Butterflies II: Torsors for 2-group stacks

TY - JOUR

T1 - Butterflies II: Torsors for 2-group stacks

AU - Aldrovandi, Ettore

AU - Noohi, Behrang

PY - 2010/10

Y1 - 2010/10

N2 - 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. Butterfly diagrams encode morphisms of 2-groups and we employ them to examine the functorial behavior of non-abelian cohomology under change of coefficients. We reinterpret the first non-abelian cohomology with coefficients in a 2-group in terms of gerbes bound by a. crossed module. Our main result is to provide a geometric version of the change of coefficients 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 discuss various commutativity conditions on cohomology induced by various degrees of commutativity on the coefficient 2-groups, as well as specific features pertaining to group extensions. (C) 2010 Elsevier Inc. All rights reserved.

AB - 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. Butterfly diagrams encode morphisms of 2-groups and we employ them to examine the functorial behavior of non-abelian cohomology under change of coefficients. We reinterpret the first non-abelian cohomology with coefficients in a 2-group in terms of gerbes bound by a. crossed module. Our main result is to provide a geometric version of the change of coefficients 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 discuss various commutativity conditions on cohomology induced by various degrees of commutativity on the coefficient 2-groups, as well as specific features pertaining to group extensions. (C) 2010 Elsevier Inc. All rights reserved.

U2 - 10.1016/j.aim.2010.03.011

DO - 10.1016/j.aim.2010.03.011

M3 - Article

VL - 225

SP - 922

EP - 976

JO - ADVANCES IN MATHEMATICS

JF - ADVANCES IN MATHEMATICS

IS - 2

ER -