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Beren Sanders | https://ncatlab.org/nlab/source/Beren+Sanders | * [homepage](http://www.math.ucla.edu/~beren/)
category:people |
Berezin integral | https://ncatlab.org/nlab/source/Berezin+integral |
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#### Super-Geometry
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#### Integration theory
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[[!include integration theory - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
A _Berezinian integral_ is an [[integration over supermanifolds|integral over a supermanifold]] that is a [[superpoint]] $\mathbb{R}^{0|q}$.
For the infinite-dimensional version see [[fermionic path integral]].
## Related entries
* [[integration]], [[integral]]
[[integration over supermanifolds]]
* [[superpoint]], [[Grassmann algebra]]
* [[determinant]]
* [[superdeterminant]]/[[Berezinian]]
* [[Ausdehnungslehre]]
## References
The concept is originally due to [[Felix Berezin]].
Exposition of the standard lore includes
* [[Andreas Wipf]], _Chapter 10: Berezin integral_ ([pdf](https://www.tpi.uni-jena.de/qfphysics/homepage/wipf/lectures/pfad/pfad10.pdf))
* [[Urs Schreiber]], [_Integration over supermanifolds_](http://www.math.uni-hamburg.de/home/schreiber/sin.pdf)
Discussion in relation to the [[Hodge star operator]]:
* [[Leonardo Castellani]], [[Roberto Catenacci]], [[Pietro Antonio Grassi]], _The Hodge Operator Revisited_ ([arXiv:1511.05105](https://arxiv.org/abs/1511.05105))
A [[general abstract]] discussion in terms of [[D-module]] theory is in
* [[Frédéric Paugam]], _Homotopical Poisson Reduction of gauge theories_ ([pdf](http://people.math.jussieu.fr/~fpaugam/documents/homotopical-poisson-reduction-of-gauge-theories.pdf))
[[!redirects Berezin integrals]]
[[!redirects Berezinian integral]]
[[!redirects Berezinian integrals]]
[[!redirects Berezin integration]]
|
Berger's theorem | https://ncatlab.org/nlab/source/Berger%27s+theorem |
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#### Riemannian geometry
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[[!include Riemannian geometry - contents]]
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#### Symplectic geometry
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[[!include symplectic geometry - contents]]
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#Contents#
* table of contents
{:toc}
## Statement
Berger's theorem says that if a [[manifold]] $X$ is
* [[simply connected topological space|simply connected]]
* neither locally a product nor a [[symmetric space]]
then the possible special holonomy groups are the following
[[!include special holonomy table]]
## References
Original article:
* {#Berger} [[Marcel Berger]], _Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes_, Bull. Soc. Math. France 83 (1955) ([doi:10.24033/bsmf.1464](https://doi.org/10.24033/bsmf.1464))
* Carlos Olmos, _A Geometric Proof of the Berger Holonomy Theorem_, Annals of Mathematics Second Series, Vol. 161, No. 1 (Jan., 2005), pp. 579-588 (10 pages) ([jstor:3597350](https://www.jstor.org/stable/3597350))
See also
* Wikipedia, _[Holonomy -- The Berger classification](https://en.wikipedia.org/wiki/Holonomy#The_Berger_classification)_
[[!redirects Berger theorem]]
|
Bergfinnur Durhuus | https://ncatlab.org/nlab/source/Bergfinnur+Durhuus |
* [personal page](https://web.math.ku.dk/~durhuus/)
* [institute page](https://www.math.ku.dk/english/staff/?pure=en/persons/23254)
* [inspire page](https://inspirehep.net/authors/1011165)
## Selected writings
On [[linear operators]] on [[Hilbert spaces]]:
* [[Bergfinnur Durhuus]], *Operators on Hilbert Space* [[pdf](https://web.math.ku.dk/~durhuus/MatFys/MatFys4.pdf), [[Durhuus-OperatorsOnHilbertSpace.pdf:file]]]
category: people
|
Berkovich space | https://ncatlab.org/nlab/source/Berkovich+space |
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#### Analytic geometry
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[[!include analytic geometry -- contents]]
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#Contents#
* table of contents
{:toc}
## Idea
__Berkovich analytic spaces__ are a version of [[analytic space]]s over [[nonarchimedean fields]]. Unlike the _rigid analytic spaces_ (see [[rigid analytic geometry]]) of Tate, which are locally defined via [[maximal spectra]] of Tate algebras glued via the [[Grothendieck topology|Grothendieck]] [[G-topology]], the Berkovich analytic spaces are actual [[topological space]] equipped with a cover by [[affinoid domains]] via the [[analytic spectrum]] construction, due to [[Vladimir Berkovich]]. This spectrum can be viewed as consisting of the data of prime ideal plus the extension of the norm to the residue field; thus the Berkovich spectrum has far more points (though fewer than, say, [[Huber's adic spaces]] which may also contain valuations of higher order).
For more background see _[[analytic geometry]]_.
## Definition of Berkovich analytic spaces
Let $k$ be a [[non-archimedean field]].
+-- {: .num_defn}
###### Definition
Given $n \in \mathbb{N}$ and positive elements $\{r_1, \cdots, r_n \in k\}$, consider the sub-[[power series]] [[associative algebra|algebra]] over $k$ of those series which [[convergence|converge]] inside the radii $k_i$, i.e. the algebra defined by
$$
\{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\}
:=
\left\{
\sum_\nu a_\nu T^\nu | \lim_{{\vert \nu\vert} \to \infty} {\vert a_\nu \vert} r^\nu = 0
\right\}
\,.
$$
This is a commutative [[Banach algebra]] over $k$ with [[norm]] ${\Vert f \Vert} = max {\vert a_\nu\vert} r^\nu$.
A **$k$-[[affinoid algebra]]** is a commutative Banach $k$-algebra $A$ for which there exists $n$ and $\{r_i\}$ as above and an [[epimorphism]]
$$
\{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\}
\to
A
$$
such that the [[norm]] on $A$ is the [[quotient norm]].
If one can choose here $r_i = 1$ for all $i$ then $A$ is called **strictly $k$-affinoid**.
The [[category]] of **$k$-[[affinoid spaces]]** is the [[opposite category]] of the category of $k$-[[affinoid algebras]] and bounded [[homomorphisms]] between them.
=--
Via the [[analytic spectrum]] $Spec_{an}$ there is a [[topological space]] associated with any $k$-affinoid space. Often this underlying topological space is referred to as _the analytic space_.
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###### Definition
An **[[affinoid domain]]** in an [[affinoid space]] $X = Spec_{an} A$ is a [[closed subset]] $V \subset X$ such that there is a [[homomorphism]] of $k$-affinoid spaces
$$
\phi : Spec_{an} A_V \to X
$$
for some $A_V$, whose [[image]] is $V$, and such that every other morphism of $k$-affinoid spaces into $X$ whose image is contained in $V$ uniquely factors through this morphism.
=--
+-- {: .num_defn}
###### Definition
A **$k$-analytic space** is a [[locally Hausdorff topological space|locally Hausdorff]] [[topological space]] $X$ equipped with an [[atlas]] by $k$-[[affinoid domains]] and [[affinoid domain embeddings]], such that their underlying [[analytic spectra]] [[topological spaces]] form a [[quasinet|net]] of [[compact subsets]] on $X$.
=--
([Berkovich 09, def. 3.1.4](#Berkovich09))
## Properties
### Cohomology
{#Cohomology}
Under some mild conditions, the algebraic and the analytic [[étale cohomology]] of Berkovich spaces agree. ([Berkovich 95](#Berkovich95))
The underlying [[topological space]] $X^{an}$ given by the [[Berkovich analytic spectrum]] has as [[singular cohomology]] the [[weight filtration|weight 0]]-cohomology of $X$ ([Berkovich 09](#Berkovich09)).
See also MO discussion [here](http://mathoverflow.net/a/171653/381).
### Local contractibility
{#LocalContractibility}
A [[complex analytic manifold]] and a _smooth_ [[complex analytic space]] is locally isomorphic to a [[polydisk]] and hence is trivially a [[locally contractible space]]. But over a [[non-archimedean field]] analytic spaces no longer need to be locally isomorphic to polydisks (but $p$-adic polydisks are still contractible ([Berkovich 90](#Berkovich90))). The following result establishes, under mild conditions, that general analytic spaces are nevertheless locally contractible.
Assume that the [[valuation]] on the ground field $k$ is nontrivial.
+-- {: .num_defn #LocallyEmbeddableInASmoothSpace}
###### Definition
A $k$-analytic space $X$ is called _locally embeddable in a smooth space_ if each point of $X$ has an [[open neighbourhood]] [[isomorphism|isomorphic]] to a strictly $k$-analytic domain in smooth $k$-analytic space.
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###### Theorem
Every $k$-analytic space which is locally embeddable in a smooth space, def. \ref{LocallyEmbeddableInASmoothSpace}, is a [[locally contractible space]].
More precisely, every point of a locally smooth $k$-analytic space has an open neighbourhood $U$ which is contractible, and which is a [[union]] $U = \cup_{i = 1}^\infty U_i$ of analytic domains.
=--
The local contractibility is [Berkovich (1999), theorem 9.1](#BerkovichContractible). The refined statment in terms of inductive systems of analytic domains is in [Berkovich (2004)](#BerkovichContractibleII).
## Examples
* [[analytic affine line]]
## Applications
* The proof of the [[local Langlands conjecture]]
for $GL_n$ by Harris–Taylor uses [[étale cohomology]] on non-archimedean analytic spaces (in the sense of Berkovich) to construct the required Galois representations over local fields.
## Related concepts
* [[non-commutative analytic space]]
* [[p-adic geometry]]
* [[G-topology]]
* [[Huber space]], [[perfectoid space]]
* [[global analytic geometry]]
## References
### Introductions and reviews
A nice survey is in
* {#LeStum12} [[Bernard Le Stum]], _One century of $p$-adic geometry -- From Hensel to Berkovich and beyond_, talk notes, June 2012 ([pdf](http://www-irma.u-strasbg.fr/IMG/pdf/NotesCoursLeStum.pdf))
A good introduction to the general idea is at the beginning of
* {#Payne13} [[Sam Payne]], _Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry_ ([arXiv:1309.4403](http://arxiv.org/abs/1309.4403))
Basic notions are listed in
* M. Temkin, _Non-archimedean analytic spaces_ ([pdf slides](http://www.math.huji.ac.il/~temkin/lectures/Non-Archimedean_Analytic_Spaces.pdf))
A review of basic definitions and facts about affinoid and rigid $k$-analytic spaces can be found in
* Gaëtan Chenevier, _lecture 5_ ([pdf](http://www.math.polytechnique.fr/~chenevier/coursIHP/chenevier_lecture5.pdf))
See also the references at [[rigid analytic geometry]].
A review of definitions and results on $k$-analytic spaces is in
* {#Berkovich98} [[Vladimir Berkovich]], _$p$-Adic analytic spaces_, in Proceedings of the International Congress of Mathematicians, Berlin, August 1998, Doc. Math. J. DMV, Extra Volume ICM II (1998), 141-151 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/ICM98_1998.pdf))
A more detailed set of lecture notes along these lines is
* {#Berkovich09} [[Vladimir Berkovich]], _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))
Introductory exposition of the Berkovich [[analytic spectrum]] is
* {#Poineau07} [[Jérôme Poineau]], _Global analytic geometry_, pages 20-23 in EMS newsletter September 2007 ([pdf](http://www.ems-ph.org/journals/newsletter/pdf/2007-09-65.pdf))
* [[Frédéric Paugam]], section 2.1.4 of_Global analytic geometry and the functional equation_ (2010) ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-global-analytic-geometry.pdf))
A exposition of examples of Berkovich spectra is in
* [[Scott Carnahan]], _Berkovich spaces I_ ([web](http://sbseminar.wordpress.com/2007/09/18/berkovich-spaces-i/))
### Original articles
* [[Vladimir Berkovich]], _Spectral theory and analytic geometry over non-Archimedean fields_, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, (1990) 169 pp.
{#Berkovich90}
* [[Vladimir Berkovich]], _Étale cohomology for non-Archimedean analytic spaces_, Publ. Math. IHES 78 (1993), 5-161.
Discussion of Berkovich09cohomology of Berkovich analytic spaces includes
* {#Berkovich95} [[Vladimir Berkovich]], _On the comparison theorem for étale cohomology of non-Archimedean analytic spaces._ Israel Journal of Mathematics 92.1-3 (1995): 45-59.
* {#Berkovich09} [[Vladimir Berkovich]], _A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures_, Algebra, Arithmetic, and Geometry. Birkhäuser Boston, 2009. 49-67.
Discussion of local contractibility of smooth $k$-analytic spaces is in
* {#BerkovichContractible} [[Vladimir Berkovich]], _Smooth $p$-adic analytic spaces are locally contractible_, Invent. Math. 137 1-84 (1999) ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Inven_1999_137_contra.pdf))
* {#BerkovichContractibleII} [[Vladimir Berkovich]], _Smooth p-adic analytic spaces are locally contractible. II_, in Geometric Aspects of Dwork Theory, Walter de Gruyter & Co., Berlin, (2004), 293-370. ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Dworkvol_2004_contraII.pdf))
and more generally in
* {#HrushovskiLoeser10} [[Ehud Hrushovski]], [[François Loeser]], _Non-archimedean tame topology and stably dominated types_ ([arXiv:1009.0252](http://arxiv.org/abs/1009.0252))
* [[Ehud Hrushovski]], [[François Loeser]], [[Bjorn Poonen]], _Berkovich spaces embed in Euclidean spaces_ ([arXiv:1210.6485](http://arxiv.org/abs/1210.6485))
### Relation to other topics
On the relation to [[buildings]]:
* Annette Werner, _Buildings and Berkovich Spaces_ ([pdf](http://www.uni-frankfurt.de/fb/fb12/mathematik/ag/personen/werner/talks/dmvmuench10.pdf))
Relation to [[integration]] theory
* [[Vladimir Berkovich]], _Integration of 1-forms on $p$-adic analytic spaces_, Princeton University Press,
Aspects of the [[homotopy theory]]/[[étale homotopy]] of analytic spaces are discussed in
* [[Aise Johan de Jong]], _Étale fundamental groups of non-archimedean analytic spaces_, Mathematica, 97 no. 1-2 (1995), p. 89-118 ([numdam](http://www.numdam.org/item?id=CM_1995__97_1-2_89_0))
Relation to [[formal schemes]]:
* Jérôme Poineau, [MO comment](http://mathoverflow.net/a/138577/381)
Discussion of Berkovich analytic geometry as [[algebraic geometry]] in the general sense of [[Bertrand Toën]] and [[Gabriele Vezzosi]] is in
* [[Oren Ben-Bassat]], [[Kobi Kremnizer]], _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))
[[!redirects Berkovich analytic space]]
[[!redirects Berkovich analytic spaces]]
[[!redirects Berkovich spaces]]
[[!redirects p-adic analytic space]]
[[!redirects p-adic analytic spaces]] |
Berkovits superstring | https://ncatlab.org/nlab/source/Berkovits+superstring |
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#### String theory
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#### Super-Geometry
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[[!include supergeometry - contents]]
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=--
#Contents#
* table of contents
{:toc}
## Idea
The formulation of the [[sigma-model]] 2d [[CFT]]s that are used in (super) [[string theory]] had originally been of two types:
1. In the [[RNS-string]] formulation the worldsheet is a [[supermanifold]], while the target space is an ordinary [[manifold]].
The advantage of this formulation is that its [[quantization]] is comparatively tractable: it is a 2-dimensional superconformal field theory, coming with a super [[vertex operator algebra]].
The disadvantage is that, as one says, "target space supersymmetry is not manifest": as described at [[string theory]], every 2d CFT gives rise to an effective background [[quantum field theory]] (that quantum field theory whose [[S-matrix]] approximates the [[S-matrix]] obtained by evaluating the given CFT correlators on all possible surfaces and summing over the conformal [[moduli space]] and all genera). One can identify explcitly in the [[vertex operator algebra]] of the 2d CFT those operators that correspond to the fields of this effective background field theory (the _vertex operators_). One finds that these fields are those of a theory of [[supergravity]] and [[super Yang-Mills theory]]. So in particular there is a [[super Lie algebra]] acting on them. But this "supersymmetry" of the effective target space theory appears like an accident if one unwraps all this: there is no structural one-line argument (known) that would guarantee that the effective background quantum field theory of a 2d super [[CFT]] is itself supersymmetric.
1. In the [[Green-Schwarz superstring]] [[sigma-model]] formulation, the [[worldsheet]] is an ordinary [[manifold]], but target space is a [[supermanifold]].
This has the immediate advantage that target space supersymmetry is now "manifest": as described at [[supergravity]] it is just the super-[[diffeomorphism]] invariance of the theory, and the [[sigma-models]] in question all manifestly have this property.
The big disadvantage is, that it is not known how to [[quantization|quantize]] this system. The reason is that the standard procedure for quantization shows that the GS-type sigma models are [[constrained mechanics|constrained]] systems with second-class constraints. Little to nothing is known how to deal with that.
Berkovits was motivated by the desire to find a formulation of the superstring sigma model CFT that would combine the advantages of both the RNS formulation and the Green-Schwarz formulation. The goal was to have a CFT that looked like a free field theory (as the RNS string does for a flat Minkowski background, but as the GS string does not) if one just looked at a small part of the range of its fields, but which was globally constrained: the spinorial worldsheet fields here are maps into a space of [[pure spinors]] that locally lools like a Euclidean space, but globally has a cone geometry (...details...).
Berkovits originally wrote down some more or less ad-hoc expressions. Later it was understood that what described is a _[[sheaf]] of vertex operator algebras_ (in general, not a [[sheaf]] but a [[stack]], in fact a [[gerbe]]) on target space: to each contractible open patch of target space is associate the [[vertex operator algebra]] of a free [[sigma-model]] CFT whose fields take values _just_ in that patch, such that on overlaps these vertex operator algebras glue in some way.
This at once made the previously mathemtically rather unjustified approach make close contact with the developing theory of [[chiral de Rham complex]], which is one of the most-studied examples of sheaves of vertex operator algebras.
## Related concepts
* [[superstring]]
* [[string scattering amplitude]]
## References
### General
The original articles:
(...)
Relation to the [[RNS string]]:
* [[Nathan Berkovits]], *Manifest Spacetime Supersymmetry and the Superstring* ([arXiv:2106.04448](https://arxiv.org/abs/2106.04448))
Relation to the [[Green-Schwarz superstring]]:
* [[Max Guillen]], *Green-Schwarz and Pure Spinor Formulations of Chiral Strings* ([arXiv:2108.11724](https://arxiv.org/abs/2108.11724))
Review:
* [[Nathan Berkovits]], [[Carlos R. Mafra]], *Pure spinor formulation of the superstring and its applications*, in *[[Handbook of Quantum Gravity]]* (2023) [[arXiv:2210.10510](https://arxiv.org/abs/2210.10510)]
Discussion of [[string scattering amplitudes]]:
* [[Carlos R. Mafra]], [[Oliver Schlotterer]], *Tree-level amplitudes from the pure spinor superstring* [[arXiv:2210.14241](https://arxiv.org/abs/2210.14241)]
Relation to $D=5$ [[holomorphic Chern-Simons theory]]:
* [[Nathan Berkovits]], *$D=5$ Holomorphic Chern-Simons and the Pure Spinor Superstring* [[arXiv:2211.06731](https://arxiv.org/abs/2211.06731)]
Relation to [[semi-topological 4d Chern-Simons theory]]:
* [[Nathan Berkovits]], Rodrigo S. Pitombo, *4D Chern-Simons and the pure spinor $AdS_5 \times S^5$ superstring* [[arXiv:2401.03976](https://arxiv.org/abs/2401.03976)]
### Sheaf ov VOAs
A good reference that explains the sheaf of vertex operator algebra perspective on the Berkovits superstring is
* [[Nikita Nekrasov]], _Lectures on curved $\beta$-$\gamma$ systems, pure spinors, and anomalies_ ([arXiv](http://arxiv.org/abs/hep-th/0511008))
The standard reference on the closely related mathematical theory of the chiral deRham complex is
* [[Vassily Gorbounov]], Fyodor Malikov, [[Vadim Schechtman]], _Gerbes of chiral differential operators_ Math. Res. Lett. __7__ (2000), no. 1, 55--66, [MR2002c:17040](http://www.ams.org/mathscinet-getitem?mr=2002c:17040), [math.AG/9906117](http://arxiv.org/abs/math/9906117); _Gerbes of chiral differential operators. II. Vertex algebroids_, Invent. Math. __155__ (2004), no. 3, 605--680, [MR2005e:17047](http://www.ams.org/mathscinet-getitem?mr=2005e:17047), [math.AG/0003170](http://arxiv.org/abs/math/0003170), [doi](http://dx.doi.org/10.1007/s00222-003-0333-4); _Gerbes of chiral differential operators. III_, in: The orbit method in geometry and physics (Marseille, 2000), 73--100, Progr. Math. __213__, Birkhäuser 2003, [MR2005a:17028](http://www.ams.org/mathscinet-getitem?mr=2005a:17028), [math.AG/0005201](http://arxiv.org/abs/math/0005201), _On chiral differential operators over homogeneous spaces_, Int. J. Math. Math. Sci. __26__ (2001), no. 2, 83--106, [MR2002g:14020](http://www.ams.org/mathscinet-getitem?mr=2002g:14020), [math.AG/0008154](https://arxiv.org/abs/math/0008154), [doi](https://doi.org/10.1155/S0161171201020051)
Here is some blog discussion about this topic:
* [[Jacques Distler]]
* _[CDO and pure spinors](http://golem.ph.utexas.edu/~distler/blog/archives/000664.html)_
* _[Nekrasov on pure spinors](http://golem.ph.utexas.edu/~distler/blog/archives/000670.html)_
* [[Urs Schreiber]]
* _[Nekrasov on pure spinor superstring](http://golem.ph.utexas.edu/string/archives/000661.html)_
_[Sheaves of CDOs](http://golem.ph.utexas.edu/string/archives/000668.html)_
Nekrasov work referred to above is partly from
* Nathan Berkovits, [[Nikita Nekrasov]], _The character of pure spinors_, Letters in Mathematical Physics (2005) 74:75--109 [doi](https://doi.org/10.1007/s11005-005-0009-7)
[[!redirects Berkovits string]]
[[!redirects pure spinor superstring]]
|
Bernard Badzioch | https://ncatlab.org/nlab/source/Bernard+Badzioch |
* [website](http://www.math.buffalo.edu/Badzioch_Bernard.html)
category: people |
Bernard Bolzano | https://ncatlab.org/nlab/source/Bernard+Bolzano | Bernard Bolzano (1789-1848) was a Bohemian mathematician, logician, and philosopher.
In his lifetime he was renowned perhaps mainly for his *Wissenschaftslehre* which puts forth an encompassing theory of science. In mathematics, much of his work became well known only after his death, but he can be regarded as an early proponent of [[epsilontic analysis]] and the introduction of rigor in analysis, as for example in the result known as the [[Bolzano-Weierstrass theorem]] (q.v.).
## References
Collected works are available at
* [Bolzano collection](http://dml.cz/handle/10338.dmlcz/400001) at the Czech Digital Mathematics Library
For biographical and other detail, see
* [Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/bolzano/)
* [Bolzano's Logic](https://plato.stanford.edu/entries/bolzano-logic/) in Stanford EP
* [MacTutor biographical sketch](http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bolzano.html)
* [Wikipedia](https://en.wikipedia.org/wiki/Bernard_Bolzano) |
Bernard de Wit | https://ncatlab.org/nlab/source/Bernard+de+Wit |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Bernard_de_Wit)
## Selected writings
On the regularized [[quantization]] of the [[supermembrane]] (the [[kappa-symmetry|kappa-symmetric]] [[Green-Schwarz sigma-model]] for the [[M2-brane]]) to the [[BFSS matrix model]]:
* {#deWitHoppeNicolai88} [[Bernard de Wit]], [[Jens Hoppe]], [[Hermann Nicolai]], _On the Quantum Mechanics of Supermembranes_, Nucl. Phys. B305 (1988) 545. ([pdf](http://pubman.mpdl.mpg.de/pubman/item/escidoc:153408:1/component/escidoc:153407/353961.pdf), [[deWitHoppeNicolai88.pdf:file]], [spire:261702](http://inspirehep.net/record/261702))
* [[Bernard de Wit]], W. Lüscher, [[Hermann Nicolai]], _The supermembrane is unstable_, Nucl. Phys. B320 (1989) 135 ([spire:266584](http://inspirehep.net/record/266584/), <a href="https://doi.org/10.1016/0550-3213(89)90214-9">doi:10.1016/0550-3213(89)90214-9</a>)
On [[gauged supergravity]] and introducing the concept of [[tensor hierarchies]]:
* [[Bernard de Wit]], [[Henning Samtleben]], _Gauged maximal supergravities and hierarchies of nonabelian vector-tensor systems_, Fortsch. Phys. 53 (2005) 442-449 ([arXiv:hep-th/0501243](https://arxiv.org/abs/hep-th/0501243))
* [[Bernard de Wit]], [[Henning Samtleben]], _The end of the $p$-form hierarchy_, JHEP 0808:015, 2008 ([arXiv:0805.4767](https://arxiv.org/abs/0805.4767))
On [[tensor hierarchies]] in [[gauged supergravity]] and possible relation to [[U-duality]] and [[M-theory]]:
* [[Bernard de Wit]], [[Hermann Nicolai]], [[Henning Samtleben]], _Gauged Supergravities, Tensor Hierarchies, and M-Theory_, JHEP 0802:044, 2008 ([arXiv:0801.1294](https://arxiv.org/abs/0801.1294))
category: people |
Bernard Julia | https://ncatlab.org/nlab/source/Bernard+Julia |
* [wikipedia entry](http://en.wikipedia.org/wiki/Bernard_Julia)
## Selected writings
Introducing [[D=11 supergravity]]:
* {#CremmerJuliaScherk78} [[Eugene Cremmer]], [[Bernard Julia]], [[Joël Scherk]], _Supergravity in theory in 11 dimensions_, Phys. Lett. B **76** (1978) 409 [<a href="https://doi.org/10.1016/0370-2693(78)90894-8">doi:10.1016/0370-2693(78)90894-8</a>]
On [[first-order formulation of gravity|first order formulation]] of [[supergravity]]:
* [[Bernard Julia]], S. Silva, *On first order formulations of supergravities*, JHEP 0001 (2000) 026 [[arXiv:hep-th/9911035](https://arxiv.org/abs/hep-th/9911035), [doi:10.1088/1126-6708/2000/01/026](https://doi.org/10.1088/1126-6708/2000/01/026)]
category: people |
Bernard Le Stum | https://ncatlab.org/nlab/source/Bernard+Le+Stum |
* [webpage](http://perso.univ-rennes1.fr/bernard.le-stum/Bienvenue.html)
## related $n$Lab entries
* [[p-adic geometry]]
* [[non-archimedean analytic geometry]]
category: people
|
Bernard Leclerc | https://ncatlab.org/nlab/source/Bernard+Leclerc | _Bernard Leclerc_ is a mathematician at Université de Caen. His specialties include [[algebraic combinatorics]] and [[representation theory]].
* [homepage](http://www.math.unicaen.fr/~leclerc)
category:people
[[!redirects B. Leclerc]] |
Bernard M.A.G. Piette | https://ncatlab.org/nlab/source/Bernard+M.A.G.+Piette |
* [Institute page](https://www.dur.ac.uk/directory/profile/?id=474)
* [spire page](https://inspirehep.net/authors/1048203)
## Selected writings
On [[Skyrmions]] constructed via [[rational maps]] from the [[complex plane]], hence [[holomorphic functions]] from the [[Riemann sphere]], to itself:
* [[Nicholas S. Manton]], [[Bernard M.A.G. Piette]], *Understanding Skyrmions using Rational Maps*, in: Casacuberta C., Miró-Roig R.M., Verdera J., Xambó-Descamps S. (eds.) European Congress of Mathematics. Progress in Mathematics, vol 201. Birkhäuser, Basel. 2001 ([doi:10.1007/978-3-0348-8268-2_27](https://doi.org/10.1007/978-3-0348-8268-2_27), [arXiv:hep-th/0008110](https://arxiv.org/abs/hep-th/0008110))
* W.T. Lin, [[Bernard M.A.G. Piette]], *Skyrmion Vibration Modes within the Rational Map Ansatz*, Phys. Rev. D77:125028, 2008 ([arXiv:0804.4786](https://arxiv.org/abs/0804.4786), [doi:10.1103/PhysRevD.77.125028](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.77.125028))
category: people
[[!redirects Bernard Piette]]
[[!redirects B.M.A.G. Piette]]
[[!redirects B. M. A. G. Piette]]
[[!redirects Bernard M. A. G. Piette]]
|
Bernard Malgrange | https://ncatlab.org/nlab/source/Bernard+Malgrange |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Bernard_Malgrange)
## Related entries
* [[Koszul-Malgrange theorem]]
* [[holomorphic vector bundle]]
category: people
|
Bernard S. Kay | https://ncatlab.org/nlab/source/Bernard+S.+Kay |
* [GoogleScholar page](https://scholar.google.com/citations?user=8M4mKZsAAAAJ&hl=en)
## Selected writings
On [[AQFT on curved spacetimes]]:
* [[Bernard S. Kay]], *Quantum Field Theory in Curved Spacetime*, Encyclopaedia of Mathematical Physics (2023) [[arXiv:2308.14517](https://arxiv.org/abs/2308.14517)]
category: people
[[!redirects Bernard Kay]] |
Bernardo Uribe | https://ncatlab.org/nlab/source/Bernardo+Uribe |
* [website](https://sites.google.com/site/bernardouribejongbloed/)
## Selected writings
Introducing [[twisted K-theory|twisted]] [[orbifold K-theory]] presented by
[[twisted bundles|twisted]] [[equivariant vector bundles]] (including their form that came to be called called *[[bundle gerbe modules]]*):
* {#LupercioUribe01} [[Ernesto Lupercio]], [[Bernardo Uribe]], *Gerbes over Orbifolds and Twisted K-theory*, Comm. Math. Phys. 245(3): 449-489. ([arXiv:math/0105039](http://arxiv.org/abs/math/0105039), [doi:10.1007/s00220-003-1035-x](https://doi.org/10.1007/s00220-003-1035-x))
On [[equivariant ordinary differential cohomology]]:
* [[Ernesto Lupercio]], [[Bernardo Uribe]], _Deligne Cohomology for Orbifolds, discrete torsion and B-fields_, in: _Geometric and topological methods for quantum field theory_ Proceedings, Summer School, Villa de Leyva, Colombia, July 9-27, 2001 ([arXiv:hep-th/0201184](https://arxiv.org/abs/hep-th/0201184), [spire:582101](https://inspirehep.net/literature/582101))
On [[transgression]] and [[higher holonomy]] in [[Deligne cohomology]] (for [[bundle gerbes with connection]]) over [[orbifolds]]:
* [[Ernesto Lupercio]], [[Bernardo Uribe]], *Holonomy for Gerbes over Orbifolds*, J. Geom.Phys. **56** (2006) 1534-1560 [[arXiv:math/0307114](https://arxiv.org/abs/math/0307114), [doi:10.1016/j.geomphys.2005.08.006](https://doi.org/10.1016/j.geomphys.2005.08.006)]
On [[twisted equivariant K-theory|twisted]] [[orbifold K-theory]]:
* [[Edward Becerra]], [[Bernardo Uribe]], _Stringy product on twisted orbifold K-theory for abelian quotients_, Trans. Amer. Math. Soc. 361 (2009), 5781-5803 ([arXiv:0706.3229](https://arxiv.org/abs/0706.3229), [doi:10.1090/S0002-9947-09-04760-6](https://doi.org/10.1090/S0002-9947-09-04760-6))
On the [[topology]] of the infinite [[unitary group]] [[U(ℋ)]] on a [[separable Hilbert space]] ([[operator topology]] and [[compact-open topology]]):
* {#EspinozaUribe} [[Jesus Espinoza]], [[Bernardo Uribe]], _Topological properties of the unitary group_, JP Journal of Geometry and Topology **16** (2014) Issue 1, pp 45-55 ([arXiv:1407.1869](https://arxiv.org/abs/1407.1869), [journal](http://www.pphmj.com/abstract/8730.htm))
and of the corresponding [[projective unitary group]] [[PU(ℋ)]]:
* [[Jesus Espinoza]], [[Bernardo Uribe]], *Topological properties of spaces of projective unitary representations*, Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 40 (2016), no. 155, 337-352 ([arXiv:1511.06785](https://arxiv.org/abs/1511.06785), [scielo:S0370-39082016000200013](http://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S0370-39082016000200013), [doi:10.18257/raccefyn.317](https://doi.org/10.18257/raccefyn.317))
On 3-[[twisted equivariant K-theory]] via the [[universal equivariant PU(H)-bundle|universal equivariant $PU(\mathcal{H})$-bundle]]:
* {#BEJU14} [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Michael Joachim]], [[Bernardo Uribe]], _Universal twist in Equivariant K-theory for proper and discrete actions_, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2014) ([arXiv:1202.1880](https://arxiv.org/abs/1202.1880), [doi:10.1112/plms/pdt061](https://doi.org/10.1112/plms/pdt061))
On [[equivariant principal bundles]] and their [[classifying spaces]]:
* [[Bernardo Uribe]], [[Wolfgang Lück]], _Equivariant principal bundles and their classifying spaces_, Algebr. Geom. Topol. 14 (2014) 1925-1995 ([arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925))
On the analog of the [[Atiyah-Hirzebruch spectral sequence]] for [[twisted equivariant K-theory]]:
* [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Bernardo Uribe]], [[Mario Velasquez]], _Segal's spectral sequence in twisted equivariant K-theory for proper and discrete actions_, Proceedings of the Edinburgh Mathematical Society **61** 1 (2018) ([arXiv:1307.1003](https://arxiv.org/abs/1307.1003), [doi:10.1017/S0013091517000281](https://doi.org/10.1017/S0013091517000281))
category: people
|
Bernd Fischer | https://ncatlab.org/nlab/source/Bernd+Fischer |
* [Wikipedia page](http://en.wikipedia.org/wiki/Bernd_Fischer)
## Related $n$Lab entries
* [[Monster group]]
* [[Moonshine]]
category: people |
Bernd Schroers | https://ncatlab.org/nlab/source/Bernd+Schroers |
Bernd Schroers is professor of [[mathematics]] at Heriot-Watt in Edinburgh.
* [webpage](http://www.hw.ac.uk/schools/mathematical-computer-sciences/staff-directory/bernd-schroers.htm)
## Selected writings
On [[Yang-Mills monopoles]] for [[gauge group]] [[SU(3)]]:
* F. A. Bais, [[Bernd Schroers]], *Quantisation of Monopoles with Non-abelian Magnetic Charge*, Nucl. Phys. B512 (1998) 250-294 ([arXiv:hep-th/9708004](https://arxiv.org/abs/hep-th/9708004))
On [[Michael Atiyah]]'s work relating to [[physics]] ([[Yang-Mills theory]], [[Skyrmions]]):
* [[Bernd Schroers]], _Michael Atiyah and Physics: the Later Years_, ([arXiv:1910.10630](https://arxiv.org/abs/1910.10630)), in: Notices of the AMS, _Memories of Sir Michael Atiyah_ ([pdf](https://www.ams.org/journals/notices/201911/rnoti-p1834.pdf), [[MemoriesOfAtiyah.pdf:file]])
## Related $n$Lab entries
* [[Chern-Simons theory]]
* [[Kaluza-Klein compactification]]
category: people |
Bernd Siebert | https://ncatlab.org/nlab/source/Bernd+Siebert |
* [institute page](https://www.math.uni-hamburg.de/home/siebert/)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Bernd_Siebert)
* [GoogleScholar page](https://scholar.google.de/citations?user=kgOhnLYAAAAJ&hl=de)
## Selected writings
Early discussion of [[quantum cohomology rings]]:
* [[Bernd Siebert]], [[Gang Tian]], *On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator*, Asian Journal of Mathematics **1** 4 (1997) 679 – 695 [[arXiv:alg-geom/9403010](https://arxiv.org/abs/alg-geom/9403010), [doi:10.4310/AJM.1997.v1.n4.a2](https://dx.doi.org/10.4310/AJM.1997.v1.n4.a2)]
category: people |
Bernd Sturmfels | https://ncatlab.org/nlab/source/Bernd+Sturmfels | Bernd Sturmfels is a professor at Berkeley, California.
* [webpage](http://math.berkeley.edu/~bernd/) |
Bernd Thaller | https://ncatlab.org/nlab/source/Bernd+Thaller |
* [home page](https://imsc.uni-graz.at/thaller/)
* [Wikipedia entry (deutsch)](https://de.wikipedia.org/wiki/Bernd_Thaller)
## Selected writings
* [[Bernd Thaller]], *The Dirac Equation*, Texts and Monographs in Physics, Springer (1992) ([doi:10.1007/978-3-662-02753-0](https://link.springer.com/book/10.1007/978-3-662-02753-0))
category: people
|
Bernhard Banaschewski | https://ncatlab.org/nlab/source/Bernhard+Banaschewski |
Bernhard Banaschewski (1926-2022)]
* [McMaster obituary](https://math.mcmaster.ca/bernhard-banaschewski-1926-2022/#:~:text=Bernhard%20Banaschewski%2C%20McKay%20Professor%20(Emeritus,of%20the%20Department%20in%201955.)
* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=13403)
## Selected writings
* {#Banaschewski_Pultr94} [[B. Banaschewski]], A. Pultr, *Variants of openness*, Appl. Cat. Struc. **2** (1994) 1-21 [[doi:10.1007/BF00873038](https://doi.org/10.1007/BF00873038)]
* {#Banaschewski_Pultr96} [[B. Banaschewski]], A. Pultr, *Booleanization*, Cah. Top. Géom. Diff. Cat. **XXXVII** **1** (1996) 41-60 [[numdam:CTGDC_1996__37_1_41_0](http://numdam.mathdoc.fr/numdam-bin/fitem?id=CTGDC_1996__37_1_41_0)]
On the [[Stone-Weierstrass theorem]] in [[constructive mathematics]]:
* [[B. Banaschewski]], [[C. J. Mulvey]], *A constructive proof of the Stone-Weierstrass theorem*, J. Pure Appl. Algebra __116__ (1997) 25-40 [<a href="https://doi.org/10.1016/S0022-4049(96)00160-0">doi:10.1016/S0022-4049(96)00160-0</a>]
On the [[Boolean algebra]] underlying Spencer Brown's "Laws of form":
* {#Banaschewski77} [[B. Banaschewski]], *On G. Spencer Brown's laws of form*, Notre Dame J. Formal Logic **18** 3 (1977) 507-509 [[doi:10.1305/ndjfl/1093888028](https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-18/issue-3/On-G-Spencer-Browns-laws-of-form/10.1305/ndjfl/1093888028.full)]
On [[algebraic closures]] of [[fields]] in the absence of the [[axiom of choice]]:
* {#Banaschewski92} [[Bernhard Banaschewski]], *Algebraic closure without choice*, Mathematical Logic Quarterly, Volume 38, Issue 1, 1992, Pages 383-385, [[doi:10.1002/malq.19920380136](https://doi.org/10.1002/malq.19920380136)]
## Related entries
* [[weakly open subtopos]]
* [[skeletal geometric morphism]]
category:people
[[!redirects B. Banaschewski]]
[[!redirects Banaschewski]]
|
Bernhard Hanke | https://ncatlab.org/nlab/source/Bernhard+Hanke |
* [webpage](https://www.uni-augsburg.de/de/fakultaet/mntf/math/prof/diff/team/bernhard-hanke/)
## Selected writings
On [[equivariant bordism homology theory]]:
* [[Bernhard Hanke]], _Geometric versus homotopy theoretic equivariant bordism_, Math. Ann. 332, 677–696 (2005) ([arXiv:math/0412550](https://arxiv.org/abs/math/0412550), [doi:10.1007/s00208-005-0648-0](https://doi.org/10.1007/s00208-005-0648-0))
category: people |
Bernhard Keller | https://ncatlab.org/nlab/source/Bernhard+Keller | _Bernhard Keller_ is a Swiss-French mathematician.
