This is a crosspost of [this MSE question](https://math.stackexchange.com/q/2994508/223002).




---


Let $A\overset{\alpha}{\rightarrow}B$ be a (Hurewicz) fibration.


* The homotopy lifting property w.r.t a fiber $\alpha ^{-1}(b)$
furnishes for each path $b\to b^\prime$ in the base a continuous map
$\alpha ^{-1}(b)\to \alpha ^{-1}(b^\prime)$. Moreover, this
assignment extends to a functor $\pi\_1B\longrightarrow
 \mathsf{hTop}$.
* On the other hand, as a fibration $\begin{smallmatrix}A\\\downarrow\\B\end{smallmatrix}$ admits a [Hurewicz connection](https://ncatlab.org/nlab/show/Hurewicz+connection) $s$. Given such a connection it is tempting to send a path $b\overset{\gamma}{\to} b^\prime $ in the base to the following set function (analogously to covering space theory) $$\alpha^{-1}(b)\longrightarrow \alpha ^{-1}(b^\prime),\quad a\mapsto \operatorname{eval}\_1s(a,\gamma).$$ I suspect this set function might be continuous, but I see no reason for it to be a homotopy equivalence, since $s(a,\gamma)$ need not be related in a nice way to lifts of opposite path $b\overset{\bar\gamma}{\leftarrow} b^\prime $.


**Questions.**


1. Is fiber transport along a Hurewicz connection continuous?
2. (Assuming continuity) Is fiber transport along a Hurewicz connection functorial?
3. (Assuming continuity) Does it coincide with the first transfer functor?
4. Suppose the fibers are all homeomorphic. Are there any interesting conditions that make the transport functor $\pi\_1B\longrightarrow \mathsf{hTop}$ lift to $\mathsf{Top}$? That is, can we obtain such homeomorphisms via transport?




---


I thought about possible functoriality of the transport along the Hurewicz connection. We want $\operatorname{eval}\_1(a,\delta \ast \gamma)=\operatorname{eval}\_1s(\operatorname{eval}\_1s(a,\gamma),\delta,)$. We may consider the concatenation $s(\operatorname{eval}\_1s(a,\gamma),\delta)\ast s(a,\gamma)$ which seems to lift $\delta \ast \gamma$, but I'm not quite sure where to go from here.



Relation between transport functor of a fibration and a Hurewicz connection on it
Let $p: E\to B$ be a map. Define $\Lambda(p) = E \times\_B B^I$; this is the space of pairs $(x,\gamma)$ consisting of a point $x\in E$ and a path $\gamma: [0,1] \to B$ such that $\gamma(0) = p(x)$. There is an evident restriction map 
$$
\rho: E^I \to \Lambda(p)
$$
where $E^I$ is the free path space of $E$. 


The map $p: E\to B$ is a Hurewicz fibration iff $\rho$ has a section (easy exercise). A choice of (continuous) section $s: \Lambda(p) \to E^I$ defines transport. 


Assume $p$ is a Hurewicz fibration.
Consider the commutative diagram
$\require{AMScd}$
$$
\begin{CD}
\Lambda(p) \times 0 @>>> E \\
@VVV @VV pV \\
\Lambda(p) \times I @>>> B
\end{CD}
$$
where the bottom arrow is given by $((x,\gamma),t) \mapsto \gamma(t)$ and
the top one is given by $((x,\gamma),0)\mapsto x$. By the lifting property, we can fill in the diagram with a map 
$$
\Lambda(p) \times I \to E \, .
$$
This continuous map defines the Hurewicz connection by taking its adjoint to get a map $\Lambda(p) \to E^I$.


On the other hand, for a given path $\gamma$ wiht $\gamma(0) = b$ and $\gamma(1) = b'$, we can restrict the displayed map $\Lambda(p) \times I \to E$ to 
$F\_b = (F\_b \times\_{\{b\}} \{\gamma\}) \times \{1\}$ (with $F\_b = p^{-1}(b)$)
to obtain a continuous map
$$
F\_b \to E
$$ 
whose image is contained in $F\_{b'}$. This map is your transport operation. This establishes (1)-(3).


As to (4), the transport functor restricts on closed based loops to a holonomy map
$$
\Omega B \to G(F)
$$
where $F$ is the fiber at the basepoint and $G(F)$ is the (group-like) topological monoid of self homotopy equivalences of $F$. This map is actually an $A\_\infty$-homomorphism. By replacing $\Omega B$ with its Moore loops instead, this will become a morphism of topological monoids.


Let us make the additional hypothesis that your Hurewicz fibration is a fiber bundle (this is actually not a restriction up to fiber equivalence). Then your transport map gives a holonomy homomorphism
$$
\Omega B \to H(F) 
$$
where $H(F)$ is the topological group of self homeomorphisms of $F$.


Your question (4) is more-or-less the condition that the connection admit a **flat reduction**, meaning this: let $H^\delta(F)$ be $H(F)$ with the discrete topology. Then we have a map of classifying spaces
$$
BH^\delta(F) \to BH(F) 
$$ and (4) is answered in the affirmative iff the classifying map for the bundle
$B\to BH(F)$ factors up to homotopy through $BH^\delta(F)$.


