On the lattices of exact and weakly exact structures
arXiv:2009.10024 · doi:10.1016/j.jalgebra.2022.07.043
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 and characterize weakly exact structures among them. We investigate when these structures on 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 is additively finite, giving a module-theoretic characterization of closed sub-bifunctors of $\mbox{Ext}^1$ among all additive sub-bifunctors.
arXiv commentv1: 41 pages. v2: 43 pages, minor edits, Definition 6.10 and Section 7.7 added, Section 7.6 expanded. v3: 40 pages, minor edits, title changed, presentation improved. Lemmas 5.16 and 5.19 and references added. v4: An abridged version accepted to Journal of Algebra
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