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(Ordω1) with parameters in P(ω1) 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(Ordω1) of Woodin's generic
absoluteness results for the Chang model L(Ordω). 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(Ordω1).Comment: - to appear on the Journal of the American Mathemtical Societ