COHOMOLOGY WITH COEFFICIENTS IN SYMMETRICAL CAT-GROUPS - AN EXTENSIONOF EILENBERG-MACLANE CLASSIFICATION THEOREM

In this paper we use Takeuchy-Ulbrich's cohomology of complexes of cat egories with abelian group structure to introduce a cohomology theory {H(n)(-,C)} for simplicial sets, or topological spaces, with coefficie nts in symmetric cat-groups C. This cohomology is the usual one when a belian groups are taken as coefficients, and the main topological sign ificance of this cohomology is the fact that it is equivalent to the r educed cohomology theory defined by a OMEGA-spectrum, {T(C, n)}, canon ically associated to C. We use the spaces T(C, n) to prove that symmet ric cat-groups model all homotopy type of spaces X with PI(i)(X) = 0 f or all i not-equal n, n + 1 and n greater-than-or-equal-to 3, and then we extend Eilenberg-MacLane's classification theorem to those spaces: [-,X] is-approximately-equal-to H(n)(-, C(X)).