# Stasheff J. Higher holonomy. Representations up to homotopy (abstract)

Given a connection for a smooth vector bundle ${\displaystyle p\colon E\to M}$, parallel transport with respect to smooth paths in the base space ${\displaystyle M}$ provides a correspondence between smooth vector bundles with flat connection on ${\displaystyle M}$ and representations of ${\displaystyle \pi _{1}(M)}$. Based in part on earlier groundbreaking work of K.T. Chen, recently this correspondence has been enhanced to the level of smooth paths (not homotopy classes) in the base space ${\displaystyle M}$ and differential graded vector bundles with generalized flat connections.