* [homepage](https://webusers.imj-prg.fr/~bernhard.keller/indexe.html)
* [quelques prépublications et publications](https://webusers.imj-prg.fr/~bernhard.keller/publ/index.html)
## Selected writings
On [[differential graded categories]]:
* [[Bernhard Keller]], _On differential graded categories_, International Congress of Mathematicians. **II** (2006) 151-190, Eur. Math. Soc., Zürich [[arXiv:math/0601185](http://arxiv.org/abs/math/0601185)]
category: people
[[!redirects B. Keller]] |
Bernhard Riemann | https://ncatlab.org/nlab/source/Bernhard+Riemann |
* [Wikipedia entry (en)](https://en.wikipedia.org/wiki/Bernhard_Riemann)
* [Wikipedia entry (fr)](http://fr.wikipedia.org/wiki/Bernhard_Riemann)
## Seleted writings
Envisioning the modern foundations of [[differential geometry]] via the notion of ([[differentiable manifold|differentiable]], [[smooth manifold|smooth]]) [[manifolds]] and of what camed to be called [[Riemannian geometry]]:
* [[Bernhard Riemann]], *[[Über die Hypothesen, welche der Geometrie zu Grunde liegen]]*, Göttingen (1845) [[doi:10.1007/978-3-642-35121-1](https://doi.org/10.1007/978-3-642-35121-1)]
Engl. transl: [[William Clifford]]: *[[On the hypotheses which underlie geometry]]*, Nature **VIII** (1873) 183-184 [[doi:10.1007/978-3-319-26042-6](https://doi.org/10.1007/978-3-319-26042-6)]
## Related entries
* [[manifold]], [[smooth manifold]]
* [[Riemannian geometry]], [[Riemannian metric]]
* [[Riemann conjecture]]
* [[Riemann surface]]
category: people
[[!redirects Riemann]] |
Bernhard Schölkopf | https://ncatlab.org/nlab/source/Bernhard+Sch%C3%B6lkopf |
* [webpage](https://www.is.mpg.de/~bs)
* [Wikipedia entry](https://www.is.mpg.de/~bs)
## Selected writings
On [[kernel methods]] in [[machine learning]]:
* Thomas Hofmann, [[Bernhard Schölkopf]], Alexander J. Smola, *Kernel methods in machine learning*, Annals of Statistics 2008, Vol. 36, No. 3, 1171-1220 ([arXiv:math/0701907](https://arxiv.org/abs/math/0701907))
On [[Bochner's theorem]] generalized to [[non-abelian groups]] in the context of [[kernel methods]]:
* [[Kenji Fukumizu]], [[Bharath Sriperumbudur]], [[Arthur Gretton]], [[Bernhard Schölkopf]], *Characteristic Kernels on Groups and Semigroups*, Advances in Neural Information Processing Systems 21 : 22nd Annual Conference on Neural Information Processing Systems 2008 ([NIPS 2008](https://neurips.cc/Conferences/2008/)), 473-480 ([mpg:5466](http://www.is.mpg.de/publications/5466), [pdf](http://www.gatsby.ucl.ac.uk/~gretton/papers/FukSriGreSch09.pdf))
On some philosophical aspects of [[statistical learning theory]]:
* [[David Corfield]], [[Bernhard Schölkopf]], Vladimir Vapnik, _Falsificationism and statistical learning theory: Comparing the Popper and Vapnik-Chervonenkis dimensions_, 2009, Journal for General Philosophy of Science 40 (1), 51-58 ([doi:10.1007/s10838-009-9091-3](https://doi.org/10.1007/s10838-009-9091-3))
category: people
|
Bernhard Ömer | https://ncatlab.org/nlab/source/Bernhard+%C3%96mer |
* [webpage](http://www.itp.tuwien.ac.at/~oemer/)
## Selected writings
On classical computational control in [[quantum computation]]:
* [[Bernhard Ömer]], *Structured Quantum Programming*, 2003/2009 ([pdf](http://www.itp.tuwien.ac.at/~oemer/doc/structquprog.pdf))
category: people
[[!redirects Bernhard Omer]] |
Bernoulli number | https://ncatlab.org/nlab/source/Bernoulli+number |
+-- {: .rightHandSide}
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###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Motivation
### In Lie theory
In [[Lie theory]], _Bernoulli numbers_ appear as [[coefficients]]
in the linear part of the [[Hausdorff series]] (and the [[recursion|recursive]]
relation for the Dynkin Lie polynomials appearing in the Hausdorff series); this has consequences in [[deformation theory]].
The (determinant of the square root of the) inverse of its generating function appears (for variables in an [[adjoint representation]]) in an expression for the [[Duflo isomorphism]] in Lie theory and in its generalizations in [[knot theory]] etc.
### In algebraic topology
In [[algebraic topology]]/[[cohomology]] Bernoulli numbers appear as the [[coefficients]] of the [characteristic series](genus#LogarithmAndCharacteristicSeries) of the [[A-hat genus]] (see there), and they (or equivalently, their [[generating functions]]) also appear in the expression for the [[Todd class]].
The Bernoulli numbers are also proportional to the constant terms of the [[Eisenstein series]] and as such appear in the exponential form of the characteristic series of the [[Witten genus]].
Finally they appear as the [[order of a group|order]] of some groups in the [[image of the J-homomorphism]] (cf. [Adams 65, section 2](#Adams65)).
Of course, all of these cases are related to [[formal group laws]]. Formal groups bear also some other connections to Bernoulli numbers and generalizations like [[Bernoulli polynomial]]s.
### Elsewhere
The [[Riemann zeta-function]] $\zeta$ at negative integral values is proportional to the Bernoulli numbers as
$$
\zeta(-n) = - \frac{B_{n+1}}{n+1}
\,.
$$
Bernoulli numbers appear also in [[umbral calculus]]. There are generalizations, for example, [[Bernoulli polynomials]].
They also have applications in analysis ([[Euler-MacLaurin formula]], with applications in numerical analysis).
## Definition
The Bernoulli numbers $B_k$ are [[rational numbers]] given by their [[generating function]], i.e. by the [[equation]] of [[functions]]/[[power series]] $x \mapsto f(x)$
$$
\frac{x}{
e^x -1
}
=
\sum_{k = 0}^{\infty} \frac{\beta_k}{k !}x^k
\,.
$$
The $k$th Bernoulli number $B_k \in \mathbb{Q}$ is, depending on convention, either equal to $\beta_k$ (or sporadically, in older literature, to $(-1)^{k-1} \beta_{2k}$).
If we take generating function
$x+\frac{x}{e^x-1}=\frac{x e^x}{e^x -1}=\frac{-x}{e^{-x}-1}$ this only changes the sign of $B_1$ as all other odd Bernoulli numbers (in standard convention) vanish.
## Properties
The __von Staudt-Clausen theorem__ states that
$$
B_{2n} + \sum_{p-1|2n} \frac{1}{p} \in\mathbb{Z}.
$$
## Related concepts
* [[modified Bernoulli number]]
* [[Bernoulli polynomial]]
* [[umbral calculus]]
## References
### General
Textbook accounts;
* Tom Apostol, Section 12.12 of: _Introduction to Analytic Number Theory_, Springer 1976 ([springer:book/9780387901633](https://www.springer.com/de/book/9780387901633))
Exposition:
* [[Pierre Cartier]], chapter 3 in: _Mathemagics_, Séminaire Lotharingien de Combinatoire 44 (2000), Article B44d ([pdf](http://www.mat.univie.ac.at/~slc/wpapers/s44cartier1.pdf), [[CartierMathemagics.pdf:file]])
* [[John Baez]], _The Bernoulli numbers_, 2003 expository notes ([pdf](http://math.ucr.edu/home/baez/qg-winter2004/bernoulli.pdf))
See also:
* Wikipedia, _[Bernoulli number](http://en.wikipedia.org/wiki/Bernoulli_number)_
* Wolfram MathWorld: [Bernoulli number](http://mathworld.wolfram.com/BernoulliNumber.html)
* Bernoulli numbers page [bernoulli.org](http://bernoulli.org)
### Lie theory, deformation theory, knot theory, geometry
* MathOverflow [Todd class and Baker-Campbell-Hausdorff, or the curious number 12](http://mathoverflow.net/questions/31972/todd-class-and-baker-campbell-hausdorff-or-the-curious-number-12)
* [[Nikolai Durov|N. Durov]], S. Meljanac, A. Samsarov, [[Z. Škoda]], _A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra_, Journal of Algebra __309__:1, pp.318-359 (2007) [math.RT/0604096](http://arxiv.org/abs/math.RT/0604096), MPIM2006-62
* Vinay Kathotia, _Kontsevich's universal formula for deformation quantization and the Campbell-Baker-Hausdorff formula, I_, [math.QA/9811174](http://arxiv.org/abs/math/9811174)
* Emanuela Petracci, _Functional equations and Lie algebras_, PhD thesis, [pdf](http://www1.mat.uniroma1.it/didattica/dottorato/TESI/ARCHIVIO/petracciemanuela.pdf)
* E. Meinrenken, _Clifford algebras and Lie theory_, Springer
* [[Anton Alekseev]], _Bernoulli numbers, Drinfeld associators, and the Kashiwara–Vergne problem_, slides, [pdf](http://www.6ecm.pl/docs/Alekseev.pdf)
* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))
([[modified Bernoulli numbers]] in the [[universal Vassiliev invariant]] of the [[unknot]])
### In QFT and string theory
Appearance of [[Bernoulli numbers]] in [[perturbative quantum field theory]] and [[string theory]]:
* [[Gerald Dunne]], [[Christian Schubert]], _Bernoulli Number Identities from Quantum Field Theory and Topological String Theory_, Communications in Number Theory and Physics, Volume 7 (2013) Number 2, 225 - 249 ([arXiv:math/0406610](https://arxiv.org/abs/math/0406610))
### In algebraic topology
In [[algebraic topology]]:
* {#Adams65} [[John Adams]], _On the groups $J(X)$ II_, Topology 3 (2) (1965) ([pdf](http://math1.unice.fr/~cazanave/Gdt/ImJ/J-II.pdf))
In the context of the [[A-hat genus]] the Bernoulli numbers are discussed in
* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], Section 10.2 of: _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))
### Formal groups and Bernoulli polynomials
* Piergiulio Tempesta, _Formal groups, Bernoulli-type polynomials and L-series_, Comptes Rendus de l Académie des Sciences - Series I - Mathematics 07/2007; 345
[doi](http://dx.doi.org/10.1016/j.crma.2007.05.016)
* Stefano Marmi, Piergiulio Tempesta, _Hyperfunctions, formal groups and generalized Lipschitz summation formulas_, Nonlinear Analysis 03/2012; 75:1768-1777 [doi](http://dx.doi.org/10.1016/j.na.2011.09.013)
### Analysis
* [[Terence Tao]]'s blog: [The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation](http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation)
[[!redirects Bernoulli numbers]] |
Bernoulli polynomial | https://ncatlab.org/nlab/source/Bernoulli+polynomial |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Bernoulli polynomials are a family of [[rational number|rational]] [[polynomials]] which generalize [[Bernoulli numbers]].
## Related concepts
* [[Bernoulli number]]
* [[modified Bernoulli number]]
## References
* Piergiulio Tempesta, _Formal groups, Bernoulli-type polynomials and L-series_, Comptes Rendus de l Académie des Sciences - Series I - Mathematics 07/2007; 345
[doi](http://dx.doi.org/10.1016/j.crma.2007.05.016)
* Stefano Marmi, Piergiulio Tempesta, _Hyperfunctions, formal groups and generalized Lipschitz summation formulas_, Nonlinear Analysis 03/2012; 75:1768-1777 [doi](http://dx.doi.org/10.1016/j.na.2011.09.013)
* Francis Clarke, _The universal von Staudt theorems_, Trans. Amer. Math. Soc. 315 (1989), 591-603 [doi](http://dx.doi.org/10.1090/S0002-9947-1989-0986687-3)
[[!redirects Bernoulli polynomials]] |
Bernstein-Sato polynomial | https://ncatlab.org/nlab/source/Bernstein-Sato+polynomial | Bernstein-Sato polynomial or $b$-function (in [[D-module]] theory)
* I.N. Bernšte ̆ın, The analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priložen. 6 (1972) 26–40 = Funct. Anal. Appl. 6 (1972) 273–285
* Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello, _Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians_, [arxiv/1105.6270](http://arxiv.org/abs/1105.6270)
|
Berry connection | https://ncatlab.org/nlab/source/Berry+connection |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Generally, a *Berry phase* is a non-trivial phase picked up by a [[quantum state]] of definite [[energy]] under [[quantum adiabatic theorem|adiabatic]] changes of the [[quantum system]]'s [[Hamiltonian]] around a [[loop]] in its parameter space.
Specifically, in [[condensed matter theory]] and for a [[crystal|crystalline]] material with a set of isolated (gapped) [[electronic band structure|electronic bands]], the [[partial derivatives]] along [[momentum]]-[[vectors]] of the corresponding [[Bloch states]], projected back onto these states, turn out to canonically define a [[connection on a vector bundle|connection]] [[differential 1-form|1-form]] on the [[vector bundles]] (over the [[Brillouin torus]]) which is spanned by these states. This is called a *Berry connection*, due to ([Berry 84](#Berry84), [Simon 83](#Simon83)). The [[parallel transport]] along this connection describes the change of [[Bloch states]] under the [[quantum adiabatic theorem|adiabatic]] change of their [[momentum]]/[[wave vector]]. The [[curvature 2-form]] of a Berry connection is accordingly called the *Berry curvature*.
The [[holonomy]] of such a Berry connection is called a *[[Berry phase]]*, in general, and a *[[Zak phase]]* ([Zak 89](#Zak89)) when evaluated along one of the non-trivial 1-[[cycles]] of the [[Brillouin torus]].
By default, this is understood to apply to the [[valence bundle]] of a [[crystal|crystalline]] material, but the construction works more generally.
{#TopologicalOrder} If in the case of a gapped [[valence bundle]], hence a [[topological insulator]]-phase, the [[holonomy]] of the Berry connection is [[non-abelian group|non-abelian]] (which may happen, as originally highlighted in [Wilczek & Zee 84](#WilczekZee1984)) then one also says that the topological phase exhibits *[[topological order]]*.
For [[semimetals]] the Berry phases of the [[valence bundle]] around their nodal loci of [[codimension]] 2 are a measure for the [[obstruction]] to adiabatically deforming the semimetal such as to open its gap closures, hence to become a ([[topological insulator|topological]]) [[insulator]] (eg. [Vanderbilt 18, 5.5.2](#Vanderbilt18)).
## Related concepts
* [[Berry's phase]] (needs to be merged)
* [[quantum adiabatic theorem]]
## References
### General
The original article:
* {#Berry84} [[Michael V. Berry]], *Quantal phase factors accompanying adiabatic changes*, Proc. R. Soc. Lond. A **392** (1984) 45–57 ([doi:10.1098/rspa.1984.0023](https://doi.org/10.1098/rspa.1984.0023), [jstor:2397741](https://www.jstor.org/stable/2397741))
The formulation in terms of [[connections]] on [[fiber bundles]] and their [[holonomy]]:
* {#Simon83} [[Barry Simon]], *Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase*, Phys. Rev. Lett. **51** (1983) 2167 ([doi:10.1103/PhysRevLett.51.2167](https://doi.org/10.1103/PhysRevLett.51.2167))
Generalization to (connections with) [[non-abelian group|non-abelian]] [[holonomies]]:
* {#WilczekZee1984} [[Frank Wilczek]], [[Anthony Zee]], *Appearance of gauge structure in simple dynamical systems*, Physical Review Letters **52** 24 (1984) 2111 $[$[doi:10.1103/PhysRevLett.52.2111](https://doi.org/10.1103/PhysRevLett.52.2111)$]$
The special case of Berry phases around the 1-cycles of a [[Brillouin torus]] -- [[Zak phases]]:
* {#Zak89} [[Joshua Zak]], *Berry’s phase for energy bands in solids*, Phys. Rev. Lett. **62** (1989) 2747 ([doi:10.1103/PhysRevLett.62.2747](https://doi.org/10.1103/PhysRevLett.62.2747))
Review and further discussion:
* [[Michael Berry]], *The quantum phase, five years after*, in: *Geometric phases in physics*, Adv. Ser. Math. Phys. **5**, World Scientific (1989) 7--28 ([[Berry-FiveYearsAfter.pdf:file]], [doi:10.1142/0613](https://doi.org/10.1142/0613))
* Ming-Che Chang, Qian Niu, *Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields*, J. Phys.: Condens. Matter **20** (2008) 193202 ([doi:10.1088/0953-8984/20/19/193202](https://iopscience.iop.org/article/10.1088/0953-8984/20/19/193202))
* Di Xiao, Ming-Che Chang, Qian Niu, *Berry Phase Effects on Electronic Properties*, Rev. Mod. Phys. **82** (2010) 1959-2007 ([arXiv:0907.2021](https://arxiv.org/abs/0907.2021), [doi:10.1103/RevModPhys.82.1959](https://doi.org/10.1103/RevModPhys.82.1959))
With focus on [[topological phases of matter]] ([[topological insulators]], [[semimetals]], etc.):
* {#Vanderbilt18} [[David Vanderbilt]], *Berry Phases in Electronic Structure Theory -- Electric Polarization, Orbital Magnetization and Topological Insulators*, Cambridge University Press (2018) ([doi:10.1017/9781316662205](https://doi.org/10.1017/9781316662205))
* [[Jérôme Cayssol]], [[Jean-Noël Fuchs]], Section IV.C of: *Topological and geometrical aspects of band theory*, J. Phys. Mater. **4** (2021) 034007 ([arXiv:2012.11941](https://arxiv.org/abs/2012.11941), [doi:10.1088/2515-7639/abf0b5](https://doi.org/10.1088/2515-7639/abf0b5))
* [[Tudor D. Stanescu]], Chapter 2 of: *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))
See also:
* Wikipedia, *[Berry connection and curvature](https://en.wikipedia.org/wiki/Berry_connection_and_curvature)*
### For anyon statistics
On [[anyon statistics|anyon phases]] (specifically in the [[quantum Hall effect]]) as [[Berry phases]] of a [[quantum adiabatic theorem|adiabatic]] transport of anyon positions:
* {#AvrosSchriefferWilczek84} [[Daniel P. Arovas]], [[John Robert Schrieffer]], [[Frank Wilczek]], _Fractional Statistics and the Quantum Hall Effect_, Phys. Rev. Lett. 53, 722 (1984) $[$[doi:10.1103/PhysRevLett.53.722](https://doi.org/10.1103/PhysRevLett.53.722)$]$
### Experimental observation
Experimental observation of [[Zak phases]]:
* Marcos Atala, Monika Aidelsburger, Julio T. Barreiro, Dmitry Abanin, Takuya Kitagawa, Eugene Demler, Immanuel Bloch: *Direct measurement of the Zak phase in topological Bloch bands*, Nature Physics **9** (2013) 795–800 ([doi:10.1038/nphys2790](https://doi.org/10.1038/nphys2790))
Proposal for experimental realization of [[Berry phases]] around [[codimension]]=2 nodal loci of (quantum simulations of time+space inversion symmetric) [[semi-metals]]:
* Dan-Wei Zhang, Y. X. Zhao, Rui-Bin Liu, Zheng-Yuan Xue, Shi-Liang Zhu, Z. D. Wang, *Quantum simulation of exotic PT-invariant topological nodal loop bands with ultracold atoms in an optical lattice*, Phys. Rev. A **93** (2016) 043617 ([arXiv:1601.00371](https://arxiv.org/abs/1601.00371), [doi:10.1103/PhysRevA.93.043617](https://doi.org/10.1103/PhysRevA.93.043617))
> (see sec II.A, these authors stand out as mentioning the relevant [[KO-theory]])
[[!redirects Berry connections]]
[[!redirects Berry curvature]]
[[!redirects Berry curvatures]]
[[!redirects Berry phase]]
[[!redirects Berry phases]]
[[!redirects Zak phase]]
[[!redirects Zak phases]]
|
Berry's phase | https://ncatlab.org/nlab/source/Berry%27s+phase |
#Contents#
* table of contents
{:toc}
## Idea
__Berry's geometric phase__ is a correction to the [[wave function]] arising in the study of adiabatic quantum systems; it has been discovered by M. V. Berry. There are analogous effects for other [[wave]] phenomena; there has been also much earlier work of Pancharatnam on the related phenomenon in optics. The origin of the Berry's phase is in nonflatness of a [[parallel transport]] which appears in the corresponding phase factors.
## Related concepts
* [[Berry connection]]
* [[quantum adiabatic theorem]]
* [[adiabatic quantum computation]]
## References
* [[Michael Berry]], _Quantal phase factors accompanying adiabatic changes_, Proc. Roy. Soc. London Ser. A 392 (1984), no. 1802, 45--57, [doi](http://dx.doi.org/10.1098/rspa.1984.0023)
* [[Michael Berry]], _The quantum phase, five years after_, in: Geometric phases in physics, 7--28, Adv. Ser. Math. Phys., 5, World Sci. Publ., Teaneck, NJ, 1989.
* Barry Simon, _Holonomy, the quantum adiabatic theorem, and Berry's phase_, Phys. Rev. Lett. __51__ (1983), no. 24, 2167--2170, [MR85e:81024](http://www.ams.org/mathscinet-getitem?mr85e:81024)
* J. M. Robbins, [[Michael Berry]], _The geometric phase for chaotic systems_, Proc. Roy. Soc. London Ser. A 436 (1992), no. 1898, 631--661, [doi](http://dx.doi.org/10.1098/rspa.1992.0039), [94a:81036](http://www.ams.org/mathscinet-getitem?mr=94a:81036)
* [[V. I. Arnold]], _Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect_,
Selecta Mathematica __1__:1, 1--19 (1995) [doi](http://dx.doi.org/10.1007/BF01614072)
* Dariusz Chruściński, Andrzej Jamioƚkowski, _Geometric phases in classical and quantum mechanics_, Progress in Math. Physics __36__, Birkhäuser 2004. xiv+333 pp. ISBN: 0-8176-4282-X
* [[Mikio Nakahara]], Chapter 10.6 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), <a href="http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf">pdf</a>)
A relation to Chern-Bott connection is explained in 4.1 of lecture notes
* [[Mauro Spera]], _Geometric methods of quantum mechanics_, J. Geometry and Symmetry in Physics __24__ 1-44 (2011) [euclid](https://projecteuclid.org/euclid.jgsp/1495677645)
* D. Rohrlich, _Berry's phase_, entry in _Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy_, ed. F. Weinert, K. Hentschel, D. Greenberger and B. Falkenburg (Springer), to appear; [arxiv/0708.3749](http://arxiv.org/abs/0708.3749)
* wikipedia: [geometric phase](http://en.wikipedia.org/wiki/Geometric_phase)
* <http://www.mi.infm.it/manini/berryphase.html>
* [[Frank Wilczek]], Alfred Shapere, _Geometric phases in physics_, World Scientific, 1989
* M. O. Katanaev, _On geometric interpretation of the Berry phase_, Rus. Phys. J. 54(2012)1082--1092, Izv. VUZov. Fizika 10(2011) 26--35 [arxiv/1212.1782](http://arxiv.org/abs/1212.1782)
* Maxim Braverman, _The Berry phase and the phase of the determinant_, [arxiv/1310.6332](http://arxiv.org/abs/1310.6332)
Review in a context of ([[adiabatic quantum computation|adiabatic]]) [[quantum computation]]:
* Jiang Zhang, Thi Ha Kyaw, Stefan Filipp, Leong-Chuan Kwek, Erik Sjöqvist, Dianmin Tong, *Geometric and holonomic quantum computation* [[arXiv:2110.03602](https://arxiv.org/abs/2110.03602)]
category: physics
[[!redirects Berry phase]]
[[!redirects Berry's geometric phase]] |
Bert Guillou | https://ncatlab.org/nlab/source/Bert+Guillou |
* [webpage](http://www.ms.uky.edu/~guillou/)
## Selected writings
On [[enriched model categories]]:
* [[Bertrand Guillou]], [[Peter May]], _Enriched model categories and presheaf categories_, New York J. Math. 26 (2020) 37–9 ([arXiv:1110.3567](http://arxiv.org/abs/1110.3567), [](https://www.emis.de/journals/NYJM/j/2020/26-3.html), [NYJM:2020/26-3](https://www.emis.de/journals/NYJM/j/2020/26-3.html))
On [[model category]]-presentations for [[Elmendorf's theorem]] in [[equivariant homotopy theory]]:
* {#Guillou06} [[Bert Guillou]], _A short note on models for equivariant homotopy theory_, 2006 ([pdf](http://www.math.uiuc.edu/~bertg/EquivModels.pdf), [[GuillouEquivariantHomotopy.pdf:file]])
On [[classifying spaces]]/[[universal principal bundles]] for [[equivariant principal bundles]]:
* {#GuillouMayMerling17} [[Bertrand Guillou]], [[Peter May]], [[Mona Merling]] _Categorical models for equivariant classifying spaces_, Algebr. Geom. Topol. 17 (2017) 2565-2602 ([arXiv:1201.5178](https://arxiv.org/abs/1201.5178), [doi:10.2140/agt.2017.17.2565](https://doi.org/10.2140/agt.2017.17.2565))
On [[equivariant homotopy theory]] and [[Elmendorf's theorem]] via [[enriched model categories]]:
* {#GuillouMayRubin13} [[Bertrand Guillou]], [[Peter May]], [[Jonathan Rubin]], _Enriched model categories in equivariant contexts_, Homology, Homotopy and Applications 21 (1), 2019 ([arXiv:1307.4488](https://arxiv.org/abs/1307.4488), [arXiv:10.4310/HHA.2019.v21.n1.a10](https://dx.doi.org/10.4310/HHA.2019.v21.n1.a10))
On [[equivariant homotopy theory]] and [[equivariant cohomology]]:
* {#Guillou06} [[Bert Guillou]], *Equivariant Homotopy and Cohomology*, lecture notes, 2020 ([pdf](http://www.ms.uky.edu/~guillou/F20/751Notes.pdf), [[GuillouEquivariantHomotopyAndCohomology.pdf:file]])
## Related $n$Lab entries
* [[equivariant homotopy theory]], [[equivariant stable homotopy theory]]
* [[tom Dieck splitting]]
* [[Mackey functor]]
* [[Borel model structure]]
category: people
[[!redirects Bertrand Guillou]] |
Bert Janssen | https://ncatlab.org/nlab/source/Bert+Janssen |
* [webpage](http://www.ugr.es/~bjanssen/)
## Selected writings
On [[brane intersection]] laws of [[M-branes]]:
* [[Eric Bergshoeff]], [[Mees de Roo]], [[Eduardo Eyras]], [[Bert Janssen]], [[Jan Pieter van der Schaar]], _Intersections involving waves and monopoles in eleven dimensions_, Class. Quantum Grav. **14** (1997) 2757 [[doi:0264-9381/14/10/005](http://iopscience.iop.org/0264-9381/14/10/005)]
On [[D8-branes]] as [[black branes]] in [[massive type IIA string theory]]:
* {#JanssenMeessenOrtin99} [[Bert Janssen]], [[Patrick Meessen]], [[Tomas Ortin]], _The D8-brane tied up: String and brane solutions in massive type IIA supergravity_, Phys. Lett. B453 (1999) 229-236 ([spire:494174](http://inspirehep.net/record/494174), <a href="https://doi.org/10.1016/S0370-2693(99)00315-9">doi:10.1016/S0370-2693(99)00315-9</a>)
## Related $n$Lab entries
* [[M-brane]]
* [[brane intersection]]
category: people |
Bert Lindenhovius | https://ncatlab.org/nlab/source/Bert+Lindenhovius |
* [institute page](https://www.jku.at/institut-fuer-mathematische-methoden-in-medizin-und-datenbasierter-modellierung/ueber-uns/team/bert-lindenhovius/)
## Selected writings
Review of the [[ADHM construction]] for [[Yang-Mills instantons]]:
* [[Bert Lindenhovius]], *Instantons and the ADHM construction*, Amsterdam (2011) [[[Lindenhovius-Instantons.pdf:file]]]
Introducing a notion of [[quantum CPOs]] for [[quantum computation]] (via [[quantum sets]] carrying [[quantum relations]]):
* [[Andre Kornell]], [[Bert Lindenhovius]], [[Michael Mislove]], *Quantum CPOs*, EPTCS **340** (2021) 174-187 [[arXiv:2109.02196](https://arxiv.org/abs/2109.02196), [doi:10.4204/EPTCS.340.9](https://doi.org/10.4204/EPTCS.340.9)]
category: people |
Bert Schroer | https://ncatlab.org/nlab/source/Bert+Schroer |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Bert_Schroer)
## Selected writings
On [[anyons]]:
* [[Klaus Fredenhagen]], Karl-Henning Rehren, [[Bert Schroer]], _Superselection sectors with braid group statistics and exchange algebras -- I: General theory_, Comm. Math. Phys. Volume 125, Number 2 (1989), 201-226. ([euclid:cmp/1104179464](http://projecteuclid.org/euclid.cmp/1104179464))
* [[Klaus Fredenhagen]], Karl-Henning Rehren, [[Bert Schroer]], _Superselection sectors with braid group statistics and exchange algebras -- II: Geometric aspects and conformal covariance_, Reviews in Mathematical PhysicsVol. 04, No. spec01, pp. 113-157 (1992) ([doi:10.1142/S0129055X92000170](https://doi.org/10.1142/S0129055X92000170) [pdf](ftp://ftp.theorie.physik.uni-goettingen.de/pub/papers/rehren/92/superselection_sectors_II_RMP.pdf))
category: people |
Bertfried Fauser | https://ncatlab.org/nlab/source/Bertfried+Fauser |
* [webpage](http://www.cs.bham.ac.uk/~fauserb/)
category: people
|
Bertram Kostant | https://ncatlab.org/nlab/source/Bertram+Kostant |
* [website](http://www-math.mit.edu/~kostant/)
* [Wikipedia entry](http://en.wikipedia.org/wiki/Bertram_Kostant)
## Selected writings
On the [[Jacobson-Morozov theorem]]:
* {#Kostant59} [[Bertram Kostant]], _The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group_, Amer. J. Math. 81 (1959), 973–1032 ([jstor:2372999](https://www.jstor.org/stable/2372999))
category: people
[[!redirects B. Kostant]] |
Bertrand Eynard | https://ncatlab.org/nlab/source/Bertrand+Eynard |
* [webpage](http://ipht.cea.fr/en/Pisp/bertrand.eynard/)
## related $n$Lab entries
* [[topological recursion]]
* [[spectral curve]]
* [[integrable system]]
category: people |
Bertrand Halperin | https://ncatlab.org/nlab/source/Bertrand+Halperin |
* [institute page](https://www.physics.harvard.edu/people/facpages/halperin)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Bertrand_Halperin)
## Selected writings
On a hierarchy of [[Laughlin wavefunctions]]:
* [[Bertrand I. Halperin]], *Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States*, Phys. Rev. Lett. **52** (1984) 1583 $[$[doi:10.1103/PhysRevLett.52.1583](https://doi.org/10.1103/PhysRevLett.52.1583)$]$
On the [[fictitious gauge field]]-[[model (in theoretical physics)|model]] for [[anyon statistics]] and the resulting [[superfluidity]]/[[superconductivity]] *of* [[anyons]]:
* [[Yi-Hong Chen]], [[Frank Wilczek]], [[Edward Witten]], [[Bertrand Halperin]], *On Anyon Superconductivity*, International Journal of Modern Physics B **03** 07 (1989) 1001-1067 (reprinted in [Wilczek 1990](#Wilczek90)) $[$[doi:10.1142/S0217979289000725](https://doi.org/10.1142/S0217979289000725), [[CWWH-AnyonSuperfluidity.pdf:file]]$]$
category: people
[[!redirects Bertrand I. Halperin]]
|
Bertrand Russel > history | https://ncatlab.org/nlab/source/Bertrand+Russel+%3E+history | |
Bertrand Russell | https://ncatlab.org/nlab/source/Bertrand+Russell |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Bertrand_Russell)
* [St. Andrews History entry](http://www-history.mcs.st-and.ac.uk/Biographies/Russell.html)
* [Stanford Encyclopedia of Philosophy](http://plato.stanford.edu/entries/russell/)
* The Bertrand Russell society, Russell's texts online, [html](http://users.drew.edu/~jlenz/brtexts.html)
## Selected writings
* _[[An Essay on the Foundations of Geometry]]_ (1897)
* _The Principles of Mathematics_ (1903) ([pdf](https://bertrandrussellsocietylibrary.org/br-pom/br-pom.pdf))
* _[[Principia Mathematica]]_ (1910) with [[Alfred Whitehead]]
* _[[Logic as the Essence of Philosophy]]_ (1914)
* _Introduction to Mathematical Philosophy_ (1919) - Russell’s informal account of the mathematics and logic of _Principia Mathematica_ - ([web](https://people.umass.edu/klement/imp/imp.html))
* _[[A History of Western Philosophy]]_ (1945)
## Quotes
{#Quotes}
{#OnEducation} On education:
From *[Free Thought and Official Propaganda](https://www.gutenberg.org/files/44932/44932-h/44932-h.htm)* (1922):
> "Meanwhile the whole machinery of the State, in all the different countries, is turned on to making defenceless children believe absurd propositions the effect of which is to make them willing to die in defence of sinister interests under the impression that they are fighting for truth and right. This is only one of countless ways in which education is designed, not to give true knowledge, but to make the people pliable to the will of their masters. Without an elaborate system of deceit in the elementary schools it would be impossible to preserve the camouflage of democracy."
> "[...]"
> "It must not be supposed that the officials in charge of education desire the young to become educated. On the contrary, their problem is to impart information without imparting intelligence. Education should have two objects: first, to give definite knowledge—reading and writing, languages and mathematics, and so on; secondly, to create those mental habits which will enable people to acquire knowledge and form sound judgments for themselves. The first of these we may call information, the second intelligence. The utility of information is admitted practically as well as theoretically; without a literate population a modern State is impossible. But the utility of intelligence is admitted only theoretically, not practically; it is not desired that ordinary people should think for themselves, because it is felt that people who think for themselves are awkward to manage and cause administrative difficulties. Only the guardians, in Plato’s language, are to think; the rest are to obey, or to follow leaders like a herd of sheep. This doctrine, often unconsciously, has survived the introduction of political democracy, and has radically vitiated all national systems of education."
## Related entries
* [[Russell's paradox]]
* [[analytic philosophy]]
* [[type theory]]
* [[assertion]]
category: people
|
Bertrand Souères | https://ncatlab.org/nlab/source/Bertrand+Sou%C3%A8res |
* [spire page](https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=a%20B.Soueres.1)
## Selected writings
On [[higher curvature corrections]] to [[D=11 supergravity]] (particularly the [[I8]]-term):
* {#SoueresTsimpis17} [[Bertrand Souères]], [[Dimitrios Tsimpis]], _The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity_, Phys. Rev. D 95, 026013 (2017) ([arXiv:1612.02021](https://arxiv.org/abs/1612.02021))
* [[Bertrand Souères]], _Supergravities in Superspace_, Lyon 2018 ([tel:01998725](https://tel.archives-ouvertes.fr/tel-01998725), [pdf](https://tel.archives-ouvertes.fr/tel-01998725/document))
category: people
[[!redirects Betrand Soueres]] |
Bertrand Toen > history | https://ncatlab.org/nlab/source/Bertrand+Toen+%3E+history | < [[Bertrand Toen]]
[[!redirects Bertrand Toen -- history]] |
Bertrand Toën | https://ncatlab.org/nlab/source/Bertrand+To%C3%ABn | <div style="float: right;margin:0 20px 10px 20px;"><img width = "350" src="http://owpdb.mfo.de/photoNormal?id=7420" alt="Gabriele Vezzosi and Bertrand Toen" /></div>
Bertrand Toën is a mathematician at Université Paul Sabatier in Toulouse.
Together with [[Gabriele Vezzosi]], Bertrand Toën
has laid foundations of what is now called [[derived geometry]].
(The picture [shows](http://owpdb.mfo.de/detail?photo_id=7420) Vezzosi on the left and Toën on the right during a research in pairs stay at [Oberwolfach](http://www.mfo.de/) in 2002).
* [web page](http://www.math.univ-toulouse.fr/~btoen/)
## Selected writings
On [[algebraic stacks]], [[derived algebraic geometry]] and [[rational homotopy theory]]:
* {#Toen} [[Bertrand Toën]], _Champs affines_, Selecta Math. new series **12** (2006), no. 1, 39-135 ([arXiv:math/0012219](https://arxiv.org/abs/math/0012219), [doi:10.1007/s00029-006-0019-z](https://doi.org/10.1007/s00029-006-0019-z))
On [[stacks]] and [[non-abelian cohomology]]:
* {#Toen02} [[Bertrand Toën]], _Stacks and Non-abelian cohomology_, lecture at _[Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory](https://www.msri.org/realvideo/index04.html)_, MSRI 2002 ([slides](http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/index.html), [ps](http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/meta/aux/toen.ps), [pdf](https://perso.math.univ-toulouse.fr/btoen/files/2015/02/msri2002.pdf))
On [[infinity-stacks]] and [[derived stacks]]:
* [[Bertrand Toën]], _Higher and derived stacks: a global overview_, In: _Algebraic Geometry Seattle 2005_, Proceedings of Symposia in Pure Mathematics, Vol. 80.1, AMS 2009 ([arXiv:math/0604504](http://arxiv.org/abs/math/0604504), [doi:10.1090/pspum/080.1](https://doi.org/10.1090/pspum/080.1))
On [[derived algebraic geometry]]:
* {#ToenVezzosi04} [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical algebraic geometry II: geometric stacks and applications_, Memoirs of the AMS **193** (2008) [[arXiv:math/0404373](http://arxiv.org/abs/math/0404373), [ams:memo-193-902](https://bookstore.ams.org/memo-193-902)]
See also
* _Vers une interprétation Galoisienne de la théorie de l’homotopie_. Cahiers de Top. et de Geo. Diff. Cat. 43, No. 4 (2002), 257-312. [pdf](https://perso.math.univ-toulouse.fr/btoen/files/2012/04/Galhom.pdf)
Discussion of [[opposite categories]] of [[commutative monoids in a symmetric monoidal category]] regarded as categories of generalized [[affine schemes]] (and [[Spec(Z)]]):
* [[Bertrand Toën]], [[Michel Vaquié]], *Au-dessous de $Spec \mathbb{Z}$*, Journal of K-Theory **3** 3 (2009) 437-500 [[doi:10.1017/is008004027jkt048](https://doi.org/10.1017/is008004027jkt048)]
## Related entries
* [[∞-stack]], [[derived stack]], [[locally constant ∞-stack]]
* [[model structure on simplicial presheaves]]
* [[model structure on cosimplicial abelian groups]]
* [[simplicial ring]], [[derived algebraic geometry]], [[complicial algebraic geometry]]
* [[nonabelian cohomology]]
* [[higher topos theory]]
* [[dg-categories]], [[derived noncommutative algebraic geometry]]
* [[Hall algebra]]
* [[Northwestern TFT Conference 2009]]
category: people
[[!redirects Bertrand Toen]]
[[!redirects Toen]]
[[!redirects Toën]]
[[!redirects B. Toën]]
[[!redirects B. Toen]] |
Bertrand's postulate | https://ncatlab.org/nlab/source/Bertrand%27s+postulate |
+-- {: .rightHandSide}
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###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
\tableofcontents
\section{Introduction}
_Bertrand's postulate_, first proven by analytic methods by Chebyshev in 1852, is one of the few simple facts providing precise bounds on the occurrence of [[prime numbers]].
