bundle gerbe

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###Context###
#### Bundles
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[[!include bundles - contents]]
=--
#### Cohomology
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[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _bundle gerbe_ is a special model for the total space [[Lie groupoid]] of a $\mathbf{B}U(1)$-[[principal 2-bundle]] for $\mathbf{B}U(1)$ the circle 2-group.

More generally, for $G$ a more general [[Lie 2-group]] (often taken to be the [[automorphism 2-group]] $G = AUT(H)$ of a [[Lie group]] $H$), a [[nonabelian bundle gerbe]] for $G$ is a model for the total space groupoid of a $G$-[[principal 2-bundle]].

The definition of _bundle gerbe_ is not in fact a special case (nor a generalization) of the definition of [[gerbe]], even though there are equivalences relating both concepts.

## Definition

A **bundle gerbe** over a [[smooth manifold]] $X$ is

* a [[surjective submersion]]

  $$
      \array{
          Y
          \\
          \downarrow^{\mathrlap{\pi}}
          \\
          X
      }
    $$

* together with a $U(1)$-[[principal bundle]]

  $$
      \array{
           L
           \\
           \downarrow^{\mathrlap{p}}
           \\
           Y \times_X Y
        }
    $$

  over the [[fiber product]] of $Y$ with itself, i.e.

  $$
    \array{
          L
          \\
          \downarrow^{\mathrlap{p}}
          \\
          Y \times_X Y &\stackrel{\overset{\pi_1}{\rightarrow}}{\underset{\pi_2}{\rightarrow}}& Y
          \\
          && \downarrow^{\mathrlap{\pi}}
          \\
          &&
          X
      }
     \,,
  $$

* an [[isomorphism]]

   $$
     \mu :
     \pi_{12}^*L \otimes
     \pi_{23}^*L
     \to
     \pi_{13}^* L
   $$

   of $U(1)$-bundles on $Y \times_X Y \times_X Y$

* such that this satisfies the evident associativity condition on
  $Y\times_X Y \times_X Y \times_X Y$.

Here $\pi_{12}, \pi_{23}, \pi_{13}$ are the three maps

$$
  Y^{[3]} \stackrel{\stackrel{\rightarrow}{\rightarrow}}{\rightarrow}
   Y^{[2]}
$$

in the [[Cech nerve]] of $Y \to X$.

In a [[nonabelian bundle gerbe]] the bundle $L$ is generalized to a [[bibundle]].

## Interpretation {#Interpretation}

A bundle gerbe may be understood as a specific model for the total space [[Lie groupoid]] of a [[principal 2-bundle]].

We first describe this Lie groupoid in

* [As a groupoid extension](#AsGroupoidExtension)

and then describe how this is the total space of a principal 2-bundle in

* [As the total space of a 2-bundle](#AsTotalSpace).

### As a groupoid extension {#AsGroupoidExtension}

Give a surjective submersion $\pi : Y \to X$, write

$$
  C(Y) := \left(
    Y \times_X Y \stackrel{\to}{\to} Y
  \right)
$$

for the corresponding [[Cech groupoid]]. Notice that this is a [[resolution]] of the [[smooth manifold]] $X$ itself, in that the canonical projection is a weak equivalence (see [[infinity-Lie groupoid]] for details)

$$
  \array{
     C(Y)
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     X
  }
  \,.
$$

The data of a bundle gerbe $(Y,L,\mu)$ induces a [[Lie groupoid]] $P_{(Y,L,\mu)}$ which is a $\mathbf{B}U(1)$-extension of $C(Y)$, exhibiting a [[fiber sequence]]

$$
  \mathbf{B}U(1) \to P_{(Y,L,\mu)} \to X
  \,.
$$

This Lie groupoid is the groupoid whose space of morphisms is the total space $L$ of the $U(1)$-bundle

$$
  P_{(Y,L,\mu)} =
  \left(
    L \stackrel{\overset{\pi_1 \circ p}{\to}}{\underset{\pi_2 \circ p}{\to}} Y
  \right)
$$

with composition given by the composite

$$
  L \times_{s,t} L
  \stackrel{\simeq}{\to}
  \pi_{12}^* L \times \pi_{23}^3* L
  \stackrel{}{\to}
  \pi_{12}^* L \otimes \pi_{23}^3* L
  \stackrel{\mu}{\to}
  \pi_{13}^* L
  \to
  L
  \,.
$$

### As the total space of a principal 2-bundle {#AsTotalSpace}

We discuss how a bundle gerbe, regarded as a [[groupoid]], is the total space of a $\mathbf{B}U(1)$-[[principal 2-bundles]].

Recall from the discussion at [[principal infinity-bundle]] that the total $G$ 2-bundle space $P \to X$ classified by a cocycle $X \to \mathbf{B} G$ is simply the [[homotopy fiber]] of that cocycle. This we compute now.

(For more along these lines see [[infinity-Chern-Weil theory introduction]]. For the analogous nonabelian case see also [[nonabelian bundle gerbe]].)

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

The Lie groupoid $P_{(Y,L,\mu)}$ defined by a bundle gerbe is in [[?LieGrpd]] the [[(∞,1)-pullback]]

$$
  \array{
     P_{(Y,L,\mu)} &\to& *
     \\
     \downarrow &\swArrow_{\simeq}& \downarrow
     \\
     X &\stackrel{g}{\to}& \mathbf{B}^2 U(1)
  }
$$

of a [[cocycle]] $[g] \in H(X,\mathbf{B}^2 U(1)) \simeq H^3(X,\mathbb{Z})$.

=--

In fact a somewhat stronger statement is true, as shown in the following proof.

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

We can assume without restriction that the bundle $L$ in the data of the bundle gerbe is actually the trivial $U(1)$-bundle $L = Y \times_X Y \times U(1)$ by refining, if necessary, the surjective submersion $Y$ by a [[good open cover]]. In that case we may identify $\mu$ with a $U(1)$-valued function

$$
  \mu : Y \times_X Y \times_X Y  \to U(1)
$$

which in turn we may identify with a smooth 2-[[anafunctor]]

$$
  \array{
    C(U) &\stackrel{\mu}{\to}& \mathbf{B}^2 U(1)
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
  \,.
$$

From here on the computation is a special case of the general theory of [[groupoid cohomology]] and the extensions classified by it.

Then recall from [[universal principal infinity-bundle]] that we model the $(\infty,1)$-pullbacks that defines principal $\infty$-bundles in terms of ordinary pullbacks of the universal $\mathbf{B}U(1)$-principal 2-bundle $\mathbf{E}\mathbf{B}U(1) \to \mathbf{B}^2 U(1)$.

We may model all this in the case at hand in terms of [[strict omega-groupoid|strict 2-groupoip]]s. Then using an evident cartoon-notation we have

$$
  \mathbf{B}^2 U(1)
  =
  \left\{
    \array{
       & \nearrow \searrow
       \\
       \bullet &\Downarrow^{\mathrlap{c \in U(1)}}& \bullet
       \\
       & \searrow \nearrow
    }
  \right\}
$$

and $\mathbf{E}\mathbf{B}U(1)$ is the 2-groupoid whose _morphisms_ are diagrams

$$
  \array{
      && \bullet
      \\
      & \nearrow &\swArrow_{c}& \searrow
     \\
     \bullet &&\to&& \bullet
  }
$$

in $\mathbf{B}^2 U(1)$ with composition given by horizontal [[pasting]]

$$
  \array{
     &&& \bullet
     \\
     & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow
    \\
    \bullet &\to&& \bullet &&\to& \bullet
  }
$$

and 2-morphisms are paper-cup diagrams

$$
  \array{
      && \bullet
      \\
      & \nearrow &\swArrow_{c}& \searrow
     \\
     \bullet &&\to&& \bullet
     \\
     & \searrow &\swArrow_{k}& \swarrow
     \\
     && \bullet
  }
  \;\;\;\;\;
  =
  \;\;\;\;\;
  \array{
      && \bullet
      \\
      & \nearrow &\swArrow_{c k}& \searrow
     \\
     \bullet &&\to&& \bullet
  }
  \,.
$$

So $\mathbf{E}\mathbf{B}U(1)$ is the Lie 2-groupoid with a single object, with  $U(1)$ worth of 1-morphisms and unique 2-morphism between these.

From this we read of that

$$
  \array{
    P_{(Y,L,\mu)} &\to& \mathbf{E} \mathbf{B}U(1)
    \\
    \downarrow && \downarrow
    \\
    C(U) &\stackrel{\mu}{\to}& \mathbf{B}^2 U(1)
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
$$

is indeed a [[pullback]] square (in the category of [[simplicial presheaves]] over [[CartSp]]). The morphisms of the pullback Lie groupoid are pairs of diagrams

$$
  \array{
      && \bullet
      \\
      & \nearrow &\swArrow_{c}& \searrow
     \\
     \bullet &&\to&& \bullet
     \\
     \\
     (x,i) &&\to&& (x,j)
  }
$$

hence form a trivial $U(1)$-bundle over the morphisms of $C(U)$, and the 2-morphims are pairs consisting of 2-morphisms

$$
  \array{
     && (x,j)
     \\
     & \nearrow &\swArrow& \searrow
     \\
     (x,i) &&\to&& (x,k)
  }
$$

in $C(U)$ and paper-cup diagrams of the form

$$
  \array{
     &&& \bullet
     \\
     & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow
    \\
    \bullet &\to&& \bullet &&\to& \bullet
    \\
    & \searrow &&\swArrow_{\mu_{i j k}(x)}&&& \swarrow
  }
  \;\;\;\;
  =
  \;\;\;\;
  \array{
      && \bullet
      \\
      & \nearrow &\swArrow_{c_1 c_2 \mu_{i j k}(x)}& \searrow
     \\
     \bullet &&\to&& \bullet
   }
$$

in $\mathbf{B}^2 U(1)$, which exhibits indeed the composition operation in $P_{(Y,L,\mu)}$.

