The aim of this short paper is two-fold: (i) to construct a TQlocalization functor on algebras over a spectral operad O, in the case where no connectivity assumptions are made on the O-algebras, and (ii) more generally, to establish the associated TQ-local homotopy theory as a left Bousfield localization of the usual model structure on O-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 O-algebras. More generally, we establish these results for TQhomology with coefficients in a spectral algebra 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 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.
TQ-homology of an O-algebra with coefficients in AIf X is an O-algebra, then we may factor the map * → X * →X ≃ − − → X as a cofibration followed by an acyclic fibration; we are using the positive flat stable model structure (see, for instance, [24]). In particular,X is a cofibrant replacement of X.