Properties
Unlike a regular manifold, a supermanifold is not entirely composed of a set of points. Instead, one takes the dual point of view that the structure of a supermanifold M is contained in its sheaf OM of "smooth functions". In the dual point of view, an injective map corresponds to a surjection of sheaves, and a surjective map corresponds to an injection of sheaves.
An alternative approach to the dual point of view is to use the functor of points.
If M is a supermanifold of dimension (p,q), then the underlying space M inherits the structure of a differentiable manifold whose sheaf of smooth functions is OM/I, where I is the ideal generated by all odd functions. Thus M is called the underlying space, or the body, of M. The quotient map OM → OM/I corresponds to an injective map M → M; thus M is a submanifold of M.
Read more about this topic: Supermanifold
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