=--

## Examples

### Equivariant bundle gerbes over the point

For $A \to \hat G \to G$ a [[group]] extension by an [[abelian group]] $G$ classified by a 2-cocycle $c$ in [[group cohomology]], which we may think of as a 2-functopr $c : \mathbf{B}\mathbf{G} \to \mathbf{B}^2 A$, the corresponding [[fiber sequence]]

$$
  A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \stackrel{c}{\to}
  \mathbf{B}^2 A
$$

exhibits $\mathbf{B}\hat G$ as the bundle gerbe over $\mathbf{B}G$ (in [[equivariant cohomology]] of the point, if you wish) with Dixmier-Douady class $c$.

### Tautological bundle gerbe

Let $X$ be a [[simply connected]] [[smooth manifold]] and $H \in \Omega^3(X)_{cl, int}$  a degree 3 [[differential form]] with integral periods.

We may think of this a cocycle in [[∞-Lie algebroid cohomology]]

$$
  H : T X \to b^2 \mathbb{R}
  \,.
$$

By a slight variant of Lie integration of oo-Lie algebroid cocycles we obtain from this a bundle gerbe on $X$ by the following construction

* pick any point $x_0 \in X$;

* let $Y = P_* X$ be the based smooth [[path space]] of $X$;

* let $L \to Y \times_X Y$ be the $U(1)$-bundle which over an element
  $(\gamma_1,\gamma_2)$ in $Y \times_X Y$ -- which is a _loop_ in $X$
  assigns the $U(1)$-[[torsor]] whose elements are equivalence class of
  pairs $(\Sigma,c)$, where $\Sigma$ is a surface cobounding the loop and
  where $c \in U(1)$, and where the equivalence relation is so that for
  any 3-ball $\phi : D^3 \to X$ cobounding two such surfaces
  $\Sigma_1$ and $\Sigma_2$ we have that $(\Sigma_1,c_1)$ is equivalent
  to $(\Sigma_2, c_2)$ the difference of the labels differs by the
  [[integral]] of the 3-form

  $$
    c_2 c_1^{-1} = \int_{D^3} \phi^* H \in \mathbb{R}/\mathbb{Z}
    \,.
  $$

* the composition operation $\pi_{12}^* L \otimes \pi_{23}^* L \to \pi_{13}^* L$ is loop-wise the evident operation that on loops removes from a figure-8 the inner bit and whch is group multiplication of the labels.

This produces a bundle gerbe whose class in $H^3(X,\mathbb{Z})$ has $[H]$ as its image in [[de Rham cohomology]].

## Related entries

* [[centrally extended groupoid]]

* [[line 2-bundle]], [[holomorphic line 2-bundle]]

* [[transgression of bundle gerbes]]

* [[connection on a bundle gerbe]]

* [[discrete torsion]]

* [[fusion bundle]]

and

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

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

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

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

especially

* [[circle n-bundle with connection]], [[ordinary differential cohomology]].

For applications in [[string theory]] see also

* [[B-field]], [[WZW model]].

##References##

The notion of _bundle gerbe_ as such was introduced in

* [[Michael Murray]], _Bundle gerbes_, J. Lond. Math. Soc. **54** (1996) pp. 403-416 ([arXiv:dg-ga/9407015](http://arxiv.org/abs/dg-ga/9407015))

Early texts also include

* [[David Chatterjee]], _On Gerbs_, 1998 ([pdf](https://people.maths.ox.ac.uk/hitchin/files/StudentsTheses/chatterjee.pdf))

(notice that the title here suppresses one "e" intentionally);

* [[Michael Murray]], [[Danny Stevenson]], _Bundle gerbes: stable isomorphism and local theory_, J.Lond.Math.Soc. **62** (2000) 925-937 arXiv:[math/9908135](https://arxiv.org/abs/math/9908135)

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

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

Reviews are in

* [[Nigel Hitchin]], _What is... a gerbe?_, Notices of the AMS **50** no. 2 (2003) pp 218-219 [pdf](http://www.ams.org/notices/200302/what-is.pdf)

* [[Michael Murray]], _An Introduction to Bundle Gerbes_, In: The Many Facets of Geometry, A Tribute to Nigel Hitchin, Edited by Oscar Garcia-Prada, Jean Pierre Bourguignon, [[Simon Salamon]], OUP, 2010. doi:[10.1093/acprof:oso/9780199534920.001.0001](http://dx.doi.org/10.1093/acprof:oso/9780199534920.001.0001), arXiv:[0712.1651](https://arxiv.org/abs/0712.1651)

* [[Severin Bunk]], _Gerbes in geometry, field Theory, and quantisation_, Complex Manifolds **8** 1 (2021) 150-182 [[arXiv:2102.10406](https://arxiv.org/abs/2102.10406)]

With emphasis on [[surface holonomy]]:

* [[Jürgen Fuchs]], [[Thomas Nikolaus]], [[Christoph Schweigert]], [[Konrad Waldorf]], _Bundle Gerbes and Surface Holonomy_ ([arXiv:0901.2085](https://arxiv.org/abs/0901.2085))

Bundle gerbes over [[Lie groups]] (cf. the [[basic bundle gerbe]]):

* {#Brylinski00} [[Jean-Luc Brylinski]], *Gerbes on complex reductive Lie groups* ([arXiv:math/0002158](https://arxiv.org/abs/math/0002158))

* {#Meinrenken02} [[Eckhard Meinrenken]], *The basic gerbe over a compact simple Lie group*, Enseign. Math. **49** 3-4 (2003) 307--333 ([arXiv:math/0209194](https://arxiv.org/abs/math/0209194), [doi:10.5169/seals-66691](https://dx.doi.org/10.5169/seals-66691))

* [[Christoph Schweigert]], [[Konrad Waldorf]], _Gerbes and Lie Groups_, In: KH. Neeb, A. Pianzola  (eds) _Developments and Trends in Infinite-Dimensional Lie Theory_, Progress in Mathematics, vol 288. Birkhäuser Boston 2011 ([arXiv:0710.5467](https://arxiv.org/abs/0710.5467), [doi:10.1007/978-0-8176-4741-4_10](https://doi.org/10.1007/978-0-8176-4741-4_10))

* {#Krepski17} [[Derek Krepski]], *Basic equivariant gerbes on non-simply connected compact simple Lie groups*, Journal of Geometry and Physics **133** (2018) 30-41 ([arXiv:1712.08294](https://arxiv.org/abs/1712.08294), [doi:10.1016/j.geomphys.2018.06.016](https://doi.org/10.1016/j.geomphys.2018.06.016))

{#ReferencesEquivariant} On [[equivariant infinity-bundle|equivariant bundle gerbes]]:

with a restrictive notion of equivariance ("strong" equivariance):

* [Brylinski 2000, from p. 28](#Brylinski00)

* [[Varghese Mathai]], [[Danny Stevenson]], *Chern character in twisted K-theory: equivariant and holomorphic cases*, Commun. Math. Phys. **236** 161-186, 2003 ([arXiv:hep-th/0201010](https://arxiv.org/abs/hep-th/0201010), [doi:10.1007/s00220-003-0807-7](https://doi.org/10.1007/s00220-003-0807-7))

* [Meinrenken 2002, Sec. 2.4](#Meinrenken02)

* [[Kiyonori Gomi]], *Reduction of strongly equivariant bundle gerbes with connection and curving* ([arXiv:math/0406144](https://arxiv.org/abs/math/0406144))

with a general notion of higher equivariance:

* [[Kiyonori Gomi]], [[Yuji Terashima]], Sec 5 of: _Discrete Torsion Phases as Topological Actions_, Commun. Math. Phys. (2009) 287: 889 ([doi:10.1007/s00220-009-0736-1](https://doi.org/10.1007/s00220-009-0736-1))

  > (this article speaks in terms of [[Deligne cohomology]])

* [[Thomas Nikolaus]], [[Christoph Schweigert]], *Equivariance In Higher Geometry*, Adv. Math., 226(4): 3367-3408, 2011 ([arXiv:1004.4558](https://arxiv.org/abs/1004.4558), [doi:10.1016/j.aim.2010.10.016](https://doi.org/10.1016/j.aim.2010.10.016))

* [[Michael Murray]], [[David Michael Roberts]], [[Danny Stevenson]], [[Raymond Vozzo]],  *Equivariant bundle gerbes*, Advances in Theoretical and Mathematical Physics 21 (2017) no. 4 pp 921-975 ([arXiv:1506.07931](https://arxiv.org/abs/1506.07931), [doi:10.4310/ATMP.2017.v21.n4.a3](https://dx.doi.org/10.4310/ATMP.2017.v21.n4.a3))

* [Krepski 2017](#Krepski17)

[[!redirects bundle gerbes]]


bundle gerbe module

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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=--
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#Contents#
* table of contents
{:toc}

## Idea

For $A$ an [[abelian group|abelian]] [[Lie group]] (often taken to be the [[circle group]] $U(1)$), a [[bundle gerbe]] on $X$ is a representation of a [[cohomology|cocycle]] $c$ in $\mathbf{H}(X,\mathbf{B}^2 A)$.

If a [[central extension]] $A \to \hat G \to G$ is given (often taken to be $U(1) \to U(n) \to P U (n)$) there is a notion of $\hat G$-[[twisted bundle]]s with twist given by $c$.

A _bundle gerbe module_ is the presentation of such a $\hat G$-[[twisted bundle]] corresponding to the presentation of the $\mathbf{B}^2 A$-cocycle by a [[bundle gerbe]].

## Definition

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

If $Y \to X$ is the [[surjective submersion]] relative to which the [[bundle gerbe]] $c$ is defined, and if

$$
  L \to Y \times_X Y
$$

is the transition [[line bundle]] of the bundle gerbe, then a **bundle gerbe module** for $c$ is a Hermitean [[vector bundle]]

$$
  E \to Y
$$

equipped with an [[action]]

$$
  \rho : \pi_2^* E \otimes L \to \pi_1^* E
$$

(where $\pi_1, \pi_2 : Y \times_X Y \to Y$ are the two projections out of the [[fiber product]])

that respects the bundle gerbe product

$$
  \mu : \pi_{12}^* L \otimes \pi_{2 3}^* L
  \to \pi_{1 3}^* L
$$

in the obvious way.

=--

When $Y = \coprod_i U_i$ comes form an an open [[cover]] $\{U_i \to X\}$ the above almost manifestly reproduces the explicit description of [[twisted bundle]]s given there.

