Some Properties of Homology and Exactness of Nomura's Homology Sequences in a Grandis Homological Category

TL;DR

The paper extends Lambek invariants to Grandis homological categories by formulating Nomura's null sequences and establishing their exactness criteria. It employs the semiexact/ex2/homological framework, including the left/right homology objects $H_{-}(f,g)$ and $H_{+}(f,g)$ and the Lambek morphism $\Lambda$, to prove that left and right homology coincide under suitable conditions. The main contributions are the construction of Nomura's null sequences in Grandis categories, criteria for their exactness, and results linking Lambek's isomorphism to these sequences, ultimately generalizing classical invariant theory to strongly non-abelian settings. These results deepen the understanding of homological algebra in non-abelian contexts and enable new applications to approximate representations and related areas of category theory.

Abstract

We consider Lambek's invariants $\text{Ker}$ and $\text{Im}$ for commutative squares in Grandis homological categories. We prove that Nomura's null sequences exist in such categories and find sufficienct conditions for their exactness. We also prove the coincidence of left and right homology in Grandis homological categories.

Theorems & Definitions (28)

• Lemma 1
• Lemma 2
• Lemma 3: $3 \times 3$ lemma
• Lemma 4
• Lemma 5
• Corollary 1
• Lemma 6
• proof
• Corollary 2
• proof