) volume 16, pages 705–726 (1960) ([doi:10.1007/BF02859738](https://doi.org/10.1007/BF02859738))

* [[Julian Schwinger]], _A Theory of the Fundamental Interactions_, Annals Phys. 2 (1957) 407-434 ([spire:12286](http://inspirehep.net/record/12286/))

Review:

* {#Pelaez16} J. R. Pelaez, _From controversy to precision on the sigma meson: a review on the status of the non-ordinary $f_0(500)$ resonance_, Physics Reports 658 (2016) 1 ([arXiv:1510.00653](https://arxiv.org/abs/1510.00653))

See also:

* Eef van Beveren, Frieder Kleefeld, George Rupp, Michael D. Scadron, _Remarks on the $f_0(400-1200)$ scalar meson as the dynamically generated chiral partner of the pion_, Mod. Phys. Lett. A17:1673, 2002 ([arXiv:hep-ph/0204139](https://arxiv.org/abs/hep-ph/0204139))

* M. Albaladejo, J.A. Oller, _On the size of the sigma meson and its nature_, Phys. Rev. D 86, 034003 – Published 3 August 2012 ([arXiv:1205.6606](https://arxiv.org/abs/1205.6606), [doi:10.1103/PhysRevD.86.034003](https://doi.org/10.1103/PhysRevD.86.034003))

* S. S. Afonin, T. D. Solomko, _The sigma meson from QCD sum rules for large-$N_c$ Regge spectra_, Eur. Phys. J. C 76, 678 (2016) ([arXiv:1608.08131](https://arxiv.org/abs/1608.08131))

* S. S. Afonin, T. D. Solomko, _The linear radial spectrum of scalar mesons within the QCD sum rules in the planar limit_, Eur. Phys. J. C 76, 678 (2016) ([arXiv:1608.08131](https://arxiv.org/abs/1608.08131))

* Physics.SE discussion, _[What the heck is the sigma (f0) 600?](https://physics.stackexchange.com/q/13458/5603)_

* Christian S. Fischer, Paul C. Wallbott, Richard Williams, Nico Santowsky, Gernot Eichmann, _The $\sigma$-meson: four-quark vs. two-quark components and decay width in a Bethe-Salpeter approach_ ([arXiv:2007.06495](https://arxiv.org/abs/2007.06495))

Inclusion of the [[sigma-meson]] into the [[Skyrme model]] for [[baryons]]:

* Thomas D. Cohen, _Explicit $\sigma$ meson, topology, and the large-$N$ limit of the Skyrmion_, Phys. Rev. D 37 (1988) ([doi:10.1103/PhysRevD.37.3344](https://doi.org/10.1103/PhysRevD.37.3344))

### In holographic QCD

Discussion in [[holographic QCD]]:

On [[sigma-mesons]] and [[omega-mesons]] mediating [[baryon]] [[interaction]], discussed in [[holographic QCD]] via [[D3-D7 brane intersections]]:

* Anatoly Dymarsky, Dmitry Melnikov, [[Jacob Sonnenschein]], _Attractive Holographic Baryons_, JHEP 06 (2011) 145 ([arXiv:1012.1616](https://arxiv.org/abs/1012.1616))

and discussed in the [[Witten-Sakai-Sugimoto model]] on [[D4-D8 brane intersections]]:

* [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], _Searching for an Attractive Force in Holographic Nuclear Physics_, JHEP 05 (2011) 058 ([arXiv:1003.2621](https://arxiv.org/abs/1003.2621))

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ω-category > history

< [[∞-category]]
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[[!redirects ∞-category -- history]]


ω-meson

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* table of contents
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## Idea

In [[nuclear physics]], specifically in the [[chiral perturbation theory]] of [[quantum chromodynamics]], the _omega-meson_ is the [[isospin]]-[[trivial representation|singlet]] [[vector meson]] [[field (physics)|field]] in the first-[[generation of fermions]], i.e. a [[bound state]] of an [[up quark]] and a [[down quark]] (a [[light meson]]), the [[chiral partner]] of the [[f1-meson]]:

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

\linebreak

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

## Properties

### Nuclear binding

Together with the [[sigma-meson]] the omega is responsible for most of the long-range [[interaction]] between [[baryons]], exhibiting the residual [[strong nuclear force]] between them (as modeled by [[Walecka model]] and [[quantum hadrodynamics]]).

