# Functoriality of parallel transport of a Hurewicz connection on a fiber bundle

Let $$A\overset{\alpha}{\rightarrow}B$$ be a Hurewicz fibration. Any Hurewicz connection defines parallel transport along curves in the base. In general, such parallel transport maps $$\alpha^{-1}(b)\to \alpha^{-1}(b^\prime)$$ are continuous homotopy equivalences, but they need not be homeomorphisms.

For instance, consider the fibration given by projecting the standard 2-simplex onto the $$x$$-axis. For a connection we can lift a path in the $$x$$-axis by composing it with the embedding of the $$x$$-axis in the 2-simplex as a "rotation" of the $$x$$-axis with $$(0,1)$$ fixed.

The pleasant thing is that a Hurewicz fibration admits a connection-independent functor $$\pi_1B\to \mathsf{hTop}$$ taking a homotopy class of paths to the homotopy class of some parallel transport along it.

I am trying to understand when a particular Hurewicz connection might induce a parallel transport functor $$\mathsf{Paths}(B)\to \mathsf{Top}$$. Here $$\mathsf{Paths}(B)$$ is the groupoid whose objects are the points of $$B$$ and whose arrows are continuous maps from an interval of the form $$[0,\ell]$$ to $$B$$. Composition of paths is defined to be associative.

Such a transport functor would force the fibers to be homeomorphic, so I might as well assume the fibration is a fiber bundle. Still though, I don't what might force the parallel transports of arbitrary Hurewicz connection on a fiber bundle to behave functorially (even to just be homeomorphisms).

The same question remains if we restrict to loops and replace the path groupoid with Moore loops.

Since we assume $$A\overset{\alpha}{\rightarrow}B$$ is a fiber bundle, pulling back along a curve gives a fiber bundle over an interval, which must be trivial, but I don't really see how this helps. (In the smooth theory, parallel transports are diffeomorphisms because they come from the flow of a vector field. I don't see an analogue here.)