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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.
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 ...
A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406
Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : E → B and a continuous map f : X → B, the pullback (formed in the category of topological spaces with continuous maps) X × B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.
An affine bundle is a fiber bundle with a general affine structure group (,) of affine transformations of its typical fiber of dimension . This structure group always is reducible to a general linear group G L ( m , R ) {\displaystyle GL(m,\mathbb {R} )} , i.e., an affine bundle admits an atlas with linear transition functions.
A morphism between algebraic varieties that is a homeomorphism between the underlying topological spaces need not be an isomorphism (a counterexample is given by a Frobenius morphism.) On the other hand, if f is bijective birational and the target space of f is a normal variety , then f is biregular.
(See also: morphism of schemes.) For a scheme Y, a scheme X over Y (or a Y-scheme) means a morphism X → Y of schemes. A scheme X over a commutative ring R means a morphism X → Spec(R). An algebraic variety over a field k can be defined as a scheme over k with certain properties. There are different conventions about exactly which schemes ...