In mathematics, an embedding (or imbedding[1]) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which
In the terminology of category theory, a structure-preserving map is called a morphism.
(On the other hand, this notation is sometimes reserved for inclusion maps.)
In general topology, an embedding is a homeomorphism onto its image.
Every map that is injective, continuous and either open or closed is an embedding; however there are also embeddings that are neither open nor closed.
is a topological space then the function is said to be locally injective at a point if there exists some neighborhood
smooth) embedding is a function for which every point in its domain has some neighborhood to which its restriction is a (topological, resp.
The inverse function theorem gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective.
Every fiber of a locally injective function
is called an immersion if its derivative is everywhere injective.
An immersion is precisely a local embedding, i.e. for any point
When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.
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
Equivalently, in Riemannian geometry, an isometric embedding (immersion) is a smooth embedding (immersion) that preserves length of curves (cf.
This justifies the name embedding for an arbitrary homomorphism of fields.
In model theory there is also a stronger notion of elementary embedding.
In domain theory, an additional requirement is that A mapping
of metric spaces is called an embedding (with distortion
One of the basic questions that can be asked about a finite-dimensional normed space
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).
that is an injective function from the underlying set of
is a function from the underlying set of an object
A factorization system for a category also gives rise to a notion of embedding.
may be regarded as the embeddings, especially when the category is well powered with respect to
As usual in category theory, there is a dual concept, known as quotient.