Topological Quillen localization of structured ring spectra
J.E. Harper, Y. Zhang
The aim of this short paper is twofold: (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 GoerssHopkins 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 semimodel structure in the sense of GoerssHopkins 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. 6991
