Cabbage is cross pollinated owing to

Cabbage is cross-pollinated owing to Self-incompatibility.
Self-incompatibility in plants is an adaptation that promotes cross-pollination. Self-incompatibility (SI) is defined as the inability to produce zygotes after self-pollination in a fertile hermaphrodite plant, which has stamens and pistils in the same flower.
Cabbage is a cross-pollinated crop because of sporophytic self-incompatibility, and honey bees play an important role in its pollination.