The snail lemma and the long homology sequence

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 $\mathbf{S}\mathbf{e}\mathbf{q}(\mathcal{A})$ of sequentiable families of arrows in a category $\mathcal{A}$ and we compare it with the category of chain complexes in $\mathcal{A}\mathrm{.}$ We apply the homotopy snail lemma to a morphism in $\mathbf{S}\mathbf{e}\mathbf{q}(\mathcal{A})$ obtaining first a six-term exact sequence in $\mathbf{S}\mathbf{e}\mathbf{q}(\mathcal{A})$ and then, unrolling the sequence in $\mathbf{S}\mathbf{e}\mathbf{q}(\mathcal{A}),$ a long exact sequence in $\mathcal{A}\mathrm{.}$ When $\mathcal{A}$ is abelian, this sequence subsumes the usual long homology sequence obtained from an extension of chain complexes.