References on Gerbes I am looking for some references related to gerbes and their differential geometry. Almost every article I have seen that is related to gerbes  there is a common reference that is Giraud's book Cohomologie non-abelienne. For me, it is not readable as I can not read french.
Only references I am familiar with are


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*https://arxiv.org/abs/math/0212266 Introduction to the language of stacks and gerbes

*https://arxiv.org/abs/math/0106083 Differential Geometry of Gerbes

*https://arxiv.org/abs/math/0611317 Notes on 1- and 2-gerbes


Out of these, only first article is more or less readable. The other two by Lawrence Breen are really not readable for me. 
Some excerpt of comments answering such a request were


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*"The standard reference is Giraud's book Cohomologie non-abelienne. This book is unreadable in the strongest possible meaning of the word unreadable." 

*"I find most of the contemporary articles in this area, which are often nonsystematic in terminology and notation, plus wave hands and use jargon on most issues, much less readable than Giraud's book."

*"See Giraud's book on nonabelian cohomology."

*"I don't think telling someone to see a dense 470 page book in French on non-abelian cohomology is a helpful comment."


I am getting demotivated and irritated by lack of notes on gerbes and even in Mathoverflow there are not so much to see. Is this out of fashion now? Are there any one else who read/work on these? Iam not looking for something in Physics perspective.
 A: My personal impression is that at least on the level of foundational theory, Higher Topos Theory of Lurie is a good source. I guess this also explain the hard time you feel finding references: Gerbes seat very naturally in the context of sheaves of spaces (in this language this is just a connected sheaf of 1-types!), and I guess that this language has not fully penetrated into standard algebraic geometry texts yet, or any subject which is not modern algebraic topology, actually. However, the situation do get better with time, and I think that gerbes will appear more in texts soon (in particular, they are not out of fashion, just sort of get revised by $\infty$-category theory). For example, I personally almost finished a paper with a whole section for gerbes-based obstruction theory in etale homotopy, so I know there's at least one text on the subject that will be on the archive soon :-)
A: The book of Giraud is a fundamental reference on the subject, but you have to be used to the language of Grothendieck. A reference more accessible, for example for a differential geometer is the chapter 5 of the book of Brylinski which deals only with commutative gerbes.
J.L Brylinski  Loop Spaces, Characteristic Classes and Geometric Quantization.
A: Urs Schreiber has written a lot
on gerbes and their applications to physics:
https://ncatlab.org/nlab/show/Urs+Schreiber
See, for instance, the expository works
“Differential cohomology in a cohesive ∞-topos”
and “Higher prequantum geometry”.
Most of his published papers (https://arxiv.org/find/math/1/au:+Schreiber_U/0/1/0/all/0/1) use gerbes in some way.
Almost all of Konrad Waldorf's papers also involve gerbes in some way,
and some of them may be more accessible to a beginner,
see, e.g., his survey with Christoph Schweigert Gerbes and Lie Groups.
A: The following reference might be helpful for you:
Hitchin, Lectures on Special Lagrangian submanifolds, $\S1$.
