Bicommutant categories from fusion categories

TL;DR

This paper demonstrates that every unitary fusion category naturally forms a bicommutant category, extending the analogy between finite-dimensional *-algebras and von Neumann algebras into higher categories.

Contribution

It proves that all unitary fusion categories are examples of bicommutant categories, providing a categorification of a classical algebraic result.

Findings

Every unitary fusion category is a bicommutant category

Categorification of von Neumann algebra characterization

Extension of algebraic analogies to higher categories

Abstract

Bicommutant categories are higher categorical analogs of von Neumann algebras that were recently introduced by the first author. In this article, we prove that every unitary fusion category gives an example of a bicommutant category. This theorem categorifies the well known result according to which a finite dimensional *-algebra that can be faithfully represented on a Hilbert space is in fact a von Neumann algebra.