  A dynamical Yang-Baxter map, introduced by Shibukawa, is a solution of the
set-theoretical analogue of the dynamical Yang-Baxter equation. In this paper,
we initiate a quiver-theoretical approach for the study of dynamical
Yang-Baxter maps. Our key observation is that the category of dynamical sets
over a set $\Lambda$, introduced by Shibukawa to establish a categorical
framework to deal with dynamical Yang-Baxter maps, can be embedded into the
category of quivers with vertices $\Lambda$. By using this embedding, we shed
light on Shibukawa's classification result of a certain class of dynamical
Yang-Baxter maps and extend his construction to obtain a new class of dynamical
Yang-Baxter maps. We also discuss a relation between Shibukawa's bialgebroid
associated to a dynamical Yang-Baxter map and Hayashi's weak bialgebra
associated to a star-triangular face model.
