I have a profinite directed graph $\Gamma$, i.e., I can think of $\Gamma$ as the inverse limit of a directed system of finite directed graphs under inclusion. To each vertex of the graph a $G$-module can be assigned ($G$ is some fixed group), and to each edge a homomorphism of associated $G$-modules.
I want to think of it as a $G$-representation of profinite quiver $\Gamma$, and see what the tools of quiver representation theory produce - irreducibles, moduli space etc. - for my choices of $\Gamma$ and $G$.
There may be cycles in my graph, so I also want to consider the case when edge homomorphisms satisfy obvious compatibility relations for the cycles.
I feel this in general should already be in the quiver literature, but I am not too familiar with it, so I haven't managed to find anything. So far I have only seen works on finite quivers and their representations on vector spaces. Can you point to some papers that come close to my situation?