## References

The concept is due to

* {#BCMMS02}  [[Peter Bouwknegt]], [[Alan Carey]], [[Varghese Mathai]], [[Michael Murray]], [[Danny Stevenson]], Section 4 of: _K-theory of bundle gerbes and twisted K-theory_ ,  Commun Math Phys, 228 (2002) 17-49 ([arXiv:hep-th/0106194](http://arxiv.org/abs/hep-th/0106194), [doi:10.1007/s002200200646](https://doi.org/10.1007/s002200200646))

recognized as equivalent to earlier discussion of [[twisted bundles]] in

* {#LupercioUribe01} [[Ernesto Lupercio]], [[Bernardo Uribe]], Prop. 7.2.2 (from v2 on) in: *Gerbes over Orbifolds and Twisted K-theory*, Comm. Math. Phys. 245(3): 449-489.  ([arXiv:math/0105039](http://arxiv.org/abs/math/0105039), [doi:10.1007/s00220-003-1035-x](https://doi.org/10.1007/s00220-003-1035-x))

both motivated by modelling [[twisted K-theory]] in terms of [[Grothendieck groups]] of [[twisted bundles]].

For more see also the references at _[[twisted vector bundle]]_.

The [[Cech cohomology|Cech cocycle]]-incarnation of bundle gerbe modules (effectively due to [Lupercio & Uribe 2001, Def. 7.2.1](#LupercioUribe01)) was then also considered in:

* {#Mackaay03} [[Marco Mackaay]], _A note on the holonomy of connections in twisted bundles_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 44 (2003) no. 1, pp. 39-62.  ([arXiv:math/0106019](http://arxiv.org/abs/math/0106019), [numdam:CTGDC_2003__44_1_39_0](http://www.numdam.org/item/?id=CTGDC_2003__44_1_39_0))

A [[splitting principle]] for bundle gerbe modules is discussed in

* {#Tomoda07} [[Atsushi Tomoda]], _On the splitting principle of bundle gerbe modules_, Osaka J. Math. Volume 44, Number 1 (2007), 231-246. ([Euclid](https://projecteuclid.org/euclid.ojm/1174324334), talk slides [pdf](http://ton.prosou.nu/official/ryousi2005.pdf))

On bundle gerbe modules with [[higher structure|higher]] [[G-structures]] (including [[String structure]], etc.) and the corresponding [[twisted Chern character]]:

* [[Fei Han]], [[Ruizhi Huang]], [[Varghese Mathai]], *Fractional structures on bundle gerbe modules and fractional classifying spaces* [[arXiv:2203.14439](https://arxiv.org/abs/2203.14439)]

[[!redirects bundle gerbe modules]]
[[!redirects gerbe module]]
[[!redirects gerbe modules]]


bundles - contents

**[[bundles]]**

* ([[parameterized stable homotopy theory|stable]]) [[parameterized homotopy theory]]

* [[fiber bundles in physics]]

* [[Chern-Weil theory]]

## Context {#sidebar_context}

* [[slice topos]], [[slice (∞,1)-topos]]

* ([[dependent linear type theory|linear]]) [[dependent type theory]]

## Classes of bundles {#sidebar_classes_of_bundles}

* [[covering space]]

* [[retractive space]]

* [[fiber bundle]], [[fiber ∞-bundle]]

  [[numerable bundle]]

  * [[sphere bundle]]

  * [[projective bundle]]

* [[principal bundle]], [[principal ∞-bundle]]

  * [[principal 2-bundle]], [[principal 3-bundle]]

  * [[circle bundle]], [[circle n-bundle with connection|circle n-bundle]]

  * [[orientation bundle]]

  * [[spinor bundle]], [[stringor bundle]]

* [[associated bundle]], [[associated ∞-bundle]]

  * [[gerbe]], [[2-gerbe]], [[∞-gerbe]],

  * [[local coefficient bundle]]

* [[vector bundle]], [[2-vector bundle]], [[(∞,1)-vector bundle]]

  [[real vector bundle|real]], [[complex vector bundle|complex]]/[[holomorphic vector bundle|holomorphic]], [[quaternionic vector bundle|quaternionic]]

  [[topological vector bundle|topological]], [[differentiable vector bundle|differentiable]], [[algebraic vector bundle|algebraic]]

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

  * [[line bundle]]

    [[complex line bundle|complex]], [[holomorphic line bundle|holomorphic]], [[algebraic line bundle|algebraic]]

    [[cubical structure on a line bundle|cubical structure]]

  * [[Chern-Weil theory]]

  * [[topological K-theory]]

  * [[tensor category]] [[Vect(X)|of vector bundles]]

    ([[VectBund]])

  * [[direct sum of vector bundles|direct sum]], [[tensor product of vector bundles|tensor product]], [[external tensor product of vector bundles|external tensor product]], [[inner product of vector bundles|inner product]]

  * [[dual vector bundle]]

  * [[stable vector bundle]]

  * [[virtual vector bundle]]

* [[bundle of spectra]]

* [[natural bundle]]

* [[equivariant bundle]]

## Universal bundles {#sidebar_universal_bundles}

* [[universal principal bundle]], [[universal principal ∞-bundle]]

* [[universal vector bundle]], [[universal complex line bundle]]

* [[subobject classifier]], [[object classifier]]

## Presentations {#sidebar_presentations}

* [[bundle gerbe]]

* [[groupal model for universal principal ∞-bundles]]

* [[microbundle]]

## Examples {#sidebar_examples}

* [[empty bundle]], [[zero bundle]]

* [[trivial vector bundle]]

* [[tangent bundle]], [[normal bundle]]

* [[tautological line bundle]]

  * [[basic line bundle on the 2-sphere]]

* [[Hopf fibration]]

* [[canonical line bundle]]

* [[prequantum circle bundle]], [[prequantum circle n-bundle]]

## Constructions {#sidebar_constructions}

* [[clutching construction]]


Burali-Forti's paradox

+-- {: .rightHandSide}
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### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

# Burali-Forti\'s paradox
* table of contents
{: toc}

## Summary

**Burali-Forti's paradox** is a [[paradox]] of naive [[material set theory|material]] [[set theory]] that was first observed by [[Cesare Burali-Forti]].  A closely related paradox that uses [[well-founded sets]] instead of [[ordinals]] is sometimes called **Mirimanoff's paradox**.

However, the paradox is not specific to material set theory and can be formulated in [[structural set theory]] or in [[type theory]].  When formulated in [[type theory]], it is often called **Girard's paradox** after [[Jean-Yves Girard]] (see at _[[type of types]]_).

## The paradox

Suppose that there were a [[set]] $Ord$ of all [[ordinal numbers]].  One could then prove that

1. The set $Ord$ is [[well-ordering|well-ordered]] by the relation $\lt$ on ordinals.
2. Thus, its [[order type]], call it say $\Omega$, is itself an ordinal number.
3. Thus $\Omega$ is an element of $Ord$, which implies $\Omega\lt\Omega$.
4. But this is provably impossible for any ordinal number.

There are many variations of the paradox, depending for instance on what precise definition of "well-ordered" (and "ordinal number") one chooses.

## In type theory: Girard's paradox
 {#GirardParadox}

As formulated in [[type theory]] by [[Jean-Yves Girard]], the Burali-Forti paradox shows that the original version of [[Per Martin-Löf]]'s type theory, which allowed a [[type of types]] $Type$ containing itself as a [[term]] $Type \colon Type$, is [[inconsistency|inconsistent]], in the precise sense that it contains (non-normalizing) [[proofs]] of [[false]]. As a result, a *type of all types* is similarly inconsistent because a type of all types would contain itself.

Moreover, by an adaptation of the proof, one can construct a [[looping combinator]] in this type theory, which implies the [[decidability|undecidability]] of type-checking.

## Set-theoretic Girard's paradox

A [[universal family of sets]] is a [[family of sets]] consisting of a set $U$ with index set $I$ and a function $E:U \to I$, such that for all sets $A$, there is a unique element $i_A \in I$ and a [[bijection]] $\delta_A:A \cong E^*(i_A)$ from $A$ to the [[fiber]] of $E$ at $i_A$.

The set-theoretic Girard's paradox states that having a [[universal family of sets]] is inconsistent.

## Related entries

* [[Russell's paradox]]

* [[Cantor's paradox]]

* [[cumulative hierarchy]]

## References

English translations of Burali-Forti's 1897 contributions can be found in

* J. van Heijenoort (ed.), _From Frege to Gödel: A Source Book in Mathematical Logic 1879-1931_ , Harvard UP 1967.

Another early reference on the set-theoretic paradoxes is

* D. Mirimanoff, _Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles_ , L'enseignement Mathématique **19** (1917) pp.37-52. ([pdf](http://retro.seals.ch/cntmng?pid=ensmat-001:1917:19::9); 19,4MB)

Girard's paradox is discussed in

* {#MartinLoef74} [[Per Martin-Löf]], section 1.9, p. 7 of  _An intuitionistic theory of types: predicative part_, In Logic Colloquium (1973), ed. H. E. Rose and J. C. Shepherdson (North-Holland, 1974), 73-118. ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926))

* [[Thierry Coquand]], _An analysis of Girard's paradox_ , pp.227-236 in _Proc. 1st Ann. Symb. Logic in Computer Science_ , IEEE Washington 1986. ([ps-draft](http://www.cse.chalmers.se/~coquand/paradox.ps))

category: paradox

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[[!redirects set-theoretic Girard paradox]]
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Burau representation

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#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
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#Contents#
* table of contents
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## Idea

A [[braid group representation]].

## Related concepts

* [[Alexander polynomial]]

## References

The original article:

* [[Werner Burau]], *Über Zopfgruppen und gleichsinnig verdrillte Verkettungen*, Abh. Math. Semin. Univ. Hambg. **11** (1935) 179–186 $[$[doi:10.1007/BF02940722](https://doi.org/10.1007/BF02940722)$]$

Review in the context of [[topological quantum computation|topological]] [[quantum computation]]:

* [[Colleen Delaney]], [[Eric C. Rowell]], [[Zhenghan Wang]], Section 2 of: *Local unitary representations of the braid group and their applications to quantum computing*, Revista Colombiana de Matemáticas **50** 2 (2017) 211 $[$[arXiv:1604.06429](https://arxiv.org/abs/1604.06429), [doi:10.15446/recolma.v50n2.62211](http://dx.doi.org/10.15446/recolma.v50n2.62211)$]$

See also:

* Wikipedia, *[Burau representation](https://en.wikipedia.org/wiki/Burau_representation)*

[[!redirects Burau representation]]


Burchnall-Chaundy theory

Burchnall-Chaundy theory refers to ramifications of the theory of commuting differential operators. Under some assumptions commuting operators have to satisfy an algebraic relation (defining therefore an algebraic curve, __Burchnall-Chaundy curve__). The main origin is 1928 paper of Burchnall and Chaundy.

