COHOMOLOGICALLY TRIVIAL INTERNAL CATEGORIES IN CATEGORIES OF GROUPS WITH OPERATIONS

We describe cohomologically trivial internal categories in the category C of groups with operations satisfying certain conditions ([15], [16]), As particular cases we obtain: if C = Gr, H-0(C, -) = 0 iff C is a connected internal category; if C = Ab, H-1(C, -) = 0 iff C is equivalent to the discrete internal category (Coker d, Coker d, 1, 1, 1, 1). We also discuss related questions concerning extensions, internal categories, their cohomology and equivalence in the category C.