five lemma

The five lemma is a workhorse: it lets you conclude that a map in the middle of a diagram is an isomorphism just by checking its neighbors. The intuition is that an exact sequence locks its terms together so tightly that if four out of five vertical comparison maps behave well, the fifth has no freedom left and must be an isomorphism too. It is the standard way to prove two constructions agree.

Statement: consider a commutative diagram with exact rows and five vertical maps m_1, m_2, m_3, m_4, m_5 between corresponding terms. If m_1, m_2, m_4, m_5 are all isomorphisms, then the middle map m_3 is an isomorphism. The proof is a diagram chase. In fact the sharper version, sometimes called the four lemma, gives more: if m_2 and m_4 are isomorphisms, m_1 is surjective and m_5 is injective, then m_3 is an isomorphism — and one can isolate injectivity and surjectivity of m_3 separately.

The five lemma is the reason naturality matters: when a construction produces a map of long exact sequences and you know it is an isomorphism at all but one term, the remaining term comes for free. It is used constantly to verify that two homology or cohomology theories coincide, that a quasi-isomorphism induces isomorphisms, and that comparison maps in spectral sequences are isomorphisms.