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Fibred 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. 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 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). ==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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fibred category」の詳細全文を読む スポンサード リンク
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