I will describe some joint work with Todorcevic on the Tukey theory of ultrafilters on the natural numbers. The notion of Tukey equivalence tries to capture the idea that two directed posets look cofinally the same, or have the same cofinal type. As such, it provides a device for a rough classification of directed sets based upon their cofinal type, as opposed to an exact classification based on their isomorphism type. This notion has recently received a lot of attention in various contexts in set theory. As background, I will illustrate the idea of rough classification with several examples, and explain how rough classification based on Tukey equivalence fits in with other work in set theory. The talk will be based on the paper Cofinal types of ultrafilters. A preprint of the paper is available on my website: http://www.math.toronto.edu/raghavan .

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Last updated: 11 Dec 2023