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Let $\{X_1 \rightrightarrows X_0\}$ be a smooth groupoid object in the category of affine schemes ($X_0 \to X_1$, $X_1 \to X_1$ and $X_1 \times_{X_0} X_1 \to X_1$ also belong to the datum). Equivalently, we have a smooth commutative Hopf algebroid. Let $X$ be the associated algebraic stack. Thus, we have a presentation $X_0 \twoheadrightarrow X$ (which is smooth, surjective and affine) with $X_0 \times_X X_0 = X_1$, and $X$ is geometric. Every geometric stack arises this way.

Question. If $Y$ is a scheme, how can we describe $X(Y)$ explicitly in terms of $Y$ and the $X_i$?

This should be somewhere in the literature? For example, if $X_0 = \mathrm{Spec}(\mathbb{Z})$, then $X_1$ is a group scheme, $X$ is its classifying stack, so that $X(Y)$ consists of $X_1$-torsors on $Y$, right?

In general, I expect that the answer will be some kind of "torsors under the groupoid". A morphism $Y \to X$ may be pulled back to $Y_0 \to X_0$, where $Y_0 \to Y$ is smooth, surjective and affine. Conversely, if $Y_0 \to Y$ is smooth, surjective and affine, and $Y_0 \to X_0$ is a morphism, then an extension to $Y \to X$ corresponds to a descent datum of $Y_0 \to X_0 \to X$ with respect to the fpqc cover $Y_0 \to Y$. But this still involves $X$. How can we get rid of $X$ in the description of morphisms $Y \to X$?

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2 Answers 2

Niels already links to references that answer the question, but I'll briefly summarize them.

If $X$ is a stack presented by a groupoid $X_1 \rightrightarrows X_0$ then a map from a scheme $S$ into $X$ induces a groupoid presentation $S \mathop{\times}_X X_1 \rightrightarrows S \mathop{\times}_X X_0$ of $S$. Conversely, suppose that we have a groupoid presentation $S_1 \rightrightarrows S_0$ of $S$ and compatible maps $S_i \rightarrow X_i$ (such that the two maps $S_1 \rightarrow S_0$ are induced by pullback from the two maps $X_1 \rightarrow X_0$) then there is an induced map $S \rightarrow X$.

The associated stack of the groupoid $X_\bullet$ is therefore the fibered category $X$ for which $X(S)$ is the category of groupoid presentations of $S$ with a map to $X_\bullet$ satisfying the above parenthetical condition.

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Thanks. Going through the construction of stackification in the book by Laumon and Moret-Bailly, it follows that $X(S)$ consists of covers $S' \to S$ and a morphism of groupoid schemes $(S' \times_S S' \rightrightarrows S') \to (X_1 \rightrightarrows X_0)$. No cartesian condition. What am I missing? Also, is there some more global description? If $X_0$ is the base scheme, we get the local description of $X_1$-torsors via cocycles $S' \times_S S' \to X_1$. But how do we get the global description as special morphisms over $S$ (with an action and local triviality)? –  Martin Brandenburg Oct 17 '13 at 12:54
    
The reason for the cartesian condition is to make the morphisms come out right. If you just look at maps from a groupoid presentation to $X_\bullet$ you can get two representations of the same map to $X$ coming from different presentations of $S$, without an isomorphism between them. You can get an example of this with $X_\bullet = [ (\text{point}) \rightrightarrows (\text{point}) ]$. –  Jonathan Wise Oct 17 '13 at 23:23
    
Does this mean that the construction of the associated stack in LMB is incomplete? Sorry, I'm confused. –  Martin Brandenburg Oct 18 '13 at 13:16
    
The definition in LMB also works, but you have to take more care in the definition of morphisms in the stack. (What is a morphism between two maps $S \rightarrow X$ that are given by different presentations of $S$?) –  Jonathan Wise Oct 18 '13 at 20:16

This is called "Stackification via torsors" in the book by Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi, Lothar Göttsche and Andrew Kresch. See Andrew Kresch's homepage:

http://www.math.uzh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1

more specifically chapter 4, \S 4.

http://www.math.uzh.ch/index.php?file&key1=5161

For sure, this is done in the stacks project

http://stacks.math.columbia.edu/

but I couldn't find the exact reference (probably the keyword is: quotient stack).

For a more historical point of view, see also the related question Algebraic stacks as (etale) groupoid alg.spaces/schemes .

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Thank you! It's a pity that the book project won't be continued, as far as I know. –  Martin Brandenburg Oct 17 '13 at 12:11

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