### Couplings

#### The $\omega$-$3 \pi$-coupling
 {#TheOmega3PiCoupling}

The [[interaction]] term of the omega-meson three [[pions]] is, in the [[Lagrangian density]], given by contraction

$$
  \omega_\mu B^\mu
$$

with the [[chiral anomaly|chirally anomalous]] [[baryon current]] $B \coloneqq \star tr( (U^{-1} d U) \wedge (U^{-1} d U) \wedge (U^{-1} d U) )$, with $U$ is the exponential of the [[pion]]-[[field (physics)|field]] ([Adkins-Nappi 84, (1) and (2)](#AdkinsNappi84), [Park-Vento 09, (5.5.43) and above (5.5.50)](#ParkVento09)).

This gives a [[decay mode]]

$$
  \omega \to \pi^+ + \pi^- + \pi^0
$$

(the "charged decay", e.g. [Rudaz 84, (2)](#Rudaz84)).

Or rather, this is the direct (contact term) decay. The net process $\omega \to 3 \pi$ is dominated by the successive decay

$$
  \omega \to \rho + \pi
  \to
  (2 \pi) + \pi
$$

#### The $\omega$-$\rho$-$\pi$-coupling
 {#TheOmegaRhoPiCoupling}

Then there is an [[omega-meson|ω]]-[[rho-meson|ρ]]-[[pion|π]]-coupling given by the anomalous part of the [[chiral perturbation theory|chiral]] [[WZW-term]]:

$$
  g_{\omega \rho \pi}
  \epsilon^{\mu \nu \kappa \lambda}
  \partial_\mu \omega_\nu  \partial_\kappa \rho_\lambda \cdot \pi
$$

(e.g. [Renard 69](#Renard69), [Meissner-Kaiser-Weise 87, (2.18)](#MeissnerKaiserWeise87) [Volkov-Ebert-Nagy 97, p. 12](#VolkovEbertNagy97), [Guetta-Singer 00, (1)](#GuettaSinger00),
[Kaiser 00, (12)](#Kaiser00), [GKSY 03, (1)](#GKSY03) [Gudino-Sanchez 12, (1)](#GudinoSanchez12))

#### The radiative decays

Then there is the "neutral decay"

$$
  \omega \to \pi^0 + \gamma
$$

seen in [[experiment]] as

$$
  \begin{aligned}
    \omega & \to \pi^0 + \gamma
    \\
    & \to \gamma + \gamma + \gamma
  \end{aligned}
$$

([Nambu 57, (a)](#Nambu57), [FFHNR 67](#FFHNR67), [Dolinsky et al. 89,
(5)](#Dolinsky89))

## Related concepts

* [[pion]], [[rho-meson]]

## References

### General

The $\omega$-meson was first postulated by

* {#Nambu57} [[Yoichiro Nambu]], _Possible Existence of a Heavy Neutral Meson_, Phys. Rev. 106, 1366 (1957) ([doi:10.1103/PhysRev.106.1366](https://doi.org/10.1103/PhysRev.106.1366))

as reviewed in

* [[Jun John Sakurai]], p. 48-49 of: _Currents and Mesons_, Chicago Lectures in Physics, based on notes by George Barry, University of Chicago Press (1969) ([ISBN: 9780226733838](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3622598.html))

  (in the context of [[vector meson dominance]])

See also:

* Wikipedia, _[Omega meson](https://en.wikipedia.org/?title=Omega_meson)_

See also

* Reinhard Alkofer, H. Reinhardt, H. Weigel, U. Zuckert, _The Isoscalar vector meson omega in the Nambu-Jona-Lasinio soliton_, Phys. Lett. B298 (1992) 132-140 ([spire:343052](http://inspirehep.net/record/343052), doi:10.1016/0370-2693(93)91719-4)

Phenomenology:

* Cheng-Qun Pang, Ya-Rong Wang, Jing-Fu Hu, Tian-Jie Zhang, Xiang Liu, _Study of the $\omega$ meson family and newly observed $\omega$-like state $X(2240)$_ ([arXiv:1910.12408](https://arxiv.org/abs/1910.12408))

