Editing Gerbe

{{Short description|Construct in mathematics}}
{{other uses}}

In [[mathematics]], a '''gerbe''' ({{IPAc-en|dʒ|ɜr|b}}; {{IPA|fr|ʒɛʁb|lang}}) is a construct in [[homological algebra]] and [[topology]]. Gerbes were introduced by [[Jean Giraud (mathematician)|Jean Giraud]] {{harv|Giraud|1971}} following ideas of [[Alexandre Grothendieck]] as a tool for non-commutative [[cohomology]] in degree 2. They can be seen as an analogue of [[fibre bundle]]s where the fibre is the [[classifying stack]] of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of [[Deformation theory|deformation]] questions especially in modern [[algebraic geometry]]. In addition, special cases of gerbes have been used more recently in [[differential topology]] and [[differential geometry]] to give alternative descriptions to certain [[cohomology classes]] and additional structures attached to them.

"Gerbe" is a French (and archaic English) word that literally means [[wheat]] [[sheaf (agriculture)|sheaf]].

== Definitions ==

===Gerbes on a topological space===

A gerbe on a [[topological space]] <math>S</math><ref>{{Cite book|url=https://www.worldcat.org/oclc/233973513|title=Basic bundle theory and K-cohomology invariants|date=2008|publisher=Springer|others=Husemöller, Dale.|isbn=978-3-540-74956-1|location=Berlin|oclc=233973513}}</ref>{{rp|318}} is a [[stack (mathematics)|stack]] <math>\mathcal{X}</math> of [[groupoid]]s over <math>S</math> that is ''locally non-empty'' (each point <math>p \in S</math> has an open neighbourhood <math>U_p</math> over which the [[Section (category theory)|section category]] <math>\mathcal{X}(U_p)</math> of the gerbe is not empty) and ''transitive'' (for any two objects <math>a</math> and <math>b</math> of <math>\mathcal{X}(U)</math> for any open set <math>U</math>, there is an open covering <math>\mathcal{U} = \{U_i \}_{i \in I}</math> of <math>U</math> such that the restrictions of <math>a</math> and <math>b</math> to each <math>U_i</math> are connected by at least one morphism).

A canonical example is the gerbe <math>BH</math> of [[principal bundles]] with a fixed [[structure group]] <math>H</math>: the section category over an open set <math>U</math> is the category of principal <math>H</math>-bundles on <math>U</math> with isomorphism as morphisms (thus the category is a groupoid). As principal bundles glue together (satisfy the descent condition), these groupoids form a stack. The trivial bundle <math>X \times H \to X</math> shows that the local non-emptiness condition is satisfied, and finally as principal bundles are locally trivial, they become isomorphic when restricted to sufficiently small open sets; thus the transitivity condition is satisfied as well.

=== Gerbes on a site ===
The most general definition of gerbes are defined over a [[Site (mathematics)|site]]. Given a site <math>\mathcal{C}</math> a <math>\mathcal{C}</math>-gerbe <math>G</math><ref name="stacks.math.columbia.edu">{{Cite web|title=Section 8.11 (06NY): Gerbes—The Stacks project|url=https://stacks.math.columbia.edu/tag/06NY|access-date=2020-10-27|website=stacks.math.columbia.edu}}</ref><ref>{{Cite book|last=Giraud, J. (Jean)|url=https://www.worldcat.org/oclc/186709|title=Cohomologie non abélienne.|date=1971|publisher=Springer-Verlag|isbn=3-540-05307-7|location=Berlin|oclc=186709}}</ref>{{rp|129}} is a category fibered in groupoids <math>G \to \mathcal{C}</math> such that

# There exists a refinement<ref>{{Cite web|title=Section 7.8 (00VS): Families of morphisms with fixed target—The Stacks project|url=https://stacks.math.columbia.edu/tag/00VS|access-date=2020-10-27|website=stacks.math.columbia.edu}}</ref> <math>\mathcal{C}'</math> of <math>\mathcal{C}</math> such that for every object <math>S \in \text{Ob}(\mathcal{C}')</math> the associated fibered category <math>G_S</math> is non-empty
# For every <math>S \in \text{Ob}(\mathcal{C})</math> any two objects in the fibered category <math>G_S</math> are locally isomorphic
Note that for a site <math>\mathcal{C}</math> with a final object <math>e</math>, a category fibered in groupoids <math>G \to \mathcal{C}</math> is a <math>\mathcal{C}</math>-gerbe admits a local section, meaning satisfies the first axiom, if <math>\text{Ob}(G_e) \neq \varnothing</math>.

