On one hand, exact structures were introduced by D. Quillen in the '70s. They can be defined as collections of short exact sequences in a fixed abelian category satisfying additional properties. On the other hand, in a recent work, A. Garver, R. Patrias, and H. Thomas introduced Jordan recoverability. Given a bounded quiver (Q,R), a full additive subcategory of rep(Q,R) is said to be Jordan recoverable if any X∈C can be recovered, up to isomorphism, from the Jordan form of its generic nilpotent endomorphisms. Such a subcategory
Nearby in the stack
C
is said to be canonically Jordan recoverable if, moreover, there exists a precise algebraic procedure that allows one to get back
X∈C
from that same Jordan form data. We introduce a new family of operators, called Gen-Sub operators
GSE
, parametrized by the exact structures
E
of abelian categories. After showing some properties of those operators in hereditary abelian categories, by focusing on the setting of modules over path algebras of type
A
quivers endowed with the diamond exact structure
E⋄
, we establish that the maximal canonically Jordan recoverable subcategories are precisely of the form