Incompatible category forcing axioms

TL;DR

This paper develops a general theory of category forcing axioms, demonstrating the existence of many incompatible axioms for certain classes, which influence the structure of the associated Chang models.

Contribution

It introduces a broad framework for category forcing axioms and proves the existence of numerous incompatible axioms for \\omega_1-suitable classes.

Findings

Existence of \\aleph_1$-many incompatible category forcing axioms.

Development of a general theory of category forcings.

Implications for the theory of Chang models.

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.