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Let : be a smooth map between (smooth) manifolds and , and suppose : is a smooth function on .Then the pullback of by is the smooth function on defined by () = (()). ...
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.
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 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.