Products in the category of $\mathbb{Z}_2 ^n$-manifolds

TL;DR

This paper establishes that the category of $Z_2^n$-manifolds has all finite products and that these manifolds can be reconstructed from their algebra of global functions, advancing the understanding of their structure.

Contribution

It proves the existence of finite products in the category of $Z_2^n$-manifolds and shows reconstruction from algebraic data, which is crucial for studying $Z_2^n$ Lie groups.

Findings

Category of $Z_2^n$-manifolds has all finite products.

Manifolds can be reconstructed from their algebra of global functions.

Addresses challenges with tensor products of structure sheaves.

Abstract

We prove that the category of $\mathbb{Z}_2 ^n$-manifolds has all finite products. Further, we show that a $\mathbb{Z}_2 ^n$-manifold (resp., a $\mathbb{Z}_2 ^n$-morphism) can be reconstructed from its algebra of global $\mathbb{Z}_2 ^n$-functions (resp., from its algebra morphism between global $\mathbb{Z}_2 ^n$-function algebras). These results are of importance in the study of $\mathbb{Z}_2 ^n$ Lie groups. The investigation is all the more challenging, since the completed tensor product of the structure sheafs of two $\mathbb{Z}_2 ^n$-manifolds is not a sheaf. We rely on a number of results on (pre)sheaves of topological algebras, which we establish in the appendix.