Editing Vector bundle (section)

==Vector bundle morphisms==
A '''[[morphism]]''' from the vector bundle {{pi}}<sub>1</sub>: ''E''<sub>1</sub> → ''X''<sub>1</sub> to the vector bundle {{pi}}<sub>2</sub>: ''E''<sub>2</sub> → ''X''<sub>2</sub> is given by a pair of continuous maps ''f'': ''E''<sub>1</sub> → ''E''<sub>2</sub> and ''g'': ''X''<sub>1</sub> → ''X''<sub>2</sub> such that
: ''g''&nbsp;∘&nbsp;{{pi}}<sub>1</sub> = {{pi}}<sub>2</sub>&nbsp;∘&nbsp;''f''
:: [[File:BundleMorphism-01.png]]
: for every ''x'' in ''X''<sub>1</sub>, the map {{pi}}<sub>1</sub><sup>−1</sup>({''x''}) → {{pi}}<sub>2</sub><sup>−1</sup>({''g''(''x'')}) [[induced map|induced]] by ''f'' is a [[linear map]] between vector spaces.

Note that ''g'' is determined by ''f'' (because {{pi}}<sub>1</sub> is surjective), and ''f'' is then said to '''cover ''g'''''.

The class of all vector bundles together with bundle morphisms forms a [[category (mathematics)|category]]. Restricting to vector bundles for which the spaces are manifolds (and the bundle projections are smooth maps) and smooth bundle morphisms we obtain the category of smooth vector bundles. Vector bundle morphisms are a special case of the notion of a [[bundle map]] between [[fiber bundle]]s, and are sometimes called '''(vector) bundle homomorphisms'''.

A bundle homomorphism from ''E''<sub>1</sub> to ''E''<sub>2</sub> with an [[Inverse element|inverse]] which is also a bundle homomorphism (from ''E''<sub>2</sub> to ''E''<sub>1</sub>) is called a '''(vector) bundle isomorphism''', and then ''E''<sub>1</sub> and ''E''<sub>2</sub> are said to be '''isomorphic''' vector bundles. An isomorphism of a (rank ''k'') vector bundle ''E'' over ''X'' with the trivial bundle (of rank ''k'' over ''X'') is called a '''trivialization''' of ''E'', and ''E'' is then said to be '''trivial''' (or '''trivializable'''). The definition of a vector bundle shows that any vector bundle is '''locally trivial'''.

We can also consider the category of all vector bundles over a fixed base space ''X''. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the [[identity function|identity map]] on ''X''. That is, bundle morphisms for which the following diagram [[commutative diagram|commutes]]:
: [[File:BundleMorphism-02.png]]

(Note that this category is ''not'' [[abelian category|abelian]]; the [[kernel (category theory)|kernel]] of a morphism of vector bundles is in general not a vector bundle in any natural way.)

A vector bundle morphism between vector bundles {{pi}}<sub>1</sub>: ''E''<sub>1</sub> → ''X''<sub>1</sub> and {{pi}}<sub>2</sub>: ''E''<sub>2</sub> → ''X''<sub>2</sub> covering a map ''g'' from ''X''<sub>1</sub> to ''X''<sub>2</sub> can also be viewed as a vector bundle morphism over ''X''<sub>1</sub> from ''E''<sub>1</sub> to the [[pullback bundle]] ''g''*''E''<sub>2</sub>.