Gerbe

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 which 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 groupsformula_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_59which 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.
Algebraic geometry.
Let formula_72 be a variety over an algebraically closed field formula_73, formula_24 an algebraic group, for example formula_75. Recall that a "G"-torsor over formula_72 is an algebraic space formula_77 with an action of formula_24 and a map formula_79, such that locally on formula_72 (in étale topology or fppf topology) formula_81 is a direct product formula_82. A G"-gerbe over "M may be defined in a similar way. It is an Artin stack formula_83 with a map formula_84, such that locally on "M" (in étale or fppf topology) formula_81 is a direct product formula_86. Here formula_87 denotes the classifying stack of formula_24, i.e. a quotient formula_89 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_83 and formula_72 are the same, but in formula_83 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_95on a scheme formula_96 has a canonical sheaf of groupoids associated to it, where on an open subset formula_97 there is a two-term complex of formula_98-modulesformula_99giving a groupoid. It has objects given by elements formula_100 and a morphism formula_101 is given by an element formula_102 such thatformula_103In order for this stack to be a Gerbe, we must have the cohomology sheaf formula_104 to 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_105 with formula_106 flat over formula_107 give additional models of gerbes which are non-strict.
Moduli stack of stable bundles on a curve.
Consider a smooth projective curve formula_108 over formula_73 of genus formula_110. Let formula_111 be the moduli stack of stable vector bundles on formula_108 of rank formula_113 and degree formula_114. It has a coarse moduli space formula_115, 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_116 the automorphism group formula_117 consists only of scalar multiplications, so each point in a moduli stack has a stabilizer isomorphic to formula_75. It turns out that the map formula_119 is indeed a formula_75-gerbe in the sense above. It is a trivial gerbe if and only if formula_113 and formula_114 are coprime.
Root stacks.
Another class of gerbes can be found using the construction of root stacks. Informally, the formula_113-th root stack of a line bundle formula_124 over a scheme is a space representing the formula_113-th root of formula_54 and is denotedformula_127pg 52 The formula_113-th root stack of formula_54 has the propertyformula_130as gerbes. It is constructed as the stackformula_131sending an formula_1-scheme formula_133 to the category whose objects are line bundles of the formformula_134and morphisms are commutative diagrams compatible with the isomorphisms formula_135. This gerbe is banded by the algebraic group of roots of unity formula_136, where on a cover formula_133 it acts on a point formula_138 by cyclically permuting the factors of formula_72 in formula_140. Geometrically, these stacks are formed as the fiber product of stacksformula_141where the vertical map of formula_142 comes from the Kummer sequenceformula_143This is because formula_144 is the moduli space of line bundles, so the line bundle formula_124 corresponds to an object of the category formula_146 (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_147 be a section. Then the formula_113-th root stack of the pair formula_149 is defined as the lax 2-functorformula_150sending an formula_1-scheme formula_133 to the category whose objects line bundles of the formformula_153and 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. Given an affine scheme formula_154, all line bundles are trivial, hence formula_155 and any section formula_156 is equivalent to taking an element formula_157. Then, the stack is given by the stack quotientformula_158withformula_159If formula_160 then this gives an infinitesimal extension of formula_161.
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.