Operads of genus zero curves and the Grothendieck–Teichmüller group

Brito, Pedro Boavida de; Horel, Geoffroy; Robertson, Marcy

doi:10.2140/gt.2019.23.299

Cited by 13 publications

(18 citation statements)

References 25 publications

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“…The proof is entirely analogous once we have an action of GT on the framed little disks operad. This is constructed in [BdBHR17,Theorem 8.4] and the effect of this action on homology is explained in [BdBHR17, Theorem 9.1].…”

Section: Main Results In the Homologically Graded Casementioning

confidence: 99%

Étale cohomology, purity and formality with torsion coefficients

Cirici¹,

Horel²

2018

Preprint

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We use Galois group actions on étale cohomology to prove results of formality for dg-operads and dg-algebras with torsion coefficients. Our theory applies, among other related constructions, to the dg-operad of singular chains on the operad of little disks and to the dg-algebra of singular cochains on the configuration space of points in the complex space. The formality that we obtain is only up to a certain degree, which depends on the cardinality of the field of coefficients.

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Section: Main Results In the Homologically Graded Casementioning

confidence: 99%

Étale cohomology, purity and formality with torsion coefficients

Cirici¹,

Horel²

2018

Preprint

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“…We claim that the resulting operad is quasi‐isomorphic to

C_*(\mathcal {D},\mathbb {F}_\ell )

. The exact same statement for the framed little disks operad is proved in [7, Theorem 9.1]. The group

\widehat{GT}

comes with a surjective map

\begin{equation*} \chi _\ell :\widehat{GT}\rightarrow \mathbb {Z}_\ell ^\times \end{equation*}

that factors the cyclotomic character of the absolute Galois group of

\mathbb {Q}

(see [38, section 3.1]).…”

Section: Main Results In the Homologically Graded Casementioning

confidence: 56%

Étale cohomology, purity and formality with torsion coefficients

Cirici

Horel

2022

Journal of Topology

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We use Galois group actions on étale cohomology to prove results of formality for dg‐operads and dg‐algebras with torsion coefficients. Our theory applies, among other related objects, to the dg‐operad of singular chains of the operad of little disks and to the dg‐algebra of singular cochains of the configuration space of points in the complex space. The formality that we obtain is only up to a certain degree, which depends on the cardinality of the field of coefficients.

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“…This follows from Theorem E.3.1.6 of [Lur], which states that the functor between the underlying ∞-categories of s Set Q and sSet KQ induced by U (which is called "Mat" by Lurie) is conservative. Another way to deduce this proposition is to show that the weak equivalences between fibrant objects in s Set Q are the π * -isomorphisms (as in the proof of Proposition 3.9 of [BHR19]) and that the the underlying group/set Uπ n (X, x) of the profinite group/set π n (X, x) agrees with π n (UX, x) for any fibrant X ∈ s Set Q and any x ∈ X 0 .…”

Section: Example: Complete Segal Profinite Spaces Vs Profinite Quasic...mentioning

confidence: 99%

Simplicial model structures on pro-categories

Blom¹,

Moerdijk²

2020

Preprint

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We describe a method for constructing simplicial model structures on ind-and pro-categories. Our method is particularly useful for constructing "profinite" analogues of known model categories. Our construction quickly recovers Morel's model structure for pro-p spaces and Quick's model structure for profinite spaces, but we will show that it can also be applied to construct many interesting new model structures. In addition, we study some general properties of our method, such as its functorial behaviour and its relation to Bousfield localization. We compare our construction to the ∞-categorical approach to ind-and pro-categories in an appendix.

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Operads of genus zero curves and the Grothendieck–Teichmüller group

Cited by 13 publications

References 25 publications

Étale cohomology, purity and formality with torsion coefficients

Étale cohomology, purity and formality with torsion coefficients

Étale cohomology, purity and formality with torsion coefficients

Simplicial model structures on pro-categories

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