In Chapter II we obtained the fundamental exact sequences of homotopy theory, namely

1. (A) the cofiber sequence,

2. (B) the exact sequence for homotopy groups of function spaces, and

3. (C) the exact sequence for relative homotopy groups.

In this chapter we derive from these exact sequences the corresponding homotopy spectral sequences. In case (A) this leads to the general form of both the Atiyah–Hirzebruch spectral sequence and the Bousfield–Kan spectral sequence, compare also Eckmann–Hilton (1966). In case (B) we obtain a far reaching generalization of the Federer spectral sequence for homotopy groups of function spaces in topology. In case (C) the homotopy spectral sequence is the general form of the homotopy exact couple in topology (considered by Massey) from which we deduce the ‘certain exact sequence of J.H.C. Whitehead’ in a cofibration category. All these results on the homotopy spectral sequences are available in any cofibration category. Moreover, a functor between cofibration categories (which carries weak equivalences to weak equivalences) induces a map between these homotopy spectral sequences compatible with the differentials. Various properties of the spectral sequences are proved in this chapter, some of them seem to be new (even for the classical topological spectral sequences).

In all cases (A), (B) and (C) we study the E2-term of the spectral sequences. For this we introduce complexes, chain complexes, and twisted chain complexes in a cofibration category. Complexes are iterated mapping cones which are obtained by a succession of attaching cones.