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We give a generalization of the notion of a Cartan-Ehresmann connection fromLie algebras to L-infinity algebras and use it to study the obstruction theoryof lifts through higher String-like extensions of Lie algebras. We find(generalized) Chern-Simons and BF-theory functionals this way and describeaspects of their parallel transport and quantization.  It is known that over a D-brane the Kalb-Ramond background field of thestring restricts to a 2-bundle with connection (a gerbe) which can be seen asthe obstruction to lifting the PU(H)-bundle on the D-brane to a U(H)-bundle. Wediscuss how this phenomenon generalizes from the ordinary central extensionU(1) -> U(H) -> PU(H) to higher categorical central extensions, like theString-extension BU(1) -> String(G) -> G. Here the obstruction to the lift is a3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified bythe first Pontrjagin class. For G = Spin(n) this obstructs the existence of aString-structure. We discuss how to describe this obstruction problem in termsof Lie n-algebras and their corresponding categorified Cartan-Ehresmannconnections. Generalizations even beyond String-extensions are thenstraightforward. For G = Spin(n) the next step is "Fivebrane structures" whoseexistence is obstructed by certain generalized Chern-Simons 7-bundlesclassified by the second Pontrjagin class.

L-infinity algebra connections and applications to String- and Chern-Simons n-transport