Linear $\mathbb{Z}_2^n$-Manifolds and Linear Actions

We establish the representability of the general linear $\mathbb{Z}_2^n$-group and use the restricted functor of points – whose test category is the category of $\mathbb{Z}_2^n$-manifolds over a single topological point – to define its smooth linear actions on $\mathbb{Z}_2^n$-graded vector spaces and linear $\mathbb{Z}_2^n$-manifolds. Throughout the paper, particular emphasis is placed on the full faithfulness and target category of the restricted functor of points of a number of categories that we are using.