"Parallel" transport - revisited

TL;DR

This paper revisits the concept of parallel transport in topological fibrations, extending it to representations of the singular simplicial set and exploring related homotopy actions, thus broadening its theoretical framework.

Contribution

It introduces a new perspective on parallel transport in topological fibrations and extends existing notions to representations of Sing(B), connecting with homotopy actions.

Findings

Extended parallel transport to topological fibrations with homotopy lifting property

Connected parallel transport with strong homotopy actions

Provided a framework linking fibrations, singular sets, and homotopy actions

Abstract

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.