Boardman-Vogt tensor product

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###Context###
#### Higher algebra
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[[!include higher algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Boardman-Vogt tensor product_ is a natural [[tensor product]] on [[symmetric operads]]. It makes the category [[Operad]] of colored [[symmetric operads]] over [[Set]] into a [[closed monoidal category]].

## Definition

All [[operads]] in the following are colored [[symmetric operads]] enriched over [[Set]], equivalently [[symmetric multicategories]].

Let $\mathcal{P}$ be an operad over a set of colors $C$, and $\mathcal{Q}$ be an operad over a set of colors $D$.

Their **Boardman-Vogt tensor product** $\mathcal{P} \otimes_{BV} \mathcal{Q}$ is the operad whose set of colors is $C \times D$, and whose operations are given by generators and relations as follows:

There is one generating operation for every pair $(p,d)$ with $p \in \mathcal{P}(c_1, \cdots, c_n; c)$ and with $d \in D$, denoted

$$
  p \otimes d
  \in
  \mathcal{P} \otimes_{BV} \mathcal{Q}(
    (c_1,d), \cdots, (c_n,d); (c,d)
  )
$$

and for each pair $(c,q)$ with $c \in C$ and $q \in \mathcal{Q}(d_1, \cdots, d_n; d)$, denoted

$$
  c \otimes q
  \in
  \mathcal{P}\otimes_{BV} \mathcal{Q}(
   (c, d_1), \cdots, (c, d_n);
   (c,d)
  )
$$

for all $n \in \mathbb{N}$. These are subject to the following relations

1. The tensor product $c \otimes (-)$ with $c \in C$ respects the composition in $\mathcal{Q}$, and the tensor product $(-) \otimes d$ with $d \in D$ respects the composition in $\mathcal{P}$ and both respect the action of the [[symmetric group]] on the operations.

   Equivalently this means that for all $c \in C$ tensoring with $c$ extends to a morphism of operads

   $$
     \mathcal{Q} \to \{c\} \otimes_{BV}\mathcal{Q}
   $$

   and for all $d \in D$ a morphism of operads

   $$
     \mathcal{P} \to \mathcal{P} \otimes_{BV} \{d\}
     \,.
   $$

1. The operations  in $\mathcal{P}$ and $\mathcal{Q}$ [[distributive law|distribute]] over each other in $\mathcal{P} \otimes_{BV} \mathcal{Q}$ in the evident sense (...).

## Properties

### Closed monoidal structure
 {#ClosedMonoidalStructure}

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

Equipped with the Boardman-Vogt tensor product, [[Operad]] is a [[closed monoidal category|closed]] [[symmetric monoidal category]].

=--

See for instance the proof provided in ([Weiss, theorem 2.22](#Weiss)).

This implies directly several useful statements about the BV-tensor product

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

* The BV tensor products preserves [[colimits]] of operads in each variable separately.

=--

We write in the following

$$
  [-,-] : Operad^{op} \times Operad \to Operad
$$

for the corresponding [[internal hom]] (leaving a subscrip "${}_{BV}$" implicit.)

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

For $P, Q \in $ [[Operad]], the internal hom operad $[P, Q]$ has

* as colors the [[algebra over an operad|P-algebras]] in $Q$;

* as unary operations the $P$-algebra homomorphisms in $Q$.

=--

See ([Weiss, lemma 2.23](#Weiss)).

We may therefore speak of $[P,Q]$ as being the **operad of $P$-algebras in $Q$.

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

Write $Set$ for the operad induced by the [[cartesian monoidal category|cartesian]] [[symmetric monoidal category]] structure on [[Set]]. Then for $P$ any operad, the vertices and unary operations of the internal hom operad $[P,Set]$ form the ordinary [[algebra over an operad|category of algebras over]] $P$ in $Set$.

=--

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

For $P_1, P_2, E \in Operad$, the category of $P_1$-algebras in $P_2$-algebras in $E$ is equivalent to the category of $P_2$-algebras in $P_1$-algebras in $E$.

=--

In view of prop. \ref{InternalHomOperadAsAlgebraOperad} this is the statement of the [[closed monoidal category|closed]] [[symmetric monoidal category|symmetric monoidal]] structure $(Operad, \otimes_{BC})$:

$$
  [P_1, [P_2, E]] \simeq [P_1 \otimes_{BV} P_2, E]
  \simeq
  [P_2 \otimes_{BV} P_1, E]
  \simeq
  [P_2, [P_1, E]]
  \,.
$$

## Examples

(...)

## Related concepts

* The Boardman-Vogt tensor product extends from operads to a tensor product on [[dendroidal sets]]. See there for more details.

* [[tensor product of monads]]

## References

The original reference is

* [[Michael Boardman]], [[Rainer Vogt]], _Homotopy invariant algebraic structures on topological spaces_, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, (1973).

A review is in

* {#Weiss} [[Ittay Weiss]], _From Operads to Dendroidal Sets_, in _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ AMS (2011) ([arXiv:1012.4315](http://arxiv.org/abs/1012.4315))

see around def. 2.21 there.


Bob Coecke

**Bob Coecke**

* [website](http://web.comlab.ox.ac.uk/people/Bob.Coecke/)

* Chief Scientist at *[Quantinuum](https://www.quantinuum.com/)*

## Selected writings

On [[quantum protocols]] in [[quantum information theory]] (such as [[quantum teleportation]]) formalized as [[string diagrams]] in the [[dagger-compact category]] of [[finite dimensional vector spaces|finite-dimensional]] [[Hilbert spaces]] ([[quantum information theory via dagger-compact categories]]):

* {#AbramskyCoecke04} [[Samson Abramsky]], [[Bob Coecke]], _A categorical semantics of quantum protocols_ , Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004) [[arXiv:quant-ph/0402130](http://arxiv.org/abs/quant-ph/0402130)]

* [[Samson Abramsky]], [[Bob Coecke]], *Abstract Physical Traces*, Theory and Applications of Categories, **14** 6 (2005) 111-124. \[tac:14-06, [arXiv:0910.3144](https://arxiv.org/abs/0910.3144)\]

* [[Bob Coecke]], *De-linearizing Linearity: Projective Quantum Axiomatics from Strong Compact Closure*, Proceedings of the [3rd International Workshop on Quantum Programming Languages (2005)](https://www.mathstat.dal.ca/~selinger/qpl2005/), Electronic Notes in Theoretical Computer Science **170** (2007) 49-72 \[doi:10.1016/j.entcs.2006.12.011, arXiv:quant-ph/0506134\]

* {#AbramskyCoecke08} [[Samson Abramsky]], [[Bob Coecke]], *Categorical quantum mechanics*, in *[[Handbook of Quantum Logic and Quantum Structures]]*, Elsevier (2008) [[arXiv:0808.1023](http://arxiv.org/abs/0808.1023), [ISBN:9780080931661](https://www.sciencedirect.com/book/9780444528698/), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)]

Survey in contrast to [[quantum logic]]:

* {#AbramskyCoecke2007} [[Samson Abramsky]], [[Bob Coecke]], *Physics from Computer Science: a Position Statement*, [International Journal of Unconventional Computing **3** 3 (2007)](https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/) $[$[pdf](https://www.cs.ox.ac.uk/files/349/YORKIJUC.pdf), [ijuc-3-3-p-179-197](https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/ijuc-3-3-p-179-197/)$]$

More on the use of [[string diagrams]] in [[quantum information theory via dagger-compact categories]]:

* [[Bob Coecke]], _Kindergarten quantum mechanics_ ([arXiv:quant-ph/0510032](http://arxiv.org/abs/quant-ph/0510032))

* {#Coecke10} [[Bob Coecke]], *Quantum Picturalism*, Contemporary Physics **51** 1 (2010) [[arXiv:0908.1787](http://arxiv.org/abs/0908.1787), [doi:10.1080/00107510903257624](https://doi.org/10.1080/00107510903257624)]

* [[Bob Coecke]], [[Ross Duncan]], *Interacting Quantum Observables: Categorical Algebra and Diagrammatics*, New J. Phys. **13** (2011) 043016 $[$[arXiv:0906.4725](http://arxiv.org/abs/0906.4725), [doi:10.1088/1367-2630/13/4/043016](https://doi.org/10.1088/1367-2630/13/4/043016)$]$

Textbook accounts:

* [[Bob Coecke]], [[Aleks Kissinger]], *Picturing Quantum Processes -- A First Course in Quantum Theory and Diagrammatic Reasoning*, Cambridge University Press (2017) [[ISBN:9781107104228](https://www.cambridge.org/ae/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/picturing-quantum-processes-first-course-quantum-theory-and-diagrammatic-reasoning?format=HB&isbn=9781107104228)]

* [[Bob Coecke]], [[Stefano Gogioso]], *Quantum in Pictures*, [Quantinuum Publications](https://www.quantinuum.com/publications) (2023) [[ISBN 978-1739214715](https://www.amazon.co.uk/dp/1739214714), [Quantinuum blog](https://www.quantinuum.com/news/quantum-in-pictures)]

  > (including the [[ZX-calculus]])

Motivation and introduction basic concepts of [[category theory]] for an audience familiar with standard [[physics]] and in particular with [[quantum mechanics]] (see also [[higher category theory and physics]]):

* [[Bob Coecke]], _Introducing categories to the practicing physicist_ ([arXiv:0808.1032](http://arxiv.org/abs/0808.1032))

* {#CoeckePractising} [[Bob Coecke]], [[Eric Oliver Paquette]], *Categories for the practising physicist*, in: *New Structures for Physics*, Lecture Notes in Physics **813**, Springer (2010) [[arXiv:0905.3010](http://arxiv.org/abs/0905.3010), [doi:10.1007/978-3-642-12821-9_3](https://doi.org/10.1007/978-3-642-12821-9_3)]

The article focuses towards the end on [[monoidal categories]], their description in terms of [[string diagrams]] and on [[quantum mechanics in terms of dagger-compact categories]].

