On the lattices of exact and weakly exact structures

Abstract

We initiate in this article the study of weakly exact structures, a generalisation 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 and characterize weakly exact structures among them.We investigate when these structures on 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 weakly idempotent complete case, we characterise 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 A is additively finite, giving a module-theoretic characterization of closed sub-bifunctors of Ext1 among all additive sub-bifunctors.