==== Motivation for gerbes on a site ====
One of the main motivations for considering gerbes on a site is to consider the following naive question: if the Cech cohomology group <math>H^1(\mathcal{U},G)</math> for a suitable covering <math>\mathcal{U} = \{U_i\}_{i \in I}</math> of a space <math>X</math> gives the isomorphism classes of principal <math>G</math>-bundles over <math>X</math>, what does the iterated cohomology functor <math>H^1(-,H^1(-,G))</math> represent? Meaning, we are gluing together the groups <math>H^1(U_i,G)</math> via some one cocycle. Gerbes are a technical response for this question: they give geometric representations of elements in the higher cohomology group <math>H^2(\mathcal{U},G)</math>. It is expected this intuition should hold for [[higher gerbe]]s.

== Cohomological classification ==
One of the main theorems concerning gerbes is their cohomological classification whenever they have automorphism groups given by a fixed sheaf of abelian groups <math>\underline{L}</math>,<ref>{{Cite web|title=Section 21.11 (0CJZ): Second cohomology and gerbes—The Stacks project|url=https://stacks.math.columbia.edu/tag/0CJZ|access-date=2020-10-27|website=stacks.math.columbia.edu}}</ref><ref name="stacks.math.columbia.edu"/> called a band. For a gerbe <math>\mathcal{X}</math> on a site <math>\mathcal{C}</math>, an object <math>U \in \text{Ob}(\mathcal{C})</math>, and an object <math>x \in \text{Ob}(\mathcal{X}(U))</math>, the automorphism group of a gerbe is defined as the automorphism group <math>L = \underline{\text{Aut}}_{\mathcal{X}(U)}(x)</math>. Notice this is well defined whenever the automorphism group is always the same. Given a covering <math>\mathcal{U} = \{U_i \to X \}_{i \in I}</math>, there is an associated class<blockquote><math>c(\underline{L}) \in H^3(X,\underline{L})</math></blockquote>representing the [[isomorphism class]] of the gerbe <math>\mathcal{X}</math> banded by <math>L</math>.

For example, in topology, many examples of gerbes can be constructed by considering gerbes banded by the group <math>U(1)</math>. As the classifying space <math>B(U(1)) = K(\mathbb{Z},2)</math> is the second [[Eilenberg–MacLane space|Eilenberg–Maclane]] space for the integers, a bundle gerbe banded by <math>U(1)</math> on a topological space <math>X</math> is constructed from a homotopy class of maps in<blockquote><math>[X, B^2(U(1))] = [X,K(\mathbb{Z},3)]</math>,</blockquote>which is exactly the third [[singular homology]] group <math>H^3(X,\mathbb{Z})</math>. It has been found<ref>{{cite arXiv|last=Karoubi|first=Max|date=2010-12-12|title=Twisted bundles and twisted K-theory|class=math.KT|eprint=1012.2512}}</ref> that all gerbes representing torsion cohomology classes in <math>H^3(X,\mathbb{Z})</math> are represented by a bundle of finite dimensional algebras <math>\text{End}(V)</math> for a fixed complex vector space <math>V</math>. In addition, the non-torsion classes are represented as infinite-dimensional principal bundles <math>PU(\mathcal{H})</math> of the projective group of unitary operators on a fixed infinite dimensional [[Separable space|separable]] [[Hilbert space]] <math>\mathcal{H}</math>. Note this is well defined because all separable Hilbert spaces are isomorphic to the space of square-summable sequences <math>\ell^2</math>.

