Canonical extension by zero of a singular cochain on a simplex basis

For each $U$-simplex $\sigma$, set $(E_A^U\eta )([\sigma ])=\eta ([\sigma :A])$ if $\sigma ({\mathrm{\Delta }}^k)\subseteq A$, and set it to zero otherwise; $\sigma :A$ is the unique continuous corestriction. Extend by the finite sum formula of [F1]. Membership in $A$ and the unique corestriction specify a function, so no selection of a vector-space basis or representatives occurs. This formula is linear in $\eta$.

On an $A$-simplex the first clause always applies, so restriction of $E_A^U\eta$ is $\eta$ on each generator and hence on every chain. If $A=\mathrm{\varnothing }$ this is the zero map; if $A=U$ it is the identity. At degree zero it extends point functions by zero, including a one-point subspace. In negative degrees define the unique zero section.

To check that this is not generally a cochain map, take $U=[0,1]$, $A=\{0\}$, and the zero-degree cochain $\eta (0)=1$. On the identity path $\sigma (t)=t$, $\delta E_A^U\eta ([\sigma ])=0-1=-1$. But $\delta \eta =0$, since every path in $A$ is constant and has equal endpoints, so $E_A^U\delta \eta ([\sigma ])=0$. Thus the two composites differ. Constant and repeated simplices are fully allowed in the formulas. All degreewise section assertions, including the zero cases, follow from the prescribed basis values and require no AC.