# Extendible and enhanced supercompact cardinals

The paper "The large cardinals between supercompact and almost-huge" (2013) by Norman Perlmutter makes the following definition:

A cardinal $$\kappa$$ is enhanced supercompact if and only if there exists a strong cardinal $$\theta \gt \kappa$$ such that for every cardinal $$\lambda \gt \theta$$, there exists a $$\lambda$$-supercompactness embedding $$j : V \to M$$ such that $$\theta$$ is strong in $$M$$.

The follow-up paper "On extensions of supercompactness" (2015) by Robert Lubarsky and Norman Perlmutter proves that enhanced supercompact cardinals and extendible cardinals are hypercompact limits of hypercompact cardinals and that, if $$\kappa$$ is extendible and there is a strong cardinal greater than $$\kappa$$, $$\kappa$$ is an enhanced supercompact limit of enhanced supercompact cardinals.

Is the existence of an extendible cardinal, not necessarily with a greater strong cardinal, stronger or weaker than the existence of an enhanced supercompact cardinal?

Suppose that there is a $$\lambda$$-supercompactness embedding $$j : V \to M$$ (that is, $$M^\lambda \subset M$$) with critical point $$\kappa$$ such that $$\theta \lt \lambda$$ is strong in both $$V$$ and $$M$$. By the $$\lambda$$-closure of $$M$$, $$j \upharpoonright V_\theta \in M$$, and $$j \upharpoonright V_\theta$$ witnesses that $$M\vDash \text{\kappa is \theta-extendible}$$. For any $$\gamma \lt \theta$$, the $$\gamma$$-extendibility of $$\kappa$$ in $$M$$ is absolute to $$V_\theta$$ (which is the true $$V_\theta$$ since $$M^\lambda \subset M$$) since $$\theta$$ is strong, and thus $$\Sigma_2$$-reflecting, in $$M$$. Thus, $$\kappa$$ is extendible in $$V_\theta$$ and, by the usual reflection argument using $$j$$, $$\kappa$$ is a limit of cardinals that are likewise extendible up to their respective next strong cardinal.