Working in a simplicial and constructive context, a new spectral system is defined that relates Serre and Eilenberg–Moore spectral sequences associated to a principal simplicial fibration. The two Eilenberg–Moore spectral sequences (the one where the homology of the fiber is the output, and the other where the homology of the base is computed) are used in our construction. Explicit computer programs are developed, enhancing the Kenzo computer algebra tool to implement that spectral system.

In their original setting, both spectral sequences and persistent homology are algebraic topology tools defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set \Z of integer numbers. Recently, generalizations of both concepts have been proposed which originate from a different choice of the set of indices of the filtration, producing the new notions of multipersistence and spectral system. In this paper, we show that these notions are related, generalizing results valid in the case of filtrations over \Z. By using this relation and some previous programs for computing spectral systems, we have developed a new module for the Kenzo system computing multipersistence. We also present a new invariant providing information on multifiltrations and applications of our algorithms to spaces of infinite type.

In this paper, we present some algorithms and programs for computing generalized spectral sequences, a useful tool in Computational Algebraic Topology which provides topological information on spaces with generalized filtrations over a poset. Our programs have been implemented as a new module for the Kenzo system and solve the classical problems of spectral sequences which are differential maps and extensions. Moreover, combined with the use of effective homology and discrete vector fields, the programs make it possible to compute generalized spectral sequences of big spaces, sometimes of infinite type.