Incompatible bounded category forcing axioms

We introduce bounded category forcing axioms for well-behaved classes $Γ$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{λ_Γ^+}$ modulo forcing in $Γ$, for some cardinal $λ_Γ$ naturally associated to $Γ$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $λ_Γ=ω$--to classes $Γ$ with $λ_Γ>ω$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions on $V$. We also show the existence of many classes $Γ$ with $λ_Γ=ω_1$, and giving rise to pairwise incompatible theories for $H_{ω_2}$.

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