Chang models over derived models with supercompact measures

Abstract

Based on earlier work of the third author, we construct a Chang-type model with supercompact measures extending a derived model of a given hod mouse with a regular cardinal $\delta$ that is both a limit of Woodin cardinals and a limit of ${<}\delta$-strong cardinals. The existence of such a hod mouse is consistent relative to a Woodin cardinal that is a limit of Woodin cardinals. We argue that our Chang-type model satisfies $\mathsf{AD}_{\mathbb{R}} + \Theta$ is regular + $\omega_1$ is ${<}\delta_{\infty}$-supercompact for some regular cardinal $\delta_{\infty}>\Theta$. This complements Woodin's generalized Chang model, which satisfies $\mathsf{AD}_{\mathbb{R}}+\omega_1$ is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals.