Brown-Peterson spectrum

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## Idea

The [[Bousfield localization of spectra|localization]] of [[complex cobordism cohomology theory]] $MU$ at a [[prime]] $p$, hence the [[p-localization]] $MU_{(p)}$, decomposes as a [[direct sum]]. The direct summands are the _Brown-Peterson spectra_.

## Definition

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###### Theorem

For each [[prime]] $p$ there is an unique commutative ring [[spectrum]] $B P$ which is a [[retract]] of $M U_{(p)}$ such that the map $MU_{(p)} \to B P$ is multiplicative and such that

1. (...)

1. (...)

1. (...)

=--

Due to ([Brown-Peterson 66](#BrownPeterson66)), recalled as ([Ravenel, theorem, 4.1.12](#Ravenel)).

## Properties

### Universal characterization

(...)

The [[formal group law]] of Brown-Peterson cohomology theory is [[universal property|universal]] for $p$-local [[complex oriented cohomology theories]] in that $\mathbb{G}_{B P}$ is universal among $p$-local, [[p-typical formal group laws]].

(...)

### Relation to $p$-typical formal groups

$B P$ is related to [[p-typical formal groups]] as [[MU]] is to [[formal groups]].

### Hopf algebroid structure on dual BP-Steenrod algebra

The structure of [[Hopf algebroid over a commutative base]]
on the dual $BP$-[[Steenrod algebra]]
$BP_\bullet(BP)$ is described by the [[Adams-Quillen theorem]].

### Relation to Adams-Novikov spectral sequence

The $p$-component of the $E^2$-term of the [[Adams-Novikov spectral sequence]] for the [[sphere spectrum]], hence the one converging to the
[[stable homotopy groups of spheres]] $\pi_\ast(\mathbb{S})$ is

$$
  Ext_{BP_\ast(BP)}(BP_\ast, BP_\ast)
  \,.
$$

recalled e.g. as [Ravenel, theorem 1.4.2](#Ravenel)

### As a CW spectrum

The spectrum $B P$ can be constructed as a [[CW spectrum]] (cf. Priddy 1980) starting from the $p$-local sphere spectrum $S^0 = X_0$ by minimally attaching cells to $X_n$ to kill $\pi_{2n+1}(X_n)$.

## Related concepts

* [[Morava E-theory]]

* [[BPR-theory]]

## References

The original article is

* {#BrownPeterson66} [[Edgar Brown]], F. P. Peterson, _A spectrum whose $\mathbb{Z}/p$ cohomology is the algebra of reduced $p$-th powers_, Topology 5 (1966) 149.

* [[Frank Adams]], part II.16 of _[[Stable homotopy and generalised homology]]_,1974

An alternate construction was noted by Priddy

* [[Stewart Priddy]], _A cellular construction of BP and other irreducible spectra_, [Math. Z.](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002423782) 173 (1980), pp 29-34.

A textbook accounts:

* {#Ravenel} [[Doug Ravenel]], section 4 ([pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel4.pdf)) in: _[[Complex cobordism and stable homotopy groups of spheres]]_ ([web](http://www.math.rochester.edu/people/faculty/doug/mu.html))

* {#Rudyak98} [[Yuli Rudyak]], Section IX.6 in: _On Thom Spectra, Orientability, and Cobordism_, Springer 1998 ([doi:10.1007/978-3-540-77751-9](https://doi.org/10.1007/978-3-540-77751-9))

The truncated version is discussed in

* [[Tyler Lawson]], [[Niko Naumann]], _Truncated Brown-Peterson spectra_ (2012) ([pdf](http://www.math.umn.edu/~tlawson/jmm-boston-handout.pdf))

On the [[Hopf invariant]] and [[e-invariant]] in [[BP-theory]]:

* Yasumasa Hirashima, _On the $BP_\ast$-Hopf invariant_, Osaka J. Math., Volume 12, Number 1 (1975), 187-196 ([euclid:ojm/1200757733](https://projecteuclid.org/euclid.ojm/1200757733))

* [[Martin Bendersky]], _The BP Hopf Invariant_, American Journal of Mathematics, Vol. 108, No. 5 (Oct., 1986) ([jstor:2374595](https://www.jstor.org/stable/2374595))

[[!redirects Brown-Peterson spectra]]

[[!redirects Brown-Peterson cohomology]]

[[!redirects Brown-Peterson cohomology theory]]
[[!redirects Brown-Peterson cohomology theories]]

[[!redirects Brown-Peterson theory]]
[[!redirects Brown-Peterson theories]]

[[!redirects BP]]

[[!redirects BP-spectrum]]
[[!redirects BP spectrum]]

[[!redirects BP-theory]]
[[!redirects BP theory]]


Brown-Stallings lemma

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## Statement

Let $X$ be a [[differentiable manifold]] ([[paracompact topological space|paracompact]]). If every [[compact topological space|compact]] [[subspace]] of $X$ is contained in an [[open subset]] [[diffeomorphism|diffeomorphic]] to [[Euclidean space]] (with its standard [[smooth structure]]), then all of $X$ is diffeomorphic to Euclidean  space: $X \simeq \mathbb{R}^n$.

See for instance [Milnor 64, lemma 3 (p. 168)](#Milnor64)

## Related entries

* [[Reeb sphere theorem]]

## References

* {#Milnor64} [[John Milnor]], _Differenial topology_, chapter 6 in T. L. Saaty (ed.) _Lectures On Modern Mathematic II_ 1964 ([web](https://archive.org/details/LecturesOnModernMathematicsIi), [pdf](https://ia801700.us.archive.org/6/items/LecturesOnModernMathematicsIi/Saaty-LecturesOnModernMathematicsIi.pdf))


BrownAHT

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This entry is about the article

* [[Ken Brown|Kenneth S. Brown]],

  _[[BrownAbstractHomotopyTheory.pdf:file]]_,

  Transactions of the American Mathematical Society,

  **186** (1973) 419-458

  ([jstor:1996573](http://www.jstor.org/stable/1996573)).

Brown was a student of [[Dan Quillen]]. He expands in this article work from his thesis on a notion of [[homotopical category]] -- called a [[category of fibrant objects]] -- with a bit less structure than that of a full [[model category]], which he uses to study "generalized [[abelian sheaf cohomology]]":

Following, or rather as a precursor of, the [[nPOV]] on [[cohomology]] he studies precisely the cohomology within [[∞-stack]] [[(∞,1)-topos]]es and, in the second part of the article, that in [[(∞,1)-category|(∞,1)-categories]] of [[(∞,1)-sheaf|(∞,1)-sheaves]] of [[spectrum|spectra]], see at _[[sheaves of spectra]]_.

As such, this article is a forerunner of the development of the full [[model category]] structures built for the purpose of [[models for ∞-stack (∞,1)-toposes|modelling ∞-stack (∞,1)-toposes]]: the [[model structure on simplicial presheaves|model structure on simplicial (pre)sheaves]] as first defined by [[Andre Joyal]] in his letter to Grothendieck and then later generalized from simplicial sheaves to [[simplicial presheaf|simplicial presheaves]] and developed in great detail by Jardine (in its injective version) and [[Dan Dugger]] (in its projective version).

[[Dugger's theorem]] that every [[combinatorial model category]] is a left [[Bousfield localization of model categories|Bousfield localization]] of the projective global [[model structure on simplicial presheaves]] was finally understood by [[Jacob Lurie]] in the book [[Higher Topos Theory]] to be the [[model category]]-theoretic version of the intrinsic [[higher category theory]] statement that every [[presentable (∞,1)-category]] is the left [[localization of an (∞,1)-category]] of an [[(∞,1)-category of (∞,1)-presheaves]], and that the left _exact_ such localizations are precisely the [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-categories of (∞,1)-sheaves]] -- the [[∞-stack]] [[(∞,1)-topos]]es.

But the [[homotopy category of an (∞,1)-category|homotopy category]] of these [[(∞,1)-topos]]es -- that category which remembers just the [[∞-stack]] [[cohomology]] classes, and not the [[cocycle]]s -- is precisely what Kenneth Brown constructed, studied and identified correctly in its nature back then in 1973.

See also at _[[Ken Brown's lemma]]_.

category: reference

[[!redirects Brown -- Abstract Homotopy Theory and Generalized Sheaf Cohomology]]
[[!redirects Brown -- Abstract Homotopy Theory and Generalized Sheaf Coho]]

[[!redirects Abstract Homotopy Theory and Generalized Sheaf Cohomology]]
[[!redirects Abstract homotopy theory and generalized sheaf cohomology]]


Brownian loop soup

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* table of contents
{:toc}

## Idea

(...)

## Related entries

* [[Brownian motion]]

* [[conformal field theory]]

* [[loop space]]

## References

The original article:

* Gregory F. Lawler, Wendelin Werner, *The Brownian loop soup*, Probab. Th. Rel. Fields **128** (2004) 565-588 [[arXiv:math/0304419](https://arxiv.org/abs/math/0304419), [doi:10.1007/s00440-003-0319-6](https://doi.org/10.1007/s00440-003-0319-6)]

Relation to [[conformal field theory]]:

* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *Scalar conformal primary fields in the Brownian loop soup* [[arXiv:2109.12116](https://arxiv.org/abs/2109.12116)]

* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *The Brownian loop soup stress-energy tensor* [[arXiv:2112.00074](https://arxiv.org/abs/2112.00074), doi:10.1007/JHEP11(2022)009]

Review:

* [[Valentino Foit]], *Brownian loops and conformally invariant systems*, talk at *[[CQTS]]* (Nov 2022) [[[Foit_CQTS-Nov2022.pdf:file]], video: [rec](https://nyu.zoom.us/rec/play/DxmoQSXAGXA331jrD4isyOJrmpvTdgmRnaDLeD3gAabO1v6zlTjYNcxPT74SS0VDtWzbiRyYQTltw9vp.dI1nqe74UCpaouu_?continueMode=true&_x_zm_rtaid=kQZZ5EfLRZiMesvQXmFYBg.1669295362714.dd968befceb1ba32e41f0349dfb7ca0f&_x_zm_rhtaid=14)]

[[!redirects Brownian loop soups]]


Brownian motion

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* table of contents
{:toc}

## Idea

Brownian motion is an example of a [[stochastic process]]. Brownian motion $B_t$ is a stochastic process on $[0,\infty)$ with the following properties:

* $B_0=0$ almost surely.

* The increments $B_t-B_s$ are independent and normally distributed $N(0,t-s)$ for $t \gt s$.

* The function $t\mapsto B_t$ is continuous.

## Related concepts

* [[Wiener measure]]

* [[Brownian loop soup]]

## References

Introduction:

* Tamas Szabados, *An elementary introduction to the Wiener process and stochastic integrals*, Studia Scientiarum Mathematicarum Hungarica **31** (1996) 249-297 [[arXiv:1008.1510](https://arxiv.org/abs/1008.1510)]

History:

* Arthur Genthon, *The concept of velocity in the history of Brownian motion -- From physics to mathematics and back*, Eur. Phys. J. H **45** (2020) 49-105 [[arXiv:2006.05399](https://arxiv.org/abs/2006.05399), [doi:10.1140/epjh/e2020-10009-8]( https://doi.org/10.1140/epjh/e2020-10009-8)]

See also:

* Wikipedia, *[Brownian motion](https://en.wikipedia.org/wiki/Brownian_motion)*

* Wikipedia, *[Wiener process](https://en.wikipedia.org/wiki/Wiener_process)*


Brown–Douglas–Fillmore theory

## Related entries

* [[Weyl–von Neumann theorem]]
* [[Calkin algebra]]
* [[self-adjoint operator]]
* [[C-star algebra]]

## References

* Brown--Douglas--Fillmore theory, _Encyclopedia of Mathematics_, [web](https://www.encyclopediaofmath.org/index.php/Brown%E2%80%93Douglas%E2%80%93Fillmore_theory)

* L.G. Brown, R.G. Douglas, P.A. Fillmore, _Unitary equivalence modulo the compact operators and extensions of C$^\ast$-algebras_, Proc. Conf. Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) , Lecture Notes in Mathematics __345__, Springer  (1973)  pp. 58–128, [MR380478](http://www.ams.org/mathscinet-getitem?mr=380478)

* L.G. Brown, R.G. Douglas, P.A. Fillmore, _Extensions of C$^\ast$-algebras and K-homology_, Ann. of Math. (2) __105__: 2  (1977)  pp. 265–324, [MR458196](http://www.ams.org/mathscinet-getitem?mr=458196), [doi](http://dx.doi.org/10.2307/1970999)

* L. G. Brown, R. G. Douglas, P. A. Fillmore, _Extensions of C$^\ast$-algebras, operators with compact self-commutators, and K-homology_, Bull. Amer. Math. Soc. __79__:973-978, 1973, [MR346540](http://www.ams.org/mathscinet-getitem?mr=346540),  [doi](http://dx.doi.org/10.1090/S0002-9904-1973-13284-7)

* J. Kaminker, C. Schochet, _K-theory and Steenrod homology: applications to the Brown–Douglas–Fillmore theory of operator algebras_, Trans. Amer. Math. Soc. __227__  (1977)  pp. 63–107, [MR431136](http://www.ams.org/mathscinet-getitem?mr=431136)

[[!redirects Brown-Douglas-Fillmore theory]]
[[!redirects Brown–Douglas–Fillmore theory]]
[[!redirects Brown--Douglas--Fillmore theory]]


BRST

This is a disambiguation and history entry. BRST is an abbreviation for Becchi-Rouet-Stora-Tyutin.

