mathematics, an embedding (or imbedding) is one instance of some mathematical structurecontained 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
injectiveand 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 of category theory, a structure-preserving map is called a morphism.
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 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 nowrap|"X" ⊆ "Y".
Topology and geometry
general topology, an embedding is a homeomorphismonto 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 injectiveand 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 setnor a closed setin "Y".
For a given space X, the existence of an embedding X → Y is a
topological invariantof 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:Let "M" and "N" be smooth manifolds and 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. homeomorphismonto 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 there is a neighborhood such that 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 theoremstates that "n" = 2"m" is enough. For example the real projective planeof 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 surfacefails 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 to be such that
* is transversal to in any point of .
The first condition is equivalent to having and . The second condition, roughly speaking, says that f(X) is not tangent to the boundary of Y.
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
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).
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.
In field theory, an embedding of a field "E" in a field "F" is a
ring homomorphismσ : "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" is isomorphicto 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 are σ-structures (also called σ-algebras in
universal algebraor models in model theory), then a map is a σ-embedding iffall the following holds:
* for every -ary function symbol and we have ,
* for every -ary relation symbol and we have iff
Here is a model theoretical notation equivalent to . In model theory there is also a stronger notion of
Order theory and domain theory
order theory, an embedding of partial orders is a function F from X to Y such that :
domain theory, an additional requirement is :
"Based on an article from FOLDOC, ."
A mapping of
metric spacesis called an "embedding"(with distortion ) if :for some constant .
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 spaceis, "what is the maximal dimension such that the Hilbert spacecan be linearly embedded into with constant distortion?"
The answer is given by
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:
concrete categorystructure, embeddings are then defined as the morphisms with injective underlying function satisfying an initiality condition
factorization system, embeddings are then defined as the morphisms in (in this case, the category is often required to be well powered with respect to ).
In most cases, concrete categories have a factorization structure where 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.
*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
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