Bundle map

In mathematics, a bundle map (or bundle morphism) is a function that relates two fiber bundles in a way that respects their internal structure. Fiber bundles are mathematical objects that locally look like a cartesian product of a base space and another space, the typical "fiber", but may have more complex global structure.

A bundle map typically consists of two functions: one between the total spaces of the bundles, and one between their base spaces, such that the diagram formed by the projections commutes. In some cases, both bundles share the same base space, and in others, the map includes a separate function between different base spaces.

There are several versions of bundle maps depending on the specific types of fiber bundles involved—for example, smooth bundles, vector bundles, or principal bundles—and on the category in which they are defined (e.g., topological spaces or smooth manifolds).

The first three sections of this article discusses general fiber bundles in the category of topological spaces, while the fourth section provides other examples.