Search results
Results From The WOW.Com Content Network
Then the set of k-rational points of the fiber product X × Y Z is easy to describe: () = () (). That is, a k-point of X × Y Z can be identified with a pair of k-points of X and Z that have the same image in Y. This is immediate from the universal property of the fiber product of schemes.
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 .
If is a morphism of , then those morphisms of that project to are called -morphisms, and the set of -morphisms between objects and in is denoted by (,). A morphism m : x → y {\displaystyle m:x\to y} in F {\displaystyle F} is called ϕ {\displaystyle \phi } -cartesian (or simply cartesian ) if it satisfies the following condition:
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.
A morphism f : X → Y is called a monomorphism if f ∘ g 1 = f ∘ g 2 implies g 1 = g 2 for all morphisms g 1, g 2 : Z → X. A monomorphism can be called a mono for short, and we can use monic as an adjective. [1] A morphism f has a left inverse or is a split monomorphism if there is a morphism g : Y → X such that g ∘ f = id X.
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
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.
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