Homotopy Fiber Sequences from a New Perspective

Quillen introduced fibration sequences in the homotopy category of a pointed model category. Up to a (non-canonical) Ho(M)-isomorphism a fibration sequence is then a Ho(M)–sequence K → F → G that is implemented by the kernel K of a fibration F → G between fibrant objects F and G. Using the fact that the loop space functor Ω^QF of a fibrant object F is a group object in the homotopy category Ho(M), Quillen shows that there is an action of Ω^QG on K which induces a connecting Ho(M)–morphism Ω^QG → K and the sequence Ω^QG → K → F is again a fibration sequence. I want to present an alternative approach to construct homotopy fiber sequences without using the concept of an action. We define a loop space functor Ω and for every morphism f : F → G we define its homotopy fiber K_f such that K_f → F → G is a homotopy fiber sequence. We get a universal connecting morphism ΩF → K_f such that ΩF → K_f → G is also a homotopy fiber sequence. It turns out that the loop space functor we define as well as the connecting homomorphism coincide with the one proposed by Quillen. At the beginning of this talk I will explain how to construct an equivalence of categories between the homotopy category of morphisms (arrows) in M, denoted by Ho(M^→) and the homotopy category of long homotopy fiber sequences, denoted by Ho(l(M)). This will lead to a canonical choice of an isomorphism in Ho(M) between two different homotopy fibers of a morphism in Ho(M).