In mathematics, a gerbe (/dʒɜːrb/; French: [ʒɛʁb]) is a construct in homological algebra and topology.
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.
As principal bundles glue together (satisfy the descent condition), these groupoids form a stack.
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.
-gerbe admits a local section, meaning satisfies the first axiom, if
Gerbes are a technical response for this question: they give geometric representations of elements in the higher cohomology group
It is expected this intuition should hold for higher gerbes.
It has been found[6] that all gerbes representing torsion cohomology classes in
In addition, the non-torsion classes are represented as infinite-dimensional principal bundles
of the projective group of unitary operators on a fixed infinite dimensional separable Hilbert space
The homotopy-theoretic interpretation of gerbes comes from looking at the homotopy fiber square
analogous to how a line bundle comes from the homotopy fiber square
There are natural examples of Gerbes that arise from studying the algebra of compactly supported complex valued functions on a paracompact space
, which is associated to the set of compact supported complex valued functions of the space
There is no need to impose the compatibility with the group structure in that case since it is covered by the definition of a stack.
Note this can be applied to the situation of comodules over Hopf-algebroids to construct algebraic models of gerbes over affine or projective stacks (projectivity if a graded Hopf-algebroid is used).
In addition, two-term spectra from the stabilization of the derived category of comodules of Hopf-algebroids
These two moduli problems parametrize the same objects, but the stacky version remembers automorphisms of vector bundles.
consists only of scalar multiplications, so each point in a moduli stack has a stabilizer isomorphic to
Another class of gerbes can be found using the construction of root stacks.
This gerbe is banded by the algebraic group of roots of unity
These stacks can be constructed very explicitly, and are well understood for affine schemes.
In fact, these form the affine models for root stacks with sections.
These and more general kinds of gerbes arise in several contexts as both geometric spaces and as formal bookkeeping tools: Gerbes first appeared in the context of algebraic geometry.
One can think of gerbes as being a natural step in a hierarchy of mathematical objects providing geometric realizations of integral cohomology classes.
Essentially they are a smooth version of abelian gerbes belonging more to the hierarchy starting with principal bundles than sheaves.
Current work by others is developing a theory of non-abelian bundle gerbes.