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First, I'm a string theory student hoping to grasp some math involved in some physics developments.

I'm hoping to read the famous Kapustin-Witten Paper "Electric-magnetic duality and the geometric Langlands program" and the related "The Yang-Mills equations over Riemann surfaces".

The following statement serves to explain the origin of my trouble: Let $G$ be a simple complex Lie group and $C$ a Riemann surface. Geometric Langlands is a set of mathematical ideas relating the category of coherent sheaves over the moduli stack of flat $G^{L}$-bundles ($G^{L}$ is the dual Langlands group of $G$) over $C$ with the category of $\mathcal{D}$-modules on the moduli stack of holomorphic $G$-bundles over $C$.

My problem: I have working knowledge of representation theory, but I'm completely ignorant about the theory of mathematical stacks and the possible strategies to begin to learn it. What specifically worries me is how much previous knowledge of $2$-categories is needed to begin.

My background: I've read Hartshorne's book on algebraic geometry in great detail, specifically the chapters on varieties, schemes, sheaf cohomology and curves. My category theory and homological algebra knowledge is exactly that needed to read and solve the problems of the aforementioned book. I'm also familiar with the identification between the topological string $B$-model branes and sheaves at the level of Sharpe's lectures.

My questions: I'm asking for your kind help to find references to initiate me on the theory of stacks given my elementary background and orientation. I'm completely unfamiliar with the literature and the pedagogical routes to begin to learn about stacks.

Any suggestion will be extremely helpful to me.

Update: I have found the paper String Orbifolds and Quotient Stacks very useful and explicit.

Ramiro Hum-Sah
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    This seems relevant. Personally I got started with Gomez's article and Olsson's book, but nowadays I treat the stacks project as the canonical reference. – Will Chen Sep 22 '20 at 22:03
  • Thanks for your kind comment Will Chen. – Ramiro Hum-Sah Sep 22 '20 at 22:09
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    You don’t need to know what a stack is to understand those papers. This will probably offend all the mathematicians here, but in this context, you can just think of them as quotient spaces where you keep track of what’s going where the action isn’t free. – Aaron Bergman Sep 22 '20 at 23:53
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    But if you really want to know, I’d suggest Vistoli’s notes: https://arxiv.org/abs/math/0412512 – Aaron Bergman Sep 22 '20 at 23:54
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    And, while I’m here, you might find Witten’s comments in section 6 of https://arxiv.org/abs/0802.0999 to understand the role of stacks in the correspondence from the physics side. – Aaron Bergman Sep 23 '20 at 00:11
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    Aaron Bergman, I sincerely appreciate the comments and references you share with me. I agree with you, It may not be crucial for a physicist to understand the precise definitions a properties of stacks to understand those papers. The problem is that I'm finding really finding terribly difficult to begin with the papers, and I suspect that the reason is probably my lack of intuition about the underlying concepts. I was not aware of the Witten's paper you share and looks extremely helpful for my purposes. Vistoli's book also is probably what I was waiting for. My sincere gratitude to you. – Ramiro Hum-Sah Sep 23 '20 at 04:35
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    @AaronBergman I can't speak for all mathematicians, but I'm not offended, since that's pretty much how I think of a stack. – Donu Arapura Sep 23 '20 at 12:57
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    @AaronBergman: Not only this description cannot offend any mathematician, but in fact, this a perfectly rigorous definition of a stack. The only two subtleties is that the spaces can be more general than manifolds (e.g., they can be infinite-dimensional), which is formally expressed using presheaves of sets, and the group that acts on a space can itself vary as you move around the space, which is formally expressed using the language of groupoids. – Dmitri Pavlov Sep 23 '20 at 20:19

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I still think that Vistoli's notes on descent, fibred categories, and stacks are the canonical non-infinity-categorical introduction:

  • Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory, arXiv:math/0412512.
David Roberts
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Harry Gindi
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