The two-square lemma
An essential step in the Snake Lemma in Homological Algebra is the construction of a connecting homomorphism. Generalizing this aspect of the Snake Lemma (in an Abelian category), the authors here give an interesting new result which they call the Two Square Lemma: Consider the commutative diagram \[ \begin{matrix} A_ {pa} && @>\psi>> && B && @>\varphi>> && C \\ &&& \overset\tau\swarrow &&&& \overset\sigma\nearrow \\ \text{\llap{\(\alpha\)}}\downarrow && Q && \text{\llap{\(\beta\)}}\downarrow && P_{pb} && \downarrow\text{\rlap{\(\gamma\)}} \\ & \overset{\tau'}\nearrow &&&& \overset{\sigma'}\swarrow \\ A' && @>>\psi'> && B' && @>>\varphi'> && C' \end{matrix} \] with exact rows and Q the pushout of \(\psi\) and \(\alpha\) and P the pullback of \(\gamma\) and \(\phi '\). Then (i) there exists a unique \(\theta\) : \(Q\to B'\) such that \(\theta \tau =\beta\), \(\theta \tau '=\psi '\); (ii) there exists a unique \(\rho\) : \(B\to P\) such that \(\sigma \rho =\phi\), \(\sigma '\rho =\beta\); (iii) there exists a unique \(\eta\) : \(Q\to P\) such that \(\eta \tau =\rho\), \(\sigma '\eta =\theta\), \(\sigma \eta \tau '=0.\) Moreover, \(\eta\) is monic if \(\psi '\) is monic, and \(\eta\) is epic if \(\phi\) is epic. As an easy application of this lemma they then obtain the Snake Lemma.
scientific article; zbMATH DE number 4146225