 Fibred category

Fibred 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 pullbacks) 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. Fibrations also play an important role in categorical type theory and theoretical computer science, particularly in models of dependent type theory.
Fibred categories were introduced by Alexander Grothendieck in Grothendieck (1959), and developed in more detail by himself and Jean Giraud in Grothendieck (1971) in 1960/61, Giraud (1964) and Giraud (1971).
Contents
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 → Y between base spaces, there is a corresponding inverse image (also called pullback) 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 → 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 setup 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 nontrivial 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 settheoretical 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 φ: F → 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 φ(x)=S and those morphisms m satisfying φ(m)=id_{S}, is called the fibre category (or fibre) over S, and is denoted F_{S}. The morphisms of F_{S} are called Smorphisms, and for x,y objects of F_{S}, the set of Smorphisms is denoted by Hom_{S}(x,y). The image by φ of an object or a morphism in F is called its projection (by φ). If f is a morphism of E, then those morphisms of F that project to f are called fmorphisms, and the set of fmorphisms between objects x and y in F is denoted by Hom_{f}(x,y). A functor φ: F → E is also called an Ecategory, or said to make F into an Ecategory or a category over E. An Efunctor from an Ecategory φ: F → E to an Ecategory ψ: G → E is a functor α: F → G such that ψ o α = φ. Ecategories form in a natural manner a 2category, with 1morphisms being Efunctors, and 2morphisms being natural transformations between Efunctors whose components lie in some fibre.
A morphism m: x → y in F is called Ecartesian (or simply cartesian) if it satisfies the following condition:
 if f: T → S is the projection of m, and if n: z → y is an fmorphism, then there is precisely one Tmorphism a: z → x such that n = m o a.
A cartesian morphism m: x → y is called an inverse image of its projection f = φ(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 → 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}.
An Efunctor between two Ecategories is called a cartesian functor if it takes cartesian morphisms to cartesian morphisms. Cartesian functors between two Ecategories F,G form a category Cart_{E}(F,G), with natural transformations as morphisms. A special case is provided by considering E as an Ecategory via the identity functor: then a cartesian functor from E to an Ecategory 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 → S a choice of an inverse image m_{f}: x_{T} → x_{S}. 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 cleaved 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 (1974) in 1960–1961.
An E category φ: F → E is a fibred category (or a fibred Ecategory, 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 o n of any two cartesian morphisms m,n in F is always cartesian. In other words, an Ecategory 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 ε 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 Ecategory, it is always possible, for each morphism f: T → S in E and each object y in F_{S}, to choose (by using the axiom of choice) precisely one inverse image m: x → 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 cleaved 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 Ecategory F specifies, for each morphism f: T → S in E, a functor f^{*}: F_{S} → F_{T}: 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 → S and g: U → T are morphisms in E, then there is an isomorphism of functors
These isomorphisms satisfy the following two compatibilities:
 for three consecutive morphisms and object the following holds:
It can be shown (see Grothendieck (1971) section 8) that, inversely, any collection of functors f^{*}: F_{S} → F_{T} together with isomorphisms c_{f,g} satisfying the compatibilities above, defines a cleaved 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 morphisms 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 o g. In other words, the compatibility isomorphisms c_{f,g} of the previous section are all identities for a split category. Thus split Ecategories 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.
Cocartesian morphisms and cofibred categories
One can invert the direction of arrows in the definitions above to arrive at corresponding concepts of cocartesian morphisms, cofibred categories and split cofibred categories (or cosplit categories). More precisely, if φ: F →E is a functor, then a morphism m: x → y in F is called cocartesian if it is cartesian for the opposite functor φ^{op}: F^{op} → E^{op}. Then m is also called a direct image and y a direct image of x for f = φ(m). A cofibred Ecategory is anEcategory such that direct image exists for each morphism in E and that the composition of direct images is a direct image. A cocleavage and a cosplitting are defined similarly, corresponding to direct image functors instead of inverse image functors.
Properties
The 2categories of fibred categories and split categories
The categories fibred over a fixed category E form a 2category Fib(E), where the category of morphisms between two fibred categories F and G is defined to be the category Cart_{E}(F,G) of cartesian functors from F to G.
Similarly the split categories over E form a 2category Scin(E) (from French catégorie scindée), where the category of morphisms between two split categories F and G is the full subcategory Scin_{E}(F,G) of Efunctors 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 Ecategories is also a morphism of Efibred categories, i.e., Scin_{E}(F,G) ⊂ Cart_{E}(F,G).