On [[quantum measurement]] formulated in [[finite quantum mechanics in terms of dagger-compact categories ]] in terms of [[Frobenius algebras]] and the [[quantum reader monad]]:

* [[Bob Coecke]], [[Duško Pavlović]], *Quantum measurements without sums*, in [[Louis Kauffman]], [[Samuel Lomonaco]] (eds.), *Mathematics of Quantum Computation and Quantum Technology*, Taylor & Francis (2008) 559-596 [[arXiv:quant-ph/0608035](https://arxiv.org/abs/quant-ph/0608035), [doi:10.1201/9781584889007](https://doi.org/10.1201/9781584889007)]

* [[Bob Coecke]], [[Eric Oliver Paquette]], *POVMs and Naimark's theorem without sums*, Electronic Notes in Theoretical Computer Science **210** (2008) 15-31 [[arXiv:quant-ph/0608072](https://arxiv.org/abs/quant-ph/0608072), [doi:10.1016/j.entcs.2008.04.015](https://doi.org/10.1016/j.entcs.2008.04.015)]

* [[Bob Coecke]], [[Eric Paquette]], [[Dusko Pavlovic]], *Classical and quantum structures* (2008) [[pdf](http://www.comlab.ox.ac.uk/files/627/RR-08-02.pdf), [[Coecke-Paquette-Pavlovic-CQS.pdf:file]]]

* [[Bob Coecke]], [[Eric Oliver Paquette]], [[Duško Pavlović]], *Classical and quantum structuralism* [[arXiv:0904.1997](https://arxiv.org/abs/0904.1997)]

* [[Bob Coecke]], [[Duško Pavlović]], [[Jamie Vicary]],  *A new description of orthogonal bases*, Mathematical Structures in Computer Science **23** 3 (2012) 555- 567 [[arXiv:0810.0812](https://arxiv.org/abs/0810.0812), [doi:10.1017/S0960129512000047](https://doi.org/10.1017/S0960129512000047)]

Introducting the [[ZX-calculus]]:

* [[Bob Coecke]], [[Ross Duncan]], *A graphical calculus for quantum observables* [[pdf](https://www.cs.ox.ac.uk/people/bob.coecke/GreenRed.pdf)]

* [[Bob Coecke]], [[Ross Duncan]], *Interacting Quantum Observables*, in *Automata, Languages and Programming. ICALP 2008*, Lecture Notes in Computer Science **5126**, Springer (2008) [[doi:10.1007/978-3-540-70583-3_25](https://doi.org/10.1007/978-3-540-70583-3_25)]

* {#CoeckeDuncan11} [[Bob Coecke]], [[Ross Duncan]], *Interacting Quantum Observables: Categorical Algebra and Diagrammatics*, New J. Phys. **13** (2011) 043016 [[arXiv:0906.4725](http://arxiv.org/abs/0906.4725), [doi:10.1088/1367-2630/13/4/043016](https://doi.org/10.1088/1367-2630/13/4/043016)]

* [[Bob Coecke]], *Basic ZX-calculus for students and professionals* [[arXiv:2303.03163](https://arxiv.org/abs/2303.03163)]

Discussion of [[DisCoPy]]:

* Giovanni de Felice, [[Alexis Toumi]], [[Bob Coecke]]. _DisCoPy: Monoidal Categories in Python_. Applied Category Theory, 2020. ([arXiv:2005.02975l](https://arxiv.org/abs/2005.02975l))

## Related entries

* [[quantum information theory via dagger-compact categories]]

  * [[dagger-category]]

  * [[classical structure]]

  * [[ZX-calculus]]

* [[quantum information theory]]

category: people


Bob Oliver

* [webpage](https://www.math.univ-paris13.fr/~bobol/)

## Selected writings

On the [[equivariant Chern character]] in [[equivariant K-theory]]:

* [[Wolfgang Lück]], [[Bob Oliver]], _Chern characters for the equivariant K-theory of proper G-CW-complexes_, In: Aguadé J., Broto C., Casacuberta C. (eds.) _Cohomological Methods in Homotopy Theory_ Progress in Mathematics, vol 196. Birkhäuser 2001 ([doi:10.1007/978-3-0348-8312-2_15](https://doi.org/10.1007/978-3-0348-8312-2_15))

category: people


Bob Walters

Bob Walters (1943-2015), often called R F C Walters (Robert Frank Carslaw Walters), taught at the University of Sydney and from 1999 at the University of Insubria.

* [Web biography](http://dscpi.uninsubria.it/staff/Walters/Biography) (not working) [InternetArchive](https://web.archive.org/web/20130207023208/dscpi.uninsubria.it/staff/Walters/Biography)

* [blog](http://rfcwalters.blogspot.de/)

* [list of publications](http://comocategoryarchive.com/People/Walters-papers/pdffiles.html) (not working)

* [list of publications at University of Sydney](https://www.maths.usyd.edu.au/ut/pub-seek.py?sau=Walters+RFC&pp=3)

* [Como Category Theory Archive](http://comocategoryarchive.com/) (not working)

## Selected writings

On [[categorical algebra|categorical]] [[universal algebra]] ([[monads]]):

* [[R. F. C. Walters]], *A categorical approach to universal algebra*, Ph.D. Thesis (1970) [[anu:1885/133321](https://openresearch-repository.anu.edu.au/handle/1885/133321)]

On [[closed bicategories]] and [[locally internal categories]]:

* {#BettiWalters1985} [[Renato Betti]], [[Robert F. C. Walters]], *Closed bicategories and variable category theory*, Universita degli Studi di Milano (1985), reprinted in: Reprints in Theory and Applications of Categories, **26** (2020) 1-27 $[$[tac:tr26](http://www.tac.mta.ca/tac/reprints/articles/26/tr26abs.html)$]$

On [[extensive categories]] and [[distributive categories]]:

* {#CarboniLackWalters93} [[Aurelio Carboni]], [[Stephen Lack]], [[Bob Walters|R. F. C. Walters]], _Introduction to extensive and distributive categories_, JPAA **84** (1993) pp. 145-158 (doi:10.1016/0022-4049(93)90035-R)

On [[Cartesian bicategories]]:

* [[Aurelio Carboni]], [[Bob Walters]], _Cartesian Bicategories, I_, Journal of Pure and Applied Algebra, Volume 49, Issues 1–2, November 1987,  (doi:10.1016/0022-4049(87)90121-6)

* [[Aurelio Carboni]], [[Max Kelly]], [[Bob Walters]], [[Richard Wood]], _Cartesian Bicategories II_, Theory and Applications of Categories, Vol. 19, 2008, No. 6, pp 93-124. ([arXiv:0708.1921](https://arxiv.org/abs/0708.1921), [tac](http://www.tac.mta.ca/tac/volumes/19/6/19-06abs.html))

*  Piergiulio Katis, Nicoletta Sabadini, [[Bob Walters]], _On the algebra of feedback and systems with boundary_, Rendiconti del Circolo Matematico di Palermo. (2) Supplemento, 64 (2000), 123–156.  ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.51.5758&rep=rep1&type=pdf))

* Piergiulio Katis, Nicoletta Sabadini, [[Bob Walters]], _Bicategories of processes_, Journal of Pure and Applied Algebra **115** (1997) 141–178 [doi:10.1016/S0022-4049(96)00012-6, [pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.99.5583&rep=rep1&type=pdf)]

* Piergiulio Katis, Nicoletta Sabadini, [[Bob Walters]]: *$Span(Graph)$: A categorical algebra of transition systems*, in Michael Johnson (ed.), *Algebraic Methodology and Software Technology, AMAST '97*, Lecture Notes in Computer Science **1349**, Springer (1997) 307–321 [[doi:10.1007/BFb0000479](https://doi.org/10.1007/BFb0000479), [ps](http://aleteya.cs.buap.mx/~jlavalle/papers/categorias/walters/spangraph.ps)]

An introductory textbook on category theory for [[computer science]]:

* [[R. F. C. Walters]], _Categories and Computer Science_, Cambridge University Press, 1992. ([doi:10.1017/CBO9780511608872](https://doi.org/10.1017/CBO9780511608872))

## Related entries:

* [[allegory]]

* [[bicategory of relations]]

category: people

[[!redirects Robert Walters]]
[[!redirects Robert F. C. Walters]]
[[!redirects R. F. C. Walters]]
[[!redirects RFC Walters]]


Bobby Acharya

* [webpage](http://users.ictp.it/~bacharya/)

## Selected writings

On [[M-theory on G2-manifolds]]:

