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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 ...
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 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
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 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.
Specifically, if V is in the fiber p −1 (x), then the fiber of S over V is V itself; thus, S has rank r = d = dim(V) and is the determinant line bundle. Now, by the universal property of a projective bundle, the injection ∧ r S → p ∗ ( ∧ r E ) {\displaystyle \wedge ^{r}S\to p^{*}(\wedge ^{r}E)} corresponds to the morphism over X :