Gerbe
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In mathematics, a gerbe is a construct in homological algebra.
History
Gerbes were invented by Giraud, as a way of generalising to higher sheaf cohomology and group cohomology groups Hi what the torsor concept does for H1. The terminology is somewhat 'twisted' (gerbe means wreath in French). The initial presentation of the idea made heavy use of category theory: a gerbe, roughly speaking, is a particular type of sheaf of categories. Later gerbes were discussed at length by Jean-Luc Brylinski.
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 pricipal bundles than sheaves. Bundle gerbes have been used in gauge theory and also string theory. Current work by others is developing a theory of nonabelian bundle gerbes.
References
- Lectures on Special Lagrangian Submanifolds, Nigel Hitchin.
- Bundle gerbes (http://arxiv.org/abs/dg-ga/9407015), Michael Murray.