Codescent theory I: Foundations

Consider a cofibrantly generated model category S, a small category C and a subcategory D of C. The category SC of functors from C to S has a model structure, with weak equivalences and fibrations defined objectwise but only on D. Our first concern is the effect of moving C, D and S. The main notion introduced here is the “D-codescent” property for objects in SC. Our program aims at reformulating as codescent statements the Conjectures of Baum–Connes and Farrell–Jones, and, in the long run, at tackling them with new methods. We set the grounds of a systematic theory of codescent, including pull-backs, push-forwards and various invariance properties.