The n-Category Café

I meant my question literally: is there a textbook/paper (preferably in English, but I suppose I can manage Russian too) where this is spelled out?

I meant my question literally: is there a textbook/paper (preferably in English, but I suppose I can manage Russian too) where this is spelled out?

But let me know what “this” is. My impression so far is that you are asking for the nonabelian Cech-cocycle description of gerbes. If so, this is in every standard text on gerbes, the first theorem to be proven about them.

For instance in Breen’s Notes on 1- and 2-gerbes it is section 5 cocycles and coboundaries for gerbes.

The cocycle description itself is equation (5.1.10), the classification theorem is mentioned and referenced on the bottom of page 14.

Or Ieke Moerdijk’s Introduction to the language of gerbes and stacks discusses the Cech-cocycle description of gerbes from page 16 on, and the classification theorem appears as theorem 3.1 on p. 21.

The statement is originally due to Giraud’s work Cohomologie non-abélienne.

In Brylinski’s book Loop spaces, characteristic classes and geometric quantization the cocycle description of gerbes is extracted in chapter 5.2 Sheaves of groupoids and gerbes and the classification theorem is theorem 5.2.8 on p. 200, 201.

The discussion of cocycles for gerbes is traditionally complicated by the fact that general sheaves of groups are used, instead of just a group, then there is the discussion of band, etc., all of which somewhat contributes to tending to hide a simple idea behind non-essential technical details.

Another thing that gerby tradition has is to express in linear formulas or rectangular diagrams what is intrinsically a nice geometric higher dimensional structure. The funny-looking nonabelian cocycle for a gerbe is really just a tetrahedron (the 3-simplex, since we are talking about a 2-cocycle) in $\mathbf{B} AUT(G)$.

I find this helpful, since it makes at once clear a lot of structure, such as for instance the nature of coboundaries. You can find these tetrahedra drawn in my work with John Baez, for instance the gerbe 2-cocycle tetrahedron is the title piece of John’s Namboodiri lecture slides Higher Categories, Higher Gauge Theory.

John recalls the theorem in question there on slide 10.

Finally, gerbes, in as far as they are nonabelian, are really objects associated to principal 2-bundles. The cocycle description of principal 2-bundles is more transparent, conceptually, as it is the 2-bundle that is associated by abstract nonsense to the 2-cocycle, whereas the gerbe comes from that only after some fiddling.

Accordingly, the nonabelian Cech cocycles in question here are discussed at length and in detail in the literature on 2-bundles by Toby Bartels, Igor Baković and Christoph Wockel. The relevant links are collected at [[principal 2-bundle]].

Finally, in case I am misunderstanding your question, maybe you are asking for literature that describes nonabelian Cech cocycles as $n$-functors out of Cech $n$-groupoids?

This is described in some detail for instance in the article that this thread here is about. Another discussion more in the style of the Lie-groupoid community is in section 2 of Ginot, Stiénon, $G$-gerbes, principal 2-group bundles and characteristic classes.

Please let me know if that serves to answer the question.