Logical relationship between supercompact and rank-into-rank cardinals

Question

It is well known that the large cardinals are ordered by logical consistency. In many cases, the logical consistency is, in fact, a direct implication - for example, Mahlo $\Rightarrow$ Inaccessible because a Mahlo cardinal is itself inaccessible and a stationary limit of inaccessibles.

My questions are regarding how exactly Cons(ZFC+Rank-into-rank) implies Cons(ZFC+Supercompact). In particular:

Q1: Is every rank-into-rank cardinal supercompact (and maybe a limit of supercompacts) ?

If not, then two more questions:

Q2: Let $j:V_\lambda \rightarrow V_\lambda$ be a rank-into-rank embedding. I am guessing that consistency of supercompacts is proved by some cardinal being "supercompact in $V_\lambda$". How is this cardinal identified ?

Q3: Would any of the known large cardinal axioms $\it{imply}$ the existence of both, a rank-into-rank embedding and a supercompact cardinal ?

Answer 1

The answer to Q1 is no. The least rank-to-rank cardinal is never supercompact. The reason is that a cardinal $\kappa$ being rank-to-rank is a $\Sigma_2$ property, being witnessed by the existence of an elementary embedding $j:V_\lambda\to V_\lambda$ with critical point $\kappa$, and any higher $V_\theta$ can see the existence of such a $j$. (See my post on local properties to get familiar with this kind of observation.) So the least such $\kappa$ will be strictly below the least $\Sigma_2$-reflecting cardinal. In particular, it will be below the first strong cardinal, and also below the first supercompact cardinal.

Nevertheless, if $\kappa$ is rank-to-rank, witnessed by $j:V_\lambda\to V_\lambda$, then $V_\lambda$ will be a model of ZFC in which $\kappa$ is supercompact. This is because the induced measures on $P_\kappa\gamma$ will exist in $V_\lambda$ for every $\gamma<\lambda$. So in $V_\lambda$, the cardinal $\kappa$ will be supercompact and therefore also a limit of supercompact cardinals in $V_\lambda$. Perhaps this is an answer to Q2.

For Q3, yes, there are natural large cardinal notions that imply both rank-to-rank and supercompactness. For example, if $\kappa$ is rank-to-rank with arbitrarily large targets, which means that there are $j:V_\lambda\to V_\lambda$ with critical point $\kappa$ and $j(\kappa)$ as large as desired, then $\kappa$ is both rank-to-rank and supercompact.

There are weaker notions with this property, and the weakest is simply the assumption that $\kappa$ rank-to-rank and supercompact.