Editing Bundle (mathematics) (section)

==Bundle objects==
More generally, bundles or '''bundle objects''' can be defined in any [[category (mathematics)|category]]: in a category '''C''', a bundle is simply an [[epimorphism]] π: ''E'' → ''B''.  If the category is not [[concrete category|concrete]], then the notion of a preimage of the map is not necessarily available. Therefore these bundles may have no fibers at all, although for sufficiently well behaved categories they do; for instance, for a category with [[pullback (category theory)|pullbacks]] and a [[initial object|terminal object]] 1 the points of ''B'' can be identified with morphisms ''p'':1→''B'' and the fiber of ''p'' is obtained as the pullback of ''p'' and π. The category of bundles over ''B'' is a subcategory of the [[slice category]] ('''C'''↓''B'') of objects over ''B'', while the category of bundles without fixed base object is a subcategory of the [[comma category]] (''C''↓''C'') which is also the [[functor category]] '''C'''², the category of [[morphism]]s in '''C'''.

The category of smooth vector bundles is a bundle object over the category of smooth manifolds in '''Cat''', the [[category of small categories]].  The [[functor]] taking each manifold to its [[tangent bundle]] is an example of a section of this bundle object.