* {#Acharya98} [[Bobby Acharya]], _M theory, Joyce Orbifolds and Super Yang-Mills_, Adv. Theor. Math. Phys. 3 (1999) 227-248 ([arXiv:hep-th/9812205](http://arxiv.org/abs/hep-th/9812205))

* {#Acharya00} [[Bobby Acharya]], _On Realising $N=1$ Super Yang-Mills in M theory_ ([arXiv:hep-th/0011089](http://arxiv.org/abs/hep-th/0011089))

* {#AcharyaSpence00} [[Bobby Acharya]], B. Spence, _Flux, Supersymmetry and M theory on 7-manifolds_ ([arXiv:hep-th/0007213](http://arxiv.org/abs/hep-th/0007213))

* {#AcharyaWitten01} [[Bobby Acharya]], [[Edward Witten]], _Chiral Fermions from Manifolds of $G_2$ Holonomy_ ([arXiv:hep-th/0109152](http://arxiv.org/abs/hep-th/0109152))

* {#AcharyaDenefHofmanLambert03} [[Bobby Acharya]], F. Denef, C. Hofman, [[Neil Lambert]], _Freund-Rubin Revisited_ ([arXiv:hep-th/0308046](http://arxiv.org/abs/hep-th/0308046))

* {#Acharya02} [[Bobby Acharya]], _M Theory, $G_2$-manifolds and four-dimensional physics_,  Classical and Quantum Gravity Volume 19 Number 22, 2002  ([doi:10.1088/0264-9381/19/22/301](https://iopscience.iop.org/article/10.1088/0264-9381/19/22/301/meta), [pdf](http://users.ictp.it/~pub_off/lectures/lns013/Acharya/Acharya_Final.pdf))

* {#AcharyaGukov04} [[Bobby Acharya]], [[Sergei Gukov]], _M theory and Singularities of Exceptional Holonomy Manifolds_, Phys.Rept.392:121-189,2004 ([arXiv:hep-th/0409191](http://arxiv.org/abs/hep-th/0409191))

* {#AcharyaBobkov08} [[Bobby Acharya]], Konstantin Bobkov, _Kähler Independence of the G2-MSSM_, HEP 1009:001,2010 ([arXiv:0810.3285](http://arxiv.org/abs/0810.3285))

* {#Acharya02} [[Bobby Acharya]], _A Moduli Fixing Mechanism in M theory_ ([arXiv:hep-th/0212294](http://arxiv.org/abs/hep-th/0212294))

* [[Bobby Acharya]], Lorenzo Foscolo, Marwan Najjar, Eirik Eik Svanes, _New $G_2$-conifolds in M-theory and their Field Theory Interpretation_ ([arXiv:2011.06998](https://arxiv.org/abs/2011.06998))

and in relation to the [[duality between M-theory and heterotic string theory]]:

* [[Bobby Acharya]], Alex Kinsella, [[David Morrison]], *Non-Perturbative Heterotic Duals of M-Theory on $G_2$ Orbifolds* ([arXiv:2106.03886](https://arxiv.org/abs/2106.03886))

On [[moduli stabilization]]:

* {#AcharyaBobkovKaneKumarVaman06} [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Diana Vaman, _An M theory Solution to the Hierarchy Problem_, Phys.Rev.Lett.97:191601,2006 ([arXiv:hep-th/0606262](http://arxiv.org/abs/hep-th/0606262))

* {#AcharyaBobkovKaneKumarShao07} [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, _Explaining the Electroweak Scale and Stabilizing Moduli in M Theory_, Phys.Rev.D76:126010,2007 ([arXiv:hep-th/0701034](http://arxiv.org/abs/hep-th/0701034))

* {#AcharyaKumarBobbkovKaneShaoWatson08} [[Bobby Acharya]], [[Piyush Kumar]], Konstantin Bobkov, [[Gordon Kane]], Jing Shao, Scott Watson, _Non-thermal Dark Matter and the Moduli Problem in String Frameworks_,JHEP 0806:064,2008 ([arXiv:0804.0863](http://arxiv.org/abs/0804.0863))

On the [[G2-MSSM]]:

* {#AcharyaKaneKumar12} [[Bobby Acharya]], [[Gordon Kane]], [[Piyush Kumar]], _Compactified String Theories -- Generic Predictions for Particle Physics_, Int. J. Mod. Phys. A, Volume 27 (2012) 1230012 ([arXiv:1204.2795](http://arxiv.org/abs/1204.2795))

* {#Acharya12} [[Bobby Acharya]], _$G_2$-manifolds at the CERN Large Hadron collider and in the Galaxy_, talk at _$G_2$-days_ (2012) ([pdf](http://www.mth.kcl.ac.uk/~tbmadsen/acharya.pdf))

* [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, _The $G_2$-MSSM - An $M$ Theory motivated model of Particle Physics_ ([arXiv:0801.0478](http://arxiv.org/abs/0801.0478))

On [[KK-compactification]] on [[compact topological space|compact]] [[Ricci flat manifolds]] and [[special holonomy]]/[[Killing spinors]]/[[supersymmetry]]:

* {#Acharya19} [[Bobby Acharya]], _Supersymmetry, Ricci Flat Manifolds and the String Landscape_ ([arXiv:1906.06886](https://arxiv.org/abs/1906.06886))

On the [[metric cone over complex projective 3-space]] and the [[twistor fibration]]:

* [[Bobby Acharya]], [[Robert Bryant]], [[Simon Salamon]], _A circle quotient of a $G_2$ cone_, Differential Geometry and its Applications Volume 73, December 2020, 101681  ([arXiv:1910.09518](https://arxiv.org/abs/1910.09518), [doi:10.1016/j.difgeo.2020.101681](https://doi.org/10.1016/j.difgeo.2020.101681))

Relating [[moduli spaces]] of [[G2-manifolds|$G_2$-manifolds]] to those of [[flat connections]] on [[tori]]:

* [[Bobby S. Acharya]], Daniel A. Baldwin, *Ricci Flat Metrics, Flat Connections and $G_2$-Manifolds* [[arXiv:2312.12311](https://arxiv.org/abs/2312.12311)]

## Related entries

* [[G2-MSSM]]

* [[M-theory on G2-manifolds]]

[[!redirects Bobby S. Acharya]]

category: people


Bochner's theorem

#Contents#
* table of contents
{:toc}

## Statement

(...)

## Related concepts

* [[Fourier transform]]

* [[harmonic analysis]]

* [[kernel method]]

## References

### For abelian groups

Textbook accounts:

* {#HewettRoss70} [[Edwin Hewett]], [[Kenneth Ross]], Section 33 of: *Abstract Harmonic Analysis*, Volume II: *Structure and Analysis for Compact Groups, Analysis on Locally Compact Abelian Groups*,  Grundlehren der mathematischen Wissenschaften **152**, Springer 1970 ([doi:10.1007/978-3-662-26755-4](https://link.springer.com/book/10.1007/978-3-662-26755-4))

See also

* Wikipedia, *[Bochner's theorem](https://en.wikipedia.org/wiki/Bochner%27s_theorem)*

### For non-abelian groups
  {#ReferencesForNonabelianGroups}

Generalization to [[non-abelian groups]]:

* [Hewett-Ross 70, Section 34](#HewettRoss70)

Discussion in the context of [[kernel methods]]:

* [[Kenji Fukumizu]], [[Bharath Sriperumbudur]], [[Arthur Gretton]], [[Bernhard Schölkopf]], *Characteristic Kernels on Groups and Semigroups*, Advances in Neural Information Processing Systems 21 : 22nd Annual Conference on Neural Information Processing Systems 2008 ([NIPS 2008](https://neurips.cc/Conferences/2008/)), 473-480 ([mpg:5466](http://www.is.mpg.de/publications/5466), [pdf](http://www.gatsby.ucl.ac.uk/~gretton/papers/FukSriGreSch09.pdf))

[[!redirects Bochner theorem]]


Bockstein homomorphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Bockstein homomorphism_ is a [[connecting homomorphism]] induced from a [[short exact sequence]] whose injective map is given by multiplication with an [[integer]].

The archetypical examples are the Bockstein homomorphisms induced this way from the short exact sequence

$$
  \mathbb{Z}
    \stackrel{\cdot 2}{\to}
  \mathbb{Z}
    \stackrel{}{\to}
  \mathbb{Z}/2\mathbb{Z}
  \,.
$$

These relate notably degree-$n$ [[cohomology]]  with [[coefficients]] in $\mathbb{Z}_2$ (such as [[Stiefel-Whitney classes]]) to cohomology with integral coefficients in degree $n+1$ (such as [[integral Stiefel-Whitney classes]]).

## Definition

Let $A$ be an [[abelian group]] and $m$ be an [[integer]]. Then multiplication by $m$

$$
  A \stackrel{m\cdot}{\to} A
$$

induces a [[short exact sequence]] of abelian groups

$$
  0\to A/A_{m-tors} \stackrel{m\cdot}{\to} A \to A/m A\to 0,
$$

where $A_{m-tors}$ is the subgroup of $m$-[[torsion subgroup|torsion elements]] of $A$,
and so a long [[fiber sequence]]

$$
\cdots \mathbf{B}^n (A/A_{m-tors}) \to \mathbf{B}^n A \to \mathbf B^n(A/ m A) \to \mathbf{B}^{n+1} (A/A_{m-tors}) \to \cdots
$$

of [[∞-groupoid]]s, where $\mathbf{B}^n(-)$ denotes the $n$-fold [[delooping]] (hence $\mathbf{B}^n A$ is the [[Eilenberg-MacLane object]] on $A$ in degree $n$).

