This talk will be an exposition of the recent work A descriptive approach to higher derived limits, joint with Nathaniel Bannister, Justin Moore, and Stevo Todorcevic (arXiv:2203.00165). The material of this paper is somewhat more ranging than its title would suggest.

At its heart is a new family of partition principles which synthesize several recent advances in the study of higher derived limits, rendering those results far more amenable to combinatorial analyses. These principles admit formulation on any directed quasi-order, and are of particular, and interrelated, interest on the quasi-orders \(({^\omega}\omega,\leq^*)\) and the ordinals \(\omega_n\).

A main implication of these principles in any case is the triviality of (higher dimensionally) coherent families of functions; we'll use any remaining time to note ways that such objects, and even higher derived limits, are closer to classical set theoretic concerns than perhaps tends to be realized.