When some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map 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.
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 f(X) ⊆ Y.
When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.
Analogously, isometric immersion is an immersion between (pseudo)-Riemannian manifolds which preserves the (pseudo)-Riemannian metrics.
In general, for an algebraic category C, an embedding between two C-algebraic structures X and Y is a C-morphism e : X → Y that is injective.
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. Hence, E is isomorphic to the subfield σ(E) of F. This justifies the name embedding for an arbitrary homomorphism of fields.
Injectivity of F follows quickly from this definition. In domain theory, an additional requirement is that
An important special case is that of normed spaces; in this case it is natural to consider linear embeddings.
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.If an incorrectly led you here, you may wish to change the link to point directly to the intended article.