arXiv subjects
Publications and source records attributed to Geoffroy Horel.
In this note we recall the construction of two chain level lifts of the gravity operad, one due to Getzler-Kapranov and one due to Westerland. We prove that these two operads are formal and that they indeed have isomorphic homology.
We prove a formality theorem for algebraic objects internal to smooth complex varieties that are not compact but whose mixed Hodge structure has a certain purity property.
We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck-Teichmüller group. Using a result of Drummond-Cole, we deduce that the Grothendieck-Teichmüller group acts nontrivially on $\overline{\mathcal{M}}_{0,\bullet+1}$, the operad of stable curves of genus zero. As a second application, we give an alternative proof that the framed little 2-disks operad is formal.
The goal of this paper is to prove an equivalence between the model categorical approach to pro-categories, as studied by Isaksen, Schlank and the first author, and the $\infty$-categorical approach, as developed by Lurie. Three applications of our main result are described. In the first application we use (a dual version of) our main result to give sufficient conditions on an $ω$-combinatorial model category, which insure that its underlying $\infty$-category is $ω$-presentable. In the second application we consider the pro-category of simplicial étale sheaves and use it to show that the topological realization of any Grothendieck topos coincides with the shape of the hyper-completion of the associated $\infty$-topos. In the third application we show that several model categories arising in profinite homotopy theory are indeed models for the $\infty$-category of profinite spaces. As a byproduct we obtain new Quillen equivalences between these models, and also obtain an example which settles negatively a question raised by Raptis.
We lift the classical theorem of Arnol'd on homological stability for configurations spaces of the plane to the motivic world. More precisely, we prove that the schemes of unordered configurations of points in the affine line satisfy stability with respect to the motivic t-structure on mixed Tate motives.
Given a limit sketch in which the cones have a finite connected base, we show that a model structure of "up to homotopy" models for this limit sketch in a suitable model category can be transferred to a Quillen equivalent model structure on the category of strict models. As a corollary of our general result, we obtain a rigidification theorem which asserts in particular that any $Θ_n$-space in the sense of Rezk is levelwise equivalent to one that satisfies the Segal conditions on the nose. There are similar results for dendroidal spaces and $n$-fold Segal spaces.
In a Brown category of cofibrant objects, there is a model for the mapping spaces of the hammock localization in terms of zig-zags of length 2. In this paper we show how to assemble these spaces into a Segal category that models the infinity-categorical localization of the Brown category.
We put a model structure on the category of categories internal to simplicial sets whose weak equivalences are reflected by the nerve functor to bisimplicial sets with Rezk's model structure. This model structure is shown to be Quillen equivalent to Rezk's model structure, thus internal category are another model for $(\infty,1)$-categories.
In this paper, we prove that the group of homotopy automorphisms of the profinite completion of the operad of little $2$-disks is isomorphic to the profinite Grothendieck-Teichmüller group. In particular, the absolute Galois group of $\mathbb{Q}$ acts faithfully on the profinite completion of $E_2$ in the homotopy category of profinite weak operads.
We give a method for computing factorization homology of $\oper{E}_n$-algebra using as an input an algebraic version of higher Hochschild homology due to Pirashvili. We then show how to compute higher Hochschild homology and cohomology when the algebra is étale in a sense that we make precise. As an application, we compute higher Hochschild cohomology of the Lubin-Tate ring spectrum.
Using factorization homology techniques, we give a simple proof of the action of Kontsevich and Soibelman operad on the pair consisting of Hochschild cohomology and Hochschild homology. The proof obviously extends to higher Hochschild cohomology and factorization (also called chiral) homology. Note that our result works in categories of chain complexes but also in categories of modules over a commutative ring spectrum.
In this paper, we describe a general theory of modules over an algebra over an operad. We also study functors between categories of modules. Specializing to the operad E_d of little d-dimensional disks, we show that each (d-1)-manifold gives rise to a theory of modules over E_d-algebras and each bordism gives rise to a functor from the category defined by its incoming boundary to the category defined by its outgoing boundary. We describe how to assemble these categories into a map from a certain operad to the operad of (infinity)-categories.