The snail lemma and the long homology sequence
Julia Ramos González, Enrico Vitale
TL;DR
This work develops a homotopical generalization of the snail lemma and introduces the category $\mathbf{Seq}(\mathcal{A})$ of sequentiable families to encode homology arrows. By applying the homotopy snail lemma to morphisms in $\mathbf{Seq}(\mathcal{A})$, it obtains a six-term exact sequence in $\mathbf{Seq}(\mathcal{A})$ and, after unrolling, a long exact sequence in $\mathcal{A}$; in abelian $\mathcal{A}$ this recovers the classical long homology sequence arising from a chain-complex extension. The paper then situates sequentiable families as a natural bridge from chain complexes to an arrow-centric homological framework, showing that, via a functor $\mathcal{F}: \mathbf{Ch}(\mathcal{A}) \to \mathbf{Seq}(\mathcal{A})$, one can translate standard homological constructions into the snail sequence setting. This yields a cohesive, higher-dimensional unifying perspective on exact sequences in homotopy theory and homological algebra, with explicit comparisons to the classical snake lemma when $\mathcal{A}$ is abelian. The framework potentially clarifies how homology information can be organized and transported through sequentiable structures beyond strict chain complexes.
Abstract
In the first part of the paper, we establish an homotopical version of the snail lemma (which is a generalization of the classical snake lemma). In the second part, we introduce the category $\mathbf{Seq}(\mathcal A)$ of sequentiable families of arrows in a category $\mathcal A$ and we compare it with the category of chain complexes in $\mathcal A.$ We apply the homotopy snail lemma to a morphism in $\mathbf{Seq}(\mathcal A)$ obtaining first a six-term exact sequence in $\mathbf{Seq}(\mathcal A)$ and then, unrolling the sequence in $\mathbf{Seq}(\mathcal A),$ a long exact sequence in $\mathcal A.$ When $\mathcal A$ is abelian, this sequence subsumes the usual long homology sequence obtained from an extension of chain complexes.