* M. K. Volkov, A. A. Pivovarov, K. Nurlan, _On the mixing angle of the vector mesons $\omega(782)$ and $\phi(1020)$_ ([arXiv:2005.00763](https://arxiv.org/abs/2005.00763))

### Decays

The direct decay $\omega \to \pi^0 + \pi^+ + \pi^-$:

* {#Rudaz84} S. Rudaz, _Anomalies, vector mesons and the $\omega \to 3 \pi$ contact term_, Phys. Lett. B 145 (1984) 281-284 ([spire:208193](https://inspirehep.net/literature/208193), doi:10.1016/0370-2693(84)90355-1)

* E. A. Kuraev, Z. K. Silagadze, _Once more about the $\omega \to 3 \pi$ contact term_, Phys. Atom. Nucl. 58:1589-1596, 1995 ([arXiv:hep-ph/9502406](https://arxiv.org/abs/hep-ph/9502406))

* M. Albaladejo, I. Danilkin, S. Gonzalez-Solis, D. Winney, C. Fernandez-Ramirez, A. N. Hiller Blin, V. Mathieu, M. Mikhasenko, A. Pilloni, A. Szczepaniak, _$\omega \to 3\pi$ and $\omega \pi^0$ transition form factor revisited_ ([arXiv:2006.01058](https://arxiv.org/abs/2006.01058))

The $\omega \pi \rho$-coupling

* {#GudinoSanchez12} D. Garcia Gudino, G. Toledo Sanchez, _The $\omega \rho \pi$ coupling in the VMD model revisited_, Int. J. Mod. Phys. A 27, 1250101 (2012) ([arXiv:1106.1467](https://arxiv.org/abs/1106.1467))

On [[Dalitz decays]] of omega-mesons:

* {#Wachs00} Mirko Wachs, _Die Selbstenergie des Omega-Mesons_, 2000 ([epda:000050](http://elib.tu-darmstadt.de/diss/000050))

* {#Berghauser10} Henning Berghäuser, _Investigation of the Dalitz decays and the electromagnetic form factors of the $\eta$ and $\pi^0$-meson_, 2010 ([spire:1358057](https://inspirehep.net/literature/1358057))

[[!include Skyrme hadrodynamics with vector mesons -- references]]

### Couplings

On [[omega-meson]] [[interactions]] and [[decay modes]]:

* Stanley M. Flatté, Darrell O. Huwe, Joseph J. Murray, Janice Button-Shafer, Frank T. Solmitz, M. Lynn Stevenson, and Charles Wohl, _Decay Properties of the $\omega$ Meson_, Phys. Rev. 145, 1050 – Published 27 May 1966 ([doi:10.1103/PhysRev.145.1050](https://doi.org/10.1103/PhysRev.145.1050))

* {#FFHNR67} M. Feldman, W. Frati, R. Gleeson, J. Halpern, M. Nussbaum, S. Richert, _Neutral Decay of the $\omega$ Meson_, Phys. Rev. 159, 1219 (1967) ([doi10.1103/PhysRev.159.1219](https://doi.org/10.1103/PhysRev.159.1219), [spire:52556](http://inspirehep.net/record/52556))

* W. Deinet A. Menzione H.Müller, H. M.Staudenmaier, S.Buniatov, D.Schmitt, _Neutral decay modes of the $\omega^0$-meson_, Physics Letters B
Volume 30, Issue 6, 10 November 1969, Pages 426-429 (doi:10.1016/0370-2693(69)90479-1)

* {#Renard69} F. M. Renard, _The reaction $e^+ e^-  \to \pi^0 + \omega(\pi^+ \pi^- \pi^0) $ and the $\omega$-$\rho$-$\pi$ coupling_, Nuovo Cimento A (1965-1970) 64, 979–984 (1969) ([doi:10.1007/BF02758844](https://doi.org/10.1007/BF02758844))

* {#VolkovEbertNagy97} M. K. Volkov, D. Ebert, M. Nagy, _Excited pions, $\rho$- and $\omega$-mesons and their decays in a chiral $SU(2) \times SU(2)$ Lagrangian_, Int. J. Mod. Phys. A13 (1998) 5443-5458 ([arXiv:hep-ph/9705334](https://arxiv.org/abs/hep-ph/9705334))

