Who needs category theory?

TL;DR

This paper argues that category theory, despite skepticism, is essential for the mathematical foundation of topological quantum computing and likely other complex computational frameworks.

Contribution

It provides a high-level explanation of why category theory is necessary in topological quantum computing, challenging the view that it can be decategorized.

Findings

Category theory is necessary for the foundation of topological quantum computing.

Decategorization is insufficient for the mathematical rigor required.

The necessity of category theory likely extends beyond quantum computing.

Abstract

In mathematical applications, category theory remains a contentious issue, with enthusiastic fans and a skeptical majority. In a muted form this split applies to the authors of this note. When we learned that the only mathematically sound foundation of topological quantum computing in the literature is based on category theory, the skeptical author suggested to "decategorize" the foundation. But we discovered, to our surprise, that category theory (or something like it) is necessary for the purpose, for computational reasons. The goal of this note is to give a high-level explanation of that necessity, which avoids details and which suggests that the case of topological quantum computing is far from unique.