* J. L. Burchnall, T.W. Chaundy, _Commutative ordinary differential operators_, Proc. London Math. Soc. 3 (21), 420–440 (1923); _Commutative ordinary differential operators_ Proc. Royal Soc. Lond. Ser. A Math. Phys. Eng. Sci. __118__, 557–583 (1928); _Commutative ordinary differential operators ii, the identity

* G. Wilson, _Commuting flows and conservation laws for Lax equations_, Math. Proc. Camb. Phil. Soc. 86 (1979) 131–143
[doi](https://doi.org/10.1017/S0305004100000700)
* I. M. Krichever, _Integration of non-linear equations by methods of algebraic geometry_ Funct. Anal, and Its Appl. 11 (1977), 15–31 (Russian), 12–26 (English); _Methods of algebraic geometry in the theory of non-linear equations_ Uspekhi Mat. Nauk 32, 6 (1977) 183–208 = Russian Math. Surveys 32, 6 (1977) 185–213.
* Emma Previato, _Multivariable Burchnall–Chaundy theory_, Phil. Trans. R. Soc. A (2008) 366, 1155–1177 [doi](https://doi.org/10.1098/rsta.2007.2064)
* Maurice J. Dupré, James F. Glazebrook, Emma Previato, _A Banach algebra version of the Sato Grassmannian and commutative rings of differential operators_, Acta Appl Math (2006) 92: 241 [doi](https://doi.org/10.1007/s10440-006-9052-3)
* James F. Glazebrook, _Banach algebras associated to Lax pairs_, Rep. Math. Phys. __75__:2 (2015) 279-301 doi; _A quasi-triangular Hopf structure derived from the Burchnall–Chaundy $C^*$-algebra_, Reports on Mathematical Physics __71__:1 (2013) 93-111 doi

[[!redirects Burchnall-Chaundy curve]]


Burnside category

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

#Contents#
* table of contents
{:toc}

## Idea

By the _Burnside category_ of a [[finite group]] $G$ one means either

1. the [[category of correspondences]] in [[finite set|finite]] [[G-sets]] (Prop. \ref{PermutativeCategoryOfFiniteGSets} below)

1. or an abelianization of that

   1. either by [[group completion]] of its monoidal [[hom-groupoids]] to an [[additive category|additive]] [[1-category]] (Def. \ref{AdditiveBurnsideCategory} below);

   1. or by forming the [[K-theory of a permutative category]] of the [[hom-groupoids]] to a [[(∞,1)-category of spectra|spectra]]-[[enriched (∞,1)-category]] (Def. \ref{SpectralBurnsideCategory} below).

The plain _[[Burnside ring]]_ is the [[endomorphism ring]] of the [[terminal object|terminal]] [[G-set]] in the [[additive category|additive]] version of the category (Example \ref{BurnsideRingIsEndomorphismRingInBurnsideCategory} below).

The [[functors]]/[[additive functors]]/[[enriched (∞,1)-functors]] from the Burnside category to, in particular, the [[category of abelian groups]]/[[(∞,1)-category of spectra]] are called _[[Mackey functors]]_.

A [[functor]] out of the Burnside category is called a _[[Mackey functor]]_, specifically if it is an [[Ab]]-[[enriched functor]] to [[Ab]] out of the [[additive category]]-version of the Burnside category. A [[spectra]]-[[enriched functor]] out of the spectral Burnside category to [[Spectra]] is called a _[[spectral Mackey functor]]_, for emphasis. These spectral Mackey functors on the Burnside category are [[equivalence of (infinity,1)-categories|equivalent]] to [[genuine G-spectra]] in [[equivariant stable homotopy theory]].

## Definition
 {#Definition}

There are various incarnations of the Burnside category as [[enriched categories]], in varying degree of sophistication of the enriching "[[cosmos]]". They are all induced from the canonical [[structure]] of [[permutative categories]] on the [[correspondences]] between two fixed [[finite set|finite]] [[G-sets]] (made explicit as Prop. \ref{PermutativeCategoryOfFiniteGSets} below). This yields the Burnside category as

1. _[enriched in permutative categories](#AsEnrichedInPermutativeCategories)_;

1. _[enriched in abelian groups](#AsEnrichedInabelianGroups)_;

1. _[enriched in spectra](#EnrichedInSpectra)_.

+-- {: .num_prop #PermutativeCategoryOfFiniteGSets}
###### Proposition
**(the [[permutative category]] of [[finite set|finite]] [[G-sets]])**

For $G$ be a [[finite group]], write $G FinSet_{sk}$ for the [[skeleton]] of the [[category]] of [[finite set|finite]] [[G-sets]].

Its [[objects]] may be identified with [[pairs]] $(n,\rho)$ consisting of a [[natural number]] $n$, reflecting the [[finite set]] $\{1,2, \cdots, n\}$, and a [[group homomorphism]] $\rho \;\colon\; G \longrightarrow S_n = Aut(\{1,2, \cdots, n\})$ from $G$ to the [[symmetric group]] on $n$ elements, reflecting the [[automorphism]] group of that finite set.

Its [[morphisms]] $(n_1,\rho_1) \overset{ f }{\longrightarrow} (n_2, \rho_2)$ are [[functions]] $f \;\colon\; \{1, 2, \cdots, n_1\} \longrightarrow \{1,2, \cdots, n_2\}$ that intertwine $\rho_1$ and $\rho_2$.

The [[coproduct]] $\sqcup$ of [[G-sets]] ([[disjoint union]]) makes this [[skeleton]] a [[permutative category]] $\big( G FinSet_{sk}, \sqcup \big)$.

In the same way its [[slice category]] over any object $X \in G FinSet_{sk}$ becomes a [[permutative category]] $\big( G FinSet_{sk}/_{X}, \sqcup \big)$ under disjoint union of [[domains]].

Similarly, the [[Cartesian product]] of finite $G$-sets can be restricted to these [[skeleta]] to produce [[bipermutative categories]] $\big( G FinSet_{sk}/_{X}, \sqcup, \times \big)$.

Finally, restriction to [[isomorphisms]] (passage to [[cores]]) yields the bipermutative [[groupoids]] $\big( Core(G FinSet_{sk})/_{X}, \sqcup, \times \big)$.

=--

(e.g. [Guillou-May 11, Def. 1.3](#GuillouMay11), [Bohmann-Osorno 14, Def. 1.4](#BohmannOsorno14))

### As enriched in permutative categories
 {#AsEnrichedInPermutativeCategories}

+-- {: .num_defn #AdditiveBurnsideCategory}
###### Definition
**([[PermCat]]-[[enriched category|enriched]] [[Burnside category]])**

The [[(2,1)-category of correspondences]] $Corr(G FinSet)$ is [[equivalence of (infinity,1)-categories|equivalent]] to the [[(2,1)-category]] whose [[objects]] are the [[finite set|finite]] [[G-sets]] and whose [[hom-categories]] are the [[permutative category|permutative]] [[cores]] of [[skeleta]] of [[slice categories]] of [[G-sets]] from Def. \ref{PermutativeCategoryOfFiniteGSets}, over the [[Cartesian product]] of [[source]] and [[domain]] [[G-sets]]:

$$
  Corr(G FinSet)\big( S_1, S_2  \big)
  \;\simeq\;
  Core(G FinSet_{sk}/_{S_1 \times S_2})
  \,.
$$

This locally skeletal [[(2,1)-category]] is the _[[PermCat]]-[[enriched category|enriched]] Burnside category_ $G Burn_{pc}$_

It may be regarded as an [[enriched category]] over the [[multicategory]] [[PermCat]] of [[permutative categories]] (a $\mathbf{PC}$-cateory in the sense of [Guillout 10](#Guillout10)).

=--

([Guillou-May 11, Def. 1.6](#GuillouMay11), [Bohmann-Osorno 14, Def. 7.1, 7.2](#BohmannOsorno14))

### As enriched in abelian groups
 {#AsEnrichedInabelianGroups}

+-- {: .num_defn #AdditiveBurnsideCategory}
###### Definition
**(additive Burnside category)**

The _additive Burnside category_ $G Burn_{ad}$ of $G$ is the [[additive category]] obtained from the [[PermCat]]-[[enriched category|enriched]] Burnside category $G Burn_{pc}$ (Def. \ref{AdditiveBurnsideCategory}) under replacing each [[hom-object|hom]]-[[permutative category]] $Core(G FinSet_{sk}/_{S_1 \times S_2})$ with its [[Grothendieck group]], hence with the [[abelian group]] which is the [[group completion]]

$$
  K \;\colon\; PermCat \overset{\text{iso classes}}{\longrightarrow} CMon \overset{K}{\longrightarrow} Ab
$$

of the [[commutative monoid]] of [[isomorphism classes]] of [[objects]] in $Core(G FinSet_{sk}/_{S_1 \times S_2})$, under [[disjoint union]]:

$$
  G Burn_{ab}
  \;\coloneqq\;
  K_{\bullet} G Burn_{pc}
  \,.
$$

=--

+-- {: .num_example #BurnsideRingIsEndomorphismRingInBurnsideCategory}
###### Example
**([[Burnside ring]] is [[endomorphism ring]] of additive Burnside category)**

The [[endomorphism ring]] of the [[terminal object|terminal]] [[G-set]] (the [[point]] $\ast$ equipped with the, necessarily, [[trivial action]]) in the additive Burnside category (Def. \ref{AdditiveBurnsideCategory}) is the _[[Burnside ring]]_ $A(G)$:

$$
  End_{G Burn_{ad}}(\ast, \ast)
  \;\simeq\;
  A(G)
$$

=--

### As enriched in spectra
 {#EnrichedInSpectra}

+-- {: .num_defn #SpectralBurnsideCategory}
###### Definition
(**[[Spectra]]-[[enriched category|enriched]] [[Burnside category]])**

Since construction of [[K-theory spectra of permutative categories]] applies  [[hom-object]]-wise to [[PermCat]]-[[enriched categories]] ([this Prop.](multifunctor#MonoidalFunctoriality))

$$
  \mathbb{K}_\bullet
  \;\colon\;
  PermCat Cat \longrightarrow Spectra Cat
$$

the image of the [[PermCat]]-[[enriched category|enriched]] [[Burnside category]] $G \mathcal{E}$ from Def. \ref{AdditiveBurnsideCategory} under forming [[hom-object]]-wise the [[K-theory spectra of permutative categories]] yields a [[Spectra]]-[[enriched category]]

$$
  G Burn_{sp}
  \;\coloneqq\;
  \mathbb{K}_\bullet G Burn_{pc}
  \,.
$$

This is called the _spectral Burnside category_.

