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Concept in category theory
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 .

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.

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.

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.

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.

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).

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. 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 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. 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 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 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. 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. 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). 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.