# 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

Out of these, only first article is more or less readable. The other two by Lawrence Breen are really not readable for me.

• "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.

• @NeilStrickland Sir, I am asking for current research trends and notes on gerbes. How is that off topic? – Praphulla Koushik Mar 31 '18 at 16:39
• Whilst perhaps not the best written question, I personally share many of the same frustrations as the OP and would be interested in seeing some experts thoughts on good references for gerbes. – Daniel Loughran Mar 31 '18 at 17:08
• @cello If that is your actual question, then I suggest you remove all the earlier quotes and stats, which just give the impression that you are complaining about something, rather than asking about something – Yemon Choi Mar 31 '18 at 17:29
• @YemonChoi I am just sharing my search. I believe it gives better idea of what is happening – Praphulla Koushik Mar 31 '18 at 17:37
• @cello I think this is a completely valid question. It would also help if you had a concrete problem or goal, as someone might know of a specific paper where this is addressed. I know a little bit about gerbes in algebraic geometry. For those you definitely don't need any $\infty$-categories. In any case, I think it would greatly help to be familiar with principal bundles and with stacks (for example at the level of Vistoli's notes on descent). – Yosemite Sam Apr 1 '18 at 2:39

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.

• YHank you, I knew this book but did not spend time on reading this. As you said, I will roughly go through chapter five and will respond.. – Praphulla Koushik Apr 1 '18 at 2:28
• To read Giraud's book, in addition to familiarity with the language of Grothendieck, you also need some familiarity with the language of France. – zzz Apr 1 '18 at 4:00
• @bianchira I agree. – Praphulla Koushik Apr 1 '18 at 7:17

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 :-)

• Thanks for the reference. I do not know if I can just start reading that book. I do not know what to respond for this. Looking forward for your paper. – Praphulla Koushik Apr 1 '18 at 2:27
• A gerbe is a connected sheaf of 1-types (otherwise you get only a stack) – Denis Nardin Apr 3 '18 at 13:44
• right, I forgot the connected thing, thanks. – S. carmeli Apr 4 '18 at 14:25

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.

• I have mentioned that I do not feel comfortable with “Physics”... can we not study gerbes just for their own sake?? Is it not interesting,?? What is your view on that? – Praphulla Koushik Apr 1 '18 at 17:02
• @cello: Much of Urs Schreiber's work studies gerbes for their own sake. As I said, gerbes and their applications to physics, not just applications of gerbes to physics. – Dmitri Pavlov Apr 1 '18 at 17:20

The following reference might be helpful for you: Hitchin, Lectures on Special Lagrangian submanifolds, $\S1$.

• Yes, even my advisor said the same thing. +1. – Praphulla Koushik Apr 5 '18 at 15:42
• I have read that first part. Is there any way I can ask you doubts separately (by email)? I have to explain so much here to ask a question and I am not sure if people here are ready to read very long question. – Praphulla Koushik Apr 8 '18 at 14:44