In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), let $({\mathcal {A}},\partial _{\bullet }),({\mathcal {B}},\partial _{\bullet }')$ and $({\mathcal {C}},\partial _{\bullet }'')$ be chain complexes that fit into the following short exact sequence:

The maps $\alpha _{*}^{}$ and $\beta _{*}^{}$ are the usual maps induced by homology. The boundary maps $\delta _{n}^{}$ are explained below. The name of the lemma arises from the "zig-zag" behavior of the maps in the sequence. A variant version of the zig-zag lemma is commonly known as the "snake lemma" (it extracts the essence of the proof of the zig-zag lemma given below).

Construction of the boundary maps

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The maps $\delta _{n}^{}$ are defined using a standard diagram chasing argument. Let $c\in C_{n}$ represent a class in $H_{n}({\mathcal {C}})$, so $\partial _{n}''(c)=0$. Exactness of the row implies that $\beta _{n}^{}$ is surjective, so there must be some $b\in B_{n}$ with $\beta _{n}^{}(b)=c$. By commutativity of the diagram,

Thus, since $\alpha _{n-1}^{}$ is injective, there is a unique element $a\in A_{n-1}$ such that $\alpha _{n-1}(a)=\partial _{n}'(b)$. This is a cycle, since $\alpha _{n-2}^{}$ is injective and

since $\partial ^{2}=0$. That is, $\partial _{n-1}(a)\in \ker \alpha _{n-2}=\{0\}$. This means $a$ is a cycle, so it represents a class in $H_{n-1}({\mathcal {A}})$. We can now define

$\delta _{}^{}[c]=[a].$

With the boundary maps defined, one can show that they are well-defined (that is, independent of the choices of c and b). The proof uses diagram chasing arguments similar to that above. Such arguments are also used to show that the sequence in homology is exact at each group.