Local Forms of Morphisms of Colored Supermanifolds

Abstract

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 (${\mathbb{Z}}_2^n={\mathbb{Z}}_2\times \dots \times {\mathbb{Z}}_2$ ($n$ 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.

21 pages. arXiv admin note: text overlap with arXiv:1608.00949