Incompatible bounded category forcing axioms

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) giving rise to pairwise incompatible theories for H-omega 2.