Threefold Nature of Graded Vector Bundles

Abstract

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.

References in corpus (9)

• Deformation quantization of Poisson manifolds, I
• Higher vector bundles and multi-graded symplectic manifolds
• Introduction to supergeometry
• Graded bundles and homogeneity structures
• Introduction to graded geometry
• Graded Geometry, $Q$-Manifolds, and Microformal Geometry
• Global Theory of Graded Manifolds
• Graded Generalized Geometry
• The Serre-Swan Theorem in supergeometry