Topological Quillen localization of structured ring spectra

Abstract

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.

Homotopy pro-nilpotent structured ring spectra and topological Quillen localization

“…Unless otherwise specified, we work with the positive flat stable model structure on Alg O [28,47]. To keep this paper appropriately concise, we freely use notation from [13,28,29]. Finally, "connected" is short for "0-connected".…”

“…In this paper we consider localization of O-algebras with respect to TQ-homology. Recall the following from [29]. Definition 2.2.…”

“…It is useful to note that the collection of TQ-local O-algebras is exactly the collection of weak TQ-fibrant objects in the TQ-local homotopy theory [29]. Intuitively, a TQ-local object in Alg O is an O-algebra X which only sees TQ-homology information.…”

“…The TQ-local homotopy theory for O-algebras is established in [29], where the upshot is that: if X is a cofibrant O-algebra, then its weak TQ-fibrant replacement X → L TQ (X) is the TQ-localization of X. By construction, the comparison map X → L TQ (X) is a cofibration that is also a TQ-equivalence such that L TQ (X) is TQ-local (Defintion 2.2).…”

“…Our main result, Theorem 1.8, is that (i) is true in general, provided that in the comparison map we replace "TQ-completion" with "TQ-localization". Our strategy of attack is to leverage the TQ-local homotopy theory of O-algebras in [29] with the fact, proved in [12], that M -nilpotent Oalgebras are TQ| Nil M -complete. Definition 1.1.…”