\section{Bertrand's postulate}
Given an [[integer]] $i \geq 1$, let $p_{i}$ denote the $i^{th}$ [[prime number|prime]]. _Bertrand's postulate_ is the following.
\begin{thm} \label{TheoremBertrandsPostulate} For any integer $n \geq 1$, we have that $p_{n+1} \lt 2p_{n}$. \end{thm}
\begin{rmk} \label{RemarkAlternativeFormulations} There are a number of equivalent formulations of Theorem \ref{TheoremBertrandsPostulate}. One is that, for any integer $n \gt 3$, there is a prime $p$ such that $n \lt p \lt 2n$. \end{rmk}
\section{Proof given Goldbach's conjecture}
{#ProofFromGoldbach}
Around 2005, it was noticed by Henry J. Ricardo and Yoshihiro Tanaka ([Ricardo 05](#Ricardo05)) that Bertrand's postulate follows from the [[Goldbach conjecture]]. We record the [[proof]] (of the formulation in Remark \ref{RemarkAlternativeFormulations}).
\begin{proof} If the [[Goldbach conjecture]] holds, there are prime numbers $p_{1}$ and $p_{2}$ such that $2n = p_{1} + p_{2}$. We must have that at least one of $p_{1}$ and $p_{2}$ is greater than or equal to $n$. Without loss of generality, suppose that $p_{1}$ has this property.
If $n$ is not prime, then it is immediate that $n \lt p_{1} \lt 2n$, as required. Suppose instead that $n$ is prime. Then $n+1$ is composite, since it must be divisible by $2$.
By Goldbach's conjecture once more, there are prime numbers $p_{1}'$ and $p_{2}'$ such that $2(n+1) = p_{1}' + p_{2}'$. As above, at least one of $p_{1}'$ and $p_{2}'$ must be greater than or equal to $n+1$. Without loss of generality, suppose that $p_{1}'$ has this property.
Then since $n+1$ is not prime, we have that $n+1 \lt p_{1}' \lt 2(n+1)$, and thus that $n \lt p_{1}' \lt 2(n+1)$.
Now, $p_{1}'$ is not equal to $2n+1$, for this would imply that $p_{2}'=1$, which is impossible. Moreover, $p_{1}'$ is not equal to $2n$, since $2n$ is not prime. We deduce that $p_{1}' \lt 2n$, as required.
\end{proof}
\section{References}
* {#Ricardo05} Henry J. Ricardo, _Goldbach's conjecture implies Bertrand's postulate_, Amer. Math. Monthly, 112, pg. 492, 2005
|
Bertrand's theorem | https://ncatlab.org/nlab/source/Bertrand%27s+theorem | Bertrand's theorem in [[classical mechanics]]: the only central forces for which all orbits are closed are $f(r) = a r$ and $f(r) = - b r^{-2}$ where $a,b\gt 0$. In both cases the orbits are ellipses, centered at origin in the first case and with the focus in the center in the second case.
* [wikipedia](http://en.wikipedia.org/wiki/Bertrand%27s_theorem) |
Bessel function | https://ncatlab.org/nlab/source/Bessel+function |
#Contents#
* table of contents
{:toc}
## Idea
(...)
## Properties
### Integral representations
{#IntegralRepresentations}
$$
\label{J0AsIntSinOfxCoshtdt}
J_0(x)
\;=\;
\frac{2}{\pi}
\int_0^\infty
\sin\left(
x \, \cosh(t)
\right)
\,
dt
\phantom{AAAA}
\text{if}
\,\,
x \gt 0
$$
([DLMF](#DLMF) [10.9.9](http://dlmf.nist.gov/10.9#E9))
$$
\label{N0AsIntSinOfxCoshtdt}
N_0(x)
\coloneqq
Y_0(x)
\;=\;
-\frac{2}{\pi}
\int_0^\infty
\cos\left(
x \,\cosh(t)
\right)
\,
dt
\phantom{AAAA}
\text{if}
\,\,
x \gt 0
$$
([DLMF](#DLMF) [10.9.9](http://dlmf.nist.gov/10.9#E9))
$$
\label{K0AsIntSinOfxCoshtdt}
K_0(x)
\;=\;
\int_0^\infty
\cos\left(
x \,\sinh(t)
\right)
\,
dt
\phantom{AAAA}
\text{if}
\,\,
x \gt 0
$$
([DLMF](#DLMF) [10.32.6](http://dlmf.nist.gov/10.32#E6))
## Related concepts
* [[special function]]
* [[causal propagator]], [[Hadamard propagator]]
## References
* Milton Abramowitz, Irene Stegun, sections 9, 10, 11 of_Handbook of mathematical functions_, 1964 ([pdf](http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf))
chapter 9: F. W. J. Olver, _Bessel functions of integer order_ ([pdf](http://www.dam.brown.edu/people/mariom/AM282-01/HANDOUTS/Abramowitz_Stegun_Chapter_9.pdf))
* G. N. Watson, _A treatise on the theory of Bessel functions_, Cambridge University Press 1966 ([web](https://archive.org/details/ATreatiseOnTheTheoryOfBesselFunctions))
* {#DLMF} Digital Library of Mathematical Functions, chapter 10 _[Bessel functions](http://dlmf.nist.gov/10)_
See also
* Wikipedia, _[Bessel function](https://en.wikipedia.org/wiki/Bessel_function)_
[[!redirects Bessel functions]]
[[!redirects Neumann function]]
[[!redirects Neumann functions]]
[[!redirects modified Bessel function]]
[[!redirects modified Bessel functions]]
|
beta decay | https://ncatlab.org/nlab/source/beta+decay |
#Contents#
* table of contents
{:toc}
## Idea
A process caused by the [[weak nuclear force]] by which a [[neutron]] decays into a [[proton]], an [[electron]] and a [[neutrino]].
$$
n^0 \mapsto p^+ + e^- + \overline{\nu}_e
$$
## References
* Wikipedia, _[Beta decay](http://en.wikipedia.org/wiki/Beta_decay)_
[[!redirects weak beta decay]] |
beta function | https://ncatlab.org/nlab/source/beta+function | > This entry is about the concept if [[quantum field theory]]. For the [[Euler beta function]], related to the [[Gamma function]], see there.
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In statistical field theory and in [[perturbative quantum field theory]], what is called the _beta function_ is the [[logarithm|logarithmic]] [[derivative]] of the [[running of the coupling constants]] under [[renormalization group flow]]. See there for more.
## Examples
### For Yang-Mills theory
In ([Metsaev-Tseytlin 88](#MetsaevTseytlin88)) the [[1-loop]] [[beta function]] for pure [[Yang-Mills theory]] was obtained as the point-particle limit of the [[partition function]] of a [[bosonic string|bosonic]] [[open string]] in a Yang-Mills [[background field]]. This provided a theoretical explanation for the observation, made earlier in ([Nepomechie 83](#Nepomechie83)) that when computed via [[dimensional regularization]] then this [[beta function]] coefficient of [[Yang-Mills theory]] vanishes in [[spacetime]] [[dimension]] 26. This of course is the critical dimension of the [[bosonic string]].
For more on this see at _[[worldline formalism]]_
## References
The original informal discussion of beta functions for [[scaling transformations]] is due to
* {#GellMannLow54} [[Murray Gell-Mann]] and F. E. Low, _Quantum Electrodynamics at Small Distances_, Phys. Rev. 95 (5) (1954), 1300–1312 ([pdf](http://www.fafnir.phyast.pitt.edu/py3765/GellManLow.pdf))
there denoted "$\psi$". The notation "$\beta$" is due to
* {#Callan70} [[Curtis Callan]], _Broken Scale Invariance in Scalar Field Theory_, Phys. Rev. D 2, 1541, 1970 ([doi:10.1103/PhysRevD.2.1541](https://doi.org/10.1103/PhysRevD.2.1541))
* {#Symanzik70} [[Kurt Symanzik]], _Small distance behaviour in field theory and power counting_, Communications in Mathematical Physics. 18 (3): 227–246 ([doi:10.1007/BF01649434](https://doi.org/10.1007/BF01649434))
Formulation in the rigorous context of [[causal perturbation theory]]/[[pAQFT]], via the [[main theorem of perturbative renormalization]], is due to
* {#BrunettiDuetschFredenhagen09} [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](https://arxiv.org/abs/0901.2038))
reviewed in
* {#Duetsch18} [[Michael Dütsch]], section 3.5.3 of _[[From classical field theory to perturbative quantum field theory]]_, 2018
Discussion for [[Yang-Mills theory]] includes
* {#Nepomechie83} R.I. Nepomechie, _Remarks on quantized Yang-Mills theory in 26 dimensions_, Physics Letters B Volume 128, Issues 3–4, 25 August 1983, Pages 177-178 Phys. Lett. B128 (1983) 177 (<a href="https://doi.org/10.1016/0370-2693(83)90385-4">doi:10.1016/0370-2693(83)90385-4</a>)
* {#MetsaevTseytlin88} [[Ruslan Metsaev]], [[Arkady Tseytlin]], _On loop corrections to string theory effective actions_, Nuclear Physics B Volume 298, Issue 1, 29 February 1988, Pages 109-132 (<a href="https://doi.org/10.1016/0550-3213(88)90306-9">doi:10.1016/0550-3213(88)90306-9</a>)
[[!redirects beta functions]]
[[!redirects beta-function]]
[[!redirects beta-functions]]
|
beta-gamma system | https://ncatlab.org/nlab/source/beta-gamma+system |
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###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What is called the _$\beta$-$\gamma$ system_ is a 2-dimensional [[quantum field theory]] defined on [[Riemann surfaces]] $X$ whose [[field (physics)|fields]] are pairs consisting of a $(0,0)$-form and a $(1,0)$-form and whose [[equations of motion]] demand these fields to be [[holomorphic differential forms]].
The name results from the traditional symbols for these fields, which are
$$
(\gamma,\beta) \in \Omega^{0,\bullet}(X) \oplus \Omega^{1, \bullet}(X)
\,.
$$
## Definition
We state the definition of the $\beta$-$\gamma$-system as a [[free field theory]] (see there) in [[BV-BRST formalism]], following ([Gwilliam, section 6.1](#Gwilliam)).
We first give the standard variant of the theory, the
* [Abelian massless theory](#AbelianMasslessTheory).
Then we consider the
* [Abelian massive theory](#AbelianMassiveTheory)
* [Holomorphic Chern-Simons theory](#HolomorphicChernSimonsTheory)
### Abelian massless theory
{#AbelianMasslessTheory}
Let $X$ be a [[Riemann surface]].
**[[kinematics]]**
* the [[field bundle]] $E \to X$ is
$$
E \coloneqq \wedge^{0,\bullet}\Gamma(T X) \oplus \wedge^{1,\bullet} \Gamma(T X)
$$
* hence the ([[abelian sheaf|abelian]]) [[sheaf]] of [[local sections]] is
$$
\mathcal{E} = \Omega_X^{0,\bullet} \oplus \Omega_X^{1, \bullet}
\,,
$$
we write
$\mathcal{E}_c \hookrightarrow \Gamma_X(E)$
for the sections of [[compact support]]
* the local pairing
$$
\langle -,-\rangle_{loc} \colon E \otimes E \to Dens_X
$$
with values in the [[density bundle]] is given by [[wedge product]] followed by projection on the $(1,1)$-forms
$$
\langle \gamma_1 + \beta_1, \gamma_2, \beta_2\rangle_{loc}
\coloneqq
(\gamma_1 \wedge \beta_2 + \gamma_2 \wedge \beta_1)_{|(1,1)}
$$
* hence the global pairing
$$
\langle -,-\rangle \colon \mathcal{E}_c \otimes \mathcal{E}_c \to \mathbb{C}
$$
is given by
$$
\langle \gamma_1 + \beta_1, \gamma_2, \beta_2\rangle_{loc}
\coloneqq
\int_{X}\left(\gamma_1 \wedge \beta_2 + \gamma_2 \wedge \beta_1\right)
$$
**[[dynamics]]**
* the [[differential operator]]
$$
Q \colon \mathcal{E} \to \mathcal{E}
$$
is the [[Dolbeault differential]] $\bar \partial$
* hence the [[elliptic complex]] of fields is
$$
(\mathcal{E}, Q) = (\Omega_X^{0,\bullet}\oplus \Omega_X^{1,\bullet}, \bar \partial)
$$
is the [[Dolbeault complex]];
* and hence the [[action functional]]
$$
S \colon \mathcal{E}_c \to \mathcal{C}
$$
is
$$
\begin{aligned}
(\gamma + \beta) & \mapsto
\frac{1}{2}\int_X \langle \gamma+ \beta, \; \bar \partial (\gamma + \beta)\rangle
\\
& = \int_X \beta \wedge \bar \partial \gamma
\end{aligned}
$$
### Abelian massive theory
{#AbelianMassiveTheory}
(...)
### Holomorphic Chern-Simons theory
{#HolomorphicChernSimonsTheory}
...[[holomorphic Chern-Simons theory]]...
## Properties
### Euler-Lagrange equations of motion
The [[equations of motion]] are
$$
\bar \partial \gamma = 0
\;\;,
\;\;
\bar \partial\beta = 0
\,.
$$
## Relation with $\sigma$-models
Consider a [[sigma-model]] $X\hookrightarrow \mathbb{R}^{1,1}$ to the [[target space]] $\mathbb{R}^{1,1}$
$$ S[q,\gamma] = \int_X \partial q \wedge \bar{\partial}\gamma, $$
which has an abelian right-moving Kac-Moody symmetry $q\mapsto q+\lambda$ with $\partial\lambda=0$. We can can consider a theory where this symmetry is promoted to a [[gauge symmetry]], i.e.
$$ S_{\mathrm{gauged}}[q,\gamma] = \int_X \partial_\beta q \wedge \bar{\partial}\gamma, $$
where $\partial_\beta q := \partial q + \beta$ where $\beta\in\Omega^{1,\bullet}(X)$ is the [[connection]].
If we choose the gauge with $q=0$, we obtain the $\beta$-$\gamma$ system with action
$$ S_{\beta\gamma}[\beta,\gamma] = \int_X \beta \wedge \bar{\partial}\gamma. $$
Thus, a $\beta$-$\gamma$ system can be interpreted as a chiral (or Kac-Moody) quotient along a null killing vector of a [[sigma-model]] with target space $\mathbb{R}^{1,1}$ ([LinRoc20](#LinRoc20)).
## Related concepts
* [[pure spinor formalism]]
## References
### General
* [[Nikita Nekrasov]], _Lectures on curved beta-gamma systems, pure spinors, and anomalies_, ([hep-th/0511008](https://arxiv.org/abs/hep-th/0511008))
* [[Anton Zeitlin]], _Beta-gamma systems and the deformations of the BRST operator_, J.Phys. A42:355401 (2009) ([doi](https://doi.org/10.1088/1751-8113/42/35/355401) [arXiv/0904.2234](https://arxiv.org/abs/0904.2234))
Discussion in the context of [[BV-quantization]] and [[factorization algebras]] is in chapter 6 of
* [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis ([pdf](https://people.math.umass.edu/~gwilliam/thesis.pdf))
{#Gwilliam}
A construction of [[chiral differential operator]]s via quantization of $\beta\gamma$ system in [[BV formalism]] with an intermediate step using factorization algebras:
* [[Vassily Gorbounov]], [[Owen Gwilliam]], Brian Williams, _Chiral differential operators via Batalin-Vilkovisky quantization_, [pdf](http://people.mpim-bonn.mpg.de/gwilliam/cdo.pdf)
* {#LinRoc20} [[Ulf Lindstrom]], [[Martin Rocek]], _$\beta$-$\gamma$-systems interacting with sigma-models_, ([arXiv:2004.06544](https://arxiv.org/abs/2004.06544v3))
### As a fractional level or logarithmic CFT
On the beta-gamma system as the [[su(2)|$\mathfrak{su}(2)$]] [[WZW model]] at fractional [[level (Chern-Simons theory)|level]] $-1/2$:
* {#LesageMathieuRasmussenSaleur02} F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, *The $\mathfrak{su}(2)_{-1/2}$ WZW model and the beta-gamma system*, Nucl. Phys. B **647** (2002) 363-403 $[$[arXiv:hep-th/0207201](https://arxiv.org/abs/hep-th/0207201), <a href="https://doi.org/10.1016/S0550-3213(02)00905-7">doi:10.1016/S0550-3213(02)00905-7</a>a$]$
and its lift to a [[logarithmic CFT]]:
* {#LesageMathieuRasmussenSaleur04} F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, *Logarithmic lift of the $\mathfrak{su}(2)_{-1/2}$ model*, Nuclear Physics B **686** 3 (2004) 313-346 [[doi:10.1016/j.nuclphysb.2004.02.039](https://doi.org/10.1016/j.nuclphysb.2004.02.039)]
|
beta-reduction | https://ncatlab.org/nlab/source/beta-reduction |
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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
# $\beta$-reduction
* table of contents
{: toc}
## Idea
In [[type theory]], $\beta$-reduction is a [[conversion rule]] of "computation", which generally replaces more complicated terms with simpler ones. It was originally identified in the [[lambda-calculus]], where it contrasts with $\alpha$-[[alpha-equivalence|equivalence]] and $\eta$-[[eta expansion|expansion]]; this is the version described below for [[function types]]. The analogous reduction for [[inductive types]] may also be known as $\iota$-reduction.
## "Definition"
In its most general form, $\beta$-reduction consists of rules which specify, for any given [[type]] $T$, if we apply an "eliminator" for $T$ to the result of a "constructor" for $T$, how to "evaluate" the result. We write
$$s \to_\beta t$$
if the term $s$ beta-reduces to the term $t$. Sometimes we write $s \to_\beta^* t$ if this reduction takes $n$ steps (leaving off the $*$ to denote $n=1$). The relation "reduces to" generates an [[equivalence relation]] on the set of terms called **beta equivalence** and often denoted $s =_\beta t$ or $s \equiv_\beta t$.
### Function types
The most common (and original) example is when $T$ is a [[function type]] $A \to B$.
In this case, the constructor of $A \to B$ is a *$\lambda$-expression*: given a term $b$ of type $B$ containing a free variable $x$ of type $A$, then $\lambda x.\, b$ is a term of type $A \to B$.
The eliminator of $A \to B$ says that given a term $f$ of type $A \to B$ and a term $a$ of type $A$, we can [[function application|apply]] $f$ to $a$ to obtain a term $f(a)$ of type $B$.
Now if we first construct a term $\lambda x.\, b\colon A \to B$, and then apply *this term* to $a\colon A$, we obtain a term $(\lambda x.\, b)(a)\colon B$. The rule of $\beta$-reduction then tells us that this term *evaluates* or *computes* or *reduces* to $b[a/x]$, the result of [[substitution|substituting]] the term $a$ for the variable $x$ in the term $b$.
See [[lambda calculus]] for more.
### Product types
Although function types are the most publicized notion of $\beta$-reduction, basically all types in type theory have a form of it. For instance, in the negative presentation of a [[product type]] $A \times B$, the constructor is an ordered pair $(a,b)\colon A \times B$, while the eliminators are projections $\pi_1$ and $\pi_2$ which yield elements of $A$ or $B$.
The beta reduction rules then say that if we first apply a constructor $(a,b)$, then apply an eliminator to this, the resulting terms $\pi_1(a,b)$ and $\pi_2(a,b)$ compute to $a$ and $b$ respectively.
## Informal usage
Informally, one sometimes speaks of a "$\beta$-reduction" of a definition or a proof to mean the elimination of levels of abstraction. For instance, if Theorem A is proven by invoking the existence of a green widget, which is proven by Lemma B, then a $\beta$-reduced proof of Theorem A would proceed instead by using the specific green widget constructed in the proof of Lemma B.
It makes some sense to call this $\beta$-reduction because under [[propositions as types]], the proof of Lemma B would be a [[term]] $\lemmab:B$, whereas the proof of Theorem A would be an application $(\lambda x.\theorema)(b)$, where $\theorema$ is the proof of Theorem A using an unspecified green widget $x$. This application $(\lambda x.\theorema)(lemmab)$ can then be literally $\beta$-reduced, in the above sense, to $\theorem[lemmab/x]$, in which the specific green widget constructed in the proof of Lemma B is used instead of the unspecified one $x$.
## Related concepts
* [[lambda-calculus]]
* [[eta-reduction]]
* [[confluent category]] / [[diamond]] / [[Church-Rosser theorem]]
[[!redirects beta reduction]]
[[!redirects beta-reduction]]
[[!redirects ∞-reduction]]
[[!redirects beta conversion]]
[[!redirects beta-conversion]]
[[!redirects ∞-conversion]]
[[!redirects beta rule]]
[[!redirects beta-rule]]
[[!redirects ∞-rule]]
[[!redirects beta equivalent]]
[[!redirects beta-equivalent]]
[[!redirects ∞-equivalent]]
[[!redirects beta equivalence]]
[[!redirects beta-equivalence]]
[[!redirects ∞-equivalence]]
[[!redirects beta reduced]]
[[!redirects beta-reduced]]
[[!redirects ∞-reduced]]
[[!redirects iota reduction]]
[[!redirects iota-reduction]]
[[!redirects ∞-reduction]]
[[!redirects iota equivalent]]
[[!redirects iota-equivalent]]
[[!redirects ∞-equivalent]]
[[!redirects iota equivalence]]
[[!redirects iota-equivalence]]
[[!redirects ∞-equivalence]]
|
beta-ring | https://ncatlab.org/nlab/source/beta-ring | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
* table of contents
{: toc}
## Idea
A **$\beta$-ring** is a [[commutative ring]], $R$, equipped with a set of operations, $\beta_H: R \to R$, indexed by subgroups of [[symmetric groups]], $S_n$, satisfying a number of conditions. They may be seen as collections of integral linear combinations of generalized symmetric powers defined on [[Burnside rings]]. The [[cohomotopy]] of a space, $\pi^0(X)$, is a $\beta$-ring ([Guillot 06, Thrm 4.5](#Guillot06)).
They are completely unrelated to [[relational beta-modules]].
## Related concepts
* [[lambda-ring]]
## References
Note that there are variations in the literature as to the definition of $\beta$-rings. For a close comparison with [[λ-rings]], see
* {#Guillot06} [[Pierre Guillot]], _Adams operations in cohomotopy_ ([arXiv:0612327](http://arxiv.org/abs/math/0612327))
Other references
* Ernesto Vallejo, _The free $\beta$-ring on one generator_, Journal of Pure and Applied Algebra 86(1), 1993, pp. 95-108, ([doi](https://doi.org/10.1016/0022-4049%2893%2990156-N))
* N.W. Rymer, Power operations on the Burnside ring, J. London Math. Sot. (2) 15 (1977) 75-80.
* E. Vallejo, Polynomial operations from Burnside rings to representation functors, J. Pure Appl. Algebra 65 (1990) 163-190.
* G. Ochoa, _Outer plethysm, Burnside rings and $\beta$-rings_, J. Pure Appl. Algebra 55 (1988), 173-195.
* I. Morris and C.D. Wensley, _Adams operations and λ-operations in β-rings_, Discrete Mathematics Volume 50, 1984, Pages 253-270, ([doi](https://doi.org/10.1016/0012-365X%2884%2990053-0))
[[!redirects beta-rings]]
[[!redirects β-rings]]
[[!redirects β-ring]]
|
Beth definability theorem | https://ncatlab.org/nlab/source/Beth+definability+theorem |
##Idea
The _Beth definability theorem_ is a classical result in the [[model theory]] of [[first-order logic]].
It is probably the oldest 'result' in model theory, as it goes back to _Alessandro Padoa_ in 1900. The first proof for the case of predicate logic appeared in Beth (1953) whereas [[Alfred Tarski]] had treated the type-logical case in 1935. Modern proofs rely on the [[Robinson consistency theorem]] or the [[Craig interpolation theorem]].
##Related Concepts
* [[definability]]
* [[first-order logic]]
* [[conceptual completeness]]
* [[Robinson consistency theorem]]
* [[Craig interpolation theorem]]
* [[descent theory]]
##References
* Evert Willem Beth, _On Padoa's method in the theory of definition_ , Indagationes Mathemathicae **15** (1953) pp.30-39.
For an English translation of Padoa's contribution and historical background information consult:
* J. van Heijenoort (ed.), _From Frege to Gödel - A Source Book in Mathematical Logic 1879-1931_ , Harvard UP 1976.
* M. Aiguier, F. Barbier, _An institution-independent Proof of the Beth Definability Theorem_ , Studia Logica **85** no. 3 (2007) pp.333-359. ([pdf](http://perso.ecp.fr/~aiguierm/publications/communications/sl07.pdf))
A categorical generalization that uses methods of [[descent theory]] appears in:
* [[Michael Makkai]], _Duality and Definability in First Order Logic_ , Memoirs AMS no. 503, Providence 1993. |
Betti number | https://ncatlab.org/nlab/source/Betti+number |
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#### Homological algebra
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[[!include homological algebra - contents]]
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=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $n \in \mathbb{Z}$, the $n$-**Betti number** of a [[chain complex]] $V$ (of [[module]]s over a [[ring]] $R$) is the [[rank]]
$$
b_n(V) := rk_R H_n(V)
$$
of its $n$th [[homology group]], regarded as an $R$-[[module]].
For $X$ a [[topological space]], its $n$th Betti number is that of its [[singular homology]]-complex
$$
b_n(V) = rk_R H_n(X, R)
\,.
$$
For $X$ moreover a [[smooth manifold]] then by the [[de Rham theorem]] this is equivalently the dimension of the [[de Rham cohomology]] groups.
## Properties
### Euler characteristic
The alternating sum of all the Betti numbers is -- if it exists -- the [[Euler characteristic]].
## Related concepts
* [[Hodge number]]
* [[etale topos]]
## References
Named after [[Enrico Betti]].
(...)
[[!redirects Betti numbers]]
|
BF-theory | https://ncatlab.org/nlab/source/BF-theory |
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#### Quantum field theory
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#### Physics
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[[!include physicscontents]]
=--
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#Contents#
* table of contents
{:toc}
## Idea
What is called **BF-theory** is a [[topological quantum field theory]] defined by an [[action functional]] $S_{BF}$ on a space of certain [[connection on a bundle|connections]] and [[differential forms]] over a [[4-manifold]] $X$, such that locally on $X$ the [[space of field histories]] is given by
1. [[Lie algebra-valued 1-forms]] $\;A$ with values in some [[Lie algebra]] $\mathfrak{g}_1$ and with [[field strength]]/[[curvature]] 2-form $F_A$;
1. [[differential 2-forms]] $\;B$ with values in some [[Lie algebra]] $\mathfrak{g}_2$,
1. together with
1. a homomorphism $\partial \colon \mathfrak{g}_2 \to \mathfrak{g}_1$
1. an [[invariant polynomial]] $\langle -,- \rangle$
by
$$
S_{BF}
\;\colon\;
(A,B) \;\mapsto\;
\int_X \langle F_A \wedge \partial B\rangle
\,.
$$
There is not much of a proposal in the literature for how to make sense of this expression globally. It has been observed that it looks like the action functional is one on [[∞-Lie algebra-valued forms]] with values in a [[strict Lie 2-algebra]] $\mathfrak{g} = (\mathfrak{g}_2 \stackrel{\partial}{\to} \mathfrak{g}_1)$.
This would suggest that the BF-action functional is to be regarded as a functional on the [[space]] ([[2-groupoid]]) of $G$-[[principal 2-bundles]]
with [[connection on a 2-bundle]], where $G = (G_2 \to G_1)$ is a [[Lie 2-group]] integrating $\mathfrak{g}$.
If one couples to the above action functional that for [[topological Yang-Mills theory]] and a [[cosmological constant]] with coefficients as in
$$
\int_X( \langle F_A \wedge B\rangle - \frac{1}{2} \langle F_A \wedge F_A\rangle - \frac{1}{2}\langle \partial B \wedge \partial B\rangle)
$$
then this is the [[generalized Chern-Simons theory]] action functional induced from the canonical [[Chern-Simons element]] on the [[strict Lie 2-algebra]] $\mathfrak{g}$. See [[Chern-Simons element]] for details.
## Applications
Much of the interest in BF-theory results from the fact that on a 4-dimensional manifold, to some extent the [[Einstein-Hilbert action]] for [[gravity]] may be encoded in BF-theory form. See [[gravity as a BF-theory]].
## References
BF theory was maybe first considered in
* [[Gary Horowitz]], _Exactly soluable diffeomorphism invariant theories_ Commun. Math. Phys. 125, 417-437 (1989)
The observation that the BF-theory action functional looks like it should be read as a functional on a space of [[∞-Lie algebra valued forms]] with values in a [[strict Lie 2-algebra]] possibly appears in print first in <a href="http://arxiv.org/PS_cache/hep-th/pdf/0309/0309173v2.pdf#page=22">section 3.9</a> of
* [[Florian Girelli]], Hendryk Pfeiffer, _Higher gauge theory -- differential versus integral formulation_, [arXiv:hep-th/0309173](http://arxiv.org/abs/hep-th/0309173)
The observation that coupled to [[topological Yang-Mills theory]] it can be read as the [[schreiber:∞-Chern-Simons theory]] action functional on [[connections on 2-bundles]] is in
* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _$L_\infty$-conections_ (<a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#SSSI">web</a>)
{#SSSI}
and a more comprehensive discussion is in section 4.3 of
* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ .
See also
* Aristide Baratin, Florian Girelli, Daniele Oriti, _Diffeomorphisms in group field theories_, Physical Review D, vol. 83, Issue 10, id. 104051, [doi](http://dx.doi.org/10.1103/PhysRevD.83.104051), [arxiv/1101.0590](http://arxiv.org/abs/1101.0590)
There is a more general BFCG action introduced by Girelli, Pfeiffer and Popescu
which has been shown to be a special case of the categorified BF-theory
in
* João Martins, Aleksandar Miković, _Lie crossed modules and gauge-invariant actions for 2-BF theories_, Adv. Theor. Math. Phys. __15__:4 (2011), 913-1199 [euclid](http://projecteuclid.org/euclid.atmp/1339438351)
Relation to [[Einstein gravity]] (in [[first-order formulation of gravity|first-order formulation]]):
* Mariano Celada, Diego González, Merced Montesinos, _BF gravity_ [[arxiv/1610.02020](https://arxiv.org/abs/1610.02020)]
* [[Alberto S. Cattaneo]], [[Leon Menger]], [[Michele Schiavina]], *Gravity with torsion as deformed BF theory* [[arXiv:2310.01877](https://arxiv.org/abs/2310.01877)]
On a version of BF-theory in [[arithmetic]] related to non-[[orientation|orientability]] of [[arithmetic schemes]]:
* [[Magnus Carlson]], [[Minhyong Kim]], _A note on abelian arithmetic BF-theory_, ([arXiv:1911.02236](https://arxiv.org/abs/1911.02236))
[[!redirects BF theory]]
|
BFSS matrix model | https://ncatlab.org/nlab/source/BFSS+matrix+model |
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#### String theory
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[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _BFSS matrix model_ ([Banks-Fischler-Shenker-Susskind 96](#BanksFischlerShenkerSusskind96), [Seiberg 97](#Seiberg97)) is the description of the [[worldline]] dynamics of interacting [[D0-branes]]. In the [[large N limit]] of a large number of [[D0-branes]] this is supposed to encode the [[non-perturbative quantum field theory|strong coupling limit]] of [[type IIA string theory]] known as _[[M-theory]]_ at least in certain corners of its [[moduli space]].
The BFSS model is a limiting case of the _[[BMN matrix model]]_, which improves on some of its shortcomings (see the [Open problems](#OpenProblems) below).
The BFSS matrix model was argued to arise in several seemingly rather different (but apparently secretly equivalent) ways:
1. as the [[worldline]] theory of a large number of [[D0-branes]] in [[type IIA string theory]],
1. as the [[Kaluza-Klein compactification]] of [[10d super Yang-Mills theory]] to 1+0 space dimensions,
1. {#AsM2Regularization} as a certain non-commutative regularization of the [[light-cone gauge quantization]] of the [[Green-Schwarz sigma-model]] for the [[M2-brane]] ([Nicolai-Helling 98](#NicolaiHelling98), [Dasgupta-Nicolai-Plefka 02](#DasguptaNicolaiPlefka02)).
In this picture matrix blocks around the diagonal correspond to blobs of [[membrane]], while off-diagonal matrix elements correspond to thin tubes of membrane connecting these blobs.
<center>
<img src="https://ncatlab.org/nlab/files/MatrixMembrane.jpg" width="500">
</center>
> graphics grabbed from [Dasgupta-Nicolai-Plefka 02](#DasguptaNicolaiPlefka02)
In any case, the BFSS matrix model ends up being a [[quantum mechanics|quantum mechanical]] system whose bosonic degrees of freedom are a set of 9+1 large [[matrices]]. These play the role of would-be [[coordinate functions]] and their [[eigenvalues]] may be interpreted as points in a [[non-commutative geometry|non-commutative]] [[spacetime]] thus defined.
There is also the [[IKKT matrix model]], which takes this one step further by reducing one dimension further down to [[D(-1)-branes]] in [[type IIB string theory]].
See also at _[[membrane matrix model]]_.
## Open problems
{#OpenProblems}
### General
{#OpenProblemsGeneral}
In the 1990s there was much excitement about the BFSS model (and then its cousin, the [[IKKT matrix model]]), as people hoped it might provide a definition of [[M-theory]], whose formulation remains [elusive](M-theory#TheOpenProblem). It is from these times that [[Edward Witten]] changed the original suggestion that "M" is for "magic, mystery and membrane" to the suggestion that it is for "magic, mystery and matrix". (See [Witten's 2014 Kyoto prize speech](M-theory#Witten14), last paragraph.)
However, while the BFSS matrix model clearly sees something M-theoretic, just as clearly it is not the full answer. Notably it needs for its definition an ambient [[asymptotic boundary|asymptotic]] [[Minkowski spacetime]] [[background field|background]], a light cone limit and a peculiar scaling of [[string coupling]] over [[string length]], all of which means that it pertains to a particular corner of a full theory.
From [Nicolai-Helling 1998, p. 2](#NicolaiHelling98):
> Despite the recent excitement, however, we do not think that M(atrix) theory and the $d= 11$ supermembrane in their present incarnation are already the final answer in the search for M-Theory, even though they probably are important pieces of the puzzle. There are still too many ingredients missing that we would expect the final theory to possess. For one thing, we would expect a true theory of quantum gravity to exhibit certain pregeometrical features corresponding to a “dissolution” of space-time and the emergence of some kind of non-commutative geometry at short distances; although the matrix model does achieve that to some extent by replacing commuting coordinates by non-commuting matrices, it seems to us that a still more radical departure from conventional ideas about space and time may be required in order to arrive at a truly background independent formulation (the matrix model “lives” in nine _flat_ transverse dimensions only). Furthermore, there should exist some huge and so far completely hidden symmetries generalizing not only the duality symmetries of extended supergravity and string theory, but also the principles underlying general relativity.
From [Mohammed-Murugan-Nastase 2010, p. 6](#MohammedMuruganNastase10):
> If Matrix theory is to correctly describe M-theory (and its dimensional reduction to type IIA string theory) then it should be able to describe all D−branes in the theory and not just D2−branes. For example, a D4−brane wrapping an $S^4$ was found in [9], following the earlier works of [10, 11], but the solution is not without several unresolved subtleties. In general, finding the complete spectrum of D−branes from Matrix
theory remains a very difficult problem. The D2− and D4−branes already found are reductions to
ten dimensions of M2− and M5−branes, and while they are a minimum necessary for the spectrum of M-theory, they are by no means sufficient. Indeed, we would also need to find a D6−brane, coming from an eleven dimensional KK monopole, and a D8−brane.
Then, even assuming that all the crucial [[generalized (Eilenberg-Steenrod) cohomology|cohomological]] aspects of [[D-brane]] and [[M-brane]] charges (in [[twisted differential K-theory]], [[twisted cohomotopy]] etc.) are secretly encoded in the matrix model, somehow, none of this is manifest, making the matrix model spit out numbers about a conceptually elusive theory in close analogy to how [[lattice QCD]] produces numbers without informing us about the actual conceptual nature of [[confinement|confined]] [[hadron]] physics.
A similar assessment has been given by [[Greg Moore]], from pages 43-44 of his _[[Physical Mathematics and the Future]]_ (2014, [here](Physical+Mathematics+and+the+Future#AGoodStartWasGivenByTheMatrixTheory)):
> A good start $[$with defining M-theory$]$ was given by the Matrix theory approach of Banks, Fischler, Shenker and Susskind. We have every reason to expect that this theory produces the correct scattering amplitudes of modes in the 11-dimensional supergravity multiplet in 11-dimensional Minkowski space - even at energies sufficiently large that black holes should be created. (This latter phenomenon has never been explicitly demonstrated). But Matrix theory is only a beginning and does not give us the whole picture of M-theory. The program ran into increasing technical difficulties when more complicated compactifications were investigated. (For example, compactification on a six-dimensional torus is not very well understood at all. $[$...$]$). Moreover, to my mind, as it has thus far been practiced it has an important flaw: It has not led to much significant new mathematics.
> If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily. But, ultimately, Physical Mathematics must return to this grand issue.
### Ground state
{#ProblemsGroundState}
There are furcher technical open issues, such as the open question whether the theory has a decent ground state the way it needs to have to make sense (see the references [below](#ReferencesGroundStateProblem)).