=--

([Guillou-May 11, Def. 1.12](#GuillouMay11), [Bohmann-Osorno 14, Def. 7.3](#BohmannOsorno14))

## Properties

### Relation to $G$-Spectra

+-- {: .num_prop #GSpectraAreSpectralMackeyFunctors}
###### Proposition
**([[G-spectra]] are spectral presheaves on the spectral Burnside category)**

There is a [[zig-zag]] of [[Quillen equivalences]]

$$
  [ G Burn_{sp}, Spectra ]
  \;\simeq_{Qu}^{zigzag}\;
  G Spectra
$$

between the [[model structure on functors|model category of]] [[Spectra]]-[[enriched presheaves]] over the [[spectral Burnside category]] from Def. \ref{SpectralBurnsideCategory} ("[[spectral Mackey functors]]") and that of [[genuine G-spectra]].

This equivalence is such that the [[spectral Mackey functor]] corresponding to a fibrant [[G-spectrum]] $E$ assigns to the [[transitive action|transitive]] [[G-set]] $G/H$ the [[fixed point spectrum]] $E^H$:

$$
  G/H
    \;\mapsto\;
  E(G/H)
    \;\coloneqq\;
  [\Sigma^\infty_G G/H, E]^G
    \;\simeq\;
  E^H
  \,.
$$

=--

([GuillouMay 11, theorem 1.13, corollary 1.14, remark 2.5](#GuillouMay11)).

## Related concepts

* [[Burnside ring]]

* [[Mackey functor]], [[equivariant stable homotopy theory]]

* [[disjunctive (∞,1)-category]]

## References

Lecture notes include:

* {#Blumberg17} [[Andrew Blumberg]], section 3.1 of _The Burnside category_, 2017 ([pdf](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf), [GitHub](https://github.com/adebray/equivariant_homotopy_theory))

The spectrally-enriched version and its role in the equivalent description of [[G-spectra]] via [[spectral Mackey functors]] is due to

* {#GuillouMay11} [[Bert Guillou]], [[Peter May]], _Models of $G$-spectra as presheaves of spectra, ([arXiv:1110.3571](http://arxiv.org/abs/1110.3571))

A beautified review is given in

* {#BohmannOsorno14} [[Anna Marie Bohmann]], [[Angélica Osorno]], section 7 of _Constructing equivariant spectra via categorical Mackey functors_, Algebraic & Geometric Topology 15.1 (2015): 537-563 ([arXiv:1405.6126](http://arxiv.org/abs/1405.6126))

making use of

* {#Guillout10} [[Bertrand Guillou]], _Strictification of categories weakly enriched in symmetric monoidal categories_, Theory Appl. Categ., 24:No. 20, 564–579, 2010 ([TAC](http://www.tac.mta.ca/tac//volumes/24/20/24-20abs.html))

[[!redirects Burnside categories]]

[[!redirects spectral Burnside category]]
[[!redirects spectral Burnside categories]]

[[!redirects additive Burnside category]]
[[!redirects additive Burnside categories]]


Burnside ring

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#### Representation theory
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[[!include representation theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

The Burnside ring $A(G)$ of a [[finite group]] $G$ is the analogue of the [[representation ring]] in the category of [[finite sets]], as opposed to the category of [[finite dimensional vector spaces]] over a field $F$.

Elements of the Burnside ring are thus formal differences of [[G-sets]] (with respect to [[disjoint union]]).

$A$ is a contravariant functor $\text{FinGrp} \xrightarrow{A} \text{AbRing}$.

## Definition

For *any* group $G$, the *Burnside [[rig]]* of $G$ is the set of [[isomorphism classes]] of the [[topos]] $FinSet^G$ of finite [[G-sets]], the [[category]] of [[permutation representations]] of $G$ on [[finite sets]], equipped with the addition operation descended from [[coproducts]] in $FinSet^G$ and the multiplication operation descended from [[products]] in $FinSet^G$.

(In fact the Burnside rig $B(G)$ is an [[exponential ring]] [[rig]], where exponentiation is derived from the [[cartesian closed category|cartesian closed structure]] of the topos. )

The *Burnside ring* $A(G)$ is then the (additive) [[group completion]] of the Burnside rig, $A(G) = \mathbb{Z} \otimes_{\mathbb{N}} B(G)$. (This tensor product in [[commutative monoids]] is the coproduct of $\mathbb{Z}$ and $B(G)$ in the category of commutative rigs, and $\mathbb{Z} \otimes_{\mathbb{N}} -$ is [[left adjoint]] to the forgetful functor from [[commutative rings]] to commutative rigs.)

More generally, any [[distributive category]] determines a Burnside rig ([Schanuel91](#Schanuel91)).

Explicitly, the Burnside ring can be seen to be the free abelian group on the set $G / H_1 , G / H_2, \ldots, G / H_t$ of $G$, where $H_{1}, \ldots, H_{t}$ are representatives of the distinct conjugacy classes of $G$, equipped with the product described in Definition \ref{BurnsideMultiplicities}.

## Properties

### Span by the coset spaces $G/H$

Since every [[finite set|finite]] [[G-set]] is a [[direct sum]] of the basic [[coset spaces]] $G/H$, for $H \hookrightarrow G$ [[subgroups]] of $G$, and since $G/H_1$ and $G/H_2$ are [[isomorphism|isomorphic]] [[G-sets]] of $H_1$ and $H_2$ are [[conjugation action|conjugate]] to each other, the Burnside ring is spanned, as an [[abelian group]] by the $[G/H]$ for $H$ ranging over [[conjugacy classes]] of subgroups.

$$
  A(G)
  \;\simeq_{\mathbb{Z}}\;
  \underset{[H \subset G]}{\oplus}
  [G/H]
  \,.
$$

Notice that the [[permutation representations]] $k[G/H]$ corresponding to these generators are precisely the [[induced representations of trivial representations]]: $k[G/H] \simeq ind_H^G(\mathbf{1})$.

### In terms of the table of marks
 {#InTermsOfTheTableOfMarks}

We discuss how the product in the Burnside ring is encoded in the [[table of marks]] of the given [[finite group]].

+-- {: .num_defn #BurnsideMultiplicities}
###### Definition
**(Burnside multiplicities)**

Given a choice of [[linear order]] on the [[conjugacy classes]] of [[subgroups]] of $G$ (for instance as in [this Lemma](table+of+marks#LinearOrderOnConjugacyClassesOfSubgroups)), we say that the corresponding _structure constants_ of the [[Burnside ring]] (or _Burnside multiplicities_) are the [[natural numbers]]

\[
  n_{i j}^\ell \;\in\; \mathbb{N}
\]

uniquely defined by the [[equation]]

\[
  \label{BurnsideStructureConstants}
  [G/H_i] \times [G/H_j]
  \;=\;
  \underset{ \ell }{\sum}
  n_{i j}^\ell [G/H_\ell]
  \,.
\]

=--

+-- {: .num_prop #BurnsideProductInTermsOfTableOfMarks}
###### Proposition
**([[Burnside ring]] product in terms of [[table of marks]])**

The Burnside ring structure constants  $\left( n_{i j}^\ell\right)$ (Def. \ref{BurnsideMultiplicities}) are equal to the following algebraic expression in the [[table of marks]] $M$ and its [[inverse matrix]] $M^{-1}$ (which exists by [this Prop.](table+of+marks#TableOfMarksIsInvertibleUpperTriangular)):

$$
  n_{i j}^l
  \;=\;
  \underset{1 \leq m \leq t}{\sum}
  M_{i m} \cdot M_{j m}
  \cdot
  (M^{-1})_{m l}
$$

where $t$ is the dimension of $M$, i.e. $M$ is a $t \times t$ matrix.
=--

For [[proof]] see at _[[table of marks]]_, [this Prop.](table+of+marks#BurnsideProductInTermsOfTableOfMarks)

### As the equivariant stable cohomotopy of the point
 {#AsTheEquivariantStableCohomotopyOfThePoint}

+-- {: .num_prop #BurnsideRingIsEquivariantStableCohomotopyOfPoint}
###### Proposition
**([[Burnside ring is equivariant stable cohomotopy of the point]])**

Let $G$ be a [[finite group]], then its [[Burnside ring]] $A(G)$ is [[isomorphism|isomorphic]] to the [[equivariant stable cohomotopy]] [[cohomology ring]] $\mathbb{S}_G(\ast)$ of the [[point]] in degree 0.

$$
  A(G) \overset{\simeq}{\longrightarrow} \mathbb{S}_G(\ast)
  \,.
$$

=--

This is originally due to [Segal 71](#Segal71), a detailed proof is given by [tom Dieck 79, theorem 7.6.7, 8.5.1](#tomDieck79). See also [Lück 05, theorem 1.13](#Lueck05), [tom Dieck-Petrie 78](#tomDieckPetrie78).

From a broader perspective, this statement is a special case of [[tom Dieck splitting]] of [[equivariant suspension spectra]] (e.g. [Schwede 15, theorem 6.14](#Schwede15)), see [there](tom+Dieck+splitting#OfFixedPointSpectraOfEquivariantSuspensionSpectra).

