Cohomologically trivial internal categories in categories of groups with operations (Q1899873)

The cohomology is that of ``Cohomology of internal categories in categories of groups with operations'' [the author, in: Proc. Int. Conf. Categorical Topology Prague 1988, World Scientific, Singapore, 270-283 (1989)], and \({\mathcal C}\) is a category of (ordinary) groups with extra primary operations of arity at most 2. For an internal category \(d_0, d_1 : D_1 \to D_0\) in \({\mathcal C}\) the author lets \(d : \text{Ker} (d_0) \to D_0\) (the restriction of \(d_1)\) denote the associated crossed module. In the presence of reasonable axioms on the operations (allowing rings and Lie algebras as examples) the author characterizes those \(D\) for which \(H^0 (D,A)\) or \(H^1 (D,A) = 0\) for all internal diagrams \(A\) or \(D\) in \({\mathcal C}\). The condition for \(H^0\) is that the module of differentials on \(\text{coker} (d)\) vanishes. The conditions for \(H^1\) require that \({\mathcal C}\) be a ``category of interest'' in the sense of \textit{G. Orzech}, and involve an abelianization functor \(S:\text{Cat} ({\mathcal C}) \to \text{Cat} ({\mathcal A} b)\). The author constructs both the differentials and the abelianization.