## Related concepts
[[!include brane matrix models -- content]]
* [[SYK model]]
* [[Myers effect]]
## References
{#References}
### General
First inkling of [[matrix models]] from the [[large N limit]] of [[QCD]]:
* {#EguchiKawai82} [[Tohru Eguchi]], [[Hikaru Kawai]], _Reduction of Dynamical Degrees of Freedom in the Large-$N$ Gauge Theory_, Phys. Rev. Lett. 48, 1063 (1982) ([spire:176459](http://inspirehep.net/record/176459), [doi:10.1103/PhysRevLett.48.1063](https://doi.org/10.1103/PhysRevLett.48.1063))
* A. Gonzalez-Arroyo, M. Okawa, _A twisted model for large $N$ lattice gauge theory_, Physics Letters B Volume 120, Issues 1–3, 6 January 1983, Pages 174-178 (<a href="https://doi.org/10.1016/0370-2693(83)90647-0">doi:10.1016/0370-2693(83)90647-0</a>)
* A. Gonzalez-Arroyo, M. Okawa, _Twisted-Eguchi-Kawai model: A reduced model for large-
$N$ lattice gauge theory_, Phys. Rev. D 27, 2397 (1983) ([doi:10.1103/PhysRevD.27.2397](https://doi.org/10.1103/PhysRevD.27.2397))
The original articles on the BFSS matrix model:
* {#BanksFischlerShenkerSusskind96} [[Tom Banks]], [[Willy Fischler]], [[Stephen Shenker]], [[Leonard Susskind]], _M Theory As A Matrix Model: A Conjecture_, Phys. Rev. D **55** (1997) [[doi:10.1103/PhysRevD.55.5112](https://doi.org/10.1103/PhysRevD.55.5112), [arXiv:hep-th/9610043](http://arxiv.org/abs/hep-th/9610043)]
* [[Leonard Susskind]], _Another Conjecture about M(atrix) Theory_ ([arXiv:hep-th/9704080](https://arxiv.org/abs/hep-th/9704080))
> (argument for [[small N limit|small N]]-validity)
and with more details on the [[discrete light front quantization]] involved:
* {#Sen97} [[Ashoke Sen]], *D0 Branes on $T^n$ and Matrix Theory*, Adv. Theor. Math. Phys. **2** (1998) 51-59 [[arXiv:hep-th/9709220](https://arxiv.org/abs/hep-th/9709220)]
* {#Seiberg97} [[Nathan Seiberg]], *Why is the Matrix Model Correct?*, Phys. Rev. Lett. **79** (1997) 3577-3580 [[arXiv:hep-th/9710009](https://arxiv.org/abs/hep-th/9710009), [doi:10.1103/PhysRevLett.79.3577](https://doi.org/10.1103/PhysRevLett.79.3577)]
* [[Adel Bilal]], *DLCQ of M-Theory as the Light-Like Limit*, Phys. Lett. B **435** (1998) 312-318 [[arXiv:hep-th/9805070](https://arxiv.org/abs/hep-th/9805070), <a href="https://doi.org/10.1016/S0370-2693(98)00811-9">doi:10.1016/S0370-2693(98)00811-9</a>]
In view of [[flat space holography]]:
* [[Leonard Susskind]], *Holography in the flat space limit*, AIP Conf.Proc. 493 (1999) 1, 98-112, ([spire](https://inspirehep.net/literature/482388), [arXiv:hep-th/9901079](https://arxiv.org/abs/hep-th/9901079), [doi:10.1063/1.1301570](https://doi.org/10.1063/1.1301570))
Review:
* [[Adel Bilal]], *M(atrix) Theory : a Pedagogical Introduction*, Fortsch. Phys. **47** (1999) 5-28 [[arXiv:hep-th/9710136](https://arxiv.org/abs/hep-th/9710136), <a href="https://doi.org/10.1002/(SICI)1521-3978(199901)47:1/3%3C5::AID-PROP5%3E3.0.CO;2-B">doi:10.1002/(SICI)1521-3978(199901)47</a>]
* [[Tom Banks]], _Matrix Theory_, Nucl. Phys. Proc. Suppl. 67 (1998) 180-224 [[arXiv:hep-th/9710231](https://arxiv.org/abs/hep-th/9710231), <a href="https://doi.org/10.1016/S0920-5632(98)00130-3">doi:10.1016/S0920-5632(98)00130-3</a>]
* [[Washington Taylor]], _M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory_, Rev. Mod. Phys. **73** (2001) 419-462 [[arXiv:hep-th/0101126](https://arxiv.org/abs/hep-th/0101126)]
* {#Ydri18} [[Badis Ydri]], _Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory_ [[arXiv:1708.00734](https://arxiv.org/abs/1708.00734)), published as: _Matrix Models of String Theory_, IOP 2018 ([ISBN:978-0-7503-1726-9](https://iopscience.iop.org/book/978-0-7503-1726-9)]
Review in the context of the [[holographic principle]]:
* [[Juan Maldacena]], *A simple quantum system that describes a black hole* [[arXiv:2303.11534](https://arxiv.org/abs/2303.11534), <a href="https://inspirehep.net/literature/2644529">inspire:2644529</a>]
A review of further developments:
* [[David Berenstein]], _Classical dynamics and thermalization in holographic matrix models_, talk at Leiden, October (2012) [[pdf](http://www.lorentzcenter.nl/lc/web/2012/514/presentations/Berenstein.pdf)]
See also
* [[Paul Townsend]], _M(embrane) theory on $T^0$_, Nucl. Phys. Proc. Suppl. **68** (1998) 11-16 [[arXiv:hep-th/9708034](http://arxiv.org/abs/hep-th/9708034)]
* [[Chris D. A. Blair]], Johannes Lahnsteiner, [[Niels A. Obers]], Ziqi Yan. *Unification of Decoupling Limits in String and M-theory*. (2023). ([arXiv:2311.10564](https://arxiv.org/abs/2311.10564))
Discussion as a solution to the open problem of defining [[M-theory]] is in
* {#Moore14} [[Gregory Moore]], section 12, p. 43-44 in: _[[Physical Mathematics and the Future]]_, talk at *[Strings 2014](http://physics.princeton.edu/strings2014/)* [[talk slides](http://physics.princeton.edu/strings2014/slides/Moore.pdf), [companion text pdf](http://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf), [[MooreVisionTalk2014.pdf:file]]]
where it says:
> {#AGoodStartWasGivenByTheMatrixTheory} A good start was given by the [[BFSS matrix model|Matrix theory]] approach of [[Tom Banks|Banks]], [[Willy Fischler|Fischler]], Shenker and [[Leonard Susskind|Susskind]]. We have every reason to expect that this theory produces the correct [[scattering amplitudes]] of modes in the [[11-dimensional supergravity]] multiplet in 11-dimensional [[Minkowski space]] - even at energies sufficiently large that [[black holes]] should be created. (This latter phenomenon has never been explicitly demonstrated). But Matrix theory is only a beginning and does not give us the whole picture of [[M-theory]]. The program ran into increasing technical difficulties when more complicated compactifications were investigated. (For example, compactification on a six-dimensional torus is not very well understood at all. $[...]$). Moreover, to my mind, as it has thus far been practiced it has an important flaw: It has not led to much significant new mathematics.
> If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of [[M-theory]] is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily. But, ultimately, Physical Mathematics must return to this grand issue.
Derivation from [[open string field theory]] is discussed in
* [[Taejin Lee]], _Covariant Open String Field Theory on Multiple D$p$-Branes_ ([arXiv:1703.06402](https://arxiv.org/abs/1703.06402))
Relation to the [[6d (2,0)-supersymmetric QFT]]:
* [[Micha Berkooz]], [[Moshe Rozali]], [[Nathan Seiberg]], _Matrix Description of M-theory on $T^3$ and $T^5$_ ([arXiv:hep-th/9704089](http://arxiv.org/abs/hep-th/9704089))
[[!include quantization of M2-brane on Minkowski spacetime to BFSS matrix model -- references]]
See also in relation to the [[ABJM model]]:
* {#MohammedMuruganNastase10} Asadig Mohammed, Jeff Murugan, [[Horatiu Nastase]], _Looking for a Matrix model of ABJM_, Phys. Rev. D82:086004, 2010 ([arXiv:1003.2599](https://arxiv.org/abs/1003.2599))
### Relation to M5-branes
Discussion of [[light cone longitudal]] [[M5-branes]] in the [[BFSS matrix model]] (for [[light cone transversal]] M5-s see at _[[BMN matrix model]]_):
* [[Tom Banks]], [[Nathan Seiberg]], [[Stephen Shenker]], _Branes from Matrices_, Nucl. Phys. B490:91-106, 1997 ([arXiv:hep-th/9612157](https://arxiv.org/abs/hep-th/9612157))
* Judith Castelino, Sangmin Lee, [[Washington Taylor]], _Longitudinal 5-branes as 4-spheres in Matrix theory_, Nucl. Phys. B526:334-350, 1998 ([arXiv:hep-th/9712105](https://arxiv.org/abs/hep-th/9712105))
(introducing the [[fuzzy 4-sphere]])
Discussion of a [[BFSS matrix model|BFSS-like]] [[matrix model]] for [[MK6-branes]]:
* [[Amihay Hanany]], Gilad Lifschytz, _M(atrix) Theory on $T^6$ and a m(atrix) Theory Description of KK Monopoles_, Nucl. Phys. B519:195-213, 1998 ([arXiv:hep-th/9708037](https://arxiv.org/abs/hep-th/9708037))
### Ground state problem
{#ReferencesGroundStateProblem}
There remains the problem of existence of a sensible ground state of the BFSS model.
* [[Bernard de Wit]], M. Luscher, [[Hermann Nicolai]], _The Supermembrane Is Unstable_, Nucl.Phys. B320 (1989) 135-159 ([spire:266584](http://inspirehep.net/record/266584), <a href="https://doi.org/10.1016/0550-3213(89)90214-9">doi:10.1016/0550-3213(89)90214-9</a>)
For a new attempt at solving this problem, and for pointers to previous attempts see
* L. Boulton, M.P. Garcia del Moral, A. Restuccia, _The ground state of the D=11 supermembrane and matrix models on compact regions_, Nuclear Physics B
Volume 910, September 2016, Pages 665-684 ([arXiv:1504.04071](https://arxiv.org/abs/1504.04071))
* L. Boulton, M.P. Garcia del Moral, A. Restuccia, _Measure of the potential valleys of the supermembrane theory_, Physics Letters B
Volume 797, 2019, 134873 ([arXiv:1811.05758](https://arxiv.org/abs/1811.05758))
### Graviton scattering
{#ReferencesGravitonScattering}
Computation of [[graviton]] [[scattering amplitudes]] with the BFSS matrix model:
* [[Katrin Becker]], [[Melanie Becker]], _A Two-Loop Test of M(atrix) Theory_, Nucl.Phys. B506 (1997) 48-60 ([arXiv:hep-th/9705091](https://arxiv.org/abs/hep-th/9705091))
* [[Katrin Becker]], [[Melanie Becker]], [[Joseph Polchinski]], [[Arkady Tseytlin]], _Higher Order Graviton Scattering in M(atrix) Theory_, Phys. Rev. D **56** (1997) 3174-3178 [[arXiv:hep-th/9706072](https://arxiv.org/abs/hep-th/9706072), [doi:10.1103/PhysRevD.56.R3174](https://doi.org/10.1103/PhysRevD.56.R3174)]
* also [Kabat-Taylor 97](#KabatTaylor97)
* M. Fabbrichesi, _Graviton scattering in matrix theory and supergravity_, in: Ceresole A., Kounnas C., [[Dieter Lüst]], [[Stefan Theisen]] (eds.) _Quantum Aspects of Gauge Theories, Supersymmetry and Unification_, Lecture Notes in Physics, vol 525. Springer, Berlin, Heidelberg ([arXiv:hep-th/9811204](https://arxiv.org/abs/hep-th/9811204))
* [[Robert Helling]], [[Jan Plefka]], Marco Serone, Andrew Waldron, *Three-graviton scattering in M-theory*, Nuclear Physics B **559** 1–2 (1999) 184-204 [[arXiv:hep-th/9905183](https://arxiv.org/abs/hep-th/9905183), <a href="https://doi.org/10.1016/S0550-3213(99)00451-4">doi:10.1016/S0550-3213(99)00451-4</a>]
* Robert Echols, _M-theory, supergravity and the matrix model: Graviton scattering and non-renormalization theorems_, PhD thesis, 1999 [[pdf](https://web.calpoly.edu/~rechols/phys403/mythesis.pdf)]
* [[Aidan Herderschee]], [[Juan Maldacena]], *Three Point Amplitudes in Matrix Theory* [[arXiv:2312.12592](https://arxiv.org/abs/2312.12592)]
In relation to the [[soft graviton theorem]]:
* Noah Miller, [[Andrew Strominger]], Adam Tropper, Tianli Wang, *Soft Gravitons in the BFSS Matrix Model* [[arXiv:2208.14547](https://arxiv.org/abs/2208.14547)]
* Adam Tropper, Tianli Wang, *Lorentz Symmetry and IR Structure of The BFSS Matrix Model* [[arXiv:2303.14200](https://arxiv.org/abs/2303.14200)]
* [[Aidan Herderschee]], [[Juan Maldacena]], *Soft Theorems in Matrix Theory* [[arXiv:2312.15111](https://arxiv.org/abs/2312.15111)]
### Black holes
Relation to [[black holes in string theory]]:
* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzschild Black Holes from Matrix Theory_, Phys.Rev.Lett.80:226-229,1998 ([arXiv:hep-th/9709091](https://arxiv.org/abs/hep-th/9709091))
* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzchild Black Holes in Matrix Theory II_, JHEP 9801:008,1998 ([arXiv:hep-th/9711005](https://arxiv.org/abs/hep-th/9711005))
* [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzschild Black Holes in Various Dimensions from Matrix Theory_, Phys.Lett.B416:62-66,1998 ([arXiv:hep-th/9709108](https://arxiv.org/abs/hep-th/9709108))
* Edi Halyo, _Six Dimensional Schwarzschild Black Holes in M(atrix) Theory_ ([arXiv:hep-th/9709225](https://arxiv.org/abs/hep-th/9709225))
* [[Gary Horowitz]], [[Emil Martinec]], _Comments on Black Holes in Matrix Theory_, Phys. Rev. D 57, 4935 (1998) ([arXiv:hep-th/9710217](https://arxiv.org/abs/hep-th/9710217))
* {#KabatTaylor97} Daniel Kabat, [[Washington Taylor]], _Spherical membranes in Matrix theory_, Adv.Theor.Math.Phys.2:181-206,1998 ([arXiv:hep-th/9711078](https://arxiv.org/abs/hep-th/9711078))
* Yoshifumi Hyakutake, _Black Hole and Fuzzy Objects in BFSS Matrix Model_
([arXiv:1801.07869](https://arxiv.org/abs/1801.07869))
* {#DuSahakian18} Haoxing Du, Vatche Sahakian, _Emergent geometry from stochastic dynamics, or Hawking evaporation in M(atrix) theory_ ([arXiv:1812.05020](https://arxiv.org/abs/1812.05020))
(combination with [[random matrix theory]])
### Relation to lattice gauge theory
Relation to [[lattice gauge theory]] and numerical tests of [[AdS/CFT]]:
* {#Joseph15} Anosh Joseph, _Review of Lattice Supersymmetry and Gauge-Gravity Duality_
([arXiv:1509.01440](https://arxiv.org/abs/1509.01440))
* Veselin G. Filev, Denjoe O'Connor, _The BFSS model on the lattice_, JHEP 1605 (2016) 167 ([arXiv:1506.01366](https://arxiv.org/abs/1506.01366))
* {#Hanada16} Masanori Hanada, _What lattice theorists can do for superstring/M-theory_, International Journal of Modern Physics AVol. 31, No. 22, 1643006 (2016) ([arXiv:1604.05421](https://arxiv.org/abs/1604.05421))
* Georg Bergner, Norbert Bodendorfer, Masanori Hanada, Stratos Pateloudis, Enrico Rinaldi, Andreas Schäfer, Pavlos Vranas, [[Hiromasa Watanabe]], *Confinement/deconfinement transition in the D0-brane matrix model -- A signature of M-theory?*, JHEP 05 (2022) 096 [[arXiv:2110.01312](https://arxiv.org/abs/2110.01312)]
### Holography
On [[AdS/CFT]] in the form of [[AdS2/CFT1]] with the [[BFSS matrix model]] on the CFT side and [[black hole in string theory|black hole-like solutions]] in [[type IIA supergravity]] on the AdS side:
* [[Juan Maldacena]], Alexey Milekhin, _To gauge or not to gauge?_, JHEP 04 (2018) 084 ([arxiv:1802.00428](https://arxiv.org/abs/1802.00428))
and concerning the analog of its [[holographic entanglement entropy]]:
* Tarek Anous, Joanna L. Karczmarek, Eric Mintun, [[Mark Van Raamsdonk]], Benson Way, _Areas and entropies in BFSS/gravity duality_ ([arXiv:1911.11145](https://arxiv.org/abs/1911.11145))
[[!redirects BFSS model]]
|
BGG resolution | https://ncatlab.org/nlab/source/BGG+resolution |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _BGG resolution_ is a certain [[projective resolution]] of certain [[representations]].
More in detail, give a finite dimensional semisimple complex [[Lie algebra]] $\mathfrak{g}$ with [[Cartan subalgebra]] $\mathfrak{h}$, and a positive [[root system]], and let $P^+\subset \mathfrak{h}^*$ be the set of dominant integral [[weights]]. Then for every $\lambda\in P^+$ consider the corresponding finite dimensional left $U(\mathfrak{g})$-[[module]] $L(\lambda)$. Certain [[resolutions]] of $L(\lambda)$ are defined in a series of papers of Bernstein, Gel'fand and Gel'fand ([BBG 7x](#BBG7x)) and are now called _BGG resolutions_. There are also generalizations, e.g. for Kac-Moody algebras.
These resolutions have a natural incarnation in terms of [[complexes]] of sections of [[tractor bundles]] over [[flag varieties]] or more generally over [[homogeneous space|homogeneous]] [[parabolic geometry|parabolic]] [[Klein geometries]]. As such, there are generalizations of the construction to more general [[parabolic geometry|parabolic]] [[Cartan geometries]], called _curved BGG sequences_. See for instance ([Calderbank-Diemer 00, theorem 3.6](#CalderbankDiemer00)).
BGG resolutions may be used to construct [[resolutions]] of sheaves of constant functions on [[Klein geometries]]/[[coset space]] $G/H$ that are more efficient (smaller) that the general resolution given by the [[de Rham complex]] (the [[Poincare lemma]]). In this way BGG resolutions are used notably for computation in [[Leray spectral sequences]] as they appear in [[Penrose transforms]] ([Baston-Eastwood 89, chapter 8](#BastonEastwood89)).
## Properties
### Curved $A_\infty$-Structure
Under a [[cup product]] the BGG sequence becomes a [[curved A-infinity algebra]]. ([Calderbank-Diemer 00, section 6](#CalderbankDiemer00))
## Related concepts
* [[Verma module]]
* [[parabolic geometry]]
## References
### Original case over flag variety
* [[eom]]: Alvany Rocha, [BGG resolution](http://www.encyclopediaofmath.org/index.php?title=BGG_resolution&oldid=14028); wikipedia, [category O](http://en.wikipedia.org/wiki/Category_O)
* {#BBG7x} I.N. Bernstein, [[I. M. Gelfand]], S. I. Gelfand, _Structure of representations generated by vectors of highest weight_, Funkts. Anal. Prilozh. __5__: 1 (1971) pp. 1–9; _A certain category of -modules_, Funkts. Anal. Prilozh. __10__: 2 (1976) pp. 1–8; _Differential operators on the base affine space and a study of $\mathfrak{g}$-modules), I.M. Gelfand (ed.), Lie groups and their representations, Proc. Summer School on Group Representations, Janos Bolyai Math. Soc.& Wiley (1975) pp. 39–64
* James E. Humphreys, _Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$_, 2008, [pdf](http://www.math.umass.edu/~jeh/bgg/main.pdf)
* M. Falk, V. Schechtman, A. Varchenko, _BGG resolutions via configuration spaces_, [arxiv/1309.7811](http://arxiv.org/abs/1309.7811)
> We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik-Solomon manifold; it allows us to compute the intersection cohomology of certain flat connections with logarithmic singularities using some Aomoto type complexes of logarithmic forms. Using this construction we realize geometrically the $sl_2$ Bernstein - Gelfand - Gelfand resolution as an Aomoto complex.
* Sergey Arkhipov, _A new construction of the semi-infinite BGG resolution_, [q-alg/9605043](http://arxiv.org/abs/q-alg/9605043)
Discussion in the context of the [[Penrose transform]] includes
* Michael Eastwood, _Variations on the de Rham complex_, Notices Amer. Math. Soc. __46__ (1999), no. 11, 1368–1376 [pdf](http://www.ams.org/notices/199911/fea-eastwood.pdf)
* {#BastonEastwood89} R. J. Baston, M. G. Eastwood, _The Penrose transform_, Oxford Univ. Press, New York, 1989; [MR92j:32112](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=1038279)
### Curved generalization to parabolic Cartan geometries
{#ReferencesCurved}
The generalization from [[coset spaces]] to [[parabolic geometry|parabolic]] [[Cartan geometries]] (curved BGG sequences) is discussed in
* {#Baston90} R. J. Baston, _Verma modules and differential conformal invariants_, J. Diff. Geom. 32 (1990) 851–898.
* [[Andreas Čap]], [[Jan Slovák]], [[Vladimír Souček]], _Bernstein-Gelfand-Gelfand sequences_, ESI preprint 722 (1999).
* [[Andreas Čap]], [[Jan Slovák]], [[Vladimír Souček]], _Bernstein-Gelfand-Gelfand sequences_, Ann. Math. (II) __154__:1 (2001) pp. 97-113 ([jstor](http://www.jstor.org/stable/3062111))
* {#CalderbankDiemer00} [[David Calderbank]], Tammo Diemer, _Differential invariants and curved Bernstein-Gelfand-Gelfand sequences_, J.Reine Angew.Math. 537 (2001) 67-103 ([arXiv:math/0001158](http://arxiv.org/abs/math/0001158))
* {#CapSoucek07} [[Andreas Čap]], [[Vladimír Souček]], _Curved Casimir operators and the BGG machinery_, SIGMA __3__, 2007, 111 ([arxiv:0708.3180](http://arxiv.org/abs/0708.3180) [doi](http://dx.doi.org/10.3842/SIGMA.2007.111))
category: representation theory
[[!redirects BGG resolutions]]
[[!redirects Bernstein-Gelfand-Gelfand resolution]]
[[!redirects BGG sequence]]
[[!redirects BGG sequences]] |
Bharath Sriperumbudur | https://ncatlab.org/nlab/source/Bharath+Sriperumbudur |
* [webpage](http://www.personal.psu.edu/bks18/)
* [webpage](https://science.psu.edu/stat/people/bks18)
## Selected writings
On [[Bochner's theorem]] generalized to [[non-abelian groups]] in the context of [[kernel methods]]:
* [[Kenji Fukumizu]], [[Bharath Sriperumbudur]], [[Arthur Gretton]], [[Bernhard Schölkopf]], *Characteristic Kernels on Groups and Semigroups*, Advances in Neural Information Processing Systems 21 : 22nd Annual Conference on Neural Information Processing Systems 2008 ([NIPS 2008](https://neurips.cc/Conferences/2008/)), 473-480 ([mpg:5466](http://www.is.mpg.de/publications/5466), [pdf](http://www.gatsby.ucl.ac.uk/~gretton/papers/FukSriGreSch09.pdf))
category: people
|
Bhargav Bhatt | https://ncatlab.org/nlab/source/Bhargav+Bhatt |
* [webpage](http://www.math.ias.edu/~bhatt/)
* [wikipedia](https://en.wikipedia.org/wiki/Bhargav_Bhatt_%28mathematician%29)
## Selected writings
On the [[Mordell conjecture]]:
* {#BhattSnowden} [[Bhargav Bhatt]], [[Andrew Snowden]] (org.): _Faltings' proof of the Mordell conjecture_ (seminar notes, 2016) [[pdf](https://web.math.princeton.edu/~takumim/Mordell.pdf), [[BhattSnowden-FaltingsProofofMordell.pdf:file]]]
On [[crystalline cohomology]] and [[de Rham cohomology]]:
* Bhargav Bhatt, [[Aise Johan de Jong]], _Crystalline cohomology and de Rham cohomology_ ([pdf](http://www.math.columbia.edu/~dejong/papers/crystalline-comparison.pdf))
On [[Witt vectors]]:
* Bhargav Bhatt, [[Peter Scholze]], _Projectivity of the Witt vector affine Grassmannian_, Invent. math. 209, 329-423 (2017) [doi](https://doi.org/10.1007/s00222-016-0710-4) [arXiv:1507.06490](https://arxiv.org/abs/1507.06490)
On [[p-adic Hodge theory|$p$-adic Hodge theory]]:
* Bhargav Bhatt, [[Matthew Morrow]], [[Peter Scholze]] *Integral $p$-adic Hodge Theory* [[arXiv:1602.03148](https://arxiv.org/abs/1602.03148)]
and on [[topological Hochschild homology]]:
* Bhargav Bhatt, [[Matthew Morrow]], [[Peter Scholze]], *Topological Hochschild homology and integral $p$-adic Hodge theory* [[arXiv:1802.03261](https://arxiv.org/abs/1802.03261)]
On [[prismatic cohomology]]:
* Bhargav Bhatt, [[Peter Scholze]], _Prisms and prismatic cohomology_, Ann. of Math. (2) 196(3): 1135-1275 [doi](https://10.4007/annals.2022.196.3.5) arXiv:[1905.08229](https://arxiv.org/abs/1905.08229)
* Bhargav Bhatt, [[Jacob Lurie]], _Absolute prismatic cohomology_, arXiv:[2201.06120](https://arxiv.org/abs/2201.06120)
## Related entries:
* [[pro-étale site]]
* [[p-adic Hodge theory]]
category: people
|
BHK interpretation | https://ncatlab.org/nlab/source/BHK+interpretation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
{#Idea}
In [[constructive mathematics]] (originally: [[intuitionistic logic]]), the *BHK interpretation* of [[logical connectives]] (due to [Kolmogorov (1932, p. 59)](#Kolmogorov32), [Troelstra (1969, §2)](#Troelstra69), see the comments on attribution [below](#Attribution)) is such that the resulting [[proposition]] is regarded as [[true]] only if it is possible to [[constructive mathematics|construct]] a [[proof]] of its [[assertion]].
For instance, to assert a [[logical conjunction]] ("and") or a [[universal quantification]] ("for all") is taken to mean to provide a proof of all the instances.
Dually but more notably, to assert a [[logical disjunction]] ("or") or an [[existential quantification]] ("exists") is taken to mean to prove one of the instances, so that there is no intuitionistic existence statement without construction of an example (the "disjunction property", see [there](intuitionistic+logic#DisjunctionProperty)).
This constructive interpretation of logical truth is the crux of the rejection of the [[principle of excluded middle]] in [[intuitionism]]/[[constructive mathematics]], for it implies that to prove $P \vee (\not P)$ (which may superficially/classically seem tautologous) one must prove $P$ or one must prove $\not P$ --- but neither proof may be known (e.g. if $P$ = [[Riemann hypothesis]]).
(Here the classical mathematician is regarded as "idealistic" in their assumption that either case must hold, even if it is impossible to tell which one.)
In short this means that a proposition is regarded a [[true]] if there is an [[algorithm]], hence a [[computable function]], to realize its [[proof]] whence one also speaks of the *[[realizability]] interpretation*.
Closely related to the point of being synonymous is the paradigm of *[[propositions as types]]* and *[[proofs as programs]]*, also known as the *[[Curry-Howard correspondence]]* in [[type theory]].
Indeed, the fully formal version of the BHK interpretation may be understood as being the [[inference rules]], specifically the [[term introduction rules]], of [[intuitionistic type theory]] (as amplified in [Girard (1989, §2)](#Girard89) and [Martin-Löf (1996, Lec 3)](#MartinLöf96)).
## Historical versions
{#HistoricalVersions}
The BHK interpretation evolved from a rather informal statement in [Kolmogorov (1932, p. 59)](#Kolmogorov32), over a more pronounced but still informal statement due to [Troelstra (1969, p. 5)](#Troelstra69), to the modern [[inference rules]] in [[intuitionistic type theory]] due to [Martin-Löf 75](Martin-Löf+dependent+type+theory#MartinLof75) (gentle exposition in [Martin-Löf (1996, Lec 3)](#MartinLöf96)):
\linebreak
From [Kolmogorov (1932, p. 59)](#Kolmogorov32):
\begin{imagefromfile}
"file_name": "KolmogorovIntroducingBHK.jpg",
"width": 540,
"unit": "px",
"margin": {
"top": -30,
"bottom": 20,
"right": 0,
"left": 10
}
\end{imagefromfile}
From [Heyting (1956, p. 97)](#Heyting56):
\begin{imagefromfile}
"file_name": "HeytingIntroducingBHK.jpg",
"width": 500,
"unit": "px",
"margin": {
"top": -30,
"bottom": 20,
"right": 0,
"left": 10
}
\end{imagefromfile}
From [Troelstra (1969, p. 5)](#Troelstra69):
\begin{imagefromfile}
"file_name": "Troelstra-IntroducingBHKInterp.jpg",
"width": 660,
"unit": "px",
"margin": {
"top": -30,
"bottom": 20,
"right": 0,
"left": 10
}
\end{imagefromfile}
From [Troelstra (1977, p. 977)](#Troelstra77):
\begin{imagefromfile}
"file_name": "Troelstra-AttributingBHK.jpg",
"width": 500,
"unit": "px",
"margin": {
"top": -30,
"bottom": 20,
"right": 0,
"left": 10
}
\end{imagefromfile}
{#FromTroelstraVanDalen88} From [Troelstra & van Dalen (1988)](#TroelstraVanDalen88):
\begin{imagefromfile}
"file_name": "Troelstra-VanDalen-BHKInterpretation.jpg",
"width": 500,
"unit": "px",
"margin": {
"top": -30,
"bottom": 20,
"right": 0,
"left": 10
}
\end{imagefromfile}
{#FromBridges} From [Bridges (1999), p. 96](#Bridges99):
\begin{imagefromfile}
"file_name": "Bridges-IntuitInterpOfConnectives.jpg",
"width": 600,
"unit": "px",
"margin": {
"top": -30,
"bottom": 20,
"right": 0,
"left": 10
}
\end{imagefromfile}
The analogous discussion for [[inference rules]] in [[intuitionistic type theory]] is then given in [Martin-Löf (1975)](#MartinLof75),
reviewed in [Girard (1989, §2)](#Girard89) and with more emphasis in [Martin-Löf (1996, Lec 3)](#MartinLöf96).
## Attribution
{#Attribution}
The idea of the interpretation is clearly expressed in [Kolmogorov (1932, p. 59)](#Kolmogorov32), though rather briefly and in unusual terminology: Instead of *propositions*, Kolmogorov speaks of *Aufgaben* (Deutsch for "tasks", but here in the sense used in math classes where it means "exercises" or "mathematical problems") in order to amplify the [[constructive mathematics|constructive]] aspects (he expands it out to *Konstruktionsaufgabe*, meaning "construction task").
While [[Arend Heyting]] may have been speaking of the idea of the "proof interpretation" of propositions since the 1930s, the first recognizable statement of what later was called the BHK interpretation is given in [Heyting (1956, §7.1.1)](#Heyting56).
The statement of the interpretation close to its modern formulation (but clearly of the same conceptual content as previously expressed in [Kolmogorov (1932, p. 59)](#Kolmogorov32) and [Heyting (1956, §7.1.1)](#Heyting56)) was then made explicit in [Troelstra (1969, §2)](#Troelstra69), where it is not attributed to anyone but presented as if the author's invention.
Later, [Troelstra (1977, §2)](#Troelstra77) credits the idea to "Brouwer-Heyting-*Kreisel* (BHK)", not mentioning Kolmogorov --- indeed [Kreisel (1965)](#Kreisel65) is the only reference that [Troelstra (1969, §2)](#Troelstra69) makes explicit use of.
(While this is the origin of the term "BHK interpretation", the adjustment/correction of its expanded version by exchanging "*K*reisel" for "*K*olmogorov" seems to be due to [Troelstra (1990, §2.1)](#Troelstra90).)
Still later, [Girard (1989, §1.2.2)](#Girard89) presents the same idea as "Heyting's semantics" (not mentioning anyone else, nor actually citing Heyting).
More discussion of this history is in *[SEP: "The Development of Intuitionistic Logic"](https://plato.stanford.edu/entries/intuitionistic-logic-development/)*.
Notice that Brouwer never explicitly formulated any interpretation of the kind that [Troelstra (1977, §2)](#Troelstra77) attributes to him, and in fact remained against all formalism his entire life. (His role in this history is to provide motivation and inspiration for Heyting and Kolmogorov.) Moreover, [Escardo & Xu (2015)](#EscardoXu15) have shown that Brouwer's famous intuitionistic theorem "all functions $\mathbb{N}^{\mathbb{N}} \to \mathbb{N}$ are continuous" is actually inconsistent under a literal version of this interpretation (i.e. without including [[propositional truncation]]).
Thus, perhaps it should be called the "Heyting-Kolmogorov" interpretation.
## Related concepts
* [[realizability topos]]
* [[computational trinitarianism]]
## References
Original articles on [[intuitionism]] and, eventually, on [[intuitionistic logic]]:
* [[Arend Heyting]], *Die intuitionistische Grundlegung der Mathematik*, Erkenntnis **2** (1931) 106-115 [[jsotr:20011630](https://www.jstor.org/stable/20011630), [pdf](http://www.psiquadrat.de/downloads/heyting1931.pdf)]
* {#Kolmogorov32} [[Andrey Kolmogorov]], *Zur Deutung der intuitionistischen Logik*, Math. Z. **35** (1932) 58-65 [[doi:10.1007/BF01186549](https://link.springer.com/article/10.1007/BF01186549)]
* [[Hans Freudenthal]], *Zur intuitionistischen Deutung logischer Formeln*, Comp. Math. **4** (1937) 112-116 [[numdam:CM_1937__4__112_0](http://www.numdam.org/item/?id=CM_1937__4__112_0)]
* [[Arend Heyting]], *Bemerkungen zu dem Aufsatz von Herrn Freudenthal "Zur intuitionistischen Deutung logischer Formeln"*, Comp. Math. **4** (1937) 117-118 [[doi:CM_1937__4__117_0](http://www.numdam.org/item/?id=CM_1937__4__117_0)]
* [[L. E. J. Brouwer]], *Points and Spaces*, Canadian Journal of Mathematics **6** (1954) 1-17 [[doi:10.4153/CJM-1954-001-9](https://doi.org/10.4153/CJM-1954-001-9)]
Early monographs:
* {#Heyting56} [[Arend Heyting]], *Intuitionism: An introduction*, Studies in Logic and the Foundations of Mathematics, North-Holland (1956, 1971) [[ISBN:978-0720422399]()]
* {#Kreisel65} [[Georg Kreisel]], Section 2 of: *Mathematical Logic*, in T. Saaty et al. (ed.), *Lectures on Modern Mathematics III*, Wiley New York (1965) 95-195
* {#TroelstraVanDalen88} [[Anne Sjerp Troelstra]], [[Dirk van Dalen]], Section 3.1 of: *Constructivism in Mathematics -- An introduction*, Vol 1, Studies in Logic and the Foundations of Mathematics **121**, North Holland (1988) [[ISBN:9780444702661](https://www.elsevier.com/books/constructivism-in-mathematics-vol-1/troelstra/978-0-444-70266-1)]
Early historical review:
* {#Troelstra90} [[Anne Sjerp Troelstra]], *On the Early History of Intuitionistic Logic*, in *Mathematical Logic*, Springer, (1990) [[doi:10.1007/978-1-4613-0609-2_1](https://doi.org/10.1007/978-1-4613-0609-2_1)]
Recognizable formulation of the so-called "BHK interretation" first appears in:
*
[Kolmogorov (1932, p. 59)](#Kolmogorov32)
(who however speaks not of propositions but of *Aufgaben*, i.e. "tasks", here in the sense of: "mathematical problems")
* [Heyting (1956, §7.1.1)](#Heyting56)
(who is maybe the first to speak of the "meaning of logical connectives") and then
* {#Troelstra69} [[Anne Sjerp Troelstra]], §2 of: *Principles of Intuitionism*, Lecture Notes in Mathematics **95** Springer Heidelberg (1969) [[doi:10.1007/BFb0080643](https://link.springer.com/book/10.1007/BFb0080643)]
(where it is presented as if the author's invention) and then recalled in
* {#Troelstra77} [[Anne Sjerp Troelstra]], §2 of: *Aspects of Constructive Mathematics*, Studies in Logic and the Foundations of Mathematics **90** 973-1052 (1977) [<a href="https://doi.org/10.1016/S0049-237X(08)71127-3">doi:10.1016/S0049-237X(08)71127-3</a>]
(where the same is now called the "Brouwer-Heyting-*Kreisel (BHK)* explanation" --- not mentioning Kolmogorov).
and later recalled in the context of [[constructive analysis]]:
* [[Douglas Bridges]], p. 96 of: *Constructive mathematics: a foundation for computable analysis*, Theoretical Computer Science **219** 1–2 (1999) 95-109 [<a href="https://doi.org/10.1016/S0304-3975(98)00285-0">doi:10.1016/S0304-3975(98)00285-0</a>]
This lead to the formulation of [[intuitionistic type theory]] in
* {#MartinLof75} [[Per Martin-Löf]], _An intuitionistic theory of types: predicative part_, in: H. E. Rose, J. C. Shepherdson (eds.), *Logic Colloquium '73, Proceedings of the Logic Colloquium*, Studies in Logic and the Foundations of Mathematics **80**, Elsevier (1975) 73-118 [<a href="https://doi.org/10.1016/S0049-237X(08)71945-1">doi:10.1016/S0049-237X(08)71945-1</a>, [CiteSeer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926)]
reviewed in
* {#Girard89} [[Jean-Yves Girard]] (translated and with appendiced by [[Paul Taylor]] and [[Yves Lafont]]), §1.2.2 of: *Proofs and Types*, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), [webpage](http://www.paultaylor.eu/stable/Proofs+Types.html), [pdf](https://www.paultaylor.eu/stable/prot.pdf)]
(where it is called *Heyting semantics*) and then in
* {#MartinLöf96} [[Per Martin-Löf]], Lecture 3 of: *On the Meanings of the Logical Constants and the Justifications of the Logical Laws*, Nordic Journal of Philosophical Logic, **1** 1 (1996) 11-60 [[pdf](http://docenti.lett.unisi.it/files/4/1/1/6/martinlof4.pdf), [[MartinLofOnTheMeaning96.pdf:file]]]
See also:
* Wikipedia, _[BHK interpretation](http://en.wikipedia.org/wiki/BHK_interpretation)_
* Stanford Encyclopedia of Philosophy, *[The Development of Intuitionistic Logic](https://plato.stanford.edu/entries/intuitionistic-logic-development/#ProoInte)*
* Wouter Pieter Stekelenburg, _Realizability Categories_, ([arXiv:1301.2134](http://arxiv.org/abs/1301.2134)).
* {#EscardoXu15} [[Martin Escardo]], Chuangjie Xu, _The inconsistency of a Brouwerian continuity principle with the Curry--Howard interpretation_, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015), Leibniz International Proceedings in Informatics (LIPIcs) *38* (2015) [[doi:10.4230/LIPIcs.TLCA.2015.153](https://doi.org/10.4230/LIPIcs.TLCA.2015.153), [pdf](http://www.cs.bham.ac.uk/%7Emhe/papers/escardo-xu-inconsistency-continuity.pdf)]
* E. G. F. Díez, _Five observations concerning the intended meaning of the intuitionistic logical constants_ , J. Phil. Logic **29** no. 4 (2000) pp.409–424 . ([preprint](https://webs.um.es/picazo/miwiki/lib/exe/fetch.php?id=inicio&cache=cache&media=five_observations_2000_jour_of_phil_logic_29_4_pp409_424.pdf))
Links to many papers on realizability and related topics may be found [here](http://www.staff.science.uu.nl/~ooste110/realizability.html).
For a comment see also
* [[Robert Harper]], _Extensionality, Intensionality, and Brouwer's Dictum_ ([web](http://existentialtype.wordpress.com/2012/08/11/extensionality-intensionality-and-brouwers-dictum/))
[[!redirects Brouwer-Heyting-Kolmogorov interpretation]]
[[!redirects Brouwer Heyting Kolmogorov interpretation]]
[[!redirects Brouwer-Heyting-Kolmogorov interpretation]]
[[!redirects Brouwer–Heyting–Kolmogorov interpretation]]
[[!redirects BHK interpretation]]
[[!redirects realizability interpretation]]
[[!redirects realisability interpretation]]
[[!redirects BHK correspondence]] |
bi- | https://ncatlab.org/nlab/source/bi- | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--
# bi-
* table of contents
{: toc}
## Meanings
In [[category theory]],
the prefix "bi-" is used in several ways with different (and not always compatible) meanings.
### Weak 2-dimensional
Here, "bi-" is used to denote a weak 2-dimensional analogue of a 1-categorical concept.
Examples:
- [[bicategory]]
- [[biadjunction]]
- [[bilimit]]
In some cases, the prefix "pseudo-" may be used instead, such as [[pseudoadjunction]] rather than [[biadjunction]]. However, be warned that in some cases, these (unfortunately) refer to different concepts: e.g. a [[pseudolimit]] is a [[bilimit]], but the converse is not always true.
### X and also co-X
Here, "bi-X" is used to denote a structure that is equipped with both X-[[structure]] and [[formal duality|dual]] co-X-structure (for some definition X) that *do not necessarily coincide*.
Examples:
- [[bicartesian category]]
- [[bireflective subcategory]]
- [[bifibrant objects]]
- [[bifibration]]
An unambiguous alternative is to simply be explicit about both structures being assumed.
### Compatible X and co-X
Here, "bi-X" is used to denote a structure that is equipped with X-structure and co-X structure (not coincident) that verify some compatibility.
Examples:
- [[bimonoid]]
- [[Hopf monad|bimonad]]
In many cases, these concepts have different, unambiguous names (such as [[Hopf monad]]) that may be used instead.
### Coincident X and co-X
Here, "bi-X" is used to denote a structure that is equipped with X-structure and co-X-structure that coincides. This is a particularly common special case of the meaning above.
Examples:
- [[bilimit]]
- [[biproduct]]
An alternative, unambiguous prefix that can be used for this meaning is "ambi-".
### Left-X and also right-X
In some settings without symmetry, "bi-X" is occasionally used to denote having X-structure on the left and also X-structure on the right (not coincident).
Examples:
- **biclosed**
- [[bimodule]]
- [[bicomodule]]
In such settings, it is unambiguous to simply elide the prefix "bi-". E.g. one can simply write "[[closed category|closed]]" to mean left- and right-closed.
Note that this is not the same as "X and co-X" (except in the [[delooping]]): a "biclosed category" with this terminology would be one that is [[closed category|closed]] and [[coclosed category|coclosed]].
Some compatibility may be required between the left and right X-structures.
### Two compatible X structures
A less common usage is the prefix "bi-" to mean two structures of the same kind that interact in some way.
Examples:
- **bimonoidal category** (i.e. [[rig category]])
However, these examples tend to have alternative, more descriptive terminology.
