Supermanifold - Batchelor's Theorem

Batchelor's Theorem

Batchelor's theorem states that every supermanifold is noncanonically isomorphic to a supermanifold of the form ΠE. The word "noncanonically" prevents one from concluding that supermanifolds are simply glorified vector bundles; although the functor Π maps surjectively onto the isomorphism classes of supermanifolds, it is not an equivalence of categories.

The proof of Batchelor's theorem relies in an essential way on the existence of a partition of unity, so it does not hold for complex or real-analytic supermanifolds.