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.
Categories: math.AG math.KT