Mackey functors for posets

Abstract We define Mackey functors over posets mimicking the classical notion and introduce a weak version of them. Then we show that they are acyclic by analyzing cofibrant and pseudo-projective objects in the category of functors indexed in a filtered poset. As application, we study homotopy colimits over posets and we give a homology decomposition for the classifying space of the Bianchi group $$\Gamma _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> .