  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.

Algebraic Topology