Abstract: We construct a DG bicategory of enhanced triangulated categories, functors, and natural transformations. We define A-infinity structures on enhanced functors. This allows us to prove an analogue of the Barr-Beck monadicity theorem for enhanced triangulated categories. Moreover, we show that there is a simple natural DG category derived equivalent to the Eilenberg-Moore category of the adjunction of enhanced functors.
We then apply this to two problems: (1). Descent for the derived categories. We take an affine cover of a variety X and show how the objects in the derived category of D(X) can be “glued” from the objects in the derived categories of open sets. E.g how the derived category of P^1 can be glued from the derived categories of two A^1s and where do the first Exts come from. (2). “Fake” descent (aka linear Koszul duality) for a regular closed embedding Z→X. Assuming a formality condition, we show that the category of objects in the derived category D(Z) with an action of the exterior algebra of the normal bundle NZ/X is equivalent to the derived category DZ(X) of objects in D(X) with the support on Z. For a point in a vector space V, this gives the classical Koszul duality between the modules over the exterior algebra of V and the symmetric algebra of V∗. Without the formality condition, DZ(X) is equivalent to the derived category of Ainfty-modules over a certain A∞ structure on the exterior algebra of NZ/X. Thus the entire information about the formal neighborhood of Z in X is encoded in the higher operations on the exterior algebra of NZ/X.
This is joint work with Rina Anno and Sergey Arkhipov.
Abstract: We explain homotopy Gerstenhaber algebras, a certain weakening of commutative algebras. In many objects case, homotopy Gerstenhaber categories can serve as a model for weak monoidality. We then report on the ongoing project with Julian Holstein, where we aim to construct an explicit homotopy Gerstenhaber structure on the category of representations up to homotopy, based on combinatorics of Hochschild polytopes.
Abstract: Given a self-dual quadratic Koszul algebra, one gets a remarkable equation on its Poincare series. A similar relation holds for a self-dual Koszul operad. We study an example coming from combinatorics of certain polytopes when the equation mentioned above turns out to be a shadow of a non-linear action of a one parameter subgroup in automorphisns of a free algebra generated by classes in the Grothendieck group of modules over the incidence algebra for the polytope. The main tool in the construction is monoidal Koszul duality.