Fibration models and localisation of categories

We define a fibration model on the basis of a Thomason model and use it to analyse the localisation of a category. We show that the model suffices to prove existence of the localisation and to develop the basic homological tools, including derived functors, homotopy pullbacks and the unstable triangulated structure. The corresponding constructions in the case of Quillen, Baues and Thomason models can be recovered as special cases.