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{{Redirect|Isometric embedding|related concepts for [[metric space]]s|isometry}}
{{For|embeddings of graphs in two-dimensional manifolds|graph embedding}}
{{Other uses}}
{{Short description|Inclusion of one mathematical structure in another, preserving properties of interest}}
In [[mathematics]], an '''embedding''' (or '''imbedding'''<ref>{{harvnb|Spivak|1999|page=49}} suggests that "the English" (i.e. the British) use "embedding" instead of "imbedding".</ref>) is one instance of some [[mathematical structure]] contained within another instance, such as a [[group (mathematics)|group]] that is a [[subgroup]].

When some object <math>X</math> is said to be embedded in another object <math>Y</math>, the embedding is given by some [[Injective function|injective]] and structure-preserving map <math>f:X\rightarrow Y</math>. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which <math>X</math> and <math>Y</math> are instances. In the terminology of [[category theory]], a structure-preserving map is called a [[morphism]].

The fact that a map <math>f:X\rightarrow Y</math> is an embedding is often indicated by the use of a "hooked arrow" ({{unichar|21AA|RIGHTWARDS ARROW WITH HOOK|ulink=Unicode}});<ref name="Unicode Arrows">{{cite web| title = Arrows – Unicode| url = https://www.unicode.org/charts/PDF/U2190.pdf| access-date = 2017-02-07}}</ref> thus: <math> f : X \hookrightarrow Y.</math> (On the other hand, this notation is sometimes reserved for [[inclusion map]]s.)

Given <math>X</math> and <math>Y</math>, several different embeddings of <math>X</math> in <math>Y</math> may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the [[natural number]]s in the [[integer]]s, the integers in the [[rational number]]s, the rational numbers in the [[real number]]s, and the real numbers in the [[complex number]]s. In such cases it is common to identify the [[Domain of a function|domain]] <math>X</math> with its [[image (mathematics)|image]] <math>f(X)</math> contained in <math>Y</math>, so that <math>X\subseteq Y</math>.

==Topology and geometry==
===General topology===

In [[general topology]], an embedding is a [[homeomorphism]] onto its image.<ref>{{harvnb|Hocking|Young|1988|page=73}}. {{harvnb|Sharpe|1997|page=16}}.</ref> More explicitly, an injective [[continuous function (topology)|continuous]] map <math>f : X \to Y</math> between [[topological space]]s <math>X</math> and <math>Y</math> is a '''topological embedding''' if <math>f</math> yields a homeomorphism between <math>X</math> and <math>f(X)</math> (where <math>f(X)</math> carries the [[subspace topology]] inherited from <math>Y</math>). Intuitively then, the embedding <math>f : X \to Y</math> lets us treat <math>X</math> as a [[subspace topology|subspace]] of <math>Y</math>. Every embedding is injective and [[continuous function (topology)|continuous]]. Every map that is injective, continuous and either [[open map|open]] or [[closed map|closed]] is an embedding; however there are also embeddings that are neither open nor closed. The latter happens if the image <math>f(X)</math> is neither an [[open set]] nor a [[closed set]] in <math>Y</math>.

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

====Related definitions====

If the domain of a function <math>f : X \to Y</math> is a [[topological space]] then the function is said to be ''{{visible anchor|locally injective at a point}}'' if there exists some [[Neighbourhood (mathematics)|neighborhood]] <math>U</math> of this point such that the restriction <math>f\big\vert_U : U \to Y</math> is injective. It is called ''{{visible anchor|locally injective}}'' if it is locally injective around every point of its domain. Similarly, a ''{{visible anchor|local topological embedding|text=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. 

Every injective function is locally injective but not conversely. [[Local diffeomorphism]]s, [[local homeomorphism]]s, and smooth [[Immersion (mathematics)|immersion]]s are all locally injective functions that are not necessarily injective. The [[inverse function theorem]] gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Every [[Fiber (mathematics)|fiber]] of a locally injective function <math>f : X \to Y</math> is necessarily a [[Discrete space|discrete subspace]] of its [[Domain of a function|domain]] <math>X.</math>

