Inmathematics, anembedding (orimbedding^[1]) is one instance of somemathematical structure contained within another instance, such as agroup that is asubgroup.

When some object${\displaystyle X}$ is said to be embedded in another object${\displaystyle Y}$, the embedding is given by someinjective and structure-preserving map${\displaystyle f:X\to Y}$. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which${\displaystyle X}$ and${\displaystyle Y}$ are instances. In the terminology ofcategory theory, a structure-preserving map is called amorphism.

The fact that a map${\displaystyle f:X\to Y}$ is an embedding is often indicated by the use of a "hooked arrow" (U+21AA ↪RIGHTWARDS ARROW WITH HOOK);^[2] thus:${\displaystyle f:X\hookrightarrow Y.}$ (On the other hand, this notation is sometimes reserved forinclusion maps.)

Given${\displaystyle X}$ and${\displaystyle Y}$, several different embeddings of${\displaystyle X}$ in${\displaystyle Y}$ may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of thenatural numbers in theintegers, the integers in therational numbers, the rational numbers in thereal numbers, and the real numbers in thecomplex numbers. In such cases it is common to identify thedomain${\displaystyle X}$ with itsimage${\displaystyle f(X)}$ contained in${\displaystyle Y}$, so that${\displaystyle X\subseteq Y}$.

Ingeneral topology, an embedding is ahomeomorphism onto its image.^[3] More explicitly, an injectivecontinuous map${\displaystyle f:X\to Y}$ betweentopological spaces${\displaystyle X}$ and${\displaystyle Y}$ is atopological embedding if${\displaystyle f}$ yields a homeomorphism between${\displaystyle X}$ and${\displaystyle f(X)}$ (where${\displaystyle f(X)}$ carries thesubspace topology inherited from${\displaystyle Y}$). Intuitively then, the embedding${\displaystyle f:X\to Y}$ lets us treat${\displaystyle X}$ as asubspace of${\displaystyle Y}$. Every embedding is injective andcontinuous. Every map that is injective, continuous and eitheropen orclosed is an embedding; however there are also embeddings that are neither open nor closed. The latter happens if the image${\displaystyle f(X)}$ is neither anopen set nor aclosed set in${\displaystyle Y}$.

For a given space${\displaystyle Y}$, the existence of an embedding${\displaystyle X\to Y}$ is atopological invariant of${\displaystyle X}$. 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${\displaystyle f:X\to Y}$ is atopological space then the function is said to belocally injective at a point if there exists someneighborhood${\displaystyle U}$ of this point such that the restriction${\displaystyle f{|}_U:U\to Y}$ is injective. It is calledlocally injective if it is locally injective around every point of its domain. Similarly, alocal (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.

Every injective function is locally injective but not conversely.Local diffeomorphisms,local homeomorphisms, and smoothimmersions are all locally injective functions that are not necessarily injective. Theinverse function theorem gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Everyfiber of a locally injective function${\displaystyle f:X\to Y}$ is necessarily adiscrete subspace of itsdomain${\displaystyle X.}$

Indifferential topology:Let${\displaystyle M}$ and${\displaystyle N}$ be smoothmanifolds and${\displaystyle f:M\to N}$ be a smooth map. Then${\displaystyle f}$ is called animmersion if itsderivative is everywhere injective. Anembedding, or asmooth embedding, is defined to be an immersion that is an embedding in the topological sense mentioned above (i.e.homeomorphism onto its image).^[4]

