On the lattices of exact and weakly exact structures

TL;DR

This paper introduces weakly exact structures as a generalization of Quillen exact structures, explores their lattice properties, and characterizes maximal structures and sub-bifunctors in additive categories.

Contribution

It defines weakly exact structures, investigates their lattice properties, and characterizes maximal structures and sub-bifunctors in additive categories.

Findings

Lattice of substructures is isomorphic to topologizing subcategories.

Existence of a unique maximal weakly exact structure.

Characterization of sub-bifunctors in additive categories.

Abstract

We initiate in this article the study of weakly exact structures, a generalization of Quillen exact structures. We introduce weak counterparts of one-sided exact structures and show that a left and a right weakly exact structure generate a weakly exact structure. We further define weakly extriangulated structures on an additive category $\mathcal{A}$ and characterize weakly exact structures among them. We investigate when these structures on $\mathcal{A}$ form lattices. We prove that the lattice of substructures of a weakly extriangulated structure is isomorphic to the lattice of topologizing subcategories of a certain functor category. In the idempotent complete case, we characterize the lattice of all weakly exact structures and we prove the existence of a unique maximal weakly exact structure. We study in detail the situation when $\mathcal{A}$ is additively finite, giving a module-theoretic characterization of closed sub-bifunctors of $\mbox{Ext}^1$ among all additive sub-bifunctors.