$n$lab articles:

* [[BV-BRST formalism]] -- the main in-depth entry on BRST formalism and its generalization "in antifield formalism", Batalin-Vilkovisky
* [[BRST complex]]
* [[BV integral]]

Need:

* [[BV algebra]]
* [[BRST symmetry]]

See also scholarpedia: [Becchi-Rouet-Stora-Tyutin_symmetry](http://www.scholarpedia.org/article/Becchi-Rouet-Stora-Tyutin_symmetry)

## Historical references

 * [[Carlo Becchi]], [[Alain Rouet]], [[Raymond Stora]], _The Abelian Higgs-Kibble Model_, Commun. math. Phys. __42__, 127—162 (1975) [euclid](http://projecteuclid.org/euclid.cmp/1103899003)

 * [[Carlo Becchi]], [[Alain Rouet]], [[Raymond Stora]], (1974) _The Abelian Higgs-Kibble Model_, Unitarity of the S Operator. Phys. Lett. B52: 344.

 * [[Carlo Becchi]], [[Alain Rouet]], [[Raymond Stora]], (1976). Renormalization of gauge theories. Ann. Phys. 98: 287.

 * [[Carlo Becchi]], [[Alain Rouet]], [[Raymond Stora]], Lectures given at the International School on Elementary Particle Physics. Baško-Polje (September 1974)

 * [[I. V. Tyutin]], (1975), _Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism_, [arXiv:0812.0580](https://arxiv.org/abs/0812.0580).

[[!redirects Becchi-Rouet-Stora-Tyutin]]

[[!redirects BRST formalism]]


BRST complex

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## Idea

What is called the "BRST complex" in the [[physics]] literature is the [[Lie infinity-algebroid|qDGCA]] which is the [[Chevalley-Eilenberg algebra]] of the $L_\infty$-[[Lie infinity-algebroid|algebroid]] which is the differential version in [[Lie theory]] of the $\infty$-groupoid

 * whose space of objects is the space of configurations/histories of a given physical system;

 * whose morphisms describe the gauge transformations between these configurations/histories;

 * whose $k$-morphisms describe the $k$-fold gauge-of-gauge transformations.

The generators of the BRST complex are called

 * in degree 0: **fields**;

 * in degree 1: **[[ghost field]]**;

 * in degree 2: **[[ghost-of-ghost fields]]**;

 * etc.

The [[cochain cohomology]] of the BRST complex is called, of course, _BRST cohomology_.

## Details

For details see at

* _[[A first idea of quantum field theory]]_ the chapter _[Gauge symmetries](A+first+idea+of+quantum+field+theory#Gauge+symmetries)_.

## Related concepts

The BRST complex described a homotopical [[quotient]] of a space by an infinitesimal [[action]]. Combined with a homotopical intersection, it is part of the [[BRST-BV complex]].

* [[gauge transformation]], [[higher gauge transformation]]

* [[ghost field]], [[ghost-of-ghost field]]

* [[antifield]], [[antighost field]]

* [[local BRST complex]]

* [[variational BRST-bicomplex]]

* [[equivariant de Rham cohomology]]

[[!include gauge field - table]]

## References

### General

The idea of "ghost" fields was introduced in

* [[Richard Feynman]], _Quantum theory of gravitation_ In: Acta physica polonica. vol 24, 1963, S. 697

and expanded on in

* [[Richard Feynman]] in [[John Wheeler]], Klauder (eds.), _Magic without Magic_, Wheeler-Festschrift (1972)

The BRST formalism originates around

* [[Carlo Becchi]], A. Rouet, [[Raymond Stora]], (1976). Renormalization of gauge theories. Ann. Phys. 98: 287,

 * [[I. V. Tyutin]], (1975), _Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism_, [arXiv:0812.0580](https://arxiv.org/abs/0812.0580).

see also the references at _[[BRST]]_.

A classical standard references for the Lagrangian formalism is

* {#Henneaux90} [[Marc Henneaux]], _Lectures on the Antifield-BRST formalism for gauge theories_, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 (doi [pdf](http://www.math.uni-hamburg.de/home/schweigert/ws07/henneaux2.pdf))

Similarly the bulk of the textbook

* [[Marc Henneaux]], [[Claudio Teitelboim]], _[[Quantization of Gauge Systems]]_

considers the Hamiltonian formulation. Chapters 17 and 18 are about the Lagrangian ("antifield") formulation, with section 18.4 devoted to the relation between the two.

The [[L-infinity algebroid]]-structure of the [[local BRST complex]] on the [[jet bundle]] is made manifest in

* {#Barnich10} [[Glenn Barnich]], _A note on gauge systems from the point of view of Lie algebroids_, in: *XXIX Workshop on Geometric Methods in Physics*, AIP Conference Proceedings, **1307** 7 (2010) [[arXiv:1010.0899](https://arxiv.org/abs/1010.0899), [doi:/10.1063/1.3527427]( https://doi.org/10.1063/1.3527427)]

Discussion with more emphasis on the applications to quantum field theory of interest is in lecture 3 of

* [[Edward Witten]], _Dynamics of Quantum Field Theory_ in vol II, starting page 1119, of [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey,  [[David Kazhdan]], [[John Morgan]], D.R. Morrison and [[Edward Witten]], eds.  _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999.

The [[perturbative quantum field theory|perturbative quantization]] of [[gauge theories]] ([[Yang-Mills theory]]) in [[causal perturbation theory]]/[[perturbative AQFT]] is discussed (for trivial [[principal bundles]] and restricted to [[gauge invariant observables]]) via [[BRST-complex]]/[[BV-formalism]] in

* {#Hollands07} [[Stefan Hollands]], _Renormalized Quantum Yang-Mills Fields in Curved Spacetime_, Rev. Math. Phys.20:1033-1172, 2008 ([arXiv:0705.3340](https://arxiv.org/abs/0705.3340))

* {#FredenhagenRejzner11a} [[Klaus Fredenhagen]], [[Kasia Rejzner]], _Batalin-Vilkovisky formalism in the functional approach to classical field theory_, Commun. Math. Phys. 314(1), 93–127 (2012) ([arXiv:1101.5112](https://arxiv.org/abs/1101.5112))

* {#FredenhagenRejzner11b} [[Klaus Fredenhagen]], [[Kasia Rejzner]], _Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory_, Commun. Math. Phys. 317(3), 697–725 (2012) ([arXiv:1110.5232](https://arxiv.org/abs/1110.5232))

* {#Rejzner13} [[Katarzyna Rejzner]], _Remarks on local symmetry invariance in perturbative algebraic quantum field theory_ ([arXiv:1301.7037](https://arxiv.org/abs/1301.7037))

* {#Rejzner13} [[Katarzyna Rejzner]], _Remarks on local symmetry invariance in perturbative algebraic quantum field theory_ ([arXiv:1301.7037](https://arxiv.org/abs/1301.7037))

* Mojtaba Taslimi Tehrani, _Quantum BRST charge in gauge theories in curved space-time_ ([arXiv:1703.04148](https://arxiv.org/abs/1703.04148))

and surveyed in

* {#Rejzner16} [[Kasia Rejzner]], section 7 of _Perturbative algebraic quantum field theory_ Springer 2016

With focus on the [[cochain cohomology]]:

* [[José Figueroa-O'Farrill]], [[Takashi Kimura]], *The cohomology of BRST complexes* (1988) [[pdf](https://www.maths.ed.ac.uk/~jmf/Research/PVBLICATIONS/homol.pdf), [[FigueroaKimuar-BRSTCohomology.pdf:file]]]

Discussion of the BRST complex of the [[bosonic string]]/for [[2d CFT]] includes

* [[Graeme Segal]], p.114 and following of _The definition of conformal field theory_ , preprint, 1988; also in [[Ulrike Tillmann]] (ed.) _Topology, geometry and quantum field theory_ , London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577. ([pdf](https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf))

Discussion of the BRST complex for the [[superstring]] (hence with the corresponding [[Lie algebroid]] being actually a [[super Lie algebroid]]) is for instance in

* [[José Figueroa-O'Farrill]], [[Takashi Kimura]], _The BRST cohomology of the NSR string: vanishing and ``no-ghost'' theorems_, Comm. Math. Phys. **124** 1 (1989) 105-132. [[euclid:cmp/1104179078](http://projecteuclid.org/euclid.cmp/1104179078)]

* [[Alexander Belopolsky]], _De Rham Cohomology of the Supermanifolds and Superstring BRST Cohomology_, Phys.Lett. B403 (1997) 47-50 ([arXiv:hep-th/9609220](http://arxiv.org/abs/hep-th/9609220))

The perspective on the BRST complex as a formal dual to a space in [[dg-geometry]] is relatively clearly stated in section 2 of

* [[Kevin Costello]], _Renormalisation and the Batalin-Vilkovisky formalism_ ([arXiv](http://arxiv.org/abs/0706.1533)).

For more along these lines see [[BV-BRST formalism]].

In relation to [[equivariant de Rham cohomology]]:

* {#Kalkman93} [[Jaap Kalkman]], _BRST model applied to symplectic geometry_, Ph.D. Thesis, Utrecht, 1993 ([arXiv:hep-th/9308132 (broken)](https://arxiv.org/abs/hep-th/9308132), [cds:9308132](http://cds.cern.ch/record/568522), [pdf](http://cds.cern.ch/record/568522/files/9308132.pdf))

* [[Jaap Kalkman]], _BRST Model for Equivariant Cohomology and  Representatives for the Equivariant Thom Class_, Comm. Math. Phys. Volume 153, Number 3 (1993), 447-463. ([euclid:1104252784](http://projecteuclid.org/euclid.cmp/1104252784))

### History

* [[Carlo Becchi]], _BRS "Symmetry", prehistory and history_ ([arXiv:1107.1070](https://arxiv.org/abs/1107.1070))

[[!redirects BRST complexes]]

[[!redirects BRST-complex]]
[[!redirects BRST-complexes]]

[[!redirects BRST differential]]
[[!redirects BRST differentials]]

[[!redirects BRST cohomology]]

[[!redirects BRST operator]]
[[!redirects BRST operators]]

[[!redirects BRST Lagrangian]]


Bruce Bartlett

Hi, I'm Bruce Bartlett, a maths lecturer at the University of Stellenbosch, South Africa.

category: people


Bruce Blackadar

Bruce Evan Blackadar is an American operator algebra theorist at the University of Nevada.

* [webpage](http://wolfweb.unr.edu/homepage/bruceb/)

* [maths genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=32249)

* [Wikipedia page](http://de.wikipedia.org/wiki/Bruce_Blackadar) (German)

There is a typo found in some bibliographies listing Blackadar\'s middle initial as 'A'.  This is not a different person.

## Selected writings

On [[K-Theory]] for [[operator algebras]]:

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_, Cambridge University Press 1986, second ed. 1999 ([pdf](http://wolfweb.unr.edu/homepage/bruceb/Book6.pdf))

On [[operator alegbras]]: [[C-star algebras|$C^\ast$-algebras]] and [[von Neumann algebras]]:

* {#Blackadar06} [[Bruce Blackadar]], *Operator Algebras -- Theory of $C^\ast$-Algebras and von Neumann Algebras*, Encyclopaedia of Mathematical Sciences **122**, Springer (2006) [[doi:10.1007/3-540-28517-2](https://doi.org/10.1007/3-540-28517-2)]

On variants of the [[Tietze extension theorem]]:

* [[Bruce Blackadar]], *Extending continuous functions* [[arXiv:1207.6147](https://arxiv.org/abs/1207.6147)]

## Related entries

* _[[K-Theory for Operator Algebras]]_

category: people

[[!redirects Bruce Even Blackadar]]
[[!redirects Bruce E. Blackadar]]
[[!redirects Bruce A. Blackadar]]
[[!redirects Bruce E Blackadar]]
[[!redirects Bruce A Blackadar]]
[[!redirects B. E. Blackadar]]
[[!redirects B. A. Blackadar]]
[[!redirects B.E. Blackadar]]
[[!redirects B.A. Blackadar]]
[[!redirects B E Blackadar]]
[[!redirects B A Blackadar]]
[[!redirects BE Blackadar]]
[[!redirects BA Blackadar]]
[[!redirects Bruce Blackadar]]


Bruce Corrigan-Salter

* [webpage](http://clas.wayne.edu/brcs)

## related $n$Lab entries

* [[right Bousfield delocalization]]

category: people


Bruce Eli Sagan

[[!redirects Bruce E. Sagan]]
[[!redirects Bruce Sagan]]

* [MSU home page](http://users.math.msu.edu/users/sagan/)

* [Wikipedia page](https://en.wikipedia.org/wiki/Bruce_Sagan)

## Selected writings

On the [[symmetric group]] and the [[representation theory of the symmetric group]]:

