Characterizations of the weakly compact ideal on $P_κλ$

Hellsten \cite{MR2026390} gave a characterization of $Π^1_n$-indescribable subsets of a $Π^1_n$-indescribable cardinal in terms of a natural filter base: when $κ$ is a $Π^1_n$-indescribable cardinal, a set $S\subseteqκ$ is $Π^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq κ$. We generalize Hellsten's characterization to $Π^1_n$-indescribable subsets of $P_κλ$, which were first defined by Baumgartner. After showing that under reasonable assumptions the $Π^1_0$-indescribability ideal on $P_κλ$ equals the minimal \emph{strongly} normal ideal $\text{NSS}_{κ,λ}$ on $P_κλ$, and is not equal to $\text{NS}_{κ,λ}$ as may be expected, we formulate a notion of $n$-club subset of $P_κλ$ and prove that a set $S\subseteq P_κλ$ is $Π^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq P_κλ$. We also prove that elementary embeddings considered by Schanker \cite{MR2989393} witnessing \emph{near supercompactness} lead to the definition of a normal ideal on $P_κλ$, and indeed, this ideal is equal to Baumgartner's ideal of non--$Π^1_1$-indescribable subsets of $P_κλ$. Additionally, as applications of these results we answer a question of Cox-Lücke \cite{MR3620068} about $\mathcal{F}$-layered posets, provide a characterization of $Π^m_n$-indescribable subsets of $P_κλ$ in terms of generic elementary embeddings, prove several results involving a two-cardinal weakly compact diamond principle and observe that a result of Pereira \cite{MR3640048} yeilds the consistency of the existence of a $(κ,κ^+)$-semimorasses $μ\subseteq P_κκ^+$ which is $Π^1_n$-indescribable for all $n<ω$.

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