Let us briefly recall what local data of gerbes , isomorphisms and 2 - isomorphisms are , for the details we refer the reader to [ 18 ] . For a given gerbe G over M , one can choose a sufficiently " good " open cover O of M that permits to extract a cocycle c ∈ A 2 ( O ) , D 2 c = 0 , in a certain way . Suppose that two gerbes G 1 and G 2 are given , and O 1 and O 2 are open covers that permit to extract cocycles c 1 and c 2 . Suppose further that A : G 1 → G 2 is an isomorphism . Then one can choose a common refinement O of O 1 and O 2 that permits to extract a cochain b ∈ A 1 ( O ) such that c 2 = c 1 + D 1 b. The cochains for isomorphisms add under the composition of these isomorphisms . Finally , if a 2isomorphism ϕ : A 1 ⇒ A 2 is given and b 1 and b 2 are cochains for A 1 and A 2 , respectively , for a suitable open cover O , one can always extract a cochain a ∈ A 0 ( O ) such that b 2 = b 1 + D 0 a. The cochains for 2 - isomorphisms add under both the horizontal and the vertical composition of 2 - isomorphisms . Conversely , one can reconstruct gerbes , isomorphisms and 2 - isomorphisms from given local data , and the two procedures are inverse to each other in an appropriate sense . In particular , they establish a bijection between H 2 ( M , D(2 ) ) and the set of isomorphism classes of gerbes over M .