In mathematics, a gerbe is a construct in homological algebra and topology. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Gerbes were introduced by Jean Giraud (Giraud 1971) following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. There is another famous Jean Giraud, Comics author Jean Giraud is a French Mathematician. Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain They can be seen as a generalization of principal bundles to the setting of 2-categories. In Mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G In Category theory, a 2-category is a category with "morphisms between morphisms" Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with 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. In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry
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Definitions
Gerbe
A gerbe on a topological space X is a stack G of groupoids 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 set V inside U such that the restrictions of a and b to V are connected by at least one morphism). Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics a stack is an Abstract entity used to formalise some of the main concepts of Descent theory. In Mathematics, especially in Category theory and Homotopy theory
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). In Mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G 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.
Examples
Algebraic geometry
- Azumaya algebras
- Deformations of infinitestimal thickenings
- Twisted forms of projective varieties
- Fiber functors for motives
Differential geometry
and 
-gerbes: Jean-Luc Brylinski's approach
History
Gerbes first appeared in the context of algebraic geometry. In Mathematics, an Azumaya algebra is a generalization of Central simple algebras to R -algebras where R need not be a field. Jean-Luc Brylinski (born in 1951 is a French mathematician Educated at the Ecole Normale Supérieure in Paris after an appointment as researcher with the C Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with They were subsequently developed in a more traditional geometric framework by Brylinski (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. In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain
A more specialised notion of gerbe was introduced by Murray and called bundle gerbes. In Mathematics, a bundle gerbe is a geometrical model of certain 1- Gerbes with connection, or equivalently of a 2-class in Deligne cohomology Essentially they are a smooth version of abelian gerbes belonging more to the hierarchy starting with principal bundles than sheaves. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G Bundle gerbes have been used in gauge theory and also string theory. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings Current work by others is developing a theory of non-abelian bundle gerbes.
References
- Giraud, Jean (1971), Cohomologie non abélienne, Springer, ISBN 3-540-05307-7
- Brylinski, Jean-Luc (1993), Loop space, characteristic classes and geometric quantization, Birkhäuser, ISBN 0-817-63644-7
External links
- Moerdijk, Ieke. Introduction to the Language of Stacks and Gerbes. Retrieved on 2007-05-20. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 325 - The First Council of Nicaea &ndash the first Ecumenical Council of the Christian Church is held
- What is a Gerbe?, by Nigel Hitchin in Notices of the AMS
- Bundle gerbes, Michael Murray. Nigel Hitchin (b 2 August 1946 in Holbrook Derbyshire) is a British Mathematician working in the fields of Differential
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