# The Abelian Group of Equivalence Classes of Gerbes

Are there any good references/explanations for understanding the "contracted product" (as Brylinski calls it) of a gerbe with abelian band? I am finding it difficult, using Brylisnki's definition. to show that the map from the equivalence class of a gerbe to the degree 2 Cech cocycle is in fact a group homomorphism.

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If you can show that the map associating to the gerbe its Cech cohomology class is a bijection, then it follows that the product of gerbes is given by the product in cohomology, and that this is a homomorphism. This however is not the answer you are wanting. :-) –  David Roberts Mar 14 '13 at 4:52
Perhaps consulting something like Danny Stevenson's thesis arxiv.org/abs/math/0004117, where the bundle gerbe case is done, will help. He also considers the gerbe associated to a bundle gerbe, which may help in translating the ideas across. –  David Roberts Mar 14 '13 at 4:54
I believe I can show the map is a bijection; maybe I am missing this fact that the bijection is good enough? It's that I want to have this theorem about the cohomology classifying these equivalence classes of gerbes (I believe due to Giraud) in terms of intersections of open sets. Brylinski's definition of the product uses local homeomorphisms, which I've dealt with before with no problems but this time translating the definition into intersections and open covers is stumping me. I'll check out that paper thanks for the suggestions as always! –  cheyne Mar 14 '13 at 15:04