  We use categorical method and birational geometry to study moduli spaces of
quiver representations. From certain "representable" functor, we construct a
birational transformation from the moduli space of representations of one
quiver to another new quiver with one vertex less. The dimension vector and the
stability for the new moduli are determined functorially. We introduce several
relative notions of stability to study such a birational transformation. The
essential case is proven to be the usual blow-ups. Moreover, we compare the
induced ample divisors for the two moduli. We illustrate this theory by various
examples.
