Gerbe

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In mathematics, a gerbe is a construct in homological algebra.

[edit] Definition

A gerbe is a stack over a topological space which is locally isomorphic to the Picard groupoid of that space. (The Picard groupoid on an open set U is the category whose objects are line bundles on U and whose morphisms are isomorphisms; cf. Picard group)

[edit] 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 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.