- Embedding
In

mathematics , an**embedding**(or**imbedding**) is one instance of somemathematical structure contained within another instance, such as a group that is asubgroup .When some object "X" is said to be embedded in another object "Y", the embedding is given by some

injective and structure-preserving map nowrap|"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 ofcategory theory , a structure-preserving map is called amorphism .The fact that a map nowrap|"f" : "X" → "Y" is an embedding is often indicated by the use of a "hooked arrow", thus: nowrap|"f" : "X" ↪ "Y".

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 number s in theinteger s, the integers in therational number s, the rational numbers in thereal number s, and the real numbers in thecomplex number s. In such cases it is common to identify the domain "X" with its image "f"("X") contained in "Y", so that then nowrap|"X" ⊆ "Y".**Topology and geometry****General topology**In

general topology , an embedding is ahomeomorphism onto its image. More explicitly, a map "f" : "X" → "Y" betweentopological space s "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 isinjective 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 anopen set nor aclosed 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 smoothmanifold s and $f:M\; o\; N$ 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 $xin\; M$ there is a neighborhood $xin\; Usubset\; M$ such that $f:U\; o\; N$ 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"=

**R**^{n}. The interest here is in how large "n" must be, in terms of the dimension "m" of "M". TheWhitney embedding theorem states that "n" = 2"m" is enough. For example thereal projective plane of dimension 2 requires "n" = 4 for an embedding. An immersion of this surface is, however, possible in**R**^{3}, and one example isBoy's surface —which has self-intersections. TheRoman surface fails to be an immersion as it contains cross-caps.An embedding is

**proper**if it behaves wellw.r.t. boundaries: one requires the map $f:\; X\; ightarrow\; Y$ 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") beRiemannian manifold s.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:$v,win\; T\_x(M)$

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

curve s (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 aring 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 . Moreover, "E" isisomorphic 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 inmodel theory ), then a map $h:A\; o\; B$ is a σ-embeddingiff all the following holds:

* $h$ isinjective ,

* for every $n$-ary function symbol $f\; insigma$ 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 $R\; insigma$ 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 $Amodels\; R\; (a\_1,ldots,a\_n)$ 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**ofpartial order s is a function F from X to Y such that :$forall\; x\_1,x\_2in\; X:\; x\_1leq\; x\_2Leftrightarrow\; F(x\_1)leq\; F(x\_2)$.

In

domain theory , an additional requirement is :$forall\; yin\; Y:\{x:\; F(x)leq\; y\}$ is

directed ."Based on an article from FOLDOC, ."

**Metric spaces**A mapping $phi:\; X\; o\; Y$ of

metric spaces is called an "embedding"(with distortion $C>0$) if :$L\; d\_X(x,\; y)\; leq\; d\_Y(phi(x),\; phi(y))\; leq\; CLd\_X(x,y)$for some constant $L>0$.**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 theHilbert 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

subobject s of a given object, thought equivalent up to an isomorphism, is small, and thus anordered 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 aclosure operator ).The kind of structures on a category allowing to define embeddings are:

* aconcrete category structure, embeddings are then defined as the morphisms with injective underlying function satisfying an initiality condition

* afactorization 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.

**ee also***

Inclusion map **References***cite book|last=Adámek|first=Jiří|coauthors=Horst Herrlich, George Strecker|title=Abstract and Concrete Categories (The Joy of Cats)|url=http://katmat.math.uni-bremen.de/acc/|origyear=1990|year=2006

*Wikimedia Foundation.
2010.*