* {#Dolinsky89} S. I. Dolinsky, et al., _Radiative Decays of $\rho$ and $\omega$ Mesons_, Z. Phys. C42 (1989) 511 ([spire:264694](http://inspirehep.net/record/264694), [doi:10.1007/BF01557655](https://doi.org/10.1007/BF01557655))

* J. T. Dakin, M. G. Hauser, M. N. Kreisler, R. E. Mischke, _Measurement of the Branching Ratios for ω Neutral Decays_, Phys. Rev. D 6, 2321 (1972) ([doi:10.1103/PhysRevD.6.2321](https://doi.org/10.1103/PhysRevD.6.2321))

* {#GuettaSinger00} Dafne Guetta, Paul Singer, _$\omega$-$\rho$ Mixing and the $\omega \to \pi \pi \gamma$ Decay_, Phys. Rev. D63 (2001) 017502 ([arXiv:hep-ph/0005059](https://arxiv.org/abs/hep-ph/0005059))

* {#Kaiser00} Roland Kaiser, equation (12) of: _Anomalies and WZW-term of two-flavour QCD_, Phys. Rev. D63:076010, 2001 ([arXiv:hep-ph/0011377](https://arxiv.org/abs/hep-ph/0011377))

* {#GKSY03} A. Gokalp, A. Kucukarslan, S. Solmaz, O. Yilmaz, _$\sigma$-Meson and $\omega$-$\rho$ mixing effects in $\omega \to \pi^+ \pi^- \gamma$ decay_, Acta Phys.Polon. B34 (2003) 4095-4104 ([arXiv:hep-ph/0306044](https://arxiv.org/abs/hep-ph/0306044))

* [[Jeffrey Harvey]], Christopher T. Hill, Richard J. Hill, Section II.B of: _Standard Model Gauging of the WZW Term: Anomalies, Global Currents and pseudo-Chern-Simons Interactions_, Phys. Rev. D77:085017, 2008 ([arXiv:0712.1230](https://arxiv.org/abs/0712.1230))

* S. Leupold, M. F. M. Lutz, _Hadronic three-body decays of light vector mesons_, Eur. Phys. J. A39:205-212, 2009 ([arXiv:0807.4686](https://arxiv.org/abs/0807.4686))

* Florian Jonas, _Measurement of $\omega$ and $\eta$ mesons via their three pion decay with ALICE in pp collisions at $\sqrt{s} = t TeV$_, 2018 ([cds:2653176](https://cds.cern.ch/record/2653176))

### In holographic QCD

The [[omega-meson]] in [[holographic QCD]] ([[Witten-Sakai-Sugimoto model]]):

* {#SakaiSugimoto05} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], Section 4.3 of: _More on a holographic dual of QCD_, Progr. Theor. Phys. 114: 1083-1118, 2005 ([arXiv:hep-th/0507073](https://arxiv.org/abs/hep-th/0507073))

### Mediating baryon interaction

On [[sigma-mesons]] and [[omega-mesons]] mediating [[baryon]] [[interaction]], discussed in [[holographic QCD]] via [[D3-D7 brane intersections]]:

* Anatoly Dymarsky, Dmitry Melnikov, [[Jacob Sonnenschein]], _Attractive Holographic Baryons_, JHEP 06 (2011) 145 ([arXiv:1012.1616](https://arxiv.org/abs/1012.1616))

[[!redirects ω-mesons]]

[[!redirects omega-meson]]
[[!redirects omega-mesons]]
[[!redirects omega meson]]
[[!redirects omega mesons]]

[[!redirects Omega-meson]]
[[!redirects Omega-mesons]]
[[!redirects Omega meson]]
[[!redirects Omega mesons]]


Е. Б. Богомольный

[[!redirects Е. Б. Богомольний]]

Evgeny Bogomolny

* [webpage](http://www.itp.ac.ru/en/persons/bogomolny-evgeny-borisovich)

## Papers and related $n$Lab entries

* [[BPS state]]
* E. B. Bogomolny, _Stability of slassical solutions_, Sov. J. Nucl. Phys. 24 (1976), 449; Yad. Fiz. 24 (1976), 861

category: people

[[!redirects Evgeny Bogomolny]]

[[!redirects Evgeny Bogomolnyj]]
[[!redirects E. B. Bogomolny]]


Павел Мнёв

[[!redirects Pavel Mnev]]