This induces, in turn, for any object $X \in \mathbf{H}$ (for $\mathbf{H}$ the ambient [[(∞,1)-topos]], such as [[Top]] $\simeq$ [[∞Grpd]]) , a long [[fiber sequence]]

$$
\cdots \mathbf{H}(X,\mathbf{B}^n (A/A_{m-tors})) \to \mathbf{H}(X,\mathbf{B}^n A) \to \mathbf{H}(X,\mathbf B^n(A/ m A)) \stackrel{\beta_m}{\to} \mathbf{H}(X,\mathbf{B}^{n+1} (A/A_{m-tors})) \to \cdots
$$

of [[cocycle]] [[∞-groupoids]].

Here the [[connecting homomorphisms]] $\beta_m$ are called the **Bockstein homomorphisms**.

Notice that often this term is used to refer only to the image of the above in [[cohomology]], hence to the image of $\beta_m$ under [[0-truncated|0-truncation]]/[[homotopy group|0th homotopy group]] $\pi_0$:

$$
 \beta_m : H^n(X,A/ m A) \to H^{n+1}(X,(A/A_{m-tors}))
  \,.
$$

## Examples
  {#Examples}

+-- {: .num_example #Mod2BocksteinIntoIntegralCohomology}
###### Example
**(mod 2 Bockstein homomorphism into [[integral cohomology]])**

The Bockstein homomorphism $\beta$ for the sequence

$$
  \mathbb{Z}
    \stackrel{\cdot 2}{\longrightarrow}
  \mathbb{Z}
    \stackrel{mod\, 2}{\longrightarrow}
  \mathbb{Z}/2\mathbb{Z}
$$

serves to define [[integral Stiefel-Whitney classes]]

$$
  W_{n+1} \coloneqq \beta w_n
$$

in degree $n+1$ from $\mathbb{Z}/2\mathbb{Z}$-valued [[Stiefel-Whitney classes]] in degree $n$.

=--

+-- {: .num_example #Mod2BocksteinIntoMod2Cohomology}
###### Example
**(first [[Steenrod square]])**

The Bockstein homomorphism for the sequence

$$
  \mathbb{Z}/2\mathbb{Z}
    \overset{\cdot 2}{\longrightarrow}
  \mathbb{Z}/4\mathbb{Z}
    \overset{mod\, 2}{\longrightarrow}
  \mathbb{Z}/2\mathbb{Z}
$$

is also called the _first [[Steenrod square]]_, denoted $Sq^1$.

This is often equivalently denoted $\beta$, as in example \ref{Mod2BocksteinIntoIntegralCohomology}. The difference between the two is just the mod-2 reduction in their codomain:

$$
  \label{Mod2BocksteinSequences}
  \array{
    \mathbb{Z}
      &\overset{\cdot 2}{\longrightarrow}&
    \mathbb{Z}
      &\overset{mod\, 2}{\longrightarrow}&
    \mathbb{Z}/2\mathbb{Z}
      &\simeq&
    \mathbb{Z}/2\mathbb{Z}
      &\overset{\beta}{\longrightarrow}&
    B \mathbb{Z}
    \\
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{\mathrlap{mod\, 4}}
      &&
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{ id }
      &&
    \downarrow^{\mathrlap{B(mod\, 2)}}
    \\
    \mathbb{Z}/2\mathbb{Z}
      &\underset{\cdot 2 }{\longrightarrow}&
    \mathbb{Z}/4\mathbb{Z}
      &\underset{mod\, 2}{\longrightarrow}&
     \mathbb{Z}/2\mathbb{Z}
      &\simeq&
     (\mathbb{Z}/4\mathbb{Z})/(\mathbb{Z}/2\mathbb{Z})
       &\underset{Sq^1}{\longrightarrow}&
     B (\mathbb{Z}/2\mathbb{Z})
  }
$$

More generally, for $p$ any [[prime number]] the multiplication by $p$ on $\mathbb{Z}_{p^2}$ induces the short exact sequence $\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/{p^2}\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}$. The corresponding Bockstein homomorphism $\beta_p$ appears as one of the generators of the mod $p$ [[Steenrod algebra]].

=--

+-- {: .num_example #IntegralSteenrodSquares}
###### Example
**([[integral Steenrod squares]])**

For [[odd natural numbers|odd]] $2n + 1 \in \mathbb{N}$ defines the [[integral Steenrod squares]] to be

$$
  Sq^{2n + 1}_{\mathbb{Z}}
  \;\coloneqq\;
  \beta \circ Sq^{2n}
  \,.
$$

By example \ref{Mod2BocksteinIntoMod2Cohomology} and by the first [[Adem relation]] $Sq^1 \circ Sq^{2n} = Sq^{2n+1}$ ([this example](Steenrod+square#CompositionWithSq1)) these indeed are lifts of the odd [[Steenrod squares]]:

$$
  (mod\, 2) \circ Sq^{2n + 1}_{\mathbb{Z}}
  \;=\;
  Sq^{2n+1}
  \,,
$$

because, by (eq:Mod2BocksteinSequences) we have

$$
  \array{
    Sq^{2n+1}_{\mathbb{Z}}
    &\colon&
    B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z})
      &\overset{Sk^{2n}}{\longrightarrow}&
    B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z})
      &\overset{ \beta }{\longrightarrow}&
    B^{\bullet + 2n + 1} \mathbb{Z}
    \\
    &&
    \downarrow^{ id }
    &&
    \downarrow^{ id }
      &&
    \downarrow^{\mathrlap{B^{k + 2 n + 1}(mod\, 2)}}
    \\
    Sq^{2n+1}
    &\colon&
    B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z})
      &\underset{Sk^{2n}}{\longrightarrow}&
    B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z})
       &\underset{  Sq^1 }{\longrightarrow}&
    B^{\bullet + 2n + 1} (\mathbb{Z}/2\mathbb{Z})
  }
$$

=--

When $A=\mathbb{Z}$, the equivalence $\vert \mathbf{B}^{n+1}\mathbb{Z} \vert \cong \vert \mathbf{B}^n U(1)\vert$ (which holds in ambient contexts such as $\mathbf{H} = $ [[ETop∞Grpd]] or [[Smooth∞Grpd]] under [[geometric realization]] $\vert - \vert : ETop \infty Grpd \stackrel{\Pi}{\to} \infty Grpd \stackrel{\simeq}{\to} Top$) identifies the morphisms $\mathbf{B}^n(\mathbb{Z}_m)\to \mathbf{B}^{n+1}\mathbb{Z}$ with the morphisms $\mathbf{B}^n(\mathbb{Z}_m)\to \mathbf{B}^{n} U(1)$ induced by the inclusion of the subgroup of $m$-th roots of unity into $U(1)$. This identifies the Bockstein homomorphism $\beta_m: H^n(X;\mathbb{Z}_m)\to H^{n+1}(X;\mathbb{Z})$ with the natural homomorphism  $H^n(X;\mathbb{Z}_m)\to H^{n}(X;U(1))$.

More in detail:

+-- {: .num_example #Mod2BocksteinAndExponentialExactSequence}
###### Example
**(mod 2 [[Bockstein homomorphism]] and the [[exponential exact sequence]])**

Let

1. $\beta \;\colon\; \mathbb{Z}/2\mathbb{Z} \longrightarrow B \mathbb{Z}$ be the ordinare Bockstein homomorphism

1. $\iota\coloneqq (\cdot \pi) \;\colon\; \mathbb{Z}/2\mathbb{Z} \hookrightarrow U(1)$ the canonical inclusion;

1. $\delta \;\colon\; U(1) \longrightarrow B\mathbb{Z}$ the classifying map.

