Schreier theory for singular extensions of categorical groups and homotopyclassification

If G is a categorical group, a G-module is defined to be a braided categori cal group (A, c) together with an action of G on (th, c). In this work we d efine the notions of singular extension of G by the G-module (A,c) and of 1 -cocycle of G with coefficients in (A, c) and we obtain. first. a bijection between the set of equivalence classes of singular extensions of G by (A, c) and the set of equivalence classes of 1-cocycles. Next, we associate to any G-module (A, c) a Kan fibration of simplicial sets phi : Ner(G, (A, c)) --> Ner(G), and then we show that there is a bijection between the set of equivalence classes of singular extensions of G by (A,c) and Gamma[Ner(G, ( A,c))/Ner(G)], the set of fibre homotopy classes of cross-sections of the f ibration phi.