$n$-representations of Quivers

Let $n\ge 2$. We introduce the notion of $n$-representations of quivers, and we explicitly provide concrete examples of $2$-representations of quivers. We establish the categories of $n$-representations and investigate kernels and cokernels in the categories of $n$-representations of quivers. Further, we construct them in terms of kernels and cokernels of morphisms in the usual categories of quiver representations. We show that every morphism in the categories of $n$-representations has a canonical decomposition. Most importantly, we prove that the categories of $n$-representations of quivers are $k$-linear abelian categories.