I will outline a definition of cohomology with coefficients in a strict omega-category due to Ross Street. This involves the notion of parity complexes and orientals; the latter are also used to define the nerve of a strict omega-category.
We can pass from the world of groups to the world of n-groupoids (or even infinity-groupoids) and still make sense of extensions as fibrations. In this world, however, "abelianness" is no longer just a property but rather a variety of possible structures, and hence an extension will be more than a mere fibration. In my talk, I will extend the notion of group extension to k-tuply groupal n-groupoids, give an algebraic description of the corresponding cohomology n-groupoids, and show how short exact sequences give rise to long exact sequences in cohomology. I will focus in particular on the simplest case that does not fit into the existing paradigm, namely, extensions of groups by braided 2-groups. This particular problem closely parallels work by Etingof, Nikshych, and Ostrik on G-extensions of braided fusion categories.