* {#Sagan01} [[Bruce Sagan]], _The symmetric group_, Springer 2001 ([doi:10.1007/978-1-4757-6804-6](https://link.springer.com/book/10.1007/978-1-4757-6804-6), [pdf](http://math.sfsu.edu/federico/Clase/RepTh/sagan.pdf))

On [[Schur functions]]:

* [[Bruce E. Sagan]], _Schur functions_, in (M. Hazewinkel, ed.) Encyclopaedia of Mathematics, Springer ([pdf](http://www.mth.msu.edu/~sagan/Papers/Old/schur.pdf), [[SaganSchurFunctions.pdf:file]])

On [[combinatorics]]:

* [[Bruce Sagan]], *Combinatorics: The Art of Counting*, Graduate studies in Mathematics **210**, AMS  ([ISBN:978-1-4704-6032-7](https://bookstore.ams.org/gsm-210) [pdf](https://users.math.msu.edu/users/bsagan/Books/Aoc/final.pdf))

On the [[hook length formula]]:

* [[Bruce Sagan]], *Probabilistic proofs of the hook length formulas involving trees*, S éminaire Lotharingien de Combinatoire 61A (2009) ([pdf](https://users.math.msu.edu/users/bsagan/Papers/Old/pph.pdf))

category:people


Bruce Westbury

* [website](http://www.warwick.ac.uk/~masdbn/home.html)

## Selected writings

On [[spherical category|spherical]] [[fusion categories]] and the [[Turaev-Viro model]]:

* [[John Barrett]], [[Bruce Westbury]], _Invariants of piecewise-linear 3-manifolds_, Trans. Amer. Math. Soc. 348 (1996), no. 10, 3997–4022, ([doi:10.1090/S0002-9947-96-01660-1](https://doi.org/10.1090/S0002-9947-96-01660-1), MR1357878)

* [[John Barrett]], [[Bruce Westbury]], _Spherical Categories_, Adv. Math. 143:357-375, 1999 ([arXiv:hep-th/9310164](http://arxiv.org/abs/hep-th/9310164), [doi:10.1006/aima.1998.1800](https://doi.org/10.1006/aima.1998.1800))

category: people


Bruck-Toyoda theorem

A [[binary algebraic structure]] $(A,\cdot)$ is __medial__ if for all $a,b,c,d\in A$ $(a\cdot b)\cdot(c\cdot d) = (a\cdot c)\cdot (b\cdot d)$.
This law is quite close to commutativity: for example, if $(A,\cdot)$ has a unit element (is a unital [[magma]]), then by [[Eckmann-Hilton argument]], it is an Abelian [[monoid]]. Without two-sided unit element, the connection with Abelian groups is somewhat more intricate

__(Bruck-Toyoda theorem)__ A [[quasigroup]] $(A,\cdot)$ is medial iff there is an Abelian group structure $(A,+)$ on $A$ and mutually commuting group [[automorphism]]s $\phi,\psi:A\to A$ and an element $h\in A$ such that
$$
a\cdot b = \phi(a)+\psi(b)+h
$$

An element $a$ in a magma $(A,\cdot)$ is left regular if the left multiplication $L_a$ by $a$ is bijective and right regular if the right multiplication $R_a$ is bijective.
The Bruck-Toyoda theorem is a direct corollary of a more general statement about (nonunital in general) magmas:

Theorem. A [[medial magma]] $(A\cdot)$ has a left regular element $f$ and a right regular element $g$ such that also $f^2$ is left regular and $g^2$ is right regular iff there exist a commutative [[semigroup]] structure $(A,+)$ on $A$, mutually commuting automorphisms $\phi,\psi:A\to A$ and a regular element $h$ in semigroup $(A,+)$ such that
$$
a\cdot b = \phi(a)+\psi(b)+h
$$

category: algebra
[[!redirects Toyoda theorem]]


Bruhat decomposition

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

#Contents#
* table of contents
{:toc}

## Definition

Given a sequence of inclusions

$$
  T \subset B \subset G
$$

where

* $G$ is a [[connected topological space|connected]] [[complex geometry|complex]] [[reductive group|reductive]] [[algebraic group]]

* $B$ is a [[Borel subgroup]]

* $T$ is a [[maximal torus]];

this induces

* the [[Weyl group]] $W_0 = N(T)/T$;

* the [[character lattice]] $\mathfrak{h}_{\mathbb{Z}}^\ast = Hom(T, \mathbb{C}^\times)$;

* the [[cocharacter lattice]] $\mathfrak{h}_{\mathbb{C}} = Hom(\mathbb{C}^\times, T)$.

* a _[[standard parabolic subgroup]]_ of $G$ is a subgroup $P_J$ including $B$ such that $G/P$ is a [[projective variety]];

   [[parabolic subgroup]] is one [[conjugation|conjugate]] to the standard parabolic subgroup.

* the [[flag variety]] $G/B$;

* the _[[partial flag varieties]]_ $G/P_J$

A **Bruhat decomposition** is, if it exists, a [[coproduct]] decomposition into a disjoint union of [[double cosets]]

$$
  G = \underset{w \in W_0}{\coprod} B w B
$$

$$
  G =  \underset{u \in W^J}{\coprod} B u P_j
$$

with

* $W_J \coloneqq \{v \in W_0 | v T  \subset P_J\}$

* $W^J \coloneqq \{coset\; representatives\; u \; of \; cosets \; in W_0/W_J\}$

into [[Schubert varieties]]

$$
  X_w = \overline{B w B} \subset G/B
$$

$$
  X_u^J = \overline{B u P_J} \subset G/P_J
  \,.
$$

## Related concepts

* [[Schubert calculus]]

## References

* Wikipedia, _[Bruhat decomposition](http://en.wikipedia.org/wiki/Bruhat_decomposition)_

[[!redirects Bruhat decompositions]]


Bruhat-Tits tree

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Examples

### The Bruhat-Tits tree of $SL(2)$

The Bruhat-Tits tree of the [[special linear group]] $SL(2)$:

> graphics from [Casselman 14](#Casselman14)

In the [[p-adic AdS/CFT correspondence|2-adic AdS/CFT correspondence]] this tree plays the role of [[anti de Sitter spacetime]]. This may be regarded (at some finite depth truncation) as a [[tree tensor network|tree]]-[[tensor network state]]

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

(As in [HMSS 16](p-adic+AdS/CFT+correspondence#HMSS16), [HLM 19](p-adic+AdS/CFT+correspondence#HLM19). But  maybe one wants the Poincaré-dual networks, instead, as in [HMPS 18](p-adic+AdS/CFT+correspondence#HMPS18)?)

## Related concepts

* [[building]]

* [[p-adic AdS/CFT correspondence]]

* [[tensor network state]], [[holographic entanglement entropy]]

## References

### General

* {#Casselman14} Bill Casselman, _The Bruhat-Tits tree of $\mathrm{SL}(2)$_, 2014 ([pdf](https://pdfs.semanticscholar.org/3cc6/b53850a03003204f1225c780cf29ab625a6c.pdf), [[CasselmanOnBruhatTitsTree2014.pdf:file]])

### In $p$-adic string theory

Suggestion that the [[disk]] [[worldsheet]] of the [[open string|open]] [[p-adic string]] is to be identified with the [[Bruhat-Tits tree]] $T_p$:

* [[Anton Zabrodin]], _Non-Archimedean strings and Bruhat-Tits trees_, Comm. Math. Phys.
Volume 123, Number 3 (1989), 463-483 ([euclid.cmp/1104178891](https://projecteuclid.org/euclid.cmp/1104178891))

Suggestion to identify the [[Bruhat-Tits tree]] $T_p$ with [[anti de Sitter spacetime]] in the [[p-adic AdS/CFT correspondence]]:

* [[Steven Gubser]], Johannes Knaute, Sarthak Parikh, Andreas Samberg, Przemek Witaszczyk, _$p$-adic AdS/CFT_, Communications in Mathematical Physics volume 352, pages 1019–1059 (2017) ([arXiv:1605.01061](https://arxiv.org/abs/1605.01061))

Discussion of Bruhat-Tits-[[tree tensor networks]] in this context of the [[p-adic AdS/CFT correspondence]]:

* {#BHLL17} Arpan Bhattacharyya, Ling-Yan Hung, Yang Lei, Wei Li, _Tensor network and ($p$-adic) AdS/CFT_, JHEP 1801 (2018) 139 ([arXiv:1703.05445](https://arxiv.org/abs/1703.05445))

* {#HLM19} Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, _$p$-adic CFT is a holographic tensor network_ ([arXiv:1902.01411](https://arxiv.org/abs/1902.01411))

[[!redirects Bruhat-Tits trees]]


Bruno Barras

* [webpage](http://www.lix.polytechnique.fr/~barras/)

category: people


Bruno Bentzen

* [webpage](https://www.brunobentzen.com/)

## Selected writings

On informal [[cubical type theory]]:

* {#Bentzen19a} [[Bruno Bentzen]], _Naive cubical type theory_, 2019 ([pdf](https://arxiv.org/pdf/1911.05844.pdf))

On the justification of [[homotopy type theory]]:

* {#Bentzen19b} [[Bruno Bentzen]], _What types should not be_, 2019 ([link](https://doi.org/10.1093/philmat/nkz014))

## Related $n$Lab entries

* [[cubical type theory]]

category: people


Bruno de Finetti

Bruno de Finetti (13 June 1906 – 20 July 1985) was an Italian probabilist and statistician, famous for his subjectivist version of [[Bayesian reasoning]].

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

category: people


Bruno Gavranović

I am a PhD student at the Mathematically Structured Programming (MSP) Group at the University of Strathclyde, Glasgow.

I am working on understanding [[machine learning]] and game theory through the lens of [[category theory]].

[My webpage](https://www.brunogavranovic.com/)

## Selected writings

On [[category theory]] for [[machine learning]]:

* G.S.H. Cruttwell, [[Bruno Gavranović]], [[Neil Ghani]], Paul Wilson, Fabio Zanasi, _Categorical Foundations of Gradient-Based Learning_, ([arXiv:2103.01931](https://arxiv.org/abs/2103.01931))

* Dan Shiebler, [[Bruno Gavranović]], Paul Wilson, _Category Theory in Machine Learning_ ([arXiv:2106.07032](https://arxiv.org/abs/2106.07032))

On [[actegories]]:

* {#CG22} [[Matteo Capucci]], [[Bruno Gavranović]], _Actegories for the working amthematician_ [[arXiv:2203.16351](https://arxiv.org/abs/2203.16351)]

On [[categorical systems theory]] ([[cybernetics]]):

* [[Matteo Capucci]], [[Bruno Gavranović]], [[Jules Hedges]], [[Eigil F. Rischel]], *Towards Foundations of Categorical Cybernetics*, EPTCS **372** (2022) 235-248 [[arXiv:2105.06332](https://arxiv.org/abs/2105.06332), [doi:10.4204/EPTCS.372.17](https://doi.org/10.4204/EPTCS.372.17)]

category: people


Bruno Kahn

* [website](https://webusers.imj-prg.fr/~bruno.kahn/)

category: people


Bruno Mera

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

* [GoogleScholar page](https://scholar.google.pt/citations?user=eS0baHEAAAAJ&hl=en)

## Selected writings

On the [[external tensor product of vector bundles]] in the [[K-theory classification of topological phases of matter]]:

* [[Bruno Mera]], *The product of two independent Su-Schrieffer-Heeger chains yields a two-dimensional Chern insulator*, Phys. Rev. B **102** 155150 (2020) [[arXiv:2009.02268](https://arxiv.org/abs/2009.02268), [doi:10.1103/PhysRevB.102.155150](https://doi.org/10.1103/PhysRevB.102.155150)]

category: people


Bruno Nachtergaele

## Selected writings

On the [[quantum adiabatic theorem]] in [[condensed matter theory]] with an eye towards [[topological phases of matter]]:

* [[Sven Bachmann]], Spyridon Michalakis, [[Bruno Nachtergaele]], Robert Sims, *Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems*, Commun. Math. Phys. **309** (2012) 835-871 ([arXiv:1102.0842](https://arxiv.org/abs/1102.0842), [doi:10.1007/s00220-011-1380-0](https://doi.org/10.1007/s00220-011-1380-0))

On [[topological phases]] (i.e. [[mass gap|gapped]] [[ground states]]) of [[spin chains]]:

* [[Houssam Abdul-Rahman]], Marius Lemm, Angelo Lucia, [[Bruno Nachtergaele]], Amanda Young,  *A class of two-dimensional AKLT models with a gap*, in: *Analytic Trends in Mathematical Physics*,  Contemporary Mathematics **741**,  AMS (2020) [[arXiv:1901.09297](https://arxiv.org/abs/1901.09297), [doi:10.1090/conm/741/14917](https://doi.org/10.1090/conm/741/14917)]

category: people


Bruno Valette > history

see *[[Bruno Valette]]*


Bruno Vallette

__Bruno Valette__ is a mathematician at Univ. Paris 13 with main interests in [[operad]] theory, its generalizations and applications, including [[homotopy algebras]] and [[deformation theory]].

