Date: 27 March 2024

Abstract: Starting from any sufficiently nice topological space X, one can produce two different (derived) stacks defined over a field k of characteristic 0: the Betti stack X_B, which bears information on the underlying homotopy type of X, and the coaffine stack cSpec(C*(X; k)) on the commutative algebra C*(X; k) of kvalued singular cochains on X, which behaves as the affinization of the Betti stack X_B.
In this talk, I shall apply the technical machinery of sheaves of (∞)categories on derived stacks as developed in Gaitsgory’s work to the case of Betti stacks and their associated coaffine stacks. We shall see how sheaves of categories on X_B are intimately related to categorified local systems over the original space X and to homotopycoherent actions of topological groups on klinear ∞categories. At the very end, I shall briefly describe how sheaves of categories on X_B and on cSpec(C*(X; k)) interact, and how such interaction can be interpreted as an instance of higher Koszul duality.
This talk is based on upcoming joint work with J. Pascaleff and N. Sibilla.

Slides: https://drive.google.com/file/d/1DGuv...

Emanuele Pavia: https://emanuelepavia.com/

LAGOON Webinar: https://sites.google.com/view/lagoonw...