| Résume | 14h-15h Sophie d'Espalungue
Boardman Vogt tensor product of operads in Cat and interchange of En structures
Iterated monoidal categories were introduced by Baltaneau, Fiedorowitzch, Schwänzl and Vogt as a categorical equivalent for iterated loop spaces and defined as algebras over operads $M_n$ with values in the category $Cat$ of categories. We construct a tensor product for operads in $Cat$ verifying $M_n \otimes M_m \cong M_{n+m}$. We equip the category of operads in $Cat$ with a model structure such that the tensor product preserves cofibrant objects, providing explicit models for homotopy n-fold monoidal structures.
15h30-16h30 Guglielmo Nocera
The E3-structure on the spherical category of a reductive group
The E3-structure on the spherical category of a reductive group Let G be a reductive group over the complex numbers, e.g. GLn,C. There is a monoidal triangulated/dg/∞-category Sph(G), called the spherical category of G, which plays an important role in the Geometric Langlands program. For example, its behaviour provides important constraints in the formulation of the Geometric Langlands Conjecture. This ∞-category is not symmetric monoidal, but it admits a t-structure whose heart is symmetric monoidal: more precisely, the heart is monoidal-equivalent to a category of representations of a group (the Langlands dual of G) with its (symmetric monoidal) tensor product. In this talk, I will present how to upgrade the existing E1-monoidal structure on Sph(G) to an E3-monoidal one, which formally recovers the symmetric monoidal structure of the heart. The construction implements ideas of Jacob Lurie and uses a strongly topologically flavoured presentation of Sph(G), namely as an ∞-category of constructible sheaves over a stratified space. If time permits, I will also briefly explain its connection to ongoing work of Campbell–Raskin, which allows to show that our E3-structure agrees with the interpretation of Sph(G) as the operadic E2-center of a derived ∞-category of representations induced by results of Bezrukavnikov–Finkelberg and Ben-Zvi–Francis–Nadler. Part of this work is joint with Morena Porzio.
16h45-17h45 Victor Pecanha Brittes
Localisations of the model structure for 2-quasi-categories
Localisations of the model structure for 2-quasi-categories 2-quasi-categories are a model for (infinity, 2)-categories introduced by Ara. They are the fibrant objects of a model structure on the category of Θ_2-sets, where Θ_2 is a 2-dimensional analog of the simplex category Δ. In this talk, we will explain how we can localise this model structure to obtain models for the homotopy theory of spaces, 2-categories and homotopy 2-types. |
