Who needs category theory?

Abstract

In computer science, category theory remains a contentious issue, with enthusiastic fans and a skeptical majority. Categories were introduced by Samuel Eilenberg and Saunders Mac Lane as an auxiliary notion in their general theory of natural equivalences. Here we argue that something like categories is needed on a more basic level. As you work with operations on structures, it may be necessary to coherently manipulate isomorphism (or more generally homomorphism) witnesses for various properties of these operations, e.g. associativity, commutativity and distributivity. A working mathematician, to use Mac Lane’s term, is well advised to be aware of the coherent witness-manipulation problem and to know that category theory is an appropriate framework to address the problem. Of course, the working mathematician in question may be a computer scientist or physicist.