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 stackss, 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 between base spaces, there is a corresponding inverse image (also called pull-back.

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.

is another map, the inverse image functors are not strictly is an object over (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 universess.

Cartesian morphisms and functor

is a functor between two categories, then the subcategory, is called the fibre category (or fibre) over-morphisms is called its projection-morphisms.

is called \phi-cartesian (or simply cartesian) if it satisfies the following condition:

1. -morphism, then there is precisely one.

A cartesian morphism is called an inverse image is called an inverse image .

.

-category-category or a category over-categories form in a natural manner a 2-category-functors whose components lie in some fibre.

-categories is called a cartesian functor, with natural transformations is called a cartesian section is denoted by

has a terminal object is a topos of arrows) 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.

is a fibred category (or a -category, or a whose codomain-category is a fibred category if inverse images always exist (for morphisms whose codomains are in the range of projection) and are transitive.

defined at the end of the previous section is an equivalence of categories and moreover surjective on objects.

, to choose (by using the axiom of choice. 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; 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.

, a functor 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 the inverse image functor is almost, then there is an isomorphism of functors

These isomorphisms satisfy the following two compatibilities:

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 equals the inverse image functor 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

is called co-cartesian if it is cartesian for the opposite functor is also called a direct image. A co-fibred 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

, where the category.

(from French catégorie scindée. Each such -categories.

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 are both cartesian and equivalences (ibid 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

given by

. The associated 2-functors from the Grothendieck construction are examples of stacks which is a functor of groupoids.

Examples

Fibered categories

1. . The cartesian arrows are the fully faithful functors.

2. Categories of arrows the category of arrows are precisely the cartesian squares precisely when fibre products.

3. Fibrations of graphs: Each directed graph as objects and the paths is a graph fibration if and only if the functor is a fibration.

4. Fibre bundles of topological spaces consisting of arrows that are projection maps of fibre bundles. A choice of a cleavage amounts to a choice of ordinary inverse image (or pull-back) functors for fibre bundles.

5. Vector bundles of real (complex) vector bundles (morphisms of vector bundles respecting the vector space-category is also fibred, and the inverse image functors are the ordinary pull-back.

6. Sheaves on topological spaces: The inverse image functors of sheaves consisting of étalé spaces of sheaves. As with vector bundles, the sheaves of groups and rings.

7. Sheaves on topoi is a topos into a split of topological spaces.

8. Quasi-coherent sheaves on schemes: Quasi-coherent sheaves form a fibred category over the category of schemes. This is one of the motivating examples for the definition of fibred categories.

9. Fibred category admitting no splitting as the morphisms, composition of morphisms being given by the group law. A group homomorphism is surjective. A splitting in this setup is a (set-theoretic) section which is also a homomorphism. But as is well known in group theory, this is not always possible (one can take the projection in a non-split group extension).

10. Co-fibred category of sheaves: The direct image functor of sheaves makes the categories of sheaves on topological spaces into a co-fibred category. The transitivity of the direct image shows that this is even naturally co-split.

Category fibered in groupoids

One of the main examples of categories fibered in groupoids comes from groupoid objects. So given a groupoid object

there is an associated groupoid object

from the yoneda embedding gives a groupoid internal to sets

, and using the Grothendieck construction. Note the fiber category over an object is just the associated groupoid from the original groupoid in sets.

Group quotient

, there is an associated groupoid object

.

Two-term chain complex

For an abelian category 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.

See also

• Grothendieck construction

• Stack (mathematics)

• Artin's criterion

• Fibration of simplicial sets

References

• .

External links

• SGA 1.VI - Fibered categories and descent - pages 119-153