A Grothendieck fibration (also called a cartesian fibration or fibered category or just a fibration) is a functor such that the fibers depend (contravariantly) pseudofunctorially on . One also says that is a fibered category over . Dually, in a (Grothendieck) opfibration the dependence is covariant.
The construction of a fibration from a pseudofunctor is sometimes called the Grothendieck construction, although fortunately (or unfortunately) Grothendieck performed many constructions. Less ambiguous terms for are the category of elements and the oplax colimit of .
Let be an arrow in . We say that is cartesian if for any arrow in and in such that , there exists a unique such that and . In other words, for any , any filling of the image of the following diagram under can be lifted up to a filling in :
We say that is a fibration if for any and , there is a cartesian arrow with . Such an arrow is called a “cartesian lifting” of to , and a choice of cartesian lifting for every and is called a cleavage. Thus, assuming the axiom of choice, a functor is a fibration iff it admits some cleavage.
As a side note, we say that is weakly cartesian if it has the property described above only when is an identity. One can prove that is a fibration if and only if firstly, it has the above property with “cartesian” replaced by “weakly cartesian,” and secondly, the composite of weakly cartesian arrows is weakly cartesian. In a fibration, every weakly cartesian lifting of to is in fact cartesian (as one can show by combining the universal properties of and of a given cartesian lifting of to ), but this is not true in general. Some sources say “cartesian” and “strongly cartesian” instead of “weakly cartesian” and “cartesian,” respectively.
is a cartesian morphism or morphism of fibrations if the top arrow takes cartesian arrows to cartesian arrows. Most frequently when considering morphisms of fibrations, the bottom arrow is an identity. A 2-cell between morphisms of fibrations is a pair of 2-cells, one lying over the other. If the bottom arrow is an identity, this means that the top 2-cell is “vertical” (its components lie in fibers).
Given a fibration , we obtain a pseudofunctor by sending each to the category of objects mapping onto and morphisms mapping onto . To obtain the action on morphisms, given an in and an object , we choose a cartesian arrow over and call its source . The universal factorization property of cartesian arrows then makes into a functor , and it is easy to verify that it is a pseudofunctor. The functor in the other direction is called the Grothendieck construction. This yields a strict 2-equivalence of 2-categories between
fibrations over , morphisms of fibrations over , and 2-cells over , and
In fact, this is an instance of the general theory of representability for generalized multicategories. There is a monad whose pseudoalgebras are pseudofunctors , and whose “generalized multicategories” are functors . Such a multicategory is “representable” precisely when it is a fibration, and moreover there is an induced monad on which is colax-idempotent and whose pseudoalgebras are precisely the fibrations.
Fibrations are a “nonalgebraic” structure, since the base change functors are determined by universal properties, hence uniquely up to isomorphism. By contrast, pseudofunctors are an “algebraic” structure, since the functors are specified, together with the necessary coherence data and axioms; the latter come for free in a fibration because of the universal property.
A stack, being a particular type of pseudofunctor, can also be described as a particular sort of fibration. This was the original application for which Grothendieck introduced the notion.
Let be the category of rings, and the category of pairs where is a ring and is a (left) -module. Then the evident forgetful functor is a fibration and an opfibration. The functors are given by restriction of scalars, is extension of scalars, and the right adjoint is coextension of scalars.
Let be any category with pullbacks and the free-living arrow, so that is the category of arrows and commutative squares in . Then the “codomain” functor is a fibration and opfibration. The cartesian arrows are precisely the pullback squares, and the functors are just given by composition. The right adjoints exist iff is locally cartesian closed. The term “cartesian” is motivated by this example, which is usually called the codomain fibration over .
Let be any category and let be the category of set-indexed families of objects of . The forgetful functor taking a family to its indexing set is a fibration; the functors are given by reindexing. They have left adjoints iff has small coproducts, and right adjoints iff has small products.
It is easy to verify that fibrations are closed under pullback in Cat, and that the composite of fibrations is a fibration. This latter property is notably difficult to even express in the language of pseudofunctors.
If is a fibration, then limits in can be constructed out of limits in and in the fiber categories. Specifically, given a diagram , let be the limit of , with projections . Then for each , let ; these objects form a diagram whose limit is the limit of . Dually, if is an opfibration, then colimits in can be constructed out of those in and in the fiber categories.
If is a fibration and admits an orthogonal factorization system , then there is a factorization system on , where is the class of cartesian arrows whose image in lies in , and is the class of all arrows whose image in lies in . A dual construction is possible if is an opfibration. If it is a bifibration (or more generally, an ambifibration? over ), then these together form a ternary factorization system.
