Butterflies II: Torsors for 2-group stacks
Ettore Aldrovandi, Behrang Noohi
We study torsors over 2-groups and their morphisms. In particular, we study the first non-abelian cohomology group with values in a 2-group, which we re-interpret in terms of gerbes bound by a crossed module, a notion originally due to Debremaeker. Butterfly diagrams encode morphisms of 2-groups and we employ them to examine the functorial behavior of non-abelian cohomology under change of coefficients. Our main result is to provide a geometric version of this map by lifting a gerbe along the ``fraction'' (weak morphism) determined by a butterfly. As a practical byproduct, we show how butterflies can be used to obtain explicit maps at the cocycle level.