basis theorem

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## Idea

The **basis theorem** for [[vector space | vector spaces]] states that every [[vector space]] $V$ admits a [[basis of a vector space|basis]], or in other words is a [[free module]] over its [[ground field]] of scalars. It is a famous classical consequence of the [[axiom of choice]] (and is equivalent to it by [Blass (1984)](#Blass84).

## Statement and proofs

+-- {: .un_theorem}
###### Basis theorem

If $V$ is a [[vector space]] over any [[field]] (or, say, a left vector space over a [[skewfield]]) $K$, then $V$ has a [[basis of a vector space|basis]].

=--

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

We apply [[Zorn's lemma]] as follows: consider the [[poset]] consisting of the linearly independent [[subsets]] of $V$, ordered by inclusion (so $S \leq S'$ if and only if $S \subseteq S'$). If $(S_\alpha)$ is a chain in the poset, then

$$S = \bigcup_\alpha S_\alpha$$

is an upper bound, for each $v$ in the span of $S$ belongs to some $S_\alpha$ and is uniquely expressible as a finite linear combination of elements in $S_\alpha$ and in any $S_\beta$ containing $S_\alpha$, hence uniquely expressible as a finite linear combination of elements in $S$. Thus the hypothesis of Zorn's lemma obtains for this poset; therefore this poset has a maximal element, say $B$.

Let $W$ be the span of $B$.  If $W$ were a proper subspace of $V$, then for any $v$ in the set-theoretic complement of $W$, $B \cup \{v\}$ is a linearly independent set (for if

$$a_0 v + a_1 w_1 + \ldots + a_n w_n = 0 \qquad (w_1, \ldots, w_n \in B)$$

we must have $a_0 = 0$ -- else we can multiply by $1/a_0$ and express $v$ as a linear combination of the $w_i$, contradicting $\neg(v \in W)$ -- and then the remaining $a_i$ are 0 since the $w_i$ are linearly independent). This contradicts the maximality of $B$. We therefore conclude that $W = V$, and $B$ is a basis for $V$.
=--

I'll write out a proof of the converse, that the axiom of choice follows from the basis theorem, as soon as I've digested it -- Todd.

+-- {: .un_theorem}
###### Equipotence of vector space bases (Steinitz)

Any two bases of a vector space are of the same cardinality.

=--

## Generalisations

Given any linearly independent set $A$ and spanning set $C$, if $A \subseteq C$, then there is a basis $B$ with $A \subseteq B \subseteq C$; the theorem above is the special case where $A = \empty$ and $C = V$.  The proof of this more general theorem is a straightforward generalisation of the proof above.

## See also

* [[vector space]]

## References

The original proof of the existence of [[Hamel bases]] (after which the concept was named) was for the case of the [[real numbers]] regarded as a [[rational vector space]] and used (not directly [[Zorn's lemma]] but) the [[well-ordering theorem]]:

* {#Hamel1905} [[Georg Hamel]], pp. 460 in: *Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: $f(x + y) = f(x) + f(y)$*, Mathematische Annalen **60** (1905) 459–462 [[doi:10.1007/BF01457624](https://doi.org/10.1007/BF01457624)]

Lecture notes:

* [[Christoph Schweigert]], *Basis und Dimension*, §2.4 in:  *Lineare Algebra*, lecture notes, Hamburg (2022) [[pdf](https://www.math.uni-hamburg.de/home/schweigert/skripten/laskript.pdf)]

See also:

* {#Blass84} [[Andreas Blass]], *Existence of bases implies the axiom of choice*, Contemporary Mathematics **31** (1984) 31-33 [[doi:10.1090/conm/031](https://doi.org/10.1090/conm/031) [pdf](https://dept.math.lsa.umich.edu/~ablass/bases-AC.pdf)]


Bastian Rieck

* [personal page](https://bastian.rieck.me/)

## Selected writings

On [[topological machine learning]]:

* {#RSL} [[Bastian Rieck]], Filip Sadlo, Heike Leitte, *Topological Machine Learning with Persistence Indicator Functions*,  In: *Topological Methods in Data Analysis and Visualization V* TopoInVis (2017) 87-101 Mathematics and Visualization. Springer, $[$[arXiv:1907.13496](https://arxiv.org/abs/1907.13496), [doi:10.1007/978-3-030-43036-8_6](https://doi.org/10.1007/978-3-030-43036-8_6)$]$

* Felix Hensel, Michael Moor, [[Bastian Rieck]], *A Survey of Topological Machine Learning Methods*, Front. Artif. Intell., **26** (2021) $[$[doi:10.3389/frai.2021.681108](https://doi.org/10.3389/frai.2021.681108), [[HenselMoorRieck-TopologicalMachineLearning.pdf:file]]$]$

category: people


Batalin-Vilkovisky integral

Similarly to the idea of Wilson efective action where some degrees of freedom are integrated out, in BV-formalism one introduces BV-effective action; using a variant of integration utilizing Lagrangean submanifolds. In this procedure BV integrals are involved.

This is related to solving quantum master equation where the "induction" data are equivalent to the data involved in [[homological perturbation theory]].

* Albert Schwarz, Geometry of BV quantization, CMP 1993

* Losev

* Cattaneo, Felder 2008

[[!redirects BV integral]]


Batalin-Vilkovisky quantization

A version of Koszul type resolutions appear in attempts to do quantizations; namely a general theory leads to singularities and new ghost and antighost fields are used to remove them. The simplest such cohomological method is BRST quantization. Much more general versions, allowing more general singularities is BV-quantization introduced by physicists Batalin and Vilkovisky in late 1980s.

See [[nlab:BV theory|BV "theory" in nlab]]

[[!redirects BV quantization]]


Batanin omega-category

I
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## Idea

A _[[Michael Batanin|Batanin]] $\omega$-category_ is a [[weak ∞-category]] defined as an [[algebra over an operad|algebra]] over a suitable contractible [[globular operad]]. So this is an [[algebraic definition of higher category]].

The definition is similar to that of [[Trimble n-category]] (which is actually a special case of a Batanin $\omega$-category) and similar to the definition of [[Grothendieck-Maltsiniotis infinity-category]].

## Morphisms

When a weak $\infty$-category is modeled as a module over an $O$-operad, morphisms of modules $F : C \to D$ will correspond to _strict_  $\infty$ functors. To get weak $\infty$-functors one has to resolve $C$.

One way to do this is described in ([Garner](#Garner)).

## References

* [[Michael Batanin]], _Monoidal globular categories as a natural environment for the theory of weak $n$-categories_ , Advances in Mathematics 136 (1998), no. 1, 39–103.

* [[Ross Street]], _The role of Michael Batanin's monoidal globular categories_, in _Higher Category Theory_, eds. E. Getzler and M. Kapranov, Contemp. Math. 230, American
Mathematial Society, Providence, Rhode Island, 1998, pp. 99--116.   ([pdf](https://www.researchgate.net/publication/2741768_The_role_of_Michael_Batanin's_monoidal_globular_categories/link/02bfe5108f456cad85000000/download))

Work towards establishing the [[homotopy hypothesis]] for Batanin $\omega$-groupoids can be found here:

* [[Clemens Berger]], _A cellular nerve for higher categories_, Advances in Mathematics 169, 118-175 (2002) ([pdf](http://math1.unice.fr/~cberger/nerve.pdf))
 {#Berger}

* [[Denis-Charles Cisinski]], _Batanin higher groupoids and
homotopy types_ ([arXiv:math/0604442](http://arxiv.org/abs/math/0604442))

A nice introduction to this subject is:

* [[Eugenia Cheng]], _Batanin omega-groupoids and the homotopy hypothesis_, ([recorded lecture](http://www.fields.utoronto.ca/audio/06-07/highercat/cheng)) from the Fields Institute Workshop on Higher Categories and their Applications, January 10, 2007.

A discussion of weak $\omega$-functors between Batanin $\omega$-categories is in

* [[Richard Garner]], _Homomorphisms of higher categories_ ([arXiv:0810.4450](http://arxiv.org/abs/0810.4450))

An application of Batanin weak $\omega$-groupoids to [[homotopy type theory]] appears in

* [[Benno van den Berg]], [[Richard Garner]], _Types are weak $\omega$-groupoids_ ([arXiv:0812.0298](http://arxiv.org/abs/0812.0298))

A discussion of weak $\omega$-functors between Batanin $\omega$-categories, and all kind of weak $n$-transformations in the spirit of Batanin approach, with an emphasis to the possibility to the existence of the weak $\omega$-category of the weak $\omega$-categories in Batanin's sense appears in

* [[Camell Kachour]], Steps toward the weak higher category of weak higher categories in the globular setting, published in *Categories and General Algebraic Structures with Applications* (2015).  ([web](https://www.researchgate.net/publication/283018471_Steps_toward_the_weak_higher_category_of_weak_higher_categories_in_the_globular_setting))

Batanin weak $\omega$-categories were further developed here:

* [[Tom Leinster]], Higher Operads, Higher Categories, London Mathematical Society Lecture Note Series **298**, Cambridge University Press, 2004.   ([arXiv:math/0305049](https://arxiv.org/abs/math/0305049))

An approach to Batanin's weak $\omega$-categories that also sets up a definition of [[Trimble n-category|Trimble infinity-category]] and a [[fundamental infinity-groupoid]] construction is here:

* [[Eugenia Cheng]] and [[Tom Leinster]], Weak ∞-categories via terminal coalgebras, ([arXiv:1212.5853](https://arxiv.org/abs/1212.5853)).

[[!redirects Batanin infinity-category]]
[[!redirects Batanin infinity-categories]]
[[!redirects Batanin weak infinity-category]]
[[!redirects Batanin infinity-categories]]
[[!redirects Batanin ∞-category]]
[[!redirects Batanin omega-categories]]
[[!redirects Batanin ∞-categories]]
[[!redirects Batanin omega-groupoid]]
[[!redirects Batanin ∞-groupoid]]
[[!redirects Batanin omega-groupoids]]
[[!redirects Batanin ∞-groupoids]]
[[!redirects Batanin weak omega-category]]
[[!redirects Batanin weak ∞-category]]
[[!redirects Batanin weak omega-categories]]
[[!redirects Batanin weak ∞-categories]]
[[!redirects Batanin weak omega-groupoid]]
[[!redirects Batanin weak ∞-groupoid]]
[[!redirects Batanin weak omega-groupoids]]
[[!redirects Batanin weak ∞-groupoids]]


Baues > history

< [[Baues]]
[[!redirects Baues -- history]]


Baues-Wirsching cohomology

##The development of the cohomology of categories.

The [[cohomology of categories|cohomology of (small) categories]] went through several preliminary stages before the  publication by Baues and Wirshing.  In fact, the actual method and ideas that they use can be found earlier in an unpublished document by Charles Wells from 1979.  That paper traces the theory back to Leech earlier in the 1970s for the related situation of  cohomology of monoids. It handles non-Abelian coefficients as well as the Abelian ones that the Baues and Wirsching theory uses.

##Idea

##References

###Early related work

* Jonathan Leech, _$\mathcal{H}$-coextensions of monoids_, vol. 1, Mem. Amer. Math. Soc, no. 157, American
Mathematical Society, 1975.

* [[Charles Wells]], _Extension theories for categories (preliminary report)_, (available from [here](http://www.cwru.edu/artsci/math/wells/pub/pdf/catext.pdf), 1979.)

