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Unter einer äquivarianten Abbildung versteht man in der Mathematik eine Abbildung, die mit der Wirkung einer Gruppe kommutiert. (de) In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation. Equivariant maps generalize the concept of invariants, functions whose value is unchanged by a symmetry transformation of their argument. The value of an equivariant map is often (imprecisely) called an invariant. In statistical inference, equivariance under statistical transformations of data is an important property of various estimation methods; see invariant estimator for details. In pure mathematics, equivariance is a central object of study in equivariant topology and its subtopics equivariant cohomology and equivariant stable homotopy theory. (en) En mathématiques, l'équivariance est une forme de symétrie de fonctions d'un espace par symétrie avec un autre (tels que les espaces symétriques). Une fonction est appelée application équivariante quand son domaine et son codomaine agissent sur le même groupe de symétrie, et quand la fonction commute avec l'action de groupe. Ainsi, appliquer une transformation de symétrie puis calculer la fonction produit le même résultat que le calcul de la fonction suivi de la transformation. Les applications équivariantes généralisent le concept d'invariants, des fonctions dont la valeur est inchangée par une transformation symétrique de son argument. La valeur d'une application équivariante est souvent (par abus) appelé un invariant. En inférence statistique, l'équivariance sous transformation statistique de données est une propriété importante de plusieurs méthodes d'estimation ; voir pour plus de détails. En mathématiques pures, l'équivariance est un objet central d'étude en et ses sous-sujets ( et ). (fr) Em matemática, uma aplicação invariante é uma função entre dois conjuntos que comuta com a ação de um grupo. Especificamente, seja G um grupo e sejam X e Y os G-conjuntos associados. Uma função f : X → Y é dita equivariante se f(g·x) = g·f(x) para todo g ∈ G e todo x em X. Note que se uma ou ambas as ações forem ações à direita a condição que define a equivariância deve ser modificada adequadamente: f(x·g) = f(x)·g ; (direita-direita)f(x·g) = g−1·f(x) ; (direita-esquerda)f(g·x) = f(x)·g−1 ; (esquerda-esquerda) Aplicações equivariantes são homomorfismos na categoria dos G-conjuntos (para um G fixado). Assim, elas também são conhecidas como G-aplicações ou G-homomorphismos. Os isomorfismos de G-conjuntos são simplesmente as aplicações equivariantes bijetoras. A condição de equivariância também pode ser entendida como por meio do seguinte diagrama comutativo. Observe que denota a aplicação que recebe um elemento e retorna . (pt) 在数学中，一个等变映射（equivariant map）是两个集合之间与群作用交换的一个函数。具体地，设 G 是一个群，X 与 Y 是两个关联的 G-集合。一个函数 f : X → Y 称为等变，如果 f(g·x) = g·f(x) 对所有 g ∈ G 与 x ∈ X 成立。注意如果其中一个或两个作用是右作用，则等变条件必须适当地修改： f(x·g) = f(x)·g ; （右-右）f(x·g) = g−1·f(x) ; （右-左）f(g·x) = f(x)·g−1 ; （左-右） 等变映射是 G-集合范畴（对一个取定的 G）中的同态。从而它们也称为 G-映射或 G-同态。G-集合的同构就是等变双射。 等变条件也能理解为下面的交换图表。注意 表示映射取元素 得到 。 (zh) |
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Unter einer äquivarianten Abbildung versteht man in der Mathematik eine Abbildung, die mit der Wirkung einer Gruppe kommutiert. (de) 在数学中，一个等变映射（equivariant map）是两个集合之间与群作用交换的一个函数。具体地，设 G 是一个群，X 与 Y 是两个关联的 G-集合。一个函数 f : X → Y 称为等变，如果 f(g·x) = g·f(x) 对所有 g ∈ G 与 x ∈ X 成立。注意如果其中一个或两个作用是右作用，则等变条件必须适当地修改： f(x·g) = f(x)·g ; （右-右）f(x·g) = g−1·f(x) ; （右-左）f(g·x) = f(x)·g−1 ; （左-右） 等变映射是 G-集合范畴（对一个取定的 G）中的同态。从而它们也称为 G-映射或 G-同态。G-集合的同构就是等变双射。 等变条件也能理解为下面的交换图表。注意 表示映射取元素 得到 。 (zh) In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation. (en) En mathématiques, l'équivariance est une forme de symétrie de fonctions d'un espace par symétrie avec un autre (tels que les espaces symétriques). Une fonction est appelée application équivariante quand son domaine et son codomaine agissent sur le même groupe de symétrie, et quand la fonction commute avec l'action de groupe. Ainsi, appliquer une transformation de symétrie puis calculer la fonction produit le même résultat que le calcul de la fonction suivi de la transformation. (fr) Em matemática, uma aplicação invariante é uma função entre dois conjuntos que comuta com a ação de um grupo. Especificamente, seja G um grupo e sejam X e Y os G-conjuntos associados. Uma função f : X → Y é dita equivariante se f(g·x) = g·f(x) para todo g ∈ G e todo x em X. Note que se uma ou ambas as ações forem ações à direita a condição que define a equivariância deve ser modificada adequadamente: f(x·g) = f(x)·g ; (direita-direita)f(x·g) = g−1·f(x) ; (direita-esquerda)f(g·x) = f(x)·g−1 ; (esquerda-esquerda) (pt) |
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Äquivariante Abbildung (de) Equivariant map (en) Équivariance (fr) Aplicação equivariante (pt) 等变映射 (zh) |
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