Let \(X\) be a topological space, \(G\) an abelian group, and \(n\geq0\). Let \(C_n(X)\) be the free abelian group on the singular \(n\)-simplices \(\sigma\colon\Delta^n\to X\), as in singular homology. A singular \(n\)-cochain of \(X\) with coefficients in \(G\) is a homomorphism \(\varphi\colon C_n(X)\to G\), and \[ C^n(X;G)\coloneqq\Hom(C_n(X),G) \] is the group of singular \(n\)-cochains [@Hat02].
Since \(C_n(X)\) is free on the singular \(n\)-simplices, a singular \(n\)-cochain is the same as an arbitrary function assigning to each singular \(n\)-simplex \(\sigma\) an element \(\varphi(\sigma)\in G\), extended linearly: \(\varphi\qty{\sum_in_i\sigma_i}=\sum_in_i\varphi(\sigma_i)\).