Topological Quillen localization of structured ring spectra

Abstract

The aim of this short paper is two-fold: (i) to construct a $\mathsf{TQ}$-localization functor on algebras over a spectral operad $\mathcal{O}$, in the case where no connectivity assumptions are made on the $\mathcal{O}$-algebras, and (ii) more generally, to establish the associated $\mathsf{TQ}$-local homotopy theory as a left Bousfield localization of the usual model structure on $\mathcal{O}$-algebras, which itself is almost never left proper, in general. In the resulting $\mathsf{TQ}$-local homotopy theory, the ''weak equivalences'' are the $\mathsf{TQ}$-homology equivalences, where ''$\mathsf{TQ}$-homology'' is short for topological Quillen homology, which is also weakly equivalent to stabilization of $\mathcal{O}$-algebras. More generally, we establish these results for $\mathsf{TQ}$-homology with coefficients in a spectral algebra $\mathcal{A} $. A key observation, that goes back to the work of Goerss-Hopkins on moduli problems, is that the usual left properness assumption may be replaced with a strong c ofibration condition in the desired subcell lifting arguments: Our main result is that the $\mathsf{TQ}$-local homotopy theory can be established (e.g., a semi-model structure in the sense of Goerss-Hopkins and Spitzweck, that is both cofibrantly generated and simplicial) by localizing with respect to a set of strong cofibrations that are $\mathsf{TQ}$-equivalences.