  We consider conditions on a $k$-graph $\Lambda$, a semigroup $S$ and a
functor $\eta : \Lambda \to S$ which ensure that the $C^*$-algebra of the
skew-product graph $\Lambda \times_\eta S$ is simple. Our results allow give
some necessary and sufficient conditions for the AF-core of a $k$-graph
$C^{*}$-algebra to be simple.
