The category of Z2 n -supermanifolds

In physics and in mathematics Z2n-gradings, n ≥ 2, appear in various fields. The corresponding sign rule is determined by the "scalar product" of the involved Z2n-degrees. The Z2n-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 Z2n-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 Z2•-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 Z2n-supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the Z2n-context.