  The well-known difficulties arising in a classification which is not
set-theoretically trivial---involving what is sometimes called a non-smooth
quotient---have been overcome in a striking way in the theory of operator
algebras by the use of what might be called a classification functor---the very
existence of which is already a surprise. Here the notion of such a functor is
developed abstractly, and a number of examples are considered (including those
which have arisen for various classes of operator algebras).
