A long exact sequence in non-abelian cohomology

Abstract

The context ,~ , in which this paper is developed is a variety of ~-groups ,[19]. This includes the most usual algebraic categories (Groups, Associative algebras .... ) in which a comprehensible and well motived non-abelian cohomology theory has been developed in dimensions ~2 using crossed modules as coefficients. All of these theories have a natural interpretation using the homotopy theory of the corresponding simplicial category, [28]: crossed modules are equivalent to internal groupoids and then, by taking nerves, to simplicial objects whose Moore complex is trivial in dimensions >i, and there are natural bijections between M2(X,~) and [F.,G(~)], the set of homotopy classes of simplicial morphisms from a free simplicial resolution of an object X into the simplicial object associated to the crossed module ~, [14],[6],[7]. In the general context S ,the concept of crossed module has been explicitely given by Ellis, [16], who shows the equivalence between the category of crossed modules in S, XM(S), the category of internal groupoids in S, GPD(S), and the category of simplicial objects in S with trivial Moore complex at dimensions >I, which we will denote by I-HYPGD(S} and will call the category of l-hypergroupoids in S , because their objects are just the internal l-hypergroupoids in the sense of Duskin-Glenn,[17]. Using the fact that Simpl(S), the category of simplicial objects in S, is a closed simplicial model category, [28], we study the sets [~.(X),B.] of homotopy classes of simplicial morphisms from the standard cotriple resolution of an object X into l-hypergroupoid B., and then apply our results to the classical non-abelian cohomology theories. In fact, we will establish some facts about the sets [S.(X),B.], where B. is an n-hypergroupoid in S, [17], which is just a simplicial object in B whose Moore complex is trivial at dimensions >n, for an arbitrary nzl. The main reason to do it is that for certain n-hypergroupoids B. these sets will be used to provide an appropriate definition of non abelian ~n with coefficients in crossed modules in S. For nzl, we will denote n-~D(S) the full subcategory of Simpl(S) whose objects are the n-hypergroupoids.. Let us note too that the use of n-hypergroupoids in non-abelian cohomology appears in some recents papers [6], [7],[14],[15] ;and also in abelian cohomology since Duskin, [13], showed that the usual monadic cohomology of an object X in B with coefficients in an internal X-module A, Hn(X,A), is isomorphic to [G.(X),K(A~X,n)] I ,the subset of [6.(X),K(A~X,n)] x whose elements are those homotopy clases of simplicial morphisms f.:G.(X) ........ ) K(A~X,n) such that ~0(f.)=Ix, where K(A~X,n) is the