###Baues-Wirsching cohomology of small categories:  Explicit references,

* [[Hans Joachim Baues]], Günther Wirsching, _Cohomology of small categories_,  J. Pure Appl. Algebra  38  (1985),  no. 2-3, 187--211, [doi](http://dx.doi.org/10.1016/0022-4049%2885%2990008-8), [MR87g:18013](http://www.ams.org/mathscinet-getitem?mr=87g:18013)

* Hans Joachim Baues, Winfried Dreckmann, _The cohomology of homotopy categories and the general linear group_,  $K$-Theory __3__  (1989),  no. 4, 307--338,  [doi](http://dx.doi.org/10.1007/BF00584524), [MR91d:18008](http://www.ams.org/mathscinet-getitem?mr=91d:18008)

* H. J. Baues, [[F. Muro]], _The homotopy category of pseudofunctors and translation cohomology_, J. Pure Appl. Algebra __211__ (2007), no. 3, 821--850, [doi](http://dx.doi.org/10.1016/j.jpaa.2007.04.008) [MR2008g:18009](http://www.ams.org/mathscinet-getitem?mr=2008g:18009)

* [[Hans Joachim Baues]], _On the cohomology of categories, universal Toda brackets and homotopy pairs_, $K$-Theory __11__ (1997), no. 3, 259--285, [doi](http://dx.doi.org/10.1023/A:1007796409912), [MR98h:55020](http://www.ams.org/mathscinet-getitem?mr=98h:55020)

* Petar Pavešić, _Diagram cohomologies using categorical fibrations_, J. Pure Appl. Algebra  112  (1996),  no. 1, 73--90, [doi]

* [[Teimuraz Pirashvili]], _On the center and Baues-Wirsching cohomology of small categories_,  Georgian Math. J. __16__  (2009),  no. 1, 131--144, [journal](http://www.reference-global.com/doi/abs/10.1515/GMJ.2009.145)

* [[Mamuka Jibladze]], [[Teimuraz Pirashvili]], _Quillen cohomology and Baues-Wirsching cohomology of algebraic theories_  Cah. Topol. Géom. Différ. Catég.  47  (2006),  no. 3, 163--205, [numdam](http://www.numdam.org/item?id=CTGDC_2006__47_3_163_0)

* [[Teimuraz Pirashvili]], María Julia Redondo, _Cohomology of the Grothendieck construction_  Manuscripta Math. __120__  (2006),  no. 2, 151--162, [doi](http://dx.doi.org/10.1007/s00229-006-0634-1)

Relation with Andre-Quillen cohomology and $(S,O)$-cohomology of Dwyer-Kan is elucidated in

* [[Hans-Joachim Baues]], David Blanc, _Comparing cohomology obstructions_, [arxiv/1008.1712](http://arxiv.org/abs/1008.1712)


Baum-Connes conjecture

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## Idea

The Baum-Connes conjecture asserts that under suitable technical conditions, the [[operator K-theory]]/[[KK-theory]] of the [[groupoid convolution algebra]] of an [[action groupoid|action]] [[topological groupoid]] $X//G$ is equivalent to the _$G$-[[equivariant cohomology|equivariant]]_ [[topological K-theory]]/[[equivariant operator K-theory]]/[[equivariant KK-theory]] of $X$. Moreover, it says that this equivalences is exhibited by the _[[analytic assembly map]]_.
(Just the [[injective map|injectivity]] of this map is related to the [[Novikov conjecture]].)

The original version of the Baum-Connes conjecture ([Baum-Connes](#BaumConnes00)) stated for a suitable [[topological group]] $G$ that with $X = E G \simeq \ast$ the point incarnated as the $G$-[[universal principal bundle]] with its free $G$-[[action]] the [[analytic assembly map]] (the $G$-equivariant [[index]] map)

$$
  K_\bullet^G(E G) \stackrel{}{\to} K_\bullet(C(\mathbf{B}G))
  \,,
$$

from the $G$-[[equivariant K-theory]] of $E G$ to the [[operator K-theory]] of the [[group algebra]], hence the [[groupoid convolution algebra]] of the [[delooping]] groupoid $\mathbf{B}G \simeq \ast // G$, is an [[isomorphism]].

This Baum-Connes conjecture is known to be true for

* [[compact topological groups]],
* [[abelian groups]],
* [[Lie groups]] with finitely many [[connected components]];
* $p$-adic groups,
*  adelic groups.

It is not known if the conjecture is true for all [[discrete groups]].

Later the statement was generalized ([Tu 99](#Tu99)) to more general groupoids.

In ([Kasparov 88](#Kasparov88)) the refinement of the [[analytic assembly map]] to [[equivariant cohomology|equivariant]] [[KK-theory]] is given, and called the _descent map_. This is of the form (recalled as [Blackadar, theorem 20.6.2](#Blackadar))

$$
  KK^G(A,B) \to KK(G \ltimes A, G \ltimes B)
$$

where on the left we have $G$-[[equivariant KK-theory]] and on the right ordinary [[KK-theory]] of [[crossed product C*-algebras]] (which by the discussion there are models for the [[groupoid convolution algebras]] of $G$-[[action groupoids]]).

This is an [[isomorphism]] at least for $G$ a [[compact topological group]] and restricted to [[operator K-theory]] (hence to the first argument being $\mathbb{C}$) and for $G$ a [[discrete group]] and restricted to [[K-homology]] (hence ot the second argument being $\mathbb{C}$). In this form this is the **Green-Julg theorem**, see [below](#GreenJulgTheorem).

## Theorems
 {#Theorems}

### Green-Rosenberg-Julg theorem

The Green-Rosenberg-Julg theorem identifies [[equivariant K-theory]] with the [[operator K-theory]] of [[crossed product algebras]].

+-- {: .num_theorem #GreenJulgTheorem}
###### Theorem
**(Green-Julg theorem)**

Let $G$ be a [[topological group]] acting on a [[C*-algebra]] $A$.

1. If $G$ is a [[compact topological group]] then the descent map

   $$
     KK^G(\mathbb{C}, A) \to KK(\mathbb{C},G\ltimes A)
   $$

   is an [[isomorphism]], identifying the [[equivariant operator K-theory]] of $A$ with the ordinary [[operator K-theory]] of the [[crossed product C*-algebra]] $G \ltimes A$.

1. if $G$ is a [[discrete group]] then the descent map

   $$
     KK^G(A, \mathbb{C}) \to KK(G \ltimes A, \mathbb{C})
   $$

   is an [[isomorphism]], identifying the equivariant [[K-homology]] of $A$ with the ordinary [[K-homology]] of the [[crossed product C*-algebra]] $G \ltimes A$.

=--

This goes back to ([Green 82](#Green82)), ([Julg 81](#Julg81)). A [[KK-theory]]-proof is in ([Echterhoff, theorem 0.2](#Echterhoff)); a textbook account is in ([Blackadar, 11.7, 20.2.7](#Blackadar)). See also around ([Land 13, prop. 41](#Land13)).

## Related theorems

* [[Atiyah-Segal completion theorem]], [[Green-Julg theorem]]

* [[delocalized equivariant cohomology]]

[[!include Segal completion -- table]]

## References

### Introductions and surveys

Introductions and surveys include

* [[Alain Valette]], _Introduction to the Baum-Connes conjecture_  ([pdf](http://www.univ-orleans.fr/mapmo/membres/chatterji/Valette.pdf))

* [[Nigel Higson]], _The Baum-Connes conjecture_ ([pdf](http://media.cit.utexas.edu/math-grasp/Higson_supplemental_1.pdf))

* [[Paul Baum]], _The Baum-Connes conjecture, localisation of categories and quantum groups_, 2008 ([pdf](http://www.mimuw.edu.pl/~pwit/TOK/sem8/files/Baum_bcclcqg.pdf))

Textbook discussion:

* {#Blackadar} [[Bruce Blackadar]], Section 11.7 and  20.2.7 of: _[[K-Theory for Operator Algebras]]_,   Cambridge University Press 1986, second ed. 1999 ([doi:10.1007/978-1-4613-9572-0](https://link.springer.com/book/10.1007/978-1-4613-9572-0), [pdf](http://wolfweb.unr.edu/homepage/bruceb/Book6.pdf))

* {#MislinValette03} [[Guido Mislin]], [[Alain Valette]], _Proper Group Actions and the Baum-Connes Conjecture_, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 ([doi:10.1007/978-3-0348-8089-3](https://link.springer.com/book/10.1007/978-3-0348-8089-3))

See also

* Wikipedia, _[Baum-Connes conjecture](http://en.wikipedia.org/wiki/Baum%E2%80%93Connes_conjecture)_

### Original articles

The original article is

* [[Paul Baum]], [[Alain Connes]], _K-theory for Lie groups and foliations_, Enseign. Math. 46 (2000), 3–42.
 {#BaumConnes00}

Proof of the conjecture for hyperbolic groups is in

* [[Vincent Lafforgue]], _La conjecture de Baum-Connes à coefficients pour les groupes hyperboliques_, Journal of Noncommutative Geometry, Volume 6, Issue 1, 2012, pages 1-197 ([arXiv:1201.4653](http://arxiv.org/abs/1201.4653))

Technical subtleties are discussed in

* [[Nigel Higson]], [[Vincent Lafforgue]], [[Georges Skandalis]], _Counterexamples to the Baum-Connes conhecture_ ([pdf](http://www.personal.psu.edu/ndh2/math/Papers_files/Higson,%20Lafforgue,%20Skandalis%20-%202002%20-%20Counterexamples%20to%20the%20Baum-Connes%20conjecture.pdf))

The generalization to [[Lie groupoids]] is due to

* [[Jean-Louis Tu]], _The Baum-Connes conjecture for groupoids_, 1999 ([[JLTBaumConnesForGroupoids.pdf:file]])
 {#Tu99}

Proofs for some cases are in

* [[Georges Skandalis]], [[Jean-Louis Tu]], G. Yu, _The coarse Baum-Connes conjecture and groupoids_ ([pdf](http://www.math.univ-metz.fr/~tu/publi/coarse.pdf))

[[KK-theory]] tools and the descent map are introduced in

* [[Gennady Kasparov]], _Equivariant KK-theory and the Novikov conjecture_, Inventiones Mathematicae, vol. 91, p.147, 1988([web](http://adsabs.harvard.edu/abs/1988InMat..91..147K))
 {#Kasparov88}

The "Green-Julg theorem" for commutative algebra and finite group is due to [[Michael Atiyah]], for commutative algebra and general group due to

* P. Green, _Equivariant if-theory and crossed product C*- algebras_, pp.  337-338 in Operator algebras and applications (Kingston, Ont., 1980), vol. 1, edited
by R. V. Kadison, Proc. Sympos. Pure Math. 38, Amer. Math. Soc, Providence, 1982.
{#Green82}

and the general case is due to an unpublished result by Green and [[Jonathan Rosenberg]] and independently due to

* P. Julg, _K-theorie equivariante et produits croises_, C. R. Acad. Sci.  Paris Ser. I Math. 292:13 1981, 629-632.
 {#Julg81}

Further discussion is in

* Siegfried Echterhoff, _The Green-Julg theorem_ [pdf](http://wwwmath.uni-muenster.de/u/paravici/Focused-Semester/lecturenotes/green-julg-Echterhoff.pdf)
 {#Echterhoff}

* Walther Paravicini, _A generalised Green-Julg theorem for proper groupoids and Banach algebras_, ([arXiv:0902.4365](http://arxiv.org/abs/0902.4365))

* V. Lafforgue, [pdf](http://www.mmas.univ-metz.fr/~gnc/bibliographie/HarmonicAnalysis/LafforgueICM.pdf)

A modification of the Baum-Connes conjecture with coefficient where many counterexamples (to the  conjecture with coefficients) are eliminated is in

* [[Paul Baum]], Erik Guentner, Rufus Willett, _Expanders, exact crossed products, and the Baum-Connes conjecture_, [arxiv/1311.2343](http://arxiv.org/abs/1311.2343)

Discussion in terms of [[localization]]/[[homotopy theory]] is in

* [[Ralf Meyer]], [[Ryszard Nest]], _The Baum-Connes conjecture via localisation of categories, Topology 45 (2006), no. 2, 209–259.

[[!redirects Green-Julg theorem]]


Baum-Douglas geometric cycle

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## Idea

A _Baum-Douglas geometric cycle_ on a given [[manifold]] $X$ is a representative for [[K-homology]] classes on $X$. It is given by a [[submanifold]] $Q \hookrightarrow X$ equipped with [[spin^c structure]] and with a [[complex vector bundle]]. The [[equivalence relation]] identifying such data that represents the same [[K-homology]] class includes a compatible [[bordism]] relation.

Viewed as a [[correspondence]] of the form

$$
  X \stackrel{}{\leftarrow} (Q,E) \to \ast
$$

a Baum-Douglas geometric cycle is a special case of the spans that represent classes in [[KK-theory]] (between manifolds) according to ([Connes-Skandalis 84, section 3](#ConnesSkandalis84)).

In [[string theory]] Baum-Douglas cycles constitute one formalization of the concept of [[D-brane]] carrying a [[Chan-Paton gauge field]] ([Reis-Szabo 05](#ReisSzabo05), [Szabo 08](#Szabo08)): the submanifold $Q$ represents the [[worldvolume]] of the D-brane and the [[complex vector bundle]] it carries the [[Chan-Paton gauge field]].

