## Basic questions about stacks 2

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I have again three basic questions about stacks.

1) When we consider categories fibered in groupoids, do we always mean small or essentially small groupoids? Especially I want to know if algebraic geometers always impose this condition when they talk about categories fibered in groupoids, especially stacks.

2) In the proof of Artin's criterion for algebraic spaces/stacks $X/S$ for every point $p \in X$ of finite type over $S$ a "local approximation" $X_p$ is constructed. Then $X = \coprod_p X_p$ does the job. But in order to show that this is actually a scheme in the given universe, we need that the points of finite type constitute a set. Perhaps I'm overlooking something trivial here, but I cannot see how we can use Artin's criterions to deduce this.

3) What is the current status of the book "Algebraic Stacks" by Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi, Lothar Göttsche und Andrew Kresch? I would love to read it as soon as it is completed.

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Regarding 1), the definition of a fibred category does not assume that the fibres are small, but in geometric examples they are (or at least are essentially small). For example, would you consider the groupoid of G-bundles on a fixed topological space to be a set? This groupoid is certainly essentially small once you have a classifying space BG. – David Roberts Jun 23 2011 at 12:42
Me again. The category GBund(X) of principal G-bundles on a fixed object X in a site (S,J) is equivalent to the hom-category Gpd_ana(X,_B_G) in the bicategory of internal groupoids, anafunctors and transformations (without loss of generality, assume S is a superextensive site - see nLab for definition - which is true in all geometric situations). Here _B_G is the groupoid with one object and morphisms G, and we consider X as a groupoid with only trivial morphisms. GBund(X) is essentially small when the axiom 'WISC' holds for J, again see nLab. But 'morally', if not actually, the 2-category.... – David Roberts Jun 23 2011 at 23:11
...of stacks in groupoids on (S,J) is equivalent to the bicategory of groupoids, anafunctors and transformations in S. Roughly speaking, all fibres of a stack are representable in this bicategory, and so given WISC, one knows that the fibres are essentially small. Oh, I should mention that really this is only expected to hold for geometric stacks (presentable by some space/scheme/what-have-you), but I guess that that is the case you are most interested in. – David Roberts Jun 23 2011 at 23:16

## 1 Answer

Regarding 3), Andrew Kresch just told me that they gave up on the project.

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5
Oh no :-( – Martin Brandenburg Jun 23 2011 at 14:14

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There is always the stack project which actively developed, available at math.columbia.edu/algebraic_geometry/stacks-git – Andrei Halanay Jun 23 2011 at 21:29
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