Higher forms of stationarity and reflection on \(P_\kappa(\lambda)\) provide natural two-cardinal analogues of classical reflection principles. This talk focuses on their interaction with strong compactness and projection/lifting phenomena in the spirit of Menas' theorem. After briefly recalling the motivation for extending stationary reflection to the two-cardinal setting, recent joint work with Hiroshi Sakai will be discussed showing that strong compactness does not suffice for \(2\)-stationarity on \(P_\kappa(\kappa^+)\). It will also be shown that Menas-type arguments behave differently for higher stationarity: while an analogue of Menas' theorem holds for \(1\)-stationarity under suitable assumptions, the corresponding lifting phenomenon for \(2\)-stationarity can consistently fail. These results provide further evidence that higher stationarity on \(P_\kappa(\lambda)\) exhibits genuinely new behavior beyond ordinary stationarity.
\begin{thebibliography}{9}
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{Joan Bagaria}.
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Transactions of the American Mathematical Society,
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\bibitem{Torres}
{M. Catalina Torres}.
\emph{Higher Stationarity and Derived Topologies on \(P_\kappa(A)\)}.
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\bibitem{TorresSakai}
{M. Catalina Torres and Hiroshi Sakai}.
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Manuscript in preparation.
\end{thebibliography}