Local Coordinate Spaces: a proposed unification of manifolds and fiber bundles, and associated machinery

TL;DR

This paper proposes a unified framework for manifolds and fiber bundles using local coordinate spaces, introducing new concepts like m-atlas and functors to bridge the two structures.

Contribution

It introduces the notions of m-atlas and local coordinate spaces, unifying manifolds and fiber bundles within a common categorical framework.

Findings

Special cases of the framework are equivalent to fiber bundles and manifolds.

Defines categories of atlases and constructs functors for the unified theory.

Provides a new notation and tools for studying local geometric structures.

Abstract

This paper presents a unified view of manifolds and fiber bundles, which, while superficially different, have strong parallels. It introduces the notions of an m-atlas and of a local coordinate space, and shows that special cases are equivalent to fiber bundles and manifolds. Along the way it defines some convenient notation, defines categories of atlases, and constructs potentially useful functors.