Homological lemmas for (non-abelian) group-like structures by diagram chasing in a self-dual context

TL;DR

This paper extends homological diagram lemmas to non-abelian group-like structures via a self-dual noetherian form, enabling dual proofs by subobject chasing. It introduces the Pyramid construction and the Homomorphism Induction Theorem to realize induced morphisms along zigzags, and verifies the framework in Słomiński algebras, illustrating applicability to semi-abelian and Grandis exact categories. It derives self-dual versions of classical lemmas (Four, Five, 3×3, Snake) and clarifies the role of the Middle 3×3 Lemma via a lattice-theoretic obstruction. It also provides exercises and discusses axiom-independence, highlighting the framework's versatility for non-abelian homological reasoning.

Abstract

Through abelian categories, homological lemmas for modules admit a self-dual treatment, where half of the proof of a lemma is sufficient to prove the full lemma. In this paper, we show how the context of a `noetherian form', recently introduced by the second and third authors, allows a self-dual treatment of these lemmas even in the case of non-abelian categories of group-like structures. This context covers a wide range of examples: module categories, the category of groups, of graded abelian groups, the categories of Lie algebras, of cocommutative Hopf algebras, the category of Heyting semilattices, of loops, the dual of the category of pointed sets, the category of modular/distributive lattices and modular connections, the category of sets and partial bijections, and many others. More generally, it includes all semi-abelian and Grandis exact categories.

Figures (3)

• Figure 1: A Pyramid of Morphisms
• Figure 2: Diagram connecting two zigzags
• Figure 3: Double Complex

Theorems & Definitions (106)

• Lemma 1.1: Restricted Modular Law, DNA IV
• proof
• Example 1.2
• Remark 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Definition 2.1
• Lemma 2.2
• proof