  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.