[[!include Segal completion -- table]]

More explicitly, this means that the Burnside ring of a group $G$ is isomorphic to the [[colimit]]

$$
  A(G) \simeq \underset{\longrightarrow}{\lim}_V [S^V,S^V]_G
$$

over $G$-[[representations]] in a complete [[G-universe]], of $G$-[[homotopy classes]] of $G$-equivariant based [[continuous functions]] from the [[representation sphere]] $S^V$ to itself ([Greenlees-May 95, p. 8](#GreenleesMay95)).

### As the endomorphism ring of the additive Burnside category

+-- {: .num_example #BurnsideRingIsEndomorphismRingInBurnsideCategory}
###### Example
**([[Burnside ring]] is [[endomorphism ring]] of [[additive Burnside category]])**

The [[endomorphism ring]] of the [[terminal object|terminal]] [[G-set]] (the [[point]] $\ast$ equipped with the, necessarily, [[trivial action]]) in the [[additive Burnside category]] $G Burn_{ad}$ is the _Burnside ring_ $A(G)$:

$$
  End_{G Burn_{ad}}(\ast, \ast)
  \;\simeq\;
  A(G)
  \,.
$$

=--

### Relation to representation ring

Let $G$ be a [[finite group]]. Consider

1. the [[Burnside ring]] $A(G)$, which is the [[Grothendieck group]] of the [[monoidal category]] $G Set$ of [[finite set|finite]] [[G-sets]];

1. the [[representation ring]] $R(G)$, which is the [[Grothendieck group]] of the monoidal category $G Rep$ of [[finite dimensional vector space|finite dimensional]] $G$-[[linear representations]].

Then then map that sends a G-set to the corresponding linear [[permutation representation]] is a [[strong monoidal functor]]

$$
  G Set \overset{\mathbb{C}[-]}{\longrightarrow} G Rep
$$

and hence induces a [[ring homomorphism]]

$$
  A(G) \overset{ \mathbb{C}[-] }{\longrightarrow} R(G)
$$

Under the identitification

1. of the [[Burnside ring]] with the [[equivariant stable cohomotopy]] of the point

   $$
     A(G) \;\simeq\; \mathbb{S}_G(\ast)
   $$

   (as [above](#AsTheEquivariantStableCohomotopyOfThePoint))

1. of the [[representation ring]] with the [[equivariant K-theory]] of the point

   $$
     R(G) \;\simeq\; K_G(\ast)
   $$

   (see [there](representation+ring#AsEquivariantKTheoryOfThePoint))

this should be the image of the initial morphism of [[E-infinity ring spectra]]

$$
  \mathbb{S} \longrightarrow KU
$$

from the [[sphere spectrum]] to [[KU]].

For details on this comparison map see at _[[permutation representation]]_, [this section](permutation+representation#ExamplesVirtualPermutationRepresentations).

## Related concepts

* [[orbit category]]

* [[Segal conjecture]]

* [[equivariant stable cohomotopy]]

* [[beta-ring]]

* [[objective number theory]]

## References
 {#References}

The Burnside product seems to first appear as equation (i) in:

* {#Burnside01} [[William Burnside]], _On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups_, Proceedings of the American Mathematical Society 1901 ([doi:10.1112/plms/s1-34.1.159](https://doi.org/10.1112/plms/s1-34.1.159))

(beware the terminology: a [[G-set]] is called a "permutation group $G$" in that article, a [[subset]] is called a "compound" and the [[Cartesian product]] of $G$-sets is called their "compounding").

It is then included (not in the first but) in the second edition (Sections 184-185) of:

* {#Burnside1897} [[William Burnside]], _Theory of Groups of Finite Order_,  Second edition Cambridge 1911 ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/burnside1911.pdf))

  reprinted by Cambridge University Press 2012 ([doi:10.1017/CBO9781139237253]( https://doi.org/10.1017/CBO9781139237253))

The term "Burnside ring" as well as "Burnside algebra" is then due to (see [NMT 04, Vol. 1 p. 60](William+Burnside#NMT04) for historical comments)

* {#Solomon67} [[Louis Solomon]], _The Burnside algebra of a finite group_, Journal of Combinatorial Theory Volume 2, Issue 4, June 1967, Pages 603-615 (doi:10.1016/S0021-9800(67)80064-4)

Modern textbook accounts and lecture notes:

* Charles Curtis, Irving Reiner, chapter XI of _Representation theory of finite groups and associative algebras_, AMS 1962

* {#tomDieck79} [[Tammo tom Dieck]], _[[Transformation Groups and Representation Theory]]_, Springer 1979

* {#tomDieck87} [[Tammo tom Dieck]], Chapter IV of: _[[Transformation Groups]]_, de Gruyter 1987 ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

* {#Bouc00} Serge Bouc, _Burnside rings_, in _Handbook of Algebra_ Volume 2, 2000, Pages 739-804 (doi:10.1016/S1570-7954(00)80043-1)

* {#tomDieck09} [[Tammo tom Dieck]], _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf))

* {#Schwede15} [[Stefan Schwede]], section 6 of _[[Lectures on Equivariant Stable Homotopy Theory]]_, 2015 ([pdf](http://www.math.uni-bonn.de/people/schwede/equivariant.pdf))

See also

* {#GreenleesMay95} [[John Greenlees]], [[Peter May]], section 2 of _Equivariant stable homotopy theory_, in I.M. James (ed.), _Handbook of Algebraic Topology_ , pp. 279-325. 1995. ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

* {#Schanuel91} [[Stephen Schanuel]], 1991, _Negative sets have Euler characteristic and dimension_, in: Proc. Como 1990. Lecture Notes in Mathematics, vol. 1488. Springer-Verlag, Berlin, pp. 379–385.

Discussion in relation to [[equivariant stable cohomotopy]] and the [[Segal-Carlsson completion theorem]] is in

* {#Segal71} [[Graeme Segal]], _Equivariant stable homotopy theory_, In Actes du Congrès International des Math ématiciens (Nice, 1970), Tome 2 , pages 59–63. Gauthier-Villars, Paris, 1971 ([[SegalEquivariantStableHomotopyTheory.pdf:file]])

* {#tomDieckPetrie78} [[Tammo tom Dieck]], T. Petrie, _Geometric modules over the Burnside ring_, Invent. Math. 47 (1978) 273-287;
chapter 10 in: _[[Transformation Groups and Representation Theory]]_  ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-geometric.pdf))

* Erkki Laitinen, _On the Burnside ring and stable cohomotopy of a finite group_,  Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 ([jstor:24491306](https://www.jstor.org/stable/24491306), [[Laitinen79.pdf:file]])

* [[Gunnar Carlsson]], _Equivariant Stable Homotopy and Segal's Burnside Ring Conjecture_,  Annals of Mathematics Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 189-224 ([jstor:2006940](https://www.jstor.org/stable/2006940), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/carlsson.pdf))

* {#GayMorrisMorris} C. D. Gay, G. C. Morris, and I. Morris, _Computing Adams operations on the Burnside ring of a finite group_, J. Reine Angew. Math., 341 (1983), pp. 87–97.

* {#Lueck05} [[Wolfgang Lück]], _The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups_ ([arXiv:math/0504051](https://arxiv.org/abs/math/0504051))

Computational aspects are discussed in

* Martin Kreuzer, _Computational aspects of Burnside rings, part I: the ring structure_, D.P. Beitr Algebra Geom (2017) 58: 427 ([doi:10.1007/s13366-016-0324-4](https://doi.org/10.1007/s13366-016-0324-4))

[[!redirects Burnside rings]]

[[!redirects burnside ring]]
[[!redirects burnside rings]]

[[!redirects Burnside rig]]
[[!redirects Burnside rigs]]

[[!redirects burnside rig]]
[[!redirects burnside rigs]]


Burnside ring is equivariant stable cohomotopy of the point

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In analogy to how the [[representation ring]] of a [[finite group]] is equivalently the [[equivariant K-theory]] of the point, so the [[equivariant stable cohomotopy]] of the point is the _[[Burnside ring]]_.

[[!include Segal completion -- table]]

## Statement

+-- {: .num_prop #BurnsideRingIsEquivariantStableCohomotopyOfPoint}
###### Proposition
**([[Burnside ring]] is [[equivariant stable cohomotopy]] of the [[point]])**

Let $G$ be a [[compact Lie group]] (for instance a [[finite group]]). Its [[Burnside ring]] $A(G)$ is [[isomorphism|isomorphic]] to the [[equivariant stable cohomotopy]] [[cohomology ring]] $\mathbb{S}_G(\ast)$ of the [[point]] in degree 0, via the Lefschwetz-Dold index:

$$
  A(G)
    \underoverset{\simeq}{LD}{\longrightarrow}
  \mathbb{S}_G(\ast)
  \,.
$$

More in detail, for $G$ a [[finite group]], this [[isomorphism]] identifies the [[Burnside character]] on the left with the [[fixed locus]]-[[degree of a continuous function|degrees]] on the right, hence for all [[subgroups]] $H \subset G$ it identifies

1. the $H$-[[Burnside marks]] $\left\vert S^H \right\vert \in \mathbb{Z}$ of [[virtual representation|virtual]] [[finite set|finite]] [[G-sets]] $S$

   (which, as $H \subset G$ ranges, completely characterize the [[G-set]], by [this Prop.](table+of+marks#BurnsideCharacterIsInjective))

1. the [[degree of a continuous function|degrees]] $deg\left( \left(LD(S)\right)^H\right) \in \mathbb{Z}$ at $H$-[[fixed points]] of representative [[equivariant Cohomotopy]] [[cocycles]] $LD(S) \colon S^V \to S^V$

   (which completely characterize the [[equivariant Cohomotopy]]-class by the [[equivariant Hopf degree theorem]], [this Prop.](Hopf+degree+theorem#EquivariantHopfDegreeTheorem))

\[
  \label{CorrespondenceOnMarksAndDegrees}
  \array{
    A(G)
    &\underoverset{\simeq}{LD}{\longrightarrow}&
    \underset{\longrightarrow_{\mathrlap{V}}}{\lim}
    \;\;
    \left(
      \pi_0
      \mathrm{Maps}^{\{0\}/}
      \left(
        S^V, S^V
      \right)^G
    \right)
    &=&
    \mathbb{S}_G(\ast)
    \\
    S &\mapsto& LD(S)
    \\
    \underset{
      \mathclap{
        \text{Burnside character}
      }
    }{
    \underbrace{
    \left(
      H \mapsto
      \left\vert
        S^H
      \right\vert
    \right)
    }
    }
    &=&
    \underset{
      \mathclap{
        \text{degrees on fixed strata}
      }
    }{
      \underbrace{
      \left(
        H
        \;\mapsto\;
        deg
        \left(
          S^{ dim\left( V^H\right) }
          \overset{\big(LD(S)\big)^H}{\longrightarrow}
          S^{ dim\left( V^H\right) }
        \right)
      \right)
      }
    }
  }
\]

For $G$ a [[compact Lie group]] this correspondence remains intact, with the relevant conditions on subgroups $H$ ([[closed subset|closed]] subgroups such that the [[Weyl group]] $W_G(H) \coloneqq N_G(H)/H$ is a [[finite group]]) and generalizing the [[Burnside marks]] to the [[Euler characteristic]] of [[fixed loci]].

