L-homologies of double complexes

TL;DR

This paper introduces L-homologies for double complexes, extending classical homologies and providing a comprehensive classification, new exact sequences, and a fibration structure linking L-homologies to the double complex category.

Contribution

It defines L-homologies for double complexes, explores their properties, classifies associated exact sequences, and establishes a fibration relating L-homologies to the double complex category.

Findings

Enumerates all L-homologies for double complexes

Provides new examples of exact sequences involving L-homologies

Establishes a faithful amnestic Grothendieck fibration

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 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 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.