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BUTTERFLIES I: MORPHISMS OF 2-GROUP STACKS Ettore Aldrovandi & Behrang Noohi
 

Summary: BUTTERFLIES I: MORPHISMS OF 2-GROUP STACKS
by
Ettore Aldrovandi & Behrang Noohi
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.
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