Obstruction Theory for Extensions of Categorical Groups

For any categorical group H, we introduce the categorical group Out(H) and then the well-known group exact sequence 1 → Z(H) → H → Aut(H)→ Out(H)→ 1 is raised to a categorical group level by using a suitable notion of exactness. Breen’s Schreier theory for extensions of categorical groups is codified in terms of homomorphism to Out(H) and then we develop a sort of Eilenberg-Mac Lane’s obstruction theory that solves the general problem of the classification of all categorical group extensions of a group G by a categorical group H, in terms of ordinary group cohomology.