Classification cohomologique et homotopique des extensions singulières de Gr-catégories (Extensions singulières de Gr-catégories)

Abstract

If G is a categorical group, a G -module is defined to be a braided categorical group ( A , c ) together with an action of G on ( A , c ) . We associate to any G -module ( A , c ) a Kan simplicial set Ner ( G ( A , c ) ) and a Kan fibration Ner ( G ( A , c ) ) → ϕ Ner ( G ) . In addition, we define the set of equivalence classes of singular extensions of G by ( A , c ) , and also a 1-cohomology set of G with coefficients in ( A , c ) . We construct bijections between these sets, and also with the set of fibre homotopy classes of cross-sections of the fibration φ.