Functional analytic issues in $\mathbb{Z}_2^n$-Geometry

TL;DR

This paper establishes the functional analytic structure of the sheaf of functions on $Z_2^n$-manifolds as a nuclear Fréchet sheaf and proves the continuity of morphism components, enabling categorical product construction.

Contribution

It introduces the nuclear Fréchet sheaf structure for $Z_2^n$-manifold function sheaves and proves the continuity of morphism components, facilitating categorical product existence.

Findings

Function sheaf is a nuclear Fréchet sheaf of $Z_2^n$-graded algebras.

Components of pullback sheaf morphisms are continuous.

Results are essential for categorical product existence in $Z_2^n$-manifolds category.

Abstract

We show that the function sheaf of a $\mathbb{Z}_2^n$-manifold is a nuclear Fr\'echet sheaf of $\mathbb{Z}_2^n$-graded $\mathbb{Z}_2^n$-commutative associative unital algebras. Further, we prove that the components of the pullback sheaf morphism of a $\mathbb{Z}_2^n$-morphism are all continuous. These results are essential for the existence of categorical products in the category of $\mathbb{Z}_2^n$-manifolds. All proofs are self-contained and explicit.