Informally, Z n 2 -manifolds are 'manifolds' with Z n 2 -graded coordinates and a sign rule determined by the standard scalar product of their Z n 2 -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 Z n 2 -manifold within a categorical framework via the functor of points. We show that it is sufficient to consider Z n 2 -points, i.e., trivial Z n 2 -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 Z n 2 -manifolds into a subcategory of contravariant functors from the category of Z n 2 -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 Z n 2 -manifolds is equivalent to the full subcategory of locally trivial functors in the preceding subcategory.

We show how the theory of Z n 2 -manifolds -which are a non-trivial generalisation of supermanifolds -may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed.

We establish the representability of the general linear Z n 2 -group and use the restricted functor of points -whose test category is the category of Z n 2 -manifolds over a single topological point -to define its smooth linear actions on Z n 2 -graded vector spaces and linear Z n 2 -manifolds. Throughout the paper, particular emphasis is placed on the full faithfulness and target category of the restricted functor of points of a number of categories that we are using.