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.