Then

$$
  \beta
  \;=\;
  \delta \circ \iota
  \,.
$$

Because

$$
  \array{
    \mathbb{Z}
      &\overset{\cdot 2}{\longrightarrow}&
    \mathbb{Z}
      &\overset{mod\, 2}{\longrightarrow}&
    \mathbb{Z}/2\mathbb{Z}
      &\simeq&
    \mathbb{Z}/2\mathbb{Z}
      &\overset{\beta}{\longrightarrow}&
    B \mathbb{Z}
    \\
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{\mathrlap{\cdot \pi}}
      &&
    \downarrow^{\mathrlap{\cdot \pi}}
      &&
    \downarrow^{\iota \coloneqq \mathrlap{\cdot \pi}}
      &&
    \downarrow^{\mathrlap{id}}
    \\
    \mathbb{Z}
      &\underset{\cdot 2 \pi}{\longrightarrow}&
    \mathbb{R}
      &\underset{mod\, 2\pi}{\longrightarrow}&
     U(1)
      &\simeq&
     \mathbb{R}/2\pi\mathbb{Z}
       &\underset{\delta}{\longrightarrow}&
     B \mathbb{Z}
  }
$$

=--

+-- {: .num_prop #SteenrodSquaresAndDBCupProductOnOddClasses}
###### Proposition
**([[Deligne-Beilinson cup product]] on odd-degree [[ordinary differential cohomology]])**

Let

$$
  \hat H
  \;\colon\;
  X \longrightarrow \mathbf{B}^{2n} U(1)_{conn})
$$

be a class in [[ordinary differential cohomology]] with underlying class in odd degree

$$
  [H]
  \;\colon\;
  X
    \overset{\hat H}{\longrightarrow}
  \mathbf{B}^{n} U(1)_{conn}
    \overset{\chi}{\longrightarrow}
  B^{2n+1} \mathbb{Z}
$$

This implies that its [[Beilinson-Deligne cup product]] with itself satisfies

$$
  \hat H \hat \cup \hat H = - \hat H \hat \cup \hat H
$$

hence

$$
  2 \hat H \hat \cup \hat H \;\simeq\; 0
$$

hence

$$
  2 [H] \cup [H] \;\simeq\; 0
$$

hence that the ordinary [[cup product]] $[H] \cup [H]$ is a 2-torsion class. Let then

$$
  j
  \;\colon\;
  \mathbf{B}^{4n+1} \mathbb{Z}/2\mathbb{Z}
    \overset{
      B^{4n+1} (\iota)
    }{\hookrightarrow}
  \flat \mathbf{B}^{4n+1}  U(1)
    \longrightarrow
  \mathbf{B}^{4n+1} U(1)_{conn}
$$

with $\iota$ from example \ref{Mod2BocksteinAndExponentialExactSequence}.

Then

$$
  \hat H  \hat \cup \hat H
  \;\simeq\;
  j Sq^{2n}([H]_{mod\,2})
  \,.
$$

This is a [[differential cohomology]]-refinement of the first [[Adem relation]] $Sq^1 \circ Sq^{2n} = Sq^{2n+1}$ on the [[Steenrod squares]] ([this example](Steenrod+square#CompositionWithSq1)) in that, by example \ref{Mod2BocksteinAndExponentialExactSequence}, its image in ordinary cohomology with coefficients in $\mathbb{Z}/2\mathbb{Z}$ is

$$
  \array{
    ([H]  \cup [H])_{mod 2}
     & \simeq &
    \underset{
      \beta
    }{
    \underbrace{
      Sq^1
    }}
    \circ Sq^{2n}([H]_{mod\,2})
    \\
    =
    \\
    [H]_{mod\, 2} \cup [H]_{mod,2}
    \\
    =
    \\
    Sq^{2n+1}([H]_{mod\, 2})
  }
  \,.
$$

=--

This was first observed in ([Gomi 08](#Gomi08)). Streamlined proofs are given in ([Bunke 12, propblem 3.106](#Bunke12), [Grady-Sati 16, prop. 22](#GradySati16)).

## Related concepts

* [[Steenrod squares]], [[cohomology operation]]

* [[Bockstein spectral sequence]]

## References

Original references include

* [[Meyer Bockstein]],

  _Universal systems of $\nabla$-homology rings_, C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942), 243–245, MR0008701

  _A complete system of fields of coefficients for the $\nabla$-homological dimension_ ,  C. R. (Doklady) Acad. Sci. URSS (N.S.) (1943),  38: 187–189, MR0009115

* [[Meyer Bockstein]],  _Sur la formule des coefficients universels pour les groupes d'homologie_ ,  Comptes Rendus de l'académie des Sciences. Série I. Mathématique (1958),  247: 396–398, MR0103918

The relation to the [[Beilinson-Deligne cup product]] is discussed in

* {#Gomi08} [[Kiyonori Gomi]], _Differential  characters  and  the  Steenrod  squares_, In Groups of diffeomorphisms, volume 52 of Adv. Stud. Pure Math., pages 297?308. Math. Soc. Japan, Tokyo, 2008

* {#Bunke12} [[Ulrich Bunke]], problem 3.106 in _Differential cohomology_ ([arXiv:1208.3961](https://arxiv.org/abs/1208.3961))

* [[Daniel Grady]], [[Hisham Sati]], prop. 22 in: _Primary operations in differential cohomology_, Adv. Math. 335 (2018), 519-562 ([arXiv:1604.05988](https://arxiv.org/abs/1604.05988), [doi:10.1016/j.aim.2018.07.019](https://www.sciencedirect.com/science/article/pii/S0001870818302676?via%3Dihub))

[[!redirects Bockstein homomorphisms]]
[[!redirects Bockstein morphism]]
[[!redirects Bockstein morphisms]]
[[!redirects Bockstein operation]]
[[!redirects Bockstein operations]]


Bockstein spectral sequence

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

#Contents#
* table of contents
{:toc}

## Idea

A kind of [[spectral sequence]] related to the [[Bockstein homomorphism]].

The Bockstein spectral sequence is traditionally defined by an [[exact couple]] in a way that does not manifestly come from a [[filtered chain complex]]. However there is a way to set it up such that it does appear as the [[spectral sequence of a filtered complex]] ([Palmieri, section 3.2](#Palmieri)).
## References

* Jerrold Siegel, _On the Construction of the Bockstein Spectral Sequence_, Transactions of the American Mathematical Society
Vol. 210, (Sep., 1975), pp. 203-224 ([JSTOR](http://www.jstor.org/stable/1997132))

* {#Palmieri} J. Palmieri, _A User's guide to the Bockstein spectral sequence_ ([pdf](http://www.math.washington.edu/~palmieri/Papers/bss.pdf))


Bodo Pareigis

__Bodo Pareigis__ is a retired Professor of Mathematics at LMU Muenchen, specialized in [[category theory]] and [[algebra]].

* [web](http://www.mathematik.uni-muenchen.de/~pareigis)

## Selected writings

On [[category theory]]:

* {#Pareigis70} [[Bodo Pareigis]], *Categories and Functors*, Pure and Applied Mathematics **39**, Academic Press (1970) [[epub:7244](https://epub.ub.uni-muenchen.de/7244)]

On [[Brauer groups]] and early discussion of [[dualizable objects]] (calling them "finite objects"):

* [[Bodo Pareigis]], *Non-additive ring and module theory IV: The Brauer group of a symmetric monoidal category*, Lecture Notes in Mathematics **549** (1976) [[doi:10.1007/BFb0077339]( https://doi.org/10.1007/BFb0077339)]

category: people


body

#Contents#
* table of contents
{:toc}

## In superalgebra

In the context of [[superalgebra]], the [[quotient]] of a superalgebra by the ideal generated by its odd part (its "[[soul]]") is called its _body_.

See at _[superalgebra -- Adjoints to inclusion of plain algebras](super+algebra#AdjointsToInclusionOfPlainAlgebra)_ and at _[[bosonic modality]]_.

This terminology is (for the special case of [[Grassmann algebras]] at least) due to [DeWitt 92, p. 1](#DeWitt92)

## References

* {#DeWitt92} [[Bryce DeWitt]], _Supermanifolds_, 1992

[[!redirects body]]
[[!redirects bodies]]


Bogdan Ganchev

* [InSpire page](https://inspirehep.net/authors/1644021)

## Selected writings

On [[type II supergravity]]-solutions corresponding to [[D1-D5-P bound states]] in non-supersymmetric generality via [[AdS3-CFT2 duality]] ("[[microstrata]]"):

* [[Bogdan Ganchev]], [[Anthony Houppe]], [[Nicholas P. Warner]], *Q-Balls Meet Fuzzballs: Non-BPS Microstate Geometries*, J. High Energ. Phys. **2021** 28 (2021)
[[arXiv:2107.09677](https://arxiv.org/abs/2107.09677)]

* [[Bogdan Ganchev]], [[Stefano Giusto]], [[Anthony Houppe]], [[Rodolfo Russo]], *$AdS_3$ holography for non-BPS geometries*, European Physical Journal C **82** 217 (2022)
 [[https://arxiv.org/abs/2112.03287](https://arxiv.org/abs/2112.03287), [doi:10.1140/epjc/s10052-022-10133-2](https://doi.org/10.1140/epjc/s10052-022-10133-2)]

* [[Bogdan Ganchev]], [[Stefano Giusto]], [[Anthony Houppe]], [[Rodolfo Russo]], [[Nicholas P. Warner]], *Microstrata* [[arXiv:2307.13021](https://arxiv.org/abs/2307.13021)]

category: people


Bogdan Nica

* [personal page](http://math.iupui.edu/~bnica/)

* [institute page](https://science.iupui.edu/people-directory/people/nica-bogdan.html)

## Selected writings

On [[geometric group theory]]:

* [[Cornelia Druţu]], [[Michael Kapovich]] (appendix by [[Bogdan Nica]]), *Geometric group theory*, Colloquium Publications **63**, AMS 2018 ([ISBN:978-1-4704-1104-6](https://bookstore.ams.org/coll-63/) [pdf](https://courses.maths.ox.ac.uk/node/view_material/35649))

category: people


Bogdan Stefanski

* [webpage](https://www.city.ac.uk/people/academics/bogdan-stefanski)

## related $n$Lab entries

* [[D-brane]], [[fractional D-brane]]

* [[boundary conformal field theory]]

* [[orientifold]]

* [[RR-field tadpole cancellation]]

category: people

[[!redirects Bogdan Stefański]]


