Construct in mathematics
In mathematics, a gerbe (; ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles 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 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.
Definitions.
Gerbes on a topological space.
A gerbe on a topological space formula_1 is a stack formula_2 of groupoids over formula_1 that is "locally non-empty" (each point formula_4 has an open neighbourhood formula_5 over which the section category formula_6 of the gerbe is not empty) and "transitive" (for any two objects formula_7 and formula_8 of formula_9 for any open set formula_10, there is an open covering formula_11 of formula_10 such that the restrictions of formula_7 and formula_8 to each formula_15 are connected by at least one morphism).
A canonical example is the gerbe formula_16 of principal bundles with a fixed structure group formula_17: the section category over an open set formula_10 is the category of principal formula_17-bundles on formula_10 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 formula_21 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. Given a site formula_22 a formula_22-gerbe formula_24 is a category fibered in groupoids formula_25 such that
Note that for a site formula_22 with a final object formula_33, a category fibered in groupoids formula_25 is a formula_22-gerbe admits a local section, meaning satisfies the first axiom, if formula_36.
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 formula_37 for a suitable covering formula_38 of a space formula_39 gives the isomorphism classes of principal formula_24-bundles over formula_39, what does the iterated cohomology functor formula_42 represent? Meaning, we are gluing together the groups formula_43 via some one cocycle. Gerbes are a technical response for this question: they give geometric representations of elements in the higher cohomology group formula_44. It is expected this intuition should hold for higher gerbes.
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 formula_45, called a band. For a gerbe formula_2 on a site formula_22, an object formula_48, and an object formula_49, the automorphism group of a gerbe is defined as the automorphism group formula_50. Notice this is well defined whenever the automorphism group is always the same. Given a covering formula_51, there is an associated classformula_52representing the isomorphism class of the gerbe formula_2 banded by formula_54.
For example, in topology, many examples of gerbes can be constructed by considering gerbes banded by the group formula_55. As the classifying space formula_56 is the second Eilenberg–Maclane space for the integers, a bundle gerbe banded by formula_55 on a topological space formula_39 is constructed from a homotopy class of maps informula_59,which is exactly the third singular homology group formula_60. It has been found that all gerbes representing torsion cohomology classes in formula_60 are represented by a bundle of finite dimensional algebras formula_62 for a fixed complex vector space formula_63. In addition, the non-torsion classes are represented as infinite-dimensional principal bundles formula_64 of the projective group of unitary operators on a fixed infinite dimensional separable Hilbert space formula_65. Note this is well defined because all separable Hilbert spaces are isomorphic to the space of square-summable sequences formula_66.
The homotopy-theoretic interpretation of gerbes comes from looking at the homotopy fiber squareformula_67analogous to how a line bundle comes from the homotopy fiber squareformula_68where formula_69, giving formula_70 as the group of isomorphism classes of line bundles on formula_1.
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 formula_39pg 3. Given a cover formula_73 of formula_39 there is the Cech groupoid defined asformula_75with source and target maps given by the inclusionsformula_76and the space of composable arrows is justformula_77Then a degree 2 cohomology class formula_78 is just a mapformula_79We can then form a non-commutative C*-algebra formula_80, which is associated to the set of compact supported complex valued functions of the spaceformula_81It has a non-commutative product given byformula_82where the cohomology class formula_83 twists the multiplication of the standard formula_84-algebra product.
Algebraic geometry.
Let formula_85 be a variety over an algebraically closed field formula_86, formula_24 an algebraic group, for example formula_88. Recall that a "G"-torsor over formula_85 is an algebraic space formula_90 with an action of formula_24 and a map formula_92, such that locally on formula_85 (in étale topology or fppf topology) formula_94 is a direct product formula_95. A G"-gerbe over "M may be defined in a similar way. It is an Artin stack formula_96 with a map formula_97, such that locally on "M" (in étale or fppf topology) formula_94 is a direct product formula_99. Here formula_100 denotes the classifying stack of formula_24, i.e. a quotient formula_102 of a point by a trivial formula_24-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 spaces of formula_96 and formula_85 are the same, but in formula_96 each point is equipped with a stabilizer group isomorphic to formula_24.
From two-term complexes of coherent sheaves.
Every two-term complex of coherent sheavesformula_108on a scheme formula_109 has a canonical sheaf of groupoids associated to it, where on an open subset formula_110 there is a two-term complex of formula_111-modulesformula_112giving a groupoid. It has objects given by elements formula_113 and a morphism formula_114 is given by an element formula_115 such thatformula_116In order for this stack to be a gerbe, the cohomology sheaf formula_117 must always have a section. This hypothesis implies the category constructed above always has objects. Note this can be applied to the situation of comodules over Hopf-algebroids to construct algebraic models of gerbes over affine or projective stacks (projectivity if a graded Hopf-algebroid is used). In addition, two-term spectra from the stabilization of the derived category of comodules of Hopf-algebroids formula_118 with formula_119 flat over formula_120 give additional models of gerbes that are non-strict.
Moduli stack of stable bundles on a curve.
Consider a smooth projective curve formula_121 over formula_86 of genus formula_123. Let formula_124 be the moduli stack of stable vector bundles on formula_121 of rank formula_126 and degree formula_127. It has a coarse moduli space formula_128, which is a quasiprojective variety. These two moduli problems parametrize the same objects, but the stacky version remembers automorphisms of vector bundles. For any stable vector bundle formula_129 the automorphism group formula_130 consists only of scalar multiplications, so each point in a moduli stack has a stabilizer isomorphic to formula_88. It turns out that the map formula_132 is indeed a formula_88-gerbe in the sense above. It is a trivial gerbe if and only if formula_126 and formula_127 are coprime.
Root stacks.
Another class of gerbes can be found using the construction of root stacks. Informally, the formula_126-th root stack of a line bundle formula_137 over a scheme is a space representing the formula_126-th root of formula_54 and is denotedformula_140pg 52 The formula_126-th root stack of formula_54 has the propertyformula_143as gerbes. It is constructed as the stackformula_144sending an formula_1-scheme formula_146 to the category whose objects are line bundles of the formformula_147and morphisms are commutative diagrams compatible with the isomorphisms formula_148. This gerbe is banded by the algebraic group of roots of unity formula_149, where on a cover formula_146 it acts on a point formula_151 by cyclically permuting the factors of formula_85 in formula_153. Geometrically, these stacks are formed as the fiber product of stacksformula_154where the vertical map of formula_155 comes from the Kummer sequenceformula_156This is because formula_157 is the moduli space of line bundles, so the line bundle formula_137 corresponds to an object of the category formula_159 (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 formula_160 be a section. Then the formula_126-th root stack of the pair formula_162 is defined as the lax 2-functorformula_163sending an formula_1-scheme formula_146 to the category whose objects are line bundles of the formformula_166and 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. Locally, we may assume formula_167 and the line bundle formula_54 is trivial, hence any section formula_169 is equivalent to taking an element formula_170. Then, the stack is given by the stack quotientformula_171withformula_172If formula_173 then this gives an infinitesimal extension of formula_174.
Examples throughout algebraic geometry.
These and more general kinds of gerbes arise in several contexts as both geometric spaces and as formal bookkeeping tools:
History.
Gerbes first appeared in the context of algebraic geometry. They were subsequently developed in a more traditional geometric framework by Brylinski . 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 Murray and called bundle gerbes. Essentially they are a smooth version of abelian gerbes belonging more to the hierarchy starting with principal bundles 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 gerbes.
References.
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