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 $\mathcal{SH}(k)$.