Search results
Results From The WOW.Com Content Network
In point set topology, one generally considers functions from topological spaces to topological spaces. If f {\displaystyle f} is a continuous function and if Y {\displaystyle Y} (or more generally, the image set f ( X ) {\displaystyle f(X)} ) is a T 1 space then every fiber is a closed subset of X . {\displaystyle X.}
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.
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:
The morphism f : Y → X has finite fibers if the fiber over each point is a finite set. A morphism is quasi-finite if it is of finite type and has finite fibers. quasi-projective A quasi-projective variety is a locally closed subvariety of a projective space. quasi-separated
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 examples, this definition reconstructs the old notion of the set of solutions of the defining equations of X with values in R. When R is a field k, X(k) is also called the set of k-rational points of X. More generally, for a scheme X over a commutative ring R and any commutative R-algebra S, an S-point of X means a morphism Spec(S) → X over R.
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