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.