There are no fewer than two closely related but somewhat different notions of gerbe in mathematics.
For a fixed topological space 
, a gerbe on 
can refer to a stack of groupoids 
on 
satisfying the properties
for 
2. given objects 
, any point 
has a neighborhood 
for which there is at least
one morphism 
in 
.
The second definition is due to Giraud (Brylinski 1993). Given a manifold 
and a Lie group 
, a gerbe 
with band 
is a sheaf of groupoids
over 
satisfying the following three properties:
1. Given any object 
of 
, the sheaf 
of automorphisms of
this object is a sheaf of groups on 
which is locally isomorphic
to the sheaf 
of smooth 
-valued functions. Such a local isomorphism 
is unique up to inner
automorphisms of 
.
2. Given two objects 
and 
of 
, there exists a surjective local homeomorphism 
such that 
and 
are isomorphic. In
particular, 
and 
are locally isomorphic.
3. There exists a surjective local homeomorphism 
such that the category 
is non-empty.
Clearly, the notion of a gerbe's band is fundamental for the second definition; though not explicitly mentioned, the band
of a gerbe 
defined by the first definition is also important (Moerdijk 2002). According to Brylinski,
gerbes whose bands 
corresponds to a Lie group 
are significant in that they give rise
to degree-2 cohomology classes
in 
, a fact utilized by Giraud
in his study of non-abelian degree-2 cohomology.