* [webpage](http://www.pdmi.ras.ru/~pmnev/)

## Selected writings

On [[BV quantization]]:

* [[Alberto S. Cattaneo]], [[Pavel Mnev]], [[Michele Schiavina]], *BV Quantization*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2307.07761](https://arxiv.org/abs/2307.07761)]

category: people


שנה טובה > history

< [[שנה טובה]]


’t Hooft operator

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

## Idea

In [[gauge theory]], where a [[Wilson line]] is a curve in ambient [[spacetime]] with a [[gauge field]] [[holonomy]] around the curve, dually a _'t Hooft operator_ is a curve with [[Dirac monopole]]-like [[singularity]] of the ambinent [[gauge field]] along it (hence they may be thought of as 1-dimensional distributions of [[magnetic charge]]).

review includes ([Kapustin-Witten 06, section 6.2](#KapustinWitten06))

At least in the additional presence of [[Higgs bundle]] fields the singularity makes the [[field strength]] [[curvature]] that of a [[differential form with logarithmic singularities]] along the specified curve

([Kapustin-Witten 06, (6.8), (6.9)](#KapustinWitten06))

## Properties

### Relation to S-duality and geometric Langlands correspondence

Under the identification of the [[geometric Langlands correspondence]] with aspects of [[S-duality]] in [[super Yang-Mills theory]], the t Hooft operators correspond to [[Hecke operator]] ([Kapustin-Witten 06, section 9](#KapustinWitten06)).

[[!include geometric Langlands QFT -- table]]

## References

The original definition is due to

* [[Gerard 't Hooft]], _On the phase transition towards permanent quark confinement_, Nuclear Physics : B, volume: 138, issue: 1 (1978), pp. 1 - 25 ([igitur](http://dspace.library.uu.nl/handle/1874/4701))

Review includes

* Wikipedia, _['t Hooft operator](http://en.wikipedia.org/wiki/'t_Hooft_operator)_

Discussion in the context of [[S-duality]] is in

* {#Kapustin06} [[Anton Kapustin]], _Wilson-'t Hooft operators in four-dimensional gauge theories and S-duality_, Phys.Rev. D74 (2006) 025005 ([arXiv:hep-th/0501015](http://arxiv.org/abs/hep-th/0501015))

and further discussion of this relating to the [[geometric Langlands correspondence]] is in

* {#KapustinWitten06} [[Anton Kapustin]], [[Edward Witten]], _Electric-Magnetic Duality And The Geometric Langlands Program_, Communications in Number Theory and Physics Volume 1 (2007) Number 1 ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))

[[!redirects 't Hooft operators]]

[[!redirects 't Hooft operator]]
[[!redirects 't Hooft operators]]

[[!redirects 't Hoof line]]
[[!redirects 't Hoof lines]]


ℓ-adic cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

$\ell$-Adic cohomology is a [[cohomology theory]] on suitable [[varieties]] which is constructed as an [[inverse limit]] of [[étale cohomology]] over different [[coefficients]].

$\ell$-Adic cohomology is a [[Weil cohomology theory]]. It is endowed with a Galois action and therefore gives us a way of obtaining [[Galois modules]] from [[algebraic varieties]].

## Definition

### Over the étale site
 {#OverEtaleSite}

Let $X$ be a (smooth?) [[proper map|proper]] [[variety]] over a field. Fix $\ell$ a [[prime number]] different from the [[characteristic]] of $k$. The $\ell$-adic cohomology is defined to be the [[cohomology theory]] on the [[étale site]] given by the [[inverse limit]] over $n \in \mathbb{N}$

$$
  H^j_{et}(X_{\overline{k}}, \mathbb{Q}_\ell)
  \coloneqq
 \lim_n
  H^j_{et}(X_{\overline{k}}, \mathbb{Z}/\ell^n\mathbb{Z})\otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_\ell
$$

of [[étale cohomology]] with [[coefficients]] in $\mathbb{Z}/\ell^n\mathbb{Z}$.

The key insights into getting finite dimensionality with coefficients in a [[field]] of [[characteristic]] $0$ when $k$ has positive characteristic is to first base change to $\overline{k}$ to make the theory "geometric." Then only work with torsion sheaves so that appropriate finiteness theorems for [[étale cohomology]] of proper varieties can be used, and then pass to the limit.

