Embedding

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.

"Isometric embedding" redirects here. For related concepts for metric spaces, see isometry.

When some object $X$ is said to be embedded in another object $Y$ , the embedding is given by some injective and structure-preserving map $f:X\rightarrow Y$ . 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 $f:X\rightarrow Y$ is an embedding is often indicated by the use of a "hooked arrow" (U+21AA ↪ RIGHTWARDS ARROW WITH HOOK);^[2] thus: $f:X\hookrightarrow Y.$ (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 $X\subseteq Y$ .

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Topology and geometry[edit]

General topology[edit]

In general topology, an embedding is a homeomorphism onto its image.^[3] More explicitly, an injective continuous map $f:X\to Y$ between topological spaces $X$ and $Y$ is a topological 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 $f:X\to Y$ lets us treat $X$ as a subspace of $Y$ . 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 that are neither open nor closed. The latter happens if the image $f(X)$ is neither an open set nor a closed set in $Y$ .

For a given space $Y$ , the existence of an embedding $X\to Y$ is a topological invariant of $X$ . This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.

$h$ is injective,

for every $n$ -ary function symbol $f\in \sigma$ and $a_{1},\ldots ,a_{n}\in A^{n},$ we have $h(f^{A}(a_{1},\ldots ,a_{n}))=f^{B}(h(a_{1}),\ldots ,h(a_{n}))$ ,

for every $n$ -ary relation symbol $R\in \sigma$ and $a_{1},\ldots ,a_{n}\in A^{n},$ we have $A\models R(a_{1},\ldots ,a_{n})$ iff $B\models R(h(a_{1}),\ldots ,h(a_{n})).$

Category theory[edit]

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 $f:A\rightarrow B$ that 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 $f$ is a morphism $fg:C\rightarrow B$ , 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.

Ambient space

Closed immersion

Cover

Dimension reduction

Flat (geometry)

Immersion

Johnson–Lindenstrauss lemma

Submanifold

Subspace

Universal space

Adámek, Jiří; Horst Herrlich; George Strecker (2006). .

Abstract and Concrete Categories (The Joy of Cats)

on the Manifold Atlas

Embedding of manifolds

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