Urs:

Thanks. I am running through the above references. In addition, I have another question: How to construct the Hurewicz connection? The first part of the scratched proof did construct the connection based upon the assumption that fibration exists from the diagram. So I am thinking how to establish the condition: how to create the connection first.

]]>Hey Tom,

maybe hold it for a second. It seems you just posted this question five times to the nForum and at least two more times (here and here) to Math.Stack exchange.

I’ll reply in this one thread here and (assuming you are not a bot) would ask you to stick to this one thread here.

Now what exactly is it you are after? The nLab page *Hurewicz connection* spells out a proof of the statement that the existence of such characterizes Hurewicz fibrations. Let us know if you find this proof needs more explanation or the like.

You may or may not profit from reading it, but you should know that there is the original article available here:

- Witold Hurewicz,
*On the concept of fiber space*, Proc. Nat. Acad. Sci. USA**41**(1955) 956–961; MR0073987 (17,519e) PNAS,pdf.

I find that a decent review of these matters is in this article here, which you can find online on GoogleBooks:

- James Eells, Jr.,
*Fibring spaces of maps*, in Richard Anderson (ed.)*Symposium on infinite-dimensional topology*, 1972

I am reading the article on Hurewicz connection

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

I have two questions:

(1) How to formally construct the Hurewicz connection for $\pi_!$ ?

(2). It has scratched an idea of proof. May I know if anyone can reference a completed formal proof to me. (a paper or an article will be great !).

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