In the very last page of Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms, the authors relate the theory of fiber bundles (and covering spaces in particular) to descent and the lifting properties defining fibrations (in topology).

Below is an excerpt from the final two pages of the paper. I don't understand section (e). In particular, I don't understand/see the connected described in the first sentence, and I also don't see what the setup for descent is.

Can someone spell things out for me? I would very much like to understand these connections but can't connect the dots myself. Exactly how is descent transforming a bunch of isomorphisms between fibers to a bundle isomorphism?

(d) Since discrete fibrations of equivalence relations play an important role in this chapter, we should say a few words about the corresponding topological notion. There are various kinds of fibrations studied in algebraic topology; they are defined via various kinds of lifting properties which, roughly speaking, "help to compare fibres". As we already mentioned, the existence of a path from $b$ to $b^\prime$ in $B$ implies the existence of a homeomorphism $\alpha^{-1}(b)=F_b\cong F_{b^\prime}$ whenever $(A,\alpha)$ is a locally trivial bundle over $B$ [...]. On the other hand, if we start with an arbitrary bundle $(A,\alpha)$, the existence of such homeomorphisms might help to prove (see (e) below) that it is locally trivial. Therefore it is good to have a geometrical method of constructing homeomorphisms $\alpha^{-1}(b)\cong \alpha^{-1}(b^\prime)$. Now, an obvious way to try is to fix a path $f$ from $b$ to $b^\prime$, and then to define the image in $\alpha^{-1}(b^\prime)$ of a point $a$ in $\alpha^{-1}(b)$ as the end of a path $g$ that is a lifting of $f$ and whose starting point is $a$.

However, this procedure can be successful only under further strong geometrical conditions on $a$, and clearly the whole story would be much simpler if $\alpha$ had the unique path lifting property, which says that the $g$ above always does exist and is uniquely determined by $f$ and $\alpha$. If $\alpha$ is a product projection $B\times F \to B$, then the uniqueness of path liftings simply means that every continuous map $f : [0,1] \to B$ is constant, and since the space studied in algebraic topology usually have open path-connected components, this essentially means that $F$ is discrete. That is constant, and since the spaces studied in algebraic topology usually have open path-connected components, this essentially means that F is discrete. That is, if we allow ourselves to require the unique path lifting property, then our theory will be applicable only to covering spaces, not to general locally trivial bundles. [...].

(e) As we have seen above, the connection between locally trivial fibre
bundles and fibrations defined via path-lifting properties (with additional conditions) is based on the problem of constructing an isomorphism $(A,\alpha)\cong(B\times F,\text{first projection})$ for some $F$ out of a collection of isomorphisms $\alpha^{-1}(b)\cong \alpha^{-1}(b^\prime)$ constructed for all pairs $b,b^\prime$ of elements in $B$, and/or on doing the same for each $E_b$ instead of $B$ (see above). This is how descent theory arrives once more, this time with $B\to \bf 1$ or $E_b\to \bf 1$ playing the role of $p:E\to B$. Since $B\to \bf 1$ and $E_b\to \bf 1$ as split epimorphisms [...] are effective descent morphisms [...], descent theory tells us that the conditions on the chosen collection of isomorphisms to be checked are [reference to cocycle condition].


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Consider a bundle $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$. Observe that a trivialization is furnished precisely by a space $F\to \bf 1$ whose pullback along $B\to \bf 1$ is isomorphic to $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$. Indeed the trivialization is the isomorphism induced by the universal property of the pullback.

On the other hand, descent data for $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$ along $B\to \bf 1$ seems to be a coherent family of isomorphisms between fibers.


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