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.
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: formula_1 (On the other hand, this notation is sometimes reserved for inclusion maps.)
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 .
Topology and geometry.
General topology.
In general topology, an embedding is a homeomorphism onto its image. More explicitly, an injective continuous map formula_2 between topological spaces formula_3 and formula_4 is a topological embedding if formula_5 yields a homeomorphism between formula_3 and formula_7 (where formula_7 carries the subspace topology inherited from formula_4). Intuitively then, the embedding formula_2 lets us treat formula_3 as a subspace of formula_4. 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 formula_7 is neither an open set nor a closed set in formula_4.
For a given space formula_4, the existence of an embedding formula_16 is a topological invariant of formula_3. This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.
Differential topology.
In differential topology:
Let formula_18 and formula_19 be smooth manifolds and formula_20 be a smooth map. Then formula_5 is called an immersion if its derivative is everywhere injective. An embedding, or a smooth embedding, is defined to be an injective immersion which 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, i.e. for any point formula_22 there is a neighborhood formula_23 such that formula_24 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 formula_25. The interest here is in how large formula_26 must be for an embedding, in terms of the dimension formula_27 of formula_18. The Whitney embedding theorem states that formula_29 is enough, and is the best possible linear bound. For example, the real projective space RP"m" of dimension formula_27, where formula_27 is a power of two, requires formula_29 for an embedding. However, this does not apply to immersions; for instance, RP2 can be immersed in formula_33 as is explicitly shown by 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 with respect to boundaries: one requires the map formula_34 to be such that
The first condition is equivalent to having formula_39 and formula_40. The second condition, roughly speaking, says that "f"("X") is not tangent to the boundary of "Y".
Riemannian and pseudo-Riemannian geometry.
In Riemannian geometry and pseudo-Riemannian geometry:
Let ("M", "g") and ("N", "h") be Riemannian manifolds or more generally pseudo-Riemannian manifolds.
An isometric embedding is a smooth embedding "f" : "M" → "N" which preserves the (pseudo-)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 formula_41 we have
Analogously, isometric immersion is an immersion between (pseudo)-Riemannian manifolds which preserves the (pseudo)-Riemannian metrics.
Equivalently, in Riemannian geometry, 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 that is injective.
Field theory.
In field theory, an embedding of a field "E" in a field "F" is a ring homomorphism .
The kernel of "σ" is an ideal of "E" which cannot be the whole field "E", because of the condition . 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 formula_43 are σ-structures (also called σ-algebras in universal algebra or models in model theory), then a map formula_44 is a σ-embedding iff all of the following hold:
Here formula_55 is a model theoretical notation equivalent to formula_56. 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 is a function "F" between partially ordered sets "X" and "Y" such that
Injectivity of "F" follows quickly from this definition. In domain theory, an additional requirement is that
Metric spaces.
A mapping formula_59 of metric spaces is called an "embedding"
(with distortion formula_60) if 
for some constant formula_62.
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 formula_63 is, "what is the maximal dimension formula_64 such that the Hilbert space formula_65 can be linearly embedded into formula_3 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 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 defining new local structures in 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.