A descriptive approach to higher derived limits

We present a new aspect of the study of higher derived limits. More precisely, we introduce a complexity measure for the elements of higher derived limits over the directed set \Omega of functions from \mathbb{N} to \mathbb{N} and prove that cocycles of this complexity are images of cochains of roughly the same complexity. In the course of this work, we isolate a partition principle for powers of directed sets and show that whenever this principle holds, the corresponding derived limit \lim\nolimits^{n} is additive; vanishing results for this limit are the typical corollary. The formulation of this partition hypothesis synthesizes and clarifies several recent advances in this area.