In the most general sense, a bundle over an object$B$ in a category$\mathcal{C}$ is simply an object $E$ of $\mathcal{C}$ equipped with a morphism in $\mathcal{C}$ from $E$ to $B$:

$\array { E \\ \downarrow^{\mathrlap{p}} \\ B }$

One often refers to such a bundle simply as $E$, even though $B$ is really part of the data.

The category of bundles over a given object $B$ is the over category$\mathcal{C}/B$. The collection of all bundles in a given category$\mathcal{C}$ therefore arranges itself into the codomain fibration$cod : [I,\mathcal{C}] \to \mathcal{C}$. 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.