  In this paper, we construct a semigroup associated to an action of countable
discrete group on a compact Hausdorff space, that can be regarded as a higher
dimensional generalization of the type semigroup. Using this generalized type
semigroup we obtain a new characterization of dynamical comparison. This
answers a question of Kerr and Schafhauser. In addition, this paper suggests a
definition of comparison for dynamical systems in which neither necessarily the
acting group is amenable nor the action is minimal.
