We discuss a class of pro-groups whose derived limits are relevant to the additivity of strong homology. These are indexed by the set of functions {}^\kappa \omega. The case of width \omega, that is, the one where \kappa = \omega, is linked to the additivity of strong homology on the class of locally compact separable metric spaces. While an equivalence was proved for \lim^1 between certain narrow and wider system, the analogous equivalence for higher limits has long been an open question. Here we present a negative answer to that question and, time permitting, some ideas toward a consistency result on simultaneous vanishing for all cardinals and all \lim^n. This is joint work with Jeffrey Bergfalk.