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Stack関連 Gerbe
”They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group.”
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URLﾘﾝｸ(en.wikipedia.org)
Gerbe
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In mathematics, a gerbe (/d???rb/; French: [???b]) 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 an analogue of fibre bundles where the fibre is the classifying stack of a group. 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.
Contents
1 Definitions
1.1 Gerbe
2 Examples
2.1 Algebraic geometry
2.2 Differential geometry
3 History
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.