  We use Chen's iterated integrals to integrate representations up to homotopy.
That is, we construct an A_infty functor from the representations up to
homotopy of a Lie algebroid to those of its infinity groupoid. This
construction extends the usual integration of representations in Lie theory. We
discuss several examples including Lie algebras and Poisson manifolds. The
construction is based on an A_infty version of de Rham's theorem due to
Gugenheim. The integration procedure we explain here amounts to extending the
construction of parallel transport for superconnections, introduced by Igusa
and Block-Smith, to the case of certain differential graded manifolds.

Differential Geometry;Algebraic Topology