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