Butterflies I: morphisms of 2-group stacks

TL;DR

This paper provides a comprehensive description of weak morphisms between 2-group stacks using butterfly diagrams, establishing a bicategory structure, and characterizing non-abelian derived categories and commutativity laws.

Contribution

It introduces butterflies as a diagrammatic tool to describe weak morphisms of 2-group stacks, extending classical results and simplifying non-abelian cohomology computations.

Findings

Complete description of weak morphisms via butterflies

Bicategory of crossed modules is fibered and biequivalent to 2-group stacks

Long exact sequence in non-abelian cohomology derived

Abstract

Weak morphisms of non-abelian complexes of length 2, or crossed modules, are morphisms of the associated 2-group stacks, or gr-stacks. We present a full description of the weak morphisms in terms of diagrams we call butterflies. We give a complete description of the resulting bicategory of crossed modules, which we show is fibered and biequivalent to the 2-stack of 2-group stacks. As a consequence we obtain a complete characterization of the non-abelian derived category of complexes of length 2. Deligne's analogous theorem in the case of Picard stacks and abelian sheaves becomes an immediate corollary. Commutativity laws on 2-group stacks are also analyzed in terms of butterflies, yielding new characterizations of braided, symmetric, and Picard 2-group stacks. Furthermore, the description of a weak morphism in terms of the corresponding butterfly diagram allows us to obtain a long exact sequence in non-abelian cohomology, removing a preexisting fibration condition on the coefficients short exact sequence.