## Related concepts

* [[motivic function]]

## References

The original articles:

* [[Paul Baum]], R. Douglas, *K-homology and index theory*, in: R. Kadison (ed.), *Operator Algebras and Applications, Proceedings of Symposia in Pure Math. **38** AMS (1982) 117-173 [[ams:pspum-38-1](https://bookstore.ams.org/pspum-38-1)]

* [[Paul Baum]], R. Douglas. _Index theory, bordism, and K-homology_, Contemp. Math. **10** (1982) 1-31

A generalization to [[twisted K-theory|twisted]] homology is discussed in section of

* [[Bai-Ling Wang]], _Geometric cycles, index theory and twisted K-homology_,  Journal of Noncommutative Geometry ([arXiv:0710.1625](http://arxiv.org/abs/0710.1625))

A generalization to geometric (co)-cocycles for KK-theory is in section 3 of

* {#ConnesSkandalis84} [[Alain Connes]], [[Georges Skandalis]], _The longitudinal index theorem for foliations_. Publ. Res. Inst. Math. Sci. 20,
no. 6, 1139–1183 (1984) ([pdf](http://www.alainconnes.org/docs/longitudinal.pdf))

More generally, a construction of general [[homology theories]] in a similar fashion is discussed in

* S. Buoncristiano, C. P. Rourke and B. J. Sanderson, _A geometric approach to homology theory_, Cambridge Univ. Press, Cambridge, Mass. (1976)

and a construction of [[bivariant cohomology theories]] in this spirit is in

* Martin Jakob, _Bivariant theories for smooth manifolds_, Applied Categorical Structures 10 no. 3 (2002)

The interpretation as a formalization of [[D-branes]] in [[string theory]] is highlighted in

* {#ReisSzabo05} [[Rui Reis]], [[Richard Szabo]], _Geometric K-Homology of Flat D-Branes_ ,Commun.Math.Phys. 266 (2006) 71-122 ([arXiv:hep-th/0507043](https://arxiv.org/abs/hep-th/0507043))

* {#Szabo08} [[Richard Szabo]], _D-branes and bivariant K-theory_, Noncommutative Geometry and Physics 3 1 (2013): 131. ([arXiv:0809.3029](http://arxiv.org/abs/0809.3029))

[[!redirects Baum-Douglas geometric cycles]]


Baum-Douglas geometric cycles

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## Idea

For a given [[manifold]] $X$ a _Baum-Douglas geoemtric cycle_ on $X$ is data consisting of a [[submanifold]] $Q \hookrightarrow X$ carrying a [[vector bundle]] $E\to X$ such that this represents a class in [[K-homology]] under a suitable [[equivalence relation]].

Viewed as a [[correspondence]] of the form

$$
  X \stackrel{}{\leftarrow} (Q,E) \to \ast
$$

a Baum-Douglas geometric cycle is a special case of the spans that represent classes in [[KK-theory]] (between manifolds) according to ([Connes-Skandalis 84, section 3](#ConnesSkandalis84)).

## References

The general idea of geometric cycles for [[homology theories]] goes back to

* [[William Fulton]], [[Robert MacPherson]], _Categorical framework for the study of singular spaces_, Memoirs of the AMS, 243, 1981

The original articles by Baum and Douglas are

* [[Paul Baum]], R. Douglas, _K-homology and index theory: Operator Algebras and Applications (R. Kadison editor), volume 38 of Proceedings of Symposia in Pure Math., 117-173,
Providence RI, 1982. AMS.

* [[Paul Baum]], R. Douglas. _Index theory, bordism, and K-homology_, Contemp. Math. 10: 1-31 1982.

The proof that these geometric cycles indeed model all of [[K-homology]] is due to

* [[Paul Baum]], [[Nigel Higson]], [[Thomas Schick]], _On the Equivalence of Geometric and Analytic K-Homology_ ([arXiv:math/0701484](http://arxiv.org/abs/math/0701484))

* Jeff Raven, _An equivariant bivariant Chern character_, PhD Thesis

A generalization to geometric (co)-cocycles for [[KK-theory]] is in section 3 of

* [[Alain Connes]], [[Georges Skandalis]], _The longitudinal index theorem for foliations_. Publ. Res. Inst. Math. Sci. 20,
no. 6, 1139–1183 (1984) ([pdf](http://www.alainconnes.org/docs/longitudinal.pdf))
 {#ConnesSkandalis84}

and for [[equivariant KK-theory]] in

* [[Heath Emerson]], [[Ralf Meyer]], _Bivariant K-theory via correspondences_, Adv. Math. 225 (2010), 2883-2919 ([arXiv:0812.4949](http://arxiv.org/abs/0812.4949))

For further such generalizations see at [[bivariant cohomology theory]].

Generalization to [[twisted K-homology]] is in

* [[Bai-Ling Wang]], _Geometric cycles, index theory and twisted K-homology_ ([arXiv:0710.1625](http://arxiv.org/abs/0710.1625))

[[!redirects Baum-Douglas geometric cycles]]


Bayes rule

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#Bayes rule#
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## Statement

In [[probability theory]], the __Bayes Rule__ (or _Bayes\'s Law_, _Bayes\' Theorem_, or another permutation) is the statement that the [[conditional probabilities]] $P(H\vert E)$ for an event $H$, assuming an event $E$ is related to the condition probability $P(E\vert H)$ of $E$ assuming $H$, and the plain probabilities $P(H)$ and $P(E)$ for $H$ and $E$ separately, by

$$
  P(H|E) \;=\; \frac{P(E|H) P(H)} {P(E)}
  \,.
$$

This follows directly from the defining formula $P(A|B) = P(A \wedge B)/P(B)$ for [[conditional probability]].  The rule may also be written in the expanded form

$$ P(H|E) = \frac{P(E|H) P(H)} {P(E|H) P(H) - P(E|\neg{H}) P(H) + P(E|\neg{H})} ,$$

which additionally uses some of the axioms of probability, or somewhere in between these two forms.

As a theorem, it is quite trivial; the point is in its application as a rule for updating the probability of some hypothesis ($H$) on the basis of some evidence ($E$) (which is key to [[Bayesianism]]), using the _prior_ probability of the hypothesis before the evidence is obtained ($P(H)$) and (in the expanded form) the conditional probabilities of obtaining that evidence in the situation where the hypothesis is true ($P(E|H)$) and in the situation where the hypothesis is false ($P(E|\neg{H})$).  (Ideally, the last two can be determined on a purely theoretical basis, but since the probability of $E$ usually depends on other hypotheses of unknown veracity, the application is not always so simple.)

## Applications

### In quantum physics

In [[quantum mechanics]], the [[collapse of the wavefunction]] may be seen as a generalization of Bayes\'s Rule to [[quantum probability theory]].  This is key to the [[Bayesian interpretation of quantum mechanics]].

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# The Bayesian interpretation of quantum physics
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## Idea

[[mathematics|Mathematically]], [[quantum mechanics]], and in particular [[quantum statistical mechanics]], can be viewed as a generalization of [[probability theory]], that is as _[[quantum probability theory]]_.  The [[Bayesianism|Bayesian]] interpretation of probability can then be generalized to a Bayesian [[interpretation of quantum mechanics]], and thus (in principle) of all [[physics]].

The Bayesian interpretation is founded on these principles:

1. Quantum states (pure or mixed) are analogous to (indeed generalizations of) probability distributions, which are to be interpreted in a Bayesian way, as indicating knowledge, belief, etc.

2. Time evolution of pure quantum states by the time-dependent [[Schroedinger equation|Schrödinger's equation]] is analogous to evolution of classical statistical systems by [[Liouville's equation]] (and von Neumann has an equation for the evolution of [[density matrices]] that generalizes both).

3. [[wave function collapse|Collapse of the wave function]] (see also [[propositions as projections]]) is analogous to (indeed a generalization of) updating a probability distribution; [[Born rule|Born's Rule]] and [[Bayes rule|Bayes' Rule]] are analogues.

One should perhaps speak of *a* Bayesian interpretation of quantum mechanics, since there are different forms of Bayesianism.  (This article has been written by an objective Bayesian who naturally thinks of Bayesian probabilities as reflecting information rather than belief, betting commitments, etc.  For other flavours of Bayesianism, substitute 'knowledge' below with a more appropriate term.)

## Formalism

There are various ways to formulate [[quantum mechanics]] mathematically, but the following elements are fairly common:

*  There is an algebra $A$ of [[observables]]; some or all of the elements of this algebra correspond to physically observable quantities.
*  There is a space $S$ of [[states]], by which we mean [[mixed states]] in general.
*  Given a state $\psi$ and an observable $O$, we obtain a [[probability distribution]] (a [[probability measure]] on the [[real line]] or something like it).

Roughly speaking, this probability distribution tells us the probability of observing the value of $O$ to take particular values given that the system has state $\psi$.  Let us write $O_*\psi(E)$ for the probability that the value of $O$ will be observed to belong to the event $E$ (which is a [[measurable set]] of real numbers when $O_*\psi$ is a probability measure on the real line) for a system in state $\psi$.

So for example, we could actually be talking about a [[classical physics|classical]] system, in which we have a [[state space]] or [[phase space]] $X$, an observable is a function on $X$, a [[pure state]] is a point in $X$, and a general state is a probability measure on $X$.

Or we could be talking about a quantum system given by a [[Hilbert space]] $H$, in which an observable is a [[self-adjoint operator]] on $H$, a pure state is a $1$-dimensional subspace of $H$, and a general state is a [[density matrix]] on $H$.

Or we could use a more abstract formulation, such as those of [[AQFT]], [[FQFT]], and so forth; but all of these still have observables and states.

See also [[JBW-algebraic quantum mechanics]] for a mathematical formalism that may be motivated by treating the probability distributions $O_*\psi$ as fundamental (among other considerations).

## Interpretation

The point of the Bayesian interpretation is that the probabilities $O_*\psi(E)$ are to be interpreted as Bayesian probabilities.  That is, they are not objective features of reality (the territory) but rather an expression of what a rational observer might know about the world (a map).

Thus, a state $\psi$ represents a _state of knowledge_ about the world.  The algebra $A$ of observables may be objective (or fully known), but the particular state $\psi$ that the system is in depends on what is known about that particular system.  There is (except in the classical case) *no* possible complete specification of the actual values of all observables, only a probabilistic specification of what one might know about them.

In the [[Schroedinger picture]] (assuming a notion of [[time]]), states [[time evolution|evolve]] deterministically, unitarily, and with conservation of [[entropy]].  (Or in the [[Heisenberg picture]], they don\'t evolve at all, with the observables evolving instead.)  But a state may change otherwise, if one\'s knowledge changes.  If this is an increase in knowledge as a result of a [[measurement]], then this change in the state may be called the 'collapse of the wavefunction'.  But this collapse takes place in the map, not the territory; unlike time evolution, it is not a physical process.

Given a specification of $S$ and $A$, it may be that there exist certain states $\psi$ in $S$ such that $O_*\psi$ is a [[delta measure]] (giving a probability of $1$ for one value and $0$ for all others) for every $O$ in $A$.  Then the system is classical.  Depending on the mathematical formalism used, such states may not actually belong to $S$ (which might, for example, own only the [[absolutely continuous measure|absolutely continuous]] probability measures in some sense, as is natural in the [[JBW-algebraic quantum mechanics|W*-algebraic approach]]); so in general, we may say that the system is __classical__ if there exists a [[net]] $(\psi_n)_n$ of states such that, for each observable $O$, the net $(O_*\psi_n)_n$ of measures on the real line [[convergence in measure|converges in measure]] to a delta measure.  (Or so I imagine; this definition might not really be general enough.)

## History

People have implied (for example at the beginning of [Caves, Fuchs, & Schack, 2001](#CFS2001)) that this is what [[Niels Bohr]] meant all along when he put forth the [[Copenhagen interpretation]] (for more on this suggestion see also at _[[Bohr topos]]_), and people have implied (in [Fuchs, 2002](#Fuchs2002)) that it is what [[Albert Einstein]] was groping towards when he attacked Bohr, and still others ([[Ray Streater]] as cited in [Wikipedia](#Wikipedia)) have implied that it is what [[John von Neumann]] was doing when he eschewed interpretation for mathematical axioms.  Any time that somebody has described a quantum state as containing information, or being given by an experimenter\'s knowledge, or being different for one observer than for another, the Bayesian interpretation is implicit.  Arguably, it is implicit in any statement that the wavefunction describes probabilities, if [[probability]] is treated as Bayesian.  However, the *explicit* exposition of this interpretation seems to have come rather late.

The earliest linking of Bayesianism to quantum states as states of knowledge, as far as I have seen, is Usenet discussion in 1994 ([Baez et al, 2003](#Baez2003)).  [[John Baez]] was promoting similar ideas the previous year ([Baez, 1993](#Baez1993)) (and this is not the origin of these ideas either), but this still allows other interpretations.  The idea of a 'Bayesian interpretation' came to prominence in 2001, drawing out of work on [[quantum information theory]] ([Caves, Fuchs, & Schack, 2001](#CFS2001)).  Further work has been done principally by [[Christopher Fuchs]].

