Incompatible category forcing axioms

Abstract

Given a cardinal $\lambda$, category forcing axioms for $\lambda$-suitable classes $\Gamma$ are strong forcing axioms which completely decide the theory of the Chang model ${\mathcal{C}}_{\lambda }$, modulo generic extensions via forcing notions from $\Gamma$. ${\mathsf{MM}}^{+++}$ was the first category forcing axiom to be isolated (by the second author). In this paper we present, without proofs, a general theory of category forcings, and prove the existence of ${\aleph }_1$-many pairwise incompatible category forcing axioms for ${\omega }_1$-suitable classes.