The homotopy-theoretic interpretation of gerbes comes from looking at the [[homotopy fiber square]]<blockquote><math>\begin{matrix}
\mathcal{X} & \to & * \\
\downarrow &  & \downarrow \\
S & \xrightarrow{f} & B^2U(1)
\end{matrix}</math></blockquote>analogous to how a line bundle comes from the homotopy fiber square<blockquote><math>\begin{matrix}
L & \to & * \\
\downarrow &  & \downarrow \\
S & \xrightarrow{f} & BU(1)
\end{matrix}</math></blockquote>where <math>BU(1) \simeq K(\mathbb{Z},2)</math>, giving <math>H^2(S,\mathbb{Z})</math> as the group of isomorphism classes of line bundles on <math>S</math>.

== Examples ==

=== C*-algebras ===
There are natural examples of Gerbes that arise from studying the algebra of compactly supported complex valued functions on a paracompact space <math>X</math><ref>{{cite arXiv |last1=Block |first1=Jonathan |last2=Daenzer |first2=Calder |date=2009-01-09 |title=Mukai duality for gerbes with connection |class=math.QA |eprint=0803.1529 }}</ref><sup>pg 3</sup>. Given a cover <math>\mathcal{U} = \{U_i\}</math> of <math>X</math> there is the Cech groupoid defined as<blockquote><math>\mathcal{G} = \left\{ \coprod_{i,j}U_{ij} \rightrightarrows \coprod U_i \right\} </math></blockquote>with source and target maps given by the inclusions<blockquote><math>\begin{align}
s: U_{ij} \hookrightarrow U_j \\
t: U_{ij} \hookrightarrow U_i
\end{align}</math></blockquote>and the space of composable arrows is just<blockquote><math>\coprod_{i,j,k}U_{ijk}</math></blockquote>Then a degree 2 cohomology class <math>\sigma \in H^2(X;U(1))</math> is just a map<blockquote><math>\sigma: \coprod U_{ijk} \to U(1)</math></blockquote>We can then form a non-commutative [[C*-algebra]] <math>C_c(\mathcal{G}(\sigma))</math>, which is associated to the set of compact supported complex valued functions of the space<blockquote><math>\mathcal{G}_1 = \coprod_{i,j}U_{ij}</math></blockquote>It has a non-commutative product given by<blockquote><math>a* b(x,i,k) := \sum_j a(x,i,j)b(x,j,k)\sigma(x,i,j,k)</math></blockquote>where the cohomology class <math>\sigma</math> twists the multiplication of the standard <math>C^*</math>-algebra product.

===Algebraic geometry ===
Let <math>M</math> be a [[algebraic variety|variety]] over an [[algebraically closed field]] <math>k</math>, <math>G</math> an [[algebraic group]], for example <math>\mathbb{G}_m</math>. Recall that a [[Torsor (algebraic geometry)|''G''-torsor]] over <math>M</math> is an [[algebraic space]] <math>P</math> with an action of <math>G</math> and a map <math>\pi:P\to M</math>, such that locally on <math>M</math> (in [[étale topology]] or [[fppf topology]]) <math>\pi</math> is a direct product <math>\pi|_U:G\times U\to U</math>. A '''''G''-gerbe over ''M''''' may be defined in a similar way. It is an [[Artin stack]] <math>\mathcal{M}</math> with a map <math>\pi\colon\mathcal{M} \to M</math>, such that locally on ''M'' (in étale or fppf topology) <math>\pi</math> is a direct product <math>\pi|_U\colon \mathrm{B}G \times U \to U</math>.<ref>{{cite journal| first1=Dan | last1 = Edidin | first2 = Brendan | last2 = Hassett|first3 = Andrew | last3 = Kresch | first4 = Angelo | last4 = Vistoli | title = Brauer groups and quotient stacks | journal = [[American Journal of Mathematics]] | year = 2001 | volume = 123 | issue = 4 | pages = 761–777 | arxiv=math/9905049 | doi=10.1353/ajm.2001.0024| s2cid = 16541492 }}</ref> Here <math>BG</math> denotes the [[classifying stack]] of <math>G</math>, i.e. a quotient <math>[ * / G ]</math> of a point by a trivial <math>G</math>-action. There is no need to impose the compatibility with the group structure in that case since it is covered by the definition of a stack. The underlying [[topological space]]s of <math>\mathcal{M}</math> and <math>M</math> are the same, but in <math>\mathcal{M}</math> each point is equipped with a stabilizer group isomorphic to <math>G</math>.

