Codescent theory - I: Foundations

Consider a cofibrantly generated model category S, a small category C and a subcategory D of C. The category S-C 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 S-C. 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. (C) 2004 Elsevier B.V. All rights reserved.