Continuing in my attempts to understand bits and pieces of Borceux and Janelidze's *Galois Theories*, I've just realized that I don't have any geometric intuition for the most convenient characterization of Galois descent for [covering morphisms](https://mathoverflow.net/questions/239994/categorification-of-covering-morphisms).


The general setting is a complete category of the form $\mathsf{Fam}(\mathsf A)$, or alternatively, a complete category with a functorial choice of connected components for each object. The geometric intuition (I hope) comes from looking at locally connected spaces and I want it to understand Galois field extensions geometrically.


The idea is that an arrow $p:E\rightarrow B$ is a morphism of Galois descent if and only if the following two conditions hold (see *Galois Theories* between Prop 6.6.6 and 6.6.7).


* $p$ is an effective descent morhpsim
* $p$ is trivialized by itself, i.e $p^\ast p$ is a trivial covering morphism (trivial fiber bundle with discrete fiber).


Of course the first condition makes sense, but the second one? What's the idea there? Why would I want $p$ to trivialize itself?


For fields all arrows are effective descent morphisms so being of Galois descent amounts to asking for the square below to be a pullback, where the top left corner is a finite coproduct of copies of $\operatorname{Spec} E$:
$$\require{AMScd} \begin{CD} \operatorname{Spec} E^{\amalg{n}} @>>> \operatorname{Spec}E \\ @VVV @VV{p}V\\ \operatorname{Spec}E @>>{p}> \operatorname{Spec} B \end{CD}$$
i.e as usual an $E$-algebra isomorphism $E\otimes\_BE\cong E^n$.


**Added.** I dug out [this](http://schauenburg.perso.math.cnrs.fr/papers/hgbgt.pdf) paper by Schauenburg on Hopf-Galois and Bi-Galois extensions, and in the long paragraph above Lemma 2.4.2 he says a lot of seemingly nice things which are way over my head. The words 'principal bundle' appear often enough to warrant some hope. In particular, it's written:



> 
> This is the algebro-geometric version of a principal fiber
>  bundle with structure group $G$, or a $G$-torsor.
> 
> 
> 


Now I do **not** know almost any algebraic geometry beyond the very basics, and I am looking for topological/geometric intuition of the original condition I asked about, so if this addition is irrelevant to the question, just ignore it :)



Geometric intuition for the condition of Galois descent
Let $G \to E \to B$ be a principal bundle. It's classified by a map $f : B \to BG$ in the sense that $E$ is the homotopy fiber of this map. This means that $E$ has a certain universal property: namely, it is the universal map to $B$ such that the composition with $f$ is equipped with a nullhomotopy. This says precisely that $E$ is the universal space over $B$ equipped with a trivialization of the pullback of the principal bundle to $E$: in other words, $E \to B$ is the universal thing that trivializes itself. 


The relationship to Galois extensions is that $K \to L$ is a finite Galois extension with Galois group $G$ iff $\text{Spec } L \to \text{Spec } K$ is a principal $G$-bundle in a suitable sense ("$G$-torsor"). 


