Di-Exact Categories and Lattices of Normal Subobjects

TL;DR

This paper explores categorical-algebraic frameworks for homological algebra, focusing on properties like self-duality and normal maps, using lattice theory to characterize these properties and providing new examples and insights.

Contribution

It introduces new characterizations and examples of homological properties, such as the Second Isomorphism Property, within the lattice-theoretic framework for pointed categories.

Findings

Counterexamples separating homological conditions

New characterizations like the Second Isomorphism Property

Lattice-theoretical expressions of homological properties

Abstract

The aim of this article is to study certain categorical-algebraic frameworks for basic homological algebra, introduced in arXiv:2404.15896, with the aim of better understanding the differences between them. We focus on homological self-duality, preservation of normal maps by dinversion and diexactness, finding counterexamples that separate any two of these conditions. On the way, we encounter new examples and new characterizations, such as the Second Isomorphism Property. We also consider homological self-duality in the context of regular categories. Our main technique is to investigate the lattice of normal subobjects of an object in any pointed category with kernels and cokernels. The category of monoidal semilattices is a context where we have easy control of these lattices. We show that properties of the homological frameworks under consideration may be expressed by means of lattice-theoretical properties, such as modularity and distributivity.