Bogdan Stoica

* [webpage](https://www.simonsfoundation.org/team/bogdan-stoica/)

* [webpage](http://theory.caltech.edu/~bstoica/About_me.html)

## Selected writings

On [[renormalization group flow]] in [[p-adic physics]] and [[Berkovich spaces]]:

* An Huang, Dan Mao, [[Bogdan Stoica]], _From p-adic to Archimedean Physics: Renormalization Group Flow and Berkovich Spaces_ ([arXiv:2001.01725](https://arxiv.org/abs/2001.01725))

On [[p-adic string theory]] and the [[Riemann zeta function]]:

* {#HuangStoicaYau19} [[An Huang]], [[Bogdan Stoica]], [[Shing-Tung Yau]], _General relativity from $p$-adic strings_ ([arXiv:1901.02013](https://arxiv.org/abs/1901.02013))

On the [[p-adic AdS/CFT correspondence]] and [[tensor networks]]:

* [[Matthew Heydeman]], [[Matilde Marcolli]], [[Ingmar Saberi]], [[Bogdan Stoica]], _Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the $AdS_3/CFT_2$ correspondence_ ([arXiv:1605.07639](https://arxiv.org/abs/1605.07639))

## related $n$Lab entries

* [[p-adic string theory]]

* [[Riemann zeta function]]

category: people


Bogoliubov's formula

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

#Contents#
* table of contents
{:toc}

## Idea

In ([[perturbative quantum field theory|perturbative]]) [[quantum field theory]], what is called _Bogoliubov's formula_, originally due to ([Bogoliubov-Shirkov 59](#BogoliubovShirkov59)) is an expression for the [[interacting quantum observables]] as the [[derivatives]] with respect to a [[source field]] of the [[generating function]] corresponding to a given [[S-matrix]].

Originally this is an ad-hoc formula, motivated by the would-be [[path integral]] picture of the [[Feynman perturbation series]] (remark \ref{InterpretationOfPerturbativeSMatrix} below) and mathematically justified by the fact that it does yield a [[causally local net of observables]] in [[causal perturbation theory]] (prop. \ref{CausalLocalityOfBogoliubovFormula} below).

Much more recently it was shown that Bogoliubov's formula indeed expresses quantum observables as systematically obtained by [[Fedosov deformation quantization]] of [[field theory]] ([Collini 16](#Collini16), [Hawkins-Rejzner 16](#HawkinsRejzner16)).

## Definition

Let $\Sigma$ be a [[spacetime]] of [[dimension]] $p + 1$ and let $E \overset{fb}{\longrightarrow} \Sigma$ be a [[field bundle]]. Let $\mathbf{L}_{free}\in \Omega^{p+1,0}_\Sigma(E)$ be a [[local Lagrangian density]] for a [[free field theory]] with [[field (physics)|fields]] of type $E$. Let $\mathcal{W}$ be the corresponding [[Wick algebra]] of [[quantum observables]] of the free field, with

$$
  LocObs(E_{\text{BV-BRST}})
    \overset{:(-):}{\longrightarrow}
  PolyObs(E_{\text{BV-BRST}})
$$

the corresponding quantization map from [[local observables]] ("[[normal ordering]]").

Let then

$$
  \mathcal{S}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})((\hbar))[ [g,j] ]
$$

be a  [[perturbative S-matrix]] scheme. Moreover let

$$
  g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [\hbar, g] ]
$$

be an [[adiabatic switching|adiabatically switched]]
[[interaction]]-[[action functional|functional]].

For $A \in \mathcal{F}_{loc}$ a [[local observable]] and $j \in C^\infty_{cp}(\Sigma)$, write

$$
  \mathcal{Z}_{g S_{int}}(j A)
  \; \coloneqq \;
  \mathcal{S}(g S_{int})^{-1} \mathcal{S}( g S_{int} + j A )
$$

for the [[generating function]] induced by the perturbative [[S-matrix]] (where the product shown by juxtaposition is that in the [[Wick algebra]], hence the [[star product]] induced by the [[Wightman propagator]]).

+-- {: .num_defn #GeneratingFunctionsForCorrelationFunctions}
###### Definition
**([[Bogoliubov's formula]])**

The perturbative [[interacting field observable]]

$$
  A_{int}
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]
$$

corresponding to a [[free field]] [[local observable]] $A \in LocObs(E_{\text{BV-BRST}})$ is the [[coefficient]] in the [[generating function]] $\mathcal{S}$ ([this def.](S-matrix#SchemeGeneratingFunction)) of the term linear in the [[source field]] strenght $j$:

$$
  A_{int}
    \;\coloneqq\;
  i \hbar \frac{d}{d  j } \mathcal{Z}_{ g S_{int}}( j  A)\vert_{j = 0}
  \,.
$$

=--

This is due to [Bogoliubov-Shirkov 59](#BogoliubovShirkov59), later named _Bogoliubov's formula_ (e.g. [Rejzner 16 (6.12)](#Rejzner16)).
Based on this [[causal perturbation theory]] was formulated in ([Epstein-Glaser 73 around (74)](#EpsteinGlaser73)). Review includes ([Dütsch-Fredenhagen 00, around (17)](#DuetschFredenhagen00)).

The assignment $A \mapsto A_{int}$ is also called the _[[quantum Møller operator]]_.

The [[coefficients]] of $A_{int}$ as a [[formal power series]] in the [[coupling constant]] and [[Planck's constant]] are called the _[[retarded products]]_.

+-- {: .num_remark #PowersInPlancksConstant}
###### Remark
**(powers in [[Planck's constant]])**

That the observables as defined in def. \ref{GeneratingFunctionsForCorrelationFunctions} indeed are [[formal power series]] in $\hbar$ as opposed to more general [[Laurent series]] requires a little argument.

The explicit $\hbar$-dependence of the perturbative [[S-matrix]] is

$$
  \mathcal{S}(g S_{int} + j A)
  =
  T \exp\left(
    \tfrac{1}{i \hbar}
     \left(
       g S_{int} + j A
     \right)
  \right)
  \,,
$$

where $T(-)$ denotes [[time-ordered products]]. The generating function

$$
  \mathcal{Z}_{S_{int}}(j A)
  \;\coloneqq\;
  \mathcal{S}(S_{int})^{-1} \star_H \mathcal{S}(g S_{int} + j A)
$$

involves the [[star product]] of the free theory (the [[normal-ordered product]] of the [[Wick algebra]]). This is a [[formal deformation quantization]] of the [[Peierls-Poisson bracket]], and therefore the [[commutator]] in this algebra is a [[formal power series]] in $\hbar$ that however has no constant term in $\hbar$ (but starts out with $\hbar$ times the [[Poisson bracket]], followed by possibly higher order terms in $\hbar$):

$$
  [L_{int},A] \;=\; \hbar(\cdots)
  \,.
$$

Now schematically the derivative of the generating function is of the form

$$
  \begin{aligned}
    A_{int}
    & \coloneqq
    i \hbar \frac{d}{d j }
    \mathcal{Z}_{ S_{int}}(j A)\vert_{j  = 0}
    \\
    & =
    \exp\left(
      \tfrac{1}{i \hbar}[g S_{int}, -]
    \right)
    (A)
  \end{aligned}
  \,.
$$

(The precise expression is given by the _[[retarded products]]_, see [Dütsch-Fredenhagen 00, prop. 2 (ii)](#DuetschFredenhagen00) [Rejzner 16, prop. 6.1](#Rejzner16), [Hawkins-Rejzner 16, cor. 5.2](#HawkinsRejzner16).)
By the above, the exponent here yields a [[formal power series]] in $\hbar$, and hence so does the full exponential.

That the quantum observables takes values in formal power series of $\hbar$ is the hallmark of [[formal deformation quantization]]. While [[Bogoliubov's formula]] proceeds from an [[S-matrix]] which is axiomatized by [[causal perturbation theory]], this suggests that it actually computes the quantum observables in a [[formal deformation quantization]] of the interacting field theory. This is indeed the case ([Collini 16](#Collini16), [Hawkins-Rejzner 16](#HawkinsRejzner16)).

For the analogous analysis of powers of $\hbar$ in the [[S-matrix]] itself, see at _[[loop order]]_ the section _[Relation to powers in Planck's constant](loop%20order#RelationToPowersInPlancksConstant)_.

=--

## Properties

### Relation to the would-be path integral

+-- {: .num_remark #InterpretationOfPerturbativeSMatrix}
###### Remark
**(interpretation of Bogoliubov's formula via a "[[path integral]]")**

In informal heuristic discussion of [[perturbative quantum field theory]] the [[S-matrix]] is thought of as a [[path integral]],
written

$$
  S\left(
    \tfrac{g}{\hbar} L_{int} + j
  \right)
  \;\overset{\text{not really!}}{=}\;
  \underset{\Phi \in \Gamma_\Sigma(E)_{asmpt}}{\int}
  \exp\left(
    \int_X
    \left(
      \tfrac{g}{i \hbar} L_{int}(\Phi) + j A(\Phi)
    \right)
  \right)  e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi]
$$

where the integration is thought to be over the [[configuration space]] $\Gamma_\Sigma(E)_{asmpt}$ of [[field (physics)|fields]] $\Phi$
(the [[space of sections]] of the given [[field bundle]])
which satisfy given asymptotic conditions at $x^0 \to \pm \infty$; and as these boundary conditions
vary the above is regarded as an integral kernel that defines the required operator in $\mathcal{W}$.

