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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.
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 } .
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 mathematics, a pullback bundle or induced bundle [1] [2] [3] is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f * E over B′. The fiber of f * E over a point b′ in B′ is just the fiber of E over f(b′).
The limit L of such a diagram is called an equalizer of those morphisms. Kernels. A kernel is a special case of an equalizer where one of the morphisms is a zero morphism. Pullbacks. Let F be a diagram that picks out three objects X, Y, and Z in C, where the only non-identity morphisms are f : X → Z and g : Y → Z.
The exterior derivative is natural in the technical sense: if f : M → N is a smooth map and Ω k is the contravariant smooth functor that assigns to each manifold the space of k-forms on the manifold, then the following diagram commutes so d( f ∗ ω) = f ∗ dω, where f ∗ denotes the pullback of f .
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.
of cochain complexes, which in turn determines the pullback homomorphism : (′;) (;) on the cohomology modules and cohomology ring. If C, C ' are singular chain complexes of spaces X, Y, then this is the pullback for singular cohomology theory.