圏同値のソースを表示

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In [[category theory]], an abstract branch of [[mathematics]], an '''equivalence of categories''' is a relation between two [[Category (mathematics)|categories]] that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.
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[[数学]]、とりわけ[[圏論]]において、'''圏同値'''（けんどうち、{{lang-en-short|equivalence of categories}}）とはふたつの[[圏_(数学)|圏]]が「本質的には同じである」という関係のことをいう。
多くの分野で圏同値の例がある。
圏同値を示すことで、対象になっている数学的な構造の間に強い相関関係があることがわかる。
場合によっては、その構造は表面的には無関係に見えるので、圏同値は有用である；
つまりある定理を異なる数学的構造の定理に「翻訳」できることがある。

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If a category is equivalent to the [[dual (category theory)|opposite (or dual)]] of another category then one speaks of
a '''duality of categories''', and says that the two categories are '''dually equivalent'''.
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もしある圏が別の圏の[[双対圏]]と圏同値ならば、ふたつの圏は'''双対同値'''と言い、'''圏双対'''について論じることができる。

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An equivalence of categories consists of a [[functor]] between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation common for [[isomorphism]]s in an algebraic setting, the composition of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be ''[[natural transformation|naturally isomorphic]]'' to its image under this composition. Thus one may describe the functors as being "inverse up to isomorphism". There is indeed a concept of [[isomorphism of categories]] where a strict form of inverse functor is required, but this is of much less practical use than the ''equivalence'' concept.
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圏同値は圏の間の「可逆な」[[関手]]から成る。
しかしながら代数的な設定の下における[[同型]]とは異なり、関手とその「逆関手」の合成が恒等写像である必要はない。
その代わりに各[[対象 (圏論)|対象]]が合成の像と[[自然変換#定義|自然同型]]であればよい。
そのため、このことはふたつの関手が「同型を除いて逆関手」であると言われたりする。
実際に{{仮リンク|圏同型|en|Isomorphism of categories}}という概念もあり、こちらは本当に関手が逆関手であることを要求するが、圏同値の概念に比べると実用性を欠く。

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==Definition==
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== 定義 ==
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Formally, given two categories ''C'' and ''D'', an ''equivalence of categories'' consists of a functor ''F'' : ''C'' → ''D'', a functor ''G'' : ''D'' → ''C'', and two natural isomorphisms ε: ''FG''→'''I'''<sub>''D''</sub> and η : '''I'''<sub>''C''</sub>→''GF''. Here ''FG'': ''D''→''D'' and ''GF'': ''C''→''C'', denote the respective compositions of ''F'' and ''G'', and '''I'''<sub>''C''</sub>: ''C''→''C'' and '''I'''<sub>''D''</sub>: ''D''→''D'' denote the ''identity functors'' on ''C'' and ''D'', assigning each object and morphism to itself. If ''F'' and ''G'' are contravariant functors one speaks of a ''duality of categories'' instead.
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形式的には、ふたつの圏 {{mvar|C}} と {{mvar|D}} の'''圏同値'''はふたつの[[関手]] {{math|''F'' : ''C'' &rarr; ''D''}}, {{math|''G'' : ''D'' &rarr; ''C''}} とふたつの自然同型 {{math|''&epsilon;'' : ''FG'' &rarr; ''I''<sub>''D''</sub>}}, {{math|''&eta;'' : ''I''<sub>''C''</sub> &rarr; ''GF''}} から成る。
ここで {{math|''FG'' : ''D'' &rarr; ''D''}}, {{math|''GF'' : ''C'' &rarr; ''C''}} はそれぞれ {{mvar|F}} と {{mvar|G}} の合成を表し、{{math|''I''<sub>''C''</sub>}}, {{math|''I''<sub>''D''</sub>}} は圏 {{mvar|C}}, {{mvar|D}} 上の[[関手#性質|恒等関手]]を表す。
もし {{mvar|F}}, {{mvar|G}} が[[関手#反変関手|反変関手]]のときは、代わりに'''圏双対'''と言う。

