Consistency strength of lifting through a lot of collapsing

Question

What is the consistency strength of the following situation?

1. $j : V \to M$ is an elementary embedding definable from parameters in $V$, with critical point $\kappa$.
2. $\mathbb P$ is a forcing that collapses all ordinals between $\kappa$ and $j(\kappa)$.
3. $j$ can be lifted through $\mathbb P$.

One can deduce that $j(\kappa)$ is regular in $V$. The only examples I know of such liftings involve almost-huge cardinals.

Answer 1

$\text{AD}^{L(\mathbb R)}$ suffices. The situation actually holds in the model $H = \text{HOD}^{L(\mathbb R)}$. We will have $\kappa = \omega_1$ and $j : H\to \text{Ult}(H,U)$ equal to the ultrapower of $H$ by the club measure $U$ over $\omega_1$ as computed in $L(\mathbb R)$ (using all functions in $L(\mathbb R)$).

For any number $n$, the $\Sigma_n$-satisfaction predicate of $L(\mathbb R)$ with ordinal parameters is definable over $H$ from its restriction to ordinals less than $\Theta$, so any subclass of $H$ that is ordinal definable over $L(\mathbb R)$ is definable from parameters over $H$. In particular, $j$ is definable from parameters over $H$.

Let $N$ be a $\mathbb P_\text{max}$-extension of $L(\mathbb R)$. Note that $H = \text{HOD}^N$ by the homogeneity and definability of $\mathbb P_\text{max}$. Let $\mathbb P\in H$ be the Vopenka algebra of $N$ for adding a subset of $\omega_2$ to $H$. There is a set $A\subseteq \omega_2$ such that $N= L[A]$, and so $N = H[G_A]$ where $G_A\subseteq \mathbb P$ is the $H$-generic ultrafilter associated to $A$.

In $N$, $\text{NS}_{\omega_1}$ is saturated. Let $G\subseteq P(\omega_1)\setminus\text{NS}_{\omega_1}$ be $N$-generic, and in $N[G]$ let $i : N\to \text{Ult}(N,G)$ be the generic ultrapower embedding associated to $G$ (using functions in $N$).

Now as usual, we cite a theorem due to Woodin: $j = i\restriction H$. This follows from Theorem 4.53 in The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal.

Now in $H$, we have the situation you were looking for with $\kappa = \omega_1.$ Note that $i(\omega_1) = (\omega_2)^N$ by saturation, which means that all $H$-cardinals between $\kappa$ and $j(\kappa)$ are collapsed to $\kappa$ in $N$. Moreover $j$ lifts through the forcing $\mathbb P$ (to $i$) by construction.