* [personal page](https://www.math.univ-paris13.fr/~vallette/index-english.html)

* [Wikipedia entry (fr)](https://fr.wikipedia.org/wiki/Bruno_Vallette)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=76014)

## Selected writings

On [[operads]]:

* [[Jean-Louis Loday]], [[Bruno Vallette]], *Algebraic Operads*, Grundlehren der mathematischen Wissenschaften **346**, Springer (2012) [[ISBN 978-3-642-30362-3](https://www.springer.com/gp/book/9783642303616), [pdf](http://irma.math.unistra.fr/~loday/PAPERS/LodayVallette.pdf)&rbrack

See also:

* Johan Leray, Bruno Vallette, _Pre-Calabi--Yau algebras and homotopy double Poisson gebras_, [arXiv:2203.05062](https://arxiv.org/abs/2203.05062)

* [[Imma Gálvez-Carrillo]], [[Andy Tonks]], Bruno Valette, _Homotopy Batalin-Vilkovisky algebras_, J. Noncomm. Geom. __6__:3, 539--602 [arXiv:0907.2246](https://arxiv.org/abs/0907.2246) [doi](https://doi.org/10.4171/JNCG/99)

* [[Bruno Valette]], _Homotopy theory of homotopy algebras_, Annales de l'Institut Fourier __70__:2 (2020)  683--738 [doi](https://10.5802/aif.3322) [Zbl:1335.18001](https://zbmath.org/?q=an:1335.18001) [arXiv:1411.5533](https://arxiv.org/abs/1411.5533)

## Related entries

* [[homotopy BV-algebra]]

category: people


Bruno Zumino

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

## Selected writings

Introducing [[supersymmetry]] for [[quantum field theory]] on 4d [[spacetime]]:

* {#WessZumino74} [[Julius Wess]], [[Bruno Zumino]], _Supergauge transformations in four dimensions_, Nucl. Phys. B70 (1974) 39; Phys. Letters 49B (1974) 52 ([doi:10.1142/9789814542340_0002](https://doi.org/10.1142/9789814542340_0002))

Introducing [[super Yang-Mills theory]]:

* {#FerraraZumino74} [[Sergio Ferrara]], [[Bruno Zumino]], _Supergauge invariant Yang-Mills theories_, Nuclear Physics B Volume 79, Issue 3, 25 September 1974, Pages 413-421 (doi:10.1016/0550-3213(74)90559-8)

On the [[current-field identity]] in [[quantum hadrodynamics]] ([[vector meson dominance]]):

* {#KrollLeeZumino76} Norman M. Kroll, T. D. Lee, [[Bruno Zumino]], _Neutral Vector Mesons and the Hadronic Electromagnetic Current_, Phys. Rev. 157, 1376 (1967) ([arXiv:10.1103/PhysRev.157.1376](https://doi.org/10.1103/PhysRev.157.1376))

On [[chiral perturbation theory]]:

* [[Sidney Coleman]], [[Julius Wess]], [[Bruno Zumino]], _Structure of Phenomenological Lagrangians. I_, Phys. Rev. 177, 2239 (1969) ([doi:10.1103/PhysRev.177.2239](https://doi.org/10.1103/PhysRev.177.2239))

* [[Curtis Callan]], Jr., [[Sidney Coleman]], [[Julius Wess]], [[Bruno Zumino]], _Structure of Phenomenological Lagrangians. II_, Phys. Rev. 177, 2247 (1969) ([doi:10.1103/PhysRev.177.2247](https://doi.org/10.1103/PhysRev.177.2247))

Introducing the [[WZW term]] of [[chiral perturbation theory]]/[[quantum hadrodynamics]] which reproduces the [[chiral anomaly]] of [[QCD]] in the [[effective field theory]] of [[mesons]] and [[Skyrmions]]:

* [[Julius Wess]], [[Bruno Zumino]], _Consequences of anomalous Ward identities_, Phys. Lett. B 37 (1971) 95-97 ([spire:67330](https://inspirehep.net/literature/67330), doi:10.1016/0370-2693(71)90582-X)

Making manifest the local [[worldsheet]] [[supersymmetry]] of the [[spinning string]] (via the supersymmetric [[Polyakov action]]):

* [[Stanley Deser]], [[Bruno Zumino]], *A complete action for the spinning string*, Physics Letters B **65** 4 (1976) 369-373 ([pdf](http://cdsweb.cern.ch/record/201648/files/CM-P00061857.pdf?version=1))

category: people


Bryan Eastin

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

## Selected writings

Introducing the [[Eastin-Knill theorem]] on [[quantum error correction]] versus universality of [[quantum computation|quantum gates]]:

* {#EastinKnill09} [[Bryan Eastin]], [[Emanuel Knill]], *Restrictions on Transversal Encoded Quantum Gate Sets*,  Phys. Rev. Lett. 102, 110502 (2009) ([arXiv:0811.4262](https://arxiv.org/abs/0811.4262))

category: people


Bryce Clarke

I am a postdoctoral researcher at the [Inria Saclay](https://www.inria.fr/en/inria-saclay-centre) in Palaiseau, France, where I am working with Gabriel Scherer and [[Noam Zeilberger]] in the PARTOUT team.

Before that, I was a PhD student at the Centre of Australian Category Theory (CoACT) and Macquarie University in Sydney, Australia, under the supervision of [[Michael Johnson]] and [[Dominic Verity]].

I am broadly interested in category theory and its applications, and so far my research has focused on [[delta lens|delta lenses]], [[Grothendieck fibration|opfibrations]], [[cofunctor|cofunctors]], and [[double categories]].

* [personal website](https://bryceclarke.github.io/)
* [Mathematics Genealogy](https://www.mathgenealogy.org/id.php?id=286575)
* [arXiv papers](https://arxiv.org/a/clarke_b_2.html)

## Selected writings

On [[lenses (in computer science)]], in particular on [[delta lenses]]:

* {#Clarke20a} [[Bryce Clarke]], _Internal lenses as functors and cofunctors_, Proceedings Applied Category Theory 2019, Electronic Proceedings in Theoretical Computer Science, **323**, 2020. ([doi:10.4204/EPTCS.323.13](http://dx.doi.org/10.4204/EPTCS.323.13))

* {#Clarke21a} [[Bryce Clarke]], _A diagrammatic approach to symmetric lenses_, Proceedings of the 3rd Annual International
Applied Category Theory Conference 2020, Electronic Proceedings in Theoretical Computer Science, **333**, 2021. ([doi:10.4204/EPTCS.333.6](https://doi.org/10.4204/EPTCS.333.6))

* {#Clarke21b} [[Bryce Clarke]], _Delta lenses as coalgebras for a comonad_, Bx 2021: 9th International Workshop on Bidirectional Transformations, CEUR Workshop Proceedings, **2999**, 2021. ([pdf](https://ceur-ws.org/Vol-2999/bxpaper2.pdf), [arXiv:2108.00390](https://arxiv.org/abs/2108.00390))

* {#Clarke22} [[Bryce Clarke]], _Limits and Colimits in a Category of Lenses_, Proceedings of the Fourth International Conference on Applied Category Theory, Electronic Proceedings in Theoretical Computer Science, **372**, 2022. ([doi:10.4204/EPTCS.372.12](https://doi.org/10.4204/EPTCS.372.12))

On [[cofunctor|cofunctors]], including their relationship to [[cleavage|split]] [[Grothendieck fibration|opfibrations]] in [[internal category]] theory, and their generalisation to [[enriched category]] theory:

* {#Clarke20b} [[Bryce Clarke]], _Internal split opfibrations and cofunctors_, Theory and Applications of Categories, **35**, 2020. ([link](http://www.tac.mta.ca/tac/volumes/35/44/35-44abs.html))

* {#ClarkeDiMeglio22} [[Bryce Clarke]] and Matthew Di Meglio, _An introduction to enriched cofunctors_, 2022. ([arXiv:2209.01144](https://arxiv.org/abs/2209.01144))

On [[optics (in computer science)]]:

* [[Bryce Clarke]], Derek Elkins, [[Jeremy Gibbons]], [[Fosco Loregian]], [[Bartosz Milewski]], Emily Pillmore, [[Mario Román]], _Profunctor optics, a categorical update_, 2020. ([arXiv:2001.07488](https://arxiv.org/abs/2001.07488))

On the [[double category]] of [[delta lenses|lenses]]:

* {#Clarke2022} [[Bryce Clarke]], _The double category of lenses_, PhD thesis, Macquarie University, 2022. ([doi:10.25949/22045073.v1](https://doi.org/10.25949/22045073.v1))

category: people


Bryce DeWitt

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

* [[Cécile DeWitt-Morette]], *The Pursuit of Quantum Gravity -- Memoirs of Bryce DeWitt from 1946 to 2004*, Springer (2011) [[doi:10.1007/978-3-642-14270-3_4](https://doi.org/10.1007/978-3-642-14270-3_4)]

## Selected writings

Early discussion of [[quantum gravity]] and introducing what came to be called the *[[Wheeler-DeWitt equation]]* (5.5):

* {#DeWitt67} [[Bryce DeWitt]], *Quantum Theory of Gravity. I. The Canonical Theory*, Phys. Rev. **160** (1967) 1113 [[doi:10.1103/PhysRev.160.1113](https://doi.org/10.1103/PhysRev.160.1113)]

On the [[many-worlds interpretation of quantum mechanics]]:

* [[Bryce S. DeWitt]], [[Neill Graham]] (eds.), *The Many-Worlds Interpretation of Quantum Mechanics*, Princeton Series in Physics, Princeton University Press (1973, 2016) [[ISBN:9780691645926](https://press.princeton.edu/books/hardcover/9780691645926/the-many-worlds-interpretation-of-quantum-mechanics), [jstor:j.ctt13x0wwk](https://www.jstor.org/stable/j.ctt13x0wwk)]

## Related entries

* [[supermanifold]]

* [[Peierls bracket]]

* [[Feynman propagator]]

* [[quantum gravity]]

category: people

[[!redirects Bryce S. DeWitt]]


BTZ black hole

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

#Contents#
* table of contents
{:toc}

## Idea

[[black holes]] in [[3d gravity]] with [[negative number|negative]] [[cosmological constant]]

## Properties

### Euclidean BTZ is hyperbolic solid torus

The [[Euclidean field theory|Euclidean]] [[BTZ black hole]] is a [[hyperbolic 3-manifold]] [[homeomorphism|homeomorphic]] to (the [[interior]] of) the [[hyperbolic solid torus]], hence to the [[knot complement]] of the [[unknot]] in the [[3-sphere]].

(e.g [Gukov 03, appendix A](#Gukov03), [Kraus 06, around Fig. 1](#Kraus06), [BKR 07, 2.1](#BKR07))

Notice that the [[volume]] of the [[hyperbolic solid torus]] is _not [[finite number|finite]]_.  Therefore this [[hyperbolic 3-manifold]] [[knot complement]] does not count as a "knot complement with hyperbolic structure" in the sense of [[William Thurston|Thurston]]'s [[classification of 3-manifolds]] (see also [this MO discussion](https://mathoverflow.net/q/302038/381)).

## Related concepts

* [[AdS3/CFT2]]

## References

### General

The original BTZ black hole with trivial internal topology is due to:

* [[Máximo Bañados]], [[Claudio Teitelboim]], [[Jorge Zanelli]], _The Black Hole in Three Dimensional Space Time_, Phys. Rev. Lett. 69 (1992) 1849-1851 ([arXiv:hep-th/9204099](https://arxiv.org/abs/hep-th/9204099))

* [[Máximo Bañados]], [[Marc Henneaux]], [[Claudio Teitelboim]], [[Jorge Zanelli]],
_Geometry of the 2+1 Black Hole_, Phys. Rev. D48: 1506-1525, 1993 ([arXiv:gr-qc/9302012](https://arxiv.org/abs/gr-qc/9302012))

The generalization to arbitrary black holes in 2+1-dimensional AdS gravity, with generally non-trivial internal topology:

* Stefan Aminneborg, [[Ingemar Bengtsson]], Dieter Brill, Soren Holst, Peter Peldan, _Black Holes and Wormholes in 2+1 Dimensions_, Class. Quant. Grav. 15 (1998) 627-644 ([arXiv:gr-qc/9707036](https://arxiv.org/abs/gr-qc/9707036))

* Stefan Aminneborg, [[Ingemar Bengtsson]], Soren Holst, _A Spinning Anti-de Sitter Wormhole_, Class. Quant. Grav. 16 (1999) 363-382 ([arXiv:gr-qc/9805028](https://arxiv.org/abs/gr-qc/9805028))

* Dieter Brill, _Black Holes and Wormholes in 2+1 Dimensions_, In: Cotsakis S., Gibbons G.W. (eds) _Mathematical and Quantum Aspects of Relativity and Cosmology_. Lecture Notes in Physics, vol 537. Springer, Berlin, Heidelberg
 ([arXiv:gr-qc/9904083](https://arxiv.org/abs/gr-qc/9904083), [doi:10.1007/3-540-46671-1_6](https://doi.org/10.1007/3-540-46671-1_6)
)

See also:

