Butterflies I: Morphisms of 2-group stacks

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.

Higher cohomologies of modules

“…If P is any braided C -fibred categorical group, then, for any object u of C , the tensor product and the associativity, commutativity, and unit constraints can be restricted to the fibre category over u so that every fibre inherits a braided categorical group structure P u D .P u ;˝; I u; a; r; l ; c/. Thus, a braided (symmetric) C -fibred categorical group is the same thing as a stack of braided (symmetric) categorical groups over the discrete site C ; see Aldrovandia and Noohi [1] and Breen [8].…”

“…I . Braided categorical groups are also called braided Gr-categories by Breen [7; 8] and braided (weak) 2-groups by Baez and Lauda [2] and Aldrovandia and Noohi [1]. A braided categorical group whose braiding is a symmetry is called a symmetric categorical group by Joyal and Street [41] and Vitale [5].…”

“…A braided crossed module .H; G; @; f ; g/ is called a symmetric crossed module by Aldrovandia and Noohi [1], or a stable crossed module by Conduché [19], whenever…”

“…Then, for any integer r 2, we have: Theorem 7.6 There are natural bijections A/ are, in general, strict. Also, we should mention the work of Breen [8] (see also Aldrovandia and Noohi [1]), where he offers an excellent discussion about the cohomology classification of symmetric and braided categorical group stacks over an arbitrary site C . However, Breen's results are far from being as explicit as ours in this work in terms of cocycles when C is discrete.…”

“…Ab is the constant functor with value Z, then the groups H n .C; A/ D Ext n Mod C .Z; A/ are the cohomology groups of the category C with coefficients in the C -module A, studied by Roos [52], Watts [59], Mitchell [46] and Baues and Wrisching [3], among other authors. In this paper we introduce, for each integer r 1, an approach for a level-r cohomology theory for C -modules by defining cohomology groups (1) H n C;r .B; A/ WD H nCr .holim ! C K.B; r C 1/; holim !…”