Notice that on the left $\mathbb{Q}_{\ell}$ is not an actual sheaf whose actual [[sheaf cohomology]] is being computed instead the expression on the left is defined by the genuine sheaf cohomology groups on the right.

### Over the pro-étale site

This is rectified by passing to [[pro-étale cohomology]]. Here $\mathbb{Q}_{\ell}$ exists as an actual sheaf and its genuine [[abelian sheaf cohomology]] is $\ell$-adic cohomology.
([Bhatt-Scholze 13, section 6.8](#BhattScholze13))

## Related concepts

* [[Weil cohomology]]

  * [[étale cohomology]], [[basics of étale cohomology]]

  * [[pro-étale cohomology]]

* [[p-adic integers]]

* [[p-adic geometry]]

* [[p-adic homotopy]]

* [[p-adic cohomology]]

* [[p-adic physics]]

## References

A textbook account is in

* [[James Milne]], section 19 of _[[Lectures on Étale Cohomology]]_ ([web] (http://www.jmilne.org/math/CourseNotes/lec.html))

Surveys include

* [[Akhil Mathew]], _$l$-adic cohomology and exponential sums_ ([web](http://amathew.wordpress.com/tag/l-adic-cohomology/))

A variant of the [[étale site]], well adapted to the needs of $\ell$-adic cohomology, the _[[pro-étale site]]_ (locally contractible in some sense) is discussed in

* Bhargav Bhatt, [[Peter Scholze]], _The pro-étale topology for schemes_ ([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))
 {#BhattScholze13}

[[!redirects l-adic cohomology]]


∞-category > history


∞-group cohomology

#Contents#
* table of contents
{:toc}

## Idea

For $\mathbf{H}$ an [[∞-topos]] and for $G \in Grp(\mathbf{H})$ an [[∞-group]] in $\mathbf{H}$, the higher analog of [[group cohomology]] of $G$ is the [[cohomology]] of the [[delooping]] object $\mathbf{B}G \in \mathbf{H}$.

Given a [[cocycle]] $\mathbf{B}G \longrightarrow \mathbf{B}^{n+1} A$ in the $\infty$-group cohomology of $G$, then its [[homotopy fiber]] (the [[principal ∞-bundle]] over $\mathbf{B}G$ that it modulates) is the corresponding [[∞-group extension]].

## Examples

* [[∞-Lie groupoid cohomology]]

## Related concepts

[[!include homotopy type representation theory -- table]]

[[!redirects infinity-group cohomology]]

[[!redirects ∞-group cocycle]]
[[!redirects ∞-group cocycles]]
[[!redirects infinity-group cocycle]]
[[!redirects infinity-group cocycles]]


∞-group ∞-ring

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[group object]] in the context of [[homotopy theory]] then the [[free construction]] of a [[ring object]] from it in [[stable homotopy category]] is the corresponding "[[group ring]]" construction generalized to homotopy theory.

## Definition

One may consider the construction at various levels of algebraic structure:

* for [[∞-groups]] and [[E-∞ rings]]: _[∞-Group E-∞ rings](InfinityGroupInfintyRings)_

* for [[H-groups]] and [[ring spectra]]: _[H-group ring spectra](#HGroupRingSpectra)_

### $\infty$-Group $\infty$-rings
 {#InfinityGroupInfintyRings}

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

Write

$$
  gl_1 \; \colon \;  CRing_\infty \to AbGrp_\infty
$$

for the [[(∞,1)-functor]] which sends a [[E-∞ ring|commutative ∞-ring]] to its [[∞-group of units]].

=--

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

The [[∞-group of units]] [[(∞,1)-functor]] of def. \ref{GroupOfUnitsFunctor} is a  right-[[adjoint (∞,1)-functor]]

$$
  CRing_\infty
  \stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}}
  AbGrp_\infty
  \,.
$$

=--

This is ([ABGHR 08, theorem 2.1/3.2, remark 3.4](#ABGHR08)).

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

The [[left adjoint]]

$$
  \mathbb{S}[-] \colon AbGrp_\infty \to CRing_\infty
$$

is a higher analog of forming the [[group ring]] of an ordinary [[abelian group]] over the [[integers]]

$$
  \mathbb{Z}[-] \colon Ab \to CRing
  \,,
$$

which is indeed [[left adjoint]] to forming the ordinary [[group of units]] of a ring.

