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The basic theorem of Giraud’s theory of gerbes says that in the sense of gerbe (as a stack), gerbes -gerbes are classified by nonabelian cohomology (usually realized as nonabelian Čech cohomology, see there for more details) with coefficients in the strict 2-group .
For instance in
this 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.
discusses the Čech-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
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 .
This may be helpful, since it makes at once clear a lot of structure, such as for instance the nature of coboundaries. One can find these tetrahedra drawn in joint work John Baez and Urs Schreiber, for instance the gerbe 2-cocycle tetrahedron is the title piece of
This recalls the theorem in question 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 (see gerbe (general idea)).
Accordingly, the nonabelian Čech cocycles in question here are discussed at length and in detail in the literature on principal 2-bundles by Toby Bartels, Igor Baković and Christoph Wockel.
Nonabelian Čech cocycles as therefore are naturally expressed -functors out of Čech -groupoids.
This is described in some detail for instance in
Another discussion more in the style of the Lie groupoid community is in section 2 of