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{{Expert-subject|Mathematics|date=November 2008}}
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{{for|other meanings|Gerbe (disambiguation)}}


In [[mathematics]], a '''gerbe''' is a construct in [[homological algebra]] and [[topology]]. Gerbes were introduced by [[Jean Giraud (mathematician)|Jean Giraud]] {{harv|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 bundle]]s to the setting of [[2-categories]]. Gerbes provide a convenient, if highly abstract, language for dealing with many types of [[Deformation theory|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.
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"Gerbe" is a French (and archaic English) word that literally means [[wheat]] [[sheaf (agriculture) | sheaf]].
 
== Definitions ==
===Gerbe===
 
A gerbe on a [[topological space]] ''X'' is a [[stack (mathematics)|stack]] ''G'' of [[groupoid]]s 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<sub>i</sub>}<sub>i</sub> of U such that the restrictions of ''a'' and ''b'' to each V<sub>i</sub> 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 algebra]]s
* Deformations of infinitesimal thickenings
* Twisted forms of projective varieties
* [[Fiber functors]] for [[motive (algebraic geometry)|motive]]s
 
===Differential geometry===
* <math>H^3(X,\mathbb{Z})</math> and <math>\mathcal{O}_X^*</math>-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 {{harv|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 [[Michael Murray (mathematician)|Murray]] and called [[bundle gerbes]]. Essentially they are a [[smooth function|smooth]] version of abelian gerbes belonging more to the hierarchy starting with [[principal bundle]]s 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 gerbe]]s.
 
== References ==
*{{citation
  | last = Giraud
  | first = Jean
  | authorlink = Jean Giraud (mathematician)
  | title = Cohomologie non abélienne
  | publisher = [[Springer Science+Business Media|Springer]]
  | year = 1971
  | isbn = 3-540-05307-7
}}.
*{{citation
  | last = Brylinski
  | first = Jean-Luc
  | authorlink = Jean-Luc Brylinski
  | title = Loop space, characteristic classes and geometric quantization
  | publisher = [[Birkhäuser Verlag]]
  | year = 1993
  | isbn = 0-8176-3644-7
}}.
 
== External links ==
* {{cite web
  | last = Moerdijk
  | first = Ieke
  | title = Introduction to the Language of Stacks and Gerbes
  | url=http://www.math.uu.nl/publications/preprints/1264.ps
  | accessdate = 2007-05-20 }}
*''[http://www.ams.org/notices/200302/what-is.pdf What is a Gerbe?]'', by [[Nigel Hitchin]] in Notices of the AMS
*''[http://arxiv.org/abs/dg-ga/9407015 Bundle gerbes]'', Michael Murray.
 
[[Category:Homological algebra]]
[[Category:Sheaf theory]]

Latest revision as of 19:57, 22 November 2014

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