We might call $\mathbb{S}[A]$ the **[[∞-group ∞-ring]]** of $A$ over the [[sphere spectrum]].

=--

### H-Group ring spectra
 {#HGroupRingSpectra}

We consider here the simpler concept after passage to [[equivalence classes]].

Recall

the [[classical homotopy category]] $(Ho(Top), \times, \ast)$ which is a [[symmetric monoidal category]] with respect to forming [[Cartesian product|Cartesian]] [[product spaces]] ([[tensor unit]] is the [[point space]])

its [[pointed objects]] version $(Ho(Top^{\ast/}), \wedge, S^0)$, which is a [[symmetric monoidal category]] with respect to [[smash product]] of [[pointed topological spaces]] ([[tensor unit]] is the [[0-sphere]])

the [[stable homotopy category]] $(Ho(Spectra), \wedge, \mathbb{S})$ which is a [[symmetric monoidal category]] with respect to the [[smash product of spectra]] ([[tensor unit]] is the [[sphere spectrum]])

There is a [[free-forgetful adjunction]]

$$
  Ho(Top^{\ast/})
    \underoverset
      {\underset{U}{\longrightarrow}}
      {\overset{(-)_+}{\longleftarrow}}
      {\bot}
  Ho(Top)
$$

The [[left adjoint]] functor $(-)_+$ adjoins basepoint. This is a [[strong monoidal functor]] (by [this example](pointed+object#WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces)) in that there is a [[natural isomorphism]]

$$
  (X \times Y)_+ \simeq X_+ \wedge Y_+
  \,.
$$

Then there is the [[stabilization]] [[adjunction]]

$$
  Ho(Spectra)
    \underoverset
      {\underset{\Omega^\infty}{\longrightarrow}}
      {\overset{\Sigma^\infty}{\longleftarrow}}
      {}
  Ho(Top^{\ast/})
$$

(by [this prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsIndeedStabilizationOfClassicalHomotopyCategory))

Again the left adjoint is a [[strong monoidal functor]] in that there is a [[natural isomorphism]]

$$
  \Sigma^\infty( (X,x) \wedge (Y,y) )
    \simeq
  \left(\Sigma^\infty (X,x)\right)
    \wedge
  \left(
    \Sigma^\infty (Y,y)
  \right)
$$

(by [this prop](Introduction+to+Stable+homotopy+theory+--+1-2#SuspensionSpectrumStructuredStrictQuillenAdjunction))

Accordingly also the composite functor

$$
  \mathbb{S}[-] \coloneqq \Sigma^\infty(-)_+
$$

given by

$$
  Ho(Spectra)
    \underoverset
      {\underset{\Omega^\infty}{\longrightarrow}}
      {\overset{\Sigma^\infty}{\longleftarrow}}
      {}
  Ho(Top^{\ast/})
    \underoverset
      {\underset{U}{\longrightarrow}}
      {\overset{(-)_+}{\longleftarrow}}
      {\bot}
  Ho(Top)
$$

are strong monoidal.

That $\mathbb{S}[-] = \Sigma^\infty((-)_+)$ is [[strong monoidal functor]] for a [[monoid]] $(A,\mu)$ in $(Ho(Top), \times , \ast)$ (an [[H-space]]), then also

$$
  \Sigma^\infty (A_+)
  \;\in\;
  Ho(Spectra)
$$

is a monoid. This is the "monoid ring spectrum" of $A$.

If $A = G$ is in fact a [[group object]] in $(Ho(Top),\times, \ast)$ (hence an [[H-group]]), then we may call the ring spectrum

$$
  \Sigma^\infty(G_+)
    \;\in \;
  Ho(Spectra)
$$

the _H-group ring spectrum_ of $G$.

