On diagram-chasing in double complexes (Q2884474)

This article explains how to prove the classical diagram lemmas of homological algebra and obtain long exact homology sequences out of double complexes through a new diagram chasing technique involving the so-called \textit{Salamander Lemma}. The idea is that certain ``short-distance'' paths in a double complex give rise to a ``short-distance'' exact sequence, which sequences may then be pasted together to yield ``long-distance'' exact sequences such as the one arising in the Snake Lemma.NEWLINENEWLINEGiven an object \(A\) in a double complex in an abelian category, the author does not only consider the horizontal homology object \(A{\bullet}\,\), but also objects \(A_{\square}\) and \({}^{\square}\!A\) which he calls the \textit{donor} and the \textit{receptor} at \(A\). Now given arrows \(A\to B\) (horizontally) and \(C\to A\), \(B\to D\) (vertically) in a double complex, the Salamander Lemma produces an exact sequence NEWLINE\[NEWLINEC_{\square} \;\to\; A{\bullet} \;\to\; A_{\square} \;\to\; {}^{\square}\!B \;\to\; B{\bullet} \;\to\; {}^{\square}\!D.NEWLINE\]NEWLINE Depending on properties of the given double complex, some of the arrows in the exact sequence may turn out to be isomorphisms, which can then be used in diagram chasing arguments. In the article this technique is developed in full and applications are given.NEWLINENEWLINEIt seems interesting to explore how much of this theory is still available outside the abelian context. A recent branch of categorical algebra studies how the validity of diagram lemmas determines properties of the surrounding category, as for instance in [\textit{G.~Janelidze} et al., J. Pure Appl. Algebra 168, No. 2--3, 367--386 (2002; Zbl 0993.18008)] and \textit{Z.~Janelidze} [Theory Appl. Categ. 23, 221--242 (2010; Zbl 1234.18009)]. Whence the question: \textit{What is a category in which the Salamander Lemma holds?}