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Fibred categories are abstract entities in mathematics used to provide a general framework for descent theory. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the idea of descent has come to stand for a very general idea extending the intuitive idea of 'gluing' in Topology. 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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space 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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, a pullback bundle or induced bundle is a useful construction in the theory of Fiber bundles Given a fiber bundle &pi: In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories Fibred categories formalise the system consisting of these categories and inverse image functors. Similar set-ups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Fibrations also play an important role in categorical type theory and theoretical computer science, particularly in models of dependent type theory. In Mathematics, Logic and Computer science, type theory is any of several Formal systems that can serve as alternatives to Naive set theory In Computer science and Logic, a dependent type is a Type which depends on a value

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). Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany There is another famous Jean Giraud, Comics author Jean Giraud is a French Mathematician.

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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position The classical examples include vector bundles, principal bundles and sheaves over topological spaces. In Mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. Another example is given by "families" of algebraic varieties parametrised by another variety. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic 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 pull-back) operation f* taking the considered objects defined on Y to the same type of objects on X. In Mathematics and related technical fields the term map or mapping is often a Synonym for function. 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 Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and 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. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories 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

f^*(g^*(z))\neq (g\circ f)^*(z).

Instead, these inverse images are only naturally isomorphic. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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 Mathematics, the idea of descent has come to stand for a very general idea extending the intuitive idea of 'gluing' 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. In Mathematics, a Grothendieck universe is a set U with the following properties If x is an element of U and if y

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)=idS, is called the fibre category (or fibre) over S, and is denoted FS. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in The morphisms of FS are called S-morphisms, and for x,y objects of FS, the set of S-morphisms is denoted by HomS(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 f-morphisms, and the set of f-morphisms between objects x and y in F is denoted by Homf(x,y). A functor φ: F → 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 φ: F → E to an E-category ψ: G → E is a functor α: F → G such that ψ o α = φ. E-categories form in a natural manner a 2-category, with 1-morphisms being E-functors and natural transformations between these being the 2-morphisms. In Category theory, a 2-category is a category with "morphisms between morphisms"

A morphism m: x → y in F is called E-cartesian (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 f-morphism, then there is precisely one T-morphism 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 FS are precisely the isomorphisms of FS. 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 x in FS by f. However, it is a direct consequence of the definition that two such inverse images are isomorphic in FT.

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 CartE(F,G), with natural transformations as morphisms. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal 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 xS in FS for each object S in E, and for each morphism f: T → S a choice of an inverse image mf: xT → xS. 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

\underset{\longleftarrow}{\mathrm{Lim}} (F/E) = \mathrm{Cart}_E(E,F).

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

\epsilon\colon\underset{\longleftarrow}{\mathrm{Lim}} (F/E) \to F_e,\qquad s\mapsto s(e)

is fully faithful (Lemma 5. In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets In Category theory, a faithful functor (resp a full functor) is a Functor which is Injective (resp 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 E-category, or a category fibred over E) if each morphism f of E has at least one inverse image, and if the composition m o n of two cartesian morphisms m,n in F is always cartesian. In other words, an E-category is a fibred category if inverse images always exists 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 Fe defined at the end of the previous section is an equivalence of categories and moreover surjective on objects. In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every

If F is a fibred E-category, it is always possible, for each morphism f: T → S in E and each object y in FS, to choose (by using the axiom of choice) precisely one inverse image m: x → y. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. The class of morphisms thus selected is called a cleavage and the selected morphisms are called the transport morphisms (of the cleavage). A fibed 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 E-category F specifies, for each morphism f: T → S in E, a functor f*: FS → FT: 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 FS 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

c_{f,g}\colon \quad g^*f^* \to (f \circ g)^*.

These isomorphisms satisfy the following two compatibilities:

  1. c_{f,\mathrm{id}_T} = c_{\mathrm{id}_S,f} = \mathrm{id}_{f^*}
  2. for three consecutive morphisms h,g,f\colon\quad V \to U \to T \to S and object x\in F_S the following holds: c_{f,g\circ h} \cdot c_{g,h}(f^*(x)) = c_{f\circ g, h}(x)\cdot h^*(c_{f,g}(x)).

It can be shown (see Grothendieck (1971) section 8) that, inversely, any collection of functors f*: FS → FT together with isomorphisms cf,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).

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 cf,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 φ: F →E is a functor, then a morphism m: x → y in F is called co-cartesian if it is cartesian for the opposite functor φop: Fop → Eop. Then m is also called a direct image and y a direct image of x for f = φ(m). A co-fibred E-category is anE-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 Fib(E), where the category of morphisms between two fibred categories F and G is defined to be the category CartE(F,G) of cartesian functors from F to G.

Similarly the split categories over E form a 2-category Scin(E) (from French catégorie scindée), where the category of morphisms between two split categories F and G is the full sub-category ScinE(F,G) 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. , ScinE(F,G) ⊂ CartE(F,G).

There is a natural forgetful 2-functor 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 2-functor i: Scin(E) → Fib(E) 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) → 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 pre-stack). In Mathematics a stack is an Abstract entity used to formalise some of the main concepts of Descent theory.

Examples

  1. 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 E-category; for an object S of E the fibre ES is the category E/S of S-objects 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. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Category theory, a branch of Mathematics, a pullback (also called a fibered product or Cartesian square) is the limit of a
  2. Fibre bundles: Fibre products exist in the category Top of topological spaces and thus by the previous example A(Top) is fibred over Top. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. If Fib is the full subcategory of A(E) consisting of arrows that are projection maps of fibre bundles, then FibS is the category of fibre bundles on S and Fib is fibred over Top. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. A a choice of a cleavage amounts to a choice of ordinary inverse image (or pull-back) functors for fibre bundles.
  3. 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 VectR (VectC) over Top (morphisms of vector bundles respecting the vector space structure of the fibres). In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added This Top-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 Fib.
  4. 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 sub-category of A(Top) consisting of etale spaces of sheaves. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. As with vector bundles, the sheaves of groups and rings also form fibred categories of Top. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real
  5. Sheaves on topoi: If E is a topos and S is an object in E, the category ES of S-objects is also a topos, interpreted as the category of sheaves on S. In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets If f: T → S is a morphism in E, the inverse image functor f* can be described as follows: for a sheaf F on ES and an object p: U → T in ET one has f*F(U) = HomT(U, f*F) equals HomS(f o p, F) = F(U). These inverse image make the categries ES into a split fibred category on E. This can be applied in paricular to the "large" topos TOP of topologicals spaces.
  6. Quasi-coherent sheaves on schemes: Quasi-coherent sheaves form a fibred category over the category of schemes. In Mathematics, especially in Algebraic geometry and the theory of Complex manifolds coherent sheaves are specific class of sheaves having In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. This is one of the motivating examples for the definition of fibred categories.
  7. 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 H-category. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector It can be checked that in this set-up all morphisms in G are cartesian; hence G is fibred over H precisely when f is surjective. A splitting in this setup is a (set-theoretic) section of f which commutes strictly with composition, or in other words a section of f which is also a homomorphism. In Category theory, a branch of Mathematics, a section is a right inverse of a morphism But as is well-known in group theory, this is not always possible (one can take the projection in a non-split group extension). Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a group extension is a general means of describing a group in terms of a particular Normal subgroup and Quotient group.
  8. Co-fibred category of sheaves: The direct image functor of sheaves makes the categories of sheaves on topological spaces into a co-fibred category. In Mathematics, in the field of Sheaf theory and especially in Algebraic geometry, the direct image functor generalizes the notion of a Section of The transitivity of the direct image shows that this is even naturally co-split.

References

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