Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images (or pull-backs) of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space X to another topological space Y is associated the pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories (over a site) with "descent". Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories. Fibred categories were introduced by, and developed in more detail by.

Background and motivations

There are many examples in topology and geometry where some types of objects are considered to exist on or above or over some underlying base space. The classical examples include vector bundles, principal bundles, and sheaves over topological spaces. Another example is given by "families" of algebraic varieties parametrised by another variety. Typical to these situations is that to a suitable type of a map f:X\to Y between base spaces, there is a corresponding inverse image (also called pull-back) operation f^* taking the considered objects defined on Y to the same type of objects on X. This is indeed the case in the examples above: for example, the inverse image of a vector bundle E on Y is a vector bundle f^*(E) on X. Moreover, it is often the case that the considered "objects on a base space" form a category, or in other words have maps (morphisms) between them. In such cases the inverse image operation is often compatible with composition of these maps between objects, or in more technical terms is a functor. Again, this is the case in examples listed above. However, it is often the case that if g:Y\to Z is another map, the inverse image functors are not strictly compatible with composed maps: if z is an object over Z (a vector bundle, say), it may well be that Instead, these inverse images are only naturally isomorphic. This introduction of some "slack" in the system of inverse images causes some delicate issues to appear, and it is this set-up that fibred categories formalise. The main application of fibred categories is in descent theory, concerned with a vast generalisation of "glueing" techniques used in topology. In order to support descent theory of sufficient generality to be applied in non-trivial situations in algebraic geometry the definition of fibred categories is quite general and abstract. However, the underlying intuition is quite straightforward when keeping in mind the basic examples discussed above.

Formal definitions

There are two essentially equivalent technical definitions of fibred categories, both of which will be described below. All discussion in this section ignores the set-theoretical issues related to "large" categories. The discussion can be made completely rigorous by, for example, restricting attention to small categories or by using universes.

Cartesian morphisms and functors

If \phi:F\to E is a functor between two categories and S is an object of E, then the subcategory of F consisting of those objects x for which \phi(x)=S and those morphisms m satisfying, is called the fibre category (or fibre) over S, and is denoted F_S. The morphisms of F_S are called S-morphisms, and for x,y objects of F_S, the set of S-morphisms is denoted by. The image by \phi of an object or a morphism in F is called its projection (by \phi). If f is a morphism of E, then those morphisms of F that project to f are called f-morphisms, and the set of f-morphisms between objects x and y in F is denoted by. A morphism m:x\to y in F is called \phi-cartesian (or simply cartesian) if it satisfies the following condition: A cartesian morphism m:x\to y is called an inverse image of its projection f=\phi(m); the object x is called an inverse image of y by f. The cartesian morphisms of a fibre category F_S are precisely the isomorphisms of F_S. There can in general be more than one cartesian morphism projecting to a given morphism f:T\to S, possibly having different sources; thus there can be more than one inverse image of a given object y in F_S by f. However, it is a direct consequence of the definition that two such inverse images are isomorphic in F_T. A functor \phi:F\to E is also called an E-category, or said to make F into an E-category or a category over E. An E-functor from an E-category \phi:F\to E to an E-category \psi:G\to E is a functor such that. E-categories form in a natural manner a 2-category, with 1-morphisms being E-functors, and 2-morphisms being natural transformations between E-functors whose components lie in some fibre. An E-functor between two E-categories is called a cartesian functor if it takes cartesian morphisms to cartesian morphisms. Cartesian functors between two E-categories F,G form a category, with natural transformations as morphisms. A special case is provided by considering E as an E-category via the identity functor: then a cartesian functor from E to an E-category F is called a cartesian section. Thus a cartesian section consists of a choice of one object x_S in F_S for each object S in E, and for each morphism f:T\to S a choice of an inverse image. A cartesian section is thus a (strictly) compatible system of inverse images over objects of E. The category of cartesian sections of F is denoted by In the important case where E has a terminal object e (thus in particular when E is a topos or the category E_{/S} of arrows with target S in E) the functor is fully faithful (Lemma 5.7 of Giraud (1964)).

