(preprint, October 2000)
This note is a continuation of \cite{Crai}, where we have showed that the classical Chern classes can be computed using ``connections up to homotopy''. First, we present a slightly different approach. Next we discuss the characteristic classes associated to flat connections up to homotopy. In contrast with \cite{Crai}, we do not only recover the classical cheracteristic classes (of flat vector bundles), but we also obtain new cohomology classes. The reason for this is that ($\mathbb{Z}_2$-graded) vector bundles may have flat connections up to homotopy, without being themselves flat. In this category fall e.g. the characteristic classes of Poisson manifolds \cite{Fer2}.
As already mentioned in \cite{Crai}, one of our motivation was the understanding of the intrinsic characteristic classes for Poisson manifolds (and algebroids) of \cite{Fer1, Fer2}, and the connection with the characteristic classes of representations \cite{Cra}. Conjecturally, Fernandes' intrinsic characteristic classes \cite{Fer1} are the characteristic classes \cite{Cra} of the ``adjoint representation''. The problem is that the adjoint representation is a ``representation up to homotopy only''. As an application, we solve this problem: we extend the characteristic classes of \cite{Cra} to representations up to homotopy, and we show that Fernandes' classes are indeed the classes associated to the adjoint representation.