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)} _n0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n3, the functor K(–,n) is right adjoint to the functor _n , where _n (X ) is defined as the fundamental groupoid of the n-loop complex ⁿ (X ). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with _i (Y)=0 for all in,n+1 and n3; and also we obtain a classification theorem for those spaces: –,Y]H ⁿ (–, _n (Y)).