## Related pages
- [[locally]]
category: disambiguation
|
bi-brane | https://ncatlab.org/nlab/source/bi-brane |
#Contents#
* table of contents
{:toc}
## Idea
A _bi-brane_ is to a [[QFT with defects|defect]] in a [[FQFT]] as a [[brane]] is to a boundary condition.
The term "bi-brane" was apparently introduced in
* Jürgen Fuchs, Christoph Schweigert, Konrad Waldorf, _Bi-branes: Target Space Geometry for World Sheet topological Defects_ ([arXiv](http://arxiv.org/abs/hep-th/0703145))
The description below approaches the concept in a slightly more abstract context.
The notion of [[brane]] and [[bi-brane]] can be made very abstract, but to get the main idea it is useful to start with considering what is usually called the _geometric case_.
Recall for instance from
* Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard J. Szabo, _D-Branes, RR-Fields and Duality on Noncommutative Manifolds_ ([arXiv](http://arxiv.org/abs/hep-th/0607020), [blog](http://golem.ph.utexas.edu/string/archives/000879.html))
that a geometric _brane_ on some space $X$ and a bundle gerbe $\mathcal{G}$, regarded as a special kind of 2-vector bundle, on $X$ is
* a map $\iota : Q \to X$
* a morphism $\sigma : 1 \to \iota^* \mathcal{G}$
from the trivial 2-vector bundle on $Q$
into the pullback of $\mathcal{G}$ to $Q$ --
this morphism is called a _gerbe module_ or _twisted vector bundle_.
If we write this more diagrammatically using the classifying (fiber-assigning) cocycle $g : X \to 2 Vect$ of $\mathcal{G}$, then this data of a brane is
a transformation
$$
\array{
&& Q
\\
& \swarrow && \searrow^{\iota}
\\
pt &&\stackrel{\sigma}{\Rightarrow}&& X
\\
& \searrow && \swarrow_{g}
\\
&& 2 Vect
}
\,.
$$
Conceived in this form the notion has an obvious generalizations:
let $X$ and $Y$ be two possibly different spaces with two possibly different 2-vector bundles on them, classified by cocycles $g_1$ and $g_2$, then a bi-brane for this situation is
* a span $ x \stackrel{\iota_1}{\leftarrow} Q
\stackrel{\iota_2}{\rightarrow} Y$;
* and a transformation between the two pulled back bundles
$$
\array{
&& Q
\\
& {}^{\iota_1}\swarrow && \searrow^{\iota_2}
\\
X &&\stackrel{\bibrane}{\Rightarrow}&& Y
\\
& {}_{g_1}\searrow &&& \swarrow_{g_2}
\\
&& 2 Vect
}
\,.
$$
The description of branes in the above diagrammatic form was first given in
* Urs Schreiber, [[quant2states.pdf:file]], talk at Fields Institute workshop _Higher categories and their applications_ (2007)
and described in more detail in
* Urs Schreiber, Konrad Waldorf, _Connections on Nonabelian Gerbes and their Holonomy_ ([arXiv](http://arxiv.org/abs/0808.1923) [blog](http://golem.ph.utexas.edu/category/2008/08/connections_on_nonabelian_gerb.html)).
The generalization to bi-branes is developed at
* [[schreiber:Nonabelian cocycles and their quantum symmetries]].
This is very closely related to the spans appearing in
* [[geometric function theory]].
The relation is discussed a bit at [this blog entry](http://golem.ph.utexas.edu/category/2009/01/benzvi_on_geometric_function_t.html#c021321).
At least some aspects of the concept have more or less implicitly been considered before, notably in the context of [[topological T-duality]]. A translation of the construction in topological T-duality to the above diagrammatic formulation was originally given [here](http://golem.ph.utexas.edu/category/2007/02/qft_of_charged_nparticle_tdual.html).
The interpretation of T-duality in terms of bi-branes is discussed in more detail in
* Gor Sarkissian, Christoph Schweigert, _Some remarks on defects and T-duality_ ([arXiv](http://arxiv.org/abs/0810.3159))
##Bi-branes motivated from 2d CFT##
[[2-spectral triple|Recall]] that from a 2-dimensional [[conformal field theory|CFT]] one induces a (generalized) _target space_ geometry in generalization of how a [[spectral triple]] induces such a generalized geometry.
From category-algebraic considerations one obtains [[defect line]]s in 2-d [[conformal field theory|CFT]], which are encoded by bimodules as [brane]s are encoded by modules. **Bi-branes** are the answer to the question: "What is the target space structure corresponding to defect lines in the 2d CFT"?
For certain 2-d CFTs based on current algebras the bi-branes corresponding to certain defect lines in these theories have been introduced and discussed in
* Jürgen Fuchs, Christoph Schweigert, Konrad Waldorf, _Bi-branes: Target Space Geometry for World Sheet topological Defects_ ([arXiv](http://arxiv.org/abs/hep-th/0703145))
##In WZW theories##
In the above article it is found that, just as symmetric conformal branes in WZW models, whose target space is a Lie group $G$, correspond to
submanifolds of $G$ given by conjugacy classes in $G$, bi-branes in WZW model correspond to [[span]]s or _correspondences_
$$
\array{
&& B
\\
& \swarrow &&\searrow
\\
G &&&& G
}
$$
where $B$ is a [[higher group character|biconjugacy class]] of $G$.
## Related concepts
* [[motivic quantization]]
[[!include field theory with boundaries and defects - table]]
## References
* [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_
[[!redirects bi-branes]] |
bi-branes > history | https://ncatlab.org/nlab/source/bi-branes+%3E+history | < [[bi-branes]]
[[!redirects bi-branes -- history]] |
bi-Heyting topos | https://ncatlab.org/nlab/source/bi-Heyting+topos | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
# Contents
* automatic table of contents goes here
{:toc}
[[!redirects bi-Heyting toposes]]
[[!redirects biHeyting topos]]
[[!redirects Bi-Heyting topos]]
[[!redirects Bi-Heyting Topos]]
[[!redirects bi-Heyting Topos]]
[[!redirects bi-heyting topos]]
##Idea
A **bi-Heyting topos** is a [[topos]] whose lattices of subobjects carry the structure of a [[co-Heyting algebra]] in addition to their normal [[Heyting algebra]] structure and thereby permit to model bi-intuitionistic logic.
##Definition
A [[topos]] $\mathcal{E}$ such that the lattice $sub(X)$ of subobjects is a [[co-Heyting algebra|bi-Heyting algebra]] for every object $X\in\mathcal{E}$ is called a [[bi-Heyting topos]].
## Examples
* [[Boolean toposes]] are bi-Heyting since their subobject lattices are Boolean algebras which are self-dual Heyting algebras.
* Presheaf toposes and their [[essential subtoposes]].
##Properties
+-- {: .num_prop}
###### Proposition
A Grothendieck topos is bi-Heyting when finite unions distribute over arbitrary intersections:
$$
S\cup (\bigcap_{i\in I} T_i)=\bigcap_{i\in I} (S\cup T_i)\quad .
$$
=--
**Proof.** Because this distributivity amounts to the preservation of limits by $T\vee$\_ and implies by the [[adjoint functor theorem]] the existence of a left adjoint \_$\backslash T$ which provides the [[co-Heyting algebra]] structure. $\qed$
+-- {: .num_prop}
###### Proposition
Let $\mathcal{E}$ be a Grothendieck topos with generating set $\{G_i | i\in I\}$. Then $\mathcal{E}$ is bi- Heyting precisely iff the lattice of subobjects of $G_i$ is a bi-Heyting algebra for all $G_i$.
=--
cf. [Borceux-Bourn-Johnstone](#BBJ04) p.350.
+-- {: .num_prop #surjective_essential}
###### Proposition
Let $f:\mathcal{E}\to\mathcal{F}$ be a surjective [[essential geometric morphism]] between Grothendieck toposes. If $\mathcal{E}$ is bi-Heyting so is $\mathcal{F}$.
=--
cf. [Borceux-Bourn-Johnstone](#BBJ04) p.351.
+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ be a small category. Then $Set^{\mathcal{C}^{op}}$ is bi-Heyting.
=--
**Proof.** Let $|\mathcal{C}|$ be the discrete category on the objects of $\mathcal{C}$. The inclusion $i:|\mathcal{C}|\hookrightarrow \mathcal{C}$ induces a surjective essential geometric morphism $Set^{|\mathcal{C}|^{op}}\to Set^{\mathcal{C}^{op}}$ and the first of these is Booelan hence bi-Heyting. By prop. \ref{surjective_essential} follows the claim. $\qed$
##Remark
* In bi-Heyting toposes the co-Heyting algebra operations are generally not preserved by [[inverse image functor|inverse image functors]], so that the co-Heyting logical operators like [[co-Heyting negation]] or [[co-Heyting boundary]] are subject to _[[de re and de dicto]]_ effects.
##Related entries
* [[co-Heyting algebra]]
* [[co-Heyting negation]]
* [[co-Heyting boundary]]
* [[bitopological space]]
* [[semi-abelian category]]
* [[mereology]]
##References
Bi-Heyting toposes are explicitly defined in
* {#BBJ04} [[Francis Borceux]], [[Dominique Bourn]], [[Peter Johnstone]], _Initial Normal Covers in Bi-Heyting Toposes_, Arch. Math. **42** (2006) pp.335-356. ([pdf](https://www.emis.de/journals/AM/06-4/johnston.pdf))
But their logical possibilities were exploited well before this e.g. in the work of Lawvere, Reyes and collaborators:
* {#Law86} [[William Lawvere]], _Introduction_ , pp.1-16 in Lawvere, Schanuel (eds.), _Categories in Continuum Physics_, Springer LNM **1174** 1986.
* {#Law91a} [[William Lawvere]], _Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes_ , pp.279-281 in A. Carboni, M. Pedicchio, G. Rosolini (eds.) , _[[Como|Category Theory - Proceedings of the International Conference held in Como 1990]]_, LNM **1488** Springer Heidelberg 1991.
* {#Reyes}[[Gonzalo E. Reyes]], Houman Zolfaghari, _Bi-Heyting Algebras, Toposes and Modalities_, J. Phi. Logic **25** (1996) pp.25-43.
|
bi-infinite sequence | https://ncatlab.org/nlab/source/bi-infinite+sequence |
#Contents#
* table of contents
{:toc}
## Definition
A **bi-infinite sequence** or **doubly infinite sequence** in a [[type]] $A$ is a [[function]] $f:\mathbb{Z} \to A$ from the [[integers]] to $A$.
## See also
* [[integers]]
* [[sequence]]
* [[doubly infinite series]]
* [[sequential spectrum type]]
## References
* Wikipedia, [Bi-infinite sequence](https://en.wikipedia.org/wiki/Bi-infinite_sequence)
[[!redirects doubly infinite sequence]] |
bi-initial object | https://ncatlab.org/nlab/source/bi-initial+object |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Bi-initial objects are the [[bicategory|bicategorical]] analogues of [[initial object|initial objects]] in categories.
## Definition
In a [[bicategory]] $A$, an [[object]] $L \in A$ is **bi-initial** (or **biinitial**) when for all $X \in A$, there is an [[equivalence of categories]] between $A(L,X)$ and the [[terminal category]] with a single object and single morphism $\mathbb{1}$.
## Related concepts
* [[initial object]]
* [[bi-terminal object]]
* [[initial object in an (infinity,1)-category|initial object in an $(\infty,1)$-category]]
## References
This concept appears among others in:
* Tslil Clingman, Lyne Moser, *2-limits and 2-terminal objects are too different* ([arXiv:2004.01313](https://arxiv.org/abs/2004.01313))
[[!redirects bi-initial objects]]
[[!redirects biinitial object]]
[[!redirects biinitial objects]]
[[!redirects bi-initial]]
[[!redirects biinitial]]
[[!redirects initial object in a bicategory]]
[[!redirects initial object in a 2-category]]
|
bi-invertible morphism | https://ncatlab.org/nlab/source/bi-invertible+morphism | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
\tableofcontents
## Idea
In [[(n,1)-categories]] and [[(infinity,1)-categories]], the usual definition of [[invertible morphism]] is too strict and would violate the [[principle of equivalence]]. There are a few options to resolve this issue: one is by using [[weak inverses]]; but another is by using bi-invertible morphisms, which are a generalization of the notion of bi-invertible function in [[type theory]].
## Definition
A morphism $f:Mor(A, B)$ in a [[(n,1)-category]] or [[(infinity,1)-category]] is **bi-invertible** if there are morphisms $\mathrm{sec}(f):Mor(B, A)$ and $\mathrm{ret}(f):Mor(B, A)$ such that $\mathrm{sec}(f)$ is a [[section]] of $f$ and $\mathrm{ret}(f)$ is a [[retraction]] of $f$, with section and retraction defined using [[path space objects]] rather than strict equality.
## Examples
An [[integers object]] in an [[(n,1)-category]] $C$ with a [[terminal object]] is defined as the (homotopy) initial object $\mathbb{Z}$ with a [[global element]] $0:Mor(1, \mathbb{Z})$ and a bi-invertible [[endomorphism]] $\mathrm{succ}:Mor(\mathbb{Z}, \mathbb{Z})$ with section $\mathrm{pred}_1:Mor(\mathbb{Z}, \mathbb{Z})$ and retraction $\mathrm{pred}_2:Mor(\mathbb{Z}, \mathbb{Z})$. This corresponds to a definition of the [[integers]] [[higher inductive type]] in the [[internal logic]] of an [[(n,1)-category]].
## See also
* [[invertible morphism]]
* [[weak inverse]]
* [[equivalence of types]]
## References
For the integers object as defined with a bi-invertible morphism in the internal logic of a [[(infinity,1)-category]] see
* [[Thorsten Altenkirch]], [[Luis Scoccola]], *The Integers as a Higher Inductive Type*, [arXiv](https://arxiv.org/abs/2007.00167)
[[!redirects bi-invertible morphism]]
[[!redirects bi-invertible morphisms]] |
bi-pointed object | https://ncatlab.org/nlab/source/bi-pointed+object |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
\tableofcontents
## Definition
In evident variation of the notion of *[[pointed objects]]*, a **bi-pointed object** in a [[category]] $V$ with [[terminal object]] $pt$ is a [[co-span]] from $pt$ to itself, i.e. a [[diagram]] of this form:
$$
\array{
&& S
\\
& {}^{\sigma_S}\nearrow && \nwarrow^{\tau_S}
\\
pt &&&& pt
}
\,.
$$
Similarly, a [[pointed object]] in a category with [[initial object]] $\varnothing$ and terminal object $pt$ may be regarded as a co-span from $\varnothing$ to $pt$.
If $V$ has in addition binary [[coproducts]] then a bi-pointed object in $V$ is the same as a [[co-span]] from $\varnothing$ to the [[coproduct]] $pt \sqcup pt$.
## Examples
* The [[subobject classifier]] $\Omega$ and the [[Sierpinski space]] $\mathbb{S}$ in the category of [[choice object|choice sets]] are bi-pointed objects.
* In general, any non-degenerate subobject classifier in a topos or pretopos is a bi-pointed object.
* Any [[interval object]] is a bi-pointed object with a [[2-morphism]] connecting the two global elements.
* The [[boolean domain]] is the [[initial object|initial]] bi-pointed object in [[Set]].
## Closed structure
From the [[bicategory]] structure on [[co-span]]s in $V$ bi-pointed objects in $V$ naturally inherit the structure of a monoidal category
$$
BiPointed(V) = End_{CoSpan(V)}(pt)
\,.
$$
Assume that the terminal object $pt$ is the tensor unit in $V$.
Then moreover, following the construction of the $V$-internal hom of [[pointed object]]s and being a special case of that of [[co-span]]s in $V$, there is an internal [[hom-object]]
${}_{pt}[X,Y]_{pt} \in Obj(V)$
of bipointed objects $X$ and $Y$ defined as the pullback
$$
\array{
{}_{pt}[X,Y]_{pt}
& \longrightarrow &
pt \sqcup pt
\\
\Big\downarrow
&&
\Big\downarrow\mathrlap{{}^{\sigma_Y \sqcup \tau_Y}}
\\
[X,Y]
&
\underset
{\sigma_X^* \times \tau_X^*}
{\longrightarrow}
&
[pt \sqcup pt,Y].}
$$
Here the [[map]]
$pt \sqcup pt \stackrel{\sigma_Y \sqcup \tau_Y}{\to}
[pt \sqcup pt,Y]$
is [[adjunct]] to
$\pt \otimes (pt \sqcup pt) \to pt \sqcup pt
\stackrel{\sigma_Y \sqcup \tau_Y}{\to} Y$.
This $V$-object ${}_{pt}[X,Y]_{pt}$ is itself naturally bi-pointed with the bi-point $pt \sqcup pt \to {}_pt[X,Y]_{pt}$ given by the morphism induced from the above pullback diagram by the commuting diagram
$$
\array{
pt \sqcup pt
&
\overset{Id}{\longrightarrow}&
pt \sqcup pt
\\
\Big\downarrow\mathrlap{{}^{\sigma_X \sqcup \sigma_X}}
&&
\Big\downarrow\mathrlap{{}^{\sigma_Y \sqcup \tau_Y}}
\\
[X,Y]
&
\underset
{\sigma_X^* \times \sigma_Y^*}
{\longrightarrow}
&
[pt \sqcup pt, Y],
}
$$
where the morphism
$pt \sqcup pt \stackrel{\sigma_X \sqcup \sigma_X}{\to}
[X,Y]
$ is [[adjunct]] to
$
X \otimes (pt \sqcup pt)
\to
pt \otimes (pt \sqcup pt)
\simeq pt \sqcup pt
\stackrel{\sigma_Y \sqcup \tau_Y}{\to}
Y
$.
## See also
* [[pointed object]]
* [[boolean domain object]]
* [[bi-pointed type]]
[[!redirects bi-pointed object]]
[[!redirects bi-pointed objects]]
[[!redirects bi-pointed set]]
[[!redirects bi-pointed sets]] |
bi-pointed type | https://ncatlab.org/nlab/source/bi-pointed+type |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_Bi-pointed types_ are [[axiom|axiomatizations]] of the [[bi-pointed objects]] in the context of [[homotopy type theory]].
## Definition
A **bi-pointed type** is a type $A$ with two points $a:A$ and $b:A$. Equivalently, it is a type $A$ with a function $\mathrm{Bool} \to A$. Examples include the [[interval type]] and the [[function type]] of the [[natural numbers type]].
## See also
* [[bi-pointed object]]
[[!redirects bi-pointed type]] |
bi-terminal object | https://ncatlab.org/nlab/source/bi-terminal+object |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Bi-terminal objects are the [[bicategory|bicategorical]] analogues of [[terminal object|terminal objects]] in categories.
## Definition
In a [[bicategory]] $A$, an object $L \in A$ is **bi-terminal** (or **biterminal**) when for all $X \in A$, there is an [[equivalence of categories]] between $A(X,L)$ and the [[terminal category]] $\mathbb{1}$ with a single object and single morphism.
## Related concepts
* [[terminal object]]
* [[bi-initial object]]
* [[terminal object in an (infinity,1)-category|terminal object in an $(\infty,1)$-category]]
## References
This concept appears among others in:
* Tslil Clingman, Lyne Moser, *2-limits and 2-terminal objects are too different* ([arXiv:2004.01313](https://arxiv.org/abs/2004.01313)).
[[!redirects bi-terminal objects]]
[[!redirects biterminal object]]
[[!redirects biterminal objects]]
[[!redirects bi-terminal]]
[[!redirects biterminal]]
[[!redirects terminal object in a bicategory]]
[[!redirects terminal object in a 2-category]]
|
biactegory | https://ncatlab.org/nlab/source/biactegory | Let $C$ be a [[monoidal category]] (in fact one can easily modify all statements to generalize all statements here to [[bicategories]]). One can consider several variants of the [[2-category]] of all categories with monoidal [[action]] of $C$, (co)lax [[monoidal functors]] and their transformations. A category with an action of $C$ is sometimes called a __$C$-[[actegory]]__. The word 'module category' over $C$ is also used, especially when the category acted upon is in addition also [[additive category|additive]], like the examples in [[representation theory]].
If a $C$-actegory is a [[categorification]] of a [[module]], then for two monoidal categories $C$ and $D$, we should categorify a [[bimodule]], which we call __$C$-$D$-biactegory__. The two actions on a usual bimodule commute; for biactegories the commuting is up to certain coherence laws, which are in fact the expression of an invertible [[distributive law]] between the two monoidal actions. The tensor product of biactegories can be defined (here the invertibility of the distributive law is needed) as a [[2-limit|bicoequalizer]] of a certain diagram.
For very basic outline see section 2 in
* [[Zoran Škoda]], Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–-202, [arXiv:0811.4770](https://arxiv.org/abs/0811.4770).
and partial writeup
* Zoran Škoda, _Biactegories_, 2006, [pdf](https://www2.irb.hr/korisnici/zskoda/biact.pdf))
In a language of "module categories", a different treatment is now available in
* Justin Greenough, Monoidal 2-structure of bimodule categories, [arxiv:0911.4979](https://arxiv.org/abs/0911.4979).
It is one of the notions discussed in the review
* Matteo Capucci, Bruno Gavranović, _Actegories for the working amthematician_, [arXiv:2203.16351](https://arxiv.org/abs/2203.16351)
|
biaction | https://ncatlab.org/nlab/source/biaction |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
A ternary [[function]] which simultaneously exhibits an [[action]] on a [[set]] from both the left and the right side.
Sets with biactions are the [[bimodule objects]] [[internalization|internal to]] [[Set]].
## Definition
Given a [[set]] $S$ and [[monoids]] $(M, e_M, \mu_M)$ and $(N, e_N, \mu_N)$, a **$M$-$N$-biaction** or **two-sided action** is a ternary function $\alpha:M \times S \times N \to S$ such that
* for all $s \in S$, $\alpha(e_M, s, e_N) = s$
* for all $s \in S$, $a \in M$, $b \in M$, $c \in N$, and $d \in N$, $\alpha\big(a, \alpha(b, s, c), d\big) = \alpha\big(\mu_M(a, b), s, \mu_N(c, d)\big)$
## Left and right actions
The [[left action|left $M$-action]] is defined as
$$\alpha_M(a, s) \coloneqq \alpha(a, s, e_N)$$
for all $a \in M$ and $s \in S$. It is a left action because
$$\alpha_M(e_M, s) = \alpha(e_M, s, e_N) = s$$
$$\alpha_M\big(a, \alpha_L(b, s)\big) = \alpha\big(a, \alpha(b, s, e_N), e_N\big) = \alpha\big(\mu_M(a, b), s, \mu_N(e_N, e_N)\big) = \alpha\big(\mu_M(a, b), s, e_N\big) = \alpha_M\big(\mu_M(a, b), s\big)$$
The [[right action|right $N$-action]] is defined as
$$\alpha_N(s, c) \coloneqq \alpha(e_M, s, c)$$
for all $c \in N$ and $s \in S$. It is a right action because
$$\alpha_N(s, e_N) = \alpha(e_M, s, e_N) = s$$
$$\alpha_N\big(\alpha_N(s, c), d\big) = \alpha\big(e_M, \alpha(e_M, s, c), d\big) = \alpha\big(\mu_M(e_M, e_M), s, \mu_N(c, d)\big) = \alpha\big(e_M, s, \mu_N(c, d)\big) = \alpha_N\big(s, \mu_N(c, d)\big)$$
The left $M$-action and right $N$-action satisfy the following identity:
* for all $s \in S$, $a \in M$ and $c \in N$, $\alpha_M\big(a, \alpha_N(s, c)\big) = \alpha_N\big(\alpha_M(a, s), c\big)$.
This is because when expanded out, the identity becomes:
$$\alpha\big(a, \alpha(e_M, s, c), e_N\big) = \alpha\big(e_M, \alpha(a, s, e_N), c\big)$$
$$\alpha\big(\mu_M(a, e_M), s, \mu_N(c, e_N)\big) = \alpha\big(\mu_M(e_M, a), s, \mu_N(e_N, c)\big)$$
$$\alpha(a, s, c) = \alpha(a, s, c)$$
## See also
* [[action]]
* [[bimodule]]
[[!redirects biactions]]
[[!redirects two-sided action]]
[[!redirects two-sided actions]]
|
biadjoint pair | https://ncatlab.org/nlab/source/biadjoint+pair | Moved to [[ambidextrous adjunction]]. To be removed. |
biadjunction | https://ncatlab.org/nlab/source/biadjunction | > This page is about the generalisation of an [[adjunction]] to [[bicategories]]. For a functor $L$ which is both a left and a right adjoint to a functor $R$, see [[ambidextrous adjunction]].
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#Contents#
* table of contents
{:toc}
## Idea
A _biadjunction_ is the "maximally weak" kind of [[2-adjunction]]: a [[categorification|higher generalization]] of the notion of [[adjunction]] from [[category theory]] to [[2-category theory]], and specifically to [[bicategories]] (or more generally internally in a [[tricategory]]. See [[2-adjunction]] for other kinds of 2-adjunction.
## Definition
### Incoherent versions
Given (possibly weak) [[2-categories]], $A$ and $C$, and (possibly weak) [[2-functors]] $F:A\to C$ and $U:C\to A$, a __biadjunction__ is given by specifying for each object $a$ in $A$ and each object $c$ in $C$ an [[equivalence of categories]] $C(F a,c)\cong A(a,U c)$, which is [[pseudonatural transformation|pseudonatural]] both in $a$ and in $c$.
By the [[Yoneda lemma for bicategories]], this is equivalent to giving pseudonatural transformations $\eta : Id_A \to U F$ and $\varepsilon : F U \to Id_C$ satisfying the [[triangle identities]] up to invertible [[modifications]]. This latter definition can be internalized in any (weak) [[3-category]], such as a [[Gray-category]] or a [[tricategory]].
### Coherent versions
Both the "equivalence of hom-categories" definition and the "unit and counit" definition have stronger "coherent" versions: We can ask the equivalences $C(F a,c)\cong A(a,U c)$ to be [[adjoint equivalences]], or for the "triangulator" modifications to satisfy the [[lax 2-adjunction#definition|swallowtail equations]]. These two conditions are roughly equivalent, and any "incoherent" biadjunction can be improved to a coherent one by altering one of the triangulators; see [Gurski](#Gurski12), [Riehl-Verity](#RV15), [Pstrągowski](#Pstrągowski14), [Riehl-Shulman](#RS17), and [this proof](http://globular.science/1512.006) in [[Globular]].
## Related concepts
* [[adjoint functor]], [[adjoint triple]], [[adjoint quadruple]]
* [[proadjoint]], [[Hopf adjunction]]
* [[2-adjunction]]
**biadjunction**, [[lax 2-adjunction]], [[pseudoadjunction]]
* [[pseudoadjunction]]
* [[biequivalence]], [[biadjoint biequivalence]]
* [[adjoint (infinity,1)-functor]]
## References
* [[John Gray]], _[[Gray-adjointness-for-2-categories|Formal category theory: Adjointness for 2-categories]]_, Lecture Notes in Mathematics __391__, Springer, Berlin, 1974.
* [[Thomas M. Fiore]], _Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory_, Memoirs of the American Mathematical Society __182__ (2006), no. 860. 171 pages, [MR2007f:18006](http://www.ams.org/mathscinet-getitem?mr=2007f:18006), [math.CT/0408298](http://arxiv.org/abs/math.CT/0408298)
* [[Nick Gurski]], _Biequivalences in tricategories_, Theory and applications of categories 2012, [journal web site](http://www.tac.mta.ca/tac/volumes/26/14/26-14abs.html), [arxiv](https://arxiv.org/abs/1102.0979)
{#Gurski12}
* [[Emily Riehl]] and [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_, 2015, [arxiv](https://arxiv.org/abs/1310.8279)
{#RV15}
* Piotr Pstrągowski, _On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis_, 2014 [arxiv](https://arxiv.org/abs/1411.6691)
{#Pstrągowski14}
* [[Emily Riehl]] and [[Mike Shulman]], _A type theory for synthetic ∞-categories_, 2017, [arxiv](https://arxiv.org/abs/1705.07442)
{#RS17}
[[!redirects biadjunctions]]
[[!redirects biadjoint]]
[[!redirects biadjoints]]
|
bialgebra | https://ncatlab.org/nlab/source/bialgebra |
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# Bialgebra
* table of contents
{: toc}
## Idea
A _bialgebra_ (or bi[[gebra]]) is both an [[algebra]] and a [[coalgebra]], where the operations of either one are [[homomorphisms]] for the other. A bialgebra structure on an [[associative algebra]] is precisely such as to make its [[category of modules]] into a [[monoidal category]] equipped with a [[fiber functor]].
A bialgebra is one of the ingredients in the concept of [[Hopf algebra]].
## Definition
A **bialgebra** is a [[monoid]] [[internalization|in]] the category of [[coalgebra|coalgebras]]. Equivalently, it is a [[comonoid]] [[internalization|in]] the category of [[algebra|algebras]]. Equivalently, it is a monoid in the category of comonoids in [[Vect]] --- or equivalently, a comonoid in the category of monoids in [[Vect]].
More generally, a **[[bimonoid]]** in a monoidal category $M$ is a monoid in the category of comonoids in $M$ --- or equivalently, a comonoid in the category of monoids in $M$. So, a bialgebra is a bimonoid in $Vect$.
## Properties
### Relation to sesquialgebras
Bialgebras are precisely those [[sesquialgebras]] $A$ whose product $A \otimes A$-$A$-[[bimodule]] is induced from an algebra [[homomorphism]] $A \to A \otimes A$ and whose unit $k$-$A$ bimodule is induced from an algebra homomorphism $A \to k$.
### Tannaka duality and categories of modules
{#TannakDuality}
The structure of a bialgebra on an [[associative algebra]] equips its [[category of modules]] with the structure of a [[monoidal category]] and a monoidal [[fiber functor]]. In fact that construction is an equivalence. This is the statement of [[Tannaka duality]] for bialgebras. For instance ([Bakke](#Bakke))
[[!include structure on algebras and their module categories - table]]
## Examples
Notions of bialgebra with further structure notably include _[[Hopf algebras]]_ and their variants.
## Related concepts
* [[bimonoid]]
* [[Gerstenhaber-Schack cohomology]], [[bialgebra cocycle]], [[weak bialgebra]], [[bialgebroid]]
* [[Hopf algebra]]
* [[quasitriangular bialgebra]]
* [[sesquialgebra]]
* [[algebra]], [[module]]
* **bialgebra**, [[2-module]]
* [[trialgebra]], [[3-module]]
## References
[[Tannaka duality]] for bialgebras
* {#Bakke} Tørris Koløen Bakke, _Hopf algebras and monoidal categories_ (2007) ([pdf](http://munin.uit.no/bitstream/handle/10037/1084/finalthesis.pdf;jsessionid=C0D15EADBDC35E93D95D2DD090411004?sequence=1))
On [[bialgebras]] in [[locally presentable categories]]:
* [[Friedrich Ulmer]]. *Bialgebras in locally presentable categories*, University of Wuppertal preprint (1977) [[pdf](/nlab/files/Bialgebras_in_locally_presentable_categories.pdf)]
[[!redirects bialgebra]]
[[!redirects bialgebras]]
[[!redirects bigebra]]
[[!redirects bigebras]]
|
bialgebra cocycle | https://ncatlab.org/nlab/source/bialgebra+cocycle |
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#### Cohomology
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[[!include cohomology - contents]]
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#### Algebra
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[[!include higher algebra - contents]]
=--
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#Contents#
* table of contents
{:toc}
## Idea
[[Shahn Majid]] has introduced a notion of __bialgebra cocycles__ which as special cases comprise [[group cohomology|group cocycles]], nonabelian Drinfel'd 2-cocycle and 3-cocycle, abelian [[Lie algebra cohomology]] and so on.
Besides this case, by "bialgebra cohomology" many authors in the literature mean the abelian cohomology ([[Ext|Ext-groups]]) in certain category of "tetramodules" over a fixed bialgebra, which will be in $n$Lab referred as [[Gerstenhaber-Schack cohomology]].
## Definition
Let $(B,\mu,\eta,\Delta,\epsilon)$ be a $k$-[[bialgebra]].
Denote $\Delta_i : B^{\otimes n}\to B^{\otimes (n+1)} := \id_B^{\otimes (i-1)}\otimes\Delta\otimes\id_B^{\otimes(n-i+1)}$, for $i = 1,\ldots, n$, and $\Delta_0 := 1_B\otimes \id_B^{\otimes n}$, $\Delta_{n+1} := \id_B^{\otimes n}\otimes 1_B$. Notice that for the compositions $\Delta_i\circ\Delta_j = \Delta_{j+1}\circ\Delta_i$ for $i\leq j$.
Let $\chi$ be an invertible element of $B^{\otimes n}$. We define the __coboundary__ $\partial\chi$ by
$$\partial \chi = (\prod_{i=0}^{i \mathrm{ even}} \Delta_i\chi) (\prod_{i=1}^{i \mathrm{ odd}} \Delta_i \chi^{-1})$$
This formula is symbolically also written as $\partial\chi = (\partial_+\chi)(\partial_-\chi^{-1})$.
An invertible $\chi\in B^{\otimes n}$ is an __$n$-cocycle__ if $\partial\chi = 1$. The cocycle $\chi$ is __counital__ if
for all $i$, $\epsilon_i\chi=1$ where $\epsilon_i =\id_B^{\otimes i-1}\otimes\epsilon\otimes\id_B^{\otimes n-i}$.
## Examples
### Low dimensions
$\chi\in H$ is a 1-cocycle iff it is invertible and [[grouplike element|grouplike]] i.e. $\Delta\chi=\chi\otimes\chi$ (in particular it is counital). A 2-cocycle is an invertible element $\chi\in H^{\otimes 2}$ satisfying
$$
(1\otimes\chi)(id\otimes\Delta)\chi = (\chi\otimes 1)(\Delta\otimes id)\chi,
$$
which is counital if $(\epsilon\otimes id)\chi = (id\otimes\epsilon)\chi = 1$ (in fact it is enough to require one out of these two counitality conditions). Counital 2-cocycle is hence the famous [[Drinfel'd twist]].
The 3-cocycle condition for $\phi\in H^{\otimes 3}$ reads:
$$
(1\otimes\phi)((id\otimes\Delta\otimes id)\phi)(\phi\otimes 1) = ((id\otimes id\otimes\Delta)\phi)((\Delta\otimes id\otimes id)\phi)
$$
A counital 3-cocycle is the famous __Drinfel'd associator__ appearing in CFT and quantum group theory. The coherence for monoidal structures can be twisted with the help of Drinfel'd associator; Hopf algebras reconstructing them appear then as quasi-Hopf algebras where the comultiplication is associative only up to twisting by a 3-cocycle in $H$.
### For particular Hopf algebras
If $G$ is a finite group and $H=k(G)$ is the
[[Hopf algebra]] of $k$-valued functions on the group, then we recover the usual notions: e.g. the 2-cocycle is a function $\chi:G\times G\to k$ satisfying the cocycle condition
$$
\chi(b,c)\chi(a,b c) = \chi(a,b)\chi(a b,c)
$$
and the condition for a 3-cocycle $\phi:G\times G\times G\to k$ is
$$
\phi(b,c,d)\phi(a,b c,d)\phi(a,b,c) = \phi(a,b,c d)\phi(a b,c,d)
$$
$n$-cocycles can be in low dimensions twisted by $(n-1)$-cochains (I think it is in this context not know for hi dimensions), what gives an equivalence relation:
For example, if $\chi\in H\otimes H$ is a counital 2-cocycle, and $\partial\gamma\in H$ a counital coboundary, then
$$ \chi^\gamma = (\partial_+\gamma)\chi(\partial_-\gamma^{-1})= (\gamma\otimes\gamma)\chi\Delta\gamma^{-1} $$
is another 2-cocycle in $H\otimes H$. In particular, if $\chi = 1$ we obtain that $\partial\gamma$ is a cocycle (that is every 2-coboundary is a cocycle).
## A dual theory
In addition to cocycles "in" $H$ as above, Majid introduced a dual version -- cocycles on $H$. The usual Lie algebra cohomology $H^n(L,k)$, where $L$ is a $k$-Lie algebra, is a special case of that dual construction.
Instead of $\Delta_i$ one uses multiplications $\cdot_i$ defined analogously ($\cdot_i$ is the multiplication in $i$-th place for $1\leq i\leq n$ and $\psi\circ\cdot_0 =\epsilon\otimes\psi$, $\psi\circ\cdot_{n+1} = \psi\otimes\epsilon$).
An $n$-cochain on $H$ is a linear functional $\psi:H^{\otimes n}\to k$, invertible in the [[convolution algebra]]. An $n$-cochain $\psi$ on $H$ is a __coboundary__ if
$$
\partial\psi = (\prod_{i=0}^{\mathrm{even}}\psi\circ \cdot_i))(\prod_{i=1}^{\mathrm{odd}}\psi^{-1}\circ\cdot_i)
$$
If $\psi\in H$ then this condition reads
$$
(\partial\psi)(a\otimes b) = \sum \psi(b_{(1)})\psi(a_{(1)})\psi^{-1}(a_{(2)}b_{(2)})
$$
and, for $\psi\in H\otimes H$, the condition is
$$
(\partial\psi)(a\otimes b\otimes c) = \sum \psi(b_{(1)}\otimes c_{(1)})\psi(a_{(1)}\otimes b_{(2)}c_{(2)})\psi^{-1}(a_{(2)}\otimes b_{(3)}c_{(3)})\psi^{-1}(a_{(3)}b_{(4)})
$$
If one looks at the [[group algebra]] $kG$ of a finite group then the cocycle conditions above can be obtained by a Hopf algebraic version of the $k$-linear extension of the cocycle conditions for the group cohomology in the form appearing in Schreier's theory of extensions.
However for all $n$ the Lie algebra cohomology also appears as a special case.
(to be completed later)
## References
* [[Shahn Majid]], _Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras_, in H.-D. Doebner, V.K. Dobrev, A.G. Ushveridze, eds., Generalized symmetries in Physics. World Sci. (1994) 13-41; ([arXiv:hep.th/9311184](https://arxiv.org/abs/hep-th/9311184))
* [[Shahn Majid]], _Foundations of quantum group theory_, Cambridge UP
[[!redirects bialgebra cocycles]] |
bialgebra ideal | https://ncatlab.org/nlab/source/bialgebra+ideal | |
bialgebroid | https://ncatlab.org/nlab/source/bialgebroid | ## Idea
A bialgebroid may be viewed as a multiobject generalization of a concept of a [[bialgebra]], or a possibly noncommutative generalization of a space-algebra dual version of the concept of an internal category in spaces.
#### Nomenclature
This entry is about "associative" bialgebroid, see also the different concept of a [[Lie bialgebroid]].
#### Motivation in Tannakian formalism
When a monoidal category has a [[fiber functor]] to a category of vector spaces over a field, one tries to "reconstruct" the category as the category of representations of the endomorphism object of a fiber functor. One often does not have a fiber functor to vector spaces but only to bimodules over some base algebra $A$. Sometimes in such cases, the object of endomorphisms of the fiber functor form a bialgebroid over $A$ and the category is equivalent to the category of representations of that bialgebroid.
## Definition
### Via monoidal categories
Given a unital (possibly noncommutative) ring $R$ an __$R$-bialgebroid__ is an $R$-$R$-bimodule $H$ (object of ${}_R \mathcal{M}_R$) equipped with a structure of a comonoid in ${}_R \mathcal{M}_R$ (i.e. an $R$-coring) and of a monoid in ${}_{R^e}\mathcal{M}_{R^e}$ (i.e. an $R^e$-ring), where $R^e = R^{op}\otimes R$ is the enveloping ring of $R$; and the structures of a monoid and a comonoid satisfy certain compatibility conditions. These compatibility conditions are equivalent to the fact that the monad ${}_{\otimes_{R^e}} H : \mathcal{M}_{R^e}\to \mathcal{M}_{R^e}$ is [[opmonoidal monad|opmonoidal]]. The category of $R$-comodules is by definition the category of comodules over the underlying $R$-coring.
