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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 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
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.
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 ...
It leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology. Formally, let f : M → N be smooth, and let ω be a smooth k-form on N. Then there is a differential form f ∗ ω on M, called the pullback of ω, which captures the behavior of ω as seen relative to f.
In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map :, the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X.
Then the pullback s * ω of a principal connection is a 1-form on with values in . If the section s is replaced by a new section sg , defined by ( sg )( x ) = s ( x ) g ( x ), where g : M → G is a smooth map, then ( s g ) ∗ ω = Ad ( g ) − 1 s ∗ ω + g − 1 d g {\displaystyle (sg)^{*}\omega =\operatorname {Ad} (g)^{-1}s^{*}\omega ...
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.