  We give a generalization of the theory of $\mathbb{Z}_2$-graded manifolds to
a theory of $\mathcal{I}$-graded manifolds, where $\mathcal{I}$ is a
commutative semi-ring with some additional properties. We prove Batchelor's
theorem in this generalized setting. To our knowledge, such a proof is still
missing except for some special cases.