Fibred categories and cloven categories

The technically most flexible and economical definition of fibred categories is based on the concept of cartesian morphisms. It is equivalent to a definition in terms of cleavages, the latter definition being actually the original one presented in Grothendieck (1959); the definition in terms of cartesian morphisms was introduced in Grothendieck (1971) in 1960–1961. An E category \phi:F\to E is a fibred category (or a fibred E-category, or a category fibred over E) if each morphism f of E whose codomain is in the range of projection has at least one inverse image, and moreover the composition m\circ n of any two cartesian morphisms m,n in F is always cartesian. In other words, an E-category is a fibred category if inverse images always exist (for morphisms whose codomains are in the range of projection) and are transitive. If E has a terminal object e and if F is fibred over E, then the functor \epsilon from cartesian sections to F_e defined at the end of the previous section is an equivalence of categories and moreover surjective on objects. If F is a fibred E-category, it is always possible, for each morphism f:T\to S in E and each object y in F_S, to choose (by using the axiom of choice) precisely one inverse image m:x\to y. The class of morphisms thus selected is called a cleavage and the selected morphisms are called the transport morphisms (of the cleavage). A fibred category together with a cleavage is called a cloven category. A cleavage is called normalised if the transport morphisms include all identities in F; this means that the inverse images of identity morphisms are chosen to be identity morphisms. Evidently if a cleavage exists, it can be chosen to be normalised; we shall consider only normalised cleavages below. The choice of a (normalised) cleavage for a fibred E-category F specifies, for each morphism f:T\to S in E, a functor ; on objects f^* is simply the inverse image by the corresponding transport morphism, and on morphisms it is defined in a natural manner by the defining universal property of cartesian morphisms. The operation which associates to an object S of E the fibre category F_S and to a morphism f the inverse image functor f^* is almost a contravariant functor from E to the category of categories. However, in general it fails to commute strictly with composition of morphisms. Instead, if f:T\to S and g:U\to T are morphisms in E, then there is an isomorphism of functors These isomorphisms satisfy the following two compatibilities: It can be shown (see Grothendieck (1971) section 8) that, inversely, any collection of functors together with isomorphisms c_{f,g} satisfying the compatibilities above, defines a cloven category. These collections of inverse image functors provide a more intuitive view on fibred categories; and indeed, it was in terms of such compatible inverse image functors that fibred categories were introduced in Grothendieck (1959). The paper by Gray referred to below makes analogies between these ideas and the notion of fibration of spaces. These ideas simplify in the case of groupoids, as shown in the paper of Brown referred to below, which obtains a useful family of exact sequences from a fibration of groupoids.

Splittings and split fibred categories

A (normalised) cleavage such that the composition of two transport morphisms is always a transport morphism is called a splitting, and a fibred category with a splitting is called a split (fibred) category. In terms of inverse image functors the condition of being a splitting means that the composition of inverse image functors corresponding to composable morphisms f, g in E equals the inverse image functor corresponding to f\circ g. In other words, the compatibility isomorphisms c_{f, g} of the previous section are all identities for a split category. Thus split E-categories correspond exactly to true functors from E to the category of categories. Unlike cleavages, not all fibred categories admit splittings. For an example, see below.

Co-cartesian morphisms and co-fibred categories

One can invert the direction of arrows in the definitions above to arrive at corresponding concepts of co-cartesian morphisms, co-fibred categories and split co-fibred categories (or co-split categories). More precisely, if \phi:F\to E is a functor, then a morphism m:x\to y in F is called co-cartesian if it is cartesian for the opposite functor. Then m is also called a direct image and y a direct image of x for f=\phi(m). A co-fibred E-category is an E-category such that direct image exists for each morphism in E and that the composition of direct images is a direct image. A co-cleavage and a co-splitting are defined similarly, corresponding to direct image functors instead of inverse image functors.

Properties

The 2-categories of fibred categories and split categories

The categories fibred over a fixed category E form a 2-category, where the category of morphisms between two fibred categories F and G is defined to be the category of cartesian functors from F to G. Similarly the split categories over E form a 2-category (from French catégorie scindée), where the category of morphisms between two split categories F and G is the full sub-category of E-functors from F to G consisting of those functors that transform each transport morphism of F into a transport morphism of G. Each such morphism of split E-categories is also a morphism of E-fibred categories, i.e.,. There is a natural forgetful 2-functor that simply forgets the splitting.

Existence of equivalent split categories

While not all fibred categories admit a splitting, each fibred category is in fact equivalent to a split category. Indeed, there are two canonical ways to construct an equivalent split category for a given fibred category F over E. More precisely, the forgetful 2-functor admits a right 2-adjoint S and a left 2-adjoint L (Theorems 2.4.2 and 2.4.4 of Giraud 1971), and S(F) and L(F) are the two associated split categories. The adjunction functors S(F)\to F and F\to L(F) are both cartesian and equivalences (ibid.). However, while their composition is an equivalence (of categories, and indeed of fibred categories), it is not in general a morphism of split categories. Thus the two constructions differ in general. The two preceding constructions of split categories are used in a critical way in the construction of the stack associated to a fibred category (and in particular stack associated to a pre-stack).

Categories fibered in groupoids

There is a related construction to fibered categories called categories fibered in groupoids. These are fibered categories such that any subcategory of \mathcal{F} given by is a groupoid denoted. The associated 2-functors from the Grothendieck construction are examples of stacks. In short, the associated functor sends an object c to the category, and a morphism d \to c induces a functor from the fibered category structure. Namely, for an object considered as an object of \mathcal{F}, there is an object where p(y) = d. This association gives a functor which is a functor of groupoids.

Examples

Fibered categories

Category fibered in groupoids

One of the main examples of categories fibered in groupoids comes from groupoid objects internal to a category \mathcal{C}. So given a groupoid object there is an associated groupoid object in the category of contravariant functors from the yoneda embedding. Since this diagram applied to an object gives a groupoid internal to sets there is an associated small groupoid \mathcal{G}. This gives a contravariant 2-functor, and using the Grothendieck construction, this gives a category fibered in groupoids over \mathcal{C}. Note the fiber category over an object is just the associated groupoid from the original groupoid in sets.

Group quotient

Given a group object G acting on an object X from, there is an associated groupoid object where is the projection on X and is the composition map. This groupoid gives an induced category fibered in groupoids denoted.

Two-term chain complex

For an abelian category \mathcal{A} any two-term complex has an associated groupoid where this groupoid can then be used to construct a category fibered in groupoids. One notable example of this is in the study of the cotangent complex for local-complete intersections and in the study of exalcomm.