Let $(C,\tau,\mathbb P)$ be a geometric context, as defined by Toen and Vezzosi. Let $(X_1\rightrightarrows X_0)$ be a groupoid object in $C$ such that the source and target morphisms are in $\mathbb P$. Is it true that every $(X_1\rightrightarrows X_0)$-torsor is representable? I found proofs for many different geometric context but it seems that there are no references for a general proof.
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$\begingroup$ Do you have some examples? And representable in which (2-)category? $\endgroup$– David Roberts ♦Commented Jun 5, 2014 at 2:28
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$\begingroup$ @DavidRoberts: For example in the following geometric context: $(C^{\infty} manifold, \mbox{open immersions},\mbox{submersions})$ all torsors are representable. Maybe I should be more precise: for me a $(X_1\rightrightarrows X_0)$-torsor over $X\in C$ is just a sheaf $P$ on $C$ with an epimorphism to $X$, a morphism to $X_0$ and an action of $X_1$ with some properties.With this definition the category of $(X_1\rightrightarrows X_0)$-torsor is always a stack. Now, a priori $P$ is not always representable in $C$, but I hope that this is true in a geometric context. $\endgroup$– D. StefaniCommented Jun 5, 2014 at 8:18
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$\begingroup$ I should have asked for references with examples, because I really want to know what the definition of representable is here. From your comment, you are saying a torsor is a principal $y(X_1) \rightrightarrows y(X_0)$ bundle over $y(X)$, where $y$ is the Yoneda embedding. And so I presume 'representable' means by an object of $C$. Correct? $\endgroup$– David Roberts ♦Commented Jun 5, 2014 at 10:20
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$\begingroup$ I think this is a fantastic question, by the way. $\endgroup$– David Roberts ♦Commented Jun 5, 2014 at 10:21
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$\begingroup$ Could you email me some references for these proofs? My email can be found at ncatlab.org/nlab/show/David+Roberts $\endgroup$– David Roberts ♦Commented Jun 5, 2014 at 23:13
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