### An explicit definition
If $A$ is an [[associative algebra]] over some [[ground field]] $k$, then a left associative $A$-bialgebroid is another associative $k$-algebra $H$ together with the following additional maps:
an algebra map $\alpha:A\to H$ called the source map,
an algebra map $\beta:A^{op}\to H$ called the target map, so that the elements of the images of $\alpha$ and $\beta$ commute in $H$, therefore inducing an $A$-bimodule structure on $H$ via the rule $a.h.b = \alpha(a)\beta(b) h$ for $a,b\in A, h\in H$; an $A$-bimodule morphism $\Delta:H\to H\otimes_A H$ which is required to be a counital coassociative comultiplication on $H$ in the monoidal category of $A$-bimodules with monoidal product $\otimes_A$. The map $H\otimes A\ni h\otimes a\mapsto \epsilon(h\alpha(a))\in $ must be a left action extending the multiplication $A\otimes A\to A$ along $\alpha\otimes id_A$. Furthermore, a compatibility between the comultiplication $\Delta$ and multiplications on $H$ and on $H\otimes H$ is required. For a noncommutative $L$ the tensor square $H\otimes_A$ is not an algebra, hence asking for a bialgebra-like compatibility that $\Delta:H\to H\otimes_A H$ is a morphism of $k$-algebras does not make sense. Instead, one requires that $H\otimes_A H$ has a $k$-subspace $T$ which contains the image of $\Delta$ and has a well-defined multiplication induced from its preimage under the projection from the usual tensor square algebra $H\otimes H$. Then one requires that the [[corestriction]] $\Delta|^T :H\to T$ is a homomorphism of unital algebras. Under these conditions, one can make a canonical choice for $T$, namely the so called Takeuchi's product $H\times_A H\subset H\otimes_A H$, which always inherits an associative multiplication along the projection from $H\otimes H$.
### Via $A\otimes A^{op}$-rings
All modules and morphisms will be over a fixed ground commutative ring $k$.
A __left $A$-bialgebroid__ is
an $A\otimes_k A^{op}$-[[ring]] $(H,\mu_H,\eta)$,
together with the $A$-bimodule map "comultiplication" $\Delta : H\to H\otimes_A H$, which is coassociative and counital with a counit $\epsilon$, such that
(i) the $A$-bimodule structure used on $H$ is $a.h.a':= s(a)t(a')h$, where $s := \eta(-\otimes 1_A):A\to H$ and $t:=\eta(1_A\otimes -):A^{op}\to H$ are the algebra maps induced by the unit $\eta$ of the $A\otimes A^{op}$-ring $H$
(ii) the coproduct $\Delta : H\to H\otimes_A H$ corestricts to the [[Takeuchi product]] and the corestriction $\Delta : H\to H\times_A H$
is a $k$-algebra map, where the Takeuchi product $H\times_A H$ has a multiplication induced factorwise
(iii) $\epsilon$ is a left [[character]] on the $A$-ring $(H,\mu_H,s)$.
Notice that $H\otimes_A H$ is in general not an algebra, just an $A$-bimodule. That is why (ii) is needed. An equivalent condition to (ii) is the following: the formula $h.(\sum_i k_i \otimes l_i) = \sum_i h_{(1)}\cdot k_i \otimes h_{(2)} \cdot l_i$ defines a well-defined action of $H$ on $H\otimes_A H$.
The definition of a __right $A$-bialgebroid__ differs by the $A$-bimodule structure on $H$ given instead by $a.h.a':= h s(a')t(a)$ and the counit $\epsilon$ is a _right_ character on the $A$-coring $(H,\mu_H,t)$ ($t$ and $s$ can be interchanged in the last requirement).
## Literature
Related notions: [[Hopf algebroid]]
#### Commutative case
The commutative case is rather classical. See for example the appendix to
* Douglas C. Ravenel, _Complex cobordism and stable homotopy groups of spheres_, Pure and Applied Mathematics __121__. Academic Press Inc., Orlando, FL, 1986.
#### Noncommutative case
The first version of a bialgebroid over a noncommutative base was more narrow:
* M. Sweedler, _Groups of simple algebras_, Publ. IHES 44:79–189, 1974, [numdam](http://www.numdam.org/item?id=PMIHES_1974__44__79_0)
A modern generality, but in different early formalism, is due to Takeuchi (who was motivated to generalize the results from the Sweedler's paper), under the name of $\times_A$-bialgebra (as it involves the $\times_A$-product, nowdays called [[Takeuchi product]]):
* M. Takeuchi, _Groups of algebras over $A \times \bar{A}$, J. Math. Soc. Japan __29__, 459–492, 1977, [MR0506407](http://www.ams.org/mathscinet-getitem?mr=0506407), [euclid](http://projecteuclid.org/euclid.jmsj/1240432948)
Lu introduces the name bialgebroid for a structure which is equivalent to the Takeuchi's $\times_A$-bialgebra (though differently axiomatized there):
* Jiang-Hua Lu, _Hopf algebroids and quantum groupoids_, Int. J. Math. __7__, 1 (1996) pp. 47-70, [q-alg/9505024](http://arxiv.org/abs/q-alg/9505024), [MR95e:16037](http://www.ams.org/mathscinet-getitem?mr=95e:16037), [doi](http://dx.doi.org/10.1142/S0129167X96000050)
Modern treatments are in
* [[Gabriella Böhm]], _Internal bialgebroids, entwining structures and corings_, [math.QA/0311244](http://arxiv.org/abs/math/0311244), in: Algebraic structures and their representations, 207–226, Contemp. Math. __376__, Amer. Math. Soc. 2005.
* [[G. Böhm]], _Hopf algebroids_, (a chapter of) Handbook of algebra, [arxiv:math.RA/0805.3806](http://arxiv.org/abs/0805.3806)
* [[Kornél Szlachányi]], _The monoidal Eilenberg–Moore construction and bialgebroids_, J. Pure Appl. Algebra __182__, no. 2–3 (2003) 287–315; _Fiber functors, monoidal sites and Tannaka duality for bialgebroids_, [arxiv/0907.1578](http://arxiv.org/abs/0907.1578)
* [[T. Brzeziński]], G. Militaru, _Bialgebroids, $\times_{R}$-bialgebras and duality_, J. Algebra 251: 279-294, 2002, [math.QA/0012164](http://arxiv.org/abs/math/0012164)
* J. Donin, A. Mudrov, _Quantum groupoids and dynamical categories_,
J. Algebra __296__ (2006), no. 2, 348–384, [math.QA/0311316](http://arxiv.org/abs/math/0311316), [MR2007b:17022](http://www.ams.org/mathscinet-getitem?mr=2201046), [doi](http://dx.doi.org/10.1016/j.jalgebra.2006.01.001); MPIM-2004-21, [dvi](http://www.mpim-bonn.mpg.de/preblob/1921) with hyperlinks, [ps](http://www.mpim-bonn.mpg.de/preblob/1922)
There is also a notion of quasibialgebroid, where the coassociativity is weakened by a [[bialgebra cocycle|bialgebroid 3-cocycle]]. See also [[Hopf algebroid]].
[[!redirects bialgebroids]]
[[!redirects associative bialgebroid]] |
Bianca Cerchiai | https://ncatlab.org/nlab/source/Bianca+Cerchiai |
* [InSpire page](https://inspirehep.net/authors/1014175)
* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=107587)
## Selected writings
On [[magic supergravity]]:
* {#CacciatoriCerchiaiMarrani12} Sergio L. Cacciatori, [[Bianca Cerchiai]], [[Alessio Marrani]], _Squaring the Magic_ [[arXiv:1208.6153](http://arxiv.org/abs/1208.6153)]
* {#CacciatoriCerchiaiMarrani2013} Sergio L. Cacciatori, [[Bianca Cerchiai]], [[Alessio Marrani]], _Magic coset decompositions_, Adv. Theor. Math. Phys. **17** 5 (2013) 1077-1128 [[doi:10.4310/ATMP.2013.v17.n5.a4](https://doi.org/10.4310/ATMP.2013.v17.n5.a4)]
Usage of full [[supergravity]] (retaining the [[gravitino]]) for [[AdS-CMT duality]], with application to [[graphene]]-like systems:
* [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], [[Mario Trigiante]], *Unconventional Supersymmetry at the Boundary of $AdS_4$ Supergravity*, J. High Energ. Phys. **2018** 7 (2018) [[arXiv:1801.08081](https://arxiv.org/abs/1801.08081), <a href="https://doi.org/10.1007/JHEP04(2018)007">doi:10.1007/JHEP04(2018)007</a>]
* [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], A. Gallerati, R. Noris, [[Mario Trigiante]], [[Jorge Zanelli]], *$\mathcal{N}$-Extended $D=4$ Supergravity, Unconventional SUSY and Graphene*, JHEP 01 (2020) 084 [[arXiv:1910.03508](https://arxiv.org/abs/1910.03508), <a href="https://doi.org/10.1007/JHEP01(2020)084">doi:10.1007/JHEP01(2020)084</a>]
Exposition:
* [[Bianca L. Cerchiai]], *Supergravity in a pencil*, [talk at](M-Theory+and+Mathematics#Cerchiai2020) *[[M-Theory and Mathematics]] [2020](M-Theory+and+Mathematics#2020)*, NYU Abu Dhabi [[[CerchiaiSlidesAtMTheoryAndMathematics2020.pdf:file]], video: [YT](https://youtu.be/xE7TmwyqqaU)]
* [[Bianca L. Cerchiai]], *Holography, Supergravity and Graphene*, talk at *106th online SIF Congress* (2020) [[pdf](https://agenda.infn.it/event/23656/contributions/120378/attachments/75347/96340/cerchiai_sif2020.pdf), [[Cerciai-SIF2020.pdf:file]]]
On [[SU(N)]]-[[Skyrmions]]:
* Pedro D. Alvarez, Sergio L. Cacciatori, Fabrizio Canfora, [[Bianca Cerchiai]], _Analytic $SU(N)$ Skyrmions at finite Baryon density_ ([arXiv:2005.11301](https://arxiv.org/abs/2005.11301))
category: people
[[!redirects Bianca Letizia Cerchiai]]
[[!redirects Bianca L. Cerchiai]] |
Bianchi identity | https://ncatlab.org/nlab/source/Bianchi+identity |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
The _Bianchi identity_ is a [[differential equation]] satisfied by [[curvature]] data.
It can be thought of as generalizing the equation $d (d A) = 0$ for a real-valued [[differential form|1-form]] $A$ to higher degree and nonabelian forms.
Generally it applies to the [[curvature]] of [[∞-Lie algebroid valued differential forms]].
## Definition
### For 2-form curvatures
Let $U$ be a [[smooth manifold]].
For $A \in \Omega^1(U)$ a differential 1-form, its [[curvature]] 2-form is the de Rham differential $F_A = d A$. The Bianchi identity in this case is the equation
$$
d F = 0
\,.
$$
More generally, for $\mathfrak{g}$ an arbitrary [[Lie algebra]] and $A \in \Omega^1(U,\mathfrak{g})$ a [[Lie-algebra valued 1-form]], its [[curvature]] is the 2-form $F_A = d A + [A \wedge A]$. The Bianchi identity in this case is the equation
$$
d F_A + [A\wedge F_A] = 0
$$
satisfied by these curvature 2-forms.
### Reformulation in terms of Weil algebras
We may reformulate the above identities as follows.
For $\mathfrak{g}$ a [[Lie algebra]] we have naturally associated two [[dg-algebra | dg-algebras]]: the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ and the [[Weil algebra]] $W(\mathfrak{g})$.
The dg-algebra morphisms
$$
\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : (A,F_A)
$$
are precisely in bijection with [[Lie-algebra valued 1-form | Lie-algebra valued 1-forms]] as follows: the [[Weil algebra]] is of the form
$$
W(\mathfrak{g}) = \wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W(\mathfrak{g})}
$$
with one copy of $\mathfrak{g}^*$ in degree 1, the other in degree 2. By the [[free construction|free]] nature of the Weil algebra, dg-algebra morphisms $\Omega^\bullet(U) \leftarrow W(\mathfrak{g})$ are in bijection to their underlying morphisms of vector spaces of generators
$$
\Omega^1(U) \leftarrow \mathfrak{g}^* : A
\,.
$$
This identifies the 1-form $A \in \Omega^1(U,\mathfrak{g})$. This extends uniquely to a morphism of dg-algebras and thereby fixes the image of the shifted generators
$$
\Omega^2(U) \leftarrow \mathfrak{g}^*[1] : F_A
\,.
$$
The _Bianchi identity_ is precisely the statement that these linear maps, extended to morphisms of graded algebra, are compatible with the differentials and hence do constitute [[dg-algebra]] morphisms.
Concretely, if $\{t^a\}$ is a dual basis for $\mathfrak{g}^*$ and $\{r^a\}$ the corresponding dual basis for $\mathfrak{g}^*[1]$ and $\{C^a{}_{b c}\}$ the structure constants of the Lie bracket $[-,-]$ on $\mathfrak{g}$, then the differential $d_{W(\mathfrak{g})}$ on the Weil algebra is defined on generators by
$$
d_{W(\mathfrak{g})} t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a
$$
and
$$
d_{W(\mathfrak{g})} r^a = C^a{}_{b c} t^b \wedge r^c
\,.
$$
The image of $t^a$ under $\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : (A,F_A)$ is the component $A^a$. The image of $r^a$ is therefore, by respect for the differential on $t^a$
$$
r^a \mapsto (F_A)^a := d A^a + \frac{1}{2}C^a{}_{b c} A^b \wedge A^c
\,.
$$
Respect for the differential on $r^a$ then implies
$$
d (F_A)^a + C^a{}_{b c} A^a \wedge (F_A)^c = 0
\,.
$$
This is the Bianchi identity.
### For curvature of $\infty$-Lie algebra valued forms.
Let now $\mathfrak{g}$ be an arbitrary [[∞-Lie-algebra]] and $W(\mathfrak{g})$ its [[Weil algebra]]. Then a collection of [[∞-Lie algebra valued differential forms]] is a [[dg-algebra]] morphism
$$
\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : A
,.
$$
It [[curvature]] is the composite of morphism of [[graded vector space]]
$$
\Omega^\bullet(U)
\stackrel{A}{\leftarrow}
W(\mathfrak{g})
\stackrel{F_{(-)}}{\leftarrow}
\mathfrak{g}^*[1]
:
F_A
\,.
$$
Since $A$ is a [[homomorphism]] of [[dg-algebra | dg-algebras]], this satisfies
$$
d_{dR} F_A + A(d_{W(\mathfrak{g})}(-)) = 0
\,.
$$
This identity is the **Bianchi identity for $\infty$-Lie algebra valued forms**.
## Related concepts
* [[Gauss law]]
* [[∞-Lie algebroid valued differential forms]]
* [[connection on an ∞-bundle]]
* [[curvature]]
* [[adjusted Weil algebra]]
* [[curvature characteristic form]]
* [[Chern-Simons form]]
## References
Named after *[[Luigi Bianchi]]*.
The Bianchi identity for [[∞-Lie algebroid valued differential forms]] is discussed in
* SSSI (<a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#SSSI">web</a>)
[[!redirects Bianchi identities]] |
Biao Lian | https://ncatlab.org/nlab/source/Biao+Lian |
* [Institute page](https://phy.princeton.edu/people/biao-lian)
## Selected writings
On [[Chern-Simons theory]] potentially applicable to a form of [[Berry curvature]] on the [[Brillouin torus]] of 3d [[topological semimetals]], with nodal lines playing the role of [[Wilson lines]]:
* [[Biao Lian]], [[Cumrun Vafa]], [[Farzan Vafa]], [[Shou-Cheng Zhang]], *Chern-Simons theory and Wilson loops in the Brillouin zone*, Phys. Rev. B **95** (2017) 094512 [[doi:10.1103/PhysRevB.95.094512](https://doi.org/10.1103/PhysRevB.95.094512)]
On [[Shoucheng Zhang]]:
* [[Biao Lian]] et al (eds). *Memorial Volume for Shoucheng Zhang*, World Scientific (2021) [[doi:10.1142/12146](https://doi.org/10.1142/12146)]
category: people
|
biased definition | https://ncatlab.org/nlab/source/biased+definition |
# Biased definitions
* table of contents
{: toc}
## Idea
Often in mathematics, when requiring some structure/operation/property/... to exist at every [[finite set|finite]] [[arity]], it suffices to require only the binary ($2$-ary) and nullary ($0$-ary) forms, from which the others follow. A definition in which only these are required is called __biased__.
For example, in defining a [[category]], one could use an "unbiased" definition in which [[composites]] of all [[list|finite sequences]] of [[morphisms]] are directly postulated, with corresponding [[associativity]] laws, but it suffices to require only binary composites and nullary composites (i.e., [[identity morphisms]]) and some particular corresponding associativity laws.
As a special case of this, we have perhaps the prototypical example of a binary/nullary pair sufficing to generate all finite instances of some structure: the [[natural numbers]] themselves (the arities of the operations that we are considering) are the [[free monoid]] on one generator, and thus are freely associatively produced from that one generator (aka, $1$) using only binary and nullary addition.
Sometimes it is too easy to write a definition that involves only the binary condition; writing an unbiased definition can make it easier to notice the corresponding nullary condition. Compare when things are [[too simple to be simple]].
## When a nullary operation does not exist
Sometimes a nullary operation does not exist but one still wants to decompose a n-ary operation into binary operations. For example, consider [[real number|the reals]], $\mathbb{R}$, as an [[lattice|unbounded lattice]] ([[top|top]], $\top$, and [[bottom|bottom]], $\bot$, do not exist) where
$\wedge =$ [[product]] $=$ [[meet]] $= infimum = min$
and
$\vee =$ [[coproduct]] $=$ [[join]] $= supremum = max$.
Here $\bigwedge(\{\})$ does not exist while
$\bigwedge(\{a\}) = a$
$\bigwedge(\{a,b\}) = a \wedge b$
$\bigwedge(\{a,b,c\}) = a \wedge (b \wedge c) = (a \wedge b) \wedge c = a \wedge b \wedge c$.
One approach is to compute in the [[extended real number|extended reals]], $\mathbb{R}_{\pm \infty}$ ($\mathbb{R}$ enlarged with $+\infty$ and $-\infty$.) Here
$\top = +\infty =$ [[terminal object]]
and
$\bot = -\infty =$ [[initial object]].
In $\mathbb{R}_{\pm \infty}$ we have the nullary $\wedge() = +\infty$ which gives:
$\bigwedge(\{\}) = +\infty$
$\bigwedge(\{a\}) = \bigwedge(\{a, +\infty\}) = a$.
Another approach is to define a special scheme for composition for when a nullary operator does not exist that instead uses a unary operator that is an identity map (or factored through one).
This approach generalizes to any [[semigroup]] or to any category with binary ([[coproduct|co]]) [[products]]. A yet more general context (possibly not fully worked out) would be a [[binary operation]] in any [[associative operad]].
[[!redirects binary/nullary pair]]
[[!redirects binary/nullary pairs]]
[[!redirects binary-nullary pair]]
[[!redirects binary-nullary pairs]]
[[!redirects binary nullary pair]]
[[!redirects binary nullary pairs]]
[[!redirects bias]]
[[!redirects biased]]
[[!redirects unbiased]]
[[!redirects biased definition]]
[[!redirects biased definitions]]
[[!redirects unbiased definition]]
[[!redirects unbiased definitions]]
|
bibundle | https://ncatlab.org/nlab/source/bibundle |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
A _bibundle_ is a ([[groupoid-principal bundle|groupoid]]-)[[principal bundle]] which is equipped with a compatible second ([[groupoid action|groupoid]]-)[[action]] "from the other side".
In particular, Lie groupoid bibundles serve to exhibit "generalized morphisms"/[[Morita morphisms]] between [[Lie groupoids]]. This is in generalization of how the [[differentiable stack]]/[[smooth groupoid]] represented by a [[Lie groupoid]] is the [[moduli stack]] for [[groupoid-principal bundles]].
Therefore groupoid bibundles play a role in [[geometry]] analogous to the role played by [[bimodules]] in [[algebra]]. In this role they were originally introduced in ([Haefliger 84](#Haefliger), [Hilsum-Skandalis 87](#HilsumSkandalis87), [Pradines 89](#Pradines89)) and accordingly they are also called _Hilsum-Skandalis maps_. Independently they were seen in [[topos theory]] ([Bunge 90](#Bunge), [Moerdijk 91](#Moerdijk91)).
Historically, a central motivation for their study has been that the [[groupoid convolution algebra]] construction sends smooth bibundles between [[Lie groupoids]] to ([[Hilbert bimodule|Hilbert-]])[[bimodules]] of the corresponding [[C-star-algebra|C-star]] [[convolution algebras]], such that [[Morita equivalence]] is respected ([Muhly-Renault-Williams 87](#MuhlyRenaultWilliams), [Landsman 00](#Landsman), [Mrčun 05](#Mrcun)). This is of relevance notably for [[KK-theory]] of Lie groupoids ([Hilsum-Skandalis 87](#HilsumSkandalis87)).
Bibundles also appear as _transition bundles_ of [[nonabelian bundle gerbes]].
## Properties
### Lie groupoid bibundles and Morita/stack morphisms
{#LieGroupoidBibundlesAndMoritaMorphisms}
We discuss how Lie groupoid bibundles correspond to [[Morita morphism]] (morphisms of [[differentiable stacks]]/[[smooth stacks]]) between the Lie groupoids.
First we set up the relevant definitions and establish our notation in
* [Lie groupoids and smooth stacks](#LieGroupoidsAndSmoothStacks)
Then we discuss smooth groupoid-principal bundles and how a Lie groupoid [[moduli stack]] for the bundles principal over it in
* [Smooth groupoid-principal bundles](#SmoothGrooupoidPrincipalBundles).
Finally we consider the corresponding smooth bibundles and how they correspond to their modulating stack morphisms in
* [Smooth groupoid-principal bibundles](#SmoothGroupoidPrincipalBibundles)
#### Lie groupoids and smooth stacks
{#LieGroupoidsAndSmoothStacks}
A _[[smooth stack]]_ or _[[smooth groupoid]]_ is a [[stack]] on the site [[SmoothMfd]] of [[smooth manifolds]] or equivalently (and often more conveniently) on its [[dense subsite]] [[CartSp]] of just [[Cartesian spaces]] $\mathbb{R}^n, n \in \mathbb{N}$ and [[smooth functions]] between them, equipped with the standard [[coverage]] of [[good open covers]].
We write
$\;\;\; $[[SmoothGrpd]] $\coloneqq Sh_{(2,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, Grpd) $
for the [[(2,1)-category]] of [[stacks]] on this site, equivalently the result of taking [[groupoid]]-valued [[presheaves]] and then [[simplicial localization|universally]] turning local (as seen by the [[coverage]]) [[equivalences of groupoids]] into global [[equivalence in an (infinity,1)-category]].
By generalizing here [[groupoids]] to general [[Kan complexes]] and [[equivalences of groupoids]] to [[homotopy equivalences]] of Kan complexes, one obtains _smooth [[∞-stacks]]_ or _[[smooth ∞-groupoids]]_, which we write
$\;\;\;$ [[Smooth∞Grpd]] $\coloneqq Sh_{(\infty,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, KanCplx) $.
We often write $\mathbf{H} \coloneqq$ [[Smooth∞Grpd]] for short.
By the [[(∞,1)-Yoneda lemma]] there is a sequence of [[full and faithful (∞,1)-functor|faithful inclusions]]
$\;\;\;$ [[SmoothMfd]] $\hookrightarrow$ [[SmoothGrpd]] $\hookrightarrow$ [[Smooth∞Grpd]].
This induces a corresponding inclusion of [[simplicial objects]] and hence of [[groupoid objects in an (∞,1)-category|groupoid objects]]
$$
LieGrpd \hookrightarrow Grpd_\infty(SmoothMfd) \hookrightarrow Grpd_\infty(Smooth\infty Grpd)
\,.
$$
For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a groupoid object we write
$$
\mathcal{G}_0 \to \mathcal{G} \coloneqq \underset{\longrightarrow}{\lim}_{n} \mathcal{G}_n
$$
for its [[(∞,1)-colimit|(∞,1)-colimiting]] [[cocone]], hence $\mathcal{G} \in \mathbf{H}$ (without subscript decoration) denotes the [[realization]] of $\mathcal{G}_\bullet$ as a single object in $\mathbf{H}$.
+-- {: .num_defn }
###### Definition
For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a [[groupoid object in an (infinity,1)-category|groupoid object]], we write $\mathcal{G} \coloneqq {\lim}_{n} \mathcal{G}_n \in \mathbf{H}$ for its **[[realization]]** and call the canonical [[1-epimorphism]]
$$
\mathcal{G}_0 \to \mathcal{G}
$$
the canonical **[[atlas]]** of this realization.
=--
+-- {: .num_example }
###### Example
For $\mathcal{G}_\bullet \in Grpd(SmoothMfd) \hookrightarrow Grpd_\infty(Smooth\infty Grpd)$ a [[Lie groupoid]], we have that
1. $\mathcal{G}_0 \in SMoothMfd \hookrightarrow Smooth\infty Gprd$ is its [[smooth manifold]] of [[objects]]
1. $\mathcal{G} \in $ [[SmoothGrpd]] $\hookrightarrow$ [[Smooth∞Grpd]] is the realization of the Lie groupoid as a [[differentiable stack]], hence as a [[smooth groupoid]]
1. $\mathcal{G}_0 \to \mathcal{G}$ is the canonically induced [[atlas]] in the traditional sense of [[geometric stack]]-theory.
=--
+-- {: .num_remark }
###### Remark
By the [[Giraud-Rezk-Lurie axioms]] of the [[(∞,1)-topos]] $\mathbf{H}$ this morphism $\mathcal{G}_0 \to \mathcal{G}$ is a [[1-epimorphism]] and its construction establishes is an [[equivalence of (∞,1)-categories]] $Grpd_\infty(\mathbf{H}) \simeq \mathbf{H}^{\Delta^1}_{1epi}$, hence morphisms $\mathcal{G}_\bullet \to \mathcal{K}_\bullet$ in $Grpd_\infty(\mathbf{H})$ are equivalently [[diagrams]] in $\mathbf{H}$ of the form
$$
\array{
\mathcal{G}_0 &\to& \mathcal{K}_0
\\
\downarrow &\swArrow& \downarrow
\\
\mathcal{G} &\to& \mathcal{K}
}
\,.
$$
=--
This is evidently more constrained than just morphisms
$$
\mathcal{G} \to \mathcal{K}$ in $\mathbf{H}
$$
by themselves. The latter are the _generalized morphisms_ or _[[Morita morphisms]]_ between the groupoid objects $\mathcal{G}_\bullet$, $\mathcal{K}_\bullet$.
+-- {: .num_defn #MoritaMorphism}
###### Definition
Given [[groupoid objects in an (∞,1)-category|groupoid objects]] $\mathcal{G}_\bullet, \mathcal{K}_\bullet \in Grpd_\infty(\mathbf{H})$, a **[[Morita morphism]]** between them is a morphism $\mathcal{G} \to \mathcal{K}$ in $\mathbf{H}$ between their [[realizations]]. A Morita morphism that is an [[equivalence in an (infinity,1)-category|equivalence]] in $\mathbf{H}$ is called a **[[Morita equivalence]]** of groupoid objects in $\mathbf{H}$.
=--
Here we want to express these Morita morphisms $\mathcal{G} \to \mathcal{K}$ in terms of bibundle objects $\mathcal{P} \in \mathbf{H}$ on which both $\mathcal{G}_\bullet$ and $\mathcal{K}_\bullet$ [[action|act]].
+-- {: .num_example}
###### Example
For $X \in \mathbf{H}$ any [[object]], its [[pair groupoid]] $Pair(X)_\bullet \in Grpd_\infty(\mathbf{H})$ is
$$
Pair(X)_n \coloneqq X^{\times^{n+1}}
\,.
$$
The [[realization]] of this is equivalent to the point
$$
Pair(X) \coloneqq \underset{\longrightarrow}{\lim}_n Pair(X)_n
\simeq *
\,.
$$
Hence all [[Morita morphisms]], def. \ref{MoritaMorphism}, to the pair groupoid are equivalent. As a groupoid object $Pair(X)_\bullet$ is non-trivial, but it is [[Morita equivalence|Morita equivalent]] to the terminal groupoid object.
=--
#### Smooth groupoid-principal bundles
{#SmoothGrooupoidPrincipalBundles}
+-- {: .num_defn #GroupoidAction}
###### Definition
For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a [[groupoid object in an (∞,1)-category|groupoid object]], $X \in \mathbf{H}$ any object equipped with a morphism $a \colon X \to \mathcal{G}_0$ to the object of objects of $\mathcal{G}$, a **$\mathcal{G}_\bullet$-[[groupoid ∞-action]]** on $X$ with **anchor** $a$ is a [[groupoid object in an (∞,1)-category|groupoid]] $(X//\mathcal{G})_\bullet$ over $\mathcal{G}_\bullet$ of the form
$$
\array{
\vdots && && \vdots
\\
\downarrow \downarrow \downarrow \downarrow
&& &&
\downarrow \downarrow \downarrow \downarrow
\\
X \underset{\mathcal{G}_0}{\times} \mathcal{G}_2
&& \to && \mathcal{G}_2
\\
\downarrow \downarrow \downarrow && &&
\downarrow \downarrow \downarrow
\\
X \underset{\mathcal{G}_0}{\times} \mathcal{G}_1
&& \to && \mathcal{G}_1
\\
\downarrow \downarrow && && \downarrow \downarrow
\\
X && \stackrel{a}{\to} && \mathcal{G}_0
}
\,,
$$
where the [[homotopy fiber products]] on the left are those of the anchor $a$ with the leftmost 0-face map $\mathcal{G}_{(\{0\} \hookrightarrow \{0, \cdots, n\})}$ and the horizontal morphisms are the corresponding [[projections]] on the second factor.
We call $(X//\mathcal{G})_\bullet$ also the **[[action groupoid]]** of the action of $\mathcal{G}_\bullet$ on $(X,a)$ and call its [[realization]] $X \to (X//\mathcal{G})$ the [[homotopy quotient]] of the action.
=--
+-- {: .num_example }
###### Example
For $\mathcal{G}_\bullet = (\mathbf{B}G)_\bullet$ the [[delooping]] of a [[group object in an (∞,1)-category|group object]], def. \ref{GroupoidAction} reduces to the definition of an [[∞-action]] of the [[∞-group]] $G$.
=--
Under this relation, the discussion of [[∞-groupoid-principal ∞-bundles]] proceeds in direct analogy with that of $G$-[[principal ∞-bundles]]:
+-- {: .num_prop }
###### Proposition
For $X \in $ [[Smooth∞Grpd]] any object, a morphism $f \colon X \to \mathcal{G}$ in $\mathbf{}H$ induces ("modulates") a $\mathcal{G}_\bullet$-[[groupoid action]], def. \ref{GroupoidAction}, on the [[homotopy pullback]] $f^\ast \mathcal{G}_0$
$$
\array{
f^* \mathcal{G}_0 &\to& \mathcal{G}_0
\\
\downarrow &pb_\infty& \downarrow
\\
X &\stackrel{f}{\to}& \mathcal{G}
\,.
}
$$
of the [[atlas]] of $\mathcal{G}$: the corresponding [[action groupoid]] is the [[Cech nerve]] of the [[projection]] $p \colon f^*\mathcal{G}_0 \to X$ (which as the [[(∞,1)-pullback]] of a [[1-epimorphism]] is itself a [[1-epimorphism]]):
$$
\array{
&& \vdots && \vdots
\\
&& \downarrow \downarrow \downarrow &&
\downarrow \downarrow \downarrow
\\
&& (f^\ast \mathcal{G}_0) \underset{\mathcal{G}_0}{\times} \mathcal{G}_1
&\to&
\mathcal{G}_1
\\
&& \downarrow \downarrow && \downarrow \downarrow
\\
&& f^* \mathcal{G}_0 &\stackrel{a}{\to}& \mathcal{G}_0
\\
&& \downarrow &pb_\infty& \downarrow
\\
(f^\ast \mathcal{G}_0)//\mathcal{G} &\simeq& X &\stackrel{f}{\to}& \mathcal{G} &\simeq& \underset{\longrightarrow}{\lim}_n \mathcal{G}_n
\,.
}
$$
=--
+-- {: .num_example #LieGroupoidPrincipalBundle}
###### Example
Let $f_\bullet \colon X_\bullet \to \mathcal{G}_\bullet$ be a morphism of 1-groupoid objects, say of [[Lie groupoids]]. Then, as discussed at _[[homotopy pullback]]_, the [[(∞,1)-pullback]] of the [[atlas]] $\mathcal{G}_0 \to \mathcal{G}$ along the [[realization]] $f$ is computed as the 1-categorical [[pullback]]
$$
\array{
&\to& (\mathcal{G}_0 \underset{\mathcal{G}}{\times} \mathcal{G}^{\Delta^1})_\bullet
\\
\downarrow &pb& \downarrow
\\
\mathcal{X}_\bullet &\to& \mathcal{G}_\bullet
}
$$
in $Sh(CartSp)^{\Delta^{op}}$. Schematically the groupoid on the right has morphisms $ \gamma_0 \to \gamma_1$ which are [[commuting diagrams]] in $\mathcal{G}$ of the form
$$
\array{
&& g
\\
& {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}}
\\
g_0 && \to && g_1
}
\,.
$$
Therefore the pullback is the sheaf of groupoids which is schematically of the form
$$
f^\ast \mathcal{G}_0
\;\simeq\;
\left\{
\array{
&& g
\\
& {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}} & && \in \mathcal{G}
\\
f(x_0) && \stackrel{f(\xi)}{\to} && f(x_1)
\\
x_0 && \stackrel{\xi}{\to}&& x_1 && \in \mathcal{X}
}
\right\}
\,.
$$
In this presentation now
$$
\mathcal{G}_1
\;\simeq\;
\left\{
\array{
&& g_0
\\
&& \downarrow^{\gamma}
\\
&& g_1
\\
& {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}}
\\
g_{1 0} && \to && g_{11}
}
\right\}
\,.
$$
and the target map $\mathcal{G}_1 \to \mathcal{G}_0$ is given by forgetting the top vertical morphism in this diagram, while the source map is given by composing (!) the top vertical morphism with the two diagonal morphism.
Pullback of these two maps induces the left and right vertical map in
$$
\array{
f^\ast \mathcal{G}_0 \underset{\mathcal{G}_0}{\times}
\mathcal{G}_1
&\to& \mathcal{G}_1
\\
\downarrow\downarrow && \downarrow \downarrow
\\
f^\ast \mathcal{G}_0 &\stackrel{a}{\to}& \mathcal{G}_0
}
\,.
$$
from
$$
f^\ast \mathcal{G}_0 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1
\;\simeq\;
\left\{
\array{
&& g_0
\\
&& \downarrow^{\gamma}
\\
&& g_1 && && \in \mathcal{G}
\\
& {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}} &
\\
f(x_0) && \stackrel{f(\xi)}{\to} && f(x_1)
\\
x_0 && \stackrel{\xi}{\to} && x_1 & & \in \mathcal{X}
}
\right\}
\,.
$$
The left one just forgets the top vertical morphism, the right one composes it with the diagonal morphisms. This composion is the $\mathcal{G}_\bullet$-action on $f^\ast \mathcal{G}_0$.
=--
#### Smooth groupoid-principal bibundles
{#SmoothGroupoidPrincipalBibundles}
Finally then for $\mathcal{X}_\bullet$ and $\mathcal{G}_\bullet$ two Lie groupoids and $f \;\colon\; \mathcal{X} \to \mathcal{G}$ a morphism in [[Smooth∞Grpd]] between the corresponding [[differentiable stacks]], we obtain first the $\mathcal{G}$-[[groupoid principal bundle]] $f^* \mathcal{G}_0 \stackrel{p}{\to} \mathcal{X}$ and then by further homotopy pullback also the left $\mathcal{X}$-[[groupoid principal bundle]] $p^* \mathcal{X}_0$:
+-- {: .num_defn }
###### Definition
For $\mathcal{X}_\bullet, \mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ two [[groupoid object in an (infinity,1)-category|groupoid objects]] and $f \colon \mathcal{X} \to \mathcal{G}$ a [[Morita morphism]] between them, def. \ref{MoritaMorphism}, we say that the corresponding **$\mathcal{X}_\bullet-\mathcal{G}_\bullet$-bibundle** $\mathcal{P}(f)$ is the $\mathcal{G}_\bullet$-[[groupoid-principal bundle]] $f^\ast \mathcal{G}_0$ pulled back to the canonical [[atlas]] of $\mathcal{X}$ and equipped with the induced $\mathcal{X}_\bullet$-[[groupoid action]]:
$$
\array{
\mathcal{P}(f)& \coloneqq& p^* \mathcal{X}_0 &\to&
f^* \mathcal{G}_0&\to& \mathcal{G}_0
\\
&& \downarrow &\pb_\infty& \downarrow^{\mathrlap{p}} &pb_\infty& \downarrow
\\
&& \mathcal{X}_0
&\to&
\mathcal{X}
&\stackrel{f}{\to}&
\mathcal{G}
}
\,.
$$
=--
+-- {: .num_remark }
###### Remark
Here the $\mathcal{G}_\bullet$-action on $\mathcal{P}(f)$ is principal over $\mathcal{X}_0$, in that the quotient map is
$$
\mathcal{P}(f) \to \mathcal{P}(f)//\mathcal{G} \simeq \mathcal{X}_0
\,,
$$
since $\mathcal{P}(f)$ is the pullback of a $\mathcal{G}_\bullet$-principal bundle (modulated by the bottom composite map in the above diagram).
On the other hand the $\mathcal{X}_\bullet$-action on $\mathcal{P}(f)$ is not principal over $\mathcal{G}_0$ -- unless $f$ is an [[equivalence in an (infinity,1)-category]] (hence a ([[Morita equivalence]]) from $\mathcal{X}_\bullet$ to $\mathcal{G}_\bullet$.) It is instead always principal over $f^\ast \mathcal{G}_0$.
=--
Thus we arrive at an equivalent, however more basic definition of Lie groupoid bibundle:
+-- {: .num_defn }
###### Definition
Given Lie groupoids $G:=G_1\Rightarrow G_0$ and $H:=H_1\Rightarrow H_0$, a $G$-$H$-**bibundle** is a principal $H$-bundle $E \xrightarrow{\pi_G} G_0$ over $G_0$ with anchor $E\xrightarrow{\pi_H} H_0$ together with a left $G$-action (see [here](http://ncatlab.org/nlab/show/groupoid+principal+bundle#lie_groupoid_principal_bundles) ) with anchor $\pi_G$, such that the two actions commute. If the $G$-action also gives rise to a principal bundle over $H_0$, then $E$ induces a [[Morita equivalence]] between $G$ and $H$ and it is sometimes called a **Morita bibundle** in this case.
=--
+-- {: .num_example }
###### Example
Given a manifold $M$, and two open covers $\{U_i\}$ and $\{V_\alpha\}$, we may form two Cech groupoids (see [here](http://ncatlab.org/nlab/show/Lie+groupoid#examples_for_lie_groupoids) ) $\sqcup U_{ij} \Rightarrow \sqcup U_i$ and $\sqcup V_{\alpha \beta} \Rightarrow \sqcup V_\alpha$. Then $\sqcup_{i, \alpha} U_i \times_{M} V_\alpha
$ (which is a common refinement of $\{U_i\}$ and $\{V_\alpha\}$) is a Morita bibundle. The actions are
$$(x_i, x_j)\cdot (x_j, x_\alpha)=(x_i, x_\alpha), \quad (x_i, x_\alpha)\cdot (x_\alpha, x_\beta)=(x_i, x_\beta). $$
Obviously these actions are free. Moreover, it is also not hard to see that $\sqcup U_i\times_M V_\alpha/\sqcup V_{\alpha \beta} = \sqcup U_i$ and $\sqcup U_i\times_M V_\alpha/\sqcup U_{ij} = \sqcup V_\alpha$. When a free action has representible quotient, it must automatically be proper.
=--
##### Composition
Given a bibundle functor $E: G\to H$ and a bibundle functor $F: H\to K$ between Lie groupoids, the composition $E\circ F: G\to K$ is the quotient manifold $E\times_{H_0} F/H_1$ equipped with the remaining $G$ and $K$ action. Here $H$ acts on $E\times_{H_0} F$ from right by $(x, y)\cdot h_1=(x\cdot h_1^{-1}, y \cdot h_1)$. It is free and proper because the right action of $H$ on $E$ is so. Then $G$ action and $K$ action descend to the quotient $E\times_{H_0} F/H_1$. Moreover, those who free and proper is (are), remains so.
