埋め込み (数学)のソースを表示

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'''埋め込み'''（うめこみ、embedding, imbedding<ref>It is suggested by {{harvnb|Spivak|1999|page=49}}, that the word "embedding" is used instead of "imbedding" by "the English", i.e. the British.</ref>）とは、[[数学的構造]]間の構造を保つような[[単射]]のことである。
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In [[mathematics]], an '''embedding''' (or '''imbedding'''<ref>It is suggested by {{harvnb|Spivak|1999|page=49}}, that the word "embedding" is used instead of "imbedding" by "the English", i.e. the British.</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 ''X'' is said to be embedded in another object ''Y'', the embedding is given by some [[injective]] and structure-preserving map {{nowrap|''f'' : ''X'' → ''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 {{nowrap|''f'' : ''X'' → ''Y''}} is an embedding is often indicated by the use of a "hooked arrow", thus: <math> f : X \hookrightarrow Y.</math>  On the other hand, this notation is sometimes reserved for [[inclusion map]]s.

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 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 (mathematics)|domain]] ''X'' with its [[image (mathematics)|image]] ''f''(''X'') contained in ''Y'', so that then {{nowrap|''X'' ⊆ ''Y''}}.
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==トポロジーと幾何学==
===位相空間論===
[[位相空間論]]において、埋め込みとは、像の上への[[同相写像]]のことである<ref>{{harvnb|Hocking|Young|1988|page=73}}. {{harvnb|Sharpe|1997|page=16}}.</ref>。つまり、[[位相空間]] ''X'' と ''Y'' の間の[[単射]][[連続写像]] ''f'': ''X'' &rarr; ''Y'' であって、（''f''(''X'') には ''Y'' の[[相対位相]]を入れて）''f'' が ''X'' と ''f''(''X'') の間の同相写像であるようなもののことである。
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 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 [[topological subspace|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 [[topological subspace|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 which 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>.
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与えられた空間 ''X'' に対し、埋め込み ''X'' &rarr; ''Y'' の存在は ''X'' の[[位相的性質]]である。これによって2つの位相空間を、一方がある空間に埋め込めて他方はできないならば、区別することができる。

===微分トポロジー===
[[微分トポロジー]]において：
''M'' と ''N'' を滑らかな[[多様体]]とし、''f'': ''M'' &rarr; ''N'' を滑らかな写像とする。このとき ''f'' が[[はめ込み]]とは、[[微分 (多様体)|微分]]がいたるところ単射であることをいう。'''埋め込み''' (embedding)、あるいは'''滑らかな埋め込み''' (smooth embedding) は、上に述べた位相的な意味で埋め込みであるような（すなわち像の上への[[同相写像]]であるような）単射はめ込みと定義される<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>。

言い換えると、埋め込みは像への[[微分同相]]であり、とくに埋め込みの像は[[部分多様体]]でなければならない。はめ込みは局所的な埋め込みである（すなわち任意の点 ''x'' &isin; ''M'' に対し、近傍 ''x'' &isin; ''U'' ⊂ ''M'' が存在して、''f'': ''U'' &rarr; ''N'' は埋め込みである）。
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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, 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]] of dimension <math>m</math> requires <math>n=2m</math> for an embedding. An immersion of this surface is, however, possible in <math>\mathbb{R}^3</math>, and one example is [[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 [[List_of_mathematical_abbreviations|w.r.t.]] [[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 ''f''(''X'') is not tangent to the boundary of ''Y''.
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===リーマン幾何学===
[[リーマン幾何学]]において：
(''M'', ''g'') と (''N'', ''h'') を[[リーマン多様体]]とする。'''等長埋め込み''' (isometric embedding) とは、滑らかな埋め込み ''f'': ''M'' → ''N'' であって[[リーマン計量|計量]]を保つもの、つまり ''g'' は ''h'' の ''f'' による{{仮リンク|引き戻し (微分幾何学)|label=引き戻し|en|pullback (differential geometry)}}に等しい、すなわち ''g'' = ''f''*''h'' であるようなもののことである。明示的には、任意の2つの接ベクトル

:<math>v,w\in T_x(M)</math>

に対し、

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

が成り立つ。
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Analogously, '''isometric immersion''' is an immersion between Riemannian manifolds which preserves the Riemannian metrics.