In other words, the domain of an embedding isdiffeomorphic to its image, and in particular the image of an embedding must be asubmanifold. An immersion is precisely alocal embedding, i.e. for any point${\displaystyle x\in M}$ there is a neighborhood${\displaystyle x\in U\subset M}$ such that${\displaystyle f:U\to N}$ 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${\displaystyle N={\mathbb{R}}^n}$. The interest here is in how large${\displaystyle n}$ must be for an embedding, in terms of the dimension${\displaystyle m}$ of${\displaystyle M}$. TheWhitney embedding theorem^[5] states that${\displaystyle n=2m}$ is enough, and is the best possible linear bound. For example, thereal projective space${\displaystyle \mathbb{R}{\mathrm{P}}^m}$ of dimension${\displaystyle m}$, where${\displaystyle m}$ is a power of two, requires${\displaystyle n=2m}$ for an embedding. However, this does not apply to immersions; for instance,${\displaystyle \mathbb{R}{\mathrm{P}}^2}$ can be immersed in${\displaystyle {\mathbb{R}}^3}$ as is explicitly shown byBoy's surface—which has self-intersections. TheRoman surface fails to be an immersion as it containscross-caps.

An embedding isproper if it behaves well with respect toboundaries: one requires the map${\displaystyle f:X\to Y}$ to be such that

• ${\displaystyle f(\mathrm{\partial }X)=f(X)\cap \mathrm{\partial }Y}$, and
• ${\displaystyle f(X)}$ istransverse to${\displaystyle \mathrm{\partial }Y}$ in any point of${\displaystyle f(\mathrm{\partial }X)}$.

The first condition is equivalent to having${\displaystyle f(\mathrm{\partial }X)\subseteq \mathrm{\partial }Y}$ and${\displaystyle f(X\setminus \mathrm{\partial }X)\subseteq Y\setminus \mathrm{\partial }Y}$. The second condition, roughly speaking, says that${\displaystyle f(X)}$ is not tangent to the boundary of${\displaystyle Y}$.

InRiemannian geometry and pseudo-Riemannian geometry:Let${\displaystyle (M,g)}$ and${\displaystyle (N,h)}$ beRiemannian manifolds or more generallypseudo-Riemannian manifolds.Anisometric embedding is a smooth embedding${\displaystyle f:M\to N}$ that preserves the (pseudo-)metric in the sense that${\displaystyle g}$ is equal to thepullback of${\displaystyle h}$ by${\displaystyle f}$, i.e.${\displaystyle g=f^{\ast }h}$. Explicitly, for any two tangent vectors${\displaystyle v,w\in T_x(M)}$ we have

${\displaystyle g(v,w)=h(df(v),df(w)).}$

Analogously,isometric immersion is an immersion between (pseudo)-Riemannian manifolds that preserves the (pseudo)-Riemannian metrics.

Equivalently, in Riemannian geometry, an isometric embedding (immersion) is a smooth embedding (immersion) that preserves length ofcurves (cf.Nash embedding theorem).^[6]

In general, for analgebraic category${\displaystyle C}$, an embedding between two${\displaystyle C}$-algebraic structures${\displaystyle X}$ and${\displaystyle Y}$ is a${\displaystyle C}$-morphism${\displaystyle e:X\to Y}$ that is injective.

Infield theory, anembedding of afield${\displaystyle E}$ in a field${\displaystyle F}$ is aring homomorphism${\displaystyle \sigma :E\to F}$.

Thekernel of${\displaystyle \sigma }$ is anideal of${\displaystyle E}$, which cannot be the whole field${\displaystyle E}$, because of the condition${\displaystyle 1=\sigma (1)=1}$. 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${\displaystyle 0}$, so any embedding of fields is amonomorphism. Hence,${\displaystyle E}$ isisomorphic to thesubfield${\displaystyle \sigma (E)}$ of${\displaystyle F}$. This justifies the nameembedding for an arbitrary homomorphism of fields.

