$${ \mathsf {TQ} }$$-completion and the Taylor tower of the identity functor

Abstract

The goal of this short paper is to study the convergence of the Taylor tower of the identity functor in the context of operadic algebras in spectra. Specifically, we show that if $A$ is a $(-1)$-connected $\mathcal{O}$-algebra with $0$-connected $\mathsf{TQ}$-homology spectrum $\mathsf{TQ}(A)$, then there is a natural weak equivalence $P_\infty$(id)$A\simeq A_\mathsf{TQ}^\wedge$ between the limit of the Taylor tower of the identity functor evaluated on $A$ and the $\mathsf{TQ}$-completion of $A$. Since, in this context, the identity functor is only known to be $0$-analytic, this result extends knowledge of the Taylor tower of the identity beyond its "radius of convergence."