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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
Pullback (differential geometry) Let be a smooth map between smooth manifolds and . Then there is an associated linear map from the space of 1-forms on (the linear space of sections of the cotangent bundle) to the space of 1-forms on . This linear map is known as the pullback (by ), and is frequently denoted by .
Pullback (category theory) In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is written.
Inverse image functor. 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.
A pullback motor (also pull back, pull back and go or pull-back) is a simple clockwork motor used in toy cars. A patent for them was granted to Bertrand 'Fred' Francis in 1952 as a keyless clockwork motor. [1][2] Pulling the car backward (hence the name) winds up an internal spiral spring; a flat spiral rather than a helical coil spring.
Base change theorems. In mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves: where. is a Cartesian square of topological spaces and is a sheaf on X . Such theorems exist in different branches ...
Pullback bundle. 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 ...
Pushforwards of measures allow to induce, from a function between measurable spaces , a function between the spaces of measures . As with many induced mappings, this construction has the structure of a functor, on the category of measurable spaces. For the special case of probability measures, this property amounts to functoriality of the Giry ...