A nilpotent Whitehead theorem for $\mathsf{TQ}$-homology of structured ring spectra

Ching, Michael; Harper, John E.

doi:10.32513/tbilisi/1538532027

Cited by 4 publications

(4 citation statements)

References 31 publications

(43 reference statements)

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“…As an application of the TQ-local homotopy theory established here, together with the completion results in [9], it is shown in [45] that every homotopy pronilpotent O-algebra is TQ-local; this improves the result in [10] that 0-connected O-algebras are TQ-complete (assuming O, R are (−1)-connected), to the much larger class of homotopy pro-nilpotent O-algebras, provided that one replaces "TQcompletion" with "TQ-localization", and is closely related to (and partially motivated by) a conjecture of Francis-Gaitsgory [14, 3.4.5]. The TQ-local homotopy theory developed here may also find potential applications for studying the closely related invariants in [17,25].…”

Section: Introductionmentioning

confidence: 83%

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Topological Quillen localization of structured ring spectra

Harper¹,

Zhang²

2019

Tbilisi Math. J.

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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.

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Section: Introductionmentioning

confidence: 83%

“…Topological Quillen homology (or TQ-homology) is the precise analog for Oalgebras of singular homology for spaces, and is also weakly equivalent to stabilization of O-algebras [2,24,36]. A useful starting point is [18,35,37], together with [1,2,3] and [10,30,31,32]; see also [8,9,15,16,23,38].…”

Section: Introductionmentioning

confidence: 99%

Topological Quillen localization of structured ring spectra

Harper¹,

Zhang²

2019

Tbilisi Math. J.

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“…Recall the TQ| Nil M -completion construction from [12]. For each n ≥ 1, τ n O is the operad associated to O where…”

Section: Proposition 28 Let Y Be a (Not Necessarily Fibrant) Object I...mentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Homotopy pro-nilpotent structured ring spectra and topological Quillen localization

Zhang

2019

Preprint

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The aim of this short paper is to show that the homotopy limit of any diagram of nilpotent structured ring spectra is TQ-local, where structured ring spectra are described as algebras over a spectral operad O; in particular, every homotopy pro-nilpotent structured ring spectrum is TQ-local. Here, TQ is short for topological Quillen homology, which is weakly equivalent to O-algebra stabilization. As an application, we simultaneously extend the previously known connected and nilpotent TQ-Whitehead theorems to a homotopy pro-nilpotent TQ-Whitehead theorem. We also compare TQ-localization with TQ-completion and show that TQ-local O-algebras that are TQ-good are TQcomplete. Finally, we show that every (−1)-connected O-algebra with a principally refined Postnikov tower is TQ-local, provided that O is (−1)-connected.

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Homotopy pro-nilpotent structured ring spectra and topological Quillen localization

Zhang

2022

J. Homotopy Relat. Struct.

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A nilpotent Whitehead theorem for $\mathsf{TQ}$-homology of structured ring spectra

Cited by 4 publications

References 31 publications

Topological Quillen localization of structured ring spectra

Topological Quillen localization of structured ring spectra

Homotopy pro-nilpotent structured ring spectra and topological Quillen localization

Homotopy pro-nilpotent structured ring spectra and topological Quillen localization

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