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Assume given an algebraic stack(*) $\mathcal{X}$ with presentation $X_0 \to \mathcal{X}$, and the corresponding groupoid $X = (X_0\times_\mathcal{X} X_0 \rightrightarrows X_0)$ in algebraic spaces (or schemes, given the right sort of algebraic stack(*)).

(*) Allow me to be a little loose with the definition. Take your favourite sort of algebraic stack if you like.

Who first proved that $\mathcal{X}$ is equivalent to the stack of $X$-torsors?

I'm interested not just for a reference, but in the technique, at least from a category-theoretic point of view.

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The notion of "torsor under a groupoid in a topos" already appears in

Breen, Lawrence Tannakian categories. Motives (Seattle, WA, 1991), 337–376, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.

precisely p.354. The related idea of stackification via torsors is also included. The result you mention seems to be only shown there for gerbes, though. I have no idea if this is the first occurrence of the idea, probably not.

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I have heard this result attributed to Kai Behrend, who apparently came up with it while writing his part of the now defunct many-authored stack book. However it is certainly possible that someone else had already noticed it.

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  • $\begingroup$ Really, that recently? I thought this was an old result. Certainly I hadn't seen it in the drafts of said book, or at least I hadn't read it in enough detail to see it. $\endgroup$
    – David Roberts ♦
    Commented Aug 30, 2011 at 9:17

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