Category forcings, MM+++, and generic absoluteness for the theory of strong forcing axioms

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 $L(Ord^omega_1)$.