FIBRED CATEGORY

Fibred categories are abstract entities in mathematics used to provide the general framework for descent theory. They formalise various situations in geometry and algebra which reproduce the certain features of the example of vector bundles on topological spaces. More precisely, 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''. Similar set-ups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared.
Fibred categories were introduced by Alexander Grothendieck in Grothendieck (1959), and developed in more detail Grothendieck (1971) in 1960/61, Giraud (1964) and Giraud (1971).

Contents

Formal definitions

References

Formal definitions

This is to include both the definition in terms of "catégories clivées" (the original definition) and the "modern" definition in terms of cartesian morphisms.

References

★ Méthode de la descente, , Jean, Giraud, Mémoires de la Société Mathématique de France,

★

★ Technique de descente et théorèmes d'existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, , Alexander, Grothendieck, Séminaire Bourbaki,

★