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A degreewise short exact sequence of cochain complexes induces a long exact sequence from a coefficient sequence in the coefficient case, and a long exact cohomology sequence generally. Its degree-one connecting homomorphism lifts a quotient cocycle, takes its coboundary, and identifies that result in the subcomplex.

A singular $n$-cochain with coefficients in an abelian group $A$ is a homomorphism from the singular chain group $C_n(X;\mathbb{Z})$ to $A$. It is specified by its values on all singular simplices, without a finite-support restriction. Its coboundary evaluates on the alternating sum of the faces. The resulting cochain complex computes singular cohomology.

For a topological space $X$ and abelian group $A$, singular cohomology is the cohomology of the cochain complex $C^n(X;A)=\mathrm{Hom}(C_n(X;\mathbb{Z}),A)$, where $C_n$ is the singular chain group. Its differential is dual to the singular boundary. This theory has relative cohomology, long exact sequences and the Excision theorem. On spaces that are not locally contractible it can differ from Čech cohomology.