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A function between topological spaces is called monotone if every fiber is a connected subspace of its domain. A function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is monotone in this topological sense if and only if it is non-increasing or non-decreasing , which is the usual meaning of " monotone function " in real analysis .
The choice of a (normalised) cleavage for a fibred -category specifies, for each morphism : in , a functor:; on objects is simply the inverse image by the corresponding transport morphism, and on morphisms it is defined in a natural manner by the defining universal property of cartesian morphisms.
In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber ...
In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings given the Euler class of the sphere bundle and vice versa.
Then there is a morphism Spec(k(y)) → Y with image y, where k(y) is the residue field of y. The fiber of f over y is defined as the fiber product X × Y Spec(k(y)); this is a scheme over the field k(y). [3] This concept helps to justify the rough idea of a morphism of schemes X → Y as a family of schemes parametrized by Y.
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a P n -bundle if it is locally a projective n -space; i.e., X × S U ≃ P U n {\displaystyle X\times _{S}U\simeq \mathbb {P} _{U}^{n}} and transition automorphisms are linear.
For every object X, there exists a morphism id X : X → X called the identity morphism on X, such that for every morphism f : A → B we have id B ∘ f = f = f ∘ id A. Associativity h ∘ (g ∘ f) = (h ∘ g) ∘ f whenever all the compositions are defined, i.e. when the target of f is the source of g, and the target of g is the source of h.
A genus fibration: of is a proper flat morphism to a smooth curve such that and all fibers of have arithmetic genus. If X {\displaystyle X} is a smooth projective surface and the fibers of f {\displaystyle f} do not contain rational curves of self-intersection − 1 {\displaystyle -1} , then the fibration is called minimal .