Having deﬁned topological spaces and manifolds, which are the setting of further considerations, it is also required to deﬁne how such spaces may be combined and attached to each other. The next deﬁnitions provide the fundamentals of these endeavours and also provide the major abstraction for the storage of values in digital computers.

Deﬁnition 39 (Fibration) A continuous mapping (Deﬁnition 30)

It is also commonly written as ﬁbration sequence

An important specialization, with an additional local requirement, can be deﬁned which has important realizations as demonstrated in Deﬁnition 52.

Deﬁnition 40 (Fiber bundle) The topological spaces , called the base space, and , called the total space, along with a surjective (Deﬁnition 23) projection (Deﬁnition 25) is known as a ﬁber bundle , if it locally satisﬁes the following condition:

For every element there exists an open neighbourhood such that the preimages of the projection , are homeomorphic to a product space , such that the following diagram commutes:

A ﬁber bundle is also commonly given as

Beside the already presented mechanisms it is also desirable to ﬁrmly establish a formal manner in which to transport properties of mappings to various topological spaces, where they have previously not been deﬁned. To provide Deﬁnition 44 with in depth backing, ﬁrst very general notions are introduced.

Deﬁnition 42 (Fiber product) Given two mappings with identical codomain

the ﬁber product over consists of two mappings

such that , which may also be expressed by saying that the following diagram commutes:

This demand ensures that a tuple is deﬁned uniquely up to an isomorphism. It is also common to ﬁnd the notation

The general notion just deﬁned can be applied to ﬁber bundles to attach ﬁbers originally situated in one topological space to another one using a simple mapping. The formalization of this is presented next.

Deﬁnition 43 (Pullback bundle) Given a ﬁber bundle and a mapping it is possible to deﬁne a ﬁberbundle, the so called pullback bundle , which uses as a base space by attaching at every element the ﬁber corresponding to the element (the position of attachment is given by the index):

In short, the pullback bundle, as sketched in Figure 4.2, is simply the ﬁber product (Deﬁnition 42) . It should not go unnoticed that this construct is compatible with sections of ﬁber bundles (Deﬁnition 41); therefore entities which appear as the section of a ﬁber bundle, such as presented in Deﬁnition 61 and Deﬁnition 62, will be pulled back and appear again as sections of the pullback bundle (Deﬁnition 43).

Considering two topological spaces and and the mappings

Deﬁnition 44 (Pullback (of functions)) The mapping resulting from a mapping between two topological spaces

The pullback of functions is a particular case of the of the pullback bundle (Deﬁnition 43), which illustrates the concept in a relatively simple fashion.