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One often does not specify all the above data. For instance, we say that the categories ''C'' and ''D'' are ''equivalent'' (respectively ''dually equivalent'') if there exists an equivalence (respectively duality) between them. Furthermore, we say that ''F'' "is" an equivalence of categories if an inverse functor ''G'' and natural isomorphisms as above exist. Note however that knowledge of ''F'' is usually not enough to reconstruct ''G'' and the natural isomorphisms: there may be many choices (see example below).
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実際には上のすべての情報が指定されないこともしばしばである。
たとえば、ふたつの圏 {{mvar|C}}, {{mvar|D}} の間に圏同値（圏双対）があるときに、圏 {{mvar|C}}, {{mvar|D}} は'''圏同値'''
（'''圏双対'''）であると言ったりする。
さらに逆関手 {{mvar|G}} や自然同型 {{math|''&epsilon;'', ''&eta;''}} が存在するときに、関手 {{mvar|F}} が圏同値であると言ったりもする。
しかし関手 {{mvar|F}} に関する知識から普通は逆関手 {{mvar|G}} と自然同型 {{math|''&epsilon;'', ''&eta;''}} を復元することはできず、いくつもの可能性が残ることがある。

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==Equivalent characterizations==
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== 特徴づけ ==
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One can show that a functor ''F'' : ''C'' → ''D'' yields an equivalence of categories if and only if it is:
* [[full functor|full]], i.e. for any two objects ''c''<sub>1</sub> and ''c''<sub>2</sub> of ''C'', the map Hom<sub>''C''</sub>(''c''<sub>1</sub>,''c''<sub>2</sub>) → Hom<sub>''D''</sub>(''Fc''<sub>1</sub>,''Fc''<sub>2</sub>) induced by ''F'' is [[surjective]];
* [[faithful functor|faithful]], i.e. for any two objects ''c''<sub>1</sub> and ''c''<sub>2</sub> of ''C'', the map Hom<sub>''C''</sub>(''c''<sub>1</sub>,''c''<sub>2</sub>) → Hom<sub>''D''</sub>(''Fc''<sub>1</sub>,''Fc''<sub>2</sub>) induced by ''F'' is [[injective]]; and
* [[essentially surjective functor|essentially surjective (dense)]], i.e. each object ''d'' in ''D'' is isomorphic to an object of the form ''Fc'', for ''c'' in ''C''.
This is a quite useful and commonly applied criterion, because one does not have to explicitly construct the "inverse" ''G'' and the natural isomorphisms between ''FG'', ''GF'' and the identity functors. On the other hand, though the above properties guarantee the ''existence'' of a categorical equivalence (given a sufficiently strong version of the [[axiom of choice]] in the underlying set theory), the missing data is not completely specified, and often there are many choices. It is a good idea to specify the missing constructions explicitly whenever possible.
Due to this circumstance, a functor with these properties is sometimes called a '''weak equivalence of categories''' (unfortunately this conflicts with terminology from homotopy theory).
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関手 {{math|''F'' : ''C'' &rarr; ''D''}} が圏同値を定める必要十分条件は以下の3条件を満たすことである。
; [[関手#関手に対する様々な条件|充満関手]]   :任意の {{mvar|C}} のふたつの対象 {{math|''c''<sub>1</sub>, ''c''<sub>2</sub>}} について、関手 {{mvar|F}} の誘導する写像 {{math|Hom<sub>''C''</sub>(''c''<sub>1</sub>, ''c''<sub>2</sub>) &rarr; Hom<sub>''D''</sub>(''Fc''<sub>1</sub>, ''Fc''<sub>2</sub>)}} は[[全射]]
; [[関手#関手に対する様々な条件|忠実関手]]   :任意の {{mvar|C}} のふたつの対象 {{math|''c''<sub>1</sub>, ''c''<sub>2</sub>}} について、関手 {{mvar|F}} の誘導する写像 {{math|Hom<sub>''C''</sub>(''c''<sub>1</sub>, ''c''<sub>2</sub>) &rarr; Hom<sub>''D''</sub>(''Fc''<sub>1</sub>, ''Fc''<sub>2</sub>)}} は[[単射]]
; [[本質的全射]] : 任意の {{mvar|D}} の対象 {{mvar|d}} は {{mvar|C}} のある対象 {{mvar|c}} の像 {{mvar|Fc}} と[[同型]]

