Algebra/Topology Seminar

Speaker: Yuqing Shi

Title: Lie Algebras and En-algebras in higher chromatic heights

Abstract:

The sequence of natural embeddings of the little cubes operads E_n−1 → E_n induces an equivalence between the ∞-category of E_∞-algebras and the inverse limits of ∞-category of E_n-algebras in a stable symmetric monoidal ∞-category C. The Koszul dual of this equivalence gives a functor U from the ∞-category of Lie algebras to an inverse limits of the ∞-category of E_n-algebras in C, where the functor from Lie algebras to E_n-algebras is a natural generalisation of taking the universal enveloping algebra and the functor from E_n-algebras to E_n−1-algebras is the Bar construction. Taking C to be the ∞-category of rational chain complexes, formality of the little cubes operads implies that U is an equivalence. In this talk I will show that U is fully faithful when we let C be the ∞-category of T(h)-local spectra for 1 ≤ h < ∞, where T(h) denotes the telescope spectrum of height h. This is our first step to generalise the rational equivalence to higher chromatic heights.