So far I have found only two possible constructions of the Leray spectral sequence for a continuous map $f : X \to Y$ between topological spaces with CW-complex structures: one through Cech complexes, and the other one more categorical through Grothendieck spectral sequences.
I would like to construct this spectral sequence directly by hand from the filtration induced by the CW-structure of $Y$, but haven't found any reference about it.
May I emphasise that I'm looking for a construction of the Leray spectral sequence (for continuous maps which are not necessarily filtrations, and not of the Leray-Serre spectral sequence (where direct constructions can be found in many places).
Does someone have an idea ? Thanks !