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 Alexander Grothendieck (1959, 1971), and developed in more detail by Jean Giraud (1964, 1971).

Background and motivations

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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) operation   taking the considered objects defined on   to the same type of objects on  . This is indeed the case in the examples above: for example, the inverse image of a vector bundle   on   is a vector bundle   on  .

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   is another map, the inverse image functors are not strictly compatible with composed maps: if   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

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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

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If   is a functor between two categories and   is an object of  , then the subcategory of   consisting of those objects   for which   and those morphisms   satisfying  , is called the fibre category (or fibre) over  , and is denoted  . The morphisms of   are called  -morphisms, and for   objects of  , the set of  -morphisms is denoted by  . The image by   of an object or a morphism in   is called its projection (by  ). If   is a morphism of  , then those morphisms of   that project to   are called  -morphisms, and the set of  -morphisms between objects   and   in   is denoted by  .

A morphism   in   is called  -cartesian (or simply cartesian) if it satisfies the following condition:

if   is the projection of  , and if   is an  -morphism, then there is precisely one  -morphism   such that  .

A cartesian morphism   is called an inverse image of its projection  ; the object   is called an inverse image of   by  .

The cartesian morphisms of a fibre category   are precisely the isomorphisms of  . There can in general be more than one cartesian morphism projecting to a given morphism  , possibly having different sources; thus there can be more than one inverse image of a given object   in   by  . However, it is a direct consequence of the definition that two such inverse images are isomorphic in  .

A functor   is also called an  -category, or said to make   into an  -category or a category over  . An  -functor from an  -category   to an  -category   is a functor   such that  .  -categories form in a natural manner a 2-category, with 1-morphisms being  -functors, and 2-morphisms being natural transformations between  -functors whose components lie in some fibre.

An  -functor between two  -categories is called a cartesian functor if it takes cartesian morphisms to cartesian morphisms. Cartesian functors between two  -categories   form a category  , with natural transformations as morphisms. A special case is provided by considering   as an  -category via the identity functor: then a cartesian functor from   to an  -category   is called a cartesian section. Thus a cartesian section consists of a choice of one object   in   for each object   in  , and for each morphism   a choice of an inverse image  . A cartesian section is thus a (strictly) compatible system of inverse images over objects of  . The category of cartesian sections of   is denoted by

 

In the important case where   has a terminal object   (thus in particular when   is a topos or the category   of arrows with target   in  ) the functor

 

is fully faithful (Lemma 5.7 of Giraud (1964)).

Fibred categories and cloven categories

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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   category   is a fibred category (or a fibred  -category, or a category fibred over  ) if each morphism   of   whose codomain is in the range of projection has at least one inverse image, and moreover the composition   of any two cartesian morphisms   in   is always cartesian. In other words, an  -category is a fibred category if inverse images always exist (for morphisms whose codomains are in the range of projection) and are transitive.

If   has a terminal object   and if   is fibred over  , then the functor   from cartesian sections to   defined at the end of the previous section is an equivalence of categories and moreover surjective on objects.

If   is a fibred  -category, it is always possible, for each morphism   in   and each object   in  , to choose (by using the axiom of choice) precisely one inverse image  . 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  ; 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  -category   specifies, for each morphism   in  , a functor  ; on objects   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   of   the fibre category   and to a morphism   the inverse image functor   is almost a contravariant functor from   to the category of categories. However, in general it fails to commute strictly with composition of morphisms. Instead, if   and   are morphisms in  , then there is an isomorphism of functors

 

These isomorphisms satisfy the following two compatibilities:

  1.  
  2. for three consecutive morphisms   and object   the following holds:  

It can be shown (see Grothendieck (1971) section 8) that, inversely, any collection of functors   together with isomorphisms   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

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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   in   equals the inverse image functor corresponding to  . In other words, the compatibility isomorphisms   of the previous section are all identities for a split category. Thus split  -categories correspond exactly to true functors from   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

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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   is a functor, then a morphism   in   is called co-cartesian if it is cartesian for the opposite functor  . Then   is also called a direct image and   a direct image of   for  . A co-fibred  -category is an  -category such that direct image exists for each morphism in   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

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The 2-categories of fibred categories and split categories

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The categories fibred over a fixed category   form a 2-category  , where the category of morphisms between two fibred categories   and   is defined to be the category   of cartesian functors from   to  .

