Summary: BUTTERFLIES I: MORPHISMS OF 2­GROUP STACKS
by
Ettore Aldrovandi & Behrang Noohi
Abstract. --- Weak morphisms of non­abelian complexes of length 2, or crossed mod­
ules, are morphisms of the associated 2­group stacks, or gr­stacks. 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 2­stack of 2­group stacks. As a consequence we
obtain a complete characterization of the non­abelian 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 2­group stacks are
also analyzed in terms of butterflies, yielding new characterizations of braided, sym­
metric, and Picard 2­group stacks. Furthermore, the description of a weak morphism
in terms of the corresponding butterfly diagram allows us to obtain a long exact se­
quence in non­abelian cohomology, removing a preexisting fibration condition on the
coe#cients short exact sequence.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1. General beginning remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2. The content of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Source: Aldrovandi, Ettore - Department of Mathematics, Florida State University

Collections: Mathematics