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This is a well known and well understood problem when the base field is $\mathbb C$. It was first studied by Chevalley and Weil (almost a century ago !) who were interested in modular curves (what els …

The answer is yes (with mild hypothesis on the space). Moreover the topological situation is simpler, and this was very likely Grothendieck's inspiration. To see this you need two facts. First take …

You can get a lot of examples by dimension shifting. Namely, consider any exact sequence of groups $$1\to K\to G \to H\to 1 \; .$$ Fix a $H$-torsor $T$. The stack $\mathcal G_T$ of liftings of the str …

A typical example from deformation theory : fix $i:X_0\to X$ of first order thickening defined by a square zero ideal $\mathcal I$, and let $\mathcal E_0$ be a locally free sheaf of finite rank on $X …

Parabolic bundles were introduced in the 70's by Mehta and Seshadri in the set up of a Riemann surface with cusps. They were trying to generalize the Narasimhan-Seshadri correspondence on a compact Ri …

In the french school, un torseur sert à tordre, a torsor is used to twist. More precisely, let $\eta$ be an object in a topos, and $G=\operatorname{Aut}(\eta)$. If $\nu$ is a form of $\eta$ (another …