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.

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 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 :)