Homotopy classification of gerbes
Algebraic Topology
2007-05-23 v1 Algebraic Geometry
Abstract
Gerbes are locally connected presheaves of groupoids. They are classified up to local weak equivalence by path components in a 2-cocycle category taking values in all sheaves of groups, their isomorphisms and homotopies. If F is a full presheaf of sheaves of groups, isomorphisms and homotopies, then [*,BF] is isomorphic to equivalence classes of gerbes locally equivalent to groups appearing in F. Giraud's non-abelian cohomology object of equivalence classes of gerbes with band L is isomorphic to morphisms in the homotopy category from the point * to the homotopy fibre over L for a map defined on BF and taking values in the classifying space for the stack completion of the fundamental groupoid of F.
Cite
@article{arxiv.math/0605200,
title = {Homotopy classification of gerbes},
author = {J. F. Jardine},
journal= {arXiv preprint arXiv:math/0605200},
year = {2007}
}