Thus bibundle functors compose to a bibundle functor, and Morita bibundles compose to a Morita bibundle.
Then we see that there is a $(2,1)$-category $BUN$ with objects Lie groupoids, 1-morphisms bibundle functors, and 2-morphisms isomorphisms of bibundles. It is $(2,1)$-category because 2-morphisms are obviously invertible. This $(2,1)$-category is equivalent to the one obtained by generalised morphism or by anafunctors.
##### Bundlisation #####
Given a strict morphism $G\xrightarrow{f} H$, then we may form a bibundle $E:= G_0\times_{f_0, H_0, t} H_1$ with right $H$ action induced by $H$-multiplication and with left $G$ action induced by $G$-action on $G_0$. Bundlisation preserves composition.
+-- {: .num_remark #TruncationOfBibundle}
###### Remark
If both [[atlases]] are [[0-truncated]] objects ([[smooth spaces]]) $\mathcal{X}_0, \mathcal{G}_0 \in Sh(CartSp) \simeq \tau_1 \mathbf{H} \hookrightarrow \mathbf{H}$, then by the [[pasting law]] for [[homotopy pullbacks]] we have that $\mathcal{P}(f)$ is [[n-truncated|(n-1)-truncated]] if $\mathcal{G}$ is [[n-truncated]].
In particular therefore the total space of a [[smooth groupoid|smooth 1-groupoid]] bibundle is 0-truncated hence is a [[smooth space]].
=--
+-- {: .num_example }
###### Example
In order to discuss Lie-groupoid bibundles we continue the discussion in example \ref{LieGroupoidPrincipalBundle} of Lie-groupoid principal bundles. Proceeding for the second homotopy pullback diagram as discussed there for the first one, one finds that the total space $\mathcal{P}$ of the bibundle is presented by the sheaf of groupoids whose schematic depiction is
$$
f^\ast \mathcal{G}_0
\;\simeq\;
\left\{
\array{
&& g
\\
& {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}} & && \in \mathcal{G}
\\
f(x_0) && \stackrel{f(\xi)}{\to} && f(x_1)
\\
x_0 && \stackrel{\xi}{\to}&& x_1
\\
& \searrow && \swarrow & && \in \mathcal{X}
\\
&& x
}
\right\}
\,.
$$
Here the vertically-running morphisms are the [[objects]] and two such are related by a [[morphism]] if they fit into a commuting diagram complete by horizontal morphisms as indicated. Since $\mathcal{X}_\bullet$ and $\mathcal{G}_\bullet$ both are groupoids, these morphisms are unique if they exist, and hence, as predicted by remark \ref{TruncationOfBibundle}, $\mathcal{P}(f)$ is 0-truncated, hence is a [[smooth space]]. Moreover, since the [[isomorphism]] [[equivalence relation]] here is free, the [[quotient]] [[smooth space]] is actually a [[smooth manifold]] (since $\mathcal{X}_\bullet$ and $\mathcal{G}_\bullet$ are [[Lie groupoids]]).
This then recovers the definition of bibundles for Lie groupoids as often found in the literature.
The right $\mathcal{G}_\bullet$-action is by precomposition of these diagram with morphisms in $\mathcal{G}$, while the left $\mathcal{X}$-action is by postcomposition with morphisms in $\mathcal{X}$.
=--
Conversely, given a $\mathcal{X}_\bullet$-$\mathcal{G}_\bullet$-Lie groupoid bibundle which is principal on the left
$$
\array{
\mathcal{P} &\to& \mathcal{B} &\to& \mathcal{G}_0
\\
\downarrow && && \downarrow
\\
\mathcal{X}_0 &\to& \mathcal{X} && \mathcal{G}
}
$$
we recover the Morita morphism $f$ that it coresponds to by the [[Giraud-Rezk-Lurie axioms]]: first $p$ is the induced map between the [[homotopy colimits]] of the Cech nerves of the two left horizontal maps
$$
\array{
\mathcal{P} &\to& \mathcal{B} &\to& \mathcal{G}_0
\\
\downarrow && \downarrow^{\mathrlap{p}} && \downarrow
\\
\mathcal{X}_0 &\to& \mathcal{X} && \mathcal{G}
}
$$
and then $f$ is similarly the map between the homotopy colimits of the Cech nerves of the two right vertical maps.
(...)
### Relation to groupoid convolution bimodules
{#RelationToGroupoidConvolutionBimodules}
There should be a [[2-functor]] from [[Lie groupoids]] to [[C-star-algebras]] and [[Hilbert C-star-bimodules]] between them given by forming [[groupoid convolution algebras]] and naturally exhibited by Lie groupoid bibundles: the [[groupoid convolution algebra]] of the total space of the bibindle becomes a [[bimodule]] over the two other groupoid convolution algebras.
Some aspects of this are in the literature, e.g. ([Mrčun 99](#Mrcun99)) for [[étale Lie groupoids]] and ([Landsman 00](#Landsman00)) for general [[Lie groupoids]]. The follwing is taken from the latter article.
+-- {: .num_defn #BundleOfVertical}
###### Definition/Notation
For $p \colon E \to X$ a [[smooth function]] between smooth manifolds, we write $T^p E \hookrightarrow T E$ for the bundle of [[vertical vector fields]], the sub-bundle of the [[tangent bundle]] of $E$ on those vectors in the [[kernel]] of the [[differentiation]] maps $d p|_{e} \colon T_e E \to T_{\tau(e)} X$.
We write ${\vert \Lambda\vert^{1/2}}(T^\tau E)$ for the bundle of [[half-densities]] on vertical vector fields.
=--
+-- {: .num_remark #ActionOnVerticalVectors}
###### Remark
Let $\mathcal{G}_\bullet$ be a [[Lie groupoid]] and let ($E \stackrel{\tau}{\to} \mathcal{G}_0, \rho)$ be a $\mathbb{G}_\bullet$-[[groupoid-principal bundle]] $E \to E//\mathcal{G}$ (with anchor $\tau$ and action map $\rho$).
Then the bundle of [[vertical vector fields]] $T^\tau E$ equipped with the anchor map $T^\tau E \stackrel{d \tau}{\to} T \mathcal{G}_0 \to \mathcal{G}_0$ inherits a canonical $\mathcal{G}_\bullet$-action itself.
The [[quotient]] map
$$
{\vert \Lambda\vert^{1/2}}(T^\tau E)/\mathcal{G} \to E/\mathcal{G}
$$
exists and is naturall a vector bundle again.
=--
+-- {: .num_defn #TwoVersionsOfHalfDensities}
###### Definition
In the situation of remark \ref{ActionOnVerticalVectors}, write
* $C^\infty_{c/G}(E, {\vert\Lambda\vert}^{1/2}(T^\tau) E)^G$
for the space of [[smooth function|smooth]] [[sections]] of the [[half-density]]-bundle of $T^\tau E$ which are $\mathcal{G}$-equivariant and which have [[compact support]] up to $\mathcal{G}$-action;
* $C^\infty_c(E/\mathcal{G}, {\vert \Lambda\vert}^{1/2}(T^\tau E))$
for the space of smooth sections with compact support of the quotient bundle.
=--
The following constructions work by repeatedly applying the following identification:
+-- {: .num_prop }
###### Proposition
In the situation of def. \ref{TwoVersionsOfHalfDensities}, there is a [[natural isomorphism]]
$$
C^\infty_{c/G}(E, {\vert\Lambda\vert}^{1/2}(T^\tau) E)^G
\simeq
C^\infty_c(E/\mathcal{G}, {\vert \Lambda\vert}^{1/2}(T^\tau E) )
\,.
$$
=--
The central definition here is now:
+-- {: .num_defn #BisectionsOnFiberProductOfTwoGActionSpaces}
###### Definition
For $(E_1, \tau_1)$, $(E_2, \tau_2)$ two principal $\mathcal{G}_\bullet$ manifolds, set
$$
(E_1, E_2)_{\mathcal{G}}
\coloneqq
C^\infty_c( E_1 \underset{\mathcal{G}_0}{\times}) E_2, {\vert\Lambda\vert^{1/2}(T^\tau E_1)} \otimes {\vert\Lambda\vert^{1/2}(T^\tau E_2)}
$$
=--
And the central fact is:
+-- {: .num_prop #TheBimdoduleConvolutionProduct}
###### Proposition
Given 3 $\mathcal{G}_\bullet$-manifolds $(E_i, \tau_i)$, $i \in \{1,2,3\}$, there is a [[smooth function]]
$$
\star
\;\colon\;
(E_1, E_2)_{\mathcal{G}}
\times
(E_2, E_3)_{\mathcal{G}}
\to
(E_1, E_3)_{\mathcal{G}}
$$
given on sections $\sigma_1, \sigma_2$ and points $(e_1, e_3)$ by
$$
\sigma_1 \star \sigma_2
\colon
(e_1, e_3)
\mapsto
\int_{\tau_2^{-1}(\tau_1 e_1)}
\sigma_1(e_1, -) \otimes \sigma_2(-,e_3)
\,,
$$
where the [[integration]] is against the [[measure]] that appears by tensoring two (of the four) [[half-densities]] in the integrand.
This operation is an associative and invoutive partial composition operation and hence defines a [[star-category]] whose [[objects]] are $\mathcal{G}_\bullet$-principal manifolds and whose spaces of morphisms are as in def. \ref{BisectionsOnFiberProductOfTwoGActionSpaces}.
=--
In particular one has the following identifications.
+-- {: .num_example #GroupoidConvolutionAlgebraFromBibundleCase}
###### Example
For $\mathcal{G}_1 \to \mathcal{G}_0$ regarded as a $\mathcal{G}_\bullet$-principal action space, there is a [[natural isomorphism]]
$$
(\mathcal{G}_1, \mathcal{G}_1)_{\mathcal{G}}
\simeq
C^\infty_c(\mathcal{G}_1, {\vert\Lambda\vert}^{1/2}(T^s \mathcal{G}_1) \otimes {\vert\Lambda\vert}^{1/2}(T^t \mathcal{G}_1))
$$
and the algebra structure on this by prop. \ref{TheBimdoduleConvolutionProduct} is isomorphic to the [[groupoid convolution algebra]] of smooth sections over $\mathcal{G}_\bullet$.
=--
More generally:
+-- {: .num_example }
###### Example
For $E \stackrel{\tau}{\to} \mathcal{G}_0$ any $\mathcal{G}$-principal manifold, we have a [[natural isomorphism]]
$$
(\mathcal{G}_1, E)_{\mathcal{G}}
\simeq
C^\infty_c(E_1, {\vert\Lambda\vert}^{1/2}(T^G E) \otimes {\vert\Lambda\vert}^{1/2}(T\tau E))
\,.
$$
=--
> We consider [[completion]] of all this to the [[C-star-algebra]] context (...)
Now we can put the pieces together and sends groupoid-bindunles to $C^\ast$-bimodules over the two [[groupoid convolution algebras]].
+-- {: .num_prop }
###### Proposition
Given two [[Lie groupoids]] $\mathcal{G}_\bullet$ and $\mathcal{K}_\bullet$ and given a [[Morita equivalence]] groupoid [[bibundle]] $E$ between them, we have
$$
N \coloneqq (\mathcal{G}_1, E)_{\mathcal{G}} \simeq (E, \mathcal{K})_{\mathcal{K}}
$$
and this identification makes $N$ into a $C^\ast(\mathcal{G}_\bullet)-C^\ast(\mathcal{K}_\bullet)$-pre-[[Hilbert bimodule]] as follows:
1. The identification $N \simeq (E, \mathcal{K}_1)_{\mathcal{K}}$ defines the right $C^\ast(\mathcal{K}_\bullet)$-action by example \ref{GroupoidConvolutionAlgebraFromBibundleCase}; and similarly the identification $N \simeq (\mathcal{G}_1, E)_{\mathcal{G}}$ defines a left $C^\ast(\mathcal{G}_\bullet)$-action.
1. The $C^\ast(\mathcal{K})$-valued [[inner product]] on $N$ is that induced by the composite
$$
(E,\mathcal{K}_1)_{\mathcal{K}}^\ast
\times
(E,\mathcal{K}_1)_{\mathcal{K}}
\stackrel{\simeq}{\to}
(\mathcal{K}_1)_{\mathcal{K}, E}
\times
(E,\mathcal{K}_1)_{\mathcal{K}}
\to
(\mathcal{K}_1, \mathcal{K}_1)_{\mathcal{K}}
\hookrightarrow
C^\ast(\mathcal{K}_\bullet)
\,.
$$
=--
## References
{#References}
### General
Groupoid bibundles were first considered for [[foliation groupoids]] in
* {#HilsumSkandalis87} [[Michel Hilsum]], [[Georges Skandalis]], _Morphismes K-orientés d'espaces de feuilles et functorialité en théorie de Kasparov_. Ann. Scient. Ec. Norm. Sup. 20 (1987), 325–390. ([numdam:ASENS_1987_4_20_3_325_0](http://www.numdam.org/item?id=ASENS_1987_4_20_3_325_0))
The generalization to arbitrary [[topological groupoids]] was considered in
* {#Haefliger84} [[André Haefliger]], _Groupoïdes d'holonomie et classifiants_, Astérisque 116 (1984), 70–97.
* {#Pradines89} [[Jean Pradines]], _Morphisms between spaces of leaves viewed as fractions_. Cahiers Top. Géom. Diff. Cat. XXX-3 (1989), 229–246 ([numdam:CTGDC_1989__30_3_229_0](http://www.numdam.org/item/?id=CTGDC_1989__30_3_229_0))
and independently in [[topos theory]] in
* {#Bunge90} [[Marta Bunge]], _An application of descent to a classification theorem. Math. Proc. Cambridge Phil. Soc. 107 (1990), 59–79.
* {#Moerdijk91} [[Ieke Moerdijk]], _Classifying toposes and foliations_. Ann. Inst. Fourier, Grenoble 41, 1 (1991), 189–209.
Groupoid bibundles are used in the context of [[groupoid convolution algebras]] as geometric analogs of [[bimodules]] in
* {#MuhlyRenaultWilliams} [[Paul Muhly]], [[Jean Renault]], and D. Williams, _Equivalence and isomorphism for groupoid $C^\ast$-algebras_, J. Operator Th. 17 (1987), 3–22.
* [[Klaas Landsman]], _The Muhly-Renault-Williams theorem for Lie groupoids and its classical counterpart_, Lett. Math. Phys. 54 (2000), no. 1, 43–59. ([arXiv:math-ph/0008005](http://arxiv.org/abs/math-ph/0008005))
{#Landsman}
{#Landsman00}
* {#Mrcun05} [[Janez Mrcun]], _Stability and invariants of Hilsum-Skandalis maps_ ([arXiv:math/0506484](http://arxiv.org/abs/math/0506484))
A review of [[Lie groupoid]]-bibundles and maps of [[differentiable stacks]] is in section 2 of
* {#Blohmann} [[Christian Blohmann]], _Stacky Lie groups_, Int. Mat. Res. Not. (2008) Vol. 2008: article ID rnn082 ([arXiv:math/0702399](http://arxiv.org/abs/math/0702399))
Discussion of [[Lie group cohomology]] and the [[string 2-group]] [[infinity-group extension]] in terms of Lie groupoid bibundles is in
* [[Chris Schommer-Pries]] _Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group_ [arXiv:0911.2483](http://arxiv.org/abs/0911.2483/)
{#Sp}
Talk notes on bibundles include
* [[Michael Murray]], _Bispaces and bibundles_ ([pdf slides](http://www.mpim-bonn.mpg.de/digitalAssets/7050_Murray.pdf))
* [[Jeffrey Morton]], ([pdf slides](http://www.math.uwo.ca/~jmorton9/seminar/gpdrep.pdf))
See also
* [[Michael Murray]], [[David Roberts]], [[Danny Stevenson]], _On the existence of bibundles_ LMS (2012)([arXiv:1102.4388](http://arxiv.org/abs/1102.4388))
### Convolution to $C^\ast$-bimodules
For groupoid bibundles between [[étale Lie groupoids]] the assignment of the [[groupoid convolution algebra]]-[[bimodule]] to them is shown to be [[functor|functorial]] in
* {#Mrcun99} [[Janez Mrcun]], _Functoriality of the bimodule associated to a Hilsum-Skandalis map_. K-Theory 18 (1999) 235–253.
For more references along these lines see for the moment at [groupoid convolution algebra -- Extension to bibundles and bimodules](http://ncatlab.org/nlab/show/category+algebra#ExtensionToBibundlesAndBimodules)
[[!redirects bibundle]]
[[!redirects bibundles]]
[[!redirects groupoid bibundle]]
[[!redirects groupoid bibundles]]
[[!redirects Hilsum-Skandalis morphism]]
[[!redirects Hilsum-Skandalis morphisms]]
[[!redirects Hilsum-Skandalis map]]
[[!redirects Hilsum-Skandalis maps]]
[[!redirects ∞-groupoid bibundle]]
[[!redirects ∞-groupoid bibundles]]
[[!redirects infinity-groupoid bibundle]]
[[!redirects infinity-groupoid bibundles]] |
bicartesian category | https://ncatlab.org/nlab/source/bicartesian+category | A [[category]] is __bicartesian__ if it is both cartesian and cocartesian, that is if both it and its [[opposite category|opposite]] may be made into [[cartesian monoidal categories]].
Note this is different than a [[semiadditive category]], i.e. a category with *coinciding* products and coproducts. Thus every semiadditive category is bicartesian, but not every bicartesian category is semiadditive.
A bicartesian category which is also [[cartesian closed category|cartesian closed]] is a __[[bicartesian closed category]]__. Bicartesian closed categories are usually *not* [[cocartesian closed category|cocartesian closed]].
## See also
* [[bicartesian preordered object]]
[[!redirects bicartesian]]
[[!redirects bicartesian categories]]
|
bicartesian closed category | https://ncatlab.org/nlab/source/bicartesian+closed+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--
A __bicartesian closed category__ is a [[cartesian closed category]] with finite [[coproducts]]. In the case where this is furthermore a [[preorder]] or [[poset]], it is called a [[Heyting prealgebra]] or [[Heyting algebra]], respectively. They provide the semantics and proof theory of [[intuitionistic logic|intuitionistic]] [[propositional logic]].
Note that a bicartesian closed category is [[bicartesian category|bicartesian]] (that is, it is both [[cartesian monoidal category|cartesian]] and [[cocartesian monoidal category|cocartesian]]), and furthermore it is cartesian closed, but it is usually *not* [[cocartesian closed category|cocartesian closed]] (as the only such category is the trivial terminal category), nor co-(cartesian closed) (i.e., the dual of a cartesian closed category; aka, cocartesian coclosed). Thus the terminology could be confusing, but since the only categories which are both cartesian closed and co-(cartesian closed) are preorders, there is not much danger.
Also note that a bicartesian closed category is automatically a [[distributive category]]. This follows since the functors $X\mapsto A\times X$ have right adjoints (by closedness), so they preserve colimits.
A bicartesian closed category is one kind of [[2-rig]].
## See also
* [[bicartesian closed preordered object]] |
bicartesian closed preordered object | https://ncatlab.org/nlab/source/bicartesian+closed+preordered+object |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
The notion of a *bicartesian closed preordered object* or *Heyting prealgebra object* is the generalization of that of *[[bicartesian closed preordered set]]* or *[[Heyting prealgebra]]* as one passes from the ambient [[category of sets]] [[internalization|into]] more general ambient [[categories]] with suitable properties.
## Definition
In a [[finitely complete category]] $C$, a **bicartesian closed preordered object** or **Heyting prealgebra object** is a [[bicartesian preordered object]]
$$(X, R, s, t, \rho, \tau_p \wedge, \top, \tau, \lambda_l, \lambda_r, \vee, \bot, \beta, \kappa_l, \kappa_r)$$
with a morphism $(-)\Rightarrow(-):X \times X \to X$ and functions
$$\epsilon_l:((* \to X) \times (* \to X)) \to (* \to R)$$
$$\epsilon_r:((* \to X) \times (* \to X)) \to (* \to R)$$
such that for all global elements $a:* \to X$ and $b:* \to X$,
* $s \circ \epsilon_l(a, b) = (a \Rightarrow b) \wedge a$
* $t \circ \epsilon_l(a, b) = b$
* $s \circ \epsilon_r(a, b) = a$
* $t \circ \epsilon_r(a, b) = b \Rightarrow (a \wedge b)$
## See also
* [[Heyting prealgebra]]
* [[bicartesian preordered object]]
* [[bicartesian closed category]]
* [[Boolean prealgebra object]]
* [[Heyting algebra object]]
[[!redirects bicartesian closed preordered objects]]
[[!redirects Heyting prealgebra object]]
[[!redirects Heyting prealgebra objects]] |
bicartesian preordered object | https://ncatlab.org/nlab/source/bicartesian+preordered+object |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
In a [[finitely complete category]] $C$, a **bicartesian preordered object** or **prelattice object** is a [[preordered object]] $X$ that is both a [[cartesian monoidal preordered object]] and a [[cocartesian monoidal preordered object]].
## See also
* [[cartesian monoidal preordered object]]
* [[cocartesian monoidal preordered object]]
* [[bicartesian closed preordered object]]
* [[lattice object]]
* [[bicartesian category]]
[[!redirects bicartesian preordered objects]]
[[!redirects prelattice object]]
[[!redirects prelattice objects]] |
bicategorical trace | https://ncatlab.org/nlab/source/bicategorical+trace |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A generalization of the notion of [[trace]] in a [[monoidal category]] to a [[bicategory]] context, based on equipping the bicategory with a "shadow" or categorified trace (including the [[trace of a category]]).
## Examples
The following traditional notions of "traces" turn out to be special cases of bicategorical traces
* All symmetric monoidal traces, such as the [[Lefschetz number]].
* The [[Reidemeister trace]].
## Related concepts
* [[trace]]
* [[higher trace]]
* [[span trace]]
## References
* [[Kate Ponto]], [[Michael Shulman]], _Shadows and traces in bicategories_ ([arXiv:0910.1306](https://arxiv.org/abs/0910.1306))
A survey is in
* [[Mike Shulman]], _Traces in monoidal categories and bicategories_ (2008) ([pdf slides](http://home.sandiego.edu/~shulman/papers/ccrtraces.pdf))
Relating the [[bicategorical trace]] to the [[Dennis trace]] on [[algebraic K-theory]]:
* {#CLMPZ20} [[Jonathan Campbell]], [[John Lind]], [[Cary Malkiewich]], [[Kate Ponto]], [[Inna Zakharevich]], _K-theory of endomorphisms, the TR-trace, and zeta functions_ [[arXiv:2005.04334](https://arxiv.org/abs/2005.04334)]
[[!redirects 2-bicategorical traces]]
[[!redirects 2-categorical trace]]
[[!redirects 2-categorical traces]]
|
bicategory | https://ncatlab.org/nlab/source/bicategory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
# Bicategories
* table of contents
{:toc}
## Idea
A **bicategory** is a particular [[algebraic definition of higher category|algebraic]] notion of _weak [[2-category]]_ (in fact, the earliest to be formulated, and still the one in most common use). The idea is that a bicategory is a category _[[weak enrichment|weakly enriched]]_ over [[Cat]]: the [[hom-objects]] of a bicategory are [[hom-category|hom-categories]], but the associativity and unity laws of [[enriched category|enriched categories]] hold only up to coherent isomorphism.
For information on [[morphisms]] of bicategories, see [[pseudofunctor]].
## Definition
A **bicategory** $B$ consists of
* A [[collection]] of __[[objects]]__ $x,y,z,\dots$, also called __$0$-cells__;
* For each pair of $0$-cells $x,y$, a [[category]] $B(x,y)$, whose objects are called __[[morphisms]]__ or __$1$-cells__ and whose morphisms are called __[[2-morphisms]]__ or __$2$-cells__;
* For each $0$-cell $x$, a distinguished $1$-cell $1_x\in B(x,x)$ called the __[[identity morphism]]__ or __identity $1$-cell__ at $x$;
* For each triple of $0$-cells $x,y,z$, a functor ${\circ}\colon B(y,z)\times B(x,y) \to B(x,z)$ called __[[horizontal composition]]__;
* For each pair of $0$-cells $x,y$, [[natural isomorphisms]] called __[[unitors]]__: $\left(
\begin{array}{rcl}
f&\mapsto&f \circ 1_x\\
\theta&\mapsto&\theta \circ 1_{1_x}
\end{array}
\right) \cong id_{B(x,y)} \cong \left(
\begin{array}{rcl}
f&\mapsto&1_y\circ f\\
\theta&\mapsto&1_{1_y} \circ \theta
\end{array}
\right):B(x,y)\rightarrow B(x,y)$
* For each quadruple of $0$-cells $w,x,y,z$, a natural isomorphism called the __[[associator]]__ between the two functors from $B(y,z) \times B(x,y) \times B(w,x)$ to $B(w,z)$ built out of ${\circ}$
such that
* The [[pentagon identity]] is satisfied by the [[associators]];
* And the triangle identity is satisfied by the [[unitors]].
If there is exactly one $0$-cell, say $*$, then the definition is exactly the same as a monoidal structure on the category $B(*,*)$. This is one of the motivating examples behind the [[delooping hypothesis]] and the general notion of [[k-tuply monoidal n-category]].
### Details
{#detailedDefn}
Here we spell out the above definition in full detail. Compare to the [detailed definition of strict $2$-category](/nlab/show/strict+2-category#detailedDefn), which is written in the same style but is simpler.
A bicategory $B$ consists of
* a collection $Ob B$ or $Ob_B$ of _objects_ or _$0$-cells_,
* for each object $a$ and object $b$, a collection $B(a,b)$ or $Hom_B(a,b)$ of _morphisms_ or _$1$-cells_ $a \to b$, and
* for each object $a$, object $b$, morphism $f\colon a \to b$, and morphism $g\colon a \to b$, a collection $B(f,g)$ or $2Hom_B(f,g)$ of _$2$-morphisms_ or _$2$-cells_ $f \Rightarrow g$ or $f \Rightarrow g\colon a \to b$,
equipped with
* for each object $a$, an _identity_ $1_a\colon a \to a$ or $\id_a\colon a \to a$,
* for each $a,b,c$, $f\colon a \to b$, and $g\colon b \to c$, a _composite_ $f ; g\colon a \to c$ or $g \circ f\colon a \to c$,
* for each $f\colon a \to b$, an _identity_ or _$2$-identity_ $1_f\colon f \Rightarrow f$ or $\Id_f\colon f \to f$,
* for each $f,g,h\colon a \to b$, $\eta\colon f \Rightarrow g$, and $\theta\colon g \Rightarrow h$, a _vertical composite_ $\theta \bullet \eta\colon f \Rightarrow h$,
* for each $a,b,c$, $f,g\colon a \to b$, $h\colon b \to c$, and $\eta\colon f \Rightarrow g$, a _left whiskering_ $h \triangleleft \eta \colon h \circ f \Rightarrow h \circ g$,
* for each $a,b,c$, $f\colon a \to b$, $g,h\colon b \to c$, and $\eta\colon g \Rightarrow h$, a _right whiskering_ $\eta \triangleright f\colon g \circ f \Rightarrow h \circ f$,
* for each $f\colon a \to b$, a _left unitor_ $\lambda_f\colon \id_b \circ f \Rightarrow f$, and an _inverse left unitor_ $\bar{\lambda}_f\colon f \Rightarrow \id_b \circ f$,
* for each $f\colon a \to b$, a _right unitor_ $\rho_f\colon f \circ \id_a \Rightarrow f$ and an _inverse right unitor_ $\bar{\rho}_f\colon f \Rightarrow f \circ \id_a$, and
* for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, an _associator_ $\alpha_{h,g,f}\colon (h \circ g) \circ f \Rightarrow h \circ (g \circ f)$ and an _inverse associator_ $\bar{\alpha}_{h,g,f}\colon h \circ (g \circ f) \Rightarrow (h \circ g) \circ f$,
such that
* for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\eta \bullet \Id_f$ and $\Id_g \bullet \eta$ both equal $\eta$,
* for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h \overset{\iota}\Rightarrow i\colon a \to b$, the vertical composites $\iota \bullet (\theta \bullet \eta)$ and $(\iota \bullet \theta) \bullet \eta$ are equal,
* for each $a \overset{f}\to b \overset{g}\to c$, the whiskerings $\Id_g \triangleright f$ and $g \triangleleft \Id_f$ both equal $\Id_{g \circ f }$,
* for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h\colon a \to b$ and $i\colon b \to c$, the vertical composite $(i \triangleleft \theta) \bullet (i \triangleleft \eta)$ equals the whiskering $i \triangleleft (\theta \bullet \eta)$,
* for each $f\colon a \to b$ and $g \overset{\eta}\Rightarrow h \overset{\theta}\Rightarrow i\colon b \to c$, the vertical composite $(\theta \triangleright f) \bullet (\eta \triangleright f)$ equals the whiskering $(\theta \bullet \eta) \triangleright f$,
* for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\lambda_g \bullet (\id_b \triangleleft \eta)$ and $\eta \bullet \lambda_f$ are equal,
* for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\rho_g \bullet (\eta \triangleright \id_a)$ and $\eta \bullet \rho_f$ are equal,
* for each $a \overset{f}\to b \overset{g}\to c$ and $\eta\colon h \Rightarrow i\colon c \to d$, the vertical composites $\bar{\alpha}_{i,g,f} \bullet (\eta \triangleright (g \circ f))$ and $((\eta \triangleright g) \triangleright f) \bullet \bar{\alpha}_{h,g,f}$ are equal,
* for each $f\colon a \to b$, $\eta\colon g \Rightarrow h\colon b \to c$, and $i\colon c \to d$, the vertical composites $\bar{\alpha}_{i,h,f} \bullet (i \triangleleft (\eta \triangleright f))$ and $((i \triangleleft \eta) \triangleright f) \bullet \bar{\alpha}_{i,g,f}$ are equal,
* for each $\eta\colon f \Rightarrow g\colon a \to b$ and $b \overset{h}\to c \overset{i}\to d$, the vertical composites $\bar{\alpha}_{i,h,g} \bullet (i \triangleleft (h \triangleleft \eta))$ and $((i \circ h) \triangleleft \eta) \bullet \bar{\alpha}_{i,h,f}$ are equal,
* for each $\eta\colon f \Rightarrow g\colon a \to b$ and $\theta\colon h \Rightarrow i\colon b \to c$, the vertical composites $(i \triangleleft \eta) \bullet (\theta \triangleright f)$ and $(\theta \triangleright g) \bullet (h \triangleleft \eta)$ are equal,
* for each $f\colon a \to b$, the vertical composites $\lambda_f \bullet \bar{\lambda}_f\colon f \Rightarrow f$ and $\bar{\lambda}_f \bullet \lambda_f\colon \id_b \circ f \Rightarrow \id_b \circ f$ equal the appropriate identity $2$-morphisms,
* for each $f\colon a \to b$, the vertical composites $\rho_f \bullet \bar{\rho}_f\colon f \Rightarrow f$ and $\bar{\rho}_f \bullet \rho_f\colon f \circ \id_a \Rightarrow f \circ \id_a$ equal the appropriate identity $2$-morphisms,
* for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, the vertical composites $\bar{\alpha}_{h,g,f} \bullet \alpha_{h,g,f}\colon (h \circ g) \circ f \Rightarrow (h \circ g) \circ f$ and $\alpha_{h,g,f} \bullet \bar{\alpha}_{h,g,f}\colon h \circ (g \circ f) \Rightarrow h \circ (g \circ f)$ equal the appropriate identity $2$-morphisms,
* for each $a \overset{f}\to b \overset{g}\to c$, the vertical composite $(\rho_g \triangleright f) \bullet \bar{\alpha}_{g,\id_b,f}$ equals the whiskering $g \triangleleft \lambda_f$, and
* for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d \overset{i}\to e$, the vertical composites $((\bar{\alpha}_{i,h,g} \triangleright f) \bullet \bar{\alpha}_{i,h \circ g,f}) \bullet (i \triangleleft \bar{\alpha}_{h,g,f})$ and $\bar{\alpha}_{i \circ h,g,f}\bullet \bar{\alpha}_{i,h,g \circ f} $ are equal.
It is quite possible that there are errors or omissions in this list, although they should be easy to correct. The point is not that one would *want* to write out the definition in such elementary terms (although apparently I just did anyway) but rather that one *can*.
## Examples
* Any [[strict 2-category]] is a bicategory in which the unitors and associator are identities. This includes [[Cat]], [[MonCat]], the algebras for any strict [[2-monad]], and so on, at least as classically conceived.
* A [[monoidal category]] $M$ may be regarded as a bicategory $B M$ with a single object $\bullet$. The objects $A$ of $M$ become 1-cells $[A]: \bullet \to \bullet$ of $B M$; these are composed across the 0-cell $\bullet$ using the definition $[A] \circ_0 [B] = [A \otimes B]$, using the monoidal product $\otimes$ of $M$. The identity 1-cell $\bullet \to \bullet$ is $[I]$, where $I$ is the monoidal unit of $M$. The morphisms $f: A \to B$ become 2-cells $[f]: [A] \to [B]$ of $B M$. The associativity and unit constraints of the monoidal category $M$ transfer straightforwardly to associativity and unit data of the bicategory $B M$. The construction is a special case of [[delooping]] (see there).
* Categories, [[anafunctor]]s, and natural transformations, which is a more appropriate definition of [[Cat]] in the absence of the [[axiom of choice]], form a bicategory that is not a strict 2-category. Indeed, without the axiom of choice, the proper notion of bicategory is [[anabicategory]].
* [[ring|Rings]], [[bimodule]]s, and bimodule homomorphisms are the prototype for many similar examples. Notably, we can generalize from rings to [[enriched category|enriched categories]].
* Objects, [[span]]s, and morphisms of spans in any category with [[pullback]]s also form a bicategory.
* The [[fundamental 2-groupoid]] of a space is a bicategory which is not necessarily strict (although it can be made strict fairly easily when the space is Hausdorff by quotienting by [[thin homotopy]], see [[path groupoid]] and [[fundamental infinity-groupoid]]). When the space is a CW-complex, there are easier and more computationally amenable equivalent strict 2-categories, such as that arising from the fundamental [[crossed complex]].
## Coherence theorems
{#Coherence}
One way to state the [[coherence theorem]] for bicategories is that every bicategory is [[equivalence of 2-categories|equivalent]] to a [[strict 2-category]]. This "[[rectification]]" is not obtained naively by forcing [[composition]] to be [[associativity|associative]], but (at least in one construction) by freely adding new composites which are strictly associative. Another way to state the coherence theorem is that every formal diagram of the constraints ([[associators]] and [[unitors]]) [[commuting diagram|commutes]].
Note that $n=2$ is the greatest value of $n$ for which every weak $n$-category is equivalent to a fully strict one; see [[semi-strict infinity-category]] and [[Gray-category]].
The [[proof]] of the coherence theorem is basically the same as the proof of the [[coherence theorem for monoidal categories]]. An abstract approach can be found in [Power 1989](#Power89). For a related statement see at *[[Lack's coherence theorem]]*.
The [[rectification]] [[adjunction]] between [[bicategories]] and [[strict 2-categories]] can be expressed in terms of a [[coreflection|coreflective]] [[triadjunction]] between [[tricategories]]; see [Campbell](#Campbell18).
\begin{tikzcd}
{2\text{-}\mathrm{Cat}} & {\mathrm{Bicat}}
\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{str}}"', shift right=2, from=1-2, to=1-1]
\arrow[""{name=1, anchor=center, inner sep=0}, shift right=2, from=1-1, to=1-2]
\arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=0, to=1]
\end{tikzcd}
Consequently, for any bicategory $B$ and 2-category $A$:
$$2\text{-}\mathrm{Cat}(\mathrm{str} B, A) \simeq \mathrm{Bicat}(B, A)$$
## Terminology
Classically, "2-category" meant [[strict 2-category]], with "bicategory" used for the weak notion. This led to the more general use of the prefix "2-" for strict (that is, strictly [[Cat]]-enriched) notions and "bi-" for weak ones. For example, classically a "2-adjunction" means a Cat-enriched adjunction, consisting of two strict 2-functors $F,G$ and a strictly Cat-natural isomorphism of categories $D(F X, Y)\cong C(X, G Y)$, while a "biadjunction" means the weak version, consisting of two weak 2-functors and a pseudo natural equivalence $D(F X, Y)\simeq C(X, G Y)$. Similarly for "2-equivalence" and "biequivalence," and "2-limit" and "bilimit."
We often use "2-category" to mean a strict or weak 2-category without prejudice, although we do still use "bicategory" to refer to the particular classical algebraic notion of weak 2-category. We try to avoid the more general use of "bi-" meaning "weak," however. For one thing, it is confusing; a "biproduct" could mean a weak [[2-limit]], but it could also mean an object which is both a product and a coproduct (which happens quite frequently in [[additive category|additive categories]]).
Moreover, in most cases the prefix is unnecessary, since once we know we are working in a bicategory, there is usually no point in considering strict notions at all. Fully weak limits are really the only sensible ones to ask for in a bicategory, and likewise for fully weak adjunctions and equivalences. Even in a strict 2-category, while we might need to say "strict" sometimes to be clear, we don\'t need to say "$2$-", since we know that we are not working in a mere category. (Max Kelly pushed this point.)
When we do have a strict 2-category, however, other strict notions can be quite technically useful, even if our ultimate interest is in the weak ones. This is somewhat analogous to the use of strict structures to model weak ones in [[homotopy theory]]; see [here](http://arxiv.org/abs/math/0702535) and [here](http://arxiv.org/abs/math/0607646) for good introductions to this sort of thing.
## Related concepts
* [[category]]
* **bicategory**
[[(infinity,2)-category]]
* [[double bicategory]] (and at [[double category#double_bicategories|double category]])
* [[tricategory]]
* [[tetracategory]]
* [[n-category]]
* [[(infinity,n)-category]]
* [[pseudofunctor]]
* [[locally univalent bicategory]]
* [[univalent bicategory]]
* [[monoidal bicategory]]
* [[closed bicategory]]
* [[initial object in a bicategory]]
* [[terminal object in a bicategory]]
* [[2-Grothendieck construction]]
Discussion about the use of the term "weak enrichment" above is at _[[weak enrichment]]_.
## References
See also the references at *[[2-category]]*.
* [[Jean Bénabou]], *Introduction to Bicategories*, Lecture Notes in Mathematics **47** Springer (1967), pp.1-77 ([doi:10.1007/BFb0074299](http://dx.doi.org/10.1007/BFb0074299))
* {#Power89} [[A. John Power]], _A general coherence result._ J. Pure Appl. Algebra 57 (1989), no. 2, 165–173. [doi:10.1016/0022-4049(89)90113-8](http://dx.doi.org/10.1016/0022-4049%2889%2990113-8) [MR0985657](http://www.ams.org/mathscinet-getitem?mr=985657)
* [[Saunders MacLane]], §XII.6 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]
* {#Campbell18} [[Alexander Campbell]], _How strict is strictification?_, [arxiv](https://arxiv.org/abs/1802.07538)
Formalization in [[homotopy type theory]] (see also at [[internal category in homotopy type theory]]):
* [[Benedikt Ahrens]], Dan Frumin, Marco Maggesi, Niels van der Weide, _Bicategories in Univalent Foundations_ ([arXiv:1903.01152](https://arxiv.org/abs/1903.01152))
[[!redirects bicategory]]
[[!redirects bicategories]]
[[!redirects bicategorical]]
[[!redirects coherence theorem for bicategories]]
|
bicategory of fractions | https://ncatlab.org/nlab/source/bicategory+of+fractions |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Much has been said about **inverting a class of morphisms** in a category (see [[localization]]), and there are many different settings in which one wants to, and can, do this. Homotopical algebra is largely concerned with how to compute the [[homotopy category]] so it is locally small. One the other hand, we have [[simplicial localization]] which retains all the homotopy information and returns an $(\infty,1)$-category.
