Cap product with cohomology written first

Let $X$ be a space and $R$ a commutative unital ring. For $p,n\ge 0$, $\phi \in C^p(X;R)$, and a singular $n$-simplex $\sigma$, define $\phi \cap \sigma =\{\begin{array}{ll}\phi (\sigma [0,\dots ,p])\sigma [p,\dots ,n],& n\ge p,\\ 0,& n<p\mathrm{.}\end{array}$ Extend $R$-bilinearly in $\phi$ and the finite chains of Singular simplices and singular chain groups with coefficients. An $R$-linear cochain is a function on simplex generators; each value multiplies one specified back-face generator. Thus the formula is well-defined on finite formal chains and is $R$-balanced in its two inputs. It gives $C^p(X;R){\otimes }_R{C}_n(X;R)⟶C_{n-p}(X;R),$ with negative chain groups zero. This is the cap product with cohomology first: evaluate on the front face and retain the back face.

The face convention is the same as Singular cup product on cochains. In terms of its Alexander–Whitney diagonal $D_X$, cap is the bidegree-$(p,n-p)$ part of $D_X$, followed by evaluation of the first factor by $\phi$. There is no additional sign in this evaluation. Subsequent boundary and projection identities use this order.

For $p=0$, this multiplies a simplex by the value of $\phi$ at its first vertex; in particular the constant value-one cochain acts as the identity on chains. For $p=n$, it returns $\phi (\sigma )$ times the last vertex, a degree-zero chain, whose boundary is zero. For $p>n$ it is zero by definition. On a degenerate simplex the same face formula applies; unnormalized chains retain these generators. Empty $X$, zero inputs and the zero ring give zero maps. The construction also applies to the higher singular simplices of a point and requires no AC.