## Related interpretations

The Bayesian interpretation fits into a broader family of [[interpretations of quantum mechanics]] known as _epistemic_ (or _$\psi$-epistemic_ to be more precise).  Although 'epistemic' literally refers to knowledge (suggesting objective Bayesianism in particular), the term may be used more broadly to distinguish from _ontic_ interpretations, in which the state $\psi$ is an objective feature of reality.  In contrast, an epistemic interpretation is one in which the state is subjective in some way (whether relative to one\'s knowledge or otherwise).  Much of what is written above applies more broadly to any epistemic interpretation, although I\'m not sure how much; at least some epistemic interpretations go on to posit a more fundamental reality (involving [[hidden-variable theory|hidden variables]]), while the Bayesian interpretation does not require this.

In the other direction, 'Quantum Bayesianism' or 'QBism' is used specifically to refer to the position advanced over the course of the 21st century principally by [[Christopher Fuchs]].  Although Fuchs\'s first papers on the subject (starting with [Caves, Fuchs, & Schack, 2001](#CFS2001)) referred explicitly to 'states of knowledge', Fuchs has since adopted an increasingly subjective approach, drawing many philosophical conclusions that go beyond mere Bayesianism.  These ideas should be attributed to QBism specifically rather than to the Bayesian interpretation generally. For a historical view of how QBism came to be distinguished from earlier Bayesian interpretations of quantum probability, See [Stacey (2019)](#Stacey2019). The technical side of this work has generally not yet taken a categorical flavor, though see [van de Wetering (2018)](#vandeWetering2018).

## Related entries

* [[interpretation of quantum mechanics]]

* [[quantum probability]]

* [[collapse of the wave function]]

## References

* {#Baez1993} [[John Baez]] (1993). This Week's Finds in Mathematical Physics (Week 27). [Web](http://math.ucr.edu/home/baez/week27.html).

* {#Baez2003} John Baez et al (2003). Bayesian Probability Theory and Quantum Mechanics. Collection of Usenet posts from 1994, with commentary. [Web](http://math.ucr.edu/home/baez/bayes.html).

* {#CFS2001} Carlton Caves, [[Christopher Fuchs]], Ruediger Schack (2001). Unknown Quantum States: The Quantum de Finetti Representation. [arXiv:quant-ph/0104088](http://arxiv.org/abs/quant-ph/0104088).

* {#Fuchs2002} [[Christopher Fuchs]] (2002). Quantum Mechanics as Quantum Information (and only a little more). [arXiv:quant-ph/0205039](https://arxiv.org/abs/quant-ph/0205039).

* {#FMS2014} [[Christopher Fuchs]], [[David Mermin]], Ruediger Schack (2014). An Introduction to QBism with an Application to the Locality of Quantum Mechanics. American Journal of Physics 82:8, 749--754. [arXiv:1311.5253](https://arxiv.org/abs/1311.5253).

* {#FSS2014} [[Christopher Fuchs]], [[Maximilian Schlosshauer]] (forward), [[Blake Stacey]] (editor) (2014). My Struggles with the Block Universe. [arXiv:1405.2390](https://arxiv.org/abs/1405.2390).

* {#Mermin2018} [[David Mermin]] (2018). Making better sense of quantum mechanics. [arXiv:1809.01639](https://arxiv.org/abs/1809.01639).

* {#vandeWetering2018} John van de Wetering (2018). Quantum Theory is a Quasi-stochastic Process Theory. EPTCS 266, 179--196. [arXiv:1704.08525](https://arxiv.org/abs/1704.08525).

* {#Stacey2019} [[Blake Stacey]] (2019). Ideas abandoned en route to QBism. [arXiv:1911.07386](https://arxiv.org/abs/1911.07386).

See also

* {#Wikipedia} Wikipedia. Quantum Bayesianism. [Web](https://en.wikipedia.org/wiki/Quantum_Bayesianism).

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Bayesian inversion

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## Idea

In elementary [[probability theory]], [[Bayes' formula]] refers to a version of the formula
$$
   P(A)\,P(B|A) \,=\, P(B)\,P(A|B)
   \,,
$$
and relates the [[conditional probability]] of $B$ given $A$ to the one of $A$ given $B$.

From the [[category-theoretic probability|category-theoretic point of view]], this formula expresses a duality, sometimes called **Bayesian inversion**, which often gives rise to a [[dagger category|dagger structure]].

## Intuition

In logical reasoning, implications in general cannot be reversed: $A \to B$, alone, does not imply $B\to A$.

In [[probability theory]], instead, conditional statements exhibit a [[duality]] which is absent in pure logical reasoning.

Consider for example a city in which *all taxis are yellow*. (The implication is $taxi\to yellow$.)
If we see a yellow car, of course we can't be sure it's a taxi. However, it's *more likely* that it's a taxi compared to a randomly colored car. This increase in likelihood is larger if the fraction of yellow cars is small. Indeed, [[Bayes' rule]] says that
$$
P(taxi|yellow) = \frac{P(yellow\,taxi)}{P(yellow)} = \frac{P(taxi)}{P(yellow)} .
$$

More generally, in a city where *most* taxis are yellow, there is a high conditional probability that a given taxi is yellow,
$$
P(yellow|taxi) .
$$
The higher this probability is, the higher is the probability that a given yellow car is a taxi, again according to [[Bayes' rule]]:
$$
P(taxi|yellow) = \frac{P(yellow\,taxi)}{P(yellow)} = \frac{P(taxi)\,P(yellow|taxi)}{P(yellow)}
$$

In [[categorical probability]], this phenomenon can be modeled by saying that to each "conditional" morphism in the form $\{taxi, not\,taxi\}\to\{colors\,of\,cars\}$ there corresponds a canonical morphism $\{colors\,of\,cars\}\to\{taxi, not\,taxi\}$, called the *Bayesian inverse*.

In some cases, this symmetry gives rise to a [[dagger category]].

## In traditional probability theory

In traditional [[probability theory]], Bayesian inversions are a special case of [[conditional probability]].
Some care must be taken to avoid dividing by zero or incurring into paradoxes via limiting procedures.

### Discrete case

In the discrete case, a [[probability distribution]] on a [[set]] $X$ is simply a function $p:X\to[0,1]$ such that
$$
\sum_{x\in X} p(x) = 1 .
$$

A [[stochastic map]] $k:X\to Y$ is a function
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
]
X\times Y \ar{r} & {[0,1]} \\
(x,y) \ar[mapsto]{r} & k(y|x)
\end{tikzcd}
such that for all $x\in X$,
$$
\sum_{y\in Y} k(y|x) = 1 .
$$
If we now equip $X$ and $Y$ with discrete probability distributions $p$ and $q$, obtaining discrete [[probability spaces]], we say that $k$ is a [[stochastic map#measurepreserving_stochastic_maps|measure-preserving stochastic map]] $(X,p)\to(Y,q)$ if for every $y\in Y$,
$$
\sum_{x} k(y|x) \, p(x) = q(y) .
$$

A **Bayesian inverse** of $k:(X,p)\to(Y,q)$ is then defined to be a measure-preserving stochastic map $k^\dagger:(Y,q)\to(X,p)$ such that for every $x\in X$ and $y\in Y$, the following [[Bayes formula]] holds.
$$
p(x) \,k(y|x) = q(y)\,k^\dagger(x|y)
$$

In the discrete case, Bayesian inverses always exist, and can be obtained by taking
$$
k^\dagger(x|y) \coloneqq \frac{p(x)\,k(y|x)}{q(y)}
$$
for those $y\in Y$ with $q(y)\ne 0$, and an arbitrary number on the $y\in Y$ with $q(y)=0$.
(To ensure the normalization condition, on such $y$ one can for example take $k^\dagger(x|y)=\delta_{x_0,x}$, where $x_0$ is a fixed element of $X$. Note that $X$ is nonempty since it admits a probability measure.)

Bayesian inverses are not unique, but they are uniquely defined on the support of $q$. That is, they are unique up to [[almost surely|almost sure]] [[equality]].

### Measure-theoretic case

The situation is more delicate outside the discrete case.
Given [[probability spaces]] $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$, and a [[Markov kernel#measurepreserving_kernels|measure-preserving Markov kernel]] $k:(X,\mathcal{A},p)\to(Y,\mathcal{B},q)$, a **[[Markov kernel#bayesian_inversion|Bayesian inverse]]** of $k$ is a Markov kernel $k^\dagger:(Y,\mathcal{B},q)\to(X,\mathcal{A},p)$ such that for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$, the following Bayes-type formula holds.
$$
\int_A k(B|x)\,p(dx) = \int_B k^\dagger(A|y)\,q(dy)
$$
As one can see from [[Markov kernel#almost_sure_equality|Markov kernel - Almost sure equality]], this formula specifies a kernel only up to [[almost surely|almost sure]] [[equality]], just as in the discrete case.

Existence, on the other hand, is more tricky. In general, a kernel $k^\dagger$ as above may fail to exist. The problem is that in order for $k^\dagger$ to be a well-defined [[Markov kernel]] $(Y,\mathcal{B})\to(X,\mathcal{A})$, we need the following two conditions:

* the map $y\mapsto k^\dagger(A|y)$ is [[measurable function|measurable]] in $y$ for all $A\in\mathcal{A}$;
* the assignment $A\mapsto k^\dagger(A|y)$ is a [[probability measure]] in $A$ for all $y\in Y$.

The first condition can be taken care of using [[conditional expectation]].
That however does not assure the second condition. It can be shown, however, that if $(X,\mathcal{A})$ is [[standard Borel]] and either

* $(Y,\mathcal{B})$ is standard Borel too

or

* the kernel $k:X\to Y$ is in the form $\delta_f$ for a measurable $f:X\to Y$,

then a Bayesian inverse always exists.

(See also [[Markov kernel#bayesian_inversion|Markov kernel - Bayesian inversion]] and [[Markov kernel#conditionals|Markov kernel - Conditionals]].)

## In categorical probability

In [[categorical probability]], Bayesian inverses are axiomatized in a way which reflects the measure-theoretic version of the concepts.
One then can choose to work in categories where such axioms are satisfied.

### In Markov categories

In [[Markov categories]], Bayesian inverses are defined in a way that parallels the construction for Markov kernels.

The abstraction of a [[probability space]] is given by an [[object]] $X$ in a [[Markov category]], together with a state $p:I\to X$. As usual, the abstraction of a kernel is a morphism $f:X\to Y$.
A **Bayesian inverse** of $f$ with respect to $p$ is a morphism $f^\dagger:Y\to X$ such that the following equation holds, where $q=f\circ p$.

\begin{tikzpicture}[%
thick,%
scale=0.8,%
every node/.style={scale=1.25},%
none/.style={fill=none,draw=none},%
morphism/.style={fill=white, draw=black, shape=rectangle},%
bn/.style={fill=black, draw=black, shape=circle, inner sep=1.8pt},%
over arrow/.style={-, black, preaction={draw=white, line width=2mm}},%
]

		\node [style=bn] (0) at (-3, -1) {};
		\node [style=bn] (1) at (3, -1) {};
		\node [style=none] (2) at (-3, -2) {};
		\node [style=none] (3) at (3, -2) {};
		\node [style=none] (4) at (-4, 0) {};
		\node [style=none] (5) at (-2, 0) {};
		\node [style=none] (6) at (2, 0) {};
		\node [style=none] (7) at (4, 0) {};
		\node [style=none] (8) at (-4, 1.5) {};
		\node [style=none] (9) at (-2, 1.5) {};
		\node [style=none] (10) at (2, 1.5) {};
		\node [style=none] (11) at (4, 1.5) {};
		\node [style=none] (16) at (-4, 2) {$X$};
		\node [style=none] (17) at (-2, 2) {$Y$};
		\node [style=none] (18) at (2, 2) {$X$};
		\node [style=none] (19) at (4, 2) {$Y$};
		\node [style=none] (20) at (0, -0.25) {$=$};

		\draw [in=165, out=-90] (4.center) to (0);
		\draw [in=15, out=-90] (5.center) to (0);
		\draw (0) to (2.center);
		\draw [in=165, out=-90] (6.center) to (1);
		\draw [in=15, out=-90] (7.center) to (1);
		\draw (1) to (3.center);
		\draw (8.center) to (4.center);
		\draw (9.center) to (5.center);
		\draw (10.center) to (6.center);
		\draw (11.center) to (7.center);

		\node [style=morphism] (12) at (-4, 0.25) {$f^\dagger_p$};
		\node [style=morphism] (13) at (4, 0.25) {$f$};
		\node [style=morphism] (14) at (-3, -2) {$q$};
		\node [style=morphism] (15) at (3, -2) {$p$};
\end{tikzpicture}

This recovers the classical probability definitions when instantiated in [[Stoch]] and its subcategories.

Just as in traditional probability, Bayesian inverses are unique only up to [[almost sure equality]].
Also, just as in traditional probability, they may fail to exist. Being an instance of [[Markov category#conditionals|conditionals]], however, they always exists when conditionals exist, such as in the category [[BorelStoch]].

