nLab 2-gerbe

Definition

Nonabelian 2-gerbes

For $\mathcal{X}$ an (∞,1)-topos, a 2-gerbe $P$ in $\mathcal{X}$ is an object which is

1. 1-connective;

2. 2-truncated.

The first condition says that it is an (∞,1)-sheaf with values in 2-groupoids. The second says that $P \to *$ is an effective epimorphism and that the 0-th homotopy sheaf is the terminal sheaf. In the literature this is often stated as saying that $P$ is a) locally connected and b) locally non-empty .

Abelian 2-gerbes

For $\mathcal{X}$ an (∞,1)-topos, an abelian 2-gerbe $P$ in $\mathcal{X}$ is an object which is

1. 2-truncated;

2. 2-connected.

Related concepts

• principal bundle / torsor / associated bundle

• principal 2-bundle / gerbe / bundle gerbe

• principal 3-bundle / 2-gerbe / bundle 2-gerbe

• principal ∞-bundle / associated ∞-bundle / ∞-gerbe

References

A comprehensive discussion of nonabelian 2-gerbes is in

• Lawrence Breen, On the classification of 2-gerbes and 2-stacks , Astérisque 225 (1994) (numdam:AST_1994__225__1_0)

A more expository discussion is in

• Lawrence Breen, Notes on 1- and 2-gerbes in John Baez, Peter May (eds.) Towards Higher Categories (arXiv:math/0611317).

Abelian 2-gerbes are a special case (see ∞-gerbe) of the discussion in section 7.2.2 of

• Jacob Lurie, Higher Topos Theory

See also

• Ettore Aldrovandi, 2-Gerbes bound by complexes of gr-stacks, and cohomology Journal of Pure and Applied Algebra 212 (2008), 994–103 (pdf)