==== From two-term complexes of coherent sheaves ====
Every two-term complex of coherent sheaves<blockquote><math>\mathcal{E}^\bullet = [\mathcal{E}^{-1} \xrightarrow{d} \mathcal{E}^0]</math></blockquote>on a scheme <math>X \in \text{Sch}</math> has a canonical sheaf of groupoids associated to it, where on an open subset <math>U \subseteq X</math> there is a two-term complex of <math>X(U)</math>-modules<blockquote><math>\mathcal{E}^{-1}(U) \xrightarrow{d} \mathcal{E}^0(U)</math></blockquote>giving a groupoid. It has objects given by elements <math>x \in \mathcal{E}^0(U)</math> and a morphism <math>x \to x'</math> is given by an element <math>y \in \mathcal{E}^{-1}(U)</math> such that<blockquote><math>dy + x = x' </math></blockquote>In order for this stack to be a gerbe, the cohomology sheaf <math>\mathcal{H}^0(\mathcal{E})</math> must always have a section. This hypothesis implies the category constructed above always has objects. Note this can be applied to the situation of [[Comodule over a Hopf algebroid|comodules over Hopf-algebroids]] to construct algebraic models of gerbes over affine or projective stacks (projectivity if a graded [[Hopf algebroid|Hopf-algebroid]] is used). In addition, two-term spectra from the stabilization of the [[derived category]] of comodules of Hopf-algebroids <math>(A,\Gamma)</math> with <math>\Gamma</math> flat over <math>A</math> give additional models of gerbes that are [[Abelian 2-group|non-strict]].

==== Moduli stack of stable bundles on a curve ====
Consider a smooth [[projective variety|projective]] [[algebraic curve|curve]] <math>C</math> over <math>k</math> of genus <math>g > 1</math>. Let <math>\mathcal{M}^s_{r, d}</math> be the [[moduli space|moduli stack]] of [[stable vector bundle]]s on <math>C</math> of rank <math>r</math> and degree <math>d</math>. It has a [[Moduli space#Coarse moduli spaces|coarse moduli space]] <math>M^s_{r, d}</math>, which is a [[quasiprojective variety]]. These two moduli problems parametrize the same objects, but the stacky version remembers [[automorphism]]s of vector bundles. For any stable vector bundle <math>E</math> the automorphism group <math>Aut(E)</math> consists only of scalar multiplications, so each point in a moduli stack has a stabilizer isomorphic to <math>\mathbb{G}_m</math>. It turns out that the map <math>\mathcal{M}^s_{r, d} \to M^{s}_{r, d}</math> is indeed a <math>\mathbb{G}_m</math>-gerbe in the sense above.<ref>{{cite journal|last1=Hoffman|first1=Norbert|year=2010|title=Moduli stacks of vector bundles on curves and the King-Schofield rationality proof|journal=Cohomological and Geometric Approaches to Rationality Problems|series=Progress in Mathematics |volume=282 |pages=133–148|doi=10.1007/978-0-8176-4934-0_5|arxiv=math/0511660|isbn=978-0-8176-4933-3|s2cid=5467668}}</ref> It is a trivial gerbe if and only if <math>r</math> and <math>d</math> are [[coprime]].

