Towards a theory of classification

TL;DR

This paper develops an abstract theory of classification functors to address complex classification problems in operator algebras, providing new tools for understanding non-trivial classification issues.

Contribution

It introduces the concept of classification functors in an abstract setting and explores examples related to operator algebras, advancing the theoretical framework.

Findings

Classification functors can resolve non-trivial classification problems.

Examples include classifications within various classes of operator algebras.

The theory provides a new perspective on handling non-smooth quotients.

Abstract

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).