of a space gives the isomorphism classes of principal -bundles over , what does the iterated cohomology functor represent? Meaning, we are gluing together the groups via some one cocycle. 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. Cohomological classification One of the main theorems concerning gerbes is their cohomological classification whenever they have automorphism groups given by a fixed sheaf of abelian groups , called a band. For a gerbe on a site , an object , and an object , the automorphism group of a gerbe is defined as the automorphism group . Notice this is well defined whenever the automorphism group is always the same. Given a covering , there is an associated classrepresenting the isomorphism class of the gerbe banded by . For example, in topology, many examples of gerbes can be constructed by considering gerbes banded by the group . As the classifying space is the second Eilenberg-Maclane space for the integers, a bundle gerbe banded by on a topological space is constructed from a homotopy class of maps inwhich is exactly the third singular homology group . It has been found that all gerbes representing torsion cohomology classes in are represented by a bundle of finite dimensional algebras for a fixed complex vector space . 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 . Note this is well defined because all separable Hilbert spaces are isomorphic to the space of square-summable sequences . The homotopy-theoretic interpretation of gerbes comes from looking at the homotopy fiber squareanalogous to how a line bundle comes from the homotopy fiber squarewhere , giving as the group of isomorphism classes of line bundles on . Examples C*-algebras There are natural examples of Gerbes which arise from studying the algebra of compactly supported complex valued functions on a paracompact space pg 3. Given a cover of there is the Cech groupoid defined aswith source and target maps given by the inclusionsand the space of composable arrows is justThen a degree 2 cohomology class is just a mapWe can then form a non-commutative C*-algebra which is associated to the set of compact supported complex valued functions of the spaceIt has a