a universal investigation of n-representations of n-quivers

We have two goals in this paper. first, we investigate and construct cofree coalgebras over n-representations of quivers, limits and colimits of n-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of n-representations of quivers. second, for any given quivers q1,q2,..., qn, we construct a new quiver q(q1,q2,...,qn) , called an nquiver, and identify each category repk(qj) of representations of a quiver qj as a full subcategory of the category repk(q(q1,q2,...,qn) ) of representations of q(q1,q2,...,qn) for every j 2 {1, 2, . . . , n}.