  In this paper we introduce the notion of a $cdp$-functor to a Waldhausen
category. We show that such functors admit extensions that satisfy the excision
property, to which we associate Euler-Poincar\'e characteristics that send the
class of a proper scheme to the class of its image. As an application, we show
that the Yoneda embedding gives rise to a monoidal proper-fibred Waldhausen
category over Noetherian schemes of finite Krull dimensions, with canonical
$cdp$-functors to its fibres.
