Aspherical abelian groupoids and their directions

From the intrinsic notion of normal subobject and abelian object in a protomodular category, the notion of abelian groupoid in a category E is introduced. A cofibration d1, the direction functor, is built up from the category of aspherical abelian groupoid in E to the category of abelian groups in E. The fibres of d1 are automatically endowed with a symmetric tensor product. The associated abelian group structure on the set of connected components of the fibre above the internal abelian group A realizes the second cohomology group H2(E,A).