  We introduce bounded category forcing axioms for well-behaved classes
$\Gamma$. These are strong forms of bounded forcing axioms which completely
decide the theory of some initial segment of the universe
$H_{\lambda_\Gamma^+}$ modulo forcing in $\Gamma$, for some cardinal
$\lambda_\Gamma$ naturally associated to $\Gamma$. These axioms naturally
extend projective absoluteness for arbitrary set-forcing--in this situation
$\lambda_\Gamma=\omega$--to classes $\Gamma$ with $\lambda_\Gamma>\omega$.
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
$\Gamma$ with $\lambda_\Gamma=\omega_1$, and giving rise to pairwise
incompatible theories for $H_{\omega_2}$.