===Differential topology===

In [[differential topology]]:
Let <math>M</math> and <math>N</math> be smooth [[manifold]]s and <math>f:M\to N</math> be a smooth map. Then <math>f</math> is called an [[immersion (mathematics)|immersion]] if its [[pushforward (differential)|derivative]] is everywhere injective. An '''embedding''', or a '''smooth embedding''', is defined to be an immersion that is an embedding in the topological sense mentioned above (i.e. [[homeomorphism]] onto its image).<ref>{{harvnb|Bishop|Crittenden|1964|page=21}}. {{harvnb|Bishop|Goldberg|1968|page=40}}. {{harvnb|Crampin|Pirani|1994|page=243}}. {{harvnb|do Carmo|1994|page=11}}. {{harvnb|Flanders|1989|page=53}}. {{harvnb|Gallot|Hulin|Lafontaine|2004|page=12}}. {{harvnb|Kobayashi|Nomizu|1963|page=9}}. {{harvnb|Kosinski|2007|page=27}}. {{harvnb|Lang|1999|page=27}}. {{harvnb|Lee|1997|page=15}}. {{harvnb|Spivak|1999|page=49}}. {{harvnb|Warner|1983|page=22}}.</ref> 
 
In other words, the domain of an embedding is [[diffeomorphism|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 <math>x\in M</math> there is a neighborhood <math>x\in U\subset M</math> such that <math>f:U\to N</math> 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 <math>N = \mathbb{R}^n</math>. The interest here is in how large <math>n</math> must be for an embedding, in terms of the dimension <math>m</math> of <math>M</math>. The [[Whitney embedding theorem]]<ref>Whitney H., ''Differentiable manifolds,'' Ann. of Math. (2), '''37''' (1936), pp. 645–680</ref> states that <math>n = 2m</math> is enough, and is the best possible linear bound. For example, the [[real projective space]] <math>\mathbb{R}\mathrm{P}^m</math> of dimension <math>m</math>, where <math>m</math> is a power of two, requires <math>n = 2m</math> for an embedding. However, this does not apply to immersions; for instance, <math>\mathbb{R}\mathrm{P}^2</math> can be immersed in <math>\mathbb{R}^3</math> as is explicitly shown by [[Boy's surface]]&mdash;which has self-intersections. The [[Roman surface]] fails to be an immersion as it contains [[cross-cap]]s.

An embedding is '''proper''' if it behaves well with respect to [[Topological manifold#Manifolds with boundary|boundaries]]: one requires the map <math>f: X \rightarrow Y</math> to be such that

*<math>f(\partial X) = f(X) \cap \partial Y</math>, and
*<math>f(X)</math> is [[Transversality (mathematics)|transverse]] to <math>\partial Y</math> in any point of <math>f(\partial X)</math>.

The first condition is equivalent to having <math>f(\partial X) \subseteq \partial Y</math> and <math>f(X \setminus \partial X) \subseteq Y \setminus \partial Y</math>. The second condition, roughly speaking, says that <math>f(X)</math> is not tangent to the boundary of <math>Y</math>.

===Riemannian and pseudo-Riemannian geometry===

In [[Riemannian geometry]] and pseudo-Riemannian geometry:
Let <math>(M,g)</math> and <math>(N,h)</math> be [[Riemannian manifold]]s or more generally [[pseudo-Riemannian manifold]]s.
An '''isometric embedding''' is a smooth embedding <math>f:M\rightarrow N</math> that preserves the  (pseudo-)[[Riemannian metric|metric]] in the sense that <math>g</math> is equal to the [[pullback (differential geometry)|pullback]] of <math>h</math> by <math>f</math>, i.e. <math>g=f^{*}h</math>. Explicitly, for any two tangent vectors <math>v,w\in T_x(M)</math> we have

:<math>g(v,w)=h(df(v),df(w)).</math>

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 of [[curve]]s (cf. [[Nash embedding theorem]]).<ref>Nash J.,  ''The embedding problem for Riemannian manifolds,'' Ann. of Math. (2), '''63''' (1956), 20–63.</ref>

==Algebra==
In general, for an [[Variety (universal algebra)|algebraic category]] <math>C</math>, an embedding between two <math>C</math>-algebraic structures <math>X</math> and <math>Y</math> is a <math>C</math>-morphism {{nowrap|<math>e:X\rightarrow Y</math>}} that is injective.

===Field theory===

In [[field theory (mathematics)|field theory]], an '''embedding''' of a [[field (mathematics)|field]] <math>E</math> in a field <math>F</math> is a [[ring homomorphism]] {{nowrap|<math>\sigma:E\rightarrow F</math>}}.

