Axiom A and supercompactness

We produce a model where every supercompact cardinal is C^ (1) -supercompact with inaccessible targets. This is a significant improvement of the main identity-crises configuration obtained in HMP and provides a definitive answer to a question of Bagaria. 19Bag. This configuration is a consequence of a new axiom we introduce -- called A -- which is showed to be compatible with Woodin's I₀ cardinals. We also answer a question of V. Gitman and G. Goldberg on the relationship between supercompactness and cardinal-preserving extendibility. As an incidental result, we prove a theorem suggesting that supercompactness is the strongest large-cardinal notion preserved by Radin forcing.