(See also [[Markov category#conditionals|Markov category - conditionals]].)

### In dagger categories

In the [[category of couplings]], the idea of Bayesian inversion is made explicit from the start by means of the [[dagger category|dagger structure]].
Given [[probability spaces]] $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$, a [[coupling (probability theory)|coupling]] between them can be seen equivalently as going from $X$ to $Y$ or from $Y$ to $X$. This duality, when the couplings are expressed via [[Markov kernels]], reflects exactly Bayesian inversion.
Therefore, at the level of joint distributions, one can consider the duality given by Bayesian inversions to be already part of the symmetry of the category.

Categorical abstractions of the category of coupling via [[dagger categories]] have therefore the concept of Bayesian inversion already built in.

## Quantum versions

(For now, see the references.)

## See also

* [[categorical probability]], [[Markov category]]

* [[transport plan]], [[category of couplings]]

* [[Markov kernel]], [[stochastic map]], [[Stoch]]

* [[conditional probability]], [[conditional expectation]]

* [[probability theory]], [[statistics]], [[Bayesian statistics]]

* [[dagger category]]

## References

For categorical probability:

* {#fritzmarkov} [[Tobias Fritz]], _A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics_, Advances of Mathematics 370, 2020. ([arXiv:1908.07021](http://arxiv.org/abs/1908.07021))

* {#cd_categories} Kenta Cho and [[Bart Jacobs]], _Disintegration and Bayesian Inversion via String Diagrams_, Mathematical Structures of Computer Science 29, 2019. ([arXiv:1709.00322](https://arxiv.org/abs/1709.00322))

* Dario Stein and [[Sam Staton]], _Probabilistic Programming with Exact Conditions_, JACM, 2023. ([arXiv](https://arxiv.org/abs/2312.17141))

* Noé Ensarguet and [[Paolo Perrone]], _Categorical probability spaces, ergodic decompositions_, and transitions to equilibrium, 2023. ([arXiv:2310.04267](https://arxiv.org/abs/2310.04267))

For the quantum case:

* {#coecke_spekkens12} [[Bob Coecke]] and Robert W. Spekkens, _Picturing classical and quantum Bayesian inference_, Synthese, 186(3), 2012. ([arXiv](https://arxiv.org/abs/1102.2368))

* {#quantum_markov} [[Arthur J. Parzygnat]], _Inverses, disintegrations, and Bayesian inversion in quantum Markov categories_, 2020. ([arXiv](https://arxiv.org/abs/2001.08375))

* {#noncommutative_bayes} [[Arthur J. Parzygnat]] and Benjamin P. Russo, _A noncommutative Bayes theorem_, Linear Algebra Applications 644, 2022. ([arXiv](https://arxiv.org/abs/2005.03886))

* {#conditional_quantum} [[Arthur J. Parzygnat]], _Conditional distributions for quantum systems_, EPTCS 343, 2021. ([arXiv](https://arxiv.org/abs/2102.01529))

* {#retrodiction} [[Arthur J. Parzygnat]], Francesco Buscemi, _Axioms for retrodiction: achieving time-reversal symmetry with a prior_, Quantum 7(1013), 2023. [arXiv](https://arxiv.org/abs/2210.13531)

* {#quantum_bayes} James Fullwood and [[Arthur J. Parzygnat]], _From time-reversal symmetry to quantum Bayes rules_, PRX Quantum 4, 2023. ([arXiv](https://arxiv.org/abs/2212.08088))

* {#noncomm_disintegrations} [[Arthur J. Parzygnat]], Benjamin P. Russo, _Non-commutative disintegrations: existence and uniqueness in finite dimensions_, Journal of Noncommutative Geometry 17(3), 2023. ([arXiv](https://arxiv.org/abs/1907.09689))

category: probability

[[!redirects Bayesian inversions]]
[[!redirects Bayesian inverse]]
[[!redirects Bayesian inverses]]
[[!redirects bayesian inversion]]
[[!redirects bayesian inversions]]
[[!redirects bayesian inverse]]
[[!redirects bayesian inverses]]


Bayesian reasoning

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### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
=--
=--

# Bayesian reasoning
* table of contents
{: toc}

## Idea

Bayesian [[reasoning]] is an application of [[probability theory]] to [[inductive reasoning]] (and [[abductive reasoning]]).  It relies on an interpretation of probabilities as expressions of an agent's uncertainty about the world, rather than as concerning some notion of objective chance in the world. The perspective here is that, when done correctly, inductive reasoning is simply a generalisation of [[deductive reasoning]], where knowledge of the truth or falsity of a proposition  corresponds to adopting the extreme probabilities $1$ and $0$.

Several advantages to this interpretation have been proposed. For one thing, it is possible to have (rational) degrees of belief about a wide range of propositions, including matters which are settled, yet unknown. For example, it is possible to speak of the probability of the value of a constant of nature falling within a given interval on the basis of various pieces of evidence, or of the outcome of a coin that has already been tossed. This interpretation of probability beyond limiting frequencies leads, so advocates such as [[Edwin Jaynes]] claim, to a more integrated treatment of probability theory and statistics.

## Dutch Book justification

It can be shown by so-called "Dutch Book" arguments, that a rational agent must set their degrees of belief in the outcomes of events in such a way that they satisfy the axioms of probability theory. The idea here is that to believe a proposition to degree $p$ is equivalent to being prepared to accept a wager at the corresponding odds. For instance, if I believe there is a 0.75 chance of rain today, then I should be prepared to accept a wager so that I receive $S$ units of currency if it does not rain, and pay out $S/3$ units if it does rain. Note that $S$ may be chosen to be negative by the bettor.

It can be shown then that such betting odds must satisfy the probability axioms, otherwise it will be possible for someone to place multiple bets which will cause the bookmaker to suffer a certain loss whatever the outcome. For example, in the case above, my degree of belief that it will _not_ rain today should be 0.25. Were I to offer, say, 0.5, someone could stake $(-3)$ units on the first bet and $(-2)$ units on the second bet, and will gain $1$ unit whether or not it rains.  Of course, real bookmakers have odds which sum to more than 1, but they suffer no guaranteed loss since clients are only allowed positive stakes.

##Cox's axioms

Some consider the reliance on the idea of the undesirability of certain financial loss to be unbefitting for a justification of what is supposed to be an extension of ordinary deductive logic ([Jaynes 2003](#Jaynes)). Axiomatisations in terms of the properties one should expect of degrees of plausibility have been given, and it can be shown from such axioms that these degrees satisfy the axioms of probability. Richard Cox is responsible for one such axiomatisation (for the moment see [Wikipedia: Cox's theorem](http://en.wikipedia.org/wiki/Cox's_theorem)).

##Conditionalizing

Using [[Bayes' Rule]], degrees of belief can be updated on receipt of new evidence.

$$
P(h|e) = P(e|h) \cdot \frac{P(h)}{P(e)},
$$

where $h$ is a hypothesis and $e$ is evidence.

The idea here is that when $e$ is observed, your degree of belief in $h$ should be changed from $P(h)$ to $P(h|e)$. This is known as **conditionalizing**. If $P(h|e) \gt P(h)$, we say that $e$ has provided **confirmation** for $h$.

Typically, situations will involve a range of possible hypotheses, $h_1, h_2, \ldots$, and applying Bayes' Rule will allow us to compare how these fare as new observations are made. For example, comparing the fate of two hypotheses,

$$
\frac{P(h_1|e)}{P(h_2|e)} = \frac{P(e|h_1)}{P(e|h_2)}\cdot \frac{P(h_1)}{P(h_2)}.
$$

How to assign prior probabilities to hypotheses when you don't think you have an exhaustive set of rivals is not obvious. When astronomers in the nineteenth century tried to account for the anomalies in the position of Mercury's perihelion, they tried out all manner of explanations: maybe there was a planet inside Mercury's orbit, maybe there was a cloud of dust surrounding the sun, maybe the power in the inverse square law ought to be (2 - $\epsilon$),... Assigning priors and changing these as evidence comes in is one thing, but it would have been wise to have reserved some of the prior for 'none of the above'.

Interestingly, one of the first people to give a qualitative sketch of how such an approach would work was [[George Polya]] in 'Mathematics and Plausible Reasoning' ([Polya](#Polya)), where examples from mathematics are widely used. The idea of a Bayesian account of plausible reasoning in mathematics surprises many, it being assumed that mathematicians rely solely on [[deductive reasoning|deduction]]. (See also Chap. 4 of [Corfield03](#Corfield).)

##Objective Bayesianism

For some Bayesians, degrees of belief must satisfy further restrictions. One extreme form of this view holds that given a particular state of knowledge, there is a single best set of degrees of belief that should be adopted for any proposition.

Some such restrictions are generally accepted. If, for example, all I know of an event is that it has $n$ possible outcomes, the objective Bayesian will apply the principle of indifference to set their degrees of belief to $1/n$ for each outcome. On the other hand, if there is background knowledge concerning differences between the outcomes, indifference need not hold. This principle of indifference can be generalized to other kinds of invariance, such as the Jeffreys prior ([wiki](http://en.wikipedia.org/wiki/Jeffreys_prior)).

Other objective Bayesian principles include maximum entropy (see [Jaynes 2003](#Jaynes)). For instance, Jaynes argues that if all that is known of a die is that the mean value of throws is equal to, say, 4, then a prior distribution over $\{1, 2, 3, 4, 5, 6\}$ should be chosen which maximizes [[entropy]], subject to the constraint that the mean is 4. Many familiar distributions are maximum entropy distributions, subject to moment constraints. For instance, the Normal distribution, $N(\mu, \sigma^2)$, is the distribution over the reals which maximises entropy subject to having mean $\mu$ and variance $\sigma^2$.

##Exchangeability

Frequentist statistics makes much use of independent and identically distributed (iid) random variables, for example in sampling situations. If, say, we were to toss a coin repeatedly and record the outcomes, the frequentist would typically understand this as sampling from a Bernoulli distribution for some fixed value $p$ of the coin showing heads. From the sample one could then calculate an estimate and confidence interval for the true value of $p$.

Many Bayesians, in particular [[Bruno de Finetti]], argue that this makes no sense since probability is not in the world, but rather it represents the strengths of our beliefs in different outcomes. Their formulation in such repeated sampling cases is to say that if our degrees of belief are such that, for all $n$, the probability we assign to any sequence of $n$ tosses is invariant under any permutation of $n$ elements, then we can represent our degrees of belief for all sequences as arising from a mixture of Bernoulli distributions for some prior distribution over the value of $p$.

More formally, given a sequence of random variables $\{X_i\}^\infty_{i = 1}$ each taking the values $0$ and $1$, de Finetti's Representation Theorem says that the sequence is exchangeable if and only if there is a random variable $\Theta: \Omega \to [0, 1]$, with distribution $\mu_{\Theta}$, such that

$$
P\{X_1=x_1,\ldots,X_n=x_n\}=\int_{[0,1]} \theta^s (1 - \theta)^{n - s} d \mu_{\Theta}(\theta),
$$

in which $s = \sum^n_{i=1} x_i$.

Often, for ease of calculation, Beta distributions are chosen as priors on $p$, which can be taken as representing one's confidence as though one had already seen a certain number of heads and tails. Bayesians are sometimes criticized for the subjectivity inherent in the choice of a prior, but in many cases, such as this one, prior distributions will eventually be 'washed out' by the weight of the evidence.  The de Finetti theorem has a generalization for multivariate distributions ([BBF](#BBF)).

Of course, exchangeability may not represent the strength of one's prior beliefs accurately. For example, in the case of coin tossing, I may have a suspicion of there being something in the tossing mechanism which would make the result of one toss depend on its predecessor. Then it would be quite reasonable for me, say, to have accorded a higher prior probability to the sequence of one hundred heads followed by one hundred tails than to some of its permutations. There are, however, generalizations of de Finetti's representation theorem to Markov chain situations ([Diaconis and Freedman](#DF80)).

## Related entries

* [[conditional expectation]]

* [[Bayesian interpretation of quantum mechanics]]

## References

### General

* {#Jaynes} [[Edwin Jaynes]], _Probability Theory: The Logic of Science_, Cambridge University Press, 2003.

* {#Polya} [[George Polya]], _Mathematics and Plausible Reasoning: Vol. II: Patterns of Plausible Inference_, Princeton University Press, 1954.

