Abstract

The notion of L-homologies (of double complexes) as proposed in this paper extends the notion of classical horizontal and vertical homologies along with two other new homologies introduced in the homological diagram lemma called the salamander lemma. We enumerate all L-homologies associated with an object of a double complex and provide new examples of exact sequences. We describe a classification problem of these exact sequences. We study two poset structures on these L-homologies; one of them determines the trivialities of horizontal and vertical homologies of an object in terms of other L-homologies of that object, whereas the second structure shows the significance of the two homologies introduced in the salamander lemma. Finally, we prove the existence of a faithful amnestic Grothendieck fibration from the category of L-homologies to a category consisting of objects and morphisms of a given double complex.

摘要

本文提出的（双重复合体）L-同调的概念扩展了经典的水平和垂直同调的概念，以及在同源图引理中引入的另两个新同调，称为the。我们列举了与双复数对象相关联的所有L-同源性，并提供了精确序列的新示例。我们描述了这些精确序列的分类问题。我们研究了关于这些L-同源性的两个球状结构；其中之一根据其他L来确定对象的水平和垂直同质性的琐碎性-该对象的同源性，而第二结构显示了the引理中引入的两种同源性的重要性。最后，我们证明了从L-同源性类别到由给定双复合物的对象和态射态组成的类别存在忠实的记忆格罗腾迪克纤维化的存在。