When some object is said to be embedded in another object , 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 and 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" (U+21AA↪RIGHTWARDS ARROW WITH HOOK); thus: (On the other hand, this notation is sometimes reserved for inclusion maps.)
Given and , several different embeddings of in 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 with its image contained in , so that .
For a given space , the existence of an embedding is a topological invariant of . This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.
If the domain of a function is a topological space then the function is said to be locally injective at a point if there exists some neighborhood of this point such that the restriction is injective. It is called locally injective if it is locally injective around every point of its domain. Similarly, a local (topological, resp. smooth) embedding is a function for which every point in its domain has some neighborhood to which its restriction is a (topological, resp. smooth) embedding.
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 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 . The interest here is in how large must be for an embedding, in terms of the dimension of . The Whitney embedding theorem states that is enough, and is the best possible linear bound. For example, the real projective space of dimension , where is a power of two, requires for an embedding. However, this does not apply to immersions; for instance, can be immersed in 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 to be such that
The kernel of is an ideal of which cannot be the whole field , because of the condition . Furthermore, any field has as ideals only the zero ideal and the whole field itself (because if there is any non-zero field element in an ideal, it is invertible, showing the ideal is the whole field). Therefore, the kernel is , so any embedding of fields is a monomorphism. Hence, is isomorphic to the subfield of . This justifies the name embedding for an arbitrary homomorphism of fields.
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 is, what is the maximal dimension such that the Hilbert space can be linearly embedded into with constant distortion?
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 which is an injective function from the underlying set of to the underlying set of and is also an initial morphism in the following sense:
If is a function from the underlying set of an object to the underlying set of , and if its composition with is a morphism , then itself is a morphism.
A factorization system for a category also gives rise to a notion of embedding. If is a factorization system, then the morphisms in may be regarded as the embeddings, especially when the category is well powered with respect to . Concrete theories often have a factorization system in which 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.