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The notion of pullback as a fiber-product ultimately leads to the very general idea of a categorical pullback, but it has important special cases: inverse image (and pullback) sheaves in algebraic geometry, and pullback bundles in algebraic topology and differential geometry. See also: Pullback (category theory) Fibred category; Inverse image sheaf
For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square. The dual concept of the pullback is the pushout .
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; [1] a three-dimensional solid bounded exclusively by faces is a polyhedron. A face can be finite like a polygon or circle, or infinite like a half-plane or plane.
Any section s of E over B induces a section of f * E, called the pullback section f * s, simply by defining (′):= (′, ((′)) ) for all ′ ′.If the bundle E → B has structure group G with transition functions t ij (with respect to a family of local trivializations {(U i, φ i)}) then the pullback bundle f * E also has structure group G.
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties.
This article begins with the simplest operations, then uses them to construct more sophisticated ones. Roughly speaking, the pullback mechanism (using precomposition) turns several constructions in differential geometry into contravariant functors.