Bundle map  In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the category of topological spaces. Then in the fourth section, some other examples will be given. Let "π""E":"E"→ "M" and "π""F":"F"→ "M" be fiber bundles over a space "M". Then a bundle map from "E" to "F" over "M" is a continuous map formula_1 such that formula_2. That is, the diagram should commute. Equivalently, for any point "x" in "M", formula_3 maps the fiber "E""x" = "π""E"−1({"x"}) of "E" over "x" to the fiber "F""x" = "π""F"−1({"x"}) of "F" over "x". Let π"E":"E"→ "M" and π"F":"F"→ "N" be fiber bundles over spaces "M" and "N" respectively. Then a continuous map formula_1 is called a bundle map from "E" to "F" if there is a continuous map "f":"M"→ "N" such that the diagram commutes, that is, formula_5. In other words, formula_3 is fiber-preserving, and "f" is the induced map on the space of fibers of "E": since π"E" is surjective, "f" is uniquely determined by formula_3. For a given "f", such a bundle map formula_3 is said to be a bundle map "covering f". It follows immediately from the definitions that a bundle map over "M" (in the first sense) is the same thing as a bundle map covering the identity map of "M". Conversely, general bundle maps can be reduced to bundle maps over a fixed base space using the notion of a pullback bundle. If π"F":"F"→ "N" is a fiber bundle over "N" and "f":"M"→ "N" is a continuous map, then the pullback of "F" by "f" is a fiber bundle "f"*"F" over "M" whose fiber over "x" is given by ("f"*"F")"x" = "F""f"("x").