Equivalence of categories (original) (raw)
Die Äquivalenz von Kategorien ist eine Beziehung, die im mathematischen Teilgebiet der Kategorientheorie zwischen zwei Kategorien bestehen kann. Zwei äquivalente Kategorien haben dieselben kategoriellen Eigenschaften. Viele wichtige mathematische Theorien behaupten die Äquivalenz zweier Kategorien.
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| dbo:abstract | Die Äquivalenz von Kategorien ist eine Beziehung, die im mathematischen Teilgebiet der Kategorientheorie zwischen zwei Kategorien bestehen kann. Zwei äquivalente Kategorien haben dieselben kategoriellen Eigenschaften. Viele wichtige mathematische Theorien behaupten die Äquivalenz zweier Kategorien. (de) In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two 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. If a category is equivalent to the opposite (or dual) of another category then one speaks ofa duality of categories, and says that the two categories are dually equivalent. 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 isomorphisms in an algebraic setting, the composite of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be 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. (en) En mathématiques, plus précisément en théorie des catégories, une équivalence de catégories est une relation qui établit que deux catégories sont "essentiellement les mêmes". C'est un foncteur entre les deux catégories, qui prend compte formellement du fait que ces catégories relèvent d'une même structure : on dit alors que les catégories sont équivalentes. À la différence de la notion d'isomorphisme de catégories, la notion d'équivalence est moins rigide, plus pratique et plus courante. La notion d'équivalence de catégories rend compte, de manière unifiée, de nombreuses dualités observées dans plusieurs pans de l'algèbre et de l'analyse. (fr) 범주론에서, 두 범주 사이의 동치(同値, 영어: equivalence (of categories))는 두 범주가 사실상 같은 구조를 지니게 하는 함자이다. 범주의 동형보다 더 약한 개념이며, 범주의 동형보다 더 널리 쓰인다. (ko) 数学、とりわけ圏論において、圏同値（けんどうち、英: equivalence of categories）とはふたつの圏が「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。 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。 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が合成の像と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。 (ja) In de categorietheorie, een abstract deelgebied van de wiskunde, is een equivalentie van categorieën een relatie tussen twee categorieën, die vaststelt dat deze categorieën "in essentie gelijk" zijn. Er zijn talrijke voorbeelden van categoriale equivalenties uit vele gebieden van de wiskunde. Het vaststellen van een equivalentie impliceert sterke gelijkenissen tussen de betrokken wiskundige structuren. In sommige gevallen kunnen deze structuren op oppervlakkig of intuïtief niveau ongerelateerd lijken; hierdoor wint de notie van equivalentie in de categorietheorie aan kracht: de notie creëert de mogelijkheid om stellingen tussen verschillende soorten van wiskundige structuren te "vertalen", wetende dat de essentiële betekenis van deze stellingen onder deze vertaling bewaard blijft. (nl) Na teoria das categorias, equivalência de categorias é o conceito "correto" para dizer se categorias são "essencialmente as mesmas". Assim como objetos numa categoria são comparados não por serem iguais ou não, mas por haver ou não isomorfismos entre eles, e são equivalentes quando são relacionadas por dois functores que são inversos a menos de isomorfismos naturais. (pt) Эквивале́нтность катего́рий в теории категорий — отношение между категориями, показывающее, что две категории «по существу одинаковы». Установление эквивалентности свидетельствует о глубокой связи соответствующих математических концепций и позволяет «переносить» теоремы с одних структур на другие. (ru) 在数学的一个抽象分支范畴论中，范畴的等价（equivalence of categories）是两个范畴间的一个关系，在这种关系之下的范畴是“本质上一样的”。从数学的许多地方都有范畴等价的例子。建立一个等价涉及展示所考虑的数学结构间很强的相似性。在许多情形，这些结构表面或直觉上看并无关联，这样就使这种概念特别有用：它提供了在不同数学结构之间翻译的可能性，本质一语是指在翻译中保持的定理。 如果一个范畴等价于另一个范畴的，则我们说“范畴的对偶性”，以及这两个范畴对偶等价。 范畴的等价由所涉范畴的一个函子组成，这个函子要求有一个“逆”函子。但与通常代数语境的同构不同，这个函子与它的逆不必是恒等映射，二只要每个对象与在此复合函子下的像。从而我们可以说这个函子是差一个同构下的逆。这实际上是的概念，其中要求逆函子的严格性质，但这比“等价”概念用得要少。 (zh) |
