Singular cochain complex with coefficients

Let $X$ be any topological space and $G$ an abelian group. Its singular cochains with coefficients in $G$ are $C^n(X;G)={\mathrm{Hom}}_{\mathbb{Z}}(C_n(X;\mathbb{Z}),G)(n\ge 0),C^n(X;G)=0(n<0),$ where the chains are those of The singular chain complex and singular homology. Addition and negatives of homomorphisms are pointwise. The coboundary is ${\delta }^n:C^n(X;G)\to C^{n+1}(X;G),{\delta }^n\phi =\phi \circ {\mathrm{\partial }}_{n+1}$ for $n\ge 0$, and is zero for $n<0$. Composition with the homomorphism ${\mathrm{\partial }}_{n+1}$ is additive, so this is a homomorphism. The sign convention is positive: there is no extra degree sign on this composite. On a singular $(n+1)$-simplex it says ${\delta }^n\phi (\sigma )={\sum }_{i=0}^{n+1}(-1{)}^i\phi (\sigma \circ {\delta }_i),$ where ${\delta }_i$ denotes the affine face map, distinct from the superscripted coboundary. In particular ${\delta }^0\phi$ on an oriented path is its endpoint value minus its initial value.

By Singular simplices and singular chain groups with coefficients, the integer chains are finite sums on the specified simplex set $S_n(X)$. Thus a cochain is equivalently an arbitrary function $S_n(X)\to G$: the extension sends ${\sum }_{\sigma }{a}_{\sigma }\sigma$ to ${\sum }_{\sigma }{a}_{\sigma }\phi (\sigma )$. The sum is finite and the formal coefficients are unique. The cochain need not have finite support.

For a commutative unital ring $R$ and an $R$-module $G$, there is a canonical identification $C^n(X;G)\cong {\mathrm{Hom}}_R(C_n(X;R),G)\mathrm{.}$ Indeed the given chain convention is $C_n(X;R)=C_n(X;\mathbb{Z}){\otimes }_{\mathbb{Z}}R$. A homomorphism $\phi$ on integer chains defines an $R$-linear map by $c\otimes r\mapsto r\phi (c)$; the defining additive and integer-balanced tensor relations are preserved. Conversely evaluate an $R$-linear map on $c\otimes 1$. These operations are inverse since $c\otimes r=r(c\otimes 1)$, and they intertwine the two coboundaries because the coefficient boundary is $\mathrm{\partial }\otimes 1$. The construction also covers the zero ring with its unital zero module.

The resulting graded group and coboundary are called the singular cochain complex. The square-zero identity follows by precomposition with ${\mathrm{\partial }}^2=0$ in the chain complex and is spelled out in the next lemma. If $X$ is empty there are no singular simplices, so every cochain group is zero; if $G=0$ the same is true. The extension and tensor formulas are uniquely specified and require no AC.