* {#Corfield} [[David Corfield]], Chap. 4  of: _Towards a Philosophy of Real Mathematics_, Cambridge University Press (2003) [[doi:10.1017/CBO9780511487576](https://doi.org/10.1017/CBO9780511487576)]

* [[Sumio Watanabe]], _Mathematical theory of Bayesian statistics_, Cambridge University Press (2018) [[ISBN:9780367734817](https://www.routledge.com/Mathematical-Theory-of-Bayesian-Statistics/Watanabe/p/book/9780367734817), [pdf](http://196.189.45.87/bitstream/123456789/36917/1/Sumio%20Watanabe_2018.pdf)]

See also

* Wikipedia, _[Bayesian inference](https://en.wikipedia.org/wiki/Bayesian_inference)_

### Bayesian reasoning and neuroscience

* David C. Knill, Alexandre Pouget: *The Bayesian Brain: The Role of Uncertainty in Neural Coding and Computation*, in Trends in Neurosciences **27** 12 (2004) 712–719 [[doi:10.1016/j.tins.2004.10.007](https://doi.org/10.1016/j.tins.2004.10.007)]

### Category-theoretic treatment

Discussion of Bayesian inference with methods from [[category theory]]:

* Kirk Sturtz, _Bayesian Inference using the Symmetric Monoidal Closed Category Structure_, ([ arXiv:1601.02593](https://arxiv.org/abs/1601.02593))

* Jared Culbertson, Kirk Sturtz, _Bayesian machine learning via category theory_, ([arXiv:1312.1445](https://arxiv.org/abs/1312.1445))

* Kotaro Kamiya, John Welliaveetil, _A category theory framework for Bayesian learning_ ([arXiv:2111.14293](https://arxiv.org/abs/2111.14293))

* [[Toby St Clere Smithe]], _Mathematical Foundations for a Compositional Account of the Bayesian Brain_ [[arXiv:2212.12538](https://arxiv.org/abs/2212.12538)]

### Exchangeability

* {#DF80} [[Persi Diaconis]] and David Freedman, "De Finetti's theorem for Markov chains." Annals of Probability, 8(1), 115-130, 1980.

* {#BBF} A. Bach, H. Blank, H. Francke, _Bose-Einstein statistics derived from the statistics of classical particles_, Lettere Al Nuovo Cimento Series 2, Volume 43, Issue 4, pp 195-198.

* {#FGP21} [[Tobias Fritz]], Tomáš Gonda, [[Paolo Perrone]], _De Finetti's Theorem in Categorical Probability_, ([arXiv:2105.02639](https://arxiv.org/abs/2105.02639))

### Bayesian inference in physics

Discussion of applications in [[astronomy]], [[cosmology]] and [[particle physics]] includes

* [[Jörg Rachen]], _Bayesian Classification of Astronomical Objects -- and what is behind it_ ([arXiv:1302.2429](http://arxiv.org/abs/1302.2429))

* [[Roberto Trotta]], _Bayesian Methods in Cosmology_, ([arXiv:1701.01467](https://arxiv.org/abs/1701.01467))

* {#Lyons13} Louis Lyons, _Bayes and Frequentism: a Particle Physicist's perspective_, ([arXiv:1301.1273](https://arxiv.org/abs/1301.1273))

* Joshua Landon, Frank X. Lee, Nozer D. Singpurwalla, _A Problem in Particle Physics and Its Bayesian Analysis_, ([arXiv:1201.1141](https://arxiv.org/abs/1201.1141))

[[!redirects Bayesian reasoning]]
[[!redirects Bayesian induction]]
[[!redirects Bayesian probability]]
[[!redirects Bayesianism]]

[[!redirects Bayesian inference]]
[[!redirects Bayesian inferences]]

[[!redirects Bayesian statistics]]


BBDG decomposition theorem

+-- {: .rightHandSide}
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###Context###
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There are various "decomposition theorems" in various fields of mathematics.  This entry will be about the **[[Beilinson]]--Bernstein--Deligne--Gabber decomposition theorem** which is probably the strongest single result of wide applicability in modern [[geometric representation theory]].

## References

* A. A. Beilinson, J. Bernstein, P. Deligne _Faisceaux pervers_, Astérisque __100__(1980).

* Mark Andrea A. de Cataldo, Luca Migliorini, _The decomposition theorem, perverse sheaves and the topology of algebraic maps_, Bull. Amer. Math. Soc. __46__ (2009), 535-633, [html abs, refs](http://www.ams.org/bull/2009-46-04/S0273-0979-09-01260-9/home.html), [pdf article](http://www.ams.org/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf)

[[!redirects decomposition theorem]]
[[!redirects the decomposition theorem]]
[[!redirects Beilinson--Bernstein--Deligne--Gabber decomposition theorem]]
[[!redirects Beilinson–Bernstein–Deligne–Gabber decomposition theorem]]
[[!redirects Beilinson-Bernstein-Deligne-Gabber decomposition theorem]]


BCJ relations

> under construction

+-- {: .rightHandSide}
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### Context
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[effective QFT]] incarnations of [[open/closed string duality]],

  relating ([[supergravity|super]]-)[[gravity]] to ([[super Yang-Mills theory|super]]-)[[Yang-Mills theory]]:

  * [[KLT relations]]

  * [[BCJ relations]]

  * [[classical double copy]]

## References

Review:

* Mathias Tolotti, _BCJ-relations in Quantum Field Theory_ ([pdf](https://openscience.ub.uni-mainz.de/bitstream/20.500.12030/1524/1/3941.pdf))

[[!redirects BCJ-relations]]

[[!redirects BCJ relation]]
[[!redirects BCJ-relation]]


BD operad

#Contents#
* table of contents
{:toc}

## Idea

The _Beilinson-Drinfeld operad_ is the [[operad]] whose [[algebras over an operad]] are [[BD algebras]].

See at _[[relation between BV and BD]]_.

## Definition

The underlying graded vector spaces are

$$
  BD \coloneqq P_0 \otimes \mathbb{R}[ [\hbar] ]
$$

and the [[differential]] is

$$
  d ( (-)\cdots (-))
  =
  \hbar \{-,-\}
  \,.
$$

(...)

## Related concepts

[[!include deformation quantization - table]]

## References

The BD operad was introduced in

* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _[[Chiral Algebras]]_

A review in the context of [[factorization algebras of observables]] is in section 2.4 of

* [[Kevin Costello]], [[Owen Gwilliam]], _Factorization algebras in perturbative quantum field theory : $P_0$-operad_ ([wiki](http://math.northwestern.edu/~costello/factorization_public.html#[[P_0%20operad]]), [pdf](http://math.northwestern.edu/~gwilliam/factorization.pdf))

[[!redirects BD-operad]]


BDR 2-vector bundle

> under construction

> for the time being this here are nothing but [references](#References) and rough notes taken in a talk long ago...

_BDR 2-vector bundles_ are a notion of [[categorified]] [[vector bundle]] motivated by the concept of [Kapranov-Voevodsy's 2-vector spaces](http://ncatlab.org/nlab/show/2-vector+space#KV2VectorSpace).

See also at _[[iterated algebraic K-theory]]_.

#Contents#
* table of contents
{:toc}

## 2-Vector bundles following Baas-Dundas-Rognes

We are looking for a generalization of the notion of [[vector bundle]] in
[[higher category theory]].

Let $X$ be a [[topological space]] and $\{U_\alpha \to X\}_{\alpha \in A}$ an [[open cover]],
where the index set $A$ is assumed to be a [[poset]].

Defition. A **charted 2-vector bundle** (i.e. a [[cocycle]] for a BDR 2-vector bundle)
of rank $n$ is

* for $\alpha \lt \beta \in A$ on $U_\alpha \cap U_\beta =: U_{\alpha\beta}$
  a matrix $(E^{\alpha\beta}_{i j})_{i,j = 1}^{n}$ of [[vector bundle]]s
  $E^{\alpha \beta}_{i j} \to U_{\alpha \beta}$
  such that the determinant of the underlying matrix of dimensions
  is $det(dim(E^{\alpha \beta}_{i j})) = \pm 1$.

* on triple overlaps $U_{\alpha \beta \gamma}$
  for  $\alpha \lt \beta \lt \gamma \in A$ we have
  [[isomorphism]]s

  $$
    \phi^{\alpha \beta \gamma}
    :
    \oplus_{j} E^{\alpha \beta}_{i j} \otimes E^{\beta \gamma}_{j k}
      \stackrel{\simeq}{\to}
      E^{\alpha \gamma}_{i k}
  $$

* such that the $\phi$ satisfy  on quadruple overlaps the evident cocycle condition
  (as described at [[gerbe]] and [[principal 2-bundle]]).

Next we need to define morphisms of such charted 2-vector bundles. These involve the
evident refinements of covers and fiberwise transformations.

Write $2Vect(X)$ for the equivalence classes of charted 2-vector bundles under these morphisms.

**Remark** If we restrict attention to $n = 1$ then this gives the same as
$U(1)$-gerbes/[[bundle gerbe]]s.

**Theorem** (Baas-Dundas-Rognes)

There exists a [[classifying space]] $\mathcal{K}(V)$
such that  for $X$ a finite [[CW-complex]] there is an [[isomorphism]]

$$
  [X, \mathcal{K}(V)] =
  {\lim_\to}_{a : Y \to X}
    Gr(2Vect(Y))
$$

between [[homotopy]] classes of continuous maps $X \to \mathcal{K}(V)$
and equivalence classes of the group completed [[infinity-stackification|stackification]] of
2-vector bundles,

where the [[colimit]] is over acyclic [[Serre fibration]]s (Note: these are not acyclic fibrations in the usual sense, rather their fibres have trivial integral [[homology]]) and $Gr(-)$ indicates
the [[Grothendieck group]] completion using the monoid structure arising from the direct sum of 2-vector bundles.

**Proof** In BDR, Segal Birthday Proceedings

**Note** $2Vect_n(X) = [X, |B Gl_n (V)| ]$.

BDR called $\mathcal{K}(V)$ the **2-K-theory** of the [[bimonoidal category]]
of [[Kapranov-Voevodsky 2-vector space]]s.

## The homotopy type of the classifying space

**Theorem** (Baas-Dundas-Rognes-Richter)

The K-theory of BDR 2-vector bundles is the [[algebraic K-theory]] of [[ku]] (see at _[[iterated algebraic K-theory]]_)

$\mathcal{K}(V) \simeq K(ku)$

Here:

* $ku$ is the connected version of the [[spectrum]] of  complex [[topological K-theory]];

* $K(-)$ denotes forming the [[algebraic K-theory]] spectrum of [[ring spectrum|ring spectra]].

So by the general formula for [[algebraic K-theory]] for [[ring spectrum|ring spectra]], this is

$$
 K(ku) \simeq \mathbb{Z} \times B Gl(ku)^+
 \,
$$

Some flavor of $\mathcal{K}(V)$.

* $V$ is the (a [[skeleton]] of the [[core]] of) of [[category]] of complex [[vector space]]s.

  * [[object]]s are [[natural number]]s, $n$ corresponding to $\mathbb{C}^n$;

  * [[morphism]]s are $Hom(k,k)  = GL(k)$ and there are no morphisms between different
    $k,l$.

This category $V$ is naturally a [[bimonoidal category]] under [[coproduct]] and
[[tensor product]] of [[vector space]]s.

K-theory is about understanding linear algebra on a ring, so we want to understand the
linear algebra of this [[monoidal category]].

We write $Mat_n(V) $ for the $n \times n$ matrices of linear isomorphisms between
finite dimensional vector spaces.

Such matrices can be multiplied using the usual formula for matrix products
on the [[tensor product]] and [[direct sum]] of vector space and linear maps.

Write $Gl_n(V)$ for the subcollection of those matrices for which the
determinant of their matrix of dimensions is $\pm 1$.

Now define

$$
  \mathcal{K}(V) = \Omega B (\coprod_{n \geq 0} B Gl_n (V))
  \,.
$$

Notice that this is a direct generalization of the corresponding formula
for the algebraic K-theory of a ring $R$,

$$
  K(R) = \Omega B (\coprod_{n \geq 0} B Gl_n(R))
  \,.
$$

## $K(ku)$ as a form of elliptic cohomology

Ausoni and Rognes compute the [[homology]] groups
(for a certain sense of homology) of $K(ku)$.

take rational homotopy

* $H^*(-, \mathbb{Q})$

  for $p$ a [[prime]], multiplying by $p$ gives an isomorphism on this.

  p = $\nu_0$

* Let $KU^*(-)$ be complex oriented [[topological K-theory]], then

  $$
    KU_* = \mathbb{Z}[u^{\pm 1}]
  $$

  for $|u| = 2$ (the [[Bott class]]) we have that multiplying by $u$ is an
  isomorphism and $u^{p-1} = \nu_1$

The $\nu_i$ come from the [[Brown-Peterson spectrum]] $B P$ and
$\pi_* BP = \mathbb{Z}_{(p)}[\nu_1, \nu_2, \cdots]$

motto: the higher the $\nu_i$ the more you detect.

