The snail lemma and the long homology sequence

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.

arXiv comment31 pages