Eilenberg-Mac Lane spaces have been recognized as important object of study in Algebraic Topology (and now Algebraic Geometry too) since the early works of Eilenberg and Mac Lane. These spaces can be loosely characterized as classifying spaces for cohomology, or equivalently as homotopy types with exactly one nonzero homotopy group.

Despite having been around for a long time, cohomology groups of Eilenberg-Mac Lane spaces are still not well understood (as opposed to the their homotopy groups).

It is classical that some of the low-dimensional cohomology groups of E-ML spaces characterize e.g. group extensions. We will sketch how in general cohomology groups of E-ML spaces relate to analogous extension problems for n-categories equipped with group-like structures, and how these relate to general homotopy n-types.