nLab
cleavage

Idea

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.

Definition

Let $p:E\to B$ be a functor. A cleavage of $p$ is a choice, for each $e\in E$ and $u:b\to p(e)$ in $B$, of a single cartesian arrow $f:e\prime \to e$ such that $p(f)=u$. Evidently if there exists a cleavage for $p$, then $p$ is a Grothendieck fibration. If $p$ 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 $f$ is an identity whenever $u$ is, the cleavage is said to be normal, and if each composite $f_2{f}_1$ is the specified lifting of $u_2{u}_1$, 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.)

Cloven fibrations vs pseudofunctors

Given a cleavage of $p$ and an arrow $u:b\prime \to b$ in $B$, there is a functor $u^{*}:p^{-1}(b)\to p^{-1}(b\prime )$ which to every object $e\in p^{-1}(b)$ assigns the domain of the specified arrow in the cleavage which is above $u$ and whose codomain is $e$. This correspondence extends to a functor, thanks to the universal property of the cartesian arrows.

The functor $u^{*}$ may be called either the inverse image functor along $u$, or the direct image functor of $u$, depending on the context; see the remarks on notation at domain opfibration. It depends on the choice of cleavage as well as on $u$, although different cleavages produce canonically naturally isomorphic functors. If one doesn’t choose a cleavage, then $u^{*}$ can still be defined as an anafunctor.

The direct image functors corresponding to varying morphisms in $B$ together form a pseudofunctor $B^{\mathrm{op}}\to \mathrm{Cat}$. 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 $B^{\mathrm{op}}\to \mathrm{Cat}$. 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.

Non-strict cleavages

There is a corresponding notion of cleavage for a Street fibration, namely a choice of, for each $e\in E$ and $u:b\to p(e)$ in $B$, a cartesian arrow $f:e\prime \to e$ in $E$ and an isomorphism $v:p(e\prime )\stackrel{\cong }{\to }b$ such that $u\circ v=p(f)$. Such a cleavage induces, for every $u:b\prime \to b$, a functor between the essential fibers of $b$ and $b\prime$, and thereby a pseudofunctor and another equivalence of 2-categories (though not a strict 2-equivalence).