L-homologies of double complexes

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