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Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : E → B and a continuous map f : X → B, the pullback (formed in the category of topological spaces with continuous maps) X × B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.
In particular, if is a diffeomorphism between open subsets of and , viewed as a change of coordinates (perhaps between different charts on a manifold ), then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate dependent) approaches to the subject.
The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as a Cartesian square. In that example, the base space of a fiber bundle is pulled back, in the sense of precomposition, above. The fibers then travel along with the points in the base space at which they are anchored: the ...
In the language of category theory, the pullback bundle construction is an example of the more general categorical pullback. As such it satisfies the corresponding universal property. The construction of the pullback bundle can be carried out in subcategories of the category of topological spaces, such as the category of smooth manifolds.
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The simplest case of a blowup is the blowup of a point in a plane. Most of the general features of blowing up can be seen in this example. The blowup has a synthetic description as an incidence correspondence.
For example, if X, Y are manifolds, R the field of real numbers, and the cohomology is de Rham cohomology, then the pullback is induced by the pullback of differential forms. The homotopy invariance of cohomology states that if two maps f, g: X → Y are homotopic to each other, then they determine the same pullback: f * = g *.
Pullback, a name given to two different mathematical processes; Pullback (cohomology), a term in topology; Pullback (differential geometry), a term in differential geometry; Pullback (category theory), a term in category theory; Pullback attractor, an aspect of a random dynamical system; Pullback bundle, the fiber bundle induced by a map of its ...