* Aritra Ghosh, Chandrasekhar Bhamidipati, _Thermodynamic geometry and interacting microstructures of BTZ black holes_ ([arXiv:2001.10510](https://arxiv.org/abs/2001.10510))

* [[Roberto Emparan]], Antonia Micol Frassino, Benson Way, _Quantum BTZ black hole_, JHEP11 (2020) 137 ([arXiv:2007.15999](https://arxiv.org/abs/2007.15999))

* Pablo Bueno, Pablo A. Cano, Javier Moreno, Guido van der Velde, *Regular black holes in three dimensions* ([arXiv:2104.10172](https://arxiv.org/abs/2104.10172))

In view of the [[cosmic censorship hypothesis]]:

* [[Roberto Emparan]], Marija Tomašević, _Strong cosmic censorship in the BTZ black hole_, JHEP 06 (2020) 038 ([arXiv:2002.02083](https://arxiv.org/abs/2002.02083))

Via [[topological M-theory]]:

* Javier Chagoya, Graciela Reyes-Ahumada, M. Sabido, _BTZ entropy from topological M-theory_ ([arXiv:2011.01094](https://arxiv.org/abs/2011.01094))

See also:

* Wikipedia, _[BTZ black hole](http://en.wikipedia.org/wiki/BTZ_black_hole)_

### Euclidean BTZ black holes

Discussion of [[Euclidean field theory|Euclidean]] [[BTZ black holes]]/[[thermal AdS3]] (the [[hyperbolic solid torus]]), partly with an eye towards [[black hole entropy]] computed via [[AdS3/CFT2]]:

* [[Kirill Krasnov]], _Holography and Riemann Surfaces_, Adv. Theor. Math. Phys. 4 (2000) 929-979 ([arXiv:hep-th/0005106](https://arxiv.org/abs/hep-th/0005106))

* [[Kirill Krasnov]], around Figure 6 of: _Analytic Continuation for Asymptotically AdS 3D Gravity_, Class. Quant. Grav. 19 (2002) 2399-2424 ([arXiv:gr-qc/0111049](https://arxiv.org/abs/gr-qc/0111049))

* [[Kirill Krasnov]], _Black Hole Thermodynamics and Riemann Surfaces_, Class. Quant. Grav. 20 (2003) 2235-2250 ([arXiv:gr-qc/0302073](https://arxiv.org/abs/gr-qc/0302073))

* {#Gukov03} [[Sergei Gukov]], Appendix A of: _Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial_, Commun. Math. Phys. 255: 577-627, 2005 ([arXiv:hep-th/0306165](https://arxiv.org/abs/hep-th/0306165))

* {#Kraus06} [[Per Kraus]], _Lectures on black holes and the $AdS_3/CFT_2$ correspondence_,  Lect. Notes Phys. 755: 193-247, 2008 ([arXiv:hep-th/0609074](https://arxiv.org/abs/hep-th/0609074))

* {#BKR07} [[Micha Berkooz]], Zohar Komargodski, Dori Reichmann, _Thermal $AdS_3$, BTZ and competing winding modes condensation_, JHEP 0712:020, 2007 ([arXiv:0706.0610](https://arxiv.org/abs/0706.0610))

* M. Cadoni, M. Melis, _Holographic entanglement entropy of the BTZ black hole_, Found. Phys. 40: 638-657, 2010 ([arXiv:0907.1559](https://arxiv.org/abs/0907.1559))

  (relation to [[holographic entanglement entropy]])

* [[Ingmar Saberi]], around Figure 4.3 of: _Knots, Trees, and Fields: Common Ground Between Physics and Mathematics_ ([doi:10.7907/Z9VX0DHZ](https://thesis.library.caltech.edu/9896), [pdf](https://thesis.library.caltech.edu/9896/1/Saberi_Ingmar_2016.pdf))

  (with generalization to [[p-adic string theory]])

See also:

* Zhen-Ming Xu, Bin Wu, Wen-Li Yang, _Thermodynamic curvature and isoperimetric inequality for the charged BTZ black hole_ ([arXiv:2002.00117](https://arxiv.org/abs/2002.00117))

  (not Euclidean, but thermodynamic)

[[!include Chern-Simons Wilson lines in AdS3-CFT2 -- references]]

### In $p$-adic AdS/CFT

Discussion of [[BTZ black holes]] via [[tensor networks]] in the [[p-adic AdS/CFT correspondence]]:

* Matthew Heydeman, [[Matilde Marcolli]], Sarthak Parikh, [[Ingmar Saberi]], _Nonarchimedean Holographic Entropy from Networks of Perfect Tensors_ ([arXiv:1812.04057](https://arxiv.org/abs/1812.04057))

* Stephen Ebert, Hao-Yu Sun, Meng-Yang Zhang, _Probing holography in $p$-adic CFT_ ([arXiv:1911.06313](https://arxiv.org/abs/1911.06313))

[[!redirects BTZ black holes]]

[[!redirects thermal AdS3]]


Bub-Clifton theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The two fundamental [[no-go theorems]] for [[hidden variable theory|hidden variable]] reconstructions of the [[quantum mechanics|quantum]] statistics, the [[Kochen-Specker theorem]] and [[Bell's theorem]], may be formulated as results about the impossibility of associating a classical [[probability space]] with a [[quantum system]], when certain constraints are placed on the [[probability measure]].

The _Bub-Clifton-Halvorson theorem_ ([Bub-Clifton 96](#BubClifton96), [Clifton-Bub-Halvorson 03](#CliftonBubHalvorson03)) on the other hand is a positive result about the possibility of associating a classical probability space with a quantum system in a given state.

Given a [[Hilbert space]] $H$ and a [[quantum observable]] represented by a [[Hermitean operator]] on this space, and given a [[pure state]], the Bub-Clifton theorem characterizes a maximal sub-[[lattice]] of the Birkhoff-vonNeumann [[Hilbert lattice]] of subspace of $H$ (the "[[quantum logic]]" of $H$) such that there are sufficiently many yes-no questions on the elements in the lattice to recover all the probabilities induced by the given pure state to compatible sets of projections ([[classical contexts]]).

A useful summary is in [Bub 09, pages 1-2](#Bub09).

## Related concepts

* [[interpretation of quantum mechanics]]

* [[beable]]

Other theorems about the foundations and [[interpretation of quantum mechanics]] include:

* [[order-theoretic structure in quantum mechanics]]

  * [[Kochen-Specker theorem]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Fell's theorem]]

* [[Gleason's theorem]]

* [[Wigner theorem]]

* [[Bell's theorem]]

* [[Kadison-Singer problem]]

## References

* {#BubClifton96} [[Jeffrey Bub]], Rob Clifton, _A Uniqueness Theorem for "No Collapse" Interpretations of Quantum Mechanics_,  Studies in the History
and Philosophy of Modern Physics 27, 181-219. (1996)

* [[Jeffrey Bub]], Rob Clifton, [[Sheldon Goldstein]], _Revised Proof of the Uniqueness Theorem for 'No Collapse' Interpretations of Quantum Mechanics_, Studies in History and Philosophy of Modern Physics 31 (2000) 95 ([arXiv:quant-ph/9910097](http://arxiv.org/abs/quant-ph/9910097))

* {#CliftonBubHalvorson03} Rob Clifton, [[Jeffrey Bub]], [[Hans Halvorson]], _Characterizing Quantum Theory in Terms of Information-Theoretic Constraints_, Foundations of Physics 33 (2003), p 1561 ([arXiv:quant-ph/0211089](https://arxiv.org/abs/quant-ph/0211089))

* {#Bub09} [[Jeffrey Bub]], _Bub-Clifton Theorem_, in Greenberger D., Hentschel K., Weinert F. (eds.) _Compendium of Quantum Physics_
Springer 2009, pp 84-86 ([web](https://link.springer.com/chapter/10.1007%2F978-3-540-70626-7_25))

A broader textbook discussion is in

* [[Jeffrey Bub]], _Interpreting the quantum world_, Cambridge University Press, Aug 26, 1999

An interpretation in [[topos theory]] is proposed in

* {#Nakayama07} Kunji Nakayama, _Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics_, Int. J. Theor. Phys. 47:2065-2094, 2008 ([arXiv:0711.2200](https://arxiv.org/abs/0711.2200))

See also

* Jonathan Bain, _Physics from Quantum Information_ ([pdf](http://faculty.poly.edu/~jbain/physinfocomp/lectures/10.PhysicsfromQI.pdf)), chapter 10 in _[Physics, Information and Computation](http://faculty.poly.edu/~jbain/physinfocomp/)_

* Wikipedia, _[CBH-Theorem](https://de.wikipedia.org/wiki/CBH-Theorem)_

[[!redirects Bub-Clifton-Halvorson theorem]]

[[!redirects CBH-theorem]]


bubble of nothing

[[!redirects Bubble of nothing]]
[[!redirects Bubble of nothing]]
[[!redirects Bubble of nothing]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A *bubble of nothing* in the sense of [Witten 82](#Witten82) is an instability of the [[Kaluza-Klein]] [[vacuum]] where a [[gravitational instanton]] mediates the decay of a $5$-dimensional [[Euclidean]] [[Schwarzschild spacetime]] into nothing:
the hole in space grows at the [[speed of light]] until it hits [[null infinity]], leading to the "annihilation" of spacetime.

## Construction of the solution

Let us start from the spacetime $(\mathbb{R}^{4}-B_R^3)\times S^1$, where $B_R^3$ is a [[ball]] of radius $R$ and $S^1$ is a circle of radius $R_5$, equipped with the following [[Euclidean]] [[Schwarzschild metric]]:
$$ g \;=\; r^2\mathrm{vol}_{S^3} + \frac{\mathrm{d}r^2}{1-\frac{R^2}{r^2}} + R_5^2 \left(1-\frac{R^2}{r^2}\right)\mathrm{d}\theta^2,$$
where $\theta$ is the coordinate on the circle and $r$ is the radius of $\mathbb{R}^{4}$.

Let the volume of the $3$-sphere be $\mathrm{vol}_{S^3} = \mathrm{d}\chi^2 + \sin(\chi)\mathrm{vol}_{S^2}$. We can now go back to the Minkowski signature by the [[Wick rotation]] $\psi := -i\chi + i\frac{\pi}{2}$. Thus, we have
$$ g \;=\; -r^2\mathrm{d}\psi^2 + \frac{\mathrm{d}r^2}{1-\frac{R^2}{r^2}} + r^2\cosh^2(\psi)\mathrm{vol}_{S^2} + R_5^2 \left(1-\frac{R^2}{r^2}\right)\mathrm{d}\theta^2.$$

## Properties

By change of coordinates $\rho := r\cosh(\psi)$ and $t:=r\sinh(\psi)$, we see that at large radius the metric goes to the [[Minkowski metric]]
$$\lim_{r\rightarrow \infty}g \;=\; -\mathrm{d}t^2 +\mathrm{d}\rho^2 + \rho^2\mathrm{vol}_{S^2}+R_5^2\mathrm{d}\theta^2.$$

However, the coordinates $(t,\rho)$ parametrize all the $4$-dimensional base manifold but the region $\{ \rho^2 - t^2 \leq R \}$ of the bubble.
The boundary of the bubble in the $4$-dimensional base space has a radius which grows with time as
$$ \rho_{\mathrm{BON}}(t) \;=\;\sqrt{R^2+t^2}.$$
Moreover, the effective size of the circle compactification will be depend on $r$ by
$$ R_{5}(r) \;=\; R_5\sqrt{1-\frac{R^2}{r^2}},$$
so that it goes to $0$ for $r\rightarrow R$ and it goes to $R_5$ for for $r\rightarrow \infty$.

## Related concepts

  * [[Coleman-De Luccia instanton]]

  * [[Kaluza-Klein theory]]

  * [[swampland]]

  * [[instanton]]

    * [[gravitational instanton]]

  * [[double dimensional reduction]]

  * [[moduli stabilization]]

  * [[landscape of string theory vacua]]

    * [[flux compactification]]

    * [[supersymmetry and Calabi-Yau manifolds]]

    * [string theory FAQ -- What does it mean to say that string theory has a "landscape" of solutions?](string%20theory%20FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions)

## References

* {#Witten82} [[Edward Witten]], _Instability of the Kaluza-Klein Vacuum_, Nucl. Phys. B195 (1982) 481-492 (doi:10.1016/0550-3213(82)90007-4).

[[!redirects Bubble of nothing]]
[[!redirects Bubble of nothing]]
[[!redirects Bubble of nothing]]


Buchstaber–Rees theorem

under construction

## Idea

Gelfand and Kolmogorov proved in 1939 that a compact Hausdorff topological space $X$ can be canonically embedded into the infinite‐dimensional vector space $C(X)^*$, the dual space of the algebra of continuous functions $C(X)$, as an "algebraic variety", specified by an infinite system of quadratic equations.

Buchstaber and Rees have extended this to all [[symmetric power]]s $Sym_n(X)$ using their notion of the Frobenius $n$‐homomorphisms.

