Cofinal types on $\omega_2$

In this paper we start the analysis of the class $\mathcal D_{\aleph_2}$, the class of cofinal types of directed sets of cofinality at most $\aleph_2$. We compare elements of $\mathcal D_{\aleph_2}$ using the notion of Tukey reducibility. We isolate some simple cofinal types in $\mathcal D_{\aleph_2}$, and then proceed to show which of these types have an immediate successor in the Tukey ordering of $\mathcal D_{\aleph_2}$.