  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.