The [[Kernel (algebra)|kernel]] of <math>\sigma</math> is an [[ideal (ring theory)|ideal]] of <math>E</math>, which cannot be the whole field <math>E</math>, because of the condition {{nowrap|<math>1=\sigma(1)=1</math>}}. 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 <math>0</math>, so any embedding of fields is a [[monomorphism]]. Hence, <math>E</math> is [[isomorphic]] to the [[Field extension|subfield]] <math>\sigma(E)</math> of <math>F</math>. This justifies the name ''embedding'' for an arbitrary homomorphism of fields.

===Universal algebra and model theory===
{{further|Substructure (mathematics)|Elementary equivalence}}
If <math>\sigma</math> is a [[signature (logic)|signature]] and <math>A,B</math> are <math>\sigma</math>-[[structure (mathematical logic)|structures]] (also called <math>\sigma</math>-algebras in [[universal algebra]] or models in [[model theory]]), then a map <math>h:A \to B</math> is a <math>\sigma</math>-embedding exactly if all of the following hold:
* <math>h</math> is injective,
* for every <math>n</math>-ary function symbol <math>f \in\sigma</math> and <math>a_1,\ldots,a_n \in A^n,</math> we have <math>h(f^A(a_1,\ldots,a_n))=f^B(h(a_1),\ldots,h(a_n))</math>,
* for every <math>n</math>-ary relation symbol <math>R \in\sigma</math> and <math>a_1,\ldots,a_n \in A^n,</math> we have <math>A \models R(a_1,\ldots,a_n)</math> iff <math>B \models R(h(a_1),\ldots,h(a_n)).</math>

Here <math>A\models R (a_1,\ldots,a_n)</math> is a model theoretical notation equivalent to <math>(a_1,\ldots,a_n)\in R^A</math>. In model theory there is also a stronger notion of [[elementary embedding]].

==Order theory and domain theory==
In [[order theory]], an embedding of [[partially ordered set]]s is a function <math>F</math> between partially ordered sets <math>X</math> and <math>Y</math> such that

:<math>\forall x_1,x_2\in X: x_1\leq x_2 \iff F(x_1)\leq F(x_2).</math>

Injectivity of <math>F</math>
follows quickly from this definition. In [[domain theory]], an additional requirement is that

:<math> \forall y\in Y:\{x \mid F(x) \leq y\}</math> is [[Directed set|directed]].

==Metric spaces==

A mapping <math>\phi: X \to Y</math> of [[metric spaces]] is called an ''embedding''
(with [[stretch factor|distortion]] <math>C>0</math>) if
:<math> L d_X(x, y) \leq d_Y(\phi(x), \phi(y)) \leq CLd_X(x,y) </math>
for every <math>x,y\in X</math> and some constant <math>L>0</math>.

=== Normed spaces ===

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]] <math>(X, \| \cdot \|)</math> is, ''what is the maximal dimension <math>k</math> such that the [[Hilbert space]] <math>\ell_2^k</math> can be linearly embedded into <math>X</math> with constant distortion?''

The answer is given by [[Dvoretzky's theorem]].

==Category theory==

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 [[monomorphism#Related concepts|extremal monomorphism]] is an embedding and embeddings are stable under [[Pullback (category theory)|pullback]]s.

Ideally the class of all embedded [[subobject]]s of a given object, up to isomorphism, should also be [[small class|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 <math>f:A\rightarrow B</math> that is an injective function from the underlying set of <math>A</math> to the underlying set of <math>B</math> and is also an '''initial morphism''' in the following sense:
If <math>g</math> is a function from the underlying set of an object <math>C</math> to the underlying set of <math>A</math>, and if its composition with <math>f</math> is a morphism <math>fg:C\rightarrow B</math>, then <math>g</math> itself is a morphism.

A [[factorization system]] for a category also gives rise to a notion of embedding. If <math>(E,M)</math> is a factorization system, then the morphisms in <math>M</math> may be regarded as the embeddings, especially when the category is well powered with respect to <math>M</math>. Concrete theories often have a factorization system in which <math>M</math> 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 (category theory)|dual]] concept, known as quotient. All the preceding properties can be dualized.

An embedding can also refer to an [[Subcategory#Embeddings|embedding functor]].

