Well, your suggestion? Could you pick up the right parent element in the ontology?
1) Classical (N. Burbaki's approach): Kantor's set theory and logic 2) Constructive: where we are standing on constructive (intuitionistic) logic 3) Univalent foundations of mathematics (a novel approach).
Even if we stick to the 1st approach only (as we did for the ontology), there are also many dualities (alternative definitions), if we apply, say, terminology from geometry or, alternatively, from set theory while describing the same math objects.
Anyway, I think the methodology, we are working on during this project, should clarify many such hidden aspects. And we expect that it will be valuable for the modern math theory itself. So, let's collaborate:)