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There is also a close relation to the concept of [[adjoint functors]]. The following statements are equivalent for functors ''F'' : ''C'' → ''D'' and ''G'' : ''D'' → ''C'':
* There are natural isomorphisms from ''FG'' to '''I'''<sub>''D''</sub> and '''I'''<sub>''C''</sub> to ''GF''.
* ''F'' is a left adjoint of ''G'' and both functors are full and faithful.
* ''G'' is a right adjoint of ''F'' and both functors are full and faithful.
One may therefore view an adjointness relation between two functors as a "very weak form of equivalence". Assuming that the natural transformations for the adjunctions are given, all of these formulations allow for an explicit construction of the necessary data, and no choice principles are needed. The key property that one has to prove here is that the ''counit'' of an adjunction is an isomorphism if and only if the right adjoint is a full and faithful functor.
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[[随伴関手]]と密接に関連する概念もある。
関手 {{math|''F'' : ''C'' &rarr; ''D''}}, {{math|''G'' : ''D'' &rarr; ''C''}} について次の3つの条件は同値である。
* 自然同型 {{math|''FG'' &rarr; ''I''<sub>''D''</sub>}}, {{math|''I''<sub>''C''</sub> &rarr; ''GF''}} が存在する
* {{mvar|F}} は {{mvar|G}} の左随伴関手で、ふたつの関手は充満かつ忠実である
* {{mvar|G}} は {{mvar|F}} の右随伴関手で、ふたつの関手は充満かつ忠実である
したがってふたつの関手の間の随伴性は「非常に弱い形の同値関係」と見ることもできる。
随伴関手の間の[[自然変換]]が与えられているとすると、これらすべての定式化から必要な情報を明示的に構成することができて、どれを選ぶか決める必要がない。
ここで証明しなければならない要となる性質は随伴の counit が同型である必要十分条件が右随伴が充満かつ忠実となることである。

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==Examples==
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== 例 ==
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* Consider the category <math>C</math> having a single object <math>c</math> and a single morphism <math>1_{c}</math>, and the category <math>D</math> with two objects <math>d_{1}</math>, <math>d_{2}</math> and four morphisms: two identity morphisms <math>1_{d_{1}}</math>, <math>1_{d_{2}}</math> and two isomorphisms <math>\alpha \colon d_{1} \to d_{2}</math> and <math>\beta \colon d_{2} \to d_{1}</math>.  The categories <math>C</math> and <math>D</math> are equivalent; we can (for example) have <math>F</math> map <math>c</math> to <math>d_{1}</math> and <math>G</math> map both objects of <math>D</math> to <math>c</math> and all morphisms to <math>1_{c}</math>.
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* 1つの対象 {{mvar|c}} と1つの[[射 (圏論)|射]] {{math|1<sub>''c''</sub>}} を持つ圏 {{mvar|C}} と2つの対象 {{math|''d''<sub>1</sub>, ''d''<sub>2</sub>}} と4つの射 (2つの恒等射 {{math|1<sub>''d''<sub>1</sub></sub>, 1<sub>''d''<sub>2</sub></sub>}} と2つの同型射 {{math|''&alpha;'' : ''d''<sub>1</sub> &rarr; ''d''<sub>2</sub>}}、{{math|''&beta;'' : ''d''<sub>2</sub> &rarr; ''d''<sub>1</sub>}}) を持つ圏 {{mvar|D}} を考える。{{mvar|C}}, {{mvar|D}} は圏同値である。たとえば、{{mvar|c}} を {{math|''d''<sub>1</sub>}} に移す関手 {{mvar|F}} と {{mvar|D}} のすべての対象を {{mvar|c}} に移し、すべての射を {{math|1<sub>''c''</sub>}} に移す関手 {{mvar|G}} を取れば良い。

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* By contrast, the category <math>C</math> with a single object and a single morphism is ''not'' equivalent to the category <math>E</math> with two objects and only two identity morphisms as the two objects therein are ''not'' isomorphic.
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* 一方、1つの対象と1つの射を持つ圏 {{mvar|C}} と2つの対象と2つの恒等射のみを持つ圏 {{mvar|E}} は {{mvar|E}} の2つの対象が同型ではないので、圏同値ではない。

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* Consider a category <math>C</math> with one object <math>c</math>, and two morphisms <math>1_{c}, f \colon c \to c</math>.  Let <math>1_{c}</math> be the identity morphism on <math>c</math> and set <math>f \circ f = 1</math>.  Of course, <math>C</math> is equivalent to itself, which can be shown by taking <math>1_{c}</math> in place of the required natural isomorphisms between the functor <math>\mathbf{I}_{C}</math> and itself.  However, it is also true that <math>f</math> yields a natural isomorphism from <math>\mathbf{I}_{C}</math> to itself.  Hence, given the information that the identity functors form an equivalence of categories, in this example one still can choose between two natural isomorphisms for each direction.

