Gerbe
Encyclopedia
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 a generalization of principal bundle
s to the setting of 2-categories. 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.
X is a stack
G of groupoid
s over X which is locally non-empty (each point in X has an open neighbourhood U over which the section category G(U) of the gerbe is not empty) and transitive (for any two objects a and b of G(U) for any open set U, there is an open covering {Vi}i of U such that the restrictions of a and b to each Vi are connected by at least one morphism).
A canonical example is the gerbe of principal bundles with a fixed structure group H: the section category over an open set U is the category of principal H-bundles on U 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 X x H over X 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.
. 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 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 gerbes.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a gerbe is a construct in homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. Gerbes were introduced by Jean Giraud
Jean Giraud (mathematician)
Jean Giraud was a French mathematician, a student of Alexander Grothendieck and the author of the book "Cohomologie non abélienne" ....
following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
in degree 2. They can be seen as a generalization of principal bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
s to the setting of 2-categories. Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. In addition, special cases of gerbes have been used more recently in differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them.
Gerbe
A gerbe on a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X is a stack
Stack (descent theory)
In mathematics a stack is a concept used to formalise some of the main constructions of descent theory.Descent theory is concerned with generalisations of situations where geometrical objects can be "glued together" when they are isomorphic when restricted to intersections of the sets in an open...
G of groupoid
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...
s over X which is locally non-empty (each point in X has an open neighbourhood U over which the section category G(U) of the gerbe is not empty) and transitive (for any two objects a and b of G(U) for any open set U, there is an open covering {Vi}i of U such that the restrictions of a and b to each Vi are connected by at least one morphism).
A canonical example is the gerbe of principal bundles with a fixed structure group H: the section category over an open set U is the category of principal H-bundles on U 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 X x H over X 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.
Algebraic geometry
- Azumaya algebraAzumaya algebraIn mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring...
s - Deformations of infinitesimal thickenings
- Twisted forms of projective varieties
- Fiber functors for motives
Differential geometry
-
and 
-gerbes: Jean-Luc BrylinskiJean-Luc BrylinskiJean-Luc Brylinski is a French-American mathematician. Educated at the Lycée Pasteur and the Ecole Normale Supérieure in Paris, after an appointment as researcher with the C. N. R. S., he became a Professor of Mathematics at Pennsylvania State University. He proved the Kazhdan–Lusztig conjectures...
's approach
History
Gerbes first appeared in the context of algebraic geometryAlgebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. 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
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
classes.
A more specialised notion of gerbe was introduced by Murray and called bundle gerbes. Essentially they are a smooth
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
version of abelian gerbes belonging more to the hierarchy starting with principal bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
s than sheaves. Bundle gerbes have been used in gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
and also string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
. Current work by others is developing a theory of non-abelian bundle gerbes.
External links
- What is a Gerbe?, by Nigel HitchinNigel HitchinNigel Hitchin is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.-Academic career:...
in Notices of the AMS - Bundle gerbes, Michael Murray.