+-- {: .num_remark #HGroupRingSpectrumSplitsAsDirectSumWithSphereSpectrum}
###### Remark
**(H-group ring spectrum is a direct sum with the [[sphere spectrum]])**

Notice that an [[H-group]] $G$ already is canonically a [[pointed object]] itself, pointed by its [[neutral element]] $e \colon \ast \to G$. Regarded as an object $(G,e) \in Ho(Top^{\ast/})$ this way then the pointed object $G_+$ above is equivalently the [[wedge sum]] of $G$ with the [[0-sphere]]:

$$
  G_+ \simeq (G,e) \vee S^0
  \,.
$$

Since $\Sigma^\infty$ preserves [[wedge sum]], this means that there is an [[isomorphism]] in $Ho(Spectra)$

$$
  \begin{aligned}
    \Sigma^\infty(G_+)
      & \simeq
     \Sigma^\infty
     \left(
      (G,e)
      \vee
       S^0
    \right)
    \\
    & \simeq
    \left( \Sigma^\infty(G,e)  \right) \vee \mathbb{S}
    \\
    & \simeq
    \left( \Sigma^\infty(G,e)  \right) \oplus \mathbb{S}
  \end{aligned}
$$

(where the last isomorphism exhibits that [[wedge sum]] is the [[direct sum]] in the [[additive category]] $Ho(Spectra)$ (by [this lemma](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryHasCoproducts))).

=--

## Examples

* The localization of the $E_\infty$-group ring $\Sigma^\infty(BU(1)_+)$ of the  [[circle 2-group]] at the [[Bott element]] $\beta$ is equivalently the [[Brown representability theorem|representing]] [[spectrum]] [[KU]] of complex [[topological K-theory]]:

  $\mathbb{S}[B U(1)][\beta^{-1}]  \simeq KU$

  This is _[[Snaith's theorem]]_.

## References

* {#ABGHR08} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra and Thom spectra_ ([arXiv:0810.4535](http://arxiv.org/abs/0810.4535))

[[!redirects H-group ring spectrum]]
[[!redirects H-group ring spectra]]

[[!redirects H-space ring spectrum]]
[[!redirects H-space ring spectra]]

[[!redirects ∞-group ∞-rings]]

[[!redirects ∞-group E-∞-ring]]
[[!redirects ∞-group E-∞-rings]]

[[!redirects ∞-group E-∞ ring]]
[[!redirects ∞-group E-∞ rings]]

[[!redirects ∞-group E-infinity-ring]]
[[!redirects ∞-group E-infinity-rings]]

[[!redirects ∞-group E-infinity ring]]
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道德經

[[!redirects 道德经]]

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### Context
#### Philosophy
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[[!include philosophy - contents]]
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# Tao Te Ching
* table of contents
{: toc}

(...)

## On nothing, being and becoming

On [[being]], [[nothing]] and [[becoming]] (English translation following [[Xiao-Gang Wen]] [here](http://physics.stackexchange.com/a/94612/5603)):

> The nameless nonbeing is the origin of universe;

> The named being is the mother of all observed things.

> Within nonbeing, we enjoy the mystery of the universe.

> Among being, we observe the richness of the world.

> Nonbeing and being are two aspects of the same mystery.

> From nonbeing to being and from being to nonbeing is the gateway to all understanding.

Notice the similarity to ([Hegel 12](#Hegel12)), for more on this see ([Hegel 27](#Hegel27)).

## Related entries

* [[ontology]], [[metaphysics]]

* [[logos (in philosophy)]], [[Nous]]

* [[hermeticism]]

## References

* Wikipedia, _[Tao Te Ching](http://en.wikipedia.org/wiki/Tao_Te_Ching)_

* Bureau of Public Secrets _[Lao Tzu: Tao Te Ching](http://www.bopsecrets.org/gateway/passages/tao-te-ching.htm)_

* {#Hegel27} [[Georg Hegel]], _[[Lectures on the Philosophy of Religion]]_, Volume II: Determinate Religion. Edited by Peter C. Hodgson; translated by R.F. Brown, P.C. Hodgson, and J.M. Stewart, with the assistance of J.P. Fitzer and H.S. Harris. Berkeley: University of California Press, 1995 (orig. 1987). (Translation of: Vorlesungen über die Philosophie der Religion.)  This extract (pp. 556-561) is from the Lectures of 1827; A. Immediate Religion, or Nature Religion; 1. The Religion of Magic; c. The State Religion of the Chinese Empire and the Dao. ([web](http://www.autodidactproject.org/quote/hegel-tao1.html))

* {#Hegel12} [[Georg Hegel]], _[[Science of Logic]]_ (1812)

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