Vol. 26, No. 1, 1968

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Postnikov-decompositions of functors

Friedrich-Wilhelm Bauer

Vol. 26 (1968), No. 1, 9–24

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 = {XX 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.

Mathematical Subject Classification
Primary: 55.40
Secondary: 18.00
Received: 25 May 1967
Published: 1 July 1968
Friedrich-Wilhelm Bauer