=--

The statement is due to [Segal 71](#Segal71), a detailed proof making manifest the correspondence (eq:CorrespondenceOnMarksAndDegrees) is given by [tom Dieck 79, theorem 8.5.1](#tomDieck79). See also [tom Dieck-Petrie 78](#tomDieckPetrie78), [Lück 05, theorem 1.13](#Lueck05).

From a broader perspective of [[equivariant stable homotopy theory]], this statement is a special case of [[tom Dieck splitting]] of [[equivariant suspension spectra]] (e.g. [Schwede 15, theorem 6.14](#Schwede15)), see [there](tom+Dieck+splitting#OfFixedPointSpectraOfEquivariantSuspensionSpectra).

## References

* {#Segal71} [[Graeme Segal]], _Equivariant stable homotopy theory_, In Actes du Congrès International des Math ématiciens (Nice, 1970), Tome 2 , pages 59–63. Gauthier-Villars, Paris, 1971

* {#tomDieckPetrie78} [[Tammo tom Dieck]], T. Petrie,  _Geometric modules over the Burnside ring_, Invent. Math. 47 (1978) 273-287 ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-geometric.pdf))

* {#tomDieck79} [[Tammo tom Dieck]], _[[Transformation Groups and Representation Theory]]_, Springer 1979

* {#Lueck05} [[Wolfgang Lück]], _The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups_ ([arXiv:math/0504051](https://arxiv.org/abs/math/0504051))

* {#Schwede15} [[Stefan Schwede]], section 6 of _[[Lectures on Equivariant Stable Homotopy Theory]]_, 2015 ([pdf](http://www.math.uni-bonn.de/people/schwede/equivariant.pdf))


Burt Ovrut

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

## Selected writings

On [[heterotic M-theory]]:

* {#LalakLukasOvrut97} Zygmunt Lalak, [[André Lukas]], [[Burt Ovrut]], _Soliton Solutions of M-theory on an Orbifold_, Phys. Lett. B425 (1998) 59-70 ([arXiv:hep-th/9709214](https://arxiv.org/abs/hep-th/9709214))

* {#Ovrut02} [[Burt Ovrut]], _Lectures on Heterotic M-Theory_, TASI 2001. 2004. 359-407  ([arXiv:hep-th/0201032](http://arxiv.org/abs/hep-th/0201032), [doi:10.1142/9789812702821_0007](https://doi.org/10.1142/9789812702821_0007))

* {#Ovrut18} [[Burt Ovrut]], _Vacuum Constraints for Realistic Heterotic M-Theories_, Symmetry 2018, 10(12), 723 ([arXiv:1811.08892](https://arxiv.org/abs/1811.08892), [doi:10.3390/sym10120723](https://doi.org/10.3390/sym10120723))

On [[string phenomenology]] in [[heterotic M-theory]]:

* {#DOPW99} [[Ron Donagi]], [[Burt Ovrut]], [[Tony Pantev]], [[Daniel Waldram]], _Standard Models from Heterotic M-theory_, Adv. Theor. Math. Phys. 5 (2002) 93-137 ([arXiv:hep-th/9912208](https://arxiv.org/abs/hep-th/9912208))

* {#DOPW00} [[Ron Donagi]], [[Burt Ovrut]], [[Tony Pantev]], [[Daniel Waldram]], _Standard Model Vacua in Heterotic M-Theory_, talk at [Strings '99](http://strings99.aei.mpg.de/), Potsdam, Germany, 19 - 24 Jul 1999 ([arXiv:hep-th/0001101](https://arxiv.org/abs/hep-th/0001101))

* Cédric Deffayet, [[Burt Ovrut]], [[Paul Steinhardt]], *Stable Vacua with Realistic Phenomenology and Cosmology in Heterotic M-theory Satisfying Swampland Conjectures* [[arXiv:2401.04828](https://arxiv.org/abs/2401.04828)]

and with emphasis on [[heterotic line bundles]] in the [[hidden sector]]:

On [[heterotic line bundles]] in the [[hidden sector]] of [[heterotic M-theory]]:

* [[Anthony Ashmore]], [[Sebastian Dumitru]], [[Burt Ovrut]], Section 4.2 of: _Line Bundle Hidden Sectors for Strongly Coupled Heterotic Standard Models_ ([arXiv:2003.05455](https://arxiv.org/abs/2003.05455))

* [[Anthony Ashmore]], [[Sebastian Dumitru]], [[Burt Ovrut]], _Explicit Soft Supersymmetry Breaking in the Heterotic M-Theory B−L MSSM_ ([arXiv:2012.11029](https://arxiv.org/abs/2012.11029))

* [[Anthony Ashmore]], [[Sebastian Dumitru]], [[Burt Ovrut]], *Hidden Sectors from Multiple Line Bundles for the B−L MSSM* ([arXiv:2106.09087](https://arxiv.org/abs/2106.09087))

* [[Sebastian Dumitru]], [[Burt A. Ovrut]], *Heterotic M-Theory Hidden Sectors with an Anomalous $U(1)$ Gauge Symmetry* ([arXiv:2109.13781](https://arxiv.org/abs/2109.13781))

On [[heterotic M-theory on ADE-orbifolds]]:

* Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Intersecting Orbifold Planes and Local Anomaly Cancellation in M-Theory_, Nucl. Phys. B554: 437-483, 1999 ([arXiv:hep-th/9903028](https://arxiv.org/abs/hep-th/9903028))

* Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Local Anomaly Cancellation, M-Theory Orbifolds and Phase-Transitions_, Nucl. Phys. B589: 269-291, 2000 ([arXiv:hep-th/0005251](https://arxiv.org/abs/hep-th/0005251))

* Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _An M-Theory Perspective on Heterotic K3 Orbifold Compactifications_, Int. J. Mod. Phys. A18:3273-3314, 2003 ([arXiv:hep-th/0010087](https://arxiv.org/abs/hep-th/0010087))

* Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Twisted Sectors and Chern-Simons Terms in M-Theory Orbifolds_, Int. J. Mod. Phys. A18: 2995-3014, 2003 ([arXiv:hep-th/0011031](https://arxiv.org/abs/hep-th/0011031))

Specifically on [[gaugino condensation]] in [[heterotic string theory]] and [[Horava-Witten theory]]:

* [[André Lukas]], [[Burt Ovrut]], [[Daniel Waldram]], _Gaugino condensation in M theory on $S^1/Z_2$_, Phys. Rev. D 57, 7529 (1998) ([arXiv:hep-th/9711197](https://arxiv.org/abs/hep-th/9711197), [doi:10.1103/PhysRevD.57.7529](https://doi.org/10.1103/PhysRevD.57.7529))

* [[André Lukas]], [[Burt Ovrut]], [[Daniel Waldram]], _Five-Branes and Supersymmetry Breaking in M-Theory_, JHEP 9904:009, 1999 ([arXiv:hep-th/9901017](https://arxiv.org/abs/hep-th/9901017))

On [[moduli stabilization]]:

* [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Burt Ovrut]], _Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories_ ([arXiv:1304.2704](http://arxiv.org/abs/1304.2704))

Discussion via [[machine learning]] of [[connection on a bundle|connections]] on [[heterotic line bundles]] over [[Calabi-Yau manifold|Calabi-Yau 3-folds]]:

* [[Anthony Ashmore]], Rehan Deen, [[Yang-Hui He]], [[Burt Ovrut]], *Machine Learning Line Bundle Connections* ([arXiv:2110.12483](https://arxiv.org/abs/2110.12483))

and for computation of [[Yukawa couplings]]:

* Andrei Constantin, Cristofero S. Fraser-Taliente, Thomas R. Harvey, [[Andre Lukas]], [[Burt Ovrut]], *Computation of Quark Masses from String Theory* \[arXiv:2402.01615\]

## Related pages

* [[heterotic M-theory]]

* [[string phenomenology]]

category: people

[[!redirects Burt A. Ovrut]]


Burt Totaro

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Burt_Totaro)

## Selected writings

Relation of [[complex cobordism cohomology]] with [[divisors]], [[algebraic cycles]] and [[Chow groups]]:

* [[Burt Totaro]], _Torsion algebraic cycles and complex cobordism_, J. Amer. Math. Soc. 10 (1997), 467-493 ([doi:10.1090/S0894-0347-97-00232-4](https://doi.org/10.1090/S0894-0347-97-00232-4))


Butcher group

#Butcher group#

The Butcher group was introduced in Butcher’s seminal work on [[Runge-Kutta method]]s. The elements of the group are (generalised) numerical schemes which contain the Runge-Kutta schemes and the group product encodes composition of numerical schemes.

Later Brouder observed that the Butcher group can be viewed as the character group of the (Butcher-)Connes-Kreimer-Hopf algebra used in the combinatorial approach to renormalisation.

The group is used in numerical analysis to deal with order conditions of numerical methods. Further, it has been discovered that the Butcher group is an [[infinite-dimensional Lie group]].

References

* The English [wikipedia](https://en.wikipedia.org/wiki/Butcher_group) article is quite detailed

* J. C. Butcher, An algebraic theory of integration methods, Math. Comp. 26 (1972), 79–106.

