I'm not quite sure if this is the right place to ask, and if this is the right way to ask, but I dare.
In philosophy (of mind, e.g.) the concept of supervenience is used:
"Supervenience [is] used to describe relationships between sets of properties in a manner which does not imply a strong reductive relationship."
That means an object might possess higher properties that depend on some base properties, but cannot be reduced to (defined by) them.
My question is: Can this situation occur in mathematics?
As I see it, for every mathematical object - be it a set with a structure or a vertex in an abstract graph or an object in a category - all its (relevant) properties are defined determined by its inner structure or its relations/morphisms to other objects. To me it seems unconceivable that a mathematical object can have any extra properties, let alone in a supervenient manner, that is, two isomorphic objects would have to share them.
But maybe I'm wrong. Can anyone point me to an example?