Generalizing in a different direction, if is a fibration and is a factorization structure for sinks on , then admits a factorization structure for sinks , where is the class of cartesian arrows whose image in lies in , and is the class of all arrows whose image in lies in . Similarly, we can lift factorization structures for cosinks along an opfibration. To lift in the “opposite” way we require more of ; see topological concrete category.
One important special case of a fibration is when each fiber is a groupoid; these correspond to pseudofunctors . These are also called categories fibered in groupoids. A fibration is fibered in groupoids precisely when every morphism in is cartesian.
Another important special case is when each fiber is a discrete category; these correspond to functors . These are also called discrete fibrations. A functor is a discrete fibration precisely when for every and there is a unique lifting of to a morphism .
We say that is an opfibration if is a fibration. These correspond to covariant pseudofunctors . A functor that is both a fibration and an opfibration is called a bifibration. It is not hard to see that a fibration is a bifibration iff each functor has a left adjoint, written or . In many cases also has a right adjoint, written or , but this is not as easily expressible in fibrational language.
Grothendieck originally called an opfibration a cofibered category, and if the fibers are groupoids a category cofibered in groupoids (SGA I, Higher Topos Theory). However, that term has fallen out of favor in the homotopy-theory and category-theory communities (though still used sometimes in algebraic geometry), because an opfibration still has a lifting property, as is characteristic of other notions of fibration, as opposed to the extension property exhibited by cofibrations in homotopy theory.
Note that an opfibration is the same as an internal fibration in the 2-category , while it is the fibrations in the 2-category which are more deserving of the name “cofibration.”
Note also that a given pseudofunctor can be represented both by a fibration and by an opfibration . However is not the opposite category of .
There is something evil about the notion of fibration, namely the requirement that for every and there exists a such that is equal, rather than merely isomorphic, to . This is connected with the fact that we use strict fibers, rather than essential fibers, and that fibrations and pseudofunctors can be recovered from each other up to isomorphism rather than merely equivalence.
While almost any fibration between “concrete” categories that arises in practice does satisfy this evil property, it is sometimes useful to have a non-evil version, for instance when working internally in a bicategory where equality of objects doesn’t even really make sense. The correct modification, first given by Ross Street, is simply to require that for any and there exists a cartesian and an isomorphism such that ; the definition of “cartesian” is unchanged; this gives the notion of Street fibration. Every equivalence of categories is a Street fibration, which is not true for the “evil” Grothendieck fibrations, but every Street fibration can be factored as an equivalence of categories followed by a Grothendieck fibration.
We might also think that it is evil to say that the target of the cartesian arrow is equal to the given object , akin to the topological distinction between Serre fibrations and Dold fibrations, where the initial point of a lifted path can only be specified up to homotopy. However, this part of the definition is really better regarded as a typing assertion, akin to saying, in the definition of a product of two objects and , that the target of the two projections and are equal to and . Moreover, any weakening along these lines would actually be equivalent to the version above: if we demand only that for any in and , there exists a cartesian with and an isomorphism in the fiber , then you can just compose with the isomorphism to get a cartesian arrow with still. The reason this doesn’t work in topology is that paths come with parametrizations, and requiring the lower triangle (in the square drawn at Dold fibration) to commute strictly prevents the reparametrization necessary to compose with a vertical homotopy.
The idea of proto-fibration is closely related to this.
In a strict 2-category , a morphism is called a fibration if for every object , is a fibration of categories, and for every morphism , the square
is a morphism of fibrations. There is an alternate characterization in terms of comma objects and adjoints, see fibration in a 2-category. The same definition works in a bicategory, as long as we use the non-evil version above. Interpreted in Cat we obtain the explicit notion we started with.
A two-sided fibration, or a fibration from to , is a span such that is a fibration, is an opfibration, and the structure “commutes” in a natural way. Such two-sided fibrations correspond to pseudofunctors . If they are discrete, they correspond to functors , i.e. to profunctors from to .
See two-sided fibration for more details.
Note that a pseudofunctor can also be represented by an opfibration and by a fibration , but there is no simple relationship between the three categories , , and .
There is an analog for multicategories. See
There is also a notion of fibration for 2-categories that has been studied by Hermida. See n-fibration for a general version.
For (∞,1)-categories the notion of fibered category is modeled by the notion of Cartesian fibration of simplicial sets. The corresponding analog of the Grothendieck construction is discussed at (∞,1)-Grothendieck construction.
There are several alternate characterizations of when a functor is a fibration, some of which are more convenient for internalization. Here we mention a few.
Since Grothendieck fibrations are a strict notion, in what follows we denote by the strict comma category (i.e. determined up to isomorphism, not merely up to equivalence) and by the strict slice 2-category (elsewhere denoted ).
A functor is a cloven fibration if and only if the canonical functor has a right adjoint in .
