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