Fibred Category - Background and Motivations

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 pull-back) 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 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.