* The category of sets and [[partial function]]s is equivalent to but not isomorphic with the category of [[pointed set]]s and point-preserving maps.<ref name="KoslowskiMelton2001">{{cite book|editor=Jürgen Koslowski and Austin Melton|title=Categorical Perspectives|year=2001|publisher=Springer Science & Business Media|isbn=978-0-8176-4186-3|page=10|author=Lutz Schröder|chapter=Categories: a free tour}}</ref>

* Consider the category <math>C</math> of finite-[[dimension of a vector space|dimensional]] [[real number|real]] [[vector space]]s, and the category <math>D = \mathrm{Mat}(\mathbb{R})</math> of all real [[matrix (mathematics)|matrices]] (the latter category is explained in the article on [[additive category|additive categories]]).  Then <math>C</math> and <math>D</math> are equivalent: The functor <math>G \colon D \to C</math> which maps the object <math>A_{n}</math> of <math>D</math> to the vector space <math>\mathbb{R}^{n}</math> and the matrices in <math>D</math> to the corresponding linear maps is full, faithful and essentially surjective.

* One of the central themes of [[algebraic geometry]] is the duality of the category of [[affine scheme]]s and the category of [[commutative ring]]s.  The functor <math>G</math> associates to every commutative ring its [[spectrum of a ring|spectrum]], the scheme defined by the [[prime ideal]]s of the ring.  Its adjoint <math>F</math> associates to every affine scheme its ring of global sections.

* In [[functional analysis]] the category of commutative [[C*-algebra]]s with identity is contravariantly equivalent to the category of [[compact space|compact]] [[Hausdorff space]]s.  Under this duality, every compact Hausdorff space <math>X</math> is associated with the algebra of continuous complex-valued functions on <math>X</math>, and every commutative C*-algebra is associated with the space of its [[maximal ideal]]s.  This is the [[Gelfand representation]].

* In [[lattice theory]], there are a number of dualities, based on representation theorems that connect certain classes of lattices to classes of [[topology|topological spaces]].  Probably the most well-known theorem of this kind is ''[[Stone's representation theorem for Boolean algebras]]'', which is a special instance within the general scheme of ''[[Stone duality]]''.  Each [[Boolean algebra (structure)|Boolean algebra]] <math>B</math> is mapped to a specific topology on the set of [[lattice theory|ultrafilters]] of <math>B</math>.  Conversely, for any topology the clopen (i.e. closed and open) subsets yield a Boolean algebra.  One obtains a duality between the category of Boolean algebras (with their homomorphisms) and [[Stone space]]s (with continuous mappings). Another case of Stone duality is [[Birkhoff's representation theorem]] stating a duality between finite partial orders and finite distributive lattices.

* In [[pointless topology]] the category of spatial locales is known to be equivalent to the dual of the category of sober spaces.

* Any category is equivalent to its [[skeleton (category theory)|skeleton]].

==Properties==
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== 性質 ==
大雑把に述べて圏同値は「圏論的な」すべての概念と性質を保つ。たとえば {{math|''F'' : ''C'' &rarr; ''D''}} が圏同値のとき次が成り立つ。

* 圏 {{mvar|C}} の対象 {{mvar|c}} が[[始対象]]（あるいは[[終対象]]、[[零対象]]）である必要十分条件は圏 {{mvar|D}} の対象 {{math|''Fc''}} がそうであることである。
* 圏 {{mvar|C}} の射 {{math|&alpha;}} が[[圏 (数学)#諸定義|単射]]（あるいは[[圏 (数学)#諸定義|全射]]、[[圏 (数学)#諸定義|同型射]]）である必要十分条件は圏 {{mvar|D}} の射 {{math|''F''&alpha;}} がそうであることである。
* 関手 {{math|''H'' : ''I'' &rarr; ''C''}} が[[極限 (圏論)#極限|極限]]（あるいは[[極限 (圏論)#余極限|余極限]]) {{mvar|l}} を持つ必要十分条件は関手 {{math|''FH'' : ''I'' &rarr; ''D''}} が極限（あるいは余極限） {{math|''Fl''}} を持つことである。これは[[等化子]]、[[直積 (圏論)|直積]]、や[[直和 (圏論)|直和]]などにも適用できる。[[核 (代数学)|核]]や[[余核]]に適用すれば圏同値 {{mvar|F}} は[[完全関手]]であることがわかる。
* 圏 {{mvar|C}} が[[デカルト閉圏|デカルト閉]]（あるいは[[トポス (数学)|トポス]]）である必要十分条件は圏 {{mvar|D}} がそうであることである。

双対性はすべての概念を「逆転」させる。始対象は終対象に、単射は全射に、核は余核に、直積は直和になど。

{{math|''F'' : ''C'' &rarr; ''D''}} を圏同値とし、{{math|''G''<sub>1</sub>}} と {{math|''G''<sub>2</sub>}} を関手 {{mvar|F}} の逆とすれば、{{math|''G''<sub>1</sub>}} と {{math|''G''<sub>2</sub>}} は自然同型である。

