This talk will provide an overview of the landscape of bicommutant categories, these are tensor categories with a strong functional-analytic flavour. I will discuss the evolution of the definition (and give the current version of the definition) and explain precisely how they categorify von Neumann algebras, in the same way a tensor category can be viewed as a categorification of an algebra. We will also introduce the string-calculus that renders the coherences in the definition transparent and workable.
The necessary background from functional analysis (in particular, operator theory) will be reviewed, and I will conclude with open questions (if waiting for the end of talk is not your style, there are 75 Open problems on André’s website).