The snail lemma in a pointed regular category (Q308152)

In this article, the authors chase subobjects and elaborate on various diagram lemmas, such as the (incomplete) snail lemma and spider lemma.NEWLINENEWLINEAs a result of their investigations, they show that in a pointed regular category where every regular epimorphism is an epimorphism, the validity of the snake lemma can be deduced from that of the snail lemma, and that in fact both lemmas are equivalent to the so-called \(3\times3\) lemma.NEWLINENEWLINEFurthermore, from these results, they prove that in a pointed regular category having cokernels of kernels, the snail lemma is equivalent to subtractivity of the category in the sense of \textit{Z. Janelidze} [Appl. Categ. Struct. 13, No. 4, 343--350 (2005; Zbl 1095.18002)].