Gerbe

In mathematics, a gerbe (dʒɜrb; ʒɛʁb) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud 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. A gerbe on a topological space is a stack of groupoids over which is locally non-empty (each point has an open neighbourhood over which the of the gerbe is not empty) and transitive (for any two objects and of for any open set , there is an open covering of such that the restrictions of and to each are connected by at least one morphism). A canonical example is the gerbe of principal bundles with a fixed structure group : the section category over an open set is the category of principal -bundles on 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 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. The most general definition of gerbes are defined over a site. Given a site a -gerbe is a category fibered in groupoids such that There exists a refinement of such that for every object the associated fibered category is non-empty For every any two objects in the fibered category are locally isomorphic Note that for a site with a final object , a category fibered in groupoids is a -gerbe admits a local section, meaning satisfies the first axiom, if .