One often refers to such a bundle simply as , even though is really part of the data.
One generally considers bundles with extra properties or structure:
The category of bundles over a given object is the over category . The collection of all bundles in a given category therefore arranges itself into the codomain fibration . As such, the descent for bundles may be expressed as monadic descent with respect to the codomain bifibration. This does in general not work inside one of the more restrictive subcategories of bundles with extra structure and property, as the push-forward operation typically does not respect these extra conditions. For more on this see monadic descent of bundles.
bundle, display map
fiber bundle / fiber ∞-bundle
A useful collection of introductory notes to fiber bundles, vector bundles and fiber bundles with connection is at