==== Root stacks ====
Another class of gerbes can be found using the construction of root stacks. Informally, the <math>r</math>-th root stack of a line bundle <math>L \to S</math> over a [[Scheme (mathematics)|scheme]] is a space representing the <math>r</math>-th root of <math>L</math> and is denoted<blockquote><math>\sqrt[r]{L/S}.\,</math><ref name=":0">{{cite arXiv|last1=Abramovich|first1=Dan|last2=Graber|first2=Tom|last3=Vistoli|first3=Angelo|date=2008-04-13|title=Gromov-Witten theory of Deligne-Mumford stacks|eprint=math/0603151}}</ref><sup>pg 52</sup> </blockquote>The <math>r</math>-th root stack of <math>L</math> has the property<blockquote><math>\bigotimes^r\sqrt[{r}]{L/S} \cong L</math></blockquote>as gerbes. It is constructed as the stack<blockquote><math>\sqrt[r]{L/S}: (\operatorname{Sch}/S)^{op} \to \operatorname{Grpd}</math></blockquote>sending an <math>S</math>-scheme <math>T \to S</math> to the category whose objects are line bundles of the form<blockquote><math>\left\{
(M \to T,\alpha_M) : \alpha_M: M^{\otimes r} \xrightarrow{\sim} L\times_ST
\right\}</math></blockquote>and morphisms are commutative diagrams compatible with the isomorphisms <math>\alpha_M</math>. This gerbe is banded by the [[algebraic group]] of roots of unity <math>\mu_r</math>, where on a cover <math>T \to S</math> it acts on a point <math>(M\to T,\alpha_M)</math> by cyclically permuting the factors of <math>M</math> in <math>M^{\otimes r}</math>. Geometrically, these stacks are formed as the fiber product of stacks<blockquote><math>\begin{matrix}
X\times_{B\mathbb{G}_m} B\mathbb{G}_m & \to & B\mathbb{G}_m \\
\downarrow & & \downarrow \\
X & \to & B\mathbb{G}_m
\end{matrix}</math></blockquote>where the vertical map of <math>B\mathbb{G}_m \to B\mathbb{G}_m</math> comes from the [[Kummer sequence]]<blockquote><math>1 \xrightarrow{} \mu_r \xrightarrow{} \mathbb{G}_m \xrightarrow{ (\cdot)^r} \mathbb{G}_m \xrightarrow{} 1</math></blockquote>This is because <math>B\mathbb{G}_m</math> is the moduli space of line bundles, so the line bundle <math>L \to S</math> corresponds to an object of the category <math>B\mathbb{G}_m(S)</math> (considered as a point of the moduli space).

===== Root stacks with sections =====
There is another related construction of root stacks with sections. Given the data above, let <math>s: S \to L</math> be a section. Then the <math>r</math>-th root stack of the pair <math>(L\to S,s)</math> is defined as the lax 2-functor<ref name=":0" /><ref name=":1">{{cite journal|last=Cadman|first=Charles|year=2007|title=Using stacks to impose tangency conditions on curves|url=https://www.charlescadman.com/pdf/stacks.pdf|journal=Amer. J. Math.|volume=129|issue=2|pages=405–427|arxiv=math/0312349|doi=10.1353/ajm.2007.0007|s2cid=10323243}}</ref><blockquote><math>\sqrt[r]{(L,s)/S}: (\operatorname{Sch}/S)^{op} \to \operatorname{Grpd}</math></blockquote>sending an <math>S</math>-scheme <math>T \to S</math> to the category whose objects are line bundles of the form<blockquote><math>\left\{
(M \to T,\alpha_M, t) :
\begin{align}
&\alpha_M: M^{\otimes r} \xrightarrow{\sim} L\times_ST \\
& t \in \Gamma(T,M) \\
&\alpha_M(t^{\otimes r}) = s
\end{align}
\right\}</math></blockquote>and morphisms are given similarly. These stacks can be constructed very explicitly, and are well understood for affine schemes. In fact, these form the affine models for root stacks with sections.<ref name=":1" />{{rp|4}} Locally, we may assume <math>S = \text{Spec}(A)</math> and the line bundle <math>L</math> is trivial, hence any section <math>s</math> is equivalent to taking an element <math>s \in A</math>. Then, the stack is given by the stack quotient<blockquote><math>\sqrt[r]{(L,s)/S} = [\text{Spec}(B)/\mu_r]</math><ref name=":1" />{{rp|9}}</blockquote>with<blockquote><math>B = \frac{A[x]}{x^r - s}</math></blockquote>If <math>s = 0</math> then this gives an infinitesimal extension of <math>[\text{Spec}(A)/\mu_r]</math>.

