We describe two ways to define higher order Toda brackets in a pointed simplicial model category \({\mathcal {D}}\): one is a recursive definition using model categorical constructions, and the second uses the associated simplicial enrichment. We show that these two definitions agree, by providing a third, diagrammatic, description of the Toda bracket, and explain how it serves as the obstruction to rectifying a certain homotopy-commutative diagram in \({\mathcal {D}}\).

中文翻译：

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高阶 Toda 括号

我们描述了两种在尖头单纯模型范畴\({\mathcal {D}}\) 中定义高阶 Toda 括号的方法：一种是使用模型分类构造的递归定义，第二种是使用相关的单纯充实。我们通过提供 Toda 括号的第三个图解描述来证明这两个定义是一致的，并解释它如何阻碍纠正\({\mathcal {D}}\) 中的某个同伦交换图。