The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds

Abstract

Informally, ℤⁿ₂-manifolds are 'manifolds' with ℤⁿ₂-graded coordinates and a sign rule determined by the standard scalar product of their ℤⁿ₂-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in integration theory. In this paper, we reformulate the notion of a ℤⁿ₂-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ℤⁿ₂-points, i.e., trivial ℤⁿ₂-manifolds for which the reduced manifold is just a single point, as 'probes' when employing the functor of points. This allows us to construct a fully faithful restricted Yoneda embedding of the category of ℤⁿ₂-manifolds into a subcategory of contravariant functors from the category of ℤⁿ₂-points to a category of Fréchet manifolds over algebras. We refer to this embedding as the Schwarz-Voronov embedding. We further prove that the category of ℤⁿ₂-manifolds is equivalent to the full subcategory of locally trivial functors in the preceding subcategory.