  We characterize cofibrant objects in the category of functors indexed in a
filtered poset and we show that these objects are acyclic. As a consequence, we
show that Mackey functors over posets are also acyclic, where we define this
type of Mackey functors mimicking the classical notion. As application, we
study homotopy colimits over posets and we give a homology decomposition for
the classifying space of the Bianchi group $\Gamma_1$.