## Related concepts

* [[Gelfand-Kolmogorov theorem]]

## References

* Victor M. Buchstaber, [[Elmer G. Rees]], _The Gelfand map and symmetric products_, Selecta Math. 8 (2002), no. 4, 523–535; ...,
Russian Math. Surveys __59__ (1(355)):125–145, 2004; _Frobenius n-homomorphisms, transfers and branched coverings_, Math. Proc. Cambr. Phil. Soc. 2007, arXiv:math.RA/0608120 .

* I. M. Gel'fand, A. N. Kolmogorov. Dokl. Akad. Nauk SSSR __22__: 11–15, 1939.

* H. M. Khudaverdian, Th. Th. Voronov, dadada, Lett. Math. Phys., 74(2):201–228, 2005; _Operators on superspaces and generalizations of the Gelfand–Kolmogorov theorem_, AIP Conf. Proc. 956, 149 (2007);[doi](http://dx.doi.org/10.1063/1.2820962); _On generalized symmetric powers and a generalization of Gel'fand-Kolmogorov-Buchstaber-Rees theorem_, [pdf](http://www.google.com/url?q=http://www.maths.manchester.ac.uk/~khudian/Research/Papers/frobenius_umn_final.pdf&sa=U&ei=rad_UrywFaWM5ATu4oG4Cw&ved=0CB0QFjAB&sig2=sRw7x42Kw2A8LUUaqeZBQQ&usg=AFQjCNFuKJfikzpCopTwYioiogx_CXb3Hw); _A short proof of the Buchstaber–Rees theorem_,   Phil. Trans. R. Soc. A  __369__, no. 1939, 1334-1345 (2011) [full](http://rsta.royalsocietypublishing.org/content/369/1939/1334.full), [pdf](http://rsta.royalsocietypublishing.org/content/369/1939/1334.full.pdf+html) (keywords: Berezinian (superdeterminant), Frobenius recursion, symmetric powers, maps of algebras, $n$-homomorphism, $p|q$-homomorphism
* Alexander Pavlov, Evgenij Troitsky, _Quantization of branched coverings_, Russian J. of Math. Phys.  18, N 3 (2011), 338-352 [arxiv/1002.3491](http://arxiv.org/abs/1002.3491)


building

#Contents#
* table of contents
{:toc}

## Idea

A **building** (also **Tits building**, **Bruhat--Tits building**) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of [[flag manifold]]s, finite [[projective plane]]s, and [[Riemannian manifold|Riemannian]] [[symmetric space]]s.

Initially introduced by [[Jacques Tits]] as a means to understand the structure of [[exceptional Lie group]]s, the theory has also been used to study the geometry and topology of [[homogeneous space]]s of p-adic Lie groups and their discrete subgroups of symmetries, in the same way that [[tree]]s have been used to study [[free group]]s.

> (for the moment the above text is simply a copy of the intro of the [wikipedia entry](http://en.wikipedia.org/wiki/Building_%28mathematics%29))

See also

* [[toddtrimble:Buildings for category theorists]]

## Introduction

This page is intended to be a supplementary exposition of the page [[toddtrimble:Buildings for category theorists]], at least for now. So we will try to explain the aspects of buildings that the kind reader would like to know in order to understand the category theoretic construction over there.
The notion of a building relies heavily on that of a [[Coxeter group]]. Then there are currently three different viewpoints of thinking about a building.
From now on, let $W$ be a Coxeter group and $S$ the set of generators, which may be finite or infinite: In case we need $S$ to be finite we will mention it explicitly. We will call the tuple $(W, S)$ a **Coxeter system** and let $l_S$ be the **length function**
 of $W$ with respect to $S$.

### Buildings from three Viewpoints

The three viewpoints are distinguished by how one thinks about chambers.

#### Simplicial Complexes

In Tits' original approach, a building is a simplicial complex satisfying additional axioms. Chambers are maximal simplices.

#### Combinatorial Approach

Note that this is not a firmly established term in the literature.
This more modern approach originated with

* [Jaques Tits: A local approach to buildings, The geometric vein (C. Davis, B.Grünbaum, and F. A. Sherk, eds.), Springer-Verlag, New York, 1981, pp. 519–547.]

Buildings are viewed as sets of chambers with a Coxeter-group-valued distance function satisfying certain axioms. Chambers are elements of an abstract set - more is not needed in the definition. As an alternative, chambers can be viewed as vertices of a graph. This is the point of view we will expand in future versions of this page.

#### Buildings as Metric Spaces

The following paper

* [M. W. Davis, Buildings are CAT(0), Geometry and cohomology in group theory
(P. H. Kropholler, G. A. Niblo, and R. Stöhr, eds.), London Mathematical
Society Lecture Note Series, vol. 252, Cambridge University Press, Cambridge,
1998, pp. 108–123.]

explains that every building has a geometric realization that admits a CAT(0) metric, which means that it has metric properties analogous to those of complete simply connected Riemannian manifolds of nonpositive curvature.

* [Wikipedia: [CAT(k) spaces] (http://en.wikipedia.org/wiki/CAT%28k%29_space)]

From this viewpoint chambers are metric spaces.

## Definition

### According to the Combinatorial Approach

This definition is due to

* [Jaques Tits: Twin buildings and groups of Kac-Moody type, Groups, combinatorics
& geometry (M. Liebeck and J. Saxl, eds.), London Mathematical Society
Lecture Note Series, vol. 165, Cambridge University Press, Cambridge, 1992,
pp. 249–286.]

Let (W, S) be a Coxeter system (see above).

**Definition:**

A **building of type (W, S)** is a pair (C, $\delta$) of a nonempty set C, whose elements are called **chambers**, and a function
$\delta$: CxC $\to$ W
subject to the following axioms:

* WD1, "identity of indiscernibles": $\delta$(C, D) = 1 iff C = D

* WD2, "triangle inequality": Let A, B, C be chambers and $\delta$(A, B) = w, $\delta$(C, A) = s, then $\delta$(C, B) = either w or sw. If in addition $l_S$(sw) = $l_S$(w) + 1, then $\delta$(C, B) = sw.

* WD3, "existence of geodesics": Let A, B be chambers and $\delta$(A, B) = w, then for any s $\in$ S there is a chamber C such that $\delta$(C, A) = s and $\delta$(C, B) = sw

This definition is equivalent from the one usually given from the simplicial viewpoint, but we will not prove that here, see e.g. the book by Abramenko and Brown in the introductory references.
In the following paragraphs we will explain a few simple consequences of this definition, and introduce concepts that will allow us to identify the chambers and their "distance" relation with vertices and edges of a graph respectively (TODO: not done yet).

But first let us note that the axioms WD1 and WD2 are somewhat similar to the axioms defining the distance function of a metric space, one notable difference is that the function $\delta$ of a building takes values in the Coxeter group W of the building rather than in the nonnegative real numbers. For this reason, from the combinatorial viewpoint, buildings are sometimes called **W-metric spaces**.

In order to distinguish the two viewpoints, which is, given their equivalence, strictly speaking not necessary, some authors will talk about **W-metric buildings** and **simplicial buildings**.

**Remark:**
If the Coxeter group W is finite, the building is called **spherical**. The reason for this is, that in this case, for $n := |S|$, the Coxeter group W has a faithful representation in the group of reflections on $\Re^n$. Reflections are in 1:1 correspondence to (linear) hyperplanes, the reflections that are images of the elements of the Coxeter group define a triangulation of the [[unit sphere]] of $\Re^n$, such that the resulting simplicial complex provides a geometric realization of the building.

## Related concepts

* [[Bruhat-Tits tree]]

* [[p-adic AdS/CFT correspondence]]

## References

### Introductory

* {#Casselman19} Bill Casselman, _Geometry of the tree_, 2019 ([pdf](https://www.math.ubc.ca/~cass/research/pdf/Tree.pdf))

* [[John Baez]], This Week's Finds in Mathematical Physics, [week 263] (http://math.ucr.edu/home/baez/week263.html) second half.

* Wikipedia on [buildings] (http://en.wikipedia.org/wiki/Bruhat-Tits_building)

An introductory textbook that starts with explaining Coxeter groups is this:

* Abramenko, Peter; Brown, Kenneth S.: Buildings. Theory and applications. (Springer 2008, [ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:pre05288866&format=complete) )

A short introduction to spherical buildings (this notion will be explained below):

* Richard M. Weiss: The Structure of Spherical Buildings. (Princeton University Press 2004, [ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1061.51011&format=complete))

### Buildings and [[Berkovich spaces]]

* Remy, Bertrand; Thuillier, Amaury; Werner, Annette, _Bruhat-Tits buildings and analytic geometry_ ([arXiv:1110.1362] (https://arxiv.org/abs/1110.1362)).

* Werner, Annette, _Buildings and Berkovich Spaces_ ([pdf](https://www.uni-frankfurt.de/50581265/dmvmuench10.pdf))

[[!redirects buildings]]

[[!redirects Tits building]]
[[!redirects Bruhat-Tits building]]
[[!redirects Bruhat–Tits building]]
[[!redirects Bruhat--Tits building]]

[[!redirects Tits buildings]]
[[!redirects Bruhat-Tits buildings]]
[[!redirects Bruhat–Tits buildings]]
[[!redirects Bruhat--Tits buildings]]


bulk

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

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] on [[manifolds with boundary]] or with [[domain walls]], [[branes]], [[QFT with defects|defects]], etc. that piece of [[worldvolume]]/[[spacetime]] away from all these singularies (the remaining [[interior]]) is often called the _bulk_ of [[worldvolume]]/[[spacetime]]. Accordingly, the sector of [[field theory]] applying to the bulk is sometimes called the _[[bulk field theory]]_, in contrast to the [[boundary field theory]].

## Related entries

* [[holographic principle]]

  [[AdS-CFT correspondence]]

[[!include field theory with boundaries and defects - table]]

[[!redirects bulks]]

[[!redirects bulk spacetime]]
[[!redirects bulk spacetimes]]

[[!redirects bulk space]]
[[!redirects bulk spaces]]


bulk field theory

+-- {: .rightHandSide}
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#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For a  [[field theory]] with [[boundary field theory|boundary]] the sector not containing boundaries is often called the corresponding _bulk field theory_.

For instance in [[AdS3-CFT2 and CS-WZW correspondence]] the [[Chern-Simons theory]] is the bulk field theory while the [[WZW model]] is the boundary field theory

## Related concept

[[!include holographic principle -- table]]

[[!redirects bulk field theories]]


bullet cluster

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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called the _bullet cluster_ is a pair of colliding [[galaxy clusters]] observed at a distance of about 3.7 billion [[light-years]].

The bullet cluster stands out in that comparison of its distribution of visible matter and non-visible gravitating matter seems to show that the centers of mass of the two components are separating during the collision. This has famously been interpreted as a strong indirect evidence for the [[dark matter]] hypothesis.

## References

* Chiara Mastropietro, Andreas Burkert, _Simulating the Bullet Cluster_,
Wiley Online Library  ([doi:10.1111/j.1365-2966.2008.13626.x](https://doi.org/10.1111/j.1365-2966.2008.13626.x))

* M.Bradač, _Dark Matter: Revealing the Invisible with 2 Cosmic Supercolliders the “Bullet Cluster” 1E0657-56 and MACSJ0025-1222_, Nuclear Physics B - Proceedings Supplements Volume 194, October 2009, Pages 17-21 ([doi:10.1016/j.nuclphysbps.2009.07.002](https://doi.org/10.1016/j.nuclphysbps.2009.07.002))

* Andrew Robertson, Richard Massey, Vincent Eke, _What does the Bullet Cluster tell us about self-interacting dark matter?_ ([arXiv:1605.04307](https://arxiv.org/abs/1605.04307))

See also

* PennState Phys010: _[Bullet cluster](https://online.science.psu.edu/phys010_archive_suwd/node/8306)_

* Wikipedia, _[Bullet cluster](https://en.wikipedia.org/wiki/Bullet_Cluster)_


bump function

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

# Contents
* table of contents
{: toc}

## Idea

A _bump function_ is a [[smooth function]] with [[compact support]], especially one that is not zero on a space that it not [[compact space|compact]].

One reason the category [[SmthMfd]] of [[smooth manifolds]] and [[smooth functions]] is important, hence one reason why [[differential geometry]] is special, is that a good supply of bump functions exists; this fact accounts for the great flexibility of smooth objects, in stark contrast to [[analytic geometry]] or [[algebraic varieties]] (since non-zero analytic or algebraic functions with compact support exist only on compact spaces). For instance, bump functions can be used to construct _[[partition of unity|partitions of unity]]_, which can in turn be used to smoothly patch together local structures into a global structure without obstruction, as for example [[Riemannian metric|Riemannian metrics]] and more generally [[inner products on vector bundles]] (see [this prop.](inner+product+on+vector+bundles#ExistenceOfInnerProductOfTopologicalVectorBundlesOverParacompactHausdorffSpaces)).

## Constructions

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

A _bump function_ is a [[function]] on [[Cartesian space]] $\mathbb{R}^n$, for some $n \in \mathbb{R}$ with values in the [[real numbers]] $\mathbb{R}$

$$
  b
    \;\colon\;
  \mathbb{R}^n
    \longrightarrow
  \mathbb{R}
$$

such that

1. $b$ is [[smooth function|smooth]];

1. $b$ has [[compact support]].

