A Universal Investigation of n-representations of n-quivers

\noindent 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 n-quiver, 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∈{1,2,…,n}.