We discuss two extensions of results conjectured by Nick Kuhn about the non-realization of unstable algebras as the mod-_p singular cohomology of a space, for _p a prime. The first extends and refines earlier work of the second and fourth authors, using Lannes’ mapping space theorem. The second (for the prime 2) is based on an analysis of the -1 and -2 columns of the Eilenberg–Moore spectral sequence, and of the associated extension.

In both cases, the statements and proofs use the relationship between the categories of unstable modules and functors between F_p-vector spaces. The second result in particular exhibits the power of the functorial approach.