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.
Fibred categories were introduced by Alexander Grothendieck (1959, 1971), and developed in more detail by Jean Giraud (1964, 1971).
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.
between base spaces, there is a corresponding inverse image (also called pull-back) operation
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.
(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.
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.
, possibly having different sources; thus there can be more than one inverse image of a given object
However, it is a direct consequence of the definition that two such inverse images are isomorphic in
A cartesian section is thus a (strictly) compatible system of inverse images over objects of
The technically most flexible and economical definition of fibred categories is based on the concept of cartesian morphisms.
whose codomain is in the range of projection has at least one inverse image, and moreover the composition
-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) precisely one inverse image
A cleavage is called normalised if the transport morphisms include all identities in
However, in general it fails to commute strictly with composition of morphisms.
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.
In terms of inverse image functors the condition of being a splitting means that the composition of inverse image functors corresponding to composable morphisms
Unlike cleavages, not all fibred categories admit splittings.
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).