A band over a fixed topological space 
is represented by an open cover 
, 
, and for each 
, a sheaf of groups 
on 
along with outer isomorphisms 
satisfying the cocycle conditions 
and 
. Here,
,
and restrictions to a finer open cover should be viewed
as defining the same band. An outer isomorphism over
an open set 
is an element of the value on
of the sheaf
associated to the presheaf of sheaf isomorphisms modulo inner
automorphisms of the target. Thus, it can be represented locally by isomorphisms
whose classes agree locally on overlaps, without requiring one sheaf isomorphism on the whole open
set.
The collection of all bands over the topological space 
with respect to a single cover 
has a natural category
structure. Indeed, if 
and 
are two bands
over 
with respect to 
, then an isomorphism 
consists of outer isomorphisms 
, understood
as elements of these associated sheaves
on 
, compatible on overlaps so that
.
The collection of all such bands and isomorphisms
thereof forms a category.
The notion of band is essential to the study of gerbes (Moerdijk 2002). In particular, for a gerbe 
over a topological space 
, one can choose an open
cover 
of 
by open subsets 
, and for each 
, an object 
which determines a sheaf
of groups 
on 
. Local isomorphisms
between the chosen objects induce sheaf isomorphisms on an open cover
of each overlap. Their classes modulo inner automorphisms
agree locally and define the outer isomorphisms 
on the whole overlap.
In some literature, an alternative definition of gerbe is used, thereby resulting in an even more specific definition of band. For example,
the associated band of some gerbe 
is sometimes assumed to be a sheaf
of Lie groups 
(Brylinski 1993), though such assumptions appear to be somewhat
rare.