Equivalently, an isometric embedding (immersion) is a smooth embedding (immersion) which 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>
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==代数学==
一般に、代数的圏 ''C'' に対して、2つの ''C''-代数構造 ''X'' と ''Y'' の間の埋め込みとは、単射 ''C''-射 ''e'': ''X'' → ''Y'' である。

===体論===
（可換）[[体論]]において、[[可換体|体]] ''E'' の体 ''F'' への'''埋め込み''' (embedding) とは、[[環準同型]] σ: ''E'' → ''F'' のことである。

σ の[[核 (代数学)|核]]は ''E'' の[[イデアル]]であり、これは条件 σ(1) = 1 により、体 ''E'' 全体ではありえない。さらに、体のイデアルは零イデアルと体自身全体しかないことはよく知られた体の性質である。したがって核は 0 であるから、体の任意の埋め込みは[[単射]]である。したがって、''E'' は ''F'' の部分体 σ(''E'') に[[同型]]である。これによって体の任意の準同型に対して''埋め込み''という呼称が正当化される。

===普遍代数学とモデル理論===
{{further2|{{仮リンク|部分構造|en|Substructure (mathematics)}}および{{仮リンク|Elementary equivalence|en|Elementary equivalence}}}}
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If σ is a [[signature (logic)|signature]] and <math>A,B</math> are σ-[[structure (mathematical logic)|structures]] (also called σ-algebras in [[universal algebra]] or models in [[model theory]]), then a map <math>h:A \to B</math> is a σ-embedding [[iff]] all the following holds:
* <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]].
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==順序理論と領域理論==
{{仮リンク|順序理論|en|Order theory}}において、[[半順序]]の埋め込みは ''X'' から ''Y'' への写像 ''F'' であって

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

を満たすもののことである。

[[領域理論]]においては、さらに次のことが要求される：

:<math> \forall y\in Y:\{x: F(x)\leq y\}</math> は[[有向集合|有向]]である。

==距離空間==
[[距離空間]]の間の写像 <math>\phi: X \to Y</math> が（distortion <math>C>0</math> の）''埋め込み''とは、
:<math> L d_X(x, y) \leq d_Y(\phi(x), \phi(y)) \leq CLd_X(x,y) </math>
がある定数 <math>L>0</math> に対して成り立つことをいう。

=== ノルム空間 ===
重要な特別な場合は[[ノルム空間]]の場合である。この場合線型埋め込みを考えるのが自然である。

有限次元[[ノルム空間]] <math>(X, \| \cdot \|)</math> について問うことのできる基本的な問題の1つは、''[[ヒルベルト空間]] <math>\ell_2^k</math> を定数 distortion で ''X'' に線型に埋め込めるような最大の次元 ''k'' は何か？''である。

答えは{{仮リンク|ドヴォレツキーの定理|en|Dvoretzky's theorem}}によって与えられる。

==圏論==
[[圏論]]において、すべての圏において適用可能な埋め込みの満足のいきかつ一般的に受け入れられている定義は存在しない。すべての同型射と埋め込みのすべての合成は埋め込みであることと、すべての埋め込みはモノ射であることは期待されるだろう。他の典型的な要求は： 任意の{{仮リンク|extremal monomorphism|en|monomorphism#Related concepts}}は埋め込みであり、埋め込みは[[引き戻し (圏論)|引き戻し]]のもとで安定である。
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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 to define new local structures on the category (such as a [[closure operator]]).

In a [[concrete category]], an '''embedding''' is a morphism ''ƒ'':&nbsp;''A''&nbsp;→&nbsp;''B'' which 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 ''ƒ'' is a morphism ''ƒg'':&nbsp;''C''&nbsp;→&nbsp;''B'', then ''g'' itself is a morphism.

