  Let $R=\mathbb{F}_p$ or a field of characteristic $0$. For each $R$-good
topological space $Y$, we define a collection of higher cohomology operations
which, together with the cohomology algebra $H^*(Y;R)$ suffice to determine $Y$
up to $R$-completion. We also provide a similar collection of higher cohomology
operations which determine when two maps $f_0,f_1: Z\to Y$ between $R$-good
spaces(inducing the same algebraic homomorphism $H^*(Y;R)\to H^*(Z;R)$) are
$R$-equivalent.
