On the existence of some kind of "universal fibre bundle" While attending an introductory course on the theory of (smooth) fibre bundles, an example I was given of (principal) bundles was that of topological coverings of a space with structure group the galois group of the covering. In this case, when the base space is nice enough, we get an universal covering space in the sense that any other covering space is covered by this universal cover.
This motivated me to ask the following question: can this be generalized? I mean, given a base space, can I construct some kind of "universal fibre bundle" which is universal in the sense that it is a fibre bundle over the base space with morphisms to every other fibre bundle over the base?
Initially, I considered the set(?) of all fibre bundles $E$ over $B$ and tried to give it an order (I wanted to apply Zorn's lemma) $E'\geq E$ iff $E'\overset{\pi}{\rightarrow}E$ is a fibre bundle and $\pi$ is a morphism of fibre bundles over $B$ but I didn't manage to prove that $E'\geq E$ and $E\geq E'$ implies $E\cong E'$.
Then I tried to consider, given $E$ and $E'$ over $B$, $E\times_B E'$ over $B$ but this is not necessarily a fibre bundle over $E$ and $E'$ (am I wrong?), so I got stuck again.
Any suggestion on how (and if) this can be done? Thanks in advance.
 A: You can get close. I claim that there is a fibration over $B$ (somewhat weaker than a fiber bundle, but a very useful notion in homotopy theory) which maps to any other fibration over $B$ up to homotopy. 
To make the statement cleaner let $B$ be a path-connected space and fix a basepoint $b$ of it. There is an extremely interesting distinguished fibration over $B$ called the path space fibration
$$\Omega B \to PB \to B$$
(here when I write $F \to E \to B$ I mean that $E$ is a fiber bundle over $B$ with fiber $F$). Here $PB$ is the space of paths $p : [0, 1] \to B$ such that $p(0) = b$, and the bundle map $\pi : PB \to B$ is the evaluation map $p \mapsto p(1)$. The fiber of the bundle map $\pi$ over $b$ is the based loop space of paths $p : [0, 1] \to B$ such that $p(0) = p(1) = b$.
If $B$ is the classifying space $BG \cong K(G, 1)$ of a discrete group $G$, then the path space fibration is more or less the universal cover of $B$. In general you can think of the process of taking the universal cover as killing $\pi_1$ but retaining all the higher homotopy groups; similarly there are higher analogues of the universal cover which kill the first $n$ homotopy groups but retain all of the higher homotopy groups. They organize themselves into a tower called the Whitehead tower, and at the very top of the Whitehead tower is the path space fibration, which has the effect of killing all of the homotopy groups of $B$. It is the "most universal cover" of $B$ in that sense. 
In particular, $PB$ is contractible. That makes it very easy to construct maps from the path space fibration to any other fibration $F \to E \to B$, up to homotopy; if $\pi : E \to B$ is the bundle map, pick any point in the preimage $\pi^{-1}(b)$ and map $PB \cong \text{pt}$ into it. More explicitly, given a path $p : [0, 1] \to B$ in $PB$, lift it up to a path $\tilde{p}$ in $E$ starting at the point you picked in $\pi^{-1}(b)$ and send $p$ to $\tilde{p}(1)$. Since paths don't lift uniquely for fibrations this isn't well-defined, but it turns out to be well-defined up to homotopy. ($E$ might be empty, but that's fine too.) 
