Being a Grothendieck fibration is a property-like structure on a functor, like the existence of limits in a category: it is defined by the existence of certain objects (in this case, cartesian morphisms) which, when they exist, are unique up to unique isomorphism. Any property-like structure can be “algebraicized” by requiring a specific choice of the objects that are required to exist; a cleavage is this “algebraicization” of being a fibration.
Let be a functor. A cleavage of is a choice, for each and in , of a single cartesian arrow such that . Evidently if there exists a cleavage for , then is a Grothendieck fibration. If is equipped with a cleavage, it said to be cloven. Conversely, if we assume the axiom of choice, then every fibered category has a cleavage.
If the cartesian arrow is an identity whenever is, the cleavage is said to be normal, and if each composite is the specified lifting of , the cleavage is said to be a splitting. Any cleavage can be modified to become normal, but not neccesarily to become split. (Any fibration is, however, equivalent to a split fibration.)
Given a cleavage of and an arrow in , there is a functor which to every object assigns the domain of the specified arrow in the cleavage which is above and whose codomain is . This correspondence extends to a functor, thanks to the universal property of the cartesian arrows.
The functor may be called either the inverse image functor along , or the direct image functor of , depending on the context; see the remarks on notation at domain opfibration. It depends on the choice of cleavage as well as on , although different cleavages produce canonically naturally isomorphic functors. If one doesn’t choose a cleavage, then can still be defined as an anafunctor.
The direct image functors corresponding to varying morphisms in together form a pseudofunctor . The inverse Grothendieck construction produces a cloven fibration from a pseudofunctor, setting up an equivalence of 2-categories (in fact, a strict 2-equivalence of strict 2-categories) between cloven fibrations and pseudofunctors . If we either assume the axiom of choice, or allow the morphisms in Cat to be anafunctors, then this extends to an equivalence between pseudofunctors and not-necessarily-cloven fibrations.
There is a corresponding notion of cleavage for a Street fibration, namely a choice of, for each and in , a cartesian arrow in and an isomorphism such that . Such a cleavage induces, for every , a functor between the essential fibers of and , and thereby a pseudofunctor and another equivalence of 2-categories (though not a strict 2-equivalence).