Central extensions of gerbes (Q986084)

A stack of groupoids on a space, \(X\), is the lax analogue of a sheaf of groupoids, and so is a geometric version of a groupoid. A gerbe is then the lax version of a connected non-empty groupoid. The notion of a central extension of groupoids is very useful in the study of obstructions to liftings of morphisms and here this is extended to handle gerbes and non-abelian cohomology. The paper starts by setting down terminology and notation for the required part of 2-category theory, then discusses pre-stacks and stacks on a space in a clear and concise way. Gerbes are then introduced and central extension of them defined, and a comparison made with central extensions of groupoids. Obstruction classes are introduced, giving the obstruction to lifting isomorphisms, so if \(F :{\mathcal G}\to{\mathcal H}\) is a weak epimorphism of gerbes forming part of a central extension, and \(i\), \(j\) are objects in \({\mathcal G}(X)\), with \(h : F(i)\to F(j)\) in \(\mathcal{H}\), then the obstruction \(c\ell^1_F(h)\in \check{H}^1(X, \operatorname{Ker}F)\) and \(h\) lifts to an isomorphism \(g : i\to j\) if, and only if, this class is trivial. A second obstruction class handles the obstruction to lifting objects. Sufficient conditions are given for the existence of such obstruction classes. Pro-nilpotent gerbes are then discussed and a final section looks at a class of example of central extensions in which a gerbe \({\mathcal G}\) is divided by its centre, \(Z({\mathcal G})\). The global objects of \({\mathcal G}/Z({\mathcal G})\) are called the fake global objects of \({\mathcal G}\). The question of when fake global objects lift to true global objects of \({\mathcal G}\) is addressed through some interesting examples and results. Further applications will be given in a forthcoming survey article by the author. (A version is on the Arxiv, see \url{http://arxiv.org/abs/0801.3233}.)