defn - nilpotent maps of commutative ring spectra

Let $f:R\to A$ be a map of $E_{\infty }$ ring spectra. The kernel of the underlying $R$-module map $Fib(\underline{f})$ lifts to a non-unital $E_{\infty }$ algebra, we denote this object by $I_f\in CAl{g}_{nu}(Mo{d}_R)$.

Let $\rho$ be a functorial tower of algebras, as in defn - functorial algebra tower.

Definition

$f$ is said to be $\rho$-nilpotent if $I_f$ is $\rho$-complete.

In the following definitions we’ll fix a $\rho$ omit it from the notation.

Definition The category of formal $R$-algebras, $\widehat{CAl{g}_R}\subset CAl{g}_R$ is full subcategory spanned by maps $u:R\to A$ such that $u$ is $\rho$-nilponent.

Definition The category of formal extensions of $A$, $\widehat{Ext{n}_A}\subset CAl{g}_{/A}$ is full subcategory spanned by maps $p:B\to A$ such that $p$ is $\rho$-nilponent.