=--

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

For every [[closed ball]] $B_{x_0}(\epsilon) = \{x \in \mathbb{R}^n \,\vert\, {\Vert x - x_0 \Vert} \leq \epsilon\} \subset \mathbb{R}^n$ there exists a bump function $b \colon \mathbb{R}^n \to \mathbb{R}$ (def. \ref{BumpFunction}) with

$$
  Supp(b) = B_{x_0}(\epsilon)
  \,.
$$

=--

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

Consider the function

$$
  \phi \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}
$$

given by

$$
  \phi(x)
    \;\coloneqq\;
  \left\{
    \array{
      \exp\left(
       \frac{1}{{\Vert x \Vert}^2 - 1}
      \right)
       & \vert & { \Vert x \Vert} \lt 1
      \\
      0 &\vert & \text{otherwise}
     }
  \right.
  \,.
$$

By construction the [[support]] of this function is the closed unit ball at the origin, $Supp(\phi) = B_0(1)$.

We claim that $\phi$ is smooth. That it is smooth away from $r \coloneqq {\Vert x \Vert} = 1$ is clear, hence smoothness only need to be checked at $r = 1$, where it amounts to demanding that all the [[derivatives]] of the exponential function vanish as $r \to 1$.

But that is the case since

$$
  \frac{d}{d r}
  \left(
    \exp\left(
     \frac
       {1}
       {
         r^2 - 1
       }
    \right)
  \right)
  =
  \frac{
    -2 r
  }
  {
    \left( r^2 - 1 \right)^2
  }
    \exp\left(
     \frac
       {1}
       {
         r^2 - 1
       }
    \right)
  \,.
$$

This clearly tends to zero as $r \to 1$. A quick way to see this is to consider the inverse function and expand the [[exponential]] to see that this tends to $\infty$ as $r \to 1$:

$$
  \frac{
    \left( 1- r^2  \right)^2
  }
  {
    2 r
  }
  \exp\left(
     \frac
       {1}
       {
         1- r^2
       }
    \right)
  =
  \sum_{n = 0}^\infty
    \frac{1}{n!}
  \frac{
    \left( 1- r^2  \right)^2
  }
  {
    2 r
  }
   \frac{1}
     {
       (1- r^2)^n
     }
$$

The form of the higher derivatives is similar but with higher inverse powers of $(r^2 -1)$ and so this conclusion remains the same for all derivatives. Hence $\phi$ is smooth.

Now for arbitrary radii $\varepsilon \gt 0$ define

$$
  \phi_\varepsilon(x) \coloneqq \phi(x/\varepsilon)
  \,.
$$

This is clearly still smooth and $Supp(\phi_{\varepsilon}) = B_0(\epsilon)$.

Finally the function $x \mapsto \phi_\varepsilon(x-x_0)$ has support the closed ball $B_{x_0}(\varepsilon)$.

=--

Define

$$
  \psi_\varepsilon
    \coloneqq
  \frac{\phi_\varepsilon}{{\|\phi_\varepsilon\|}_1}
$$

so that the family $(\psi_\varepsilon)_\varepsilon$ is a smooth [[approximation to the identity]] (of convolution, the [[Dirac distribution|Dirac functional]]), compactly supported on the closed ball of radius $\varepsilon$ and having an $L^1$-norm equal to $1$. Then, it is standard that for any pair $K \subset V$ with $K$ [[compact subspace|compact]] and $V$ [[open subspace|open]] in a cartesian space $\mathbb{R}^n$, we can choose an open $U$ containing $K$ and with compact [[closure]] contained in $V$, and then taking the [[convolution]] product
$$\psi_\varepsilon \ast \chi_U$$
of $\psi_\varepsilon$ with the [[characteristic function]] $\chi_U$, for $\varepsilon$ sufficiently small, we obtain a smooth function equal to $1$ on $K$ and equal to $0$ outside $V$.

By performing the above construction in [[charts]], we obtain, in any [[smooth manifold]], a smooth function equal to $1$ on any [[compact subspace]] $K$ and equal to $0$ outside any [[neighbourhood]] $V$ of $K$.  (This is a smooth [[completely regular space|regularity property]].)

## Properties

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

[Smooth manifolds admit smooth partitions of unity](partition+of+unity#ExistenceOnSmoothManifolds)

=--

+-- {: .num_prop}
###### Proposition
**([[Paley-Wiener-Schwartz theorem]])**

For $n \in \mathbb{N}$ the vector space $C^\infty_c(\mathbb{R}^n)$ of bump functions on [[Euclidean space]] $\mathbb{R}^n$ is (algebraically and topologically) [[isomorphism|isomorphic]], via the [[Fourier transform]], to the space of [[entire functions]] $F$ on $\mathbb{C}^n$ which satisfy the following estimate: there is a [[positive number|positive]] [[real number]] $B$ such that for every [[integer]] $N \gt 0$ there is a real number $C_N$ such that:

$$
  \underset{\xi \in \mathbb{C}^n}{\forall}
  \left(
    {\Vert  F(\xi) \Vert}
    \le C_N (1 + {\vert \xi\vert })^{-N} \exp{ (B \; |Im(\xi)|)}
  \right)
  \,.
$$

=--

## References

Textbook account:

* {#Lee12} [[John Lee]], Prop. 2.25 in: *Introduction to Smooth Manifolds*, Springer (2012) [[doi:10.1007/978-1-4419-9982-5](https://doi.org/10.1007/978-1-4419-9982-5), [book webpage](https://sites.math.washington.edu/~lee/Books/ISM/), [pdf](https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth_manifolds.pdf)]

See also:

* Wikipedia, _[Bump function](http://en.wikipedia.org/wiki/Bump_function)_

A different example is examined in detail in

* Juan Arias de Reyna, _An infinitely differentiable function with compact support: Definition and properties_, arXiv:[1702.05442](https://arxiv.org/abs/1702.05442) (English translation of: _Definición y estudio de una función indefinidamente diferenciable de soporte compacto_, Rev. Real Acad. Ciencias 76 (1982) 21-38 available from [Universidad de Sevilla repository](http://hdl.handle.net/11441/28557))

One can also build bump functions that are _nowhere_ analytic (as opposed to merely on the boundary of the support, as above) using, for instance, the Fabius function:

* Wikipedia, [_Fabius function_](https://en.wikipedia.org/wiki/Fabius_function)

[[!redirects bump function]]
[[!redirects bump functions]]

[[!redirects smooth bump function]]
[[!redirects smooth bump functions]]


bunched logic

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

# Bunched logic
* table of contents
{:toc}

## Idea
 {#Idea}

In [[formal logic]], and more specifically in [[sequent calculus]]/[[natural deduction]]/[[type theory]], a **bunched logic** is a logical [[theory]] in which the [[formulas]] appearing in any [[context]] are not necessarily just a [[list]] but have some additional structure, usually that of [[tree]] whose nodes are then the eponymous "bunches":

Where a classical [[context]] is traditionally presented as a comma-separated [[list]] of [[propositions]] [[propositions as types|and/or]] [[types]], e.g.

$$
  \Gamma
  ,\;
  X
  ,\;
  Y
  \;\;\;
  ctxt
  \,,
$$

which classically may always be thought of as a ([[dependent pair type|dependent]]) classical [[product type]]

$$
  \Gamma
  \times
  X
  \times
  Y
  \;\;\;
  ctxt
  \,,
$$

so a context in bunched logic is allowed, more generally, to be a nested list of *[[structural rule|structural]] [[logical connective|connectives]]*, iterating between this comma-denoted conjunction and another connective typically denoted by a semicolon, e.g.:

$$
  A,\;
  B,\;
  \big(
    C;\;
    (D,\,E);
    \;
    F
  \big),\;
  (G;\,H)
  \;\;\;\;
  ctxt
  \,.
$$

The original and common motivation for such "bunched contexts" is the existence of further ([[function types]] and) [[product types]] *in addition* to the classical ones; such as in [[linear type theories]] with their additional ([[linear implication]] $(-)\multimap (-)$ and) [[linear conjunction]] $(-)\otimes (-)$, and more generally in the [[internal logic]] of a [[doubly closed monoidal category]], whence bunched contexts as above may usefully be thought of as types formed consecutively from [[cartesian products]] and another [[tensor product]]:

$$
  A
  \times
  B
  \times
  \big(
    C
    \otimes
    (D \times E)
    \otimes
    F
  \big)
  \times
  (G \otimes H)
  \;\;\;\;
  ctxt
  \,.
$$

> (NB: In saying this we are reversing the role of commas and semicolons as compared to the original literature on bunched logic, so that the comma retains it original meaning as the classical connective.)

Accordingly, the bunched context indicates a variation of [[structural rules]]: The classical product $\times$ will admit the [[contraction rule]] and/or the [[weakening rule]], while $\otimes$ may not.

Notice that the general phrase *bunched logic* is not entirely standard, and the word "bunches" has been used with more than one logic of this form.

The notion (together with its [[categorical semantics]] in [[doubly closed monoidal categories]]) originates with the logic of "bunched implication" (BI) and its "$\alpha \lambda$-calculus" (a bunched variation of the [[lambda-calculus]]) due to [[O'Hearn & Pym (1999)](#O'HearnPym99), [Pym (2002)](#Pym2002), see also [O'Hearn (2003)](#O'Hearn03).

{#Unsatisfactory} But aspects of the proposal of [O'Hearn & Pym (1999)](#O'HearnPym99) and [Pym (2002)](#Pym2002) have remained unsatisfactory, see [Pym (2008), Errata to Pym (2002)](#Pym08Errata) and [Riley(2022)](#Riley22Thesis), [pp. 19](https://mvr.hosting.nyu.edu/pubs/thesis.pdf#page=29) and Rem. 1.4.1 ([p. 64](https://mvr.hosting.nyu.edu/pubs/thesis.pdf#page=64)):

> [[Riley 2022, pp. 19](https://mvr.hosting.nyu.edu/pubs/thesis.pdf#page=29)]: [Since] context manipulations are not recorded in the terms, typechecking a term can be difficult. First, it is not always obvious when and how the contraction rule has to be applied, and secondly, it is not always obvious how the context has to be split into two pieces to apply $\multimap$-ELIM. Besides the challenge of typechecking, the combination of the weakening rule and the rules for the monoidal unit also breaks soundness for the intended semantics.

> The various extensions of the αλ-calculus that have since appeared [BO06; CPR08; SS04; Atk04] do not resolve these issues. It is also not so clear how to add [[dependent types]] to this system and how dependency should behave between bunches.

Further developments had not resolved the fundamental issue:

> {#UnsatisfactorySchöppStark} [[Riley 2022, p. 91](https://mvr.hosting.nyu.edu/pubs/thesis.pdf#page=101):] Schöpp and Stark’s Bunched Dependent Type Theory [[Schöpp & Stark (2004)](#SchöppStark04)] [admits] no dependency across bunches. The corresponding monoidal pair type therefore requires the input types to be closed. [...] the monoidal unit is the terminal object. This is certainly not the case in [linear] models of interest.

In reaction to this, [Riley (2022)](#Riley22Thesis) proposes an enhanced form of bunched type theory which is claimed to fix all these problems at least in the case that the second structural connective $\otimes$ is a [[multiplicative conjunction]] in a [[linear type theory]], hence for a form of [[dependent linear type theory]]. The key new ingredients in [Riley (2022)](#Riley22Thesis) to make this work are:

1. a [[classical modality]] $\natural$ which from any "parameterized linear type" projects out the underlying classical type;

2. a decoration of bunched contexts by "color palettes" which keep explicit track of the bunching/tree structure.

  (On the other hand, this second point leads to a proliferation of [[inference rules]], but maybe that's what it takes.)

## Examples

### Bunched implications

In **bunched implications** logic ([O'Hearn & Pym (1999)](#O'HearnPym99)), the semicolon allows contraction and weakening while the comma does not.  This allows defining both an [[additive conjunction]] and a [[multiplicative conjunction]], internalizing the semicolon and the comma respectively, such that both [[distributive law|distribute]] over the "" [[additive disjunction]] and moreover both come with a corresponding implication: the "additive" [[implication]] and the [[multiplicative implication]].

### Relevance logic

Some forms of [[relevance logic]] can be presented with a bunched sequent calculus that is similar to BI, but in which the comma also has contraction (though not weakening) and there is no additive implication.  See for instance ([Mints](#Mints)).

### Classical bunched implication

A logic of **classical bunched implication** is like BI but with arbitrary bunches on the right as well as on the left.  On the right, the semicolon represents the [[additive disjunction]] and the comma represents a [[multiplicative disjunction]], and there are both an additive and a multiplicative [[negation]] that move formulas back and forth.  Both negations are "classical" with respect to their corresponding connectives, e.g. we have $\sim\sim A \multimap A$ (where $\sim$ is the multiplicative negation and $\multimap$ the multiplicative implication) and also $\neg\neg A \to A$ (where $\neg$ is the additive negation and $\to$ the additive implication).  See [Pym 2002](#Pym2002) and [this discussion](https://nforum.ncatlab.org/discussion/4004/paraconsistent-logic/?Focus=57557#Comment_57557).

