A representation formula for maps on supermanifolds

We analyze the notion of morphisms of rings of superfunctions which is the basic concept underlying the definition of supermanifolds as ringed spaces (i.e., following Berezin, Leites, Manin, etc.). We establish a representation formula for all (pull-back) morphisms from the algebra of functions on an ordinary manifolds to the superalgebra of functions on an open subset of a superspace. We then derive two consequences of this result. The first one is that we can integrate the data associated with a morphism in order to get a (nonunique) map defined on an ordinary space (and uniqueness can be achieved by restriction to a scheme). The second one is a simple and intuitive recipe to compute pull-back images of a function on a manifold M by a map from a superspace to M.