nLab gerbe (in nonabelian cohomology)

This is a sub-entry of

• gerbe .

Idea

Recall that principal bundles are classified by nonabelian cohomology in degree 1 with coefficients in a group $G$.

A central motivation for the introduction of the notion of gerbe was to find an analog of this statement for nonabelian cohomology in degree 2.

The basic theorem of Giraud’s theory of gerbes says that in the sense of gerbe (as a stack), gerbes $G$-gerbes are classified by nonabelian cohomology (usually realized as nonabelian ?ech cohomology, see there for more details) with coefficients in the strict 2-group $AUT(G) = Aut_{Grpd}(\mathbf{B}G)$.

References

For instance in

• Breen Notes on 1- and 2-gerbes (pdf)

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.

Or

• Ieke Moerdijk Introduction to the language of gerbes and stacks (pdf)

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

• Giraud, Cohomologie non-abélienne

In

• Brylinski 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 $\mathbf{B} AUT(G)$.

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

• John Baez, Namboodiri lecture slides: Higher Categories, Higher Gauge Theory, III (pdf)

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 $n$-functors out of Čech $n$-groupoids.

This is described in some detail for instance in

• Urs Schreiber, Konrad Waldorf, Connections on non-abelian gerbes and their cohomology (arXiv)

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 (pdf)

Giraud’s gerbes as being the objects classified by nonabelian cohomology are recovered from general principles:

• Alexander Campbell, A higher categorical approach to Giraud’s non-abelian cohomology, PhD thesis, Macquarie University (2016) [hdl:1959.14/1261186]