As stated in this [paper][1], a longstanding open problem in set theory is to construct a canonical inner model for supercompactness.  In general there are various ways of spelling out what a “canonical inner model of large cardinal axiom A” means.  The linked [paper][1], as well as other work of Woodin, suggests the following weak thesis:

**Thesis 1:** If there are canonical inner models of supercompactness, at least one of them should be a *weak extender model* (as defined in [1]).

As it is said in the last paragraph on page 16 in the linked paper, supported by Example 27, the exhibition of a weak extender model of supercompactness is not sufficient to solve the inner model problem for supercompacts.  Of course , $V$ is trivially such a model. This is why Thesis 1 is a weak thesis.

Now, due to [difficulties][2] involved in spelling out in a general context what a canonical inner model is, the consensus view seems to be that a “canonical inner model of large cardinal axiom A” should be a proper-class iterate of a mouse.  A mouse is a small iterable model of a sufficient fragment of ZFC + A, which is canonical because of the Comparison Lemma.  In particular, it makes sense to speak of “the smallest mouse for ZFC + A”. For example $M_1^\sharp$ is the smallest mouse above a Woodin cardinal, and $M_1$ is the canonical inner model of one Woodin cardinal obtained by iterating the top measure of $M_1^\sharp$ out of the ordinals.  So based these general considerations, we have:

**Thesis 2:** All canonical inner models of sufficiently large cardinals are iterates of mice.

The low-level versions of these theses are in harmony.  For example, assuming the existence of a measurable and a sharp for measurability, a weak extender model for measurability is also an iterate of a mouse.  But they seem to come into conflict around supercompactness.  It is shown in [1] that if $\delta$ is extendible and $M$ is a weak extender model for the supercompactness of $\delta$, then there is no $j : M \to M$ with critical point $\geq \delta$.  On the other hand, iterates of mice (through all ordinals) are, by their construction, self-embeddable with arbitrarily high critical point.

**Question:** How do we resolve the “paradox” and clarify the inner model problem?


  [1]: https://arxiv.org/abs/1605.00613
  [2]: https://mathoverflow.net/questions/322462/does-inner-model-theory-seek-canonical-models-for-large-cardinals