If we have a 2-category with a notion of weak equivalence, one could localize the underlying 1-category in a way hopefully compatible with the 2-arrows, or extend the result fully into the 2-dimensional setting. In general this will require bicategories, and is the subject of [Pronk 96](#Pronk96).
## Definition
Let $B$ be a [[bicategory]] with a class $W$ of 1-[[morphism|cells]]. $W$ is said to **admit a right calculus of fractions** if it satisfies the following conditions
* [2CF1.] $W$ contains all equivalences
* [2CF2.]
* a) $W$ is closed under composition
* b) If $a\in W$ and a iso-2-cell $a \stackrel{\sim}{\Rightarrow} b$ then $b\in W$
* {#2CF3} [2CF3.] For all $w: A' \to A$, $f: C \to A$ with $w\in W$ there exists a 2-commutative square
$$
\begin{matrix}
P& \stackrel{g}{\to} & A' \\
v \downarrow&\Rightarrow &\, \downarrow w\\
C &\underset{f}{\to} & A
\end{matrix}
$$
with $v\in W$.
* [2CF4.] If $\alpha: w \circ f \Rightarrow w \circ g$ is a 2-cell and $w\in W$ there is a 1-cell $v \in W$ and a 2-cell $\beta: f\circ v \Rightarrow g \circ v$ such that $\alpha\circ v = w \circ \beta$. Moreover: when $\alpha$ is an iso-2-cell, we require $\beta$ to be an isomorphism too; when $v'$ and $\beta'$ form another such pair, there exist 1-cells $u,\,u'$ such that $v\circ u$ and $v'\circ u'$ are in $W$, and an iso-2-cell $\epsilon: v\circ u \Rightarrow v' \circ u'$ such that the following diagram commutes:
$$
\begin{matrix}
f \circ v \circ u & \stackrel{\beta\circ u}{\Rightarrow} & g\circ v \circ u \\
f\circ \epsilon \Downarrow \simeq &&
\simeq \Downarrow g\circ \epsilon \\
\\
f\circ v' \circ u' &\underset{\beta'\circ u'}{\Rightarrow}& g\circ v' \circ u'
\end{matrix}
$$
If $B$ is a category, then these axioms reduce to the ones of Gabriel and Zisman for a [[calculus of fractions]].
Given such a setup, Pronk constructs the [[localization]] of $B$ at $W$ and the universal functor sending elements of $W$ to equivalences.
## Example
Let $S$ be a category with binary products and pullbacks together with a class of [[davidroberts:class of admissible maps|admissible maps]] $E$.
+-- {: .num_theorem}
######Theorem:
The 2-categories $Cat(S)$ and $Gpd(S)$ of categories and groupoids internal to $S$ admit [[bicategory of fractions|bicategories of fractions]] for the class of $E$-[[davidroberts:weak equivalence|equivalences]].
=--
The resulting localization is equivalent to the bicategory of [[anafunctor|anafunctors]] in $S$. For details, see [Roberts (2012)](#Roberts12).
\begin{example}
**(Grothendieck toposes as a bicategory of fractions of localic groupoids)**\linebreak
The category of etale-complete [[localic groupoids]] (with open source and target maps) admits a [[bicategory of fractions]] at open essentially surjective fully faithful functors.
The resulting bicategory is equivalent to the bicategory of [[Grothendieck toposes]], [[geometric morphisms]], and [[natural isomorphisms]].
See Theorem 7.7 in [Moerdijk](#Moerdijk).
\end{example}
## Related entries
* [[localization of a 2-category]]
* [[category of fractions]]
* [[localization]]
* [[bicategory]]
* [[anafunctor]]
* [[orbifold]]
* [[stack]]
* [[étendue]]
## References
* {#Moerdijk} [[Ieke Moerdijk]], _The classifying topos of a continuous groupoid. I_, Transactions of the American Mathematical Society, Volume 310, Number 2, December 1988.
* O. Abbad, [[Enrico Vitale|E. M. Vitale]], _Faithful calculus of fractions_ , Cah. Top. Géom. Diff. Catég. **54** No. 3 (2013) 221-239. ([preprint](perso.uclouvain.be/enrico.vitale/FrazioniFedeli.pdf))
* E. Vitale, _Bipullbacks and categories of fractions_, [pdf](https://perso.uclouvain.be/enrico.vitale/BPBc.pdf)
* [[Dorette A. Pronk]], _Etendues and stacks as bicategories of fractions_, Comp. Math. **102** 3 (1996) pp.243-303. ([numdam:CM_1996__102_3_243_0](http://www.numdam.org/item/?id=CM_1996__102_3_243_0))
* {#Roberts12} [[David Roberts]], _Internal categories, anafunctors and localisations_, TAC **26** (2012) pp.788-829. ([pdf](http://www.tac.mta.ca/tac/volumes/26/29/26-29.pdf)) {#Roberts12}
* M. Tommasini, _A bicategory of reduced orbifolds from the point of view of differential geometry_ , arXiv:1304.6959 (2013). ([pdf](http://arxiv.org/pdf/1304.6959))
* M. Tommasini, _Some insights on bicategories of fractions I_ , arXiv:1410.3990 (2014). ([pdf](http://arxiv.org/pdf/1410.3990))
* M. Tommasini, _Some insights on bicategories of fractions II_ , arXiv:1410.5075 (2014). ([pdf](http://arxiv.org/pdf/1410.5075))
* M. Tommasini, _Some insights on bicategories of fractions III_ , arXiv:1410.6395 (2014). ([pdf](http://arxiv.org/pdf/1410.6395))
See also:
* {#Renaudin06} [[Olivier Renaudin]], Section 1.2 of: *Plongement de certaines théories homotopiques de Quillen dans les dérivateurs*, Journal of Pure and Applied Algebra Volume 213, Issue 10, October 2009, Pages 1916-1935
([arXiv:math/0603339](https://arxiv.org/abs/math/0603339), [doi:10.1016/j.jpaa.2009.02.014](https://doi.org/10.1016/j.jpaa.2009.02.014))
[[!redirects bicategories of fractions]]
[[!redirects bicategorical localization]]
[[!redirects bicategorical localizations]]
|
bicategory of maps | https://ncatlab.org/nlab/source/bicategory+of+maps |
# Bicategory of maps
* table of contents
{: toc}
## Definition
If $K$ is a [[bicategory]], then a [[morphism]] $f \colon a \to b$ is called a **map** if it has a [[right adjoint]] $f^* \colon b \to a$. (This is in slight contrast to the common usage of "map" to denote simply a [[morphism]] in any category.)
The bicategory $Map K$ is the [[locally full sub-2-category]] of $K$ determined by the maps.
## Examples
* In the bicategory [[Rel]] of [[sets]] and [[relations]], a relation is a map if and only if it is the [[graph of a function]]. Consequently, $Map Rel$ is equivalent to [[Set]].
* Similarly, if $C$ is a category with finite [[limits]], then there is a bicategory $Span C$ of [[spans]] in $C$. The bicategory $Map Span C$ is equivalent to $C$.
* In the bicategory [[Prof]] of [[categories]] and [[profunctors]] (perhaps [[enriched category|enriched]]), if $B$ is a [[Cauchy complete category]], then a profunctor $A\to B$ is a map if and only if it is represented by a functor $A\to B$. If $B$ is not Cauchy complete, then maps $A\to B$ correspond to functors from $A$ to the Cauchy completion of $B$.
## Properties
If every map in $K$ is [[comonadic morphism|comonadic]] and $Map K$ has a terminal object, then $Map K$ is equivalent to a $1$-category. If in addition $K$ is a [[cartesian bicategory]] and every [[comonad]] in $K$ has an [[Eilenberg--Moore object]], then $K$ is [[biequivalence|biequivalent]] to $Span Map K$, $Map K$ having finite limits. The converse is true if pullback squares in $Map K$ satisfy the Beck--Chevalley condition in $K$, i.e. if their [[mates]] are invertible (see [\[LWW10\]](#LWW10)).
$Map K$ is a [[regular category]] if and only if $K$ is a unitary tabular [[allegory]], equivalently a [[bicategory of relations]] in which every [[coreflexive morphism]] [[split idempotent|splits]]. In that case $Rel Map K \simeq K$.
Similarly, $Map K$ is a [[topos]] if and only if $K$ is a unitary tabular power allegory.
## Maps and equipments
A [[2-category equipped with proarrows]] is, by definition, a bijective-on-objects [[pseudofunctor]] $K\to M$ such that the image of every arrow in $K$ is a map in $M$. Equivalently, therefore, it is a bijective-on-objects pseudofunctor $K\to Map M$.
Hence the inclusion $Map M \to M$ is the "universal" proarrow equipment that can be constructed with a given bicategory $M$ as its bicategory of proarrows. More precisely, there is a forgetful functor from $Equip$ to $Bicat$ which remembers only the bicategory $M$ of proarrows, and the assignment of $M$ to $Map M \to M$ is its right adjoint.
+--{: .query}
[[Mike Shulman]]: This is obviously morally true, but I can't be bothered right now to check which 1-, 2-, or 3-categories of equipments and bicategories one has to use to make it precisely correct.
=--
A lot of work in bicategories that makes use of maps can easily be reformulated in a proarrow equipment. One advantage of proarrow equipments over bicategories of maps is they can distinguish between a category and its Cauchy completion (as objects of [[Prof]]).
## See also
* [[category of maps]]
## References
* Carboni, Walters, _Cartesian bicategories I_, JPAA 49, 1987.
{: #CW87 }
* Lack, Walters, Wood, _Bicategories of spans as cartesian bicategories_, TAC 24(1), 2010.
{: #LWW10 }
[[!redirects bicategories of maps]]
[[!redirects map in a bicategory]]
[[!redirects maps in a bicategory]]
|
bicategory of relations | https://ncatlab.org/nlab/source/bicategory+of+relations |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Bicategories of relations
* table of contents
{: toc}
## Idea
A *bicategory of relations* is a [[(1,2)-category]] which behaves like the 2-category of internal [[relations]] in a [[regular category]]. The notion is due to Carboni and Walters.
## Definition
[Note: in this article, the direction of composition is diagrammatic (i.e., "anti-Leibniz").
A **bicategory of relations** is a [[cartesian bicategory]] which is [[locally posetal 2-category|locally posetal]] and moreover in which for every object $X$, the [[diagonal morphism|diagonal]] $\Delta\colon X\to X\otimes X$ and [[codiagonal]] $\nabla\colon X\otimes X\to X$ satisfy the [[Frobenius condition]]:
$$\nabla\Delta = (1\otimes \Delta)(\nabla \otimes 1).$$
Of course, in the locally posetal case the definition of cartesian bicategory simplifies greatly: it amounts to a symmetric monoidal [[Pos]]-[[enriched category]] for which every object carries a commutative [[comonoid]] structure, for which the structure maps $\Delta_X: X \to X \otimes X$, $\varepsilon_X: X \to 1$ are [[left adjoint]] 1-morphisms, and for which every morphism $R: X \to Y$ is a colax morphism of comonoids in the sense that the following inequalities hold:
$$R \Delta_Y \leq \Delta_{X \otimes X} (R \otimes R), \qquad R \varepsilon_Y \leq \varepsilon_X.$$
We remark that such structure is unique when it exists (being a cartesian bicategory is a property of, as opposed to structure on, a bicategory). The [[tensor product]] $\otimes$ behaves like the ordinary product of [[relation]]s. Note this is not a cartesian product in the sense of the usual universal property; nevertheless, it is customary to write it as a product $\times$, and we follow this custom below. It does become a cartesian product if one restricts to the subcategory of left adjoints (called _maps_), which should be thought of as functional relations.
## Properties
We record a few consequences of this notion.
+-- {: .num_prop}
######Proposition
(Separability condition) $\Delta\nabla = 1$.
=--
+-- {: .proof}
######Proof
In one direction, we have the 2-cell $1 \leq \Delta\nabla$ which is the unit of the adjunction $\Delta \dashv \Delta_* = \nabla$. In the other direction, there is an adjunction $\varepsilon \dashv \varepsilon_*$ between the counit and its dual, and the unit of this adjunction is a 2-cell $1 \leq \varepsilon\varepsilon_*$, from which we derive
$$\Delta\nabla = \Delta\Delta_* \leq \Delta (1 \times \varepsilon)(1 \times \varepsilon_*)\Delta_* = 1$$
where the last equation follows from the equation $\Delta(1 \times \varepsilon) = 1$ and its dual $(1 \times \varepsilon_*)\Delta_* = 1$.
=--
+-- {: .num_prop}
######Proposition
(Dual Frobenius condition) $\nabla \Delta = (\Delta \times 1)(1 \times \nabla)$.
=--
+-- {: .proof}
######Proof
The locally full subcategory whose 1-cells are _maps_ (= left adjoints) has finite products, in particular a symmetry involution $\sigma$ for which
$$\Delta_X \sigma_{X X} = \Delta_X$$
Additionally, the right adjoint $\sigma_{X Y *}$ is the inverse $\sigma_{X Y}^{-1} = \sigma_{Y X}$. Hence the equation above is mated to $\sigma_{X X} \nabla_X = \nabla_X$, and we calculate
$$\array{
\nabla_X \Delta_X & = & \sigma_{X X}\nabla_X \Delta_X \sigma_{X X} \\
& = & \sigma_{X X} (1 \times \Delta_X)(\nabla_X \times 1) \sigma_{X X} \\
& = & (\Delta_X \times 1_X) \sigma_{X \times X, X} \sigma_{X \times X, X} (1_X \times \nabla_X) \\
& = & (\Delta_X \times 1_X) (\sigma_{X X} \times 1) (1 \times \sigma_{X X}) (1 \times \nabla_X) \\
& = & (\Delta_X \times 1_X)(1_X \times \nabla_X)
}$$
which gives the dual equation.
=--
+-- {: .num_theorem}
######Theorem
In a bicategory of relations, each object is dual to itself, making the bicategory a compact closed bicategory.
=--
+-- {: .proof}
######Proof
Both the unit and counit of the desired adjunction $X \dashv X$ are given by equality predicates:
$$\eta_X = (1 \stackrel{\varepsilon_*}{\to} X \stackrel{\Delta}{\to} X \times X) \qquad \theta_X = (X \times X \stackrel{\nabla}{\to} X \stackrel{\varepsilon}{\to} 1)$$
One of the triangular equations follows from the commutative diagram
$$\array{
X & \stackrel{1 \times \varepsilon_*}{\to} & X \times X & \stackrel{1 \times \Delta}{\to} & X \times X \times X \\
& {}_1\searrow & \downarrow \mathrlap{\nabla} & & \downarrow \mathrlap{\nabla \times 1} \\
& & X & \overset{\Delta}{\to} & X \times X \\
& & & {}_{1}\searrow & \downarrow \mathrlap{\varepsilon \times 1} \\
& & & & X
}$$
where the square expresses a Frobenius equation. The other triangular equation uses the dual Frobenius equation.
=--
The compact closure allows us to define the opposite of a relation $R: X \to Y$ as the 1-morphism mate:
$$R^{op} = (Y \stackrel{1_Y \times \eta_X}{\to} Y \times X \times X \stackrel{1_Y \times R \times 1_X}{\to} Y \times Y \times X \stackrel{\theta_Y \times 1_X}{\to} X)$$
In this way, a bicategory of relations becomes a [[dagger-compact category|†-compact]] $Pos$-enriched category.
Recall (again) that a **map** in the bicategory of relations is the same as a 1-cell that has a right adjoint.
+-- {: .num_prop #strict}
###### Proposition
If $f \colon X \to Y$ is a map, then $f$ is a strict morphism of comonoids.
=--
+-- {: .proof}
###### Proof
If $g$ is the right adjoint of $f$, we have $\Delta_X g \leq (g \times g)\Delta_Y$ since $g$, like any morphism, is a lax comonoid morphism. But this 2-cell is mated to a 2-cell $(f \times f)\Delta_X \to \Delta_Y f$, inverse to the 2-cell $\Delta_Y f \to (f \times f)\Delta_X$ that comes for free. So $f$ preserves comultiplication strictly. A similar argument shows $f$ preserves the counit strictly.
=--
+-- {: .num_prop}
###### Proposition
If $f \colon X \to Y$ is a map, then $f^{op}: Y \to X$ equals the right adjoint $f_\ast$.
=--
+-- {: .proof}
###### Proof
The proof is much more perspicuous if we use [[string diagram|string diagrams]]. But the key steps are given in two strings of equalities and inequalities. The first gives a counit for $f \dashv f^{op}$, and the second gives a unit. We have
$$\array{
f f^{op} & \stackrel{0}{=} & f \circ (\varepsilon_Y \times 1)(\nabla_Y\times 1)(1 \times f \times 1)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \\
& \stackrel{1}{=} & (\varepsilon_Y \times 1)(\nabla_Y \times 1)(1 \times f \times f)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \\
& \stackrel{2}{=} & (\varepsilon_Y \times 1)(\nabla_Y \times 1)(1 \times \Delta_Y)(1 \times f)(1 \times \varepsilon_X_\ast)\\
& \stackrel{3}{\leq} & (\varepsilon_Y \times 1)(\nabla_Y \times 1)(1 \times \Delta_Y)(1 \times \varepsilon_Y_\ast) \\
& \stackrel{4}{=} & (\varepsilon_Y \times 1)\Delta_Y \nabla_Y(1 \times \varepsilon_Y_\ast) \\
& \stackrel{5}{=} & 1_Y
}$$
where (0) uses the definition of $f^{op}$, (1) uses properties of monoidal categories, (2) uses the fact that $f$ strictly preserves comultiplication, (3) is mated to the fact that $f$ laxly preserves the counit, (4) is a Frobenius condition, and (5) uses comonoid and dual monoid laws. We can also "almost" run the same calculation backwards to get the unit:
$$\array{
1_X & \stackrel{0}{=} & (\varepsilon_X \times 1)\Delta_X \nabla_X(1 \times \varepsilon_X_\ast) \\
& \stackrel{1}{=} & (\varepsilon_X \times 1)(\nabla_X \times 1)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \\
& \stackrel{2}{=} & (\varepsilon_Y \times 1)(f \times 1)(\nabla_X\times 1)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \\
& \stackrel{3}{\leq} & (\varepsilon_Y \times 1)(\nabla_Y \times 1)(f \times f \times 1)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \\
& \stackrel{4}{=} & (\varepsilon_Y \times 1)(\nabla_Y \times 1)(1 \times f \times 1)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \circ f \\
& \stackrel{5}{=} & f^{op} f
}$$
where (0) uses comonoid and dual monoid laws, (1) uses a Frobenius condition, (2) uses the fact that $f$ preserves the counit, (3) is mated to the fact that $f$ laxly preserves comultiplication, (4) uses properties of monoidal categories, and (5) uses the definition of $f^{op}$.
=--
In fact, what this proof really proves is a converse of the earlier [proposition](#strict):
+-- {: .num_prop}
###### Proposition
If $f$ is a strict comonoid morphism, then $f$ has a right adjoint: $f \dashv f^{op}$.
=--
+-- {: .num_prop}
######Proposition
If $f, g: X \to Y$ are maps and $f \leq g$, then $f = g$. Thus, the locally full subcategory whose morphisms are maps is locally discrete (the hom-posets are discrete).
=--
+-- {: .proof}
######Proof
A 2-cell inequality $\alpha: f \leq g$ is mated to a inequality $\alpha_*: g_* \leq f_*$. On the other hand, whiskering $1_Y \times \alpha \times 1_X$ with $1_Y \times \eta_X$ and $\theta_Y \times 1_X$, as in the construction of opposites above, gives $\alpha^{op}: f^{op} \leq g^{op}$. Since $f^{op} = f_*$ and $g^{op} = g_*$, we obtain $f^{op} = g^{op}$, and because $f^{op op} = f$, we obtain $f = g$.
=--
## The Beck-Chevalley condition
Bicategories of relations $\mathbf{B}$ satisfy a Beck-Chevalley condition, as follows. Let $Prod(B_0)$ denote the free category with finite products generated by the set of objects of $\mathbf{B}$. According to the results at [[free cartesian category]], $Prod(B_0)$ is finitely complete.
Since $Map(\mathbf{B})$ has finite products, there is a product-preserving functor $\pi: Prod(B_0) \to \Map(\mathbf{B})$ which is the identity on objects. Again, according to the results of [[free cartesian category]], we have the following result.
+-- {: .num_lemma}
###### Lemma
If a diagram in $Prod(B_0)$ is a pullback square, then application of $\pi$ to that diagram is a pullback square in $Map(\mathbf{B})$.
=--
Let us call pullback squares of this form in $Map(\mathbf{B})$ _product-based_ pullback squares.
+-- {: .num_prop}
######Proposition
Given a product-based pullback square
$$\array{
W & \stackrel{h}{\to} & X \\
k \downarrow & & \downarrow f \\
Y & \underset{g}{\to} & Z
}$$
in $Map(\mathbf{B})$, the Beck-Chevalley condition holds: $h_* k = f g_*$.
=--
See Brady-Trimble for further details. The critical case to consider is the pullback square
$$\array{
X & \overset{\Delta}{\to} & X \times X \\
\mathllap{\Delta} \downarrow & & \downarrow \mathrlap{\Delta \times 1} \\
X \times X & \underset{1 \times \Delta}{\to} & X \times X \times X
}$$
where the Beck-Chevalley condition is exactly the Frobenius condition.
One way of interpreting this result is by viewing $\mathbf{B}$ as a hyperdoctrine or monoidal fibration over $Prod(B_0)$, where the fiber over an object $B$ is the local hom-poset $\hom(B, 1)$. Each $f: A \to B$ in the base induces a pullback functor, by precomposing $R: B \to 1$ with $\pi(f): A \to B$ in $Map(\mathbf{B})$. Existential quantification is interpreted by precomposing with right adjoints $\pi(f)_*$. The Beck-Chevalley condition exerts compatibility between quantification and pullback/substitution functors.
## Relation to allegories
Any bicategory of relations is an [[allegory]]. Recall that an allegory is a $Pos$-enriched $\dagger$-category whose local homs are meet-semilattices, satisfying Freyd's modular law:
$$R S \cap T \leq (R \cap T S^{op})S$$
A proof of the modular law is given in the blog post by R.F.C. ("Bob") Walters referenced below.
In fact, we may prove a little more:
+-- {: .num_theorem}
######Theorem
The notion of bicategory of relations is equivalent to the notion of unitary pretabular allegory.
=--
+-- {: .proof}
######Proof
A bicategory of relations has a unit $1$ in the sense of allegories:
* $1$ is a partial unit: we have $\varepsilon_1 = id_1: 1 \to 1$, and for any $R: 1 \to 1$ we have $R = R\varepsilon_1 \leq \varepsilon_1 = id_1$.
* Any object $X$ is the source of a map $\varepsilon_X: X \to 1$, which being a map is entire. Thus $1$ is a unit.
A bicategory of relations is also pretabular, for the maximal element in $\hom(X, Y)$ is tabulated as $(\pi_X)_* \pi_Y$, where $\pi_X$, $\pi_Y$ are the product projections for $X \times Y$.
In the other direction, suppose $\mathbf{A}$ is a unitary pretabular allegory. There is a faithful embedding, which preserves the unit, of $\mathbf{A}$ into its coreflexive splitting, which is unitary and tabular, and hence equivalent to $Rel$ of the regular category $Map(\mathbf{A})$. By the proposition below, then, $\mathbf{A}$ is a full sub-2-category of a bicategory of relations. Because the product of two Frobenius objects is again Frobenius, it suffices to show that $\mathbf{A}$ is closed under products in its coreflexive splitting. The inclusion of $\mathbf{A}$ into the latter preserves the unit and the property of being a map, and hence preserves top elements of hom sets, while any allegory functor must preserve tabulations. So the tabulation $(\pi^{X Y}_X, \pi^{X Y}_Y)$ of $\top_{X Y}$ in $\mathbf{A}$ is a tabulation of $\top_{1_X 1_Y}$ in the coreflexive splitting, and hence $1_{X \times Y} \cong 1_X \times 1_Y$.
=--
+-- {: .num_prop}
###### Proposition
If $C$ is a [[regular category]], then the bicategory $Rel C$ of [[Rel|relations]] in $C$ is a bicategory of relations.
=--
+-- {: .proof}
###### Proof
By theorem 1.6 of Carboni--Walters, $\mathbf{A}$ is a [[Cartesian bicategory]] if:
* $Map(\mathbf{A})$ has finite products.
* $\mathbf{A}$ has local finite products, and $id_1$ is the top element of $\mathbf{A}(1,1)$.
* The tensor product defined as
$$ R \otimes S = (p R p_*) \cap (p' S p'_*) $$
is functorial, where $p$ and $p'$ are the appropriate product projections.
The first two are obvious, and for the third we may reason in the [[internal language]] of $C$. Clearly
$$ 1 \otimes 1 = [x = x \wedge x' = x'] = 1 $$
The formula whose meaning is by $R S \otimes R' S'$ is
$$ (\exists y. R x y \wedge S y z) \wedge (\exists y'. R' x' y' \wedge S' y' z') $$
and we may use [[Frobenius reciprocity]] to get
$$
\begin{aligned}
& \exists y. (R x y \wedge S y z \wedge y^* \exists y'. R' x' y' \wedge S' y' z') \\
& \equiv \exists y. (R x y \wedge S y z \wedge \exists y'. y^* (R' x' y' \wedge S' y' z')) \\
& \equiv \exists y, y'. R x y \wedge S y z \wedge R' x' y' \wedge S' y' z' \\
\end{aligned}
$$
which is the meaning of $(R \otimes R')(S \otimes S')$. Finally, the Frobenius law is
$$ [\exists x'. (x_1, x_2) = (x', x') = (x_3, x_4)]
= [\exists x'. (x_1, x') = (x_3, x_3) \wedge (x_2, x_2) = (x_4, x')] $$
which follows from transitivity and symmetry of equality.
=--
## See also
Other attempted axiomatizations of the same idea "something that acts like the category of relations in a regular category" include:
* [[allegories]]
* [[1-category equipped with relations]]
## Related concepts
* [[category of correspondences]]
* [[(infinity,n)-category of correspondences]]
## References
* [[Aurelio Carboni]], [[Bob Walters]], _Cartesian Bicategories, I_, [article](https://doi.org/10.1016/0022-4049%2887%2990121-6)
* [[Aurelio Carboni]], [[Max Kelly]], [[Bob Walters]], [[Richard Wood]], _Cartesian Bicategories II_, ([arXiv:0708.1921](https://arxiv.org/abs/0708.1921))
* [[Bob Walters]], [blog post](http://rfcwalters.blogspot.com/2009/10/categorical-algebras-of-relations.html), showing that any bicategory of relations is an [[allegory]]. Indeed, a bicategory of relations is equivalent to a unitary pretabular allegory.
* [[Evan Patterson]], _Knowledge Representation in Bicategories of Relations_, ([arXiv:1706.00526](https://arxiv.org/abs/1706.00526))
This article shows how one can model RDF (Resource Description Framework) and parts of OWL (Ontology Web Language) in bicategories of relations, whose internal logic is [[regular logic]]. He ends by showing how one can extend these to [[distributive bicategories of relations]] whose internal logic is [[coherent logic]], which is equivalent in expressivity to first order logic.
[[!redirects bicategories of relations]] |
BICEP2 | https://ncatlab.org/nlab/source/BICEP2 |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_BICEP2_ (for "Background Imaging of Cosmic Extragalactic Polarization") is the name of an astrophysical [[experiment]] (their [webpage](http://www.cfa.harvard.edu/CMB/bicep2/)) which released its data in March 2014. The experiment claims to have detected a pattern called the "B-mode" in the polarization of the [[cosmic microwave background]] (CMB).
### Claims
This data, if confirmed, would have been likely due to a [[gravitational wave]] mode created during the period of [[cosmic inflation]] by a [[vacuum fluctuation]] in the [[field (physics)|field]] of [[gravity]] which then at the era of decoupling left the characteristic B-mode imprint on the CMB. This fact alone was regarded as further strong evidence for the already excellent experimental evidence for [[cosmic inflation]] as such (competing models did not predict such gravitational waves to be strong enough to be detectable in this way).
What would have singled out the BICEP2 result over previous confirmations of cosmic inflation is that the data also gives a quantitative value for the [[energy]] scale at which cosmic inflation happened (the [[mass]] of the hypothetical [[inflaton]]), namely at around $10^{16}$[[GeV]]. This is noteworthy as being only two order of magnituded below the [[Planck scale]], and hence 12 or so orders of magnitude above energies available in current accelerator [[experiments]] (the [[LHC]]). Also, it is at least a curious coincidence that this is precisely the hypothetical [[GUT]] scale.
It was thought that this value rules out a large number of variant models of [[cosmic inflation]] and favors the model known as _[[chaotic inflation]]_.
### Problems
Further measurement by the [Planck collaboration](#Planck) showed that much of the signal claimed by BICEP2 is naturally explained just by galactic dust forground, see [below](#Resonaances). Further measurements seem to be needed to clarify the situation.
## Related entries
## References
* [BICEP2 webpage](http://www.cfa.harvard.edu/CMB/bicep2/)
* [general BICEP webpage](http://bicep.caltech.edu/public/
* {#Planck} [Planck collaboration webpage](http://sci.esa.int/planck/)
* {#Resonaances} Resonaances, _[BICEP: what was wrong and what was right](http://resonaances.blogspot.de/2014/09/bicep-what-was-wrong-and-what-was-right.html)_
* _A Joint Analysis of BICEP2/Keck Array and Planck Data_ ([arXiv:1502.00612](http://arxiv.org/abs/1502.00612))
[[!redirects bicep2]]
[[!redirects BICEP2 experiment]]
[[!redirects bicep2 experiment]] |
bicharacteristic flow | https://ncatlab.org/nlab/source/bicharacteristic+flow |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Let $X$ be a [[smooth manifold]] and let $D$ be a [[differential operator]] on (smooth [[sections]] of) the [[trivial line bundle]] over $X$ (or more generally a [[properly supported pseudo-differential operator]]). Then the [[principal symbol]] $q(D)$ of $D$ is equivalently a [[smooth function]] on the [[cotangent bundle]] $T^\ast X$ (by [this example](symbol+of+a+differential+operator#BicharacteristicFlow)). With the [[cotangent bundle]] canonically regarded as a [[symplectic manifold]], let
$$
v_{q(D)} \in \Gamma\left(T\left(T^\ast X\right) \right)
$$
be the corresponding [[Hamiltonian vector field]].
+-- {: .num_defn #BicharacteristicFlow}
###### Definition
The _bicharacteristic flow_ of $D$ is the [[Hamiltonian flow|Hamiltonian]] [[flow of a vector field|flow]] of the [[Hamiltonian vector field]] $v_{q(D)}$ inside the submanifold defined by $q = 0$. Moreover:
1. A single [[flow line]] in $T^\ast X$ is called a _bicharacteristic strip_ of $D$,
1. the [[projection]] of such to a [[curve]] in $X$ is called a _bicharacteristic curve_.
1. The [[relation]] $C$ on $T^\ast X$ given by
$$
\left((x_1,k_1) \sim (x_2, k_2)\right)
\;\coloneqq\;
\left(
q(x_i,k_i) = 0
\;\;\text{and}\;\;
(x_1,k_1) \,\text{is connected to}\,
(x_2,k_2) \,\text{by a bicharacteristic strip}
\right)
$$
is called the _bicharacteristic relation_.
=--
## Examples
### Of the Klein-Gordon operator
{#OfTheKleinGordonOperator}
+-- {: .num_example #BicharachteristicFlowOfKleinGordonOperator}
###### Example
**(bicharacteristic curves of [[wave operator]]/[[Klein-Gordon operators]] are the [[lightlike]] [[geodesics]])**
Let $(X,g)$ be a [[Lorentzian manifold]] and let $D \coloneqq \Box_g - m^2$ be its [[wave operator]]/[[Klein-Gordon operator]].
Then the bicharacteristic curves of $D$ (def. \ref{BicharacteristicFlow}) are precisely the [[lightlike]] [[geodesics]] of $(X,e)$, and the bicharacteristic strips are precisely these geodesices with their [[cotangent vectors]].
Accordingly two cotangent vectors are bicharacteristically related $(x_1,k_1) \sim (x_2,k_2)$ precisely if there is a [[lightlike]] [[geodesic]] connecting the points, with $k_1$ and $k_2$ the corresponding cotangents, hence one the result of [[parallel transport]] of the other along the geodesic.
=--
([Radzikowski 96, prop. 4.2 and below (6)](#Radzikowski96))
Specifically on [[Minkowski spacetime]]:
+-- {: .num_example}
###### Example
**([[bicharacteristic flow]] of [[Klein-Gordon operator]] on [[Minkowski spacetime]])**
Let $\mathbb{R}^{p,1}$ be [[Minkowski spacetime]] of [[dimension]] $p+1$ consider the [[Klein-Gordon operator]]
$$
D
\;=\;
\eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu}
-
\left(\tfrac{m c}{\hbar}\right)^2
\,.
$$
Its [[principal symbol]] is the function
$$
\array{
T^\ast \mathbb{R}^{p,1}
&\overset{q}{\longrightarrow}&
\mathbb{R}
\\
(x,k) &\mapsto& \eta^{\mu \nu} k_\mu k_\nu
}
$$
Hence $q(k) = 0$ is the condition that the [[wave vector]] $k$ be [[lightlike]].
The [[Hamiltonian vector field]] corresponding to $q$ is
$$
\begin{aligned}
v_q
& =
-\tfrac{1}{2} \eta^{\mu \nu} k_\mu \partial_{x^\nu}
\\
& =
-\tfrac{1}{2} k^\mu \partial_{x^\mu}
\end{aligned}
$$
in that
$$
\begin{aligned}
\iota_{v_q} d k_\mu \wedge d x^\mu
&=
\tfrac{1}{2} \eta^{\mu \nu} k_\mu d k_\mu
\\
& =
d q(k)
\end{aligned}
$$
It follows that the [[bicharacteristic curves]] are precisely the [[lightlike]] [[curves]]
$$
\array{
\mathbb{R} &\overset{\gamma_k}{\longrightarrow}& \mathbb{R}^{p,1}
\\
\tau &\mapsto& (\gamma^\mu(0) + \tau k^\mu)
}
$$
and the corresponding [[bicharacteristic strips]] are these with their lightlike contangent vector constantly carried along
$$
\array{
\mathbb{R} &\overset{\gamma_k}{\longrightarrow}& T^\ast\mathbb{R}^{p,1}
\\
\tau &\mapsto& \left((\gamma^\mu(0) + \tau k^\mu),(k_\mu)\right)
}
$$
=--
## Properties
### Propagation of singularities
The _[[propagation of singularities theorem]]_ says that the [[wave front set]] of a [[distribution|distributional]] solution to the [[differential equation]] of a sufficiently nice [[differential operator]] (or generally of a [[properly supported pseudo-differential operator]]) is preserved by the bicharacteristic flow.
## References
* {#DuistermaatHoermander72} [[Johann Duistermaat]], [[Lars Hörmander]], _Fourier integral operators II_, Acta Mathematica 128, 183-269, 1972 ([Euclid](https://projecteuclid.org/euclid.acta/1485889724))
Review in the context of the [[free field|free]] [[scalar field]] on [[globally hyperbolic spacetimes]] (with $Q$ the [[wave operator]]/[[Klein-Gordon operator]]) is in
* {#Radzikowski96} [[Marek Radzikowski]], _Micro-local approach to the Hadamard condition in quantum field theory on curved space-time_, Commun. Math. Phys. 179 (1996), 529–553 ([Euclid](http://projecteuclid.org/euclid.cmp/1104287114))
[[!redirects bicharacteristic flows]]
[[!redirects bicharacteristic strip]]
[[!redirects bicharacteristic strips]]
[[!redirects bicharacteristic curve]]
[[!redirects bicharacteristic curves]]
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bicommutant theorem | https://ncatlab.org/nlab/source/bicommutant+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
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[[!include higher algebra - contents]]
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# Contents
* table of contents
{:toc}
## Idea
The _bicommutant theorem_ characterizes concrete [[von Neumann algebras]] as those concrete $C^*$-[[C-star algebra|algebras]] ($C^*$-algebras of [[bounded operator]]s on some [[Hilbert space]]) that are the [[commutant]]s of their own commutants.
## Statement
The __bicommutant theorem__ (as known as the _double commutant theorem_, or _von Neumann's double commutant theorem_) is the following result:
+-- {: .un_thm}
###### Theorem
Let $A \subseteq B(H)$ be a sub-$*$-[[star-algebra|algebra]] of the algebra of [[bounded linear operator|bounded linear operators]] on a [[Hilbert space]] $H$. Then $A$ is a [[von Neumann algebra]] (and therefore also a $C^*$-[[C-star algebra|algebra]]) in $H$ if and only if $A = A''$, where $A'$ denotes the [[commutant]] of $A$.
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Notice that the condition of $A$ being a von Neumann algebra (being closed in the [[weak operator topology]]; "weak" here can be replaced by "strong", "ultrastrong", or "ultraweak" as described in [[operator topology]]), which is a [[topology|topological]] condition, is by this result equivalent to an algebraic condition (being equal to its bicommutant).
## References
* Wikipedia article: [bicommutant theorem] (http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem)
[[!redirects bicommutant theorem]]
[[!redirects double commutant theorem]]
[[!redirects von Neumann double commutant theorem]]
[[!redirects von Neumann's double commutant theorem]]
[[!redirects von Neumann's double commutant theorem]]
[[!redirects von Neumann\'s double commutant theorem]]
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bicomodule | https://ncatlab.org/nlab/source/bicomodule | Let $C,D$ be [[comonoid]]s in a [[monoidal category]] $A = (A,\otimes,1)$. A $C$-$D$ __bicomodule__ is an object $M$ in $A$, with left $C$-[[coaction]] $\lambda_C : M\to C\otimes M$ and right $D$-coaction $\rho_D: M\to M\otimes D$ which commute in the sense that
$$
(\lambda_C\otimes id_D)\circ\rho_D = (id_C\otimes \rho_D)\circ \lambda_C.
$$
Typical cases are when $A$ is the category of $k$-[[modules]] where $k$ is a [[commutative unital ring]] (the comonoids are then $k$-[[coalgebras]]), and the more general case of bicomodules over [[corings]], where $A$ is the category of $k$-bimodules where $k$ is a possibly [[noncommutative ring]].
There is an operation of [[cotensor product]] for bicomodules over coalgebras/corings; however it is not associative in general, unlike the [[tensor product]] of bimodules over rings!
[[!redirects bicomodule]]
[[!redirects bicomodules]]
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bicompact space | https://ncatlab.org/nlab/source/bicompact+space |
# Bicompact spaces
* table of contents
{: toc}
## Definitions
A __bicompact space__ is a [[bitopological space]] both of whose underlying [[topological spaces]] are [[compact spaces]], often assumed to be [[compact Hausdorff space|Hausdorff]].
$Bi Comp$ is the [[category]] whose [[object|objects]] are bicompact spaces and whose [[morphism|morphisms]] are [[bicontinuous maps]].
## References
* [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and concrete categories: the joy of cats_. [free online](http://katmat.math.uni-bremen.de/acc/acc.pdf)
[[!redirects bicompact space]]
[[!redirects bicompact spaces]]
[[!redirects bicompact bitopological space]]
[[!redirects bicompact bitopological spaces]]
[[!redirects BiComp]]
[[!redirects Bi Comp]]
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bicomplete category | https://ncatlab.org/nlab/source/bicomplete+category | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
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[[!include category theory - contents]]
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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# Complete categories
* table of contents
{: toc}
## Definition
A [[category]] $C$ is **bicomplete** if it is both a [[complete category]] as well as a [[cocomplete category]], hence if it has all [[small limit|small]] [[limit|limits]] and [[colimits]]: that is, if every [[small diagram|small]] [[diagram]]
$$ F: D \to C$$
where $D$ is a [[small category]] has a [[limit]] and a [[colimit]] in $C$.
## Related concepts
* [[M-complete category]]
[[!redirects bicomplete categories]] |
biconditional | https://ncatlab.org/nlab/source/biconditional |
biconditional "$\Leftrightarrow$" is [[conditional]] in both directions "$\Rightarrow$", "$\Leftarrow$": [[equivalence]]
[[!redirects biconditionals]]
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