Christian Ausoni figured out something that implies that

"$K(ku)$ detects as much in the [[stable homotopy category]] as
any other form of [[elliptic cohomology]]."

## From gerbes to 2-vector bundles

It is hard to directly construct charted 2-vector bundles. We have more examples
of [[gerbe]]s. So we want to get one from the other.

**Example**
We have

$$
  \mathbb{S}^1 = \mathbb{C}P^1 \subset \mathbb{C}P^\infty = B U(1) = K(\mathbb{Z},2)
$$

(see [[Eilenberg-MacLane space]] and [[classifying space]])

$$
  \mathbb{S}^3 = \Sigma \mathbb{S}^2 \to \Sigma B U(1) \subset
  \Sigma B U \to B B U_{\otimes}
  \subset units(ku)
  \to B GL(ku)
$$

using $\Sigma B U(1) \to B BU(1) \to B B U_{\otimes}$
we can take a $U(1)$-[[gerbe]] classified by maps into $B^2 U(1)$ and
induce from it the associated 2-vector bundle.

the canonical map

$$
  \mathbb{S}^3 \to K(\mathbb{Z},3)
$$

may be thought of as classifying the gerbe called the [[magnetic monopole]]-gerbe

Postcomposing with $\mu : K(\mathbb{Z},3) \to K(ku)$
we have

Fact: $\mu$ gives a generator in $\pi_3 K(\mathbb{Z},3) = H^3(\mathbb{S}^3)$

**Theorem** (Ausoni-Dundas-Rognes)

$$
  j(\mu) = 2 \zeta - \nu
$$

in $\pi_3(K(ku))$

so regarded as a 2-vector bundle $\mu$ is not a generator.

ADR: $\zeta$ is "half a monopole".

$$
  \pi_3(K(ku))  = \mathbb{Z} \oplus \mathbb{Z}/24 \mathbb{Z}
$$

(the first summand is $\zeta$, the second $\nu$).

Thomas Krogh has an [[orientation]] theory for 2-vector bundles which says that $j(\nu)$
is not orientable.

## 2K-theory of bimonoidal categories {#2K-theory}

Let $(R, \oplus, \otimes, 0,1, c_{\oplus})$
 be a [[bimonoidal category]], i.e. a [[categorification|categorified]] [[rig]].

This can be broken down as

1. $(R, \oplus, 0 , c_{\oplus})$ a permutative category, a categorified abelian [[monoid]];

1. $(R , \otimes, 1)$ is a  [[monoidal category]], assumed to be _strict_ monidal in the following;

1. a distributivity law.

**Examples**

1. $E = Core$[[FinSet]],
   the [[core]] of the
   category of finite sets and morphisms only between sets of the
   same cardinality.

   In the [[skeleton]], objects are [[natural number]]s $n \in \mathb{N}$,
   $\oplus$ and $\otimes$ is addition and multiplication on $\mathbb{N}$,
   respectively. Here $c_{\oplus}$ is the evident natural [[isomorphism]]
   between direct sums of finite sets.

1. $V = Core$[[Vect]] the [[core]]
   of the category of finite dim vector spaces, with morphisms only between
   those of the same dimension.

1. ...

**Definition** For $R$ a bimonoidal category, write $Mat_n(R)$
for the $n \times n$ matrices with entries morphisms in $R$. Then matrix multiplication is defined using the bimonoidal structure on $R$. This gives a weak [[monoid]] structure.

Let $Gl_n(R)$ be the category of weakly invertible such matrices. This is the [[full subcategory]] of $Mat_n(R)$. We get a diagram of [[pullback]] squares

$$
  \array{
    Gl_n(R) &\hookrightarrow& Mat_n(R)
    \\
    \downarrow && \downarrow
    \\
    Gl_n(\pi_0(R))&\to& Mat_n(\pi_0(R))
    \\
    \downarrow && \downarrow
    \\
    Gl_n(Gr(\pi_0(R)))&\to& Mat_n(Gr(\pi_0(R)))
  }
  \,,
$$

where $Gr(-)$ denotes [[Grothendieck group]]-completion.

**Definition** (Baas-Dundas-Rognes, 2004)

For $R$ a [[bimonoidal category]], the **2K-theory** of $R$ is

$$
  \mathcal{K}(R) :=  \Omega B \coprod_{n \geq 0} | B Gl_n(R) |
$$

where the $\Omega$ is forming [[loop space]], the leftmost $B$ is forming classifying space of a category and the inner $B$ is a flabby version of classifying space of a category.

This can also be written

$$
  \cdots \simeq \mathbb{Z} \times |B Gl_n(R)|^+
  \,.
$$

Here $B_q Gl_n(R)$ is a [[simplicial category]]

...

**Theorem** (Baas-Dundas-Rognes)

Let $R$ be a [[small category|small]] [[Top]]-[[enriched category|enriched]] [[bimonoidal category]]
 such that

1. $R$ is a [[groupoid]];

1. for all $X \in R$ we have that $X \oplus (-)$ is [[faithful functor|faithful]].

Then $\mathcal{K}(R) \simeq K(H R)$ is the ordinary [[algebraic K-theory]] of the [[ring spectrum]] $H R$.

Notice that for $H R$ to be a spectrum we only need the additive structure $(R, \oplus, 0, c_{\oplus})$.  The point is that the other monoidal structure $\otimes$ indeed makes this a [[ring spectrum]]. This is a not completely trivial statement due to a bunch of people, involving [[Peter May]] and Elmendorf-Mandell (2006).

**Examples**

1. For the category $R := E = Core(FinSet)$
   of finite sets as above we have that $H E$ is
   the [[sphere spectrum]].

1. For $R := V = Core(FinDimVect)$ the [[core]] of [[complex vector spaces|complex]] [[finite dimensional vector spaces]] we have $H V$ is the complex [[K-theory spectrum]].

1. For $V_{\mathbb{R}}$ analogously we get the real K-theory spectrum.

So by the above theorem

1. $\mathcal{K}(E) \simeq K(S) \simeq A(*)$

1. $\mathcal{K}(V) \simeq K(ku)$;

1. etc.

**Remarks**

1. A. Osono: The equivalence $\mathcal{K}(R) \simeq K(H R)$ of [[topological space]]s is even an equivalence of [[infinity loop space]]s;

1. Application of that:

   a) for $E$ a [[ring spectrum]]: find a model

   $$
     E \simeq H R(R)
   $$

   and use the equivalence $\mathbb{K}(R) \simeq K(H R E)$ to understand
   _arithmetic_ properties of $E$.

   b) Often one knows $K(H(R))$ via calculations. here $\mathcal{K}(R)$ might
   help to get some deeper understanding.

   c) Theorem ([[Birgit Richter]]): for $R$ a bimonoidal category with
   anti-involution, then you get an involution of $\mathcal{K}(R)$.

## Related concepts

* [[red-shift conjecture]]

## References
 {#References}

The original articles on BDR 2-vector bundles are

* [[Nils Baas]], [[Ian Dundas]],  [[John Rognes]], _Two-vector bundles and forms of elliptic cohomology_, in: _Topology, geometry and quantum field theory_, volume 308 of London Math. Soc. Lecture Note Ser., pages 18–45.
Cambridge Univ. Press, Cambridge, (2004).

Their [[classifying spaces]] are discussed in

* [[Nils Baas]], [[Ian Dundas]], [[Birgit Richter]], [[John Rognes]], _Ring completion of rig categories_ ([arXiv:0706.0531](http://arxiv.org/abs/0706.0531))

  (a previous version of this carried the title _Two-vector bundles define a form of elliptic cohomology_)

* [[Nils Baas]], M. Böckstedt, [[Tore Kro]], _Two-Categorical Bundles and Their Classifying Spaces_ (2008) ([arXiv:math/0612549](http://arxiv.org/abs/math/0612549)), .

Divisibility of the gerbe on the 3-sphere, seen as a 2-vector bundle is in

* [[Christian Ausoni]], [[Bjørn Ian Dundas]] and [[John Rognes]],  _Divisibility of the Dirac magnetic monopole as a two-vector bundle over the three-sphere_, Doc. Math. 13, 795-801 (2008). ([dml:45387](https://eudml.org/doc/45387), [pdf](http://www.math.univ-paris13.fr/~ausoni/papers/ADR08.pdf))

Orientation of BDR 2-vector bundles is discussed in

* Thomas Kragh, _Orientations and Connective Structures on 2-vector Bundles_ Mathematica Scandinavica, 113 (2013) no 1, ([journal](http://www.mscand.dk/article/view/15482)), ([arXiv:0910.0131](http://arxiv.org/abs/0910.0131))

[[!redirects BDR 2-vector bundles]]


beable

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## Idea

The term _beable_ was proposed in ([Bell 75](#Bell75)) as a replacement in [[quantum physics]] of the traditional term _[[observable]]_. While the latter is typically given a precise mathematical meaning which does express the way in which the [[physical system]] may be, the word "observable" alludes to the complicated and subtle issue of something (a [[quantum measurement]] device) or even somebody (a concious experimentor) "observing" these ways of the system to be. The point of the term "beable" is to help conceptually cleanly separate the [[being]] of quantum systems from whatever it means to observe them.

In ([Bell 75](#Bell75)), to further justify this, recourse is made to [[Niels Bohr]]'s expressed view that whatever [[quantum physics]] really is, it must be possible to communicate statements about it in terms of classical [[logic]]. Hence beables mostly refer to sets of commuting operators, [[classical contexts]]. This same argument was much later used to motivate [[Bohr toposes]]. In this context, the [[Kochen-Specker theorem]] proves the absence of certain global sections (Proposition 1). Halvorson and Clifton define a beable subalgebra (p9) with respect to a given state $\rho$. We first consider the important special case of the _definite algebra_ $D_\rho$ for a state $\rho$. This is the subalgebra on which $\rho$ is dispersion free. This gives, for each commutative subalgebra of $C\subset D_\rho$, a choice of an element of its spectrum. These choices are compatible. Hence, we obtain not a global section, but only local section. The general notion of beable subalgebra only requires $\rho$ to be a _mixture_ of dispersion-free states.

In practice of theoretical physics most everytime one writes "observable" it should, by this logic, rather be "beable". While this might be reasonable, as a convention of conversation it has not been picked up.

## Related concepts

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

## References

The original discussion of beables in the context of [[causal locality]] in [[AQFT]] is in

* [[John Bell]], _The theory of local beables_, [Epistemological Letters](http://en.wikipedia.org/wiki/Epistemological_Letters) (1975) ([pdf](http://cds.cern.ch/record/980036/files/197508125.pdf))
 {#Bell75}

and further in

* [[John Bell]], _Beables for quantum field theory_ (1984) ([pdf](http://prac.us.edu.pl/~ztpce/QM/Bell_beables.pdf))

A survey is in

* Scholarpedia, _[Bell's theorem - beables](http://www.scholarpedia.org/article/Bell's_theorem#beable)_

A more technical development can be found here:

* [[Hans Halvorson]], Rob Clifton,  _Maximal Beable Subalgebras of Quantum-Mechanical Observables_ ([arXiv:quant-ph/9905042](http://arxiv.org/abs/quant-ph/9905042))

Discussion in the context of [[interpretation of quantum mechanics]] includes

* Yuichiro Kitajima, _Interpretations of Quantum Mechanics in Terms of Beable Algebras_, ([pdf](http://link.springer.com/article/10.1007/s10773-005-4051-0))

[[!redirects beables]]


beauty > history


Beck module

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## Idea

One usually defines [[cohomology]] with respect to some [[coefficient]] objects:

* For [[group cohomology]] of a [[group]] $G$, the coefficients come from a (left) $G$-[[module]].
* For [[Lie algebra cohomology]] of a [[Lie algebra]] $\mathfrak{g}$, the coefficients come from a (left) $\mathfrak{g}$-module.
* For [[Hochschild cohomology]] of an [[associative algebra]] $A$, the coefficients come from an $A$-[[bimodule]].

**Beck modules** are a simultaneous generalisation of all three types of module.

## Definition

Let $\mathcal{C}$ be a [[category]] with [[pullback]]s and let $A$ be an object in $\mathcal{C}$. A **Beck module** over $A$ is an [[abelian group object]] in the slice category $\mathcal{C}_{/ A}$. In particular, if $A$ is the terminal object, this reduces to the notion of an abelian group object in $\mathcal{C}$. We write $\mathbf{Ab}(\mathcal{C}_{/ A})$ for the category of Beck modules over $A$.

([Beck 67, def. 5](#Beck67))

## Properties

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

Let $\mathcal{C}$ be an [[effective regular category]] (resp. [[locally presentable category]]) and let $A$ be an object in $\mathcal{C}$. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is an [[abelian category]] (resp. locally presentable category).

