nLab
2-gerbe

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

Nonabelian 2-gerbes

For 𝒳\mathcal{X} an (∞,1)-topos, a 2-gerbe PP in 𝒳\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*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 PP is a) locally connected and b) locally non-empty .

Abelian 2-gerbes

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

  1. 2-truncated;

  2. 2-connected.

References

A comprehensive discussion of nonabelian 2-gerbes is in

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

A more expository discussion is in

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

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)

Revised on June 29, 2012 14:04:01 by Urs Schreiber (89.204.138.61)