# Dictionary Definition

embed v : fix or set securely or deeply; "He
planted a knee in the back of his opponent"; "The dentist implanted
a tooth in the gum" [syn: implant, engraft, imbed, plant] [also: embedding, embedded]embedding See embed

# User Contributed Dictionary

## English

### Noun

- A map which maps a subspace (smaller structure) to the whole space (larger structure).
- To inject, insert a code (malicious) in other code or into the operating system.

#### Translations

Mathematics

- German: Einbettung, Einschluss
- Russian: вложе́ние (vložénie)
- Swedish: inbäddning

Computer

- German: Einbinden

### Verb

embedding- present participle of embed

# Extensive Definition

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.

When some object X is said to be embedded in
another object Y, the embedding is given by some injective and
structure-preserving map . 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 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 then .

## Topology and geometry

### General topology

In general
topology, an embedding is a homeomorphism onto its
image. More explicitly, a map f : X → Y between topological
spaces X and Y is an embedding if f yields a homeomorphism
between X and f(X) (where f(X) carries the subspace
topology inherited from Y). Intuitively then, the embedding f :
X → Y lets us treat X as a subspace
of Y. Every embedding is injective and
continuous. Every map that is injective, continuous and either
open or
closed
is an embedding; however there are also embeddings which are
neither open nor closed. The latter happens if the image f(X) is
neither an open set nor a
closed
set in Y.

For a given space X, the existence of an
embedding X → Y is a topological
invariant of X. This allows two spaces to be distinguished if
one is able to be embedded into a space which the other is
not.

### Differential topology

In differential
topology: Let M and N be smooth manifolds and f:M\to N be a
smooth map, it is called an immersion
if the derivative
of f is everywhere injective. Then an embedding, or a smooth
embedding, is defined to be an immersion which is an embedding in
the above sense (i.e. homeomorphism onto its
image).

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 x\in M there is
a neighborhood x\in U\subset M such that f:U\to N is an
embedding.)

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

An important case is N=Rn. The interest here is
in how large n must be, in terms of the dimension m of M. The
Whitney
embedding theorem states that n = 2m is enough. For example the
real
projective plane of dimension 2 requires n = 4 for an
embedding. An immersion of this surface is, however, possible in
R3, and one example is Boy's
surface—which has self-intersections. The Roman
surface fails to be an immersion as it contains
cross-caps.

An embedding is proper if it behaves well
w.r.t.
boundaries: one requires the map f: X \rightarrow Y to be such
that

- f(\partial X) = f(X) \cap \partial Y, and
- f(X) is transversal to \partial Y in any point of f(\partial X).

The first condition is equivalent to having
f(\partial X) \subseteq \partial Y and f(X \setminus \partial X)
\subseteq Y \setminus \partial Y. The second condition, roughly
speaking, says that f(X) is not tangent to the boundary of Y.

### 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

- v,w\in T_x(M)

we have

- g(v,w)=h(df(v),df(w)).\,

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

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.
Moreover, 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 A,B are σ-structures
(also called σ-algebras in universal
algebra or models in model
theory), then a map h:A \to B is a σ-embedding iff all the following holds:

- h is injective,
- for every n-ary function symbol f \in\sigma and a_1,\ldots,a_n \in A^n, we have h(f^A(a_1,\ldots,a_n))=f^B(h(a_1),\ldots,h(a_n)),
- for every n-ary relation symbol R \in\sigma and a_1,\ldots,a_n \in A^n, we have A \models R(a_1,\ldots,a_n) iff B \models R(h(a_1),\ldots,h(a_n)).

Here A\models R (a_1,\ldots,a_n) is a model
theoretical notation equivalent to (a_1,\ldots,a_n)\in R^A. In
model theory there is also a stronger notion of elementary
embedding.

## Order theory and domain theory

In order
theory, an embedding of partial
orders is a function F from X to Y such that :

\forall x_1,x_2\in X: x_1\leq x_2\Leftrightarrow
F(x_1)\leq F(x_2).

In domain
theory, an additional requirement is :

\forall y\in Y:\ is directed.

Based on an article from FOLDOC, used
by permission.

## Metric spaces

A mapping \phi: X \to Y of metric
spaces is called an embedding (with distortion C>0) if

- L d_X(x, y) \leq d_Y(\phi(x), \phi(y)) \leq CLd_X(x,y)

### 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
(X, \| \cdot \|) is, what is the maximal dimension k such that the
Hilbert
space \ell_2^k can be linearly embedded into X with constant
distortion?

The answer is given by Dvoretzky's
theorem.

## Category theory

In category
theory, it is not possible to define an embedding without
additional structures on the base category. However, in all
generality, it is possible to define what properties should satisfy
a class of embeddings in a given category.

In all cases, the class of embeddings should
contain all isomorphisms. Most of the time, embeddings are required
to be stable under composition and be monic. Other typical
requirements are: any extremal monomorphism is an
embedding and embeddings are stable under pullbacks.

A common property of embeddings is that the class
of all embedded subobjects of a given object,
thought equivalent up to an isomorphism, is 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).

The kind of structures on a category allowing to
define embeddings are:

- a concrete category structure, embeddings are then defined as the morphisms with injective underlying function satisfying an initiality condition
- a factorization system (E,M), embeddings are then defined as the morphisms in M (in this case, the category is often required to be well powered with respect to M).

In most cases, concrete categories have a
factorization structure (E,M) where M is the class of embeddings
defined by the concrete structure. 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.

## See also

## References

- Abstract and Concrete Categories (The Joy of Cats)">http://katmat.math.uni-bremen.de/acc/|origyear=1990|year=2006}}

embedding in Catalan: Immersió
(matemàtiques)

embedding in German: Einbettung

embedding in Spanish: Encaje (matemática)

embedding in French: Plongement

embedding in Italian: Immersione
(matematica)

embedding in Hebrew: שיכון (מתמטיקה)

embedding in Polish: Zanurzenie
(matematyka)