Obstruction Theory for Extensions of Categorical Groups

Abstract

Abstract. For any categorical group H, we introduce the categorical group Out(H) and then the well-known group exact sequence 1 → Z(H ) → H → Aut(H ) → Out(H ) → 1 is raised to a categorical group level by using a suitable notion of exactness. Breen's Schreier theory for extensions of categorical groups is codified in terms of homomorphism to Out(H) and then we develop a sort of Eilenberg-Mac Lane obstruction theory that solves the general problem of the classification of all categorical group extensions of a group G by a categorical group H, in terms of ordinary group cohomology.

Equivariant Extensions of Categorical Groups

“…If G is a -graded categorical group then Ker G is a -categorical group and, conversely, any -categorical group H determines a -graded categorical group H gr whose objects are the same as those of H and whose arrows of grade σ are the arrows f : σ X → Y in H. These two constructions determine actually [ [2] associated to any categorical group H. This graded categorical group provides the key to codify the equivariant extensions we study in next section. The objects of Out (H) are the equivalences of the categorical group H. A premorphism of grade σ ∈ , from T to T , is a triple (A, ϕ A , σ ) where A ∈ H and ϕ A : σ T → i A T is a monoidal natural transformation (where σ T and i A are the equivalences of H given respectively, for every object…”

“…In this section we develop a theory of extensions of -categorical groups which extends both categorical group extensions theory [1,11,2] and equivariant group extensions theory [4]. This general theory provides, as an example, a theory of equivariant extensions of a -group G by a -crossed module of groups L and, in particular, when L is the crossed module H → Aut(H ) given by inner automorphisms, one obtains a definition of a three-dimensional cohomology of a -group G with coefficients in a non-abelian -group H .…”

“…We should remark that Schreier's theories for the classification of the sets of extensions Ext(G, H) and Ext (G, H ) were respectively established in [ [2]) and Out (dis H ) is just the -graded categorical group Hol H (see [4]), both theories appear as particular cases of the following theorem.…”

“…In [1] L. Breen developed a theory of extensions of categorical groups generalizing the classical Schreier's theory [12] on the classification of group extensions with non-abelian kernel and, more recently, an obstruction theory for these categorical group extensions has been also showed in [2]. The cohomology sets defined by Breen to codify the set of equivalence classes of categorical group extensions provided then, as a particular instance, a suitable definition of a 3-cohomology of a group G with coefficients in a non-abelian group H .…”

Split extensions, semidirect product and holomorph of categorical groups

“…Then, they prove that equivalence classes of split extensions bijectively correspond to equivalence classes of actions of H on G (where an action of H on G is defined as a monoidal functor H → Eq(G), Eq(G) being the categorical group of monoidal autoequivalences of G). The quoted paper [15] is one of a series of papers devoted to extensions and derivations in the context of categorical groups, see [6,7,8,9,10,11,12], [15,16,17,18], [23]; these papers contain a number of examples showing that categorical groups provide a unifying framework for studying classical homological algebra. Despite its interest, the paper [15] suffers from a mixture of one-dimensional and two-dimensional arguments, and the aim of this note is to put some of the results by Garzón and Inassaridze into their proper 2-categorical context.…”