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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.
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 ...
This linear map is known as the pullback (by ), and is frequently denoted by . More generally, any covariant tensor field – in particular any differential form – on N {\displaystyle N} may be pulled back to M {\displaystyle M} using ϕ {\displaystyle \phi } .
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.
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 ...
One can define the pullback of vector-valued forms by smooth maps just as for ordinary forms. The pullback of an E-valued form on N by a smooth map φ : M → N is an (φ*E)-valued form on M, where φ*E is the pullback bundle of E by φ. The formula is given just as in the ordinary case. For any E-valued p-form ω on N the pullback φ*ω is ...
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 *.
The afore-mentioned flat base change result is in fact a special case since for g flat the homotopy pullback (which is locally given by a derived tensor product) agrees with the ordinary pullback (locally given by the underived tensor product), and since the pullback along the flat maps g and g' are automatically derived (i.e., =). The ...