CUBE AXIOM AND QUILLENS FUNCTORS

The two definitions of the Lusternik and Schnirelmann category, due to Whitehead and Ganea, conceptually differ from each other. The first a uthor showed that they exist within Quillen's model categories, and co incide when a further non autodual axiom is assumed, namely the cube a xiom. Here we extend this study within model categories which are not necessarily proper and do not satisfy the cube axiom. For this, the gl obal hypothesis of this axiom is transformed into a condition on a cla ss of morphisms, namely the cube maps. We then study the image of cube maps by a Quillen's couple of adjoint functors. We finally consider t he example of the chain of functors appearing in rational homotopy.