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| rdfs:comment | Die Äquivalenz von Kategorien ist eine Beziehung, die im mathematischen Teilgebiet der Kategorientheorie zwischen zwei Kategorien bestehen kann. Zwei äquivalente Kategorien haben dieselben kategoriellen Eigenschaften. Viele wichtige mathematische Theorien behaupten die Äquivalenz zweier Kategorien. (de) 범주론에서, 두 범주 사이의 동치(同値, 영어: equivalence (of categories))는 두 범주가 사실상 같은 구조를 지니게 하는 함자이다. 범주의 동형보다 더 약한 개념이며, 범주의 동형보다 더 널리 쓰인다. (ko) 数学、とりわけ圏論において、圏同値（けんどうち、英: equivalence of categories）とはふたつの圏が「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。 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。 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が合成の像と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。 (ja) In de categorietheorie, een abstract deelgebied van de wiskunde, is een equivalentie van categorieën een relatie tussen twee categorieën, die vaststelt dat deze categorieën "in essentie gelijk" zijn. Er zijn talrijke voorbeelden van categoriale equivalenties uit vele gebieden van de wiskunde. Het vaststellen van een equivalentie impliceert sterke gelijkenissen tussen de betrokken wiskundige structuren. In sommige gevallen kunnen deze structuren op oppervlakkig of intuïtief niveau ongerelateerd lijken; hierdoor wint de notie van equivalentie in de categorietheorie aan kracht: de notie creëert de mogelijkheid om stellingen tussen verschillende soorten van wiskundige structuren te "vertalen", wetende dat de essentiële betekenis van deze stellingen onder deze vertaling bewaard blijft. (nl) Na teoria das categorias, equivalência de categorias é o conceito "correto" para dizer se categorias são "essencialmente as mesmas". Assim como objetos numa categoria são comparados não por serem iguais ou não, mas por haver ou não isomorfismos entre eles, e são equivalentes quando são relacionadas por dois functores que são inversos a menos de isomorfismos naturais. (pt) Эквивале́нтность катего́рий в теории категорий — отношение между категориями, показывающее, что две категории «по существу одинаковы». Установление эквивалентности свидетельствует о глубокой связи соответствующих математических концепций и позволяет «переносить» теоремы с одних структур на другие. (ru) 在数学的一个抽象分支范畴论中，范畴的等价（equivalence of categories）是两个范畴间的一个关系，在这种关系之下的范畴是“本质上一样的”。从数学的许多地方都有范畴等价的例子。建立一个等价涉及展示所考虑的数学结构间很强的相似性。在许多情形，这些结构表面或直觉上看并无关联，这样就使这种概念特别有用：它提供了在不同数学结构之间翻译的可能性，本质一语是指在翻译中保持的定理。 如果一个范畴等价于另一个范畴的，则我们说“范畴的对偶性”，以及这两个范畴对偶等价。 范畴的等价由所涉范畴的一个函子组成，这个函子要求有一个“逆”函子。但与通常代数语境的同构不同，这个函子与它的逆不必是恒等映射，二只要每个对象与在此复合函子下的像。从而我们可以说这个函子是差一个同构下的逆。这实际上是的概念，其中要求逆函子的严格性质，但这比“等价”概念用得要少。 (zh) In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two 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. (en) En mathématiques, plus précisément en théorie des catégories, une équivalence de catégories est une relation qui établit que deux catégories sont "essentiellement les mêmes". C'est un foncteur entre les deux catégories, qui prend compte formellement du fait que ces catégories relèvent d'une même structure : on dit alors que les catégories sont équivalentes. À la différence de la notion d'isomorphisme de catégories, la notion d'équivalence est moins rigide, plus pratique et plus courante. (fr) |
| rdfs:label | Äquivalenz (Kategorientheorie) (de) Equivalence of categories (en) Équivalence de catégories (fr) 圏同値 (ja) 범주의 동치 (ko) Equivalentie (categorietheorie) (nl) Equivalência de categorias (pt) Эквивалентность категорий (ru) 范畴的等价 (zh) |
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