=--

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

If $\mathcal{C}$ is a effective regular category (resp. locally presentable category), then so is $\mathcal{C}_{/ A}$. Thus, the claim reduces to the fact that the category of abelian group objects in an effective regular category (resp. locally presentable category) is an abelian category (resp. locally presentable category).

=--

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

Let $\mathcal{C}$ be an effective regular category with [[filtered colimits]] and let $A$ be an object in $\mathcal{C}$. If filtered colimits in $\mathcal{C}$ preserve finite [[limits]], then $\mathbf{Ab}(\mathcal{C}_{/ A})$ (is an abelian category and) satisfies axiom AB5.

=--

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

The forgetful functor $\mathcal{C}_{/ A} \to \mathcal{C}$ [[created limit|creates]] pullbacks and filtered colimits, so filtered colimits in $\mathcal{C}_{/ A}$ also preserve finite limits. The forgetful functor $\mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ creates limits and filtered colimits, so filtered colimits in $\mathbf{Ab}(\mathcal{C}_{/ A})$ preserve kernels. In view of the earlier proposition, it follows that $\mathbf{Ab}(\mathcal{C}_{/ A})$ satisfies axiom AB5.

=--

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

Let $\mathcal{C}$ be a locally finitely presentable effective regular category and let $A$ be an object in $\mathcal{C}$. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is a [[Grothendieck category]].

=--

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

Combine the two propositions above.

=--

## Derivations

Let $\mathcal{C}$ be a category with pullbacks, let $A$ be an object in $\mathcal{C}$, and let $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ be the forgetful functor. Given a Beck module $M$ over $A$, an $M$-valued **derivation** of $A$ is a morphism $1_A \to U M$ in $\mathcal{C}_{/ A}$, where $1_A$ is the terminal object in $\mathcal{C}_{/ A}$, and we write
$$Der (A, M) = \mathcal{C}_{/ A} (1_A, U M)$$
for the set of $M$-valued derivations of $A$. The **Beck module of differentials** over $A$ is an object $\Omega_A$ in $\mathbf{Ab}(\mathcal{C}_{/ A})$ [[representable functor|representing]] the functor $Der (A, -) : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathbf{Set}$.

The Beck module $\Omega_A$ is not guaranteed to exist in general. When the functor $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ has a left adjoint, $\Omega_A$ is simply the value of the left adjoint at $1_A$.

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

Let $\mathcal{C}$ be a locally presentable category and let $A$ be an object in $\mathcal{C}$. Then the forgetful functor $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ has a left adjoint.

=--

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

The forgetful functor $\mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ creates limits and $\kappa$-filtered colimits (for some $\kappa$ large enough), so we may apply the accessible [[adjoint functor theorem]].

=--

## Examples

### Beck modules over associative algebras
 {#OverAssociativeAlgebras}

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

Let $\mathcal{C}$ be the category of (not necessarily commutative) [[rings]] and let $A$ be a ring. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is equivalent to the category of $A$-bimodules.

=--

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

Let $\epsilon : B \to A$ be ring homomorphism. To give it the structure of a Beck module over $A$, we must give ring homomorphisms $\eta : A \to B$ and $\mu : B \times_A B \to B$ such that $\epsilon \circ \eta = id_A$, $\epsilon (\mu (b_0, b_1)) = \epsilon (b_0) = \epsilon (b_1)$, as well as various other equations. Given elements $b_0, b_1, b_2, b_3$ of $B$ such that $\epsilon (b_0) = \epsilon (b_2)$ and $\epsilon (b_1) = \epsilon (b_3)$, we have the following interchange law:
$$\mu (b_0 + b_1, b_2 + b_3) = \mu (b_0, b_2) + \mu (b_1, b_3)$$
Hence, if $a = \epsilon (b_1) = \epsilon (b_2)$,
$$\mu (b_1, b_2) = \mu (\eta (a) + b_1 - \eta (a), \eta (a) + b_2 - \eta (a)) = \mu (\eta (a), \eta (a)) + \mu (b_1 - \eta(a), b_2 - \eta (a)) = \eta (a) + \mu (b_1 - \eta(a), b_2 - \eta (a))$$
but $\epsilon (b_1 - \eta (a)) = \epsilon (b_2 - \eta (a)) = 0$, so
$$\mu (b_1 - \eta(a), b_2 - \eta(a)) = \mu (0, b_2 - \eta (a)) + \mu (b_1 - \eta (a), 0) = b_2 - \eta (a) + b_1 - \eta (a)$$
and we conclude that
$$\mu (b_1, b_2) = b_1 - \eta (\epsilon (b_1)) + b_2 = b_1 + b_2 - \eta (\epsilon (b_2))$$
and in particular, $\mu$ is entirely determined by $\eta$ and $\epsilon$. We also have the following interchange law,
$$\mu (b_0 b_1, b_2 b_3) = \mu (b_0, b_2) \mu (b_1, b_3)$$
and in particular,
$$\mu (\eta (\epsilon (b_2)) b_1, b_2 \eta (\epsilon (b_1))) = \mu (\eta (\epsilon (b_2)), b_2) \mu (b_1, \eta (\epsilon (b_1))) = b_2 b_1$$
hence,
$$b_2 b_1 = \eta (\epsilon (b_2)) b_1 - \eta (\epsilon (b_2 b_1)) + b_2 \eta (\epsilon (b_1))$$
so if $\epsilon (b_1) = \epsilon (b_2) = 0$, then $b_2 b_1 = 0$.

Let $M = \ker \epsilon$. The above shows that the internal abelian group structure on $B$ restricts to the pre-existing abelian group structure on $\ker \epsilon$. In addition, the homomorphism $\eta : A \to B$ gives $M$ the structure of an $A$-bimodule, and we see that $B$ is naturally isomorphic to the [[square-0 extension]] ring $A \oplus M$, with componentwise addition and the multiplication given below,
$$(a_0, m_0) \cdot (a_1, m_1) = (a_0 a_1, a_0 m_1 + m_0 a_1)$$
regarded as a Beck module over $A$ by defining $\epsilon : A \oplus M \to A$, $\eta : A \to A \oplus M$, and $\mu : (A \oplus M) \times_A (A \oplus M) \to A \oplus M$ as follows:
$$\epsilon (a, m) = a$$
$$\eta (a) = (a, 0)$$
$$\mu ((a, m_0), (a, m_1)) = (a, m_0 + m_1)$$
Thus, we have an equivalence between $\mathbf{Ab}(\mathcal{C}_{/ A})$ and the category of $A$-bimodules, as claimed.

=--

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

Let $A$ be a ring. Then the Beck module $\Omega_A$ is isomorphic to the $A$-bimodule of [[Kähler differentials]] (relative to $\mathbb{Z}$).

=--

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

Let $M$ be an $A$-bimodule, regard $A \oplus M$ as a ring as above, and let $\epsilon : A \oplus M \to A$ be the obvious projection. A ring homomorphism $\phi : A \to A \oplus M$ satisfying $\epsilon \circ \phi = id_A$ is the same thing as an additive homomorphism $\delta : A \to M$ satisfying the following equations,
$$\delta (a_0 a_1) = \delta (a_0) a_1 + a_0 \delta (a_1)$$
i.e. a derivation $A \to M$ (over $\mathbb{Z}$). Thus, the Beck module $\Omega_A$ has the same universal property as the $A$-bimodule of Kähler differentials.

=--

### Beck modules over groups

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

Let $\mathcal{C}$ be the category of (not necessarily abelian) [[groups]] and let $G$ be a group. Then $\mathbf{Ab}(\mathcal{C}_{/ G})$ is equivalent to the category of left $G$-modules.

=--

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

Let $\epsilon : H \to G$ be group homomorphism. To give it the structure of a Beck module over $G$, we must give group homomorphisms $\eta : G \to H$ and $\mu : H \times_G H \to H$ such that $\epsilon \circ \eta = id_A$, $\epsilon (\mu (h_0, h_1)) = \epsilon (h_0) = \epsilon (h_1)$, as well as various other equations. Given elements $h_0, h_1, h_2, h_3$ of $H$ such that $\epsilon (h_0) = \epsilon (h_2)$ and $\epsilon (h_1) = \epsilon (h_3)$, we have the following interchange law:
$$\mu (h_0 h_1, h_2 h_3) = \mu (h_0, h_2) \mu (h_1, h_3)$$
and in particular,
$$\mu (\eta (\epsilon (h_2)) h_1, h_2 \eta (\epsilon (h_1))) = \mu (\eta (\epsilon (h_2)), h_2) \mu (h_1, \eta (\epsilon (h_1))) = h_2 h_1$$
but on the other hand, if $g = \epsilon (h_1) = \epsilon (h_2)$, then
$$\mu (h_1, h_2) = \mu (\eta (g) \eta (g)^{-1} h_1, \eta (g) \eta (g)^{-1} h_2) = \mu (\eta (g), \eta (g)) \mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2) = \eta (g) \mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2)$$
and writing $e$ for the unit of $G$ and $H$, we have $\epsilon (\eta (g)^{-1} h_1) = \epsilon (\eta (g)^{-1} h_2) = e$, hence
$$\mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2) = \mu (e, \eta (g)^{-1} h_2) \mu (\eta (g)^{-1} h_1, e) = \eta (g)^{-1} h_2 \eta (g)^{-1} h_1$$
so we conclude that
$$\mu (h_1, h_2) = h_2 \eta (g)^{-1} h_1$$
and in particular, $\mu$ is entirely determined by $\eta$.

Let $M = \ker \epsilon$. The above shows that the internal abelian group structure on $B$ restricts to the pre-existing group structure on $\ker \epsilon$. (In particular, $M$ is an abelian group!) We make $M$ into a left $G$-module as follows:
$$g \cdot m = \eta (g) m \eta (g)^{-1}$$
We can then construct the [[semi-direct product]] $M \rtimes G$, which has the following multplication:
$$(m_0, g_0) \cdot (m_1, g_1) = (m_0 \eta (g_0) m_1 \eta (g_0)^{-1}, g_0 g_1)$$
There is a group homomorphism $M \rtimes G \to H$ defined by $(m, g) \mapsto m \eta (g)$, and it is bijective: surjectivity is clear, and injectivity is a consequence of the fact that $M \cap \operatorname{im} \eta = \{ e \}$. We may regard $M \rtimes G$ as a Beck module over $G$ by defining $\epsilon : M \rtimes G \to G$, $\eta : G \to M \rtimes G$, and $\mu : (M \rtimes G) \times_G (M \rtimes G) \to M \rtimes G$ as follows:
$$\epsilon (m, g) = g$$
$$\eta (g) = (0, g)$$
$$\mu ((m_0, g), (m_1, g)) = (m_1 m_0, g)$$
Thus, we have an equivalence between $\mathbf{Ab}(\mathcal{C}_{/ G})$ and the category of left $G$-modules, as claimed.

=--

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

Let $G$ be a group and let $M$ be a left $G$-module. Under the above identification of Beck modules over $G$ with left $G$-modules, $M$-valued derivations of $G$ are precisely crossed homomorphisms $G \to M$, i.e. maps $\delta : G \to M$ satisfying the following equation:
$$\delta (g_0 g_1) = \delta (g_0) + g_0 \cdot \delta (g_1)$$

=--

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

Let $\epsilon : M \rtimes G \to G$ be the evident projection. A group homomorphism $\phi : G \to M \rtimes G$ such that $\epsilon \circ \phi = id_G$ is the same thing as a map $\delta : G \to M$ satisfying the equation below,
$$(\delta (g_0), g_0) \cdot (\delta (g_1), g_1) = (\delta (g_0 g_1), g_0 g_1)$$
which is equivalent to the defining equation for crossed homomorphisms.

=--

## The tangent category

One may assemble the individual categories of Beck modules over the objects of $\mathcal{C}$ into a category fibred over $\mathcal{C}$, called the [[tangent category]].

## References

The concept is due to

* {#Beck67} [[Jon Beck]], _Triples, algebras and cohomology_, Ph.D. thesis, Columbia University, 1967, Reprints in Theory and Applications of Categories, No. 2 (2003) pp 1-59 ([TAC](http://www.tac.mta.ca/tac/reprints/articles/2/tr2abs.html))

and was popularized in

* {#Quillen70} [[Daniel G. Quillen]], _On the (co-)homology of commutative rings_, in Proc. Symp. on Categorical Algebra, 65 – 87, American Math. Soc.,  1970.

See also

* [[Michael Barr]], _Acyclic models_, Chapter 6, §1.

An application to [[knot theory]] is given in

* [[Markus Szymik]], _Alexander-Beck modules detect the unknot_, Fund. Math. 246 (2019) 89-108.

[[!redirects Beck modules]]