With the perturbative S-matrix informally thought of as a path integral this way,  the the Bogoliubov formula in def. \ref{GeneratingFunctionsForCorrelationFunctions}
similarly would have the following interpretation:

$$
  A_{int}
  \;\overset{\text{not really!}}{=}\;
  \frac{
    \int
    j A(\Phi)
    \exp\left(
      \int_X
      \left(
        \tfrac{g}{i \hbar} L_{int}(\Phi)
      \right)
    \right)  e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi]
  }
  {
    \int
    \exp\left(
      \int_X
      \left(
        \tfrac{g}{i \hbar} L_{int}(\Phi)
      \right)
    \right)  e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi]
  }
$$

If here we were to regard the expression

$$
  \mu(\Phi)
  \;\overset{\text{not really}}{\coloneqq}\;
  \frac{
    \exp\left(
      \int_X
      \left(
        \tfrac{g}{i \hbar} L_{int}(\Phi)
      \right)
    \right)  e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi]
  }
  {
    \int
    \exp\left(
      \int_X
      \left(
        \tfrac{g}{i \hbar} L_{int}(\Phi)
      \right)
    \right)  e^{\tfrac{1}{i \hbar}\int_X L_{free}(\phi) }D[\phi]
  }
$$

as a "complex probability measure" on the the configuration space of fields, then this formula
would express the [[expectation value]] of the functional $A$ under this measure:

$$
  A_{int}
    \overset{\text{not really!}}{=}
   [A]_{\mu} = \int  A(\Phi) \, \mu(\Phi)
  \,.
$$

=--

### Causal locality of interacting field quantum observables

+-- {: .num_prop #CausalLocalityOfBogoliubovFormula}
###### Proposition
**([[causal locality]])**

As the spacetime support varies, the algebras of [[interacting field quantum observables]] spanned via the Bogoliubov formula consistitute a [[causally local net of observables]], hence an instance of [[perturbative AQFT]].

=--

([Dütsch-Fredenhagen 00, section 3](#DuetschFredenhagen00), following [Brunetti-Fredenhagen 99, section 8](#BrunettiFredenhagen99),
[Il'in-Slavnov 78](#IlinSlavnov78))

For **[[proof]]** see [this prop.](S-matrix#PerturbativeQuantumObservablesIsLocalnet) at _[[S-matrix]]_.

## Related concepts

* [[quantum Møller operator]]

* [[interacting field algebra]]

[[!include products in pQFT -- table]]

## References

The formula originates in

* {#BogoliubovShirkov59} [[Nikolay Bogoliubov]], [[Dmitry Shirkov]], _Introduction to the Theory of Quantized Fields_, New York (1959)

Its rigorous discussion in terms of [[causal perturbation theory]] is due to

* {#EpsteinGlaser73} [[Henri Epstein]], [[Vladimir Glaser]], _[[The Role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211 ([Numdam](http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0 ))

The observation that the Bogoliubov formula yields a [[causally local net of quantum observables]] is due to

* {#IlinSlavnov78} V. A. Il'in and D. S. Slavnov, _Observable algebras in the S-matrix approach_, Theor. Math. Phys. 36 (1978) 32. ([spire](http://inspirehep.net/record/135575), [doi](http://dx.doi.org/10.1007/BF01035870))

then rediscovered in

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661, 2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))

and made more explicit in

* {#DuetschFredenhagen00} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion_, Commun.Math.Phys. 219 (2001) 5-30 ([arXiv:hep-th/0001129](https://arxiv.org/abs/hep-th/0001129))

The recognition of the Bogoliubov formula as the result [[formal deformation quantization]] and specifically of [[Fedosov deformation quantization]] is due to

* {#Collini16} [[Giovanni Collini]], section 2.2 of _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

* {#HawkinsRejzner16} [[Eli Hawkins]], [[Kasia Rejzner]], _The Star Product in Interacting Quantum Field Theory_ ([arXiv:1612.09157](https://arxiv.org/abs/1612.09157))

Review includes

* {#DuetschFredenhagen00} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion_, Commun.Math.Phys. 219 (2001) 5-30 ([arXiv:hep-th/0001129](https://arxiv.org/abs/hep-th/0001129))

* {#Rejzner16} [[Katarzyna Rejzner]], _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([pdf](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

[[!redirects Bogoliubov formula]]


Bogomolny equation

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

#Contents#
* table of contents
{:toc}

## Idea

The _Bogomolny equations_ are [[differential equations]] re-expressing the [[equations of motion]] of [[self-dual Yang-Mills theory]] on 4-dimensional [[Minkowski spacetime]] for [[time]]-independent solutions in terms of dimensional reductions $D = 3$ [[Euclidean space]].

## Related concepts

* [[Nahm transform]]

* [[Yang-Mills instanton]]

* [[magnetic monopole]]

* [[moduli space of monopoles]]

## References

* [[Michael Atiyah]], [[Nigel Hitchin]], Equation (2.1) in: _The geometry and dynamics of magnetic monopoles_  M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988 ([jstor:j.ctt7zv206](https://www.jstor.org/stable/j.ctt7zv206))

See also

* Wikipedia, _[Bogomolny equations](https://en.wikipedia.org/wiki/Bogomolny_equations)_

[[!redirects Bogomolny equations]]


Bogomolov-Tian-Todorov theorem

## References

* Donatella Iacono, [[Marco Manetti]], _An algebraic proof of Bogomolov-Tian-Todorov theorem_ ([arXiv:0902.0732](https://arxiv.org/abs/0902.0732))
* [[Ludmil Katzarkov]], [[Maxim Kontsevich]], [[Tony Pantev]], _Bogomolov–Tian–Todorov theorems for Landau–Ginzburg models_,  J. Differential Geom. 105(1): 55--117 [doi](https://doi.org/10.4310/jdg/1483655860)


Bohmian mechanics

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

#Contents#
* table of contents
{:toc}

## Idea

What is called _Bohmian mechanics_ or _de Broglie-Bohm theory_ is a rewriting of the [[Schrödinger equation]] of [[quantum mechanics]] in a way that makes it look -- away from the [[zero locus]] of its solutions -- like discribing a [[diffusion]] process subject to an unusual (for ordinary diffusion) potential. There is a [[stochastic process]] modelling this diffusion ([Nelson 66](#Nelson66)) and the point of view expressed by _Bohmian mechanics_ is to regard this as a [[hidden variable theory]] of [[quantum mechanics]].

Specifically, for the simple [[mechanical system]] of a [[particle]] of [[mass]] $m$ propagating on the [[real line]] $\mathbb{R}$ and subject to a [[potential]] $V \in C^\infty(\mathbb{R})$, the [[Schrödinger equation]] is the [[differential equation]] on [[complex number|complex]]-valued [[functions]] $\Psi \colon \mathbb{R}\times \mathbb{R} \to \mathbb{C}$ given by

$$
  i \hbar
  \frac{\partial}{\partial t}
  \Psi
  =
  \frac{\hbar^2}{2m}
  \frac{\partial^2}{\partial^2 x} \Psi
  +
  V \Psi
  \,,
$$

where $\hbar$ denotes [[Planck's constant]].

By the nature of [[complex numbers]] and by the discussion at _[[phase and phase space in physics]]_, it is natural to parameterize $\Psi$ -- away from its [[zero locus]] -- by a [[complex phase]] function

$$
  S \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}
$$

and an [[absolute value]] function $\sqrt{\rho}$

$$
  \sqrt{\rho} \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}
$$

which is positive, $\sqrt{\rho} \gt 0$, as

$$
  \Psi
    \coloneqq
  \exp\left(\frac{i}{\hbar} S / \hbar\right) \sqrt{\rho}
  \,.
$$

Entering this Ansatz into the above [[Schrödinger equation]], that [[complex number|complex]] equation becomes equivalent to the following two [[real number|real]] equations:

$$
  \frac{\partial}{\partial t} S
  =
  -
  \frac{1}{2m} \left(\frac{\partial}{\partial x}S\right)^2
  -
  V
  +
  \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}}\frac{\partial^2}{\partial^2 x} \sqrt{\rho}
$$

and

$$
  \frac{\partial}{\partial t} \rho
  =
  -
  \frac{\partial}{\partial x}
  \left(
    \frac{1}{m} \left(\frac{\partial}{\partial x}S\right) \rho
  \right)
  \,.
$$

Now in this form one may notice a similarity of the form of these two equations with other equations from [[classical mechanics]] and [[statistical mechanics]]:

1. The first equation is similar to the [[Hamilton-Jacobi equation]] that expresses the classical [[action functional]] $S$ and the [[canonical momentum]]

   $$
     p \coloneqq \frac{\partial}{\partial x} S
   $$

   except that in addition to the ordinary [[potential energy]] $V$ there is an additional term

   $$
    Q
      \coloneq
    \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}}\frac{\partial^2}{\partial^2 x} \sqrt{\rho}
   $$

   which is unlike what may appear in an ordinary [[Hamilton-Jacobi equation]]. The perspective of Bohmian mechanics is to regard this as a correction of [[quantum physics]] to classical [[Hamilton-Jacobi theory]], it is then called the _quantum potential_. Notice that unlike ordinary potentials, this "quantum potential" is a function of the density that is subject to the potential. (Notice that this works only away from the [[zero locus]] of $\rho$.)

