In the first part of the paper, we establish a version of the snail lemma (which is a generalization of the classical snake lemma) for categories with a structure of nullhomotopies. This lemma allows us to construct a six-term exact sequence in a (sufficiently nice) category with a structure of nullhomotopies associated to a morphism in the category. In the second part, we introduce the category with nullhomotopies Seq(A) of sequentiable families of arrows in a category A and we apply the homotopy snail lemma to a morphism in Seq(A) obtaining first a six-term exact sequence in Seq(A) and then, unrolling the sequence in Seq(A), a long exact sequence in A. We then compare the category Seq(A) of sequentiable families with the category Ch(A) of chain complexes in A and prove that, when A is abelian, the long exact sequence built using the snail lemma subsumes the usual long exact sequence of homology obtained from an extension of chain complexes. This result suggests that the category Seq(A), which has a nicer structure of nullhomotopies than that of Ch(A), could provide a useful alternative to chain complexes for the study of homological algebra.
Keywords: homology, exact sequence, homotopy kernel, snail lemma, sequentiable family
2020 MSC: 18G50, 18G35, 18N40
Theory and Applications of Categories, Vol. 45, 2026, No. 13, pp 417-460.
Published 2026-02-27.
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