==See also==
*[[Embedding (machine learning)]]
*[[Ambient space]]
*[[Closed immersion]]
*[[Cover (algebra)|Cover]]
*[[Dimensionality reduction]]
*[[Flat (geometry)]]
*[[Immersion (mathematics)|Immersion]]
*[[Johnson–Lindenstrauss lemma]]
*[[Submanifold]]
*[[Subspace (topology)|Subspace]]
*[[Universal spaces in the topology and topological dynamics|Universal space]]

==Notes==
{{reflist}}

== References ==
{{refbegin}}
* {{cite book|last1=Bishop|first1=Richard Lawrence|author-link1=Richard L. Bishop|last2=Crittenden|first2=Richard J.|title=Geometry of manifolds|publisher=Academic Press|location=New York|year=1964|isbn=978-0-8218-2923-3}}
* {{cite book|last1=Bishop|first1=Richard Lawrence|author1-link=Richard L. Bishop|last2=Goldberg|first2=Samuel Irving|title=Tensor Analysis on Manifolds|publisher=The Macmillan Company|year=1968|edition=First Dover 1980|isbn=0-486-64039-6|url=https://archive.org/details/tensoranalysison00bish}}
* {{cite book|last1=Crampin|first1=Michael|last2=Pirani|first2=Felix Arnold Edward|author-link2=Felix Pirani|title=Applicable differential geometry|publisher=Cambridge University Press|location=Cambridge, England|year=1994|isbn=978-0-521-23190-9|url-access=registration|url=https://archive.org/details/applicablediffer0000cram}}
*{{cite book|title = Riemannian Geometry|first=Manfredo Perdigao | last = do Carmo |author-link=Manfredo do Carmo | year = 1994|publisher=Birkhäuser Boston |isbn=978-0-8176-3490-2}}
* {{cite book|last=Flanders|first=Harley|author-link=Harley Flanders|title=Differential forms with applications to the physical sciences|publisher=Dover|year=1989|isbn=978-0-486-66169-8}}
* {{Cite book| last1=Gallot | first1=Sylvestre | author1-link=Sylvestre Gallot | last2=Hulin | first2=Dominique |author2-link=Dominique Hulin| last3=Lafontaine | first3=Jacques | title=Riemannian Geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | isbn=978-3-540-20493-0 | year=2004}}
* {{cite book|first1=John Gilbert|last1=Hocking|first2=Gail Sellers|last2=Young|title=Topology|year=1988|orig-year=1961|publisher=Dover|isbn=0-486-65676-4|url=https://archive.org/details/topology00hock_0}}
*{{cite book|last=Kosinski|first=Antoni Albert|year=2007|orig-year=1993|title=Differential manifolds|location=Mineola, New York|publisher=Dover Publications|isbn=978-0-486-46244-8}}
*{{Cite book| isbn = 978-0-387-98593-0 | title = Fundamentals of Differential Geometry | last1 = Lang | first1 = Serge |author-link1=Serge Lang| year = 1999 |publisher=Springer|location=New York| series = Graduate Texts in Mathematics}}
*{{cite book|last1=Kobayashi|first1=Shoshichi|author-link1=Shoshichi Kobayashi|last2=Nomizu|first2=Katsumi|author-link2=Katsumi Nomizu| title = Foundations of Differential Geometry, Volume 1| publisher=Wiley-Interscience |location=New York| year=1963}}
* {{cite book|first=John Marshall|last=Lee|authorlink = John M. Lee|title=Riemannian manifolds|publisher=Springer Verlag|year=1997|isbn=978-0-387-98322-6}}
* {{cite book| first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997| isbn = 0-387-94732-9}}.
* {{cite book|last=Spivak|first=Michael|author-link=Michael Spivak|title=A Comprehensive introduction to differential geometry (Volume 1)|year=1999|orig-year=1970|publisher=Publish or Perish|isbn=0-914098-70-5}}
* {{cite book| first=Frank Wilson| last = Warner |authorlink  = Frank Wilson Warner| title = Foundations of Differentiable Manifolds and Lie Groups | publisher = Springer-Verlag, New York | year = 1983| isbn = 0-387-90894-3}}.
{{refend}}

== External links ==
*{{cite book|last=Adámek|first=Jiří|author2=Horst Herrlich |author3=George Strecker |title=Abstract and Concrete Categories (The Joy of Cats)|url=http://katmat.math.uni-bremen.de/acc/|year=2006}}
* [http://www.map.mpim-bonn.mpg.de/Embedding Embedding of manifolds] on the Manifold Atlas
{{set index article}}

[[Category:Abstract algebra]]
[[Category:Category theory]]
[[Category:General topology]]
[[Category:Differential topology]]
[[Category:Functions and mappings]]
[[Category:Maps of manifolds]]
[[Category:Model theory]]
[[Category:Order theory]]