The factor set of Gr-categories of the type $(Π,A)$

Any $\Gamma$-graded categorical group is determined by a factor set of a categorical group. This paper studies the factor set of the group $\Gamma$ with coefficients in the categorical group of the type $(\Pi,A).$ Then, an interpretation of the notion of $\Gamma-$operator $3-$cocycle is presented and the proof of cohomological classification theorem for the a $\Gamma-$graded Gr-category is also presented.