Codescent theory I: Foundations

Consider a cofibrantly generated model category $S$, a small category $C$ and a subcategory $D$ of $C$. We endow the category $S^C$ of functors from $C$ to $S$ with a model structure, defining weak equivalences and fibrations 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 long-term program aims at reformulating as codescent statements the Conjectures of Baum-Connes and Farrell-Jones, and at tackling them with new methods. Here, we set the grounds of a systematic theory of codescent, including pull-backs, push-forwards and various invariance properties.