Does anyone have any experience teaching stacks over the category of manifolds to students whose background is, say, a semester-long course on manifolds? Does anyone know of any publicly available notes on the subject, preferably in English? [My French is limited to the knowledge of the alphabet :). I can read Russian.]
I am aware of a paper by Behrend and Xu, Metzler's paper in the arxiv, and notes by Heinloth. Hepworth has a nice exposition of vector fields on stacks, but his papers are rather terse. Vistoli's notes on descent are quite nice, but are clearly aimed at algebraic geometers. And there differences between the categories of manifolds and schemes — fiber products of manifolds are badly behaved, for one thing.
The challenges in teachign such a course seem many. For one thing I don't know how to talk about stacks without getting into 2-category theory. And most differential geometers don't know much of 1-category theory. But I don't want to start with a crash course on category theory.
Best Answer
I had a good experience with Heinloth's notes. I tried to explain the two-categorical stuff in the example of the stack of principal $G$-bundles. For example, a nice way to understand 2-pull-backs is to calculate $G\cong *\times_{BG}*$ explicitly. And of course, orbifolds and gerbes, e.g. of $Spin^{c}$-reductions of a $Spin^{c}$-principal bundle a provide examples accessible to differential geometers.