Functoriality in resolution of singularities

Algorithms for resolution of singularities in characteristic zero are based on Hironaka's idea of reducing the problem to a simpler question of desingularization of all "idealistic exponent" (or "marked ideal"). How can we determine whether two marked ideals are equisingular in the sense that they can be resolved by the same blowing-tip sequences? We show there is a desingularization functor defined on the category of equivalence classes of marked ideals and smooth morphisms, where marked ideal, are "equivalent" if they have the same sequences of "test transformations". Functoriality in this sense realizes Hironaka's idealistic exponent philosophy. We use it to show that the recent algorithms for desingularization of marked ideals of Wlodarczyk and of Kollar coincide with our own, and we discuss open problems.