  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 $\Fp$-vector spaces. The second result in
particular exhibits the power of the functorial approach.