First, recall that the strict slice 2-category has objects the functors , as morphisms the commuting triangles
and as 2-cells the natural transformations such that .
Next, recall that the comma category has objects the triples , with . Let denote the projection .
The canonical morphism is simply the inclusion functor of identity maps .
Somewhat imprecisely, seeing both categories and as sitting over means that functors between those should be the identity on the component, and natural transformations should have the identity as their component.
To give an adjunction it suffices to give, for each in , an object in such that and an arrow in that is universal from to . For the adjunction to live in we must have that , so the universal arrow must be of the form
and thus amounts to a choice of in such that .
The universal property of tells us that for any other morphism in from some to , i.e., for any and any pair making the square
commute, there is a unique map in such that the above square factors in as
In other words, the universal property provides a unique such that and , which exactly asserts that is a cartesian lift of .
So the existence of a right adjoint to means precisely that for each morphism a choice is given of a cartesian lift of , which means in turn that is a cloven fibration.
The following discussion brings out some interesting points about the equivalence between fibrations and pseudofunctors.
Sridhar Ramesh: I have a (possibly stupid) question about the nature of this equivalence. I assume the idea here is that moving from a cloven fibration to the corresponding pseudofunctor is in some sense “inverse” to carrying out the Grothendieck construction in the other direction. But, in trying to get a good intuition for the nuances of non-splittable fibrations, I seem to be stumbling upon just in what sense this is so. For example, consider the nontrivial group homomorphism from Z (integer addition) to Z_2 (integer addition modulo 2); this gives us a non-splittable fibration (and, for that matter, an opfibration), for which a cleavage can be readily selected. No matter what cleavage is selected, the corresponding (contravariant) pseudofunctor from Z_2 to Cat, it would appear to me, is the one which sends the unique object in Z_2 to the subcategory of Z containing only even integers (let us call this 2Z), and which sends both of Z_2’s morphisms to identity; thus, it is actually a genuine functor, and indeed, a “constant” functor. Applying the Grothendieck construction now, I would seem to get back the projection from Z_2 X 2Z onto Z_2. But can this really be equivalent to the fibration I started with? After all, Z and Z_2 X 2Z are very different groups. So either “equivalence” means something trickier here than I realize, or I keep making a mistake somewhere along the line. Either way, it’d be great if someone could help me see the light.
Mike Shulman: Good question! I think the missing subtlety is that a pseudofunctor is not uniquely determined by its action on objects and morphisms, even if its domain is a mere category (or a mere group); there are also natural coherence isomorphisms to take into account. For instance, if is the nonidentity element of , then , so even if acts by the identity on , a pseudofunctor also contains the additional data of a natural automorphism of the identity of , i.e. a (central) element of . If you start from , then depending on your cleavage your element can be anything that’s 2 mod 4, while if you start from , your element can be anything that’s 0 mod 4. Finally, there is a pseudonatural equivalence between two such pseudofunctors just when their corresponding elements differ by a multiple of 4, so you get exactly two equivalence classes of pseudofunctors, corresponding to the two groups and . Of course we are reproducing the classification of group extensions via group cohomology.
By the way, this sort of thing (by which I mean, the cohomology class that classifies some categorical structure arising as the trace of a coherence isomorphism) happens in lots of other places too. For instance, a 2-group is classified by a group , an abelian group , an action of on , and an element in . If you replace a 2-group by a skeletal one, then is the group of objects (which is strictly associative and unital, by skeletality), is the group of endomorphisms of the unit, and the action is defined by “whiskering”. The cohomology class comes from the associator isomorphism, which can (and often must) still be nontrivial even though the multiplication is “strictly associative” at the level of objects (by skeletality).
Toby: So the multiplication is strictly associative, but the -group itself is not a strict -group, since it uses a different associator than the identity. As in the example of the pseudofunctor from to , there is some additional structure here which is not trivial, even though it seems like it could be.
Sridhar Ramesh: Ah, of course, that’s what I was missing. Thanks, both of you; that clears it all up.
Another important early reference is
A concise and didactic introduction to fibrations can be found in ch.11 of
More thorough is ch.8 of
The elephant contains a lot of basic information and some good intuition1:
A standard reference that focusses on the interplay with type theory is
Bénabou’s perspective on fibrations and some results related to Moens’ thesis are documented in the highly recommendable lecture notes:
Bénabou en personne on the role of fibrations in the foundations of category theory and on differences to indexed categories:
The following lecture notes stress the original perspective of algebraic geometry:
For the use of fibrations in homotopy see e.g.
André Joyal’s take on fibrations can be found here:
The Wikipedia entry on fibered categories is okay, and contains a number of other references:
But beware that Johnstone uses the non-standard words “prone” and “supine” where most people say “cartesian” and “opcartesian” morphism! ↩