There is a natural forgetful 2functor i: Scin(E) → Fib(E) 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 2functor i: Scin(E) → Fib(E) admits a right 2adjoint S and a left 2adjoint 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) → F and F → L(F) are both cartesian and equivalences (ibid.). However, while their composition S(F) → L(F) 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 prestack).
Examples
 Categories of arrows: For any category E the category of arrows A(E) in E has as objects the morphisms in E, and as morphisms the commutative squares in E (more precisely, a morphism from (f: X → T) to (g: Y → S) consists of morphisms (a: X → Y) and (b: T → S) such that bf = ga). The functor which takes an arrow to its target makes A(E) into an Ecategory; for an object S of E the fibre E_{S} is the category E_{/S} of Sobjects in E, i.e., arrows in E with target S. Cartesian morphisms in A(E) are precisely the cartesian squares in E, and thus A(E) is fibred over E precisely when fibre products exist in E.
 Fibre bundles: Fibre products exist in the category Top of topological spaces and thus by the previous example A(Top) is fibred over Top. If Fib is the full subcategory of A(Top) consisting of arrows that are projection maps of fibre bundles, then Fib_{S} is the category of fibre bundles on S and Fib is fibred over Top. A a choice of a cleavage amounts to a choice of ordinary inverse image (or pullback) functors for fibre bundles.
 Vector bundles: In a manner similar to the previous examples the projections (p: V → S) of real (complex) vector bundles to their base spaces form a category Vect_{R} (Vect_{C}) over Top (morphisms of vector bundles respecting the vector space structure of the fibres). This Topcategory is also fibred, and the inverse image functors (are the ordinary pullback functors for vector bundles. These fibred categories are (nonfull) subcategories of Fib.
 Sheaves on topological spaces: The inverse image functors of sheaves make the categories Sh(S) of sheaves on topological spaces S into a (cleaved) fibred category Sh over Top. This fibred category can be described as the full subcategory of A(Top) consisting of etale spaces of sheaves. As with vector bundles, the sheaves of groups and rings also form fibred categories of Top.
 Sheaves on topoi: If E is a topos and S is an object in E, the category E_{S} of Sobjects is also a topos, interpreted as the category of sheaves on S. If f: T → S is a morphism in E, the inverse image functor f^{*} can be described as follows: for a sheaf F on E_{S} and an object p: U → T in E_{T} one has f^{*}F(U) = Hom_{T}(U, f^{*}F) equals Hom_{S}(f o p, F) = F(U). These inverse image make the categories E_{S} into a split fibred category on E. This can be applied in particular to the "large" topos TOP of topological spaces.
 Quasicoherent sheaves on schemes: Quasicoherent sheaves form a fibred category over the category of schemes. This is one of the motivating examples for the definition of fibred categories.
 Fibred category admitting no splitting: A group G can be considered as a category with one object and the elements of G as the morphisms, composition of morphisms being given by the group law. A group homomorphism f: G → H can then be considered as a functor, which makes G into a Hcategory. It can be checked that in this setup all morphisms in G are cartesian; hence G is fibred over H precisely when f is surjective. A splitting in this setup is a (settheoretic) section of f which commutes strictly with composition, or in other words a section of f which is also a homomorphism. But as is wellknown in group theory, this is not always possible (one can take the projection in a nonsplit group extension).
 Cofibred category of sheaves: The direct image functor of sheaves makes the categories of sheaves on topological spaces into a cofibred category. The transitivity of the direct image shows that this is even naturally cosplit.
See also
 Grothendieck construction
References
 Giraud, Jean (1964). "Méthode de la descente". Mémoires de la Société Mathématique de France 2: viii+150.
 Giraud, Jean (1971). Cohomologie non abélienne. Springer. ISBN 3540053077
 Grothendieck, Alexander (1959). "Technique de descente et théorèmes d'existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats". Séminaire Bourbaki 5 (Exposé 190): viii+150.
 Gray, John W. (1966). "Fibred and cofibred categories". Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965). Springer Verlag. pp. 21–83.
 Brown, R., "Fibrations of groupoids", J. Algebra 15 (1970) 103132.
 Grothendieck, Alexander (1971). "Catégories fibrées et descente". Revêtements étales et groupe fondamental. Springer Verlag. pp. 145–194.
 Jacobs, Bart (1999). Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, Elsevier. ISBN 0444501703. http://www.cs.ru.nl/B.Jacobs/CLT/bookinfo.html.
 Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory, arXiv:math.AG/0412512.
 Fibred Categories à la Bénabou, Thomas Streicher
 An introduction to fibrations, topos theory, the effective topos and modest sets, Wesley Phoa
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