In \cite{ Covolo:2016}, \cite{ Covolo:2012} and \cite{ Poncin:2016}, we introduced the category of colored supermanifolds ($\mathbb{ Z}_2^n$-super\-ma\-ni\-folds or just $\mathbb{ Z}_2^n$-manifolds ($\mathbb{ Z}_2^n=\mathbb{ Z}_2\times\ldots\times\mathbb{ Z}_2$ ($n$ times))), explicitly described the corresponding $\mathbb{ Z}_2^n$-Berezinian and gave first insights into $\mathbb{ Z}_2^n$-integration theory. The present paper contains a detailed account of parts of the $\mathbb{ Z}_2^n$-differential calculus and of the $\mathbb{ Z}_2^n$-variants of the trilogy of local theorems, which consists of the inverse function theorem, the implicit function theorem and the constant rank theorem.
Categories: math.DG