On the lattices of exact and weakly exact structures

Abstract

The study of exact structures on an additive category A is closely related to the study of closed additive sub-bifunctors of the maximal extension bifunctor Ext 1 on A. We initiate in this article the study of "weakly exact structures", which are the structures on A corresponding to all additive subbifunctors of Ext 1 . We introduce weak counter-parts of one-sided exact structures and show that a left and a right weakly exact structure generate a weakly exact structure. We 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 sub-structures of a weakly extriangulated structure is isomorphic to the lattice of topologizing subcategories of a certain abelian category. In the idempotent complete case, this characterises the lattice of all weakly exact structures. We study in detail the situation when A is additively finite, giving a module-theoretic characterization of closed sub-bifunctors of Ext 1 among all additive sub-bifunctors.