### Dependent versions

Bunched logics are also used to combine linear type theories with [[dependent type theories]].  See some of the references at [[dependent linear type theory]].

## Categorical semantics

Bunched logics naturally have semantics in categories with more than one [[monoidal structure]], so that a bunch such as $(A,(B;C),((D,E);F))$ can be interpreted as $A \otimes (B\boxtimes C) \otimes ((D\otimes E)\boxtimes F)$.  Frequently (e.g. if one kind of bunch admits contraction and weakening) one of the two monoidal structures is a [[cartesian monoidal category|cartesian]] one.  A typical and motivating example of a model for BI is the [[category of presheaves]] $[C^{op},Set]$ over a monoidal category $C$, which comes equipped both with the ordinary [[ccc structure on presheaves]] as well as the closed monoidal structure given by [[Day convolution]].

## Related pages

* [[linear logic]]

* [[doubly closed monoidal category]]

* [[relevance logic]]

* [[separation logic]]

* [[display logic]]

* [[substructural logic]]

* [[monoidal topos]]

## References

Precursors:

* {#Mints} G. E. Mints. *Cut-elimination theorem for relevant logics*, Zap. Nauchn. Sem. LOMI, 1972,	Volume 32, Pages 90–97. ([math-net.ru](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=znsl&paperid=2569&option_lang=eng)).
An English translation appears in the Journal of Soviet Mathematics 6 (1976) 422-8 [[doi:10.1007/BF01084083](http://doi.org/10.1007/BF01084083)]

The original articles:

* {#O'HearnPym99} [[Peter O'Hearn]], [[David J. Pym]], *The Logic of Bunched Implications*, The Bulletin of Symbolic Logic **5** 2 (1999) 215-244  [[pdf](http://www.lsv.ens-cachan.fr/~demri/OHearnPym99.pdf), [doi:10.2307/421090](https://doi.org/10.2307/421090)]

* {#Pym2002} [[David Pym]], *The Semantics and Proof Theory of the Logic of Bunched Implications*, Applied Logic Series **26**, Springer (2002) [[doi:10.1007/978-94-017-0091-7](https://doi.org/10.1007/978-94-017-0091-7), [GoogleBooks](https://books.google.com/books/about/The_Semantics_and_Proof_Theory_of_the_Lo.html?id=0bAfqhzDuOcC&redir_esc=y)]

  {#Pym08Errata} Errata and Remarks for *The Semantics and Proof Theory of the Logic of Bunched Implications* (2008) [[pdf](https://www.cantab.net/users/david.pym/BI-monograph-errata.pdf), [[Pym-BI-Errata.pdf:file]]]

  > [[p. 5](https://ncatlab.org/nlab/files/Pym-BI-Errata.pdf#page=5)] "Items 3, 4, and 5 in the errata for Chapter 12 and 4, 5 in Chapter 13, whilst necessary to fix a technical problem, leave us in a conceptually and computationally undesirable situation: we have no clean characterization of a useful notion of well-formedness, leading to obvious computational difficulties in, say, theorem proving. We conclude that such a highly multiplicative system is far too complex, at least in its current formulation, to be of much value. All of the applications of multiplicative quantifiers known to-date require much simpler systems. It is the author's recommendation that readers should not devote very much of their resources to studying the system of predicate BI discussed in these chapters"

* {#O'Hearn03} [[Peter O'Hearn]], *On bunched typing*, Journal of Functional Programming **13** 4 (2003) 747-796 [[doi:10.1017/S0956796802004495](https://doi.org/10.1017/S0956796802004495)]

Review and further development:

* Bodil Biering, *On the logic of bunched implications -- and its relation to separation logic*, MSc thesis, Copenhagen (2004) [[[Biering-BunchedLogic.pdf:file]]]

* Brotherston and Calcagno, *Classical BI: Its Semantics and Proof Theory*, [arxiv](http://arxiv.org/abs/1005.2340)

Discussion of bunched logic with the [[linear logic]]-part regarded as a [[quantum logic]]:

* [[Li Zhou]], [[Gilles Barthe]], [[Justin Hsu]], [[Mingsheng Ying]], [[Nengkun Yu]], *A Quantum Interpretation of Bunched Logic for Quantum Separation Logic*, LICS '21: Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science  75 (2021) 1–14 [[arXiv:2102.00329](https://arxiv.org/abs/2102.00329), [doi:10.1109/LICS52264.2021.9470673](https://doi.org/10.1109/LICS52264.2021.9470673)]

A bunched [[dependent type theory]], to some extent:

* {#SchöppStark04} [[Ulrich Schöpp]], [[Ian Stark]], *A Dependent Type Theory with Names and Binding*, in *Computer Science Logic. CSL 2004*, Lecture Notes in Computer Science **3210** (2004) 235–249 [[doi:10.1007/978-3-540-30124-0_20](https://doi.org/10.1007/978-3-540-30124-0_20), [web](https://homepages.inf.ed.ac.uk/stark/names+binding.html)]

Further refinement to *dependent bunches*  for a [[dependent linear types|dependent linear]] [[homotopy type theory|homotopy types]]:

* {#Riley22Thesis} [[Mitchell Riley]], *A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory*, PhD Thesis (2022) [[doi:10.14418/wes01.3.139](https://doi.org/10.14418/wes01.3.139), [ir:3269](https://digitalcollections.wesleyan.edu/object/ir%3A3269), [pdf](https://mvr.hosting.nyu.edu/pubs/thesis.pdf)]

* {#Riley22} [[Mitchell Riley]], *Linear Homotopy Type Theory*, talk at: [HoTTEST Event for Junior Researchers 2022](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest_junior_2022.html)  (Jan 2022) [slides: [pdf](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Riley-2022-01-20-HoTTEST.pdf), video: [YT](https://www.youtube.com/watch?v=o2oWhHabjdM)]

* [[Mitchell Riley]], *Dependent Type Theories à la Carte*, talk at *[[CQTS]] Initial Researcher's Meeting* (Sep 2022) [[[CQTS-InitialResearcherMeeting-Riley-220913.pdf:file]]]

[[!redirects bunched logics]]
[[!redirects bunch]]
[[!redirects bunches]]
[[!redirects bunched implication]]
[[!redirects bunched implications]]
[[!redirects BI]]
[[!redirects structural connective]]
[[!redirects structural connectives]]

[[!redirects bunched type theory]]
[[!redirects bunched type theories]]


bundle

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

#Contents#
* table of contents
{:toc}

## Definition

In the most general sense, a **bundle** over an [[object]] $B$ in a [[category]] $\mathcal{C}$ is simply an object $E$ of $\mathcal{C}$ equipped with a [[morphism]] in $\mathcal{C}$ from $E$ to $B$:
$$ \array { E \\ \downarrow^{\mathrlap{p}} \\ B } $$

One often refers to such a bundle simply as $E$, even though $B$ is really part of the data.

For $x \in B$ a [[generalized element]] of $B$, the [[fiber]] $E_x$ of the bundle over $x$ is the [[pullback]] $x^* E$.

Given two bundles $E_1$ and $E_2$ over $B$, then a _morphism of bundles over $B$_ is a [[morphism]] $E_1 \to E_2$ which makes this [[commuting diagram|diagram commute]]:

$$
  \array{
    E_1 && \longrightarrow && E_2
    \\
    & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}}
    \\
    && B
  }
  \,.
$$

This way bundles over $B$ form a [[category]], also called the _[[slice category]]_ $\mathcal{C}_{/B}$ of $\mathcal{C}$ over $B$.

## Bundles with extra structure and properties

One generally considers bundles with extra properties or structure:

* [[fiber bundle]],

* [[microbundle]]

* [[principal bundle]]

* [[vector bundle]],

* [[connection on a bundle|bundle with connection]]

  * [[Simons-Sullivan structured bundle]]

* etc.

## The collection of all bundles

The category of bundles over a given object $B$ is the [[over category]] $\mathcal{C}/B$. The collection of all bundles in a given [[category]] $\mathcal{C}$ therefore arranges itself into the [[codomain fibration]] $cod : [I,\mathcal{C}] \to \mathcal{C}$. As such, the [[descent]] for bundles may be expressed as [[monadic descent]] with respect to the codomain [[bifibration]]. This does in general not work inside one of the more restrictive subcategories of bundles with extra structure and property, as the push-forward operation typically does not respect these extra conditions. For more on this see [monadic descent of bundles](http://ncatlab.org/nlab/show/monadic+descent#ForCodomainFibs).

## Related concepts

* Bundles tend to have [[sheaf|sheaves]] of local [[section]]s.

* In [[physics]], [[gauge field]]s may be described in terms of [[connection on a bundle|bundles with connection]].

## Related concepts

* **bundle**, [[display map]]

* [[sub-bundle]]

* [[retractive space]]

* **[[fiber bundle]]** / [[fiber ∞-bundle]]

  * [[associated bundle]]

  * [[vector bundle]]

    * [[line bundle]]

    * [[stable vector bundle]]

  * [[sphere bundle]]

  * [[natural bundle]]

  * [[equivariant bundle]]

  * [[bundle of spectra]]

  * [[fiber bundles in physics]]

* [[principal bundle]] / [[torsor]] / [[associated bundle]]

* [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]

* [[principal 3-bundle]] / [[bundle 2-gerbe]]

* [[principal ∞-bundle]] / [[associated ∞-bundle]]

* [[(∞,1)-vector bundle]] / [[(∞,n)-vector bundle]]

  * [[spherical fibration]]

## References

Introductory notes on [[fiber bundles]], [[vector bundles]] and [[connections on bundles]]:

* Steven Bradlow, [Vector bundles and introduction to gauge theory](http://cwillett.imathas.com/bundles/)

See also many references at *[[fiber bundles in physics]]*.

[[!redirects bundles]]

[[!redirects bundle morphism]]
[[!redirects bundle morphisms]]

[[!redirects bundle homomorphism]]
[[!redirects bundle homomorphisms]]


bundle 2-gerbe

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

#Contents#
* table of contents
{:toc}

## Idea

A _bundle 2-gerbe_ is a special presentation of the total space of a $\mathbf{B}^2 U(1)$-[[principal 3-bundle]], where $\mathbf{B}^2 U(1)$ is the circle Lie 3-group.

A [[connection on a bundle 2-gerbe]] is a special cocycle representative for [[circle n-bundles with connection]], hence for degree 4 [[Deligne cohomology]], hence for degree 4 [[Cheeger-Simons differential character]]s.

The definition is built by iteration on the definition of [[bundle gerbe]]:

a bundle 2-gerbe over a manifold $X$ is

* a surjective [[submersion]] $Y \to X$;

* on the [[fiber product]] $Y \times_X Y$ a [[bundle gerbe]] $\mathcal{L} \to Y\times_X Y$;

* a morphims of bundle gerbes $\pi_1^* \mathcal{L} \otimes\pi_2^* \mathcal{L} \to \pi_1^* \mathcal{L}$;

* which is associative up to a choice of coherent [[2-morphism]]s.

## Examples

* [[Chern-Simons 2-gerbe]]

## Related concepts

* [[principal bundle]] / [[torsor]] / [[associated bundle]]

* [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]

* [[principal 3-bundle]] / [[2-gerbe]] / **bundle 2-gerbe**

* [[principal ∞-bundle]] / [[associated ∞-bundle]] / [[∞-gerbe]]

## References

Bundle 2-gerbes were briefly introduced in

* [[Alan Carey]], [[Michael Murray]], B.L. Wang, _Higher bundle gerbes and cohomology classes in gauge theories_, Journal of Geometry and Physics
Volume 21, Issue 2, January 1997, Pages 183-197 ([arXiv:hep-th/9511169](http://arxiv.org/abs/hep-th/9511169), doi:10.1016/S0393-0440(96)00014-9)

and further developed in

* [[Danny Stevenson]], _Bundle 2-gerbes_, Proceedings of the London Mathematical Society, Volume 88, Issue 2, March 2004, Pages 405–435 ([arXiv:math/0106018](http://arxiv.org/abs/math/0106018), [doi:10.1112/S0024611503014357]( https://doi.org/10.1112/S0024611503014357)),

drawing on ideas from Stevenson's PhD thesis ([arXiv:math/0004117](http://arxiv.org/abs/math/0004117)).

A general picture of bundle $n$-gerbes (with connection) as [[circle n-bundle with connection|circle (n+1)-bundles with connection]] classified by [[Deligne cohomology]] is in

* [[Pawel Gajer]], _Geometry of Deligne cohomology_, Invent. Math., 127(1):155-207 (1997) ([arXiv:alg-geom/9601025](http://arxiv.org/abs/alg-geom/9601025), [doi:10.1007/s002220050118](https://doi.org/10.1007/s002220050118))

A model for the [[supergravity C-field]] in terms of [[nonabelian bundle 2-gerbes]]:

* [[Paolo Aschieri]], [[Branislav Jurčo]], *Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory*, JHEP 0410:068 (2004) [[arXiv:hep-th/0409200](https://arxiv.org/abs/hep-th/0409200), [doi:10.1088/1126-6708/2004/10/068](https://doi.org/10.1088/1126-6708/2004/10/068)]

[[!redirects bundle 2-gerbes]]