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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 ...
Each component in a Wardley map is plotted according to its position in two dimensions: in the vertical dimension or y-axis, the component's position within the value chain, corresponding to its visibility to the end-user (whether customer, consumer, business, government or other);
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′).
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A homotopy pullback (or homotopy fiber-product) is the dual concept of a homotopy pushout. It satisfies the universal property of a pullback up to homotopy. [ citation needed ] Concretely, given f : X → Z {\displaystyle f:X\to Z} and g : Y → Z {\displaystyle g:Y\to Z} , it can be constructed as
The pullback of a vector bundle is a categorical pullback. It is the pullback of the diagram f p M--->N<----VN Where VN is the vector bundle over N. So if we replace VN by T*N we see that a pullback of a differential form is a general pullback. So they are more than loosely related. I disagree.