Monoidally graded manifolds

TL;DR

This paper generalizes the concept of graded manifolds from the classical rac{rac{1}{2}}-grading to a broader rac{rac{1}{2}}-grading over a semi-ring, establishing foundational theorems in this new context.

Contribution

It introduces rac{rac{1}{2}}-graded manifolds over a semi-ring and proves Batchelor's theorem in this generalized setting, filling a gap in the theory.

Findings

Established a generalized theory of graded manifolds

Proved Batchelor's theorem for rac{rac{1}{2}}-graded manifolds

Extended classical results to a broader algebraic framework

Abstract

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.