Description:

Title: A point-set model for Lurie's bar-cobar adjunction

Abstract: This is a report of joint work in progress with Victor Roca i Lucio and Sinan Yalin. The bar-cobar adjunction is an adjunction between augmented DG algebras and coaugmented conilpotent DG coalgebras, dating back to the origins of homological algebra and algebraic topology. One aspect of "Koszul duality" is the fact that it becomes close to an equivalence, once one inverts quasi-isomorphisms. More recently, Jacob Lurie introduced a very general "bar-cobar adjunction" between E_1-algebras and E_1-coalgebras in a monoidal \infty-category satisfying mild hypotheses. However, if one specializes Lurie's bar-cobar adjunction to the \infty-category of chain complexes, then one does NOT recover the classical bar-cobar adjunction, for multiple reasons: (i) Lurie's bar functor is left adjoint, and the classical algebraic bar functor is right adjoint. (ii) Lurie's adjunction does not involve any conilpotence hypothesis. (iii) DG coalgebras localized at quasi-isomorphisms do not present the \infty-category of E_1-coalgebras in the \infty-category of chain complexes. This raises the question of what the two adjunctions have to do with each other. We explain how to think about the DG case of Lurie's adjunction in more classical language: one needs to firstly work with A_\infty-coalgebras, and secondly take a completion of the cobar functor.