  We show that a $KK$-equivalence between two unital $C^*$-algebras produces a
correspondence between their DG categories of finitely generated projective
modules which is a $\mathbf{K}_*$-equivalence, where $\mathbf{K}_*$ is
Waldhausen's $K$-theory. We discuss some connections with strong deformations
of $C^*$-algebras and homological dualities. Motivated by a construction of
Cuntz we associate a pro $C^*$-algebra to any simplicial set. We show that this
construction is functorial with respect to proper maps of simplicial sets, that
we define, and also respects proper homotopy equivalences. We propose to
develop a noncommutative proper homotopy theory in the context of topological
algebras.
