This is joint work with Stevo Todorcevic.
Let U and V be ultrafilters on countable base sets. We say that V is Tukey reducible to U ("V is below U") if there is a "Tukey map" g: V -> U, meaning that g maps unbounded subsets of V to unbounded subsets of U. Equivalently, there is a "cofinal" map f: U -> V which maps cofinal subsets of U to cofinal subsets of V. Tukey reducibility on ultrafilters is a generalization of Rudin-Keisler reducibility.
We present a canonization of cofinal maps from a p-point into another ultrafilter as monotone continuous functions, and some analogues for ultrafilters with similarities to p-points. We also give some results on the structure of the Tukey types of ultrafilters on omega and FIN, concentrating on p-points, selective ultrafilters, and ultrafilters with similar properties.