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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.

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One suggestion would be to limit yourself to orbifolds; then there are more resources available and teaching this to geometric topology students is not too difficult (in my experience). –  Misha Jan 10 '13 at 18:03
    
Have looked at Weimin Chen's paper "A homotopy theory of orbispaces"? front.math.ucdavis.edu/0102.5020 –  Liviu Nicolaescu Jan 10 '13 at 18:57
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@Misha Thank you for the suggestion. But presenting orbifolds as topological spaces with extra structure kind of defeats the purpose of explaining how to think of them as stacks, doesn't it? –  Eugene Lerman Jan 10 '13 at 19:23
    
@Liviu I had, when it first came out. I don't understand it. –  Eugene Lerman Jan 10 '13 at 19:25
    
Hi Eugene. I have given some informal lectures about differentiable stacks to differential geometers a couple of times. I might be able to find some handwritten notes of mine from this. I also spend a good 100 pages or so giving a careful introduction to them in my thesis. (You can find a copy on my webpage: people.mpim-bonn.mpg.de/carchedi) –  David Carchedi Jan 25 '13 at 0:50
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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.

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