Higher cohomology operations and R-completion

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.