In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
Contents
- General topology
- Differential topology
- Riemannian geometry
- Algebra
- Field theory
- Universal algebra and model theory
- Order theory and domain theory
- Metric spaces
- Normed spaces
- Category theory
- References
When some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map f : X → Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In the terminology of category theory, a structure-preserving map is called a morphism.
The fact that a map f : X → Y is an embedding is often indicated by the use of a "hooked arrow", thus:
Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In such cases it is common to identify the domain X with its image f(X) contained in Y, so that X ⊆ Y.
General topology
In general topology, an embedding is a homeomorphism onto its image. More explicitly, an injective continuous map
For a given space
Differential topology
In differential topology: Let
In other words, an embedding is diffeomorphic to its image, and in particular the image of an embedding must be a submanifold. An immersion is a local embedding (i.e. for any point
When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.
An important case is
An embedding is proper if it behaves well with respect to boundaries: one requires the map
The first condition is equivalent to having
Riemannian geometry
In Riemannian geometry: Let (M,g) and (N,h) be Riemannian manifolds. An isometric embedding is a smooth embedding f : M → N which preserves the metric in the sense that g is equal to the pullback of h by f, i.e. g = f*h. Explicitly, for any two tangent vectors
we have
Analogously, isometric immersion is an immersion between Riemannian manifolds which preserves the Riemannian metrics.
Equivalently, an isometric embedding (immersion) is a smooth embedding (immersion) which preserves length of curves (cf. Nash embedding theorem).
Algebra
In general, for an algebraic category C, an embedding between two C-algebraic structures X and Y is a C-morphism e:X→Y which is injective.
Field theory
In field theory, an embedding of a field E in a field F is a ring homomorphism σ : E → F.
The kernel of σ is an ideal of E which cannot be the whole field E, because of the condition σ(1) = 1. Furthermore, it is a well-known property of fields that their only ideals are the zero ideal and the whole field itself. Therefore, the kernel is 0, so any embedding of fields is a monomorphism. Hence, E is isomorphic to the subfield σ(E) of F. This justifies the name embedding for an arbitrary homomorphism of fields.
Universal algebra and model theory
If σ is a signature and
Here
Order theory and domain theory
In order theory, an embedding of partial orders is a function F from X to Y such that:
In domain theory, an additional requirement is:
Metric spaces
A mapping
for some constant
Normed spaces
An important special case is that of normed spaces; in this case it is natural to consider linear embeddings.
One of the basic questions that can be asked about a finite-dimensional normed space
The answer is given by Dvoretzky's theorem.
Category theory
In category theory, there is no satisfactory and generally accepted definition of embeddings that is applicable in all categories. One would expect that all isomorphisms and all compositions of embeddings are embeddings, and that all embeddings are monomorphisms. Other typical requirements are: any extremal monomorphism is an embedding and embeddings are stable under pullbacks.
Ideally the class of all embedded subobjects of a given object, up to isomorphism, should also be small, and thus an ordered set. In this case, the category is said to be well powered with respect to the class of embeddings. This allows to define new local structures on the category (such as a closure operator).
In a concrete category, an embedding is a morphism ƒ: A → B which is an injective function from the underlying set of A to the underlying set of B and is also an initial morphism in the following sense: If g is a function from the underlying set of an object C to the underlying set of A, and if its composition with ƒ is a morphism ƒg: C → B, then g itself is a morphism.
A factorization system for a category also gives rise to a notion of embedding. If (E, M) is a factorization system, then the morphisms in M may be regarded as the embeddings, especially when the category is well powered with respect to M. Concrete theories often have a factorization system in which M consists of the embeddings in the previous sense. This is the case of the majority of the examples given in this article.
As usual in category theory, there is a dual concept, known as quotient. All the preceding properties can be dualized.
An embedding can also refer to an embedding functor.