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

TL;DR

This paper develops the theory of linear actions of the general linear ${\mathbb Z}_2^n$-group on ${\mathbb Z}_2^n$-graded structures, establishing representability and functorial properties within this algebraic framework.

Contribution

It introduces the representability of the general linear ${\mathbb Z}_2^n$-group and defines its smooth linear actions using a restricted functor of points, emphasizing categorical properties.

Findings

Proves the representability of the general linear ${\mathbb Z}_2^n$-group.

Defines smooth linear actions on ${\mathbb Z}_2^n$-graded vector spaces.

Analyzes the full faithfulness of the functor of points.

Abstract

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.