Towards a taxonomy of atlases and of morphisms between them

TL;DR

This paper develops a unified categorical framework for atlases of manifolds and fiber bundles, highlighting their structural similarities and defining functors between their categories.

Contribution

It introduces a formal, unified categorical approach to atlases for manifolds and fiber bundles, expanding on previous work with detailed definitions and notation.

Findings

Categories of atlases are formally defined.

Functors between atlas categories are constructed.

Provides a unified view of manifolds and fiber bundles.

Abstract

Manifolds and fiber bundles, while superficially different, have strong parallels; in particular, they are both defined in terms of equivalence classes of atlases or in terms of maximal atlases, with the atlases treated as mere adjuncts. This paper presents a unified view of atlases for manifolds and fiber bundles as mathematical entities in their own right. It defines some convenient notation, defines categories of atlases and defines functors among them. The paper "Local Coordinate Spaces: a proposed unification of manifolds with fiber bundles, and associated machinery" (Arxiv:1801.05775) introduced some of the ideas presented here, but many of the details are not needed there. This paper fleshes out the concepts in more detail than would be relevant there.