and more, in that it not only classifies every class, but also gives a distinguished (canonical) representative for each object in the class.

Formally, a canonicalization with respect to an equivalence relation R on a set S is a mapping c:S->S such that for all s, s₁, s₂ ? S:
#c(s) = c(c(s)) (idempotence),
#s₁ R s₂ if and only if c(s₁) = c(s₂) (decisiveness),