If${\displaystyle \sigma }$ is asignature and${\displaystyle A,B}$ are${\displaystyle \sigma }$-structures (also called${\displaystyle \sigma }$-algebras inuniversal algebra or models inmodel theory), then a map${\displaystyle h:A\to B}$ is a${\displaystyle \sigma }$-embedding exactly if all of the following hold:

• ${\displaystyle h}$ is injective,
• for every${\displaystyle n}$-ary function symbol${\displaystyle f\in \sigma }$ and${\displaystyle a_1,\dots ,a_n\in A^n,}$ we have${\displaystyle h(f^A(a_1,\dots ,a_n))=f^B(h(a_1),\dots ,h(a_n))}$,
• for every${\displaystyle n}$-ary relation symbol${\displaystyle R\in \sigma }$ and${\displaystyle a_1,\dots ,a_n\in A^n,}$ we have${\displaystyle A\models R(a_1,\dots ,a_n)}$ iff${\displaystyle B\models R(h(a_1),\dots ,h(a_n)).}$

Here${\displaystyle A\models R(a_1,\dots ,a_n)}$ is a model theoretical notation equivalent to${\displaystyle (a_1,\dots ,a_n)\in R^A}$. In model theory there is also a stronger notion ofelementary embedding.

Inorder theory, an embedding ofpartially ordered sets is a function${\displaystyle F}$ between partially ordered sets${\displaystyle X}$ and${\displaystyle Y}$ such that

${\displaystyle \mathrm{\forall }{x}_1,x_2\in X:x_1\le x_2⟺F(x_1)\le F(x_2).}$

Injectivity of${\displaystyle F}$follows quickly from this definition. Indomain theory, an additional requirement is that

${\displaystyle \mathrm{\forall }y\in Y:\{x\mid F(x)\le y\}}$ isdirected.

A mapping${\displaystyle \varphi :X\to Y}$ ofmetric spaces is called anembedding(withdistortion${\displaystyle C>0}$) if

${\displaystyle L{d}_X(x,y)\le d_Y(\varphi (x),\varphi (y))\le CL{d}_X(x,y)}$

for every${\displaystyle x,y\in X}$ and some constant${\displaystyle L>0}$.

An important special case is that ofnormed spaces; in this case it is natural to consider linear embeddings.

One of the basic questions that can be asked about a finite-dimensionalnormed space${\displaystyle (X,\Vert \cdot \Vert )}$ is,what is the maximal dimension${\displaystyle k}$ such that theHilbert space${\displaystyle {\ell }_2^k}$ can be linearly embedded into${\displaystyle X}$ with constant distortion?

The answer is given byDvoretzky's theorem.

Incategory 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: anyextremal monomorphism is an embedding and embeddings are stable underpullbacks.

Ideally the class of all embeddedsubobjects of a given object, up to isomorphism, should also besmall, and thus anordered 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 aclosure operator).

In aconcrete category, anembedding is a morphism${\displaystyle f:A\to B}$ that is an injective function from the underlying set of${\displaystyle A}$ to the underlying set of${\displaystyle B}$ and is also aninitial morphism in the following sense:If${\displaystyle g}$ is a function from the underlying set of an object${\displaystyle C}$ to the underlying set of${\displaystyle A}$, and if its composition with${\displaystyle f}$ is a morphism${\displaystyle fg:C\to B}$, then${\displaystyle g}$ itself is a morphism.

Afactorization system for a category also gives rise to a notion of embedding. If${\displaystyle (E,M)}$ is a factorization system, then the morphisms in${\displaystyle M}$ may be regarded as the embeddings, especially when the category is well powered with respect to${\displaystyle M}$. Concrete theories often have a factorization system in which${\displaystyle M}$ 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 adual concept, known as quotient. All the preceding properties can be dualized.

An embedding can also refer to anembedding functor.

• Embedding (machine learning)
• Ambient space
• Closed immersion
• Cover
• Dimensionality 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).
• Embedding of manifolds on the Manifold Atlas

Index of articles associated with the same name

Thisset index article includes a list of related items that share the same name (or similar names).
If aninternal link incorrectly led you here, you may wish to change the link to point directly to the intended article.

• Set index articles
• Abstract algebra
• Category theory
• General topology
• Differential topology
• Functions and mappings
• Maps of manifolds
• Model theory
• Order theory

• Articles with short description
• Short description is different from Wikidata
• All set index articles