Elliptic characterization and unification of Oka maps

We generalize our elliptic characterization of Oka manifolds to Oka maps. The generalized characterization can be considered as an affirmative answer to the relative version of Gromov's conjecture. As an application, we unify previously known Oka principles for submersions; namely the Gromov type Oka principle for subelliptic submersions and the Forstneri\v{c} type Oka principle for holomorphic fiber bundles with CAP fibers. We also establish the localization principle for Oka maps which gives new examples of Oka maps.