Let $\mathsf C$ be a category and $G$ an internal group. Suppose $\mathsf C$ is finitely complete, so that $\pi_2:G\times B\to B$ is an internal group in $\mathsf C_{/B}$ for every $B$. A $G$-bundle over $B$ is a $\pi_2$-action in $\mathsf C_{/B}$.
Say a $G$-bundle is a trivial $G$-torsor over $B$ if it's isomorphic to the multiplicative action of $\pi_2$ on itself. A $G$-bundle $\alpha:A\to B$ over $B$ is a trivial $G$-torsor over $B$ iff it has a global section and the canonical map $G\times A\to A\times _B A$ is an isomorphism of the bundles $\pi_2:G\times A\to A$ and $\alpha^\ast\alpha$.
Proposition 4.43 of Vistoli's notes on descent gives an analogous characterization for locally triviail $G$-torsors where "locally" is expressed in terms of a subcanonical site structure on the underlying category, using lemma 4.44.
I want to define "locally" in terms of descent: say a $G$-bundle $\alpha:A\to B$ is a $G$-torsor if it admits an effective descent morphism $p$ pulling it back to a trivial $G$-torsor. I then expect the following conditions to be equivalent for a $G$-bundle to be a $G$-torsor.
There's an effective descent morphism $p$ such that:
- $p^\ast \alpha $ admits a global section;
- the canonical map $G\times A\to A\times _B A$ is an isomorphism of the bundles $\pi_2:G\times A\to A$ and $\alpha^\ast\alpha$.
That is, a locally trivial torsor still trivializes itself. Are the conditions indeed equivalent? How to prove the equivalence?
