Simplicial cohomology with coefficients in symmetric categorical groups

In this paper we introduce and study a cohomology theory {H-n(-, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A, n)}(ngreater than or equal to0), which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n greater than or equal to 3, the functor K(-, n) is right adjoint to the functor p(n), where p(n)(X.) is defined as the fundamental groupoid of the n-loop complex Omega(n)(X.). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with pi(i)(Y) = 0 for all i not equal n, n + 1 and n greater than or equal to 3; and also we obtain a classification theorem for those spaces: [-, Y] congruent to H-n(-, p(n)(Y)).