==== Examples throughout algebraic geometry ====
These and more general kinds of gerbes arise in several contexts as both geometric spaces and as formal bookkeeping tools:
* [[Azumaya algebra]]s
* Deformations of infinitesimal thickenings
* Twisted forms of projective varieties
* [[Fiber functor]]s for [[motive (algebraic geometry)|motive]]s

===Differential geometry===
* <math>H^3(X,\mathbb{Z})</math> and <math>\mathcal{O}_X^*</math>-gerbes: [[Jean-Luc Brylinski]]'s approach

== History ==
{{more citations needed section|date=January 2021}}

Gerbes first appeared in the context of [[algebraic geometry]]. They were subsequently developed in a more traditional geometric framework by Brylinski {{harv|Brylinski|1993}}.  One can think of gerbes as being a natural step in a hierarchy of mathematical objects providing geometric realizations of integral [[cohomology]] classes.

A more specialised notion of gerbe was introduced by [[Michael Murray (mathematician)|Murray]] and called [[bundle gerbes]]. Essentially they are a [[smooth function|smooth]] version of abelian gerbes belonging more to the hierarchy starting with [[principal bundle]]s than sheaves. Bundle gerbes have been used in [[gauge theory]] and also [[string theory]]. Current work by others is developing a theory of [[non-abelian bundle gerbe]]s.

== See also ==
*[[Twisted sheaf]]
*[[Azumaya algebra]]
*[[Twisted K-theory]]
*[[Algebraic stack]]
*[[Bundle gerbe]]
*[[String group]]

== References ==
{{reflist}}
*{{citation 
  | last = Giraud
  | first = Jean
  | author-link = Jean Giraud (mathematician)
  | title = Cohomologie non abélienne
  | publisher = [[Springer Science+Business Media|Springer]]
  | year = 1971
  | isbn = 3-540-05307-7
 }}.
*{{citation
  | last = Brylinski
  | first = Jean-Luc
  | author-link = Jean-Luc Brylinski
  | title = Loop space, characteristic classes and geometric quantization
  | publisher = [[Birkhäuser Verlag]]
  | year = 1993
  | isbn = 0-8176-3644-7
  | url-access = registration
  | url = https://archive.org/details/loopspacescharac0000bryl
  }}.

== External links ==

=== Introductory articles ===

* [[arxiv:math/0312175|Constructions with Bundle Gerbes]] - Stuart Johnson
* ''[[arxiv:math/0402318|An Introduction to Gerbes on Orbifolds]]'', Ernesto Lupercio, Bernado Uribe.
* ''[https://www.ams.org/notices/200302/what-is.pdf What is a Gerbe?]'', by [[Nigel Hitchin]] in Notices of the AMS
* ''[[arxiv:dg-ga/9407015|Bundle gerbes]]'', Michael Murray.
* {{cite web
  | last = Moerdijk
  | first = Ieke|authorlink = Ieke Moerdijk
  | title = Introduction to the Language of Stacks and Gerbes
  | url=http://www.math.uu.nl/publications/preprints/1264.ps 
  | access-date = 2007-05-20 }}

=== Gerbes in topology ===

* [[arxiv:math/0401283v1|Homotopy theory of presheaves of simplicial groupoids]], Zhi-Ming Luo

=== Twisted K-theory ===

* [[arxiv:hep-th/0106194v2|Twisted K-theory and K-theory of bundle gerbes]]
* Twisted Bundles and Twisted K-Theory - Karoubi

=== Applications in string theory ===
*[[arxiv:hep-th/9503208|Stable Singularities in String Theory]] - contains examples of gerbes in appendix using the Brauer group
*[[arxiv:hep-th/0012164|Branes on Group Manifolds, Gluon Condensates, and twisted K-theory]]
*[[arxiv:math/9907034|Lectures on Special Lagrangian Submanifolds]] - Very down-to earth introduction with applications to Mirror symmetry
*[[arxiv:math/0209194|The basic gerbe over a compact simple Lie group]] - Gives techniques for describing groups such as the [[String group]] as a gerbe

[[Category:Homological algebra]]
[[Category:Sheaf theory]]