# Algebraic stacks as (etale) groupoid alg.spaces/schemes

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 alg. 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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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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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. –  David Roberts Aug 30 '11 at 9:17