Topological Quillen localization of structured ring spectra

J.E. Harper, Y. Zhang

The aim of this short paper is two-fold: (i) to construct a $\TQ$-localization functor on algebras over a spectral operad $\capO$, in the case where no connectivity assumptions are made on the $\capO$-algebras, and (ii) more generally, to establish the associated $\TQ$-local homotopy theory as a left Bousfield localization of the usual model structure on $\capO$-algebras, which itself is almost never left proper, in general. In the resulting $\TQ$-local homotopy theory, the ``weak equivalences'' are the $\TQ$-homology equivalences, where ``$\TQ$-homology'' is short for topological Quillen homology, which is also weakly equivalent to stabilization of $\capO$-algebras. More generally, we establish these results for $\TQ$-homology with coefficients in a spectral algebra $\capA$. 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 cofibration condition in the desired subcell lifting arguments: Our main result is that the $\TQ$-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 $\TQ$-equivalences. 

Tbilisi Mathematical Journal, Vol. 12(3) (2019), pp. 69-91