Embedding on arvau

Wikipedia / Embedding

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.

, the embedding is given by some injective are instances. In the terminology of category theory, a structure-preserving map is called a morphism.

is an embedding is often indicated by the use of a "hooked arrow" (); (On the other hand, this notation is sometimes reserved for inclusion maps.)

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 with its image.

Topology and geometry

General topology

In general topology, an embedding is a homeomorphism onto its image. More explicitly, an injective continuous between topological spaces is a topological embedding carries the subspace topology as a subspace. Every embedding is injective and continuous. Every map that is injective, continuous and either open or closed is neither an open set nor a closed set.

is a topological invariant. This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.

Related definitions

is a topological space then the function is said to be if there exists some neighborhood is injective. It is called if it is locally injective around every point of its domain. Similarly, a is a function for which every point in its domain has some neighborhood to which its restriction is a (topological, resp. smooth) embedding.

Every injective function is locally injective but not conversely. Local diffeomorphisms, local homeomorphisms, and smooth immersionss are all locally injective functions that are not necessarily injective. The inverse function theorem gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Every fiber is necessarily a discrete subspace of its domain

Differential topology

In differential topology be smooth manifolds is called an immersion if its derivative is everywhere injective. An embedding, or a smooth embedding, is defined to be an immersion that is an embedding in the topological sense mentioned above (i.e. homeomorphism onto its image).

In other words, the domain of an embedding is diffeomorphic to its image, and in particular the image of an embedding must be a submanifold. An immersion is precisely a local embedding is an embedding.

When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.

. The Whitney embedding theorem is enough, and is the best possible linear bound. For example, the real projective space as is explicitly shown by Boy's surfacewhich has self-intersections. The Roman surface fails to be an immersion as it contains cross-caps.

An embedding is proper if it behaves well with respect to boundaries to be such that

, and
is transverse.

Riemannian and pseudo-Riemannian geometry

In Riemannian geometry be Riemannian manifolds or more generally pseudo-Riemannian manifolds. An isometric embedding that preserves the (pseudo-)metric is equal to the pullback we have

Analogously, isometric immersion is an immersion between (pseudo)-Riemannian manifolds that preserves the (pseudo)-Riemannian metrics.

Equivalently, in Riemannian geometry, an isometric embedding (immersion) is a smooth embedding (immersion) that preserves length of curves (cf. Nash embedding theorem).

Algebra

In general, for an algebraic category-morphism that is injective.

Field theory

In field theory, an embedding of a field is a ring homomorphism .

The kernel is an ideal, because of the condition , so any embedding of fields is a monomorphism is isomorphic to the subfield. This justifies the name embedding for an arbitrary homomorphism of fields.

Universal algebra and model theory

is a signature-structures-algebras in universal algebra or models in model theory-embedding exactly if all of the following hold:

is injective,
,

. In model theory there is also a stronger notion of elementary embedding.

Order theory and domain theory

In order theory, an embedding of partially ordered sets such that

follows quickly from this definition. In domain theory, an additional requirement is that

is directed.

Metric spaces

of metric spaces is called an embedding (with distortion) if

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 is, such that the Hilbert space with constant distortion?

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 pullbackss.

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 defining new local structures in the category (such as a closure operator).

In a concrete category, an embedding and is also an initial morphism itself is a morphism.

A factorization system 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.