??so consider for example a directed graph, and consider the sublattice of its subobject lattice generated by subobjects that "occur as pieces of irredundant covers" ... ??or something ???...

??perhaps any "edge piece" qualifies?? ???something about cover by the edge pieces ... ??

??vertex qualifies if ... ???if what???

???something about the "element poset" of a presheaf ... ???and sub-poset of it consisting of just those elements that ... ??? ???or something ??

??what about idempotence or otherwise of functorial process on posets (or something...) here???... and so forth ... ???...

??hmm, bit consider for example representable presheaf on _comm monoid_ ... or something ... ???maybe something about considering also how presheaf may fit into other presheaves, in trying to decide what the "good" pieces are??? ... and so forth ... ????...