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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 idea behind the pullback is essentially the notion of precomposition of one function with another. However, by combining this idea in several different contexts, quite elaborate pullback operations can be constructed. This article begins with the simplest operations, then uses them to construct more sophisticated ones.
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 ...
Oral literature is especially rich in chiastic structure, possibly as an aid to memorization and oral performance. In Homer's Iliad and Odyssey, for instance, Cedric Whitman finds chiastic patterns "of the most amazing virtuosity" that simultaneously perform both aesthetic and mnemonic functions, permitting the oral poet easily to recall the basic structure of the composition during ...
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 of this diagram is called the J th power of X and denoted X J. Equalizers. If J is a category with two objects and two parallel morphisms from one object to the other, then a diagram of shape J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer of those morphisms. Kernels.
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 ...
A sentence diagram is a pictorial representation of the grammatical structure of a sentence. The term "sentence diagram" is used more when teaching written language, where sentences are diagrammed. The model shows the relations between words and the nature of sentence structure and can be used as a tool to help recognize which potential ...