Threefold Nature of Graded Vector Bundles

Graded vector bundles over a given $\mathbb{Z}$-graded manifold can be defined in three different ways: certain sheaves of graded modules over the structure sheaf of the base graded manifold, finitely generated projective graded modules over the algebra of global functions on the base graded manifold, or locally trivial graded manifolds with a suitable linear structure. We argue that all three approaches are the same. More precisely, the respective categories are proved to be equivalent.