The category of $\mathbb{Z}_2^n$-supermanifolds

In Physics and in Mathematics $\mathbb{Z}_2^n$-gradings, $n>1$, appear in various fields. The corresponding sign rule is determined by the `scalar product' of the involved $\mathbb{Z}_2^n$-degrees. The $\mathbb{Z}_2^n$-Supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise. In this article we develop the foundations of the theory: we define $\mathbb{Z}_2^n$-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 $\mathbb{Z}_2^\bullet$-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 $\mathbb{Z}_2^n$-supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the $\mathbb{Z}_2^n$-context.