Isbell's subfactor projections in a noetherian form

TL;DR

This paper revisits Isbell's 1979 work on subfactors and projections, correcting an error, extending results to noetherian forms, and clarifying the canonical nature of key isomorphisms in the context of semi-abelian and exact categories.

Contribution

It extends Isbell's subfactor results to noetherian forms, corrects an error in the refinement theorem, and clarifies the canonicity of related isomorphisms.

Findings

Extension of Isbell's results to noetherian forms

Correction of an error in the refinement theorem

Establishment of canonicity of key isomorphisms

Abstract

In this paper, we revisit the 1979 work of Isbell on subfactors of groups and their projections, which he uses to establish a stronger formulation of the butterfly lemma and its consequence, the refinement theorem for subnormal series of subgroups. We point out an error in the second part of Isbell's refinement theorem, but show that the rest of his results can be extended to the general self-dual context of a noetherian form, which includes in its scope all semi-abelian categories as well as all Grandis exact categories. Furthermore, we show that Isbell's formulations of the butterfly lemma and the refinement theorem amount to canonicity of isomorphisms established in these results.