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When understanding about gerbe over a manifold $X$ from Hitchin - Lectures on special Lagrangian submanifolds it is said that

We are basically in gerbe territory (for smooth manifolds) if any one of the following is being considered

  • a cohomology class in $H^3(X,\mathbb{Z})$
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In similar manner, When reading about gerbes over stacks what kind of cohomology do we come across? Can some one give me some outline of how and what cohomology comes in when studying about gerbes over stacks?

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Gerbes over stacks are classified by (∞,1)-sheaf cohomology.

Concretely, one can implement it as a derived mapping space in the model category of simplicial presheaves. See, for instance, Jardine's book Local Homotopy Theory, Springer Monographs in Mathematics, Springer-Verlag New York 2015, doi:10.1007/978-1-4939-2300-7.

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  • $\begingroup$ Stacks are also called $2$-sheaves (I do not really know what they mean when they say $2$-sheaf).. so, it is only natural to expect some relation with some kind of sheaf cohomology.. I am not at all familiar with what is $(\infty,1)$-sheaf cohomology or what is local homotopy theory... Thanks for the reference, I will check that.. I see that chapter 9 in that book discusses about stacks and gerbes.. by any chance do you have time to say roughly what it is about, may be little more than what you have already said in the answer.. $\endgroup$
    – Praphulla Koushik
    Commented Sep 3, 2018 at 4:05
  • $\begingroup$ @PraphullaKoushik: 2-sheaves are typically understood as sheaves of categories, unlike stacks, which are (typically) sheaves of groupoids. Jardine's book is an essentially an exposition of modern machinery to manipulate gerbes and similar objects. Chapter 9 treats gerbes using this machinery. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 3, 2018 at 12:26
  • $\begingroup$ Can you let me know where can I find a precise statement of how gerbe over stack are classified by $(\infty,1)$-sheaf cohomology and how? Are there any ways to learn this with out using homotopy theory and all that $\endgroup$
    – Praphulla Koushik
    Commented Feb 27, 2019 at 2:37
  • $\begingroup$ @PraphullaKoushik: Chapter 9 in Jardine's “Local Homotopy Theory” and Section 7.2.2 in Lurie's “Higher Topos Theory” both explain this. I'm not sure what “without homotopy theory” could mean here, considering that (∞,1)-sheaf cohomology is a part of homotopy theory. $\endgroup$
    – Dmitri Pavlov
    Commented Feb 27, 2019 at 16:39
  • $\begingroup$ I do not have enough background. I am trying to learn things... I will come back to this :) :) Thanks.. $\endgroup$
    – Praphulla Koushik
    Commented Feb 27, 2019 at 17:02
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I think you would profit more from studying the first five sections of these notes by Larry Breen

Notes on 1- and 2-Gerbes, in: Baez J., May J. (eds) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications 152 (2010) pp 193-235, doi:10.1007/978-1-4419-1524-5_5 arXiv:math/0611317

Abstract: The aim of these notes is to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. They are for the most part based on the author’s texts [1–4]. Our main goal is to describe the construction which associates to a gerbe or a 2-gerbe the corresponding non-abelian cohomology class.

While it is true that this is all subsumed by Lurie's work, and is also covered by Jardine's work (less general than Lurie's), it is not what you need to read when starting out, if all you want is to understand gerbes, and not $\infty$-stacks that are (higher) gerbes.

Breen's work is a synthesis of work by Debremaeker (in this thesis), Duskin (An outline of non-abelian cohomology in a topos. I. The theory of bouquets and gerbes), Ulbrich (On the correspondence between gerbes and bouquets), and some of his own (and perhaps a couple more that I forgot). Ultimately the roots of the idea go back to Giraud, but I don't recommend reading his book as his cohomology theory is the wrong one (see this answer I once wrote).

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  • $\begingroup$ This is just the ordinary theory of gerbes on manifolds, spaces, schemes, what-have-you. But then the version to work over stacks is the groupoid-equivariant analogue of the cohomology that Breen discusses. I may advertise also this joint paper arxiv.org/abs/1506.07931 that takes a calculational/example-driven approach, as well as the more conceptual arxiv.org/abs/1004.4558 $\endgroup$
    – David Roberts ♦
    Commented Mar 3, 2019 at 0:41
  • $\begingroup$ I read that notes more than once but did not feel comfortable.. Then I read Behrend and ping xu’s Notes.. I will give one more try as you said... yes, i remember you saying more than once not to read Giraud’s book... thanks for links... I will see one by one... $\endgroup$
    – Praphulla Koushik
    Commented Mar 3, 2019 at 0:43
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    $\begingroup$ They are much, much easier to digest than Lurie is... $\endgroup$
    – David Roberts ♦
    Commented Mar 3, 2019 at 1:26
  • $\begingroup$ :D I felt so :D $\endgroup$
    – Praphulla Koushik
    Commented Mar 3, 2019 at 2:14

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