$\mathrm{\infty }$-Categorical Approaches to Hodge-Iwasawa Theory II: $\mathrm{\infty }$-Categorical and Derived Hodge-Iwasawa Modules

This paper is a discussion on $\mathrm{\infty }$-categorical approaches to Hodge-Iwasawa Theory, which was initiated in our project on the $\mathrm{\infty }$-categorical approaches to Hodge-Iwasawa Theory. The theory aims at the serious unification of $p$-adic Hodge Theory and $p$-adic Iwasawa Theory, by taking deformation of Hodge-theoretic constructions along some consideration in Iwasawa Theory beyond the Iwasawa deformation of certain motives in the general sense. The Hodge modules in our current consideration will be essentially within $\mathrm{\infty }$-categorical derived categories of inductive Banach modules and $\mathrm{\infty }$-categorical derived categories of condensed solidification of certain topological modules.