Category forcings, $MM^{+++}$, and generic absoluteness for the theory of strong forcing axioms

TL;DR

This paper introduces a new category of stationary set preserving forcings, explores its properties under large cardinal assumptions, and establishes generic absoluteness results for the theory of certain inner models, advancing the understanding of strong forcing axioms.

Contribution

It defines a category of stationary set preserving forcings, introduces the axiom $MM^{+++}$, and proves generic invariance of the theory of $L(Ord^{ ext{ extomega}_1})$ under these forcings.

Findings

The cutoff of the category at a supercompact limit produces a forcing that forces $MM^{++}$.

Models obtained by known methods collapse a superhuge cardinal and relate to presaturated towers.

The theory of $L(Ord^{ ext{ extomega}_1})$ is generically invariant under forcings preserving $MM^{+++}$.

Abstract

We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces $MM^{++}$ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of $MM^{++}$ collapsing a superhuge cardinal to become the second uncountable cardinal produces a model in which the cutoff of the category of stationary set preserving forcings at any rank initial segment of the universe of large enough height is forcing equivalent to a presaturated tower of normal filters. We let $MM^{+++}$ denote this statement and we prove that the theory of $L(Ord^{\omega_1})$ with parameters in $P(\omega_1)$ is generically invariant for stationary set preserving forcings that preserve $MM^{+++}$. Finally we argue that the work of Larson and Asper\'o shows that this is a next to optimal generalization to the Chang model $L(Ord^{\omega_1})$ of Woodin's generic absoluteness results for the Chang model $L(Ord^{\omega})$. It remains open whether $MM^{+++}$ and $MM^{++}$ are equivalent axioms modulo large cardinals and whether $MM^{++}$ suffices to prove the same generic absoluteness results for the Chang model $L(Ord^{\omega_1})$.