* Ch. Brouder, _Runge-Kutta methods and renormalization_, Europ. Phys. J. C12 (2000) 512–534

* Charles Brouder, _Trees, renormalization and differential equations_, BIT Numerical Mathematics, 44 (3): 425–438 [doi](https://doi.org/10.1023%2FA%3A1007523409317)

* Alain Connes, Dirk Kreimer,_Lessons from quantum field theory: Hopf algebras and spacetime geometries_, Letters in Mathematical Physics, 48: 85–96, 1999 [doi](https://dx.doiorg/10.1023/A:1007523409317)

* G. Bogfjellmo, [[Alexander Schmeding]]:  The Lie group structure of the Butcher group, Foundations of Computational Mathematics, ([arXiv:1410.4761
](https://arxiv.org/abs/1410.4761))


butterfly

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

For $G$ and $H$ two strict [[2-group]]s, their [[delooping]]s are strict one-object [[2-groupoid]]s $\mathbf{B}G$ and $\mathbf{B}H$ and a general morphism $f : G \to H$ of 2-groups is by definition a morphism

$$
 \mathbf{B}f : \mathbf{B}G \to \mathbf{B}H
$$

i.e. a [[2-functor]] -- in general a weak one.

A _butterfly diagram_ is a way to describe such weak 2-functors in terms of morphisms between the ordinary [[group]]s appearing in the [[crossed module]]s corresponding to $G$ and $H$.

There is a more general version describing weak morphisms among internal crossed modules in [[semi-abelian categories]].

## Definition

A __butterfly__ or __papillon__ is a [[crossed profunctor]]

$$
  \mathbb{X} \to \mathbb{Y}
$$

between [[crossed module]]s $\mathbb{X}=(\partial_X : X_1\to X_0)$ and $\mathbb{Y}=(\partial_Y:Y_1\to Y_0)$ (actions suppressed from the notation), given by a diagram of [[group]]s

$$
  \array{
    X_1 &&&& Y_1
    \\
    & \searrow^{\mathrlap{x_1}} & & {}^{\mathllap{y_1}}\swarrow
    \\
    {}^{\mathllap{\partial_X}}\downarrow && P &&
    \downarrow^{\mathrlap{\partial_Y}}
    \\
    & \swarrow_{\mathrlap{x_0}} && {}_{\mathllap{y_0}}\searrow
    \\
    X_0 &&&& Y_0
  }
$$

satisfying the properties of a [[crossed profunctor]], and in addition such that the NE-SW sequence is exact i.e. a (nonabelian in general) [[group extension]] sequence.

## Related concepts

Butterflies corresponds to weak functors between the corresponding $2$-[[2-group|groups]]. A butterfly is flippable, or reversible, if both diagonals are [[group extension]]s. There is also a straightforward generalization for $2$-group [[stack]]s.

Under the correspondence between [[crossed module]]s and [[internal category|categories internal]] to [[Grp]], butterflies are precisely the saturated [[anafunctor|anafunctors]] internal to $Grp$, using the [[Grothendieck pretopology]] of surjective homomorphisms.

In fact under the same correspondence, butterflies correspond to
locally representable [[profunctor|profunctors]], while split butterflies give representable profunctors. This extends to the intrinsic setting, where the base category is semi-abelian (see references below).

## References

Butterfly between strict 2-groups have been introduced in

* [[Behrang Noohi]], _On weak maps between 2-groups_, [arXiv](http://arxiv.org/abs/math/0506313)

* [[Behrang Noohi]], E. Aldrovandi, _Butterflies I: morphisms of 2-group stacks_, [arXiv](http://arxiv.org/abs/0808.3627), Advances in Mathematics, 221, (2009), 687–773.

* [[Behrang Noohi]], E. Aldrovandi, _Butterflies II: Torsors for 2-group stacks_, [arXiv](http://arxiv.org/abs/0909.3350),Advances in Mathematics, 225, (2010), 922-976.

Notice that a "[[torsor]] over a [[2-group]] [[stack]]" is another term for [[principal 2-bundle]] ([[n-truncated object of an (infinity,1)-category|2-truncated]] [[principal ∞-bundle]]) in a [[(∞,1)-topos]] of [[∞-stack]]s over some [[site]].

* Omar Abbad, Sandra Mantovani, Giuseppe Metere, Enrico M. Vitale, _Butterflies in a semi-Abelian context_, [arxiv/1104.4275](http://arxiv.org/abs/1104.4275)

[[!redirects pappilon]]
[[!redirects papillon]]


BV-algebra

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

#Contents#
* table of contents
{:toc}

## Idea

For the actual relation to [[BV-complexes]] see at _[[relation between BV and BD]]_.

## Definition

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

A **Batalin-Vilkovisky algebra** or **BV-algebra** for short is

* a [[Gerstenhaber algebra]] $(A, \cdot, [-,-])$

* equipped with a unary [[linear operator]] $\Delta : A \to A$ of the same degree as the bracket

* such that

  1. $\Delta$ is a [[derivation]] for $[-,-]$;

  1. $[-,-]$ is the failure of $\Delta$ being a derivation for $\cdot$:

     $$
        [-,-] = \Delta \circ (-\cdot -) - (\Delta(-) \cdot - ) - (- \cdot \Delta(-))
      \,.
     $$

=--

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

A **$(n+1)$-BV algebra** is a similar structure with a BV-operator being of degree $n$ if $n$ is odd, and of degree $n/2$ if it is even.

=--

See ([Cohen-Voronov, def. 5.3.1, theorem 2.1.3](#CohenVoronov)) for details.

## Properties

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

The [[operad]] for BV-algebras is the [[homology]] of the [[framed little 2-disk operad]].

=--

This is due to ([Getzler](#Getzler))

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

The [[homology]] of an [[algebra over an operad]] over the [[framed little n-disk operad]] has a natural structure of an $(n+1)$-BV-algebra.

=--

This appears as ([CohenVoronov, theorem 5.3.3](#CohenVoronov)). The full [[homology]] of the [[framed little n-disk operad]] is described by ([SalvatoreWahl, theorem 5.4](#SalvatoreWahl)).

## Examples

* On a [[smooth manifold]] $X$ with a [[volume form]] $\omega$ the operation of contraction with the volume form gives an [[isomorphism]] between [[differential form]]s and [[multivector field]]s. Under this isomorphism the de Rham differential on forms becomes a BV-operator on [[multivector field]]s. See there for more details.

Multivector fields may be identified with [[Hochschild cohomology]] in good cases (the [[Hochschild-Kostant-Rosenberg theorem]]). So the next example is a generalization of the previous one:

* {#ExampleHochschildCohomology} under good conditions, [[Hochschild cohomology]] (see there for details) carries a BV-algebra structure (eg. [Felix, Thomas & Vigué-Poirrier 2004](#FelixThomasVigue-Poirrier04); [Menichi 2009](#Menichi09); [Tradler 2008](#Tradler08))

## Related concepts

* [[Batalin-Vilkovisky module]]

* [[Poisson n-algebra]]

* [[homotopy BV-algebra]]

  * [[dg-BV-algebra]]

* [[BD-algebra]]

## References

The identification of BV-algebras as algebras over the homology of the framed little disk operad is due to

* [[Ezra Getzler]], _Batalin-Vilkovisky algebras and two-dimensional topological field theories_ , Comm. Math. Phys. 159 (1994), no. 2, 265–285. ([arXiv](http://arxiv.org/abs/hep-th/9212043))
{#Getzler}

The generalization to higher dimensional framed little disks is discussed in

* {#CohenVoronov} [[Ralph Cohen]], [[Alexander Voronov]], _Notes on string topology_ ([arXiv:math/0503625](http://arxiv.org/abs/math/0503625))

* {#SalvatoreWahl} [[Paolo Salvatore]], [[Nathalie Wahl]], _Framed discs operads and Batalin-Vilkovisky algebras_ ([pdf](http://www.math.ku.dk/~wahl/SalvatoreWahl.pdf))

There are examples coming from Lagrangian intersection theory:

* Vladimir Baranovsky, Victor Ginzburg, _Gerstenhaber-Batalin-Vilkoviski structures on coisotropic intersections_, [arxiv/0907.0037](http://arxiv.org/abs/0907.0037)

The BV-algebra structure on [[multivector fields]] on an [[oriented]] [[smooth manifold]] is discussed for instance in:

* {#CattaneoFiorenzaLongoni} [[Alberto Cattaneo]], [[Domenico Fiorenza]], Riccardo Longoni, Section 2 of: _On the Hochschild-Kostant-Rosenberg map for graded manifolds_, Int. Math. Res. Not. 2005, no. 62, 3899-3918 ([arXiv:math/0503380](http://arxiv.org/abs/math/0503380))

and

* [[Claude Roger]], p. 6 of: *Gerstenhaber and Batalin-Vilkovisky algebras*, Archivum mathematicum, Volume 45 (2009), No. 4 ([pdf](http://www.emis.de/journals/AM/09-4/roger.pdf))

The BV-algebra structure on  [[Hochschild cohomology]]:

* {#FelixThomasVigue-Poirrier04} [[Yves Félix]], [[Jean-Claude Thomas]], [[Micheline Vigué-Poirrier]]: *The Hochschild cohomology of a closed manifold* Publ. Math. IHÉS Sci. **99** (2004) 235-252 [[numdam:PMIHES_2004__99__235_0/](http://www.numdam.org/item/PMIHES_2004__99__235_0)]

* {#Menichi09} [[Luc Menichi]], *Batalin-Vilkovisky algebra structures on Hochschild cohomology*, Bulletin de la Société Mathématique de France, **137** 2 (2009) 277-295
[[arXiv:0711.1946](https://arxiv.org/abs/0711.1946), [numdam:BSMF_2009__137_2_277_0/](http://www.numdam.org/item/BSMF_2009__137_2_277_0/), [pdf](http://math.univ-angers.fr/perso/lmenichi/BV_Hochschild.pdf)]

* {#Tradler08} [[Thomas Tradler]], *The BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products*, Annales de l'institut Fourier **58** 7 (2008) 2351-2379 [[arXiv:math/0210150](http://arxiv.org/abs/math/0210150)]

There is a prominent class of examples coming from [[Lie-Rinehart algebras]]:

* [[Johannes Huebschmann]], _Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras_, Annales de l'institut Fourier __48__:2 (1998) 425-440 [eudml](https://eudml.org/doc/75288)

[[!redirects BV-algebras]]

[[!redirects Batalin-Vilkovisky algebra]]
[[!redirects Batalin-Vilkovisky algebras]]