The category of (Zn2)-supermanifolds

In physics and in mathematics -gradings, n ≥ 2, appear in various fields. The corresponding sign rule is determined by the "scalar product" of the involved -degrees. The -supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (respectively, odd) coordinates do not necessarily commute (respectively, anticommute) pairwise. In this article we develop the foundations of the theory: we define -supermanifolds and provide examples in the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the class of -supermanifolds is closed with respect to the tangent and cotangent functors. We explain that any n-fold vector bundle has a canonical "superization" to a -supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the -context.