Codescent theory I: Foundations

Given a cofibrantly generated model category $S$, like the category of topological spaces, together with a small category $C$ and a subcategory $D$ of $C$, we establish the existence of a model structure on the category $S^C$ of functors from $C$ to $S$, where the weak equivalences and the fibrations are tested objectwise, but only on $D$. We prove the basic results on the behaviour of this model structure with respect to $C$, $D$ and $S$. We introduce the notion of ``$D$-codescent'' and present the first elementary results. One of the goals of this series is a reformulation of the Isomorphism Conjectures of Farrell-Jones and Baum-Connes in $K$-theory, as a codescent statement.