Common mathematical expressions for a characterization of X in terms of P include "P is necessary and sufficient for X", and "X holds if and only if P".
The first type of statement says in different words that the extension of P is a singleton set, while the second says that the extension of Q is a single equivalence class (for isomorphism, in the given example — depending on how up to is being used, some other equivalence relation might be involved).
A reference on mathematical terminology notes that characteristic originates from the Greek term kharax, "a pointed stake":From Greek kharax came kharakhter, an instrument used to mark or engrave an object.
The objects can be considered the extensions of the world, while the features are expressions of the intensions.
A continuing program of characterization of various objects leads to their categorization.