Higher cup-i products

Work over ${\mathbb{F}}_2$ and use the natural higher diagonals $D_i$ of Natural higher diagonal approximations. For $a\in C^p(X;{\mathbb{F}}_2)$, $b\in C^q(X;{\mathbb{F}}_2)$, and $i\ge 0$, their cup-$i$ product is the cochain of degree $p+q-i$ defined by

$$(a{⌣}_{i}b)(c):=(a\otimes b)(D_{i}c),c\in C_{p+q-i}(X;{\mathbb{F}}_2)\mathrm{.}$$

Set $a{⌣}_{i}b=0$ when $i<0$ or $p+q-i<0$. Since $D_0$ is the Alexander--Whitney diagonal, $a{⌣}_{0}b$ is exactly the singular cup product of Singular cup product on cochains, with the same front-face/back-face order.

For a subspace $A\subseteq X$, the carrier property of $D_i$ implies that the formula restricts to relative cochains. More precisely, either of

$$C^p(X,A)\otimes C^q(X)⟶C^{p+q-i}(X,A),C^p(X)\otimes C^q(X,A)⟶C^{p+q-i}(X,A)$$

is well defined: on a chain in $A$, both tensor factors of $D_i$ lie in $A$, so the relative factor evaluates to zero. The construction uses only the fixed $D_i$ and evaluation, and hence makes no choice. It includes the empty space, zero cochains, one-point spaces, and degenerate singular simplices.