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 p:EBp\colon E\to B be a functor. A cleavage of pp is a choice, for each eEe\in E and u:bp(e)u\colon b\to p(e) in BB, of a single cartesian arrow f:eef:e'\to e such that p(f)=up(f) = u. Evidently if there exists a cleavage for pp, then pp is a Grothendieck fibration. If pp 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 ff is an identity whenever uu is, the cleavage is said to be normal, and if each composite f 2f 1f_2 f_1 is the specified lifting of u 2u 1u_2 u_1, the cleavage is said to be a splitting. Any cleavage can be modified to become normal, but not necessarily to become split. (Any fibration is, however, equivalent to a split fibration.)

Cloven fibrations vs pseudofunctors

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

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

The direct image functors corresponding to varying morphisms in BB together form a pseudofunctor B opCatB^{op}\to 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 opCatB^{op} \to 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 eEe\in E and u:bp(e)u\colon b\to p(e) in BB, a cartesian arrow f:eef:e'\to e in EE and an isomorphism v:p(e)bv\colon p(e') \xrightarrow{\cong} b such that uv=p(f)u\circ v = p(f). Such a cleavage induces, for every u:bbu\colon b'\to b, a functor between the essential fibers of bb and bb', and thereby a pseudofunctor and another equivalence of 2-categories (though not a strict 2-equivalence).

Last revised on August 9, 2016 at 07:54:37. See the history of this page for a list of all contributions to it.