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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 .
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.
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.
A morphism f : X → Y is called an isomorphism if there exists a morphism g : Y → X such that f ∘ g = id Y and g ∘ f = id X. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f. Inverse morphisms, if they exist, are unique.
(iii) means that each geometric fiber of f is a nonsingular variety (if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. If S is the spectrum of an algebraically closed field and f is of finite type, then one recovers the definition of a nonsingular variety.
In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is of finite type and satisfies any of the following equivalent conditions: [1] Every point x of X is isolated in its fiber f −1 (f(x)). In other words, every fiber is a discrete (hence finite) set.
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 ...
It is also commonly called an algebraic fiber space, as it is an analog of a fiber space in algebraic topology. By the Stein factorization, any surjective projective morphism is a contraction morphism followed by a finite morphism. Examples include ruled surfaces and Mori fiber spaces.