diagram chasing in homological algebra
salamander lemma $\Rightarrow$
four lemma $\Rightarrow$ five lemma
snake lemma $\Rightarrow$ connecting homomorphism
and
nonabelian homological algebra
The $3 \times 3$-lemma or nine lemma is one of the basic diagram chasing lemmas in homological algebra.
Let
be a short exact sequence of chain complexes. Then if two of the three complexes $A_\bullet, B_\bullet, C_\bullet$ are exact, so is the remaining third.
Let
be a commuting diagram in some abelian category such that each of the three columns is an exact sequence. Then
If the two bottom rows are exact, then so is the top.
If the top two rows are exact, then so is the bottom.
If the top and bottom rows are exact and $A \to C$ is the zero morphism, then also the middle row is exact.
A proof by way of the salamander lemma is spelled out in detail at Salamander lemma - Implications - 3x3 lemma.
An early appearance of the $3 \times 3$-lemma is as lemma (5.5) in
In
it appears as exercise 1.3.2.
The sharp $3 \times 3$-lemma appears as lemma 2 in
Also lemma 3.2-3.4 of
Discussion of generalization to non-abelian categories is in
Marino Gran, Diana Rodelo, Goursat categories and the $3 \times 3$-lemma, Applied Categorical Structures, Vol. 20, No 3, 2012, 229-238. (journal, pdf slides)
Marino Gran, Zurab Janelidze and Diana Rodelo, $3 \times 3$ lemma for star-exact sequences, Homology, Homotopy and Applications, Vol. 14 (2012), No. 2, pp.1-22. (journal)
Dominique Bourn, $3 \times 3$-lemma and protomodularity, Journal of Algebra, Volume 236, Number 2, 15 February 2001 , pp. 778-795(18)