{{math|''F'' : ''C'' &rarr; ''D''}} を圏同値とし、圏 {{mvar|C}} が[[前加法圏]]（あるいは{{仮リンク|加法圏|en|additive category}}、[[アーベル圏]]）ならば関手 {{mvar|F}} が加法的になるようにして圏 {{mvar|D}} もそうなる。一方、加法的圏の間の圏同値は加法的でなければならない。（最後の主張は前加法圏の間では正しいとは限らない。）

圏 {{mvar|C}} の'''自己同値'''とは圏同値 {{math|''F'' : ''C'' &rarr; ''C''}} のことである。圏 {{mvar|C}} の自己同値は自然同型な自己同値を同一視することによって合成に関して[[群 (数学)|群]]をなす。この群は本質的に圏 {{mvar|C}} の「対称性」を捉えている。（注意：もし {{mvar|C}} が[[小圏]]でなければ、圏 {{mvar|C}} の自己同値は[[集合]]ではなく[[クラス (集合論)|クラス]]をなすかもしれない。）

<!--As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If ''F'' : ''C'' → ''D'' is an equivalence, then the following statements are all true:
* the object ''c'' of ''C'' is an [[initial object]] (or [[terminal object]], or [[zero object]]), [[if and only if]] ''Fc'' is an [[initial object]] (or [[terminal object]], or [[zero object]]) of ''D''
* the morphism α in ''C'' is a [[monomorphism]] (or [[epimorphism]], or [[isomorphism]]), if and only if ''Fα'' is a monomorphism (or epimorphism, or isomorphism) in ''D''.
* the functor ''H'' : ''I'' → ''C'' has [[limit (category theory)|limit]] (or colimit) ''l'' if and only if the functor ''FH'' : ''I'' → ''D'' has limit (or colimit) ''Fl''. This can be applied to [[equaliser (mathematics)|equalizers]], [[product (category theory)|product]]s and [[coproduct]]s among others. Applying it to [[kernel (category theory)|kernel]]s and [[cokernel]]s, we see that the equivalence ''F'' is an [[Regular category#Exact sequences and regular functors|exact functor]].
* ''C'' is a [[cartesian closed category]] (or a [[topos]]) if and only if ''D'' is cartesian closed (or a topos).

Dualities "turn all concepts around": they turn initial objects into terminal objects, monomorphisms into epimorphisms, kernels into cokernels, limits into colimits etc.

If ''F'' : ''C'' → ''D'' is an equivalence of categories, and ''G''<sub>1</sub> and ''G''<sub>2</sub> are two inverses of ''F'', then ''G''<sub>1</sub> and ''G''<sub>2</sub> are naturally isomorphic.

If ''F'' : ''C'' → ''D'' is an equivalence of categories, and if ''C'' is a [[preadditive category]] (or [[additive category]], or [[abelian category]]), then ''D'' may be turned into a preadditive category (or additive category, or abelian category) in such a way that ''F'' becomes an [[additive functor]]. On the other hand, any equivalence between additive categories is necessarily additive. (Note that the latter statement is not true for equivalences between preadditive categories.)

An '''auto-equivalence''' of a category ''C'' is an equivalence ''F'' : ''C'' → ''C''. The auto-equivalences of ''C'' form a [[group (mathematics)|group]] under composition if we consider two auto-equivalences that are naturally isomorphic to be identical. This group captures the essential "symmetries" of ''C''. (One caveat: if ''C'' is not a small category, then the auto-equivalences of ''C'' may form a proper [[class (set theory)|class]] rather than a [[Set (mathematics)|set]].)

== See also ==
* [[Equivalent definitions of mathematical structures]]

==References==
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== 参考文献 ==
{{Reflist}}

*{{SpringerEOM|title=Equivalence of categories|urlname=Equivalence_of_categories}}
*{{cite book|last=Mac Lane|first=Saunders|authorlink=ソーンダース・マックレーン|title=Categories for the working mathematician|year=1998|publisher=Springer|location=New York|isbn=0-387-98403-8|pages=xii+314|url={{google books|6KPSBwAAQBAJ|Categories for the working mathematician|plainurl=yes}}}}

{{圏論}}

{{デフォルトソート:けんとうち}}
[[Category:圏論]]
[[Category:同値 (数学)]]
[[Category:数学に関する記事]]