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Butterflies II: Torsors for 2group stacks Ettore Aldrovandi

Summary: Butterflies II: Torsors for 2­group stacks
Ettore Aldrovandi
Department of Mathematics, Florida State University
1017 Academic Way, Tallahassee, FL 32306­4510, USA
Behrang Noohi
Department of Mathematics, King's College London
Strand, London WC2R 2LS, UK
We study torsors over 2­groups and their morphisms. In particular,
we study the first non­abelian cohomology group with values in a 2­group.
Butterfly diagrams encode morphisms of 2­groups and we employ them to
examine the functorial behavior of non­abelian cohomology under change
of coe#cients. We re­interpret the first non­abelian cohomology with co­
e#cients in a 2­group in terms of gerbes bound by a crossed module. Our
main result is to provide a geometric version of the change of coe#cients
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. In addition, we dis­


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


Collections: Mathematics