There are several variations on the definition of “graded manifold”, and it can be a bit confusing for a newcomer. The goal of this post is to describe one definition that rules them all. This definition hopefully sheds some light on the relationships between the various types of graded manifolds.
I thought I had seen the definition in Kostant’s early treatise “Graded manifolds, graded Lie theory, and prequantization,” but, looking back at the paper now, I can’t find the definition there. In any case, I don’t claim any originality here. If anyone can remind me of the correct reference, I would appreciate it.