Gerbe

In mathematics, a gerbe (/dʒɜːrb/; French: [ʒɛʁb]) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (Giraud 1971) following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as a generalization of principal bundles 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.

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

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 covering {V_i}_i of U such that the restrictions of a and b to each V_i 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.

Examples

Algebraic geometry

• Azumaya algebras
• Deformations of infinitesimal thickenings
• Twisted forms of projective varieties
• Fiber functors for motives

Differential geometry

• $H^3(X,\mathbb{Z})$ and $\mathcal{O}_X^*$-gerbes: Jean-Luc Brylinski's approach

History

Gerbes first appeared in the context of algebraic geometry. 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.

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

• Giraud, Jean (1971), Cohomologie non abélienne, Springer, ISBN 3-540-05307-7.
• Brylinski, Jean-Luc (1993), Loop space, characteristic classes and geometric quantization, Birkhäuser Verlag, ISBN 0-8176-3644-7.

External links

• Moerdijk, Ieke. "Introduction to the Language of Stacks and Gerbes". Retrieved 2007-05-20.
• What is a Gerbe?, by Nigel Hitchin in Notices of the AMS
• Bundle gerbes, Michael Murray.