A [[factorization system]] for a category also gives rise to a notion of embedding. If (''E'',&nbsp;''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&nbsp;''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 (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]].
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== 関連項目 ==
*{{仮リンク|閉沈め込み|en|Closed immersion}}
*{{仮リンク|被覆 (代数学)|label=被覆|en|Cover (algebra)}}
*{{仮リンク|Dimension reduction|en|Dimension reduction}}
*[[沈め込み]]
*{{仮リンク|ジョンソン・リンデンシュトラウスの補題|en|Johnson–Lindenstrauss lemma}}
*[[部分多様体]]
*[[相対位相|部分空間]]

== 脚注 ==
{{reflist}}

== 参考文献 ==
* {{cite book|ref=harv|last1=Bishop|first1=Richard Lawrence|authorlink1=リチャード・ローレンス・ビショップ|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|ref=harv | last1=Bishop|first1=R.L.|author1-link=リチャード・ローレンス・ビショップ|last2=Goldberg|first2=S.I. | title = Tensor Analysis on Manifolds| publisher=The Macmillan Company | year=1968|edition=First Dover 1980|isbn=0-486-64039-6}}
* {{cite book|ref=harv|last1=Crampin|first1=Michael|last2=Pirani|first2=Felix Arnold Edward|title=Applicable differential geometry|publisher=Cambridge University Press|location=Cambridge, England|year=1994|isbn=978-0-521-23190-9}}
*{{cite book|ref=harv|title = Riemannian Geometry|first=Manfredo Perdigao | last = do Carmo |authorlink=Manfredo do Carmo | year = 1994|isbn=978-0-8176-3490-2}}
* {{cite book|ref=harv|last=Flanders|first=Harley|title=Differential forms with applications to the physical sciences|publisher=Dover|year=1989|isbn=978-0-486-66169-8}}
* {{Cite book|ref=harv | last1=Gallot | first1=Sylvestre | last2=Hulin | first2=Dominique | 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|ref=harv|first1=John Gilbert|last1=Hocking|first2=Gail Sellers|last2=Young|title=Topology|year=1988|origyear=1961|publisher=Dover|isbn=0-486-65676-4}}
*{{cite book|ref=harv|last=Kosinski|first=Antoni Albert|year=2007|origyear=1993|title=Differential manifolds|location=Mineola, New York|publisher=Dover Publications|isbn=978-0-486-46244-8}}
*{{Cite book|ref=harv | isbn = 978-0-387-98593-0 | title = Fundamentals of Differential Geometry | last1 = Lang | first1 = Serge |authorlink1=Serge Lang| year = 1999 |publisher=Springer|location=New York| series = Graduate Texts in Mathematics}}
*{{cite book|ref=harv|last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = Foundations of Differential Geometry, Volume 1| publisher=Wiley-Interscience |location=New York| year=1963}}
* {{cite book|ref=harv|first=John|last=Lee|title=Riemannian manifolds|publisher=Springer Verlag|year=1997|isbn=978-0-387-98322-6}}
* {{cite book|ref=harv| 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|ref=harv|last=Spivak|first=Michael|authorlink=Michael Spivak|title=A Comprehensive introduction to differential geometry (Volume 1)|year=1999|origyear=1970|publisher=Publish or Perish|isbn=0-914098-70-5}}
* {{cite book|ref=harv| first = F.W. | last = Warner | title = Foundations of Differentiable Manifolds and Lie Groups | publisher = Springer-Verlag, New York | year = 1983| isbn = 0-387-90894-3}}.

== 外部リンク ==
*{{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

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{{DEFAULTSORT:うめこみ}}
[[Category:抽象代数学]]
[[Category:圏論]]
[[Category:位相空間論]]
[[Category:微分位相幾何学]]
[[Category:関数]]
[[Category:多様体の写像]]
[[Category:モデル理論]]
[[Category:順序理論]]
[[Category:数学に関する記事]]