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I have a few somehow related questions:

Question 1. What do we expect for an inner model $\mathcal{K}$ to be a core model for a supercompact cardinal? What properties should it have, and what should be its relation with $V$.

The reason I am asking this question, is that we can not expect such a model be $L-$like, for example square principles should fail in it, from somewhere above, ....

My second question is a bout a concept which seems to play an essential role in core model theorey, namely the notion of iteration trees.

Question 2. Can one explain me in a very short way, what are ``iteration trees'' and what is their relation with core model constructions?

As far as I understand the iterability in core model constructions is important.

Question 3. How can one expect to construct a core model for supercompact cardinals? Do we expect iteration trees play any role in the construction, and if yes, how?

My finial last question is about forcing and core models. Most of known core models are known to be invariant under set generic extensions, in the sense that if $V[G]$ is a set generic extension of $V$, then $\mathcal{K}_V=\mathcal{K}_{V[G]},$ where $\mathcal{K}_V$ is the corresponding core model as computed in $V$. This is the case for example for $L$, or Dodd-Jensen core model, ...

Question 4. Is there any known core model which is not invariant under set forcing extensions? What if we consider cardinal preserving (or even tame ) class generic extensions?

Of course core models like $L$, Dodd-Jensen core model, ... are invariant under cardinal preserving class generic extensions.

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    $\begingroup$ For Q4: It is an expected property that for core models there is a $\Sigma_2$ formula, defining $K$ under an anti-large cardinal hypothesis, which is generically absolute. See the introduction of math.berkeley.edu/~steel/papers/knombljune2013.pdf For Q3: it is not yet known how to construct a core model for supercompact cardinals. There are some partial results of Woodin on this topic. If the core model for a supercompact can be constructed then a core model for all large cardinals can be constructed. For Q2: Iteration trees provide a way to... $\endgroup$ Jan 19, 2015 at 2:29
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    $\begingroup$ compare inner models with large cardinals above a Woodin. But serious problems arise at the level of a supercompact cardinals. There are results of Woodin which states that comparison fails. $\endgroup$ Jan 19, 2015 at 2:37
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    $\begingroup$ Actually I should be saying more w.r.t iteration trees: they are necessary above a Woodin because you want to preserve the generators. Linear iteration for these models will move the generators, so the way out is to take an extender from a model and to apply it to the earliest model in the tree to which it makes sense to apply it to (even though the extender may not belong to that model, the models may agree about the power set of the c.p. That way you get a tree of models. $\endgroup$ Jan 19, 2015 at 2:44

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