Similarly the split categories over   form a 2-category   (from French catégorie scindée), where the category of morphisms between two split categories   and   is the full sub-category   of  -functors from   to   consisting of those functors that transform each transport morphism of   into a transport morphism of  . Each such morphism of split  -categories is also a morphism of  -fibred categories, i.e.,  .

There is a natural forgetful 2-functor   that simply forgets the splitting.

Existence of equivalent split categories

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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   over  . More precisely, the forgetful 2-functor   admits a right 2-adjoint   and a left 2-adjoint   (Theorems 2.4.2 and 2.4.4 of Giraud 1971), and   and   are the two associated split categories. The adjunction functors   and   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

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There is a related construction to fibered categories called categories fibered in groupoids. These are fibered categories   such that any subcategory of   given by

  1. Fix an object  
  2. The objects of the subcategory are   where  
  3. The arrows are given by   such that  

is a groupoid denoted  . The associated 2-functors from the Grothendieck construction are examples of stacks. In short, the associated functor   sends an object   to the category  , and a morphism   induces a functor from the fibered category structure. Namely, for an object   considered as an object of  , there is an object   where  . This association gives a functor   which is a functor of groupoids.

Examples

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Fibered categories

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  1. The functor  , sending a category to its set of objects, is a fibration. For a set  , the fiber consists of categories   with  . The cartesian arrows are the fully faithful functors.
  2. Categories of arrows: For any category   the category of arrows   in   has as objects the morphisms in  , and as morphisms the commutative squares in   (more precisely, a morphism from   to   consists of morphisms   and   such that  ). The functor which takes an arrow to its target makes   into an  -category; for an object   of   the fibre   is the category   of  -objects in  , i.e., arrows in   with target  . Cartesian morphisms in   are precisely the cartesian squares in  , and thus   is fibred over   precisely when fibre products exist in  .
  3. Fibre bundles: Fibre products exist in the category   of topological spaces and thus by the previous example   is fibred over  . If   is the full subcategory of   consisting of arrows that are projection maps of fibre bundles, then   is the category of fibre bundles on   and   is fibred over  . A choice of a cleavage amounts to a choice of ordinary inverse image (or pull-back) functors for fibre bundles.
  4. Vector bundles: In a manner similar to the previous examples the projections   of real (complex) vector bundles to their base spaces form a category   ( ) over   (morphisms of vector bundles respecting the vector space structure of the fibres). This  -category is also fibred, and the inverse image functors are the ordinary pull-back functors for vector bundles. These fibred categories are (non-full) subcategories of  .
  5. Sheaves on topological spaces: The inverse image functors of sheaves make the categories   of sheaves on topological spaces   into a (cleaved) fibred category   over  . This fibred category can be described as the full sub-category of   consisting of étalé spaces of sheaves. As with vector bundles, the sheaves of groups and rings also form fibred categories of  .
  6. Sheaves on topoi: If   is a topos and   is an object in  , the category   of  -objects is also a topos, interpreted as the category of sheaves on  . If   is a morphism in  , the inverse image functor   can be described as follows: for a sheaf   on   and an object   in   one has   equals  . These inverse image make the categories   into a split fibred category on  . This can be applied in particular to the "large" topos   of topological spaces.
  7. 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.
  8. Fibred category admitting no splitting: A group   can be considered as a category with one object and the elements of   as the morphisms, composition of morphisms being given by the group law. A group homomorphism   can then be considered as a functor, which makes   into a  -category. It can be checked that in this set-up all morphisms in   are cartesian; hence   is fibred over   precisely when   is surjective. A splitting in this setup is a (set-theoretic) section of   which commutes strictly with composition, or in other words a section of   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).
  9. 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

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One of the main examples of categories fibered in groupoids comes from groupoid objects internal to a category  . 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  . This gives a contravariant 2-functor  , and using the Grothendieck construction, this gives a category fibered in groupoids over  . Note the fiber category over an object is just the associated groupoid from the original groupoid in sets.

Group quotient

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Given a group object   acting on an object   from  , there is an associated groupoid object

 

where   is the projection on   and   is the composition map  . This groupoid gives an induced category fibered in groupoids denoted  .

Two-term chain complex

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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

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References

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  • Giraud, Jean (1964). "Méthode de la descente". Mémoires de la Société Mathématique de France. 2: viii+150.
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