1. The second equation has the form of the [[continuity equation]] of the [[flow]] expressed by $\frac{1}{m}p$.

The perspective of Bohmian mechanics is to regard this equivalent rewriting of the [[Schrödinger equation]] as providing a [[hidden variable theory]] formulation of [[quantum mechanics]].

The bulk of the discussion of Bohmian mechanics in the literature revolves around [[philosophy|philosophical]] implications of this perspective, e.g. ([Stanford Enc. Philosph](#SEP)).

## Related concepts

* [[interpretation of quantum mechanics]]

## References

Named after:

* [[David Bohm]], *A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I*, Phys. Rev. **85** (1952) 166 [[doi:10.1103/PhysRev.85.166](https://doi.org/10.1103/PhysRev.85.166)]

* [[David Bohm]], *A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. II*, Phys. Rev. **85** (1952) 180 [[doi:10.1103/PhysRev.85.180](https://doi.org/10.1103/PhysRev.85.180)]

See also:

* Wikipedia, *[De Broglie–Bohm theory](https://en.wikipedia.org/wiki/De_Broglie–Bohm_theory)*

* {#SEP} Stanford Encyclopedia of Philosophy, _[Bohmian mechanics](http://plato.stanford.edu/entries/qm-bohm/)_

The formulation of the exotic diffusion process described by the phase part of the Schrödinger equation as a [[stochastic process]] is due to

* {#Nelson66} [[Edward Nelson]], _Derivation of the Schrödinger equation from Newtonian mechanics_, Phys. Rev. 150, 1079–1085, 1966

* [[Edward Nelson]], _Quantum Fluctuations_, Princeton University Press, Princeton. 1985

A review of this relating to Bohmian mechanics:

* Guido Bacciagaluppi, _A Conceptual Introduction to Nelson's
Mechanics_ [[pdf](http://philsci-archive.pitt.edu/8853/1/Nelson-revised.pdf)]

The surreal trajectory problem is pointed out in

* B.-G. Englert, M. O. Scully, G. Süssmann, H. Walther, *Surrealistic Bohm trajectories*, Z. Naturforsch. **47**a  (1992) 1175-1186

The following two articles offer solution to the surreal trajectory problem:

* D. H. Mahler et al. _Experimental nonlocal and surreal Bohmian trajectories_, Science Advances __2__:2, e1501466 (2016) [[doi:10.1126/sciadv.1501466](https://doi.org/10.1126/sciadv.1501466)]

* B. J. Hiley, R. Callaghan, O. Maroney, Quantum trajectories, real, surreal or an approximation to a deeper process? [[quant-ph/0010020](http://arxiv.org/abs/quant-ph/0010020)]

category: physics

[[!redirects pilot-wave theory]]
[[!redirects Bohmian quantum mechanics]]


Bohr compactification

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

#Contents#
* table of contents
{:toc}

## Idea

_Bohr compactification_ is a kind of [[compactification]] of a [[topological group]] to one that is [[compact topological space|compact]] [[Hausdorff topological space|Hausdorff]].

## Definition and existence

Let $i: CHGrp \to TopGrp$ denote the full embedding of the category of compact Hausdorff topological groups into the category of all topological groups. Let $G$ be a topological group.

+-- {: .num_defn}
###### Definition
The *Bohr compactification* $Bohr(G)$ is a representing object for the functor $TopGrp(G, i-): CHGrp \to Set$.
=--

+-- {: .num_theorem}
###### Theorem
The Bohr compactification exists for any topological group $G$.
=--

+-- {: .proof}
###### Proof
We verify that the hypotheses of the general [[adjoint functor theorem]] are satisfied. Certainly $CHGrp$ is complete, and $i: CHGrp \to TopGrp$ preserves small limits since products and equalizers in $CHGrp$ are formed just as they are in $TopGrp$. We must now show there is a small solution set, i.e., a [[weakly initial set]] for the comma category $G \downarrow i$.

Let $I = [0, 1]$ be the unit [[interval]] with its standard topology. There is a canonical map $u_G: G \to I^{Top(G, I)}$ where $u: id \to M$ is the unit of the monad $M = I^{Top(-, I)}$ on $Top$; note $u_X$ is a closed embedding if $X$ is compact Hausdorff (see the discussion [here](http://ncatlab.org/nlab/show/compactum#StoneCechCompactification)). We claim that the collection of closed subspaces of $I^{Top(G, I)}$ that come equipped with topological group structures is a solution set. Indeed, for an arbitrary CH group $K$ and continuous homomorphism $f: G \to K$, the closure of the image $\widebar{f(G)}$ is a compact Hausdorff subgroup of $K$ through which $f$ factors (according to the lemma that follows), and this is a closed subspace of $M(f)^{-1}(u_K)$ in the pullback diagram

$$\array{
M(f)^{-1}(u_K) & \hookrightarrow & I^{Top(G, I)} \\
\downarrow & & \downarrow_{M(f)} \\
K & \underset{u_K}{\hookrightarrow} & I^{Top(K, I)}
}$$

and therefore $\widebar{f(G)}$ occurs as a closed subspace of $I^{Top(G, I)}$, as claimed.
=--

+-- {: .num_lemma}
###### Lemma
If $H$ is a topological group and $J \subseteq H$ is a subgroup, then the topological closure $\widebar{J} \subseteq H$ is also a subgroup. (The same is true for any finitary algebraic theory in place of the theory of groups.)
=--

+-- {: .proof}
###### Proof
For $X, Y$ arbitrary topological spaces and subsets $A \subseteq X$, $B \subseteq Y$, it is elementary that $\widebar{A \times B} = \widebar{A} \times \widebar{B}$. Hence for any operation $m: H^n \to H$ that $J$ is closed under, $\widebar{J}^n = \widebar{J^n}$ is contained in $m^{-1}(\widebar{J})$ (by continuity of $m$), so that $m$ restricts to an operation $\widebar{J}^n \to \widebar{J}$ as required.
=--

+-- {: .num_remark}
###### Remark
The Bohr compactification admits a more elegant construction if $G$ is a topological *abelian* group: if $S = S^1$ is the unit [[circle]], then $Bohr(G)$ may be taken to be the closure of the image of $G$ under the canonical map $G \to S^{TopAb(G, S)}$. The argument is that it suffices to consider only compact Hausdorff abelian groups $K$, where $K$ embeds as a closed subgroup of $S^{TopAb(K, S)}$ by [[Pontryagin duality]]; using a pullback similar to the above, the argument is easily completed. The description simplifies further if $G$ is a locally compact Hausdorff abelian group; here $Bohr(G)$ is the Pontryagin dual of the discretization of the Pontryagin dual of $G$.
=--

## Example

A MathOverflow question from 2011 asks whether, for $G$ a compact Hausdorff group, can $\mathbb{Z}$ appear as a quotient of $G$ *considered as an abstract group*?

A truly simple answer was [given](http://mathoverflow.net/questions/80966/morphism-from-a-compact-group-to-z#) by Sean Eberhard using the Bohr compactification, as follows. A quotient $p: G \to \mathbb{Z}$ admits a section $i: \mathbb{Z} \to G$ which extends to the Bohr compactification $\widehat{i}: Bohr(\mathbb{Z}) \to G$. The composite $p \widehat{i}: Bohr(\mathbb{Z}) \to \mathbb{Z}$ is still surjective as its restriction along the unit $\mathbb{Z} \to Bohr(\mathbb{Z})$ is the identity. According to the remark above, $Bohr(\mathbb{Z})$ is the Pontryagin dual of the discrete group $S^1 \cong \mathbb{R}_{disc} \oplus \mathbb{Q}/\mathbb{Z}$, so $Bohr(\mathbb{Z}) \cong \mathbb{R}_{disc}' \oplus (\mathbb{Q}/\mathbb{Z})'$. Now the Pontryagin dual of a discrete torsionfree group such as $\mathbb{R}_{disc}$ is divisible (more generally, if $Q$ is an injective module over a commutative ring and $F$ is flat, then $\hom(F, Q)$ is also injective), so the restriction of $p \widehat{i}: Bohr(\mathbb{Z}) \to \mathbb{Z}$ to the summand $\mathbb{R}_{disc}'$ is zero. But the restriction to the other summand $(\mathbb{Q}/\mathbb{Z})' \cong \widehat{\mathbb{Z}} = \prod_p \mathbb{Z}_p^\wedge$ is also zero, as the further restriction to $\mathbb{Z}_2^\wedge$ is zero by 3-divisibility, and the restriction to $\prod_{p \neq 2} \mathbb{Z}_p^\wedge$ is zero by 2-divisibility.

## Related concepts

* [[compactification]]

* [[one-point compactification]]

* [[Stone-Cech compactification]]

* [[end compactification]]

* [[Kaluza-Klein compactification]]

## References

* Wikipedia, _[Bohr compactification](http://en.wikipedia.org/wiki/Bohr_compactification)_

[[!redirects Bohr compactifications]]