CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Given a continuous map $\pi : E\to B$ of topological spaces, one constructs the mapping cocylinder $Cocyl(\pi)$ as the pullback
where $\mathcal{P}(B)$ is the path space in Top, the space of continuous paths $u:[0,1]\to B$ in $B$, and where $\mathcal{P}(B)\to B$ is the map sending a path $u$ to its value $u(0)$. The cocylinder can be realized as a subspace of $E\times \mathcal{P}(B)$ consisting of pairs $(e,u)$ where $e\in E$ and $u:[0,1]\to \mathcal{P}(B)$ are such that $\pi(e)=u(0)$.
A Hurewicz connection is any continuous section
of the map $\pi_!:\mathcal{P}(E)\to Cocyl(\pi)$ given by $\pi_!(u)=(u(0),\pi\circ u)$.
A map $\pi:E\to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$.
To see that consider the following diagram
where $pr_E: Cocyl(\pi)\to E$ is the restriction of the projection $E\times B^I\to E$ to the factor $E$ and the map $Cocyl(\pi)\times I\to B$ is the evaluation $(e,u,t)\mapsto u(t)$ for $(e,u)\in Cocyl(\pi)$. The right-hand square is commutative and this square defines a homotopy lifting problem. If $\pi$ is a cofibration this universal homotopy lifting problem has a solution, say $\tilde{s}:Cocyl(p)\times I\to E$. By the hom-mapping space adjunction (exponential law) this map corresponds to some map $s:Cocyl(\pi)\to \mathcal{P}(E)$. One can easily check that this map is a section of $\pi_!$.
Conversely, let a Hurewicz connection $s$ exist, and fill the right-hand square of the diagram with diagonal $\tilde{s}$ obtained by hom-mapping space adjunction. Let the data for the general homotopy lifting problem be given: $\tilde{f}:Y\to E$, $F:Y\times I\to B$ with $F_0 = p\circ \tilde{f}:Y\to E$; let furthermore $F':Y\to \mathcal{P}(B)$ be the map obtained from $F$ by the hom-mapping space adjunction. By the universal property of the cocylinder (as a pullback), there is a unique mapping $\theta: Y\to Cocyl(\pi)$ such that $pr_{\mathcal{P}(B)}\circ\theta=F':Y\to \mathcal{P}(B)$ and $pr_E\circ\theta =\tilde{f}:Y\to E$. Now notice that the by composing the horizontal lines we obtain $\tilde{f}$ upstairs and $F$ downstairs, hence the external square is the square giving the homotopy lifting problem for this pair. The lifting is then given by $\tilde{s}\circ (\theta\times id_I):Y\times I\to E$. Simple checking finishes the proof.
Of course there are many other equivalent characterizations of Hurewicz fibrations.
If $\pi:E\to B$ is a covering space where $B$ is Hausdorff, then $\pi_!$ is a homeomorphism; thus in that case the Hurewicz connection is unique.
If $\pi$ is a smooth principal bundle equipped with a distribution of horizontal spaces forming an Ehresmann connection, then one can define a corresponding “smooth” Hurewicz connection in the sense that the Ehresmann connection provides a continuous choice of smooth path lifting, with prescribed initial point, of a smooth path in the base. This can be expressed in terms as a continuous section of $\pi_!^{smooth}:\mathcal{P}^{smooth}(E)\to Cocyl^{smooth}(\pi)$ where the subspaces of smooth paths are used.
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