  Quivers over a fixed base set form a monoidal category with tensor product
given by pullback. The quantum Yang-Baxter equation, or more properly the braid
equation, is investigated in this setting. A solution of the braid equation in
this category is called a "solution" for short. Results of
Etingof-Schedler-Soloviev, Lu-Yan-Zhu and Takeuchi on the set-theoretical
quantum Yang-Baxter equation are generalized to the context of quivers, with
groupoids playing the r\^ole of groups. The notion of "braided groupoid" is
introduced. Braided groupoids are solutions and are characterized in terms of
bijective 1-cocycles. The structure groupoid of a non-degenerate solution is
defined; it is shown that it is braided groupoid. The reduced structure
groupoid of a non-degenerate solution is also defined. Non-degenerate solutions
are classified in terms of representations of matched pairs of groupoids. By
linearization we construct star-triangular face models and realize them as
modules over quasitriangular quantum groupoids introduced in recent papers by
M. Aguiar, S. Natale and the author.
