Categorification of the Leray spectral sequence

Abstract

The first part of this thesis concerns the categorification of differentials in the Leray spectral sequence. Given a continuous map of topological spaces ( f \colon X \rightarrow Y ) and a sheaf of abelian groups ( G ), the Leray spectral sequence takes the form[ E_2^{p,q} := H^p(Y, R^q f_* G) \Rightarrow H^{p+q}(X, G). ] We provide a categorification of certain differentials in this spectral sequence, specifically those of the form [ d_2 \colon H^0(Y, R^q f_* G) \rightarrow H^2(Y, R^{q-1} f_* G), ] by constructing a gerbe that encodes the first obstruction to a lifting problem. The differential ( d_2 ) is first described at the level of Čech cocycles. Then, for a class ( \alpha \in H^0(Y, R^q f_* G) ), we construct the sheaf of local lifts, which takes values in ( q )-groupoids. Taking the 1-truncation of this object yields an ( R^{q-1} f_* G )-banded gerbe whose cohomology class represents the image of ( \alpha ) under ( d_2 ). Finally, we show that this construction is compatible with the additive structure on cohomology. Second part concerns it with the existence of non-cellular objects in the motivic stable homotopy category $\mathcal{SH}(k)$.