Extension theory and the calculus of butterflies

Abstract

This paper provides a unified treatment of two distinct viewpoints concerning the classification of group extensions: the first uses weak monoidal functors, the second classifies extensions by means of suitable H2 -actions. We develop our theory formally, by making explicit a connection between (non-abelian) G-torsors and fibrations. Then we apply our general framework to the classification of extensions in a semi-abelian context, by means of butterflies between internal crossed modules. As a main result, we get an internal version of Dedecker's theorem on the classification of extensions of a group by a crossed module. In the semi-abelian context, Bourn's intrinsic Schreier–Mac Lane extension theorem turns out to be an instance of our Theorem 6.3. Actually, even just in the case of groups, our approach reveals a result slightly more general than classical Schreier–Mac Lane theorem

Fibred-categorical obstruction theory

“…Otherwise, one can see this isomorphism as a natural consequence of a more sophisticated general argument, which will be only outlined here. As it was observed in [35] (see also Section 6.2 of [19]), given a C-module B with action ξ, the Baer sum endows the groupoid OPEXT(Gp)(C, B, ξ) with a natural symmetric monoidal structure which makes it a categorical group ( [34], see also Section 7.1), since every object is invertible up to isomorphism; the identity object is the canonical semi-direct product extension determined by ξ. In fact, one can prove that the group π 0 (OPEXT(Gp)(C, B, ξ)) of the connected components of this categorical group is isomorphic to H 2 (C, B, ξ), and that the abelian group π 1 (OPEXT(Gp)(C, B, ξ)) of the automorphisms of the identity object is in fact isomorphic to Z 1 (C, B, ξ).…”

“…In order to understand the solution we propose, let us look at the point of view on the obstruction problem for (crossed) extensions of groups adopted in [19]. The category XExt(Gp) of crossed extensions of groups is equipped with a functor Π : XExt(Gp) → Mod(Gp), which sends each crossed extension…”

Discrete and Conservative Factorizations in Fib(B)

“…So, one may look for a finer reflection, where fibres are turned into groupoids. This gives a richer structure which is at the base, for example, of the interpretation given in [1] of Schreier-Mac Lane Theorem on the classification of group extension and its further generalizations (see Proposition 2.7 in [1]).…”

“…Like L, also the 2-functors R and I on C//B, defined by the corresponding comma squares in (1), can be endowed with a structure of 2-monad, which is colax-idempotent in the case of R and pseudo-idempotent in the case of I . In both cases, these structures restrict to strict 2-monads (R, v, n, ρ) and (I , i, l, ι) on C/B.…”

On the classification of Schreier extensions of monoids with non-abelian kernel

“…The same kind of result was then extended to other algebraic structures, such as associative algebras [12] and Lie algebras [13] over a field, rings [15], categories of interest [19], categorical groups [11,7]. A categorical approach to this problem was initiated by Bourn in [1] and then generalized in [6,2,9,8] to the context of semi-abelian [14] action accessible [3] categories.…”