The purpose of a
Postnikov-decomposition of a covariant functor Φ :K→ C is to approximate Φ by
simpler functors Ψ : K → C (so called Postnikov-functors for Φ) which reflect certain
properties of Φ but are simpler to handle. There is always a functor transformation
T : Φ → Ψ and a “coincidence-category” LΨ= {X∣X ∈ K,T : Φ(X) ≈ Ψ(X)}. A
Postnikov-resolution A is a family of Postnikov-functors for Φ. There is always a
maximal Postnikov-resolution. Under certain conditions each object X ∈ K
can be decomposed into a family of objects {XΨ} where XΨ∈ LΨ and
Ψ ∈ A. The whole theory can be dualized and one gets the theory of dual
Postnikov-resolutions. Let π = Φ be the homotopy-functor, π = {πm}; it turns out,
that the maximal Postnikov-decomposition of π gives exactly the classical
theory of Postnikov-complexes of topological spaces or Kan-complexes. A
similar result is proved for Φ = H = {Hm}, the homology functor and dual
Postnikov-resolutions as well as for dual Postnikov-resolutions of π and their
connection to Cartan-Serre-fibrations.
