Hurewicz connection (changes) in nLab

Context

Topology

• Setup and definition
• Characterization of Hurewicz fibrations
• Special cases and properties
• References

Definition

A Hurewicz connection is any continuous section

s:Cocyl(π)→𝒫(E)s:Cocyl(\pi)\to \mathcal{P}(E)

of the map π !:𝒫(E)→Cocyl(π)\pi_!:\mathcal{P}(E)\to Cocyl(\pi) given by π !(u)=(u(0),π∘u)\pi_!(u)=(u(0),\pi\circ u).

Theorem

A map π:E→B\pi:E\to B is a Hurewicz fibration iff there exists at least one Hurewicz connection for π !\pi_!.

Proof

To see that consider the following diagram

Y →θ Cocyl(π) →pr E E σ 0↓ σ 0↓ ↓π Y×I →θ×I Cocyl(π)×I →ev B\begin{matrix} Y& \stackrel{\theta}\to & Cocyl(\pi) &\overset{pr_E}\to & E\\ \sigma_0\downarrow&&\sigma_0\downarrow&&\downarrow \pi\\ Y\times I&\stackrel{\theta\times I}\underset{}{\to}& Cocyl(\pi)\times I& \underset{ev}\to &B \end{matrix}

where pr E:Cocyl(π)→Epr_E: Cocyl(\pi)\to E is the restriction of the projection E×B I→EE\times B^I\to E to the factor EE and the map Cocyl(π)×I→BCocyl(\pi)\times I\to B is the evaluation (e,u,t)↦u(t)(e,u,t)\mapsto u(t) for (e,u)∈Cocyl(π)(e,u)\in Cocyl(\pi). The right-hand square is commutative and this square defines a homotopy lifting problem. If π\pi is a fibration this universal homotopy lifting problem has a solution, say s˜:Cocyl(p)×I→E\tilde{s}:Cocyl(p)\times I\to E. By the hom-mapping space adjunction (exponential law) this map corresponds to some map s:Cocyl(π)→𝒫(E)s:Cocyl(\pi)\to \mathcal{P}(E). One can easily check that this map is a section of π !\pi_!.

Conversely, let a Hurewicz connection ss exist, and fill the right-hand square of the diagram with diagonal s˜\tilde{s} obtained by hom-mapping space adjunction. Let the data for the general homotopy lifting problem be given: f˜:Y→E\tilde{f}:Y\to E, F:Y×I→BF:Y\times I\to B with F 0=p∘f˜:Y→EF_0 = p\circ \tilde{f}:Y\to E; let furthermore F′:Y→𝒫(B)F':Y\to \mathcal{P}(B) be the map obtained from FF by the hom-mapping space adjunction. By the universal property of the cocylinder (as a pullback), there is a unique mapping θ:Y→Cocyl(π)\theta: Y\to Cocyl(\pi) such that pr 𝒫(B)∘θ=F′:Y→𝒫(B)pr_{\mathcal{P}(B)}\circ\theta=F':Y\to \mathcal{P}(B) and pr E∘θ=f˜:Y→Epr_E\circ\theta =\tilde{f}:Y\to E. Now notice that the by composing the horizontal lines we obtain f˜\tilde{f} upstairs and FF downstairs, hence the external square is the square giving the homotopy lifting problem for this pair. The lifting is then given by s˜∘(θ×id I):Y×I→E\tilde{s}\circ (\theta\times id_I):Y\times I\to E. Simple checking finishes the proof.