High stationarity and derived topologies on $\mathcal{p}_{\kappa}(a)$

Let $\kappa$ be an uncountable regular cardinal, $\kappa \subseteq A$. We study a notion of $n$-stationarity on $\mathcal{P}_{\kappa}(A)$. We construct a sequence of topologies $\langle \tau_0, \tau_1, \dots \rangle $ on $\mathcal{P}_{\kappa}(A)$ characterising the simultaneous reflection of a pair of $n$-stationary sets in terms of elements in the base of $\tau_n$. This result constitutes a complete generalisation to the context of $\mathcal{P}_{\kappa}(A)$ of Bagaria's prior characterisation of $n$-simultaneous reflection in terms of derived topologies on ordinals.