"Parallel" transport - revisited

Parallel transport in a fibre bundle with respect to smooth paths in the base space B have recently been extended to representations of the smooth singular simplicial set Sing_{smooth}(B). Inspired by these extensions,I revisit the development of a notion of parallel' transport in the topological setting of fibrations with the homotopy lifting property and then extend it to representations of Sing(B) on such fibrations. Closely related is the notion of (strong or infty’) homotopy action, which has variants under a variety of names.