Issues of set theoretic compactness frequently arise when considering questions about derived functors. In particular, the non-vanishing of such derived functors is often witnessed by a concrete combinatorial instance of set theoretic incompactness. In this talk, we will discuss some recent results about the derived functors of the inverse limit functor.

We will focus on a specific inverse system of abelian groups, $\mathbf{A}$, that arose in Marde\v{s}i\'{c} and Prasolov's work on the additivity of strong homology and has since arisen independently in a number of contexts. Our main result states that, relative to the consistency of a weakly compact cardinal, it is consistent that the n-th derived limits $\lim^n \mathbf{A}$ vanish simultaneously for all $n \geq 1$. We will sketch a proof of this fact and then discuss the extent to which certain generalizations of this result can hold. This is joint work with Jeffrey Bergfalk.