The
pullback of a function f: X → Y along a function g: Z → Y is defined as the function (g o f)^-1: Z → X, which maps elements from Z to their preimages under f.
函数f: X → Y的拉回是通过函数g: Z → Y定义的,它被定义为函数(g o f)^-1: Z → X,它将Z中的元素映射为其在f下的原像。
In algebraic geometry, the
pullback of a sheaf F on a scheme Y along a morphism f: X → Y is the inverse image sheaf f^*F, which assigns to each open subset U of X the set of sections of F over f^{-1}(U).
在代数几何中,函数f: X → Y沿方案Y上的束F的拉回是逆像束f^*F,它将X的每个开子集U分配给F在f^{-1}(U)上的截面集。
The
pullback of a differential form ω on a manifold Y along a smooth map f: X → Y is defined as the differential form f^*ω on X, which at each point x in X is given by (f^*ω)_x = ω_{f(x)}.
函数f: X → Y沿流形Y上的微分形式ω的拉回定义为X上的微分形式f^*ω,在X的每个点x处给出的是(f^*ω)_x = ω_{f(x)}。
In category theory, the
pullback of two morphisms f: X → Z and g: Y → Z with common codomain is the limit of the diagram formed by X, Y, and Z with arrows f and g, and is denoted as X ×_Z Y.
在范畴论中,两个同尾的形态f: X → Z和g: Y → Z的拉回是由X、Y和Z形成的图式的极限,箭头为f和g,记为X ×_Z Y。
The
pullback of a vector bundle E → Y along a continuous map f: X → Y is the vector bundle f^*E → X, which is the fiber product of X and Y over E.
连续映射f: X → Y沿向量丛E → Y的拉回是向量丛f^*E → X,它是X和Y在E上的纤维积。
In functional analysis, the
pullback of a bounded linear operator T: V → W along an invertible linear operator S: U → V is defined as the composition S^-1 o T o S: U → W.
在泛函分析中,有界线性算子T: V → W沿可逆线性算子S: U → V的拉回定义为复合算子S^-1 o T o S: U → W。
The
pullback of a Lie group action G × M → M along a smooth map f: N → M is defined as the Lie group action G × N → N given by (g,n) ↦ f(g.n), where g.n denotes the action of g on n.
微分流形N → M的光滑映射f沿李群作用G × M → M的拉回定义为李群作用G × N → N,由(g,n) ↦ f(g.n)给出,其中g.n表示g对n的作用。
In algebraic topology, the
pullback of a continuous map f: X → Z along another continuous map g: Y → Z is the space X ×_Z Y, which is the set of all pairs (x,y) ∈ X × Y such that f(x) = g(y).
在代数拓扑中,连续映射f: X → Z沿另一连续映射g: Y → Z的拉回是空间X ×_Z Y,它是所有满足f(x) = g(y)的对(x,y) ∈ X × Y的集合。
The
pullback of a vector field V on a manifold Y along a smooth map f: X → Y is the vector field f^*V on X, which is defined by (f^*V)_x = V_{f(x)} for all x ∈ X.
微流形Y上的微分方程V沿光滑映射f: X → Y的拉回是在X上定义的微分方程f^*V,由(f^*V)_x = V_{f(x)}对所有x ∈ X给出。
In algebraic number theory, the
pullback of a Galois extension K/L along a field homomorphism φ: M → L is the Galois extension M ×_L K, which is the compositum of M and K over L.
在代数数论中,伽罗华扩张K/L沿域同态φ: M → L的拉回是伽罗华扩张M ×_L K,它是M和K在L上的合成。
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