A band over a fixed topological space  is represented by a cover , , and for each , a sheaf of groups  on  along with outer automorphisms  satisfying the cocycle conditions  and . Here, restrictions of the cover  to a finer cover  should be viewed as defining the exact same band.

The collection of all bands over the 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 automorphisms  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). 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 together form a sheaf of groups  on . One can then consider a collection of sheaf isomorphisms  between any two groups  and  which forms a collection of well-defined outer automorphisms.

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.

REFERENCES:

Brylinski, J. Loop Spaces, Characteristic Classes, and Geometric Quantization. Boston, MA: Birkhäuser, 1993.

Moerdijk, I. "Introduction to the Language of Stacks and Gerbes." 2002. http://arxiv.org/abs/math/0212266.