An infinity-categorical perspective on spectral sequences

Abstract

Spectral sequences have been used as important computational tools in algebraic topology and homological algebra. In recent years, these tools have been described in the setting of infinity-categories. We focus on the approach using 'décalage'. This has the advantage that it describes multiplicative structures on spectral sequences in a categorical way. We apply these results to the Leray-Serre-Atiyah-Hirzebruch spectral sequence. Given a fibration, this spectral sequence calculates the generalized cohomology of the total space from the cohomology of the base space with coefficients in the generalized cohomology of the fiber. Using this new approach we can show that the spectral sequence admits a multiplicative structure.