Towards a theory of classification

The well-known difficulties arising in a classification which is not set-theoretically trivial—involving what is sometimes called a nonsmooth 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). 1. The purpose of this note is to propose an approach to the general question of classification. Except in the simplest cases (sets, vector spaces, finite simple groups!) it is not possible even in principle to label the isomorphism classes of a given class of mathematical objects in a reasonable way. Even when this is possible, one is often interested in more than just when two objects are isomorphic. (For instance, one might be interested in when one object is isomorphic to a subobject of another.) (This is obviously important for sets. For finite simple groups it is an open problem.) Perhaps, rather than labels for isomorphism classes, what one really wants, given a category, is a functor—distinguishing isomorphism classes—from this category into some other, simpler, category. (In other words, still labels for objects in the given category, but with isomorphic objects no longer required to have the same The research of the author was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. AMS 2000 Mathematics Subject Classification. Primary: 18A22, 46L35, 46M15; Secondary: 19K14, 19K35, 20E36.